Advances in Optics: Reviews
Book Series, Volume 5Sergey Y. Yurish
EditorAdvances in Optics: Reviews
Book Series, Volume 5International Frequency Sensor Association PublishingSergey Y. Yurish
EditorAdvances in Optics: Reviews
Book Series, Vol. 5

Published by International Frequency Sensor Association (IFSA) Publishing, S. L., 2021
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ISBN: Print: 978-84-09-34834-3
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Acknowledgments
As Editor I would like to express my undying gratitude to all authors, editorial staff,
reviewers and others who actively participated in this book. We want also to express our
gratitude to all their families, friends and colleagues for their help and understanding.

Contents

Contents
Contributors................................................................................................................... 17
Preface ............................................................................................................................ 21
1. Physical Optics........................................................................................................... 23
1.1. Wave Equations of the Electromagnetic Field ................................................................. 23
1.1.1. The Maxwell’s Equations ...................................................................................................... 23
1.1.2. Waves in a Linear, Isotropic, and Homogeneous Medium .................................................... 24
1.1.3. Wave Speed of Light and the Index of Refraction ................................................................. 25
1.1.4. Scalar Time Harmonic Waves ............................................................................................... 25
1.1.5. The Energy Law of the Electromagnetic Field: Pointing Vector .......................................... 26
1.1.6. Vectorial Nature of the Electromagnetic Field ..................................................................... 27
1.1.7. Electromagnetic Theory of Radiation ................................................................................... 29
1.1.7.1. Radiation from an Accelerating Charge ..................................................................... 29
1.1.7.2. Radiation from an Oscillating Dipole ......................................................................... 30
1.1.8. Evanescent Waves ................................................................................................................. 30
1.1.9. Laws of Reflection and Refraction ........................................................................................ 33
1.1.9.1. Laws of Reflection and Refraction for Homogeneous Dielectric Media .................... 33
1.1.9.2. Fresnel’s Formulas ..................................................................................................... 35

1.2. Scalar Theory of Diffraction ............................................................................................ 37
1.2.1. The Rayleigh-Sommerfeld Formulation of Diffraction ......................................................... 38
1.2.2. Paraxial Approximations for Diffraction .............................................................................. 39
1.2.2.1. Fresnel Diffraction ..................................................................................................... 40
1.2.2.2. Fraunhofer Diffraction ................................................................................................ 41
1.2.3. Diffraction with Nonmonochromatic Light ........................................................................... 43

1.3. Coherence ......................................................................................................................... 44
1.3.1. Correlation Function: Definition .......................................................................................... 45
1.3.2. The Concept of Coherence of Light....................................................................................... 45

1.4. Fourier Theory and Fourier Optics ................................................................................... 50
1.4.1. Fourier Theory...................................................................................................................... 50
1.4.1.1. Analysis of Periodic Functions: Fourier Series .......................................................... 50
1.4.1.2. Analysis of Non-periodic Functions: Fourier Transform ............................................ 51
1.4.2. Fourier Optics....................................................................................................................... 52
1.4.2.1. The Concept of Angular Spectrum ............................................................................. 52
1.4.2.2. Fourier Transform of a Positive Lens ......................................................................... 54
1.4.2.3. Fourier Optics for Analyzing Diffraction Gratings ..................................................... 59
1.4.2.4. Fourier Optics for Imaging under Coherent Illumination ........................................... 64
1.4.2.5. Fourier Optics for Imaging under Incoherent Illumination ......................................... 67
1.4.2.6. Fourier Transform Spectroscopy ................................................................................ 68

1.5. Optical Interferometry ...................................................................................................... 69
1.5.1. Superposition of Waves ......................................................................................................... 70
1.5.1.1. General Principle ........................................................................................................ 70
1.5.1.2. Superposition of Waves with Same Frequencies ........................................................ 70
1.5.1.3. Standing Waves .......................................................................................................... 72
1.5.1.4. Superposition of Waves with Diffident Frequencies: Group Velocity,
Phase Velocity ............................................................................................................ 72
1.5.2. Superposition of Light Waves: Optical Interferometry ......................................................... 75
1.5.2.1. Interference Phenomena of Monochromatic Light ..................................................... 76

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Advances in Optics: Reviews. Book Series, Vol. 5
1.5.2.2. Interference Phenomena of Partial Coherent Light ..................................................... 81
1.5.2.3. Michelson Stellar Interferometer ................................................................................ 85

1.6. Polarization of Light ......................................................................................................... 86
1.6.1. Generation of Polarization Light .......................................................................................... 86
1.6.1.1. Polarization by Reflection from Dielectric Surfaces ................................................... 87
1.6.1.2. Polarization through Diattenuation............................................................................. 88
1.6.1.3. Polarization by Scattering ........................................................................................... 89
1.6.1.4. Polarization by Birefringence ..................................................................................... 89
1.6.1.5. Phase Shift by Birefringence ...................................................................................... 93
1.6.1.6. Optical Activity .......................................................................................................... 94
1.6.2. Matrix Treatment for Completely Polarized Light ................................................................ 95
1.6.2.1. Mathematical Representation of Completely Polarized Light: Jones Vector .............. 95
1.6.2.2. Mathematical Representation of Polarizers: Jones Matrices ....................................... 98
1.6.3. Basic Theory of Polarization of Stochastic Light Fields ..................................................... 100
1.6.3.1. Correlation Matrix of a Light Field .......................................................................... 101
1.6.3.2. Completely Polarized, Unpolarized, and Partially Polarized Light ........................... 101

References.............................................................................................................................. 104
Appendix. Derivation of the Rayleigh-Sommerfeld Formula ................................................ 104

2. CPU Technology Trend and Optical Interconnects ............................................. 109
2.1. Introduction..................................................................................................................... 109
2.2. CPU Design and Fabrication .......................................................................................... 110
2.3. Smartphone SoC Trend ................................................................................................... 118
2.4. x86 CPU Technology and Trend .................................................................................... 125
2.5. Supercomputer Architecture ........................................................................................... 136
2.6. Optical Network-on-chip Architecture ........................................................................... 143
2.7. Key Elements toward Optically Interconnected CPU ..................................................... 148
2.8. Conclusion ...................................................................................................................... 160
References.............................................................................................................................. 161

3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix
Polarimetry with Full Poincaré Beams and a CCD Camera ............................. 163
3.1. Introduction..................................................................................................................... 163
3.2. Polarimetry with a Full Poincaré Beam and a CCD Camera .......................................... 164
3.3. Noise in CCD Images ..................................................................................................... 167
3.4. Strategies for Obtaining the Mueller Matrix ................................................................... 172
3.4.1. Center Point Strategy .......................................................................................................... 172
3.4.2. Nearest Neighbor Strategy .................................................................................................. 173

3.5. Results from the Simulation............................................................................................ 174
3.6. Conclusions..................................................................................................................... 176
Acknowledgements ................................................................................................................ 176
References.............................................................................................................................. 176

4. The Optical Bidirectional Lambertian Conductance Law with Applications ... 179
4.1. Prologue .......................................................................................................................... 179
4.2. Introduction to Solar Concentrators and Their Characterization .................................... 185
4.3. Theory of the Bidirectional Lambertian Irradiation Method (BLIM) ............................. 188
4.4. Experimental ................................................................................................................... 192

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Contents
4.4.1. Optical Elements under Test ............................................................................................... 192
4.4.2. Simulation Methods ............................................................................................................ 194
4.4.3. Measurement of the Inverse Radiance ................................................................................ 196
4.4.4. Test of the Optical Bidirectional Lambertian Conductance (OBLC) Law .......................... 198

4.5. Optical Conductance Simulation Results ....................................................................... 202
4.5.1. Nonimaging Solar Concentrators (CPC) ............................................................................ 202
4.5.2. Light Cones (LC) ................................................................................................................ 206
4.5.3. Solar Tunnels (ST) .............................................................................................................. 210

4.6. Exit Radiance Simulation Results .................................................................................. 213
4.6.1. Nonimaging Solar Concentrators (CPC) ............................................................................ 213
4.6.2. Light Cones (LC) ................................................................................................................ 215
4.6.3. Solar Tunnels (ST) .............................................................................................................. 221

4.7. Short Discussion of Practical Applications. ................................................................... 222
4.7.1. Net Flux Crossing a Solar Tunnel ....................................................................................... 223
4.7.2. Light Sources from CPCs and LCs ..................................................................................... 225
4.7.3. Bent Solar Tunnels .............................................................................................................. 226

4.8. Some Insights into the Law of Optical Lambertian Conductance .................................. 229
4.9. Conclusions .................................................................................................................... 230
Acknowledgements ............................................................................................................... 232
References ............................................................................................................................. 232

5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging .................... 237
5.1. Introduction .................................................................................................................... 237
5.2. Theory of Aperiodic Zone Plates ................................................................................... 239
5.2.1. Aperiodic Sequence............................................................................................................. 239
5.2.2. Method of Constructing Aperiodic Zone Plates .................................................................. 239
5.2.3. Theory of Diffraction .......................................................................................................... 241

5.3. Aperiodic Zone Plates for Optical Tweezers .................................................................. 243
5.3.1. Fractal Zone Plate .............................................................................................................. 243
5.3.2. Fibonacci Zone Plate .......................................................................................................... 249
5.3.3. Thue-Morse Zone Plate....................................................................................................... 250

5.4. Aperiodic Zone Plates for Optical Imaging .................................................................... 252
5.4.1. Clear Imaging ..................................................................................................................... 252
5.4.2. Colourful Imaging............................................................................................................... 257

5.5. Conclusion ..................................................................................................................... 262
Acknowledgments ................................................................................................................. 262
References ............................................................................................................................. 263

6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical
Aberrations in the Centered Conical Optical Systems with Highand Super-high Numerical Apertures .................................................................. 267
6.1. Introduction .................................................................................................................... 267
6.2. Theoretical Basis ............................................................................................................ 268
6.2.1. Paraxial Matrix Transformations and Axial Chromatic Aberration ................................... 268
6.2.1.1. Paraxial Matrix Transformations from Object to Image Space in a Single
Optical Surface .......................................................................................................... 268

9Advances in Optics: Reviews. Book Series, Vol. 5
6.2.1.2. Exact Paraxial Matrix Transformations in a Centered Optical System of ConicalSurfaces. Invention of Exact Analytical Dependences between Constructive
Parameters and Paraxial Characteristics .................................................................... 270
6.2.1.3. Axial Image Chromatic Aberration .......................................................................... 274
6.2.1.4. Simultaneous Correction of Axial Image Chromatic Aberration and Guaranteeof Preset Focal Length in a Centered Optical System ................................................ 276
6.2.2. Rigorous Analytical Function of the On-axes Spherical Aberration in a Centered
Optical System Containing Conical Surfaces...................................................................... 277
6.2.2.1. Sagittal Radius and Sagittal Radius-coefficient in a Single Conical Surface ............ 277
6.2.2.2. Connections between the Parameters of the Tangential Rays and the Paraxial
Matrix Transformations on a Single Conic Surface ................................................... 282
6.2.2.3. Paraxial Matrix Coefficients Linked to the Real Tangential Rays in a System
of Centered Conic Surfaces ....................................................................................... 285
6.2.2.4. Geometrical Interpretation of the Paraxial Image Function and Its Connection
with On-axis Spherical Aberration ............................................................................ 287

6.3. Simple Examples of Analytical Aberrations Correction for an Infinitely DistantObject............................................................................................................................. 288
6.3.1. Analytical Correction of Axial Chromatic Aberration Together with Preset Focal
Length in a System of Three Surfaces (Case of the so Called Doublets) ............................. 289
6.3.2. Analytical Correction of On-axis Spherical Aberration in a Systems of Two
and Three Surfaces.............................................................................................................. 291
6.3.2.1. Systems of Two Surfaces .......................................................................................... 291
6.3.2.2. Multi-conical Surfaces in the Centered Optical Systems .......................................... 292
6.3.2.3. Examples of On-axis Spherical Aberration Correction in the Systemsof Two Surfaces ........................................................................................................ 293
6.3.2.4. Examples of On-axis Spherical Aberration Correction in the Systemsof Three Surfaces ...................................................................................................... 297

6.4. Conclusions..................................................................................................................... 298
References.............................................................................................................................. 299

7. Ultra-low Light Absorption Measurement in the Problem of Determining
Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path
Interferometry ........................................................................................................ 301
7.1. Introduction..................................................................................................................... 301
7.2. Relevance ........................................................................................................................ 303
7.3. State of the Art ................................................................................................................ 305
7.4. Time-resolved Photothermal Common-path Interferometry Scheme ............................. 311
7.5. The Prospect of Further Increasing of the TPCI Scheme Sensitivity .............................. 322
7.6. Conclusions..................................................................................................................... 328
Acknowledgements ................................................................................................................ 328
References.............................................................................................................................. 329

8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size
(GLMT for Light-sheet) ......................................................................................... 333
8.1. Introduction..................................................................................................................... 333
8.2. Description of Light-sheet .............................................................................................. 335
8.2.1. Angular Spectrum ............................................................................................................... 335
8.2.1.1. Airy Light-sheet ........................................................................................................ 338
8.2.1.2. Bessel Pincer-sheet ................................................................................................... 339
8.2.1.3. Gaussian Light-sheet ................................................................................................ 340

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8.2.2. The BSCs (Beam Shape Coefficients) of Arbitrary Laser Light-sheet ................................. 341
8.2.2.1. Expansion of the Incident and Scattering Fields ....................................................... 341
8.2.2.2. The Derivation of General BSCs of Laser Light-sheet ............................................. 343
8.2.2.3. Particular Structural Beam Laser Light-sheet ........................................................... 349

8.3 Light Scattering of Laser Light-sheet by Dielectric Sphere of Arbitrary Size ............... 351
8.3.1. Far-field Scattering Intensity .............................................................................................. 351
8.3.2. Scattering, Extinction and Absorption Efficiencies ............................................................. 352
8.3.3. Radiation Force and Torque ............................................................................................... 353

8.4. Numerical Results and Discussions ............................................................................... 354
8.4.1. Airy light-sheet.................................................................................................................... 354
8.4.1.1. Far-field Scattering Intensity .................................................................................... 355
8.4.1.2. Scattering, Extinction and Absorption Efficiencies .................................................. 357
8.4.1.3. Radiation Force and Torque ..................................................................................... 360
8.4.2. Bessel Pincer Light Sheet.................................................................................................... 365
8.4.2.1. Far-field Scattering Intensity .................................................................................... 365
8.4.2.2. Scattering, Extinction and Absorption Cross Section ............................................... 369
8.4.2.3. Radiation Force and Torque ..................................................................................... 373
8.4.3. Gaussian Light-sheet .......................................................................................................... 383
8.4.3.1. Far-field Scattering Intensity .................................................................................... 383
8.4.3.2. Scattering, Extinction and Absorption Efficiencies .................................................. 385
8.4.3.3. Radiation Force and Torque ..................................................................................... 387

8.5. Conclusions .................................................................................................................... 389
Acknowledgements ............................................................................................................... 390
References ............................................................................................................................. 391

9. Laser Characterizations for Optical Wireless Communications ....................... 395
9.1. Introduction to Optical Wireless Communications ........................................................ 395
9.2. Underwater Wireless Optical Communication ............................................................... 397
9.3. Chaos Based Secure Optical Communications .............................................................. 398
9.4. Vertical-cavity Surface-emitting Lasers (VCSELs) ....................................................... 399
9.4.1. Optical and Electrical Properties of VCSELs ..................................................................... 400
9.4.1.1. Light-current (L-I) Curve Characteristics ................................................................. 400
9.4.1.2. Polarization Switching.............................................................................................. 402

9.5. OWC Based on VCSELs ................................................................................................ 403
9.6. Nonlinearity of VCSELs ................................................................................................ 404
9.7. VCSELs-based Chaos Dynamics ................................................................................... 405
9.7.1. Chaos Synchronization in VCSEL ....................................................................................... 406

9.8. Theoretical Analysis of Lasers Dynamics ...................................................................... 407
9.8.1. Rate Equations Describing Carriers and Photons Density Dynamics ................................ 407

9.9. Conclusions .................................................................................................................... 409
Acknowledgements ............................................................................................................... 410
References ............................................................................................................................. 410

10. Compact Solar-pumped Lasers ............................................................................ 417
10.1. Introduction .................................................................................................................. 417
10.1.1. Solar-pumped Lasers and Their Possible Applications .................................................... 417
10.1.2. Brief History of SPLs ........................................................................................................ 417
10.1.3. Stance and Orientation of This Work ................................................................................ 418

11Advances in Optics: Reviews. Book Series, Vol. 5

10.2. Background of the Concept Design .............................................................................. 418
10.2.1. Preferable Oscillation Wavelength of SPLs ...................................................................... 418
10.2.2. Incident Power Density Required to Oscillate SPLs ......................................................... 420
10.2.3. Preferable Shape of LMs for SPLs .................................................................................... 422
10.2.4. Theoretical Limit of Solar Concentration ......................................................................... 424
10.2.5. Size of Images of the Sun and Concentration Ratios by Solar Concentrators ................... 425
10.2.5.1. Convex Lens ........................................................................................................... 425
10.2.5.2. Parabolic Concave Mirror ....................................................................................... 426
10.2.6. Merits of Miniaturization of SPLs ..................................................................................... 427
10.2.7. Direct Solar Radiation and Diffuse Solar Radiation ......................................................... 429

10.3. Compact Solar-Pumped Fiber Laser (SPFL) System .................................................... 432
10.3.1. Choice of Laser Mediums (LMs) ....................................................................................... 432
10.3.1.1. Shape of LMs.......................................................................................................... 432
10.3.1.2. Glass LMs ............................................................................................................... 433
10.3.2. Solar Concentrator............................................................................................................ 434
10.3.3. SPL Resonator Based on a Double Cladding Optical Fiber ............................................. 436
10.3.4. Outdoor Oscillation Performance of the Compact SPFL .................................................. 437

10.4. Improvement of Spectral Matching Efficiency by Cr Codoping .................................. 439
10.5. Compact Solar-Pumped Micro-Rod Laser (μSPL) System .......................................... 441
10.6. Continuous Oscillation of μSPL for Over 6.5 Hours Tracking the Sun ........................ 446
10.7. Effect of Cr Content in a Cr-Codoped Nd:YAG Transparent Ceramic Laser
Rod for μSPLs................................................................................................................ 450
10.8. For Improvement of Mode-matching Efficiency .......................................................... 453
10.9. Evaluation of Energy Transfer Efficiency from Cr 3+ to Nd3+ in μSPL in Outdoor
Operation ....................................................................................................................... 457
10.10. Coordinated Solar Tracking of an Array of μSPLs to Harvest Larger Amountof Solar Energy .............................................................................................................. 461
10.11. SPL-PV Combined System and Its Application to Optical Wireless Power
Transmission .................................................................................................................. 463
10.12. Conclusion .................................................................................................................. 464
Acknowledgments ................................................................................................................. 465
References.............................................................................................................................. 465

11. Sub-micron Direct Silicon Processing by Microsphere Focused
Femtosecond Infrared Laser ................................................................................. 473
11.1. Introduction................................................................................................................... 473
11.2. Laser-material Interactions ........................................................................................... 474
11.2.1. Gaussian Beam Propagation ............................................................................................ 474
11.2.2. Ablation Mechanisms ........................................................................................................ 477
11.2.3. Simulations ........................................................................................................................ 478

11.3. Experimental Details ..................................................................................................... 479
Acknowledgements ................................................................................................................ 484
References.............................................................................................................................. 484

12. Mid Infrared Quantum Cascade Laser Reflection-absorption
Spectroscopy at the Grazing Angle Incidence .................................................... 487
12.1. Introduction................................................................................................................... 487
12.2. Quantum Cascade Lasers .............................................................................................. 488
12.3. Grazing Angle Reflection-Absorption Infrared Spectroscopy (RAIRS) ....................... 489
12.3.1. Design of the Optical Setup ............................................................................................... 489

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12.3.2. Advantages over Conventional Methods ........................................................................... 490

12.4. Sample Preparation for Trace Detection Analyses ....................................................... 491
12.4.1. Deposition Techniques ...................................................................................................... 491
12.4.2. Effect of the Substrate: Specular and Diffuse Reflectance Measurements ........................ 492
12.4.3. Sample Preparation of HEs and APIs ............................................................................... 492

12.5. Trace Detection Analyses Using QCL-GAP ................................................................ 494
12.5.1. Detection of HEs ............................................................................................................... 494
12.5.2. Detection of API................................................................................................................ 498

12.6. Improving Detection Performance ............................................................................... 499
12.6.1. Development of Spectral Library ...................................................................................... 499
12.6.2. MVA Routines ................................................................................................................... 503
12.6.2.1. Data Preprocessing ................................................................................................. 503
12.6.2.2. Data Exploration..................................................................................................... 504
12.6.2.3. Data Discrimination................................................................................................ 506
12.6.3. Fourier Transform Data Preprocessing Algorithm........................................................... 507

12.7. Conclusions .................................................................................................................. 508
Acknowledgments ................................................................................................................. 508
References ............................................................................................................................. 508

13. Advances in Linewidth Compression, Linewidth Measurement, Noise
Characterization, Wavelength Switching, and Wavelength Tuningof Lasers .................................................................................................................. 511
13.1. Introduction .................................................................................................................. 511
13.2. Laser Dynamics ............................................................................................................ 512
13.3. Classification of Lasers ................................................................................................ 512
13.4. Lasers and Systems ...................................................................................................... 512
13.4.1. Semiconductor Lasers ....................................................................................................... 513
13.4.2. Fiber Lasers ...................................................................................................................... 513
13.4.3. Lasers in Optical Fiber Communication Systems ............................................................. 513
13.4.3.1. Lasers and Optical Communication System Technological Evolution ................... 514
13.4.4. Lasers in Optical Fiber Sensing Systems .......................................................................... 515
13.4.4.1. Lasers and Optical Sensing System Technological Evolution ................................ 516

13.5. Linewidth Compression ............................................................................................... 518
13.5.1. Whispering Gallery Mode (WGM) .................................................................................... 519
13.5.2. Parallel Feedback Mechanism .......................................................................................... 522
13.5.3. Optical Self-Injection Feedback........................................................................................ 524
13.5.4. Microfibers ....................................................................................................................... 526
13.5.5. Electrical Feedback Control ............................................................................................. 527
13.5.6. Rayleigh Backscattering ................................................................................................... 529

13.6. Linewidth Measurement ............................................................................................... 538
13.6.1. Heterodyne Detection ....................................................................................................... 538
13.6.2. Delayed Self-Heterodyne Detection .................................................................................. 539
13.6.3. Delayed Self-Homodyne Detection ................................................................................... 540
13.6.4. Amplitude Difference Comparison of Coherent Envelope ................................................ 541

13.7. Noise Characterization ................................................................................................. 544
13.7.1. Digital Cross Correlation ................................................................................................. 544
13.7.2. Michelson Interferometer .................................................................................................. 545
13.7.3. Cross-Spectrum Method.................................................................................................... 546

13.8. Wavelength Switching and Wavelength Tuning .......................................................... 547

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Advances in Optics: Reviews. Book Series, Vol. 5
13.8.1. Mechanical Switching/Tuning ........................................................................................... 547
13.8.2. Acousto-optic Switching .................................................................................................... 560
13.8.3. All Optical Tuning ............................................................................................................. 563
13.8.4. Thermal Tuning ................................................................................................................. 565

13.9. Conclusion .................................................................................................................... 569
Acknowledgements ................................................................................................................ 569
References.............................................................................................................................. 570

14. Coherence Characterization of DSR-like Square Pulses in Passively
Mode-locked Fiber Lasers .................................................................................... 575
14.1. Introduction................................................................................................................... 575
14.2. Experimental Setup ....................................................................................................... 577
14.3. DSR-like Square Pulses ................................................................................................ 578
14.4. Coherence Characterization of Square Pulses with MZI Technique............................. 581
14.5. Coherence Characterization of Square Pulses with DFT Technique ............................ 584
14.6. Conclusion .................................................................................................................... 588
References.............................................................................................................................. 588

15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials
and Numbers ........................................................................................................... 593
15.1. Introduction................................................................................................................... 593
15.2. Fourier Series ................................................................................................................ 594
15.2.1. Fourier Series Coefficients................................................................................................ 594
15.2.2. Formula for Obtaining Fourier Series of Functions ......................................................... 596
15.2.3. Fourier Series of Even and Odd Functions ....................................................................... 597
15.2.4. From Fourier Series to Fourier Transform ....................................................................... 597
15.2.5. Obtaining Fourier Series of Current Functions ................................................................ 598
15.2.5.1. Periodic Function Built from Delta-like Function
 a ( x)   ( x),  a  x  a, P  2a ................................................................................ 598
15.2.5.2. The Saw-tooth Function Built from f ( x)  x, 0  x  a ..................................... 598
15.2.5.3. Periodic Function Defined from the Heaviside Function ........................................ 600
15.2.5.4. The Square Wave Function SW ( x) ....................................................................... 600
15.2.5.5. The Periodic Function Constructed from x .......................................................... 601
15.2.5.6. Fourier Series of a Function Corresponding to a Displaced Graph ......................... 602
15.2.5.7. Fourier Series of a Function Corresponding to a Symmetrised Graph .................... 603
15.2.5.8. The Periodic Function Built from f ( x)  x2 , 0  x  a, P  a ......................... 603
15.2.5.9. The Ramp Function Built fromf ( x)  0,  a  x  0, f ( x)  x,0  x  a,

2a ............................................................................................................................605
15.2.5.10. The Parabolic Function f ( x)  x2 , a  x  a, P  2a .................................606
15.2.5.11. The Periodic Function Built from f ( x)  x3 , 0  x  a, P  a ....................... 608
15.2.6. Polynomial Equivalent to an Individual Series in a Fourier Series .................................. 609
15.2.7. Obtaining Bernoulli Polynomials as Fourier Series ......................................................... 610
15.2.8. Obtaining Powered of Pi and Even Positive Zeta Functions ............................................ 615
15.2.9. Fourier Series of Polynomials........................................................................................... 615

15.3. Remarks and Conclusion .............................................................................................. 616
Acknowledgements ................................................................................................................ 617
References.............................................................................................................................. 617

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Contents

16. Obtaining Fourier Transforms of Functions by Differential Calculus............. 619
16.1. Introduction .................................................................................................................. 619
16.2. Differential Operators Realizing Fourier Transform .................................................... 620
16.2.1. Introduction to Operator Calculus.................................................................................... 620
16.2.2. Operators Realizing the Fourier Transform ..................................................................... 622
16.2.3. The Fourier Transform of f ( x)  1 .................................................................................. 625
16.2.4. Geometric Signification of a Convolution Product ........................................................... 626
16.2.5. Fourier Transform of a Convolution Product ................................................................... 627

16.3. General Properties of the Fourier Transform ............................................................... 627
16.3.1. The Primordial Property ................................................................................................... 628
16.3.2. The Convolution Product .................................................................................................. 628
16.3.3. The double Fourier Transform.......................................................................................... 628
16.3.4. The Inverse Transform ...................................................................................................... 628
16.3.5. Transform of Complex Conjugate ..................................................................................... 628
16.3.6. Transform of Translated Functions Following (16.11) ..................................................... 628
16.3.7. Transform of a Dilated Function ...................................................................................... 629
16.3.8. Transform of a Sum of Two Functions .............................................................................. 629

16.4. Table of Fourier Transform of Differential Operators ................................................. 629
16.5. Table of Fourier Transform of Functions ..................................................................... 630
16.6. Remarks and Conclusion .............................................................................................. 633
Acknowledgments ................................................................................................................. 633
References ............................................................................................................................. 633

Index ............................................................................................................................. 635

15

Contributors

Contributors
Nikolai Andreev
Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanova St,
Nizhny Novgorod, 603950, Russia
Alpan Bek
Department of Physics, Middle East Technical University, 06800 Ankara, Turkey
E-mail: bek@metu.edu.tr
Edwin Caballero-Agosto
Department of Chemistry, University of Puerto Rico, Mayagüez
John R. Castro-Suárez
Department of Chemistry, University of Puerto Rico, Mayagüez
Shubo Cheng
School of Physics and Electronics, Central South University, Changsha 410083,
China
Charles Ciret
Laboratoire de Photonique d’Angers, Université d’Angers, 2 Bd Lavoisier, 49045,
Angers cedex 01, France
Francheska M. Colón-González
Department of Chemistry, University of Puerto Rico, Mayagüez
Annette M. Colón-Mercado
Department of Chemistry, University of Puerto Rico, Mayagüez
Ling Fan
School of science, Beijing Jiaotong University, Beijing, China
Email: lfan@bjtu.edu.cn
Nataly J. Galán-Freyle
Department of Chemistry, University of Puerto Rico, Mayagüez
Shuhong Gong
Xidian University, Xi’an 710071, China
Juan C. González de Sande
Department of Materials Science, Universidad Politécnica de Madrid, Spain
Kazuo Hasegawa
The Graduate School for the Creation of New Photonics Industries (GPI),
Hamamatsu, Shizuoka, Japan
Toyota Central R&D Labs., Inc., Nagakute, Aichi, Japan
Samuel P. Hernandez-Rivera
Department of Chemistry, University of Puerto Rico, Mayagüez
17Advances in Optics: Reviews. Book Series, Vol. 5

Boian A. Hristov
Institute of Optical Materials and Technologies, Bulgarian Academy of Sciences,
Sofia, Bulgaria
Fırat İdikut
Department of Physics, Middle East Technical University, 06800 Ankara, Turkey
Paul Ikechukwu Iroegbu
School of Microelectronics and Communication Engineering, Chongqing
University, Chongqing, 400044, China
Key Laboratory of Optoelectronic Technology and Systems (Education Ministryof China), Chongqing University, Chongqing, 400044, China
Email: paul.iroegbu@gmail.com
Hiroshi Ito
Institute of Materials Innovation, Institutes of Innovation for Future Society,
Nagoya University, Japan
Burcu Karagöz
Department of Physics, Middle East Technical University, 06800 Ankara, Turkey
Meriem Kemel
Laboratoire de Photonique d’Angers, Université d’Angers, 2 Bd Lavoisier, 49045,
Angers cedex 01, France
Renxian Li
School of Physics and Optoelectronic Engineering, Xidian University, Xi’an
710071, China
Collaborative Innovation Center of Information Sensing and Understanding, Xidian
University, Xi’an 710071, China
Min Liu
School of Microelectronics and Communication Engineering, Chongqing
University, Chongqing, 400044, China
Key Laboratory of Optoelectronic Technology and Systems (Education Ministryof China), Chongqing University, Chongqing, 400044, China
Alexandre Makarov
Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanova St,
Nizhny Novgorod, 603950, Russia
Shintaro Mizuno
Toyota Central R&D Labs., Inc., Nagakute, Aichi, Japan
Tomoyoshi Motohiro
Institute of Materials Innovation, Institutes of Innovation for Future Society,
Nagoya University, Japan
Ahmed Nady
Laboratoire de Photonique d’Angers, Université d’Angers, 2 Bd Lavoisier, 49045,
Angers cedex 01, France
18

Contributors

Salam Nazhan
Department of Communication, College of Engineering, University of Diyala, Iraq
Leonardo C. Pacheco-Londoño
Department of Chemistry, University of Puerto Rico, Mayagüez
Sahnggi Park
Oprocessor Inc., Boston, MA, USA, E-mail: sahnggi@gmail.com
Gemma Piquero
Department of Materials Science, Universidad Politécnica de Madrid, Spain
Antonio Parretta
Physics and Earth Science Department, University of Ferrara, Academy of Sciencesof Ferrara, Italy
Mohamed Salhi
Laboratoire de Photonique d’Angers, Université d’Angers, 2 Bd Lavoisier, 49045,
Angers cedex 01, France
François Sanchez
Laboratoire de Photonique d’Angers, Université d’Angers, 2 Bd Lavoisier, 49045,
Angers cedex 01, France
Massimo Santarsiero
Department of Materials Science, Universidad Politécnica de Madrid, Spain
Georges SemaanLaboratoire de Photonique d’Angers, Université d’Angers, 2 Bd Lavoisier, 49045,
Angers cedex 01, France
Do Tan Si
HoChiMinh-city Physical Association, Vietnam
E-mail: tansi_do@yahoo.com
Ningning Song
Xidian University, Xi’an 710071, China
Juan C. Suárez-Bermejo
Department of Materials Science, Universidad Politécnica de Madrid, Spain
Han SunXidian University, Xi’an 710071, China
Yasuhiko Takeda
Toyota Central R&D Labs., Inc., Nagakute, Aichi, Japan
Huan Tang
Xidian University, Xi’an 710071, China
Shaohua Tao
School of Physics and Electronics, Central South University, Changsha 410083,
China
19

Advances in Optics: Reviews. Book Series, Vol. 5

Karla M. Vázquez-Vélez
Department of Chemistry, University of Puerto Rico, Mayagüez
Vladimir Villanueva-López
Department of Chemistry, University of Puerto Rico, Mayagüez
Ksenia Vlasova
Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanova St,
Nizhny Novgorod, 603950, Russia
Bing Wei
Xidian University, Xi’an 710071, China
Bojian Wei
Xidian University, Xi’an 710071, China
Qun Wei
Xidian University, Xi’an 710071, China
Tian Xia
School of Physics and Electronics, Central South University, Changsha 410083,
China
Liu Yang
Xidian University, Xi’an 710071, China
Shu Zhang
Xidian University, Xi’an 710071, China
Sijiong Zhang
Nanjing Institute of Astronomical Optics and Technology, Nanjing, China
University of Chinese Academy of Sciences, Beijing, China
E-mail: sjzhang@niaot.ac.cn
Tao Zhu
Key Laboratory of Optoelectronic Technology and Systems (Education Ministryof China), Chongqing University, Chongqing, 400044, China

20

Preface

Preface
It is my great pleasure to introduce the fifth volume from our popular open access Book
Series ‘Advances in Optics: Reviews’ started by the IFSA Publishing in 2018. The Vol. 5of this Book Series is also published as an Open Access Book in order to significantly
increase the reach and impact of this volume, which also published in two formats:electronic (pdf) with full-color illustrations and print (paperback).
‘Advances in Optics: Reviews’ Book Series is a comprehensive study of the field of optics,
which provides readers with the most up-to-date coverage of optics, photonics and lasers
with a good balance of practical and theoretical aspects. Directed towards both physicists
and engineers this Book Series is also suitable for audiences focusing on applications of
optics. A clear comprehensive presentation makes these books work well as both ateaching resources and a reference books. The book is intended for researchers andscientists in physics and optics, in academia and industry, as well as postgraduate students
Like the first four volumes of this Book Series, the fifth volume also has been organized
by topics of high interest to offer a fast and easy reading of each topic, every chapter inthis book is independent and self-contained. All chapters have the same structure: first an
introduction to specific topic under study; second particular field description including
sensing or/and measuring applications. Each of chapter is ending by well selected list ofreferences with books, journals, conference proceedings and web sites.
The fifth volume is devoted to optics, lasers, optical communication and networks, andwritten by 53 authors from academia and industry from 12 countries: Belgium, Bulgaria,
China, France, Italy, Iraq, Japan, Russia, Spain, Turkey, USA and Vietnam. But it is nota simple set of reviews. As usually, each chapter contains the extended state-of-the-art
followed by new, unpublished before, obtained by the authors results.
This book ensures that our readers will stay at the cutting edge of the field and get theright and effective start point and road map for the further researches and developments.
By this way, they will be able to save more time for productive research activity andeliminate routine work.
I shall gratefully receive any advices, comments, suggestions and notes from readers tomake the next volumes of ‘Advances in Optics: Reviews’ Book Series very interesting anduseful.
Dr. Sergey Y. Yurish
EditorIFSA Publishing

Barcelona, Spain
21

Chapter 1. Physical Optics

Chapter 1

Physical Optics
Sijiong Zhang and Ling Fan1

1.1. Wave Equations of the Electromagnetic Field
1.1.1. The Maxwell’s Equations
In this section, we will summarize the theory of the electromagnetic wave starting fromMaxwell’s equations that lead to the equations of electromagnetic waves. The standard
Maxwell’s macroscopic equations in a medium are written
𝛻 · 𝑫 = 𝜌,

(1.1)

𝛻 · 𝑩 = 0,

(1.2)
𝜕𝑫

𝛻 × 𝑯 = 𝑱 + 𝜕𝑡 ,
𝛻 × 𝑬 = −

𝜕𝑩
,
𝜕𝑡

(1.3)
(1.4)

where E and D are the electric field vector and the electric displacement vector
respectively; B and H the magnetic induction vector and the magnetic field vector,
respectively, and ρ and J the free charge density and the electric current density vector
generated by free charges, respectively. And ∇·and ∇×, respectively, stand for thedivergence and curl operators. The equation (1.1) is Gaussian Law for electric fields, theequation (1.2) Gaussian Law for magnetic fields, the equation (1.3) Ampere’s Law with𝜕𝑫
the displacement current 𝜕𝑡 introduced by James C. Maxwell, and the equation (1.4)
Faraday’s Law.

Sijiong Zhang
Nanjing Institute of Astronomical Optics and Technology, University of Chinese Academy of Sciences,
China

23

Advances in Optics: Reviews. Book Series, Vol. 5

1.1.2. Waves in a Linear, Isotropic, and Homogeneous Medium
In this chapter, we mainly consider about electromagnetic fields within a medium that islinear, isotropic, homogeneous, and non-conducting, in which free charge density ρ andfree current density J are all zeros. The linearity of the medium is the property in which
the response of the medium to an applied electromagnetic field is proportional to the field.
An isotropic medium refers to one in which the properties of the medium in all directions
are the same. A homogeneous medium is the one in which its index of refraction is
identical at every point. Such a medium with the relative permittivity 𝜖 and the relative
permeability 𝜇 can be described by material equations
𝑫 = 𝜖𝜖0 𝑬, 𝑩 = 𝜇𝜇0 𝑯,(1.5)

where 𝜖0 and 𝜇0 are the vacuum permittivity and the vacuum permeability respectively;
𝜖 and 𝜇 are generally dependent on the frequency of the electromagnetic wave, which isthe dispersive property of the medium.
On inserting the equation (1.5) into the equations (1.1) to (1.4), and considering ρ = 0 and
J = 0, we get the following equations
𝛻 · 𝑬 = 0,

(1.6)

𝛻 · 𝑯 = 0,

(1.7)
𝜕𝑬

𝛻 × 𝑯 = 𝜖𝜖0 𝜕𝑡 ,
𝛻 × 𝑬 = −𝜇𝜇0

𝜕𝑯
𝜕𝑡

(1.8)
(1.9)

On taking the curl of both sides of the equation (1.9) and inserting the equation (1.8) into
it, we obtain𝛻 × (𝛻 × 𝑬) = −𝜇𝜇0𝜕𝜕2 𝑬(𝛻 × 𝑯) = −𝜖𝜇𝜖0 𝜇0 2
𝜕𝑡
𝜕𝑡

(1.10)

By invoking the formula 𝛻 × (𝛻 × 𝑬) = 𝛻 (𝛻 · 𝑬) − 𝛻 2 𝑬 , where 𝛻 2 is Laplace’s
operator, and considering the equation (1.6), then the equation (1.10) becomes
𝜕2 𝑬

𝛻 2 𝑬 = 𝜖𝜇𝜖0 𝜇0 𝜕𝑡 2

(1.11)

The above equation is a vector wave equation for the electric field. Furthermore,1
let 𝑣 2 = 𝜖𝜇𝜖 𝜇 , which 𝑣 is the wave speed for the wave equation (1.11). Then we get a
0 0standard electromagnetic wave equation as𝛻 2𝑬 =
24

1 𝜕2 𝑬𝑣 2 𝜕𝑡 2(1.12)

Chapter 1. Physical Optics

This is a fundamental and starting formula for studying physical optics or wave optics ina linear, isotropic, and homogeneous dielectric medium.
1.1.3. Wave Speed of Light and the Index of Refraction
From the formula (1.11), we know that wave speed in the medium is𝑣 =In free space, the wave speed 𝑐 =

1
𝜖𝜇𝜖
√
0 𝜇0

1
√𝜖0 𝜇0(1.13), which is just the light speed in vacuum. Basedon this point, Maxwell thought of light as an electromagnetic wave. According to the𝑐definition of the index of refraction 𝑛 = 𝑣 and the expression (1.13), such that we can getthat the index of refraction of the medium is 𝑛 = √𝜖𝜇. In optics, it is generally valid that𝜇 ≈ 1 (the reason is to be explained in Section 1.1.4), so that the index of refraction of themedium can be written 𝑛 = √𝜖.
1.1.4. Scalar Time Harmonic Waves
When the Cartesian components of the light field 𝐄, 𝐇 are not coupled, the equation (1.12)
can be reduced to a scalar wave equation
1 𝜕2 𝑈𝛻 2 𝑈 = 𝑣 2 𝜕𝑡 2 ,(1.14)

where U is a scalar light field that represents any Cartesian components of E and H. In
optics the scalar U just stands for the electric field. The reason for U representing for theelectric field is that in view of atomic physics the electric field of light influences theatoms much stronger than the magnetic field does. In the optical frequency domain, thepower of magnetic diploe radiation is basically equivalent to that of electric quadrupole
radiation which is much weaker than that of electric dipole radiation that is the dominatedcontribution to the radiation of atoms [12].
For most of our work on optics, the scalar wave equation (1.14) is accurate enough todescribe optical phenomena provided that the diffracting structures and the observation
distance are large compared with the wavelength of light. Attention is now focused on thecase of a monochromatic wave. By means of the Fourier transform theory, thegeneralization to polychromatic or nonmonochromatic waves can be done by Fourier
synthesis by means of the results of the monochromatic waves, which will be discussed
in Section 1.2.
For a monochromatic wave, its analytic expression can be written𝑈(𝒓, 𝑡) = 𝑈(𝒓)𝑒 −𝑗𝜔𝑡 ,(1.15)

25

Advances in Optics: Reviews. Book Series, Vol. 5

where r is the spatial coordinate in the three dimensional space, t time, 𝜔 the angularfrequency of light, and U(r) the complex amplitude of the monochromatic wave.
On substituting the equation (1.15) into (1.14), it follows that U(r) obeys the Helmholtz
equation𝛻 2 𝑈(𝒓) + 𝑘 2 𝑈(𝒓) = 0,

(1.16)

2𝜋

where k is called the wave number, given by k = 𝜆 , and 𝜆 is the wavelength of light inthe medium. Equation (1.16) is used as a starting point to study the diffraction
theory of light.
There are many different solutions of the equation of vector waves (1.12) for different
situations. Particularly, we provide two useful solutions without proof, plane waves andspherical waves, which are often used in this chapter.
The electric field of a plane wave can be expressed as an analytic function
𝑬 = 𝑬𝟎 𝑒 𝑗(𝒌∙𝒓−𝜔𝑡) , where 𝒌 is the wave-vector in the medium through which light is
propagating, 𝒓 the spatial position, and 𝐸0 the amplitude vector of the wave. The planewaves are the basis for Fourier optics. We should emphasize that the phase of the plane
wave is denoted as the form (𝒌 ∙ 𝒓 − 𝜔𝑡). As a result of this sign convention of theright-handed Cartesian coordinate adopted, the electric field 𝑬𝟎 𝑒 𝒋(𝒌∙𝒓−𝝎𝒕+𝜹) lags theelectric field 𝑬𝟎 𝑒 𝑗(𝒌∙𝒓−𝜔𝑡) by δ if the additional phase δ > 0; conversely, if the additional
phase δ < 0 the electric field 𝑬𝟎 𝑒 𝒋(𝒌∙𝒓−𝝎𝒕+𝜹) leads the electric field 𝐸0 𝑒 𝑗(𝒌∙𝒓−𝜔𝑡) by |δ|.
We should aware of that the formalism of representation for polarization of light isdependent on the notation of the phase, which will be discussed in the following
Section 1.6.
The electric field of a spherical wave can be expressed as an analytic function
𝑬
𝐸 = 𝑟𝟎 𝑒 𝑗(𝒌∙𝒓−𝜔𝑡) , where 𝑟 = |𝒓| stands for the distance from the origin, and themeaning of the other quantities in the expression is same as those for the plane wave. This
wave represents one propagating outwards from the origin, if wavenumber is positive.
The spherical waves are building blocks for Huygens’ principle.
1.1.5. The Energy Law of the Electromagnetic Field: Pointing Vector
An electromagnetic field is a wave, and so is light. The wave can transport energy. Theenergy flux density vector, representing the power passing through unit area, is called
Poynting vector expressed as
𝑺 = 𝑬 × 𝑯
The light intensity, 𝐼, is defined as the time averaged power crossing a unit area.
1

𝑇/2

𝐼 = 𝑇 ∫−𝑇/2 𝑆(𝑡) 𝑑𝑡 ≡< 𝑆(𝑡) >,26(1.17)

Chapter 1. Physical Optics

where T is the response time of an optical detector.
In the following, we will give out the concrete expression of the light intensity of a planewave. On inserting the expression 𝑬 = 𝑬𝟎 𝑒 𝑗(𝒌∙𝒓−𝜔𝑡) into Maxwell’s equations (1.6) to𝜕(1.9), and considering that 𝛻 ≡ 𝑖𝒌, 𝜕𝑡 ≡ −𝑖𝜔, we get the following equations
𝒌 · 𝑬 = 0, 𝒌 · 𝑯 = 0, 𝒌 × 𝑯 = −𝜔𝜖𝜖0 𝑬, 𝒌 × 𝑬 = 𝜔𝜇0 𝑯(1.18)

From the above equations, it is easy to know that 𝑫, 𝑯, and 𝒌 form a right-handed
mutually orthogonal triad, so do 𝑬, 𝑩, and 𝒌. For an isotropic medium, 𝑬, 𝑯, and 𝒌 also
form a right-handed mutually orthogonal triad, and the electromagnetic wave istransverse. Furthermore, the magnetic field for the plane wave can be derived as
𝑯 =1
𝑬 𝑒 𝑗(𝒌∙𝒓−𝜔𝑡) , where that 𝑍
𝑍 𝟎

|𝑬|

= |𝑯| =

|𝒌|
𝜖𝜖0 𝜔

1
𝜇0
√
𝜖
𝜖0
√

=

=

𝑍0
is the impedance of𝑛the light field in the medium, further the index of refraction of the medium 𝑛 = √𝜖 and𝜇the impedance of the light field in vacuum 𝑍0 = √ 𝜖 0 . Then the intensity of light for a0plane wave is
1

𝑇

1

1

𝐼 = 𝑇 ∫2𝑇 𝐸0 𝑐𝑜𝑠(𝜔𝑡) 𝐻0 𝑐𝑜𝑠(𝜔𝑡) 𝑑𝑡 = 2𝑍 𝐸02 = 2𝑍 𝑛𝐸02 ,−20(1.19)
1

where 𝐸0 is the magnitude of 𝑬𝟎 . To get the equation (1.19) the condition 𝑇 ≫ 𝜔 isrequired, which is always valid because of the very high frequency, 𝜔, of the light field.
From the expression (1.19), we know that the light intensity is proportional to the electricfield squared. An optical detector can just detect the energy of light, such its detected
result is proportional to the electric field squared. Based on this reason, the optical detector
is called a square-law detector.
1.1.6. Vectorial Nature of the Electromagnetic Field
As stated previously, the electromagnetic fields are vectorial and transverse vectors, so
that, at every point in the wave, the electric field E, the magnetic field H, and Poynting
vector S forms a mutually perpendicular triad. The direction of the electric field E is
known as the polarization of the wave.
Depending on how the electric field is oriented, the polarized light can be classified into
three types of polarization: elliptical polarization, and its degenerate cases, i.e., circularpolarization and linear polarization.
Linear polarization: the electric field of light keeps the fixed direction in a single plane
being perpendicular to the direction of propagation, as shown in Fig. 1.1.
Circular polarization: two orthogonal components of the electric field of light are equal inmagnitude, but have a phase difference of π/2. The resulting electric field then rotates in27

Advances in Optics: Reviews. Book Series, Vol. 5

a circle around the direction of propagation. Depending on the rotation sense, there exists
left- or right-hand circularly polarized light, as shown in Fig. 1.2.

Fig. 1.1. Schematic of electric vectors of the linearly polarized light with different orientations.
The light propagates out of the diagram along z-axis.

Fig. 1.2. Schematic of electric vectors of the left- and right-circularly polarized light.
The light propagates out of the diagram along z-axis.

Elliptical polarization: two orthogonal components of the electric field of light are notequal in magnitude, and a phase difference is different from zero. The resulting electricfield then rotates in an ellipse around the direction of propagation. This is the most generaldescription of polarized light, and circular and linear polarized light can be viewed as the
special cases of elliptically polarized light, as shown in Fig. 1.3.

Fig. 1.3. Schematic of electric vectors of the left- and right-elliptically polarized light.
The light propagates out of the diagram along z-axis.

28

Chapter 1. Physical Optics

General treatment for completely polarized, unpolarized and partially polarized light will
be discussed in Section 1.6.
1.1.7. Electromagnetic Theory of Radiation
A linearly accelerating charge and an oscillating dipole are two important types of sources
for radiation in optics and we will briefly summarize them here.

1.1.7.1. Radiation from an Accelerating Charge
A charged particle moving at uniform velocity, according to Ampere’s law, just produces
a constant magnetic field. From the equation (1.9), a constant magnetic field cannot induce
an electric field, which means that a moving particle with a constant velocity does notradiate electromagnetic waves. However, if the particle accelerates, the magnetic field has
a time derivative that induces an electric field. As a result, the accelerated particle
generates radiation. For a particle of charge q moving with velocity 𝒗(𝑡) and thenacceleration 𝒗̇ (𝑡) (the top dot refers to derivative), the far field of radiation at the spatial
position r is𝑞

𝑬 = 4𝜋𝜖 𝑟𝑐 2 [𝒓̂ × (𝒓̂ × 𝒗̇ )],0(1.20)

where 𝒓̂ is the unit vector along the direction of r, 𝑟 = |𝒓|, representing the magnitude ofr and its value being large comparing with the wavelength of light, and c the speed oflight. From the equation (1.20), the electric field of radiation is always perpendicular tothe direction of 𝒓̂, and E, 𝒓̂, and 𝒗̇ are coplanar, as shown in Fig. 1.4.

Fig. 1.4. Schematic of spatial relationship of a triad of E, 𝒓̂, and 𝒗̇ .

From the Fig. 1.4 and the formula (1.20), we know that the magnitude of the electric field𝜋
is maximum when 𝜃 = 2 ; and the magnitude of the electric field is minimum
when 𝜃 = 0.
29

Advances in Optics: Reviews. Book Series, Vol. 5

1.1.7.2. Radiation from an Oscillating Dipole
The radiative source most frequently encountered in optics is a periodically oscillating
dipole that can radiate light, for example, in the scattering theory. Let a point charge q has
a position vector with time, 𝐴𝑐𝑜𝑠(𝜔𝑡)𝒛̂, A being oscillating amplitude of the dipole, 𝜔representing oscillatory frequency of a dipole with the direction 𝒛̂. The moment of thedipole is 𝒑 = 𝑞𝐴 𝑐𝑜𝑠(𝜔𝑡)𝒛̂. The acceleration of the dipole is𝒑̈𝒗̇ (𝑡) = −𝐴𝜔2 𝑐𝑜𝑠(𝜔𝑡)𝒛̂ = 𝑞,
where the double dot on the top of 𝒑 represents the second derivative with respect to time.
From the equation (1.20), we obtain that the far field of radiation at the spatial
position r is
𝑬 =1
[𝒓̂ × (𝒓̂ × 𝒑̈ )]4𝜋𝜖0 𝑟𝑐 2(1.21)

From the above equation, we can see that the electric field of radiation is always
perpendicular to the direction of 𝒓̂, and 𝑬, 𝒓̂, and 𝒑̈ are coplanar, as shown in Fig. 1.5 (a).
And the same as the case of radiation from an accelerating charge, the magnitude of
position vector 𝒓 is also large comparing to the wavelength of light.

Fig. 1.5. The spatial relationship of a triad of E, 𝒓̂, and 𝒑̈ (a), and the intensity radiation patternof an oscillating dipole as a function of angle 𝜃 (b).

As the case of the radiation from an accelerating charge, for the radiation from an
oscillating dipole the magnitude of the electric field is maximum when 𝜃 = 𝜋/2; and themagnitude of the electric field is minimum when 𝜃 = 0, shown as in Fig. 1.5(b).
For these two cases, radiation field E is always perpendicular to 𝒓̂ . This property of
perpendicularity means that the light wave is a transverse wave.
1.1.8. Evanescent Waves
When light in a dielectric medium with the index of refraction 𝑛1 is incident onto aninterface of a different dielectric material with a lower index of refraction 𝑛2 , at an angle
30

Chapter 1. Physical Optics

𝜃1 , as shown in Fig. 1.6. The propagation of light, in view of geometrical optics, obeys
Snell’s law, 𝑛1 𝑠𝑖𝑛 𝜃1 = 𝑛2 𝑠𝑖𝑛 𝜃2 , where 𝜃2 is the refracted angle.
For an incident angle 𝜃1 greater than the critical angle defined as the following𝑛𝑛1𝜃𝑐 = 𝑠𝑖𝑛−1 ( 2 ),(1.22)

the total internal reflection occurs. Although all of the incident energy is reflected,
Maxwell’s equations predict the existence of an electromagnetic field in the less dense
medium 𝑛2 with the intensity decaying exponentially away from the interface. This field
is known as the evanescent wave.

Fig. 1.6. Schematic of the light refraction on the interface.

The existence of the evanescent wave can be appreciated by considering the transmitted
plane wave of light:
𝑬𝒕 = 𝑬𝒕𝟎 𝑒 𝑗(𝒌𝑡 ∙𝒓−𝜔𝑡) ,(1.23)

where 𝑬𝒕 and 𝑬𝒕0 are the electric vector of the transmitted light wave and its amplitude
vector respectively; 𝒌𝑡 is the wave-vector of the transmitted light wave, 𝒓 is the position
vector, and ω is the angular frequency of light. According to the coordinate axes shownin Fig. 1.6, we have
(1.24)

𝒌𝑡 ∙ 𝒓 = 𝑘𝑡 𝑥 𝑠𝑖𝑛 𝜃2 + 𝑘𝑡 𝑧𝑐𝑜𝑠𝜃2𝑛2 𝑠𝑖𝑛2 𝜃1

Combining Snell’s law with (1.24), we have 𝑐𝑜𝑠𝜃2 = √1 − 1 𝑛2

. When incident

2

angle 𝜃1 is greater than the critical angle 𝜃𝑐 , 𝑐𝑜𝑠𝜃2 becomes pure imaginary and we can𝑛2 𝑠𝑖𝑛2 𝜃1write 𝑐𝑜𝑠𝜃2 = 𝑗√ 1 𝑛2
2

− 1. Thus the term (𝒌𝑡 ∙ 𝒓) can be written.
31Advances in Optics: Reviews. Book Series, Vol. 5
𝑛2 𝑠𝑖𝑛2 𝜃1
− 1,
𝑛22

𝒌𝑡 ∙ 𝒓 = 𝑘𝑡 𝑥 𝑠𝑖𝑛 𝜃2 + 𝑗𝑘𝑡 𝑧 √ 1(1.25)

2𝜋

where 𝑘𝑡 = 𝜆 𝑛2 is the magnitude of 𝒌𝑡 , 𝜆 is the wavelength of light in the vacuum
space. On inserting the equation (1.25) into the equation (1.23) and omitting the time
factor, we have2

𝑬𝒕 = 𝑬𝒕𝟎 𝑒 𝑗𝑘𝑡𝑥 𝑠𝑖𝑛 𝜃2 𝑒𝑛 𝑠𝑖𝑛2 𝜃−𝑘𝑡 𝑧√ 1 2 1 −1𝑛2

(1.26)

From the expressions (1.19) and (1.26), the intensity of light field, 𝐼 ∝ |𝑬𝒕 |2 , can be
written in the second medium𝑧

𝐼 = 𝐼0 𝑒 −𝑑 ,(1.27)

where 𝐼0 ∝ |𝑬𝒕𝟎 |2 is the intensity of light field of the incident on the interface, and apenetration depth given by𝑑 =𝜆4𝜋√𝑛12 𝑠𝑖𝑛2 𝜃1 −𝑛22

(1.28)

This evanescent wave propagates parallel to the interface and has a penetration depth, 𝑑,on the order of the wavelength of the illuminating light, λ. However, light is not actuallyreflected at the exact position of the interface, but rather penetrates a distance 𝑑 into themedium with the lower index of refraction by optical tunneling. As a result, the reflectedbeam of light is shifted along the interface by a small amount 𝑥 = 2𝑑𝑡𝑎𝑛𝜃1 , which is
known as the Goos-Hänchen Shift as shown in Fig. 1.7.

Fig. 1.7. Schematic of the Goos-Hänchen Shift.

The evanescent wave in the second medium, or optical tunnelling, can be detected. As
shown in Fig. 1.8, two same right angle prisms are placed with their long faces close
32

Chapter 1. Physical Optics

tighter but not in actual contact. Light from a source S is found to be partially transmitted
into the second right-angle prism, the amount of light transmitted depending on theseparation of the prism faces.

Fig. 1.8. Schematic of the effect of optical tunnelling.

The total internal reflection and the evanescent wave have been exploited to apply to many
fields, for example, optical fibers and the evanescent wave microscopy. Optical fibers use
the total internal reflection at the interface between core and cladding media to transfer
energy along to the fiber. The evanescent wave microscopy or total internal reflection
microscopy (TIRM) can increase the resolution of images.
At last, we will discuss the laws of reflection and refraction, encountered in geometrical
optics, on a plane boundary between two homogeneous dielectric media using Maxwell’sequations.
1.1.9. Laws of Reflection and Refraction
The phenomena of reflection and refraction are very important in optics, especially they
are the physical basis for instrumental optics. When a plane wave is incident onto a planeboundary between two homogeneous dielectric media with different indices, this wave ispartly transmitted into the second medium and partly reflected back into the first medium.
Here we are going to derive, by using macroscopic Maxwell’s equations, the laws ofreflection and refraction on an abrupt plane boundary between two homogeneous
dielectric media.
1.1.9.1. Laws of Reflection and Refraction for Homogeneous Dielectric MediaAs it is shown in Fig. 1.9, a plane wave, 𝑬𝐼 = 𝑬0𝐼 𝑒 𝑗(𝒌𝐼∙𝒓−𝜔𝑡) , at the incident angle 𝜃𝐼 , isincident on the interface between the medium with the index of refraction 𝑛1 and the one
with the index of refraction 𝑛2 . The reflected wave, 𝑬𝑅 = 𝑬0𝑅 𝑒 𝑗(𝒌𝑅∙𝒓−𝜔𝑡) , is reflectedback into the first medium at the reflected angle 𝜃𝑅 . And the refracted wave,
𝑬𝑇 = 𝑬0𝑇 𝑒 𝑗(𝒌𝑇 ∙𝒓−𝜔𝑡) , is transmitted into the second medium at the refracted angle is 𝜃𝑇 .
The plane specified by the incident wave vector 𝒌𝐼 and the normal z-axis of the boundaryis called the plane of incidence.

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Advances in Optics: Reviews. Book Series, Vol. 5

At any point on the boundary between the two media, for example, the meeting point of
three light rays at the boundary, as shown in Fig. 1.9, the phases of three light waves are
the same at any time instant. We therefore get the following expression.
(1.29)

𝒌𝐼 ∙ 𝒓 = 𝒌𝑅 ∙ 𝒓 = 𝒌𝑇 ∙ 𝒓,
where 𝒓 = (𝑥, 𝑦, 0) represents any point on the boundary, 𝑘𝐼 = |𝒌𝐼 | =2𝜋𝑘𝑅 = |𝒌𝑅 | = 𝜆2𝜋𝑛1 , and 𝑘 𝑇 = |𝒌𝑇 | = 𝜆

2𝜋
𝜆

𝑛1 ,

𝑛2 . We write three wave vectors in theform of components, respectively, as 𝒌𝐼 = (𝑘𝐼𝑥 , 𝑘𝐼𝑦 , 𝑘𝐼𝑧 ), 𝒌𝑅 = (𝑘𝑅𝑥 , 𝑘𝑅𝑦 , 𝑘𝑅𝑧 ), and𝒌𝑻 = (𝑘 𝑇𝑥 , 𝑘 𝑇𝑦 , 𝑘 𝑇𝑧 ). On inserting them into the equation (1.29), we have𝑥𝑘𝐼𝑥 + 𝑦𝑘𝐼𝑦 = 𝑥𝑘𝑅𝑥 + 𝑦𝑘𝑅𝑦 = 𝑥𝑘 𝑇𝑥 + 𝑦𝑘 𝑇𝑦(1.30)

Because the equations (1.30) are valid for all values 𝑥 and 𝑦, the following equations must
be valid.

Fig. 1.9. Reflection and refraction of a plane wave.

𝑘𝐼𝑥 = 𝑘𝑅𝑥 = 𝑘 𝑇𝑥 , 𝑘𝐼𝑦 = 𝑘𝑅𝑦 = 𝑘 𝑇𝑦

(1.31)The equations (1.31) shows that both 𝒌𝑅 and 𝒌𝑇 lie the plane of incidence. Based on this𝟐𝝅
result and Fig. 1.9, we can, by trigonometry, obtain that 𝒌𝐼 = 𝝀 𝑛1 (𝑠𝑖𝑛𝜃𝐼 , 0, −𝑐𝑜𝑠𝜃𝐼 ),𝟐𝝅

𝟐𝝅

then 𝒌𝑅 = 𝝀 𝑛1 (𝑠𝑖𝑛𝜃𝑅 , 0, 𝑐𝑜𝑠𝜃𝑅 ) , and 𝒌𝑇 = 𝝀 𝑛2 (𝑠𝑖𝑛𝜃𝑇 , 0, −𝑐𝑜𝑠𝜃𝑇 ) . On
combining them with the equation (1.31), such that the equation (1.31) can be explicitly
written as follows.
2𝜋
𝜆𝑛1 𝑠𝑖𝑛𝜃𝐼 =2𝜋
𝜆To simplify the above equations, we have34

2𝜋

𝑛1 𝑠𝑖𝑛𝜃𝑅 , 𝜆 𝑛1 𝑠𝑖𝑛𝜃𝐼 =2𝜋
𝜆𝑛2 𝑠𝑖𝑛𝜃𝑇

Chapter 1. Physical Optics

𝑠𝑖𝑛𝜃𝐼 = 𝑠𝑖𝑛𝜃𝑅 or 𝜃𝐼 = 𝜃𝑅 ,(1.32)

𝑛1 𝑠𝑖𝑛𝜃𝐼 = 𝑛2 𝑠𝑖𝑛𝜃𝑇(1.33)The equation (1.32), together with the statement that the reflected wave vector 𝒌𝑅 lies in
the plane of incidence is just the law of reflection; the equation (1.33), together with thestatement that the refracted wave vector 𝒌𝑇 lies in the plane of incidence is just the law ofrefraction.
1.1.9.2. Fresnel’s Formulas
We now consider the amplitudes of the reflected and the refracted waves. And assume
that the two media are homogeneous, isotropic, of zero conductivity, and non-magnetic(𝜇 = 1). At an abrupt boundary between two media, the boundary conditions must beobeyed between the fields on the two sides. In words, the parallel components of the fields
𝑬 and 𝑯 to the boundary are equal on the two sides, whereas the normal components of
𝑫 and 𝑩 to the boundary are also continuous. Detailed derivation of the boundary
conditions, readers can refer to the reference [1].
We resolve each field vector into the parallel components (𝑝) and the normal components(𝑠) to the plane of incidence. Then the fields 𝑬 and 𝑯 can be expressed as 𝑬 = 𝑬𝒑 + 𝑬𝒔 ,and 𝑯 = 𝑯𝒑 + 𝑯𝒔 , as shown in Fig. 1.10. By means of the equation (1.18) and theexpression of the plane wave, we have
𝑯𝒑 =1
𝒌 × 𝑬𝒔 , 𝑯𝒔
𝜔𝜇0

=

𝟏
𝒌 × 𝑬𝒑 ,𝝎𝝁𝟎

(1.34)

where 𝒌 is the wave vector of the plane wave in the medium that light is traveling. Its
2𝜋
2𝜋
magnitude in the medium 𝑛1 is 𝜆 𝑛1 , and 𝜆 𝑛2 in the medium 𝑛2 .

Fig. 1.10. Conventions for directions of field vectors, black dots stand for the normal components
of field vectors pointing out the diagram.

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Advances in Optics: Reviews. Book Series, Vol. 5

The components of the electric vector of the incident field then are
𝐸𝐼𝑥 = |𝑬𝐼𝑝 |𝑐𝑜𝑠𝜃𝐼 , 𝐸𝐼𝑦 = −|𝑬𝐼𝒔 |,𝑘(1.35)𝑘

𝐻𝐼𝑥 = − 𝜔𝜇𝐼 |𝑬𝐼𝑠 |𝑐𝑜𝑠𝜃𝐼 , 𝐻𝐼𝑦 = − 𝜔𝜇𝐼 |𝑬𝐼𝑝 |00(1.36)

The components of the electric vector of the refracted field then are
𝐸𝑇𝑥 = |𝑬𝑇𝑝 |𝑐𝑜𝑠𝜃𝑇 , 𝐸𝑇𝑦 = −|𝑬𝑇𝒔 |,𝑘(1.37)𝑘𝐻𝑇𝑥 = − 𝜔𝜇𝑇 |𝑬𝑇𝑠 |𝑐𝑜𝑠𝜃𝑇 , 𝐻𝑇𝑦 = − 𝜔𝜇𝑇 |𝑬𝑇𝑝 |00(1.38)

The components of the electric vector of the reflected field then are
𝐸𝑅𝑥 = |𝑬𝑅𝑝 |𝑐𝑜𝑠𝜃𝑅 , 𝐸𝑅𝑦 = −|𝑬𝑅𝒔 |,𝑘(1.39)

𝑘𝐻𝑅𝑥 = 𝝎𝝁𝑅 |𝑬𝑅𝑠 |𝑐𝑜𝑠𝜃𝑅 , 𝐻𝑅𝑦 = 𝝎𝝁𝑅 |𝑬𝑅𝒑 |𝟎𝟎(1.40)

Application of boundary conditions to the equation (1.34), we can obtain the following
expressions.
𝐸𝐼𝑥 + 𝐸𝑅𝑥 = 𝐸𝑇𝑥 , 𝐸𝐼𝑦 + 𝐸𝑅𝑦 = 𝐸𝑇𝑦 ,(1.41)

𝐻𝐼𝑥 + 𝐻𝑅𝑥 = 𝐻𝑇𝑥 , 𝐻𝐼𝑦 + 𝐻𝑅𝑦 = 𝐻𝑇𝑦(1.42)

Combining the equations (1.35) to (1.42) with algebraic manipulations and the law ofreflection (1.32) and of refraction (1.33), we can get Fresnel’s formulas.
|𝑬|sin(𝜃 −𝜃 )𝑟𝒔 = |𝑬𝑅𝒔| = − 𝑠𝑖𝑛(𝜃𝐼 +𝜃𝑇 ),
𝑰𝒔

𝑟𝑝 =

𝐼

|𝑬𝑅𝒑 ||𝑬𝐼𝒑 ||𝑬|𝑡𝑎𝑛(𝜃 −𝜃 )= 𝑡𝑎𝑛(𝜃𝐼 +𝜃𝑇 ),(1.44)

2𝑐𝑜𝑠𝜃𝐼 𝑠𝑖𝑛𝜃𝑇,𝑠𝑖𝑛 (𝜃𝐼 +𝜃𝑇 )(1.45)

𝐼

𝑡𝒔 = |𝑬𝑇𝒔| =
𝑰𝒔

𝑡𝑝 =

|𝑬𝑇𝒑 ||𝑬𝐼𝒑 |(1.43)

𝑇𝑇

2𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃𝐼𝑇= 𝑠𝑖𝑛(𝜃 +𝜃 )𝑐𝑜𝑠
(𝜃 −𝜃 )𝐼

𝑇

𝐼

𝑇

(1.46)We exclude the cases of total external and internal reflections, then 𝜃𝐼 and 𝜃𝑇 are real.
The transmitted coefficients 𝑡𝑠 in the equation (1.45) and 𝑡𝑝 in the equation (1.46) are also
real and positive. Consequently, the phase of the refracted wave is equal to that of theincident wave, i. e., they are in phase. For the reflected wave, if the second medium is
optically denser than the first (𝑛2 > 𝑛1 ), then 𝜃𝑇 < 𝜃𝐼 . According to the equations (1.43),
𝑟𝒔 < 0. For the normal component of the electric vector, the phases of the reflected wave
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Chapter 1. Physical Optics

and the incident wave therefore differ by 𝜋, which is always valid. Under the same𝜋circumstances if 𝜃𝐼 + 𝜃𝑇 > 2 , then 𝑟𝑝 < 0. In this case the phases of the reflected wave𝜋and the incident wave also differ by 𝜋 . However, when 𝜃𝐼 + 𝜃𝑇 < 2 , the previous
conclusion is not valid. Similar analyses apply when the second medium is optically rarer
than the first (𝑛2 < 𝑛1 ).
Especially, if we change the order of 𝜃𝐼 and 𝜃𝑇 in the equations (1.43) and (1.44), we
know that the sign of 𝑟𝑠 and of 𝑟𝑝 will be changed to the opposite one. This can be also
thought of as the 𝜋 phase shift of 𝑟𝑠 and of 𝑟𝑝 relative to each own original phase,respectively. As shown in Fig. 1.11, we now consider a situation that a parallel plane plateis put in a medium with the uniform index of refraction, for example, in the air,encountered in interferometry. The angles of incidence and refraction of ray 1 are,respectively, 𝜃𝐼 and 𝜃𝑇 at the first surface of the plate whereas the angles of incidence andrefraction of ray 2 are, respectively, 𝜃𝑇 and 𝜃𝐼 at the second surface of the plate. The
second reflection with respect to the first one is equivalent to swap over the order of 𝜃𝐼
and 𝜃𝑇 in the equations (1.43) and (1.44). Such that the relative phase shift between theray 1 and the ray 2 generated by just reflections is 𝜋.

Fig. 1.11. Light reflections from a parallel plane plate for the same incident ray.

Based on the above discussion, we know that it is a misleading concept for theconventional statement that a light wave reflected from an optically denser medium
undergoes a phase shift of half of a wavelength.

1.2. Scalar Theory of Diffraction
Diffraction is any deviation of the light rays from rectilinear paths, predicted bygeometrical optics, as they pass through apertures or around obstacles in a uniform
medium. It is easy to observe fringes caused by diffraction when the size of the apertureor obstacle is of the same order of magnitude as the wavelength of the incident light wave.
The essential features of diffraction phenomena are diffraction fringes which can bequalitatively explained with help of Huygens-Fresnel’s principle. Based on this principle,
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Advances in Optics: Reviews. Book Series, Vol. 5

the diffraction fringes, in fact, result from the interference of all waves originating from acontinuous wavefront. The distinction between diffraction and interference is somewhat
vague. In general, the term diffraction refers to the case where all the superposed waves
come from a continuous wavefront, and the mathematical formulation of the light field
superposed is integral; the term interference refers to the case where separable waves aresuperposed, and the mathematical formulation of the light field superposed is summation.
In this section we try to treat the scalar theory of diffraction in a more quantitative way.
Rigorous solutions of diffraction problems encountered in optics are very rare. Becauseof mathematical difficulties, approximate approaches must be invoked in most of practicalproblems. Of these the Rayleigh-Sommerfeld approximating formula is adequate for thetreatment of the majority of problems encountered in instrumental optics. We will, startingfrom Helmholtz equation (1.16), obtain the Rayleigh-Sommerfeld formula that casts
Huygens’ principle onto a firm mathematical basis. Furthermore, the paraxial formula of
the Fresnel diffraction and of the Fraunhofer diffraction is derived from this formuladepending on the different propagation distances of light.
1.2.1. The Rayleigh-Sommerfeld Formulation of Diffraction
Typically, the optical layout of a plane screen that is opaque except for the open aperture
S in a uniform medium is shown in Fig. 1.12. Suppose that the light field 𝑈(𝑃1 ) withwavelength 𝜆 in the aperture S of the plane z1 is known，we want to seek the diffracted
light field 𝑈(𝑃) on the right-hand plane z2 with some distance from the plane z1.

Fig. 1.12. Optical layout of a plane screen for diffraction.

Starting from the Helmholtz equation (1.16) for a monochromatic electromagnetic wave,
we can solve the scalar wave equation by applying Green’s theorem and boundary
conditions, provided that:a) The aperture S is large in comparison to the light wavelength considered;38

Chapter 1. Physical Opticsb) The distance of the plane for observing the diffracted field from the aperture S islarge comparing to the size of the aperture.
The detailed derivation of the following formula (1.47) is given in the appendix at the endof this chapter. Here we just give out its final expression of Rayleigh-Sommerfeld
diffraction formula as follows.
𝑈(𝑃) =
where 𝑘 =1
𝑒 𝑗𝑘𝑟
𝑈(𝑃1 )
𝑐𝑜𝑠(𝒏, 𝒓)𝑑𝑆,
∬
𝑆
𝑗𝜆
𝑟

(1.47)

2𝜋
is the wavenumber of light in the medium, 𝑟 the magnitude of the position
𝜆vector 𝒓 from the point 𝑃 to the point 𝑃1 ， 𝒏 is the outward normal direction at each pointon the aperture S， and 𝑐𝑜𝑠(𝒏, 𝒓) the cosine value of the counter clockwise angle from 𝒓
to 𝒏. The integral area is the light-transmission portion of the aperture represented byformula (1.47) is known as Rayleigh-Sommerfeld formula for diffraction.
The formula (1.47) can be pictorially explained as follows. It expresses the observed light
𝑒 𝑗𝑘𝑟
𝑈(𝑃 )
with the amplitude 1𝑟
𝜆1
originating from secondary source located at every point 𝑃1 within the aperture S. in the𝑗𝜋formula cancels out the Gouy’s phase shift 2 gained by each secondary wavelet in thefield 𝑈(𝑃) as a superposition of diverging spherical waves

propagation of the wavelet. The 𝑐𝑜𝑠(𝒏, 𝒓) indicates the pattern of directivity for eachsecondary wavelet. Base on this explanation, we can know that the formula (1.47) is justthe mathematical expression of Huygens-Fresnel’s principle.
Let ℎ(𝑃, 𝑃1 ) =1 𝑒 𝑗𝑘𝑟𝑐𝑜𝑠(𝒏, 𝒓), then the formula (1.47) can be written𝑗𝜆 𝑟𝑈(𝑃) = ∬𝑆 ℎ(𝑃, 𝑃1 )𝑈(𝑃1 )𝑑𝑆(1.48)

The above formula says the light field 𝑈(𝑃) on the plane z2 is a linear superposition oflight field 𝑈(𝑃1 ) on the plane z1. In the point of view for a linear system, ℎ(𝑃, 𝑃1 ) is called
the impulse response for the light propagation (or diffraction) from the plane z1 to theplane z2. The occurrence of this linear superposition is not surprise. It is a result of linearityof the Helmholtz equation (1.16).
1.2.2. Paraxial Approximations for Diffraction
Although the Rayleigh-Sommerfeld formula gives a solution to the diffraction problem ingeneral form, it is still difficult to calculate diffraction fields with this formula in practicalproblems. Attention is now turned to further approximations to the Rayleigh-Sommerfeld
formula. The approximations commonly made are Fresnel and Fraunhofer ones that havecomparatively simple mathematical manipulations. These two types of approximations
can be obtained depending on different approximating conditions that will be, in detail,elaborated as follows.
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Before performing further approximations to the Rayleigh-Sommerfeld formula, it isconvenient for setting up the Cartesian coordinates. As shown in Fig. 1.13, the diffracting
aperture lies in the (x1, y1) plane and is illuminated in the +z direction. We are going tocalculate the light field on the (x, y) plane that is parallel to the (x 1, y1) plane and at
distance z from it along z-axis. Let the light field of point 𝑃1 on the (x1, y1) plane be𝑈(𝑥1 , 𝑦1 , 0). While the light wave propagates a certain distance, the light field at point 𝑃
on the observation plane (x, y) is 𝑈(𝑥, 𝑦, 𝑧).

Fig. 1.13. Diffraction geometry.

For the above Cartesian coordinates the Rayleigh-Sommerfeld diffraction formula (1.47)
can be written as
𝑈(𝑥, 𝑦, 𝑧) =

1
𝑒 𝑗𝑘𝑟
𝑈(𝑥
,
𝑦
,
0)
𝑐𝑜𝑠𝜃𝑑𝑥1 𝑑𝑦1 ,
∬
1
1
𝑗𝜆 𝑆
𝑟

(1.49)𝑧where 𝑟 = √(𝑥 − 𝑥1 )2 + (𝑦 − 𝑦1 )2 + 𝑧 2 , and 𝑐𝑜𝑠𝜃 = 𝑟.
1.2.2.1. Fresnel Diffraction
When the distance z is much greater than the size of the diffracting aperture and the lightwavelength, the 𝑟 appearing at the denominators in the formula (1.49) can be replaced byz and the result is accurate enough. The distance 𝑟 appearing in 𝑒 𝑗𝑘𝑟 for the formula (1.49)
cannot be approximated to just the first order term because the wavenumber 𝑘 is verylarge. It can be approximated to the second order terms according to Taylor’s expansion
as follows.
1

𝑟

𝑥−𝑥 2
𝑦−𝑦 2 2= √(𝑥 − 𝑥1 )2 + (𝑦 − 𝑦1 )2 + 𝑧 2 = 𝑧 [1 + ( 1 ) + ( 1 ) ] ≈
𝑧
𝑧
1 𝑥−𝑥

2

1 𝑦−𝑦

2

1

𝑥−𝑥

2

𝑦−𝑦

2 2

≈ 𝑧 [1 + 2 ( 𝑧 1 ) + 2 ( 𝑧 1 ) − 8 [( 𝑧 1 ) + ( 𝑧 1 ) ] ]40(1.50)

Chapter 1. Physical Optics

On inserting the above different approximations to 𝑟′𝑠 in the formula (1.49), its
approximating formulation for the light field at the point P on the observation surface
therefore becomes
𝑒 𝑗𝑘𝑧

𝑘

𝑈(𝑥, 𝑦, 𝑧) = 𝑗𝜆𝑧 ∬ 𝑈(𝑥1 , 𝑦1 , 0)𝑒 𝑗2𝑧[(𝑥−𝑥1)2 +(𝑦−𝑦1 )2 ]𝑑𝑥1 𝑑𝑦1(1.51)

The above formula (1.51) is called the Fresnel diffraction approximation, or the Fresnel
diffraction
integral.
To
get
the
formula(1.51),
thecondition21
𝑥−𝑥1 2𝑦−𝑦1 2𝑘𝑧)+()≪ 1 should be satisfied. However, the validity of the Fresnel
[(
]
8
𝑧
𝑧

diffraction approximation does not need so strict condition. When the diffraction distanceis greater than about 50 , fairly accurate results can be obtained by the Fresnel diffraction
approximation. In what follows we will qualitatively justify its legitimacy [16].
1

2

𝑥−𝑥

𝑦−𝑦

2 2

For the convenience of statement, we let 𝜀 = 8 𝑘𝑧 [( 𝑧 1 ) + ( 𝑧 1 ) ] . When the point(𝑥1 , 𝑦1 ) explores the diffracting aperture and the distance √(𝑥1 − 𝑥)2 + (𝑦1 − 𝑦)2 isgreater than some value, the function 𝜀 then changes by many wavelengths due to thelarge number 𝑘. Such that the exponential 𝑒 −𝑗𝜀 in the integral oscillates many times withalternatively positive and negative signs, and therefore the contributions to the integral
from this region of the diffraction aperture will be negligible. However, when the point(𝑥1 , 𝑦1 ) on the diffracting aperture is the critical one [1], i.e., 𝜀 is stationary, then the
integrand varies much more slowly, so that it significantly contributes the integral ofdiffraction. Based on the previous analysis, we know that the integral value is substantiallycontributed by a small region around the critical point (𝑥1 , 𝑦1 ), which is equivalent to theoriginal diffracting aperture becoming the very small region around the critical point.
Therefore, the condition 𝜀 ≪ 1 can be satisfied even if the diffraction distance z is justfew of tens wavelengths.
Equations (1.51) can also be re-written in an equivalent form as the following.
𝑈(𝑥, 𝑦, 𝑧) ==𝑘𝑘𝑒 𝑗𝑘𝑧 𝑗 𝑘 (𝑥 2 +𝑦 2 )𝑗 (𝑥1 2 +𝑦1 2 )−𝑗 (𝑥𝑥1 +𝑦𝑦1 )
2𝑧
2𝑧
𝑧
𝑒
𝑈(𝑥
,
𝑦
,
0)]
𝑒
𝑑𝑥1 𝑑𝑦1
[𝑒
∬
1
1
𝑗𝜆𝑧

(1.52)

We can see that, apart from the complex term outside the integration sign, the field𝑘2

2

𝑈(𝑥, 𝑦, 𝑧) is the Fourier transform of the term 𝑒 𝑗2𝑧(𝑥1 +𝑦1 ) 𝑈(𝑥1 , 𝑦1 , 0), evaluated at
𝑥 𝑦
two-dimensional spatial frequencies (𝜆𝑧 , 𝜆𝑧). Therefore, Fresnel diffraction integral canbe calculated using the fast Fourier transform (FFT).
1.2.2.2. Fraunhofer Diffraction
In the condition of the Fresnel diffraction, 𝑟 in phase factor 𝑒 𝑗𝑘𝑟 was approximated to thesecond order terms as
41

Advances in Optics: Reviews. Book Series, Vol. 5
1

𝑥−𝑥

2

= 𝑧+

𝑥 2 +𝑦 2
𝑥12 +𝑦12
𝑥𝑥 +𝑦𝑦+− 1𝑧 1
2𝑧
2𝑧

𝑦−𝑦

2

𝑟 = 𝑧 {1 + [( 1 ) + ( 1 ) ]}2𝑧𝑧
That is𝑟 = 𝑧+(𝑥−𝑥1 )2 +(𝑦−𝑦1 )22𝑧

If the propagation distance is further increased, and the third term of the above expression
𝑥 2 +𝑦 2

can be ignored, i.e., 𝑘 12𝑧 1 ≪ 1 , which means that the wavefront of light on thediffracting aperture is a plane. Then the formula (1.49) can be further approximated as
2

𝑈(𝑥, 𝑦, 𝑧) =2

𝑒 𝑗𝑘𝑧 𝑗𝑘𝑥 +𝑦
2𝑧
𝑒
𝑗𝜆𝑧

∬𝑆 𝑈(𝑥1 , 𝑦1 , 0)𝑒−𝑗𝑘

𝑥𝑥1 +𝑦𝑦1𝑧

𝑑𝑥1 𝑑𝑦1(1.53)

The above formula (1.53) is called the Fraunhofer diffraction approximation, or the
Fraunhofer diffraction integral. The Fraunhofer diffraction approximation condition is𝑥 2 +𝑦 2that 𝑘 12𝑧 1 ≪ 1 . This condition required z be extremely large, so that Fraunhofer
diffraction is also called the far field diffraction that is just a special case of Fresnel
diffraction.
The Fraunhofer diffraction integral (1.53) shows that the diffraction field on the plane(x, y) is the Fourier transformation of the input light field on the plane (x1, y1), except for
one complex factor associated with (x, y). The Fraunhofer diffraction field is very far
away from the diffracting aperture. For easy observations, we can put a converging lensnear to the diffracting aperture between it and the diffraction field, such that the
Fraunhofer diffraction pattern is formed onto the focal plane of the converging lens thanks
to its imaging property. This is just about the starting point of Fourier optics. The
Fraunhofer diffraction is, therefore, the physical basis for Fourier optics of a positive lens.
Relying on different approximations taken for the distance 𝑟 , we can delimit thediffraction zone into three regions. The first is the very near zone to the aperture, where
the sharp image of the aperture, predicted by geometrical optics, can be formed [16]. The
second is the very far zone, where the diffraction of the aperture is governed by Fraunhofer
diffraction pattern. The third is in between zone, where the diffraction pattern of theaperture can be predicted by Fresnel diffraction. Therefore, Fresnel diffraction patterns is
a transition one between the patterns governed by geometrical optics at one extreme andFraunhofer diffraction patterns at the other.
Fresnel and Fraunhofer diffraction formulas derived from the Rayleigh-Sommerfeld
integral through the paraxial approximations at different distances are valid only in certain
regions, not very close to the input aperture plane. The Rayleigh-Sommerfeld integral isvalid very close to the aperture, but not valid to the distance of the order of one
wavelength. The valid regions for the Rayleigh-Sommerfeld integral, Fresnel, andFraunhofer approximations are shown in Fig. 1.14.
42

Chapter 1. Physical Optics

Fig. 1.14. Three diffraction regions.

1.2.3. Diffraction with Nonmonochromatic LightIn above, we just considered the Rayleigh-Sommerfeld diffraction formula (1.47) which
will serve a fundamental physical law governing the propagation of monochromatic light.
In addition, it can be used to find a similar relation for the propagation of
nonmonochromatic light. Let us now turn to investigate the diffraction case in which theincident light wave is nonmonochromatic. As shown in Fig. 1.13, we denote thenonmonochromatic light field as an analytic function 𝑈(𝑃, 𝑡) at point 𝑃, and express it ina non-rigorous way using Fourier synthesis as follows∞

̃(𝑃, 𝑓) 𝑒 −𝑗2𝜋𝑓𝑡 𝑑𝑓,
𝑈(𝑃, 𝑡) = ∫−∞ 𝑈(1.54)

̃(𝑃, 𝑓) thewhere 𝑓 is the temporal frequency of a monochromatic light wave, and 𝑈
Fourier transform of the light field 𝑈(𝒓, 𝑡) into the frequency domain. Let thẽ(𝑃1 , 𝑓), then we have the following equation
monochromatic light field at point 𝑃1 be 𝑈
̃(𝑃, 𝑓).
according to Rayleigh-Sommerfeld diffraction formula (1.47) for 𝑈
̃(𝑃, 𝑓) = 1 ∬ 𝑈
̃(𝑃1 , 𝑓) 𝑒
𝑈
𝑆
𝑗𝜆

𝑗𝑘𝑟

𝑟

𝑐𝑜𝑠(𝒏, 𝒓)𝑑𝑆

(1.55)

On inserting the equation (1.55) into the equation (1.54), we have1

∞ 𝑓̃(𝑃1 , 𝑓)𝑈(𝑃, 𝑡) = 𝑗 ∫−∞ 𝑣 [∬𝑆 𝑈𝑒𝑟
𝑗2𝜋𝑓( −𝑡)𝑣𝑟𝑐𝑜𝑠(𝒏, 𝒓)𝑑𝑆] 𝑑𝑓 =𝑟∞1
̃(𝑃1 , 𝑓)𝑒 −𝑗2𝜋𝑓(𝑡−𝑣) 𝑑𝑓 ] 1 𝑐𝑜𝑠(𝒏, 𝒓)𝑑𝑆
= 2𝜋𝑣 ∬𝑆 [∫−∞(−𝑗2𝜋𝑓)𝑈𝑟
1(1.56)

𝑓

In the equation (1.56), we have used the relation 𝜆 = 𝑣 , where 𝑣 is the speed of light inthe medium. We differentiate the formula (1.54), the following expression can be obtained∞𝜕̃(𝑃1 , 𝑓) 𝑒 −𝑗2𝜋𝑓𝑡 𝑑𝑓
𝑈(𝑃1 , 𝑡) = ∫−∞(−𝑗2𝜋𝑓)𝑈𝜕𝑡(1.57)
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Advances in Optics: Reviews. Book Series, Vol. 5

We further have𝑟∞𝜕𝑟
̃(𝑃1 , 𝑓) 𝑒 −𝑗2𝜋𝑓(𝑡−𝑣) 𝑑𝑓
𝑈
(𝑃
,
𝑡
−
)
=
(−𝑗2𝜋𝑓)𝑈
∫
1
−∞
𝜕𝑡
𝑣

(1.58)

Combining the equation (1.56) and (1.58), we have the diffraction formula for
quasi-monochromatic light as follows.
1

1 𝜕

𝑟

(1.59)

𝑈(𝑃, 𝑡) = 2𝜋𝑣 ∬𝑆 𝑟 [𝜕𝑡 𝑈 (𝑃1 , 𝑡 − 𝑣)] 𝑐𝑜𝑠(𝒏, 𝒓)𝑑𝑆𝑟

In the above expression, is the time of light propagation from the point 𝑃1 to the point𝑣
𝑃 , i.e., the delayed time. This means that the light field of the point 𝑃 at time 𝑡 is𝑟contributed by the light field of the point 𝑃1 at the previous time 𝑡 − .
𝑣

1.3. Coherence
In the previous sections of this chapter, we discussed light that behaves in a totally
predictable way for all time instants 𝑡 and positions 𝒓 , i.e., the light fields are
deterministic. However, perfectly deterministic light never exists in actual physicalrealities. For actual light there is a range of frequencies 𝛥𝑓. If the center frequency of light𝛥𝑓
is 𝑓, and≪ 1, the light with this feature is called quasi-monochromatic.
𝑓

The bandwidth of a light source is mainly caused by the following several physicalmechanisms:
1) A finite lifetime of the higher energy level leads to this state not just an exact energy
level but with a range of energies according to the uncertainty principle. This energy
uncertainty results in the natural linewidth of light which is a fundamental linewidth;
2) High temperature involved in a light source results in a largely thermal motion of theatoms. The light emitted from atoms moving towards or away from the observer willproduce a change in frequency due to the Doppler effect. The Maxwell’s distributionof the speed of the thermal motion of atoms, therefore, broadens the spectral
linewidth;
3) In a light source, especially for a gas source, there exist collisions between the various
molecules. The collision will interrupt the wave train so that the phase correlation
between the emitted light before and after the collision is destroyed. In Fourier theory,the collision then broadens the spectral linewidths.
In practice, light with a finite bandwidth about the average (or center) optical frequency
is, therefore, not fully predictable, such the real light has certain randomness. In addition,
the phases of the light field (even if it is monochromatic) emitting from different parts ofan extended light source are not in unison, so that the field of light still has a certain
randomness. The phase randomness of the light field results in the different degrees ofcoherence for light fields.
44

Chapter 1. Physical Optics

To deal, quantitatively and clearly, with the randomness of a light field, the statistical
approach must be invoked. The statistical approach needed for describing coherence is theaverage intensity and the correlation functions.
1.3.1. Correlation Function: Definition
We take the complex functions of space and time to define the correlation function. Let𝑈(𝒓1 , 𝑡1 ) and 𝑈(𝒓2 , 𝑡2 ) be randomly complex functions at two different positions ofspace 𝒓 and time 𝑡, for example, and suppose that the observation time for these twofunctions is very long comparing to the varying time of two functions. Furthermore, we
suppose that 𝑈(𝒓1 , 𝑡1 ) and 𝑈(𝒓2 , 𝑡2 ) are stationary, at least wide-sense stationary, andergodic processes, and let 𝑡1 = 𝑡 and 𝑡2 = 𝑡 + 𝜏. The correlation function of 𝑈(𝒓1 , 𝑡1 )and 𝑈(𝒓2 , 𝑡2 ) is defined as〈𝑈 ∗ (𝒓1 , 𝑡)𝑈(𝒓2 , 𝑡 + 𝜏)〉,(1.60)

where 〈∙〉 stands for the ensemble average, * the conjugate complex, and 𝜏 is the time shift.
Based on the assumption of ergodicity, the definition can be further written as1

𝑇/2〈𝑈 ∗ (𝒓1 , 𝑡)𝑈(𝒓2 , 𝑡 + 𝜏)〉 = lim ∫−𝑇/2 𝑈 ∗ (𝒓1 , 𝑡)𝑈(𝒓2 , 𝑡 + 𝜏)𝑑𝑡
𝑇𝑇→∞

(1.61)For convenience of statement, we simply denote two functions 𝑈(𝒓1 , 𝑡1 ) and 𝑈(𝒓2 , 𝑡2 ),respectively, as 𝑈1 and 𝑈2 . If 𝑈1 and 𝑈2 are different functions, the correlation function
is called cross-correlation one; if 𝑈1 with 𝑈2 are the same functions, the correlation
function is called autocorrelation one. For most of time, if 𝑈1 and 𝑈2 tend to take both
positive values together or negative values together, its correlation function will take on a
positive value, being positively correlated. If 𝑈1 and 𝑈2 tend to take the values of opposite
signs together for most of time, its correlation function will take on a negative value, being
negatively correlated. For all time if 𝑈1 and 𝑈2 randomly take both positive and negativevalues, such that their product will be as often positive as negative, and its correlation
function will be negligible, being uncorrelated. The correlation function, therefore, is agood way to describe the degree of resemblance of the two functions in a single number.
In terminology when 𝑈1 and 𝑈2 are correlated, we say that 𝑈1 and 𝑈2 are statisticallysimilar.
Particularly, the autocorrelation of a function 𝑈(𝒓, 𝑡) is mathematically expressed as1

𝑇/2

Γ(𝒓, 𝜏) = 〈𝑈 ∗ (𝒓, 𝑡)𝑈(𝒓, 𝑡 + 𝜏)〉 = 𝑙𝑖𝑚 𝑇 ∫−𝑇/2 𝑈 ∗ (𝒓, 𝑡)𝑈(𝒓, 𝑡 + 𝜏)𝑑𝑡
𝑇→∞

(1.62)

𝛤(𝜏) describes the resemblance of 𝑈(𝒓, 𝑡 + 𝜏) to 𝑈(𝒓, 𝑡) both in magnitude and in phase.
1.3.2. The Concept of Coherence of LightIn the previous subsection, we defined the correlation of two general functions varying
with time. We now turn to the correlation properties of a light field. In optics, we refer to45Advances in Optics: Reviews. Book Series, Vol. 5

the correlation between two values of a light field with different temporal and spatial
coordinates as the coherence of the light field. Let functions 𝑈(𝒓1 , 𝑡1 ) and 𝑈(𝒓2 , 𝑡2 )particularly represent two values of a light field, as shown in Fig. 1.15, then the coherence
between 𝑈(𝒓1 , 𝑡1 ) and 𝑈(𝒓2 , 𝑡2 ) is 〈𝑈 ∗ (𝒓1 , 𝑡1 )𝑈(𝒓2 , 𝑡2 〉 which means that, given
𝑈(𝒓1 , 𝑡1 ) , what extent of information for 𝑈(𝒓2 , 𝑡2 ) can be obtained. The better thecoherence is, much information for 𝑈(𝒓2 , 𝑡2 ) can be retrieved.

Fig. 1.15. Two values, 𝑈(𝒓1 , 𝑡1 ) and 𝑈(𝒓2 , 𝑡2 ), of a light field.

If we confine the light field to be stationary, at least wide sense, then we can write that𝑡1 = 𝑡 and 𝑡2 = 𝑡 + 𝜏, we define the complex degree of coherence of a stationary lightfield as𝛾(𝒓1 , 𝒓2 , 𝜏) =〈𝑈 ∗ (𝒓1 ,𝑡)𝑈(𝒓2 ,𝑡+𝜏)〉∗(𝒓√〈𝑈 1 ,𝑡)𝑈(𝒓1 ,𝑡)〉√〈𝑈 ∗ (𝒓2 ,𝑡)𝑈(𝒓2 ,𝑡)〉(1.63)

Invoking the Schwarz inequality, we know that 0 ≤ |𝛾(𝒓1 , 𝒓2 , 𝜏)| ≤ 1. According to theproperties of the correlation function, we can get the following conclusions. If|𝛾(𝒓1 , 𝒓2 , 𝜏)| = 0, the light field is completely uncorrelated, i.e. incoherent light;
if |𝛾(𝒓1 , 𝒓2 , 𝜏)| = 1 , the light field is completely correlated, i.e. coherent light; if
0 < |𝛾(𝒓1 , 𝒓2 , 𝜏)| < 1, the light field is partially correlated, i.e. partially coherent light. In
the real world, there only exists partially coherent light, and the other two extreme cases
are just ideal situations.
a) Temporal coherence
Temporal coherence measures the coherence between 𝑈(𝒓, 𝑡1 ) and 𝑈(𝒓, 𝑡2 ), i. e. between
two field values of light at the same spatial point r but different times, 〈𝑈 ∗ (𝒓, 𝑡1 )𝑈(𝒓, 𝑡2 〉.
Temporal coherence can be used to define a coherence time 𝜏𝑐 , in which time duration we
can think that 𝑈(𝒓, 𝑡1 ) and 𝑈(𝒓, 𝑡2 ) have nearly the same behaviors. Its reciprocal is the1bandwidth Δ𝑓 of a quasi-monochromatic light wave, i.e. 𝛥𝑓 = 𝜏 .
𝑐

The Wiener-Khintchine theorem
A major practical importance of the autocorrelation function defined by the equation(1.62) is that there exists a very useful relationship associated with power spectral density.
46

Chapter 1. Physical Optics

The Wiener-Khintchine theorem says that the autocorrelation function,Γ(𝒓, 𝜏), and thecorresponding power spectral density, 𝑆(𝒓, 𝑓), form a Fourier transform pair between time
domain 𝜏 and frequency domain 𝑓∞(1.64)∞(1.65)

𝑆(𝒓, 𝑓) = ∫−∞ 𝛤(𝒓, 𝜏)𝑒 𝑗2𝜋𝑓𝜏 𝑑𝜏,𝛤(𝒓, 𝜏) = ∫−∞ 𝑆(𝑟, 𝑓)𝑒 −𝑗2𝜋𝑓𝜏 𝑑𝑓

We follow the Goodman’s approach [8] to prove the above relations. Because of thestationarity of random light field 𝑈(𝒓, 𝑡), its Fourier transform does not exist. For defining
its Fourier transform, i.e. the spectrum in frequency domain 𝑓, we perform its Fouriertransform as∞𝑡𝑢 𝑇 (𝒓, 𝑓) = ∫−∞ 𝑟𝑒𝑐𝑡( 𝑇1 )𝑈(𝒓, 𝑡1 )𝑒 −𝑗2𝜋𝑓𝑡1 𝑑𝑡1 ,∞𝑡𝑢 𝑇 (𝒓, 𝑓)∗ = ∫−∞ 𝑟𝑒𝑐𝑡( 𝑇2 )𝑈(𝒓, 𝑡2 )∗ 𝑒 𝑗2𝜋𝑓𝑡2 𝑑𝑡2 ,(1.66)𝑡where 𝑟𝑒𝑐𝑡(𝑇) is a temporal rectangle function with its length 𝑇, * stands for the complex
conjugate. The power spectral density can be defined as<𝑢𝑇 ∗ (𝒓,𝑓)𝑢𝑇 (𝒓,𝑓)>
𝑇
𝑇→∞

𝑆(𝒓, 𝑓) = 𝑙𝑖𝑚(1.67)

On substituting (1.66) into (1.67), we get<𝑢𝑇 ∗ (𝒓,𝑓)𝑢𝑇 (𝒓,𝑓)>=
𝑇
𝑇→∞

𝑆(𝒓, 𝑓) = 𝑙𝑖𝑚
∞

1

𝑡

𝑡

= 𝑙𝑖𝑚 𝑇 ∬−∞ 𝑟𝑒𝑐𝑡( 𝑇1 ) 𝑟𝑒𝑐𝑡( 𝑇2 ) < 𝑈 ∗ (𝒓, 𝑡2 )𝑈(𝒓, 𝑡1 ) > 𝑒 𝑗2𝜋𝑓(𝑡2 −𝑡1) 𝑑𝑡1 𝑑𝑡2𝑇→∞

Let 𝑡1 = 𝑡 and 𝑡2 = 𝑡 + 𝜏, the above expression becomes
𝑆(𝒓, 𝑓) =
1

∞

𝑡

𝑡+𝜏

= 𝑙𝑖𝑚 𝑇 ∬−∞ 𝑟𝑒𝑐𝑡(𝑇) 𝑟𝑒𝑐𝑡( 𝑇 ) < 𝑈 ∗ (𝒓, 𝑡 + 𝜏)𝑈(𝒓, 𝑡) > 𝑒 𝑗2𝜋𝑓𝜏 𝑑𝑡 𝑑𝜏𝑇→∞(1.68)

We now interchange orders of integration with respect to 𝑑𝜏 and the limit, and have
1

∞

𝑡

𝑡+𝜏

𝛤(𝒓, 𝜏) = 𝑙𝑖𝑚 𝑇 ∫−∞ 𝑟𝑒𝑐𝑡(𝑇)𝑟𝑒𝑐𝑡( 𝑇 ) < 𝑈 ∗ (𝒓, 𝑡 + 𝜏)𝑈(𝒓, 𝑡) > 𝑑𝑡
𝑇→∞

(1.69)

On inserting the expression (1.69) into (1.68), we have the Wiener-Khintchine theorem(1.64). This theorem is the theoretical basis for the Michelson Fourier transform
spectroscopy.

47

Advances in Optics: Reviews. Book Series, Vol. 5b) Spatial coherence
Spatial coherence measures the coherence between 𝑈(𝒓1 , 𝑡) and 𝑈(𝒓2 , 𝑡), i. e. between
two field values of light at the different spatial points with the same times,〈𝑈 ∗ (𝒓1 , 𝑡) 𝑈(𝒓2 , 𝑡)〉. By means of the spatial coherence, one can define a coherent area
(in two-dimensional space) or a coherent volume (in three-dimensional space). In these
coherent regions, 𝑈(𝒓1 , 𝑡) and 𝑈(𝒓2 , 𝑡) can be roughly thought of as having the same
behaviors.
The van Cittert-Zernike theorem
This theorem reveals that when light propagates in space even a spatially incoherent
source will generate a field that can become spatially coherent over a large region of space.
This theorem is the spatial equivalent of the Wiener-Khintchine theorem (1.64). The vanCittert-Zernike theorem is as follows
𝛾(𝑃1 , 𝑃2 ) =𝑒 −𝑗𝜑∬ 𝐼(𝑥1 ,𝑦1 )𝑑𝑥1 𝑑𝑦1∬ 𝐼(𝑥1 , 𝑦1 )𝑒 −𝑗𝑘(𝑝𝑥1 +𝑞𝑦1 ) 𝑑𝑥1 𝑑 𝑦1

(1.70)In what follows we will demonstrate for what every quantity in (1.70) stands. As shownin Fig. 1.16, in the plane denoted by (𝑥, 𝑦) parallel to the source plane S and far from it,
there are the field points 𝑃1 and 𝑃2 , the van Cittert-Zernike theorem, expressed by theequation (1.70), reveals the relationship between the intensity distribution of the lightsource and the complex degree of coherence of the light field in the far field. In the source
plane we choose a Cartesian coordinate system with its own origin and denote by (𝑥1 , 𝑦1 )the coordinates of a point in the source plane; in the plane (𝑥, 𝑦) of the far region of thesource we choose a Cartesian coordinate system with its own origin, and with 𝑥 and𝑦-axes parallel to the 𝑥1 and 𝑦1 -axes. Let us denote that the Cartesian coordinates of the
field points 𝑃1 and 𝑃2 , respectively, (𝑋1 𝑌1 ) and (𝑋2 𝑌2 ), and the distance between theorigin of the source and the origin of the plane (𝑥, 𝑦) is 𝑅. The distance between the point
𝑆 and the point 𝑃1 is 𝑅1 ; the distance between the point 𝑆 and the point 𝑃2 is 𝑅2 .
According to the Rayleigh-Sommerfeld diffraction formula, the light field at the point 𝑃1
from the field 𝑈(𝑆) of the source point 𝑆 is𝑈(𝑃1 ) =1
𝑒 𝑗𝑘𝑅1
𝑈(𝑆)
𝑑𝑆
∫
𝑗𝜆
𝑅1

(1.71)

𝑒 𝑗𝑘𝑅2𝑑𝑆
𝑅2(1.72)

Similar to 𝑈(𝑃2 ).
1

𝑈(𝑃2 ) = 𝑗𝜆 ∫ 𝑈(𝑆)In above two expressions, 𝜆 is the center wavelength of the quasi-monochromatic light
2𝜋
wave, 𝑘 = 𝜆 , and the direction factors have been omitted because 𝑅1 and 𝑅2 are verylarge in comparison with the size of the source 𝑆.

48

Chapter 1. Physical Optics

According to the definition of the complex degree of coherence of the expression (1.63),
we obtain the following formula for the equal-time complex degree of coherence of lightbetween two field points 𝑃1 and 𝑃2 .
𝛾(𝑃1 , 𝑃2 ) =

1
√𝐼(𝑃1

∫ 𝐼(𝑆))𝐼(𝑃 )
2

𝑒 𝑗𝑘(𝑅2 −𝑅1 )
𝑑𝑆,
𝑅1 𝑅2

(1.73),

where 𝐼(𝑆) is the intensity of the point 𝑆 in the source.

Fig. 1.16. Notation for the van Cittert-Zernike theorem.

And we can further get, from the equations (1.71) and (1.72), that 𝐼(𝑃𝑖 ) = ∫

𝐼(𝑆)
𝑑𝑆
𝑅𝑖2

(𝑖 = 1,2) which are, respectively, light intensities at two field points 𝑃1 and 𝑃2 .
From Fig. 1.16, we have
𝑅12 = (𝑋1 − 𝑥1 )2 + (𝑌1 − 𝑦1 )2 + 𝑅 2,
𝑅22 = (𝑋2 − 𝑥1 )2 + (𝑌2 − 𝑦1 )2 + 𝑅 2(1.74)The linear dimensions of the light source and the distance between 𝑃1 and 𝑃2 are small
comparing to the distance R, so that
𝑅1 = 𝑅 +(𝑋1 −𝑥1 )2 +(𝑌1 −𝑦1 )2
, 𝑅2
2𝑅

= 𝑅+

(𝑋2 −𝑥1 )2 +(𝑌2 −𝑦1 )2
2𝑅(1.75)In above expressions, we just retained the leading terms of the power-series expansions.
It follows from equations (1.75) that𝑅2 − 𝑅1 =(𝑋2 2 +𝑌2 2 )−(𝑋1 2 +𝑌1 2 )
2𝑅

𝑥 (𝑋2 −𝑋1 )+𝑦1 (𝑋2 −𝑋1 )𝑅− 1(1.76)We set49

Advances in Optics: Reviews. Book Series, Vol. 5

𝑝 =
𝜑 =

𝑋2 −𝑋1,𝑞
𝑅=

𝑌2 −𝑌1,
𝑅

𝑘[(𝑋22 +𝑌22 )−(𝑋12 +𝑌12 )]2𝑅(1.77)(1.78)

On inserting the formulas (1.76), (1.77), and (1.78) into the formula (1.73), we get the vanCittert-Zernike theorem as the formula (1.70).
The van Cittert-Zernike theorem says the equal-time degree of coherence 𝛾(𝑃1 , 𝑃2 ),excepting a phase factor, is equal to the normalized Fourier transform of the intensity
distribution across the light source. This theorem provides a basis for the technique ofaperture synthesis.
In summary, Coherence of light is classified into temporal and spatial coherence forconvenience, not for the deep physical difference. Temporal coherence relates directly tothe bandwidth of the light source; Spatial coherence to its finite extent of light source.
Their applications will be discussed in Section 1.5.

1.4. Fourier Theory and Fourier Optics
Fourier Optics is a branch of optical physics that Fourier analysis methods were applied
into light and optical systems to calculate the propagation of light and analyze theproperties of optical systems. Under certain conditions, optical systems are linear andinvariant ones that can easily be processed by Fourier analysis methods. Although thetargets processed in Fourier optics are still the propagation of light, interference, imaging,and information processing etc., application of Fourier analysis methods into optics has
enabled us to address optical issues in an easy and physically insightful way in the spatial
frequency domain.
This section will first describe the concept of the Fourier transform. Second, the conceptof the angular spectrum, a positive lens, and grating diffraction are discussed by Fourier
analysis method. Third, the imaging of coherent and incoherent illumination through theoptical system is discussed in the approach of Fourier optics. Finally, a typical opticalsystem for application of Fourier optics, the Fourier Transform Spectrometer, will be
studied.
1.4.1. Fourier Theory
1.4.1.1. Analysis of Periodic Functions: Fourier Series
The idea of Fourier series is to express a periodic function by the linear combination of aseries of orthogonal functions that form a set of basis in a function space. For Fourier
series, its basis is a set of trigonometric functions or complex exponential functions
relying on the representation formalism of Fourier series.
50

Chapter 1. Physical Optics

Suppose that the period of a periodic function 𝑔(𝑥) is , and furthermore 𝑔(𝑥) has limited
extreme points and discontinuity points and is absolutely integrable. Then 𝑔(𝑥) can be
expanded into Fourier series.
𝑔(𝑥) =𝑎0+ ∑∞𝑛 = 1[𝑎𝑛 𝑐𝑜𝑠(2𝜋𝑛𝑓0 𝑥) + 𝑏𝑛 𝑠𝑖𝑛(2𝜋𝑛𝑓0 𝑥)],2

(1.79)

1

where 𝑓0 = 𝜏 , being called the fundamental frequency of 𝑔(𝑥), and n’s are called theorders of the terms that are harmonics.
𝑎𝑛 =

2 𝜏
∫ 𝑔(𝑥) 𝑐𝑜𝑠(2𝜋𝑛𝑓0 𝑥) 𝑑𝑥
𝜏 0, 𝑏𝑛 =2 𝜏∫ 𝑔(𝑥) 𝑠𝑖𝑛(2𝜋𝑛𝑓0 𝑥) 𝑑𝑥,
𝜏 0

𝑎𝑛 , 𝑏𝑛 are called Fourier coefficients.
The expression (1.79) shows that the periodic function 𝑔(𝑥) can be represented as a linear
combination of infinite cosine and sine functions that have different frequencies.
According to the Euler’s formula 𝑒 𝑗𝑥 = 𝑐𝑜𝑠𝑥 + 𝑗𝑠𝑖𝑛𝑥 , the Fourier series in thetrigonometric form can be rewritten as the exponential series.
𝑗2𝜋𝑛𝑓0 𝑥
𝑔(𝑥) = ∑∞,𝑛 = −∞ 𝑐𝑛 𝑒
1(1.80)

𝜏

where 𝑐𝑛 = 𝜏 ∫0 𝑔(𝑥)𝑒 −𝑗2𝜋𝑛𝑓0 𝑥 𝑑𝑥.
The expression (1.80) shows that a periodic function 𝑔(𝑥) can be represented as a linear
combination of infinite complex exponential functions that have different frequencies.
The Fourier coefficients 𝑐𝑛 are the weight coefficients for each complex exponentialfunction.
The above two kinds of series expansions of a periodic function are called Fourier series.
A function with a period of  can be expanded into a Fourier series. Its fundamental
1𝑛frequency 𝑓0 = 𝜏 , the harmonic frequencies 𝑛𝑓0 = 𝜏 , and the frequency interval of the1

adjacent frequencies is ∆𝑓 = 𝑓0 = 𝜏 .
1.4.1.2. Analysis of Non-periodic Functions: Fourier TransformIf a function 𝑔(𝑥) is a non-periodic one, it can be regarded as a periodic function with an
infinite period. When 𝜏 → ∞，∆𝑓 can be expressed as 𝑑𝑓. Discrete frequencies 𝑛𝑓0 turnto a continuous frequency 𝑓. The sum of polynomials in (1.80) can be rewritten into an
integral as the below.
∞𝑔(𝑥) = ∫−∞ 𝐺(𝑓) 𝑒 𝑗2𝜋𝑓𝑥 𝑑𝑓,(1.81)

where51

Advances in Optics: Reviews. Book Series, Vol. 5∞

𝐺(𝑓) = ∫−∞ 𝑔(𝑥) 𝑒 −𝑗2𝜋𝑓𝑥 𝑑𝑥(1.82)The expressions (1.81) and (1.82) are called a Fourier transform pair. The formula (1.82)
of 𝐺(𝑓) obtained by 𝑔(𝑥) is called the Fourier transform or Fourier analysis and can be,
in shorthand, denoted as 𝐺(𝑓) = 𝐹{𝑔(𝑥)}. The formula (1.81) of 𝑔(𝑥) obtained by 𝐺(𝑓)is called the Inverse Fourier transform or Fourier synthesis and can be, in shorthand,denoted as 𝑔(𝑥) = 𝐹 −1 {𝐺(𝑓)}. Furthermore 𝐺(𝑓) is also called the spectrum of 𝑔(𝑥),and 𝑔(𝑥) the original function of 𝐺(𝑓).
Because two-dimensional spatial functions are common in optics ， the definition of
two-dimensional Fourier transform is given below. Let 𝑔(𝑥, 𝑦) be the original function
and its spectrum function 𝐺(𝑓𝑥 , 𝑓𝑦 ).
The Fourier transform of the function 𝑔(𝑥, 𝑦) is∞

𝐺(𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝑔(𝑥, 𝑦)} = ∫ ∫−∞ 𝑔(𝑥, 𝑦)𝑒 −𝑗2𝜋(𝑓𝑥 𝑥+𝑓𝑦 𝑦) 𝑑𝑥𝑑𝑦,(1.83)

where 𝑓𝑥 , 𝑓𝑦 are spatial frequencies. The Inverse Fourier transform of the function
𝐺(𝑓𝑥 , 𝑓𝑦 ) is∞𝑔(𝑥, 𝑦) = 𝐹 −1 {𝐺(𝑓𝑥 , 𝑓𝑦 )} = ∫ ∫−∞ 𝐺(𝑓𝑥 , 𝑓𝑦 )𝑒 𝑗2𝜋(𝑓𝑥 𝑥+𝑓𝑦 𝑦) 𝑑𝑓𝑥 𝑑𝑓𝑦(1.84)The mutual Fourier transform between the function 𝑔(𝑥, 𝑦) and the function 𝐺(𝑓𝑥 , 𝑓𝑦 ) is
usually denoted as 𝑔(𝑥, 𝑦) ⇔ 𝐺(𝑓𝑥 , 𝑓𝑦 ).
Fourier transform has some basic mathematical properties that are described by thetheorem of linearity, of similarity, of phase shift, of the convolution, and of correlation.
Readers can consult reference [6]. These theorems can greatly simplify the manipulationsof the Fourier transform.
1.4.2. Fourier Optics
1.4.2.1. The Concept of Angular Spectrum
The spatial distribution of light field obeys the Helmholtz equation (1.16). Plane waves
are the simplest solution to this equation. Plane waves have the same amplitude and phasein the plane perpendicular to the propagation direction of the light wave. The scalar lightfield of a monochromatic plane wave with the amplitude 𝐴 can be mathematically denoted
as 𝑢(𝒓, 𝑡) = 𝐴𝑒 𝑗(𝒌∙𝒓−𝜔𝑡) . The factor related to its spatial position is 𝑢(𝒓) = 𝐴𝑒 𝑗𝒌∙𝒓 ,where 𝒓 = (𝑥, 𝑦, 𝑧) is the spatial coordinate in the three-dimensional space, 𝒌 the
2𝜋 ̂wave-vector in the medium through which light is propagating, given by 𝒌 = 𝜆 𝒌, 𝜆 thê
̂ aswavelength of the light wave, and 𝒌 the unit vector of the wave-vector 𝒌. We denote 𝒌
52

Chapter 1. Physical Optics

̂ = (𝑐𝑜𝑠𝛼, 𝑐𝑜𝑠𝛽, 𝑐𝑜𝑠𝛾). Then the expression 𝑢(𝒓) = 𝐴𝑒 𝑗𝒌∙𝒓 canits direction cosines 𝒌
be rewritten as
𝑈(𝑥, 𝑦, 𝑧) = 𝐴𝑒 𝑗𝒌∙𝒓 = 𝐴𝑒 𝑗𝑘(𝑥 𝑐𝑜𝑠 𝛼+𝑦 𝑐𝑜𝑠 𝛽+𝑧 𝑐𝑜𝑠 𝛾)(1.85)For most of optical systems, the object and image planes are perpendicular to the opticalaxis. We are concerned with the distribution of light fields in these planes. Let the opticalaxis as z-axis, we discuss the distribution of light field in the 𝑧 = 𝑧1 plane. Inserting𝑧 = 𝑧1 into formula (1.85), we have2

2

𝑈(𝑥, 𝑦) = 𝐴𝑒 𝑗𝑘𝑧1 √1−𝑐𝑜𝑠 𝛼−𝑐𝑜𝑠 𝛽 𝑒 𝑗𝑘(𝑥 𝑐𝑜𝑠 𝛼+𝑦 𝑐𝑜𝑠 𝛽)(1.86)For the above formula, we have used the relation 𝑐𝑜𝑠 2 𝛼 + 𝑐𝑜𝑠 2 𝛽 + 𝑐𝑜𝑠 2 𝛾 = 1. For the22plane 𝑧1 , 𝐴𝑒 𝑗𝑘𝑧1 √1−𝑐𝑜𝑠 𝛼−𝑐𝑜𝑠 𝛽 is a complex constant just depending on 𝑧1 but irrelevantof 𝑥 and 𝑦, and we denote it as 𝑈0 , so that the expression (1.86) becomes𝑈(𝑥, 𝑦) = 𝑈0 𝑒 𝑗𝑘(𝑥 𝑐𝑜𝑠 𝛼+𝑦 𝑐𝑜𝑠 𝛽)(1.87)The isophase line equation of the plane wave in the 𝑧 = 𝑧1 plane is𝑘(𝑥 𝑐𝑜𝑠 𝛼 + 𝑦 𝑐𝑜𝑠 𝛽) = 𝐶, where 𝐶 is a constant. Each 𝐶 value corresponds to a contour
line representing an isophase one. These isophase lines are parallel to each other. Fromthe equation (1.87), it is obvious that the period of the isophase lines is 2𝜋. For a group ofisophase lines whose 𝐶 values differ by 2𝜋 , the spatial period between them in the
direction of light wave propagation is wavelength λ, and then the spatial periods in 𝑥 and𝑦 directions are, respectively, as shown in Fig. 1.17.
2𝜋

𝜆

2𝜋

𝜆

𝑑𝑥 = 𝑘𝑐𝑜𝑠𝛼 = 𝑐𝑜𝑠𝛼, 𝑑𝑦 = 𝑘𝑐𝑜𝑠𝛽 = 𝑐𝑜𝑠𝛽

Fig. 1.17. Spatial periods in 𝑥 and 𝑦 directions.

Then the spatial frequencies corresponding to 𝑥 and 𝑦 directions are, respectively
53Advances in Optics: Reviews. Book Series, Vol. 5
1𝑓𝑥 = 𝑑 =
𝑥

𝑐𝑜𝑠𝛼
, 𝑓𝑦
𝜆

1

= 𝑑 =
𝑦

𝑐𝑜𝑠𝛽
𝜆

(1.88)

On substituting the equation (1.88) into the equation (1.87), we can get the complex
amplitude distribution of a plane wave expressed by spatial frequency
𝑈(𝑥, 𝑦) = 𝑈0 𝑒 𝑗2𝜋(𝑓𝑥 𝑥+𝑓𝑦 𝑦)(1.89)

Comparing the kernel, 𝑒 𝑗2𝜋(𝑓𝑥 𝑥+𝑓𝑦 𝑦) , of the inverse Fourier transform (1.84) with theexpression of the plane wave (1.89), we can see that the kernel has the same form as theexpression of plane wave (1.89), i.e., the kernel in the inverse Fourier transform is a plane
wave with a unit amplitude in the direction of 𝑐𝑜𝑠 𝛼 = 𝜆𝑓𝑥 ， 𝑐𝑜𝑠 𝛽 = 𝜆𝑓𝑦 . If 𝑔(𝑥, 𝑦) isa light field distribution on some plane, it can be regarded as the weighted superpositionof numerous plane waves in different directions. In other words, Fourier transform is aprocess of decomposing a complex light wave into a series of plane waves with different
spatial frequencies. The weight factor 𝐺(𝑓𝑥 , 𝑓𝑦 ) is the spectrum of the light field
distribution 𝑔(𝑥, 𝑦). Since 𝑓𝑥 =𝑐𝑜𝑠𝛼 𝑐𝑜𝑠𝛽

𝑐𝑜𝑠𝛼
，𝑓𝑦
𝜆

=

𝑐𝑜𝑠𝛽
,
𝜆

𝐺(𝑓𝑥 , 𝑓𝑦 ) can be written as

𝐺 ( 𝜆 , 𝜆 ), which means that the propagation direction of one plane wave component
is proportional to its spatial frequency, therefore it is called the angular spectrum of thelight field distribution 𝑔(𝑥, 𝑦).
1.4.2.2. Fourier Transform of a Positive Lens
Lenses are the most common elements in optical systems. Here we will in detail study apositive lens that plays the role in Fourier optics. In Section 1.2, we qualitatively explainedthat a positive lens could perform Fourier transform due to the property of Fraunhofer
diffraction. In the following, we will rigorously prove in the case for the thin lens that thepositive lens can perform Fourier transform under certain conditions. We will firstly
discuss the phase transformation of the lens, and then the property of the Fourier transformof the lens. We suppose that there is no energy loss when light passes through the lens,such that the lens only changes the phase distribution of the incident light wave.
Fig. 1.18 shows the imaging of a point light source by a positive lens. Without loss ofgenerality, we just consider the case that the lens is in air. If the diffraction and aberrationsof the lens with a limited aperture are not considered, the object point 𝑃 is ideally imaged
at the point 𝑃′ through a positive thin lens. Therefore, the optical path lengths of all lightrays from the object point 𝑃 to the image point 𝑃′ are the same. Furthermore, δ0 and δ are
optical path lengths of the on-axis and off-axis light ray from the plane 𝑈𝑙 , lying on thefront surface of the lens, to the plane 𝑈𝑙′ , lying on the behind surface of the lens, and thegeometric meaning of all other quantities is shown in Fig. 1.18. Thus we have thefollowing equation
𝑑0 + 𝛿0 + 𝑑𝑖 = √𝑑02 + (𝑥 2 + 𝑦 2 ) + 𝛿 + √𝑑𝑖2 + (𝑥 2 + 𝑦 2 )54(1.90)

Chapter 1. Physical Optics

Further manipulations we get the expression for 𝛿 − 𝛿0 as follows
𝛿 − 𝛿0 = 𝑑0 + 𝑑𝑖 − √𝑑02 + (𝑥 2 + 𝑦 2 ) − √𝑑𝑖2 + (𝑥 2 + 𝑦 2 )(1.91)In the above formula, the term, 𝛿 − 𝛿0 , stands for the variation of the optical path lengthof the off-axis light ray with respect to the on-axis light ray.

Fig. 1.18. Imaging scheme of a lens.

Expanding the terms in the square root sign of the formula (1.91) into Taylor’s series up
to the second order terms in the paraxial approximation, the formula (1.91) can be
simplified as the following.
𝛿 − 𝛿0 = −(𝑥 2 +𝑦2 )2𝑑0−(𝑥 2 +𝑦 2 )
2𝑑𝑖

1
2

1
𝑑𝑖

= − (𝑥 2 + 𝑦 2 ) ( +1)𝑑0(1.92)The sign convention of the right-handed Cartesian coordinate system has been applied in
the equations (1.90) to (1.92).
Therefore, under the condition of the paraxial approximation the variation of the optical
path length caused by the lens is given by the formula (1.92). Invoking the imaging
property of the lens, we know that the object distance 𝑑0 and the image distance 𝑑𝑖 follow
11
1
the imaging formula 𝑑 + 𝑑 = 𝑓 , where 𝑓 is the focal length of the lens, so that the𝑖0transmittance function of the lens can be expressed as𝑡𝑙 (𝑥, 𝑦) = 𝑒−𝑗𝑘

𝑥2 +𝑦22𝑓,(1.93)

where 𝑘 is the wavenumber.
Although the above formula is obtained in the case of a positive lens, it is also applicable
to a negative lens. Under the condition of the paraxial approximation, the lens without
loss is just a phase converter in the optical system.
We will demonstrate that a lens performs Fourier transform in the following two cases:
the object is in front of the lens, and the object is behind the lens.
55

Advances in Optics: Reviews. Book Series, Vol. 5

As shown in Fig. 1.19, the focal length of a positive lens in air is f. The object with theamplitude transmissivity, 𝑡(𝑥0 , 𝑦0 ), is placed at a finite distance 𝑑 in front of the lens andnormally illuminated by a plane wave with the amplitude 𝐴. The light field just behind theobject is then 𝑈0 (𝑥0 , 𝑦0 ) = 𝐴𝑡(𝑥0 , 𝑦0 ).

Fig. 1.19. Fourier transform of a lens for 𝑈1 .

According to the Fresnel diffraction formula (1.52), the light field distribution at the front
surface, 𝑈1 , of the lens is
𝑈1 (𝑥1 , 𝑦1 ) =𝑘
𝐴𝑒 𝑗𝑘𝑑𝑗 [(𝑥1 −𝑥0 )2 +(𝑦1 −𝑦0 )2 ]
2𝑑
𝑡(𝑥
,
𝑦
)𝑒
𝑑𝑥0 𝑑𝑦0 ,
∬
0
0
𝑗𝜆𝑑

(1.94)

where (𝑥1 , 𝑦1 ) is the coordinate in the plane, 𝑈1 , of the lens. By means of thetransmittance function (1.94) of the lens, the light field distribution just behind the lens,𝑈2 , can be obtained as
𝑈2 (𝑥1 , 𝑦1 ) = 𝑈1 (𝑥1 , 𝑦1 )𝑒

𝑥2 +𝑦21
2𝑓−𝑗𝑘 1(1.95)

Due to the thin lens, the light ray coordinates at the front and back surfaces of the lens areboth (𝑥1 , 𝑦1 ). From the back surface, 𝑈2 , of the lens to the back focal plane, with help of
the Fresnel diffraction formula again, then the complex amplitude distribution of the lightwave on the back focal plane is𝑘𝑈𝑓 (𝑥, 𝑦) =𝑗 [(𝑥−𝑥1 )2 +(𝑦−𝑦1 )2 ]𝑒 𝑗𝑘𝑓
𝑈2 (𝑥1 , 𝑦1 )𝑒 2𝑓
𝑑𝑥1 𝑑𝑦1 ,∬𝑗𝜆𝑓(1.96)

where (𝑥, 𝑦) are the coordinates of the back focal plane. By taking (1.95) into (1.96), we
can get
𝑒 𝑗𝑘𝑓

𝑗

𝑘

𝑈𝑓 (𝑥, 𝑦) = 𝑗𝜆𝑓 𝑒 2𝑓

(𝑥 2 +𝑦2 )∞∫ ∫−∞ 𝑈1 (𝑥1 , 𝑦1 ) 𝑒−𝑗

2𝜋(𝑥 𝑥+𝑦1 𝑦)
𝜆𝑓 1

𝑑𝑥1 𝑑𝑦1(1.97)

Obviously, the integral in the above formula is the Fourier transform of 𝑈1 (𝑥1 , 𝑦1 ). Bymeans of the shorthand notation of the Fourier transform, the expression (1.97) canbe written
56

Chapter 1. Physical Optics𝑘

𝑈𝑓 (𝑥, 𝑦) = =𝑒 𝑗𝑘𝑓 𝑗2𝑓(𝑥 2 +𝑦 2 )𝑒𝐹{𝑈1 (𝑥1 , 𝑦1 )},𝑗𝜆𝑓(1.98)

According to the convolution theorem, from the expression (1.94) we have2

2

𝐹{𝑈1 (𝑥, 𝑦)} = 𝐴𝑇(𝑓𝑥 , 𝑓𝑦 )𝑒 𝑗𝑘𝑑 𝑒 −𝑗𝜋𝜆𝑑(𝑓𝑥 +𝑓𝑦 ) ,where 𝑇(𝑓𝑥 , 𝑓𝑦 ) is the Fourier transform of the transmittance function 𝑡(𝑥0 , 𝑦0 ) , and𝑥𝑦𝑓𝑥 = 𝜆𝑓, 𝑓𝑦 = 𝜆𝑓.
On inserting the above expression into the equation (1.98), and considering𝑥𝑦𝑓𝑥 =, 𝑓𝑦 = , then we have the following expression.
𝜆𝑓

𝜆𝑓

𝑘

𝑈𝑓 (𝑥, 𝑦) =𝑑

𝐴𝑒 𝑗𝑘(𝑑+𝑓) 𝑗2𝑓(1−𝑓)(𝑥 2 +𝑦 2 )𝑥 𝑦𝑒
𝑇 (𝜆𝑓 , 𝜆𝑓)𝑗𝜆𝑓(1.99)

From this formula, we can get the following conclusions. If a lens is placed at a finite
distance behind the object, the product of the quadratic phase factor and the Fouriertransform of the object's transmittance function can be obtained on the back focal planeof the lens, such that it is an inexact Fourier transform. If the object is just lying on thefront focal plane of the lens, i.e. 𝑑 = 𝑓，and the quadratic phase factor disappears, theformula (1.99) can be simplified as
𝑈𝑓 (𝑥, 𝑦) =𝐴𝑒 𝑗2𝑘𝑓𝑥 𝑦
𝑇( , )
𝑗𝜆𝑓
𝜆𝑓 𝜆𝑓(1.100)

The above formula means that if the object is placed on the front focal plane of the lens,and the light field on the back focal plane is the exact Fourier transform of the light fieldtransmitted through the object.
If the object is placed behind the lens, it is still illuminated normally by a plane wave withamplitude 𝐴, as shown in Fig. 1.20. Under a paraxial approximation, the light wave
𝐴𝑓

𝑘

2

2

𝐴𝑓

illuminating the object is 𝑑 𝑒 −𝑗2𝑑(𝑥0 +𝑦0 ) ., in which the factor 𝑑 is the amplitude of thespherical wave impinging on the object, due to the fact that the linear dimension of thecircular converging bundle of rays has been reduced by this factor. Then the amplitude of
the wave transmitted by the input could be written

𝐴𝑓 −𝑗 𝑘 (𝑥0 2 +𝑦02 )𝑒 2𝑑𝑡(𝑥0 , 𝑦0 ).
𝑑

The propagation of the light wave from the object to the back focal plane can be calculatedwith help of the Fresnel diffraction formula (1.52). Thus, the field distribution on the focal
plane behind the lens can be obtained.
𝐴𝑓

𝑘

2

2

𝑥

𝑦

𝑈𝑓 (𝑥, 𝑦) = 𝑗𝜆𝑑2 𝑒 𝑗2𝑑(𝑥 +𝑦 ) 𝑇 (𝜆𝑑 , 𝜆𝑑),(1.101)57

Advances in Optics: Reviews. Book Series, Vol. 5𝑥𝑦where 𝑇 (𝜆𝑑 , 𝜆𝑑) is the spectrum of the object's transmittance function 𝑡(𝑥0 , 𝑦0 ). The
expression (1.101) shows that when the object is placed behind the lens, up to thequadratic phase factor, the light field on the back focal plane of the lens is the Fourier𝑥𝑦transform of the light field from the object. In this case, 𝑓𝑥 = 𝜆𝑑, 𝑓𝑦 = 𝜆𝑑 the size of thelight spot on the back focal plane can be changed by changing d. In the optical information
processing, this flexibility will bring a lot of convenience.

Fig. 1.20. Fourier transform of a lens for 𝑈0 .

In summary, no matter is the object in front of, against, or behind the lens at a finite
distance from it, the light field distribution on the back focal plane of the lens is, up to aquadratic phase factor, the Fourier transform of the field distribution transmitted throughthe object. If the intensity distribution of the light field on the back focal plane of the lensis just concerned, the quadratic phase factor disappears for the intensity. Therefore, thepower spectrum of the light field from the object can be obtained on the back focal planeof the lens.
Furthermore, with help of the concept of the angular spectrum in the formula (1.88), theFourier transform of the light field transmitted through the object is, in essence, todecompose the light field into plane waves with different directions. Then these planewaves with different directions are converged to different positions to form diffraction
spots on the back focal plane of the lens. The higher the frequency is, the larger thediffraction angle with respect to the optical axis of the lens, as shown in Fig. 1.21. The
intensity of each diffraction spot is proportional to the square of its own Fourier
coefficient. The Fourier spectrum of the light field from the object appears directly on theback focal plane of the lens, so that the back focal plane of the lens is often called Fourier
spectrum plane.
In the above discussion, the limitation of lens aperture is ignored. This limited aperture of
the lens may cause the distortion of the object spectrum, because the lens is actually alow-pass filter: the low frequency signal can pass through, the cut-off frequency one canpartially pass through, and the high frequency one cannot pass through. Therefore, due tothe influence of the limited aperture of the lens, the object spectra cannot be fully obtained58

Chapter 1. Physical Optics

on the back focal plane. The higher the frequency is, the greater the spectrum loss. Thisphenomenon is called vignetting effect.

Fig. 1.21. Optical system for Fourier transform.

1.4.2.3. Fourier Optics for Analyzing Diffraction Gratings
A diffraction grating is an important optical element in optical instruments or opticalinformation processing systems. Any object with the spatial periodic structure can be used
as a grating. Gratings can modulate the amplitude or phase of a light wave periodically.
Next, we will discuss two kinds of plane diffraction gratings: cosine andRonchi’s gratings.
As shown in Fig. 1.22, the transmittance function in the plane (𝑥0 , 𝑦0 ) for a cosine
1
amplitude grating with an infinite length and the grating constant is mathematically𝑓0

expressed as1

1

𝑡(𝑥0 , 𝑦0 ) = 2 + 2 𝑐𝑜𝑠( 2𝜋𝑓0 𝑥0 )(1.102)

Fig. 1.22. A cosine amplitude grating.

If the grating is placed on the front focal plane, (𝑥0 , 𝑦0 ), of the lens, a plane light beamwith the amplitude 𝐴 is normally incident on it, then the diffraction light field on the back59

Advances in Optics: Reviews. Book Series, Vol. 5

focal plane, (𝑥, 𝑦),of the lens with the focal length 𝑓 can be calculated by the equation(1.100). On inserting the transmittance function (1.102) into (1.100), then we have
𝑈𝑓 (𝑥, 𝑦) ==𝐴𝑒 𝑗2𝑘𝑓
𝐴𝑒 𝑗2𝑘𝑓 1 1
𝐹{𝑡(𝑥0 , 𝑦0 )} =𝐹 { + 𝑐𝑜𝑠( 2𝜋𝑓0 𝑥0 )} =
𝑗𝜆𝑓
𝑗𝜆𝑓
2 2

𝐴𝑒 𝑗2𝑘𝑓
𝑦
1
𝑥
1
𝑥
1
𝑥
𝛿 (𝜆𝑓) [2 𝛿 (𝜆𝑓) + 4 𝛿 (𝜆𝑓 − 𝑓0 ) + 4 𝛿 (𝜆𝑓 + 𝑓0 )] =𝑗𝜆𝑓=

𝐴𝑒 𝑗2𝑘𝑓1
11
𝛿(𝑓𝑦 ) [ 𝛿(𝑓𝑥 ) + 𝛿(𝑓𝑥 − 𝑓0 ) + 𝛿(𝑓𝑥 + 𝑓0 )]
𝑗𝜆𝑓
2
4
4

The above formula indicates that a cosine amplitude grating with an infinite length
illuminated by a plane wave can obtain three δ functions at different positions on the focal
plane behind the lens, as shown in Fig. 1.23. The first term in the square bracket
corresponds to the zero order component of diffraction at x = 0; the second term to the +1
order of diffraction; the third term to -1 order of diffraction. The zero order component
has no dispersion, and the other two components have dispersion.

Fig. 1.23. Diffraction distribution of a cosine amplitude grating.

The transmittance function of the grating can be rewritten as the following form by Euler’s
11
11
formula 𝑡(𝑥0 , 𝑦0 ) = 2 [1 + 𝑐𝑜𝑠( 2𝜋𝑓0 𝑥1 )] = 2 + 4 𝑒 𝑗2𝜋𝑓0 𝑥1 + 4 𝑒 −𝑗2𝜋𝑓0 𝑥1 .
It can be seen that the light field behind the grating contains three plane wave components,1
1
1
represents the plane wave propagating along the optical axis, and 4 𝑒 𝑗2𝜋𝑓0 𝑥1 , 4 𝑒 −𝑗2𝜋𝑓0 𝑥1
2
a pair of inclined plane waves with a certain angle between the propagation direction andthe optical axis and symmetrical about the optical axis. The three plane waves correspondto the frequency components of 0, 𝑓0, −𝑓0 in the complex amplitude distribution. The lensconverges the three plane waves on the back focal plane to form three focal points at
different positions, namely the above δ functions, which are called 0-order, 1-order and-1-order components, respectively, as shown in Fig. 1.24.
60

Chapter 1. Physical Optics

If the size of the grating discussed above is limited, and assume that the grating is a squarewith the side length L, as shown in Fig. 1.25, then the transmittance function can be
expressed as
1

1

𝑥

𝑦

𝑡(𝑥0 , 𝑦0 ) = [2 + 2 𝑐𝑜𝑠( 2𝜋𝑓0 𝑥0 )] 𝑟𝑒𝑐𝑡( 𝐿0 )𝑟𝑒𝑐𝑡( 𝐿0 )(1.103)

If the grating is placed on the front focal plane, (𝑥0 , 𝑦0 ), of the lens with the focal length
𝑓 and illuminated normally by a plane wave with the amplitude 𝐴, the light field on theback focal plane, (𝑥, 𝑦), of the lens is
𝑈𝑓 (𝑥, 𝑦) ==𝐴𝑒 𝑗2𝑘𝑓
𝐹{𝑡(𝑥0 , 𝑦0 )} =𝑗𝜆𝑓

𝐴𝑒 𝑗2𝑘𝑓
1 1
𝑥0
𝑦0
𝐹 {[ + 𝑐𝑜𝑠( 2𝜋𝑓0 𝑥0 )] 𝑟𝑒𝑐𝑡( )𝑟𝑒𝑐𝑡( )}
𝑗𝜆𝑓
2 2
𝐿
𝐿

Fig. 1.24. Grating diffraction.

Fig. 1.25. Cosine amplitude grating with the finite aperture.

By substituting the Fourier transforms of 𝑐𝑜𝑠( ∗), and 𝑟𝑒𝑐𝑡(∗), furthermore applying theconvolution theorem and the scaling transform theorem of the Fourier transform,
we can get
61Advances in Optics: Reviews. Book Series, Vol. 5

𝑈𝑓 (𝑥, 𝑦) =𝐴𝑒 𝑗2𝑘𝑓 1
1
1{[2 𝛿(𝑓𝑥 ) + 4 𝛿(𝑓𝑥 − 𝑓0 ) + 4 𝛿(𝑓𝑥 + 𝑓0 )] ∗∗ 𝐿2 𝑠𝑖𝑛𝑐(𝐿𝑓𝑥 )} 𝑠𝑖𝑛𝑐(𝐿𝑓𝑦 ) =𝑗𝜆𝑓
𝐴𝑒 𝑗2𝑘𝑓 𝐿2
1
1= 𝑗𝜆𝑓 2 𝑠𝑖𝑛𝑐(𝐿𝑓𝑦 ) {𝑠𝑖𝑛𝑐(𝐿𝑓𝑥 ) + 2 𝑠𝑖𝑛𝑐[𝐿(𝑓𝑥 − 𝑓0 )] + 2 𝑠𝑖𝑛𝑐[𝐿(𝑓𝑥 + 𝑓0 )]} ===𝐴𝑒 𝑗2𝑘𝑓 𝐿2
𝑦
𝑥
1
𝑥
1
𝑥
𝑠𝑖𝑛𝑐(𝐿 𝜆𝑓) {𝑠𝑖𝑛𝑐(𝐿 𝜆𝑓) + 2 𝑠𝑖𝑛𝑐 [𝐿(𝜆𝑓 − 𝑓0 )] + 2 𝑠𝑖𝑛𝑐 [𝐿(𝜆𝑓 + 𝑓0 )]}𝑗𝜆𝑓 2

In the above formula the notation, ∗∗, represents the operation of convolution. This
formula shows that due to the limitation of diffraction grating size, the complex amplitudeof the light field on the back focal plane with the focal length 𝑓 is no longer thesuperposition of three δ functions, but of three 𝑠𝑖𝑛𝑐 functions with the same width anddifferent position. The intensity distribution of the light field on the back focal plane is𝐼(𝑥, 𝑦) = 𝑈𝑓 (𝑥, 𝑦)𝑈𝑓 ∗ (𝑥, 𝑦) =
𝐴𝐿2

2

𝐿𝑦

𝐿𝑥

1

𝐿

1

𝐿

2

= (2𝜆𝑓) 𝑠𝑖𝑛𝑐 2 (𝜆𝑓) {𝑠𝑖𝑛𝑐(𝜆𝑓) + 2 𝑠𝑖𝑛𝑐 [𝜆𝑓 (𝑥 − 𝑓0 𝜆𝑓)] + 2 𝑠𝑖𝑛𝑐 [𝜆𝑧 (𝑥 + 𝑓0 𝜆𝑓)]}

For general gratings, the size of a grating is much larger than the grating constant, i.e.
22𝜆𝑓
𝐿 ≫ 𝑓 , then 𝑓0 𝜆𝑓 >> 𝐿 . Here we should note that 𝑓0 represents frequency and 𝑓 the0

focal length of the lens. In this case, the value of overlapped part of 𝑠𝑖𝑛𝑐 functions issmall, so the cross multiplication terms in the above formula can be ignored, then we havethe following formula.
𝐴𝐿2

2

𝐿𝑦

𝐼(𝑥, 𝑦) = (2𝜆𝑓) 𝑠𝑖𝑛𝑐 2 (𝜆𝑓)⋅
𝐿𝑥

1

𝐿

1

𝐿

⋅ {𝑠𝑖𝑛𝑐 2 (𝜆𝑓) + 4 𝑠𝑖𝑛𝑐 2 [𝜆𝑓 (𝑥 − 𝑓0 𝜆𝑓)] + 4 𝑠𝑖𝑛𝑐 2 [𝜆𝑓 (𝑥 + 𝑓0 𝜆𝑓)]}(1.104)

And the diffraction pattern is as shown in Fig. 1.26.

Fig. 1.26. Light intensity distribution from a cosine grating.

Next we discuss Ronchi amplitude grating. The transmittance function for a Ronchi
grating with grating constant 𝑑 is shown in Fig. 1.27.
62

Chapter 1. Physical Optics

If its size is a square with side length L, the transmittance function can be written as
𝑥

1

𝑥

𝑥

𝑦

𝑡(𝑥0 , 𝑦0 ) = {𝑟𝑒𝑐𝑡 ( 𝑎0 ) ∗∗ [𝑑 𝑐𝑜𝑚𝑏 ( 𝑑0 )]} 𝑟𝑒𝑐𝑡 ( 𝐿0 ) 𝑟𝑒𝑐𝑡 ( 𝐿0 )(1.105)In the above formula comb(∗) is called the comb or shah function [7]. Its concrete𝑥expression is that 𝑐𝑜𝑚𝑏 ( 𝑑0 ) = 𝑑∑𝛿(𝑥0 − 𝑛𝑑) , 𝑛 = 0,1,2, … ∞ . And the Fourier
1
𝑑

𝑥
𝑑

transform of 𝑐𝑜𝑚𝑏 ( 0 ) is1𝑛𝐹[∑𝛿(𝑥0 − 𝑛𝑑)] = 𝑑 ∑𝛿 (𝑓𝑥 − 𝑑) = ∑𝛿(𝑑𝑓𝑥 − 𝑛) = 𝑐𝑜𝑚𝑏(𝑑𝑓𝑥 )Fig. 1.27. Ronchi’s grating.

In the formula (1.105), and 𝑟𝑒𝑐𝑡(∗) is called the rectangular function or
the normalized boxcar function.
When the grating is placed on the front focal plane, (𝑥0 , 𝑦0 ), of the lens with the focal
length 𝑓 and illuminated normally by a plane wave with the amplitude 𝐴, the light fieldon the back focal plane, (𝑥, 𝑦), of the lens is
𝑈𝑓 (𝑥, 𝑦) =𝐴𝑒 𝑗2𝑘𝑓
𝐹{𝑡(𝑥0 , 𝑦0 )} =𝑗𝜆𝑓=

𝐴𝑒 𝑗2𝑘𝑓[{𝑎[𝑠𝑖𝑛𝑐(𝑎𝑓𝑥 )][𝑐𝑜𝑚𝑏(𝑑𝑓𝑥 )]} ∗∗ 𝐿2 𝑠𝑖𝑛𝑐(𝐿𝑓𝑥 )]𝑠𝑖𝑛𝑐(𝐿𝑓𝑦 ) =𝑗𝜆𝑓=

𝐴𝑒 𝑗2𝑘𝑓 𝑎𝐿2𝑦𝑛𝑎
𝑥𝑛𝑠𝑖𝑛𝑐 (𝐿 𝜆𝑓) ∑∞𝑛 = −∞ 𝑠𝑖𝑛𝑐 ( 𝑑 ) 𝑠𝑖𝑛𝑐 [𝐿 (𝜆𝑓 − 𝑑 )]𝑗𝜆𝑓
𝑑(1.106)In the above formula the notation, ∗∗ , represents the operation of convolution. The
complex amplitude of the light field on the focal plane behind Ronchi’s grating is the𝑛superposition of infinite sinc(x) functions with the same width and the translation, 𝑑 ,respectively, where 𝑛 is an integer called the diffraction order. Similar to the cosine
amplitude grating, the cross multiplication terms can be ignored, and the intensity
distribution of the light field on the back focal plane is63Advances in Optics: Reviews. Book Series, Vol. 5

𝐼(𝑥, 𝑦) = 𝑈𝑓 (𝑥, 𝑦)𝑈𝑓 ∗ (𝑥, 𝑦) =
𝑎𝐴𝐿2

2

𝑦

𝑛𝑎

𝑥

𝑛

2
2
= ( 𝜆𝑓𝑑 ) 𝑠𝑖𝑛𝑐 2 (𝐿 𝜆𝑓) ∑∞𝑛 = −∞ 𝑠𝑖𝑛𝑐 ( 𝑑 ) 𝑠𝑖𝑛𝑐 [𝐿 (𝜆𝑓 − 𝑑 )](1.107).

The light intensity distribution of plane(𝑥, 0) is shown in Fig. 1.28, only limited numberof diffraction orders are shown.

Fig. 1.28. Light intensity distribution from a Ronchi’s grating.

1.4.2.4. Fourier Optics for Imaging under Coherent Illumination
In the following two sections, Fourier optics is used to analyze the imaging properties of
optical systems illuminated, respectively, under coherent and incoherent light. Most
optical systems can be regarded as linear and spatially invariant systems. We can describe
the imaging characteristics of the system not only through its point spread function (theimpulse response) in the space domain but also through its transfer function in thefrequency domain. If the transfer function of the optical system is known, the spectrum of
the image can be obtained by multiplying the transfer function with the spectrum of theobject or its ideal image.
Under illumination of coherent light, the imaging system is linear to the complex
amplitude.
In the actual imaging systems, the size of the incident beam is limited by the entrance
pupil of the optical system, and the size of the outgoing beam is restricted by the exit pupilof the optical system. The imaging process can be described as follows: the light wave
passes from the object surface to the entrance pupil, from the entrance pupil to the exit
pupil through the optical system, and then from the exit pupil to the image plane, as shownin Fig. 1.29.
64Chapter 1. Physical Optics

In the ideal imaging case, the spherical wavefront emitted by the point light source will
become a convergent spherical wavefront centered on its ideal image point after passing
through the system. This convergent spherical wavefront can be taken as a reference one
for evaluating wave aberrations for a practical system. The difference between the actual
wavefront from a practical system and the reference wavefront is called the wave
aberration that can assess the imaging quality of the optical system.

Fig. 1.29. An imaging system.

The general exit pupil function is defined as
𝐻(𝑥1 , 𝑦1 ) = 𝑐𝑖𝑟𝑐 (where 𝑐𝑖𝑟𝑐 (√𝑥12 +𝑦12𝜌√𝑥12 +𝑦12
𝜌) 𝑒 𝑗𝜑(𝑥1 ,𝑦1 ),(1.108)) is the circular function with radius ρ, and 𝜑(𝑥1 , 𝑦1 ) is the wave

aberration. The distribution of light field on the exit pupil can be expressed as the productof exit pupil function (1.108) and the ideal convergent spherical wave (1.93)
𝐻′(𝑥1 , 𝑦1 ) = 𝐻(𝑥1 , 𝑦1 )𝑒−𝑗𝑘(𝑥 2 +𝑦12 )
2𝑧0 1

,

(1.109)where 𝑧0 is the distance from the exit pupil to the image plane. The propagation of lightwave from the exit pupil to the image plane generally meets the Fresnel diffraction
condition. On inserting the formula (1.109) into the Fresnel diffraction formula (1.52), we
can get the light field distribution of the image plane as follows
𝑒 𝑗𝑘𝑧0𝑘𝑗 (𝑥 2 +𝑦 2 )𝑝(𝑥, 𝑦) = 𝑗𝜆𝑧 𝑒 𝑧00𝑘𝑗 (𝑥𝑥1 +𝑦𝑦1 )∬ 𝐻(𝑥1 , 𝑦1 )𝑒 𝑧0

𝑑𝑥1 𝑑𝑦1

In the above formula, the coefficient in front of the integral does not affect the relative
intensity distribution of the image plane and can be ignored. Then we have65Advances in Optics: Reviews. Book Series, Vol. 5𝑘𝑗 (𝑥𝑥1 +𝑦𝑦1 )𝑝(𝑥, 𝑦) = ∬ 𝐻(𝑥1 , 𝑦1 )𝑒 𝑧0(1.110)𝑑𝑥1 𝑑𝑦1

The formula (1.110) represents the complex amplitude distribution of the light field on
the image plane after the spherical wave emitted by a point object passes through theoptical system. 𝑝(𝑥, 𝑦) is called the Amplitude Point Spread Function (APSF) of the
optical system. It is the Fourier transform of the exit pupil function 𝐻(𝑥1 , 𝑦1 ), which only
depends on the properties of the optical system.
Assuming that the object distribution is 𝑜(𝑥0 , 𝑦0 ), the complex amplitude distribution of
the geometrical optical ideal image on the image plane is𝑔𝑖 (𝑥, 𝑦) =where 𝑀 =

𝑥
𝑥0

=

1
𝑜(𝑥, 𝑦),𝑀(1.111)𝑦is the absolute value of the magnification of the image. The ideal
𝑦0

image is the same as the object, but the coordinates are scaled.
The imaging system can be regarded as a linear and spatially invariant system withdiffraction, and the form of APSF does not change with its position in the paraxial regime.
By means of the properties of the linear system, the image can be written as the
convolution between the ideal image and APSF as follows.
∞𝑔(𝑥, 𝑦) = ∬−∞ 𝑔𝑖 (𝜉, 𝜂)𝑝(𝑥 − 𝜉, 𝑦 − 𝜂)𝑑𝜉𝑑𝜂 ,(1.112)

Performing Fourier transform on the both sides of the above formula and invoking theconvolution theorem of Fourier transform, we have
(1.113)𝐺(𝑓𝑥 , 𝑓𝑦 ) = 𝐺𝑖 (𝑓𝑥 , 𝑓𝑦 )𝑇(𝑓𝑥 , 𝑓𝑦 ),where 𝐺(𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝑔(𝑥, 𝑦)} , 𝐺𝑖 (𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝑔𝑖 (𝑥, 𝑦)} , 𝑇(𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝑝(𝑥, 𝑦)} ,𝑥yand 𝑓𝑥 = 𝜆𝑧 ，𝑓𝑦 = 𝜆𝑧 .
00𝑇(𝑓𝑥 , 𝑓𝑦 ) is the Fourier transform of APSF 𝑝(𝑥, 𝑦). By using the expression (1.107) we
can get 𝑇(𝑓𝑥 , 𝑓𝑦 ) as follows𝑘𝑗 (𝑥𝑥1 +𝑦𝑦1 )𝑇(𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝑝(𝑥, 𝑦)} = 𝐹 {∬ 𝐻(𝑥1 , 𝑦1 )𝑒 𝑧0

𝑑𝑥1 𝑑𝑦1 }= 𝐻(−𝜆𝑧0 𝑓𝑥 , −𝜆𝑧0 𝑓𝑦 )If we ignore the immaterial negative sign in the above expression, then we have𝑇(𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝑝(𝑥, 𝑦)} = 𝐻(𝜆𝑧0 𝑓𝑥 , 𝜆𝑧0 𝑓𝑦 ),(1.114)𝑇(𝑓𝑥 , 𝑓𝑦 ) is called the Amplitude Transfer Function (ATF) of the optical system, which
has the same form as the exit pupil function of the optical system. The imaging systemgenerally acts as a low-pass filter due to the Amplitude Transfer Function.
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Chapter 1. Physical Optics

1.4.2.5. Fourier Optics for Imaging under Incoherent Illumination
Under the illumination of the coherent light, the imaging process is the superposition ofcomplex amplitudes. There is a very simple and direct relationship between the exit pupil
and the amplitude transfer function (ATF). When the incoherent light is used forillumination, the imaging process is the superposition of intensity. Although the transfer
function of the imaging system is still determined by the exit pupil function, therelationship between them is not straightforward.
The intensity distribution of the image is the time average of the instantaneous intensity
in detection duration. The intensity distribution can be obtained by squaring the coherent
complex amplitude distribution of the optical system (1.112), and then by time average,
the incoherent intensity distribution on the image plane of the optical system can beobtained∞2

𝐼(𝑥, 𝑦) = 〈|∬−∞ 𝑔𝑖 (𝜉, 𝜂)𝑃(𝑥 − 𝜉, 𝑦 − 𝜂)𝑑𝜉𝑑𝜂 | 〉,(1.115)where 〈∙〉 is the time average or ensemble average. By exchanging the order of the integral
and the time average in the above formula, we can get∞

𝐼(𝑥, 𝑦) = ∬ ∬−∞〈𝑔𝑖 (𝜉, 𝜂)𝑔𝑖 ∗ (𝜉′, 𝜂′)〉 𝑃(𝑥 − 𝜉, 𝑦 − 𝜂)𝑃∗ (𝑥 − 𝜉′, 𝑦 − 𝜂′)𝑑𝜉𝑑𝜂𝑑𝜉′𝑑𝜂 ′(1.116)
With help of the properties of completely incoherent light, we have〈𝑔𝑖 (𝜉, 𝜂)𝑔𝑖 ∗ (𝜉′, 𝜂′)〉 = 𝐼𝑖 (𝜉, 𝜂)δ(𝜉 − 𝜉′, 𝜂 − 𝜂′),where 𝐼𝑖 (𝜉, 𝜂) is the intensity distribution of the geometrical optical ideal image. The
point spread function p (x, y) is only related to the properties of the system, so that the
average does not affect its value. On inserting the above formula into the formula (1.116),
we have the imaging formula for an inherent optical system.
∞

𝐼(𝑥, 𝑦) = ∬−∞ 𝐼𝑖 (𝜉, 𝜂)|𝑃(𝑥 − 𝜉, 𝑦 − 𝜂)|2 𝑑𝜉𝑑𝜂 == 𝐼𝑖 (𝑥, 𝑦) ∗ |𝑃(𝑥, 𝑦)|2

(1.117)

The above formula shows that the incoherent intensity distribution of image can be written
as the convolution of the ideal image intensity distribution 𝐼𝑖 (𝑥, 𝑦) with incoherent point
spread function |𝑃(𝑥, 𝑦)|2 . Performing Fourier transform on both sides of the above
formula, we haveΠ(𝑓𝑥 , 𝑓𝑦 ) = Π𝑖 (𝑓𝑥 , 𝑓𝑦 )𝑇𝑜 (𝑓𝑥 , 𝑓𝑦 ),(1.118)whereΠ(𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝐼(𝑥, 𝑦)}, Π𝑖 (𝑓𝑥 , 𝑓𝑦 ) = 𝐹{𝐼𝑖 (𝑥, 𝑦)},67

Advances in Optics: Reviews. Book Series, Vol. 5

and 𝑇𝑜 (𝑓𝑥 , 𝑓𝑦 ) = 𝐹{|𝑃(𝑥, 𝑦)|2 }. 𝑇𝑜 (𝑓𝑥 , 𝑓𝑦 ) is called the non-normalized Optical Transfer
Function (OTF) of the incoherent illumination that is a comprehensive parameter forevaluating the quality of the optical system.
According to the correlation theorem of Fourier transform and the formula (1.114), we
can obtain the relationship between the optical transfer function and the exit pupil function
as follows
𝑇𝑜 (𝑓𝑥 , 𝑓𝑦 ) = 𝐹{|𝑃(𝑥, 𝑦)|2 } = 𝐹{𝑃(𝑥, 𝑦)𝑃∗ (𝑥, 𝑦)} = 𝐹{𝑃(𝑥, 𝑦)} ⊗ 𝐹{𝑃∗ (𝑥, 𝑦)} == 𝐻(𝜆𝑧0 𝑓𝑥 , 𝜆𝑧0 𝑓𝑦 ) ⊗ 𝐻 ∗ (𝜆𝑧0 𝑓𝑥 , 𝜆𝑧0 𝑓𝑦 ),(1.119)where ⊗ stands for the correlation operation. The above formula shows that the opticaltransfer function 𝑇𝑜 (𝑓𝑥 , 𝑓𝑦 ) is the autocorrelation of the exit pupil
function 𝐻(𝜆𝑧0 𝑓𝑥 , 𝜆𝑧0 𝑓𝑦 ). If the exit pupil function is known, the optical transfer function
can be calculated by the expression (1.119).
In general, OTF is a complex, and its modulus is called the Modulation Transfer Function(MTF), which represents the ratio of the contrast of the actual image to the contrast of theideal image at a particular frequency. The argument of the OTF is called the Phase
Transfer Function (PTF), which represents the phase shift of each frequency componentof the actual image distribution relative to the ideal image distribution.
There is a very important concept, resolution, in an optically imaging system. The
resolution of the optical system is dependent on the size of its aperture stop and thecoherence of the illumination. There exist the simple expressions of the resolution for theperfect coherence and perfect incoherence of the illumination. However, the partiallycoherent illumination is complicated and the results can only be given in rather generalterms. Here we do not discuss them further. Interested readers can consult references[8] and [10].
1.4.2.6. Fourier Transform Spectroscopy
Fourier transform spectroscopy is a technique for the measurement of the its spectrum ofa light source, which is based on the temporal coherence of light and the tool of the Fouriertransform. The schematic of the layout of the Fourier transform spectrometer is shown in
Fig. 1.30, and its core part is the Michelson’s interferometer with a movable mirror in one
arm. The basic principle is as follows. The movable mirror is scanned a series of distances,and the corresponding intensity at each scanned position is recorded by a point detector.
Then the spectrum of the light source can be obtained through the Fourier transform of aseries of recorded intensities.
For simplicity, suppose that light emitted from a source is monochromatic, and theintensities of two light waves arriving at the point detector are the same, denoted as 𝐼0 (𝑘).
From the following Section 1.5 for interferometry to be discussed, the interferometric
2𝜋
intensity on the point detector can be expressed as 2𝐼0 (𝑘)[1 + cos(𝑘𝑙)], where 𝑘 = 𝜆 ,68Chapter 1. Physical Optics

𝜆 is the wavelength of light, and 𝑙 is the difference between optical paths of two lightwaves passing through the two arms of the Michelson interferometer. Since light emitted
from an actual source includes light with different wavelengths, the intensity on the point
detector will be the superposition of intensities for different wavelengths, and let1
wavenumber 𝑝 = 𝜆 , such that the total intensity on the point detector is as follows∞

𝐼(𝑙) = ∫0 2𝐼0 (𝑝)[1 + cos(2𝜋𝑝𝑙)] 𝑑𝑝 =∞= 2𝐼 + ∫0 𝐼0 (𝑝)(𝑒 𝑗2𝜋𝑝𝑙 + 𝑒 −𝑗2𝜋𝑝𝑙 )𝑑𝑘,

(1.120)∞where 𝐼 = ∫0 𝐼0 (𝑝) 𝑑𝑝.

Fig. 1.30. Illustration of Fourier transform spectrometer.

If let 𝐼𝑡 (𝑙) = 𝐼(𝑙) − 2𝐼, then we have from (1.117)∞∞

𝐼𝑡 (𝑙) = ∫0 𝐼0 (𝑝)(𝑒 𝑗2𝜋𝑝𝑙 + 𝑒 −𝑗2𝜋𝑝𝑙 )𝑑𝑝 = ∫−∞ 𝐼0 (𝑝)𝑒 𝑗2𝜋𝑝𝑙 𝑑𝑝(1.121)This equation indicates that 𝐼𝑡 (𝑙) and 𝐼0 (𝑝) are a Fourier transform pair. Thus, the power
spectrum of the light source 𝐼0 (𝑝) can be got by the Fourier transform of 𝐼𝑡 (𝑙), and∞𝐼0 (𝑝) = ∫−∞ 𝐼𝑡 (𝑙)𝑒 −𝑗2𝜋𝑝𝑙 𝑑𝑙(1.122)The difference, 𝑙, of the optical paths is proportional to the displacement of the movablemirror in Fig. 1.30 he mirror, the power spectrum of the light source under measurement
can be determined.

1.5. Optical Interferometry
In Section 1.3 we have discussed the coherence of light, which is manifested byinterferometric phenomena. We know that that when light waves are superimposed, thefield vibration is strengthened at some positions, while weakened at some other positions.
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Advances in Optics: Reviews. Book Series, Vol. 5

This phenomenon is called the interference of light waves, which obeys the superposition
principle of light waves. When light beams superposed satisfy certain conditions,interferometric fringes are stable over certain time duration. In this section, thesuperposition principle for any types of waves is first introduced, then the interferometricphenomenon of light waves is explained, and at final, some examples for light interference
are given.
1.5.1. Superposition of Waves
1.5.1.1. General Principle
The principle of superposition is that when two or more waves meet at a point, the field
vibration at that point is equal to the sum of the field vibrations of the individual waves.
Furthermore, the propagation of waves at the meet point is independent of each other, andthe characteristics of a series of waves will not be changed by the existence of other waves.
After two (or more) waves meet in space, each wave still keeps its own characteristics offrequency, wavelength and vibration direction, and continues to propagate along theoriginal direction. This is called the principle of independent propagation of waves.
The above two principles are not universal. Generally, for a physical system, if thedifferential equation describing the change of physical quantity is linear, thecorresponding physical quantity satisfies the principle of superposition. However, for verystrong waves, the principle of superposition fails.
When two waves meet and superimpose in space, the composite waves is often very
complicated. For simplicity, we just consider the superposition of two simple harmonic
waves with the same vibration direction, and consider the superposition of two harmonic
waves with the same frequency and different frequencies, respectively.
1.5.1.2. Superposition of Waves with Same Frequencies
As shown in Fig. 1.31, two plane waves with the same vibration direction are emitted
from wave sources 𝑆1 and 𝑆2 , both of which have the same frequency 𝜔. They propagate
in the same medium and have the same wavenumber 𝑘. Let the initial phase of the wave
source be 𝜑1 and 𝜑2 , and meet at a point P after propagating the distance of 𝑟1 and 𝑟2
respectively.

Fig. 1.31. Supposition of two waves.

70

Chapter 1. Physical Optics

The amplitudes of the two waves at the point 𝑃 are 𝐴1 and 𝐴2 . The vibration
expressions are
𝑈1 (𝑡) = 𝐴1 cos( 𝑘𝑟1 − 𝜔𝑡 + 𝜑1 ),𝑈2 (𝑡) = 𝐴2 cos( 𝑘𝑟2 − 𝜔𝑡 + 𝜑2 )(1.123)

Because of the same frequency, the composite vibration at the point 𝑃 is still a simple
harmonic vibration
𝑈 = 𝑈1 + 𝑈2 = 𝐴 cos( 𝜔𝑡 + 𝜑)

(1.124)

The amplitude 𝐴 of the composite vibration is determined by the following equation.
𝐴 = √𝐴1 2 + 𝐴2 2 + 2𝐴1 𝐴2 𝑐𝑜𝑠∆𝜑(1.125)

On taking the formula (1.125) squared, we have following expression.
𝐼 = 𝐼1 + 𝐼2 + 2√𝐼11 𝐼2 𝑐𝑜𝑠∆𝜑(1.126)

In formulas (1.125) and (1.126), ∆𝜑 is the phase difference of two waves at the point 𝑃∆𝜑 = 𝑘(𝑟2 − 𝑟1 ) + (𝜑2 − 𝜑1 ),(1.127)

where 𝐼1 = 𝐴1 2 and 𝐼2 = 𝐴2 2 are, respectively, the intensities of waves 𝑈1 (𝑡) and𝑈1 (𝑡) at the point 𝑃 , (𝜑2 − 𝜑1 ) is the initial phase difference of two waves,and 𝑘(𝑟2 − 𝑟1 ) is the phase difference caused by different wave paths from two wave
sources to the point 𝑃. For a given point 𝑃, the wave path difference 𝑟2 − 𝑟1 is certain. If
the changing time of the initial phase difference (𝜑2 − 𝜑1 ) is much longer than thedetection time of the detector, stable fringes can be formed at the point 𝑃.
When the phase difference at a space point satisfies∆𝜑 = 𝑘(𝑟2 − 𝑟1 ) + (𝜑2 − 𝜑1 ) = ±2𝑚𝜋, 𝑚 = 0,1,2 ⋯ ,(1.128)the amplitude and intensity at a point have a maximum value of𝐴𝑚𝑎𝑥 = 𝐴1 + 𝐴2 , 𝐼𝑚𝑎𝑥 = 𝐼1 + 𝐼2 + 2√𝐼11 𝐼2(1.129)When the phase difference at a point satisfies∆𝜑 = 𝑘(𝑟2 − 𝑟1 ) + (𝜑2 − 𝜑1 ) = ±(2𝑚 + 1)𝜋, 𝑚 = 0,1,2 ⋯ ,(1.130)the amplitude and intensity at a point have a minimum value of𝐴𝑚𝑖𝑛 = |𝐴1 − 𝐴2 |, 𝐼𝑚𝑖𝑛 = 𝐼1 + 𝐼2 − 2√𝐼11 𝐼2(1.131)71

Advances in Optics: Reviews. Book Series, Vol. 5

1.5.1.3. Standing Waves
When two coherent waves with the same amplitude propagate in opposite directions, astanding wave is formed. Suppose that two coherent plane waves with the same amplitude
propagating along the positive and negative directions of z-axis as the following
expressions 𝑈1 (𝑧, 𝑡) = 𝐴 cos( 𝑘𝑧 − 𝜔𝑡), 𝑈2 (𝑧, 𝑡) = 𝐴 cos( 𝑘𝑧 + 𝜔𝑡).
The superposition wave function of these two waves is(1.132)

𝑈 = 𝑈1 + 𝑈2 = 2𝐴 cos(𝑘𝑧) cos(𝜔𝑡)This is the expression of a standing wave, where the space variable z and the time variablet are separated. The time function cos 𝜔𝑡 shows that all points vibrate harmonically at thesame frequency. The space function 2𝐴 cos(𝑘𝑧) shows that the amplitudes of each point
are different and distributed according to the cosine function.
At the position of |cos(𝑘𝑧)| = 1, the maximum amplitude is 2𝐴, and its coordinate is𝜆2

𝑧 = 𝑚 , 𝑚 = 0, ±1, ±2, ⋯These positions of the maximum amplitude are called antinodes.
At the position of |cos(𝑘𝑧)| = 0, the minimum amplitude is zero, and its coordinate is
𝜆4𝑧 = ±(2𝑚 + 1) ， 𝑚 = 0, ±1, ±2, ⋯These positions with the smallest amplitude are called nodes. The positions of antinodes
and nodes of the standing wave are fixed, and the distance between adjacent antinodes or
𝜆nodes is .
2

At every moment, a standing wave has a certain waveform, but this waveform does not
move with time like a traveling wave. The standing wave propagates neither the vibration
state nor the energy. This is because the average energy flux density of two travelingwaves forming the standing wave is equal and opposite, so the average energy flux densityof the composite wave is zero. The standing wave has stable energy state, called the energy
steady state.
1.5.1.4. Superposition of Waves with Diffident Frequencies:
Group Velocity, Phase Velocity
The plane wave with the frequency of 𝜔 propagates along z-axis, and the wave function
can be written𝑧𝑈(𝑧, 𝑡) = 𝐴 cos( 𝑘𝑧 − 𝜔𝑡)] = 𝐴𝑐𝑜𝑠𝜔 (𝑡 − 𝑣),
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Chapter 1. Physical Optics

where 𝐴 is the amplitude, 𝜔 is the angular frequency, and 𝑣 is the speed of wave,2𝜋𝜔𝑘 = 𝜆 = 𝑣 wavenumber or spatial frequency. For the constant phase (𝑘𝑧 − 𝜔𝑡), its𝑑𝑧moving speed is 𝑑𝑡 = 𝑣, which is called phase velocity.
For the superposition of two simple harmonic waves with the same frequency, the same
vibration direction and the same wave velocity 𝑣, the composite wave is still the simple
harmonic wave with the same frequency and propagates with the phase velocity 𝑣 .
However, the superposition of simple harmonic waves with different frequencies, thecomposite wave is no longer simple harmonic, which is generally a more complex wave.
Next, we will discuss this situation. Suppose that the two plane waves propagate along thez-axis and the wave function is as follows.
𝑈1 (𝑧, 𝑡) = 𝐴 cos( 𝑘1 𝑧−𝜔1 𝑡), 𝑈2 (𝑧, 𝑡) = 𝐴 cos(𝑘2 𝑧 − 𝜔2 𝑡),
2𝜋

𝜔

2𝜋

𝜔

1

1

2

2

where 𝑘1 = 𝜆 = 𝑣 1 ，𝑘2 = 𝜆 = 𝑣 2 ，𝜔1 and 𝜔2 are the frequencies of the wave, 𝜆1
and 𝜆2 the wavelengths of two waves, which are very close, and 𝑣1 and 𝑣2 the phasevelocities of two waves. The composite wave of these two waves is𝑈(𝑧, 𝑡) = 𝑈1 (𝑡) + 𝑈2 (𝑡) = 𝐴[cos(𝑘1 𝑧 −𝜔1 𝑡) + cos( 𝑘2 𝑧 − 𝜔2 𝑡)]
With help of trigonometric formulas, the above formula can be expressed as𝑈(𝑧, 𝑡) = 2𝐴 cos( 𝑘𝑚 𝑧−𝜔𝑚 𝑡) cos( 𝑘̅ 𝑧 − 𝜔̅𝑡),(1.134)

𝜔 −𝜔
𝑘 −𝑘
𝜔 +𝜔
𝑘 +𝑘
where 𝜔𝑚 = 1 2 2 , 𝑘𝑚 = 1 2 2, 𝜔
̅ = 1 2 2，and 𝑘̅ = 1 2 2. Because that 𝜔1 and𝜔2 are near the same, such that 𝜔𝑚 ≪ 𝜔1 or 𝜔2 and 𝜔̅ ≈ 𝜔1 or 𝜔2 . The composite wave
expression (1.134) can be regarded as a plane wavecos( 𝑘̅ 𝑧 − 𝜔̅𝑡) with a slowly changing
amplitude 2𝐴 cos( 𝑘𝑚 𝑧−𝜔𝑚 𝑡) . The waveform of the composite wave is shown in
Fig. 1.32. In the figure above, two simple harmonics with similar frequency are
represented by solid line and dotted line respectively. The figure below shows thesuperposition of the two waves. The solid line represents the waveform with the frequencyof 𝜔̅, and the dotted line represents the waveform with the slowly varying amplitude. Dueto the change of amplitude, the composite wave forms wave packet in shape.

There are two propagation velocities in the wave packet. One is the phase propagation
velocity of the high frequency wave in the composite wave. From (1.134), let𝜔̅𝑡 − 𝑘̅ 𝑧 = constant, the phase velocity of the wave packet can be obtained𝑑𝑧̅𝜔

𝑣𝑝 = 𝑑𝑡 = 𝑘̅

(1.135)Since 𝜔̅ ≈ 𝜔1 or 𝜔2 and 𝑘̅ ≈ 𝑘1 or 𝑘2 , this phase velocity is approximately equal to thephase velocities of the two plane waves that consist of the wave packet.

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Advances in Optics: Reviews. Book Series, Vol. 5

Fig. 1.32. Wave packet.

The other velocity is the group velocity of the wave packet, that is, the moving velocityof the point with the maximum amplitude of the composite wave. According to (1.134),the change of amplitude is 2𝐴 cos( 𝜔𝑚 𝑡 − 𝑘𝑚 𝑧), let 𝜔𝑚 𝑡 − 𝑘𝑚 𝑧 = constant, the moving
velocity of the maximum amplitude point of the wave packet can be obtained𝑑𝑧𝜔𝑣𝑔 = 𝑑𝑡 = 𝑘 𝑚 ,𝑚(1.136)

The formula (1.136) is the group velocity of the wave packet formed by the superpositionof two plane waves with different frequencies. The wave energy is proportional to thesquare of the amplitude, so that the group velocity represents the propagation velocity of
the wave energy. If the wave packet is composed of multiple frequency components, thegroup velocity should be the moving velocity of the point with the maximum amplitude,
and the maximum amplitude appears at the same phase position of all plane waves. Let
the phase of any plane wave be 𝜙 = 𝜔𝑡 − 𝑘𝑧, and the extremum point of the same phasecan be expressed as𝑑𝜙𝑑𝜔
= 𝑡 𝑑𝑘 − 𝑧 = 0
𝑑𝑘(1.137)

Then the moving velocity (group velocity) of the point with the maximum amplitude of
the wave packet is
𝑣𝑔 =

𝑧
𝑡

=

𝑑𝜔
𝑑𝑘

(1.138)For a plane wave, there only exists the phase velocity not the group velocity that
propagates the signal. In order to transmit the signal, the wave must be modulated, which
involves the complex wave composed of more than one frequency wave, that is, the signal
can only propagate with the complex wave, and its propagation velocity is that of wave
packet movement.
If we take 𝜔 = 𝑘𝑣, where 𝑣 is phase velocity, into (1.138), we get
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Chapter 1. Physical Optics

𝑣𝑔 =and considering 𝑘 =2𝜋，and 𝑑𝑘
𝜆𝑑𝜔𝑑𝑣= 𝑣+𝑘 ,𝑑𝑘𝑑𝑘
2𝜋

= − 𝜆2 𝑑𝜆, we furthermore have𝑣𝑔 = 𝑣 − 𝜆𝑑𝑣𝑑𝜆(1.139)The formula (1.139) indicates that the relationship between group velocity and phasevelocity is related to the dispersion of the medium. When the medium dispersion is small,the group velocity and phase velocity are almost the same, and the wave packet has astable shape. When the dispersion is large, the group velocity and phase velocity may bevery different. If the complex wave propagates in the medium with large dispersion, thewave packet will gradually flatten and eventually disappear due to the significant
difference of phase velocity of each frequency. At this scenario, the concept of groupvelocity is meaningless.
1.5.2. Superposition of Light Waves: Optical Interferometry
In Section 1.5.1.3, we discussed that if the two waves satisfy the coherent conditions, as
discussed in Section 1.3, they will interfere when they are superposed, and the composite
amplitude and its intensity will form a stable distribution in space, with the light field
vibration strengthened in some locations and weakened in some locations.
The reason can be understood from the luminous characteristics of ordinary light source.
The emission mechanism of an ordinary light source is due to the spontaneous emissionof atoms. The atoms in the high energy state will spontaneously transit to the low energy
state and emit light waves. The duration of each emission is about 10−8 s, and the lengthof the wave train is less than 3 m. There are a large number of atoms in the light source,
and the transitions occur randomly. The wave trains emitted by one atom successively andthose emitted by different atoms are completely independent. Even if the frequencies of
the wave trains emitted by ordinary light sources are the same, their initial phasedifference is changing rapidly, so that the composite vibration distribution is also changing
rapidly. The response speed of common detectors including human eyes is limited. As aresult, the light intensity detected is the average value in the response time of the detector.
The intensity expression (1.126) for the composite light field is averaged in the responsetime interval τ of the detector.
1 𝜏
1 𝜏𝐼̄ = 𝜏 ∫0 𝐼 𝑑 𝑡 = 𝐼1 + 𝐼2 + 2√𝐼1 𝐼2 𝜏 ∫0 𝑐𝑜𝑠∆𝜑𝑑𝑡(1.140)Because the initial phase difference of the two wave trains changes rapidly and irregularly,
it experiences all possible values from 0 to 2π for many times in the time interval τ. The
integral of the third interference term in the above formula is equal to zero, that is, theinstantaneous interference term is averaged out by the average effect of the detector. The
average intensity of the composite vibration in the time interval τ is75

Advances in Optics: Reviews. Book Series, Vol. 5

𝐼̄ = 𝐼1 + 𝐼2

(1.141)In this case, the average intensity of the composite light field is equal to the sum of theintensities of the individual fields, and there is no interference phenomenon. Usually thelight waves from two independent light sources or different parts of the same light source
do not satisfy the coherent condition.
1.5.2.1. Interference Phenomena of Monochromatic Light
In order to produce observable interference pattern by using an ordinary light source(suppose that the light source was monochromatic), we can use some optical methods todivide a light wave train emitted by one luminescent atom into two beams. Such two
divided light beams have the same initial phases. Then the phase difference between themis only determined by the difference of two different optical paths through which lightbeams propagate.
The methods splitting a light wave from an ordinary source includes wavefront division
and amplitude division.
Wavefront division
The wavefront of the same light beam is divided into two parts, and the two divided beams
are overlapped after passing through different paths to produce interference. Young's
experiment is the most famous example of wavefront division interferometry.
The Young's experimental setup is shown in Fig. 1.33. After the light from the point lightsource S0 on the optical axis passes through two pinholes S1 and S2 with equal sizes andsymmetry about the optical axis, the interference fringes can be seen on the screen withthe distance D from the two pinholes. Because the two light waves emitted from S1 and S2
are separated from the same wavefront, and S1 and S2 are equidistant from S0, they can beused as two coherent light sources with the same phases. The light intensity distributionof a point P near the optical axis on the observation screen is
𝐼 = 𝐼1 + 𝐼2 + 2√𝐼1 𝐼2 𝑐𝑜𝑠 𝛿,(1.142)where 𝛿 is the phase difference between S1 to P and S2 to P, and 𝛿 = 𝑘(𝑟2 − 𝑟1 ). Taking
it into the above formula, we have
𝐼 = 𝐼1 + 𝐼2 + 2√𝐼1 𝐼2 𝑐𝑜𝑠 [2𝜋(𝑟2 −𝑟1 )]𝜆(1.143)The light intensity of the point P on the surface depends on the optical path difference of
two light waves at this point. Let the coordinates of point P be (x, y, D), as shown in theFig. 1.33, the distances from S1 and S2 to P are, respectively
𝑑 2
2

𝑑 2
2

𝑟1 = √(𝑥 + ) + 𝑦 2 + 𝐷 2, 𝑟2 = √(𝑥 − ) + 𝑦 2 + 𝐷 2,
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Chapter 1. Physical Optics

where 𝑑 is the distance between S1 and S2. From the above two formulas, we can get
2𝑥𝑑
𝑟12 − 𝑟22 = 2𝑥𝑑. Such the optical path difference is |𝑟1 − 𝑟2 | = 𝑟 +𝑟 . If the distance of12

the screen is far enough to make D»d, and the observation range is small, i.e. D»x,y, there
𝑥𝑑

will be 𝑟1 + 𝑟2 ≈ 2𝐷, then |𝑟1 − 𝑟2 | = 𝐷 . Bringing it into (1.143), we obtain
2𝜋𝑥𝑑
)
𝜆𝐷

𝐼 = 𝐼1 + 𝐼2 + 2√𝐼1 𝐼2 𝑐𝑜𝑠 ((1.144)

Fig. 1.33. Young's experimental setup.

The formula (1.144) indicates that the interference pattern of Young's experiment near thez-axis on the screen is a series of parallel fringes. The direction of fringes is perpendicular
to the line connecting S1 and S2. The fringes are distributed according to the cosine
𝑚𝜆𝐷
function, as shown in Fig. 1.34. When 𝑥 = 𝑑 (𝑚 = 0, ±1, ±2, ⋯ ) on the screen,
there is the maximum light intensity 𝐼 = 𝐼1 + 𝐼2 + 2√𝐼1 𝐼2 , the bright fringe. When
1 𝜆𝐷

𝑥 = (𝑚 + 2) 𝑑 (𝑚 = 0, ±1, ±2, ⋯ ) , there is the minimum light intensity
𝐼 = 𝐼1 + 𝐼2 − 2√𝐼1 𝐼2, the dark fringe.

Fig. 1.34. Fringes from Young's experiment.

The wavefront division method requires the size of the source be small enough. A point
or line light source is usually used. Because the light wave emitted by each point on thelight source will form its own set of interference fringes, the positions of fringes
77Advances in Optics: Reviews. Book Series, Vol. 5

corresponding to different points will shift, and the larger the light source is, the greater
the movement will be, which will cause the fringes to blur until they are invisible.
Other devices for generating coherent light by wavefront division include Fresnel biprism,
Fresnel double-sided mirror, and Lloyd's mirror, etc.
Fresnel biprism is composed of two thin prisms with a very small vertex angle with thecommon bottom, as shown in Fig. 1.35. A beam of light from the point source is refracted
by the biprism and divided into two beams, which is equivalent to the two beams from thetwo virtual images S1 and S2 formed by the prism. Therefore, they can be regarded as thecoherent light source and produce interference fringes in the overlapping area.

Fig. 1.35. Fresnel biprism.

Fresnel double-sided mirror is composed of two mirrors with a small angle, as shown in
Fig. 1.36. The light wave emitted by the point light source S is reflected by the
double-sided mirror M1 and M2 to form two beams, which irradiate the screen to produce
interference fringes. It can be regarded as the two beams of light emitted by the two virtual
images S1 and S2 formed by S in the double-sided mirror, so that S1 and S2 are equivalent
to a pair of coherent light sources.

Fig. 1.36. Fresnel double-sided mirror.

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Chapter 1. Physical Optics

In the Lloyd’s mirror experiment as shown in Fig. 1.37, the point source S1 is placed quite
far away from the mirror MM and close to the mirror plane. One part of the light wave
emitted by S1 directly irradiates the screen E, and the other part of the light wave isincident on the mirror MM at nearly 90 degrees (grazing incidence), and then it isreflected by the mirror MM to reach the screen E. The two beams come from the samelight wave by dividing its wavefront, so that they are coherent. The corresponding
coherent light source is S1 and its virtual image S2 in the mirror.

Fig. 1.37. Lloyd’s mirror.

In the experiment, when the screen E is moved to contact with one end M of the mirror,
the distance of the two beams from S1 and S2 to the point M is the same, so that the pointM seems to be the center of the bright fringe, but it is actually the center of the dark fringe.
The reason for this phenomenon, readers can consult to the discussion about Fresnel’s
formulas in Section 1.2.
Amplitude division
A beam of light incident on a transparent dielectric film is successively reflected andrefracted by the upper and lower surfaces to form the multiple beams of reflected light (orrefracted light). They can be regarded as coherent light beams formed by dividing theamplitude of the incident light. The basic principle of the amplitude division method will
be discussed by taking the interference of parallel thin films as an example.
As shown in Fig. 1.38, there is a uniform transparent parallel film with refractive index 𝑛
and thickness ℎ, which is placed in a transparent medium with refractive index 𝑛′ . The
monochromatic light with wavelength 𝜆 is incident on the upper surface of the film, andthe incident angle is 𝑖 and the refraction angle 𝛾. A pair of coherent parallel beams 1 and
2 is produced after the incident light is reflected by the upper and lower surfaces of thefilm. The two reflected beams can only overlap at infinity. If a convergent lens is used,
they can be superimposed on the focal plane to produce interference fringes. For an
arbitrary point P on the focal planeI  I1  I 2  2 I1 I 2 cos  ,79Advances in Optics: Reviews. Book Series, Vol. 5

where 𝛿 = 𝑘(𝑟2 − 𝑟1 ) is the phase difference between the two beams from point A topoint P，which depends on the optical path difference 𝑟2 − 𝑟1 . The paths of the two
reflected beams are ADP and ACBP respectively, and BD is the vertical line perpendicular
to beam 1. According to the equal optical path property of the lens, 𝐷𝑃 = 𝐵𝑃. The optical
path difference between the two beams is 𝑟2 − 𝑟1 = (𝐴𝐶 + 𝐶𝐵)𝑛 − 𝐴𝐷𝑛 . From thegeometric relationship in Fig. 1.38, we haveℎ𝐴𝐶 = 𝐶𝐵 = 𝑐𝑜𝑠 𝛾 , 𝐴𝐷 = 𝐴𝐵 𝑠𝑖𝑛 𝑖 = 2ℎ 𝑡𝑎𝑛 𝛾 𝑠𝑖𝑛 𝑖

Fig. 1.38. Parallel thin films.

Because two beams of light are reflected by the film surfaces, and the refractive index of
the film is different from that of the surrounding medium, the additional optical path
𝜆difference 2 between two beams should also be considered (see discussion about Fresnel’s
formulas in Section 1.1). By means of the refraction law, the optical path difference canbe expressed as𝑟2 − 𝑟1 = 2ℎ𝑛 𝑐𝑜𝑠 𝛾 +𝜆2= 2ℎ√𝑛2 − 𝑛′2 𝑠𝑖𝑛2 𝑖 +𝜆2(1.145)The distribution of interference fringes is determined by the optical path difference, andthen by 𝑛, ℎ, and 𝑖 in thin film interference. If 𝑛 and ℎ are fixed, i.e., for a uniform parallel
film, the optical path difference is only determined by the incident angle. The incident
light with the same incident angle 𝑖 forms interference fringes of the same order. Theinterference fringes of different orders will be formed when the reflected light interfereswith the incident light at different incident angles. The interference formed by the parallel
film with the same thickness is called equal inclination interference. If 𝑛 and 𝑖 are fixed,
i.e., a beam of parallel light is incident on the film with varying thickness, the optical path
difference is only related to the film thickness ℎ. When the film thickness is the same, thesame order of interference fringes is formed; when the film thickness is different, thedifferent orders of interference fringes are formed. This kind of interference is called equal
thickness interference.
80Chapter 1. Physical Optics

It can be seen from the above discussions that the formation of interference fringes in theamplitude division method is not related to the size of the light source, which is differ
from wavefront division method. Different points in the source emit light waves in all
directions, and the light beams with the same angle have the same optical path difference,
which will form interference fringes of the same order and the same position on the screen.
After superposition, the contrast of the fringes can be increased. Therefore, this method
can use the extended light source to produce clearer and brighter interference fringes.
However, the fringes formed from an extended light source are localized.
The above two methods of forming stable fringes with coherent light generated byordinary light source have been introduced. Next we will define the contrast, or visibility,of interference fringes as follows.
𝐼−𝐼

𝑉 = 𝐼𝑚𝑎𝑥 +𝐼𝑚𝑖𝑛 ,
𝑚𝑎𝑥

𝑚𝑖𝑛

(1.146)

where 𝐼𝑚𝑎𝑥 and 𝐼𝑚𝑖𝑛 are the maximum intensity and the minimum intensity of the fringe,
which can be obtained from the equations (1.129) and (1.131). When 𝐼𝑚𝑖𝑛 = 0, visibility
𝑉 = 1, the contrast is the biggest, and the fringe is clearest. When 𝐼𝑚𝑖𝑛 = 𝐼𝑚𝑎𝑥 , 𝑉 ≈ 0,
the fringes are indistinct or even illegible. If the equations (1.129) and (1.131) are brought
into the equation (1.146), we can also get
𝑉

2𝐴1 𝐴2= 𝐴2 +𝐴
2
1
2

=

𝐴
𝐴2

2 1
𝐴 2
𝐴2

1+( 1 )

(1.147)

The above formula shows that the amplitude ratio will affect the visibility of interference
fringes. For an ideal monochromatic coherent light source, when the amplitudes of two
beams are equal, the visibility of interference fringes is the largest.
1.5.2.2. Interference Phenomena of Partial Coherent Light
In Section 1.5.2.1, we analyzed the interference of completely coherent light, however inpractice light from the source is partially coherent. The light emitted by the source is not
an ideal single wavelength, but has a certain bandwidth; the luminous area of the lightsource has a certain size, rather than an ideal point light source. This section will discuss
how these two factors affect the visibility of interference fringes.
As mentioned above, the interference generated by an ordinary light source is actually the
result of superposition of wavelet trains separated from the same wave train emitted byatoms. Due to the short duration of the atomic luminescence, which is the time about∆𝜏 = 10−8 s, the length of the wave train 𝐿 = 𝑐∆𝜏 is less than 3 m (c is the light speed
in vacuum). Taking Young's experiment as an example, two wave trains are divided bydouble slits. When their optical path difference reaches the point P satisfies |𝑟2 − 𝑟1 | < 𝐿,i.e. the time difference satisfies ∆𝑡 < ∆𝜏 , they can meet and interfere, as shown in
Fig. 1.39 (a). If the optical path difference is too large at P, then |𝑟2 − 𝑟1 | > 𝐿, twowavelets from the same wave train can not meet at this point and will not interfere, as
81Advances in Optics: Reviews. Book Series, Vol. 5

shown in Fig. 1.39 (b). The duration, ∆𝜏, of the wave train can be called coherent time,further denoted as 𝜏𝑐 . If the two beams of light interfere, the maximum optical path
difference should be less than the wave train length. The length of wave train can be called
coherent length 𝐿𝑐 = 𝑐𝜏𝑐 . The coherence length 𝐿𝑐 and coherence time 𝜏𝑐 measure thetemporal coherence of the light source from the different dimensional quantities and they
are in fact equivalent.

abFig. 1.39. Two wavelets from the same wave train.

The coherence length is related to the monochromaticity of light. The plane wave is an
infinite wave train with a single frequency, while the actual light source emits a wave train
with a limited length. According to Fourier transform, it is impossible to have a single
wavelength (or frequency) for a finite length wave. A wave train with a finite length
always contains a certain range of wave length ∆𝜆, or frequency range ∆𝜈，as shown in
Fig. 1.40. The intensity at the center wavelength is 𝐼0 , and the wavelength range ∆𝜆 where𝐼
the intensity is reduced to 20 is usually taken as the line width. The smaller the line widthof the light source, the better the monochromaticity is.

Fig. 1.40. Actual light source with the wavelength range ∆𝜆.

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Chapter 1. Physical Optics

Obviously, light with different wavelengths in the line width will produce their own
interference fringes due to different optical path differences. Because of incoherence oflight with different wavelengths, the total light intensity overlapped by different fringes
with different wavelengths is just addition of the light intensities of fringes of various
wavelengths, which will affect the contrast of fringes. As shown in Fig. 1.41, the lower
Δ𝜆curve shows the distribution of interference fringes with wavelengths of 𝜆 − 2 , 𝜆,Δ𝜆𝜆 − 2 . Because the width of the fringes is proportional to the wavelength, the same orderinterference fringes with different wavelengths have different positions. With the increaseof the order, the misplacement of different wavelength fringes becomes more and more
serious, and these fringes will overlap to reduce the contrast. The upper curve is the total
intensity of incoherent superposition. It can be seen that with the increase of the order, thecontrast of interference fringes decreases gradually, and finally the fringes disappear.

Fig. 1.41. Light intensity of fringes of various wavelengths.

According to the condition of bright fringes, the optical path difference of the m-order𝑥𝑑bright fringe is |𝑟1 − 𝑟2 | = 𝐷 = 𝑚𝜆 (𝑚 = 0, ±1, ±2, ⋯ ). If the line width of the lightis ∆𝜆, the position where the interference fringes disappear should be the position whereΔ𝜆the m + 1 order bright fringe of the light with the wavelength of 𝜆 − 2 coincide with theΔ𝜆m order bright fringe of the light with the wavelength of 𝜆 + 2 . The reason of fringe
disappearance under this condition can be appreciated as the follows. If this condition issatisfied, it is equivalent to that the phases of light fields from all wavelengths of light
experience all possible values from 0 to 2π, so that the fringes disappear due to the average
effect of the detector. Then this condition can be expressed in the mathematical form asΔ𝜆Δ𝜆follows: (𝜆 − ) (𝑚 + 1) = (𝜆 + ) 𝑚.
2

2

Δ𝜆

From the above formula, we have 𝑚Δ𝜆 = 𝜆 − 2 .
𝜆Since Δ𝜆 ≪ 𝜆, in the above formula we can ignore Δ𝜆 and obtain that 𝑚 = Δ𝜆.
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Advances in Optics: Reviews. Book Series, Vol. 5

The maximum optical path difference, i.e. the coherence length, should meet that𝜆2

𝐿𝑐 = 𝜆𝑚 = Δ𝜆(1.148)From the above formula, we can know the coherence length 𝐿𝑐 of the light source isinversely proportional to the line width Δ𝜆. The smaller the Δ𝜆 is, the larger 𝐿𝑐 , and thebetter the monochromaticity, that is, the easier interference occurs.
Even if the ideal monochromatic light source has the infinite time coherence, however,for an extended light source the interference fringes generated by different points on thesource are still not completely coherent and their superposition will also affect the fringe
contrast. This is illustrated by Young's experiment.
As shown in Fig. 1.42, if the point light source S is on the optical axis and the distancesfrom 𝑆 to the double slit 𝑆1 and 𝑆2 (the distance between 𝑆1 and 𝑆2 is 𝑑) are equal, thelight waves emitted from the double slits have no initial phase difference, and the𝑏zero-order bright fringe is at point 𝑂 on the optical axis. When 𝑆 moves down 2, there isa certain initial phase difference in the two beams emitted from the double slit. At thistime, the zero-order bright fringe will move up to 𝑂′ above the axis. The distance between
𝑂 and 𝑂′ is𝑏(1.149)𝑥 = 2𝑅 𝐷

The width of fringes produced by different parts of the light source with the wavelengthof λ is the same. The width of fringes can be calculated by the formula of bright fringes
∆𝑥 =

𝐷𝜆
𝑑

If the width of the light source is b, the fringes generated by each part of the extended
source are displaced, and the dislocation position is determined by the formula (1.149).
Each part of the light source is incoherent, and the total intensity of the fringes is the resultof incoherent superposition of several displaced fringes. This causes the contrast of thefringes to decrease. When the fringe generated by the edge point moves half of the fringe
width relative to the fringe generated by the center point, i.e.
𝑥 =

𝑏
𝐷
2𝑅

=

𝐷𝜆
2𝑑

(1.150)The bright fringes of the interference fringes generated by the center of the light sourcecoincide with the dark fringes generated by the edge of the light source, and the fringes
will be blurred. If we denote this particular width of the light source as 𝑏0 , according toequation (1.150), the width, 𝑏0, of the light source S is𝑅

𝑏0 = 𝑑 𝜆84(1.151)

Chapter 1. Physical Optics

When the size if the light source is 𝑏0 , fringes will disappear. It shows that only when thewidth of the light source satisfies 𝑏 < 𝑏0 , or when the distance of double slit satisfies
𝑅𝜆𝑑 < , the interference fringes can be observed. The influence of the width of the light𝑏0

source or the distance between two slits on the interference fringes is called spatial
𝑅𝜆coherence, and the critical value of the distance between two slits 𝑑0 =is called the𝑏0

spatial coherence length of light.

Fig. 1.42. Young's interference experiment.

From the formula (1.151) the angular diameter β of the light source can be expressed in
the spatial coherence length 𝑑0 and the wavelength of light 𝜆 as follows.
𝛽 =

𝜆
𝑑0

(1.152)

1.5.2.3. Michelson Stellar InterferometerAs shown in the equation (1.152), the angular diameter of the light source can be measured
by the interferometric method. In 1920s, Michelson and Pease demonstrated
experimentally that the angular diameter of a star might be obtained with help of theinterferometer as shown schematically in Fig. 1.43.
The basic idea for measuring the angular diameter of a star can be understood as follows.
Light from the star illuminates the outer mirrors M1 and M2, and then is reflected onto
the two inner mirrors M3 and M4. Furthermore, light is further reflected from the mirror
M3 and M4 and brought to the focal plane of the telescope that is attached to theinterferometer. A detector is placed at the back focal plane of the telescope. The distance
between the mirror M1 and the mirror M2 in the interferometer could be symmetricallychanged freely in the direction joining M1 and M2, and the inner mirrors M3 and M4 are
fixed. On the detector a diffraction image of the star, crossed by fringes generated by thetwo interfering beams, could be obtained.
For a certain size of a star, suppose that its angular diameter is 𝛽. According to theequation (1.152), when the distance between the mirror M1 and the mirror M2 is somevalue, suppose 𝑑0 , the visibility of the fringes on the detector is just zero. Such the angular85

Advances in Optics: Reviews. Book Series, Vol. 5

diameter of the star can be approximately calculated according to the formula (1.152). For
the circular aperture of the optical system, the angular diameter of the star can be
accurately calculated by means of the following formula.
𝜆𝛽 = 1.22 𝑑0(1.153)

In essence, Michelson’s stellar interferometer is just Young’s double slit setup.

Fig. 1.43. Michelson’s stellar interferometer.

1.6. Polarization of Light
The concept of polarization of an electromagnetic wave was introduced in Section 1.1.5,
and furthermore the various types of polarized plane waves were briefly described. In thecurrent section, the several methods for producing polarized light are going to be
described, the mathematical representation of polarized light and of polarizers will be
explained, and finally the basic theory of the polarization of stochastic light fields is
introduced.
There are various applications of polarized light for different fields. For example,
communication and detection systems, liquid crystal display technology, optical
microscopy, optical spectroscopy, geology material science, astronomy, and so on.
1.6.1. Generation of Polarization Light
When light acts on any matter whose optical properties are asymmetric along transverse
directions to the light propagation vector, the polarized light will be produced. In
Section 1.1, we know that light is a transverse wave. Based on this property of light waves,
one of asymmetrical examples along transverse directions to the light propagation vector
is light traveling through anisotropic matters, which can provide a means for generating
polarized light. In essence, the process of polarizing light is taking unpolarized light and86

Chapter 1. Physical Optics

extracting from it a polarized light by means of asymmetric properties of matter due tothe transverse property of light. In the following, several most important approaches forgenerating polarized light are discussed in detail.
1.6.1.1. Polarization by Reflection from Dielectric Surfaces
Light reflected from a dielectric interface becomes partially polarized if the angle ofincidence is nonzero, which is asymmetric with respect to the local normal of the dielectricinterface in the nonzero angle of incidence. When the angle of incidence is at Brewster’sangle, the reflected light becomes the completely polarized one being perpendicular to theplane of incidence (S-polarization, here S is the initial of German word ‘Senkrecht’,meaning perpendicular). And the refracted light becomes partially polarized light that ispolarized more in the plane of incidence than in the perpendicular plane to the plane ofincidence (i.e. more P-polarization than S-polarization, here P is the initial of German
word ‘parallele’).
In Fig. 1.44, the dielectric interface lies in the x-z plane, and z-axis is also the normal of
the interface surface. 𝑛1 is the index of refraction of the first medium through which theincident light propagates, and 𝑛2 the index of refraction for the second medium throughwhich the refracted light propagates. When the reflected and refracted light rays areperpendicular to each other, the reflected light becomes S-polarization, and the refracted
light becomes partially polarized.
As shown in Fig. 1.44, 𝜃𝐵𝑖 and 𝜃𝐵𝑟 are the angle of incidence and of reflection,
respectively, and 𝜃𝐵𝑡 is the refracted angle. At Brewster’s angle of incidence, or
polarization angle, we have the following condition
𝜃𝐵𝑟 + 𝜃𝐵𝑡 =𝜋2(1.154)

Fig. 1.44. Polarization by reflection and refraction through an interface.

Snell’s law says that𝑠𝑖𝑛𝜃𝐵𝑖𝑠𝑖𝑛𝜃𝐵𝑡𝑛= 𝑛2 ,
1(1.155)87

Advances in Optics: Reviews. Book Series, Vol. 5

and further the reflection law gives the expression that 𝜃𝐵𝑟 = 𝜃𝐵𝑖 . Taken all expressions
together, we get Brewster’s law as𝑛𝑡𝑎𝑛𝜃𝐵𝑖 = 𝑛2
1

(1.156)

For unpolarized light being incident onto the interface between two different media at
Brewster’s angle 𝜃𝐵𝑖 , the reflected light has pure S-polarization, and the refracted light ispartially polarized with more P-polarization than S-polarization. The physical mechanism
for S-polarization of the reflected light can be qualitatively explained as follows. Whenlight is incident onto the interface between two media, it is absorbed by atoms in thesecond medium, and due to the transverse property of light waves, dipoles in these atoms
will execute the forced oscillation in the direction perpendicular to the traveling directionof the refracted light. These oscillating dipoles re-emit light to form both the reflected andrefracted light waves. From radiation formula (1.21), we know that the P-polarization in
the reflected light just stems from the dipole oscillations caused by the P-polarization of
the refracted light and the electric field does not exist in the direction of the dipole
oscillations. When light strikes onto the interface at the Brewster’s angle, the reflected
and refracted light rays are mutually perpendicular. As a result, the direction of the dipole
oscillations of the atoms in the second medium is the same direction as the reflected light,
so that there is no P-polarization in the reflected light, just S-polarization left. The lightpolarization for Brewster reflection is the same as the light polarizations along y- andz-axes for light scattering as shown in Fig. 1.46, to be discussed in Section 1.6.1.4.
Polarization by reflection at Brewster’s angle is very sensitive to surface quality andstructure, which is exploited in ellipsometry for investigating surfaces.
1.6.1.2. Polarization through Diattenuation
Some materials, natural or synthetic, absorb light with different polarizations by different
proportions. This asymmetric property of selective absorption of the material is referred
to as diattenuation that can be used to polarize light. The most common material, Polaroid,
uses this property to polarize light and make a polarizer. The polarizer easily transmits
light with electric vibrations along the direction perpendicular to the direction ofabsorption. This preferred transmission direction is called the transmission axis (TA) of
the polarizer. For an ideal polarizer, the transmitted light is linearly polarized in the
direction of transmission axis. When a polarizer is used to test the state of polarization oflight, the polarizer with this purpose is called an analyzer.
Malus’ law says that the light intensity transmitted through a polarizer and an analyzer
with an angle θ between their transmission axes is given by
𝐼 = 𝐼0 cos 2 𝜃,

(1.157)

where 𝐼0 ∝ 𝐸02 (𝐸0 is the amplitude of the incident field) is the light intensity incident
on the polarizer, and I is the light intensity transmitted through the analyzer.
88Chapter 1. Physical Optics

Malus’ law is easily understood through Fig. 1.45 with the help of the relationshipbetween the light intensity and the electric field defined by the formula (1.19).

Fig. 1.45. Schematic of Malus’ law.

1.6.1.3. Polarization by Scattering
Light scatterings occur whenever there is an irregular distribution of minute particles or
the fluctuations of the density of medium through which light is traveling. There are many
ways for light scattering, to mention a few, for example, dust particles whose size iscompared with the light wavelength, the particles in a colloidal, and air molecules.
We take light scattering from one air molecule, as shown in Fig. 1.46, to elaborate themechanism for the production of polarized light from unpolarized light by means ofradiation theory in Section 1.1. According to formula (1.21), we know that the orientationof the electric vector of the scattered light follows the dipole radiation pattern, and theelectric vectors of the scattered light are perpendicular to the propagation direction, as
shown in Fig. 1.46. If the incident wave is unpolarized, the scattered light in the forward
x-axis direction is completely unpolarized; off that x-axis it is partially polarized,
becoming increasingly more polarized as the angle increases away from x-axis. For y-axis
direction or z-axis direction of observation that is normal to x-axis of the incident beam,
the light is completely linearly polarized. In y-axis and z-axis directions, if light were notcompletely linearly polarized, as shown in Fig. 1.46, the radiation formula (1.21) would
not be satisfied.
1.6.1.4. Polarization by Birefringence
Crystalline substances whose lattice atoms were not completely symmetrically arrayed
are optically anisotropic. Their optical properties are not the same in all directions withina given crystal. Birefringence is associated with anisotropy in the binding forces between
the atoms forming a crystal. Mechanically, we can visualize that the atoms are boundedby springs with different stiffness in different crystalline directions. As a result, thedifferent polarizations of the electric field lead to the different indices of refraction inanisotropic materials, which, therefore, results in the phenomenon of double refraction orbirefringence in optically anisotropic media.
89Advances in Optics: Reviews. Book Series, Vol. 5

Fig. 1.46. Schematic of polarization by scattering.

For uniaxial crystals (hexagonal, tetragonal, and trigonal crystal), there is one direction sothat any light in that direction in the crystal has the same index of refraction, regardless of
its state of polarization. This direction is called the optic axis (OA) that is also thesymmetric axis of the crystal. The remaining crystal systems (orthorhombic, monoclinic,
and triclinic) have two optic axes and are said to be biaxial. Any plane that contains the
optic axis is referred to as the principal plane, and the principal plane being normal to oneof the cleavage surfaces of the crystal is called the principal section.
In the following, we focus our discussion on the phenomena of light traveling throughuniaxial crystals. When there exists an angle between the propagation direction and theoptic axis, for one part of the light wave, its electric field is normal to the optic axis, andfor the other part of the light wave, its electric field is parallel to the optic axis. The lightwave with its electric field being everywhere perpendicular to the optic axis will propagate
in the crystal in all directions with the same speed, as they would be in an isotropic
medium. This wave is called the ordinary- or o-ray. The light wave with its electric fieldparallel to the optic axis will propagate in the crystal in a speed different from the speedof o-ray. This wave is called the extraordinary- or e-ray.
Thanks to the anisotropy or asymmetry of the crystal, the propagation speed of e-ray andof o-ray in the crystal is different. Therefore, the index of refraction 𝑛𝑜 of the ordinary ray
is different from 𝑛𝑒 of the extraordinary ray. The ordinary and extraordinary rays
traveling along the optic axis have the same indices of refraction no = ne because theelectric field is always perpendicular to the optic axis in such a case. The difference of theindices of refraction ∆𝑛 = 𝑛𝑒 − 𝑛𝑜 between the ordinary and the extraordinary rays is
maximized in the direction of propagation perpendicular to the optic axis. This difference∆𝑛 is a measure of the birefringence, If ∆𝑛 < 0, the crystal is negative uniaxial; If∆𝑛 > 0, the crystal is positive uniaxial. Then the speed of the extraordinary ray and of the𝑐𝑐ordinary ray are 𝑣𝑒 = 𝑛 and 𝑣𝑜 = 𝑛 , respectively, here 𝑐 is the speed of light in𝑒0vacuum. As shown in Fig. 1.47, if we choose Cartesian coordinate system having its
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Chapter 1. Physical Optics

z-axis along the optic axis, after time 𝑡 the equation for the spheroid of the e-ray wavelet
𝑥 2 +𝑦 2𝑧2+ 𝑣 2 = 𝑡 2 , and the equation for the sphere of the o-ray wavelet can𝑣𝑒2𝑜𝑥 2 +𝑦 2 +𝑧 2
2
be written=𝑡. Drawings (a) and (b) in Fig. 1.47 schematically demonstrate𝑣𝑜2

can be written

the Huygens’ wavelets for e-ray and o-ray in the negative and positive uniaxial media,
respectively. In Fig. 1.48, birefringence is shown for a uniaxial crystal in drawing (a). The
black dots in the drawing stand for the polarization perpendicularly pointing out the
principal section (o-ray), and the arrow lines represent the polarization in the principal
section (e-ray). Based on Huygens’ wavelets for e-ray and o-ray, in the drawing (b) we
can qualitatively explains the formation of o-ray and e-ray in a negative uniaxial crystal
using Huygens’ principle (1.47) that was discussed in Section 1.2.

Fig. 1.47. Huygens’ wavelets in uniaxial media: negative crystal (a), positive crystal (b).

Fig. 1.48. Polarization by birefringence in a principal section (a); Huygens principleof birefringence (b).

It will now be a simple matter, in principle, to make some sort of linear birefringent
polarizers. Any scheme that can get the aim for angular separation between the e-ray andthe o-ray can be taken as a polarizer, all schemes exploiting the fact that 𝑛𝑜 ≠ 𝑛𝑒 . There
are various devices that can accomplish the physical separation of e-ray from o-ray, for91

Advances in Optics: Reviews. Book Series, Vol. 5

example, Wollaston prism, Glan-Foucault prism. These prisms are also named aspolarizing beam-splitting prisms.
Wollaston prism (Fig. 1.49)

Fig. 1.49. Wollaston prism.

A Wollaston prism is an optical device, invented by William H. Wollaston that separates
unpolarized light into two linearly polarized outgoing beams with mutually perpendicularpolarization directions. The Wollaston prism consists of two orthogonal prisms ofbirefringent material – typically a uniaxial material such as calcite. These prisms are
cemented together on their base (traditionally with Canada balsam) to form two right
triangle prisms with perpendicular optic axes. The o-ray in the first part of the prism
vibrating in the direction perpendicular to the diagram, as shown in Fig. 1.49, becomesthe e-ray in the second part, and the e-ray in the first part of the prism vibrating in theplane of the diagram becomes the o-ray in the second part of the prism. In this way, thetwo further spatially separated beams with mutually orthogonal polarization emerge on
the right side of the prism.
Glan-Foucault prism (Fig. 1.50)

Fig. 1.50. The Glan-Foucault prism.

The Glan-Foucault prism consists of two right-angled calcite prisms spaced with an air
gap and can produce and analyze a polarized plane wave, as shown in Fig. 1.50. The
Glan-Foucault prism is suitable for a broad spectral range. The incoming unpolarized ray
is incident on the surface normally, and the electric field of light travelling into the prism
can be resolved into components that are either completely parallel or perpendicular to theoptic axis. The two orthogonally polarized components of light are angularly separated by92

Chapter 1. Physical Optics

the birefringence property of the calcite. The angle-of-incidence on the calcite-air
interface is needed to arrange in such a way that for the o-ray totally internal refection
occurs, but not for the e-ray.
1.6.1.5. Phase Shift by Birefringence
The characteristics of optically anisotropic media can be used to manipulate the phaseshift between the o-ray and the e-ray so as to produce the other types of polarized lightbeam from an incident linearly polarized beam.
A device called the waveplate, or the phase shifter, can shift the phase between two
perpendicular polarization components of a light wave. A typical waveplate is abirefringent crystal cut into a plate with a carefully chosen orientation and thickness. For
convenience, we define the fast and slow axis for a waveplate. When the speed of the
e-ray is faster than that of the o-ray, we call the direction of the optic axis the fast axis,
and the direction perpendicular to it is the slow axis, i.e., the polarization direction of the
e-ray is the fast axis, and the polarization direction of the o-ray is the slow axis. When the
speed of the e-ray is slower than that of the o-ray, the situation is reversed.
If the optic axis is parallel to the surface of the waveplate and light is incident on the plate
at normal to the surface, as shown in Fig. 1.51, both o- and e- rays are going to propagate
in the same direction, but with the different indices of refraction. A phase shift ∆𝜙
between o- and e- rays is∆𝜙 =2𝜋(|𝑛0 − 𝑛𝑒 |)𝑑,𝜆(1.158)where 𝜆 is the wavelength of the incident light, 𝑛0 and 𝑛𝑒 are, respectively, the indices ofrefraction of o- and e- rays in the waveplate, and d is the thickness of the plate.

Fig. 1.51. Schematic of phase shift by birefringence.

1) Quarter-wave plate𝜋

If the thickness d of the plate is such as to make ∆𝜙 = 2 , it is a quarter-wave plate(QWP). This plate is referred to as the zero-order plate. For the manufacturing engineering
93Advances in Optics: Reviews. Book Series, Vol. 5𝜋reason the plate is made thicker for the higher order m, satisfying ∆𝜙 = (4𝑚 + 1) 2 ,where m is integers. A quarter-wave plate could convert linearly polarized light into
elliptically polarized light when the incident light polarized a certain angle 𝜃 with respect
to the fast axis of the quarter-wave plate. Specially, the polarization conversion from linear
to circular occurs on passage through a quarter-wave plate when the polarization directionof the incident light makes an angle of 45𝑜 with respect to the fast axis of the quarterwave plate. Conversely, when the fast and slow axes of a quarter-wave plate and the axesof the ellipse coincide, elliptically polarized light will be converted linearly on light
passing through a quarter-wave plate.
2) Half-wave plate
If the thickness d of the plate is such as to make ∆𝜙 = 𝜋, it is a half-wave plate (HWP).
These are called the zero-order plates. Because such plates are extremely thin, its thickness
being of order of one wavelength, in practice the plate is made thicker for the higher orderm, satisfying ∆𝜙 = (2𝑚 + 1)𝜋 , where m is integers. Suppose that the polarization
direction of an incident beam of linearly polarized light makes some angle 𝜃 with respect
to the fast axis, as shown in Fig. 1.52. When the light emerges from the plate, thepolarization direction of light will have rotated through 2𝜃, but the light emerged fromthe plate is still linear polarized. When the polarization direction of the incident lightmakes an angle of 45𝑜 with respect to the fast axis of the half-wave plate, the linearlypolarized light coming out from the plate is perpendicularly polarized to the polarization
direction of the incident light.

Fig. 1.52. Schematic of a half-wave plate.

1.6.1.6. Optical Activity
Optical activity is the capability of a material to rotate the plane of polarization of light
passing through it. If we are looking towards the light source, the plane of polarization
appears to have revolved clockwise, the material is referred to dextrorotatory; if looking
in the direction of the light source the plane of polarization appears to have revolved
counterclockwise, the material is levorotatory.
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1.6.2. Matrix Treatment for Completely Polarized LightIn above, we have discussed the physical mechanism for the generation of polarized lightand several corresponding devices. In what follows, we will elaborate mathematicalrepresentation of completely polarized light and of polarizers using Jones formalism.
Jones formalism is the mathematical method of describing polarized light in terms ofamplitudes and phases. Here we focus on Jones algebra of polarization for a
monochromatic light wave. For nonmonochromatic light, its Jones vector is the same asthat of monochromatic light, which is also the starting point for the other statisticalmethods for describing polarization of light and of polarizers. We will not discuss thissubject further. The interested readers can consult reference [13].
1.6.2.1. Mathematical Representation of Completely Polarized Light: Jones Vector
As shown in Fig. 1.53, the x and y complex field components 𝐸𝑥 and 𝐸𝑦 for a
monochromatic light wave traveling in the z-direction with amplitudes 𝐸𝑥0 and 𝐸𝑦0 , andinitial phases 𝜑𝑥 and 𝜑𝑦 , according to the convention introduced in Section 1.1, can be
written
𝐸𝑥 = 𝐸𝑥0 𝑒 𝑗(𝑘𝑧−𝜔𝑡+𝜑𝑥 ), 𝐸𝑦 = 𝐸𝑦0 𝑒 𝑗(𝑘𝑧−𝜔𝑡+𝜑𝑦 )(1.159)The electric field denoted by the above expressions can represent the different polarization
states of light.

Fig. 1.53. Cartesian coordinate system.

If we ignore the common factors in the expressions (1.159), and write them in a vector
form, we can get the following form.
𝐸𝑥0 𝑒 𝑗𝜑𝑥[]
𝐸𝑦0 𝑒 𝑗𝜑𝑦(1.160)This vector is also called Jones vector that describes the polarization state of light.

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Advances in Optics: Reviews. Book Series, Vol. 5

Let us now give out the particular forms of Jones vectors for the linear, circular, andelliptical polarizations. As stated in Section 1.1, there are three cases (a), (b), and (c) for
linear polarizations. The corresponding normalized Jones vector is as the following,
respectively:
Case (a)1[ ] linear polarization along x-axis;
0Case (b)0[ ] linear polarization along y-axis;
1
Case (c)𝑐𝑜𝑠𝛼[] linear polarization along at angle 𝛼 with respective x-axis.
𝑠𝑖𝑛𝛼
Suppose that normalized Jones vector taking the following form
1
√𝐸𝑥0 2 +𝐸𝑦0 2

[

𝐸𝑥0

𝜋

𝐸𝑦0 𝑒 𝑗 2

(1.161)],

where 𝐸𝑥0 and 𝐸𝑦0 represent the amplitudes of x and y components of the electric
vibration. This Jones vector refers to elliptically polarized light with counterclockwise
rotation. We will demonstrate why the expression (1.161) stands for counterclockwise
rotation of polarization.
According to the expressions (1.159) and (1.161), the real x and y components of theelectric field can be written𝜋

𝐸𝑥 = 𝐸𝑥0 𝑐𝑜𝑠(𝑘𝑧 − 𝜔𝑡), 𝐸𝑦 = 𝐸𝑦0 𝑐𝑜𝑠 (𝑘𝑧 − 𝜔𝑡 + 2 ) = −𝐸𝑦0 𝑠𝑖𝑛(𝑘𝑧 − 𝜔𝑡)First, let 𝑧 = 0, 𝑡 = 0, then 𝐸𝑥 = 𝐸𝑥0, 𝐸𝑦 = 0; second, take a little bit later time Δ𝑡𝜋(suppose that 𝜔𝛥𝑡 < ) and at the same spatial point 𝑧 = 0 , then we get
2
𝐸𝑥 = 𝐸𝑥0 𝑐𝑜𝑠(𝜔𝛥𝑡) > 0, and 𝐸𝑦 = 𝐸𝑦0 𝑠𝑖𝑛(𝜔𝛥𝑡) > 0. According to the right-handed
Cartesian coordinate system shown in Fig. 1.53, we know that electric field of light isrotating counterclockwise at an angular frequency of 𝜔. The same analysis can be applied
to the elliptically polarized light with clockwise rotation.
If Jones vector takes the following form
1
√𝐸𝑥0 2 +𝐸𝑦0 2

[

𝐸𝑥0

𝜋

𝐸𝑦0 𝑒 −𝑗 2]

(1.162)This means that the state of polarization for light is elliptically polarized light withclockwise rotation.
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Chapter 1. Physical Optics

If 𝐸𝑥0 = 𝐸𝑦0 , the cases represented by (1.161) and (1.162) are, respectively, circularly
polarized light with counterclockwise and clockwise rotations. For these cases, we write
their Jones vectors in a normalized form as follows.
The normalized form of the Jones vector for counterclockwise rotation, viewed
head-on, is
1 1
[ ]
√2 𝑗

(1.163)This mode is called left-circularly polarized (LCP) light.
The normalized form of the Jones vector for clockwise rotation, viewed head-on, is
1 1
[ ]
√2 −𝑗

(1.164)This mode is called right-circularly polarized (RCP) light.
1
0
For the linear polarization along x-axis [ ] and the linear polarization along y-axis [ ],01they are orthogonal to each other and therefore can be used as a basis for the Jones vectorof any polarization state. And they are also orthogonal to each other between the
1 1left-circularly polarized (LCP) light [ ]and the right-circularly polarized (RCP) light√2 𝑗
1 1[ ] and likewise they can be used as a new basis for the Jones vector of any√2 −𝑗polarization state. Suppose that in the linear polarization basis the x and y components of𝑎𝑥
a polarization state are 𝑎𝑥 and 𝑎𝑦 and denote as [𝑎 ], and in the circular polarization basis𝑦the LCP and RCP components of a polarization state are, respectively, 𝑎𝐿 and 𝑎𝑅 and𝑎𝐿denote as [𝑎 ], such that the transformation from the linear polarization basis to the𝑅𝑎𝐿
1 1 −𝑗 𝑎𝑥
circular polarization basis is as follows: [𝑎 ] = 2 [] [ ].
1 𝑗 𝑎𝑦√𝑅
To get the above expression, a polarization vector is decomposed onto the basis of thelinear polarization and of the circular polarization, respectively, such that an equation
relating the components in the different base is obtained. Solving the equation, the abovetransformation expression can be obtained.
It should be emphasized that Jones vectors representing different polarized states are notunique. It is obvious that any Jones vector can be multiplied a real or complex prefactor
the polarization mode represented by this vector is not changed, just changing amplitude
or/and promoting the phase of each polarization component by the same value.

97

Advances in Optics: Reviews. Book Series, Vol. 5

1.6.2.2. Mathematical Representation of Polarizers: Jones Matrices
Any optical device that can transmit light and modify the state of polarization can serve
as a polarizer. The action of the polarizer on light can generally described by a 2×2 Jones
matrix,
𝑎
[
𝑐

𝑏
],
𝑑

(1.165)where the matrix entries a, b, c, and d indicate the effects of polarizers on the state ofpolarization of the light traveling through them. In what follows, we give the Jones
matrices of linear polarizers, waveplates, and rotators.
In order to study the matrix forms for polarizers we have the following assumption that is
generally legitimate for the common scenarios encountered if the intensity of light is notvery high.
Assumption: The components of the electric field, 𝐸𝑥′ and 𝐸𝑦′ , of a light beam emerging
from a polarizer are linearly related to the components of the electric field, 𝐸𝑥 and 𝐸𝑦 , ofthe incident light beam, as shown in Fig. 1.54.
This relationship can be expressed in the matrix form as:𝐸𝑥′𝑎[ ′] = [
𝐸𝑦
𝑐

𝑏 𝐸𝑥
][ ]
𝑑 𝐸𝑦

(1.166)

Fig. 1.54. Schematic of a polarizing element.

a) Ideal linear polarizers
A linear polarizer doesn't affect the vibration direction of either 𝐸𝑥 or 𝐸𝑦 . What this meansis that 𝐸𝑥′ is just related to 𝐸𝑥 , and has nothing to do with 𝐸𝑦 , and the same as ycomponent. For the linear polarizer, the Jones matrices are:for the transmission axis (TA) of a linear polarizer along x-axis
1
[
0
98

0
],
0

(1.167)

Chapter 1. Physical Optics

for the transmission axis (TA) of a linear polarizer along y-axis
0 0(1.168)[],0 1
for the transmission axis (TA) of a linear polarizer at angle 𝜃 with respect to x-axis
2[ 𝑐𝑜𝑠 𝜃𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃

sin 𝜃 cos 𝜃 ]𝑠𝑖𝑛2 𝜃

(1.169)Next we will in detail illustrate how to get the above Jones matrix for the situation thatthe transmission axis of a linear polarizer makes an angle 𝜃 with respect to x-axis. First,
let light polarized along the same direction as the TA go through the polarizer, then its
𝑐𝑜𝑠𝜃
Jones vector is []. Second, let light with the polarization direction perpendicular to
𝑠𝑖𝑛𝜃
−𝑠𝑖𝑛𝜃
the TA go through the polarizer, then its Jones vector is []. Then, we can get the𝑐𝑜𝑠𝜃
following expressions
𝑐𝑜𝑠𝜃
𝑎
[
] = [
𝑠𝑖𝑛𝜃
𝑐

𝑏 𝑐𝑜𝑠𝜃 0𝑎][], [ ] = [𝑑 𝑠𝑖𝑛𝜃 0𝑐𝑏 −𝑠𝑖𝑛𝜃][]𝑑 𝑐𝑜𝑠𝜃

(1.170)

Solving the above equations with four unknowns a, b, c, and d, we can get the Jones matrix(1.169) for the transmission axis (TA) of a linear polarizer making an angle 𝜃 with respect
to x-axis. For the special case TA at 𝜃 = 45𝑜 , the Jones matrix is
1 1
[
2 1

1
]
1

(1.171)b) Waveplates
𝐸𝑥0 𝑒 𝑗𝜑𝑥
Suppose that the light field incident on the waveplate is []. According to the𝐸𝑦0 𝑒 𝑗𝜑𝑦property of the waveplate, the emerging light field behind it can be expressed as
𝐸𝑥0 𝑒 𝑗(𝜑𝑥 +𝛿𝑥 )[] , where 𝛿𝑥 and 𝛿𝑦 are, respectively, the phase delays of x and y
𝐸𝑦0 𝑒 𝑗(𝜑𝑦 + 𝛿𝑦 )components of the light field. Then we have𝐸𝑥0 𝑒 𝑗(𝜑𝑥 +𝛿𝑥 )𝑎[𝑗(𝜑𝑦 + 𝛿𝑦 ) ] = [ 𝑐
𝐸𝑦0 𝑒

𝑗𝜑
𝑏 𝐸𝑥0 𝑒 𝑥]][
𝑑 𝐸𝑦0 𝑒 𝑗𝜑𝑦(1.172)

Then the matrix form representing a waveplate is
𝑎
[
𝑐

𝑗𝛿𝑥
𝑏
] = [𝑒
𝑑
0

0 ]
𝑒 𝑗𝛿𝑦

(1.173)

From this general form of Jones matrix for a phase retarder, we can get the Jones matrix
of a quarter-wave plate and of a half-wave plate as followings.
99

Advances in Optics: Reviews. Book Series, Vol. 51) Quarter-wave plates
𝑒 −𝑗𝜋⁄4 [1
0

0
1], fast axis is in x direction, 𝑒 𝑗𝜋⁄4 [𝑗00], fast axis is in y direction.
−𝑗2) Half-wave plates
𝑒 −𝑗𝜋⁄2 [1 0
1], fast axis is in x direction, 𝑒 𝑗𝜋⁄2 [
0 −1
0

0
], fast axis is in y direction.
−1c) Rotators𝑐𝑜𝑠𝛼
For a linearly polarized light with the normalized field [] incident on a rotator of𝑠𝑖𝑛𝛼
cos(𝛼 + 𝛽)angle 𝛽, the emerging field is []. Moreover, the emerging and incident fields
sin(𝛼 + 𝛽)satisfy the following relationship,[cos(𝛼 + 𝛽)𝑎
] = [sin(𝛼 + 𝛽)𝑐𝑏 𝑐𝑜𝑠𝛼][]𝑑 𝑠𝑖𝑛𝛼(1.174)

We can get that 𝑎 = 𝑐𝑜𝑠𝛽, 𝑏 = −𝑠𝑖𝑛𝛽, 𝑐 = 𝑠𝑖𝑛𝛽, and 𝑑 = 𝑐𝑜𝑠𝛽, so that the Jones𝑐𝑜𝑠𝛽 −𝑠𝑖𝑛𝛽matrix of the rotator is [].
𝑠𝑖𝑛𝛽 𝑐𝑜𝑠𝛽
We have got all Jones matrices for three types of polarizers. If the light with the electric𝐸𝑥field [𝐸 ] is sent through a train of polarizers or optical systems, the output light field𝑦𝐸𝑥′[ ′ ] is given by matrix multiplication:
𝐸𝑦𝐸𝑥′𝑎[ ′] = [ 𝑛
𝐸𝑦
𝑐𝑛

𝑏𝑛
𝑎
]…[ 2
𝑑𝑛
𝑐2

𝑏2 𝑎1
][
𝑑2 𝑐1

𝑏1 𝐸𝑥
] [ ],
𝑑1 𝐸𝑦(1.175)where every 2×2 matrix represents the corresponding polarizer.
1.6.3. Basic Theory of Polarization of Stochastic Light Fields
In the Subsection 1.6.2 we just described the cases of monochromatically polarized lightby Jone’s method. This subsection deals with, in a general method, the problem ofcharacterizing the polarization of stochastic light fields that exist in practice. The reason
for the generation of stochastic light fields in nature has been explained in Section 1.3.
Here we will generally describe the completely polarized, unpolarized, and partiallypolarized light by means of the degree of polarization.

100

Chapter 1. Physical Optics

1.6.3.1. Correlation Matrix of a Light Field
Let us consider a light beam that propagates along z direction, as shown in Fig. 1.53, andsuppose that the light fields are statistically stationary, at least at wide sense. Let 𝐸𝑥 (𝑡)and 𝐸𝑦 (𝑡) be the components of the electric field of light at the point O in the beam.
The second-order correlation properties of the electric field at the point O can becharacterized by a 2×2 polarization matrix as the following.
< 𝐸𝑥∗ (𝑡)𝐸𝑥 (𝑡) > < 𝐸𝑥∗ (𝑡)𝐸𝑦 (𝑡) >
𝐽𝑥𝑥
𝐽 = [
] = [
∗
∗
𝐽𝑦𝑥
< 𝐸𝑦 (𝑡)𝐸𝑥 (𝑡) > < 𝐸𝑦 (𝑡)𝐸𝑦 (𝑡) >

𝐽𝑥𝑦
],
𝐽𝑦𝑦

(1.176)

where the asterisk denotes the complex conjugate, and < ∗ > ensemble average. The
matrix J is an equal-time correlation one whose elements can be used to describe the stateof polarization of a light field.
It is clear that 𝐽𝑥𝑥 and 𝐽𝑦𝑦 are always real and nonnegative, and 𝐽𝑥𝑦 and 𝐽𝑦𝑥 mutuallycomplex conjugate. Thus 𝐽 is a Hermitian matrix and can be written in the form
𝐽𝑥𝑥
𝐽 = [
𝐽𝑦𝑥

𝐽𝑥𝑦
]
𝐽𝑦𝑦

(1.177)

Furthermore, with help of Schwarz’s inequality, we get that
｜𝐽𝑥𝑦 ｜ ≤ √𝐽𝑥𝑥 𝐽𝑦𝑦(1.178)Therefore the determinant of 𝐽 is nonnegativedet(𝐽) = 𝐽𝑥𝑥 𝐽𝑦𝑦 − ｜𝐽𝑥𝑦 ｜2 ≥ 0(1.179)The trace of 𝐽 is equal to the average intensity, I, of the light wave.
tr(𝐽) = 𝐽𝑥𝑥 + 𝐽𝑦𝑦 = 𝐼(1.180)We can characterize the correlation between the x and y components of the electric fieldby the correlation coefficient
𝑗𝑥𝑦 =

𝐽𝑥𝑦
√𝐽𝑥𝑥 𝐽𝑦𝑦(1.181)

According to the inequality (1.178), we have 0 ≤ 𝑗𝑥𝑥 ≤ 1.
1.6.3.2. Completely Polarized, Unpolarized, and Partially Polarized Light
1) For completely polarized light, |𝑗𝑥𝑦 | = 1. This condition means that the statisticaldependence between x and y polarization components is completely correlated, which is101

Advances in Optics: Reviews. Book Series, Vol. 5

referred to as a completely polarized light. For this case, the correlation matrix J can be
written𝐽𝑥𝑥
𝐽 = [√𝐽𝑥𝑥 𝐽𝑦𝑦 𝑒 −𝑗𝜑√𝐽𝑥𝑥 𝐽𝑦𝑦 𝑒 𝑗𝜑
𝐽𝑦𝑦],

(1.182a)where 𝜑 is the phase difference between x and y components and a real number. We
express the formula (1.182a) as a general form as follows.
𝐴
𝐽 = [ ∗𝐶

𝐶],
𝐵

(1.182b)where * is the complex conjugate. From the matrix (1.182a) for a completely polarized
light we know that 𝐴 ≥ 0, 𝐵 ≥ 0, and 𝐴𝐵 − 𝐶𝐶 ∗ = 0.
2) For unpolarized light, |𝑗𝑥𝑦 | = 0. This condition means that the statistical dependence
between x and y polarization components is completely uncorrelated, therefore, the lightfield is called unpolarized because no any direction of the polarization component being
dominant. For this case, the correlation matrix J can be written
1
𝐽 = 𝐽𝑥𝑥 [00]1(1.183)3) Partially polarized light and the degree of polarization.
In the above, we just described two extreme cases of the completely polarized andunpolarized light, expressed by |𝑗𝑥𝑦 | = 1 and |𝑗𝑥𝑦 | = 0, respectively. In what follows,
it can be shown that any light beam, at each spatial point, can be expressed uniquely as
the sum of the completely polarized and unpolarized light beams.
First we introduce the concept of the degree of polarization. It is the ratio of the intensityof the polarized part to the total intensity of a light beam, denoted as ℘. The polarization
matrix J is a Hermitian matrix and semidefinite positive. From matrix theory, a unitarymatrix transformation can transform J into a diagonal matrix as𝜎
𝛴 = [ 100],
𝜎2(1.184)

where 𝜎1 and 𝜎2 (𝜎1 ≥ 𝜎2 ) are the nonnegative real-valued eigenvalues of J. We further
write the above diagonal matrix as
𝜎
[ 1
0

0
𝜎
] = [ 2𝜎200𝜎 − 𝜎2
]+[ 1
𝜎2
0

0
]
0

(1.185)

From the formulas (1.182b) and (1.183), we know that the first matrix on the right handside represents unpolarized light with average intensity 2𝜎2 , whereas the second one
represents the linearly polarized light of intensity 𝜎1 − 𝜎2 . Thus the light beam with102

Chapter 1. Physical Optics

arbitrary polarization can be expressed as a sum of completely polarized and unpolarizedcomponents. According to the definition of the degree of polarization, it is𝜎 −𝜎℘ = 𝜎1 +𝜎2
1(1.186)

2

The degree of polarization can be expressed more explicitly in terms of the elements ofthe polarization matrix J. From the matrix theory, 𝜎1 and 𝜎2 can be gotten by solving thefollowing characteristic equation.
(1.187)det[𝐽 − 𝛴𝕀] = 0,

where 𝕀 represents a 2×2 unit matrix. The solution of the above quadratic in 𝛴 yields
1

𝑑𝑒𝑡(𝐽)(1.188)

𝜎1,2 = 2 𝑡𝑟(𝐽) [1 ± √1 − 4 (𝑡𝑟(𝐽))2 ]Thus the degree of polarization, in terms of elements of the matrix J, can be expressed as𝑑𝑒𝑡(𝐽)(1.189)℘ = √1 − 4 (𝑡𝑟(𝐽))2

That ℘ = 0 stands for unpolarized light, that ℘ = 1 for completely polarized light andthat 0 < ℘ < 1 for partially polarized light. The monochromatic plane wave expressedby the formula (1.159) is a completely polarized light one. For a quasi-monochromaticbeam, if the ratio of the amplitudes of the two components of 𝐸𝑥 (𝑡) and 𝐸𝑦 (𝑡) and the
𝐸 (𝑡)

𝐸 (𝑡)

𝑥

𝑥

difference between their phases are all constants, i.e. |𝐸𝑦(𝑡)| = 𝐶1 and arg(𝐸𝑦(𝑡)) = 𝐶2 ,where 𝐶1 and 𝐶2 are constants, and arg() represents the phase difference, the degree ofpolarization ℘ of the quasi-monochromatic beam is still one. This result implies that forthis situation the conventional experiment, employing only a compensator and a polarizer
for determining the elements of the polarization matrix, cannot distinguish between arandom quasi-monochromatic light beam and a strictly monochromatic beam.
Representation of the Jones’s vector and of the polarization matrix for the light field isused to process interference effects, i.e. the combination of coherent light beams. For
incoherent light, the Stokes vector and Mueller matrices are used to describe it. Here wewill not discuss it further. Interested readers can consult the reference [5]In most of literatures, the subjects of coherence and polarization of light have been
generally treated separately. However, both of them are manifestations of the correlations
between fluctuations in light beams. Coherence of light describes the correlation between
fluctuations at two or more points in space; Polarization of light is a description ofcorrelation between fluctuating components of the electric field at a single point in space.
The unified theory of coherence and polarization is, in detail, explained in thereference [15].

103Advances in Optics: Reviews. Book Series, Vol. 5

References[1]. M. Born, E. Wolf, Principles of Optics, Cambridge University Press, 1999.
[2]. G. Brooker, Modern Classical Optics, Oxford University Press, 2003.
[3]. G. Fowles, Introduction to Modern Optics, Dover Publications, 1989.
[4]. O. K. Ersoy, Diffraction, Fourier Optics, and Imaging, John Wiley and Sons Inc., 2007.
[5]. D. Goldstein, Polarized Light, 2nd Ed., CRC Press, 2011.
[6]. G. Giusfredi, Physical Optics: Concepts, Optical Elements, and Techniques, Springer, 2019.
[7]. J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1996.
[8]. J. W. Goodman, Statistical Optics, John Wiley and Sons Inc., 2015.
[9]. E. Hecht, Optics, Pearson, 2017.
[10]. A. Lipson, S. Lipson, L. Lipson, Optical Physics, Cambridge University Press, 2011.
[11]. M. V. Klein, Optics, John Wiley and Sons Inc., New York, 1970.
[12]. L. D. Landau, E. M. Lifshits, The Classical Theory of Fields, Pergamon, 1975.
[13]. E. L. O'Neill, Introduction to Statistical Optics, Addison-Wesley, Reading, MA, 1963.
[14]. F. Pedrotti, L. Pedrotti, Introduction to Optics, Cambridge University Press, 2018.
[15]. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light, Cambridge
University Press, 2007.
[16]. S. Zhang, C. Li, S. Li, Understanding Optical Systems through Theory and Case Studies,
SPIE, 2017.

Appendix. Derivation of the Rayleigh-Sommerfeld FormulaThis appendix tries to derive the Rayleigh-Sommerfeld formula (1.47) in Section 1.2 inthe vein of Chapter 3 of the reference [7].
Any monochromatic optical field 𝑈(𝒓) propagating in a uniform medium obeys theHelmholtz equation (1.16), rewritten as follows.
𝛻 2 𝑈(𝒓) + 𝑘 2 𝑈(𝒓) = 0(1.190)

Solving this partial differential equation with certain boundary conditions, we can get the
optical field 𝑈(𝒓) at a spatial position 𝒓 of observation. In what follows, we will elucidate
how to get Rayleigh-Sommerfeld formula (1.47) from the Helmholtz equation (1.190).
A1. Green's Theorem
Calculation of the light field 𝑈(𝒓) at a spatial position 𝒓 of interest can be accomplishedwith the help of Green’s theorem. Let the light field 𝑈(𝒓) and an auxiliary function 𝐺(𝒓)be any two complex-valued functions of position 𝒓, and 𝑆 be a closed surface surrounding
a volume V. If 𝑈(𝒓), 𝐺(𝒓), and their first and second partial derivatives are single-valued
and continuous in the volume V and on surface S, Green’s theorem can be statedas follows∭𝑉 (𝑈𝛻 2 𝐺 − 𝐺𝛻 2 𝑈)𝑑𝑉 = ∬𝑆 (𝑈
104

𝜕𝐺
𝜕𝑈
− 𝐺 ) 𝑑𝑆,𝜕𝑛
𝜕𝑛

(1.191)

Chapter 1. Physical Optics
𝜕

where 𝜕𝑛 represents a partial derivative in the outward normal direction 𝒏 at a point on
surface S, and for simplicity we omit the spatial position vector 𝒓 in functions U and G.
This theorem is the starting point for solving the Helmholtz equation of scalar diffraction
theory. In what follows we will address how to choose the function G and a closed surface
S for applying this theorem to diffraction problems.
A2. The Integral Theorem of Helmholtz and Kirchhoff
As shown in Fig. 1.55, let 𝑃 be the point of observation, 𝑆 an arbitrary closed surface
enclosing 𝑃, and 𝒏 the outward normal direction of the surface 𝑆. Our goal is to express
the optical field at 𝑃 in terms of the values and derivatives of the optical field on thesurface 𝑆. We now apply Green’s theorem and choose an auxiliary function G as aunit-amplitude spherical wave from the point 𝑃 to an arbitrary point 𝑃1 in thevolume V, i.e.,
𝐺(𝑃) =

𝑒 𝑗𝑘𝑟,
𝑟(1.192)where 𝑟 is the length of the 𝒓 pointing from 𝑃 to 𝑃1 . As stated above, the function 𝐺 must
be continuous within the enclosed volume V. For this reason, a spherical surface 𝑆𝜀 withsmall radius 𝜖 is inserted about the point 𝑃. In this way, Green’s theorem can be appliedin which the volume of integration for 𝑉′ lies between 𝑆 and 𝑆𝜀 , and the surface ofintegration is on the composite surface 𝑆 ′ = 𝑆𝜀 + 𝑆.
Within the volume 𝑉′, The Green function 𝐺 also satisfies the Helmholtz equation
𝛻 2 𝐺 + 𝑘 2𝐺 = 0(1.193)Fig. 1.55. Surface of integration.

On inserting the two the Helmholtz equations (1.190) and (1.193) into Green’s theorem
𝜕𝐺𝜕𝑈(1.191), we get ∬𝑆 (𝑈 𝜕𝑛 − 𝐺 𝜕𝑛 ) 𝑑𝑆 = 0, or
105Advances in Optics: Reviews. Book Series, Vol. 5
𝜕𝐺

𝜕𝑈

𝜕𝐺

𝜕𝑈

(1.194)− ∬𝑆 (𝑈 𝜕𝑛 − 𝐺 𝜕𝑛 ) 𝑑𝑆 = ∬𝑆 (𝑈 𝜕𝑛 − 𝐺 𝜕𝑛 ) 𝑑𝑆𝜀

For a point 𝑃1 on the composite surface 𝑆 ′ , according to the expression of the Greenfunction, we have its derivative as follows
𝜕𝐺(𝑃1 )
1 𝑒 𝑗𝑘𝑟
=
cos
(𝒏,
𝒓
)
(𝑗𝑘
−
) 𝑟 ,
𝜕𝑛
𝑟

(1.195)where cos(𝒏, 𝒓 ) stands for the cosine of the angle between the outward normal 𝒏 and 𝒓
joining 𝑃 to 𝑃1 . For the case of 𝑃1 on 𝑆𝜀 , cos(𝒏, 𝒓 ) = −1 , and 𝐺(𝑃1 ) =𝜕𝐺(𝑃1 )
1
𝑒 𝑗𝑘𝜖
=
(
−
𝑗𝑘)
. Let 𝜖 become arbitrarily small, then𝜕𝑛𝜖𝜖lim ∬𝑆 (𝑈

𝜖→0

𝜀

𝑒 𝑗𝑘𝜖
and
𝜖

𝜕𝐺
𝜕𝑈
1
𝑒 𝑗𝑘𝜖
𝜕𝑈(𝑃) 𝑒 𝑗𝑘𝜖− 𝐺 ) 𝑑𝑆 = 𝑙𝑖𝑚4𝜋𝜖 2 [𝑈(𝑃) ( − 𝑗𝑘)−] = 4𝜋𝑈(𝑃)
𝜕𝑛
𝜕𝑛
𝜖
𝜖
𝜕𝑛
𝜖
𝜖→0

Substituting this result into (1.192), we have
𝑈(𝑃) =

1𝑒 𝑗𝑘𝑟 𝜕𝑈𝜕 𝑒 𝑗𝑘𝑟−𝑈( )] 𝑑𝑆[∬
4𝜋 𝑆 𝑟 𝜕𝑛𝜕𝑛𝑟(1.196)

The formula (1.196) is called the integral theorem of Helmholtz and Kirchhoff, whichstates that the light field at any point 𝑃 can be expressed in terms of the boundary valuesof the optical field on any closed surface surrounding the point 𝑃.
A3. The Kirchhoff Formulation of Diffraction by a Planar Screen
Consider now the problem of diffraction of light by an aperture in an infinite opaque with
an open aperture Σ as shown in Fig. 1.56. A monochromatic light wave is assumed tobeing incident on the screen and the aperture from the left. We try to seek the light fieldat the point 𝑃 behind the aperture 𝑆 through applying the integral theorem of Helmholtz
and Kirchhoff (1.196). In this case, the closed surface 𝑆 is a composite one with 𝑆1 + 𝑆2 ,as shown in Fig. 1.56. The plane surface 𝑆1 lies directly behind the diffracting screen, andit connects to a large spherical cap, 𝑆2 , of radius 𝑅 and centered at the observation point
𝑃. Thus applying the formula (1.196) to this case, we have𝑒 𝑗𝑘𝑟 𝜕𝑈

1

𝜕

𝑒 𝑗𝑘𝑟

(1.197)

𝑈(𝑃) = 4𝜋 ∬𝑆 +𝑆 [( 𝑟 ) 𝜕𝑛 − 𝑈 𝜕𝑛 ( 𝑟 )] 𝑑𝑆
1

2

We further examine the integration on the surface 𝑆2 , i.e. the following integral
𝑒 𝑗𝑘𝑅 𝜕𝑈

𝜕

𝑒 𝑗𝑘𝑅

𝜕

𝑒 𝑗𝑘𝑅

1 𝑒 𝑗𝑘𝑅

𝑒 𝑗𝑘𝑅∬𝑆 [( 𝑅 ) 𝜕𝑛 − 𝑈 𝜕𝑛 ( 𝑅 )] 𝑑𝑆 , and 𝜕𝑛 ( 𝑅 ) = (𝑗𝑘 − 𝑅) 𝑅 ≈ 𝑗𝑘 𝑅 , where the2last approximation is valid provided very large 𝑅. The above integral can therefore be
reduced to
𝑒 𝑗𝑘𝑅 𝜕𝑈𝜕𝑒 𝑗𝑘𝑅

∬𝑆 [( 𝑅 ) 𝜕𝑛 − 𝑈 𝜕𝑛 ( 𝑅 )] 𝑑𝑆 = ∬𝛺2

106

𝑒 𝑗𝑘𝑅 𝜕𝑈
(𝜕𝑛 − 𝑗𝑘𝑈) 𝑅 2 𝑑𝛺,𝑅(1.198)

Chapter 1. Physical Optics
𝑒 𝑗𝑘𝑅

where Ω is the solid angle subtended 𝑆2 at 𝑃. The quantity |𝑈| is uniformly bounded𝑅
on 𝑆2 . Furthermore, let the light field have the property, known as the Sommerfeld
𝜕𝑈radiation condition, lim 𝑅 (𝜕𝑛 − 𝑗𝑘𝑈) = 0. Then the integral (1.198) will vanish. We𝑅→∞

can get that the light field at the point 𝑃 from the expression (1.197) is as follows
𝑈(𝑃) =1
𝜕𝑈
𝜕𝐺∬ [𝐺 𝜕𝑛 − 𝑈 𝜕𝑛] 𝑑𝑆
4𝜋 𝑆1(1.199)Fig. 1.56. Surface of integration.

In the above expression, we have used the formula (1.192). As shown in Fig. 1.56, thescreen 𝑆1 is opaque, except for the open aperture denoted as Σ.
A4. The Rayleigh-Sommerfeld Formulation of Diffraction
The conditions for validity of the equation (1.199) are:
1. The scalar theory of diffraction holds;
2. Both 𝑈 and 𝐺 satisfy the Helmholtz equation (1.190);
𝜕𝑈− 𝑗𝑘𝑈) = 0, is satisfied.
𝜕𝑛

3. The Sommerfeld radiation condition, which 𝑙𝑖𝑚 𝑅 (𝑅→∞

The light field 𝑈 satisfies the Helmholtz equation (1.190), and we choose the Green’s
function expressed by (1.200) in the following, which Green's function 𝐺 is generated
together by a point source located at 𝑃, and a second point source 𝑃̃ at a mirror positionof 𝑃 relative to the screen as shown in Fig. 1.57.
𝐺 =

𝑒 𝑗𝑘𝑟
𝑒 𝑗𝑘𝑟̃
−𝑟𝑟(1.200)Obviously such a function 𝐺 vanishes on the aperture Σ, and satisfies the Helmholtz
equation. Furthermore Kirchhoff boundary conditions are modified as follows:
1. Across the aperture Σ, the field distribution 𝑈 is exactly the same as it would be in theabsence of the screen;107

Advances in Optics: Reviews. Book Series, Vol. 5

2. Over the portion of 𝑆1 that lies in the geometrical shadow of the screen, the field𝜕𝑈
distribution 𝑈 is identically zero, but 𝜕𝑛 ≠ 0 that is one of Kirchhoff boundary conditions.

Fig. 1.57. Geometry of a point source 𝑃 and its mirror point source 𝑃̃ .

Such the light field at observation point 𝑃, according to the formula (1.199), is reduced to1𝜕𝐺(1.201)

𝑈(𝑃) = − 4𝜋 ∬𝛴 [𝑈 𝜕𝑛] 𝑑𝑆
This solution is called the first Rayleigh-Sommerfeld solution.
𝜕𝐺

Using (1.200), we can get 𝜕𝑛 on the aperture Σ𝜕𝐺𝜕𝑛

1 𝑒 𝑗𝑘𝑟̃

1 𝑒 𝑗𝑘𝑟= 𝑐𝑜𝑠(𝒏, 𝒓) (𝑗𝑘 − 𝑟) 𝑟 − 𝑐𝑜𝑠(𝒏, 𝒓̃) (𝑗𝑘 − 𝑟̃ ) 𝑟̃

(1.202)Further we have 𝑟 = 𝑟̃ , and 𝑐𝑜𝑠(𝒏, 𝒓) = −𝑐𝑜𝑠(𝒏, 𝒓̃). Therefore the expression can be
written𝜕𝐺𝜕𝑛

1 𝑒 𝑗𝑘𝑟.
𝑟𝑟= 2𝑐𝑜𝑠(𝒏, 𝒓) (𝑗𝑘 − )1

For 𝑘 ≫ 𝑟, the above expression can be approximated as follows
𝜕𝐺𝜕𝑛= 2𝑗𝑘𝑐𝑜𝑠(𝒏, 𝒓)

𝑒 𝑗𝑘𝑟
𝑟

(1.203)On substituting (1.203) into the equation (1.201), we get the Rayleigh-Sommerfeld
formula as follows
1

𝑒 𝑗𝑘𝑟

𝑈(𝑃) = 𝑗𝜆 ∬𝛴 𝑈(𝑃1 ) 𝑟 𝑐𝑜𝑠(𝒏, 𝒓)𝑑𝑆
This is just the formula (1.47) in Section 1.2.
108

(1.204)

Chapter 2. CPU Technology Trend and Optical Interconnects

Chapter 2

CPU Technology Trend and Optical
Interconnects
Sahnggi Park1

2.1. Introduction
Computer CPUs are a measure that reveals the highest level of technology that humans
can attain. Any type of intelligent models and programs that can be imagined by humans
are affected by the CPU performance. This article describes the technology, characteristics
and market trend of the latest CPUs used in smartphones, computers, and supercomputers
released in 2020 as well as the optical network-on-chip architectures proposed byresearchers of major corporations and universities.
A CPU is the device that executes programs that list a large number of instructions. The
Section 2.2 describes CPU design and fabrication. The CPU design corresponds tocreating an instruction set and converting all logic functions contained in the instruction
to transistors. CPU fabrication is the work of drawing all the transistors andcommunication lines in photomasks and carrying out the CMOS production processaccording to the drawing [1-3].
The system on chip (SoC) technology, the internal configuration of SoCs, and the typesof ARM company licenses are described in the Section 2.3, and the latest 2020 SoC
performances and characteristics, manufacturers, market shares, and the smartphonedisassembly layouts are also described [4-7].
The internal architecture of Intel CPUs, the communication line width and communication
speeds between registers and caches, and between the L1, L2, and L3 caches, are describedin the Section 2.4. The ring bus and mesh network which are the communication systems
inside Intel CPUs are also described in the same chapter. Supercomputer server
configuration and 3D torus and fat-tree optical network architectures are described in the
Section 2.5. Furthermore, the technology and characteristics of x86 CPU and

Sahnggi Park
Oprocessor Inc., Boston, MA, USA

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supercomputers developed by China for IT technology independence are also
described [8-15].
Major corporations and universities have proposed the network architectures that can be
adopted when optical communication technology would be implemented to the CPU chip.
Among them, the frequently cited ATAC architecture of MIT and the Corona Architectureof HP are outlined in the Section 2.6. In addition, the main reasons why these architectures
have not been realized are described [16-20]. It would be the most realizable to adopt thetechnology and components utilized in supercomputers, the vertical cavity surface
emitting laser (VCSEL) and photodiode (PD) and the fat-tree optical network, while
replacing only the optical fiber with SiON/SiN optical waveguides, to implement optical
network-on-chip to the Intel server CPUs. For this, three key technologies and details
proven through experiments are described in the Section 2.7 [21-24].

2.2. CPU Design and Fabrication
For CPU design, it is essential to understand the structures of programming languages. As
shown in Fig. 2.1, C++, Python, and Java are used in coding. Once the compilation isfinished, it is converted into assembly language. The assembly language is a language that
represents an instruction set architecture (ISA) such as ARM, x86, MIPS and RISC-V,
which is converted into a machine language such as 10101 by an assembler. Although
CPU recognizes machine language when programs are executed, instruction sets
expressed in the assembly language play an important role in actual CPU design.

Fig. 2.1. Programming language level.

The x86 belongs to the complex instruction set computer (CISC), which has been
extensively developed by Intel and is frequently called the x86 instruction set. While each
instruction in the reduced instruction set computer (RISC) has a fixed-length (typically 4
bytes), which limits the expression of instructions, and there is not a large number of110Chapter 2. CPU Technology Trend and Optical Interconnects

instructions, the x86 instruction set has a variable length (typically 1-16 bytes), allowing
for a free expression of instructions, and has a much larger number of total instructions.
Although it is confidential to find out the instruction set of ARM or x86, the similar
RISC-V details are freely available and has a total number of about 100 instructions, andthe x86 instruction set is known to have thousands of instructions.
As shown in Fig. 2.2, a processor is largely composed of the logic part, register part andcache part. The logic part performs arithmetic calculations using the given function anddata, and the register and cache parts store the instructions and data. The register partstores instructions and data that are currently undergoing calculations, and the cache partstores instructions and data that are finished with the calculation or are waiting to the L1,
L2, L3 caches. The functions to be processed and the necessary data are designated in the
instructions received by the register, and after the calculation is done in the logic part
corresponding to each function, the processed result goes to the register for the next
calculation or are stored in the cache, and if the order of priority is very low, it is sent to
the main memory (DRAM).

Fig. 2.2. Schematic diagram of the processor.

The logic part is composed of transistors of logic functions such as AND, OR, NOT,
NAND, NOR. A transistor, nMOS or pMOS, is a switch to turn the current on/off between
the source and drain according to the voltage applied to the gate. Fig. 2.3 is a conceptual
diagram of 1 bit NOT function formed on a silicon wafer. Each instruction contains
functions, each of which contains many logic functions, and it performs 64-bit
calculations through complex connections and arrangements and then sends the results to
the register. For an x86 instruction set, due to the thousands of instructions, the logic parthas an assigned area for transistors corresponding to each instruction, and the register andinput/output lines are connected through a complex connecting structure. To arrange theentire instruction set, a CPU contains about 1~10 billion transistors. Between the register,
the logic and cache parts is the control unit (CU) that controls the transmission of
instructions and data, and all transmission signals go through it.
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Advances in Optics: Reviews. Book Series, Vol. 5

Fig. 2.3. Schematic diagram of NOT Function.

The register and cache parts are composed of SRAM transistors to store instructions anddata. Fig. 2.4 shows the basic structure of an SRAM storing a 1-bit signal. The two NOT
functions are connected to each other to form a structure that connects output to input andinput to output. When the word line (WL) signal turns off after a 1-bit signal is input tothe bit line (BL), the signal goes to the input of the NOT function on the right and comesout reversed as output. The signal goes back to the input of the NOT function on the left
and reverses back to its original state and goes back to the input of the NOT function on
the right, maintaining the state of storage through permanent repeating cycles. The registerand the L1, L2, and L3 caches are assigned to their corresponding locations according to
the CPU design with the determined number of bytes and are connected to the 64-bit bus
line in the CPU.

Fig. 2.4. 1-bit storage of SRAM circuit.

The description of CPU design and manufacturing is outlined as follows. As technology
advances, instructions are changed or instructions with new functionalities are added tothe existing instruction set. Once the final instruction set is determined, logic functions
that calculate the function of each instruction are placed and connecting lines are drawn
to the transistor. Since the logic part of the CPU must be able to execute all of the
instructions, for x86, logic functions and transistors corresponding to the thousands of
instructions need to be placed and connected. Registers, caches, control units, and bus
lines must also be drawn according to the determined specifications. For this work, eachof the components is drawn using the high-level synthesis tools owned by each company
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Chapter 2. CPU Technology Trend and Optical Interconnects

and the work is divided and executed by many engineers. A lot of manpower and time are
required especially for simulation and verification due to the loss of billions of dollars that
occurs when a bug is confirmed after production has begun. Once this work is finished,
the work of drawing the mask layer for the actual chip manufacturing begins. At this time,
if a manufacturing request is made to a foundry company, such as TSMC, the mask layer
must be drawn according to the technical specifications provided by the foundry company.
For example, when using the 7 nm technology of a foundry company, the gate width ofthe corresponding transistor must meet this specification. Once the basic specifications
are established, the mask layer work is automatically transferred using a computer
program. When the mask layer drawing is finished, it is sent to the foundry company to
manufacture the chip.
Fig. 2.5 is a cross-sectional view of a Broadwell CPU chip manufactured by Intel in 2013.
It used the 14 nm process and applied the finFET transistor technology. The finFET
transistor is formed on the bottommost silicon layer. Although the gate width is 14 nm, it
is known that the gate pitch is 70 nm, the fin pitch is 42 nm, and the minimum pitch of themetal line is 52 nm; these are the parts that directly affect the number of transistors per
area. The number of transistors integrated per area increases gradually according to thetechnology node but the width of the metal line, in order to maintain the electricalcharacteristics of the signal transmission line, cannot decrease in proportion. Therefore,
the number of metal layers has been increasing three-dimensionally, and Fig. 2.5 shows
the cross-sectional view of the first chip with 13 layers. As of 2019, Intel is applying a
10 nm technology to manufacture the latest CPU chips, and the metal layer is known tobe 13 layers. TSMC, which manufactures the latest CPU chips for AMD, Apple, Huawei
and Qualcomm, announced that they will introduce the 5 nm process and use 15 metal
layers. Samsung is known to have used the 8 nm process for the production of Exynos
chips that were sold in 2019 and introduced the 7 nm process in the second half of 2019.

Fig. 2.5. Cross section of Broadwell CPU chip (Source: https://www.chipworks.co.uk).

When a user runs a program through Windows or Android OS, the program stored on thehard disk is brought to the main memory (DRAM). A program is a structure in which
many instructions are sequentially arranged and machine signals such as 101010 are
113Advances in Optics: Reviews. Book Series, Vol. 5transmitted. Each instruction is brought to the CPU sequentially and stored in the L1 cachein a waiting state. When it is their turn, it is sent to the decoder and after the instruction
content is analyzed, it is transferred to the register. In addition, if the function included in
the instruction needs data, data is transferred from the hard disk to the CPU register orcache. When both instruction and data are prepared, they are sent to the corresponding
logic part to undergo calculations. Then, the results are stored in the register or the cache.
Processes like this can be seen as a diagram in Fig. 2.6.

Fig. 2.6. Pipeline diagram.

Instead of executing one instruction and storing the result data and then executing the next
instruction, the process is divided into stages: the fetch stage that receives the instructions,the decode stage, the execute stage where data is sent to the logic part to calculate thefunction, and the writeback stage where the calculated results are stored. It is more like amachine that is assembled as it moves along a conveyor belt. As multiple instructions arein the waiting state, when the first instruction enters the decode stage after the fetch stage,
the second instruction enters the fetch stage. When the first instruction enters the executestage, then in sequential order, the next instructions enter their respective next stages.
Processes like this are called pipelines. The technology in which one core in the CPU
operates two or more pipelines is called the superscalar technology, and such CPU iscalled a superscalar processor. The components that operate a pipeline are called threads.
In order to operate two or more threads, the register, logic part and cache are shared andas many decoders as the threads are placed and operated individually. When one coreexecutes two instructions on two treads at the same time, since they share the logic part,
if one instruction is a real number calculation, then the other instruction should be an
integer calculation. All recent Intel and AMD CPUs have two threads per core and arecalled hyper threads. For Quad-core CPUs, there are four physical cores, but OS
recognizes it as eight.
One of the key technologies in CPU design is parallel computing technology. Parallel
computing has a variety of parallel computing methods and is normally divided into
bit-level, instruction-level, task-level, data-level parallel computing. All CPUs used in thelatest smartphones and PCs support 64 bit. In CPU, 64 bit means that there are at least
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Chapter 2. CPU Technology Trend and Optical Interconnects

64 signal lines between the register and logic or cache, and that the 64 bit signals, 1 bit
per line, are simultaneously transmitted and calculated or stored during one clock cycle.
That is, bit-level parallel computing indicates that 64 bit of data is processed in parallel at
the same time. For a 32 bit computer, it calculates and stores by dividing the 64 bit data
into two. Therefore, a 64 bit computer has a processing speed that is twice as fast as a
32 bit computer through bit-level parallel computing. If parallel execution occurs in a
program with many instructions sequentially listed, each core executes multiple
instructions simultaneously, this is called instruction-level parallel computing. Large
programs consist of many tasks and each task has a large number of instructions. The CPU
distributes each task to each core or thread for parallel execution and this is calledtask-level parallel computing. In special programs, many data can be processed by one
instruction. Let's say there is a program that calculates the sum (A + B = C) of Vector
A = [1, 2, … 10000] and B = [10001, 1002, …, 20000]. Suppose that 10,000 components
are divided into 1,000 pieces and distributed over 10 cores to calculate the final sum. The
ten cores execute the same one instruction and all data is input regularly in a sequence(SIMD, single instruction multiple data). This type of calculation is called data-level
parallel computing, the instruction is called vector instruction and the processor is called
vector processor [1, 2].
An ARM CPU or x86 CPU is a processor for general function calculation, with logic
functions corresponding to hundreds or thousands of instructions arranged in the logic
part. When a large data is processed with a small number of instructions withgeneral-purpose CPU, such as data-level parallel processing, the logic functions that
correspond to other remaining instructions will be inactive and the CPU performance
won't be utilized properly. In order to solve this problem, various accelerator processors
are being developed. Typical examples are a graphics processing unit (GPU), an AI
processor or a neural processing unit (NPU), an image processing unit (IPU), an image
signal processor (ISP) or a digital signal processor (DSP). Using GPU as an example,
there are hundreds of cores in the GPU and each core consists of independent logic partsand resisters that can process a small number of instructions. By properly distributing thelarge data, hundreds of cores process the instructions in parallel at the same time. Since
the design and manufacturing process of the accelerator processors is the same as that ofgeneral-purpose CPUs, the CPUs for smartphones and PCs are in the form of a system onchip (SoC) in which the CPU, GPU, IPU, and NPU are manufactured as one die andintegrated on the same chip.
While the clock frequency of CPU is fixed in the range of 1.5 ~ 4.0 GHz, CPU's processing
speed of instructions is continuously increasing due to the advancement of parallel
computing technology. In other words, because instructions are executed in parallel at thesame time, instructions per cycle (IPC) continues to increase. While cycles per
instructions (CPI) or IPC is used as an indicator of CPU speed, for x86 CPU that has
different lengths of instructions, floating point operations per second (FLOPS), which isused for supercomputers, are recently used by ARM CPU and x86 CPU to express the
speed. Floating point represents real number values in a format such as +1.12345×108 andthe name was given because the decimal point can move freely. If this number is expressed
as a binary number, it will be in a format such as +10101010×210101 (not the same numeric115

Advances in Optics: Reviews. Book Series, Vol. 5

value as the former). 64 bit data is called double precision (precision below the decimal
point) and 1 bit is assigned to the +/- position, 52 bit to the 10101010 position and 11 bit
to the 10101 quotient position to express the number. 32 bit data is called single precisionand 1 bit, 23 bit, and 8 bit are assigned to each respective position. The operation used to
calculate FLOPS is called fused multiply-add (FMA) in which two numbers are multipliedand another number is added to the sum, like (A × B) + C, then finally, the number isrounded up to determine the last digit. It includes 2 operations of multiplication andaddition. During one clock cycle, 64 bit data A and B are sent to the logic function fromthe cache to be multiplied, then value C that was stored in the register and the result of theformer are sent to the addition function to find the sum. When one core completes thisoperation, it becomes two FLOPS and when the clock frequency is 3 GHz, then the core
speed becomes 6 Giga FLOPS. If the CPU has four cores, it will have a speed indicatorof approximately 24 Giga FLOPS. In addition, it will have a speed indicator of 48 Giga
FLOPS based on single precision data calculation for the same CPU. In reality, the FLOPS
indicator is not directly proportional to the number of cores but depends on the CPU
design. It is obtained by executing a benchmark program that measures FLOPS.
In order to increase the processing speed of CPU, technology is developing to allow manyof the hardware elements in the CPU to participate as much as possible in the executionof work without being idle each hour. If the order of instructions written in the programis A-B-C, then B uses the result from A after A is executed, but on the other hand, if C
can be executed through cache data regardless of the result of A, then the CPU order ofprocess becomes A-C-B, which is different from the program order. That is, if the optimalorder for the whole program is found and executed, CPUs with this capability are marked
out-of-order in its performance indication. In some cases, it proceeds by selecting either
task 1 or task 2 depending on a condition like the if function in the program. This type offunction's instruction is called branch instruction and it takes up about 20 % of the
program. In order to increase the processing speed, CPU starts operating by determining
the thread or core that is expected to run the task by predicting it before the function's
selection result is provided. If the prediction is correct, the process continues and if it iswrong, the work is discarded and it starts the selected task again. This type of function iscalled branch prediction and it requires a hardware installation, which is called a branchpredictor. Prediction programs like this are designed to increase its accuracy by learning
through the program and they are said to have a rate of accuracy over 90 %. When the
program starts running, it conducts speculation and scheduling to utilize all the necessary
elements in the optimal condition, so the necessary scheduler has to be installedon the CPU.
One of the most essential technology in enabling parallel computing, out-of-order, branchprediction, and scheduling is cache coherence. Coherence is a perfect coordination state,with no dissonance or inconsistency between the components that are connected and move
together for each process. Take a CPU with four cores as an example. Each core owns L1and L2 caches privately while L3 cache is shared. From the L1 and L2 caches belonging
to core 1, only core 1 can read to and write from L1. Although core 1 has priority over L2,
it is possible for cores 2~4 to read from and also write to L2. L3 can be read from andwritten to with the same qualification without special priority. Table 2.1 shows the storage
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Chapter 2. CPU Technology Trend and Optical Interconnects

capacity and latency, which is the time it takes to read and write, for each storage location.
Let's say that data A is stored at a specific address in L3 and core 1, core 2, and core 3
read value A and store it in their respective L1 cache. If core 1 changes data A from its
L1 cache data into B, it must notify the change to all the other cores and share the fact thatdata A is no longer valid. Data A located at the L3 address must also change from A to B.
If cache data is not correctly coordinated between the cores, some cores will input A in
the formula where B should be entered or vice versa, and the entire program will be inerror. Cache coherence is a state in which cache data is quickly and accurately coordinatedto prevent errors, even when all the cores or threads read and write numerous times. For
cache coherence, communication speed between cores plays a very important role. As thenumber of cores increases, the communication in the CPU using electrical signals is
expected to reach its limit. Therefore, researches to introduce optical signals, under thename of silicon photonics, began in the mid-2000s with the participation of world's
leading corporations and research institutes, but it has not been successful yet.
Table 2.1. Latency and capacity by data storage location.
Component
L1 cacheL2 cacheL3 cache
MemorySSD storage
HDD storage

Example latency
1 ns
3 ns
10 ns
100 ns
30,000 ns
15,000,000 ns

Example capacity
64 kB per core
256 kB per core
20,480 kB
64 GB
100 GB
1 TB

Various protocols have been developed and are used for cache coherence and a
representative protocol would be MESI (modified, exclusive, shared, invalid) protocol
used by Intel, shown in Table 2.2. When core 1 reads or writes data A from L3, core 1
goes into an exclusive state and uses the corresponding address in L3 exclusively, and all
existing data in the remaining cores becomes invalid and cannot be used. When core 2
and core 3 send a signal (called snoopy signal) to read data A over the snoopy bus, thecache data of core 1 is transmitted through the data bus. At this time, all cores are in ashared state. After that, if core 3 changes data A to B, core 3 goes into a modified state
and the other cores become invalid and the existing data A cannot be used. If core 3 writes
data B on the corresponding address of L3, core 3 becomes exclusive and thecorresponding address of L3 can only be used by core 3. Then the remaining cores will
go into an invalid state. If core 4 sends the read signal over snoopy bus, it receives data Bof core 3 through the data bus, and at this time all cores are in a shared state. Data on thedata bus is sent to all the cores, so core 1 and core 2 can each store data B in their cache.
When a perfect cache coherence is accomplished, CPUs with multiple cores run the
program as if it had only one large core [1-3].
The market size and share of ARM CPU and x86 CPU according to 2020 statistics areshown in Fig. 2.7. Although parts of the Tablet, Cell-phone App and Embedded areas use
x86 CPU, most of them use ARM CPU. So, the sum of the three areas can be estimated117

Advances in Optics: Reviews. Book Series, Vol. 5

to be the share of ARM CPU. It can be seen that the share of ARM CPU is approximately
50 %. According to the 2012 statistics by the same agency, the share of pure x86 CPU
decreased from 64 % to 50 % in 2020. The growth of ARM CPU market share is expectedto accelerate even more considering that Intel has stopped the production of Atom CPU
and Apple has announced that they will be using ARM CPU instead of Intel CPU for their
Mac-Book computers from 2020. The total market size is about 79.3 billion dollars.
Table 2.2. MESI protocol.
Core1 Core2 Core3 Core4
E
I
I
I
Read/Write
A
S
S
S
S
Read
A
A
A
I
I
M
I
Modified
A
A
B
I
I
E
I
Write
A
A
B
S
S
S
S
Read
B
B

Fig. 2.7. Market Share of ARM CPU and x86 CPU(source: 2020 edition of IC Insights’ McClean Report).

2.3. Smartphone SoC Trend
The latest processor models released on ARM's homepage include the Cortex-A78,
Cortex-A78AE, and Cortex-X1. Cortex-A78 was released and adopted in 2020 by thesmartphones with the best specifications. Cortex-A78AE has functions improved in theautomotive enhancement area from Cortex-A78. It is known that Cortex-X1 was focusedon maximizing performance with compromising on higher power usage and a larger die
area. The features of Cortex-A78 published on the homepage are shown in Table 2.3.
Depending on the manufacturing process and technology node, it can have a clock
frequency of up to 3.0 ~ 3.3 GHz. In addition to the general register and logic parts, each118

Chapter 2. CPU Technology Trend and Optical Interconnects

core has a cryptography unit that enhances the security function and a neon/floating point
unit for vector computation. Cores consisting of big-littles can have up to 4 bigs and4 littles [4-6].
The ARM processor has a different characteristic than x86. In the case of octa-core CPU
with 8 cores, 4 big cores of the latest specification with high computing speed andhigh-power consumption and 4 little cores with relatively slower computing speed but
high-power efficiency are arranged in 1:1 pairs. Big cores are used when a highperformance of CPU is required, and the small cores are used when a lower performancecan be used and when it is trying to reduce power consumption. Therefore, the actual
number of cores used corresponds to a quad-core. In 2017, ARM announced the DynamIQ
technology that they developed as a technology to efficiently operate the workload anddata sharing between the accelerator processors inside a SoC. Depending on the type ofthe license that was signed, this technology has been applied to the latest SoC. DynamIQ
technology allows for various configurations such as big-little as well as big-middle-little.
For Qualcomm's Snapdragon 888, it is configured as 1-3-4. The ARM processor models
compatible with DynamIQ technology are Cortex-A55, Cortex-A75~A78 and Cortex-X1.
Cortex-A55 is configured as little and the rest are used as middle or big [6].
Table 2.3. Features of Cortex-A78.
ARMv8.2-A (cryptography extensions, RAS extensions),
ARMv8.3-A
Max. CPU clock
3.0 GHz in phones and 3.3 GHz in tablets/laptops
Out of order pipeline, Superscalar, Neon/Floating Point Unit,
Microarchitecture
Cryptography Unit
1. 64 bit program registers
64 bit registers 31, 128 bit registers 32
Registers
2. 32 bit program registers
32 bit 13, 64 bit 32
L1 cache: I and D 64 KB,
Memory system
L2 cache: 256~512 KB,
L3 cache: optional, 512 KB~4 MB
ISA version

ARM licenses apply various options depending on the company they are signing with,and though it is difficult for a third party to understand in detail, the licenses are roughlyclassified into three types (see Table 2.4). In a cortex license, ARM provides the
instruction set and CPU core that completed the circuit design, simulation, and verification
work. The partner company takes on the mask work necessary for manufacture, which
includes the design of the above CPU and of the other components (e.g. GPU, NPU, etc.,)
included in the SoC. Huawei CPUs are known to belong to this type. In an architecture
license, ARM provides only the instruction set while the core design, verification andother follow-up work for manufacture are done by the partner company. Since 2016, the'Built on ARM Cortex' license became available in which the partner company can request
their requirements to reflect in the instruction set. It is said that the customized ISA cannot
be provided to other companies. The range of technology provision of ARM goes up to119Advances in Optics: Reviews. Book Series, Vol. 5

the stage of completed circuit design and verification, like the cortex license. Other than
this simple classification, there are many details per each type or partner. Apple, Samsung,and Qualcomm belong to the architecture license type, and Qualcomm has since been
known to have changed to the 'Built on ARM Cortex' license. In addition, the architecture
license cannot use the DynamIQ technology provided by ARM. For the two licenses otherthan the cortex license, the partner's brand name may be used for the processor.
Table 2.4. License types of ARM.

Cortex License

Architecture License‘Built on ARM Cortex’
licensePartner licenses complete microarchitecture design.
CPU differentiation through- flexible configuration options- wide implementation envelope with different
process technologies
Partner designs complete CPU microarchitecture from scratch.
Freedom to develop any design- must conform to the rules and programmers’ model of a given
architecture- Must pass ARM architecture validation to preserve software
compatibility
A new engagement model for Cortex-A partners, announced
in Feb. 2016.
Partners work with ARM to develop a custom RTL database.

Currently, in 2020, the four companies with the highest technology and market shareamong SoC suppliers, including smartphone CPUs that purchased the IP license fromARM CPU are Qualcomm, Samsung, Apple, and Huawei. The market share ofsmartphone SoCs in the second quarter of 2020 according to the data from a statistics
company (Statista) is shown in Fig. 2.8. Qualcomm has 32 %, Huawei 22 %, Apple 19 %,and others 27 %. From the percentage of others, Samsung and MediaTek have the largest
share. HiSilicon is a subsidiary of Huawei specializing in design, and MediaTek is a SoC
design specialist company in Taiwan and supplies SoC for mid to low priced smartphones.
In order to identify the highest technology as of 2020, it is necessary to analyze the SoCsof the four companies above. Table 2.5 shows the SoC model names and features of thefour companies that were released in 2020, and the 2020 smartphone models that applied
SoCs. Each SoC has CPUs consisting of six to eight cores, GPUs, AI engines, ISPs, 4G
or 5G modems, and memory controllers integrated on a chip about 1 cm2 in size, with thetotal number of 8~10 billion transistors.
Qualcomm is the company with the highest level of technology among the four
companies. Although it does not produce its own smartphone models, many companies
including Samsung have released their latest smartphones in 2021 using Qualcomm's SoC.
Qualcomm designed all the devices with its own technology except for the CPU. For theCPU, the company brand Kryo 680 was used under ARM's 'Built on ARM Cortex' license.
Qualcomm Kryo 680 SoCs were manufactured by using the Samsung foundry with 5 nm
technology node in 2020.
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Fig. 2.8. ARM CPU market share by company (source: Statista 2020).
Table 2.5. Features of 2020-2021 smartphone SoC.
Qualcomm
2021 SoC
2021 Top
deviceFeatures2021 SoC
2021 Top
deviceFeatures

Snapdragon 888
Galaxy S21, S21+, S21 Ultra, Mi 11, iQOO 7, OnePlus 9
CPU
1 × Kryo 680 Prime (Cortex X1-based), 2.84 GHz, L2 cache 1024 KB
3 × Kryo 680 Gold (Cortex-A78-based), 2.42 GHz, L2 cache 3×512 KB
4 × Kryo 680 Silver (Cortex-A55-based), 1.8 GHz, L2 cache 4×128 KB
L3 cache 4 MB
GPU: Adreno 660 (840 MHz)DSP: Hexagon 780 (AI engine)
ISP: Triple 14-bit Spectra 580 ISP
Modem: Internal LTE Modem (4G): DL 2500 Mbit/s, UL 316 Mbit/s
Internal 5G Modem: DL 7500 Mbit/s, UL 3000 Mbit/s
Memory controller: LPDDR4X 4-channel 16-bit (64-bit) 3200 MHz(51.2 GB/s) Samsung 5 nm Technology nodeSamsung
Exynos 2100
Galaxy S21, S21+, S21 Ultra
CPU
1 × Cortex X1, 2.91 GHz, L2 cache 512 KB
3 × Cortex-A78, 2.81 GHz, L2 cache 3×512 KB
4 × Cortex-A55, 2.20 GHz, L2 cache 4×64 KB
L3 cache 4 MB
GPU: Mali G78MP14, 854 MHz
NPU: Triple-core 26 TOPS
ISP: 8 K@30 fps & 4 K@150 fps
Modem: Internal LTE(4G), DL 3000 Mbit/s, UL = 422 Mbit/s,
Internal 5G Exynos DL 7350 Mbit/s, UL 3670 Mbit/s.
Memory controller: LPDDR4X Quad-channel 16-bit (64-bit) 3200 MHz(51.2 GB/s)
Samsung 5 nm Technology node121Advances in Optics: Reviews. Book Series, Vol. 5

Table 2.5. Features of 2020-2021 smartphone SoC (continued).
Huawei2020 SoC
2020 Top
deviceFeatures2020 SoC
2020 Top
deviceFeatures

Kirin 9000
Mate 40 Pro, Mate 40 RS, Mate X2
CPU
1 × Cortex-A77, 3.13 GHz, L2 cache N/A
3 × Cortex-A77, 2.54 GHz, L2 cache N/A
4 × Cortex A55, 2.04 GHz, L2 cache N/A
L3 cache: No information available
GPU: Mali G78MP24, 759 MHz, 2331 GFLOPs
NPU: 2 + 1 Da Vinci NPU
ISP: 8 K @30 fps, 4 K @60 fps
Modem: Integrated Balong 5G Modem, DL 7500 Mbit/s, Upload 3500 Mbit/s
Memory controller: LPDDR4X Quad-channel 16-bit 2133 MHz (34.13 GB/s)
TSMC 5 nm Technology nodeApple
A14 Bionic
iPhone 12, iPhone 12 Pro, iPhone 12 Pro Max
CPU
2 Firestorm, 2.99 GHz
4 Icestorm, 1.82 GHz
L1, L2 cache: No information available, L3 8 MB
GPU: Apple Custom GPU (quad-core)
NPU: Neural Engine (octa-core) + AMX Blocks (dual-core))
ISP: 4 K @60 fps
Modem: Intel Gigabit-Class LTE (4G), Qualcomm X55 (5G)
Memory controller: LPDDR5 Dual-channel 64-bit (42.7 GB/s)
TSMC 5 nm, Die size: 88 mm²
OS: iOS 14.4.1

Samsung established Samsung’s Austin R&D Center (SARC) in Austin, Texas in 2010
and recruited former AMD, Intel, and ARM engineers to design SoCs. In addition,
Samsung established Advanced Computing Lab (ACL) in San Jose, California in 2017and is developing GPUs by introducing MediaTeks' GPU technology. Exynos 9820 and
9825 in 2019, predecessor of Exynos 2100 have used the M4 core which has its own brand
name. Exynos cores and GPUs use ARM designs, but other devices, such as NPU, ISP,
and 5G modems, have been developed using the company's own technology. Fig. 2.9
shows the market share statistics of smartphone manufacturers. Samsung has been theworld's largest smartphone manufacturer from 2012. Huawei surpassed Apple in 2018 andbecame the second-largest smartphone manufacturer. The Da Vinci NPU and 5G modem
were considered to be superior to those of Qualcomm and Samsung. The 5G modem of
Kirin990 is the first to be integrated on the SoC. Although having made the boldest
investments, Huawei’s technology development may be in difficulty due to the trade
conflict between US and China, of which effect starts to appear in the second
quarter of 2020.
122Chapter 2. CPU Technology Trend and Optical Interconnects

While Android OS smartphones account for about 90 % of the market share, the market
share of Apple, which solely uses iOS, has been decreasing rapidly. It is difficult tomaintain a competitive edge as a single company that is responsible for iOS development,
SoC design, and smartphone production. The company's technology of CPU, GPU(PowerVR company license) and modem which are the main devices of SoC are relatively
low and smartphone production is done by consignment assembly. Therefore, other thaniOS, it lacks competitiveness in the remaining areas. For 5G modem, while the threecompeting companies finished development in the first half of 2019 and released
smartphones with the 5G modem attached in the second half of 2019, Apple, which
acquired 4G modem supplies from Intel, terminated its supply contract with Intel whenIntel failed to keep the promise to finish developing the 5G chip by the second half of2019. As Apple was looking for a third-party 5G chip supplier and Intel had given up on
the 5G chip development, in July 2019, Apple ultimately bought Intel's modem businessfor about 1 billion dollars and started its own development in the second half of the year.
As of September 2019, the companies that have completed the development of the
5G modem chip are the three companies mentioned above and MediaTek. It is known that
Spreadtrum, now renamed Unisoc, is preparing to integrate 5G chips on low-end phones.
Huawei and Unisoc are virtually owned by China and their technology can be seen as
shared. As of March 2021, Apple’s latest device, iPhone 12 Pro Max has a 5G Modem
provided by Qualcomm.

Fig. 2.9. Quarterly smartphone market share (source: Statista 2021).

Fig. 2.10 shows the image of the main PCB of Samsung Galaxy S10 5G released in 2019.
Major technology analysis companies have uploaded on the internet the teardown photo,along with the cost analysis of the major components. This image is one of those photos
taken from a company, Techinsights. As a package on package (PoP), there is theSamsung SoC Exynos9829 and the main memory, 8 GB SDRAM. There are also theSamsung Exynos 5100 external 5G modem, 256 GB flash memory, and WiFi/Bluetoothmodule. The PCB consists of two parts, one large and one small. The large main PCB has
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devices attached to the front and back, and only the front of the main PCB is shownin the image.

Fig. 2.10. Samsung Galaxy S10 5G (Reprinted with permission from Techinsights(https://www.techinsights.com/ko)).

Fig. 2.11 shows the front and back of the main PCB of the Apple iPhone Xs Max, released
in 2018. Like the Samsung product, the PCB consists of two parts and Intel's 4G LTE
modem is on the smaller part, which is not shown in the figure. Apple SoC A12 Bionicand 4 GB SDRAM of Micron are integrated by the PoP technology and there's Sandisk's
256 GB flash memory. Fig. 2.12 is the image without the SDRAM on an Apple SoC A12
Bionic released in 2018. The locations of the big cores, little cores, caches, GPUs, andneural engines are displayed.

Fig. 2.11. Apple iPhone Xs Max (Reprinted with permission from Techinsights(https://www.techinsights.com/ko)).

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Fig. 2.12. Apple SoC A12 Bionic Image (Reprinted with permission from Techinsights(https://www.techinsights.com/ko)).

Fig. 2.13 shows the front view of the Huawei Mate 20X (5G)'s main PCB, which was
released in 2019. The Mate 30Pro (5G) product released in the second half of 2019 has
the 5G modem integrated on the SoC, but this product has an external 5G. Huawei SoC
Kirin 980 and SK Hynix 8 GB SDRAM are integrated through the PoP technology andthere's Samsung's 256 GB flash memory.

Fig. 2.13. Huawei Mate 20X (5G) (Reprinted with permission from Techinsights(https://www.techinsights.com/ko)).

2.4. x86 CPU Technology and Trend
Intel, AMD, and VIA are the three companies that currently design and manufacture x86
CPUs. There were about 10 companies in the past but by the 2000s, the companies either
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merged with the above three companies or lost their market competitiveness and stopped
production. VIA was founded in Taiwan in 1987, and mainly designed and manufactured
chipsets for motherboards. From the late 1990s, VIA merged with US processor
companies such as IDT and Cyrix and started the x86 CPU business. In cooperation withthe municipal government of Shanghai, China, VIA established a venture company,
Zaoxin, in 2013. Starting with ZX-A in 2014, it produced ZX-E and KX-6000 in 2019.
The performance of KX-6000 is close to that of the 7th generation Intel i5 produced byIntel in 2016 and it is mainly sold in mainland China. In order to end the dependency on
the US for IT technology, China is pursuing strategic projects at the national level. VIAdeveloped the x86 instruction set through the merger of American companies andproduction is seen to be done by TSMC. According to Statista's 2019 statistics, Intel andAMD's market share was 77 % and 23 %, respectively, in the 1st quarter of 2019. It isdifficult to find the market share of VIA's CPU.
The CPUs for laptops, desktops and servers by performance are on the homepages of Inteland AMD. Since there is not a big difference between the CPUs for a desktop and a server
other than the number of cores, the CPU features and internal communication structuresof a laptop and server is analyzed here. The general features are shown in Table 2.6. Intel'smobile CPU i7 is a SoC with multiple devices integrated on one chip, as shown in
Fig. 2.14. Four cores, GPU and system agent are connected by ring bus. The system agentsinclude IPU, NPU (GNA), integrated memory controller (IMC), display and thunderbolt
ports. Fig. 2.15 shows the overlay of colors on a real die image to show the placement ofeach device. The die size is 122.52 mm2.
Table 2.6. x86 CPU features.

FeaturesCore(Thread)Clock(overclock)L1/L2/L3GPU
IPU/NPU
TDP
Tech. node(Die. mm2)
Launched

126

Intel
i7(Laptop)
Xeon Platinum
9282
1065G7

AMD
Ryzen73780U

EPYC 7742

48

56 1124864128

1.3 GHz(3.9 GHz)

2.6 GHz(3.8 GHz)

2.3 GHz(4.0 GHz)

I32 KB, D48 KB/
512 KB/8 MB

32 KB/1 MB/
77 MB

I64 KB, D32 KB/
512 KB/4 MBIntel Gen11
4th Gen/
GNAv1.0
15 W
10 nm(122.52)
2019. 8.

external

AMD Vega11

2.25 GHz(3.4 GHz)
I64 KB,
D32 KB/
512 KB/
256 MB
external

external

external

external

400 W
14 nm
(2 X 694)
2019. Q2

15 W
Global Found. 12
nm
2019. 10.

225 W
TSMC 7 nm
2019. 8.

Chapter 2. CPU Technology Trend and Optical Interconnects

Fig. 2.14. Internal architecture diagram of Intel mobile CPU i7 (source:
https://en.wikichip.org/wiki/WikiChip).

Fig. 2.15. Actual image of Intel mobile CPU i7 (source: https://en.wikichip.org/wiki/WikiChip).

Fig. 2.16 is an architecture diagram illustrating the ring bus. The ring bus is a two-ringarchitecture that runs in both directions. Each ring has a data bus consisting of 256 wires
and sub communication lines of snoop bus, request bus, and acknowledgment bus. There
are approximately 1,000 total communication lines going in both directions. The ring bus
is placed on the top layer of the 13 metal layers in the chip manufacturing process. The
core and ring bus use the same clock frequency and the data bus transmits 32 B (256 bit)per cycle, which is a 96 GB/s transmission speed per ring for 3.0 GHz CPU. As 64 bytes
of cache is stored per address block and data is transmitted in block units, the minimumtransmission unit is 64 bytes of data transmitted over two cycles. If there are four cores,
there are six access points on the ring and it moves by one access point per cycle. Whencore 1 transmits data to L3 of core 4, it has to pass over three transmission sections andwill have a transmission latency of 3 cycles. For ring bus, the latency not only continues
to increase as the number of cores increases but it also has the disadvantage of dependency
on transmission location. All of Intel's mobile CPU released after 2015 transmits at a speedof 32 B/cycle through the 256 wires in the ring bus at all times when data is transmitted
from the L2 of a core to L3, or from L3 to L2, or from GPU to CPU. L3 has a total of127

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8 MB of storage and 2 MB is allocated per core. Instruction cache L1 has a storage
capacity of 32 KB, data cache L1 has 48 KB, and L2 cache has 512 KB.

Fig. 2.16. Ring bus architecture diagram.

The internal architecture of each core is shown in Fig. 2.17. Since Intel uses the same corearchitecture from mobile to server, the core's internal architecture is nearly identicalwithin the product family of the same period, regardless of CPU performance. When aprogram is executed, the program is transferred from the hard disk to main memory(DRAM), and the instructions pass through L2 or L3 according to the order written in the
program, and then arrives at the L1 instruction cache. There are hundreds of instructions
stored on L1 on standby. The communication line from L1 to the predecoder consists of
128 wires and transmits at a speed of 16 B (128 bit)/cycle. As x86 instruction lengths varyfrom 1 byte to 16 bytes, at least 1 ~ 16 instructions are sent in one cycle. The predecoder
fetches instructions, predecodes the boundaries between multiple instructions and carries
out fusion of instructions, where it fuses two instructions that can be combined to run at
one time in succession. The completed 16 B signal is divided into several macrooperations (MOPs) and is transmitted to the decoder. The following step by step processes
are divided into several branches for transmission and each step occurs in 1 cycle. The
decoder reads and translates MOPs and converts them into 5 micro-operations (μOP) andsends them to the next step.
All μOPs have the same length and a regular signalling system. While ARM instruction
sets have a uniform length of 4 bytes, x86 varies in length and goes through a complicatedprocess, omitted in Fig. 2.17, in order to reach the decode step. There are also many
fused-μOPs of two instructions. Although there were 4 μOPs in Broadwell products, fromSkylake products, there are 5 μOPs and they are transmitted to the ReOrder Buffer. In
addition, until the decode step, two threads operate independently in rotation by cycle.
That is, there exist two predecoders and decoders and if the instruction of thread 1 goes
through the first predecoder and decoder in one cycle, then the instruction of thread 2 goes
through the second predecoder and decoder in the next cycle. From the ReOrder buffer,128

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the two threads use the same hardware for each step. Some of the μOps that have finishedthe decoding process are stored in the L0 cache and join the ReOrder buffer in the order
determined by the branch predictor unit (BPU). The communication line between L0
cache and ReOrder buffer has a high priority of 64 B/cycle.

Fig. 2.17. Internal core architecture diagram.

From the L1 cache to the decode step, each instruction is entered in the order written in
the program, but the ReOrder buffer collects 224 μOps and proceeds out-of-order byreassigning priority. When five μOPs enter the ReOrder buffer, they are renamed in theentry, allocated their order and direction and the completed μOps go into retirement. New
μOps can enter only when retirement occurs.
When μOPs are sent in fives to the scheduler in the new order, the scheduler saves up to
97 μOPs and performs the calculation by sending it to the arithmetic logic unit (ALU)
through each register according to the time allocation and order. If data is needed, it isretrieved from the L1 data cache. When calculation finishes, the outcome is either stored
in the register of the scheduler or in the L1 data cache based on the need for reuse. The
completed μOPs remain in the ReOrder buffer if they need to be reused but otherwise go
into retirement. There is a total of 512 communication lines that transmits data from L1
data cache to the logic part and the speed is 64 B/cycle (1,028 lines and a speed of
128 GB/cycle for 2017 server CPU). There are 256 communication lines for storing data
and the communication speed is 32 B/cycle (512 lines and a speed of 64 GB/cycle for2017 server CPU). There are 512 communication lines from the L2 cache to each L1 cacheand the speed is 64 B/cycle. The process of starting at the L1 instruction cache to storing
the ALU calculation in the L1 data cache is done in a pipeline format. The steps shown in
Fig. 2.17 are pipeline steps and the total number of steps is 14 for the fastest route and 19
for the slowest route. Fig. 2.17 shows multiple steps condensed into each step.

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Two methods are typically used to speed up CPU computation. The first is to increase the
IPC by increasing the number of parallel lines shown in Fig. 2.17. The second is toincrease the clock frequency of the CPU. In 14~19 pipeline steps, the slowest step
determines the highest CPU clock frequency, so the execution speed of all steps should
increase evenly. To achieve this, the communication line speed per step and the switching
speed of the transistors need to increase. The switching speed of transistors is inversely
proportional to the area of the gate since it depends on the speed that the gate is charged.
As the technology node gets smaller, the switching speed increases. As of 2020, Intel
produces the latest products using the 10 nm process at the company's fab and AMD
produces its latest products using the TSMC 7 nm process.
There used to be a north bridge that included the CPU to DRAM and display interface anda south bridge that connected the other peripheral devices. But the north bridge has been
integrated into the CPU chip and only the south bridge (PCH, platform control hub)remains. For mobile products, the CPU and PCH are mounted on the same board andpackaged as seen in Fig. 2.18. The signal between CPU and PCH is transferred by anon-package interconnect (OPI) on the board. An OPI has the same transmission speed as
the four lanes of PCIe 3.0. Each lane has a transmission speed of 8 Gb/s. The maximum
transmission speed for each direction is 32 Gb/s, four lanes from CPU to PCH and 4 lanesin the opposite direction. PCH connects to the Internet, audio, SATA, USB, printer, etc.
As can be seen in the image, the size is 53.76 mm2.

(a)

(b)

Fig. 2.18. Intel i7 SoC interface (a) architecture diagram, (b) actual image of package(source: https://en.wikichip.org/wiki/WikiChip).

The layout of Intel's latest server CPU, Xeon Platinum 9282, is shown in Fig. 2.19.
28 cores, 2 DDR4s (3 channels each), 2 IMCs, 4 PCIe3.0 (16 lanes each), 3 UPIs (20 laneseach), and 1 PCIe3.0 (4 lanes, used for OPI) are arranged on one die as shown in the
figure. It is a CPU with 56 cores, a combination of two dies in one package. Intel uses amesh interconnect architecture as shown in Fig. 2.19(a) for desktop and server productswith more than 10 cores instead of a ring bus. There are bi-directional half rings arranged
along the boundaries of each device in x and y directions. The mesh lines are on the topof the 13 metal layers and the devices are on the layer below. Therefore, the mesh lines
actually pass over the top of each device and they do not overlap. As shown in Fig. 2.19(b),every device has a converged mesh stop (CMS) at its access point that manages signal
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connections and routing. Each core has an L3 cache of 1.375 MB allocation and acaching/home agent (CHA). When its core requests data from the L3, it is delivered to thecore by sending it to the CHA. If it's requested by a different device, it is delivered to the
CMS through the CHA. Through the cache coherency control and routing function, CMS
sends the data to the address of the destination within the specified time slot by selecting
the fastest route. For example, when core 05 sends data to core 51, it moves 4 points inthe y-direction, then it switches paths and moves 5 points in the x-direction. Except forthe cores, there are 7 access points connected to the mesh. In this case, there are no L3
and CHA, and the signal is sent directly to the CMS to go on the mesh. In 2017, Intel
introduced the mesh architecture to its products, but the details such as the width of thecommunication line (number of wires) and communication speed have not been disclosed
yet. Fig. 2.20 is a die image and its size is 694 mm2.

(a)

(b)

Fig. 2.19. Intel server CPU Xeon Platinum, (a) Layout, (b) Core Detail Diagram (source:
https://en.wikichip.org/wiki/WikiChip).

Fig. 2.20. Actual image of Intel server CPU Xeon Platinum (source:
https://en.wikichip.org/wiki/WikiChip).

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Table 2.8 shows the core communication speeds of 2019 CPUs of AMD and ARM in
comparison to Intel CPUs. While the communication speed from the instruction cache IL1
to the predecoder of Intel and ARM core is 16 GB/s, 128 lines, that of AMD core is twice
as fast. The communication that stores data from the scheduler register to data cache 1(DL1) is 64 GB/s, 512 lines for Intel's server CPU, but that of AMD and ARM core is half
as much. The communication of retrieving data from DL1 to ALU is 128 GB/s, 1024 lines
for Intel while that of AMD and ARM core is ¼ as much. The speed and communicationarchitecture of other communication lines are shown in Table 2.7. Compared to Intel, themicroarchitecture and interconnect architecture of AMD and ARM cores are not disclosed
in detail.
Table 2.7. 2019 CPU core internal communication.
Core internalIntel (Server) AMD (Server) ARM
communication
IL1 to Predecode
16 GB/s
32 GB/s
16 GB/s
DL1 to ALUStore 64 GB/s
32 GB/s
32 GB/sor Register
Read 128 GB/s
32 GB/s
32 GB/s
IL1 to L2
64 GB/s
32 GB/s
32 GB/s
DL1 to L2
64 GB/s
32 GB/s
64 GB/s
L2 to L3
32 GB/s
32 GB/s
64 GB/s
Ling BusInterconnect
Infinity Fabric DynamIQ
Or MeshNumber of lines 16 GB/s: 128, 32 GB/s: 256, 64 GB/s: 512

There are three types of server CPU dies for Intel: 10 cores (LCC, low count cores),18 cores (HCC, high count cores), and 28 cores (XCC, extreme count cores). One of thethree types of CPU dies are used to place two dies on one package board and then a socket
is attached. In Fig. 2.21, four dies are tied together in pairs to create two sockets, and thefour dies are connected through UPI links. In Fig. 2.19, one die has three UPI links(20 lanes each) and six DDR4 channels (two 3-channels). Each of the three UPI links ofthe dies is used to connect the total of four dies in Fig. 2.21. Also, the 6 channels of DDR4
are connected to a DRAM through the mother board. Fig. 2.21(b) shows the actual
packaged image. Fig. 2.22 shows the method of connecting a total of 8 dies using the same
method. On a server computer or a Supercomputer, up to 28×8 = 224 Intel cores can beloaded on one board. Like this, the mesh interconnect and UPI links are currently
connected by electric communication but optical communication is the most effective
way. This has been the goal for related fields for the last 15 years.
The ultra-path interconnect (UPI), an upgrade of the quick path interconnect (QPI)
developed by Intel, is a point-to-point communication architecture that has been applied
to chips since 2017. The communication characteristics of QPI are as follows. In one QPIlink are 42 lanes and each lane consists of two communication lines. The same signal issent to the two lines and the differential signal is found to remove noise. Therefore, there
are a total of 84 signal lines in one QPI link. Of the 42 lanes, 21 are used for transmission
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for the clock signal, 4 lanes are used for the control signal and 16 lanes are used for theactual data transmission. The UPI in Fig. 2.19 has 20 unidirectional lanes, but it can beassumed that the characteristics are almost the same as QPI. When the clock signal
changes from 0 to 1 and from 1 to 0, 1 bit is sent each, and so 2 bits (double data rate) aresent per 1 cycle. The maximum transmission speed for one lane is 10.4 GT/s (giga transfer
per second). It means that since there are 2 transfers per one cycle, the maximum linkfrequency can be up to 5.2 GHz. If the CPU frequency is 3.2 GHz and the link is
synchronized to the CPU frequency, then the transmission speed per lane is 6.4 Gb/s. The
unidirectional transmission speed of 16 lanes is 102.4 Gb/s and it can send and receive
simultaneously. The CPU die has four PCIe 3.0 and each PCIe has 16 unidirectional lanes(32 lanes in both directions). The transmission method is the same as QPI and the
maximum transmission speed per lane is 8 GT/s. That is, the maximum link frequency is
up to 4.0 GHz, and when the link frequency is synchronized to the CPU 3.2 GHz, it has
the same transmission speed as QPI. VCSEL, which is commercially available for salecurrently in 2019, has a direct modulation speed of 50 GHz. It can transmit 120 Gb/s when
it uses differential signals, which is 12 times faster than QPI. Even when 25 GHz VCSEL
is used, which has higher stability, the transmission speed is six times faster. The heatgeneration rate is typically 2 mW per Gb/s for electric signals, but it is about 0.08 mW for
optical signals, which is about 20 times less.

(a)

(b)

Fig. 2.21. 2 Socket connection of server CPU (a) Architecture, (b) Image(source: https://en.wikichip.org/wiki/WikiChip).

Fig. 2.22. 8 connected server CPU die architecture(source: https://en.wikichip.org/wiki/WikiChip).

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The structure diagram of the socket (socket model: LGA2011-3) used in Intel CPU i7 isshown in Fig. 2.23. There are 2,011 solder balls on the bottom of the socket and theyelectrically contact the PCB which has the same number of contact arrangements. On topof the socket are 2,011 contact arrangements that connect to the bottom side of processor
substrate. The socket is bonded to the PCB through the surface mount technology (SMT)
process. A solder ball is composed of 3 % silver (Ag), 0.5 % copper (Cu) and 96.5 % tin(Sn), and has a melting point of 217 °C. It adheres at a higher temperature than the melting
point through thermal reflow. Normally, the packaging process refers to the process ofelectrically contacting the CPU die to the processor substrate and coating a protective
film. In the case of Intel, the surface of the CPU die is turned upside down to face thesubstrate surface, and the substrate electrodes that has the solder pad and the CPU
electrodes are bonded by a heat-treatment process (flip-chip bond). The substrate has thenecessary electrical circuit and electrode pad at the bottom side where it corresponds tothe contact points of the socket. The CPU die is in contact with the lid and a thermalinterface material (TIM). The lid is designed to function as an integrated heat spreader(IHS) as well as a protective case. The top of the lid is in contact with a heat sink blockand TIM and there is usually a fan over it. The lower electrodes of the substrate and theupper electrodes of the socket come in contact with a mechanical force applied by anindependent loading mechanism (ILM), a type of a clamp. In addition to applying force,
ILM has the function of aligning the substrate to the correct position of the socket. The
processor substrate is made out of fiber-reinforced resin and the lid is nickel-plated
copper, including TIM, and each has standards for area and thermal properties.

(a)

(b)

Fig. 2.23. Intel CPU package structure, (a) Structure diagram, (b) Cross-section diagram.

One of the biggest reasons for slow CPU performance or failures is the temperature of theCPU. Several techniques are used to maintain the temperature properly. The CPU die
contains at least four digital thermal sensors (DTS). The case temperature (Tcase) ismonitored by connecting a thermocouple (metal temperature sensor) to the center of the
CPU's case lid (IHS). The experimental values of the power used by the CPU, the DTS
temperature (TDTS), and the case temperature for the Intel i7 products are as follows.
Tcase = 0.17×P + 43.3 and TDTS = 0.398×P + 43.3. The thermal design power (TDP) of
the CPU is the power consumption amount that is set at the time of CPU design, and it is
the amount of power used when all the cores of the CPU execute at the clock frequency
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set at the time of design. This is the power that is consumed when CPU is used at peak
performance under normal conditions without using turbo boost. The TDP of Intel
i7-5960X Extreme Edition is 140 W. In this case, the die temperature is T DTS = 99.02 °C
and the case lid is Tcase = 67.1 °C. In other words, when the die temperature is about
100 °C, the case lid is 67.1 °C and CPU should always be below this temperature to ensurea stable lifespan and no calculation errors. Intel CPUs have a built-in thermal control
circuit (TCC) and a register for program execution. If the temperature gets higher than theset value, besides maximally rotating the fan, a program that forcibly lowers thetemperature is executed using two methods. The first method drops the clock frequencyof the CPU to slow down the execution speed. The second method repeats the on and off
states of the CPU clock at about 32 ms. If this fails, the CPU is forcibly stopped [8-12].
Recently, compared to computer CPUs, smartphone SoC package technology has been
improving greatly and there are many investments being made. Examples include Package
on Package (PoP) technology and 2.5D and 3D package technology (see Fig. 2.24). PoP
is the technology currently used by most smartphone SoC manufacturers, as shown in Fig.
2.24(a). The SoC die and the memory (SDRAM) die are electrically connected to eachpackage substrate and processed with a protective film. Then the two package substrates
are stacked above and below to connect the electrodes using a ball grid array (BGA)
method. In general, SoC connects to the package substrate by flip-chip bond and thememory die connects to the package substrate by the wire-bonding method. In the 2.5D
package technology, there is a silicon interposer plate between the package substrate andthe die, and the SoC die and memory die are connected to the top of the plate by
flip-chip bond. The interposer plate is also connected to the package substrate by the BGA
method. Unlike plastic substrates, silicon plates use a CMOS process that allows muchfiner wires to be stacked. The bottom electrodes are connected to the top electrodes usinga through silicon via (TSV) process, and this is the same for a 3D package. The 3D
package connects die 1 and die 2 by the flip-chip bond process, forms a TSV on die 1 andconnects to the lower electrodes of die 1. Then, the die 1 is connected to the package
substrate by the BGA method. Though there is an advantage in the communication speed
due to the higher package density and shorter connecting lines, the disadvantage is thatthe metal line design has to be shared between different manufacturers. The TSV process
involves not only simply forming holes but also the processes that increase the
manufacturing cost, such as thinning the wafer (approximately 650→100 μm) andapplying thin wafers to the CMOS process line. Therefore, 2.5D and 3D packages are notyet commonly used.

(a)

(b)

(c)

Fig. 2.24. SoC package structure, (a) POP, (b) 2.5D, (c) 3D.

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2.5. Supercomputer Architecture
As of November 2020, the top performance supercomputer rankings 1, 2, and 3 were
Fujitsu’s Fugaku (442 PFlops), IBM's Summit (148.6 PFlops) and Sierra (94.64 PFlops),
and in fourth place was Sunway TaihuLight (93.01 PFlops) of China. Out of the total
500 rankings, 59.8 % or 299 units were manufactured by the three Chinese companies
Lenovo, Inspur, and Sugon. American companies HP, Cray, etc. manufactured 21.6 % or
108 units. China invested a large budget ($200B) for IT technology independence. Basedon the five-year unit step-by-step technology development strategy that started in 2001,
China holds considerable technology for designing and manufacturing their own CPUs,
along with supercomputers. In particular, the processor mounted on the Sunway
TaihuLight uses the product (Sunway SW26010 1.45 GHz) developed by the Chinese
researchers through technology that they accumulated for approximately 15 years. Its
detailed performance and architecture are discussed in the next section. IBM was theworld's largest supplier of supercomputers, but the company's overall share decreased
significantly since selling its business division that used Intel processors in its server
manufacturing business to Lenovo, a Chinese company, in 2014, and only using its own
brand IBM Power9 line processors. Among the top 500 supercomputers, Intel processors(Xeon E5, Xeon Gold, etc.) account for about 91.8 %, which shows that a vast majority
are using Intel products. In the 500 top supercomputers, China has 214 units, the US has
113 units, Japan has 34 units, France has 18 units, and Germany has 17 units.
There are about 100,000 ~ 1,000,000 computer servers in a supercomputer that areconnected by an optical communication network and a CPU in the server consists oftypically 10~100 cores. Therefore, one supercomputer has about 1~10 million cores. Each
server CPU is designed to enable its own cache coherence and each server memory(DRAM) is accessible to all other servers and is designed in a non-uniform memory access(NUMA) network to share data with each other.
The network topology used in supercomputers mostly adopts the fat-tree networkarchitecture or 3D torus network (a type of mesh network) architecture. Most of the
top-ranking supercomputers have adopted the fat-tree architecture which is advantageousin connecting more servers. The communication path between any two servers is the same,so the latency to retrieve data from another server is uniform. But there is difficulty inadding servers later to upgrade the supercomputer. And if one of the switching devices on
the upper level fails, it has a large effect on the whole. The 3D torus architecture, otherthan the advantages over the fat-tree architecture, is beneficial for programs where data
locality between adjacent servers is important, and because of the efficient communicationarchitecture, the computation speed is faster with a small number of cores.
The supercomputer “Gordon,” installed at the San Diego Supercomputer Center (SDSC)
with a $200M grant from the National Science Foundation (NFS), began service in late2011 and was ranked 48th (285.8 TFlops) at the time. Cray manufactured it by upgrading
the previous model (CS300-AC) and adopted the 3D torus network architecture. Gordonhas 1,024 compute nodes (servers primarily used for program executions) and 64 I/O
nodes (servers primarily used for data input and output), and each node is connected to
64 Infiniband network switches.
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Fig. 2.25 shows the internal architecture of a compute node and I/O node. The computenode has two CPU sockets with 8 cores connected to QPI 2 links on top of one
motherboard as shown in the figure. Each CPU is connected to a 64 GB DRAM by
4 channels and to a chipset by OPI. The 80 GB SSD (hard disk memory) is connected tothe chipset by SATA. The host channel adapter (HCA) is a card type that slides into themotherboard slot. It is connected to the CPU through PCle3.0 and has two opticalcommunication access terminals. The I/O also has a similar architecture and the detailed
features are shown in the figure. Table 2.8 shows the number of communication lines
between each device, the communication speed, and detailed characteristics.

(a)

(b)

Fig. 2.25. Gordon's server, (a) Compute node, (b) I/O node.
Table 2.8. Internal communication width and speed of server.
DevicesCPU

Compute node
I/O nodeIntel Xeon E5-2670, 2.6 GHz Intel Xeon X5650, 2.67 GHz
DDR3 1600,
DDR3 1333,
DRAM
8 channels, 64 GB
6 channels, 48 GB
Hard disk
SSD 80 GB
SSD 4.8 TB
CPU to CPU
QPIx2, 20.8 GB/s
QPI×2, 21.36 GB/s
32 lines/ch,
32 lines/ch,
CPU to DRAM
8 ch 102.4 GB/s
6 ch 64 GB/s
CPU to HCA
PCIe3 16 ch 10.4 GB/s
PCIe3 16 ch 10.68 GB/s
CPU to chipset
OPI 4 ch, 2.6 GB/s
OPI 4 ch, 2.67 GB/s

The network switch uses a Mellanox product with 36 ports as shown in Fig. 2.26(b). It
can switch freely between 36 optical communication terminals and has 90 ns switchinglatency. It is powered by the built-in dual-core x86 CPU and switch chip. For the active
optical cable (AOC), 200 Gb/s is currently the highest communication speed product (seeTable 2.9). Four VCSELs with a 56 Gb/s modulation rate connects to four multi-mode
fibers to send signals. Because it is a duplex, it is actually composed of 8 optical fibers,
8 VCSELs, and 8 PDs, and the maximum speed is 200 Gb/s in each direction. These
optical cables follow the standard specification for package size and pin architecture. The
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standard specification is named Small Form-factor Pluggable (SFP) for 1 lane or Quad
Small Form-factor Pluggable (QSFP) for four lanes. The Gordon system used the QDR
40 Gb/s QSFP, the latest product at the time of manufacture. The optical connection
connects through the electric signal pin on the other side of the optical fiber when it isplugged into the HCA card and Infiniband switch port. As of November 2020, 154 units
from the top 500 supercomputers use the Infiniband switch while 251 units use the Gigabit
Ethernet switch. Percentage by performance shows 40 % for Infiniband and 19.2 % for
Gigabit Ethernet, showing that Infiniband is advantageous for higher performances. For
Mellanox products, the switching latency for Gigabit Ethernet is 300 ns, which isdisadvantageous compared to Infiniband. However, the advantage is that connection
between the internal and external networks is easy. The optical cables are able to connect
with the same QSFP standard and instead of HCA, it uses an Ethernet card.

(a)

(b)

(c)

(d)

Fig. 2.26. (a) HCA, (b) Infiniband 36-port switch, (c) Gigabit Ethernet 32-port switch,(d) HDR 200 Gb/s AOC.
Table 2.9. AOC Communication Speed.
AOC SFPQSFP nm/distanceSDR 2.5 Gb/s 10 Gb/s 850/500 m
850/400 m
DDR 5 Gb/s 20 Gb/s
1310/10 km
850/300 m
QDR 10 Gb/s 40 Gb/s
1310/10 km
850/300 m
FDR 14 Gb/s 56 Gb/s
1310/10 km
850/100 m
EDR 25 Gb/s 100 Gb/s
1310/10 km
HDR 50 Gb/s 200 Gb/s 850/100 m

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Gordon's 3D torus network is shown in Figs. 2.27. 18 out of 36 Infiniband ports connect
to 16 compute nodes and 2 I/O nodes, and the remaining 18 ports connect to other
Infinibands. From the 18 ports, 3 ports are allocated to each of the 6 directions of +x, -x,+y, -y, +z, and -z. Since each port is connected through QDR 40 Gb/s QSFP, there is acommunication speed of 40 Gb/s between each node and the Infiniband, and acommunication speed of 120 Gb/s between the Infinibands. The +x at the edge of 3D torus
connects to the -x at the opposite edge, and the y and z directions connect in the same way
as well so that all of the Infinibands have the same connecting architecture. Therefore, all
the nodes have the same connecting architecture regardless of location. When this type of
connection occurs in 2D, it forms a torus structure (a hollow donut structure) and this iswhere the name comes from.

(a)

(b)
Fig. 2.27. (a) Gordon 3D torus network, (b) Gordon network architecture.

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In the Gordon supercomputer, HCA1 of each node forms one 3D torus and HCA2 forms
another 3D torus, so it has two independent 3D torus architectures. This not only doubles
the communication speed between nodes but also increases safety against accidents by
operating a different torus network when one torus network fails. The 3D torus
architecture is a mesh network architecture and it communicates between two nodes byskipping over each of the multiple switching points along the shortest path. Therefore,
communication time is advantageous when two nodes are adjacent, but it isdisadvantageous when they are far apart. Either one supernode, consisting of 32 adjacentcompute nodes and 2 I/O nodes, executes the program alone or the 32 total supernodes
carry out the computation in parallel. As shown in Fig. 2.27(b), 64 I/O nodes are connectedto the system storage of 4 PB and each node is connected to an external Ethernet network
through multiple Ethernet servers. The theoretical maximum computational speed of theGordon supercomputer is 336.1 TFLOPs and the actual measurement is 285.8 TFLOPs.
In addition to China's smartphone SoC (Kirin990) with the ARM license and x86
CPU(ZX-E, KX-6000) produced in collaboration with a Taiwanese company, therepresentative CPUs developed using pure Chinese technology are Loongson and
Sunway. From the 10th five years (2001~2005) to the 12th five years (2011~2015) of
CPU development using China's own technology, towards the end in 2015, theperformance of the above two processors reached the level of application to commercial
products. Loongson was developed as a national project (863 programs) by the Instituteof Computing Technology (ICT) under the Chinese Academy of Sciences and is now
produced and sold by Loongson Technology, a company established throughpublic-private collaboration. The latest products released in 2019 is the Loongson
3B4000, 2.0 GHz, 64 bit (4 cores). Its reported theoretical maximum performance (peak
FLOPS) is 128 GFLOPS. For comparison, the maximum performance for the 2011 Intel
Xeon E5-2670 (8 cores) used by Gordon, after conversion to 4 cores, is 83.25 GFLOPS.
The Loongson processor used LoongISA that they developed based on the MIPS
instruction set, and for the four instructions they lacked, they purchased the license from
MIPS Technologies. Since it is based on MIPS, it cannot use Windows OS of x86 CPU
or Android OS of ARM CPU but has to use Linux. To compensate for this disadvantage,
multiple translation instructions were added to LoongISA so that it can retrieve Windows
and Android OS programs on the Linux OS. Since 2015, it is known to be used in national
facilities of China where security is important such as national defense, communications,e-government, transportation, etc. In addition, it developed a product (Loongson X-CPU)that applied radiation hardening by design (RHBD) to use for satellites and it has been
mounted on the BeiDou satellite, which was launched on March 30, 2015. The Sunway
processor was developed primarily for national defense use at the National High
Performance IC (Shanghai) Design Center, and the latest product, 2016 Sunway
SW26010, is used by the processor of supercomputer 'Sunway TaihuLight'.
Sunway TaihuLight was manufactured by a Chinese research institute, National ResearchCenter of Parallel Computer Engineering & Technology (NRCPC) and is installed at the
National Supercomputing Center in Shanghai. It has 40,960 nodes (10,649,600 cores)
based on a three-layer fat-tree network architecture. Each node consists of a Sunway
SW26010 processor with four core groups (CG) and 260 cores, as shown in Fig. 2.28.
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Each core group has a management processing element (MPE) of one big core andcomputing processing elements (CPE) of 64 small cores and a memory controller (MC)as shown in Fig. 2.28(b). The MPE core each has 32 KB of L1 data cache and instruction
cache and 256 KB of L2 cache. For the CPE, there is a common L1 data cache of 64 KB
and an L1 instruction cache of 16 KB for the 64 cores. Each core group has 8 GB
SDRAM each.
The four core groups are connected through a network-on-chip bus (128 bit) and they
connect to the fiber optic terminals by a system interface (SI). The SI uses PCIe 3.0 and
its maximum transmission speed is 16 GB/s. The transmission speed for Sunway's
frequency of 1.45 GHz is 5.8 GB/s. It claims that, though the instruction set is based on
RISC, it uses the ISA and architecture of its own development and has not disclosed thedetails. The MPE core has two pipelines and the CPE core has a single pipeline. Both havethe 64-bit, SIMD, out-of-order features. When comparing the performance andarchitecture with an Intel processor, it can be assumed that it combined one quad-core
CPU and a GPU with 256 processing elements and that instruction sets necessary for thechanged architecture based on RISC ISA were added. Each pipeline has a computation
speed of 8 FLOPS/cycle. Since the core frequency is 1.45 GHz, the theoretical maximum
computation speed per node is (16 FLOPS ×4 + 8 FLOPS ×256) ×1.45 = 3.0624 TFLOPS.
The value of 125.4 PFLOPS is for all 40,960 nodes and the value measured by Linpack
Benchmark is 93.01 PFLOPS. It was ranked number one in 2018 and currently
ranks fourth.

(a)

(b)

Fig. 2.28. Sunway SW26010 processor composition, (a) 250 cores in 1 die,(b) Individual group architecture.

A fat-tree network architecture has a tree-shaped structure as shown in Fig. 2.29(a) andthe name comes from the increased connection lines as the level gets higher. A common
tree structure has the same number of communication lines regardless of the level, so thehigher levels don't have enough communication lines. A non-blocking network has thesame number of lines that go up and down at the switch. In this case, all the signal input
at the switch are output simultaneously without waiting time in the buffer. Most
supercomputers and data centers have the non-blocking architecture.
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(a)

(b)

(c)

Fig. 2.29. Fat tree network architecture.

A three-layer fat-tree architecture with switches that have four ports each is shown in
Fig. 2.29(b). Normally, the upper layer is called the core, the middle layer is called theaggregation, the lower layer is called the edge, and the server is called the leaf. The left
two nodes can reach each other only through the edge switch between them, the four nodes
can reach one another only through the aggregation switch and the rest needs to go throughthe core switch. The characteristic of the fat-tree is that when a path goes through the coreswitch, all of the paths have the same length. If one path fails, there are many bypass paths
and the path lengths are all the same. When configuring a three-layer fat-tree with thesame switches and n number of ports, typically the core switch can connect up to (n/2)2,
the aggregation and edge switches can connect up to n2/2 and the server can connect up
to 2(n/2)3. If a switch with 64 ports is used, more than 130,000 servers can be used.
Fat-trees are advantageous in connecting large numbers of servers but is very
disadvantageous in adding new servers after it has been completed. For 3D torus, there
are no upper and lower levels, so a new server can be added by opening the last line andit can be connected in the same method as before. By internally connecting the four
switches inside the dotted boxes to create one 16-port switch and four 8-port switches, atwo-layer fat-tree network can be created as shown in Fig. 2.29(c). Although it looks
different externally, it is the same network architecture. Network device companies
internally connect multiple 16-port, 32-port or 36-port switches and sell switch devices
with a large number of ports such as 64 ports, 128 ports, and 256 ports.
Sunway has purchased switch chips and HCAs from Mellanox. It is assumed that thecustom switch devices were manufactured using imported switch chip and HCA. Sunway
TaihuLight has a fat-tree architecture as shown in Fig. 2.30. The upper core layer has
40 cabinets and each cabinet has 4 supernodes. Each supernode has 32 boards and eachboard has eight connected nodes. To have a non-blocking architecture, each board has
switches for 16 ports and 8 of these are connected to the nodes and the other 8 areconnected to the supernode switches. The supernode switch has 512 ports and 256 of them
are connected downwards and the other 256 are connected upwards to the cabinet switch.
The cabinet switch has 1,024 ports and is dispersed and connected downward to
160 supernodes. The 256 downward ports of a supernode are dispersed and connected to
128 boards. The entire network is broken down into a central switch network, management
network, and storage network and the switches of each layer are connected to either the
management server or the storage system (see reference [15] for more details). In addition,
they connect to the external Ethernet through the management server. EDR 100 Gb/s
QSFP is used for the optical cable. It took $270M USD to build this supercomputer, which
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was divided between the Central Chinese government, the province of Jiangsu and thecity of Wuxi [13-15].

Fig. 2.30. Sunway TaihuLight network architecture.

2.6. Optical Network-on-chip Architecture
Since the early 2000s, leading semiconductor companies such as Intel, IBM, HP, Sun
Microsystems, Samsung, and others have started conducting silicon photonics research inorder to implement optical communication technology to core-to-core or CPU-to-memorycommunication on semiconductor chips. In particular, companies that focus on designingand manufacturing CPUs have conducted competitive large-scale research projects.
Universities and companies have also proposed the network architecture for the opticalcommunication technology to be applied for. Among them, the frequently cited ATAC
architecture of MIT and the Corona Architecture of HP are outlined [16-19].
ATAC Architecture: This is the optical communication network architecture for CPU
chip application proposed by MIT silicon photonics research team in 2010. Opticalcommunication technology was applied to a CPU chip consisting of 1,024 cores and its
cache coherence and CPU performance were compared to those of electricalcommunication. It targeted relatively low-performance cores with a clock frequency of
1 GHz, in-order instruction, and 16 nm technology application. This was to minimize thefootprint of a single core in order to integrate 1,024 cores on a chip with an area of about
400 mm2.
Chip composition: As shown in Fig. 2.31, there are 64 clusters in total and each cluster
has 16 cores. There is a memory controller in each cluster and they are connected to the
DRAM at 40 Gb/s.
Network: The 16 cores of each cluster are connected to an electrical signal-based meshnetwork (EMesh) like Intel's mesh network. The 64 clusters are connected to the ring
optical communication network (ONet) as shown in Fig. 2.31(a). Each cluster has a hubthat connects electrical signals to optical signals and optical signals to electrical signals.
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For example, when core 1 transmits data to core 4 within the cluster, the signal goes
through the EMesh. When core 1 transmits data to core 1024 in a different cluster, it first
sends the signal to the hub of cluster 1 through the EMesh, then it converts to an opticalsignal and gets sent to the hub of cluster 64. Bnet is used to convert the optical signal into
an electrical signal and transmits it from the hub to core 1024. ATAC designers introduced
Bnet to simplify the transmission process by omitting the routing function for the signal
going from the hub to the core. Bnet only has a broadcasting function that delivers signals
to all the cores. The 16 cores are divided into two groups and when the specific digit of
the signal is an odd number, it gets sent to group 1 and when it is an even number, it getssent to group 2.

(a)

(b)

Fig. 2.31. ATAC network architecture, (a) 64 clusters, (b) Internal architecture of cluster (Source:
George Kurian et al., “ATAC: A 1000-Core Cache-Coherent Processor with On-Chip Optical
Network,” PACT’10, Sept.11–15, 2010).

Communication line composition: The number of data signal communication lines of
EMesh and Bnet is 128 and 128-bit signals are transmitted in parallel per clock cycle.
ONet has more than 129 optical waveguides in total including 128 waveguides for data,
one optical waveguide for the control signal and many waveguides for meta-signal. Thesewaveguides pass through each cluster in a ring shape as shown in Fig. 2.31(a). One opticalwaveguide has 64 wavelengths and one wavelength is allocated to each cluster. Therefore,
each cluster transmits 128-bit data signal in parallel one wavelength at a time to
128 optical waveguides. For example, when data is transmitted from cluster 1 to cluster
64, the signal is sent on wavelength 1 of 128 optical waveguides. Clusters 2 ~ 64 receive
the signal of wavelength 1 each by 1/63 of the strength and all clusters except cluster 64
ignore the signal and only cluster 64 takes in the transmitted data. Multiple clusters cansend signals to cluster 64 since they use different wavelengths from one another.
Latency: The time it takes to transmit a signal for each of the communication line is shownin Table 2.10. In the case of EMesh, after 1 cycle at the router, 1 cycle is added for eachhopping. 6 hopping is needed to start at core 1 and to arrive at core 16. In comparison,
Onet requires 3 cycles across the board. In other words, 1 cycle for E/O conversion,
1 cycle for O/E conversion and 1 cycle for waveguide signal transmission regardless of
distance. BNet requires 1 cycle for O/E conversion and 1 cycle is added for every hopping.
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Since the CPU clock is 1 GHz, the communication speed of the electric cable is 1 Gb/s,
and in the case of the optical waveguides, the total communication speed per opticalwaveguide is 64 Gb/s since there are 64 wavelengths.
Table 2.10. Communication line width and latency of ATAC.
In-order, 1 GHz, 1024 cores,
16 nm technology
EMesh Hop Latency
2 cycles (router delay – 1, link traversal – 1)
ONet Hop Latency 3 cycles(E/O + O/E conversion – 2, link traversal – 1)
BNet Hop Latency
2 cycles(O/E conversion – 1, link traversal – 1)
EMesh
128-bit wideONet128-bit wideBNet128-bit wide
64 memory controllers,
Memory Bandwidth
5 GB/s per controllerCore model

Optical device composition: The light source is outside of the CPU chip and shoots acontinuous wave (cw) signal through the optical fiber into the power optical waveguide
inside the CPU chip. Then, it connects from the power optical waveguide to the ONet
optical waveguide through the ring resonators. Excluding the meta signal, there are at least
129 optical waveguides with 64 wavelengths each, requiring a minimum of 64 external
light sources and a maximum of 8,192 external light sources. Each cluster has one
modulator per waveguide to send signals on the allocated wavelengths and a Ge-based
Electro-Absorption (EA) modulator. In this case, there are at least 129 modulators per
cluster and more than 8,256 modulators on the chip as a whole. For detectors, Ge-based
ones are used and at least 8,127 detectors per cluster and 520,000 detectors on the chip asa whole need to be integrated. The ring resonator has the function of connecting modulated
signals to optical waveguides and the function of dividing the optical signals of eachwavelength into 1/63 strengths. Therefore, as many ring resonators are integrated as themodulators and detectors. In addition, more ring resonators are added according to thenumber needed to connect the cw signals from the power optical waveguide to the ONet
optical waveguide. ATAC designers assumed the ONet optical device to be integrated on
a SOI substrate optical network chip separate from the CPU chip and joined to the CPU
chip using 3D package technology.
The CPU performance difference of connecting 64 clusters by ONet opticalcommunication and by Emesh electric communication of 256-bit width was comparedusing 3 benchmark application programs of Splash2, Parsec, and Synthetic. The cachecoherence protocol for this calculation used the ACKwise protocol proposed by ATAC
designers and two other existing protocols. In conclusion, ACKwise protocol based on
optical communication showed 60 % to 2.5 times better performance than any otherprotocol combinations based on electric communication. Since this difference inperformance is based on the same core performance, it is a significant difference as it
shows the difference in the cache coherence based on optical communication. There isalso a greater significance in that it notably reduces heat generation and enables
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communication in a range that is nearly impossible with Emesh through opticalcommunication [16].
Corona Architecture: This is the optical communication network architecture for CPU
chip proposed by the HP research team and participants from two universities in 2008.
The details of the CPU chip composition and network architecture are as follows [17].
CPU Chip Composition: Each cluster has four cores and a total of 64 clusters areconnected by optical communication. The clock frequency of the core is 5 GHz with two
L1 caches allocated per core and one L2 cache per four cores. Each cluster has a memory
controller and is connected to the DRAM by optical communication. Assuming 4 threads
per core, the CPU is set to run 256 cores and 1,024 threads.
Optical Communication Network: The 64 clusters are connected by the Corona network
optical communication. There isn't an explanation for the communication between thefour cores inside the cluster but it can be regarded as the same bus type electriccommunication as a commercial CPU. There is a total of 260 optical waveguides
including 256 optical waveguides for data transmission, 1 broadcast optical waveguide,2 arbitration optical waveguides, and 1 clock signal optical waveguide. One opticalwaveguide carries independent signals of 64 wavelengths. Each cluster is allocated four
optical waveguides (256 transmission channels) for data transmission. The four opticalwaveguides start at the corresponding cluster and pass through 63 clusters to return to theoriginal position and takes on a broken ring architecture that does not continue. All
clusters start and end with four optical waveguides of this type.
For example, when cluster 1 transmits data to cluster 60, the cw signal is divided from thepower optical waveguide to the optical waveguide allocated to cluster 1 to transmit the cw
signal to cluster 60. It has already informed all clusters that cluster 1 will transmit data tocluster 60 and so all the remaining clusters except for cluster 60 turns their ring resonators
off. The cw signal that started at cluster 1 arrives straight at cluster 60. Then cluster 60
uses the ring resonator modulator on the cw that was sent to generate data and sends itback to cluster 1.
Arbitration is the act of creating an order to prevent multiple clusters from sending dataat the same time to the same cluster. From the two arbitration waveguides, 64 wavelengths
are allocated to waveguide 1 and 1 wavelength is allocated to waveguide 2. The two
waveguides have an unbroken ring-type architecture. 1 wavelength of the 64 are allocatedper cluster. Arbitration is executed before cluster 1 sends data to cluster 60, and at thistime, cluster 1 sends a token signal to the wavelength of cluster 60. When cluster 60
receives the token signal, token issuance of all other clusters stops. At this time, necessary
notification proceeds in the form of broadcast through waveguide 2 (one wavelength). To
give equal opportunities, arbitration gives the opportunity to cluster 1 and once cluster 1's
data transmission ends or it gives up, the next opportunity is given to cluster 2, and then3, 4,...., until 64 in sequential order (Round Robin Method). In Corona networks, one
wavelength is allocated to one waveguide for clock signal transmission and64 wavelengths are allocated to one waveguide for the broadcast signal. This is separate
from the arbitration broadcast and it used for general-purpose broadcast signalling.
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Optical Device Composition: Corona designers use external light sources like the ATAC
and in this case, it would require a minimum of 64 and a maximum of 16,514 lasers. A
ring resonator modulator is used for signal generation and each cluster needs
16,257 modulators that include 256×63 for data transmission, 65 for arbitration and 64 forbroadcast, which is a total of 1.04 million ring resonator modulators. For detectors, a totalof 20,000 are needed for all the clusters including 256 for data transmission, 65 forarbitration, 1 for broadcast and 1 for clock per cluster. Ring resonator filters forwavelength division multiplexing (WDM) require a total of 1.1 million filters, including
the combined number of modulators and detectors and 386 for power divergence.
Corona Performance: The target operating speed of the CPU is 10 TFLOPS, the numberof data communication lines between clusters is 256 (256 bit-wide), and thecommunication speed per line is 10 Gb/s. The optical communication in the CPU has atotal bandwidth of 20 TB/s and a total bandwidth of 10 TB/s between the CPU and the
DRAM. The power consumption of the CPU was calculated as 82 W and the area as
423 mm2. From this, the power consumption for optical communication was calculated as
39 W. To compare the performance of optical communication and electriccommunication, simulations were performed using synthetic benchmark and SPLASH2
benchmark. In conclusion, it showed a performance difference of 2~6 times. Also, thepaper reported that the Corona optical network architecture was the unique solution to not
only the heat generation (power wall) but also the communication barrier (bandwidthwall) between the cores and between the CPU and memory.
Causes of Failure: A number of optical communication-based network architectures for
CPU application have been proposed, but they have not reached the stage of experimental
integration and testing on actual chips. Rather, studies have been gradually fading out
since 2015. This is because the factors that lower its feasibility have been highlighted
more as research accumulated.
First is the disadvantage of using SOI wafers. Currently, wafers used in CPU and memory
chip production are ordinary bulk silicon. The fact that electronic devices and opticaldevices can be manufactured with the same CMOS process in the same wafer, which
motivated the research of silicon photonics, did not have many benefits in reality. Whenusing a silicon optical waveguide, the optical waveguide and the optical device should bearranged in the same silicon layer as the gates of the CPU [20]. This is not only
disadvantageous in increasing the gate density of the CPU, but also makes it nearly
impossible to design metal layers (13 layers) of electronic devices. Contrary to themotives, for most proposed architectures, it is assumed that optical devices and CPU were
manufactured on different wafers and bonded in a 3D package.
Second, the transmission loss of the silicon optical waveguide is large. Silicon has a highrefractive index (n = 3.45), which is advantageous in making the size of the opticalwaveguide as small as possible, but disadvantageous in designing the low transmission
loss of waveguide. Although ATAC assumed 0.3 dB/cm, the minimum value reported in
the paper was 1.7 dB/cm [18], but in typical CMOS process, it is measured greater than
2 dB/cm. The optical waveguide length in ATAC is approximately 8 cm and 16 cm for147

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Corona, of which the transmission loss will be 16~32 dB. It becomes impossible for
63 clusters to receive data at 1/63 strength.
Third, the yield and performance of optical modulators, detectors, and ring resonators arepoor. ATAC uses Ge-based optical modulators and detectors, while Corona uses ring
resonator-based optical modulators and Ge-based detectors. For the network to function,
the failure rate of each device needs to be lower than the surplus device rate (generally
10 %). The average yield has to be over 90 %. When manufacturing modulators anddetectors by epitaxial growth of Ge materials on silicon wafers, it is possible for a few
selected devices to reach their target performance, but it is very difficult to obtain auniform yield by the necessary ratio. Particularly, it becomes nearly impossible when it isaccompanied by fabrication-attributed statistical errors such as the resonance wavelengthof the ring resonator. In this case, micro-heaters needs to be attached locally and heated
for wavelength tuning. Heating 500,000 resonators for ATAC and 1.1 million resonators
for Corona takes away the effective value of introducing optical signals. Also, theinsertion loss of Ge-based optical modulator was assumed to be 1 dB for ATAC but thecurrently reported insertion loss of the device is about 3~6 dB, which is a significant
difference.
Fourth is the light source. Intel has developed in collaboration with a research team at UC
Santa Barbara an on-chip laser by joining an InP wafer to a silicon wafer. However, it has
yield problems like the Ge-based optical device. ATAC and Corona architectures were
presumed to use external light sources, but it is difficult to integrate hundreds to thousandsof lasers in the alignment of optical fiber arrays with optical waveguide arrays on a chip.
It is also difficult to secure space to attach them on the side of the chip when considering
the cross-sectional area of optical fibers. The coupling loss measured when aligning one
optical fiber to a silicon optical waveguide is reported to be a minimum of 1~2 dB, but it
is only accomplished by modifying or adding a special structure at the inlet of the opticalwaveguide. In general, coupling loss is measured 3~6 dB or more. When integrating tensof optical fibers in an array, it is nearly impossible to align all the optical fibers with auniform coupling loss below 3 dB to single-mode silicon waveguide arrays.

2.7. Key Elements toward Optically Interconnected CPU
In fact, the 4 causes of failure may all be attributed to the use of silicon as the opticalwaveguide. Among the materials used in the CMOS process, there are no materials otherthan silicon nitride (Si3N4), silicon oxynitride (SiOxNy), silicon dioxide (SiO2), and silicon(Si) that can be used to an optical waveguide. Since silicon dioxide (SiO2) is used for the
cladding of the waveguide and if we don't use silicon, then the remaining candidate
materials are only silicon nitride (Si3N4) or silicon oxynitride (SiOxNy). Although there
are many papers that analyze optical devices using these two materials as opticalwaveguides, there are no papers that suggest or test a network architecture that can be
integrated on a CPU chip using these two materials. Here, it is suggested that the 4 causesof failure can be overcome by the key elements, three technologies, and a possible networkarchitecture based on these two materials.
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Silicon nitride or silicon oxynitride film deposition in the CMOS process is done by a low
pressure chemical vapor deposition (LPCVD) or plasma enhanced CVD (PECVD)
process. LPCVD can deposit films of silicon nitride with a precise stoichiometric ratio(3:4 of Si3N4). The deposition temperature ranges from 425-900 °C. The stoichiometricfilm used in the CMOS process is deposited most frequently around 770 °C and has arefractive index of 2.0. The transmission loss for the optical waveguide is also very low
and a measurement < 0.1 dB/cm is reported. The disadvantage is that cracks occur in thefilm when the thickness is over 200 nm due to tensile stress. The melting point ofaluminium and copper used for the metal layers in the CMOS process are 660 °C and1,084 °C, respectively. At high temperatures, copper becomes an impurity source thatpenetrates into other thin films, and for aluminium, the deposition temperature is higher
than its melting point. Therefore, LPCVD thin film is used only up to the first metal layer.
The PECVD film can be deposited at 200~400 °C and fast at 780 nm/min (LPCVD
4.8 nm/min). Therefore, it is advantageous for the formation of thick films and is used
after the first metal layer. Though PECVD silicon nitride film doesn't fit a stoichiometricratio, it has relatively low stress and can be deposited up to several micrometers without
cracks. In order to integrate the optical and electronic devices on the same wafer, it is mostadvantageous to form the optical waveguide on the top protective film layer shown in
Fig. 2.5. Therefore, the first cause of failure, the same layer of optical devices with electric
devices, can be overcome by using a PECVD silicon nitride or oxynitride film.
There are many papers on the formation and characterization of optical waveguides using
PECVD silicon nitride or oxynitride films. The ratio of the injection gas can be adjusted
when depositing silicon oxynitride (SiOxNy) to have a specific refractive index of thefilm. As shown in Fig. 2.32, when the nitrogen component in the film is 0 %, the refractive
index (1.45) is the same as that of PECVD oxide (SiO2) and as the nitrogen component
increases, the refractive index increases linearly to 2.0 [21]. According to reference [22],the refractive index and transmission loss of PECVD SiON film are n = 1.68 and
0.3 ± 0.15 dB/cm, respectively, for the wavelength 1310 nm and the optical waveguideshown in Fig. 2.33(a). According to reference [23], the refractive index is n = 1.515 andtransmission losses are 0.2 dB/cm in TE mode and 0.3 dB/cm in TM mode, for the
wavelength 850 nm and the optical waveguide shown in Fig. 2.33(b). Therefore, thetransmission loss of PECVD silicon oxynitride optical waveguide is about 0.3 dB/cm,
which is the same as the transmission loss of the optical waveguide set in the ATAC
architecture. The second cause of failure, large transmission loss of silicon waveguide,can be overcome.
Although silicon has active optical properties, the performance and yield of the
manufactured devices are far below the level for implementing a network-on-chip. The
technology of VCSEL had developed fast and reached the level of 5 GHz directmodulation in the early 2000s, and from mid 2010s, 50 GHz products becamecommercialized, and as shown in Table 2.9, it came to a level where HDR products werealready being used for supercomputer production. The performance is superior to opticalmodulators implemented by silicon optical waveguides. Therefore, it is more efficient interms of area and power to construct the light source and the optical modulator with thesame device using VCSEL rather than constructed with different devices. The third and149

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fourth causes of failure, the low yield and poor performance of optical devices andexternal light sources, can be overcome by using VCSEL and the three technologies
developed and verified experimentally by Oprocessor, Inc. (start-up company founded by
the author).

Fig. 2.32. Nitrogen component ratio and refractive index graph (source: M. I. Alayo et al.,‘Deposition and characterization of silicon oxynitride for integrated optical applications,
Journal of Non-Crystalline Solids, 2004, pp. 76-80).

(a)

(b)

Fig. 2.33. SiON optical waveguide, (a) n = 1.68, (b) n = 1.515 (Source: [a] Richard Jones et al.,“Integration of SiON gratings with SOI,” Conference: Group IV, Photonics, 2005, [b] Tai
Tsuchtza, ‘Low-loss Silicon Oxynitride Waveguides and Branches for the 850 nm Wavelength
Region’, Japanese Journal of Applied Physics, vol. 47, no. 8, 2008, pp. 6739-6743).

Among the methods that can inject light into PECVD silicon oxynitride opticalwaveguide, it is difficult to use the methods that were widely used in silicon photonics,
butt coupling and grating coupling. For a direct modulated VCSEL, butt coupling can only
generate signals at the edges of the chip, so it is impossible to generate signals from coresthat are not located at the edge. It is also difficult to secure space to attach them on theside of the chip. For grating coupling, the difference of refractive index between silicon
oxynitride (SiON) and silicon dioxide (SiO2) is not only small (Δn = 0.1 to 0.55), whichleads to low coupling rate, it also creates statistical errors like the ring resonator.
Therefore, it is difficult to overcome the yield problem. Excluding the two methods, theonly possible method is to use the prism.
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Fig. 2.34 uses a prism method to measure the transmission loss of light passing through asilicon oxynitride material. The light was refracted by the prism with applying the index
matching oil on the surface of the prism and silicon oxynitride material [24]. In order to
use the prism method at the network architecture, the following three technologies mustbe achieved.

Fig. 2.34. Transmission loss measurement of SiON material (Source: B. S. Sahu et al., ‘Influenceof hydrogen on losses in silicon oxynitride planar optical waveguides’,Semicond. Sci. Technol., Vol. 15, 2000).

First, an adhesive technology that is transparent, satisfies the refractive index matchingcondition, maintains its adhesive strength at 200-400 °C, and is not affected by solventsused in the CMOS cleansing process such as BOE, isopropanol, acetone, etc. needs to beachieved. There are no commercial adhesives that are transparent with a refractive index
greater than 1.6. Adhesives that maintain its strength in cleansing solvents and at 400 °C
have not been reported in papers. Therefore, it is necessary to develop a technology thatcan bond a prism to the surface of an optical waveguide without using a commercial
adhesive.
Second, microlenses that can produce collimated light under the conditions of Fig. 2.35
need to be achieved. Microlens consisting of quartz and silicon materials are commerciallyavailable, but their refractive indices are not suitable for the current purpose. Third, aspecial package technology for VCSEL and PD needs to be achieved. The current
commercial package is a hermetically sealed structure that gets filled with nitrogen or
argon gas after a specific distance between the laser light exit and the lens is maintained.
This type of packaging is overly bulky, and dozens to hundreds of VCSELs and PDs
cannot be integrated on the CPU chip. The area and height of a single optical device after
packaging has to be at most <0.7 mm2 and <1.0 mm, respectively. Oprocessor, Inc. hasdeveloped an adhesive technology, a microlens, and a package technology that satisfies
the requirements.

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Fig. 2.35. Conceptual diagram of optical signal connection on SiON optical waveguide.

Fig. 2.36(a) is a micrograph of a GaP prism bonded to the surface of a SiON opticalwaveguide and (b) is an enlarged photograph of the sideview. The adhesion characteristics
between the prism and wafer are shown in Table 2.11. It had a transmissivity of over
99 % at the thickness 2 μm required for adhesion, and the refractive index was measured
up to 1.72 depending on the heat treatment conditions. When force was applied from theside to observe adhesion strength, the prism broke more times than it fell off.

(a)

(b)

Fig. 2.36. Real Images of Prism Bonded to SiON Optical Waveguide.
Table 2.11. Adhesion characteristics of prism.
Specification
Refractive index 1.69~1.72 at 850 nm
Transmission> 99 % for 2 μm thickness at 850 nm
High temperature
400 C (compatible with CMOS metal process)resistance
Solvent resistance BOE, Aceton, Isopropanol, etc. (unaffected by alcoholic cleaning)
adhesion strength > 1 N/mm2 (not detachable before breaking GaP prism at 25 C)

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The radiation angle of VCSEL in the air is typically in the range of 20~40°. The microlens
must have the right value of refractive index to collimate light emitted by the median valueof 30°. Fig. 2.37 shows a scanning electron microscope (SEM) image of the microlens
array (a) and an enlarged image (b) of one. The diameter of the microlens is 30 μm.

(a)

(b)

Fig. 2.37. SEM images of microlens.

After attaching the microlens to the 25 GHz VCSEL chip, the shape of the output light
was observed as shown in Fig. 2.38. The picture was taken by placing an infrared card
7 cm vertically from the VCSEL. Since the distance from the VCSEL to the opticalwaveguide is <1 mm, the collimated light proves a sufficient performance of the lens.

(a)

(b)

Fig. 2.38. Output light images of VCSEL before and after microlens attachment.

The test results of the microlenses can be used to complete a package. Fig. 2.39 (a) shows
the package structure. The lens parts were made separately and then attached to the
VCSEL in the final step. Fig. 2.39 (b) shows a micrograph of the VCSEL packaged withthe structure. The base is 1.2×1.0 mm2 wide and 640 μm high. It does not show theminimum size because it was made for experimental verification of optical coupling.
Since the size of VCSEL is 0.25×0.25 mm2, < 0.25 mm2 is possible to achieve.
As shown in Fig. 2.40 (a), the prisms were attached to both ends of the SiON opticalwaveguide, and the VCSEL module was attached to a prism on one side. An experiment
was conducted to measure the intensity of VCSEL light entering the optical waveguideand coming out of the opposite prism. Fig. 2.40 (b) is a micrograph showing theexperimental laser light leaving the opposing prism after entering the optical waveguide.
figure (c) is a micrograph showing the laser light leaving the opposite prism while the153Advances in Optics: Reviews. Book Series, Vol. 5

scattered laser light cannot be seen due to the higher ambient brightness. The laser lightis strongly emitted from the center of the inclined side of the prism despite the partiallybroken state of the prism. The average value of the coupling loss measured from multiple
modules and optical waveguides is 3.54 dB, which is the value without anti-reflection
coating (AR) applied to the prism. The theoretical gain when AR coating is applied on
both sides of the prism is 2.6 dB, which results in a coupling loss of about 0.94 dB. It is amuch better value than the coupling loss of 3 dB set by ATAC. If the incident light iscollimated and the refractive index matching condition is met correctly, the theoreticalminimum value is close to 0 dB, excluding the reflection loss of prism.

(a)

(b)

Fig. 2.39. A package module of VCSEL, (a) Package structure, (b) Actual image.

(a)

(b)

(c)

Fig. 2.40. Laser light transmission experiment, (a) Experimental scheme, (b) Image of input
and output light, (c) Image of output light on the opposite prism.

In the above experiment, the alignment tolerance that causes 1 dB loss was measured
18.9 μm parallel to the optical waveguide and 7.2 μm perpendicular to the optical
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waveguide. The alignment tolerance of the perpendicular direction can be adjusted freely
by using an optical waveguide wider than the optical waveguide (30 μm wide) used in themeasurement. Therefore, the alignment tolerance parallel to the optical waveguide isimportant and the measured value (18.9 μm) is within the range that enables automationin mass production.
In the study of silicon photonics, ring resonators are assumed to be used for wavelength
division multiplexing filter (WDM filter) to transmit multiple optical wavelengths in one
optical waveguide. However, the ring resonator filter has not been used as a commercial
product for on-chip nor off-chip due to its poor performance. Currently, the WDM filter
with the most verified performance among the commercially available products is the thinfilm filter in which two thin films with different refractive indices are alternately stacked.
Fig. 2.41(a) shows the schematic diagram of four optical wavelengths transmitted througha single optical waveguide by the coating of WDM thin film filters on the bottom sides ofthe prisms. For example, λ3 passes through the prism coated with the filter of thecorresponding wavelength and enters into the waveguide, and it is reflected off the bottom
sides of the prisms coated with other wavelength filters and proceeds along the waveguideand is emitted from the prism of the corresponding wavelength. Fig. 2.41(b) graphicallyshows the structure of a WDM filter and the light path. Fig. 2.42 shows the reflection andtransmission graphs derived from theoretical calculation using TiO2 and Ta2O5 films,
which are often used to design WDM filters.

(a)

(b)

Fig. 2.41. 4-channel WDM, (a) Light input and output scheme, (b) Thin film filter and light path.

(a)

(b)

Fig. 2.42. Theoretical graphs for (a) reflection, and (b) transmission.

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The refractive index of the prism is nprism = 3.16 for 850 nm, that of the TiO2 is nH = 2.5086
and that of Ta2O5 is nL = 2.0908. When the prism angle is θ, the light that entersperpendicularly to the inclined side enters the bottom side as θ and proceeds as ф, asshown in Fig. 2.41(b). The thickness of each film is determined so that the path at thecorresponding angle is λ/4, and the TiO2 film is denoted as TH and the Ta2O5 film isdenoted as TL. Fig. 2.42 (a) shows the reflection spectrum calculated when T H and TL
films are repeated in alternation 8 times [(TH TL)8]. Fig. 2.42 (b) is the transmission
spectrum calculated when a spacer with a width of 4λ is inserted between the two
reflecting mirrors formed by repeating the TH and TL films 8 times [(TH TL)8(TH TH)8(TLTH)8]. Such calculations are technically very mature and used to produce commercial
products. The thin film filters of commercial products meet the WDM performancerequired by ATAC and Corona architectures.
As shown in Fig. 2.19, Intel's server CPU die is 694 mm2 with a maximum of 28 coresconnected to the mesh network through electrical communication. Desktop CPU die is~174 mm2 with a maximum of 8 cores connected by a ring bus. When opticalcommunication is introduced to server CPU, 9 cores are connected by a ring bus and
36 cores are divided into 4 clusters that are connected by optical communication as shownin Fig. 2.43. The ring bus uses the electrical communication like Intel's desktop CPUs.
Optical communication has a point-to-point crossbar network. That is, each cluster has apoint-to-point communication with the remaining three clusters. Therefore, two or moreclusters can send data to the same cluster at the same time and does not require arbitration.
In addition, all optical communication lines have the same latency (3 cycles). If the CPU
supports 3 GHz clock frequency and 64 bit process, the minimum bandwidth percommunication line is 192 Gb/s. Each optical line in Fig. 2.43 can be assumed the
bandwidth of 256 Gb/s unidirectionally and 512 Gb/s bidirectionally, which requires
10 optical waveguides and 10 VCSELs and 10 PDs of 50 Gb/s. This is twice as fast asthat of ATAC. Each cluster connected with the other three clusters needs 30 opticalwaveguides and 15 VCSELs and 15 PDs of 50 Gb/s. Since the individual package area of
VCSEL and PD is <1 mm2, the area per cluster for VCSELs and PDs is <30 mm2 which
is much smaller than the Intel's 8 core CPU area 174 mm2.

Fig. 2.43. A server CPU with 36 cores (Blue line: ring bus electrical communication, red line:
point-to-point optical communication).

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Fig. 2.44 shows a candidate package structure to be adopted if optical communicationlines are arranged on the top layer of CPU in Fig. 2.5. CPU is flip-chip bonded on theprocessor substrate, and the part corresponding to the optical device is cut out and thewires and electrodes are arranged appropriately at the substrate.

Fig. 2.44. Flip-chip bond package after processor substrate processing.

In Fig. 2.43, there are a total of six optical lines or 60 optical waveguides. The aggregate
bandwidth is 3.072 Tb/s (6×512 Gb/s). Based on this, the power consumption between
electrical and optical communications is compared. For electrical communication, it
consumes about 2 mW per 1 Gb/s, and there is an average of 5 hops for 6×6 electric meshnetwork, which leads to 30 W total (3072 Gb/s × 5 × 2 mW/Gb/s). For opticalcommunication, it consumes about 0.08 mW per 1 Gb/s, and there is 1 hopping per
communication, which leads to 0.25 W total (3072 Gb/s × 1 × 0.08 mW/Gb/s).
Considering TDP 200 W for Intel’s 28 core CPU, 30 W takes a significant portion thatcan limit CPU performance. Since optical communication has 3 cycles of latency per
50 Gb/s (2 cycles for O/E and E/O conversion and 1 cycle for traversal), a total of
184 cycles of latency occurs (3072 Gb/s × 3 / 50 Gb/s). For electrical communications,
there is an average of 5 cycles of latency per 3 Gb/s (5 hops per communication and
3 GHz clock frequency), resulting in a total of 5,120 cycles of latency(3072 Gb/s × 5/3 Gb/s). In other words, a time gain of approximately 5,000 cycles isobtained, which is reflected in the cache coherence.
Fig. 2.45 shows an example of a fat-tree network architecture connecting 1,024 cores by
optical communication. Each part (P1, P2, etc.) represents a cluster consisting of eightcores which may include a memory controller or a GPU that connects to the opticalcommunication. The controller unit (CU) is a 32-port switch (16 ports down, 16 ports up)
with the same function as the InfiniBand of a supercomputer. There are 8 CUs, each CU
having 16 parts, and the total number of cores is 1,024 (8×16×8). The eight cores in eachpart are connected by the same electrical ring bus as Intel's desktop CPUs. If the CPU
supports 3 GHz and 64 bit, the optical lines are assumed a bandwidth of 512 Gb/s
bidirectionally as before. Communication between all CUs is allocated two optical lines,
and four optical lines are allocated between adjacent CUs. For example, the bandwidth of
2,048 Gb/s bidirectionally is allocated between CU1 and CU2, and bandwidth of
1,024 Gb/s bidirectionally is allocated between CU1 and CU3 ~ CU8. Only the opticallines starting from CU1 are shown in the Fig. 2.45, but the same optical lines repeat for157

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CU2 to CU8. The CPU cores, the optical waveguides, and the VCSELs/PDs are arranged
vertically on different layers and can overlap in the same area. Therefore, if the CPU isdesigned as the area of eight cores < 50 mm2, then each part having eight cores can containits optical waveguides and optical devices within its area, and the total size of the wafer
containing 1,024 cores is calculated 8 cm × 8 cm (50 mm2×1024/8 = 6400 mm2). The
current supercomputer with the highest performance has a fat-tree architecture withbidirectional bandwidth 200 Gb/s. Our network architecture not only has a bandwidth thatis at least twice as fast but also maintains a line width of 64 bit/cycle between all cores.
The aggregate bandwidth for all the optical communication lines is 98 Tb/s
(8×16×512+32×1024). The aggregate bandwidth of ATAC is 8.192 Tb/s and that ofCorona is 160Tb/s. Since the bandwidth for individual communication lines is twice as
fast, most of the remaining characteristics are similar to or superior to ATAC except for
the latency that occurs from passing through the electrical switch, CU once (3 cycles) or
twice (6 cycles). As before, based on the aggregate bandwidth, the power consumption
and latency between optical and electrical communications are compared. For electricalcommunication, when 1,024 cores are connected by a 32×32 metric mesh network, anaverage of 32 hops/communication is assumed. Except for the average of 4 hops frommovement within the cluster, about 28 hops correspond to the optical communication
distance. For optical communication, there are 2 or 3 hops for traversals through opticallines. Since latency has a minimum of 7 cycles (2 cycles for 2 O/E, 2 cycles for 2 E/O,
2 cycles for 2 traversals, 1 cycle for 1 routing within a CU) and a maximum of 12 cycles(3 cycles for 3 O/E, 3 cycles for 3 E/O, 2 cycles for 2 traversals, 2 cycles for 2 routing,
2 cycles for a far traversal from CU to CU), an average of 9 cycles can be assumed. The
power consumption for electrical communication is 5,488 W (98 Tb/s × 28 × 2 mW/Gb/s)
and for optical communication 23.5 W (98 Tb/s × 3 × 0.08 mW/Gb/s). The aggregate
latency for electrical communication is 915,000 cycles (98 Tb/s × 28 / 3 Gb/s) and for
optical communication 18,000 cycles (98 Tb/s × 9 / 50 Gb/s). More than these numericaldifferences, the greater difference is that, while the heat generation for the opticalcommunication is within the controllable range, it is out of range for the electricalcommunication as considering the wafer size and the heat dissipation rate, which was
already experienced in supercomputers.

Fig. 2.45. Fat-tree network with 1,024 cores.

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Chapter 2. CPU Technology Trend and Optical Interconnects

Fig. 2.46 shows a 2.5D package of CPU with the interposer chip replaced by an optical
network chip (ONC), as compared with Fig. 2.24. The 2.5D interposer chip made of asilicon wafer is manufactured by the CMOS process. The optical waveguide and opticaldevices are formed on the bottom side of the interposer, and the flip-chip bond electrodes
are formed on the top side, and the top and bottom sides are connected by TSVs. On thetop, the necessary chips such as CPU, DRAM, GPU, and SoC are integrated by flip-chipbonding. For the bottom side, the proper heights and placements for the optical devices
are designed on the package substrate. The interposer upgraded as an optical network chipis manufactured independently by a third party company. When the corresponding chips
are supplied by CPU and memory companies, they can be integrated on the ONC at the
package stage. In addition to x86 server CPUs, as technology and demand grow,smartphone SoCs can also be integrated on the ONC to communicate optically with eachother. In case that the yield of individual core is 90 % in CPU manufacturing, theprobability that all 28 cores will be successful drops to 5 %. Therefore, it is much more
economical to manufacture and integrate CPU chips with 4~8 cores than to manufacture
CPU chips with tens to hundreds of cores.

Fig. 2.46. 2.5D package and optical network chip.

As shown in Table 2.12, Intel's 28-core server CPU ranges from $10,000~$18,000 and
8-core CPU for desktop ranges from $440~$600. The production cost for opticallyconnected 32-cores of CPU is estimated. For 2.2 GHz and 64 bit CPU, since the minimum
bandwidth for an optical line is 140.8 Gb/s, 25 Gb/s per lane and 6 lanes for 150 Gb/s may
be used for the configuration shown in Fig. 2.43. Based on the current prices inTable 2.12, the cost for optical devices with six lanes is $245 (duplex four lanes$162.9 × 6/4). The total cost of the optical devices for six lines is $1,470. Subtracting thecost of the AOC's package, the actual cost is assumed at most $1000. Excluding VCSELs,
PDs, and prisms, the price of an ONC wafer is estimated to be less than $1,000. The opticalwaveguide, LD/PD driver, TIA, TSV and electrical lines are arranged on the ONC wafer
by the CMOS process. Assuming that the 8 core CPU die before package costs $300, theprice for 4 chips, 32 cores is $1200. As added together, the cost is at most $3,200 whichis less than 1/3 of the 28-core server CPU that is currently on sale in 2021. Considering
the power consumption and latency, it has superior performance and pricecompetitiveness. For smartphones, numerous electric devices on the motherboard such as
SoC, 5G modem, SDRAM, external GPU, and NPU can be placed on the ONC. This will
not only be advantageous in the overall hardware size but also in price competitiveness.
As you can see from ATAC, Corona, and the above calculations, 4~16 are suitable
numbers for cores. Connecting multiple clusters through optical communication is
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beneficial in increasing CPU performance and lowering costs, and this is how
supercomputers are already being designed.
Table 2.12. Intel CPU and 100 Gb/s QSFP.
Intel CPU
100 Gb/s QSFP
CPU
i9-9980HK Xeon Platinum 8276M Form factor QSFP28 AOC Duplex
Number of cores
828
Wavelength
850 nmFrequency
2.4 GHz
2.2 GHz
Data rate/lane
25 Gb/s
Photolithography
14 nm
14 nm
Length25 m
Price as of 2021
$ 583
$ 11,722
Price as of 2021
$ 162.9

2.8. Conclusion
In order to improve the performance of CPUs, various technologies such as many cores,
multi-threads, parallel computing, superscalar, vector, out-of-order, branch prediction,and scheduling are applied to the CPU design. In particular, one of the most essential
technologies that enable parallel computing is cache coherence. Communication speeds
between cores play an important role for cache coherence, and a lot of research projects
to introduce optical signals into CPUs has been actively conducted, but there have notbeen successful results yet.
Since the early 2000s, leading semiconductor companies such as Intel, IBM, HP, Sun
Microsystems and Samsung have invested a large number of research funds and humanresources to apply the optical communication technology to core-to-core or CPU-tomemory communication of semiconductor chips. Universities and corporations such as
MIT, Columbia University, HP and Sun Microsystems have proposed the networkarchitecture for which optical communication technology will be applied, but it has notreached a status of experimentally integrating and testing on actual chips. The main causeof failure is that a silicon material has been used as the optical waveguide.
Among the materials used in the CMOS process, there are no materials other than silicon
nitride (Si3N4), silicon oxynitride (SiOxNy), silicon dioxide (SiO2), and silicon (Si) thatcan be used as an optical waveguide. Since silicon dioxide (SiO2) is used for the claddingof the waveguide and if we don't use silicon, then the remaining candidate materials are
only silicon nitride (Si3N4) or silicon oxynitride (SiOxNy). It is suggested that the causesof failure can be overcome by the three technologies and a possible network architecture
based on these two materials.
To implement the optical network architecture on CPU chip, it would be the most
realizable to use VCSELs, PDs, and the fat-tree network architecture that are alreadyutilized in supercomputers with replacing just the optical fiber with PECVD SiN/SiON
optical waveguide. As this architecture is implemented in the 32 core server CPU with the
2.5D ONC package, it would allow a price that is 1/3 lower than the current price of Intel's
server 28-core CPU. The three technologies which are necessary to realize the architecture160

Chapter 2. CPU Technology Trend and Optical Interconnects

have been developed and experimentally verified by Oprocessor Inc. If technology
advances in the direction expected by optical network designers, then in addition to the
CPU and memory, ONC will be the most important component of computers.

References[1]. D. Page, A Practical Introduction to Computer Architecture, Springer, New York, NY, 2009.
[2]. M. Brorsson, Multi-core and many-core processor architectures, Chapter 2, in Programming
Many-Core Chips (A. Vajda, Ed.), Springer Science+Business Media, Berlin, Germany,
2011.
[3]. J. Aad van der Stee, Overview of Recent Supercomputers, NCF/HPC Research, Netherlands,
2008.
[4]. ARM Architecture Reference Manual; ARMv8, for ARMv8-A Architecture Profile, ARM
Company, 2013-2019.
[5]. ARM Compiler Version 6.6; User Guide, ARM Company, 2013-2016.
[6]. E. Blemd, J. Menon, K. Sankaralingam, A detailed analysis of contemporary ARM and x86
architecture, in Proceedings of the 19th IEEE International Symposium on High Performance
Computer Architecture (HPCA’13), 2013.
[7]. A. La Manna, Outlook for 3D and 2.5D chips in smartphones, Chip Scale Review, Vol. 19,
Issue 2, Mar. 2015, pp. 9-11.
[8]. J. Turley, Introduction to Intel Architecture, White Paper, Intel Corporation, 2014.
[9]. An Introduction to the Intel QuickPath Interconnect, White Paper, Intel Corporation, 2009.
[10]. A. Kumar, New Intel Mesh Architecture: The ‘Superhighway’ of the Data Center, White
Paper, Intel Corporation, 2017.
[11]. M. Berktold, CPU Monitoring with DTS/PECI. White Paper, Intel Corporation, 2010.
[12]. Thermal/Mechanical Specification and Design Guide (TMSDG); Intel Core i7 Processor
Family for the LGA2011-3 Socket. White Paper, Intel Corporation, August 2014.
[13]. S. Strande, M. Mclaughlin, Building a Data – Intensive Supercomputer Architecture for the
National Research Community. White Paper, Cray Inc., 2013.
[14]. W. Hu, Y. Zhang, J. Fu, An introduction to CPU and DSP design in China, Science China
Information Sciences, Vol. 59, Jan. 2016, 012101.
[15]. J. Dongarra, Report on the Sunway TaihuLight System, Tech Report UT-EECS-16-742,
University of Tennessee, 2016.
[16]. G. Kurian, J. E. Miller, J. Psota, J. Eastep, J. Liu, J. Michel, L. C. Kimerling, A. Agarwal,
ATAC: A 1000-Core cache-coherent processor with on-chip optical network, in Proceedingsof the 19th International Conference on Parallel Architecture and Compilation Techniques(PACT’10), September 11-15, 2010, pp. 477-488.
[17]. D. Vantreas, R. Schreiber, M. Monchiero, M. McLaren, N. P. Jouppi, M. Fiorentino,
A. Davis, N. Binkert, R. G. Beausoleil, J. Ho Ahn, Corona: System implications of emerging
nanophotonic technology, in Proceedings of the 35th Annual International Symposium on
Computer Architecture (ISCA’08), 2008, pp. 153-164.
[18]. A. Biberman, K. Preston, G. Hendry, N. Sherwood-droz, J. Chan, J. S. Levy, M. Lipson,K. Bergman, Photonic network-on-chip architectures using multilayer deposited siliconmaterials for high-performance chip multiprocessors, ACM Journal on Emerging
Technologies in Computing Systems, Vol. 7, Issue 2, 2011, 7.
[19]. A. V. Krishnamoorthy, X. Zheng, J. Lexau, I. Shubin, Computer systems based on silicon
photonic interconnects, Proceedings of the IEEE, Vol. 97, Issue 7, Jul. 2009, pp. 1337-1361.
[20]. C. Sun, M. T. Wade, Y. Lee, J. S. Orcutt, L. Alloatti, M. S. Georgas, A. S. Waterman,
J. M. Shainline, R. R. Avizienis, S. Lin, B. R. Moss, R. Kumar, F. Pavanello, A. H. Atabaki,

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H. M. Cook, A. J. Ou, J. C. Leu, Y.-H. Chen, K. Asanović, R. J. Ram, M. A. Popović,
V. M. Stojanović, Single-chip microprocessor that communicates directly using light, Nature,
Vol. 528, Issue 24, Dec. 2015, pp. 534-538.
[21]. M. I. Alayo, D. Criado, L. C. D. Goncalves, I. Pereyra, Deposition and characterization ofsilicon oxynitride for integrated optical applications, Journal of Non-Crystalline Solids,
Vols. 338-340, pp. 76-80.
[22]. R. Jones, O. Cohen, H. Chan, D. Rubin, A. Fang, M. Paniccia, Integration of SiON gratings
with SOI, in Proceedings of the IEEE International Conference on Group IV Photonics 2 nd
(GFP’05), Sept. 2005, pp. 192-194.
[23]. T. Tsuchizawa, T. Watanabe, K. Yamada, H. Fukuda, S. Itabashi, J. Fujikata, A. Gomyo,
J. Ushida, D. Okamoto, K. Nishi, K. Ohashi, Low-loss silicon oxynitride waveguides andbranches for the 850 nm wavelength region, Japanese Journal of Applied Physics, Vol. 47,
Issue 8, Aug. 2008, pp. 6739-6743.
[24]. B. S. Sahu, O. P. Agnihotri, S. C. Jain, R. Mertens, I. Katok, Inﬂuence of hydrogen on losses
in silicon oxynitride planar optical waveguides, Semicond. Sci. Technol., Vol. 15,
pp. L1-L11.

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Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré
Beams and a CCD Camera

Chapter 3

Effects of Intensity Inaccuracy
Measurements in Mueller Matrix
Polarimetry with Full Poincaré Beams
and a CCD Camera
Juan C. Suárez-Bermejo, Juan C. González de Sande,
Massimo Santarsiero and Gemma Piquero1

3.1. Introduction
Polarimetry is a well-established technique that has a large number of applications. The
primary goal of polarimetry is the characterization of the optical behavior of samples whenlight is transmitted through or reflected by the surface of the sample [1]. Different
materials, for example, technological materials or biological tissues, can interact with lightin different ways. From the changes produced in the polarization state of the beam, theactual optical properties of the samples can be obtained. In this way, it is possible toidentify some characteristics or properties of the sample that may be related to the specificbehavior of the technological material or to some pathologies in biological tissues [2, 3].
The state of polarization of a light beam can be described by its Stokes parameters thatare usually arranged in a 4×1 column vector [1, 4]. These parameters are measurable
quantities. If a polarized light beam passes through a given sample, the state ofpolarization changes depending on the sample characteristics. The linear optical sample
̂ , that relates the Stokes
characteristics can be represented by a 4×4 Mueller matrix, 𝑀
vector of the input and of the output beam. The sample is usually tested with a uniformly
and totally polarized input beam with a Stokes vector that is sequentially varied to reachat least four linearly independent states of polarization to measure the sample Mueller
matrix [1, 5-9].
Many nonuniformly totally polarized beams contain four or more independent states ofpolarization [10, 11], so they can be used as parallel polarization state generator (PSG) in

Juan C. Suárez-Bermejo, Department of Materials Science, Universidad Politécnica de Madrid, Spain

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a Mueller matrix polarimetric system [12-14]. In particular, the so-called full Poincaré
beams (FPBs) present all possible states of polarization across their transverse section [15,16]. A simple method to obtain a specific FPB has been recently described [17, 18] andthis beam has been used to determine the Mueller matrix of a sample by measuring thechange of the state of polarization at only four specific points of its transverse section byusing a commercial Stokes polarimeter [14, 19].
This FPB has been proposed for Mueller matrix polarimetry by measuring the change ofthe state of polarization in the whole transverse section of the beam by means of a CCD
camera instead of measuring it at only four points with a commercial polarimeter [20].
The states of polarization before and after the sample are measured by means of apolarization state analyzer (PSA) consisting of a quarter wave phase plate and a linear
polarizer that are positioned before the CCD camera in six different configurations. An
image of the transmitted intensity is recorded for each of the PSA configurations with andwithout sample. From these recorded images, many combinations of at least four different
pixels can be used to determine the Mueller matrix of the sample [20].
The goal of this work is to present different strategies for recovering the Mueller matrix
of a sample from simulated intensity images. A comparison among different strategies toprocess this set of data will be presented. This work will allow to determine the best
strategy and to obtain an estimation of the errors in the measured Mueller matrix when
only the inaccuracy of the intensity values at each pixel is considered as the main
source of error.
This section constitutes the Introduction. In Section 3.2 the FPB Polarimetry method is
described. To study the errors of the intensity measurements, the characterization anddifferent tests carried out for our CCD camera are described in Section 3.3. In Section 3.4
the different strategies with the considered errors in the measurements are presented. In
Section 3.5 the main results and discussion is presented through the comparison among
the previous forms of data treatment. Finally, the main conclusions are remarked in
Section 3.6.

3.2. Polarimetry with a Full Poincaré Beam and a CCD Camera
The Stokes vector at a given point in the transverse section of a light beam, 𝑆⃗𝑖𝑛 (𝑥, 𝑦), canbe obtained by measuring the light intensity at such a point after a linear polarizer (P) with
its transmission axis oriented at four different angles, namely, 0°, 45°, 90°, 135° relative
to the x-axis of a suitable reference system (they will be denoted as I0, I45, I90, I135,
respectively); and after the combination of a quarter wave phase plate (QWP) with its fastaxis along the horizontal axis and the linear polarizer oriented at 45° and 135° (denoted
as IQWP,45 and IQWP,135) [4, 21]. This system constitutes an optimum PSA when six intensity
measurements are taken [1, 22] and is sketched in Fig. 3.1(a). By recording these
intensities with a CCD camera, a map of the Stokes parameters across the section of anonuniformly polarized beam can be obtained as

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Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré
Beams and a CCD Camera𝑖𝑛 (𝑥,
𝐼0𝑖𝑛 (𝑥, 𝑦) + 𝐼90
𝑦)𝑆0𝑖𝑛 (𝑥, 𝑦)𝑖𝑛 (𝑥,𝑖𝑛 (𝑥,𝑖𝑛 (𝑥,
𝐼0𝑦) − 𝐼90 𝑦)𝑆𝑦)𝑆⃗𝑖𝑛 (𝑥, 𝑦) ≡ 1𝑖𝑛=.
𝑖𝑛 (𝑥,𝑖𝑛 (𝑥,
𝐼45 𝑦) − 𝐼135𝑦)
𝑆2 (𝑥, 𝑦)𝑖𝑛
𝑖𝑛[𝑆3𝑖𝑛 (𝑥, 𝑦)][𝐼QWP,45 (𝑥, 𝑦) − 𝐼QWP,135 (𝑥, 𝑦)](3.1)

Fig. 3.1. Scheme of Mueller matrix determination using a Full Poincaré beam and a CCD camera.
(a) Parallel polarization state generator (PSG) by means of a specific FPB and polarimetriccharacterization through a PSA; (b) theoretical images for six specific configurations of the PSA;(c) the same images as in (b) with added Gaussian noise (AGN); (d) noisy Stokes parameter maps
for the FPB; (e) polarimetric characterization of the generated FPB after passing through a sample
̂𝑇 ; (f) theoretical images for six configurations of the PSA;
represented by a Mueller matrix 𝑀(g) the same images as in (f) with AGN; (h) noisy Stokes parameter maps for the output beam.

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In the case of a FPB, the whole Poincaré sphere [1, 4], is mapped at least once across thebeam section. If this light beam passes through a given sample, the state of polarization ateach point changes depending on the sample characteristics. Then the sample can be tested
for all polarization states at once when a FPB impinges on it. The Stokes parameters of
the output beam, 𝑆⃗𝑜𝑢𝑡 (𝑥, 𝑦), can also be obtained by measuring six images as for the input
beam [see Fig. 3.1 (e)]. The linear optical characteristics of the sample can be represented
̂ , that relates the Stokes vector of the input and output beam
by a 4×4 Mueller matrix, 𝑀aŝ 𝑆⃗𝑖𝑛 (𝑥, 𝑦).
𝑆⃗𝑜𝑢𝑡 (𝑥, 𝑦) = 𝑀(3.2)

An advantage of using a FPB as input beam is that all input states of polarization can betested at the same time. By repeating Eq. (3.2) at four points whose states of polarization
are linearly independent, the Mueller matrix of the sample can be recovered [14, 19, 20].
However, if the Stokes parameters are measured with a CCD camera before and after the
sample, many combinations of four different pixels can be used to determine the Mueller
matrix of the sample.
The theoretical Stokes vector at any point (𝑟, 𝜃) of the beam section at the exit of auniaxial crystal illuminated with a divergent quasi-monochromatic beam [see Fig. 3.1(a)]can be approximated by [14, 19]1−[cos2 2𝜃 + sin2 2𝜃 cos 𝛿(𝑟)]𝑆⃗(𝑟, 𝜃) = 𝑆0 (𝑟, 𝜃) × [],−sin2𝜃 cos2𝜃[1 − cos 𝛿(𝑟)]sin2𝜃 sin 𝛿(𝑟)(3.3)

being 𝛿(𝑟) a phase given by𝑟2𝑙𝛿(𝑟) = 𝑘(𝑛𝑒 − 𝑛𝑜 ) ,(3.4)

where 𝑘 is the vacuum wave number, 𝑙 is the crystal length, and 𝑛𝑜 and 𝑛𝑒 are theordinary and extraordinary refractive indexes of the uniaxial crystal, respectively.
It has been shown that for this particular beam, all possible totally polarized states ofpolarization can be found across the beam section inside a semicircle with radius [14, 17].
𝜋𝑙𝑟𝑀 = √𝑘|𝑛 −𝑛 |.
𝑒

𝑜

(3.5)

A possible way to recover the Mueller matrix consists in using the Stokes vector values
at four different pixels (the same set for the input and output maps). The selected pixels
must be chosen in such a way that the four corresponding input Stokes vectors be linearly
independent (we recall that the coordinates of a point across the beam transverse section
presenting a specific state of polarization can be obtained using specific analyticalexpressions [14, 17]). The optimum choice of the input states of polarization is such that166

Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré
Beams and a CCD Camera

they form a regular tetrahedron inscribed in the Poincaré sphere [see Fig. 3.2(a)] [22]. The
corresponding polarization states are found at the locations shown in Fig. 3.2(b) [14, 17].

(a)

(b)

Fig. 3.2. (a) Poincaré sphere with a set of optimum polarization states for polarimetricmeasurements; (b) Polarization map across the beam section and points where the polarization
states represented in a) are found. Red (blue) ellipses represent right-handed (left-handed)polarization states. The semicircle has radius rM given in Eq. (3.4). All possible polarization states
are found in this semicircle.

A continuous rotation of the tetrahedron around the s1 axis yields a movement of the threepoints out of the center along the green ovoids in Fig. 3.2(b). The recovered Mueller
matrix can be found by repeating the inversion of Eq. (3.2) for many sets of four selected̂𝑆 ,pixels and averaging results. The differences between the calculated Mueller matrix, 𝑀
̂and that corresponding to the ideal sample, 𝑀𝑇 , will be evaluated.
For this approach, a critical point is the inaccuracy of the intensity measurement at eachpixel of the CCD camera. In the next section, we analyze the noise introduced by the CCD
camera and justify the Gaussian distribution considered for the added noise in thesimulations.

3.3. Noise in CCD Images
Noise in a camera arises from the aggregate spatial and temporal variations in the captured
images. If a stabilized light source is used, the measured signals for every individual pixel
vary according to different sources of noise. The main sources of noise due to the CCD
camera are the dark shot noise, read noise, photon shot noise, and fixed pattern noise [23].
The shot noise is due to the current that appears even in the case of no photons reaching
the camera, it has thermal origin and is uncorrelated to the signal level. The read noise isrelated to the production of the electronic signal in the CCD camera, depends on the sensor
and electronics design, is more important for faster CCD pixel clock rates, but it isindependent of the signal level and temperature of the sensor. The photon shot noise is167

Advances in Optics: Reviews. Book Series, Vol. 5

associated with the statistical uncertainty due to the arrival of individual photons to thedetector, it depends on the signal level for every individual pixel, but independent of thesensor temperature. Finally, the fixed pattern noise has its origin in the spatial nonuniformity of the pixel array in the CCD sensor, it is independent of the signal level ortemperature of the sensor, and usually negligible for high quality sensors. The total
effective noise per pixel could be estimated as the quadrature sum of the noise
sources [24, 25].
When acquiring images with the CCD camera, several adjustments can be done toimprove the quality of the images while they are being captured by the sensor and control
software of the CCD camera, previously to other treatments of data.
Here we analyze the noise in the acquisition of images by means of a Spiricom SP620U
CCD camera RoHS, with 4.4 µm square pixels. It has an active area of 7.1 mm wide by
5.4 mm height, and a number of 1600 × 1200 active pixels, that is, 1.92 Mpx. The
spectral response is in the range of 190 – 1100 nm. The minimum system dynamic range
is 62 dB, and linearity with power is ±1 %. Saturation intensity in optimal conditions –camera is set to full resolution at maximum frame rate and exposure time, running CW at
632.8 nm wavelength – is 2.2 µW/cm2.
Background noise due to the faint light in the room – coming from the computer screen
and other dim sources but shutting down the laser output – has been recorded andsubtracted from every recorded image for subsequent computations. The average powerof the background noise is approximately 62 levels out of a maximum 4096 at saturation
with a standard deviation of 2 levels. It can be observed that this background is less than
2 % of the maximum level and is quite uniform across the whole CCD.
Afterwards, to estimate the image errors, the configuration shown in Fig. 3.1(a) without
the QWP and with the transmission axis of the linear polarizer P oriented vertically (90º)
is used to acquire a set of 25 images of the beam cross-transverse section. Thisconfiguration of the PSA is selected because is the one for which the light received by the
CCD camera when measuring the Stokes parameters of the input FPB is maximum.
Selected frame rate has been the lowest value allowed by the equipment (1.850 Hz) tominimize the errors due to the recording of the images. Exposure time of 40 ms has beenused to get the maximum intensity level without reaching saturation in the sensor. All
images show quite similar intensity patterns.
The next step is to choose one out of the 25 stored images and compare it with the averageof the remaining 24 images. This procedure is iterated 25 times, selecting a different image
for every iteration, and comparing with the average of the remaining ones. One of these
iterations is presented in Fig. 3.3, where the left-hand side shows one individual registeredimage and the right-hand side shows the average of the 24 remaining images. For bothcases, the background has been subtracted.
The accumulated noise at each pixel, computed as the difference between one of theimages and the average of the 24 remaining images is shown in Fig. 3.4. Here we show afront view of these differences along either the 1200 rows or 1600 columns of the CCD
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Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré
Beams and a CCD Camera

sensor, so the maximum and minimum difference of the pixels located on each column
can be easily visualized. It can be observed that the computed noise level is higher for
higher intensity values registered at every individual pixel (in the 400-800 row range and
700-1100 columns range). It should be noted that the computed noise includes not only
the noise due to the CCD camera but also the possible temporal fluctuations of the lightsource intensity at each pixel.

Fig. 3.3. Example of the procedure followed to estimate the image noise: (a) individual recorded
image and (b) average image obtained from the 24 remaining images.

Fig. 3.4. Difference at each individual pixel between the first recorded image and the averageof the 24 remaining ones, (a) Row view, (b) Column view.

A histogram of the differences shown in Fig. 3.4 has been done for the 1.92 Mpx present
in each image and is shown in Fig. 3.5. The best fit to a single Gaussian distribution (bluesolid line), has been obtained and some deviation from the experimental data can beobserved, specially for the tails. The R-square parameter of this fit is 0.964. However,when a superposition of two Gaussian distributions is considered (red dashed line), thehistogram is accurately fitted, with an adjusted R-square parameter of 0.998 for thisexample. Similar results are obtained for the rest of the images.
The fact that a better fit is obtained with two Gaussian distributions could be due to acorrelation between the differences and the intensity level measured at each pixel.
Fig. 3.6(a) shows a representation of the difference level versus the intensity levelmeasured at each pixel. That is, the average intensity level measured at each pixel(intensity value shown in the right-hand side of Fig. 3.3) is represented along thehorizontal axis and the difference between the intensity at that pixel in the image of the169Advances in Optics: Reviews. Book Series, Vol. 5

left-hand side of Fig. 3.3 and the average intensity is represented along the vertical axis.
It can be observed that the distribution of these differences broadens for higher and higher
mean measured intensities, so the behavior is slightly different depending on suchintensity level. There is a slight skew in the average towards negative values (respect zerodifference level) for higher intensity levels.

Fig. 3.5. Histogram of the number of pixels with a given level of difference between the firstrecorded image and the average of the 24 remaining ones. Black dots are the total number of pixels
whose intensity difference is within a given range. Solid line: best fit to a Gaussian distribution.
Dashed line: best fit to a superposition of two Gaussian distributions.

From the graph in Fig. 3.6(a), a ±2 % wide band around a given intensity level has been
selected for several measured intensity levels (500, 1000, 1500, 2000, 2500, 3000, and3300). The vertical shaded bands indicate the one around 500 and 3300 intensity levels,
respectively. A histogram of the differences has been built for each of these bands.
Fig. 3.6(b) shows the obtained histograms for the bands corresponding to 500 and
3300 measured levels (both histograms include pixels where the measured intensity is
within 2 % of the selected center value). The histogram for each band is fitted very wellby a single Gaussian distribution. The R-values of the fits are always over 0.993.
The standard deviations for the fits in the right-hand side of Fig. 3.6 are 6.5 (500-levelband) and 21 (3300-level band), respectively. This process is repeated for each of theseven mentioned bands. The results of the fits are shown in Fig. 3.7. In the left side ofFig. 3.7, the standard deviation is represented versus the intensity level whereas in theright side, the standard deviation normalized to the intensity level is represented as afunction of the intensity level.
It is observed that the standard deviation grows steadily with the level of the measured
intensity as it could be expected. The dashed line in Fig. 3.7(a) is the best linear fit
to this data.
Finally, the behavior followed by the normalized value of the standard deviation of theintensity level of the band for each band is depicted in Fig. 3.7(b). It can be observed that
this relative standard deviation is around or below 0.01.
170Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré
Beams and a CCD Camera

Fig. 3.6. (a) Difference between the first recorded image and the average of the 24 remaining ones
versus the intensity level measured at each pixel; (b) Histogram of the number of pixels with agiven level of difference between the first recorded image and the average of the 24 remaining ones
for pixels whose measured intensity level is within 2 % of 500; (c) same as (b) for 3300 intensity
level band. Black dots are the total number of pixels whose intensity difference is within a given
range. Solid line is the best fit to a Gaussian distribution.

Fig. 3.7. Absolute (a) and relative (b) standard deviation for each intensity level band. The blueline is (a) best linear fit to the data (b) a guide for the eye.

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This number will be used in the simulations presented in the following section. We will
try to estimate the expected errors in the determination of the Mueller matrix of a sample,
when the polarimetric measurements are carried out using a FPB as a parallel PSG, the
PSA sketched in Fig. 3.1, and a CCD camera with similar characteristics than the studied
in this work for. It should be noted that the simulation results that will be presented in thefollowing section, including levels of noise above and below obtained for this camera.

3.4. Strategies for Obtaining the Mueller Matrix
Besides the errors due to imperfections or errors in the positioning or the optical elementsof the PSA (polarizer and quarter wave phase plate), an important source of error is theinaccuracy in the measurement of the light intensity at each pixel of the acquired images,[22, 26-28]. To estimate how this source of error affect to the Mueller matrix
measurement, the following simulations will be carried out. The intensity maps that would
be obtained when the considered FPB passes through the PSA for six different
configurations are theoretically calculated [see Fig. 3.1 (b)]. This calculation is carried out
by using Eq. (3.3) and the corresponding Mueller matrix of the specific PSA
configuration, considering that 𝑆0 (𝑟, 𝜃) presents a Gaussian shape with a spot size twice
the value of 𝑟𝑀 [see Eq. (3.5)]. Then, a random Gaussian noise is added to each of thesecalculated images [see Fig. 3.1(c)]. The amplitude of this added Gaussian noise (AGN) at
each point presents a standard deviation that is proportional to the calculated intensity atsuch point. From these noisy images, the maps of Stokes parameters for the input beam
are calculated [see Fig. 3.1 (d)]. The same procedure is followed when a sample (with â𝑇 ) is inserted before the PSA [see Fig. 3.1 (e)-(h)]. Finally, fromknown Mueller matrix 𝑀
̂𝑆 , isthe input and output Stokes parameter maps, the Mueller matrix of the sample, 𝑀evaluated. This simulation process has been repeated Ns = 5 times and the average of the
final results has been taken. A quantitative estimate of the errors in the determination of
Mueller matrix is given as the root mean square of the elements of the differencê𝑇 − 𝑀
̂𝑆 .
𝑀
Two different approaches were considered for recovering the Mueller matrix of thesample from the simulated Stokes parameters: the center point and the nearest neighbour
strategy. Let us briefly describe both approaches.
3.4.1. Center Point Strategy
The first approach is an adaptation of the procedure followed in reference [14, 20]. Fourpixels are chosen in such a way that the four corresponding input Stokes vectors
correspond to the vertices of a tetrahedron inscribed in the Poincaré sphere [red dots in
Fig. 3.2(a)]. The process is repeated for several orientations of the tetrahedron.
Note that, for the chosen configuration of the FPB, the linear vertical polarization state,
given by the Stokes vector
𝑆⃗𝑖𝑛 (𝑟0 , 𝜑0 ) = [1, −1, 0, 0]𝑇 ,
172

(3.6)

Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré
Beams and a CCD Camera

can be found at the FPB axis. This point is always taken because the intensity is maximum
there. Several sets of the three remaining points across the transverse plane of the beam
have been selected in such a way that their polarization states correspond to rotations of
the tetrahedron inscribed in the Poincaré sphere around the axis s1 [see Fig. 3.2 (a)][14, 17, 19, 22]. Due to its symmetry, the rotation of the tetrahedron can be limited to the𝜋 𝜋interval [− 6 , 6 ) .
The coordinates of the four considered points are mapped on the CCD images (red pixels
in Fig. 3.8, for one of the choices), both for the input beam [without the sample and with
AGN, Fig. 3.1 (d)], and for the output beam [after passing through the sample, and with adifferent pattern of AGN, Fig. 3.1 (h)]. Theoretical matrix for every ideal sample iscompared to the matrix obtained from simulations and the differences are derived for̂ . An average estimate of the error is obtained from 15 analyzed sets,
every element of 𝑀
corresponding to 15 equally spaced tetrahedron orientations, and the uncertainties of thesevalues are computed.

Fig. 3.8. Red squares represent pixels that contain the point where the ideal FPB presents the statesof polarization corresponding to the vertexes of the tetrahedron of Fig. 3.1a. They are the ones used
for the center point strategy. Green squares together the center one represent the 3×3 nearest
neighbor. Any combination of four of these pixels around the red ones is used for the nearest
neighbor strategy. Shaded semicircle represents a possible area of the beam cross section where all
possible states of polarization are found.

3.4.2. Nearest Neighbor Strategy
The second strategy uses a wider set of points, all located around each of the center pointconsidered in the previous case. A total of 𝑁 × 𝑁 nearest neighbor pixels are consideredaround every center point (see Fig. 3.8). Mueller matrices are obtained for all possible
four-point sets (input and output) that can be chosen from the 𝑁 2 × 4 pixels included in
the analysis. That means a total of (𝑁 2 )4 possible combinations. For all of them, thecondition number for the equation system formed by evaluating Eq. (3.2) at four different
locations is computed and the resulting Mueller matrix is discarded if such condition
number [1] is over 20. An average value is then computed from all valid cases, for the
15 analysed orientations of the tetrahedron in the Poincaré sphere.

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Advances in Optics: Reviews. Book Series, Vol. 5

3.5. Results from the Simulation
Mueller matrices are obtained from the simulated Stokes input and output vectors for four
different standard samples: air, a linear polarizer with horizontal transmission axis (LP0)and with its transmission axis at 60º with respect to the horizontal (LP60), and a quarterwave phase retarder with its fast axis along the horizontal direction.
Fig. 3.9 shows the root mean square value of the difference between the mean matrix
obtained for the Ns simulations at each orientation of the tetrahedron and the theoretical
̂𝑆 〉 − 𝑀
̂𝑇 ), for each of the four samples considered.
one, 𝑟𝑚𝑠(〈𝑀Fig. 3.9. Individual and mean values of the rms of differences between the Mueller matrix obtained
from numerical simulation and theoretical ones, for four different ideal samples, assuming
a normalized standard deviation of the AGN amplitude  = 0.01. The four samples are air, a linear
polarizer with horizontal transmission axis (LP0) and with its transmission axis at 60º with respect
to the horizontal (LP60), and a QWP retarder with its fast axis along the horizontal direction.

The value of the root mean square difference is calculated as𝑆
𝑇 2̂𝑆 〉 − 𝑀
̂𝑇 ) = 1 √∑3𝑖,𝑗 = 0(𝑚𝑟𝑚𝑠(〈𝑀
̅ 𝑖𝑗
− 𝑚𝑖𝑗
) ,
4

(3.7)

𝑆
where 𝑚̅ 𝑖𝑗
are the mean values of the Mueller matrix elements obtained from the𝑇simulation and 𝑚𝑖𝑗
are the theoretical ones.

For these simulations, a Gaussian noise with amplitude  = 0.01 of the maximum intensityof the beam has been added to each pixel of the images in Fig. 3.1b and Fig. 3.1f. The
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Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré
Beams and a CCD Camera

results for both the center point and nearest neighbor strategies are presented. In all cases,
the nearest neighbor strategy renders a smaller value of the error compared to the center
point strategy. It is worth noting that the orientation of the tetrahedron around the 𝑠1 axis
has no influence on the obtained results.
When the maximum amplitude of the added Gaussian noise (characterized by theparameter ) is varied, the uncertainty in the errors predicted by both strategies alsochanges. Fig. 3.10 shows the average values and the error bars for seven values of  when
the sample is air: the beam does not go through any sample at all but two different
Gaussian noises are added to the images in Fig. 3.1 (b).

Fig. 3.10. Mean values of the rms of the differences between the Mueller matrix obtained
from numerical simulation and the theoretical one, for both center point and nearest neighborstrategies when the considered sample is air. Uncertainties are indicated by vertical bars.

A comparative graph for the four samples is included in Fig. 3.11(a), for the center point
strategy. The general trend is the same for all of them: the error in the determination of
the elements of the Mueller matrix grows linearly as the Gaussian noise is increased.
However, quantitatively, the errors for the case of a linear polarizer with horizontaltransmission axis are less than a half of the other three considered samples.
Similarly, a comparative graph for the four samples following the nearest neighborstrategy is shown in Fig. 3.11(b). In this case, the error in the determination of the elementsof the Mueller matrix does not grow linearly as the Gaussian noise is increased. Calculated
errors are a minimum for the case of a linear polarizer (0º), followed by the linear polarizer(60º). Mean errors in this strategy are very similar for the cases of air (no sample) andquarter wave plate linear retarder.
The final goal of these simulations is to assess the differences in the mean errors that arê during experimental
expected using the first or the second strategy for the evaluation of 𝑀
measurement using real samples. Comparison of the values in Fig. 3.11(a)-(b) for both
strategies, gives a reduction in errors using the nearest neighbor strategy, compared to thecenter point strategy. For all cases, the nearest neighbor strategy is superior in terms oferror reduction. A value of 80 % in error reduction can be noted for situations where the175Advances in Optics: Reviews. Book Series, Vol. 5

Gaussian noise parameter is  ≤ 0.02, for any of the samples. However, if higher levels ofnoise are present in the CCD images, the error reduction starts to drop steadily.

(a)

(b)

Fig. 3.11. Mean values of the rms of the differences between the Mueller matrix obtained
from numerical simulation and the theoretical ones, for (a) the center point strategy,and (b) nearest neighbor strategy.

3.6. Conclusions
In the present work, it is proposed to use a full Poincaré beam together with a simple
polarization state analyzer and a CCD camera for determining the Mueller matrix of an
unknown sample. Two different strategies are tested, which we have called the center
point strategy or nearest neighbor strategy, respectively.
From the presented simulations, it can be deduced that the information collected from theimages taken with a CCD camera for six configurations of the PSA could be useful to
reduce the errors due to the inaccuracy of the intensity measurement. The results obtained
from these simulations will be useful for deciding a measurement strategy during theexperimental development of this polarimetric method.

Acknowledgements
This work has been partially supported by Spanish Ministerio de Economía yCompetitividad under project PID2019104268GB-C21.

References[1]. R. Chipman, W-S. Lam, G. Young, Polarized Light and Optical System, CRC Press, New
York, 2018.
[2]. V. Tuchin, L. Wang, D. Zymnayakov, Optical Polarization in Biomedical Applications,
Springer, 2006.
[3]. M. Wakaki, Optical Materials and Applications, CRC Press, Boca Raton, 2013.
[4]. D. Goldstein, Polarized Light, Marcel Dekker Inc., 2003.

176

Chapter 3. Effects of Intensity Inaccuracy Measurements in Mueller Matrix Polarimetry with Full Poincaré
Beams and a CCD Camera[5]. R. Azzam, Mueller-matrix measurement using the four-detector photopolarimeter, Optics
Letters, Vol. 11, Issue 5, 1986, pp. 270-272.
[6]. D. Goldstein, Mueller matrix dual-rotating retarder polarimeter, Optics Letters, Vol. 31,
Issue 31, 1992, pp. 6676-6683.
[7]. J. Bueno, Polarimetry using liquid-crystal variable retarders: theory and calibration, Journalof Optics A: Pure and Applied Optics, Vol. 2, Issue 3, 2000, 216.
[8]. K. Twietmeyer, R. Chipman, Optimization of Mueller matrix polarimeters in the presence oferror sources, Optics Express, Vol. 16, Issue 15, 2008, pp. 11589-11603.
[9]. R. Azzam, Stokes-vector and Mueller-matrix polarimetry, Journal of the Optical Society ofAmerica A, Vol. A 33, Issue 7, 2016, pp. 1396-1408.
[10]. G. Piquero, R. Martínez-Herrero, J.C.G. de Sande, M. Santarsiero, Synthesis andcharacterization of non-uniformly totally polarized light beams: tutorial, Journal of the
Optical Society of America A, Vol. 37, Issue 4, April 2020, pp. 591-605.
[11]. J. C. G. de Sande, G. Piquero, J. Suarez-Bermejo, M. Santarsiero, Beams with propagationinvariant transverse polarization pattern, arXiv.org, arXiv:2102.00024v, 29 January 2021.
[12]. J. C. G. de Sande, M. Santarsiero, G. Piquero, Spirally polarized beams for polarimetry
measurements of deterministics and homogeneous samples, Optics and Lasers inEngineering, Vol. 91, 2017, pp. 97-105.
[13]. J. C. G. de Sande, G. Piquero, M. Santarsiero, Polarimetry with azimuthally polarized light,
Optics Communications, Vol. 410, 2018, pp. 961-965.
[14]. J. Suárez-Bermejo, J. C. G. de Sande, M. Santarsiero, G. Piquero, Mueller matrix polarimetry
using full Poincaré beams, Optics and Lasers in Engineering, Vol. 122, 2019, pp. 134-141.
[15]. A. Beckley, T. Brown, M. Alonso, Full Poincaré beams, Optics Express, Vol. 18, Issue 10,2010, pp. 10777-10785.
[16]. E. J. Galvez, S. Khadka, W. H. Schubert, S. Nomoto, Poincaré-beam patterns produced by
nonseparable superpositions of Laguerre-Gauss and polarization modes of light, Applied
Optics, Vol. 51, 2012, pp. 2925-2934.
[17]. G. Piquero, L. Monroy, M. Santarsiero, M. Alonzo, J. C. G. de Sande, Synthesis of full
Poincaré beams by means of uniaxial crystals, Journal of Optics, Vol. 20, Issue 6, 2018,065602.
[18]. G. Piquero, J. Vargas-Balbuena, Non-uniformly polarized beams across their transverse
profiles: An introductory study for undergraduate optics courses, European Journal of
Physics, Vol. 25, 2004, pp. 793-800.
[19]. J. Suarez-Bermejo, J. C. G. de Sande, M. Santarsiero, G. Piquero, Analysis of the errors inpolarimetry with full Poincaré beams, in Proceedings of the Photonics & Electromagnetics
Research Symposium – Spring (PIERS-SPRING’19), Rome, 2019, pp. 1-7.
[20]. J. Suarez-Bermejo, J. C. G. de Sande, M. Santarsiero, G. Piquero, Simulation of Mueller
matrix polarimetry with full Poincaré beams and a CCD camera, in Proceedings of the 3rdInternational Conference on Optics, Photonics and Lasers (OPAL'20), Tenerife, 2020.
[21]. R. Chipman, Polarimetry, in Handbook of Optics, McGraw-Hill, 2010.
[22]. D. Layden, F. Wood, I. Vitkin, Optimum selection of input polarization states in determining
the sample Mueller matrix: a dual photoelastic polarimeter approach, Optics Express,
Vol. 20, Issue 18, 2012, pp. 20466-20481.
[23]. K. Irie, A. E. McKinnon, K. Unsworth, I. M. Woodhead, A technique for evaluation of CCD
video-camera noise, IEEE Transactions on Circuits and Systems for Video Technology,
Vol. 18, Issue 2, 2008, pp. 280-284.
[24]. L. Chen, X. Zhang, J. Lin, D. Sha, Signal-to-noise ratio evaluation of a CCD camera, Optics& Laser Technology, Vol. 41, Issue 5, 2009, pp. 574-579.
[25]. J. Dai, F. Goudail, M. Boffety, J. Gao, Estimation precision of full polarimetric parameters in
the presence of additive and Poisson noise, Optics Express, Vol. 26, Issue 26, 2018,
pp. 34081-34093.

177

Advances in Optics: Reviews. Book Series, Vol. 5[26]. J. Zallat, M. Stoll, Optimal configurations for imaging polarimeters: impacts of image noiseand systematic errors, Journal of Optics A: Pure and Applied Optics, Vol. 8, Issue 9, 2006,
pp. 807-814.
[27]. G. López-Morales, M. M. Sánchez-López, A. Lizana, I. Moreno, J. Campos, Mueller matrix
polarimetric imaging analysis of optical components for the generation of cylindrical vector
beams, Crystals, Vol. 10, Issue 12, 2020, 1155.
[28]. N. C. Bruce, J. M. López-Téllez, O. G. Rodríguez-Herrera, Permitted experimental errors for
optimized variable-retarder Mueller-matrix polarimeters, Optics Express, Vol. 410, 2018,
pp. 13693-13704.

178

Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

Chapter 4

The Optical Bidirectional LambertianConductance Law with Applications
Antonio Parretta1

4.1. Prologue
The study of the optical properties of a generic element, or system, when subjected toLambertian irradiation, that is when each point of its optical terminal is source of lightwith constant and unpolarized radiance, is not common in the literature. However, this has
been one of our topic themes of research at the University of Ferrara for several years,
under the direction of Prof. Giuliano Martinelli (†), aimed at the design, fabrication andoptical characterization of photovoltaic type solar concentrators (SC) [1-5]. The use ofLambertian radiation in solar concentrators is unusual, as these devices are designed toconcentrate the quasi-collimated direct component of solar radiation. However, it
happened that, on the occasion of a work session dedicated to the optical characterizationof a truncated and squared compound parabolic solar concentrator [6], I illuminated, by
chance, one of them from the inside with a laser beam, while its exit opening, at thatmoment, was closed by a white Lambertian diffuser (see Fig. 4.1(a)). On that occasion, I
observed that the laser light, reflected by a small portion of the diffuser, after crossing thesolar concentrator and “exiting” from its “entrance” opening, was projected on thelaboratory wall, forming an image with a particular shape with square contours, the sameof the entrance opening itself. After this first observation, I made other experiments to
explain the meaning of the luminous figure projected on the laboratory wall. In themeantime, I realized a new optical configuration to better study the phenomenon, placinga punched white screen between the laser and the concentrator, to avoid any interferencewith the path of light (see Fig. 4.1(b)).
Furthermore, I expanded the laser beam with a beam expander in order to illuminate theentire surface of the diffuser facing inside the concentrator (see Fig. 4.2(b)). Fig. 4.1(b)shows the luminous figure formed by the reflected light on the holed screen.

Antonio Parretta
Physics and Earth Science Department, University of Ferrara, Academy of Sciences of Ferrara, Italy

179

Advances in Optics: Reviews. Book Series, Vol. 5a)b)Fig. 4.1. (a) A truncated and squared compound parabolic concentrator (TS-CPC), tappedwith a white diffuser (ld), is illuminated by a laser beam; (b) Image formed on the punchedscreen by the reflected laser light.

a)b)Fig. 4.2. (a) After being expanded by the beam expander (be), the laser beam crosses the punchedplanar screen (ps) and strikes on the full exposed area of the diffuser (ld);(b) Control of the expanded laser beam size.

After thinking to this phenomenon, it became clear that the luminous figure on the screen,
image of the angular distribution of the radiance of the emitted light, corresponded to the
angular distribution of the optical efficiency that the concentrator would show if it were
illuminated, from the entrance opening, with a collimated light. Thanks to this discovery,
and after understanding the importance it could have on the work of opticalcharacterization of solar concentrators, the configuration of the optical apparatus was
further improved until it assumed the final aspect shown in Fig. 4.3(a). Here, theLambertian light to be used to feed the concentrator is produced by a lamp illuminating afirst integrating sphere which in turn illuminates a second integrating sphere, having an
opening directly coupled to the exit opening of the solar concentrator. The use of a double
integrating sphere is necessary to supply the SC by a well-shaped Lambertian, unpolarized
light with 90° divergence. The light exiting from the input opening of the SC is thenprojected on a far, white screen normally used to project slides. The luminous figure
diffused by the white screen is then recorded by a CCD aligned with the solar concentrator.
The elaboration of the CCD image (see Fig. 4.3(b)), by an appropriate software [6], allows
finally to derive the angular distribution of radiance and therefore the angular distributionof the optical efficiency. To get the absolute optical efficiency, it is necessary to know the180

Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

optical efficiency at 0°, the on-axis efficiency. This is obtained by elaborating the image
recorded after facing the CCD directly on the solar concentrator, as shown in Fig. 4.4 (a).
Fig. 4.4 (b) shows an example of image produced by projecting the inverse light from theRondine® concentrator [7-14].

a)b)Fig. 4.3. (a) Final version of the configuration of the optical apparatus for recording the radianceof the emitted light in the inverse way. The lamp (ls) is a source of white light;(b) Image of the luminous figure projected on the planar screen (ps).

The on-axis optical efficiency is obtained by the ratio between the average intensity of thefull image in the yellow contour and the average intensity of the Lambertian source in the
red contour [6]. This technique has been successfully applied to all lab-madeconcentrators, which were characterized in few days, when, adopting the traditional
method, it would have taken more than a month [6, 15-28]. This, in short, is the story I
lived putting together the Lambertian light and a solar concentrator. But this story does
not finish here, because it gave rise to the idea of investigating the optical behavior of asolar concentrator when it is irradiated with Lambertian light from one of its openings,
regardless of the study of its optical efficiency. And this has given rise to the definition of
new optical quantities related to the transmission, reflection and absorption of light[29-37]. From here it was easy to imagine a further development of the theory in which
the two openings of the SC are simultaneously irradiated with Lambertian light [6, 29].
Hence the idea of formulating an optical law that relates the net flux crossing the SC withthe differential radiance at its ends through a quantity defined as "optical conductance"[6, 29]. But this study does not consider only solar concentrators, or at least not only solar
concentrators with a canonical form, because the law of optical conductance has auniversal character and can be applied to any optical system for which the principle of
optical reversibility is valid [38, 39]. In this work, I have chosen, therefore, some simple
optical elements to apply the law of optical conductance. Some of them are non-imaging
solar concentrators of the CPC type, well designed to concentrate light within a definedangle with maximum efficiency; others, simpler, are the so-called “light cones” (LC),
often used in refractive solar concentrators as secondary concentrating elements or for
reducing the dispersion of rays in the focal region when an accidental misalignment of thesolar tracker could lead to a damage to the walls. Other elements, such as “solar tunnels”(ST), are considered here because they allow us to go beyond the field of solar
concentration and to explore other fields such as that of the transport of sunlight fromoutdoor to indoor. Last but not least, I thought about the use of some of the mentioned
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Advances in Optics: Reviews. Book Series, Vol. 5

optical elements as light sources when irradiated by Lambertian light in the reverse way.
What kind of light do they emit? How their radiance is distributed in space? Can they beused in the field of lighting? I will try to answer these questions in the second part of thiswork. However, before presenting the results of optical simulations, it is useful to reportin the following section the fundamental concepts that are the basis of the operation of
canonical solar concentrators, because the actual work originates from them, and because
their knowledge will allow us to better understand everything that will be exposed later.

a)b)Fig. 4.4. (a) Experimental configuration used for the measurement of the on-axis optical efficiency
of a solar concentrator. Unlike the configuration of Fig. 4.3 (a), now the CCD camera is alignedwith the optical axis and facing the concentrator inlet opening, from which the inverse light exits;(b) Image of the inlet opening, with the two contours used to calculate the on-axis efficiency.

List of Symbols (as they appear in the text):
 𝐶𝑔𝑒𝑜 , geometrical concentration ratio;
 𝐶𝑜𝑝𝑡 , optical concentration ratio;
 𝐸𝑖𝑛 , input irradiance;
 𝐸𝑜𝑢𝑡 , output irradiance; 𝜂𝑜𝑝𝑡 , 𝜂𝑑𝑖𝑟 , optical efficiency under collimated radiation;
 𝐴𝑖𝑛 , 𝐴1 , inlet opening area;
 𝐴𝑜𝑢𝑡 , 𝐴2 , outlet opening area;  𝑖𝑛 , flux at input;  𝑜𝑢𝑡 , flux at output;  𝑆 , angular divergence of Sun;  𝑎𝑐𝑐 , acceptance angle;
 𝑛𝑖𝑛 , index of refraction at input;
 𝑛𝑜𝑢𝑡 , index of refraction at output;  , polar angle;
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Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

 , azimuthal angle;
 𝐿𝑑𝑖𝑟 , direct radiance; 𝐿𝑖𝑛𝑣 , inverse radiance;𝑖𝑛 𝛷𝑑𝑖𝑟, direct flux at input;𝑜𝑢𝑡 𝛷𝑑𝑖𝑟, 𝛷𝑑𝑖𝑟, transmitted direct flux;𝑖𝑛 𝛷𝑖𝑛𝑣, inverse flux at input;𝑜𝑢𝑡 𝛷𝑖𝑛𝑣, 𝛷𝑖𝑛𝑣, transmitted inverse flux;

 𝐿𝑖𝑛𝑣 , transmitted inverse radiance;  𝑙𝑎𝑚𝑏
𝑑𝑖𝑟 , direct Lambertian transmittance;  𝑙𝑎𝑚𝑏
𝑖𝑛𝑣 , inverse Lambertian transmittance; ̅̅̅̅̅
𝐿𝑜𝑢𝑡 , average direct radiance at output;
𝑑𝑖𝑟



̅̅̅̅̅
𝐿𝑜𝑢𝑡
𝑖𝑛𝑣 , average inverse radiance at output;  𝑙𝑎𝑚𝑏
𝑑𝑖𝑟 , direct Lambertian reflectance;  𝑙𝑎𝑚𝑏
𝑑𝑖𝑟 , direct Lambertian absorbance;  𝑙𝑎𝑚𝑏
𝑖𝑛𝑣 , inverse Lambertian reflectance;  𝑙𝑎𝑚𝑏
𝑖𝑛𝑣 , inverse Lambertian absorbance;𝑛𝑒𝑡 𝛷𝑑𝑖𝑟, direct net flux;𝑛𝑒𝑡 𝛷𝑖𝑛𝑣, inverse net flux; 𝛥𝛷, net flux;
 𝛥𝐿, difference of radiance; Q, transmitted heat; H, heat flux; 𝐶𝑡ℎ , thermal conductance; 𝑅𝑡ℎ , thermal resistance; 𝛥𝑇, difference of temperature; I, electrical current; 𝑅𝑒𝑙 , electrical resistance; G, electrical conductance; 𝛥𝑉, difference of electrical potential;𝑙𝑎𝑚𝑏 𝐺𝑑𝑖𝑟, direct Lambertian conductance;𝑙𝑎𝑚𝑏 𝐺𝑖𝑛𝑣, inverse Lambertian conductance;

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 𝐺 𝑙𝑎𝑚𝑏 , optical (bidirectional) Lambertian conductance; 𝐴1 , area of input (1) terminal; 𝐴2 , area of output (2) terminal;  𝑙𝑎𝑚𝑏
12 , direct Lambertian transmittance;  𝑙𝑎𝑚𝑏
21 , inverse Lambertian transmittance; 𝑙𝑎𝑚𝑏 , Lambertian transmittance;𝑙𝑎𝑚𝑏 𝐺𝑠𝑝𝑒𝑐, specific Lambertian conductance; a, radius of inlet opening; a’, radius of outlet opening; f, focal length of CPC parabolic profile; l, length (of CPCs, LCs or STs);
 𝑅𝑆 , surface reflectance of the optical element; 𝑅𝑠𝑝ℎ , surface reflectance of the integrating sphere;  𝑀 , angular aperture of the spherical screen; R, radius of the spherical screen; 𝛥𝛺, elemental solid angle;
 𝛥, elemental flux;
 𝛥𝑆, elemental area; 𝐸𝑝𝑟𝑜 , projected irradiance; 𝐿1 , radiance at optical terminal (1);
 𝐿2 , radiance at optical terminal (2); 𝐸𝑑𝑖𝑟 , direct irradiance;
 𝐸𝑑𝑖𝑓𝑓 , diffuse irradiance;
 𝐸𝑑𝑖𝑟,⊥ , normal direct irradiance;
 𝐸𝑑𝑖𝑓𝑓,⊥ , normal diffuse irradiance; 𝑑𝑖𝑟 , direct flux; 𝑑𝑖𝑓𝑓 , diffuse flux;
 𝐿𝑑𝑖𝑟 , direct radiance;
 𝐿𝑑𝑖𝑓𝑓 , diffuse radiance; AM, Air Mass.

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Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

4.2. Introduction to Solar Concentrators and Their Characterization
A photovoltaic solar concentrator is generally composed of a series of units assembled insuch a way as to efficiently transmit the solar radiation to a certain surface. Each elementof the system is composed of an optical part that collects the solar radiation, mainly thatof the direct component of the Sun, through an input optical terminal, and conveys it withincreased flux density on an output optical terminal, where a photovoltaic receiver,consisting of one or more solar cells transforming the concentrated solar radiation into
electricity, is placed. A fundamental part of the photovoltaic concentrator (CPV) is
therefore the optical unit, transmitting the solar radiation from the input terminal to the
output terminal with the maximum possible efficiency. The areas of the two terminals
determine the geometric concentration ratio, 𝐶𝑔𝑒𝑜 , that is the maximum achievable ratio
between the average density of the outgoing flux and the density of the incoming flux.
This ratio is called the optical concentration ratio, 𝐶𝑜𝑝𝑡 . We will have therefore:
𝐶𝑔𝑒𝑜 ≥ 𝐸𝑜𝑢𝑡 ⁄𝐸𝑖𝑛 = 𝐶𝑜𝑝𝑡

(4.1)

The equal sign holds when all the radiation facing the input terminal is transferred to the
output terminal, that is when the optical efficiency, 𝜂𝑜𝑝𝑡 = 1.0. We will have therefore:
𝐶𝑜𝑝𝑡 = 𝐸𝑜𝑢𝑡 ⁄ 𝐸𝑖𝑛 = 𝜂𝑜𝑝𝑡  𝐶𝑔𝑒𝑜 = 𝜂𝑜𝑝𝑡  (𝐴𝑖𝑛 ⁄ 𝐴𝑜𝑢𝑡 ),(4.2)

where 𝐴𝑖𝑛 and 𝐴𝑜𝑢𝑡 are the areas of the input and output terminals, respectively, and:𝜂𝑜𝑝𝑡 =  𝑜𝑢𝑡 ⁄  𝑖𝑛 ,(4.3)

with  𝑜𝑢𝑡 and  𝑖𝑛 outgoing and incoming fluxes (in W), respectively. Ultimately, we
can write:
𝐶𝑜𝑝𝑡 = 𝐸𝑜𝑢𝑡 ⁄ 𝐸𝑖𝑛 = ( 𝑜𝑢𝑡 ⁄ 𝑖𝑛 )  (𝐴𝑖𝑛 ⁄ 𝐴𝑜𝑢𝑡 )(4.4)

The optical unit, however, must not only transfer the radiation with the maximum
efficiency; it must also transfer it uniformly onto the PV receiver, because this is thecondition that minimize the ohmic losses [3, 40-43], maximizing the conversion efficiency
with the same incident flux. This, for example, is not a stringent condition in the case ofthermodynamic solar concentrators. However, we will not dwell here on the functional
conditions of the PV receiver, because they are beyond the subject of this work, allcentered on the study of the optical properties.
Solar concentrators essentially work with the direct component of solar radiation, that iswith the radiation emitted by the solar disk, and then are operational only when the sky isclear. Direct solar radiation can be compared to a quasi-parallel light beam, with a small
angular divergence,  𝑆 =  0.27°, emitted by an unpolarized Lambertian source. The solar
concentrators, in turn, are characterized by a certain acceptance angle,  𝑎𝑐𝑐 , generally
defined as the angle at which the solar radiation is collected with an efficiency of 50 %[44-52]. Acceptance angles  𝑎𝑐𝑐 are almost always much greater than the angulardispersion  𝑆 . It is important to consider a fundamental relationship between acceptance185

Advances in Optics: Reviews. Book Series, Vol. 5

angle and optical concentration ratio, 𝐶 𝑜𝑝𝑡 , when these conditions apply: concentrators inair (𝑛 𝑖𝑛 = 1, 𝑛 𝑜𝑢𝑡 = 1), as will be all those tested in the present work, 𝜂𝑜𝑝𝑡 = 1 (ideal
concentrator) and maximum divergence (90°) for the outgoing rays:
𝐶𝑜𝑝𝑡 = 1⁄(sin 𝑎𝑐𝑐 )2(4.5)

Eq. (4.5) establishes a maximum limit to the optical solar concentration, given by theangle of acceptance of the concentrator, even if the solar angular dispersion  𝑆 is verysmall: the optical concentration ratio 𝐶 𝑜𝑝𝑡 is independent of  𝑆 , but depends only on 𝑎𝑐𝑐 . Recurring  𝑎𝑐𝑐 values for a PV concentrator range from about 20° downwards toabout 2.5°, corresponding to optical concentrations from about 10× up to about 500× [53].
People do not operate at higher concentration levels, even if, in theory, the directcomponent of the Sun could be concentrated up to:
𝐶𝑜𝑝𝑡 = 1⁄(sin 𝑆 )2 ≈ 46.000(4.6)

If one tried to overcome this limit by designing a concentrator with ratio(𝐴𝑖𝑛 ⁄𝐴𝑜𝑢𝑡 ) > 𝐶𝑜𝑝𝑡 , it would be useless because it would automatically reduce the optical
efficiency (see Eq. (4.4)). Although concentrating sunlight is advantageous for the
purpose of reducing the size of the photovoltaic receiver, and therefore its cost, it is clearthat the more sunlight is concentrated, the smaller will be its acceptance angle, andtherefore the more precise and more expensive will have to be the pointing system, that is
the solar tracker. Furthermore, the temperature of the receiver would rise with increasing
concentration, as dictated by the Stefan-Boltzmann law, and then the tracking system
would have to be designed increasingly sophisticated for the heat dissipation. This is what
happens when the SC is irradiated by the direct solar radiation.
The entry on the scene of an unpolarized Lambertian radiation, extended to an angle of90°, occurs with the introduction of the Inverse Lambertian Irradiation Method (ILIM),
already illustrated in the prologue, which is part of an optical characterization techniqueof the optical unit of a PV concentrator 1 . This characterization technique has been
extensively illustrated in previous works [6, 15-19, 22-29, 32-37] and here I limit myself
to summarizing its functioning. As we have previously seen, the optical figure of a SC is
its optical efficiency curve defined with respect to a parallel light beam oriented at a
certain polar angle  and azimuthal angle  with respect to the optical axis of theconcentrator (for optical systems with cylindrical symmetry it is sufficient the use of only
the polar angle  ). So, it comes down to finding the 𝜂𝑜𝑝𝑡 () function. Well, it has been
demonstrated [6, 29], as discussed in the Prologue, that the profile of this efficiency is thesame as that of the radiance emitted by the concentrator from its entrance opening when
it is illuminated in the opposite direction, that is, on its exit opening, with an unpolarized
and uniform Lambertian light with 90° divergence. If then the measurement of the profile

1 Some acronyms used here for the optical methods or for the optical quantities are slightly different fromthose used so far in that I have added “I” as “Irradiation”. Other optical quantities are defined here for the
first time.

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Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applicationsof the inverse radiance is added to that of the on-axis efficiency, the absolute curve of the
optical efficiency will be completely determined. The concept underlying the ILIM
method can be summarized as follows: if the SC receives light well from a certain spatial
direction (, ) when operating in the direct mode, it will emit light equally well from theentrance opening towards the same direction (, ) when operating in the inverse mode.
The inverse mode is applied irradiating the exit aperture with uniform and unpolarized
Lambertian light, and the only required condition is the validity of the optical reversibility
principle for the tested optical system [38, 39].
The ILIM is then applied simply by projecting the inverse light on a white, Lambertian
screen (see Fig. 4.3), recording the intensity map and from this, by the use of a correction
function [6], translating it into an optical efficiency map, centered on the optical axis, andfrom which it can be extracted the profile along any chosen azimuthal direction, if theconcentrator is not of cylindrical symmetry. If the concentrator has cylindrical symmetry,
then the intensity map will be first symmetrized around the optical axis itself before
transforming it into an optical efficiency map. The ILIM method thus proves to be very
simple to apply and very fast. It has been an important innovation in the field of opticalcharacterization of SCs because the traditional method, called DLIM (Direct Lambertian
Irradiation Method) [15-17, 28, 30, 31, 33, 34, 37] is much more elaborate, and its
experimental apparatus more expensive: it involves the measurement of the incoming fluxof a collimated beam and its outgoing flux through a series of discrete measurements at
different polar  and, eventually, azimuthal angles , the more numerous the higher the
angular resolution of the optical efficiency to be achieved. The latter is obtained, for eachpoint, by the ratio between the outgoing and the incoming flux, respectively. Therefore,
the DILM provides a series of discrete values, while the ILIM method produces, with asingle measurement, a continuous map of values, whose resolution depends only on the
resolution of the webcam, or CCD camera, used to record the intensity map of the reverse
light projected on the Lambertian screen. It is useful to remember here that the ILIM
method has undergone an evolution to eliminate the limitations related to the distance
between the inverse source and the screen. This is referred to as the “Parretta-Herrero
method” (PH-Method) [22, 27].
This is a summary of the state of the art of the optical characterization of a SC, and theuse of a Lambertian source is made only for this purpose, which becomes propaedeutic towhat will be explained below. At this point we open a new research scenario which
consists in the use of a Lambertian radiation as a source for which to study different, newoptical properties of the SC. Basically, the collimated light beam is replaced by an
unpolarized Lambertian light of 90° angular divergence, and with respect to this we will
study all the optical properties we have already studied for the collimated incoming beams[6, 29]. New optical quantities will be defined, the most important of which is the lighttransmission property summarized in the quantity: “Optical Bidirectional LambertianConductance” (OBLC), or simply “Optical Lambertian Conductance” (OLC), from which
the corresponding law is derived. All this will be investigated not only on the innovative
solar concentrators, like the nonimaging concentrators of the CPC type, and also on thesimpler light cones, but will be extended to a new category of optical elements, the solar
tunnels, generally used to transfer sunlight from outdoors to indoors.
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4.3. Theory of the Bidirectional Lambertian Irradiation Method (BLIM)
When a solar concentrator (SC) is irradiated at its input opening from all the directions inspace, we say that it is subjected to Lambertian irradiation with 90° angular divergence,then all the points of the opening are source of unpolarized light with constant radiance
𝐿𝑑𝑖𝑟 . Let us consider, for simplicity, only solar concentrators with cylindrical symmetry;
all the optical quantities will be expressed only as a function of the polar angle . In thefollowing, when necessary, the dependence of the optical quantities on the azimuthal𝑖𝑛angle  will be suitably specified. The flux incident on the entrance opening, 𝛷𝑑𝑖𝑟, andthat transmitted by the exit opening, 𝛷𝑑𝑖𝑟 , can be expressed, respectively, as follows:
𝜋

2𝜋

𝑖𝑛
𝛷𝑑𝑖𝑟
= 𝐿𝑑𝑖𝑟 ∙ 𝐴𝑖𝑛 ∙ ∫0 𝑑𝜑 ∙ ∫02 𝑑 ∙ sin ∙ cos =  ∙ 𝐴𝑖𝑛 ∙ 𝐿𝑑𝑖𝑟 ,

(4.7)

𝜋⁄2

(4.8)


𝛷𝑑𝑖𝑟
= 2 𝜋 ∙ 𝐴𝑖𝑛 ∙ 𝐿𝑑𝑖𝑟 ∙ ∫0𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝑑𝑖𝑟 (𝜃),where 𝐴𝑖𝑛 is the SC input area and 𝑑𝑖𝑟 (𝜃) is the optical efficiency of the SC when it isirradiated by collimated light. This method of irradiating a SC is the Direct Lambertian
Irradiation Method (DLIM). Similar equations give the incident and transmitted fluxes,
𝑖𝑛

𝛷𝑖𝑛𝑣
and 𝛷𝑖𝑛𝑣, when a Lambertian source of radiance 𝐿𝑖𝑛𝑣 is applied to the outlet
opening.
/2

2𝜋

𝑖𝑛
𝛷𝑖𝑛𝑣
= 𝐿𝑖𝑛𝑣 ∙ 𝐴𝑜𝑢𝑡 ∙ ∫0 𝑑𝜑 ∙ ∫0
𝜋⁄2


𝛷𝑖𝑛𝑣
= 2 𝜋 ∙ 𝐴𝑖𝑛 ∙ ∫0𝑑 ∙ sin ∙ cos =  ∙ 𝐿𝑖𝑛𝑣 ∙ 𝐴𝑜𝑢𝑡 ,𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝐿𝑖𝑛𝑣 (𝜃),(4.9)
(4.10)

where 𝐴𝑜𝑢𝑡 is the area of the SC exit opening and 𝐿𝑖𝑛𝑣 (𝜃) is the radiance of light inversely
emitted towards direction (𝜃). This method of irradiating a SC is the Inverse Lambertian
Irradiation Method (ILIM). For DLIM and ILIM, the ratio of output to input fluxes gives
the “direct Lambertian transmittance”  𝑙𝑎𝑚𝑏
𝑑𝑖𝑟 , and the “inverse Lambertian transmittance”
𝑙𝑎𝑚𝑏 𝑖𝑛𝑣 , respectively:𝜋⁄2 ⁄ 𝑖𝑛 𝑙𝑎𝑚𝑏= 𝛷𝑑𝑖𝑟𝛷𝑑𝑖𝑟 = 2 ∙ ∫0𝑑𝑖𝑟𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝑑𝑖𝑟 (𝜃),𝜋⁄2 ⁄ 𝑖𝑛 𝑙𝑎𝑚𝑏= 𝛷𝑖𝑛𝑣
𝛷𝑖𝑛𝑣 = (2 ∙ 𝐶𝑔𝑒𝑜 ⁄𝐿𝑖𝑛𝑣 ) ∙ ∫0
𝑖𝑛𝑣

(4.11)𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝐿𝑖𝑛𝑣 (𝜃), (4.12)

where 𝐿𝑖𝑛𝑣 (𝜃) is the radiance of light emitted by the entrance opening of the SC. It must
be considered, of course, that, although the SC is fed with Lambertian light, thetransmitted one has not necessarily a constant radiance (this topic will be fully covered in
Section 4.6), and then we will consider an average value of it:

𝑜𝑢𝑡
̅̅̅̅̅
𝐿
𝑑𝑖𝑟 = 𝛷𝑑𝑖𝑟 ⁄( ∙ 𝐴𝑜𝑢𝑡 ) =𝜋⁄2= 2 ∙ 𝐿𝑑𝑖𝑟 ∙ 𝐶𝑔𝑒𝑜 ∙ ∫0188𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝑑𝑖𝑟 (𝜃),(4.13)

Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications
𝜋 ⁄2

𝑜𝑢𝑡
̅̅̅̅̅
𝐿
𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝐿𝑖𝑛𝑣 (𝜃)𝑖𝑛𝑣 = 𝛷𝑖𝑛𝑣 ⁄( ∙ 𝐴𝑖𝑛 ) = 2 ∙ ∫0

(4.14)

It can be demonstrated [6, 29] that, as long as the reversibility principle applies [38, 39],
the two transmittances are related as follows:𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏= 𝐴𝑖𝑛 ⁄𝐴𝑜𝑢𝑡 = 𝐶𝑔𝑒𝑜 ,𝑖𝑛𝑣 ⁄ 𝑑𝑖𝑟(4.15)

where 𝐶𝑔𝑒𝑜 is the geometrical concentration ratio. Eq. (4.15) tells us that the“transparency” of the SC under Lambertian light is not symmetrical, being 𝐶𝑔𝑒𝑜 times
higher when the SC is irradiated in the reverse way. There are other new optical quantities
that can be defined when we irradiate the SC with Lambertian light. Two of them arise
from the direct irradiation method (DLIM) and represent, one the fraction of the input flux
back reflected from the inlet opening,  𝑙𝑎𝑚𝑏
𝑑𝑖𝑟 , and the other the fraction of the input flux
absorbed by the inner surface of the SC,  𝑙𝑎𝑚𝑏
𝑑𝑖𝑟 , which are, respectively:𝑖𝑛 𝑙𝑎𝑚𝑏= 𝛷𝑑𝑖𝑟 ⁄𝛷𝑑𝑖𝑟= 2 ∙ ∫0𝑑𝑖𝑟

𝜋⁄2𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝑑𝑖𝑟 (𝜃),(4.16)𝜋⁄2𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝑑𝑖𝑟 (𝜃)(4.17)

 ⁄ 𝑖𝑛 𝑙𝑎𝑚𝑏= 𝛷𝑑𝑖𝑟𝛷𝑑𝑖𝑟 = 2 ∙ ∫0𝑑𝑖𝑟

We define these two quantities related to DLIM, respectively: the “direct Lambertian𝑙𝑎𝑚𝑏reflectance”  𝑙𝑎𝑚𝑏
𝑑𝑖𝑟 , and the “direct Lambertian absorptance”  𝑑𝑖𝑟 .
In a similar way we define the two quantities that arise from the inverse Lambertian
irradiation of the SC (ILIM): the “inverse Lambertian reflectance”  𝑙𝑎𝑚𝑏𝑖𝑛𝑣 , and the𝑙𝑎𝑚𝑏“inverse Lambertian absorptance”  𝑖𝑛𝑣 , given by, respectively:𝑖𝑛 𝑙𝑎𝑚𝑏= 𝛷𝑖𝑛𝑣 ⁄𝛷𝑖𝑛𝑣= 2 ∙ ∫0𝑖𝑛𝑣

𝜋⁄2𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝑖𝑛𝑣 (𝜃),𝜋⁄2 ⁄ 𝑖𝑛 𝑙𝑎𝑚𝑏= 𝛷𝑖𝑛𝑣
𝛷𝑖𝑛𝑣 = 2 ∙ ∫0𝑖𝑛𝑣𝑑𝜃 ∙ sin𝜃 ∙ cos𝜃 ∙ 𝑖𝑛𝑣 (𝜃)(4.18)(4.19)The theory of the Lambertian irradiation of a SC, discussed so far, can be pushed further
forward by imagining to feed the SC simultaneously on both openings with two different
Lambertian sources, of radiance 𝐿𝑑𝑖𝑟 and 𝐿𝑖𝑛𝑣 , respectively (see Fig. 4.5).
Putting 𝐿 = 𝐿𝑑𝑖𝑟 − 𝐿𝑖𝑛𝑣 , we can write for the net flux through the SC in the direct
direction:
𝑜𝑢𝑡
𝑜𝑢𝑡
𝑖𝑛
𝑖𝑛
𝑙𝑎𝑚𝑏
𝑙𝑎𝑚𝑏
 =  𝑛𝑒𝑡
𝑑𝑖𝑟 =  𝑑𝑖𝑟 −  𝑖𝑛𝑣 =  𝑑𝑖𝑟   𝑑𝑖𝑟 −  𝑖𝑛𝑣   𝑖𝑛𝑣 = …
𝑖𝑛𝑖𝑛… =  𝑙𝑎𝑚𝑏𝑑𝑖𝑟  ( 𝑑𝑖𝑟 − 𝐶𝑔𝑒𝑜   𝑖𝑛𝑣 ) = …
𝑙𝑎𝑚𝑏… =  𝑙𝑎𝑚𝑏𝑑𝑖𝑟  (  𝐿𝑑𝑖𝑟  𝐴𝑖𝑛 − 𝐶𝑔𝑒𝑜    𝐿𝑖𝑛𝑣  𝐴𝑜𝑢𝑡 ) = (  𝐴𝑖𝑛   𝑑𝑖𝑟 )  𝐿, (4.20)

and, putting 𝐿 = 𝐿𝑖𝑛𝑣 − 𝐿𝑑𝑖𝑟 , for the net flux through the SC in the reverse direction:
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Advances in Optics: Reviews. Book Series, Vol. 5
𝑜𝑢𝑡
𝑜𝑢𝑡
𝑙𝑎𝑚𝑏
 =  𝑛𝑒𝑡
𝑖𝑛𝑣 =  𝑖𝑛𝑣 −  𝑑𝑖𝑟 = (  𝐴𝑜𝑢𝑡   𝑖𝑛𝑣 )  𝐿

(4.21)

Fig. 4.5. It is represented schematically the bidirectional Lambertian irradiation (BLI), applied
to a cylindrically symmetric solar concentrator (SC), sum of the direct Lambertian irradiation(DLI) with radiance 𝐿𝑑𝑖𝑟 and the inverse Lambertian irradiation (ILI) with radiance 𝐿𝑖𝑛𝑣 .

Eqs. (4.20) and (4.21), taking account of Eq. (4.15), give the same result and are similar
to the formula used for describing the heat transmission, 𝑄 ⁄𝑡 = 𝐻 = (𝐶𝑡ℎ  𝑆)
𝑇 = (𝑆⁄𝑅𝑡ℎ ) 𝑇, where Q is the heat transmitted over time t, H the heat flux (W), 𝐶𝑡ℎthe thermal conductance (𝑊 ⁄𝑚2  𝐾), S the heat-crossed area, 𝑅𝑡ℎ the thermal resistance( 𝑚2  𝐾⁄ 𝑊 ), and 𝑇 the temperature difference, or the current transmission,
𝐼 = (1⁄𝑅𝑒𝑙 ) 𝑉 = 𝐺  𝑉 , where 𝑅𝑒𝑙 is the electrical resistance, G the electricalconductance and 𝑉 the potential difference. It is natural, therefore, to define a "Direct
Lambertian Conductance" and an "Inverse Lambertian Conductance" of the SC,
respectively, as follows:
𝑙𝑎𝑚𝑏
𝐺𝑑𝑖𝑟
= (  𝐴𝑖𝑛   𝑙𝑎𝑚𝑏𝑑𝑖𝑟 ),(4.22’)𝑙𝑎𝑚𝑏
𝐺𝑖𝑛𝑣= (  𝐴𝑜𝑢𝑡   𝑙𝑎𝑚𝑏𝑖𝑛𝑣 )(4.22’’)

Being the two conductances equal from Eq. (4.15), we can define an "Optical
Bidirectional Lambertian Conductance" (OBLC), or simply an “Optical LambertianConductance”, OLC, which can be referred indifferently to the entrance or to the exit
opening of the SC and can be put in the form:
𝐺 𝑙𝑎𝑚𝑏 = (  𝐴)  𝑙𝑎𝑚𝑏 (sr  m2),

(4.23)

that is: “Lambertian Conductance” = “étendue” x “Lambertian transmittance”. By
comparing Eq. (4.15) with Eqs. (4.22), we observe that the asymmetry of the Lambertiantransmittance is cancelled out by the symmetry of the Lambertian conductance. By virtueof the symmetry between the input and output terminals of the SC from the point of viewof the Lambertian conductance, we can look at the SC as a generic optical element, a box
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Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

with an unknown internal structure, provided with a first optical terminal (1) and a second
optical terminal (2) (see Fig. 4.6). Its Lambertian optical conductance, equal in both
directions 1  2 and 2  1, can then be expressed as:
𝐺 𝑙𝑎𝑚𝑏 =   𝐴1   𝑙𝑎𝑚𝑏=   𝐴2   𝑙𝑎𝑚𝑏1221 ,(4.24)

where  𝑙𝑎𝑚𝑏and  𝑙𝑎𝑚𝑏12
21 are the Lambertian optical transmittances in directions 12 and21, respectively. The optical element of Fig. 4.6 is crossed by a net flux 𝛷𝑛𝑒𝑡 , indirection 1  2 if 𝐿 > 0 or in direction 2  1 if 𝐿 < 0, given by:

 𝑛𝑒𝑡 = 𝐺 𝑙𝑎𝑚𝑏  (𝐿1 − 𝐿2 ) = 𝐺 𝑙𝑎𝑚𝑏  𝐿

(4.25)

Fig. 4.6. It is represented a generic optical element bidirectionally irradiated with radiances 𝐿1and 𝐿2 at the optical terminals (1) and (2).

The Lambertian conductance depends on the shape of the optical element, on its internalsurface properties and on its dimensions. Now, however, it would be better to deal with aquantity characterizing the optical conductance properties of an element regardless of its
absolute dimensions, which would allow us to make a comparison between elements ofdifferent shape and surface state more effectively. We then introduce a quantity that is
independent of the size of one of the two optical terminals, for example the terminal (1),which in solar concentrators corresponds to the entrance opening of the light, and whichis larger than the terminal (2), that of the exit of concentrated light. We then define the𝑙𝑎𝑚𝑏"Specific Optical Bidirectional Lambertian Conductance", 𝐺 𝑠𝑝𝑒𝑐
, (SOBLC), or simply
the “Specific Optical Lambertian Conductance”, SOLC, which refers to the surface unitof the terminal (1) and which is equal to:𝑙𝑎𝑚𝑏
𝐺 𝑠𝑝𝑒𝑐= 𝐺 𝑙𝑎𝑚𝑏 ⁄𝐴1 =    𝑙𝑎𝑚𝑏
= (⁄𝐶𝑔𝑒𝑜 )   𝑙𝑎𝑚𝑏1221

(4.26)

𝑙𝑎𝑚𝑏
𝐺 𝑠𝑝𝑒𝑐will have the dimension of steradian (sr). If we now refer to the thermal equivalent
𝑙𝑎𝑚𝑏of optical conduction, we can equate 𝐺 𝑠𝑝𝑒𝑐with the thermal conductance, because thisis defined specific with respect to the area crossed by the heat, while, if we refer to theelectrical equivalent of optical conduction, we can equate 𝐺 𝑙𝑎𝑚𝑏 with the electricalconductance, which does not refer to unit of area. In the following, when we will have todeal with optical elements of different shapes and sizes, we will compare them starting
𝑙𝑎𝑚𝑏from the quantity 𝐺 𝑠𝑝𝑒𝑐
. Then, to know the effective optical conductance of an element,𝑙𝑎𝑚𝑏
we will just multiply 𝐺 𝑠𝑝𝑒𝑐 by the surface area of the terminal (1), 𝐴1 .

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Advances in Optics: Reviews. Book Series, Vol. 5

4.4. Experimental
4.4.1. Optical Elements under Test
The optical simulations were carried out on three types of optical elements: i) Nonimaging
solar concentrators of the type CPC (Compound Parabolic Concentrators) [46, 50-52, 55];ii) Light cones (LC); iii) Solar tunnels (ST). All them have a cylindrical symmetry. The
Lambertian transmittance and the specific Lambertian conductance depend on their shapeand surface state, whereas the Lambertian conductance depends also on their dimensions.
The Three-dimensional CPCs (3D-CPCs) are well-known nonimaging reflective
concentrators with parabolic profile, developed by Winston to efficiently collectČerenkov radiation in high energy experiments [52]. Since then, the nonimaging CPCs
have been widely used to concentrate sunlight [46, 50, 51]. The 3D-CPCs arecharacterized by a step-like optical transmission efficiency curve, allowing the efficientcollection of collimated light from 0° to a maximum polar angle, called the acceptanceangle 𝜃𝑎𝑐𝑐 . In this chapter we will consider only 3D-CPCs with the inlet and outlet
openings in the air and with ideal shape, which means that the maximum divergence ofthe transmitted rays will be of 90° when the 3D-CPC is irradiated at entrance opening bya collimated beam incident at 𝜃 = 𝜃𝑎𝑐𝑐 . A 3D-CPC is characterized by the following
parameters: a = radius of entrance opening; 𝑎′ = radius of exit opening; l = length;𝜃𝑎𝑐𝑐 = acceptance (or tilt) angle; f = focal length of the parabolic profile. An ideal(canonical) 3D-CPC is completely determined by two of the above five parameters,
related by the following basic relationships:
𝑓 = 𝑎′ (1 + sin 𝑎𝑐𝑐 ),(4.27)

𝑎′ = 𝑎 sin 𝑎𝑐𝑐 ,(4.28)𝑙 = (𝑎 + 𝑎′ ) ctg 𝑎𝑐𝑐

(4.29)In this work, we analyze in detail four different CPCs, with 𝑎𝑐𝑐 = 2.5°, 5°, 10° and 20°.
Other CPCs are used only for comparison purpose. All the significant parameters of these
CPCs are reported in Table 4.1. The reported acceptance angles refer to an ideal specularsurface (𝑅𝑆 = 100 %), while for diffusive surfaces the acceptance angles are reported inTable 4.3. The optical elements reported in Table 4.1 are only models and theirdimensions, chosen arbitrarily, should not necessarily be associated with real components.
As it can be seen, the inlet and outlet openings are only a few millimeters in radius, but
the entry window opening radius is of approximately 12 mm for all the elements, which
is a measure I have used in the past for CPCs and which, for convenience, I have kept inthis work. All the optical quantities do not depend on the size of the element, but on its
shape, if we exclude the optical Lambertian conductance. On the other hand, the opticalLambertian conductance can be brought back to the desired real value by applying a scale
factor on the area of the inlet opening. We have also added the areas of the entrance andexit openings in Table 4.1, which can be used to calculate the optical Lambertian
conductance 𝐺 𝑙𝑎𝑚𝑏 when the Lambertian transmittance is known. Fig. 4.7(a) shows, as
an example, as the open CPC10, with diffuse reflectance 𝑅𝑆 = 50 %, appears when192

Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

raytraced under an inverse irradiation with Lambertian light of 1k rays. The different
colours of the rays are the effect of the different attenuation after impact on the absorbing
surface. The red rays, the less attenuated ones, are transmitted at lower angles just because
they undergo less internal reflections, while the blue ones, the more attenuated, aretransmitted and reflected at high angles because they carry out more internal reflections.
Table 4.1. Significant parameters of some CPCs used in the simulations. The acceptance angles
are referred to specularly mirrored surface.
𝜃𝑎𝑐𝑐 (°)
a (m)a’ (m)f (m)l (m)𝐴1 (m2 )𝐴2 (m2 )Cgeo

CPC2.5

CPC5

CPC10

CPC20

CPC30

CPC40

2.5
12.03410−3
0.524910−3
0.547810−3
287.6510−3
454.9610−6
0.865610−6
525.585
12.03410−3
1.048810−3
1.140210−3
149.5410−3
454.9610−6
3.455910−6
131.6510
12.03210−3
2.089310−3
2.452110−3
80.08610−3
454.810−6
13.71410−6
33.16

20
12.03410−3
4.115910−3
5.52310−3
44.37110−3
454.9610−6
5.32210−5
8.55

30
12.010−3
6.010−3
9.010−3
31.17710−3
452.3910−6
113.1010−6
4.0

40
12.03410−3
7.73510−3
12.70710−3
23.56010−3
454.9610−6
187.9810−6
2.42

Y

X

Z
X

(a)

Z

(b)
Y

X

Z

(c)
Fig. 4.7. a) Inverse Raytracing of the CPC10 with Lambertian light; b) Direct raytracingof the LC3 with Lambertian light; c) Raytracing of the ST3 with Lambertian light. The surface
reflectance is 𝑅𝑆 = 50 % diffuse for all the elements.

The light cones (LCs) have the shape of truncated cones and can be considered as very
simple solar concentrators. In this chapter, we will test four series of LCs with fixed inlet
opening radius 𝑎 = 12 mm, outlet opening radius 𝑎′ = 6 mm (1st series), 𝑎′ = 5 mm
(2nd series), 𝑎′ = 4 mm (3rd series) and 𝑎′ = 3 mm (4th series). The length of the LCs isvariable and depends on the needs that arise during the simulations. The parameters ofsome LCs are reported in Table 4.2.
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Table 4.2. Significant parameters of some LCs used in the simulations. The acceptance angles
are referred to specularly mirrored surface. Next to the name of the light cone the seriesin Roman numerals is indicated.

𝜃𝑎𝑐𝑐 (°)
a(m)
a’(m)
l(m)
𝐴1 (m2 )𝐴2 (m2 )Cgeo
𝜃𝑎𝑐𝑐 (°)
a(m)
a’(m)
l(m)
𝐴1 (m2 )𝐴2 (m2 )Cgeo
𝜃𝑎𝑐𝑐 (°)
a(m)
a’(m)
l(m)
𝐴1 (m2 )𝐴2 (m2 )Cgeo

LC1(I)
27.2

LC2(I)
29.3

LC3(I)
29.7

LC4(I)
30.0

LC5(II)
27.8

LC6(II)
23.0

12.010−3
6.010−3
12.010−3
4.523910−4
1.13110−4
4.0
LC7(II)
24.2

12.010−3
6.010−3
24.010−3
4.523910−4
1.13110−4
4.0
LC8(II)
24.6

12.010−3
6.010−3
48.010−3
4.523910−4
1.13110−4
4.0
LC9(III)
33.8

12.010−3
6.010−396.010−3
4.523910−4
1.13110−4
4.0
LC10(III)
16.7

12.010−3
5.010−3
12.010−3
4.523910−4
7.85410−5
5.76
LC11(III)
18.8

12.010−3
5.010−3
24.010−3
4.523910−4
7.85410−5
5.76
LC12(III)
19.2

12.010−3
5.010−3
48.010−3
4.523910−4
7.85410−5
5.76
LC13(IV)
39.4

12.010−3
5.010−3
144.010−3
4.523910−4
7.85410−5
5.76
LC14(IV)
20.0

12.010−3
4.010−3
12.010−3
4.523910−4
5.02610−5
9.0
LC15(IV)
13.0

12.010−3
4.010−3
24.010−3
4.523910−4
5.02610−5
9.0
LC16(IV)
14.0

12.010−3
4.010−3
48.010−3
4.523910−4
5.02610−5
9.0

12.010−3
4.010−396.010−3
4.523910−4
5.02610−5
9.0

12.010−3
3.010−3
12.010−3
4.523910−4
2.82710−5
16.0

12.010−3
3.010−3
24.010−3
4.523910−4
2.82710−5
16.0

12.010−3
3.010−3
48.010−3
4.523910−4
2.82710−5
16.0

12.010−3
3.010−396.010−3
4.523910−4
2.82710−5
16.0

Fig. 4.7(b) shows as it appears the open LC3, with diffuse reflectance 𝑅𝑆 = 50 %, whenraytraced under a direct irradiation with Lambertian light of 1k rays. The sameconsiderations made previously apply; moreover, it is possible to note the scarcity of thetransmitted rays and the abundance of reflected rays when the irradiation is of direct type,
due to the narrowing of the section in the direct direction. The solar tunnels (STs) aresimple cylinders with opening radius 𝑟1 = 12 mm and variable length l. We have simulated
the following STs: ST1 (l = 12 mm); ST2 (l = 24 mm); ST3 (l = 48 mm); ST4(l = 96 mm). All the STs have: 𝐴1 = 𝐴2 = 4.524  10−4 m2 and 𝐶𝑔𝑒𝑜 = 1.0 .
Fig. 4.7(c) shows as it appears the solar tunnel ST3, with diffuse reflectance 𝑅𝑆 = 50 %,when raytraced under Lambertian light of 1 k rays. The appearance of rays is similar tothat of the LC3.
4.4.2. Simulation Methods
In each simulation, a certain number of rays was chosen to obtain optical maps with asufficiently high S/N ratio. Each ray was assigned arbitrarily the power of 1 mW. The
optical simulations, in fact, have the sole purpose of demonstrating the validity of the194Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

hypotheses put forward in this paper and, as regards references to real situations from thepoint of view of dimensions and flux densities, we will expressly refer to Section 4.7.
The optical simulations were made by using a commercial optical code [56]. For
simulating the Lambertian transmittance, 𝑙𝑎𝑚𝑏 , one of the schemes of Fig. 4.8 can be𝑙𝑎𝑚𝑏used; that in Fig. 4.8 (a) for measuring  𝑙𝑎𝑚𝑏
12 , and that in Fig. 4.8 (b) for measuring  21 .
The two quantities are related by Eq. (4.15), and then it is sufficient to measure just oneof them, but measuring both allows to confirm the exposed theory. The chosen opticalelement, a light cone (LC1), is closed on both openings by two ideal absorbers, (a 1) and(a2). In Fig. 4.8(a), a Lambertian source, of flux 1,𝑖𝑛 , is imposed on the opening (w1), ofarea 𝐴1 . A part of the flux, 1 is back reflected and measured by (a1), a part, 𝑆 , isabsorbed by the internal surface, and a part, 2, is transmitted to opening (w2), of area
𝐴2 . We have then for the incident radiance at opening (w1), 𝐿1 , for the direct lambertian𝑙𝑎𝑚𝑏transmittance,  𝑙𝑎𝑚𝑏
, respectively:
12 , and for the direct lambertian conductance, 𝐺 12
𝐿1 = 1,𝑖𝑛 /(  𝐴1 ),(4.30)

 𝑙𝑎𝑚𝑏= 2 /1,𝑖𝑛 ,12(4.31)𝑙𝑎𝑚𝑏
𝐺 12= (  𝐴1 )  𝑙𝑎𝑚𝑏12(4.32)The same scheme is adopted to calculate  𝑙𝑎𝑚𝑏(see Fig. 4.8(b)), with the difference that21
now the Lambertian source, of flux 2,𝑖𝑛 and radiance 𝐿2 , is placed on the opening (w2).

Fig. 4.8. Schematic representation of the DLIM for measuring  𝑙𝑎𝑚𝑏(a) and of the ILIM
12
for measuring  𝑙𝑎𝑚𝑏(b).
21

We then have, in an equivalent manner to the above, the quantities:
𝐿2 = 2,𝑖𝑛 /(  𝐴2 ),(4.33)

 𝑙𝑎𝑚𝑏= 1 /2,𝑖𝑛 ,21(4.34)𝑙𝑎𝑚𝑏
𝐺 21= (  𝐴2 )  𝑙𝑎𝑚𝑏21

(4.35)As an example, we report below the calculation of the direct and inverse Lambertiantransmittances,  𝑙𝑎𝑚𝑏and  𝑙𝑎𝑚𝑏of the light cone (LC1), of l = 12 mm length, which will
12
21
be used in the next section to demonstrate the validity of the OLC law (Eq. (4.25)). In
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Advances in Optics: Reviews. Book Series, Vol. 5

Fig. 4.9 are shown the raytracings of DLIM simulations. By applying Eq. (4.31), we obtain
different values of  𝑙𝑎𝑚𝑏
12 , which tend, at increasing the number of rays in the raytracing
process, clearly to the value of 0.25 (see Fig. 4.10). So, as  𝑙𝑎𝑚𝑏
= 1.0 (no rays absorbed
21
due to 𝑅𝑆 = 100 %, no rays reflected due to the shape of LC and to the specular surface),and  𝑙𝑎𝑚𝑏
= 0.25, we can confirm the validity of Eq. (4.15):
12
𝑙𝑎𝑚𝑏
𝑙𝑎𝑚𝑏
= 1⁄0.25 = 4.0 = 𝐴𝑖𝑛 ⁄𝐴𝑜𝑢𝑡 = 𝐶𝑔𝑒𝑜
𝑖𝑛𝑣 ⁄𝑑𝑖𝑟

(4.15)

Fig. 4.9. Example of raytracing of the light cone (LC1) irradiated on the opening (w 1)by a Lambertian source, following the DLIM for measuring  𝑙𝑎𝑚𝑏
12 ; (a) 𝑅𝑆 = 100 % specular;(b) 𝑅𝑆 = 80 % specular.
0,254

0,254

0,252

0,252

0,250

0,250

0,248

0,248

12lamb

0,246

0,246

Light cone LC1
DLIM

0,244

0,244

0,242

0,242

0,240

0,240

0,238

0,238

0,236

0,236

0,234

0,234

0,232

0,232
0,0

500,0k

1,0M

1,5M

2,0M

2,5M

3,0M

3,5M

4,0M

Number of rays, N

Fig. 4.10. Dependence of  𝑙𝑎𝑚𝑏on the number of rays used in the simulation process. It is
12interesting to note the peculiar trend of the result, with an overcoming of the equilibrium value
and a subsequent exponential decay. A good accuracy is achieved already with 200 k rays.

4.4.3. Measurement of the Inverse Radiance
For the measurement, through optical simulation, of the radiance coming out from one ofthe openings of the optical element, when the opposite opening is irradiated by Lambertianlight, I have adopted the configuration shown in Fig. 4.11. In this figure, it is shown theexample of the irradiation of a solar concentrator SC in the inverse way. The light196

Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

projected by SC is intercepted by an absorbing spherical screen (ss), of radius R, centeredon the center O of the opening aperture of the SC. The angular aperture of the sphericalscreen,  𝑀 , is adjusted according to the light emission characteristics. For specularsurfaces a limited portion of the screen is sufficient, while for diffusive surfaces it isnecessary to set up a maximum aperture (90°). The radius R of the spherical screen (ss)has been chosen one hundred times greater than that of the diameter of the exit apertureof the rays, in order to have a high angular resolution on the measured radiance [6]. The
irradiance produced on the screen in the  direction is 𝐸( ). We then consider the surface
element 𝛥𝑆 on the screen in the  direction. It forms, with respect to the center O, the solid
angle 𝛥𝛺. The flux emitted by SC within this angle 𝛥𝛺 is given by 𝛥. The radianceemitted by SC in direction  is then given by:
𝐿𝑜𝑢𝑡 () = 𝛥 ⁄ (𝐴1  𝑐𝑜𝑠  𝛥𝛺),(4.36)

where 𝐴1 is the area of the entrance opening of the SC. We have then:𝐿𝑜𝑢𝑡 () = 𝐸( ) 𝛥𝑆 ⁄ (𝐴1  𝑐𝑜𝑠  (𝛥𝑆⁄𝑅 2 )) = 𝐸( ) 𝑅 2 ⁄𝐴1  𝑐𝑜𝑠

(4.37)

Fig. 4.11. Scheme describing the emission of light, of radiance 𝐿𝑜𝑢𝑡 (), from the entrance
opening of the SC, irradiated in inverse way by a Lambertian source of radiance 𝐿𝑖𝑛𝑣 , collectedby a spherical screen (ss).

The radiance 𝐿𝑜𝑢𝑡 () can then be derived from the measurement of 𝐸(). The opticalsimulation code, however, does not show the function 𝐸(), but its projection on thex/y plane, equal to: 𝐸𝑝𝑟𝑜 () = 𝐸() / cos. Then, the program shows a circular map ofradius 𝑅 sin 𝑀 of only the portion of 𝐸() bounded by the angle  𝑀 ,. From Eq. (4.37)
we obtain for the emitted radiance:
𝐿𝑜𝑢𝑡 () = 𝐸𝑝𝑟𝑜 ( )  𝑐𝑜𝑠  𝑅 2 ⁄𝐴1  𝑐𝑜𝑠 = 𝐸𝑝𝑟𝑜 ( )  𝑅 2 ⁄𝐴1(4.38)

Being 𝐸𝑝𝑟𝑜 () proportional to 𝐿𝑜𝑢𝑡 (), its map gives the map of the emitted radiance up
to the constant 𝑅 2⁄𝐴1 . Since all the elements considered have a cylindrical symmetry, the𝐿𝑜𝑢𝑡 () can be symmetrized with respect to the optical axis to improve thesignal-to-noise ratio.
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Advances in Optics: Reviews. Book Series, Vol. 5

From the symmetrized map, any radial profile is then extracted which, normalized at = 0°, provides the normalized 𝐿𝑛𝑜𝑟𝑚 (), which will be reported in detail for thevarious optical elements in Section 4.6. Fig. 4.12 shows an example of raytracing of thelight cone LC2 for the measurement of the emitted radiance under reverse Lambertian
irradiation. (a) Specular surface 𝑅𝑆 = 100 %; (b) diffusive surface 𝑅𝑆 = 100 %. The
beneficial effect of rotational symmetry is shown in Fig. 4.13.

Fig. 4.12. Example of raytracing of the LC2 light cone for measuring the exit radianceunder inverse Lambertian irradiation. (a) Specular surface; (b) diffused surface.

(a)

(b)

Fig. 4.13. Map of the irradiance on the spherical screen of LC2 after projection on the x/y plane:(a) without rotational symmetry; (b) with rotational symmetry.

4.4.4. Test of the Optical Bidirectional Lambertian Conductance (OBLC) Law
To apply the theory of BLIM, the configurations illustrated in Fig. 4.14 were adopted. In
Fig. 4.14(a) the optical element (LC), with specular surface reflectance 𝑅𝑆 = 100 %, isclosed at its inlet opening (w1), of area 𝐴1 , with an ideal absorber (a1), while its outlet
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Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

opening (w2), of area 𝐴2 , is coupled to an integrating sphere (is), with 𝑅𝑠𝑝ℎ = 99 %diffuse surface reflectance, provided with another ideal absorber (a4) on the opening (w4),of area 𝐴4 . The ideal absorbers (a1) and (a4) measure, respectively, the inverse flux 1 on
the inlet opening (w1) of (LC), and the flux 4 = 𝐸4  𝐴4 , where 𝐸4 is the uniform
irradiance incident on the internal surface of the sphere. At the center of the integrating
sphere, an ideal diffusive cone was placed in order to diffuse the light coming out of the(LC), which is not Lambertian but rather directional (see Section 4.6), and which would
disturb the measurement of 𝐸4 . On window (w1), a Lambertian light source, of radiance
𝐿1 = 1,𝑖𝑛 /(  𝐴1 ), is applied, whose flux 1,𝑖𝑛 is partly reflected by (LC) and thenabsorbed by (a1) as 1 , and partly transmitted by (LC) to the integrating sphere as 𝑠𝑝ℎ ;this last flux being uniformly distributed on the sphere surface with irradiance 𝐸4 . Being
𝑅𝑆 = 100%, no flux is absorbed by the (LC). On the opening (w2) the radiance
𝐿2 = 𝐸2 / = 𝐸4 / is produced. Since both the Lambertian conductance, after themeasurements described in Fig. 4.8(a), and the radiance difference 𝐿 = 𝐿1 − 𝐿2 , are
known, the net flux 𝛷 𝑛𝑒𝑡 crossing the (LC) can be obtained applying Eq. (4.32).

Fig. 4.14. Schematic representation of the bidirectional Lambertian irradiation method (BLIM)
applied to a light cone (LC). (a) A Lambertian source is applied to the inlet opening of the opticalelement (LC) and its outlet opening is coupled to an integrating sphere; (b) A Lambertian source
is applied to the outlet opening of the LC and its inlet opening is coupled to an integrating sphere.

Fig. 4.14(b) shows the configuration used when applying an inverse light source on the(LC) outlet opening (w2), of flux 2,𝑖𝑛 and radiance 𝐿2 = 2,𝑖𝑛 /(  𝐴2 ). In absence ofabsorbed flux, and thanks to the particular shape of the truncated cone and to the specularsurface, we have no reflected flux from the (LC) in the direction 1  2, and then the total
flux 2,𝑖𝑛 is transmitted to (w1) and to the (is), where it is uniformly distributed on its
surface with irradiance 𝐸3 . On the opening (w1) the radiance 𝐿1 = 𝐸1 / = 𝐸3 / isproduced. Since both the Lambertian conductance, after the measurements described inFig. 4.8 (b), and the radiance difference 𝐿 = 𝐿1 − 𝐿2 , are known, the net flux 𝛷𝑛𝑒𝑡
crossing the (LC) in the 2  1 direction can be obtained applying Eq. (4.35). An exampleof 𝛷𝑛𝑒𝑡 calculation is reported below for the light cone (LC1) of l = 12mm length,
irradiated in direct mode (DLIM) (see Fig. 4.14(a)). The radii of window (w4) and of thesphere are 10 mm and 200 mm, respectively. The results of a raytracing are: 1,𝑖𝑛 =50 W; 1 = 37.776 W; 4 = 0.6784 W; flux on the sphere surface, 𝑠𝑝ℎ = 10.893 W.
We obtain: 𝐿1 = 3.5181104 W/m2sr; 𝐿2 = 687.32 W/m2sr; 𝐿 = 3.4494  104 W/m2sr.
Since all inverse rays are transmitted, we have:  𝑙𝑎𝑚𝑏
= 1,  𝑙𝑎𝑚𝑏
=1/𝐶𝑔𝑒𝑜 = 0.25, and
21
12
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Advances in Optics: Reviews. Book Series, Vol. 5

therefore: 𝐺 𝑙𝑎𝑚𝑏 =   𝐴2 =   𝐴1  0.25 = 3.5530510-4 m2sr. From Eq. (4.25) we obtainfor the net flux crossing the (LC1): 𝛷𝑛𝑒𝑡 = 12.256 W. To check the accuracy of this value,
it is sufficient to make the sum of the direct flux (taken positive) and inverse flux (taken
negative) crossing the opening (w1). In fact, as the (LC1) is not absorbing, the net flux
must necessarily be: 𝛷𝑛𝑒𝑡 = 1,𝑖𝑛 − 1 = 12.224 W, which results in good agreementwith the 12.256 W value obtained applying Eq. (4.25) (difference of only 0.3 %). The
resulting difference is due to the limited number of rays (50 k) supported by the opticalcode and to numerical approximations. There is no way to assign an error to thesemeasurements, other than to increase the number of rays in raytracing, but it happens that
this number has an intrinsic limit in the optical code (see the result of Fig. 4.5).
Another example of 𝛷𝑛𝑒𝑡 calculation is reported here for the same light cone (LC1) asabove, irradiated in reverse mode (ILIM) (see Fig. 4.14(b)). The results of a raytracing
are: 2,𝑖𝑛 = 40 W; 2 = 0.80689 W; 3 = 2.074 W; 𝑠𝑝ℎ = 35.024 W. To calculate 𝐿1 ,we use the flux 𝑠𝑝ℎ on the sphere surface free from the windows (w1) and (w3), and ofarea 𝐴𝑠𝑝ℎ . We have therefore: 1 = (𝑠𝑝ℎ 100)(𝐴1 ⁄𝐴𝑠𝑝ℎ ). We obtain: 1 = 3.157 W,
𝐿1 = 1⁄( 𝐴1 ) = 2.2213103 W/m2sr; 𝐿2 = 1.1258105 W/m2sr; 𝐿 =‒ 1.1036  104 W/m2sr. We have, as previously: 𝐺 𝑙𝑎𝑚𝑏 = 3.5530510-4 m2sr. From
Eq. (4.25) we obtain for the net flux crossing the (LC1) in direction 2  1: 𝛷𝑛𝑒𝑡 =39.211 W. To check the accuracy of this value, it is sufficient to make the relative sum of
the direct and inverse fluxes crossing the opening (w2). As the (LC1) is not absorbing, thenet flux must necessarily be: 𝛷𝑛𝑒𝑡 = 2,𝑖𝑛 − 2 = 39.193 W, in good agreement withthe 39.211 W value (difference of only 0.05%). Like above, the small difference is due tothe limited number of rays (40k) supported by the optical code and to numericalapproximations. It is possible to adopt a different configuration to apply the BLIM
method, that is to optically couple the two openings of the (LC) with two integrating
spheres, as shown in Fig. 4.15. This method, however, does not allow to verify theaccuracy of the 𝛷𝑛𝑒𝑡 value calculated by applying Eq. (4.25), as it is not possible to place
an ideal absorber on one of the openings of the (LC). To apply a Lambertian source on
one of the openings of the (LC), one integrating sphere must be illuminated by introducing
a collimated light source (ls) which impacts on a diffuser placed at the center of the sphere.
In Fig. 4.15 the (ls) is introduced in the left sphere, in this way replicating theconfiguration of Fig.4.14 (a). The results of a raytracing are: 𝑅𝑠𝑝ℎ1 = 𝑅𝑠𝑝ℎ2 = 99 %diffuse; 1,𝑖𝑛 = 30 W; 3 = 1.6757 W; 4 = 0.027329 W; flux on the sphere (is1)surface, 𝑠𝑝ℎ1 = 26.150 W; flux on the sphere (is2) surface, 𝑠𝑝ℎ2 = 0.50121 W. By
calculating the radiance 𝐿1 from 𝐸3 = 3 /𝐴3 and the radiance 𝐿2 from 𝐸4 = 4 /𝐴4 , weobtain: 𝐿1 =1.697835103 W/m2sr and 𝐿2 = 27.69 W/m2sr; 𝐿 = 1.670145103 W/m2sr.
𝑙𝑎𝑚𝑏
We have again:  𝑙𝑎𝑚𝑏
= 0.25, and then: 𝐺 12
=   𝐴1  0.25 = 3.5530510-4 m2sr.
12
𝑙𝑎𝑚𝑏
From Eq. (4.25) we obtain for the net flux crossing the (LC1): 𝛷𝑛𝑒𝑡 = 𝐺 12
 𝐿 =
0.59341 W. If, as an alternative, we calculate 𝐿1 starting from the irradiance 𝐸𝑠𝑝ℎ1 =𝑠𝑝ℎ1 /𝐴𝑠𝑝ℎ1 on the (is1) surface, where 𝐴𝑠𝑝ℎ1 is the area of the net (is1) surface, and 𝐿2
starting from the irradiance 𝐸𝑠𝑝ℎ2 = 𝑠𝑝ℎ2 /𝐴𝑠𝑝ℎ2 on the (is2) surface, where 𝐴𝑠𝑝ℎ2 is thearea of the net (is2) surface, we would obtain: 𝐿1 = 1.65849103 W/m2sr; 𝐿2 =𝑙𝑎𝑚𝑏31.7665 W/m2sr; 𝐿 = 1.670145  103 W/m2sr, and finally: 𝛷𝑛𝑒𝑡 = 𝐺 12
 𝐿 =
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Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

0.57798 W. The discrepancy between 𝛷𝑛𝑒𝑡 = 0.59341 W and 0.57798 W (~3%) is to be
attributed to the raytracing process, whose result is closely related to the number of rays
used, which cannot be pushed beyond a certain value, the processing limit of the opticalcode. Fig. 4.16 shows an example of raytracing with few rays of the BLIM method appliedwith two integrating spheres and illuminating the left sphere.

Fig. 4.15. Schematic representation of the bidirectional Lambertian irradiation method (BLIM)
applied to a light cone (LC). The left sphere is illuminated by the collimated light source (ls).

Fig. 4.16. Example of raytracing of the BLIM method applied by using two integrating spheres.
The higher density of rays can be seen on the left sphere, where the light source (s l) is located.

It is interesting to see what happens when the same light source (ls), with the same
intensity (30 W), is placed in the right sphere (is2), that is if the net flux through (LC)changes or not (see Fig. 4.17). The results of a raytracing are: 3 = 0.031466 W; 4 =
1.6704 W; 𝑠𝑝ℎ1 = 0.45363 W; 𝑠𝑝ℎ2 = 26.199 W. The flux distribution is now almost
mirrored. By calculating the radiance 𝐿1 from 𝐸3 = 3 /𝐴3 and the radiance 𝐿2 from
𝐸4 = 4 /𝐴4 , we obtain: 𝐿1 = 31.88175 W/m2sr and 𝐿2 = 1.69247103 W/m2sr; 𝐿 =
𝑙𝑎𝑚𝑏
𝑙𝑎𝑚𝑏
𝐿1 − 𝐿2 = ‒1.66059103 W/m2sr. We have again: 𝐺 21
= 𝐺 12= 3.55305 10-4 m2sr.
𝑙𝑎𝑚𝑏
From Eq. (4.25) we obtain for the net flux crossing the (LC1): 𝛷𝑛𝑒𝑡 = 𝐺 12
 𝐿 =‒0.59002 W. The minus sign indicates that the flux is flowing in the 2  1 direction. If
we calculate 𝐿1 starting from the irradiance 𝐸𝑠𝑝ℎ1 = 𝑠𝑝ℎ1 /𝐴𝑠𝑝ℎ1 on the (is1) surface,and 𝐿2 starting from the irradiance 𝐸𝑠𝑝ℎ2 = 𝑠𝑝ℎ2 /𝐴𝑠𝑝ℎ2 on the (is2) surface, we would
obtain: 𝐿1 =28.7702 W/m2sr; 𝐿2 = 1.66048103 W/m2sr; 𝐿 = ‒1.631712  103 W/m2sr,
𝑙𝑎𝑚𝑏and finally: 𝛷𝑛𝑒𝑡 = 𝐺 21
 𝐿 = ‒ 0.57975 W. The difference between
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𝛷𝑛𝑒𝑡 = ‒0.59002 W and ‒0.57975 W is of ~2%. We obtain finally that, moving from theconfiguration of Fig. 4.15 to that of Fig. 4.17, that is reversing the orientation of the (LC),and applying at input the same flux (30W), the net flux flowing through the (LC) in thetwo directions is practically the same.

Fig. 4.17. Schematic representation of the bidirectional Lambertian irradiation method (BLIM)
applied to a light cone (LC). The right sphere is illuminated by the collimated light source (ls).

4.5. Optical Conductance Simulation Results
4.5.1. Nonimaging Solar Concentrators (CPC)
The optical simulations of this section were made for specular or Lambertian diffusive
internal surface, with total reflectivity for monochromatic light ranging from 10 % to
100 %. The direct Lambertian transmittance,  𝑙𝑎𝑚𝑏
12 , of a CPC with 𝑎𝑐𝑐 = 10° (CPC10),
as function of reflectance of a specular surface, simulated following the DLIM describedin Fig. 4.8 (a), is reported, in percent, in Fig. 4.18. It is relatively low, but increases
monotonically with increasing surface reflectance 𝑅𝑆 , as expected, On the same graph, it
is reported the inverse Lambertian transmittance  𝑙𝑎𝑚𝑏21 /𝐶𝑔𝑒𝑜 , simulated following the
ILIM described in Fig. 4.8 (b), which perfectly overlaps  𝑙𝑎𝑚𝑏
12 , demonstrating the validityof Eq. (4.15). For a specular surface with 𝑅𝑆 = 100 %, all rays incident within 𝑎𝑐𝑐 = 10°2-2cross the CPC, giving the maximum value of  𝑙𝑎𝑚𝑏
12 : sin 𝑎𝑐𝑐 = 3.0154 10 , as obtained
from Eq. (4.11) integrating the angle  from 0° to 10°. Such a low optical transmittancevalue is not surprising, as the CPC10 only transmits light incident within 10° of the opticalaxis, very little compared to light at input having 90° divergence. The corresponding-3
OBLC, 𝐺 𝑙𝑎𝑚𝑏 , has the same behavior of  𝑙𝑎𝑚𝑏
12 , being just (  𝐴1 ) = 1.4294 10 times𝑙𝑎𝑚𝑏 12 (see Eq. (4.22’)).
Although solar concentrators normally work with specular surface, it can be interesting todivert our attention from their application and consider the effect that also a diffusive
surface can have on its transmittance properties. In Fig. 4.18 it is shown that a diffusive
surface strongly hinders the transmission of Lambertian light up to the highest surface
reflectance values and that, also with 𝑅𝑆 = 100 %, the diffusive surface exhibits aLambertian transmittance lower than that of a specular surface. The simulated direct202

Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

Lambertian reflectances  𝑙𝑎𝑚𝑏and absorbances  𝑙𝑎𝑚𝑏of the CPC10 with specular and1212diffusive surface, following DLIM, are shown in Fig. 4.19. We note that the absorbances
dominate the reflectances at low-medium values of 𝑅𝑆 . Furthermore, the Lambertianreflectance and absorbance curves show an opposite behavior, which is naturalconsidering the low level of Lambertian transmittance. The Lambertian reflectance andabsorbance referred to specular surface remain flat up to high values of 𝑅𝑆 and thenrapidly converge to ~ 1 and to ~ 0, respectively. Those referring to diffusive surface, on
the contrary, converge to ~ 1 and to ~ 0, respectively, in a softer way. Similar trends for𝑙𝑎𝑚𝑏
the quantities  𝑙𝑎𝑚𝑏and  𝑙𝑎𝑚𝑏have been observed for the CPC2.5, CPC5 and
12 ,  1212
CPC20. In the following we will report separately the single optical quantities to see theeffect that the shape of the different CPCs has on them.

CPC10, DLIM and ILIM
tau_dir_spec
tau_dir_diff
tau_inv_spec(/Cgeo)
tau_inv_diff(/Cgeo)

3,012lamb, 21lamb/Cgeo (%)

2,5

2,0

spec
1,5

1,0

diff
0,5

0,0
0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.18. Direct and inverse (/Cgeo) Lambertian transmittance, in percent, of the CPC10 vs.
the reflectance, in percent, of specular and diffusive surface.
100

12lamb (%), 12lamb (%)

80

CPC10, DLIM
rho_dir_spec
alpha_dir_spec
rho_dir_diffalpha_dir_diff

60

40

20

0

0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.19. Direct Lambertian reflectance and absorbance, in percent, of the CPC10 vs.
the reflectance, in percent, of specular and diffusive surface. Method: DLIM.

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Advances in Optics: Reviews. Book Series, Vol. 5𝑙𝑎𝑚𝑏
We start with the most relevant quantity, the specific Lambertian conductance, 𝐺 𝑠𝑝𝑒𝑐,
𝑙𝑎𝑚𝑏whose knowledge allows us to derive the Lambertian conductance 𝐺of an elementof the same shape and of any size, simply multiplying it by the area of the entrance
𝑙𝑎𝑚𝑏opening, or to derive  𝑙𝑎𝑚𝑏
simply dividing it by . The 𝐺 𝑠𝑝𝑒𝑐
calculated applying the
12
DLIM method are reported in Fig. 4.20 for four tested CPCs. We can note, apart the𝑙𝑎𝑚𝑏increasing of 𝐺 𝑠𝑝𝑒𝑐with 𝑅𝑆 already anticipated by Fig. 4.18, also its evident increase
moving from low-𝑎𝑐𝑐 to high-𝑎𝑐𝑐 CPCs, a natural trend considering that the increase of
𝑎𝑐𝑐 means greater dimensions of the exit opening with respect to the entrance opening,𝑙𝑎𝑚𝑏
and then higher optical transparency. We note further the fact that the increasing of 𝐺 𝑠𝑝𝑒𝑐with 𝑅𝑆 following a near-linear trend tends to diverge the curves. But there is another𝑙𝑎𝑚𝑏interesting data to be analyzed in the 𝐺 𝑠𝑝𝑒𝑐
curves, namely the value reached by the
curves for 𝑅𝑆 = 100 %. At this condition, considering that the surface is specular, theinverse Lambertian transmittance  𝑙𝑎𝑚𝑏must necessarily be equal to 1. In fact, due to the21
shape of any CPC, any ray direct from the exit to the entrance opening crosses the CPC
𝑙𝑎𝑚𝑏undisturbed. So, as  𝑙𝑎𝑚𝑏
= 1, we have 𝐺 𝑙𝑎𝑚𝑏 =   𝐴2 and 𝐺 𝑠𝑝𝑒𝑐
= 𝜋⁄𝐶𝑔𝑒𝑜 . These
21
are, effectively, the values that we find in Fig. 4.20 taking into account the 𝐶𝑔𝑒𝑜 values
reported in Table 4.1.

0,40
CPC20
CPC10
CPC5
CPC2.5Specular surface

0,35
0,30

Gspeclamb (sr)

0,25
0,20
0,15
0,10
0,05
0,00
0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.20. The specific bidirectional Lambertian conductance of four tested CPCs as functionof reflectance of a specular surface.

Fig. 4.21 reports the direct Lambertian reflectance  𝑙𝑎𝑚𝑏of the four CPCs. It is
12interesting to note that, at high 𝑅𝑆 values, the reflectance increases as 𝑎𝑐𝑐 increases, thatis as the exit opening increases, which seems strange at a first glance. In reality, an
increase in 𝑎𝑐𝑐 involves a reduction in length of the CPC and then a lower optical pathof light, a lower number of internal reflections, a lower surface absorption and, ultimately,a higher reflectance, being reflectance and absorbance almost complementary (seeFig. 4.22) due to the relatively low values of transmittance.
204Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

100
CPC20
CPC10
CPC5
CPC2.5Specular surface

12lamb (%)

80

60

40

20

0

0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.21. The direct Lambertian reflectance of the four tested CPCs, as function of reflectanceof a specular surface.
100

80
CPC2.5
CPC5
CPC10
CPC20
Specular surface

12lamb (%)

60

40

20

0

0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.22. The direct Lambertian absorbance of the four tested CPCs, as function of reflectanceof a specular surface.

Although the use of diffusive surfaces is not contemplated in traditional solar
concentrators, be they thermal, thermodynamic or photovoltaic, it may still be interesting
to see if the use of diffusive surface can give rise to unexpected results that can eventually
be exploited for other types of application. Indeed, an interesting result arises from thecomparison of Lambertian reflectance and of Lambertian absorbance in the various tested
CPCs. The two quantities are reported in Figs. 4.23 and 4.24, respectively.
The unexpected and surprising result is the invariance, respect to 𝑎𝑐𝑐 , of the curves ofLambertian reflectance and of Lambertian absorbance. This property seems therefore
intrinsic to the particular shape of these nonimaging solar concentrators.
205

Advances in Optics: Reviews. Book Series, Vol. 5

100
CPC20
CPC10
CPC5
CPC2.5
Diffusive surface

12lamb (%)

80

60

40

20

0
0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.23. The direct Lambertian reflectance of the four tested CPCs, as function of reflectanceof a diffusive surface.
100

80

12lamb (%)

60

CPC2.5
CPC5
CPC10
CPC20
Diffusive surface

40

20

0

0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.24. The direct Lambertian absorbance of the five tested CPCs, as function of reflectanceof a diffusive surface.

4.5.2. Light Cones (LC)In this section we report only the results obtained by simulating the first series of LCs (seeTable 4.2). The simulated direct Lambertian transmittance, reflectance and absorbance ofa light cone (LC2) of 24 mm length, for specular and diffuse surface, are shown in
Fig. 4.25. Given the geometric similarity between a CPC and a truncated cone (both havean inlet opening greater than the outlet opening and the cross section of the element isreduced monotonously going from the inlet to the outlet opening), the optical quantities
shown in Fig. 4.25 for the light cone LC2 resemble those shown in Figs. 4.18 and 4.19 for
the CPC10. That is, we find a Lambertian transmittance that grows with 𝑅𝑆 for both aspecular and a diffusive surface, where the second remains lower than the first; then wefind a Lambertian reflectance that grows with 𝑅𝑆 for both a specular and a diffusive
surface, where however the second remains higher than the first one, that is, there is asituation opposite to that observed for transmittance; finally we find a Lambertian206

Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

absorbance that slowly decreases with the increase of 𝑅𝑆 both with a specular and adiffusive surface, where the second remains lower than the first one, as it happens in thecase of transmittance. We again find, as we found for the CPC10, that Lambertianreflectance shows qualitatively an opposite behavior respect to that of Lambertianabsorbance. We find a similar behavior in the other light cones, LC1, LC3 and LC4, andthen we do not report here graphs similar to that of Fig. 4.25. However, as made before
describing the CPCs, we find it useful to compare the same optical quantity in the four𝑙𝑎𝑚𝑏different light cones. The specific bidirectional Lambertian conductance, 𝐺 𝑠𝑝𝑒𝑐,
calculated applying the DLIM method, of the light cones LC1, LC2, LC3 and LC4 vs. the𝑙𝑎𝑚𝑏
𝑅𝑆 of a specular surface, is reported in Fig. 4.26. We note, apart the increasing of 𝐺 𝑠𝑝𝑒𝑐with 𝑅𝑆 already seen for the CPCs in Fig. 4.20, also its evident increase at decreasing the
LC length.

12lamb, 12lamb, 12lamb, (%)

100

80
LC2_DLIM
tau_spec
rho_specalpha_spec
tau_diff
rho_diff
alpha_diff

60

40

20

0
0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.25. Lambertian transmittance, reflectance and absorbance of the LC2, of 24 mm length, areshown as function of specular and diffuse surface reflectance.
0,9
LC1
LC2
LC3
LC4
Specular surface

0,8
0,7

Gspeclamb (sr)

0,6
0,5
0,4
0,3
0,2
0,1
0,0
0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.26. The specific bidirectional Lambertian conductance of the four tested light cones isreported as function of reflectance of the specular surface. The low index "spec" in the symbol
𝑙𝑎𝑚𝑏
𝐺 𝑠𝑝𝑒𝑐means "specific" and not "specular".

207

Advances in Optics: Reviews. Book Series, Vol. 5
𝑙𝑎𝑚𝑏
But the most interesting aspect to note in the 𝐺 𝑠𝑝𝑒𝑐
curves is their convergence to thesame maximum value in correspondence to 𝑅𝑆 = 100 %, a value around 0.8 sr. The
𝑙𝑎𝑚𝑏
explanation of this result is the same used to comment the 𝐺 𝑠𝑝𝑒𝑐
's results of the CPCs,
namely that, considering that the surface is specular and that the inverse Lambertian𝑙𝑎𝑚𝑏transmittance  𝑙𝑎𝑚𝑏must necessarily be equal to 1, we finally obtain 𝐺 𝑠𝑝𝑒𝑐
= 𝜋⁄𝐶𝑔𝑒𝑜 .
21
But now, differently from the CPCs, 𝐶𝑔𝑒𝑜 is the same for all the light cones, and then the𝑙𝑎𝑚𝑏
𝐺 𝑠𝑝𝑒𝑐at 𝑅𝑆 = 100 % must be equal to 𝜋⁄4 ≃ 0.7854. We can affirm, therefore, that the
optical "transparency" to Lambertian light of a light cone with a surface equal to an ideal
mirror (𝑅𝑆 = 100 %) is independent of its length when the dimensions of the entrance andexit opening are kept unchanged. This is a really interesting result because not so intuitive.
Things are different when the inner surface of the light cones is diffusive and Lambertian,as it is shown in Fig. 4.27. In this regard, I remember that a Lambertian diffusive surface
is such when the total reflectance is independent of the incidence angle of the light and𝑙𝑎𝑚𝑏when the radiance of the diffused light is constant [57, 58]. The quantity 𝐺 𝑠𝑝𝑒𝑐
now
tends, as 𝑅𝑆 tends to 100 %, to different values for the different models of light cones.
Fig. 4.28 shows the direct Lambertian reflectance  𝑙𝑎𝑚𝑏as function of 𝑅𝑆 of a specular
12
surface for the four light cones. The direct Lambertian absorbance  𝑙𝑎𝑚𝑏is shown in
12
Fig. 4.29. It is interesting to observe their opposite behavior and the convergence of all
curves, for both  𝑙𝑎𝑚𝑏and  𝑙𝑎𝑚𝑏
12 , at the two ends of the range of values of 𝑅𝑆 . Within12
𝑙𝑎𝑚𝑏these two extremes, the  12 and  𝑙𝑎𝑚𝑏curves are clearly distinguishable and well
12
separated. If we now turn to consider diffusive surfaces, their effect is to compact the
curves, even if not to overlap them as it has been observed with the CPCs concentrators.
The curves of  𝑙𝑎𝑚𝑏and  𝑙𝑎𝑚𝑏for diffusive surface are shown in Figs. 4.30 and 4.31,
1212respectively.

0,7

LC1
LC2
LC3
LC4
Diffusive surface

0,6

Gspeclamb (sr)

0,5
0,4
0,3
0,2
0,1
0,0
0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.27. The specific bidirectional Lambertian conductance of the four tested light cones isreported as function of reflectance of the diffusive surface.

208

Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications
80

LC1
LC2
LC3
LC4
Specular surface

12lamb (%)

60

40

20

0

0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.28. Direct Lambertian reflectance of the four tested light cones as function of reflectanceof a specular surface.
100

80

12lamb (%)

60
LC4
LC3
LC2
LC1
Specular surface

40

20

0

0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.29. Direct Lambertian absorbance of the four tested light cones as function of reflectanceof a specular surface.
100
LC1
LC2
LC3
LC4
Diffusive surface

80

12lamb (%)

60

40

20

0

0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.30. Direct Lambertian reflectance of the four tested light cones as function of reflectanceof a diffusive surface.

209Advances in Optics: Reviews. Book Series, Vol. 510080

12lamb (%)

60

40

LC4
LC3
LC2
LC1
Diffusive surface

20

0

0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.31. Direct Lambertian absorbance of the four tested light cones as function of reflectanceof a diffusive surface.

4.5.3. Solar Tunnels (ST)The simulated direct Lambertian transmittance, reflectance and absorbance of a solar
tunnel (ST3) of 48 mm length, for specular and diffuse surface, are shown in Fig. 4.32.

100

80
ST3, DLIM
tau_spec
rho_specalpha_spec
tau_diff
rho_diff
alpha_diff

12lamb (%)

60

40

20

0

0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.32. Lambertian transmittance, reflectance and absorbance of the solar tunnel ST3,of 48 mm length, as function of specular and diffuse surface reflectance.

From Fig. 4.32 we first note, as expected, that the specular Lambertian reflectance is
identically equal to zero, since there is no possibility for a light beam to be reflected back.
A notable result is that the Lambertian absorbance is practically the same for specular anddiffusive surface; soon we will see if this also happens for the other STs. We also notethat the Lambertian transmittance for specular surface is significantly greater than that for
diffusive surface, a result that has been reported for all the tested optical components,
because the diffusive surface facilitates the back reflection of the rays. In the case of CPCsand LCs, however, the back reflection of light rays was also favored by the particular
210

Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

profile of the surface, while in this case, with a flat profile of the surface, the only meansthat allows the back reflection of light rays is just the surface when it is diffusive. Similar
behaviors have been found with the other solar tunnels. So, let us see how the single
optical quantities change passing from one solar tunnel to another. Let us start, as we havedone so far, by considering the specific Lambertian conductance, which summarizes thetransmissive properties of the optical element with respect to a Lambertian irradiation,
and which is proportional to both the Lambertian transmittance and conductance, which𝑙𝑎𝑚𝑏
we can therefore avoid plotting. The specific Lambertian conductance, 𝐺 𝑠𝑝𝑒𝑐
, calculated
applying the DLIM method, of the solar tunnels ST1, ST2, ST3 and ST4 vs. the 𝑅𝑆 for a𝑙𝑎𝑚𝑏specular surface, is reported in Fig. 4.33. We note the increasing of 𝐺 𝑠𝑝𝑒𝑐
at increasingof 𝑅𝑆 and at decreasing of the ST length, as we have observed in both CPCs and LCs. It
𝑙𝑎𝑚𝑏
is interesting to note the convergence of 𝐺 𝑠𝑝𝑒𝑐to the same maximum value of  (sr) for
𝑅𝑆 = 100 %. In this case, in fact, as 𝐶𝑔𝑒𝑜 = 1.0 and the surface is specular, the Lambertiantransmittance is unitary in the two directions, and then we have:
𝑙𝑎𝑚𝑏
𝑙𝑎𝑚𝑏
𝐺 𝑠𝑝𝑒𝑐=    𝑙𝑎𝑚𝑏
= (⁄𝐶𝑔𝑒𝑜 ) 
=    𝑙𝑎𝑚𝑏
= 
12
21
21

(4.39)

3.5
ST1
ST2
ST3
ST4
Specular surface

3.0

Gspeclamb (sr)

2.5

2.0

1.5

1.0

0.5

0.0
0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.33. Specific bidirectional Lambertian conductance of the four tested solar tunnels
𝑙𝑎𝑚𝑏
as function of reflectance of the specular surface. The low index "spec" of 𝐺 𝑠𝑝𝑒𝑐
means "specific".

Things are different when the inner surface of the solar tunnels is diffusive and𝑙𝑎𝑚𝑏Lambertian, as it is shown in Fig. 4.34. We again note the increasing of 𝐺 𝑠𝑝𝑒𝑐at𝑙𝑎𝑚𝑏increasing of 𝑅𝑆 and at decreasing of the ST length, but the quantity 𝐺 𝑠𝑝𝑒𝑐
now tends, as
𝑅𝑆 tends to 100 %, to different values for the different models of solar tunnels.
Fig. 4.35 shows the direct Lambertian reflectance  𝑙𝑎𝑚𝑏of the four solar tunnels as
12function of 𝑅𝑆 for specular and diffusive surfaces. The data for the specular surface werereported equally although identically null. The direct Lambertian absorbance  𝑙𝑎𝑚𝑏is12
shown in Fig. 4.36 for both specular and diffusive surfaces. For the Lambertian reflectanceat diffusive surface, we now find that it starts the same, near zero, for the four STs at low
values of 𝑅𝑆 , but then diverges as 𝑅𝑆 increases, differently from what has been observed211

Advances in Optics: Reviews. Book Series, Vol. 5

both in the CPCs and in the LCs. We have no more the convergence of curves at the two
extremes of the range of 𝑅𝑆 values. A similar but opposite behavior is found in theLambertian absorbance (Fig. 4.36), which starts the same, zero, for the four STs at
𝑅𝑆 = 100 % and then diverges as 𝑅𝑆 decreases. What we now observe is that the diffusive
surface no longer compacts the Lambertian reflectance or absorbance curves. In particular,
we note an interesting result, that is that the Lambertian absorbance changes very little
when passing from the specular to the diffusive surface, while remaining well different in
the different solar tunnels. If we now want to see more accurately what happens to theabsorbance curves, we note that, starting from ST1 and going towards the longer solar
tunnels, we find the following: the absorbance curve for diffusive surface exceeds that for
specular surface in ST1; the same happens in the ST2; in ST3 the two curves overlap;
finally, in the ST4 the two curves swap places. So, there is a precise evolution of the two
curves of  𝑙𝑎𝑚𝑏as the length of the solar tunnel increases.
12
2,5

ST1
ST2
ST3
ST4
Diffusive surface

2,0

Gspeclamb (sr)

1,5

1,0

0,5

0,0

0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.34. The specific Lambertian conductance of the four tested solar tunnels is reported
as function of reflectance of the diffusive surface.

80
ST4_diff
ST3_diff
ST2_diff
ST1_diff
ST4_spec
ST3_spec
ST2_spec
ST1_spec

7060

12lamb (%)

50

diff

40
30
20

spec

10
0
-10
0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.35. Direct Lambertian reflectance of the four tested solar tunnels as function of reflectanceof specular and diffusive surface.

212

Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications
100

80

12lamb (%)

60

40alpha_diff_ST4
alpha_spec_ST4
alpha_diff_ST3
alpha_spec_ST3
alpha_diff_ST2
alpha_spec_ST2
alpha_diff_ST1
alpha_spec_ST1

20

0

0

20

40

60

80

100

Surface reflectance, RS (%)

Fig. 4.36. Direct Lambertian absorbance of the four tested solar tunnels as function of reflectanceof specular and diffusive surfaces.

4.6. Exit Radiance Simulation Results
In the Prologue, I explained that, applying to one of the two openings of the opticalelement a Lambertian radiation, the radiation coming out from the opposite opening is notnecessarily Lambertian. The inverse Lambertian method, ILIM, in fact, is based precisely
on the fact that the angular distribution of the radiance leaving the entrance opening whenthe exit one is irradiated with Lambertian light, is the same as that of the angular
distribution of the light collecting efficiency of a collimated beam incident on the entrance
opening. And this efficiency distribution is certainly restricted in angular terms to theextent that the acceptance angle is defined, which, as we know, is always less than about
20°. Hence, if we consider a SC, whose surface is certainly specular, its inverse radiancewill be very different from the incident Lambertian one, which has an angular divergenceof 90°. If the SC had a diffusive surface, then the inverse radiance exiting the entrance
opening could have a higher angular divergence, up to 90°. In this Section, I am interested
to exploit how is the radiance distributed out of the inlet opening of an optical element,
when the exit opening is inversely irradiated with a Lambertian radiation. The aim is toconsider the optical element as a source of light, whose emitted radiance will be
distributed in space differently according to the properties imposed on the internal surface.
The reason why we will only examine the radiance exiting the entrance opening is thatthe inverse Lambertian transmittance  𝑙𝑎𝑚𝑏is 𝐶𝑔𝑒𝑜 times larger than the direct one,21𝑙𝑎𝑚𝑏
 12 ; and this is an important aspect if we want to have a source that emits light withhigh efficiency. For all the optical elements, we will consider surfaces of specular anddiffusive type with 𝑅𝑆 = 100 %.
4.6.1. Nonimaging Solar Concentrators (CPC)
The angular distribution of the radiance emitted by four CPCs is shown in Fig. 4.37. As
already discussed in Section 4.2, these concentrators have an optical efficiency, with213

Advances in Optics: Reviews. Book Series, Vol. 5

respect to a collimated beam entering the aperture (1), of step-like shape, and thisefficiency curve can be obtained, thanks to the ILIM method, by irradiating the aperture(2) (the smallest one) with Lambertian light and measuring the radiance exiting theaperture (1), which is what we did with the simulations shown in Fig. 4.37. The radianceis step-like shaped and the on / off transition occurs exactly at the angle of acceptance𝑎𝑐𝑐 = 2.5°, 5°, 10° and 20° for the CPCs: CPC2.5, CPC5, CPC10 and CPC20,
respectively. Similar step-like curves were obtained with the CPC30 and CPC40. The exit
radiance curves were obtained with a specular 𝑅𝑆 = 100 %, which is the condition
very close to that in which solar concentrators operate. Real reflective solar concentrators
work with slightly lower reflectances (about 95 %), but this does not change the results
much. Here we will always report the normalized output radiance, 𝐿𝑛𝑜𝑟𝑚 () (seeSection 4.4.3), because we are mainly interested in studying its angular distribution, morethan its absolute intensity. The actual value of radiance at 0° will depend on the flux
entered into the aperture (2) that is on the imposed inverse radiance 𝐿𝑖𝑛𝑣 , but this is anarbitrary choice.
1,2

CPC20
CPC10
CPC5
CPC2.5

Normalized radiance

1,0Inverse irradiation, RS = 100% specular

0,8

0,6

length

0,4

0,2

0,0
-30

-20

-10

0

Exit angle, 1,out (°)

10

20

30

Fig. 4.37. Angular distribution of the exit radiance from four CPCs with specular surface,when irradiated by a Lambertian radiation incident on the exit aperture.

Here we are considering solar concentrators which are designed to minimize the reflection
losses, and then the internal surface is necessarily specular. However, it is interesting tosee how the angular distribution of radiance changes if the surface is diffusive. In order
not to field too many parameters, I have chosen to consider only a diffusive Lambertiansurface with 𝑅𝑆 = 100 %. Fig. 4.38 shows the radiance exiting all the CPCs of Table 4.1
with diffusive surface. The first thing to note is the loss of the step-like profile and theemergence of a Lorentzian-type shape that extends up to 90° The measured acceptanceangles for six CPCs are reported in Table 4.3, and are compared to the ones for specularsurface. From Table 4.3 we note the widening of the curve for CPC2.5, CPC5 and CPC10,
but the shrinking of the curve for the CPC20, CPC30 and CPC40 when moving from thespecular to the diffusive surface.

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Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

Inverse irradiation. RS=100%diffuse

Normalized radiance

1,0

CPC40
CPC30
CPC20
CPC10
CPC5
CPC2.50,80,6length
0,4

0,2

0,0
-80

-60

-40

-20

0

20

40

60

80

Exit angle, 1,out (°)

Fig. 4.38. Angular distribution of the exit radiance from the CPCs with diffusive surface,when irradiated by a Lambertian radiation incident on the exit aperture.
Table 4.3. Acceptance angles of the various CPCs with specular and diffusive surface
with 𝑅𝑆 = 100 %.

Element
CPC2.5
CPC5
CPC10
CPC20
CPC30
CPC40

𝒂𝒄𝒄 (°)
Specular Diffusive
2.5
4.1
5.0
8.1
10.0
11.6
20.0
19.0
30.0
27.0
40.0
35.2

𝜟𝒂𝒄𝒄 (%)
Spec  Diff
+64
+62
+16
‒5
‒10
‒12

4.6.2. Light Cones (LC)The choice of studying the angular distribution of the inverse radiance emitted by light
cones characterized by the same entrance and exit apertures, that is by the same 𝐶𝑔𝑒𝑜 , butof different length, has led to very interesting and unexpected results. Already the analysisof the radiance emitted by the four LCs studied in Section 4.5.2, LC1-LC4, of which wehave learned about their optical transmission and conduction properties, has shown a firstinteresting result (see Fig. 4.39). The radiance profile for LC1 is very particular, as it istriangular in shape. Moving to the longer CPCs, the profile narrows at the sides, while itwidens at the top, flattening and becoming of the step-like type, particularly for the LC4,
whose profile closely resembles that of a CPC with specular surface and𝑎𝑐𝑐 = 30°. Here it must be considered that, if an LC4 has a step-like profile similar toan ideal CPC, this means that, if it were used as a solar concentrator, it would show an
optical efficiency curve equal to that of the emitted radiance, that is step-like, because thisis what the ILIM method teaches, and therefore similar to that of an ideal CPC.
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Advances in Optics: Reviews. Book Series, Vol. 5

Furthermore, it has been found that the on-axis direct optical efficiency of the LC4,
𝜂𝑑𝑖𝑟 (0°), is equal to unity, which causes the LC4 to behave almost exactly like a CPC30,
that is a CPC with 𝑎𝑐𝑐 = 30°, even if with different dimensional values. The otherinteresting aspect that arises from Fig. 4.39 is the fact that the width at half height of thecurves, that is the acceptance angle, remains almost constant and around 30°. Probably
the almost constant 𝑎𝑐𝑐 is due to the fact that these light cones have the same 𝐶𝑔𝑒𝑜 = 4.
It would be interesting, therefore, to deepen these results, to find out how things go withlight cones still characterized by the same 𝐶𝑔𝑒𝑜 , but different from 4. So I thought toincrease the range of optical elements that characterize the first series with 𝐶𝑔𝑒𝑜 = 4, ofdimensions: a = 12, a' = 6, l = variable (which from now on we will indicate with the triadof values: (a, a', l) = (12, 6, l)), by introducing three other series of elements, the second
one with dimensional parameters (a, a', l) = (12, 5, l) and 𝐶𝑔𝑒𝑜 = 5.76, the third one withdimensional parameters (a, a', l) = (12, 4, l) and 𝐶𝑔𝑒𝑜 = 9, and the forth one withdimensional parameters (a, a', l) = (12, 3, l) and 𝐶𝑔𝑒𝑜 = 16. Since Fig. 4.39 shows with
LC1 (12, 6, 12) a triangular shape, which becomes a step-like shape as the length
increases, I have expanded this first series of elements, adding light cones shorter thanLC1 and longer than LC4. If we start from low values of length and gradually increase it,
we find a very wide and trapezoidal curve which tends to shrink gradually, keeping thesides of the trapezium almost parallel, until it becomes a triangular curve incorrespondence of LC1 (see Fig. 4.40a). There is therefore a first phase in which the curve
narrows, while maintaining a trapezoidal shape, and LC1 is the final element of thisshrinking process, with triangular shape.

Fig. 4.39. Angular distribution of the exit radiance from the light cones with specular surface,when irradiated by a Lambertian radiation incident on the exit aperture.

Starting from LC1, the increase in the length of the light cone does not change the widthof the curve very much, rather it transforms into an increasingly squared curve (seeFig. 4.40b). This curve will tend to a perfect rectangular shape by increasing the length
arbitrarily, because, having imposed a unitary and specular surface reflectance, no losses
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Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

are expected during the path of the light. Before moving on to the results obtained withthe other three series of optical elements, it is worthwhile to focus on some aspects ofthese curves that deserve to be analyzed. The first one is the trend of the width at halfheight of the curves, that is of 𝑎𝑐𝑐 , as a function of the length of the light cone. There is
therefore a first phase in which 𝑎𝑐𝑐 is reduced, which we can call the “trapezoidal phase”,
ending with the triangular curve. Then there is the “squaring phase”, which begins withthe triangular curve and ends with a quasi-squared curve. This phase is characterized byan almost constant 𝑎𝑐𝑐 , which must be related to the 𝐶𝑔𝑒𝑜 of the series. It is precisely thefact that these two phases of the curve exist, with the transition at the triangular curve, that
I show the radiance profiles in two separate graphs, one for the trapezoidal phase (see Fig.
4.40 a) and the other for the squaring phase (see Fig. 4.40 b). What happens to the otherseries is qualitatively the same as that described so far, with the difference that the valueof the length of the light cone changes at the trapezoidal / squaring transition (see Figs.
4.41, 4.42 and 4.43). We call this length as the “critical length”, 𝑙𝐶 (𝐶𝑔𝑒𝑜 ) a function of
𝐶𝑔𝑒𝑜 only. At the critical length, we then identify a “critical acceptance angle”, also afunction of 𝐶𝑔𝑒𝑜 : 𝑎𝑐𝑐,𝐶 (𝐶𝑔𝑒𝑜 ).

1,0

1,0
length0,8Normalized radiance

Normalized radiance

0,8

0,6

0,4

1st series
LC(12/6/8)
LC(12/6/9)
LC(12/6/12)
Rs=100% spec.

0,2

length0,6
1st series
LC(12/6/144)
LC(12/6/96)
LC(12/6/48)
LC(12/6/24)
LC(12/6/12)
Rs=100% spec.

0,4

0,2

0,0

0,0
-80

-60

-40

-20

0

20

Exit angle, 1,out (°)

(a)

40

60

80

-60

-40

-20

0

20

40

60

Exit angle, 1,out (°)(b)

Fig. 4.40. Profiles of the normalized inverse radiance of the first series of LCs with dimensions(12/6/l). (a) “Trapezoidal” phase with length varying from 8 to 12 mm; (b) “Squaring” phasewith length varying from 12 to 144 mm.

In Fig. 4.44 are reported the values of 𝑎𝑐𝑐 as function of the LC length for the four seriesof simulations. As we can see, the trend of 𝑎𝑐𝑐 is the same for the four series. We observe
at first a sharp drop of the angle (the trapezoidal phase), the reaching of a minimum
followed by a small and quick recovery (the transition phase), and finally a stabilization,
albeit associated with a very slight growth (the squaring phase). Significant parameters
can be extracted in correspondence of the minimum of the curves, to which we associate
the critical acceptance angle 𝑎𝑐𝑐,𝐶 and the critical length 𝑙𝐶 , and in correspondence ofhigh LC lengths, to which we associate, for simplicity, the angle to infinity, (𝐼𝑁𝐹 in the figure), which however can be only approximated. The three significant
parameters for the four series of simulations on the light cones are reported in Fig. 4.45
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Advances in Optics: Reviews. Book Series, Vol. 5

as function of 𝐶𝑔𝑒𝑜 . From Fig. 4.45 we see that the critical acceptance angle decreases as
the geometric concentration ratio of the light cone increases; the same trend is observed
for the angle to infinity, which is slightly higher than it. The critical length, on the other
hand, definitely increases as the geometric concentration ratio increases. These results canbe advantageously exploited if, for example, a light source with constant radiance withina certain exit angle is to be realized. If a tighter exit angle is desired, the geometricconcentration ratio and also the length of the LC will have to be increased. A discussion
about this application is reported in Section 4.7.2.

1,0

1,0
length0,8Normalized radiance

Normalized radiance

0,8

0,6
2nd series
LC(12/5/10)
LC(12/5/12)
LC(12/5/16)
LC(12/5/18)
Rs=100% spec.

0,4

0,2

length0,6
2nd series
LC(12/5/144)
LC(12/5/72)
LC(12/5/48)
LC(12/5/36)
LC(12/5/24)
LC(12/5/20)
LC(12/5/18)
Rs=100% spec.

0,4

0,2

0,0

0,0

-50

-60

-40

-20

0

20

40

-40

-30

-20

60

-10

0

10

20

30

40

50

Exit angle, 1,out (°)

Exit angle, 1,out (°)

(a)

(b)

Fig. 4.41. Profiles of the normalized inverse radiance of the second series of LCs
with dimensions (12/5/l). (a) “Trapezoidal” phase with length varying from 10 to 18 mm;(b) “Squaring” phase with length varying from 18 to 144 mm.

1,0

1,0

0,8

0,8

Normalized radiance

Normalized radiancelength0,60,4
3rd series
LC(12/4/12)
LC(12/4/16)
LC(12/4/20)
LC(12/4/24)
Rs=100% spec.

0,2

length0,6

3rd series
LC(12/4/144)
LC(12/4/96)
LC(12/4/48)
LC(12/4/36)
LC(12/4/24)
Rs=100% spec.

0,4

0,2

0,0

0,0

-60

-40

-20

0

Exit angle, 1,out (°)

(a)

20

40

60

-40

-30

-20

-10

0

10

20

30

40

Exit angle, 1,out (°)(b)

Fig. 4.42. Profiles of the normalized inverse radiance of the third series of LCs with dimensions(12/4/l). (a) “Trapezoidal” phase with length varying from 12 to 24 mm; (b) “Squaring” phasewith length varying from 24 to 144 mm.

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Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

1,0

1,0
lengthNormalized radiance

Normalized radiance

0,8

0,6

0,4

4th series
LC(12/3/16)
LC(12/3/20)
LC(12/3/24)
LC(12/3/36)
Rs=100% spec.

0,2

0,8

length0,64th series
LC(12/3/144)
LC(12/3/96)
LC(12/3/72)
LC(12/3/48)
LC(12/3/36)
Rs=100% spec.

0,4

0,2

0,0

0,0

-50

-40

-30

-20

-10

0

10

20

30

40

50

-30

-20

-10

0

10

20

30

Exit angle, 1,out (°)

Exit angle, 1,out (°)

(a)

(b)

Fig. 4.43. Profiles of the normalized inverse radiance of the fourth series of LCs with dimensions(12/3/l). (a) “Trapezoidal” phase with length varying from 16 to 36 mm; (b) “Squaring” phasewith length varying from 36 to 144 mm.
70
Light cones
1st series (Cgeo = 4)

Acceptance angle, acc (°)

60

2nd series (Cgeo = 5,76)50

3rd series (Cgeo = 9)
4th series (Cgeo = 16)

40
30
20
10
0
0

20

40

60

80

100

120

140

160

Length, l (mm)Fig. 4.44. Acceptance angles of the four series of simulations as function of LC length.

But there is one last aspect to consider: the curves of normalized radiance, in the “squaring
phase”, all intersect on a common point, which has as coordinates a certain angle of light
output, 1,𝑜𝑢𝑡 , and a certain level of normalized radiance. I don't know the meaning to thispoint, and therefore I limit myself to reporting the coordinates for the four series ofmeasures: 1st series: (30.2°; 0.46), 2nd series: (25.0°; 0.44); 3rd series: (20.0°; 0.40);
4th series: (14.8°; 0.42). As we can see, these points, where the curves converge, lie below
the point where the acceptance angle is measured (at half height of the curve), and havean angular coordinate slightly higher than 𝑎𝑐𝑐,𝐶 and  , , and a coordinate of normalizedradiance less than 0.5.
219

40

40

35

35

30

30

25

25

20

20

15

15
Angle to infinity
Critical angle
Critical length

105lC (mm)acc,C (°), INF (°)

Advances in Optics: Reviews. Book Series, Vol. 5

10
5

0

0
4

6

8

Cgeo

10

12

14

16

Fig. 4.45. The three significant parameters extracted by the curves of exit radiancefrom the four series of simulations on the light cones.

If we want to see now what are the effects that a diffusive surface has on the distributionof the exit radiance, we limit ourselves to examine only the light cones LC1-LC4, whose
normalized radiance curves in presence of a specular surface are shown in Fig. 4.39. We
find a substantial change in the curves, as shown in Fig. 4.46. At first, we notice that the
curves extend up to 90° angles, as always happens in the presence of diffusive surfaces,
and that their flat top decreases in width as the length increases. Also, the width of theradiance curves at half height narrows as the length of the LC increases, similarly to what
happened for the CPCs, although these two elements are of different shape. The
acceptance angle values measured for the radiance curves of Fig. 4.46 are reported inTable 4.4, compared with those relating to the specular surface. We find that the diffusive
surface has the effect of significantly widening the LC1 curve; the enlargement is reducing
moving to LC2 and transforms in a narrowing for LC3 and LC4. A similar trend, even if
less pronounced, is found in the CPCs (see Table 4.3). Summarizing, the diffusive surface,in the light cones, has shown a very strong effect on the acceptance angle, much morethan in the CPCs.
Table 4.4. Acceptance angles of four LCs with specular and diffusive surface.

𝒂𝒄𝒄 (°)
Element
LC1
LC2
LC3
LC4

220

Specular
27.2
29.3
29.7
30.0

Diffusive
51.0
34.2
20.2
10.7

𝜟𝒂𝒄𝒄 (%)
Spec  Diff
+87.5
+16.7
-32.0
-64.3

Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

Fig. 4.46. Angular distribution of the exit radiance from the light cones with diffusive surface,when irradiated by a Lambertian radiation incident on the exit aperture.

4.6.3. Solar Tunnels (ST)The inverse radiance curves for the STs are quite particular. For specular surface, theradiance curves overlap exactly. They are all of the Lambertian type with 90° angulardivergence. The curves shown in Fig. 4.47 show a drop in intensity from 70°, while we
expect to see a flat curve up to 90°. In reality this is an effect of the limited number of rays
used for the raytracing, as already observed in a previous work [31]. This has been easily
demonstrated by placing a Lambertian source at the input window of the SC of Fig. 4.11.
The profile of the emitted radiance results exactly the same of that reported in Fig. 4.47.

Normalized radiance

1,0

0,9

ST4
ST3
ST2
ST1
Inverse irradiation
RS = 100% specular

0,8

0,7

0,6

0,5
-100

-80

-60

-40

-20

0

20

40

60

80

100

Exit angle, 1,out (°)

Fig. 4.47. Angular distribution of the exit radiance from the solar tunnels with specular surface,when irradiated by a Lambertian radiation incident on the exit aperture.

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Advances in Optics: Reviews. Book Series, Vol. 5

This is the only case where the inverse radiance is Lambertian, like the applied one. Whenthe surface is diffusive, the radiance curves change drastically and the outgoing radiation
is no longer Lambertian (see Fig. 4.48). The curves, no longer flat, take on a shape that,being almost semi-circular for ST1, gets closer and closer to a Lorentzian one, with a
gradual narrowing of the curve. The acceptance angles measured for the radiance curves
with specular and diffuse surface are reported in Table 4.5, where we find that all thecurves of the STs with diffusive surface undergo a relative reduction in width that isincreasingly accentuated as the length increases.

Fig. 4.48. Angular distribution of the exit radiance from the solar tunnels with diffusive surface,when irradiated by a Lambertian radiation incident on the exit aperture.

Table 4.5. Acceptance angles of the various STs with specular and diffusive surface.

𝒂𝒄𝒄 (°)
Element
ST1
ST2
ST3
ST4

Specular
90
90
90
90

Diffusive
61.5
44.4
24.9
14.4

𝜟𝒂𝒄𝒄 (%)
Spec  Diff
-31.7
-50.7
-72.3
-84.0

4.7. Short Discussion of Practical Applications.
Although the present work is substantially theoretical, we can identify two topics on which
to carry out practical considerations. The first concerns the calculation of the net flux
crossing a solar tunnel, an element widely used in practice, to be installed on the roof of abuilding to convey sunlight from the outside to the internal environment. The second
concerns the use of the optical elements CPCs or LCs as light sources to create particular
light beams.
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Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

4.7.1. Net Flux Crossing a Solar Tunnel
An example of a solar tunnel applied to the collection of sunlight to be sent to a domestic
environment is shown in Fig. 4.49. A transparent and diffusive dome (dm) is placed
horizontally on the roof of a house. A solar tunnel, connected to the dome, crosses
vertically the roof and reaches the inside of the house, in a generic room. At the end of thesolar tunnel there is a diffuser (df), which distributes the sunlight all around. The room ismodelled as a homogeneous box whose internal surface has a reflectance 𝑅𝑤 . To apply
the law of Lambertian conductance we must evaluate the solar radiance 𝐿1 incident on thedome and the radiance 𝐿2 present inside the box and incident on the diffuser of the solar
tunnel. Even if it is the diffuser that emits most of the light, this one, once distributed
inside the room, produces a diffused light that hits the diffuser with an 𝐿2 radiance.

Fig. 4.49. Simplified scheme of a room illuminated by the light diffused from the solar tunnel(ST) and coming from the sunlight incident on the dome (dm).

The dome is irradiated by the light diffused from the sky and by the direct light of the Sun.
As a first approximation, the light transferred by the dome to the solar tunnel can betransformed, due to the multiple reflections undergone, into diffused light whose flux must
be equal to the total incident flux from the sky and the solar disk. At this point, we mustevaluate the radiance produced by the direct component of the solar irradiance, 𝐸𝑑𝑖𝑟,⊥ , andof the radiance produced by the diffuse component on the horizontal plane, 𝐸𝑑𝑖𝑓𝑓 , forsome sky conditions. Table 4.6 shows the direct and diffuse irradiances, very approximate,evaluated by our previous measurements [59]. The flux from the direct component of solar
radiation, expressed as diffuse light, becomes:

 𝑑𝑖𝑟 = 𝐴1  𝐸𝑑𝑖𝑟,⊥  𝑐𝑜𝑠 =   𝐴1  𝐿𝑑𝑖𝑟 ,(4.40)

where 𝐴1 is the cross-section area of the solar tunnel. The flux from the diffuse componentof solar radiation is:223

Advances in Optics: Reviews. Book Series, Vol. 5

 𝑑𝑖𝑓𝑓 =   𝐴1  𝐿diff(4.41)In Eq. (4.41) the diffuse component of solar radiation is approximated to be of constantradiance. The total radiance due to the direct and diffuse components of solar radiation
becomes:
𝐿 1 = (𝐸𝑑𝑖𝑟,⊥  𝑐𝑜𝑠) ⁄ + 𝐸𝑑𝑖𝑓𝑓,⊥ ⁄ == 𝐸𝑑𝑖𝑟,⊥ ⁄ (  𝐴𝑀) + 𝐸𝑑𝑖𝑓𝑓,⊥ ⁄

(4.42)

where we have placed: 𝑐𝑜𝑠 = 1/𝐴𝑀, with AM = Air Mass [60]. Following now what
is reported in Table 4.6, we can calculate the total external radiance 𝐿 1 incident on thedome for the four examined sky conditions. The results of 𝐿 1 , calculated for the standard
condition AM = 1.5, are reported in Table 4.7.
Table 4.6. Irradiance (in W/m2) from the direct and the diffuse components of solar radiation.

Component
DirectDiffuseClear sky
~ 900
~ 100

Atmospheric conditions
Partially cloudy Cloudy
~ 800
~ 400
~ 200
~ 300

Covered0~ 200

Table 4.7. The two components of the external radiance, and the total one, are calculated(in W / (m2  sr) starting from the corresponding estimated irradiances (in W / m2), and reported
as a function of atmospheric conditions.
Sky Condition
Clear skyPartially cloudy
CloudyCovered𝑬𝒅𝒊𝒓,⊥(W/m2)
~ 900
~ 800
~ 400
~0

𝑬𝒅𝒊𝒇𝒇,⊥(W/m2)
~ 100
~ 200
~ 300
~ 200

𝑳𝒅𝒊𝒓
(W/m2 sr)
~ 190
~ 170
~ 85
~0

𝑳𝒅𝒊𝒇𝒇
(W/m2 sr)
~ 30
~ 65
~ 95
~ 65

𝑳𝟏
(W/m2 sr)
~ 220
~ 235
~ 180
~ 65

 𝒏𝒆𝒕
(W)
~ 20
~ 21
~ 16
~6

The internal radiance, 𝐿 2 , can be evaluated by making the following consideration. We
set the average illuminance level in the room at 500 lx, which is high enough to read abook [61]. Simplifying sunlight as monochromatic light in the green (555 nm), theilluminance of 500 lx corresponds to an irradiance of 500/683 ≈ 0.7 (W/m2), and then toa radiance 𝐿 2 ≈ 0.7 /  ≈ 0.2 W / (m2  sr). This is the average radiance incident on thewalls of the room. Taking into account the low average reflectance of the walls of ahabitable room, the radiance incident on the diffuser (df) would be even smaller than
0.2 W / (m2  sr). We thus find that the internal radiance 𝐿 2 is incomparably smaller than
𝐿 1 for any sky condition (see Tab. 4.7), and then we can neglect it. If we now apply thelaw of optical Lambertian conductance, we find for the net flux through the ST:

 𝑛𝑒𝑡 = 𝐺 𝑙𝑎𝑚𝑏  (𝐿1 − 𝐿2 ) ≈ 𝐺 𝑙𝑎𝑚𝑏  𝐿1
224

(4.43)

Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

To calculate the 𝐺 𝑙𝑎𝑚𝑏 conductance, we choose a real solar tunnel of the same shape as
the ST4, that is with a length / diameter ratio = 4, but with a diameter of 20 cm (insteadof 24 mm), and then of a length of 20  4 = 80 cm. We choose a specular surface with
98 % reflectivity, obtained covering the internal surface of the solar tunnel with the 3 M
𝑙𝑎𝑚𝑏Radiant Mirror polymeric film. From Fig. 4.33 we obtain for ST4: 𝐺 𝑠𝑝𝑒𝑐
≈ 2.8 sr. The
𝑙𝑎𝑚𝑏
𝑙𝑎𝑚𝑏
Lambertian conductance becomes therefore: 𝐺= 𝐴1  𝐺 𝑠𝑝𝑒𝑐 ≈ 0.09 (m2  sr).
From Eq. (4.43) we then obtain the net flux  𝑛𝑒𝑡 (in W) crossing the solar tunnel in theoutdoor  indoor direction, reported in Table 4.7 as a function of the sky conditions.
4.7.2. Light Sources from CPCs and LCs
The optical elements CPCs and LCs present very interesting inverse radiancedistributions, so they can be used to realize light sources. CPCs with a specular surface,for example, if inversely irradiated with Lambertian light, are able to emit a bundle oflight of constant radiance within a well-defined angle, equal to 𝑎𝑐𝑐 (see Fig. 4.37).
Fig. 4.50(a) shows the schematic of a light source realized making use of a CPC. Light
cones with a specular surface, too, are suitable to emit light with a step-like shaped
radiance within a defined angle (see Fig. 4.39), if an appropriate length of the LC ischosen. Fig. 4.50(b) shows the schematic of a light source realized making use of the LC4
or longer LCs.

Fig. 4.50. Configurations used to make light sources with constant radiance within a definedangle. (a) CPC supplied with light from a single integrating sphere; (b) LC suppliedwith a double integrating sphere. (lp) = lamp; (bf) = baffle; (is) = integrating sphere.

To realize a light source with CPC or LC elements, however, it is necessary to supply the
opening (2), the smaller one of the CPC or the LC, with a Lambertian light. This can beobtained as we have already done in Fig. 4.3 to study the emission of inverse radiance. In
Fig. 4.50(a) a possible configuration is shown, in which a lamp source (lp) illuminates anintegrating sphere (is), inside which a baffle (bf) allows to distribute the light well inside
the (is). The opening of (is) from which Lambertian light is emitted is then connected withthe opening (2) of the CPC, which is thus able to emit light with constant radiance within
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Advances in Optics: Reviews. Book Series, Vol. 5

the acceptance angle. Alternatively, the lamp (lp) can supply two integrating spheres, (is1)and (is2), to obtain a better Lambertian distribution at the output of (is2), as shown in
Fig. 4.50 (b). The internal surface of the two spheres must be highly reflective, at least
95 %, in order not to lose too much light by absorption. It is sufficient to make a multilayer
coating of BaSO4 [6, 19].
4.7.3. Bent Solar Tunnels
In Section 4.5.3 we have studied the optical transmission properties of some linear solar
tunnels, in particular the ST4 of 96 mm long. In this section I want to expand the study tobent solar tunnels, which can be encountered in practice. In particular, I am interested toknow the effects that the presence of one, two or three bends has on the opticaltransmission properties of a linear ST. Therefore, starting from the ST4 as a basic unit, I
modeled a solar tunnel made up of two ST4 joined by a 90° bend, a solar tunnel made upof three ST4 joined by two 90° bends, and finally a solar tunnel made up of four ST4
joined by three 90° bends (see Fig. 4.51). These three bent STs were then compared withthree linear STs of the same length, in terms of Lambertian absorbance and transmittance.
In order to consider only STs that have potential practical applications, I have chosen to
study only mirror surfaces with high reflectivity, for example from 80 % to 100 %. In Fig.
4.51 the three types of bent STs are shown after a short raytracing done with a surface of
80 % reflectivity. The Lambertian source is applied to the openings (1) and the transmitted
flux is measured at the openings (2). The color along the tubes indicates the intensity of
the light, which fades from red to blue.

Fig. 4.51. Examples of raytracing of three bent solar tunnels (ST4×2 (a); ST4×3 (b); ST4×4 (c))
used for simulating the Lambertian absorbance, transmittance and conductance. (1) is theentrance opening of the Lambertian source; (2) is the exit opening where
the transmitted rays are collected and measured.

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Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

The bent STs consist of two, three or four ST4s (l = 96 mm each) and one, two or threebends, respectively. Each bend at 90° engages a length 𝑙 =   𝑅, where 𝑅 = 12 mm is
the tunnel radius. Ultimately, the three tunnels of Fig. 4.51 (a), (b) and (c) will have alength of 229.7 mm, 363.4 mm and 497.1 mm, respectively. The effect of length of the𝑙𝑎𝑚𝑏straight solar tunnels on the Lambertian absorbance  𝑙𝑎𝑚𝑏𝑑𝑖𝑟 and transmittance  𝑑𝑖𝑟 canbe seen in Fig. 4.52, where the three STs, of increasing length, are indicated as ST4×2,
ST4×3 and ST4×4. They were simulated using the direct Lambertian irradiation method(DLIM) (see Section 4.4.2). In Fig. 4.52 the Lambertian reflectance is missing because,
being the surface specular, we have no rays reflected to inputs (1). The trend of  𝑙𝑎𝑚𝑏𝑑𝑖𝑟and  𝑙𝑎𝑚𝑏shows,
asexpected,
that
theincreaseofthesurfacereflectance
𝑅
from
80
%
𝑆
𝑑𝑖𝑟
to 100 % involves a reduction of the Lambertian absorbance, down to 0 %, and an increaseof the Lambertian transmittance, up to 100 %. Opposite is the effect of the increase in thelength of the solar tunnel on the two quantities (see the arrows in Fig. 4.52). Fig. 4.52
shows that the Lambertian transmittance is very low, close to zero, even though thesurface reflectivity is quite high as 80 %. However, it must be considered that theLambertian irradiation involves many rays close to the perpendicular to the wall surface,
and this in turn involves a large number of reflections inside the tunnel.

100

100

length80alpha(ST4x4)
alpha(ST4x3)
alpha(ST4x2)
tau(ST4x2)
tau(ST4x3)
tau(ST4x4)

60

40

60

40

20

dirlamb (%)

dirlamb (%)

80

20length

0

0
80

85

90

95

100

Surface reflectance, RS (%)

Fig. 4.52. Lambertian absorbance and transmittance, as a function of surface reflectance,
for three straight solar tunnels of different length.

This is why the absorption on the wall can be high even though the reflectivity is high.
Similar trends occur with the bent solar tunnels of Fig. 4.51 (see Fig. 4.53). Now, theLambertian absorbance curves are a little shifted upwards and those of Lambertiantransmittance are a little shifted downwards, due to the effects of the bends. We find that,even in the presence of bends in the tunnels, we have no back reflected rays. The average
percentage loss observed for the Lambertian transmittance due to the presence of one, twoor three bends, is: ≈ 4 %, ≈ 5 % and ≈ 6 %, respectively. This loss is not so relevant forhigh transmittance values, rather it is for low ones, for which there may be an absolute
loss of ≈ 50 %. Naturally, these losses constitute gains for the Lambertian absorbance,since the two quantities are complementary. From the Figs. 4.52 and 4.53 we can see that,to have a Lambertian transmittance well above 50 %, it is necessary to have solar tunnels
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with very high surface reflectance, at least of 98 %, as in fact reported for the opticalcharacteristics of commercial solar tunnels (see also the 3M Radiant Mirror cited in
Section 4.7.1). From Fig. 4.52 we can see again that the effect that the solar tunnel
elongation has on the Lambertian transmittance and absorbance is less when moving from
ST4×3 to ST4×4, compared to when moving from ST4×2 to ST4×3. The same trend isobserved in Fig. 4.53 when moving from two to three bends, compared to when movingfrom one to two bends. Finally, we report in Fig. 4.54 the direct Lambertian conductance,
𝑙𝑎𝑚𝑏
𝐺 𝑑𝑖𝑟
, calculated for real bent solar tunnels with internal radius of 10 cm, the samedescribed in Section 4.7.1, and of the same shape as the model solar tunnels illustrated inFig. 4.51. These real tunnels, taking into account the due proportions, will have lengths
of: 1.914 m, 3.028 m and 4.142 m.
length

100

100

80

alpha(3 bends)
alpha(2 bends)
alpha(1 bend)tau(1 bend)tau(2 bends)
tau(3 bends)

60

40

60

dirlamb (%)

dirlamb (%)

80

40

20

20
length

0

0
80

85

90

95

100

Surface reflectance, RS (%)

Fig. 4.53. Lambertian absorbance and transmittance, as a function of surface reflectance,
for three bent solar tunnels of different length.

10

Gdirlamb (m2*sr)

8

ST4x2(1 bend)
ST4x3(2 bends)
ST4x4(3 bends)
R(tunnel) = 10 cm

6length
4

2

0
80

85

90

95

100

Surface reflectance, RS (%)

Fig. 4.54. Lambertian conductance, as a function of surface reflectance,
for three bent real solar tunnels of different length.

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Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

A more effective way of studying how 90° bends change the Lambertian transmittanceproperties of a solar tunnel is to start from a solar tunnel of a certain length and then insert
one, two and three bends along its path, at equal distances, keeping its total length thesame. Starting from the usual solar tunnel model with a radius of 12 mm, the total lengthchosen is 497.1 mm, that of a solar tunnel made up of four ST4 joined by three 90° bends(see Fig. 4.51(c)). Considering that this bent solar tunnel and the straight one, of the samelength, were already been simulated, it remains to simulate two other solar tunnels with
one and two bends, as function of the reflectance of a specular surface from 80 % to
100 %. We report here only the effect that the bends have on the Lambertian opticalconductance of the straight solar tunnel. In this way the real dimensions of the ST do notcount, but only its shape, that is the ratio between its total length and the internal radius.
In Fig. 4.55 it is shown the relative Lambertian optical conductance of model solar tunnels
with R = 12 mm and l = 497.1 mm, equivalent to real solar tunnels with R = 10 cm andl = 4.142 m. The graph of Fig. 4.55 confirms what found previously, namely that the main
effect on the reduction of conductance is given by the first bend, while the following two
have a gradually smaller effect. Furthermore, the reflectance of the surface plays afundamental role, since even with relatively high reflectances of 90 % strong reductions
in conductance are obtained, and it is necessary to rise to 98 % the reflectance value tocontain bending losses below 10 %.

100

Gdirlamb (rel) (%)

80

60

ST(straight)
ST(1 bend)ST(2 bends)
ST(3 bends)
R(tunnel) = 10 cml(tunnel) = 4.142 m

40

20

0
80

85

90

95

100

Surface reflectance, RS (%)

Fig. 4.55. Optical Lambertian conductance of solar tunnels with same length, provided with one,
two or three bends, calculated respect to that of a straight solar tunnel as a functionof surface reflectance.

4.8. Some Insights into the Law of Optical Lambertian ConductanceEq. (4.25):

 𝑛𝑒𝑡 = 𝐺 𝑙𝑎𝑚𝑏  (𝐿1 − 𝐿2 ) = 𝐺 𝑙𝑎𝑚𝑏  𝐿,(4.25)
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Advances in Optics: Reviews. Book Series, Vol. 5

where 𝐺 𝑙𝑎𝑚𝑏 is given by:
𝐺 𝑙𝑎𝑚𝑏 =   𝐴1   𝑙𝑎𝑚𝑏=   𝐴2   𝑙𝑎𝑚𝑏1221 ,(4.24)

 𝑙𝑎𝑚𝑏= 𝐶𝑔𝑒𝑜   𝑙𝑎𝑚𝑏21
12 ,(4.15)

and:is a generic expression that does not take into account the wavelength of the Lambertianlight. If then we still hold expression (4.15) valid for any wavelength, we can write:𝑙𝑎𝑚𝑏 𝑙𝑎𝑚𝑏21 (𝜆) = 𝐶𝑔𝑒𝑜   12 (𝜆),(4.44)In this case, Eq. (4.25) becomes:

 𝑛𝑒𝑡 (𝜆) = 𝐺 𝑙𝑎𝑚𝑏 (𝜆)  [𝐿1 (𝜆) − 𝐿2 (𝜆)](4.45)

However, we can introduce a more general form of Eq. (4.45), which takes into accountthe fact that the angular divergence of the incoming light into (1) and (2) optical terminals
can be different from the linear angle ±/2, and in general equal to ± (°). The law ofLambertian optical conductance becomes:(, 𝜆)  [𝐿1 (𝜆) − 𝐿2 (𝜆)] 𝑛𝑒𝑡 (, 𝜆) =   𝑠𝑖𝑛2 ()  𝐴1   𝑙𝑎𝑚𝑏12

(4.46)(, 𝜆)
𝐺 𝑙𝑎𝑚𝑏 (, 𝜆) =   𝑠𝑖𝑛2 ()  𝐴1   𝑙𝑎𝑚𝑏12

(4.47)

with:

The net flux expression does not take the canonical form of Eq. (4.25) when the lightentering the optical terminals is distributed on a spectrum s, even if a term equal to thedifference between the two radiances appears in it. The net flux takes in fact the form:(𝜆)  [𝐿1 (𝜆) − 𝐿2 (𝜆) ]} 𝑛𝑒𝑡 (𝑠) =   𝐴1  {∫𝑠 𝑑𝜆   𝑙𝑎𝑚𝑏12

(4.48)

4.9. Conclusions
This work originates from previous research in which the effects of the Lambertian
irradiation, mainly with 90° angular divergence, of a solar concentrator (SC), wereexperimented. In particular, the inverse Lambertian irradiation, that is the irradiation of
the opening normally used as exit opening of the concentrated light, has given rise to a
new technique for characterizing the solar concentrators, called ILIM (Inverse Lambertian
Irradiation Method). In previous works, this method was identified with the acronym ILM(Inverse Lambertian Method). The theoretical study that followed this technique has led
to the definition of several new optical quantities. The reason is simple: SCs are used toconcentrate a quasi-collimated bundle of light, and therefore the use of a Lambertian230

Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications

radiation, instead of a collimated one, has found a free field from which new opticalproperties have arisen. In this chapter, the previous theoretical work is further developed
up to the formalization of an optical law, called the “Optical Bidirectional LambertianConductance” law, or simply the “Optical Lambertian Conductance” law, which allows
to treat in a new way the issues of optical transmittance of a solar concentrator. Basically,
the Lambertian radiation, when applied to optical systems for which the principle of
optical reversibility applies, allows us to introduce the concept of optical conductance,that is common in the field of heat and electrical transmission. Just as, in the field of heattransmission, thermal conductivity relates the heat flux to a temperature difference, andin electrical transmission the electrical conductance relates the current flow to the electric
potential difference, in the Lambertian optical transmission of light the bidirectional
Lambertian conductance relates a flux of radiation to a difference of radiance. The law ofbidirectional Lambertian conductance is demonstrated in this work using simple opticalelements such as light cones, shaped as truncated cones. Other optical elements studied
here have been nonimaging solar concentrators of the CPC type (Compound Parabolic
Concentrators) and solar tunnels, of cylindrical shape. For each of these elements, chosenwith different characteristics of shape and size, numerous optical simulations were
performed in which their optical transmittance, reflectance and absorbance were
calculated under direct and inverse Lambertian irradiation, and as a function of thereflectivity of the specular or diffusive internal surface. Among the relevant results, notobvious, we can mention the invariability of reflectance and absorbance between thevarious ideal CPCs of different acceptance angles, when their surface is diffusive, while
this invariability in the light cones is less pronounced. The Lambertian opticalconductance was then calculated on these optical elements, and is used in the law of
conductance to calculate the net flux crossing the optical element when the difference inradiance at its optical terminals is known. A separate study consisted in the simulation of
the angular distribution of the inverse radiance, that is the radiance emitted when thesmallest aperture (of CPCs or LCs) or one of the two apertures of STs, is irradiated with
Lambertian light with 90° angular divergence. In this regard, it should be noted that theother condition that a Lambertian light beam must satisfy, in order to reproduce the datareported in this work, is that of being unpolarized light. Results of this type for CPCs arewidely known, and here I recall the fact that the emitted inverse radiance has the typicalstep-like profile. Regarding the LCs, it has been found that, in a series of themcharacterized by the same geometric concentration ratio, but of different lengths, theyshow, beyond a critical length, an angular distribution of the emitted inverse radiance witha step-like profile, as in the CPCs, while maintaining almost unchanged, and this is thenovelty, the angular interval of the distribution. This behavior has been confirmed by otherthree series of simulations. No relevant results were found for the solar tunnels. The lastpart of this work was devoted to possible practical applications of the theoretical andsimulated results. An application that refers to the just mentioned law consisted in thecalculation of the net flux crossing a solar tunnel applied on the roof of a building andconnected with an internal environment. The net flux was estimated for a conventional
solat tunnel exposed to four different sky conditions, from clear to full covered. The
second application concerned the use of the results obtained on the angular distribution of
the inverse radiance. The simulation results allowed us to design light sourcescharacterized by a constant radiance and with the bundle of light contained within a very231

Advances in Optics: Reviews. Book Series, Vol. 5

precise angular limit. These light sources can be created with CPCs or LCs suitably
equipped with a conventional light source and one or two integrating spheres. As a last
application, we have studied the effects that one, two or three bends at 90° have on theLambertian absorptance, transmittance and conductance of solar tunnels.

Acknowledgements
I dedicate this work to the memory of Prof. Giuliano Martinelli, former director of the
Physics Department of the University of Ferrara, and then Director of the Sensors andSemiconductors Laboratory, who, with his great generosity, foresight and managerial
skills, opened his laboratories to other research institutions, allowing me to work withhim, as an ENEA researcher associated at his university. Special thanks, never enough,go to my wife Antonietta, who, in the difficult period of pandemic, supported me morally,
allowing me to dedicate all my energies exclusively to this work.

References[1]. D. Vincenzi, A. Busato, M. Stefancich, G. Martinelli, Concentrating PV system based on
spectral separation of solar radiation, Physica Status Solidi, Vol. 206, Issue 2, 2009,
pp. 375-378.
[2]. D. Vincenzi, S. Baricordi, M. Occhiali, M. Stefancich, A. Parretta, G. Martinelli,
Measurement of Sun-Tracking Accuracy and Solar Irradiance through Multispectral Imaging,in Proceedings of Imaging and Applied Optics Congress, 2010, pp. 1-3.
[3]. D. Vincenzi, M. Stefancich, S. Baricordi, M. Pasquini, F. Gualdi, G. Martinelli, A. Parretta,
A. Antonini, Effects of Irradiance Distribution Unevenness on the Ohmic Losses of CPV
Receivers, in Proceedings of the 24th European Photovoltaic Solar Energy Conference andExhibition (EU PVSEC’09), Hamburg, Germany, 21-25 September 2009, pp. 725-728.
[4]. A. Parretta, G. Martinelli, M. Stefancich, D. Vincenzi, R. Winston, Modelling of CPC-based
photovoltaic concentrator, Proceedings of SPIE, Vol. 4829, Issue 1, 2002, pp. 1045-1047.
[5]. G. Martinelli, M.A. Butturi, C. Malagù, M. Stefancich, D. Vincenzi, A. Antonini, F. Bizzi,
A. Parretta, R. Winston, Composite dish systems for high efficiency concentrator
photovoltaics, in Proceedings of the 4th ISES Europe Solar Congress (EuroSun’02), Bologna,
Italy, 23-26 June 2002.
[6]. A. Parretta, Optical Methods for the characterization of PV solar concentrators, Chapter 13,
in Advances in Optics: Reviews (S. Y. Yurish, Ed.), Vol. 3, IFSA Publishing S. L., 2018,
pp. 316-416.
[7]. A. Antonini, M. A. Butturi, P. Di Benedetto, D. Uderzo, P. Zurru, E. Milan, M. Stefancich,
M. Armani, A. Parretta, N. Baggio, Rondine® PV concentrators: Field results anddevelopments, Progress in Photovoltaics, Vol. 17, 2009, pp. 451-459.
[8]. A. Antonini, M. A. Butturi, P. Di Benedetto, D. Uderzo, P. Zurru, E. Milan, M. Stefancich,
A. Parretta, N. Baggio, Rondine PV concentrators: Field results and innovations, inProceedings of the 24th European Photovoltaic Solar Energy Conference (EU PVSEC’09),
Hamburg, Germany, 21-25 Sept. 2009.
[9]. A. Antonini, M. A. Butturi, P. Di Benedetto, D. Uderzo, P. Zurru, E. Milan, A. Parretta,
N. Baggio, Rondine PV concentrators: Field results and progresses, in Proceedings of the 34th
IEEE Photovoltaic Solar Energy Conference (PVSC’09), Philadelphia, US, 7-12 June 2009,
pp. 001043-001047.

232

Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications[10]. A. Antonini, M. A. Butturi, P. Di Benedetto, E. Milan, D. Uderzo, P. Zurru, M. Stefancich,
A. Parretta, M. Armani, N. Baggio, Field testing per la tecnologia di concentratori a disco e
per moduli “Rondine”, in Proceedings of the “ZeroEmission Rome 2008” Conference, Rome,
Italy, 1-4 October 2008 (in Italian).
[11]. A. Antonini, M. A. Butturi, P. Di Benedetto, D. Uderzo, P. Zurru, E. Milan, M. Stefancich,
M. Armani, A. Parretta, N. Baggio, Development of "Rondine" concentrating PV module –field results and progresses, in Proceedings of the 23rd European Photovoltaic Solar Energy
Conference (EU PVSEC’08), Valencia, Spain, 1-5 Sept. 2008.
[12]. A. Antonini, M. Stefancich, P. Zurru, M. Butturi, P. Di Benedetto, D. Uderzo, E. Milan,
G. Martinelli, M. Armani, A. Parretta, N. Baggio, L. Secco, Experimental results of a new PV
concentrator system for silicon solar cells, in Proceedings of the 22nd European Photovoltaic
Solar Energy Conference (EU PVSEC’07), Milan, Italy, 3-7 September 2007.
[13]. A. Antonini, M. Butturi, P. Di Benedetto, P. Zurru, D. Sartore, In-field performances of firstcommercial Rondine® PV concentrators, in Proceedings of the 26th European Photovoltaic
Solar Energy Conference (EU PVSEC’11), Hamburg, Germany, 5-9 September 2011,
pp. 640-643.
[14]. A. Antonini, M. Butturi, P. Di Benedetto, E. Milan, D. Uderzo, P. Zurru, D. Sartore, GEN2
Rondine® PV concentrators, in Proceedings of the 25th European Photovoltaic Solar Energy
Conference (EU PVSEC’10), Feria Valencia, Valencia, Spain, 6-10 September 2010,
pp. 990-993.
[15]. A. Parretta, A. Antonini, M. Butturi, E. Milan, P. Di Benedetto, D. Uderzo, P. Zurru, Opticalmethods for indoor characterization of small-size solar concentrators prototypes, Advances inScience and Technology, Vol. 74, 2010, pp. 196-204.
[16]. A. Parretta, F. Aldegheri, D. Roncati, C. Cancro, R. Fucci, Optical efficiency of "Ph oCUS"
C-Module concentrators, in Proceedings of the 26th European Photovoltaic Solar Energy
Conference (EU PVSEC’11), Hamburg, Germany, 5-9 Sept. 2011, pp. 658-663.
[17]. A. Parretta, L. Zampierolo, G. Martinelli, A. Antonini, E. Milan, C. Privato, Methods ofcharacterization of solar concentrators, in Proceedings of the 25th European Photovoltaic
Solar Energy Conference (EU PVSEC’10), Feria Valencia, Valencia, Spain, 6-10 Sept. 2010.
[18]. A. Parretta, A. Antonini, M. A. Butturi, P. Di Benedetto, E. Milan, M. Stefancich, D. Uderzo,
P. Zurru, D. Roncati, G. Martinelli, M. Armani, How to "display" the angle-resolved
transmission efficiency of a solar concentrator reversing the light path, in Proceedings of the
23rd European Photovoltaic Solar Energy Conference (EU PVSEC’08), Feria Valencia,
Valencia, Spain, 1-5 Sept. 2008, pp. 95-98.
[19]. E. Bonfiglioli, Sviluppo di metodi per la caratterizzazione ottica indoor di concentratori solari,
Graduate Thesis in Physics and Astrophysics, Università degli Studi di Ferrara, Ferrara, Italy,
2007-2008.
[20]. A. Antonini, M. Stefancich, J. Coventry, A. Parretta, Modelling of compound parabolic
concentrators for photovoltaic applications, International Journal of Optics and Applications,Vol. 3, Issue 4, 2013, pp. 40-52.
[21]. A. Parretta, G. Martinelli, M. Stefancich, D. Vincenzi, R. Winston, Modelling of CPC-based
photovoltaic concentrator, Proceedings of SPIE, Vol. 4829, 2003, pp. 1045-1047.
[22]. A. Parretta, A. Antonini, M. Butturi, P. Zurru, Optical simulation of Rondine® solar
concentrators by two inverse characterization methods, Journal of Optics, Vol. 14, 2012,
125704.
[23]. A. Parretta, A. Antonini, E. Bonfiglioli, M. Campa, D. Vincenzi, G. Martinelli, Il metodo
inverso svela le proprietà dei concentratori solari, PV Technology, Vol. 3, Agosto/Settembre
2009, pp. 58-64 (in Italian).
[24]. A. Parretta, A. Antonini, E. Milan, M. Stefancich, G. Martinelli, M. Armani, Optical
efficiency of solar concentrators by a reverse optical path method, Optics Letters, Vol. 33,
2008, pp. 2044-2046.

233

Advances in Optics: Reviews. Book Series, Vol. 5[25]. A. Parretta, A. Antonini, M. Stefancich, G. Martinelli, M. Armani, Optical characterizationof CPC concentrator by an inverse illumination method, in Proceedings of the 22nd European
Photovoltaic Solar Energy Conference (EU PVSEC’07), Fiera Milano, Milan, Italy,
3-7 Sept. 2007.
[26]. A. Parretta, A. Antonini, M. Stefancich, G. Martinelli, M. Armani, Inverse illumination
method for characterization of CPC concentrators, Proceedings of SPIE, Vol. 6652, 2007,
665205.
[27]. A. Parretta, F. Aldegheri, A. Antonini, M. Butturi, P. Zurru, Optical simulation of PV solar
concentrators by two inverse characterization methods, International Journal of Optics andApplications, Vol. 2, Issue 5, 2012, pp. 62-71.
[28]. A. Parretta, F. Aldegheri, C. Cancro, R. Fucci, G. Graditi, R. Schioppo, Opticalcharacterization of “Ph oCUS” refractive solar concentrators, International Journal of Optics
and Applications, Vol. 4, Issue 4A, 2014, pp. 12-24.
[29]. A. Parretta, Optics of solar concentrators. Part I: Theoretical models of light collection,
International Journal of Optics and Applications, Vol. 3, Issue 4, 2013, pp. 27-39.
[30]. A. Parretta, A. Antonini, Optics of solar concentrators. Part II: Models of light collection of
3D-CPCs under direct and collimated beams, International Journal of Optics andApplications, Vol. 3, Issue 5, 2013, pp. 72-102.
[31]. A. Parretta, Optics of solar concentrators. Part III: Models of light collection of 3D-CPCsunder direct and Lambertian irradiation, International Journal of Optics and Applications,Vol. 5, Issue 3, 2015, pp. 82-102.
[32]. A. Parretta, A. Antonini, Optics of solar concentrators. Part IV: Maps of transmitted andreflected rays in 3D-CPCs, International Journal of Optics and Applications, Vol. 7, Issue 1,
2017, pp. 13-30.
[33]. A. Parretta, L. Zampierolo, D. Roncati, Theoretical aspects of light collection in solar
concentrators, in Proceedings of the Conference on Optics for Solar Energy (SOLAR’10), The
Westin La Paloma, Tucson, AZ, USA, 7-10 June 2010, p. STuE1.
[34]. A. Parretta, E. Bonfiglioli, L. Zampierolo, M. Butturi, A. Antonini, Optical characterizationof Rondine® PV solar concentrators, International Journal of Optics and Applications,Vol. 4, Issue 4a, 2014, pp. 25-52.
[35]. A. Parretta, L. Zampierolo, A. Antonini, E. Milan, D. Roncati, Theory of "inverse method"
applied to characterization of solar concentrators, in Proceedings of the 25th European
Photovoltaic Solar Energy Conference (EU PVSEC’10), Feria Valencia, Valencia, Spain,
6-10 September 2010.
[36]. A. Parretta, D. Roncati, Theory of the "inverse method" for characterization of solar
concentrators, in Proceedings of the Conference Optics for Solar Energy (SOLAR’10), The
Westin La Paloma, Tucson, AZ, USA, 7-10 June 2010, p. STuE2.
[37]. A. Parretta, G. Martinelli, A. Antonini, D. Vincenzi, Direct and inverse methods ofcharacterization of solar concentrators, in Proceedings of the Conference Optics for Solar
Energy (SOLAR’10), The Westin La Paloma, Tucson, AZ, USA, 7-10 June 2010, p. STuA1.
[38]. R. C. Jones, On reversibility and irreversibility in optics, Journal of Optical Society ofAmerica, Vol. 43, 1953, pp. 138-144.
[39]. Z. Knittl, The principle of reversibility and thin film optics, Journal of Modern Optics,
Vol. 9, Issue 1, 1962, pp. 33-45.
[40]. A. Antonini, M. Stefancich, D. Vincenzi, C. Malagù, F. Bizzi, A. Ronzoni, G. Martinelli,Contact grid optimization methodology for front contact concentration solar cells, Solar
Energy Materials and Solar Cells, Vol. 80, 2003, pp. 155-166.
[41]. D. Vincenzi, M. Stefancich, S. Baricordi, M. Gualdi, G. Martinelli, A. Parretta, A. Antonini,
Effects of irradiance distribution unevenness on the ohmic losses of CPV receivers, inProceedings of the 24th European Photovoltaic Solar Energy Conference (EU PVSEC’09),
2009, pp. 725-728.

234

Chapter 4. The Optical Bidirectional Lambertian Conductance Law with Applications[42]. E. Franklin, J. Coventry, Effects of highly non-uniform illumination distribution on electricalperformance of solar cells, in Proceedings of the ANZSES Solar Annual Conference,
Newcastle, Australia, 2002.
[43]. R. Leutz, A. Suzuki, A. Akisawa, T. Kashiwagi, Flux uniformity and spectral reproduction insolar concentrators using secondary optics, in Proceedings of the ISES Solar World Congress(SWC’01), Adelaide, 2001, pp. 775-784.
[44]. A. Luque, S. Hegedus, Handbook of Photovoltaic Science and Engineering, John Wiley &Sons Ltd., Chichester, UK, 2011.
[45]. A. Luque, V. M. Andreev, Concentrator Photovoltaics, Springer-Verlag, Berlin, 2007.
[46]. R. Winston, J.C. Miñano, P. Benítez, Nonimaging Optics, Elsevier Academic Press, 2005.
[47]. A. Martí, A. Luque, Next Generation Photovoltaics, Institute of Physics, Bristol, UK, 2004.
[48]. V. M. Andreev, V. A. Grilikhes, V. D. Rumyantsev, Photovoltaic Conversion of Concentrated
Sunlight, John Wiley & Sons, Chichester, UK, 1997.
[49]. A. Luque, Solar Cells and Optics for Photovoltaic Concentrators, Institute of Physics, Bristol,
UK, 1989.
[50]. W. T. Welford, R. Winston, High Collection Nonimaging Optics, Academic Press, San
Diego, 1989.
[51]. W. T. Welford, R. Winston, The Optics of Nonimaging Concentrators, Light and Solar
Energy, Academic Press, 1978.
[52]. R. Winston, Principles of solar concentrators of a novel design, Solar Energy, Vol. 16, 1974,
pp. 89-95.
[53]. L. D. Partain, Solar Cells and Their Applications, John Wiley & Sons Inc., New York, USA,1995.
[54]. P. Sansoni, F. Francini, D. Fontani, Optical characterization of solar concentrator, Optics andLasers in Engineering, Vol. 45, 2007, pp. 351-359.
[55]. A. Parretta, P. Morvillo, C. Privato, G. Martinelli, R. Winston, Modelling of 3D-CPCs forconcentrating photovoltaic systems, in Proceedings of the Conference PV in Europe – FromPV Technology to Energy Solutions, Rome, Italy, 7-11 Oct. 2002, pp. 547-550.
[56]. Lambda Research Corporation, http://lambdares.com/[57]. A. Parretta, M. L. Addonizio, Optical and structural characterization of diffuse reflectancestandards, International Journal of Optics and Applications, Vol. 5, Issue 2, 2015, pp. 33-49.
[58]. A. Parretta, G. Calabrese, About the definition of "multiplier" of an integrating sphere,
International Journal of Optics and Applications, Vol. 3, Issue 6, 2013, pp. 119-124.
[59]. A. Parretta, A. Sarno, L. Vicari, Effects of solar irradiation conditions on the outdoor
performance of photovoltaic modules, Optics Communications, Vol. 153, 1998, pp. 153-163.
[60]. S. R. Wenham, M. A. Green, M. E. Watt, R. Corkish, A. Sproul, Applied Photovoltaics,
Routledge, 2012.
[61]. M. A. Green, Solar Cells, The University of New South Wales, Kensington, New South Wales,AU, 1992.
[62]. A. V. Arecchi, T. Messadi, R. J. Kashel, Field Guide to Illumination, SPIE Press, Bellingham,
Washington, USA, 2007.

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Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging

Chapter 5

Aperiodic Zone Plates for Optical Tweezers
and Optical Imaging
Tian Xia, Shubo Cheng, Shaohua Tao1

5.1. Introduction
Since the aperiodic sequences were introduced to design new kinds of zone plates (ZPs)[1], the aperiodic zone plates have found many special applications, e.g. lithograph [2],X-ray microscope [3], optical encryption [4], THz technology [5], optical tweezers [6]and optical imaging [7, 8].
The aperiodic zone plates have a series of foci along the optic axis, which form multiple
optical traps used to trap particles at the different planes simultaneously [9]. Although asingle beam designed by a beam shaping algorithm can generate multiple traps in threedimensional space, distortion made by reflection, refraction, or absorption of a trapped
particle would affect the subsequent propagation of the beam. To trap multiple particles
separately distributed in three-dimensional space, one can utilize a beam with tailorable
three-dimensional intensity distribution and the self-reconstruction property [10, 11]. The
Bessel beams and the Airy beams were used to simultaneously trap multiple particles [1214], but positions of the trapped particles could not be customized owing to the nondiffracting intensity profiles of the beams in the propagation direction. Recently, fractal
zone plates (FZPs) were proposed to form two-dimensional photonic structures [1]. The
fractal pattern has the self-similarity when the FZP is illuminated with a plane wave [15].
Besides having one main focus and other weaker foci, i.e., the so-called major foci, an
FZP beam also has many subsidiary foci around the main focus and major foci along theoptical axis. Moreover, as focal lengths of multiple foci provided by an FZP beam aredetermined by the structural parameters of the FZP, the axial distances between adjacentfoci of the beam can be finely controlled with a fractional number of the segments of thefractal structure [16]. Thus, with the FZP beams one can realize not only the simultaneous
trapping of multiple particles in different planes but also the customized trapping locationsof trapped particles. Although optical trapping with a beam generated by the Devil's lensShaohua Tao
School of Physics and Electronics, Central South University, Changsha 410083, China

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characterized with a fractal structure was realized [17], the simultaneous manipulation ofparticles at different axial positions has not been reported.
The aperiodic zone plates have all kinds of foci used for optical imaging. There are somezone plates which generate the high-intensity foci, i.e., a conventional composite Fresnel
zone plate consists of a central zone plate with first-order focal length and outer zones
with third-order focal length, which generate the high-intensity focus with the improved
resolution [18]. A composite photon sieve can also suppress sidelobes, and produces afocus with slightly better resolution [19]. Moreover, composite fractal zone plates havethe single high-intensity main foci with many subsidiary foci [20]. Composite fractional
fractal zone plates generate the tailorable main foci with high intensity and subsidiary foci[21]. In addition, composite Thue-Morse zone plates generate two high-intensity foci withmany subsidiary foci, which are applicable to produce two images with the low chromaticaberration [22]. Furthermore, modified Thue-Morse zone plates based on the Thue-Morsesequence of S = 3 provide two high-intensity foci located at the positions satisfying the
fixed ratio of 3/5 [23]. Another interesting way to produce the high-intensity foci is toimplement devil's lenses, which are the fractal zone plates with the gradient phases andhave the high-intensity main foci with many subsidiary foci [24]. Moreover, kinoform
generalized mean lenses generate twin high-intensity main foci with the generalized mean
and the suppressed high-order foci along the optic axis [25]. Nevertheless, for the above
composite zone plates, the first-order diffraction areas of the inner and outer parts are notcoincident. Some radially modulated zone plates are used to acquire annular beams at the
focal planes. For example, Fresnel zone plates with radially shifted phases create annularbeams with designed radii at the focal planes [26]; radial phase modulated spiral zone
plates generate optical vortices with tailorable radii at the focal plane [27]; annular beams
produced by modified spiral zone plates have tailorable radii and segmented phasegradients [28]. However, these zone plates cannot produce high-intensity foci at the focal
planes used for generating the clear images. Some photon sieves generate the first-orderfoci with suppressed high-order foci [29]. For example, fractal photon sieves have thelow-intensity high-order foci and single main foci with many subsidiary foci in the firstorder focal area, which are applicable to produce the colourful images with the low
chromatic aberration [30]; lacunar fractal photon sieves whose lacunarity is used to adjust
the intensities of axial foci can also provide the self-similar foci in the first-orderdiffraction area and suppressed high-order foci [31]; generalized Fibonacci photon sieves
based on the different initial seeds and iteration rules generate two equal-intensity fociwith many ratios in the first-order focal area and low-intensity high-order foci [32].
However, the high-order foci for photon sieves still have low intensities, and these photon
sieves cannot generate the first and second order foci only. The single first-order foci canbe created by many single-focus zone plates. There are stagger-arranged zones for novel
single-focus zone plates (SFZPs) [33] and single-focus photon sieves [34], whosetransmittances are cosinusoidal to produce the first-order foci only. Annulus-sectorelement coded Gabor zone plates consist of annulus-sector-shaped nanometer structure
apertures, so that this kind of zone plates have a cosinusoidal transmittance to generate
the first-order foci only [35]. A single-focus spiral zone plate utilises the modified
transmittance function based on the two-parameter modified sinusoidal apodization
window to realise the single first-order focus along the optic axis [36]. Nevertheless, these238

Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging

zone plates cannot generate the single main foci with many subsidiary foci and two equalintensity main foci with many subsidiary foci used to create the single and two colourful
images, respectively.
In this chapter, the aperiodic zone plates are used for optical tweezers and optical imaging.
Firstly, FZP beam can simultaneously trap multiple particles positioned in different focal
planes of the FZP beam. The dynamic manipulation of microparticles in optical tweezers
will be implemented by changing Ns of the fractional FZPs. The symmetrical major fociaround the main focus of the Thue-Morse zone plate (TMZP) [37] have the same intensity
so that the microparticles can be trapped stably. Fibonacci zone plate (FiZP) [38] cansimultaneously trap particles positioned in two different planes and move particles freely
in the focal plane. The experimental results demonstrate that the FZP beam, the fractional
FZP beam, the TMZP beam and the FiZP beam can achieve the simultaneous
manipulation of particles at different axial positions. Secondly, generalized composite
aperiodic zone plates (GCAZPs) are proposed to generate clearer images at focal planes.
A modified single-focus fractal zone plate (MSFFZP) is proposed to generate the single
main focus with many subsidiary foci or two equal-intensity main foci with many
subsidiary foci, which are used to achieve optical colourful imaging. It is proved in theexperiments that the GCAZP has the clearer images at the focal planes, and the MSFFZP
produces the single or two colourful images with the low chromatic aberrations at the
focal planes.

5.2. Theory of Aperiodic Zone Plates
5.2.1. Aperiodic Sequence
Aperiodic sequences have self-similar properties and no connection with disorder.
Compared with the periodic sequences, the aperiodic sequences have more special regularproperties. The aperiodic sequences are a kind of substitution sequences, and constructedby unified substitution rules, which make the aperiodic sequences have the self-similarity.
The aperiodic sequence is first an alphabetic system, which consists of a certain numberof letters. Different letters are corresponding to the different substitution rules substituting
the letters into the strings of letters. Based on the substitution rule, the initial sequence issubstituted to generate the aperiodic sequence. Thus, this unified substitution rule makes
the aperiodic sequence have the self-similarity. For example, the initial seeds of the Cantor,
Thue-Morse and Fibonacci sequences are all ‘A’. The corresponding substitution rules are{A→ABA, B→BBB}, {A→AB, B→BA} and {A→AB, B→A}, respectively.
5.2.2. Method of Constructing Aperiodic Zone Plates
The aperiodic zone plate can be constructed by the transmittance function q   [37]NS    j  1 2  dS 
q     tS , j  rect 
,
dS
j 1



(5.1)
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Advances in Optics: Reviews. Book Series, Vol. 5

where    r a  , r is the radial coordinate of the zone plate, a is its maximum value,2 0,1 is the normalized square radial coordinate, S is the order, and NS is the total

count of the elements for the modified Fibonacci sequence of the S-th order. In Eq. (5.1),

tS , j is “1” or “0” when the j-th element of the corresponding modified Fibonacci sequence
is “A” or “B”, respectively. Thus, the sequence of each order consists of NS equal parts,and each part has the same length of dS  1 NS . rect[] presents the rectangular function,
which is defined as0rect t   1

t  0.5t  0.5.

(5.2)

The binary functions of the amplitude-only and phase-only zone plates can be calculated
by q   and q     , respectively.

The aperiodic square zone plate can be generated with a transmission function p(x, y) [39]written as
N




x
y
p  x, y    t j [rect 
rect 
 2a j / N  2a j / N 
j 1








x
y
 rect 
rect 
],
 2a ( j  1) / N  2a ( j  1) / N 





(5.3)

where N is the total number of zones, the length of the outer zone is 2a, (x, y) is the
Cartesian coordinate, and tj is “1” or “0” when the j-th element of the corresponding
sequence is “A” or “B”, respectively.
The aperiodic lens is defined with a linear variation between 0 and 2 at each subinterval{AB} of the aperiodic sequence, being 0 otherwise, and can be calculated in Eq. (5.4).

m 1 n m 1

)/( 
)
2  ( F ( )  
k
k
k
0

m 1
n
 
k
k,
others(5.4)

where k is the number of elements of the aperiodic sequence. m and n are the ordinal
numbers of two elements A and B for each subinterval {AB} of the corresponding
aperiodic sequence, respectively. The transmittance function t (r, ) of the aperiodic lensis defined byq( , S )  exp[iF ( )],

240

(5.5)

Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging

The binary transmittance function tFSZP(r2, θ) of the aperiodic spiral zone plate can becalculated by Eq. (5.6).

 r2
1
(2

2
j
)


L


 (3  2 j )1t FSZP (r 2 , )  
,
f
0
else


where     3 / 2, f 

(5.6)

a2
, the wavelength λ is set as 532 nm, (r, θ) (2 L1  2 M  3)

represents the polar coordinate, L1 presents the topological charge and j presents thesequence number of the high-transmittance spiral zone. In fact, j is equal to the sequencenumber of the element “1” for the aperiodic sequence with the maximum of M. It should
be noted that j is equal to -L1, -(L1-1), …-1 or 0 to make the first spiral zone complete.
The transmittance of the spiral aperiodic zone plate can be obtained by [40]t ( , )  q( )exp[im ],(5.7)

where m represents the topological charge.
5.2.3. Theory of Diffraction
The diffraction distributions of aperiodic zone plates can be calculated by two means. One
method is the Fresnel integral formula to find the analytical solution, and the other method
is the angular plane-wave spectrum theory or the fast Fourier transform-based FresnelKirchhoff integral to find the numerical solution.
The Fresnel approximation expressed in Eq. (5.8) [24] is used to calculate the crosssection intensity distribution illuminated with the monochromatic planar wave.
 2 
I  z, r    z 

22

  2   2 r0 r 0 p  r0  exp  i  z r0 J 0   z  r0 dr0 ,
a

(5.8)

where the rotation invariant transmittance function of the aperiodic zone plate is p(r0), r0
is the radial coordinate of zone plate, λ and z present the wavelength and axial distance,
respectively, r is radius at the diffraction plane, and J0() the Bessel function of the firstkind of order zero.
After the pupil function p(r0) in Eq. (5.8) is Hankel transformed, and Eq. (5.9) can beobtained [24].





2

I (u, v)  4 2u 2  q( )exp( i 2 u ) J m 4  uv d  ,
1

0

(5.9)

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Advances in Optics: Reviews. Book Series, Vol. 5

where u  a 2 （/ 2 z）is the reduced axial coordinate, v = r/a is the normalized transverse
coordinate. When v is equal to zero, the axial intensity distribution can be given in Eq.
(5.10),I (u )  4 2u 2  q( )exp( i 2 u ) d 
1

2(5.10)

0

Combining Eqs. (5.1) and (5.10) one can obtain Eq. (5.11)
2

NS

I (u )  i  tS , j (exp(i 2 ujN s )  exp(i 2 u ( j  1) N s ))(5.11)j 1

Eq. (5.11) can be used to calculate the intensity at the arbitrary axial position.
The transmittance of the aperiodic zone plate can be expressed by a matrix of Tp
comprising 0 and  . The angular plane-wave spectrum theory expressed in Eqs. (5.12)
and (5.13) [41] is used to analyze the focusing properties of the aperiodic zone plate,
E p  iFT  FT Tp  H  ,(5.12)22

1  x  y H ( x, y )  exp i 2 z 2        L L 

(5.13)In Eq. (5.12), E p is the complex amplitude of the diffracted beam, FT , iFT , and H
represent the Fourier transform, the inverse Fourier transform, and the transfer function,
respectively. In Eq. (5.13), z ,  , and d are the propagation distance, the wavelength oflight, and the maximum size of the ZP, respectively. x and y are the dimensionlesscoordinates of the sampling grids.
The diffraction complex amplitude U ( R, , z) of the phase-only aperiodic zone plate
expressed by a matrix of t (r, ) comprising 0 and π can be calculated by the fast Fouriertransform-based Fresnel-Kirchhoff integral expressed in Eq. (5.14) [42].

U ( R, , z)  FT -1  FT [exp(i  t (r, ))]FT [h(  )],(5.14)

where ( R, , z), FT , FT -1 and h(  )  exp(ikz) / (i z)  exp(ik  2 / (2 z)) present thecylindrical coordinate at the diffraction plane z, the Fourier transform, the inverse Fouriertransform and the free-space impulse response [43], respectively.

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Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging

5.3. Aperiodic Zone Plates for Optical Tweezers
5.3.1. Fractal Zone Plate
As an example, the construction procedure of a regular Cantor fractal structure is shownin Fig. 5.1(a) [1]. The segment is divided into an odd number of segments 2N-1 and thesegments in the even-number positions are removed. Fig. 5.1(b) shows the associated FZP
generated from the fractal structure with S = 3.

Fig. 5.1. (a) Schematic of the one-dimensional fractal structure based on the Cantor sets, and (b)the associated FZP generated from the one-dimensional fractal structure with S = 3.

The focal length of the main focus of an FZP beam can be expressed by Eq. (5.15)
f (a, λ, N, S) = a2·[λ·(2N – 1)S]-1,(5.15)

where a is the radius of the FZP, λ is the wavelength of the illuminating light, N is thenumber of the segments forming the fractal structure, and S is the fractal stage.
In Eq. (5.15), when the FZP is illuminated with a collimated laser beam with a wavelengthof 532 nm, the focal length of the main focus of the FZP with N = 3 and S = 3 is f = 222
mm. The axial irradiance of the FZP is shown in Fig. 2(a), where multiple foci withinternal fractal properties along the z direction can be observed, and the main focus has
the highest intensity and a distance of ~222 mm from the FZP. Furthermore, the additionalfoci surrounding the main focus are at distances of ~213 mm and 230 mm, respectively.
Fig. 5.2(b) shows the calculated intensity distributions of the FZP beam in the x-z plane.
In Fig. 5.2 the peaks around the main focus are marked as the first and second subsidiaryfoci, respectively.
In the trapping experiments polystyrene beads with a diameter of about 3 μm were
immersed in deionized water of refractive index of 1.33 and used as the manipulating
objects. The binary phase-only FZP with N = 3 and S = 3 was loaded onto a spatial light
modulator (SLM) to generate the FZP beam. We had added phase distribution of a blazed
grating to the phase-only FZP for an off-axis output. The computer-generated hologramis shown in Fig. 5.3(a). The CCD-captured intensity distributions of the first subsidiary243

Advances in Optics: Reviews. Book Series, Vol. 5

focus, the main focus, and the second subsidiary focus are shown in Fig. 5.3(b-d),
respectively.

Fig. 5.2. (a) Axial irradiance of the FZP with N = 3 and S = 3, and (b) the calculated intensity
distribution of the FZP in the x-z plane, where the main focus and the two additional subsidiaryfoci are illustrated.

Fig. 5.3. (a) Phase distribution of the FZP of N = 3 and S = 3, and (b-d) CCD-captured intensity
distributions of the first subsidiary focus, the main focus, and the second subsidiary focus of the
FZP beam at propagating distances of 21 cm, 22 cm, and 23 cm, respectively.

In order to conveniently observe the 3D trapping of particles in multiple focal planes inthe experiment, we arrange the optical path shown in Fig. 5.4(a), where the beam enters
the sample cell at a small oblique angle. In Fig. 5.4(a) the dashed green line represents thecollinear arrangement of optical components, and the red solid line shows theconfiguration of our trapping experiment. Fig. 5.4(a) shows that the focus of the objective
locates just in the sample cell and the first subsidiary focus is on the focal plane of theobjective, so the first subsidiary focus in the cell can be clearly observed. The observed
first subsidiary focus in the view field of the CCD camera is shown in the right bottom of
the square exposure region in Fig. 5.4(b). It is worth mentioning that the relative distance
between the objective and the sample cell was fixed in the optical trapping. When thesample cell and the objective is shifted upward synchronously about 180 μm, the main
focus shown in Fig. 5.4(c) can be observed clearly in the view field. Although the beam244

Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging

has been shifted relatively toward the bottom of the sample cell, a dim image of the firstsubsidiary focus on the right of the main focus can still be observed in Fig. 5.4(c). The
sample cell and the objective are further shifted upward synchronously until the second
subsidiary focus appears clearly. The CCD-captured image of the second subsidiary focus
is shown in Fig. 5.4(d). Similarly, a dim image of the main focus exists on the right of thesecond subsidiary focus. From Figs. 5.4(b-d) we can also observe that the locations of thethree foci shift toward a fixed direction when the sample cell and the objective shift
upward synchronously. Obviously, the displacement of the foci in the view field isresultant from the oblique incidence of the beam ray. When the first subsidiary focus isobserved clearly in the view field and then the sample cell and the objective is shifted
upward synchronously about l μm, the second subsidiary focus can be observed clearly. If
the second subsidiary focus shifts a distance of d μm from the first subsidiary focus in theview field, the incident angle of the FZP beam on the sample cell can be expressed
approximately by arctan(d/l).

Fig. 5.4. (a) Schematic of the experimental setup, and (b-d) captured CCD frames of intensity
distributions of the first subsidiary focus, the main focus, and the second subsidiary focus,
respectively. The dashed arrows in (b-d) represent the shifting directions of the foci withsynchronous shifts of the sample cell and the objective.

In the experiment the sample cell and the objective were shifted upward synchronously
until the first subsidiary focus was observed clearly in the view field of the CCD camera.
Then the sample cell moved horizontally until particles appeared in the view field. When
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the laser power was about 800 mW, the polystyrene bead marked as 1 was trapped by the
first subsidiary focus, and the trapping images are shown in Fig. 5.5(a-c). The particles
highlighted with the white dashed rectangles are used as the reference. When the particle
1 had been trapped by the first subsidiary focus, the sample cell and the objective were
shifted slowly upward synchronously. In the mean time we observed that the particle 1
trapped by the first subsidiary focus was shifted toward the bottom of the sample cell withthe first subsidiary focus, and then the CCD images of the trapped particle 1 became
blurred. The trapping images are shown in Fig. 5.5(c, d). When the sample cell and theobjective were shifted upward synchronously until the main focus appeared on the focal
plane of the objective, a cluster of particles marked as 2 was trapped. The trapping images
are shown in Fig. 5.5(d-f). Then the sample cell and the objective were further shifted
upward synchronously, we observed that the cluster of trapped particles was shifted
toward the bottom of the sample cell with the main focus, and the particle 1 trapped by
the first subsidiary focus was also attracted to the main focus for the higher intensity ofthe main focus. CCD images of the particle 1 trapped by the main focus are shown in
Fig. 5.5(f-h). The sample cell and the objective were further shifted upward synchronously
until the second subsidiary focus appeared on the focal plane of the objective. At that time
the particle marked as 3 was trapped by the second subsidiary focus. The trapping images
are shown in Fig. 5.5(g-i).

Fig. 5.5. Experimental demonstration of simultaneous trapping with three foci of the FZP beam.
(a-c) Trapping of the particle marked as 1 with the first subsidiary focus, (d-f) trapping of a clusterof particles marked as 2 with the main focus, and (g-i) trapping of the particle marked as 3 with thesecond subsidiary focus and the simultaneous trapping of the particles marked as 1 and 2 with themain focus, respectively.

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Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging

From Fig. 5.5(i) we can find that the cluster of particles is still trapped by the main focus
in the bottom of the sample cell. Due to the limited depth of the sample cell and shortwork distance of the objective, when the second subsidiary focus is located on the focal
plane of the objective, the first subsidiary focus has moved out of the sample cell.
Nonetheless, we can observe simultaneous trapping of particles by the second subsidiaryfocus and the main focus in the view field from Fig. 5.5(i). Hence, particles can be
manipulated with the multiple foci in different planes. In the trapping experiments, thedistance between the trapping positions of the first subsidiary focus and the second focus
in the view field of the CCD camera was about 25 μm. When the first subsidiary focus
appeared clearly in the view field, the sample cell and the objective were shifted upward
synchronously about 380 μm until the second subsidiary focus appeared. Thus, theincident angle on the sample cell was about 0.066 radian.
Figs. 5.6(a) and (b) show an example FZP with N = 3 and S = 3, and an example fractional
FZP with N = 2.9 and S = 3, respectively. The axial locations of multiple foci and theaxial distance between two neighbouring foci of a fractional FZP beam can be precisely
adjusted by modifying the fractional number N.

Fig. 5.6. FZPs with (a) N = 3 and S = 3 and (b) N = 2.9 and S = 3. The bright and dark segments
correspond to the transmittances of 1 and 0, respectively.

A collimated laser beam with a wavelength of 532 nm was applied in the following
simulations and trapping experiments. The radii of the FZPs are set as 256 × 15 μm. The
main foci in Figs. 5.7(a) and (b) have the calculated distances of 221.7 mm and 250.7 mmfrom the FZPs, respectively. The peaks around the respective main focus are marked as
the first subsidiary focus and the second one, respectively. The average axial distance
between two neighbouring foci (i.e., the main focus and subsidiary foci) of the FZP beamwith N = 3 and S = 3 is about 8.5 mm and the counterpart of the fractional FZP with
N = 2.9 and S = 3 is about 10.3 mm. It can be seen in Figs. 5.7(c) and (d) that when N ischanged with a fractional increment, both the axial locations and the axial spacings of thefoci of the FZP beam can be more precisely controlled. Thus, microparticles can be
dynamically manipulated by changing Ns of the fractional FZPs.
In fact, N can be modified with a variation as small as 0.001. For example, the main focusof the fractional FZP with N = 3.001 and S = 3 has shifting displacements of 0.2 mm in
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free space and 8 μm in the sample cell from that of the FZP with N = 3 and S = 3,
respectively. Similarly, the corresponding subsidiary foci also have smaller shifting
displacements. For comparisons, N can be modified with an increment of 1, e.g., fromN = 2 to N = 3. The main focus of the FZP with N = 2 and S = 3 has shifting displacementsof 804.9 mm in free space and 32.2 mm in the sample cell from that of the FZP with
N = 3 and S = 3, respectively. Thus, microparticles can be dynamically manipulated bymodifying N with an especially small fractional variation. For convenience we used thesecond subsidiary foci of the fractional FZPs to trap microparticles dynamically. Firstly,
the FZPs with N = 3.001, 3.002, and 3.003 were loaded on the SLM sequentially every
5 seconds. When N was increased, the second subsidiary focus of the corresponding
fractional FZP was shifted toward the bottom of the sample cell. At this time, it can beseen in Fig. 5.8(c)-(e) that the trapped particle becomes blurred. Then, the FZPs with
N = 3.002, 3.001, and 3.0 were loaded on the SLM sequentially every 5 seconds. The
second subsidiary focus of the corresponding fractional FZP would be shifted toward the
top of the sample cell. The dynamic trappings are shown in Fig. 5.8 (f)-(h), where thetrapped microparticle moves toward the original location and is in focus. Thus, thepositions of the trapped particles can be dynamically controlled.

Fig. 5.7. Axial irradiance of the FZPs with (a) N = 3, S = 3 and (b) N = 2.9, S = 3. Plots of (c)the axial location of the main focus from the SLM and (d) axial distance between the main focus
and the second subsidiary focus of an FZP beam with S = 3 versus N.

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Fig. 5.8. Experimental demonstration of dynamic trapping with the second subsidiary foci of thefractional FZPs with N = 3, 3.001, 3.002, and 3.003, as well as S = 3, respectively. (a) The second
subsidiary focus of the FZP with N = 3 and S = 3, and (b) the trapped microparticle in the view
field of the CCD camera. (c)-(e) Trapped particle becomes blurred with dynamic changing of thefractional FZPs with N increasing from 3.001 to 3.003. (f)-(h) Trapped particle becomes clear again
with N decreasing from 3.002 to 3.

5.3.2. Fibonacci Zone Plate
In Fig. 5.9, the tilted dashed line arrow shows the movement direction of the focus. The
main reason is that the beam striking the CCD is titled.

Fig. 5.9. (a) Hologram of FiZP with optical grating phase; (b) the first prime focus captured by
CCD; (c) the second prime focus captured by CCD.

In Fig. 5.10, the circle presents the main focal position, and the titled dashed arrow shows
the movement direction of the focus. During the trapping process with the total time of 1
minute, the particle marked by the circle is from blurry to clear. When the particle istrapped stably, the particle is the clearest and on the focal plane.
When the objective and sample cell move up 100 μm, the second focus is at the focal
plane of the objective. In Fig. 5.11, the circle and titled dashed arrows present the second
main focal position, and the movement direction of the second focus. As the depth of thesample cell is 100 μm, the particle gets rid of the trapping force of the first main focus,
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and is trapped by the second main focus. In Fig. 5.12, the dashed box presents the staticreferenced particles, and the horizontal and titled dashed arrows show the movement
directions of the particle and focus, respectively. During the process when the main focus
moves left 20 μm, the particle is trapped stably all the time and moves with the movementof the focus.

Fig. 5.10. Longitudinal capture of particle by the first prime focus of the FiZP beam.

Fig. 5.11. Longitudinal capture of the particle by the second prime focus of FiZP beam.

Fig. 5.12. Capture and move of the particle in horizontal plane.

5.3.3. Thue-Morse Zone Plate
In Fig. 5.13(e) and (f) v (v = x/a) is the normalized transverse coordinate. The main focal
distance of the equivalent periodic ZP can be written as f = a2/(λ2S), where λ is the
wavelength of incident light and λ = 532 nm, a = 1.15 mm is the radius of the TMZP, S is
the order of the adoptive TM sequence. The foci of a TMZP are located around the axial
point z = a2/(λ2S) symmetrically and the symmetrical foci have the same intensity.
As the TMZP beams are found to have the self-reconstruction property, the trapped
particles in a TMZP beam will not affect the subsequent propagation of the beam. Thus,
the marked foci in Figs. 5.14(b) and (c) with the Arabic numerals are used to trap
microparticles. Due to the oblique incidence of the beam ray, there exists a displacement250

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between the adjacent foci in the view field and the trapped microparticles by the different
foci can also be seen in the view field. But when a trapped microparticle can be seenclearly in the focal plane of the objective, the others microparticles trapped in the rest fociare out of focus and cannot been clearly. The microparticle marked as 1 was trapped at
the focal plane of the objective and the CCD-captured image is clear. The other marked
microparticles were trapped in different planes and out of focus. Furthermore, in Figs.
5.15(a)-(f) the trapped microparticles at the foci marked as 1, 2, and 3 lined up in the view
field, which can be marked with a dashed red line. In the figures the microparticle circled
with the white dashed box is used as a reference. The results demonstrated that the trapped
microparticles are not only trapped stably, but also can be manipulated simultaneously
along a defined route in different planes. In fact, all the foci of the TMZP can trap andmanipulate microparticles simultaneously in multiple planes, which will realize the 3D
optical tweezers. Compared with the FZP, the experimental results demonstrate that the
TMZP can be applied in 3D optical tweezers and the microparticles will be trapped morestably in the TMZP beam for the symmetry foci with the same intensity.

Fig. 5.13. (a) and (b) The phase-only TMZP of order 6 and the corresponding equivalent periodic
ZP, respectively; (c) and (d) the normalized axial irradiance versus the axial coordinate u for the
TMZP of order 6 and the corresponding equivalent periodic zone plate with a = 1.15 mm,
respectively; (e)-(f) the transverse irradiance provided by the TMZP of order 6 and thecorresponding equivalent periodic zone plate with a = 1.15 mm, respectively.

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Fig. 5.14. (a) The phase-only TMZP of order 6 with a blazed grating, (b) the axial intensity
distribution of the phase-only TMZP of order 6, (c) the axial intensity contour of the TMZP
shown in (a) and (d) the optical path of the optical tweezers system.

Fig. 5.15. (a)-(f) Manipulations of the trapped microparticles (1, 2, and 3) highlighted with thewhite arrows along the direction marked with the black arrows in the three different planes.

5.4. Aperiodic Zone Plates for Optical Imaging
5.4.1. Clear Imaging
For simplicity, the GCAZP based on the Fibonacci sequence is taken for example toillustrate the construction method of the GCAZP. The GCAZP based on the Fibonaccisequence is composed of the FiZP with (r0 = 0, R = αa, S) and the Fibonacci radial zone252

Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging

plate (RZP) with (r0 = βa, R = a, S). r0 and R respectively present the original radius andthe outermost radius of the FiZP or the Fibonacci RZP, α is a positive number between 0
and 1/ 2, β is a positive number between 1/ 2 and 1, and S presents the order of the
Fibonacci sequence. It should be noted that α and β satisfy the equation α2 + β2 = 1, andthese zone plates are designed on the devices with the same size of a × a.
The transmittance function qi   of the FiZP with (r0 = 0, R = αa, S), i.e., the inner partof the GCAZP based on the Fibonacci sequence, can be calculated by Eq. (5.16),
M   j  1 2  dS 
qi     tS , j  rect 
,
dS
j 1



(5.16)

where    r / ( a )  represent the normalized square radial coordinate, r presenting theradial coordinate of the ZP ranges from 0 to αa, when the j-th element of the Fibonacci
2sequence of S is “A” or “B”, the corresponding tS , j is “1” or “0”, and d S  1 M .
The transmittance function qo   of the Fibonacci RZP with (r0 = βa, R = a, S), i.e., theouter part of the GCAZP based on the Fibonacci sequence, can be calculated by Eq. (5.17),M   j  1 2  dS 
qo     tS , j  rect 
,
dS
j 1







(5.17)

where    r a 2 - 2 /  2 is the normalized square radial coordinate, which has thevalue between 0 and   1- 2 /  2 , and r is equal or greater than r0.
It should be noted that the transmittance function of the GCAZP based on the Fibonaccisequence can be expressed as qGCAZP    qi    qo   , where    r a  is thenormalized square radial coordinate. The FiZP with (r0 = 0, R = αa, S) generates thefirst-order focus located at the axial position z = (αa)2/(λM). λ and M are the wavelengthand the number of the elements of the Fibonacci sequence of order S, respectively. The
Fibonacci RZP with (r0 = βa, R = a, S) generates the first-order focus located at the axial
position z = (1-β2)‧a2/(λM). Thus, as the result of the equation α2+β2 = 1, the FiZP with (r0= 0, R = αa, S) and the Fibonacci RZP with (r0 = βa, R = a, S) generate the first-order
focus with the same axial position. That is to say, the inner and outer parts have thecoincident first-order diffraction area. In addition, the first-order focal axial position of
the GCAZP can be adjustable by designing α and β. For simplicity, a, λ, α and β are set as
3.84 mm, 532 nm, 1/ 2 and 1 / 2, respectively. Fig. 5.16 illustrates the construction
method of the GCAZP based on the Fibonacci sequence.
2

The modulation transfer function (MTF) can be obtained by the Fourier transform of thecorresponding point spread function, which is calculated by the Fresnel diffraction
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integral formula [42]. It can be found in Fig. 5.17 (a) that the thick light and black lines
are above the thin light and black lines, respectively. Thus, the images located atz = 330.05 and 532.82 mm generated by the GCAZP based on the Fibonacci sequence of
S = 8 are clearer than those of the FiZP of S = 8, respectively. In Fig. 5.17 (b), the thicksolid line represents the MTF at the main focal plane located at z = 495 mm of the GCAZP
based on the Cantor sequence of S = 3, respectively. The thin solid line represents the
MTF at the main focal plane located at z = 495 mm of the FZP of S = 3, respectively. It
can be found in Fig. 5.17 (b) that the thick solid line is above the thin solid line. Thus, theimage located at z = 495 mm generated by the GCAZP based on the Cantor sequence of
S = 3 is clearer than that located at z = 495 mm of the FZP of S = 3. In Fig. 5.17(c), thethick light and black lines represent the MTFs at two focal planes located at z = 329.34
and 632.24 mm of the GCAZP based on the Thue-Morse sequence of S = 5, respectively.
The thin light and black lines present the MTFs at two focal planes located at z = 329.34
and 632.24 mm of the TMZP of S = 5, respectively. It can be found in Fig. 5.17(c) thatthe thick light and black lines are above the thin light and black lines, respectively. Thus,
the images located at z = 329.34 and 632.24 mm generated by the GCAZP based on theThue-Morse sequence of S = 5 are clearer than those located at z = 329.34 and 632.24 mmof the TMZP of S = 5, respectively. Thus, the images generated by the GCAZP based on
the different aperiodic sequences are clearer than those of the corresponding common
aperiodic zone plate, respectively.

Fig. 5.16. Binary profiles of (a) the FiZP with (r0 = 0, R = a / 2, S = 6), (b) the Fibonacci RZPwith (r0 = a / 2, R = a, S = 6) and (c) the corresponding GCAZP.

An imaging experimental system is shown in Fig. 5.18. A laser beam (λ = 532 nm)
illuminated a beam expender, a lens and a target object (a letter U) at the focal plane of
the lens, and then passed a collimator (a focal length of 550 mm and a diameter of
55 mm), a polarizer, a SLM (CAS MICROSTAR, TSLM07U-A, 1920 × 1080 pixels,
8.5 µm × 8.5 µm /pixel, and transmissive type) with the holograms of the above zone
plates, and an analyzer, and was finally captured by a Complementary Metal Oxide
Semiconductor (CMOS, Mindvision, MV-UBS300C).
After the target object is removed from a light path, the focal intensity distribution can be
captured. It can be seen in Figs. 5.19 (a), (b), (f) and (g) that the GCAZP based on the
Fibonacci sequence produce two main foci with higher intensity than the corresponding
FiZP. It can be seen in Figs. 5.19 (c) and (h) that the GCAZP based on the Cantor sequence
produce the single main focus with higher intensity than the corresponding FZP. It can beseen in Figs. 5.19 (d), (e), (i) and (j) that the GCAZP based on the Thue-Morse sequence
produce two main foci with higher intensity than the corresponding TMZP. Thus, the254

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GCAZPs based on the different aperiodic sequences can generate the main foci withhigher intensity than the corresponding common aperiodic zone plates. It should be notedthat as the diffraction of the SLM, the intensity distribution in the shorter diffraction
distance has higher intensities than the same intensity distribution in the longer diffraction
distance. Thus, it can be found in Fig. 5.19 that for the bifocal zone plates, the measured
focus with the short focal length has higher intensity than that with the long focal length.

Fig. 5.17. (a) MTFs of the GCAZP based on the Fibonacci sequence of S = 8 and the FiZP of
S = 8 at two main focal planes; (b) MTFs of the GCAZP based on the Cantor sequence of S = 3
and the FZP of S = 3 at the main focal plane; (c) MTFs of the GCAZP based on the Thue-Morsesequence of S = 5 and the TMZP of S = 5 at two main focal planes.

Fig. 5.18. Experimental setup of the imaging system.

After the target object is added to the light path, the generated images of these zone plates
can be captured. It can be seen that the letters in Figs. 5.20 (a) and (b) have higher contrast
than those in Figs. 5.20 (f) and (g), respectively. The main reason is that the GCAZP basedon the Fibonacci sequence has twin foci with higher intensity than the corresponding
FiZP. Thus, intensity differences between the letters and background in Figs. 5.20 (a) and(b) are bigger than those in Figs. 5.20(f) and (g), respectively. Therefore, the letters in
Figs. 5.20 (a) and (b) are clearer. It can be seen that the image in Fig. 5.20 (c) have higher
contrast than that in Fig. 5.20(h). The main reason is that the GCAZP based on the Cantor
sequence has the single main focus with higher intensity than the corresponding FZP.
Thus, it can be seen that intensity difference between the letter and background in
Fig. 5.20 (c) is bigger than that in Fig. 5.20 (h), so that the letter in Fig. 5.20 (c) is clearer
than that in Fig. 5.20 (h). It can be seen that the images in Figs. 5.20 (d) and (e) are clearer
than those in Figs. 5.20 (i) and (j), respectively. The main reason is that the GCAZP based255

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on the Thue-Morse sequence has twin foci with higher intensity than the corresponding
TMZP. Thus, intensity differences between the letters and background in Figs. 5.20 (d)and (e) are bigger than those in Figs. 5.20 (i) and (j), respectively. Therefore, the letters in
Figs. 5.20 (d) and (e) are clearer. Thus, the GCAZPs have clearer images than thecorresponding common aperiodic zone plates.

Fig. 5.19. Intensity distributions at z ≈ 330 and 530 mm of (a) and (b) the GCAZP based on the
Fibonacci sequence of S = 8. Intensity distributions at z ≈ 330 and 530 mm of (f) and (g) the FiZP
of S = 8. Intensity distributions at z ≈ 495 mm of (c) the GCAZP based on the Cantor sequence of
S = 3 and (h) the FZP of S = 3. Intensity distributions at z ≈ 330 and 630 mm of (d) and (e) the
GCAZP based on the Thue-Morse sequence of S = 5. Intensity distributions at z ≈ 330 and
630 mm of (i) and (j) the TMZP of S = 5.

Fig. 5.20. Captured images at z ≈ 330 and 530 mm of (a) and (b) the GCAZP based on the Fibonaccisequence of S = 8. Captured images at z ≈ 330 and 530 mm of (f) and (g) the FiZP of S = 8.
Captured images at z ≈ 495 mm of (c) the GCAZP based on the Cantor sequence of S = 3 and (h)the FZP of S = 3. Captured images at z ≈ 330 and 630 mm of (d) and (e) the GCAZP based on theThue-Morse sequence of S = 5. Captured images at z ≈ 330 and 630 mm of (i) and (j) the TMZPof S = 5.

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5.4.2. Colourful Imaging
The mth high-transmission zone for the MSFFZP consists of the upper and bottom half
high-transmission zones. The inner and outer radii of the mth upper half high-transmission
zone for the MSFFZP are rm1 and rm2 calculated by Eqs. (5.18) and (5.19) [33],respectively. The inner and outer radii of the mth bottom half high-transmission zone for
the MSFFZP are rm3 and rm4 calculated by Eqs. (5.20) and (5.21) [33], respectively.
rm1 

f

[arccos(1  2 )  2m ]

rm 2 

f
[arccos(1  2 )  2m + ]

rm 3 

f
[arccos(3  2 )  2m ]

rm 4 

f
[arccos(3  2 )  2m + ]

0  ,0  ,    2 ,

    2 ,(5.18)(5.19)(5.20)(5.21)

where θ is the azimuthal angle, λ presents a wavelength and α is a number between 0 and2, which is used to adjust the widths of high-transmission zones. Thus, for the MSFFZP
of the radius a, the corresponding focal length f can be calculated by the expressionf  a2 / [(M +3) ]. The MSFFZP is fractal according to the Cantor sequence of order S,
which have the total elements of 3S. M  2n  3S represents the total number of hightransmission and low-transmission zones in the upper or bottom half part of the MSFFZP,
where n is a positive integer. Moreover, n high-transmission and low-transmission zones
or 2n low-transmission zones of the MSFFZP from inside out are corresponding to thesingle element, i.e., “1” or “0” of the Cantor sequence of S from left to right in turn,respectively. For simplicity, λ is set as 550 nm and the MSFFZP of a = 3.456 mm and M= 54, which is based on the Cantor sequence of S = 2 with nine elements [1], is taken for
example to illustrate the focusing properties of the MSFFZP. Thus, the corresponding f
calculated by the expression is 381 mm. It should be noted in this chapter that three hightransmission and low-transmission zones or six low-transmission zones of the MSFFZP
from inside out are corresponding to the single element, i.e., “1” or “0” of the Cantor
sequence of S = 2 from left to right in turn, respectively.
It can be seen in Figs. 5.21(a-c) that the MSFFZPs with different values of α have twofoci with the focal lengths f = 381 mm and f2 = 457.2 mm, respectively. Thus, the MSFFZP
can generate two foci with many subsidiary foci along the optic axis. Moreover, we canfind that the foci at z = 381 mm in Figs. 5.21(a) and (b) have higher intensities than thoseat z = 457.2 mm, respectively, and the focus at z = 457.2 mm in Fig. 5.21(c) has higher
intensity than that at z = 381 mm. Therefore, two foci with many subsidiary foci of the
MSFFZP may have the unequal intensities. The suppression ratio of the MSFFZP is
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main foci. In Fig. 5.21(a), there are the high-intensity first, second and fourth order focifor the MSFFZP, whose low second-order and fourth-order suppression ratios are -8.489
and -0.492 dB, respectively. Thus, besides the first-order focus, the MSFFZP in Fig.
5.21(a) have the second and fourth order foci. In Fig. 5.21(b), there are significantly lower
intensities for the second and fourth order foci, whose main foci have the high suppression
ratios of -32.261 and -31.515 dB, respectively, so there is the first-order focus only. In
Fig. 5.21(c), the fourth-order focus has significantly lower intensity, but the second-order
focus has high intensity, and the corresponding main focal suppression ratios are -17.055
and -2.811 dB, respectively. Therefore, the fourth-order suppression ratio is high, but thesecond-order suppression ratio is low. There are the first order focus and the second order
focus along the optic axis only in Fig. 5.21(c), and the other high order diffraction foci aresuppressed. Thus, different values of α, i.e., the different widths of high-transmission
zones, have influence on the number of the high-order diffraction foci.

Fig. 5.21. (a-c) Axial irradiances of the MSFFZPs of α = 0.2, 1 and 1.614, respectively.
2

The intensity non-uniformity of twin foci is defined as  =  I i  I
i 1

I , where I i and I

are the intensity of the i-th focus and the average intensity of two foci, respectively. We
can find in Fig. 5.22 (a) that with the increase of α, the intensity of the focus at z = 381
mm firstly increases and then decreases, and the intensity of the focus at z = 457.2 mm
also firstly increases and then decreases. Moreover, it can be also seen in Fig. 5.22 (a) thatwhen α is equal to 1.03, the intensity of the focus at z = 381 mm has the maximum.
Therefore, the single main focus with highest intensity is produced by the MSFFZP of α
= 1.03 shown in Fig. 5.23(a). We can find in Fig. 5.22 (b) that with the increase of α,  oftwin foci firstly decreases and then increases. In addition, it can be also seen in Fig. 5.22(b) that when α is equal to 1.614,  of twin foci has the minimum. Thus, the MSFFZP of
α = 1.614 shown in Fig. 5.23 (b) has two equal-intensity foci located at z = 381 and457.2 mm. The second main focus locates at z = 457.2 mm. Moreover, the MSFFZP of
α = 1.03 produces the single main focus with many subsidiary foci. The dense data in
Fig. 5.22 have been zoomed in, and the small sub-figures are plotted to show the results.

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The tristimulus values, X, Y and Z can be calculated by Eq. (5.22). A horizontal
chromaticity coordinate x and a vertical chromaticity coordinate y can be calculated by
Eq. (5.23).
2

X ( z )   I ( z;  ) S ( ) xd  ,1

2

Y ( z )   I ( z;  ) S ( ) yd  ,(5.22)

1

2

Z ( z )   I ( z;  ) S ( ) zd  ,1where (λ1, λ2) shows the wavelengths from 380 to 780 nm with a 10 nm interval, S(λ) is aspectral distribution of the standard illuminant C, and x, y, z  presents sensitivity

functions (CIE 1931).

Fig. 5.22. (a) Relationships between α and intensities of two foci located at z = 381
and 457.2 mm produced by the MSFFZPs of different values of α;(b) Intensity non-uniformity of corresponding two foci.

Fig. 5.23. (a) and (b) Phase distributions of the MSFFZPs of α = 1.03 and 1.614, respectively.

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X,X Y  Z .
YyX Y  Zx

(5.23)In the simulations, the wavelengths from 380 to 780 nm with a 10 nm interval are used to
calculate the chromaticity coordinate of the chromaticity diagram in Fig. 5.24 (c). The
horizontal coordinates show the normalized axial distance normalized by the focal lengthof 381 mm of the SFZP under the illumination with the wavelength of 550 nm. The
triangles, circles and squares in Fig. 5.24 (c) are corresponding to those in Fig. 5.24 (b),respectively. It can be seen in Fig. 5.24 (c) that the distances between the circles and thepoint C (0.31006, 0.31616) for the SFZP and the MSFFZP of α = 1.03 are 0.318 and 0.202,
respectively. Moreover, the distances between the point C and the circles, which arecorresponding to two foci located at z = 381 and 457.2 mm for the MSFFZP of α = 1.614,
respectively, are 0.255 and 0.087, respectively. Thus, the distances between the circles
and the point C for the MSFFZPs of α = 1.03 and 1.614 are shorter than that for the SFZP.
When the distance between the point in the chromaticity diagram and the point C
representing the white illuminant is shorter, the corresponding colourful image has thelower chromatic aberration [3]. Therefore, compared with the SFZP, the MSFFZP of
α = 1.03 has the single image with the lower chromatic aberration at the single main focal
plane, and the MSFFZP of α = 1.614 generates two images with the lower chromaticaberration at two main focal planes simultaneously. Thus, a distance of 0.318, which islargest, imply that the corresponding colourful image of the SFZP has high chromaticaberration. The MSFFZP has one or two colourful images with low chromatic aberration
at the focal planes. However, the SFZP has the colourful image with high chromaticaberration at the focal plane.
The experimental schematic is shown in Fig. 5.25. Firstly, a beam (λ = 650, 532 or
450 nm) is expended by a beam expander, and then passes a lens, a target object at the
focal plane of the lens, a collimator (a focal length of 550 mm and a diameter of 55 mm),a polarizer, a splitting prism and a SLM (CAS MICROSTAR, FSLM-2K55-P,
1920 × 1080 pixels, 6.4 μm × 6.4 μm /pixel, and reflective type). Then, the reflective beam
passes the splitting prism, and a Complementary Metal Oxide Semiconductor (CMOS,
Mindvision, MV-UBS300C) camera is used to record the transverse intensity distribution.
The original target is one of the horizontal lines of the group and element (2, 1) on
the USAF 1951 resolution test chart, whose other transparent parts are covered withthe printing paper. It should be noted be that for the phase-only SLM, the product of the
wavelength and maximum phase is constant approximately. When λ is equal to 532 nm,
the maximum phase is equal to 2π. Thus, the maximum phases of λ = 650 and 450 nm are
1.637π and 2.364π, respectively. The grey values of Gray images are proportional tothe phases of the SLM approximately. Thus, for the SLM at λ = 650, 532 and 450 nm,
the phase of π is corresponding to the grey values of 156, 127 and 108, respectively.
The phase profiles at the different wavelengths are shown in Fig. 5.26. The grey level
images of each line have the same grey bar.

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Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging

Fig. 5.24. (a) Normalized axial intensity distributions under the illumination with thewavelengths 450, 550 and 650 nm, (b) the normalized axial illuminance Y and (c) the
chromaticity diagram of the SFZP, the MSFFZP of α = 1.03 and 1.614, respectively.

Fig. 5.25. Experimental system of the imaging system.

It can be seen in Fig. 5.27 that there are the clearest images at the main focal planes forthe above zone plates, and there are not the clear images at the other focal planes for the
SFZP of a = 3.456 mm, but there are the clear images at the subsidiary focal planes for
the MSFFZPs. Moreover, the MSFFZP of α = 1.03 has the colourful image at z = 395 mm,
and the MSFFZP of α = 1.614 has two colourful images at z = 395 and 475 mm,
respectively. However, the SFZP has not the colourful images at z = 395 mm. The main
reason is that the SFZP has the single focus with the short focal length, but the subsidiaryfoci extend the focal length of the single main focus and two main foci for the MSFFZPs
of α = 1.03 and 1.614, respectively. Thus, the MSFFZP has one or two colourful images
at the focal planes.
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Advances in Optics: Reviews. Book Series, Vol. 5

Fig. 5.26. Gray level images of (a-c) the SFZP of a = 3.456 mm, the MSFFZPs of α = 1.03 and1.614 at the wavelength λ = 650 nm, respectively, and (d-f) the above zone plates at the wavelength
λ = 532 nm, respectively, and (g-i) the above zone plates at the wavelength λ = 450 nm, respectively.

5.5. Conclusion
In this chapter, we have provided an overview about the aperiodic zone plates based on
optical tweezers and optical imaging. The method of constructing aperiodic zone plates
and the theory of the diffraction are illustrated. We have proved that the FZP beam cansimultaneously trap multiple particles positioned in different focal planes, and the
dynamic manipulation of microparticles in optical tweezers is implemented by changing
Ns of the fractional FZPs. Moreover, the symmetrical major foci around the main focusof the TMZP are used to trap particles stably and the Fibonacci zone plate cansimultaneously trap particles positioned in two different planes and move particles freely
in the focal plane. It is also proved in the experiments that the GCAZP has the clearer
images at the focal planes, and the MSFFZP produces the single or two colourful images
with the low chromatic aberrations at the focal planes. Aperiodic zone plates have a wide
range for practical applications and can offer great potential developments of opticaltweezers and optical imaging in the future.

Acknowledgments
The research was financially supported by the National Natural Science Foundation ofChina (NSFC) (Grant No. 11674401), the Natural Science Foundation of Hunan Province,
China (Grant No. 2019JJ40358), and the Fundamental Research Funds for the Central
Universities of Central South University, China (Grant No. 2020zzts043).
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Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging

Fig. 5.27. (a-c) Captured images at z = 395 mm for the SFZP of a = 3.456 mm at λ = 650, 532
and 450 nm, respectively; (d) The corresponding synthetic colourful image. (e-g) Captured images
at z = 395 mm for the MSFFZP of α = 1.03 at λ = 650, 532 and 450 nm, respectively;(h) The corresponding synthetic colourful image; (i-k) Captured images at z = 395 mm for the
MSFFZP of α = 1.614 at λ = 650, 532 and 450 nm, respectively; (l) The corresponding synthetic
colourful image. (m-o) Captured images at z = 475 mm for the MSFFZP of α = 1.614 at λ = 650,
532 and 450 nm, respectively; (p) The corresponding synthetic colourful image.

References[1]. G. Saavedra, W. D. Furlan, J. A. Monsoriu, Fractal zone plates, Optics Letters, Vol. 28,
Issue 12, 2003, pp. 971-973.
[2]. R. Menon, D. Gil, G. Barbastathis, H. I. Smith, Photon-sieve lithography, Journal of the
Optical Society of America A, Vol. 22, Issue 2, 2005, pp. 342-345.
[3]. Y. Wang, W. Yun, C. Jacobsen, Achromatic Fresnel optics for wideband extreme-ultraviolet
and X-ray imaging, Nature, Vol. 424, Issue 6944, 2003, pp. 50-53.
[4]. J. F. Barrera, M. Tebaldi, D. Amaya, W. D. Furlan, J. A. Monsoriu, N. Bolognini, R. Torroba,
Multiplexing of encrypted data using fractal masks, Optics Letters, Vol. 37, Issue 14, 2012,
pp. 2895-2897.

263

Advances in Optics: Reviews. Book Series, Vol. 5[5]. S. Wang, X. C. Zhang, M. P. Maley, M. F. Hundley, L. N. Bulaevskii, A. Koshelev,
A. J. Taylor, Terahertz technology: Terahertz tomographic imaging with a Fresnel lens,
Optics & Photonics News, Vol. 13, Issue 12, 2002, p. 59.
[6]. S. Cheng, X. Zhang, W. Ma, S. Tao, Fractal zone plate beam based optical tweezers, Scientific
Reports, Vol. 6, Issue 1, 2016, 34492.
[7]. T. Xia, S. Cheng, S. Tao, W. Yu, Imaging properties of generalized composite aperiodic zone
plates, Optics Express, Vol. 28, Issue 18, 2020, pp. 27181-27195.
[8]. T. Xia, S. Cheng, S. Tao, W. Yu, Colourful imaging and self-reconstruction propertiesof modified single-focus fractal zone plates, Optics Express, Vol. 28, Issue 25, 2020,
pp. 37827-37843.
[9]. S. B. Cheng, S. H. Tao, X. Y. Zhang, W. Z. Ma, Optical tweezers with fractional fractal zone
plate, IEEE Photonics Journal, Vol. 8, Issue 5, 2016, pp. 1-7.
[10]. Z. Bouchal, J. Wagner, M. Chlup, Self-reconstruction of a distorted nondiffracting beam,
Optics Communications, Vol. 151, 1998, pp. 207-211.
[11]. A. J. Cox, J. D'Anna, Constant-axial-intensity nondiffracting beam, Optics Letters, Vol. 17,
Issue 4, 1992, pp. 232-234.
[12]. J. Arlt, V. Garces-Chavez, W. Sibbett, K. Dholakia, Optical micromanipulation using a Bessel
light beam, Optics Communications, Vol. 197, 2001, pp. 239-245.
[13]. J. A. Rodrigo, M. Angulo, T. Alieva, Dynamic morphing of 3D curved laser traps for
all-optical manipulation of particles, Optics Express, Vol. 26, Issue 14, 2018,
pp. 18608-18620.
[14]. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides,
Z. G. Chen, Trapping and guiding microparticles with morphing autofocusing Airy beams,
Optics Letters, Vol. 36, Issue 15, 2011, pp. 2883-2885.
[15]. J. A. Davis, L. Ramirez, J. A. R. Martin-Romo, T. Alieva, M. L. Calvo, Focusing propertiesof fractal zone plates: experimental implementation with a liquid-crystal display, Optics
Letters, Vol. 29, Issue 12, 2004, pp. 1321-1323.
[16]. S. H. Tao, B. C. Yang, H. Xia, W. X. Yu, Tailorable three-dimensional distribution of laser
foci based on customized fractal zone plates, Laser Physics Letters, Vol. 10, Issue 3, 2013,
035003.
[17]. J. Pu, P. H. Jones, Devil's lens optical tweezers, Optics Express, Vol. 23, Issue 7, 2015,
pp. 8190-8199.
[18]. M. J. Simpson, A. G. Michette, Imaging properties of modified Fresnel zone plates, Optica
Acta International Journal of Optics, Vol. 31, Issue 4, 1984, pp. 403-413.
[19]. C. Q. Xie, X. L. Zhu, H. L. Li, L. N. Shi, Y. H. Wang, Feasibility study of hard-x-ray
nanofocusing above 20 keV using compound photon sieves, Optics Letters, Vol. 35, Issue 23,
2010, pp. 4048-4050.
[20]. F. Giménez, W. D. Furlan, A. Calatayud, J. A. Monsoriu, Multifractal zone plates, Journal of
the Optical Society of America A, Vol. 27, Issue 8, 2010, pp. 1851-1855.
[21]. T. Xia, S. B. Cheng, S. H. Tao, An arbitrarily designed main focus with high intensity
generated by a composite fractional fractal zone plate, Optics Communications, Vol. 430,
2019, pp. 348-351.
[22]. W. Ma, S. Tao, S. Cheng, Composite Thue-Morse zone plates, Optics Express, Vol. 24,
Issue 12, 2016, pp. 12740-12747.
[23]. T. Xia, S. Cheng, J. Yan, S. Tao, Modified Thue-Morse zone plates with arbitrarily designed
high-intensity twin main foci, Laser Physics, Vol. 27, Issue 12, 2017, 125001.
[24]. J. A. Monsoriu, W. D. Furlan, G. Saavedra, F. Giménez. Devil’s lenses, Optics Express,
Vol. 15, Issue 21, 2007, pp. 13858-13864.
[25]. T. Xia, S. Cheng, S. Tao, Two high-intensity foci with the generalized mean generated by akinoform generalized mean lens, Optik, Vol. 175, 2018, pp. 99-104.

264

Chapter 5. Aperiodic Zone Plates for Optical Tweezers and Optical Imaging[26]. A. Sabatyan, B. Meshginqalam, Generation of annular beam by a novel class of Fresnel zone
plate, Applied Optics, Vol. 53, Issue 26, 2014, pp. 95995-6000.
[27]. A. Sabatyan, Z. Behjat, Radial phase modulated spiral zone plate for generation andmanipulation of optical perfect vortex, Optical & Quantum Electronics, Vol. 49, Issue 11,2017, 371.
[28]. T. Xia, S. Cheng, S. Tao, An annular beam with segmented phase gradients generated by amodified spiral zone plate, Journal of Optics, Vol. 21, Issue 11, 2019, 115602.
[29]. C. Q. Xie, X. L. Zhu, L. N. Shi, M. Liu, Spiral photon sieves apodized by digital prolate
spheroidal window for the generation of hard-x-ray vortex, Optics Letters, Vol. 35, Issue 11,2010, pp. 1765-1767.
[30]. F. Giménez, J. A. Monsoriu, W. D. Furlan, A. Pons, Fractal photon sieve, Optics Express,
Vol. 14, Issue 25, 2006, pp. 11958-11963.
[31]. F. Giménez, W. D. Furlan, J. A. Monsoriu, Lacunar fractal photon sieves, Optics
Communications, Vol. 277, Issue 1, 2007, pp. 1-4.
[32]. J. Ke, J. Zhang, Generalized Fibonacci photon sieves, Applied Optics, Vol. 54, Issue 24, 2015,
pp. 7278-7283.
[33]. X. Chen, X. Wang, Single-focus x-ray zone plate by stagger arrangement of zones, Optics
Express, Vol. 21, Issue 17, 2013, pp. 20005-20014.
[34]. Y. G. Cheng, J. P. Zhu, S. Hu, L. X. Zhao, W. Yan, Y. He, W. B. Jiang, J. Liu, Focusing
properties of single-focus photon sieve, IEEE Photonics Technology Letters, Vol. 29, Issue 3,
2016, pp. 275-278.
[35]. L. Wei, L. Kuang, W. Fan, H. Zang, L. Cao, Y. Gu, X. Wang, Annulus-sector-element coded
Gabor zone plate at the x-ray wavelength, Optics Express, Vol. 19, Issue 22, 2011,
pp. 21419-21424.
[36]. Y. H. Liang, E. L. Wang, Y. L. Hua, C. Q. Xie, T. C. Ye, Single-focus spiral zone plates,
Optics Letters, Vol. 42, Issue 13, 2017, pp. 2663-2666.
[37]. V. Ferrando, F. Giménez, W. D. Furlan, J. A. Monsoriu, Bifractal focusing and imaging
properties of Thue-Morse zone plates, Optics Express, Vol. 23, Issue 15, 2015,
pp. 19846-19853.
[38]. J. A. Monsoriu, A. Calatayud, L. Remón, W. D. Furlan, G. Saavedra, P. Andrés, Bifocal
Fibonacci diffractive lenses, IEEE Photonics Journal, Vol. 5, Issue 3, 2013, 3400106.
[39]. A. Calatayud, V. Ferrando, F. Giménez, W. D. Furlan, G. Saavedra, J. A. Monsoriu, Fractal
square zone plates, Optics Communications, Vol. 286, 2013, pp. 42-45.
[40]. A. Calatayud, V. Ferrando, L. Remón, W. D. Furlan, J. A. Monsoriu, Twin axial vortices
generated by Fibonacci lenses, Optics Express, Vol. 21, 2013, pp. 10234-10239.
[41]. S. H. Tao, X. C. Yuan, J. Lin, R. E. Burge, Sequence of focused optical vortices generated bya spiral fractal zone plate, Applied Physics Letters, Vol. 89, Issue 3, 2006, 031105.
[42]. A. Sabatyan, L. Elahi, FFT-based convolution algorithm for fast and precise numericalevaluating diffracted field by photon sieve, Optik, Vol. 124, Issue 21, 2013, pp. 4960-4962.
[43]. J. W. Goodman, Introduction to Fourier Optics, Roberts and Company Publishers, 2005.

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Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrationsin the Centered Conical Optical Systems with High and Super-high Numerical Apertures

Chapter 6

Rigorous Analytical Correction of Axial
Chromatic and On-axis Spherical
Aberrations in the Centered Conical Optical
Systems with High and Super-high Numerical
Apertures
Boian A. Hristov1

6.1. Introduction
Still the syntheses of optical systems is both an art and a science [1, pp. 3-4], [2, p. 409].
There is no procedure that will lead from a set of specifications to the successful designof an optical system [2, pp. 409-410].
From long ago it is well known that the designers use modern software and optimization
programs. However these programs require a suitable starting design which is vitally
important for a good end results [2]. Now-a-days there is no suitable computer program
to create a useful lens design in the absence of the guidance of a skilled optical designer.
In [3] and [4] we developed a rigorous analytical aberrations theory of centered opticalsystems containing spherical surfaces. It permits to correct rigorously aberrations for astrictly set aperture angles or view angles. But we are not able to correct strictly
aberrations on the whole aperture or on the whole field of view. That’s why this theory
does not allow to design diffraction-limited systems with high and super-high numericalapertures.
In Section 6.2 we present the paraxial matrix transformations in a centered optical systems
and find the rigorous analytical dependences between constructive parameters, paraxial
characteristics and axial chromatic aberrations. We show how to correct strictly the axial
chromatic aberration together with preset paraxial characteristics using some constructive

Boyan Hristov
Institute of Optical Materials and Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

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parameters as free parameters. Further, we analyze the analytical connections between theparameters of real tangential rays in the system, paraxial matrix coefficients, sagittal
radius of the conic surfaces and the so-called sagittal radius-coefficient (that is the ratioof the vertex to the sagittal radius). On this base we find rigorous analytical equation forcorrection of on-axis spherical aberration. It is important to note that the correction carries
out for each aperture ray from zero to preset maximum aperture, using only one asphericalsurface. So we develop a reliable algorithm for simultaneous correction of on-axis
spherical and axial chromatic aberrations together with preset paraxial characteristics inthe centered optical systems which contain conic surfaces. As a result, we calculate
rigorously the coordinates of aspherical (or so-called multi-conical) surface and slope
angle for each surface point. Thus we have all data for production of the aspherical opticalcomponent using CNC machines. There is no need to use optimization programs for thepresentation of aspherical surfaces with the help of conic constant and polynomials(see [5]).
In Section 6.3 we show variety of diffraction-limited lenses with high and super-highnumerical apertures that have better image quality and higher numerical apertures thanthe best available up- to-now (see [6-11]).

6.2. Theoretical Basis
6.2.1. Paraxial Matrix Transformations and Axial Chromatic Aberration
6.2.1.1. Paraxial Matrix Transformations from Object to Image Space in a Single
Optical Surface
Let us consider a single refractive (or reflective) conical surface having serial number j ina rotationally symmetric optical system composed of 𝑘 conical surfaces. It has a vertex
radius rj and separates two media of refraction (or reflection) index nj on the left and index
nj+1 on the right.
It is well known [2, p. 38] that the dependence between the object and paraxial image
positions on a single spherical or conical surface is.
𝑛𝑗+1
′
𝑧𝑜𝑗

−

𝑛𝑗
𝑧𝑜𝑗

=

𝑚𝑜𝑗
𝑟𝑗

,

(6.1)

where zoj is the axial distance from the surface vertex to the object, zoj’ is the axial distancefrom the surface vertex to the paraxial image and m0j is the following refraction (orreflection) index difference
𝑚0𝑗 = 𝑛𝑗+1 − 𝑛𝑗

(6.2)

The sign conventions for the radii, thicknesses and distances, which we shell observe here,
are the same as it is in [2, pp. 35-36].
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Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrationsin the Centered Conical Optical Systems with High and Super-high Numerical Apertures

Transforming Eq. (6.1) for the axial image distance zoj’ we get
𝑟 ∗𝑛

∗𝑧

′
𝑧𝑜𝑗
= 𝑚 𝑗 ∗𝑧𝑗+1+𝑟 0𝑗
∗𝑛
𝑎𝑗

0𝑗

𝑗

(6.3)

𝑗

The axial image distance can be represented as a bilinear function
𝐴 ∗𝑧 +𝐵′𝑧𝑜𝑗= 𝐶 𝑎𝑗∗𝑧 0𝑗+𝐷𝑎𝑗,(6.4)

𝐴𝑎𝑗 = 𝑟𝑗 𝑛𝑗+1 ; 𝐵𝑎𝑗 = 0; 𝐶𝑎𝑗 = 𝑚0𝑗 ; 𝐷𝑎𝑗 = 𝑟𝑗 𝑛𝑗

(6.5)

𝑎𝑗

0𝑗

𝑎𝑗

where

This bilinear function can be described by the following quadratic matrix Gaj
𝐺𝑎𝑗 =

𝐴𝑎𝑗
𝐶𝑎𝑗

𝐵𝑎𝑗 𝑟𝑗 ∗ 𝑛𝑗+1
= 𝑚
𝐷𝑎𝑗
0𝑗

0
𝑟𝑗 ∗ 𝑛𝑗(6.6)

The determinant Detaj of the matrix Gaj is
𝐷𝑒𝑡𝑎𝑗 = 𝐴𝑎𝑗 ∗ 𝐷𝑎𝑗 − 𝐵𝑎𝑗 ∗ 𝐶𝑎𝑗 = 𝑛𝑗 ∗ 𝑛𝑗+1 ∗ 𝑟𝑗2(6.7)

The matrix Gaj we shell call an axial surface transformation matrix (ASTM) of a single
surface or a single axial transformation matrix. The coefficients Aaj, Baj, Caj and Daj weshell call the ASTM-coefficients respectively.
Note that equations from (6.1) to (6.7) are correct in the case when the origins of the object
and image coordinate system coincide with surface vertex. If we want to relocate theorigin of the image coordinate system at a distance dj (for example to the next surface inthe centered optical system), we use a quadratic matrix Dj having the following matrix
coefficients:
𝐷𝑗 =

1 −𝑑𝑗
0
1

(6.8)

The matrix Dj we will call an axial relocation matrix (ARM) of the dj thickness. The
distance z’dj in the relocated at a distance dj coordinate system is𝑧′𝑑𝑗 = 𝑧′𝑜𝑗 − 𝑑𝑗

(6.9)

Substituting z’oj from Eq. (6.3) in Eq. (6.9) we get′
𝑧𝑑𝑗=((𝑛𝑗+1 ∗ 𝑟𝑗 − 𝑑𝑗 ∗ 𝑚0𝑗 ) ∗ 𝑧𝑗 − 𝑛𝑗 ∗ 𝑟𝑗 ∗ 𝑑𝑗 )⁄(𝑚0𝑗 ∗ 𝑧𝑗 + 𝑟𝑗 ∗ 𝑛𝑗 ) (6.10)

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Advances in Optics: Reviews. Book Series, Vol. 5

It can be proved that the axial image distance z’dj in the relocated coordinate system isrigorously described with the help of the matrix Gdj that is a matrix product of the jthsurface ATM and ARM of the jth thickness, i.e.
𝐺𝑑𝑗 = 𝐷𝑗 ∗ 𝐺𝑎𝑗 =

1
0

𝐴𝑑𝑗
0
=
𝑛𝑗 ∗ 𝑟𝑗 𝐶𝑑𝑗−𝑑𝑗 𝑛𝑗+1 ∗ 𝑟𝑗
x 𝑚
0𝑗
1

𝐵𝑑𝑗
𝐷𝑑𝑗

(6.11)The matrix Gdj we call transformation-relocation matrix (TRM). The corresponding to theTRM bilinear function is𝑧′𝑑𝑗 =

𝐴𝑑𝑗 ∗𝑧0𝑗 +𝐵𝑑𝑗

(6.12)

𝐶𝑑𝑗 ∗𝑧0𝑗 +𝐷𝑑𝑗

If we calculate TRM Gdj from Eq. (6.11) we obtain the following coefficients
𝐴𝑑𝑗 = 𝑛𝑗+1 ∗ 𝑟𝑗 -𝑑𝑗 ∗ 𝑚0𝑗 ; 𝐵𝑑𝑗 = -𝑛𝑗 ∗ 𝑟𝑗 ∗ 𝑑𝑗 ; 𝐶𝑑𝑗 = 𝑚0𝑗 ; 𝐷𝑑𝑗 = 𝑛𝑗 ∗ 𝑟𝑗 ,(6.13)

Comparing the coefficients Aaj, Baj, Caj, Daj from Eq. (6.13) with coefficients from bilinear
function in Eq. (6.10) we can see that they are exactly the same. The matrix Gdj describes
exactly the axial image distance z’dj in the relocated to the next surface coordinate system.
6.2.1.2. Exact Paraxial Matrix Transformations in a Centered Optical System
of Conical Surfaces. Invention of Exact Analytical Dependences between
Constructive Parameters and Paraxial Characteristics
Let us discuss the first surface in the centered optical system and the first relocation to thesecond surface at a distance d1.
Using Eqs. (6.11, 6.12 and 6.13) for the first surface we write
𝐺𝑑1 = 𝐷1 𝐺𝑎1 =

𝐴
0
= 𝑑1
𝑛1 ∗ 𝑟1 𝐶𝑑1

1 −𝑑1 𝑛2 ∗ 𝑟1
x
𝑚01
0
1

𝐵𝑑1
,
𝐷𝑑1

(6.14)

where
𝐴𝑑1 = 𝑛2 ∗ 𝑟1 -𝑑1 ∗ 𝑚01 ,𝐵𝑑1 = - 𝑛𝑗 ∗ 𝑟1 ∗ 𝑑1 , 𝐶𝑑1 = 𝑚01 , 𝐷𝑑1 = 𝑛1 ∗ 𝑟1

(6.15)The matrix Gd1 describes simultaneously the axial transformation of the first surface andthe relocation at a distance d1.
The corresponding to the first TRM bilinear function is
𝐴

∗𝑧 +𝐵

′
𝑧𝑑1
= 𝐶 𝑑1∗𝑧01+𝐷𝑑1
𝑑1

270

01

𝑑1

(6.16)

Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrationsin the Centered Conical Optical Systems with High and Super-high Numerical Apertures

Here zo1 is the object distance measured along the optical axis.
It is clear that image distance z’d1 for the first surface is equal to the object distance zo2 forthe second surface, i.e.
′
𝑧𝑑1
= 𝑧02

(6.17)

Applying Eq. (6.3) for the second surface and taking account of Eq. (6.17) we obtain′𝑧02=

𝐴 ∗𝑧 +𝐵𝑛3 ∗𝑟2 𝑑1 01 𝑑1

𝐶𝑑1 ∗𝑧01 +𝐷𝑑1
𝐴𝑑1 ∗𝑧01 +𝐵𝑑1
𝑚02
+𝑛2 ∗𝑟2
𝐶𝑑1∗𝑧01 +𝐷𝑑1

𝐴

∗𝑧

+𝐵

= 𝐶𝑎2∗𝑧01+𝐷𝑎2
𝑎2

01

𝑎2

(6.18)

Transforming the right equality in Eq. (6.18) the coefficients Aa2, Ba2, Ca2, Da2 became asfollows
𝐴𝑎2 = 𝑛3 ∗ 𝑟2 ∗ (𝑛2 ∗ 𝑟1 − 𝑑1 ∗ 𝑚01 ),(6.19)

𝐵𝑎2 = − 𝑛1 ∗ 𝑛3 ∗ 𝑟1 ∗ 𝑟2 ∗ 𝑑1 ,(6.20)

𝐶𝑎2 = 𝑚01 (𝑛2 ∗ 𝑟2 − 𝑑1 ∗ 𝑚02 ) + 𝑛2 ∗ 𝑟1 ∗ 𝑚02 ,(6.21)

𝐷𝑎2 = 𝑛1 ∗ 𝑟1 (𝑛2 ∗ 𝑟2 − 𝑚02 ∗ 𝑑1 ),(6.22)

One can see from Eqs. (6.19)-(6.22) that every axial matrix coefficient Aa2, Ba2, Ca2, Da2
is linear function separately to each constructive parameter r1, r2, d1 in the systemof two surfaces.
It can be proved that the quadratic matrix ga2 in Eq. (6.23) describes exactly the bilinear
function (6.18) without any approximations
𝑔𝑎2 = 𝐺𝑎2 ∗ 𝐷1 ∗ 𝐺𝑎1(6.23)

Multiplying the three matrices in Eq. (6.23) we receive exactly the same matrix
coefficients as they are in Eqs. (6.19)-(6.22).
The matrix ga2 describes exactly the axial image distance z’02 in the relocated to the second
surface vertex coordinate system. In other words Eq. (6.23) is a matrix formula to calculate
axial matrix coefficients Aa2, Ba2, Ca2, Da2 for an optical system of two conical surfaces
when the origin of the object coordinate system coincides with the vertex of the firstsurface and the origin of the image coordinate system coincides with the vertex of the last
surface.
By analogy we can derive an exact matrix equation for an optical system of k conicalsurfaces and namely
𝑔𝑎𝑘 = 𝐺𝑎𝑘 ∗ ∏1𝑗=𝑘−1 𝐷𝑗 ∗ 𝐺𝑎𝑗 =

𝐴𝑎𝑘
𝐶𝑎𝑘

𝐵𝑎𝑘
𝐷𝑎𝑘

(6.24)

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The matrix gak we shell call axial transformation matrix for the whole system i.e. ATMS.
It is easy to prove that the axial matrix coefficients Aak, Bak, Cak, Dak of the system arelinear functions to each radius or axial thickness. But they are quadratic functions to eachindex of refraction except the first and the last one.
The matrix gak corresponds to the next bilinear function of the axial image distance withthe same axial matrix coefficients
𝐴

∗𝑧

+𝐵

′
𝑧0𝑘
= 𝐶𝑎𝑘∗𝑧01+𝐷𝑎𝑘
𝑎𝑘

01

𝑎𝑘

(6.25)The back focal length (bfl) of an optical system can be found as a limit of z’0k (seeEq. (6.25)) when z01 is tending to infinity
𝑏𝑓𝑙 =

𝐴𝑎𝑘
⁄𝐶
𝑎𝑘

(6.26)

If the axial image distance z’ok is tending to infinity (in other words the denominator of
Eq. (6.25) tends to zero i.e. (Cak* z01+ Dak = 0) we can find the front focal length (ffl) as
𝑓𝑓𝑙 = −

𝐷𝑎𝑘⁄𝐶
𝑎𝑘

(6.27)

Further on we shell derive formulas for the effective focal length f’ and for the lateral (ortransverse) magnification M in an optical system of k surfaces.
Using Eqs. (6.24)-(6.27) we can calculate exactly the four axial matrix coefficients theaxial image distance for any object distance, the back focal length and the front focal
length in each centered optical system of conical surfaces. Moreover Eqs. (6.24) and(6.25) have a remarkable capacity to give us an exact analytical relations between theparaxial characteristics (effective focal length, back focal length, front focal length andmagnification) of optical system and its constructive parameters, such as vertex radiuses,
thicknesses and refraction (or reflection) indexes– for example see Eqs. (6.19)-(6.22).
Note that the axial object distance in Eq. (6.25) is measured from the first surface’s vertex
to the object and the axial image distance is measured from the last surface’s vertex to theaxial image.
Let’s transpose the object coordinate system from the first surface’s vertex to the front
focus, i.e.
𝑧01 = 𝑧 −

𝐷𝑎𝑘
⁄𝐶
𝑎𝑘

(6.28)In addition the image coordinate system is transposed from the last surface vertex to theback focus

272

Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrationsin the Centered Conical Optical Systems with High and Super-high Numerical Apertures′𝑧𝑜𝑘= 𝑧′ +𝐴𝑎𝑘⁄𝐶
𝑎𝑘(6.29)In Eq. (6.28) z is the distance from the front focus to the object and in (6.29) z’ is the
distance from the back focus to the axial image.
Substituting z01 from (6.28) and z’0k from (6.29) in Eq. (6.25) after rearranging we have−𝑧𝑧 ′ =(𝐴𝑎𝑘 ∗ 𝐷𝑎𝑘 − 𝐵𝑎𝑘 ∗ 𝐶𝑎𝑘 )⁄𝐶 2
𝑎𝑘(6.30)

It can be proved that the axial determinant Detak of the matrix gak from Eq. (6.24), is2∏𝑘𝑗=2 𝑛𝑗2
𝐷𝑒𝑡𝑎𝑘 = 𝐴𝑎𝑘 ∗ 𝐷𝑎𝑘 − 𝐵𝑎𝑘 ∗ 𝐶𝑎𝑘 = 𝑛1 ∗ 𝑛𝑘+1 ∗ ∏𝑘𝑗=1 𝑟𝑎𝑗(6.31)

Eq. (6.31) shows that the axial determinant in each optical system of conical surfaces is
independent of axial distances between the optical surfaces. It depends on vertex radii ofthe system and refraction indices only. Using Newtonian [2, p. 30] equation and Eq. (6.31)
we can rewrite Eq. (6.30) as−𝑧 ∗ 𝑧 ′ =(𝑛1 ∗ 𝑛𝑘+1 ∗ ∏𝑘𝑗=1 𝑟𝑗2 ∏𝑘𝑗=2 𝑛𝑗2 )𝑛1 ∗ (𝑓 ′ )2⁄⁄ 2
𝑛𝑘+1 =
𝐶(6.32)

𝑎𝑘

Here f’ is the effective focal length of the optical system.
From Eq. (6.32) we can find the effective focal length as follows
𝑓′ =∏𝑘𝑗=1 𝑟𝑎𝑗 ∏𝑘+1
𝑗=2 𝑛𝑗⁄𝐶𝑎𝑘(6.33)

By means of Eq. (6.33) we transform (6.32) in the form−𝑧 ′ = 𝑓 ′ ∗ ∏𝑘𝑗=1 𝑟𝑗 ∗ 𝑛𝑗 ⁄𝑧 ∗ 𝐶𝑎𝑘(6.34)

It is well known that the fraction f’/z is nothing but the lateral (or transverse) magnification(M) of an optical system, i.e.
−𝑧 ′ = 𝑀 ∗ ∏𝑘𝑗=1 𝑟𝑗 ∗ 𝑛𝑗 ⁄𝐶𝑎𝑘

(6.35)

Now we replace z’0k from (6.25) in (6.29). After that we substitute 𝑧 ′ from Eq. (6.29) in(6.35). In addition after rearranging of Eq. (6.35) we obtain rigorous formula for a lateralmagnification M in a centered optical systems of conical surfaces
𝑀 = 𝑛𝑘+1 ∗ ∏𝑘𝑗=1 𝑛𝑗∗ 𝑟𝑗 ⁄𝑛1∗ (𝐶𝑎𝑘 ∗ 𝑧01 + 𝐷𝑎𝑘 )(6.36)

If we want to design a centered optical system, composed of conical surfaces, with presetparaxial parameters (effective focal length, lateral magnification, back focal length or
front focal length) we use Eqs. (6.26), (6.27), (6.33) and (6.36).
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Below we show a simple example how to find an exact analytical connection between theconstructive parameters and paraxial characteristics of a single lens.
For example we want to design a lens with preset effective focal length f’2 and back focal
length bfl2. As a free parameters we use the two axial radii r1 and r2 respectively. Using
Eqs. (6.33) and (6.26) for the case 𝑘 = 2 it can be written
𝑓2′ = 𝑟1 ∗ 𝑟2 ∗ 𝑛2 ∗ 𝑛3 ⁄𝐶𝑎2 ,(6.37)

𝑏𝑓𝑙2 = 𝐴𝑎2 ⁄𝐶𝑎2

(6.38)

Excluding Ca2 from Eqs. (6.37) and (6.38) with the help of (6.19) we get
𝑟1 = 𝑑1 ∗ 𝑚01 ∗ 𝑓2′ ⁄𝑛2 ∗ (𝑓2′ − 𝑏𝑓𝑙2 )(6.39)

Now if we substitute r1 from Eqs. (6.39) in (6.37) or (6.38) we find𝑟2 = 𝑏𝑓𝑙2 ∗ 𝑑1 ∗ 𝑚01⁄(𝑛2 ∗ (𝑓2′ − 𝑏𝑓𝑙2 ) − 𝑑1 )(6.40)

Eqs. (6.39) and (6.40) allows us to find the numerical values of the two radii r1 and r2 sothat the two preset paraxial characteristics f’2 and bfl2 are satisfied.
Let’s the constructive parameters and preset paraxial characteristics of the lens are:n1 = n3 = 1; n2 = 1.8; d1 = 20 mm; f’2 = 100 mm; bfl2 = 60 mm;
According to Eqs. (6.39) and (6.40) we calculate: r1 = 22.222222 mm;
r2 = 18.461538 mm. If we wont to have bfl2 = 140 mm then the radii are:r1 = -22.222222 mm; r2 = -24.347826 mm.
In conclusion of Section 6.2.1.2 we claim that:•Independently from the number of surfaces in the centered optical system, Eq.
(6.24) gives a possibility through analytical computations to generate rigorous
analytical dependences between the paraxial matrix coefficients and constructive
parameters. If the latter are known, we can calculate the exact values of the matrix
coefficients for the whole system;•

Using Eqs. (6.24)-(6.27), (6.33) and (6.36) we can generate exact analyticaldependences between the paraxial characteristics and constructive parameters. If
the latter are known, we can calculate the exact values of the paraxial
characteristics for the whole system.

6.2.1.3. Axial Image Chromatic Aberration
It is well known that the index of refraction varies as a function of wavelength of light.
For a given object distance z01 the axial chromatic aberration is a longitudinal variation ofimage position with wavelength. We define the axial chromatic aberration as a difference
between the axial image distances of any two wavelength.
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Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrationsin the Centered Conical Optical Systems with High and Super-high Numerical Apertures

Let us consider a spectral wavelength range with left and right ends. We shell denote theindices of refraction nj for the right wavelength with additional lower index r (i.e. njr) andfor the left end with l (i.e. njl) respectively.
In accordance with Eq. (6.25) we write for the right wavelength
𝐴𝑟 ∗𝑧01 +𝐵𝑟𝑧′𝑟 =𝐶𝑟 ∗𝑧01 +𝐷𝑟,(6.41)

where Ar, Br, Cr and Dr are axial matrix coefficients for the right wavelength in the opticalsystem with k conical surfaces and z’r is the axial image distance for the right wavelength.
By analogy for the left spectral end we have𝐴 ∗𝑧 +𝐵𝑧′𝑙 = 𝐶𝑙∗𝑧01+𝐷𝑙𝑙

01(6.42)𝑙

Here Al, Bl, Cl and Dl are axial matrix coefficients for the left wavelength and z’l is theaxial image distance for the same wavelength.
Taking into account Eqs. (6.41) and (6.42) the axial chromatic aberration chr is
𝐴 ∗𝑧

+𝐵

𝐴 ∗𝑧

+𝐵

𝑢 ∗𝑧 2 +𝑢 ∗𝑧 +𝑢

1 01
0𝑐ℎ𝑟 = 𝑧𝑟′ − 𝑧𝑙′ = 𝐶𝑟∗𝑧01+𝐷𝑟 − 𝐶𝑙∗𝑧01+𝐷𝑙 = 𝑣2 ∗𝑧01
2 +𝑣 ∗𝑧 +𝑣
𝑟

01

𝑟

𝑙

01

𝑙

2

01

1

01

0

(6.43)

Eq. (6.43) shows that the axial chromatic aberration is a quadratic fractional function tothe object distance z01. Using the Eq. (6.43) after symbolic computation of matrix
coefficients we can find the coefficients u2, u1, uo,υ2, υ1, υ0
𝑢2 = 𝐴𝑟 ∗ 𝐶𝑙 − 𝐴𝑙 ∗ 𝐶𝑟 ,(6.44)

𝑢1 = 𝐴𝑟 ∗ 𝐷𝑙 − 𝐴𝑙 ∗ 𝐷𝑟 + 𝐵𝑟 ∗ 𝐶𝑙 − 𝐵𝑙 ∗ 𝐶𝑟 ,(6.45)

𝑢0 = 𝐵𝑟 ∗ 𝐷𝑙 − 𝐵𝑙 ∗ 𝐷𝑟 ,(6.46)

𝑣2 = 𝐶𝑟 ∗ 𝐶𝑙 ,(6.47)

𝑣1 = 𝐶𝑟 ∗ 𝐷𝑙 + 𝐶𝑙 ∗ 𝐷𝑟 ,(6.48)

𝑣0 = 𝐷𝑟 ∗ 𝐷𝑙

(6.49)

It is evident from Eqs. (6.44)-(6.49) that each coefficient of u2, u1, uo,υ2, υ1 and υ0 is aquadratic function to each constructive parameter (rj and dj) except the last vertex
radius rk.
Eq. (6.43) describes the axial chromatic aberration for any object distances z01 rigorously
without any approximations.
For the object distance tending to infinity we denote the axial chromatic aberration
as chr ∞.
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It is easy to find it as a limit of the function (6.43)
𝑢𝑐ℎ𝑟∞ = 𝑣2 =2

𝐴𝑟 ∗𝐶𝑙 −𝐴𝑙 ∗𝐶𝑟
𝐶𝑟 ∗𝐶𝑙(6.50)The right equality of Eq. (6.50) describes analytically the axial chromatic aberration for
an infinitely distant object. To correct the axial chromatic aberration in this case it isnecessary to satisfy the next condition𝑢2 = 𝐴𝑟 ∗ 𝐶𝑙 − 𝐴𝑙 ∗ 𝐶𝑟 = 0(6.51)For the satisfaction of Eq. (6.51) in general we have to solve one quadratic equation in
respect to one vertex radius rj or to one axial thickness dj of the optical system.
There is one special case when we use as a free parameter the last vertex radius of thesystem. Then the correction of the chromatic aberration reduces to a solution of one linear
equation. In these case Eq. (6.51), after analytical symbolic computations of matrix
coefficients Ar, Cl, Al, Cr has the shape𝑢2 = 𝑒2 ∗ 𝑟𝑘2 + 𝑒1 ∗ 𝑟𝑘 = 0(6.52)

Grouping the coefficient u2 in respect to the different degrees of rk-1 we can find theanalytical form of e2 and e12
𝑒2 = 𝑒22 ∗ 𝑟𝑘−1
+ 𝑒21 ∗ 𝑟𝑘−1 + 𝑒20 ,(6.53)2𝑒1 = 𝑒12 ∗ 𝑟𝑘−1
+ 𝑒11 ∗ 𝑟𝑘−1 + 𝑒10,(6.54)and the analytical form (or numerical value) of the sub-coefficients e22, e21, e20,
e12, e11, e10.
6.2.1.4. Simultaneous Correction of Axial Image Chromatic Aberration
and Guarantee of Preset Focal Length in a Centered Optical System
Below we describe a special case for a correction of axial chromatic aberration at infinitely
distant object and providing of preset focal length, when we use the two last vertex radii
as a free parameters. For the chromatic aberration correction we use equations (6.52),(6.53) and (6.54).
To provide the preset value of the focal length we formulate with the help of Eq. (6.33) a
special function Ff
𝐹𝑓 = 𝑓 ′ ∗ 𝐶𝑎𝑘 − ∏𝑘𝑗=1 𝑟𝑎𝑗 ∏𝑘+1
𝑗=2 𝑛𝑗 = 0(6.55)

After symbolic computation of Cak with the help of Eq. (6.24) and its substitution in
Eq. (6.55) we obtain
𝐹𝑓 = ℎ1 ∗ 𝑟𝑘 + ℎ0 = 0,276

(6.56)

Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrationsin the Centered Conical Optical Systems with High and Super-high Numerical Apertures

Grouping the coefficients h1 and h0 in respect to the different degrees of rk-1 we can findthe analytical form of h1 and h0and the analytical form (or numerical values) of the
sub-coefficients h11, h10, h01, hoo.
ℎ1 = ℎ11 ∗ 𝑟𝑘−1 + ℎ10 ,(6.57)ℎ0 = ℎ01 ∗ 𝑟𝑘−1 + ℎ00(6.58)For a simultaneous satisfaction of Eqs. (6.52) and (6.56) we use Sylvester’s matrices to
exclude the unknown parameter rk. After excluding of rk we find the determinant of the
Sylvester’s matrix dsm as follows
32
𝑑𝑠𝑚 = 𝑉3 ∗ 𝑟𝑘−1
+ 𝑉2 ∗ 𝑟𝑘−1
+ 𝑉1 ∗ 𝑟𝑘−1 + 𝑉0

(6.59)Here
𝑉3 = 𝑒22 ∗ ℎ01 − 𝑒12 ∗ ℎ11 ,(6.60)

𝑉2 = 𝑒21 ∗ ℎ01 − 𝑒12 ∗ ℎ10 − 𝑒11 ∗ ℎ11 + 𝑒22 ∗ ℎ00 ,(6.61)

𝑉1 = 𝑒20 ∗ ℎ01 − 𝑒11 ∗ ℎ10 − 𝑒10 ∗ ℎ11 + 𝑒21 ∗ ℎ00 ,(6.62)

𝑉0 = 𝑒20 ∗ ℎ00 − 𝑒10 ∗ ℎ10(6.63)

Knowing the sub-coefficients e22, e21, e20,e12, e11, e10, hoo, h11, h10, h01, hoo we can find thenumerical values of V3, V2, V1, V0. Now we solve Eq. (6.59) in respect to rk-1.
It is well known that the set of roots of Eq. (6.59), as roots of Sylvester’s matrix
determinant, contains in itself all common roots of the equation system (6.56) and (6.52).
Finally through a consecutive checking of the whole set of roots of Eq. (6.59) we find allmathematically existing common roots of the equation system (6.56) and (6.52).
6.2.2. Rigorous Analytical Function of the On-axes Spherical Aberration
in a Centered Optical System Containing Conical Surfaces
The discovery of an exact analytical connection between the sagittal radius in every pointof the conical surface, the vertex radius, the incidence and slope angles of the rays isextremely important for developing of an exact analytical aberrations (such as on-axes
spherical aberration, optical path difference, sagittal and tangential curvatures,astigmatism, distortion and field curvature) theory.
6.2.2.1. Sagittal Radius and Sagittal Radius-coefficient in a Single Conical Surface
Sagittal radius in each conical surface point by definition is the distance (along the normal)
from the point itself to the intersection of the normal with the optical axes. Let us consider
a conical surface together with a Descartes coordinate system zy located in the surface
vertex. The axes z coincides with surface axes. In this case the general equation for a conic
section of the surface having a conic constant K and vertex radius r is277

Advances in Optics: Reviews. Book Series, Vol. 5

𝑦 2 − 2 ∗ 𝑟 ∗ 𝑧 + (𝐾 + 1) ∗ 𝑧 2 = 0(6.64)The sagittal radius rs in each surface section point can be presented as ratio between the
y-coordinate of the point and sine of the normal slope φ.
(6.65)

𝑟𝑠 = 𝑦/ 𝑠𝑖𝑛 𝜑

Now we present the sine of the angle thru the cotangent and substitute it in Eq. (6.65)
𝑠𝑖𝑛 𝜑 = 1/√1 + 𝑐𝑜𝑡 2 𝜑(6.66)

𝑟𝑠 = 𝑦 ∗ √1 + 𝑐𝑜𝑡 2 𝜑(6.67)Then

On the other hand the derivative𝑑𝑦from Eq. (6.64) is𝑑𝑧𝑑𝑦𝑟−𝑧∗(𝐾+1)= 𝑐𝑜𝑡 𝜑 =𝑑𝑥
𝑦(6.68)

The combination between Eqs. (6.67), (6.68) and (6.64) after rearranging leads to(6.69)

𝑟𝑠 = √𝑟 2 − 𝐾 ∗ 𝑦 2
Finally for the vertex radius r we find(6.70)

𝑟 = 𝑟𝑠 ∗ √1 + 𝐾 ∗ 𝑠𝑖𝑛2 𝜑𝑟

The ratio 𝑟 between the vertex and sagittal radii we shell call a sagittal radius-coefficientρs. Then

𝑠

𝜌𝑠 =

𝑟
= √1 + 𝐾 ∗ 𝑠𝑖𝑛2 𝜑𝑟𝑠(6.71)

So we found a formula for the sagittal radius-coefficient ρs that will play a fundamentalrole in the rigorous analytical aberrations theory.
𝑟3

As it is known [2, p. 515] the tangential radius rt in a conic surface is equal to 𝑟𝑡 = 𝑟𝑠2 .
By placing r from the Eq. (6.70) in the previous formula we obtain
𝑟3

𝑟3

𝑟

𝑟

𝑠

𝑠

𝑠
𝑠
𝑟𝑡 = 𝑟𝑠2 = 𝑟 2 ∗𝜌
2 = 𝜌2 = 𝜌3𝑠𝑠

The ratio between the vertex and tangential radii
coefficient ρt i.e.
𝑟𝑟𝑟𝑡

we shell call a tangential radius-𝜌𝑡 = 𝑟 = 𝜌𝑠3 = (1 + 𝐾 ∗ 𝑠𝑖𝑛2 𝜑) ∗ √1 + 𝐾 ∗ 𝑠𝑖𝑛2 𝜑𝑡

278(6.72)

Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrationsin the Centered Conical Optical Systems with High and Super-high Numerical Apertures

So we found exact and simple formulas for the sagittal and tangential radii (rs and rt) andalso for the sagittal and tangential radius-coefficients (ρs and ρt).
The next task is to find exact dependence between the z-coordinate of a surface point andthe sagittal radius-coefficient in the same point. For this purpose we use the known
formula [2, p. 514].

𝑧=𝑟−√𝑟 2 −(𝐾+1)∗𝑦 2(6.73)

𝐾+1

Substituting r from the Eq. (6.70) and y from (6.65) in (6.73) we get𝑧=𝑟𝑠 ∗𝜌𝑠 −√𝑟𝑠2 ∗𝜌𝑠2 −(𝐾+1)∗𝑟𝑠2 ∗𝑠𝑖𝑛2 𝜑
𝐾+1= 𝑟𝑠 ∗𝜌𝑠 −√𝜌𝑠2 −(𝐾+1)∗𝑠𝑖𝑛2 𝜑𝐾+1(6.74)

It can be proved that the last radicand in Eq. (6.74) is equal to√𝜌𝑠2 − (𝐾 + 1) ∗ 𝑠𝑖𝑛2 𝜑 = cos 𝜑(6.75)As a result we obtain𝜌 −𝑐𝑜𝑠 𝜑𝑧 = 𝑟𝑠 ∗ 𝑠 𝐾+1

𝑠𝑖𝑛2 𝜑(6.76)= 𝑟𝑠 ∗ 𝜌 +𝑐𝑜𝑠 𝜑𝑠

The ratio between the z-coordinate and sagittal radius𝑧𝑟𝑠

we will call a relative

z-coordinate zs.
𝑧𝜌𝑠 −𝑐𝑜𝑠 𝜑𝑠𝐾+1

𝑧𝑠 = 𝑟 =𝑠𝑖𝑛2 𝜑= 𝜌 +𝑐𝑜𝑠 𝜑𝑠(6.77)

So with the help Eqs. (6.65), (6.71) and (6.76) we can correctly describe the conic surface
using the sagittal radius-coefficient 𝜌𝑠 , the conic constant K and the slope of the normal

𝜑 as intermediate parameters.
If we transform Eq. (6.70) the sagittal radius becomes𝑟𝑠 =𝑟√1+𝐾∗𝑠𝑖𝑛2 𝜑(6.78)

We estimate the sagittal radius as a function of the parameters φ and K at constant vertex
radius r = 100 mm. The result is presented in Figs. 6.1, 6.1a, 6.1b.
On Fig. 6.1 we show five graphics of the sagittal radius for different values of K that
correspond as follows: number 1 to K = 100, number 2 to K = 10, number 3 to K = 3,
number 4 to K = 1, and number 5 to K = 0.2 respectively. The sagittal radius is always
equal to the vertex radius at φ = 0. The bigger is the conic constant, the smaller the sagittal
radiuses are in the whole range of the slope angles. When the conic constant tends to zero279

Advances in Optics: Reviews. Book Series, Vol. 5(i.e. the conic surface becomes sphere) then the function of rs is turning in a straighthorizontal line. This means that the sagittal radius is constant and equal to the vertex one
in the whole range of the slope angles from zero to 90 deg.

Fig. 6.1. Sagittal radius rs as a function of the slope φ in an oblate spheroid with vertex radius𝑟 = 100 mm and conic constant 0 < K < infinity.

On Fig. 6.1a we present an example for an elliptical surface. Here the sagittal radius isbigger than the vertex radius and it increases significantly for 40 < φ < 85 deg.
On Fig. 6.1b we show an analogical graphic of the sagittal radius for a paraboloid. The
special thing here is that the sagittal radius (as well as the two coordinates 𝑦 and z) tends
rapidly to infinity for φ tending to 90 deg. If we go back to Eq. (6.78) we will see thatradius-coefficient ρs in this case is zero. Now we nullify the radius-coefficient and find
1 + 𝐾 ∗ 𝑠𝑖𝑛2 𝜑 =0,−1

𝑠𝑖𝑛 𝜑∞ = √ 𝐾(6.79)(6.80)

Analyzing the Eq. (6.80) we can say:280•

If the conic constant is equal to -1 the sagittal radius tends to infinity in aparaboloid at φ = 90 deg only;•

When the conic constant is in the range of -1>K> -∞ the sagittal radius approaches
infinity for a strictly defined angle φ∞ at each value of the conic constant(see Fig. 6.2). The angle φ∞ we will call boundary slope. In these cases ofhyperboloids the slope angle φ varies from zero to the boundary slope.

Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrationsin the Centered Conical Optical Systems with High and Super-high Numerical Apertures

Fig. 6.1a. Sagittal radius rs as a function of the slope 𝜑 in an ellipsoid with vertex radius𝑟 = 100 mm and conic constant K = -0.8.

Fig. 6.1b. Sagittal radius rs as a function of the slope φ in a paraboloid with vertex radius𝑟 = 100mm and conic constant K = -1.

Fig. 6.2. The boundary slope φ∞ as function of the conic constant K.

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On the Fig. 6.2a. a graphic for a sagittal radius as a function of the slope angle in a
hyperboloid with a conic constant K = -2 is presented. In this case the boundary slope is
45 deg and the sagittal radius is approaching infinity at φ = 45 deg.

Fig. 6.2a. The sagittal radius as function of the slope φ in a hyperboloid with vertex radius𝑟 = 100 mm and conic constant K = -2.

6.2.2.2. Connections between the Parameters of the Tangential Rays and theParaxial Matrix Transformations on a Single Conic Surface
We demonstrate an analytical connection between the parameters of the incident andrefracted tangential rays and paraxial matrix transformations on a single conic surface. On
the Fig. 6.3 is shown a conic section of the conical surface S with the index j and vertex
in the point V. The indices of refraction before and after the surface are nj and nj+1. The
surface point N has coordinates yj and zj. The tangential ray R (it can be a chief ray or
aperture one) falls on the surface at point N with an incidence angle εj and refraction angle
ε’j. The incident ray R and the refracted one R’ form with the optical axis angle wj, wj+1
and cross the axis at points P and P’ respectively. The normal to the conic surface at N
crosses the optical axis at the point C under the angle φj. The rectangular projection of N
on the axis z is in the point No. It can be proved that the axial distances VP and VP’ areequal to
𝑉𝑃 = 𝛼𝑗 ∗ 𝑟𝑠𝑗 ,(6.81)

𝑉𝑃′ = 𝛽𝑗 ∗ 𝑟𝑠𝑗 ,(6.82)

𝛼𝑗 = 𝑧𝑠𝑗 + ∆𝑗 ∗ 𝑐𝑜𝑠 𝑤𝑗 ,(6.83)

𝑠𝑖𝑛 𝜑∆𝑗 = 𝑠𝑖𝑛 𝑤𝑗 =,(6.84)

𝛽𝑗 = 𝑧𝑠𝑗 + 𝛿𝑗 ∗ 𝑐𝑜𝑠 𝑤𝑗+1 ,(6.85)𝑗282

𝑠𝑖𝑛(𝑤𝑗 −𝜀𝑗 )𝑠𝑖𝑛 𝑤𝑗Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrationsin the Centered Conical Optical Systems with High and Super-high Numerical Apertures𝛿𝑗 =𝑠𝑖𝑛 𝜑𝑗𝑠𝑖𝑛 𝑤𝑗+1=

𝑠𝑖𝑛(𝑤𝑗+1 −𝜀𝑗′ )𝑠𝑖𝑛 𝑤𝑗+1(6.86)

Fig. 6.3. Conic surface transformation of an incident tangential ray into a refracted ray and theirconnections with the paraxial transformations.

It is obvious from Eqs. (6.83) and (6.85) that the variables αj and βj depend on the indicesof refraction (nj, nj+1), ray’s parameters (εj, wj, φj) and the sagittal radius coefficient ρs.
Note that zsj in Eqs. (6.83) and (6.85) is the familiar to us already relative z-coordinate(see Eq. (6.77)).
Let the points A0 and A’0 be the object and paraxial image points respectively. The object
and image distances are zoj and z’oj. The two distances are measured from the point V along
the optical axis. Further on we will use Eq. (6.4) and rewrite the paraxial matrix
coefficients from Eq. (6.5) according to Eq. (6.71) i.e.
𝐴𝑗 = 𝑟𝑠𝑗 ∗ 𝜌𝑗 ∗ 𝑛𝑗+1 , 𝐵𝑗 = 0, 𝐶𝑗 = 𝑚𝑗0 , 𝐷𝑗 = 𝑟𝑠𝑗 ∗ 𝜌𝑗 ∗ 𝑛𝑗

(6.87)

Now we relocate the object coordinate system from the vertex V to point P and the image
coordinate system to point P’, respectively. Then the object distance zpj (measured fromthe point P) in the new object coordinate system is
𝑧𝑝𝑗 = 𝑧0𝑗 − 𝑟𝑠𝑗 ∗ 𝛼𝑗 ,(6.88)and the paraxial image distance z’pj (measured from the point P’) in a new image
coordinate system becomes
′
′
𝑧𝑝𝑗
= 𝑧0𝑗
− 𝑟𝑠𝑗 ∗ 𝛽𝑗(6.89)
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According to Eqs. (6.88), (6.89) and (6.3) the paraxial image distance turns into
𝐴 ∗𝑧

+𝐵

∗𝑟

′
𝑧𝑝𝑗
= 𝑟𝑠𝑗 ∗ 𝐶 𝑝𝑗∗𝑧 𝑝𝑗+𝐷 𝑝𝑗∗𝑟 𝑠𝑗 ,(6.90)

𝐴𝑝𝑗 = 𝑛𝑗+1 ∗ 𝜌𝑗 − 𝑚𝑗0 ∗ 𝛽𝑗 ,(6.91)

𝐵𝑝𝑗 = 𝑛𝑗+1 ∗ 𝜌𝑗 ∗ 𝛼𝑗 − 𝛽𝑗 ∗ 𝐷𝑝𝑗 ,(6.92)

𝑝𝑗

𝑝𝑗

𝑝𝑗

𝑧𝑗

Here

𝐶𝑝𝑗 = 𝑚𝑗0 , (6.93)
𝐷𝑝𝑗 = 𝑛𝑗 ∗ 𝜌𝑗 + 𝑚𝑗0 ∗ 𝛼𝑗(6.94)

Substituting the parameters αj, βj and zsj from Eqs. (6.83), (6.85) and (6.77) in Eq. (6.91).
(6.92), (6.93) and (6.94) after boring transformations and rearrangements (but without any
approximations) we obtain
𝐴𝑝𝑗 = (𝑛𝑗+1 ∗ 𝜌𝑗2 + 𝑝1𝑗 ∗ 𝜌𝑗 + 𝑝0𝑗 ) ∗ (𝜌𝑗 + 𝑐𝑜𝑠 𝜑𝑗 ),(6.95)

𝐵𝑝𝑗 = 𝑞3𝑗 ∗ 𝜌𝑗3 + 𝑞2𝑗 ∗ 𝜌𝑗2 + 𝑞1𝑗 ∗ 𝜌𝑗 + 𝑞0𝑗 ,(6.96)

2

𝐶𝑝𝑗 = 𝑚𝑗0 ∗ (𝜌𝑗 + 𝑐𝑜𝑠 𝜑𝑗 ) ,(6.97)

𝐷𝑝𝑗 = (𝑛𝑗 ∗ 𝜌𝑗2 + 𝑙1𝑗 ∗ 𝜌𝑗 + 𝑙0𝑗 ) ∗ (𝜌𝑗 + 𝑐𝑜𝑠 𝜑𝑗 )(6.98)The sub-coefficients p1j, p0j, q3j, q2j, q1j, qoj, l1j and loj in the Eqs. (6.95), (6.96), (6.97) and(6.98) depend on the indices of refractions and the ray parameters only. They are shownin Table 6.1.
Table 6.1. Formulas for the sub-coefficients in the paraxial matrix coefficientsof a single conic surface.

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The coefficients Apj, Bpj, Dpj are polynomial of third degree in respect to the sagittal radiuscoefficient, while Cpj is of second degree.
We proved that the paraxial matrix Gpj in the Eq. (6.99) describes correctly the bilinear
function in Eq. (6.90)
𝐺𝑝𝑗 =

𝐴𝑝𝑗
𝐶𝑝𝑗

𝐵𝑝𝑗
𝐷𝑝𝑗

(6.99)

Because Eq. (6.90) describes correctly the paraxial image distance trough the parametersof the real tangential ray in one conic surface we will call the matrix Gpj by ‘surface
paraxial real ray transformation matrix’ (SPRRTM).
If the surface is spherical, the matrix coefficients in Eq. (6.99) became much simpler as
they are in Eqs. (6.95) -(6.98) and namely
𝐴𝑝𝑗 = 𝑛𝑗+1 − 𝑚0𝑗 ∗ 𝛿0𝑗 ,(6.100)

𝐵𝑝𝑗 = 𝑛𝑗+1 ∗ ∆0𝑗 − 𝐷𝑗 ∗ 𝛿0𝑗 ,(6.101)

𝐶𝑝𝑗 = 𝑚0𝑗 ,(6.102)

𝐷𝑝𝑗 = 𝑛𝑗 + 𝑚0𝑗 ∆0𝑗(6.103)Here∆0𝑗 = 1 −𝑠𝑖𝑛 𝜀𝑗𝑠𝑖𝑛 𝑤𝑗;𝑠𝑖𝑛 𝜀𝑗′𝛿𝑜𝑗 = 1 − 𝑤

𝑗+1(6.104)(6.105)6.2.2.3. Paraxial Matrix Coefficients Linked to the Real Tangential Rays
in a System of Centered Conic Surfaces
If we add to one conic surface having an index (j) another surface with an index (j+1)
then the paraxial image from the first surface appears as an object to the second one. In
this case we can write′
𝑧𝑝(𝑗+1) = 𝑧𝑝𝑗(6.106)Further we analyze the optical system of k surfaces. Applying Eq. (6.90) for the firstsurface we have
𝐴𝑝1 ∗𝑧𝑝1 +𝐵𝑝1 ∗𝑟𝑠1′
𝑧𝑝1
= 𝑟𝑠1 ∗ 𝐶

𝑝1 ∗𝑧𝑝1 +𝐷𝑝1 ∗𝑟𝑧1

(6.107)

Taking into account Eqs. (6.106) and (6.90) for the second surface we write
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𝐴𝑝1 ∗𝑧𝑝1 +𝐵𝑝1∗𝑟𝑠1

𝑧′𝑝2 = 𝑟𝑠2 ∗𝐴𝑝2 ∗𝑟𝑠1 ∗𝐶 ∗𝑧 +𝐷 ∗𝑟 +𝐵𝑝2 ∗𝑟𝑠2
𝑝1 𝑝1 𝑝1 𝑧1
𝐴𝑝1 ∗𝑧𝑝1 +𝐵𝑝1∗𝑟𝑠1
𝐶𝑝2 ∗𝑟𝑠1 ∗𝐶 ∗𝑧 +𝐷 ∗𝑟 +𝐷𝑝2 ∗𝑟𝑧2
𝑝1 𝑝1 𝑝1 𝑧1𝐴 ∗𝑧 +𝑟 ∗𝐵= 𝑟𝑠2 ∗ 𝐶𝑝∗𝑧𝑝1+𝑟𝑠1∗𝐷𝑝
𝑝

𝑝1

𝑠1

𝑝

(6.108)After performing the right equality in Eq. (6.108) and adequate arrangement we find
𝐴𝑝 = 𝐴𝑝1 ∗ 𝐴𝑝2 ∗ 𝑥𝑠1 + 𝐵𝑝2 ∗ 𝐶𝑝1 ,(6.109)𝐵𝑝 = 𝐵𝑝1 ∗ 𝐴𝑝2 ∗ 𝑥𝑠1 + 𝐵𝑝2 ∗ 𝐷𝑝1 ,(6.110)𝐶𝑝 = 𝐴𝑝1 ∗ 𝐶𝑝2 ∗ 𝑥𝑠1 + 𝐷𝑝2 ∗ 𝐶𝑝1,

(6.111)

𝐷𝑝 = 𝐵𝑝1 ∗ 𝐶𝑝2 ∗ 𝑥𝑠1 + 𝐷𝑝1 ∗ 𝐷𝑝2 ,(6.112)𝑟(6.113)where
𝑥𝑠1 = 𝑟𝑠1
𝑠2

Now let us assign to the paraxial transformation for the first surface (see Eq. (6.107)) the
next quadratic matrix Gp1 and for the second one the matrix Gp2 then
𝐺𝑝1 =
𝐺𝑝2 =

𝐴𝑝1
𝐶𝑝1

𝐵𝑝1
,
𝐷𝑝1

𝐴𝑝2 ∗ 𝑥𝑠1
𝐶𝑝2 ∗ 𝑥𝑠1

(6.114)

𝐵𝑝2
𝐷𝑝2(6.115)The paraxial coefficients Ap, Bp, Cp, Dp for the two surfaces in the right equality ofEq. (6.108) can be fined through the matrix gp as a matrix product of the two matrices Gp1
and Gp2, i.e.
(6.116)

𝑔𝑝 = 𝐺𝑝2 ∗ 𝐺𝑝1
By analogy, for a system of k surfaces we prove that
𝑔𝑝𝑘 = ∏2𝑗=𝑘 𝐺𝑝𝑗 ∗ 𝐺𝑝1(6.117)The matrix gpk is so called paraxial real ray transformation matrix of the system of kcentered conic surfaces, i.e. (PRRTMS).
In these case the matrix Gp1 is the same as it is in the Eq. (6.114) and Gpj for k ≥ j = 2 isas follows
𝐺𝑝𝑗 =𝐴𝑝𝑗 ∗ 𝑥𝑠(𝑗−1)
𝐶𝑝𝑗 ∗ 𝑥𝑠(𝑗−1)

𝐵𝑝𝑗
𝐷𝑝𝑗

(6.118)The ratio between the two adjacent sagittal radii in an optical system we will call sagittal
relative parameter xsj
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Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrationsin the Centered Conical Optical Systems with High and Super-high Numerical Apertures𝑟

𝑥𝑠𝑗 = 𝑟 𝑠𝑗

(6.119)𝑠(𝑗+1)The matrix gpk from (6.117) belongs to the next bilinear function
𝐴∗𝑧𝑝1 +𝐵𝑝𝑘 ∗𝑟𝑠1′
𝑧𝑝𝑘
= 𝑟𝑠𝑘 ∗ 𝐶 𝑝𝑘∗𝑧
𝑝𝑘

𝑝1 +𝐷𝑝𝑘 ∗𝑟𝑧1

(6.120)Here z’pk is the paraxial image distance. It is measured from the axial intersection point of
the tangential ray after the last surface to the paraxial image. The object distance is zp1 andrsk, rs1 are the last and the first sagittal radii respectively in the optical system.
Using Eq. (6.117) after symbolic computations we find in analytical form (see for example
Eqs. (6.109)-(6.112)) the four paraxial matrix coefficients for a system of k surfaces. Eachmatrix coefficient is a linear function of each relative parameter. Furthermore you cancalculate, if it is necessary, the numerical values of the coefficients or their detached parts.
This gives a possibility to satisfy preset conditions to the system using the sagittal relative
parameters or sagittal radius-coefficients as a free parameters.
For an infinity distant object the image distance 𝑧′𝑝𝑘∞ becomes
𝐴

′
𝑧𝑝𝑘∞
= 𝑟𝑠𝑘 ∗ 𝐶𝑝𝑘
𝑝𝑘(6.121)

It can be proved that the axial distance dj between the surface with number j and the one
with number (j+1) in a system of conic surfaces is
𝑑𝑗 = 𝑟𝑠𝑗 ∗ 𝛽𝑗 − 𝑟𝑠(𝑗+1) ∗ 𝛼𝑗+1 ,(6.122)and for a system of spherical surfaces it becomes
𝑑𝑗 = 𝑟𝑗 ∗ 𝛿0𝑗 − 𝑟𝑗+1 ∗ ∆0(𝑗+1)(6.123)Knowing the matrix coefficients one can estimate the paraxial image distance in a systemof k conic surfaces using the paraxial image function (see Eqs. (6.120) and (6.121)).
The Eqs. (6.117) - (6.123) are the basic formulas both for the symbolic paraxial
computations and for the numeric paraxial calculations in a centered optical system.
6.2.2.4. Geometrical Interpretation of the Paraxial Image Function
and Its Connection with On-axis Spherical Aberration
So far, we found rigorous analytical connections between the paraxial matrix coefficientsand parameters of the real tangential rays which pass throughout the optical system. One
can consider these real rays as chief rays or aperture ones. Let us analyze the latter case.
By definition the longitudinal spherical aberration is a variation of focus with aperture. In
this sense the distance zp1 (see Eq. (6.120)) is in fact the longitudinal object spherical287

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aberration dz with the opposite sign, while the distance z’pk is the longitudinal image
spherical aberration dz’ with the opposite sign too′
𝑑𝑧 = −𝑧𝑝1 , 𝑑𝑧 ′ = −𝑧𝑝𝑘

(6.124)Using the Eq. (6.124) we can rewrite (6.120) and (6.121) as follows
𝑑𝑧 ′𝑑𝑧𝑟′ = 𝑟𝑠𝑘𝐴∗𝑑𝑧𝑟 −𝐵= −𝐶𝑝𝑘 ∗𝑑𝑧 +𝐷𝑝𝑘 ,𝑝𝑘𝑟(6.125)

𝑝𝑘

𝐴

′
𝑑𝑧𝑟∞
= − 𝐶𝑝𝑘(6.126)𝑝𝑘′

Here dz’r and 𝑑𝑧𝑟∞ are the so called image relative aberrations. The ratio‘relative object aberration’ dzr.

𝑑𝑧
𝑟𝑠1

we will call

It is evident from Eq. (6.125) that the relative image aberration dz’r is bilinear
transformation of the relative object aberration dzr. If the object space of the optical system
is aberration free, then the relative image aberration for an object at finite distances isequal to𝐵𝑝𝑘𝑑𝑧𝑟′ = − 𝐷𝑝𝑘(6.127)To remove or correct the on-axis image aberration at finite object distance we have tonullify the numerator of Eq. (6.127) i.e.
𝐵𝑝𝑘 = 0(6.128)For an infinitely distant object the correction of spherical aberration according
Eq. (6.126) is
𝐴𝑝𝑘 = 0(6.129)It is clear from Eqs. (6.128), (6.129), (6.109), (6.110), that the on-axis spherical aberration
correction both for a finite object and for an infinitely distant object leads to solution of
one linear equation in respect to one arbitrarily chosen relative parameter of the system or
to solution of one cubic equation (see Eqs. (6.95) and (6.96)) in respect to one arbitrarily
chosen sagittal radius-coefficient.

6.3. Simple Examples of Analytical Aberrations Correction
for an Infinitely Distant Object
In this paragraph we intentionally show ones of the simplest examples for the sole purposeof clear distinguishing the analytical computations from the numerical calculations in thepractical applications of our rigorous analytical aberration theory.
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6.3.1. Analytical Correction of Axial Chromatic Aberration Together with Preset
Focal Length in a System of Three Surfaces (Case of the so Called Doublets)
Let the constructive parameters of the doublet be: three axial radii, two thicknesses andtwo glasses with indices of refraction for the design wavelength n2, n3. The doublet isimmersed in air i.e. n1 = n4 = 1.
To correct the axial chromatic aberration we have to satisfy Eqs. (6.51) or (6.52) using
Eqs. (6.53) and (6.54). In our case for k = 3 they are𝑢2 = 𝑒2 ∗ 𝑟32 + 𝑒1 ∗ 𝑟3 = 0,

(6.130)𝑒2 = 𝑒22 ∗ 𝑟22 + 𝑒21 ∗ 𝑟2 + 𝑒20 ,(6.131)𝑒1 = 𝑒12 ∗ 𝑟22 + 𝑒11 ∗ 𝑟2 + 𝑒10(6.132)The indices of refraction for the left spectral end are n2l, n3l and for the right end are n2r,n3r
respectively. At this stage we do not know the numerical values of the sub-coefficients(e22, e21, e20, e12, e11, e10). Below we shell show how to do this procedures. First we
formulate the two ARM D1 and D2, the three ASTM G1r, G2r, G3r and ATMS gr for theright end of the wavelength spectrum and similarly for the left end in the system
as follows
𝐷1 = [1 − 𝑑1 ; 01]; 𝐷2 = [1 − 𝑑2 ; 01],𝐺1𝑟 = [𝑛2𝑟 ∗ 𝑟1 0; 𝑛2𝑟 − 1𝑟1 ]; 𝐺2𝑟 = [𝑛3𝑟 ∗ 𝑟2 0; 𝑛3𝑟 − 𝑛2𝑟 𝑟2 ∗ 𝑛2𝑟 ],𝐺3𝑟 = [𝑟3 0; 1 − 𝑛3𝑟 𝑟3 ],𝑔3𝑟 = 𝐺3𝑟 ∗ 𝐷2 ∗ 𝐺2𝑟 ∗ 𝐷1 ∗ 𝐺1𝑟 ; 𝐴3𝑟 = 𝑔3𝑟 (1,1); 𝐶3𝑟 = 𝑔3𝑟 (2,1),𝐺1𝑙 = [𝑛2𝑙 ∗ 𝑟1 0; 𝑛2𝑙 − 1𝑟1 ]; 𝐺2𝑙 = [𝑛3𝑙 ∗ 𝑟2 0; 𝑛3𝑙 − 𝑛2𝑙 𝑟2 ∗ 𝑛2𝑙 ],𝐺3𝑙 = [𝑟3 0; 1 − 𝑛3𝑙 𝑟3 ],𝑔3𝑙 = 𝐺3𝑙 ∗ 𝐷2 ∗ 𝐺2𝑙 ∗ 𝐷1 ∗ 𝐺1𝑙 ; 𝐴3𝑙 = 𝑔3𝑙 (1,1); 𝐶3𝑙 = 𝑔3𝑙 (2,1)
After symbolic computation of above shown matrices gr and gl we receive the four matrix
coefficients A3r, C3r, A3l, C3l. Placing the latter coefficients in Eq. (6.130) after grouping
on r3 we find the coefficients e2 and e1. Now we can group each coefficient on r2 andcalculate the numerical values of the sub-coefficients e22, e21, e20, e12, e11, e10 for the systemof three surfaces. It is clear that the just listed coefficients depend on r1, d1, d2, n2r, n3r, n2l
and n3l.
By analogy, we form the three ASTM and ATMS for the design wavelength in the systemof three surfaces. To guarantee the preset focal length f’ we have to satisfy Eq. (6.56).
After grouping on r3 and r2 we can calculate the numerical values of the
sub-coefficients h11, h10, h01, h00. The just listed coefficients depend on r1, d1, d2, n2, n3.
Further we proceed to search the common roots of the Eqs. (6.52) and (6.56). For thispurpose we shell use Eq. (6.59) which in the case of three surfaces is289

Advances in Optics: Reviews. Book Series, Vol. 5

𝑑𝑠𝑚 = 𝑉3 ∗ 𝑟23 + 𝑉2 ∗ 𝑟22 + 𝑉1 ∗ 𝑟2 + 𝑉0

(6.133)Knowing the numerical values of all sub-coefficients we calculate coefficients
𝑉3 , 𝑉2 , 𝑉1 , 𝑉0 in Eq. (6.133). Now we solve Eq. (6.133) in respect to r2 and check which of
the roots for r2 satisfy simultaneously the Eqs. (6.52) and (6.56).
Varying the lens parameters r1, d1, d2 and indices of refraction of the two glasses we canfind all mathematically existing doublets with corrected axial chromatic aberration andpreset focal length.
In Table 6.2 we present a very small sample of calculations for the doublets of
Schott-glass N-PK51 and Ohara-glass S-NPH2. The doublets have effective focal lengthof 20 mm and are corrected for two wavelengths 486 and 656 nm. In this table are shown
the preset parameters r1, d1, d2 and calculated results for the two radii r2 and r3. In the lastcolumn is shown the residual of the axial chromatic aberration (𝑐ℎ𝑟∞) according
Eq. (6.50).
Table 6.2. Doublets with effective focal length of 20 mm and corrected axial chromaticaberration for wavelengths 486 and 656 nm at object in infinity.

290

𝒓1 ,mm𝒅1 , mm𝒅2 , mm

𝒓2 , mm

𝒓3 , mm

𝒄𝒉𝒓∞, mm982-34.06914

829.4354

7.1e-15983-38.24711

355.8315-1.77e-15

984-43.55748

219.0632

5.33e-15

10

94-26.3199-74.22728

3.55e-151095-28.81828-80.90122-1.78e-15

10

9

6

-31.85959- 89.60543-1.78e-15

12

10

2

-15.5144-26.28416-3.55e-15

12

10

3

-16.13771-26.49421-3.55e-15

12

10

4

-16.83818-26.71472

3.55e-15

16

11

3

-11.62321-16.1855-5.33e-15

16

11

4

-11.72118-16.17704

0

16

11

5

-11.82786-16.16852

0

18

13

5

-10.10692-13.86927-1.78e-15

18

13

6

-10.05058-13.85794-1.78e-15

18

13

7

-10.07277-13.8467-1.78e-15

Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrationsin the Centered Conical Optical Systems with High and Super-high Numerical Apertures

The research shows that the residual chromatic abberations for the two wavelengths areless than 10-14 mm. We saw above that all calculations are analytical and in this sense“exact”. That is why we claim that the correction is rigorous analytical correction.
6.3.2. Analytical Correction of On-axis Spherical Aberration
in a Systems of Two and Three Surfaces
6.3.2.1. Systems of Two Surfaces
Let us discuss a simple system of two surfaces immersed in air. It has three constructive
parameters (two vertex radii r1, r2 and axial distance d1). The index of refraction is n2. Let
us imagine that the optical designer wants to design a lens with preset focal length f’2,
back focal length bfl2, image numerical aperture NA and corrected on-axis sphericalaberration for a given wavelength so that the residual aberration on the whole aperture isdiffraction limited. To achieve the above requirements first we use Eqs. (6.39) and (6.40)
in order to satisfy the requirements for f’2 and bfl2. In such a way we find the precise valuesof the vertex radii r1 and r2. Note that the two radii are calculated at concrete values of d1.
This means that the vertex of the first surface is situated at given axial distance d1 fromthe vertex of the second surface. In our case we will call the first vertex as a common
vertex for all points of the first aspherical surface. To correct the on-axis sphericalaberration we use the sagittal radius-coefficient ρ1 of the first surface as a free parameterto satisfy Eq. (6.129). For this purpose we write the surface paraxial real raytransformation matrices (SPRRTM) for the first aspherical surface Gp1 and for the second
spherical surface Gp2:
𝐺𝑝1 = [𝐴𝑝1 𝐵𝑝1 ; 𝐶𝑝1 𝐷𝑝1 ](6.134)The matrix coefficients for the first aspherical surface Ap1, Bp1, Cp1, Dp1 in Eq. (6.134) andits sub-coefficients (see Table 6.1) are described by Eqs. (6.95), (6.96), (6.97), (6.98) atindex j = 1.
𝐺𝑝2 = [𝐴𝑝2 ∗ 𝑥𝑠1 𝐵𝑝2 ; 𝐶𝑝2 ∗ 𝑥𝑠1 𝐷𝑝2 ](6.135)The matrix coefficients for the second spherical surface Ap2, Bp2, Cp2, Dp2 and its
sub-coefficients (see Eqs. (6.104), (6.105)) are described by Eqs. (6.100), (6.101), (6.102),(6.103) at index j = 2.
After that the paraxial real ray transformation matrix of the system gp2 (according
Eq. (6.117)) became
𝑔𝑝2 = 𝐺𝑝2 ∗ 𝐺𝑝1 ; 𝐴𝑝𝑘 = 𝑔𝑝2 (1,1)(6.136)

Performing the symbolic computations according Eqs. (6.134)-(6.136) we receive for Apk
a polynomial of the third degree in respect to the sagittal radius-coefficient ρ1 and namely
32
𝐴𝑝𝑘 = 𝑈3 ∗ 𝜌𝑠1+ 𝑈2 ∗ 𝜌𝑠1+ 𝑈1 ∗ 𝜌𝑠1 + 𝑈0 = 0(6.137)291

Advances in Optics: Reviews. Book Series, Vol. 5

The analytical formulas for the coefficients of Eq. (6.137) are
𝑈3 = 𝑛2 ∗ 𝐴𝑝2 ∗ 𝑥𝑠1 ,(6.138)

𝑈2 = 𝐵𝑝2 ∗ 𝑚02 + 𝐴𝑝2 ∗ (𝑝11 + 𝑛2 ∗ 𝑐𝑜𝑠 𝜑1 ) ∗ 𝑥𝑠1 ,(6.139)𝑈1 = 𝐵𝑝2 ∗ 𝑚01 ∗ 𝑐𝑜𝑠 𝜑1 + 𝐴𝑝2 ∗ (𝑝01 + 𝑝11 ∗ 𝑐𝑜𝑠 𝜑1 ) ∗ 𝑥𝑠1 , (6.140)
𝑈0 = 𝐵𝑝2 ∗ 𝑚01 ∗ 𝑐𝑜𝑠 2 𝜑1 + 𝐴𝑝2 ∗ 𝑐𝑜𝑠 𝜑1 ∗ 𝑥𝑠1(6.141)The latter coefficients depend on the tangential ray parameters ε1, ε2, w1, w2, w3, ε‘1, ε’2, n2
and on the relative parameter xs1.
By definition the numerical aperture NA for our case is
𝑁𝐴 = 𝑠𝑖𝑛 𝑤3𝑚𝑎𝑥(6.142)

We find the aperture angle w3max from Eq. (6.142) at given NA. The back focal length,
w3max and r2 are already known, so we can find the refracted and incidence angles on thesecond surface ε’2, ε2 and w2. The angle w1 between the aperture ray and optical axis for
an infinitely distant object is zero. Knowing w2 and w1 we find the exact value of theincidence and refraction angles ε1 and ε’1. In this way we calculate all sub-coefficients andmatrix coefficients for the first and second surface. As a result we can calculate thenumerical values of U3, U2, U1 and U0. Now we solve the Eq. (6.137) and find its rootsfor ρs1. After that it is easy to calculate the sagittal radius rs1 from Eq. (6.78), the relative
z-coordinate zs1 from the third equality of Eq. (6.77), the coordinates y1,z1 from Eq. (6.65)
and Eq. (6.76), and finally the conic constant K1. Thus we calculated the parameters(K1, y1, z1,rs1) of one point which is infinitely small part of one conic surface according
Eq. (6.64). The parameters calculated above refer to the intersection point of the aperture
ray and the first aspherical surface. Varying the aperture angle w3 from zero to the w3max,
it is easy to find all points of the first aspherical surface so that the on-axis sphericalaberration is corrected on the whale aperture. But the studies have shown that the conic
constant in each point is different as well as the vertex of the conic surface of each point
is different from the common vertex. The latter means that the 𝑧-coordinate of each pointneeds to be corrected so that the vertex of each point to coincides with the common vertex.
To do this we need one additional free parameter. How to avoid this problem you can see
in the next subheading.
6.3.2.2. Multi-conical Surfaces in the Centered Optical Systems
Eqs. (6.137)-(6.141) show that the relative parameter xs1 can be used to solve the problem
mentioned above. The combination between Eqs. (6.122) and (6.123) give us thepossibility to write in this case a formula for d1
𝑑1 = 𝑟𝑠1 ∗ 𝛽1 − 𝑟2 ∗ ∆02(6.143)Using Eqs. (6.85) and (6.86) for the case of j = 1 and Eq. (6.104) for the case of j = 2 after
substitution of β1 and ∆02 in (6.143) we can convert Eq. (6.143) into292

Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrationsin the Centered Conical Optical Systems with High and Super-high Numerical Apertures
𝐿 ∗(𝜌1 +𝑐𝑜𝑠 𝜑1 )𝑥𝑠1 = 1𝑎1 ∗𝜌1 +𝑏1(6.144)The parameters a1, b1 can be found in Table 6.1 and parameter L1 is𝑑

𝐿1 = ∆02 + 𝑟1
2

(6.145)The Eqs. (6.144) and (6.145) provide the condition each point of the first asphericalsurface to be an infinitely small part of a conic surface whose vertex coincides with thecommon vertex.
Now we put xs1 from Eq. (6.144) in the coefficients U3, U2, U1 and U0 of Eq. (6.137)
and get432
𝐴𝑝𝑘 = 𝑇4 ∗ 𝜌𝑠1+ 𝑇3 ∗ 𝜌𝑠1+ 𝑇2 ∗ 𝜌𝑠1+ 𝑇1 ∗ 𝜌𝑠1 + 𝑇0 = 0(6.146)

In fact the Eq. (6.146) is a sufficient condition to find the common roots of equation
system (6.137) and (6.144). Now varying the aperture angle w3 from zero to the w3max, we
find the sagittal radius-coefficient ρs1 from Eq. (6.146) and rs1, zs1, y1, z1, K1 as it was
mentioned above. So we form a surface in the plane of y1 and z1 that we will call ‘multiconical surface’.
Generally speaking, the multi-conical surface is a locus of points each of which is a pointfrom a conic surface that has a common vertex but a different conic constant.
6.3.2.3. Examples of On-axis Spherical Aberration Correctionin the Systems of Two Surfaces
We used the described in Subsections 6.3.2.1, 6.3.2.2 algorithm for the sphericalaberration correction and designed a series of lenses maid of fused silica with high andsupper-high apertures (see Table 6.3) for high power Nd:YAG laser applications. The firstsurface of the lenses is multi-conical and the second is spherical or plane. The lenses havethe next characteristics and parameters: D - diameter; EFL – effective focal length;
NA – numerical aperture; r1 – vertex radius of the first surface; r2 – radius of the sphericalsurface; d1 – axial thickness; CW – correction wavelength; te – edge thickness. All thelenses of the Table 6.3 are diffraction limited for the correction wavelength and have highor supper-high apertures. The supper-high precision multi-conical designs decrease laser
spot size and maintain high power per area. The comparative analysis shows that ourlenses maintain a power per area about two times better than the best existing ones
(see [6-9]).
On the Fig. 6.4 we show the transverse on-axis spherical aberration for the correction
wavelength of 350 nm. The lens characteristics and parameters are: r1 = 9.63 mm;
r2 – plane; d1 = 12.5 mm; D = 25 mm; EFL = 21 mm; NA = 0.65. The residual aberration
is estimated to be less than 1. 10-14 mm.
293

Advances in Optics: Reviews. Book Series, Vol. 5

In Table 6.4 We show a series of super-high numerical aperture diffraction limited
plan-convex lenses of N-SF5 SCHOTT- glass. The designations here are the same as in
the Table 6.3. The numerical apertures of the lenses are in the range of NA = 0.9 to
NA = 0.97. The half of them are corrected for wavelength 587.6 nm and the other – for
780 nm. The diameters vary from 25 mm to 75 mm and the effective focal lengths are
between 11.5 mm and 37.5 mm. The residuals of on-axis spherical aberration for thecorrected wavelengths in all lenses are less than 5.10-14 mm i.e. they have a diffractionlimited image quality for the corrected wavelengths.
On the Fig. 6.5. We show the transverse on-axis spherical aberration as function of theaperture angle in a lens with the next characteristics and parameters: D = 50 mm;
EFL = 22.5 mm; NA = 0.95; r1 = 15.136 mm; d = 27.4 mm; te = 1.93 mm. We see thatthe aberration is very small and it is less than 1.5.10-14 mm.
Table 6.3. High and super-high aperture diffraction limited lenses of FUSED SILICA for highpower Nd:YAG laser applications.
D, mm

EFL, mmNA

CW, nm

r1,mm

r2, mmd1, mmte, mm

16

13.5

0.65

355

6.19

∞

9

2.36

16

13.5

0.65

532

6.19

∞

9

2.35

16

13.5

0.65

1064

6.19

∞

9

2.35

25

18

0.7

355

8.9

-59.35

13.5

2.5

25

18

0.7

632

8.9

-59.35

13.5

2.39

25

18

0.7

1064

8.9

-59.35

13.5

2.31

25

16

0.75

355

7.89

-44.5

14.4

2.32

25

16

0.75

532

7.89

-44.5

14.4

2.13

25

16

0.75

1064

7.89

-44.5

14.4

1.99

23

15

0.8

355

7.33

-40

15

3.73

23

15

0.8

532

7.33

-40

15

3.08

23

15

0.8

1064

7.33

-40

15

2.98

25

16

0.85

355

7.65

-57.5

16.5

2.66

25

16

0.85

532

7.65

-57.5

16.5

2.46

25

16

0.85

1064

7.65

-57.5

16.5

2.34

25

16.5

0.95

532

7.565

∞

20

2.19

294

Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrationsin the Centered Conical Optical Systems with High and Super-high Numerical Apertures

Fig. 6.4. Transverse on-axis spherical aberration as function of the aperture angle
in a plan-convex lens of FUSED SILICA corrected for 350 nm with NA = 0.65.
Table 6.4. Super-high numerical aperture diffraction limited lenses of N-SF5 SCHOTT- glass.

NA

CW, nm

r1, mm

r2, mmd, mm

te, mm

25

EFL,
mm
12.5

0.90

587.6

8.409

plane

13.3

1.89

25

12.5

0.90

780

8.409

plane

13.3

1.77

25

11.5

0.95

587.6

7.737

plane

14

1.49

25

11.5

0.95

780

7.737

plane

14

1.66

25

11.5

0.97

587.6

7.737

plane

15

2.36

25

11.5

0.97

780

7.737

plane

15

2.61

30

15

0.90

587.6

15.6

plane

15.6

1.96

30

15

0.90

780

15.6

plane

15.6

1.81

30

14

0.95

587.6

9.418

plane

17

2.03

30

14

0.95

780

9.418

plane

17

1.96

30

13

0.97

587.6

8.75

plane

17.7

1.76

30

13

0.97

780

8.75

plane

18

1.77

40

20

0.90

587.6

13.454

plane

20.2

2.08

40

20

0.90

780

13.454

plane

20.2

1.88

40

18

0.95

587.6

12.11

plane

22.4

1.97

40

18

0.95

780

12.11

plane

22.8

2.02

40

18

0.97

587.6

12.11

plane

23

2.49

40

18

0.97

780

12.11

plane

23.6

2.63

50

24

0.90

587.6

16.145

plane

25.6

1.91

D, mm

295

Advances in Optics: Reviews. Book Series, Vol. 5

Table 6.4. Continued.

NA

CW, nm

r1, mm

r2, mmd, mm

te, mm

50

EFL,
mm
24

0.90

780

16.145

plane

25.6

1.62

50

22.5

0.95

587.6

15.136

plane

27.4

1.93

50

22.5

0.95

780

15.136

plane

27.8

1.91

50

22

0.97

587.6

14.8

plane

28

1.94

50

22

0.97

780

14.8

plane

28.4

1.89

60

30

0.90

587.6

20.181

plane

29

1.98

60

30

0.90

780

20.181

plane

29

1.68

60

27

0.95

587.6

18.163

plane

32.4

1.90

60

27

0.95

780

18.163

plane

33

1.98

60

26.5

0.97

587.6

17.827

plane

33

1.90

60

26.5

0.97

780

17.827

plane

33.4

1.78

75

37.5

0.90

587.6

25.227

plane

38

4.02

75

37.5

0.90

780

25.227

plane

38

3.65

D, mm

On the Figs. 6.6 and 6.7 we show similar diffraction-limited lenses with numericalapertures NA = 0.97 and different diameters and correction wavelengths. The residual
spherical aberrations in these lenses are very small too.

Fig. 6.5. Transverse on-axis spherical aberration as function of the aperture angle in a lensof glass N-SF5 with D = 50 mm, NA = 0.95 and corrected wavelength 587.6 nm.

296

Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrationsin the Centered Conical Optical Systems with High and Super-high Numerical Apertures

Fig. 6.6. Transverse on-axis spherical aberration as function of the aperture angle in a lensof glass N-SF5 with D = 25 mm, NA = 0.97 and corrected wavelength 780 nm.

Fig. 6.7. Transverse on-axis spherical aberration as function of the aperture angle in a lensof glass N-SF5 with D = 25 mm, NA = 0.97 and corrected wavelength 780 nm.

6.3.2.4. Examples of On-axis Spherical Aberration Correctionin the Systems of Three Surfaces
By analogy with the Subsections 6.3.1, 6.3.2.1 and 6.3.2.2 we designed series of doubletswith corrected on-axis spherical aberration for 587.6 nm and chromatic aberration for
486 nm and 656 nm. The firs surface of the doublets is multi-conical. On Fig. 6.8 we show
the transverse on-axis spherical aberration for a doublet with the next characteristics andconstructive parameters: EFL = 15 mm; D = 20 mm; NA = 0.75; r1 = 7.58 mm;
r2 = 203.435 mm; r3 = 48.243 mm; d1 = 10.7 mm; d2 = 8.1 mm. The on-axis sphericalaberration on the whole aperture is less than 4.10-14 mm. The doublet has on-axis adiffraction-limited image quality for the correction wavelength of 587.6 nm. The
maximum on-axis spherical aberration for 486 nm is less than 0.0088 mm and for 656 nm
it is less than 0.006 mm. The comparative analysis shows that this doublet has a numericalaperture which is one and a half times larger than the best existing ones
(see [10] and [11]).
297

Advances in Optics: Reviews. Book Series, Vol. 5

Fig. 6.8. Transverse on-axis spherical aberration as function of the aperture angle in a doubletof Schott-glass N-PK51 and Ohara-glass S-NPH2 with EFL = 15 mm, D = 20 mm, NA = 0.75.

6.4. Conclusions
Based on the above presented text, tables and figures we can make the following
conclusions:298•

We found rigorous analytical dependences between the paraxial characteristics
and constructive parameters in centered optical systems by the help of matrix
transformations. This gives us the possibility to design optical systems with presetparaxial characteristics strictly calculating the necessary values of theconstructive parameters such as vertex radii, axial thicknesses and refraction
indices. The differences between the preset values of the paraxial characteristics
and those, evaluated through our rigorous algorithms, constructive parameters,
are less than 1.10-14 mm;•

By analogy we found rigorous dependences between the axial chromaticaberrations and constructive parameters. Thus one can design optical systems
with preset paraxial characteristics together with corrected axial chromaticaberrations without any optimization programs. The residual axial chromaticaberration between the preset left and right wavelengths in most cases is less than
1. 10-14 mm (see the last column in Table 6.2);•

Formulas have been developed, which connect analytically the parameters of thereal tangential rays with the sagittal radius-coefficient of the conical surface inthe centered optical systems. They gave us the possibility to create a reliablerigorous algorithm for on-axis spherical aberration correction on the wholeaperture, using one multi-conical surface only. The residual on-axis transverse
spherical aberration in most cases is less than 5.10-14 mm;•

The developed above theoretical base allows the creation of simple computer
program that guarantees: rigorous correction of on-axis spherical aberration,correction of axial chromatic aberration and guarantee preset paraxial

Chapter 6. Rigorous Analytical Correction of Axial Chromatic and On-axis Spherical Aberrationsin the Centered Conical Optical Systems with High and Super-high Numerical Aperturescharacteristics in optical systems independently from the number of surfaces. The
program calculates strictly all technological data (such as y-coordinates and zcoordinates, slope of the normal in each point of the surface, etc.) for the
production of optical components with aspherical profiles using CNC machines.
Neither an optimization software nor a skilled or experienced designer are needed for thedesign of diffraction-limited (for one wavelength) optical systems with high or super-highnumerical apertures.

References[1]. J. Sasian, Introduction to Lens Design, Cambridge University Press, New York, Melbourne,
New Delhi, Singapore, 2019.
[2]. W. J. Smith, Modern Optical Engineering, McGraw Hill, New York, Chicago, San Francisco,Lisbon, London, Madrid, Mexico City, Milan, New Delhi, San Juan, Seoul, Singapore,
Sydney & Toronto, 2008.
[3]. B. A. Hristov, Development of optical design algorithms on the base of the exact (all orders)geometrical aberration theory, Proceedings of SPIE, Vol. 8167, 2011, 81670E.
[4]. B. A. Hristov, Exact analytical theory of aberrations of centered optical systems, Optical
Review, Vol. 20, Issue 5, 2013, pp. 395-419.
[5]. C. Chen, Methods of solving aspheric singlets and cemented doublets with given primaryaberrations, Applied Optics, Vol. 53, Issue 29, 2014, pp. 202-212.
[6]. Precision UV Fused Silica Aspheric Lenses, http://www.asphericon.com[7]. Fused Silica Aspheres, http://www.edmundoptics.com[8]. NIR Ultrafast Optimized Fused Silica Plano-Convex Lenses, http://www.newport.com[9]. Aspheric Lenses, http://www.optoaxis.com[10]. Aberration Corrected Lenses, http://www.Ulooptics.com[11]. Precision Aspherized Achromatic Lens, http://www.edmundoptics.com

299

Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry

Chapter 7

Ultra-low Light Absorption Measurementin the Problem of Determining Chemical
Impurities Concentrations in Quartz Glasses
and Synthetic Crystalline Quartz Using
Time-resolved Photothermal Common-path
Interferometry
Ksenia Vlasova, Alexandre Makarov and Nikolai Andreev1

7.1. Introduction
The problem of detecting low concentrations of chemical elements in various transparent
materials by thermo-optical absorption measurement has a long history [1]. The
possibility to measure concentration of inclusions by measuring the absorption of thematerial under consideration from connected with the resonant behavior of the spectral
dependence of the inclusion absorption and its linear dependence on the inclusion
concentration.
An example of such measurements in ultrapure quartz glasses (UQGs) is described
elsewhere [2]. In this paper the results of optical radiation absorption measurements wereused to calculate the OH group concentration. The sample transmission was measured at
the maximum of the absorption line of the OH group (2720 nm) associated with thevibrations of this group in quartz glass. The measurement was performed by means ofFourier transform infrared (FTIR) spectroscopy with an absorption detection limit ofabout 10-2 cm-1. To measure the quartz matrix contribution, a thermal radiation sourcewith a continuous spectrum in the resonance absorption region was exploited. Measuring
the spectral absorption dependence outside the resonant absorption line of the inclusion
made it possible to calculate accurately the quartz glass matrix contribution byinterpolation of results to the resonant absorption region. In this example, the possibility

Ksenia Vlasova
Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia

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Advances in Optics: Reviews. Book Series, Vol. 5of measuring a low concentration of an inclusion is related to the strong absorption at thecenter resonance frequency of its absorption contour, which is available for measurementby the traditional direct method.
However, e. g., for metallic inclusions, the absorption spectrum of which lies in the visibleor near-infrared range, the absorption value is no greater than 10-5 cm-1 – 10-7 cm-1 at theconcentrations of interest of ~10 ppb – 1 ppt. In this case, indirect thermooptical methods
are exploited to measure absorption. The idea of thermooptical methods is to heat the test
sample with a focused laser beam at the resonant frequency of an absorbing inclusion andobserving the refractive index response of the medium resulting from changes in its
temperature. This response is detected by a probe laser beam with a wavelength different
from the heating laser wavelength. Fig. 7.1 shows an example of one of the most widelyused sensitive absorption measurement schemes based on a thermo-optical method called
Photothermal Common-path Interferometry (PCI) [3]. This scheme has the following
features: 1. The probe laser beam crosses the region of the heated by the pump laser test
sample at a certain angle. The angle is selected from consideration of the suppression
degree of the heating radiation penetration into the probe beam recording channel. 2. The
pulse-periodic structure of the heating radiation is created by a mechanical modulator in
the form of an opaque rotating disc with holes cut therein for the CW radiationtransmission. 3. Changes in the probe beam spatial structure are detected by means of adiaphragm set up in front of the photodiode in a certain cross-section after the sample, anda lock-in amplifier. 4. A common feature of lock-in amplifying is the absence of
information on the detected signal temporal form.

Fig. 7.1. Schematic block diagram of the PCI method for absorption measurement.

Obviously, the sensitivity of such a method depends, among other things, on the dn/dT
value of the test sample. Therefore, at present, the method has found the greatest
application in the measurement of inclusion concentrations in liquids having high dn/dT
values. Detectable levels from ppb (10-9) to ppt (10-12) weight fractions have been achieved
when measuring the absorption values at the level of 10-9 cm-1 at wavelengths close to theresonant frequencies of absorbing chemical elements [1]. In this method, the inclusion
concentration calculation from the result of measuring the absorption at the resonant
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Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry

frequency is organically related to the separation of the UQG matrix contribution fromthe measured absorption.
In this chapter, we will discuss the determination of inclusion concentrations in quartz
glasses and optical crystals. The specificity of the problem of determining low inclusion
concentrations by thermooptical measurements in these dielectrics is related to the low
value of their dn/dT, which differs by about one or two orders of magnitude from thecorresponding values in liquids. E. g., for quartz glass dn/dT = 0.96×10-5 K-1 [4] is about
80 times less than for CS2 (dn/dT = -8×10-4 K-1) and about 60 times less than for CCl4(dn/dT = -6.1×10-4 K-1).

7.2. Relevance
To date, the problem of detecting inclusions in different materials remains relevant fromthe point of view of studying the influence of impurities on the properties of manufactured
materials and, ultimately, their quality control.
We will discuss the possibility of applying a thermo-optical method for measuring low
absorption to determine the inclusion concentrations in UQG and synthetic crystalline
quartz (SCQ) which are widely used in different fields of science and technology.
UQG underlies all information technologies in the modern world. UQGs are used in the
production of optical fibers [5], UV radiation projection optics [6, 7], the design of UV
lasers which play an important role in the production of printed circuit boards [8-10] andintegrated electronic microcircuits [11, 12]. The quality of UQGs is also important in the
production technology of large integrated circuits. In this case, UQGs are employed forthe manufacture of crucibles in which silicon melts [13].
SCQs grown by hydrothermal method can be utilized in the manufacture of quartz glasses
as a raw material. In addition, SCQs are also exploited in laser technology for the
manufacture of large-aperture polarizing optical elements and windows, as well as lenses
for the output stages of high-power CW lasers [14]. In both cases, low absorption at thelaser radiation wavelength is required, which is directly related to the concentration of
impurities of different chemical elements.
In addition to quality control of SCQs, the concentration measurement of polluting
chemical elements is also relevant in the process of development of the technology forgrowing crystals intended frequency conversion of high-power CW lasers using nonlinear
optics methods, because the efficiency of nonlinear optical processes is limited byparasitic radiation heating of crystals [15] due to the presence of such inclusions.
Obviously, one of the necessary conditions for improving the ultrapure materials
production technology is the possibility of measuring impurity concentrations at a level
better than the minimal level achieved at this stage of production technology development.
The state of this issue for UQG can be understood from the information provided, for
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Advances in Optics: Reviews. Book Series, Vol. 5

example, on the website of Heraeus company, the world leader in UQG production [16].
Fig. 7.2, taken from this website, shows a table of UQG characteristics (F series glasses).
It can be seen that the concentration of polluting chemical elements is below the detection
limit of one of the most sensitive Inductively Coupled Plasma Mass Spectrometry(ICP-MS) method [17] (see “Trace impurities”).

Fig. 7.2. Table of typical material parameters for quartz glasses of various Heraeus brands.
Source: [16].

On the other hand, the production of glass with trace impurities below ICP-MS detection
limit does not mean at all that there is no need to continue improving the UQG purification
technology. As an example in which a further decreasing in the concentration of impurities
relative to the achieved level is relevant, we can mention the technology of growing
ultrapure silicon crystals for microelectronics. UQG is used for the manufacture of
crucibles in which silicon melts to grow crystalline silicon for the production of large
integrated circuits and solar cells. The quality of the crystals grown critically depends on
the impurities concentrations in the crucible material [13, 18, 19]. The concentration valuelevel (e.g., for iron), affects the quality of the grown crystals, is below the sensitivity of
the ICP-MS and Laser Ablation Inductively Coupled Plasma Mass Spectrometry (LAICP-MS) methods, which does not exceed 10 ppb [17, 20].
Thus, the insufficient sensitivity of methods for determining the impurities concentrationin UQGs produced today makes it necessary to increase the sensitivity of methods fordetermining the concentration of different chemical elements in UQGs и SCQs.

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Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry

7.3. State of the Art
In this chapter, we will discuss the possibility of impurity concentration measurements inquartz glass and crystalline quartz close to the detection limits, and for some chemicalelements significantly exceeding them, of one of the most sensitive and widely used
LA-ICP-MS method. This possibility is associated with ultra-low absorption
measurements at discrete wavelengths ranging from near UV to IR utilizing athermooptical method.
From the above example of OH group concentration measurement, it is clear that theabsorption coefficients of both the UQG matrix and a chemical element at the resonant
frequency need to be known in order to calculate the concentration exploiting results ofabsorption measurement. These values are calculated allowing for the results ofmeasurements at several frequencies. In the above example, a source of thermal radiation
with a continuous spectrum in the resonant absorption region of the desired inclusion was
employed to measure the contribution of the UQG matrix. In the case of utilizing laser
sources generating at a certain wavelength, such measurements are performed using
radiation with an additional test wavelength, shifted from the line center to the region of
weaker ion absorption. Further, the actual ion absorption is calculated under theassumption of a weak wavelength dependence of the UQG matrix absorption. In a more
complex case of a notable variation in the UQG matrix absorption at the width of the line
contour, the absorption is measured at one more wavelength, shifted from the line centerin the opposite direction from the first one to construct an absorption value linear
interpolation to the line center.
Thus, it is easy to understand that in order to correctly calculate the chemical element
concentration, the absorption measurement must be accurate enough to reflect thevariations made by the chemical element in the matrix absorption. If the absorption
coefficient is changed in such a way that the contribution made by the inclusion in the
UQG absorption is less than the random deviations of the measured values, then it
becomes impossible to calculate the desired concentration. Therefore, the parameter
determining the measurement accuracy is the ratio of the signal level to its noise
component level.
The successful solution of the task is related to the possibility of the thermooptical method
to provide measurements at the absorption level of the UQG matrix with a signal-to-noise
ratio large enough to reliably detect the variations in the absorption of desired chemicalelement.
Exploiting the data presented in the literature, we will estimate the UQG matrix absorption
level and the acceptable noise level for these measurements. For the evaluation of the
UQG matrix absorption in the spectral region of the absorption lines of metal ions, thetheoretical estimates presented in Fig. 7.3 can be employed (Fig. 3.3 from [21]).
We obtain that the absorption coefficient of the UQG matrix varies in the range from
4×10-6 cm-1 at the wavelength of 400 nm (resonant for Fe3+ ions) to 0.9×10-7 cm-1 at the305Advances in Optics: Reviews. Book Series, Vol. 5

wavelength of 1100 nm (resonant for Fe2+ ions). I. e., the absorption measurement methodshould allow the absorption measurement of 10-7 cm-1 with a sufficiently high signal-tonoise ratio. To estimate the ratio tolerable value, one should consider the characteristic
values of the minimum concentrations of some chemical elements detected by
LA-ICP-MS in UQG (Fig. 7.4).

Fig. 7.3. The calculated attenuation spectra for some of the loss mechanisms contributing to theoverall fiber attenuation (dashed and dotted lines). Source: [21].

Fig. 7.4. The LA-ICP-MS detection limit for determining the concentration of inclusions
in quartz glass. Source: [17].

It follows from the Fig. 7.4 that the detection limit of different chemical elements in UQG
varies widely from 18 ppb for Al, Ti, and Fe to 1.5 ppt for Cu.
The LA-ICP-MS detection limit for polluting inclusions in SCQ, as shown in Table 7.1,
is much worse, ranging from 14 ppm for Ca to 10 ppb for Pb. This detection limit is evenworse for ICP-MS.
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Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry

Using the presented data, it is possible to calculate the changes that are made by somechemical elements into the UQG absorption, if the absorption of these elements at a given
concentration is known from literature. The absorption values of some of the mostcommon metal ion inclusions are presented in the first three columns of Table 7.2. The
data are given for resonant frequency and inclusion concentration of 1 ppb. The fourthcolumn shows the LA-ICP-MS detection limits of concentrations of the elements (datataken from Fig. 7.4). The fifth column shows the calculation results of the element
absorption values at concentrations corresponding to the LA-ICP-MS detection limit fromthe third column. The sixth one shows a theoretical estimate of the UQG matrix absorption
coefficients at the corresponding wavelengths.
Table 7.1. ICP-MS and LA-ICP-MS detection limits (ppm) for different chemical elements
in crystalline quartz [22-24].

ICP-MS [22]LA-ICP-MS [23]LA-ICP-MS [24]

Li
0.5
0.55
0.22

B
1.28

Na
100
5.5
6.4

Al
100
6
6

K
100
3.2
2.3

Ca
100
14
29

Ti
5
1.2
0.66

Mn
0.3
0.07

Fe
1
1.6
0.52

Pb
0.01
-

Table 7.2. Absorption of some of the most common metal ions (first column) at a concentrationof 1 ppb (third column) at resonance frequency (second column) and corresponding absorption
coefficient (fifth column) at the detection limit concentration (fourth column). The data of the first
three columns are from [21], the data of the fourth column are from Fig. 7.2, the data of the fifth
column are the result of the calculation based on the data of the third and fourth columns. The sixth
column presents a theoretical estimate of the UQG matrix absorption coefficients [21].

Ion

Peak
wavelength(nm)

Absorption
coefficient,
one part in
109 (dB/km)

LA-ICPMS
detection
limit (ppb)

Cr3+
C2+
Cu2+
Fe2+
Fe3+
Ni2+
Mn3+
V4+

625
685
850
1100
400
650
460
725

1.6
0.1
1.1
0.68
0.15
0.1
0.2
2.7

0.1
1.5·10-3
17.8
3.16
0.75
0.013

Absorption
coefficients at
concentration
equal todetection limit(×106 cm-1)
0.37
0.0039
27.8
1.1
0.34
0.08

Theoreticalestimate of the
UQG matrix
absorption(×106 cm-1)
≈1.4
≈1
0.26
0.09
≥3
≈1.2
≥2.8
0.69

We now collate the sensitivity of the LA-ICP-MS method and widely used PCI method,
utilizing the fifth column data. To estimate PCI method detection limit of absorption
measurements in UQG, we employ the measurement data of Suprasil 300 glass provided
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Advances in Optics: Reviews. Book Series, Vol. 5

by Heraeus, the world's leading manufacturer of UQGs. According to the company's data
(see Fig. 7.5, taken from [25]), the absorption coefficient of Suprasil 300 is
𝛼1064 ≈ (2-8)×10-7 cm-1 or 5×10-7 cm-1 ± 60 % at a wavelength of 1064 nm. The scatter ofvalues most likely indicates that the measurements were performed at the extreme levelsensitivity. I. e., according to the accepted definition, the detection limit (the absorption
value obtained under experimental conditions when the values measured are equal to theirnoise component amplitude) will be 𝛼1064 ≈ 5×10-7 cm-1.

Fig. 7.5. Absorption in quartz glasses of different Heraeus brands at a wavelength of 1064 nm.
Source: [25].

It is now possible to compare the absorption values (fifth column) of the elements having
concentration corresponding to the detection limit with the value of the PCI method
detection limit when measuring the absorption of 0.5×10-6 cm-1. In this case, without any
generality loss, we assume that the sensitivity of the PCI absorption measurements at the
wavelengths of the centers of the absorption lines of the ions under consideration does notdepend on the wavelength.
The comparison shows that among the considered ions, only two can change the UQG
absorption by the values available for detection by the PCI method, which has thesignal-to-noise ratio equal to 1 when measuring the absorption ~0.5×10-6 cm-1. These are
Fe2+ (detection limit is 18 ppb) and Fe3 (detection limit is 3.2 ppb) ions. In the presenceof Fe2+ and Fe3+ ions with the minimum detectable concentration by mass spectroscopy
methods, the variation in the UQG matrix absorption should be 50 times and 12 times,
respectively, higher than the UQG matrix absorption. It follows that PCI concentration
measurements are available for these elements, since when measuring total absorption,
the signal-to-noise ratio will be 50/1 in the first case and 12/1 in the second. All otherconsidered ions will vary the UQG absorption by values that are beyond the PCI
measurement capabilities.
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Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry

In addition to metal ions presence, the UQG absorption can increase due to the presenceof the OH groups. In the spectral range of absorption of metal ions, the third vibration
overtone of OH group (absorption maximum at a wavelength of 940 nm in SiO2 matrix)
will distort the UQG matrix absorption value. The absorption value contributed by OH
groups to the UQG matrix absorption can be obtained employing the dependence of the
OH-related absorption on the wavelength shown in Fig. 7.6 [26]. From the data in the
figure, we obtain that the absorption is 1050 Db/km or 1050 2.3×10-6 cm-1 =2.4×10-3 cm-1 at a wavelength of 940 nm (the third overton of the vibrational line of
2720 nm) at the OH group concentration of 700 ppm. E. g., if the OH concentration in the
Suprasil
300
glass
is
0.2
ppm,
then
the
absorption willbe
2.4×10-3 cm-1×0.2/700 = 6.9×10-7 cm-1. This absorption is slightly higher than the PCI
method detectable level, and under some conditions, the result of such a measurement canbe utilized to determine the concentration of about 0.2 ppm.

Fig. 7.6. Absorption spectral dependence for silica fibers with high (700 ppm, F100) and low(0.2 ppm, F300) concentrations of OH groups. Source: [26].

The above analysis shows that in some cases the absorption values detected by the PCI
method can be determined by inclusions of chemical elements, the concentration of whichlies beyond the detection limits of mass spectroscopy methods.
However, the main problem with thermooptical measurements is that the concentration
measurements of even one impurity requires an absorption measurement at least two
wavelengths (as, e.g., in the case of the OH group concentration measuring [2] describedabove). But the case where the inclusion absorption spectrum is determined by an isolated
absorption line is unique. Real glasses contain many chemical elements with overlapping
contours of the absorption spectrum. So, when discussing the issue of the quality of UQGs,
an estimate of the acceptable concentrations of more than ten chemical elements is given(see Fig. 7.7) [27, 28].
Thus, in general, the determination of impurity concentrations involves the problem ofcalculating the concentration of N chemical elements by measuring the absorption at a309

Advances in Optics: Reviews. Book Series, Vol. 5

number of selected wavelengths. This raises the question of the optimal choice of a set ofwavelengths, improving the concentration calculation accuracy, based on the wavelength
dependences of the absorption of all considered elements and the results of measuring the
UQG absorption at the selected wavelengths.
Generally, the problem of measuring the concentrations of the chemical elements is
integrated and it consists of three, in fact, independent components, the successful solutionof which influences the solution of general problem.

Fig. 7.7. ICP-MS for quartz glasses of different Heraeus brands. Source: [27].

First, the dependences of the absorption coefficients of the considered chemical elementson the wavelength and on their concentration in UQG should be known. The possibilityof obtaining such dependences for UQG was demonstrated elsewhere [29], where theavailability of the controlled doping procedure with different chemical elements up toabsorption levels ≥0.01 cm-1 in the process of UQG fabricating was shown. This makes it
possible to measure the absorption of this level utilizing traditional Fourier spectroscopy
method. Spectral dependences of absorption contours were obtained for a number ofchemical elements. The resonance values of the absorption coefficients of some chemicalelements in UQG, which we employed in the Table 7.2, are given in [21]. It should be
noted that we are not aware of the dependences of the absorption coefficients of theconsidered chemical elements on the wavelength and on their concentration in SCQ, aswell as in nonlinear crystals. Moreover, the procedure for measuring them in SCQ isapparently very problematic due to the impossibility of incorporating foreign chemicalelements into the crystal structure during the crystal growth.
Second, a detailed solution to the mathematical problem of calculating the concentrationof N chemical elements by absorption measurement at a number of selected wavelengths
is needed. This problem is not trivial and it requires serious consideration. We have notfound the solution to this problem in the literature. Although there are studies with similar
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Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry

content and formulations of the problem. E. g., the spectral dependence of the transmission
associated with the concentration of the Fe2+ ion was calculated from the measured
spectral dependence of the near-IR region transmission of silicate glass with unknown
concentrations of inclusions [30]. Both the transmission wavelength of the OH group andthe silicate glass matrix, which has a complex chemical composition, were exploited in
the calculations. Also, for silicate glass the spectral dependencies related to Fe3+, Fe2+, andOH groups and glass matrix were calculated by measured spectral transmission in thevisible and UV spectral regions [31, 32].
Third, a physical tool is needed for sufficiently accurate absorption measurements at a
certain number of selected wavelengths, determined by solving a mathematical problem.
It is quite obvious that from the point of view of the formulated problem, solving the firsttwo problems is not of great importance without a sufficiently reliable prospect of a
successful solution to the problem of creating a physical tool for measuring ultra-low
absorption with sufficient accuracy at an independent set of wavelengths. The
measurement accuracy requirement can only be met with a sufficiently high sensitivity,
i.e., at a sufficiently low detection limit value relative to the measured absorption value.
The ratio of the detection limits of the mass spectroscopy method and the PCI method
indicates the prospects of further development of the PCI method in the direction ofincreasing the sensitivity in the interests of the task at hand.

7.4. Time-resolved Photothermal Common-path Interferometry Scheme
It can be seen from the above analysis that the PCI method for measuring absorption is
one of the most sensitive of the known thermooptical methods and it can be useful inmeasuring the concentrations of different inclusions in UQGs. However, its sensitivity isclearly not enough for a comprehensive solution to the problem of determining theconcentration at the ICP-MS detection level.
Nevertheless, as we have shown, the given sensitivity of absorption measurements using
the PCI scheme is not the limit for thermooptical methods. In our recent papers [33-35],
a modification of this scheme, called Time-resolved Photothermal Common-path
Interferometry (TPCI), has been introduced. The sensitivity of this scheme in the current
implementation is two orders of magnitude higher than its PCI prototype. Fig. 7.8 showsthe electron-optical block diagram of the experimental setup.
The operation principle of this scheme has been discussed in detail elsewhere [33-35].
Here we will only point out the main features of the TPCI scheme that distinguish it fromits predecessor PCI scheme, which made it more sensitive and functional.
First of all, it is the heating and probe beams coaxiality in the test samples. In the PCI
scheme, the beams are crossed at a low angle (see Fig. 7.1) to get rid of the influence ofpowerful heating radiation on the measuring photodiode [36]. Although the angle
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intersection of the beams improves the spatial resolution of the absorption measurement
scheme, it significantly reduces sensitivity. The possibility of implementing a coaxial
scheme is associated with the using of a space-frequency filter. It includes an interference
M3 mirror for the heating laser radiation initial filtering, a DPr dispersion prism and a M5
mirror (schlieren), which cuts off radiation at a wavelength of 1070 nm. This filter
effectively filters out heating radiation and, in coaxial geometry, makes it possible toachieve its suppression coefficient of 10-9 on the photodiode.

Fig. 7.8. Electron-optical block diagram of the TPCI method.

The coaxial configuration enables the use of video cameras (not shown in the figure) tocontrol and adjust the alignment of the beams at the initial scheme setup and in the caseof its misalignment. It ensures the reproducibility of the measurement results.
In addition, the coaxial propagation of beams allowed us to develop a theory linking themagnitude of the signal observed during TPCI measurements with all existing parameters,
both the test sample and the measuring setup. The obtained analytical formulae make itpossible to calculate the required absorption in the test sample from the results of similar
measurements in a reference sample with known absorption. These formulae wereobtained by asymptotic expansions with respect to the dimensionless small parameters ofsolutions of the following problems: 1) Theory of deformation in a given nonuniform
temperature field arising when the sample is heated by focused laser radiation, and thecalculation of a thermo-optical response, i.e., spatially inhomogeneous change in thematerial refractive index when exposed to the heat of focused radiation; 2) Theory ofprobe beam diffraction on the spatial-temporal refractive index inhomogeneity, associated312Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry

with the arising stresses, and 3) Interference of the constant component of the probe beamwith its time-varying component; this interference forms the measured photodiode signal.
The accuracy of the calculations is sufficient for practical purposes, which is due to theextreme smallness of the dimensionless parameters, in order of magnitude equal to ~10-4.
The details can be found elsewhere [34, 35]. The obtained analytical formulae, given
below, make it possible to use any optical sample with known absorption andthermo-optical coefficients as a reference sample for calibration.
An important conclusion from the developed theory [34, 35, 37] is that the allowance forthe influence of spatially inhomogeneous stresses arising in a local heated region leads toa replacement of the dn/dT value by the so-called thermo-optical parameter P [38] in thecommonplace expressions for the thermo-optical response, which are valid for liquids andgases [1]. This parameter depends on the sample physical parameters and, if the sample
is anisotropic, on its orientation relative to the propagation direction and radiation
polarization. Thus, the expression for the measured signal equal to the diaphragmaveraged time-varying component of the probe radiation power P~ can be written as(7.1)

𝑃~ (𝑡) = 𝐾 ∗ 𝑃 ∗ 𝛥𝑇(𝑡)Here, ΔT is the temperature change in the heating beam center, equal to [34]𝑊 (𝑡)𝑇(𝑡) = 𝜋𝑎2ℎ∗𝑐∗𝜌 αℎ ,

(7.2)

ℎ

in the absence of heat diffusion from the heated area. Here, αℎ is the absorption (cm-1) of
the test sample at the heating laser wavelength; 𝑊ℎ (𝑡) is the current heating radiationenergy density; c is the heat capacity; ρ is the test sample density; ah is the heating beam
radius in the sample at the 1/e intensity level.
The expression for the thermo-optical parameter obtained in case of isotropic sample [35]𝜕𝑛𝑃 = ( ) −𝜕𝑇 𝜌

𝛼𝑇 ∗𝑛30 1+𝜈
41−𝜈

∗ (𝑝11 +𝑝12 )

(7.3)

𝜕𝑛

Here, (𝜕𝑇 )𝜌 is the temperature change in the refractive index at a constant density, αT isthe thermal expansion coefficient, 𝑝11 and 𝑝12 are the components of the photoelasticity
tensor, and ν is the Poisson's ratio.
The expression for the thermo-optical parameter obtained for the case where the radiation
propagates along the crystallographic axis C for a crystal sample of class 32 (symmetryof crystalline alpha-quartz)
𝑇(𝐶 +𝐶 )∗𝛼 +𝐶 ∗𝛼 𝑇
𝜕𝑛𝑜𝑝 +𝑝)𝜌 − 𝑛03 ∗ 11 12 ∗ 11 12 ⏊ 13 ∥ ,
𝜕𝑇
4
𝐶11

𝑃 = (

(7.4)

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Advances in Optics: Reviews. Book Series, Vol. 5
𝑇where С11 and С12 are the stiffness tensor coefficients, 𝛼⏊
is the thermal expansion
coefficient across the C axis, 𝛼∥𝑇 is the thermal expansion coefficient along the C axis.

Formulae for calculating the thermo-optical parameter for other configurations of themutual arrangement of the crystal axes and the direction of radiation propagation can befound elsewhere [35].
Function K, which allows for the beam configuration shown in Fig. 7.9, has the form
𝐾 = 𝑃𝑝𝑟 ∗𝑧ℎ𝑝𝑟 22
2∗𝑎𝑝𝑟

𝑧 exp[𝛼𝑒𝑓𝑓 ∗(𝑧−𝑧0 )]∗ [exp[−𝑅 2 ∗𝑓1(𝑧1 ,𝑧,𝐿,𝐿𝑠𝑎𝑚𝑝𝑙𝑒 )]−1]𝑑𝑧)],𝑓2(𝑧1 ,𝑧,𝐿,𝐿𝑠𝑎𝑚𝑝𝑙𝑒 )∗𝑓3(𝑧1 ,𝑧,𝐿,𝐿𝑠𝑎𝑚𝑝𝑙𝑒 )0∗ [−𝐼𝑚 (∫𝑧 1(7.5)

where Ppr is the time-independent probe beam power, R is the radius (normalized to ah) of
the projection of the diaphragm installed in front of the photodiode. The functions𝑓𝑖(𝑧1 , 𝑧, 𝐿, 𝐿𝑠𝑎𝑚𝑝𝑙𝑒 ) are the complex functions of real variables, the meaning anddesignations of which are shown in Fig. 7.9.

𝑓1(𝑧1 , 𝑧, 𝐿, 𝐿𝑠𝑎𝑚𝑝𝑙𝑒 ) =1−2i∗[(z−z1 )−

L∗n
zh_pr

]+B(z)

+

2
𝑎ℎ
2∗𝑎2
𝑝𝑟
𝑎2 LsampleL∗n1−i( −Δpr ∗ 2ℎ +)zpr
𝑎𝑝𝑟 2∗zpr

,

(7.6)

𝑓2(𝑧1 , 𝑧, 𝐿, 𝐿𝑠𝑎𝑚𝑝𝑙𝑒 ) =
=

2
2
𝑎2ℎ
𝑎2
𝜆ℎ ∗𝑀2
∗
(1
+
[
𝑧
+
𝛥
]
∗
(
)
) + (1 + 𝑖 ∗ 𝑧 ∗ 𝑎2ℎ ),𝑝𝑟
2∗𝑎2𝑝𝑟
𝜆𝑝𝑟
𝑝𝑟

(7.7)

𝑓3(𝑧1 , 𝑧, 𝐿, 𝐿𝑠𝑎𝑚𝑝𝑙𝑒 ) =𝐿∗𝑛= (1 − 𝑖 ∗ [ 𝑧

𝑝𝑟

𝑎2

𝑎2

𝐿

𝐿∗𝑛

𝑠𝑎𝑚𝑝𝑙𝑒− 𝛥𝑝𝑟 ∗ 𝑎2ℎ + 2∗𝑧]) + 2∗𝑎ℎ2 (−2 ∗ 𝑖 ∗ [𝑧 − 𝑧1 − 𝑧
𝑝𝑟

𝑝𝑟

𝜆 ∗𝑀2

2

𝐵(𝑧) =ℎ_𝑝𝑟

𝑝𝑟(1+[𝑧+𝛥𝑝𝑟 ] ∗( ℎ𝜆𝑝𝑟

2𝛥0_𝑝𝑟 ∗𝑛𝛼𝑒𝑓𝑓 = (𝛼ℎ +

𝑧ℎ_𝑝𝑟

𝑎𝑝𝑟

𝛼𝑝𝑟
2

,

) ∗ 𝑧ℎ_𝑝𝑟 ,

𝐿

,

(7.9)

(7.10)(7.11)

𝑧0 = − 𝑠𝑎𝑚𝑝𝑙𝑒 − 𝛥𝑝𝑟 ,(7.12)

𝐿𝑠𝑎𝑚𝑝𝑙𝑒(7.13)

2∗𝑧ℎ_𝑝𝑟

𝑧1 =
314

(7.8)

𝑎2

) )∗(1+𝑖∗𝑧∗ 2ℎ )

2
2 2
𝑎2
ℎ ∗(1+[𝑧+𝛥 ]2 ∗(𝜆ℎ ∗𝑀 ) )+((1+𝑖∗𝑧∗ 𝑎ℎ )
𝑝𝑟
2
2
𝜆𝑝𝑟
2∗𝑎𝑝𝑟
𝑎𝑝𝑟

𝛥𝑝𝑟 =

] + 𝐵(𝑧))2∗𝑧ℎ_𝑝𝑟− 𝛥𝑝𝑟Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry

Here, 𝛼ℎ and 𝛼𝑝𝑟 are the absorption coefficients (cm-1) of the heating and probe radiation,
taking into account of which is essential for the calibration sample. 𝑀2 is the beam quality
factor describing the difference between the waist length of a real beam and the diffractive
Rayleigh length of a Gaussian beam. In our experiments, the heating beam 𝑀2 value
was 1.08.

Fig. 7.9. The layout of caustics of the heat and probe coaxial beams in the sample: 1 is the test
sample, 2 is the caustic of the probe beam with power Ppr (r, z), and 3 is the caustic of the heat beamwith power Ph (r, z, t). D is the projection of the diaphragm installed in front of the photodiode.
Source: [35].

The above analytical expression for the K function can be calculated in an elementary way
utilizing any interactive computing system of the Mathcad type for a personal computer.
Fig. 7.10 shows an example of a theoretical calculation for the dependence of the powerof the probe beam time-varying component on the diaphragm size before the photodiodeand the distance L (see Fig. 7.9). K8 glass and heavy water were exploited as samples.
Matching of theoretical curves with experimental points for K8 glass is ensured by asuitable choice of a calibration coefficient of proportionality between the theoreticallydetermined time-varying component of the probe beam power, which falls on thephotodiode cathode through the diaphragm, and the measured signal. The compliance ofexperimental points obtained for heavy water with theoretical curves (Fig. 7.10) was
achieved by using the same calibration factor in the recalculation procedure.
Thus, the theoretical calculation of the coefficient, which depends both on the sample
material properties and dimensions and on the experimental setup parameters, makes it
possible to significantly simplify the calibration process and use the same reference
sample with known material parameters and absorption for all measurements. Thiseliminates the need to perform additional measurements due to IR Fourier spectrometer[3] or conduct the sample doping procedure, which in general is not possible for allmaterials [39].
Another important advantage of the TPCI scheme is the temporal resolution of the signal
obtained from the photodiode, which makes it possible to record in the observed signal
315

Advances in Optics: Reviews. Book Series, Vol. 5

the component associated with the refractive index nonlinearity in the field of the heating
beam. It allows to separate it from the thermooptical contribution, which significantly
increases the validity of the results of calculating the absorption value when processing
the shape of the observed oscillograms recorded at the limiting sensitivities.

Fig. 7.10. Pulse amplitude of the time-varying component of the probe beam observedon the oscilloscope and the corresponding theoretical calculation depending on the distance L (cm)of the aperture projection to the output ends of the following samples: 4-cm-long K8 glass (solid
curves) and heavy water with a layer thickness of 1.8 cm (dashed curves). Experimental points:
the diamond ◊ is the diaphragm diameter of 0.1 cm, the square □ is the diameter of 0.13 cm,
the plus sign + is the diameter of 0.18 cm, and the circle ○ is the diameter of 0.052 cm. The aperturemagnification factor in the projection section is 6.7. The calibration coefficient of the y axis is
the same for all curves. Source: [34].

Fig. 7.11 shows for comparison two normalized oscillograms obtained with the KU-1
quartz glass with the absorption of ≥ 10-5 cm-1 and UQG Suprasil 311 with the absorptionof 2.6×10-6 cm-1. The difference between the shapes of the presented oscillograms is
associated with a notable contribution of the refractive index surge in Suprasil 311 due tothe refractive index nonlinearity in the heating laser field in comparison with the signal
thermooptical component. It is obvious that the separation of the contributions of different
mechanisms of the refractive index change is especially important in measurements at thelimiting sensitivities.
Fig. 7.12 shows an example of such measurements conducted for SCQ (LLC “Quartz
Technology Company” [40]) with an absorption of ≈10-7 cm-1. The oscillograms show atendency of temporal saturation of the thermooptical signal due to heat diffusion from theheated area. For this particular example, taking this effect into account gives a two-foldcorrection that increases the calculated absorption. In addition, a signal surge is observed
due to the refractive index change in the field of a heating laser rectangular pulse.
Separation of the effects is performed by measuring the thermooptical contribution at the
moment immediately after the end of the rectangular heating pulse. Due to the decreasein signal because of saturation, the signal-to-noise ratio at the measurement point
is about 25/1.
316

Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry

Fig. 7.11. Oscillograms of typical signals obtained by measuring the absorption coefficientin quartz glasses: Suprasil 311, KU-1; cooling – decrease of signal due to heat diffusion
after the end of heating. Source: [35].

Fig. 7.12. Oscillograms of pulse signals obtained by observing the absorption coefficient.
The absorption coefficient is estimated in the range of 10 -7 – 5×10-7 cm-1 in synthetic crystalline
quartz. Source: [35].

The described measurement procedure in the TPCI scheme became possible as a result of
the implementation of electronic temporal modulation of the heating radiation power,
which is performed by modulating the pump diode currents in a heating CW Yb-doped
fiber laser. Electronic modulation ensures the practical absence of a jitter whensynchronizing heating pulses and digital oscilloscope sweeps. This allows averaging overa large number of pulses (≥104) in a repetitively sequence of heating radiation pulses.
Averaging leads to the suppression of the signal noise component and increases themeasurement sensitivity by two orders of magnitude. Furthermore, the absence of theinfluence of jitter on the oscillogram shape at large averaging numbers gives anundistorted signal from the photodiode, which makes it possible to correctly separate thecontributions of absorption and the refractive index nonlinearity. Electronic modulation
317

Advances in Optics: Reviews. Book Series, Vol. 5

also makes it possible to promptly adjust the duration and repetition rate of heating pulses
in conditions when the effect of heat diffusion from the heated area is significant. Thisfunction is very useful for measurements in quartz glasses and quartz crystals at extreme
sensitivities.
In addition, the use of a digital oscilloscope allows a detailed study of changes in theoscillogram with an accuracy much greater than 1/50 of the signal amplitude. This
property is based on the ADC increasing resolution effect when averaging noisy signals[41, 42]. The effect is associated with the ratio of the ADC bit width (8 bits for a standard
ADC) and the CPU bit capacity (usually 64 bits). This property is especially importantwhen measuring small deviations of the absorption of UQG from the absorption of thematrix caused by the presence of a foreign chemical element with a concentration at thedetection limit level of the mass spectroscopy method.
The possibilities of measurements using this scheme are demonstrated in Fig. 7.13, whichshows the variations in the absorption coefficient in the UQG samples. Various
experimental points were obtained by moving the samples relative to the heating beamaxis. The size of the spatial area of the absorption coefficient measurement wasdetermined by the size of the heating beam waist, which was 3 cm × 10-2 cm. The directionof the heating beam propagation was along the layers formed in the Suprasil 300 (Heraeus)
and SK-1310 (Ohara) glasses production process utilizing the vapor-phase hydrolysis
technology: outside vapor deposition and vapor axial deposition, respectively. In
particular, it follows from the figure, that the absorption coefficient can differ greatly in
different regions of the sample made of such glasses. The relative root-mean-square(RMS) deviation of the absorption values was ±70 % for Suprasil 300 glass, and ±50 %for SK-1310 glass. Here, in contrast to the indicated glasses, the 3D optically isotropic
Suprasil 311 glass demonstrates a significantly smaller spread of the absorption values
with an relative RMS deviation of the absorption values of ±3.4 %.

Fig. 7.13. Variations in the measuring absorption when the samples are displacedin the transverse direction with respect to the laser heating beam, with a step of 64.5 μm.
Source: [35].

318Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry

During the measurement process, the most transparent SK-1310 was found to containareas with a minimum absorption close to 10-7 cm-1. This leads to the important conclusionthat the absorption of the UQG matrix does not exceed 10-7 cm-1 at a wavelength of
1070 nm. The conclusion confirms the theoretical estimate of UQG absorption given
in Fig. 7.3.
This scheme has been used to demonstrate the absorption measurement in the mosttransparent UQGs at a level of 10-7 cm-1 at a wavelength of 1070 nm. In thesemeasurements the heating pulse duration was τp = 1.24 ms, which was 2.4 times less thanthe diffusion time from the heated area of the UQG
𝜏𝑑𝑖𝑓𝑓 =𝑐∗𝜌∗𝑎ℎ 24∗𝛬= 3 𝑚𝑠(7.14)

in our experiments (here 𝑐 is the heat capacity, 𝜌 is the density, and 𝛬 is the thermal
conductivity) [43]. The temporal shape of the observed thermooptical signal almost does
not differ qualitatively from the shape for Suprasil 311 glass (Fig. 7.11), obtained with aheating pulse duration of 100 μs. This is due to the relatively weak influence of the heatdiffusion effect at such a temporal relation τp and τdiff.
In contrast to the PCI method, for which the measurement of the absorption coefficient of
5×10-7 cm-1 is given with a spread of ±60 %, the absorption coefficient of 10-7 cm-1 wasmeasured with a signal-to-noise ratio of 50/1 in these experiments. I.e., the abovemeasurement was performed with a high excess above the detection limit.
As noted above, the limiting sensitivity of the TPCI scheme for a given heating pulse
power is realized at its durations close to the time τdiff, when the distortion of the signal
shape associated with thermal diffusion begins to appear, but the corrections associatedwith its influence are not too large, and taking them into account does not introduce anotable error in absorption coefficient determination [44]. An example of the result ofsuch measurements is shown in Fig. 7.12. The measurements were conducted with the
SCQ, the heat diffusion time was τdiff = 670 μs, and the heating pulse duration wasτp = 450 μs. The signal-to-noise ratio at the measurement point was about 25/1. The
correction due to heat diffusion is easily found experimentally, and the error associatedwith such procedures is at an acceptable level.
For UQG [44], a situation similar to that shown in Fig. 7.12 can be realized with a heating
pulse duration τp = 2 ms and an absorption 𝛼h = 5×10-8 cm-1. But it should be noted that
we do not know UQGs with such an absorption level, and, therefore, this situation has notbeen realized experimentally. The above estimate was obtained by recalculating theresults of the experiment carried out for the SCQ. In this case, without any generality loss,
we assumed that the geometric parameters of the beams and samples of the UQG and
SCQ, as well as the K function of the samples, were the same. The pulses shown in theFig. 7.12 were obtained with the mutual orientation of the crystal and the polarization
direction of the beams, for which the thermo-optical parameters P of the SCQ and UQG
were approximately the same (≈0.96·10-5 K-1) [35]. As follows from (7.1), in order for319

Advances in Optics: Reviews. Book Series, Vol. 5such pictures to be observed on the oscilloscope screen, it is necessary that the timedependences of temperature ΔT (t) be similar in measurements in SCQ and UQG. For thispurpose, it is necessary to increase the duration of the rectangular heating pulse in UQG
to 0.67 of the heat diffusion time equal to 3 ms, i.e. up to 2 ms. In this case, the pulse
energy will increase by 1.6 times, and taking into account the ratio of the values of the
product with c×ρ, the height of the pulses will be preserved with a decrease in theabsorption coefficient to 𝛼h = 5×10-8 cm-1. Then, taking into account that the noise
component of the measured signal is associated with the noise characteristics of the probe
laser, as well as with the choice of the scheme parameters for detecting the probe radiation
time-varying component (i.e. the noise component does not depend on the physicalparameters of the samples), we find out that the signal-to-noise ratio when measuring thethermo-optical component will remain 25/1. Measurement with such a heating pulse
duration in real UQGs with an absorption of 10-7 cm-1 will lead to an increase in the signal
on the oscilloscope screen by a factor of 2 while maintaining the level of the noise
component. That is, formally, the signal-to-noise ratio will be 50/1. This is still not enoughto detect an increase in the UQG matrix absorption at a wavelength of 850 nm due to thepresence of Cu2+ ions with a concentration at the detection limit level of the mass
spectrometric method, which is about 4×10-9 cm-1, i.e. 1/50 of the matrix absorption.
Further appropriate to make the explanations regarding measuring accuracy using the
TPCI scheme. This issue has two aspects.
As mentioned above, Fig. 7.13 shows the results of the change in the absorption
coefficient when the sample was displaced across the laser beams. In this series ofmeasurements, the accuracy of relative measurements of absorption coefficient at
different sample positions is determined by the accuracy of pulse height measurement on
a digital oscilloscope screen. This accuracy can be estimated by considering the shape of
the averaged pulse, shown, for example, in Fig. 7.12. It can be seen that after averaging
over the number of pulses N ~ 104 in a pulse-periodic sequence, the pulse has a noise
component ~50/1 in this case. Hence, the accuracy of the pulse amplitude measurement
can be estimated as ±1 %. That is, the accuracy of relative measurements of absorption
coefficient for different positions of the sample is of the order of 1-2 %.
The question of the accuracy of measuring the absolute absorption coefficients values ismore complicated. This measurement error has several components, the main of which
are associated with the measurement calibration method we use [34]. In order to calibrate
we use calculations based on the theory of the probe beam diffraction on time-varying andspatial-heterogeneous deformations due to heating the test sample by pulsed laser
radiation. Comparison of calculations with experimental results is shown in Fig. 7.10.
This allows us to compare the oscilloscope readings obtained for the test sample with thereadings obtained for the reference sample, which has different geometric and physicalparameters from the test one (linear thermal expansion coefficient, piezo-opticalcoefficients, refractive index, dn/dT). These physical parameters, measured with a certain
accuracy, are given in various papers. For a reference sample made of K8 glass (BK-7
analog), manufactured in the USSR, we use the data of the reference book [45], in whichthese values are given with an accuracy of ±5 % guaranteed by the USSR standards of320

Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry

measurements. In addition, inaccuracies in some of our measurements contribute to theerror in the absolute values of absorption. One of them, for example, measuring themagnification factor of the projection lens L3, shown in Fig. 7.8, as well as measuring thediameter of the diaphragm installed in front of the photodiode. In addition, inaccuraciesof our settings for the location of the samples, as well as the photodiode diaphragm relative
to the optical elements of the scheme, contribute to the error. There are also errors
associated with inaccuracy of determining the parameters of laser beams. All of theseerrors are not of a fundamental nature and minimized by a more careful approach when
conducting experiments. The calculation of the resulting error is beyond the scope of
published results (although it is important when comparing the results of independent
absorption measurements in different samples). The aim of our work is to demonstrate thehigh sensitivity of measurements using the proposed scheme. Nevertheless, now weestimate the absolute systematic error (which is the same for all points in Fig. 7.13) of themeasured absorption values as ±15 % in our experiments.
In addition, there is still the problem of measuring ultra-low absorption at a freely chosen
set of wavelengths, dictated by the problem of obtaining the highest accuracy ofconcentration calculation. The problem is that the reported sensitivities of the TPCI
scheme were obtained using a CW single mode fiber laser (1070 nm) as a heating one. It
operated in a pulse-periodic mode by modulating the current of the pump diodes. The
pulsed radiation peak power was ≤ 100 W, the average power was ≤ 2 W. Such lasers,
generating repetitively pulsed radiation with a peak power sufficient for measurement
using this method (from 10 W to 100 W), operate at their frequencies unrelated to thevalues from Table 7.2. The choice of such lasers is limited and, in addition, these lasers
are rather bulky and expensive to exploit in one device.
It would be possible to provide a sufficiently wide choice of discrete heating laser
frequencies using single-mode CW diode lasers currently available on the laser market[46]. For these lasers, in addition to a wide range of generated wavelengths, there is thepossibility of electronic control to form radiation in the form of sequences of pulses witha duration from 10 to 1000 μs and a repetition rate from 10 to 100 Hz, which are optimalfor measurements in UQGs and SCQs.
The problem of the practical implementation of the proposed solution to utilize diodelasers is the low peak power of pulses generated by them, ranging from 0.1 to 1 W, morethan two orders of magnitude lower than the peak power of heating lasers used in the TPCI
method. This will lead to a proportional decrease in the absorption detection level in theimplemented TPCI scheme.
Based on the studies [14, 35, 44], where absorption measurement 10-7 cm-1 with asignal-to-noise ratio of 50/1 and 100 W of peak heating power was realized, it can beconcluded that the use of a 1 W diode laser will reduce the sensitivity of the scheme by
two orders of magnitude, and the possibility of correct concentration determination will
be possible only for isolated iron atoms.

321Advances in Optics: Reviews. Book Series, Vol. 5

7.5. The Prospect of Further Increasing of the TPCI Scheme Sensitivity
We will further show that the scheme has great potential for increasing sensitivity, as thelevel achieved is not a limit and can be increased by more than two orders of magnitude
by optimization. This possibility opens the way to measuring concentrations and detecting
traces of polluting chemical elements with a better sensitivity than the most sensitive mass
spectroscopy method.
First of all, we note that in this optical scheme, the measurement sensitivity is limited notby the technical possibilities of amplifying the electric signal from the photodiode, but by
the noise level of different nature, in which the signal observed on the oscilloscope screendrowns. On the other hand, the signal level observed at a given absorption is related to theset of optical scheme parameters, which influences the K function. This function is a factor
in the formula for determining the power of the time-varying component: P~ = K×P×ΔT.
The above ratio clearly shows that an increase in the signal level is possible by optimizing
the function K value with respect to the parameters of the probe beam at the given
parameters of the heating beam. These include the waist radius apr, the mutual distanceΔpr between the waist of the probe and heating beams, and also the probe laser power Ppr.
Calculations have shown that the function K value optimization depending on the probe
beam waist radius and the distance of the waist center leads to a 4.5-fold increase in its
value relative to that realized in the presented scheme. In this case, the optimal probe beamgeometry will be close to that described elsewhere [47]. In the cited paper, the ratio of thesample length and the size of the heating beam waist differs from ours and corresponds tothe situation of a thin thermal lens.
Further, the most obvious way to increase the signal level, as follows from the above
formula for P~, is to increase the probe laser power Ppr. Theoretically, from the conditionof smallness of the distortion it introduces into the sample heated region parameters, thisincrease is limited by the value of the heating pulse peak power, which is about 1 W in ascheme with semiconductor lasers. I. e., a 10-fold increase in the probe laser power to
20 mW is theoretically acceptable.
However, as our recent studies have shown, such an increase in the probe laser power
requires a special study of the noise component of its power. As an example, below is a
study of a commercial He-Ne laser, which we exploited as a probe laser in our scheme.
This laser with a power Plaser = 2 mW operated at a wavelength of 630 nm. According todata specifications, the power of the variable noise component should have been
Pnoise = 10-3×Plaser in the frequency band up to 50 MHz. The gain was limited to the
50 kHz band in our scheme, i.e., Pnoise was supposed to be three orders of magnitude lessthan the specifications value.
To study the effect of noise on the sensitivity in our measurements and the prospects for
reducing this influence, we have studied the power of the laser radiation noise component.
For this purpose, laser radiation was sent directly to the photodiode unit shown in Fig. 7.8,322

Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry

bypassing all the optical scheme elements. The constant current component through thediode was changed by means of light filters.
Fig. 7.14 shows the oscillogram of the time-varying component of a signal for all
implemented values of the constant component. The signal had a clear periodic structurewith a 50 Hz network voltage, indicating the purely technical nature of the observed noise.
The RMS deviation value of voltage fluctuations was 1.5×10-4 of the constant component,
which means that the specification level corresponding to the ratio Pnoise = 10-6 ×Plaser was
exceeded by 150 times. In this regard, we have made some modernization of the laser tube
power supply. As a result, the RMS noise deviation decreased by approximately 7 times.
The corresponding oscillogram is shown in Fig. 7.15.
Further research was undertaken into the nature of these noises. Fig. 7.16 shows thephotodiode electrical circuit. The measurement showed that the quantum efficiency of our
photodiode was 70 % at the probe radiation frequency, i.e., 1.4 photons of probe radiation
were required for each electron transferring in the circuit.

Fig. 7.14. Typical oscillogram of the time-varying component of the signal from the photodiodeilluminated by a probe laser.

In the study of the noise source of this circuit, the output signal was amplified with anamplification coefficient Kamp = 104 (80 dB) in a band from 16 Hz to 50 kHz. The noise
power was measured employing a digital oscilloscope, due to which, by mathematicalprocessing, the RMS deviation of the signal noise component σ was calculated at a timeinterval of 4 ms. There were several sources of noise in the scheme. 1. Amplifier noise.
2. Output resistance thermal noise. The sum of the magnitudes of these noises does notdepend on the power of the probe laser radiation incident on the photodiode. According
323

Advances in Optics: Reviews. Book Series, Vol. 5

to our measurements, the noise corresponds to the RMS deviation σ1+2 = 34 mV. 3. Shotnoise of a laser-illuminated photodiode associated with the discrete nature of the flowing
current. The resulting fluctuations of the flowing current can be estimated exploiting asimple formula obtained by solving the Bernoulli mathematical problem [48].

Fig. 7.15. Oscillogram of the time-varying component of the signal from the photodiodeilluminated by the probe laser radiation after the power supply upgrade.

Fig. 7.16. Photodiode electrical circuit.
̅̅̅̅̅
𝛥𝐼2
𝑒∗𝛥𝜈
= 𝐼 ,
𝐼2

(7.15)

or for the magnitude of the noise voltage at the load resistance expressed through thevoltage constant component V√𝛥𝑉 2 𝑠ℎ𝑜𝑡 ~𝛽 ∗ √𝑉 ∗ 𝑅𝑙𝑜𝑎𝑑 ∗ 𝑒 ∗ 𝛥𝜈324

(7.16)

Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry(Here, 𝑒 = 1.6 × 10−19 C is the electron charge, I is the current through the diode causedby laser radiation, and 𝛥𝜈 = 5 × 104 Hz is the frequency band of the fluctuating current,
the β coefficient was introduced to obtain a correspondence between the RMS deviation
experimental values and its theoretical values obtained using the formula (7.3) below,
which describes the dependence of the measured noise RMS deviation on the probe laser
power incident on the photodiode). The RMS deviation value of the shot noise depends
on the constant component of the voltage on the load, which occurs when the diode isilluminated by laser radiation. This voltage was 140 mV when measured in a real scheme.
The formula implies the relation 𝜎3 = β · 19 𝑚𝑉. 4. The noise component of the laser
radiation power, proportional to the constant component of the voltage on the load. The
measurement result was 𝜎4 = 8 𝑚𝑉 at a voltage of 140 mV. I. e., the relation
σ4(V) = V*8/140 is valid. It should be note that even after upgrading the power supply, theratio of the laser noise power corresponding to the RMS deviation σ4 = 11 mV to the laser
power corresponding to 140 mV constant component will be equal to 6×10-6, i.e., 6 times
worse than the value according to the specification data of this commercial laser. Allowing
for the statistical independence of the noise sources, the following formula is valid for
calculating the RMS deviation of the total noise σsum
𝜎𝑠𝑢𝑚 2 = 𝛴𝜎𝑖 2 ,(7.17)

where σi is the RMS deviation of different sources of noise components. Taking the above
data into account, a formula describing the change in the total RMS deviation with anincrease in the probe laser power was obtained
𝑉 2 ∗0.0082

𝜎𝑡𝑒𝑜𝑟 = √ 0.1402 + 0.0342 + 𝑉 ∗ 4 ∗ 𝑒 ∗ 𝛥𝜈 ∗ 3300 ∗ 𝑅𝑙𝑜𝑎𝑑 2(7.18)The validity of this ratio has been verified experimentally. The voltage of the constantcomponent V was varied due to the light filters shown in Fig. 7.16. Fig. 7.17 shows theratio of the experimental values of the total RMS deviation and the values obtained fromformula (7.18). The best agreement between the data of the given formula and theexperimental data is obtained at β = 2. In this case, the standard deviation of theexperimental values from the theoretical ones is ±1.8 %, which indicates that the valuesof all noise components were correctly taken into account in formula (7.18).
As we noted above, the most obvious way to increase the signal level, as follows from theformula for P~, is to increase the probe laser power Ppr. However, as the laser powerincreases, the RMS deviation of the signal noise component increases due to the laser
noise and laser radiation-induced shot noise of the diode. Based on the above results ofthe study of the probe radiation noise component, it is possible to determine the signal-tonoise ratio increasing, which will occur with an increase in the probe laser power passing
through the photodiode diaphragm. Fig. 7.16 shows the dependence of the ratio of thevalid signal to the RMS deviation of the noise component with an increase in the constantcomponent of the signal from the photodiode associated with the power of the laser
radiation. Sto noise is:325

Advances in Optics: Reviews. Book Series, Vol. 5
𝐴∗𝑉

𝑆𝑡𝑜 𝑛𝑜𝑖𝑠𝑒 = 𝜎

𝑡𝑒𝑜𝑟

,

(7.19)

where A = 0.72 is the experimentally determined coefficient of coupling of the signal
value in measurements with maximum sensitivity with the value of the constantcomponent of the voltage at the photodiode. Fig. 7.18 shows that an increase in theconstant component of the output voltage V to 1.5 V (it corresponds to an increase in theprobe laser power from 2 mW to 20 mW) will increase the signal-to-noise ratio up to
about 7.3, i.e., about 3 times. In addition, the value of 𝑆𝑡𝑜_𝑛𝑜𝑖𝑠𝑒 after averaging overN = 104 pulses, as shown by the experiment, will increase by 56 times, i.e., it will grow to408. In turn, the implementation of the optimal probe beam geometry, which increases thesignal at a given absorption by 4.5 times, will increase the value of 𝑆𝑡𝑜_𝑛𝑜𝑖𝑠𝑒 to ~ 1840.
This signal-to-noise ratio corresponds to measuring the UQG absorption coefficient equal
to 5×10-8 cm-1. When measuring the deviations of the UQG absorption coefficient fromthe absorption level of the glass matrix ~2.6×10-7 cm-1, the 𝑆𝑡𝑜 𝑛𝑜𝑖𝑠𝑒 value will increase
five times.

Fig. 7.17. Ratio of the experimental values of the total RMS deviation σexp and the values σteor
obtained from (7.18).

Fig. 7.18. Dependence of the ratio of the valid signal to the RMS deviation of the noise
component with an increase in the constant component of the signal from the photodiode V,
associated with the laser radiation power.

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Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry

If single-mode diode lasers are exploited as heated lasers with a peak power 100 times
less, 𝑆𝑡𝑜_𝑛𝑜𝑖𝑠𝑒 will be ~90. The deviation of absorption due to the presence of copper ionsCu2+ with a concentration at the detection limit level (1.2 ppt) will be 4×10-9 cm-1, i.e.,~10-2 of the UQG matrix absorption. In this case, the RMS deviation of the noise
component voltage will be approximately equal to the deviation caused by the presenceof copper ions, which, in fact, means that the measurement will correspond to the detection
limit level.
Thus, it is possible to summarize the analysis of the possibilities for increasing thesensitivity of the TPCI scheme in the problem of measuring the concentration of chemical
inclusions in UQGs. Implementing the above-described changes in the case of theutilization of diode lasers as sources of heating radiation will make it possible to measureconcentrations of polluting chemical elements in UQGs having absorption lines over theentire generation range of diode lasers, with a detection sensitivity equal (e. g., for the
Cu2+ ion) or significantly exceeding (e. g., for the Fe3+ ion) LA-ICP-MS detection limit.
On the other hand, in the TPCI method, it is possible to exploit heating radiation sources
with a peak power of pulsed radiation of ~ 100 W and an average power of ~1 W,
operating at a given set of wavelengths, dictated by the problem of obtaining the highest
accuracy in calculating concentrations from the data on absorption measurements. This
will lead to an exhaustive solution of the problem of measuring the concentration ofchemical elements in UQG at a level of ~10 – 1 ppt.
The creation of such radiation sources is possible by efficient nonlinear optical frequency
conversions of laser radiation of CW fiber lasers operating in a repetitively pulsed modewith a peak pulse power of ~1 kW. The idea of obtaining a wide range of generated
wavelengths utilizing nonlinear optical conversions is to combine the effects of stimulated
Raman and Brillouin scattering with methods of frequency multiplication and summationin nonlinear crystals [49, 50]. A wide range of frequency shifts implemented employing
SRS in crystals and SBS in liquids enables to fill the frequency interval in the visible andIR range quite tightly. The possibility of achieving the efficiency of such processes atpulsed powers of ~600 W in nonlinear crystals was demonstrated by the example offrequency doubling of fiber laser radiation [15]. The efficiency of frequency conversion
exploiting SRS and SBS processes in resonators and optical fibers has been demonstrated
in many papers [51-57].
It is worth making a remark about the accuracy of measuring the calibration coefficients
when utilizing different diode lasers generating at different frequencies. Obviously, when
implementing high-sensitive measurements of inclusion concentrations, it is necessary toensure a sufficiently high accuracy of measuring the ratios of the calibration coefficientsfor various lasers used. When changing the diode laser, the calibration factor may vary
due to the variation in the parameter of the heating laser beam (waist length, beam waist
width, and waist position), as well as due to the heating laser power. Although thesechanges are taken into account in calculations using the above formulae for calculating
the K function, however, errors associated with inaccuracy in determining theseparameters will lead to an increase in the minimum detectable concentration of inclusions.
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Advances in Optics: Reviews. Book Series, Vol. 5

Minimum detectable concentrations for different chemical elements vary greatly due todifferent absorption coefficients. The known values of the coefficients for some chemicalelements at a concentration of 1 ppb are shown in Table 7.2. Calculation of the limiting
concentrations available for measurement requires special experiments to study the
specific values of the indicated errors and the development of special calibration
procedures to minimize them. This work is currently at the stage of calculations.

7.6. Conclusions
We have presented a brief overview of the modern problem of measuring ultra-low
concentrations of inclusions in UQG and SCQ. Further improvement of the technology
for the production of these materials requires the development of measurement methods,
since the currently existing measurement limits for the concentrations of various impurity
elements do not make it possible to determine their content in the purest samples.
As an alternative to the mass spectroscopy methods exploited today, we have proposed totake advantage of the possibilities of measuring ultra-low absorption using the new TPCI
method [34, 35] to determine the inclusion concentrations polluting ultrapure glasses andcrystals. The solution to the range of problems required to implement this proposal isclosest to completion for ultrapure quartz glasses. In crystals, this proposal should befurther analyzed in terms of obtaining calibration spectral absorption profiles of various
chemical elements in crystals of interest.
Allowing for the achieved sensitivity in measuring the absorption in UQGs, we analyzed
the nature of noise in the measuring scheme and calculated the optical scheme optimal
geometry. As a result, we have shown that the scheme sensitivity implemented so far
opens up the possibility of measuring absorption at almost any given set of wavelengths
in the optical and near-IR ranges by utilizing diode lasers as heating lasers.
The given analysis of the concentration values of inclusions of chemical elements having
absorption lines in the entire generation range of diode lasers showed that exploiting the
TPCI scheme for measuring absorption at different wavelengths makes it possible tomeasure inclusion concentrations with a recording sensitivity equal to (e. g., Cu2+ ions) or
significantly exceeding (e. g., Fe3+ ions) LA-ICP-MS detection limit.
We hope that the presented study will serve as an impetus for the further development of
this direction of measuring the concentrations of various extraneous chemical elements.
This will expand the possibilities for improving the technology for the production ofultrapure quartz glasses and crystals.

Acknowledgements
The research was supported by the Ministry of Science and Higher Educationof the Russian Federation, state assignment for the Institute of Applied
Physics RAS, project № 0035-2019-0012.
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Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry

References[1]. R. D. Snook, R. D. Lowe, Thermal lens spectrometry – A review, Analyst, Vol. 120,
Issue 8, 1995, pp. 2051-2068.
[2]. B. Sloots, Measuring the low oh content in quartz glass, Vibrational Spectroscopy, Vol. 48,
Issue 1, 2008, pp. 158-161.
[3]. M. Leidinger, S. Fieberg, N. Waasem, F. Kühnemann, K. Buse, I. Breunig, Comparative study
on three highly sensitive absorption measurement techniques characterizing lithium niobate
over its entire transparent spectral range, Optics Express, Vol. 23, Issue 17, 2015,
pp. 21690-21705.
[4]. Findlight Web Portal, https://www.findlight.net/front-media/products/datasheet/SuprasilUVL-.pdf[5]. Heraeus Web Portal, https://www.heraeus.com/en/hca/products_hca/optical_fiber_1/
product_optical_fibers_1/products_optical_fiber_hca.html[6]. Heraeus Web Portal, https://www.heraeus.com/en/hca/products_hca/optics_1/uv_2/
optical_materials_uv_applications_hca.html[7]. Leoni Web Portal, https://www.leoni-fiber-optics.com/en/products-and-services/rawmaterials/fused-silica/[8]. Industrial Laser Solutions Web Portal, https://www.industrial-lasers.com/cutting/article/16484876/printed-circuit-board-processing-with-uv-lasers[9]. X. C. Wang, H. Y. Zheng, High quality laser cutting of electronic printed circuit board
substrates, Circuit World, Vol. 35, Issue 4, 2009, pp. 46-55.
[10]. W. Lei, S. Raman, C. Chow, UV Laser Solutions for Electronic Interconnect and Packaging,in Proceedings of the 6th International Conference on Electronic Packaging Technology(ICEPT’05), Shenzhen, China, 30 August-2 September, 2005, pp. 1-7.
[11]. Y. Vladimirsky, Vacuum Ultraviolet Spectroscopy. In: II. Experimental Methods in the
Physical Sciences (J. A. Samson, D. L. Ederer, Eds.), Academic, New York, 1998.
[12]. L. E. Murr, Photolithography applied to Integrated Circuit (IC) microfabrication, in
Handbook of Materials Structures, Properties, Processing and Performance, Springer
International Publishing, 2015.
[13]. G. Stokkan, D. S. Marisa, R. Søndenå, M. Juel, A. Autruffe, K. Adamczyk, H. V. Skarstad,K. E. Ekstrøm, M. S. Wiig, Ch. Ch. You, H. Haug, M. M‘Hamdi, Impurity control in highperformance multicrystalline silicon, Physica Status Solidi (A): Applications and Materials,
Vol. 214, Issue 7, 2017, 1700319.
[14]. K. V. Vlasova, A. N. Konovalov, A. I. Makarov, N. F. Andreev, I. E. Kozhevatov, D. E. Silin,
Synthetic crystalline quartz as an optical material for power optics, Radiophysics andQuantum Electronics, Vol. 62, Issue 6, 2019, pp. 490-498.
[15]. N. F. Andreev, K. V. Vlasova, V. S. Davydov, S. M. Kulikov, A. I. Makarov, S. A. Sukharev,
G. I. Freidman, S. V. Shubin, Cascaded frequency doublers for broadband laser radiation,
Quantum Electronics, Vol. 42, Issue 10, 2012, pp. 887-898.
[16]. Heraeus
Web
Portal,https://www.heraeus.com/media/media/hca/doc_hca/
products_and_solutions_8/optical_fiber/Fiber_Tubes_EN.pdf[17]. J. S. Becker, D. Tenzler, Studies of LA-ICP-MS on quartz glasses at different wavelengths ofa Nd:YAG laser, Fresenius Journal of Analytical Chemistry, Vol. 370, Issue 5, 2001,
pp. 637-640.
[18]. R. Kvande, L. Arnberg, C. Martin, Influence of crucible and coating quality on the propertiesof multicrystalline silicon for solar cells, Journal of Crystal Growth, Vol. 311, Issue 3, 2009,
pp. 765-768.

329

Advances in Optics: Reviews. Book Series, Vol. 5[19]. E. Olsen, E. J. Ovrelid, Silicon nitride coating and crucible – Effects of using upgradedmaterials in the casting of multicrystalline silicon ingots, Progress in Photovoltaics, Vol. 16,
Issue 2, 2008, pp. 93-100.
[20]. K. Dash, S. Thangavel, S. M. Dhavile, K. Chandrasekaran, S. C. Chaurasia, Multichannel
vapor phase digestion (MCVPD) of high purity quartz powder and the determination of trace
impurities by ICP-AES and ICP-MS, Atomic Spectroscopy, Vol. 24, Issue 4, 2003,
pp. 143-148.
[21]. J. M. Senior, Optical Fiber Communications: Principles and Practice, 3 rd Ed., Pearson, 2008.
[22]. A. Müller, P. M. Ihlen, J. E. Wanvik, B. Flem, High-purity quartz mineralisation in kyanite
quartzites, Mineralium Deposita, Vol. 42, Issue 5, 2007, pp. 523-535.
[23]. G. Morteani, F. Eichinger, J. Götze, A. Tarantola, A. Müller, Evaluation of the potential of
the pegmatitic quartz veins of the Sierra de Comechigones (Argentina) as a source of highpurity quartz by a combination of LA-ICP-MS, ICP, cathodoluminescence, gas
chromatography, fluid inclusion analysis, Raman and FTIR spectroscopy, in Quartz:
Deposits, Mineralogy and Analytics (J. Götze, R. Möckel, Eds.), Springer, Berlin,
Heidelberg, 2012.
[24]. B. Flem, A. Müller, In situ analysis of trace elements in quartz using laser ablation inductively
coupled plasma mass spectrometry, in Quartz: Deposits, Mineralogy and Analytics (J. Götze,R. Möckel, Eds.), Springer, Berlin, Heidelberg, 2012.
[25]. F. Nürnberg, M. Stamminger, B. Kühn, M. Altwein, R. Takke, IR grade fused silica forhigh-power laser applications, in Proceedings of the International Conference on Optics andOptoelectronics (ICOL’14), Dehradun, India, 5-8 March, 2014.
[26]. O. Humbach, H. Fabian, U. Grzesik, U. Haken, W. Heitmann, Analysis of oh absorption
bands in synthetic silica, Journal of Non-Crystalline Solids, Vol. 203, 2007, pp. 19-26.
[27]. Heraeus
Web
Portal,https://www.heraeus.com/media/media/hca/doc_hca/
products_and_solutions_8/tubes/Quartz_Tubes_EN.pdf[28]. Ohara Web Portal, https://oharacorp.com/pdf/SK1310.pdf[29]. P. C. Schultz, Optical absorption of the transition elements in vitreous silica, Journal of theAmerican Ceramic Society, Vol. 57, Issue 7, 1974, pp. 309-313.
[30]. E. N. Boulos, L. B. Glebov, T. V. Smirnova, Absorption of iron and water in the Na2O-CaOMgO-SiO2 glasses. I. Separation of ferrous and hydroxyl spectra in the near IR region,
Journal of Non-Crystalline Solids, Vol. 221, Issues 2-3, 1997, pp. 213-221.
[31]. L. Glebov, E. Boulos, Absorption of iron and water in the Na2O-CaO-MgO-SiO2 glasses.
II. Selection of intrinsic, ferric, and ferrous spectra in the visible and UV regions, Journal of
Non-Crystalline Solids, Vol. 242, Issues 1, 1998, pp. 49-62.
[32]. L. Glebov, E. Boulos, Absorption Spectra of Iron and Water in Silicate Glasses, inProceedings of the International Commission on Glass Conference (ICG’2000), The
Netherlands, 2000, S4-2.
[33]. K. V. Vlasova, A. I. Makarov, N. F. Andreev, Short-pulse photothermal common-pathinterferometry for absorption measurement in transparent isotropic dielectrics, Prikladnaya
Fizika, Vol. 2, 2017, pp. 79-86 (in Russian).
[34]. K. V. Vlasova, A. I. Makarov, N. F. Andreev, A. Yu. Konstantinov, High-sensitive absorption
measurement in transparent isotropic dielectrics with time-resolved photothermal commonpath interferometry, Applied Optics, Vol. 57, Issue 22, 2018, pp. 6318-6328.
[35]. K. V. Vlasova, A. I. Makarov, N. F. Andreev, High-sensitive absorption measurement inultrapure quartz glasses and crystals using time-resolved photothermal common-passinterferometry and its possible prospects, Journal of Applied Physics, Vol. 129, Issue 4, 2021,
043101.
[36]. A. Alexandrovski, M. Fejer, R. Route, Optical Absorption Measurements in Sapphire,
Stanford Photo-Thermal Solutions (SPTS), https://www.stan-pts.com/G000242-00.pdf

330Chapter 7. Ultra-low Light Absorption Measurementin the Problem of Determining Chemical Impurities Concentrations in Quartz Glasses and Synthetic
Crystalline Quartz Using Time-resolved Photothermal Common-path Interferometry[37]. K. V. Vlasova, A. I. Makarov, N. F. Andreev, A. Yu. Konstantinov, Theory of short-pulse
photothermal common-path interferometry in isotropic dielectrics, Uspekhi Prikladnoi Fiziki,
Vol. 5, Issue 4, 2017, pp. 313-328 (in Russian).
[38]. Yu. A. Ananiev, N. I. Grishmanova, Deformation of active elements and thermo-opticalconstants of Nd-glass, Journal of Applied Spectroscopy, Vol. 12, 1970, pp. 668-691.
[39]. A. Alexandrovski, A. S. Markosyan, H. Cai, M. M. Fejer, Photothermal measurements ofabsorption in LBO with a "proxy pump" calibration technique, Proceedings of SPIE,
Vol. 10447, 2017, 104471D.
[40]. LLC Quartz Technology Web Portal, http://quartztech.ru/eng.html[41]. Dr. Lindsay Wilson’s personal website, http://www.imajeenyus.com/electronics/
20120908_improving_adc_resolution/index.shtml[42]. Microchip Web Portal, https://www.microchip.com/wwwAppNotes/AppNotes.aspx?appnote= en591540[43]. J. Shen, R. D. Lowe, R. D. Snook, A model for CW laser induced mode-mismatcheddual-beam thermal lens spectrometry, Chemical Physics, Vol. 165, Issue 2, 1992,
pp. 385-396.
[44]. K. Vlasova, A. Makarov, N. Andreev, A. Konovalov, Minimum absorption coefficientavailable for measurements using time-resolved photothermal common-path interferometryon the example of synthetic crystalline quartz, Sensors & Transducers, Vol. 233, Issue 5,
May 2019, pp. 6-14.
[45]. UdSSR-DDR, Optisches Glas, VEB Jenaer Glaswerk Schott&Gen, 1978 (in German).
[46]. Laser Diode Source Web Portal, https://www.laserdiodesource.com/laser-diode-wavelength/
635nm-780nm-mw/600nm-795nm[47]. T. Berthoud, N. Delorme, P. Mauchien, Beam geometry optimization in dual-beam thermal
lensing spectrometry, Analytical Chemistry, Vol. 57, Issue 7, 1985, pp. 1216-1219.
[48]. S. M. Rytov, Introduction to Statistical Radiophysics. Part 1, Nauka, 1976 (in Russian).
[49]. G. I. Freidman, G. A. Pasmanik, A. A. Shilov, D. G. Peterson, Some new possibilities todesign frequency-tunable lidars in the IR (2-14 μm), Proceedings of SPIE, Vol. 2366, 1995,
pp. 403-407.
[50]. G. A. Pasmanik, E. J. Shklovsky, G. I. Freidman, V. V. Lozhkarev, A. Z. Matveyev,
A. A. Shilov, I. V. Yakovlev, D. G. Peterson, J. K. Partin, Development of eye-safe IR lidar
emitter and detector technologies, Proceedings of SPIE, Vol. 3065, 1997, pp. 286-293.
[51]. R. J. Williams, J. Nold, M. Strecker, O. Kitzler, A. McKay, T. Schreiber, R. P. Mildren,
Efficient Raman frequency conversion of high-power fiber lasers in diamond, Laser &Photonics Reviews, Vol. 9, Issue 4, 2015, pp. 405-411.
[52]. T. T. Basiev, V. V. Osiko, A. M. Prokhorov, E. M. Dianov, Crystalline and fiber Raman
lasers, in Solid-State Mid-Infrared Laser Sources. Topics in Applied Physics(I. T. Sorokina, K. L. Vodopyanov, Eds.), Vol. 89, Springer, Berlin, Heidelberg, 2003.
[53]. E. M. Dianov, M. V. Grekov, I. A. Bufetov, S. A. Vasiliev, O. I. Medvedkov,
V. G. Plotnichenko, V. V. Koltashev, A. V. Belov, M. M. Bubnov, S. L. Semjonov,
A. M. Prokhorov, CW high power 1.24 µm and 1.48 µm Raman lasers based on low loss
phosphosilicate fiber, Electronics Letters, Vol. 33, Issue 18, 1997, pp. 1542-1544.
[54]. A. A. Kaminskii, C. L. McCray, H. R. Lee, S. W. Lee, D. A. Temple, T. H. Chyba,
W. D. Marsh, J. C. Barnes, A. N. Annanenkov, V. D. Legun, H. J. Eichler, G. M. A. Gad, K.
Ueda, High efficiency nanosecond Raman lasers based on tetragonal PbWO4 crystals, Optics
Communications, Vol. 183, Issues 1-4, 2000, pp. 277-287.
[55]. H. Huang, H. Wang, S. Wang, D. Shen, Designable cascaded nonlinear optical frequency
conversion integrating multiple nonlinear interactions in two KTiOAsO4 crystals, Optics
Express, Vol. 26, Issue 2, 2018, pp. 642-650.

331

Advances in Optics: Reviews. Book Series, Vol. 5[56]. M. Heinzig, G. Palma-Vega, T. Walbaum, T. Schreiber, R. Eberhardt, A. Tünnermann,
Diamond Raman oscillator operating at 1178 nm, Optics Letters, Vol. 45, Issue 10, 2020,
pp. 2898-2901.
[57]. M. Heinzig, G. Palma-Vega, B. Yildiz, T. Walbaum, T. Schreiber, A. Tünnermann,
Continuous-wave cascaded second Stokes diamond Raman laser at 1477 nm, Optics Letters,Vol. 46, Issue 5, 2021, pp. 1133-1136.

332

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)

Chapter 8

Light-sheet Scattering by a Dielectric Sphereof Arbitrary Size (GLMT for Light-sheet)
Renxian Li, Bing Wei, Shuhong Gong, Qun Wei, Ningning Song,
Shu Zhang, Bojian Wei, Han Sun, Huan Tang and Liu Yang1

8.1. Introduction
The light-sheet has been used in many fields because of its excellent characteristics suchas only imaging the sample on the focal plane of the objective lens and slight phototoxicity[1], especially super-resolution imaging and optical manipulation. A light-sheet can beapplied to imaging microscope [2], flow visualization [3], particle sizing [4] and opticalclearing [5], etc. Since the light-sheet has many applications in the field of high-resolution
imaging and optical manipulation, it is necessary for us to study the scattering and themechanical effect theory of the light-sheet by the particles. Based on actual needs, thischapter will derive the basic theory of scattering and mechanical effects of sphericalparticles illuminated by the light-sheet. Airy light-sheet [6-8], Bessel pincer light-sheet[9, 10] and Gaussian light-sheet [11, 12] are three typical laser light-sheets. A light-sheet
is equivalent to a two-dimensional state of the corresponding beam in space, at the sametime retains part of the beam’s properties, for example, the non-diffraction [13, 14] andself-healing [15, 16] characteristics of the Airy beam and Bessel beam. We first derive theangular spectrum of Airy light-sheet, Bessel pincer light-sheet, and Gaussian light-sheet,and then generalize them to arbitrary light-sheet. After that, based on the advantages ofthe light-sheet mentioned above, this chapter will use Generalized Lorenz-Mie theory(GLMT) to study the interaction between three kinds of light-sheets and a sphericalparticle of arbitrary size, including scattering characteristics (scattering intensity,
scattering, absorption, and extinction efficiencies) and mechanical effect(optical force andtorque). We will focus on negative force and torque.
Whether it is super-resolution imaging or optical manipulation of the light-sheet, the firstproblem to be solved is to study the scattering of the light-sheet by the particles. Therefore,
it is necessary to develop the theory of the scattering of light by particles. In 2017, Mitri

Renxian Li
School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, Collaborative
Innovation Center of Information Sensing and Understanding, Xidian University, Xi’an 710071, China

333

Advances in Optics: Reviews. Book Series, Vol. 5[17] introduced the nonparaxial Bessel pincer light-sheet and calculated the radiation
component of the electric field of this light-sheet. In the same year, Mitri [18]demonstrated an auto-focusing Bessel pincer light-sheet in his article, and analyzed theradiation components and propagation characteristics of the light-sheet. Cao et al. [19]used Mie theory and plane-wave spectrum to study the effect of the sphere radius, beamwidth, and other parameters on the angular distribution of the scattering Airy light-sheet.
Pan et al. [20] used the above method to numerically study the scattering of an Airy lightsheet illuminating a chiral sphere. Zhai et al. [11] used numerical methods to study thelight scattering characteristics of a sphere in a Gaussian light-sheet. The results show thatthe scattered light contains the structure and optical information of the sphere. GLMT isa rigorous theory that can calculate the scattering of arbitrary shape beams byhomogeneous spheres of any size [9]. Later, GLMT has been also used to calculate thescattering problem of various types of scatters [21]. In the framework of GLMT, vector
spherical wave functions (VSWFs), beam shape coefficients (BSCs), and angular
spectrum decomposition method (ASDM) are crucial. Because arbitrary TE and TM
polarized light-sheet can be equivalent to the superposition of countless different planewaves in the framework of the ASDM method, and then derived from the combination of
VSWFs and BSCs. Finally, we also derive the expressions of the scattering intensity,
scattering characteristics (such as extinction Qext , absorption Qabs and scattering Qsca
efficiencies, etc.), force and torque of any light-sheet illuminating a spherical particle ofarbitrary size. The energy efficiency reflects the scatter’s ability to transform the incident
energy to scattering, extinction, or absorption energy.
The manipulation of tiny particles by light can be used to detect, manipulate and locate
biological systems. Because of its minimal damage to control particles, it can be used in
the fields of cell biology [22], single molecule [23] and physics [24]. In 2016, Mitri [25]studied the mechanical effects of a non-paraxial Airy “acoustical-sheet” on a 2D circular
structure. Numerical results show that the “acoustical-sheets" can move a particle in all
directions. In 2017, Mitri [26] used the dipole approximation method to study the force
and torque generated by the Bessel pincer light-sheet illuminating the subwavelength
absorption sphere coated by a plasma layer under vacuum conditions. Numerical results
show that during plasmon resonance, coating the optimal thickness of the plasma layer on
the surface of the small sphere can enhance some components of the force and torque ofBessel pincer. The theoretical model of electromagnetic radiation force and torque on an
ideal conducting cylinder is developed by Mitri [27], and the numerical calculation of theforce and torque is carried out. Numerical calculation results show that the longitudinal
and transverse components of the radiation force vector are positive. Song et al. [28]calculated the time-averaged optical spin torque (OST) applied by an Airy light-sheet on
a sphere of arbitrary size. The mechanical effect of light is closely related to thecharacteristics of the incident light-sheet. The direction of force and torque can be changed
by changing the parameters and the polarization of the beam [29]. In 2021, Mitri [30]
introduced the circularly-polarized Airy light-sheet spinner tweezers, and considered theirinteraction with a sub-wavelength sphere. This kind of light-sheet generates longitudinal
and transverse spin torques at the same time, which makes the particle rotate
counterclockwise or clockwise due to different positions. The optical torque is caused by
the transfer of the angular momentum of the incident light-sheet to the particles [31]. In
334

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)the following sections, we mainly study the optical spin torque (OST) exerted by alight-sheet on a spherical particle, which can cause the particle to rotate around its centerof mass [32].
In this chapter, we have calculated the effects of parameters of TE and TM polarized
light-sheet (the transverse scale ys , the modulation parameter  , the scaling parameter0 , beam order l , and waist radius k0 ) and particle parameters (the size parameter kaof the dielectric sphere) on the scattering field, efficiencies, force, and torque,
respectively. The research content of other sections are as follows. In Section 8.2, we will
introduce the light-sheet theoretically. The first is the expansion of the angular spectrumof the TE and TM polarized light-sheets (Airy, Bessel, and Gaussian light-sheet). Then,the internal and scattered fields are also expressed in terms of BSCs and VSWFs using
the multipole expansion. The general BSCs of laser light-sheet is derived from the
expanded term. At the end of this section, the BSCs of Airy, Bessel, and Gaussian
light-sheets are derived. In Section 8.3, we derive the expressions of the scattering(far-field scattering intensity, Qext , Qabs , and Qsca ) and mechanical properties (opticalforce and torque) of the laser light-sheet illuminating a dielectric sphere of arbitrary size.
In Section 8.4, we numerically simulate the scattering and mechanical properties of Airy,
Bessel, and Gaussian light-sheets. In Section 8.5, the theoretical derivation and numericalsimulation are summarized.

8.2. Description of Light-sheet
8.2.1. Angular Spectrum
The focal point of a polarized laser-sheet is ( y0 , z0 ), , and the direction of its wave vector
is in the same direction as the z  axis in the half free space z > z0 of yz plane. The TE
and TM-polarized laser-sheets are discussed in this paper, and the incident electric fieldTE= H xTE = 0),
of TE-polarized laser sheet vibrates along the x  direction ( E TE
y = Ez
however the incident magnetic field of TM-polarized laser sheet vibrates along the= H zTM = ExTM = 0). By ignoring the contribution of the evanescentx  direction ( H TMywaves [33, 34], the unique non-zero component ExTE of the TE-polarized electric field canbe expressed by the angular spectrum of the plane wave,


ExTE ( y, z ) = k  2 AxTE (q)e

ik [ qy  p ( z  z0 )]

dq,

(8.1)

2

1

where k = (  ) 2  / c is the wavenumber of surrounding medium, q = sin  sin  ,p = cos (Fig. 8.1), AxTE (q) is the angular spectrum, and can be given by the Fouriertransform of the initial electric field in the z = z0 plane
335

Advances in Optics: Reviews. Book Series, Vol. 5

AxTE (q) =1 2
 ikqydy
 E x ( y , z 0 )e2  2

(8.2)

Fig. 8.1. Definition of k , r , and the corresponding angles.

The electromagnetic field’s rectangular component can be expressed as [35-38, 33]


ExTE ( y, z ) = k  2 AxTE (q)e

ik [ qy  p ( z  z0 )]

dq,

(8.3)

2

E TE
y ( y , z ) = 0,

(8.4)

EzTE ( y, z) = 0,

(8.5)

H xTE ( y, z ) = 0,

(8.6)

H TE
y ( y, z ) =k 2ik [ qy  p ( z  z0 )]TEdq, mAx ( q )e


Z0 2

(8.7)

k 2 TEik [ qy  p ( z  z0 )]dq,
 qAx ( q )e
Z 0  2(8.8)

H zTE ( y, z ) = 

1with the eit time dependence suppressed, Z 0 = (  /  ) 2 = E0 / H 0 .
In general, the TE-polarized electromagnetic fields can be expressed in vector form as
TE
ETE ( y, z ) = ExTE e x  E TEy e y  Ez e z= k  2 ATE (q)e


336

2

i[ qy  p ( z  z0 )]

dq,

(8.9)

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)with
ATE (q) = AxTE (q)e x  AyTE (q)e y  AzTE e z =1 2 TE ikqydy
 E ( y , z 0 )e2  2

1 2 TETETE
 ikqy=dy
 [ E x ( y , z0 )e x  E y ( y , z0 )e y  E z ( y , z0 )e z ]e2  2(8.10)

Similarly, the magnetic fields can also be expressed as
TE
HTE ( y, z ) = H xTE ( y, z )e x  H TEy ( y , z )e y  H z ( y , z ) e z(8.11)

When it is TM-polarized light, H xTM of the magnetic field vibrating in the x direction
can be expressed as [33, 34],
H xTM ( y, z ) =1
ik [ qy  m ( z  z0 )]k  2 AxTM (q)e
dq,

Z0
2

(8.12)

where AxTM (q) is the angular spectrum of H xTM , and can be obtained from
AxTM (q) =1 2 TM ikqydy
 H x ( y , z 0 )e2  2(8.13)

Similarly, its rectangular component can be expressed as [35-38, 33]
H xTM ( y, z ) =1
ik [ qy  m ( z  z0 )]k  2 AxTM (q)edq,
Z0  2(8.14)H TMy ( y , z ) = 0,(8.15)

H zTM ( y, z) = 0,(8.16)

ExTM ( y, z ) = 0,(8.17)
ETMy( y, z ) = k  2 mAxTM (q)e




dq,

(8.18)

2

EzTM ( y, z ) = k  2 qAxTM (q)e

ik [ qy  m ( z  z0 )]ik [ qy  m ( z  z0 )]dq(8.19)

2

In general, the TM-polarized electromagnetic fields can be expressed in vector form as
337

Advances in Optics: Reviews. Book Series, Vol. 5


TM
TM
2
ETM ( y, z ) = ExTM e x  E TM
( q )ey e y  Ez e z = k   A= k  2 (me y  qe z ) A eTMxik [ qy  m ( z  z0 )]dq

2

ik [ qy  m ( z  z0 )](8.20)

dq,

2

with
ATM (q) = (me y  qe z ) AxTM = (me y  qe z )1 2 TM ikqydy
 H x ( y , z 0 )e


2 2

(8.21)

Similarly, the magnetic fields can also be expressed as

H( y, z) = H xTM ( y, z)e x(8.22)

8.2.1.1. Airy Light-sheet
It is assumed that the polarized type is TE, the initial transverse electric field of the Airylaser-sheet in the source plane z = z0 , shown in Fig. 8.2, is described as [39-41]

Fig. 8.2. The graphical representation of the interaction of an Airy light-sheet propagating along
the z -axis with a sphere.

 y  y0   ( y  y0 ) 
ETE ( y, z0 ) = e x ExTE ( y, z0 ) = e x E0 Ai  exp ,ys
 ys 


(8.23)

where E0 is the electric field amplitude, Airy function is denoted by Ai(), modulation
parameter is  , and transverse scale is the ys .
Substituting Eq. (8.23) into Eq. (8.10),
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Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)

ATE (q) = e x AxTE (q) = e x y  y0   ( y  y0 )  ikqy
1
E0  2 Ai 
exp  e dy2ys
 ys 

2(8.24)

Considering the relation [42] it 3 F (t ) =  2 Ai( x) exp( ixt )dx = exp  

 3 
2

(8.25)

Eq. (8.24) can be analytical solved, then the vector angular spectrum of Airylight-sheet is
ATE (q) = e x AxTE (q) = e x y  y0   ( y  y0 )   ikqy
1
E0  2 Ai 
exp  e dy2ys
 ys 2

 (a  ikys q )3   ikqy0y= e x s E0 exp 
e
2
3



(8.26)

When the polarization is TM, similarly, the vector angular spectrum of Airy light-sheet is

ATM (q) = e x

 (  ikys q)3  ik y y0ys
H 0 exp 
e
2
3



(8.27)

8.2.1.2. Bessel Pincer-sheet
Considering a Bessel pincer light-sheet with the parameters scaling parameter 0 , beam
integer order n. . The initial transverse electric field (TE polarization) in the source planez = z0 , shown in Fig. 8.3, isETE  y, z0  = e x ExTE ( y, z0 ) = e x E0 jn 0 k ( y  y0 )  ,(8.28)

where spherical Bessel function of first kind is denoted by jn   . The angular spectrum
calculated by Eq. (8.28) is
ATE (q) = e x AxTE (q)  e x
= ex





n

  ikqy
Re  q  i  02  q 2  e 0 ,220  qE0
k 0n
1
E0  2 jn  0 k ( y  y0 )  e ikqy dy2
2(8.29)

where Re denotes the real part of the complex. Similarly, the vector angular spectrum of
TM-polarized Bessel pincer light-sheet is339

Advances in Optics: Reviews. Book Series, Vol. 5

ATM (q) = e xk 0nn ikqy  q  i  02  q 2  e 0220  q 

H0(8.30)

Fig. 8.3. The graphical representation of the interaction of a Bessel pincer-sheet propagating
along the z -axis with a sphere.

8.2.1.3. Gaussian Light-sheet
We first discuss the case of traverse electric (TE) waves of Gaussian light-sheet. In theinitial plane z = z0 , the electric field shown in Fig. 8.4, can be expressed as  y  y0 2 E ( y, z0 ) = e x E ( y, z0 ) = e x E0 exp  02 
TETE
x(8.31)

Fig. 8.4. The graphical representation of the interaction of a Gaussian light-sheet propagating
along the z-axis with a sphere.

After the Fourier transform of Eq. (8.31), the angular spectrum of is340

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)  y  y0 2   ikqy
1
2
A  q  = e x A (q )  e x
E0  exp   e dy2  2
02 

TE

TE
x

 2
  ikqy= e x E0 0 exp   0 k 2 q 2  e 0
2 
 4


(8.32)In the case of traverse magnetic (TM) waves of Gaussian light-sheet, the angular
spectrum is

ATM  q  = e x H 0 2
 ikqy0
exp   0 k 2 q 2  e 0
2 
 4


(8.33)

8.2.2. The BSCs (Beam Shape Coefficients) of Arbitrary Laser Light-sheet
8.2.2.1. Expansion of the Incident and Scattering Fields
According to the generalized Lorenz-Mie theory (GLMT), incident field can be expanded
by VSWFs as [43]


n

pol
(1)

Ei (r , , ) =   iEmn  pmpol, n N (1)mn ( kr )  qm , n M mn ( kr )  ,(8.34)

n =1 m =  n

H i (r, , ) = 



k

n

  E q N (kr)  p M (kr)  ,n =1 m =  n

mn

pol
m,n

(1)
mn

pol
m ,n

(1)
mn

(8.35)

where pmpol, n and qmpol, n are the Beam shape Coefficients (BSCs), pol denotes different
polarized case, i.e., TE or TM .
1/2

 2n  1 (n  m)!Emn = i E0  , n(n  1) (n  m)!n

(8.36)

the VSWFs are defined as
L(mnj ) (k , r ) =1
 (mnj ) (k , r )kz ( j ) (kr )=  mn (cos )e  i mn (cos  )e  nexp(im ) kr
1 d ( j) zn (kr )  exp(im ),er Pnm (cos )k dr 

(8.37)

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Advances in Optics: Reviews. Book Series, Vol. 5

M (mnj ) (kr ) =   r (mnj ) (k , r ) = i mn (cos )e   mn (cos )e  zn( j ) ( kr )exp(im ),N (mnj ) (kr ) =(8.38)1
  M (mnj ) (kr )k=  mn (cos )e  i mn (cos  )e 

1 d rzn( j ) ( kr )  exp(im ) kr dr (8.39)z ( j ) (kr )er n(n  1) Pnm (cos  ) n
exp(im ),krwhere (mnj ) (k , r ), scalar wave function, is the solution of Helmholtz equation, thezn( j ) (kr ) is the spherical Bessel function, Pnm (cos ) denotes the associated Legendre
function, defined as
Pnm ( x) =

m
nm
n
1
2 m/ 2 d2
2 m / 2 d Pn ( x )


1

x
x

1
=
1

x
  dxnm      dxm ,2n n!

(8.40)

anddPnm (cos )m m
 mn  cos  =Pn (cos ),  mn  cos  =
,
sin 
d

(8.41)

similar as Eqs. (8.34) and (8.35), the internal and scattered field can be expanded as


n

pol
(1)

E1 (r , , ) =   iEmn  d mpol, n N (1)mn ( kr )  cm , n M mn ( kr )  ,(8.42)

n =1 m =  n

H1 (r , , ) = 



k1

n

  E c N (kr)  d M (kr)  ,(1)
mn

(8.43)pol(3)
Es (r , , ) =   iEmn  ampol,n N (3)mn ( kr )  bm , n M mn ( kr )  ,(8.44)

1 n =1 m =  n



mn

pol
m,n

(1)
mn

pol
m ,n

n

n =1 m =  n

H s ( r , ,  ) =

k



n

  E b N (kr)  a M (kr)  ,n =1 m =  n

mn

pol
m,n

(3)
mn

pol
m,n

(3)
mn

(8.45)

where ampol, n and bmpol,n are scattering field coefficients, cmpol,n and d mpol, n are internal field
coefficients.

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Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)

8.2.2.2. The Derivation of General BSCs of Laser Light-sheet
According to the theory of electromagnetic field, the electric and magnetic fields satisfy
the boundary condition of continuous tangential component on the surface of sphericalparticlesEi + Es   er = E1  er , Hi + Hs   er = H1  er

(8.46)The scattering field (ampol,n , bmpol,n ) and internal jjfield ( cmpol,n , dmpol,n ) coefficients can be obtainedby substituting the incident field, internal field and scattered field into Eq. (8.46)

ampol,n = an pmpol,n , bmpol,n = bn qmpol,n,cmpol,n = cn qmpol,n , d mpol,n = d n pmpol,n(8.47)

where an , bn , cn , dn is the traditional Mie scattering coefficients [44]ms n (ms x) n' ( x)   n ( x) n' (ms x)
an =ms n (ms x) n' ( x)   n ( x) n' (ms x) (m x) n' ( x)  ms n ( x) n' (ms x)
bn = n s
 n (ms x) n' ( x)  ms n ( x) n' (ms x)

,

'
'
m

(
x
)

(
x
)

m

(
x
)

(
x
)
c =
s n
n
s n
n
 n  (m x) ' ( x)  m  ( x) ' ( m x)
n
s
n
s n
n
s


ms n ( x) n' ( x)  ms n ( x) n' ( x)
d n = m  (m x) ' ( x)   ( x) ' (m x)
s n
s
n
n
n
s


(8.48)

where x = ka, ms = ks / k ,  n ( x) = xjn ( x), n ( x) = xhn(1) ( x), k s is the wavenumber of thespherical particle, jn ( x) is the spherical Bessel function of the first kind. According to[45], BSCs can be obtained from:
pmpol,n =2
i1 n 1/ 2 kr
 mn
er  E(r , , ) Pnm (cos )eim sin  d d ,



=0

=0
4 E0
jn (kr )(8.49)

qmpol,n = 2
i  n Z 1/ 2 kr
 mn
er  H(r , , ) Pnm (cos )eim sin  d d ,



=0

=0
4 E0
jn (kr )(8.50)

where
 2n  1 (n  m)!  n(n  1) (n  m)! mn = (8.51)

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Advances in Optics: Reviews. Book Series, Vol. 5

An arbitrary beam propagating along z -axis with a focus point can be expressed using
the vector angular spectrum of the plane waves

Ε( x, y, z ) = 





 x
 A ( k x , k y , z0 ) e 

 

=k 
2

2



i  k x  k y y  k z z  z0 




 /2

 A( ,  )e

ikr sin  sin  cos     ikr cos  cose

 =0 0dk x dk ye

 ik cos  z0

sin  cos  d d (8.52)

Corresponding magnetic field are

H ( x, y , z ) =1i0 

  E( x, y, z )(8.53)

Substituting Eq. (8.52) into Eq. (8.49), considering the relations

er = sin  cose x  sin  sin e y  cos e z ,(8.54)A( ,  ) = Ax ( ,  )e x  Ay ( ,  )ey  Az ( ,  )e z ,(8.55)

then pmpol, n is

pmpol, ne r  A( ,  ) k 2 Pnm (cos )e  im
2 2  / 2  ikr sin  sin  cos  

ik
cos

z

 ikr cos  cos0
= C0     eee d d  d  d 
 =0  =0  =0 0
sin  cos  sin = C0 



2

2

 /2

   d d  d  d   =0  =0  =0 0 sin  cos  Ax ( ,  )  sin  sin  Ay ( ,  )  cos  Az ( ,  )  
ikr sin  sin  cos     ikr cos  cos   ik cos  z0
2
m

im

,
 k Pn (cos )e eee sin  cos  sin 



where
C0 =

i1 n 1/ 2 kr
 mn
4 E0
jn (kr )(8.56)

Considering the following mathematical relations,iJ m1 ( x)ei  iJ m1 ( x)e i cos  2

ixcos (   )  im e sin   d =  i m eim  J m 1 ( x)ei  J m 1 ( x)e i 0 e 2 J m ( x)

1



344

(8.57)

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)Thenpmpol, n=  i m C0 k 2 



2

 /2

  d d  d  e

 ik cos  z0

 =0  =0  =0

sin  cos  e  im Pnm (cos )eikr cos cos sin  sin  Ax ( ,  ) iJ m 1 (kr sin  sin  )ei  iJ m 1 (kr sin  sin  )e  i  

 

i


i

   sin  Ay ( ,  )  J m 1 (kr sin  sin  )e  J m 1 ( kr sin  sin  )e    cos Az ( ,  )  2 J m (kr sin  sin  ) Note that the angular spectrum Az can be expressed by Ax and AykykAz ( ,  ) =   x Ax ( ,  )  Ay ( ,  ) 
kz
 kz

sin

cos

sin

sin 


 
Ax ( ,  ) 
Ay ( ,  ) 
cos

cos




Then

pmpol,n=  i m C0 k 2 



2

 /2

=0

=0

=0

d d  d  e  

 ik cos  z0

sin  e  im 

iJ m 1 (kr sin  sin  )ei cos  sin  Ax ( ,  ) i iJ m 1 (kr sin  sin  )e cos  sin  Ax ( ,  )   J (kr sin  sin  )ei cos  sin  Ay ( ,  ) Pnm (cos )eikr cos cos sin   m 1 i  J m 1 (kr sin  sin  )e cos  sin  Ay ( ,  )  2sin  cos  cos J ( kr sin  sin  ) A ( ,  ) 
m
x



2sin

sin

cos

J
(
kr
sin

sin

)
A

m
y ( ,  ) (8.58)

Considering the relation [46],J m 1 (kr sin  sin  ) =m
J m (kr sin  sin  )  J m' (kr sin  sin  ),kr sin  sin 

J m 1 (kr sin  sin  ) =m
J m (kr sin  sin  )  J m' (kr sin  sin  ),kr sin  sin (8.59)

dJ m (kr sin  sin  )dJ m ( kr sin  sin  )1=d (kr sin  sin  )kr cos  sin d(8.60)

andJ m' (kr sin  sin  ) =The Eq. (8.59) becomes

345

Advances in Optics: Reviews. Book Series, Vol. 5m
J m ( kr sin  sin  ) kr sin  sin 
dJ m ( kr sin  sin  )1
,
kr cos  sin dJ m 1 ( kr sin  sin  ) =J m 1 ( kr sin  sin  ) =m

J m ( kr sin  sin  ) kr sin  sin 
dJ m ( kr sin  sin  )1

kr cos  sin d(8.61)

Substituting Eq. (8.61) into Eq. (8.58), after a series of derivation,
pmpol, n=  i m C0 k 2 



2

 /2

=0

=0

=0

d d  d  e  

 ik cos  z0e  im sin 1kriei cos  Ax ( ,  )mJ m ( kr sin  sin  ) Pnm (cos  )eikr cos cos
 iei sin  A ( ,  ) dJ m (kr sin  sin  ) P m (cos  )eikr cos  cos

x
n


d
 ie  i cos  A ( ,  ) mJ ( kr sin  sin  ) P m (cos  )eikr cos  cos

x
m
n


  i
dJ m (kr sin  sin  ) m
ikr cos  cos Pn (cos  )e ie sin  Ax ( ,  )

d
 i

m
ikr cos  cos  e cos  Ay ( ,  )mJ m ( kr sin  sin  ) Pn (cos  )edJ (kr sin  sin  ) m ei sin  Ay ( ,  ) mPn (cos  )eikr cos  cosd e  i cos  A ( ,  ) mJ (kr sin  sin  ) P m (cos  )eikr cos  cos

y
m
n


dJ m (kr sin  sin  ) m  i

ikr cos  cos Pn (cos  )e e sin  Ay ( ,  )dmikr cos  cos   2sin  sin  cos  cos  krAx ( ,  ) J m ( kr sin  sin  ) Pn (cos  )e 2sin  sin  sin  cos  krA ( ,  ) J ( kr sin  sin  ) P m (cos  )eikr cos  cos 
y
m
n



(8.62)
Defining

C = J m (kr sin  sin  ) Pnm (cos )eikr cos cos

(8.63)Sod  J m (kr sin  sin  ) ddC  ikr cos cose
/ Pnm (cos )  ikr sin  cos  J m (kr sin  sin  )d(8.64)

Substituting Eq. (8.64) into Eq. (8.62), after a series of derivation,346

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)

pmpol,n=  i m C0 k 2 

2

 /2

=0

=0

d d  e  ik cos  z0e im 2

1

kr

dC
 

  =0 sin  d d i sin   Ax ( ,  )cos   Ay ( ,  )sin   







sin

C
d

m
cos

A
(

,

)cos


A
(

,

)sin


x
 y

  =0(8.65)

Since




=0

sin 

dC
dPnm (cos  )
d 
nm
d =
sin

C
d

=
2
i
j
(
kr
)
,

n
d
d  =0d sin  C d = 2i

nm

=0

jn (kr ) Pnm (cos  )(8.66)The BSCs pmpol, n becomes
pmpol,n= 4 i n C0 k 2 

2

 /2

 d d  e =0  =0

 ik cos  z0e im

jn (kr )kri mn  cos    Ax ( ,  )cos   Ay ( ,  )sin   sin    mn  cos   cos   Ay ( ,  )cos   Ax ( ,  )sin   

(8.67)

After substituting Eq. (8.56) into Eq. (8.67), the final expression of pmn is

pmpol,n =k 2 1/2 2  /2
 ik cos  z0  im mn   d d  eesin  

=0

=0
E0

  mn  cos    Ax cos   Ay sin    i mn  cos   cos   Ay cos   Ax sin   

(8.68)The qmpol, n can be directly obtained from Eq. (8.68) by interchanging the functions

 mn  cos  and  mn  cos  qmpol,n =k 2 1/2 2  /2
 ik cos  z0  im mn   d d  eesin  

=0

=0
E0

  mn  cos    Ax cos   Ay sin    i mn  cos   cos   Ay cos   Ax sin   

(8.69)

Eqs. (8.68) and (8.69) are general expressions for any vector beam, and can be applied toany beam by changing the angular spectrum Ax and Ay .

347

Advances in Optics: Reviews. Book Series, Vol. 5

Assuming the laser light-sheet is TE-polarization, there are Ax  0 and Ay = 0. And
considering the difference between structured beam and light-sheet, removing k andsin 
pmTE, n =  cos   cos 
k 1/ 2 2  / 2
 ik cos  z0  im mn   d d  eeAx    mn

 =0  =0E0 i mn  cos   cos  sin    cos   2 e im cos  d  =0
k 1/ 2
 ik cos  z0
 mn
= mn  d eAx   

2
 =0
 im
E0
sin  d   i mn  cos   cos   =0e(8.70)

 /2

Considering  = 2, the expressions of the BSCs pmTE, n is

  ip 2 2 ikpze mn (cos  ) AxE ( )e 0 cos  d 0k 1/ 2 pmTE, n = i  mn
 ip  0E0e 2  mn (cos  ) AxE ( )e  ikpz0 cos  d 
 2





(8.71)

Similarly, the BSCs qmTE, n and pmTM, n , qmTM, n are given, respectively, as  ip 2 2 ikpze mn (cos  ) AxE ( )e 0 cos  d 0k 1/ 2 qmTE,n = i  mn
 ip  0,E0e 2  mn (cos  ) AxE ( )e  ikpz0 cos  d 
 2





(8.72)

  ip 2 2 ikpze mn (cos  ) AxH ( )e 0 cos  d 


0
k

1/ 2 
pmTM,n =  mn
 ip  0
,

ikpz
H0
e 2  mn (cos  ) AxH ( )e 0 cos  d 
 2





(8.73)

  ip 2 2 ikpze mn (cos  ) AxH ( )e 0 cos  d 0k 1/ 2 
qmTM,n =  mn  ip
0H0e 2  mn (cos  ) AxH ( )e  ikpz0 cos  d 
 2



(8.74)

Eqs. (8.71)-(8.74) are general expressions of BSCs for any vector laser light-sheet, andthe BSCs of arbitrary light-sheet can be given by changing the vector angular spectrum
AxE ( ) and AxH ( ), respectively.
348

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)

8.2.2.3. Particular Structural Beam Laser Light-sheet
Substituting Eq. (8.26) into Eqs. (8.71) and (8.72), the BSCs of TE-polarized Airylight-sheet are  ip 2 2 (  ikys q )3 
e

(cos

)
exp
exp  ik ( qy0  pz0 )  cos  d 
mn




0
3



1/ 2 y s pmTE, n = ik  mn 
 , (8.75)
32  ip 0


(


iky
q
)
s
e 2   mn (cos  ) exp 
exp  ik ( qy0  pz0 )  cos  d 




3
2




  ip 2 2 (  ikys q )3 
exp  ik ( qy0  pz0 )  cos  d e 0  mn (cos  ) exp 

3



1/ 2 y s qmTE, n = ik  mn
 
 (8.76)3
2  ip 0 (  ikys q ) 2e   mn (cos  ) exp  exp  ik ( qy0  pz0 )  cos  d 


3
2





Similarly, the BSCs pmTM, n and qmTM, n are of TM-polarized Airy light-sheet are given,
respectively, as  ip 2 2 (  ikys q )3 
exp  ik ( qy0  pz0 )  cos  d e 0  mn (cos  ) exp 

3



1/ 2 y s 
pmTM, n =  k  mn 
 , (8.77)
3
2  ip 0 (  ikys q ) 2e   mn (cos  ) exp  exp  ik ( qy0  pz0 )  cos  d 


3
2




  ip 2 2 (  ikys q )3 
e

(cos

)
exp
exp  ik ( qy0  pz0 )  cos  d 
mn




0
3



1/ 2 y s 
qmTM, n =  k  mn 
 , (8.78)
3
2  ip 0 (  ikys q ) 2e    mn (cos  ) exp  exp  ik ( qy0  pz0 )  d 


3
2





Similarly, the BSCs of Bessel pincer-sheet are







n
  ip 2 2e   mn (cos  ) q  i  02  q 2
exp  ik ( qy0  pz0 )  cos  d 

1/ 20
ik  mnpmTE, n =
 
,
n
2
2
n
k 0  0  q  ip 2 022e   mn (cos  ) q  i  0  q
exp  ik ( qy0  pz0 )  cos  d 



2

(8.79)







n
  ip 2 2

2
2
e

(cos

)

q

i


q
exp  ik ( qy0  pz0 )  cos  d 
0

1/ 2

0 mn
ik  mn
qmTE, n =
 
,
n
2
2
n
k 0  0  q  ip 2 022e   mn (cos  ) q  i  0  q
exp  ik ( qy0  pz0 )  cos  d 



2

(8.80)
349

Advances in Optics: Reviews. Book Series, Vol. 5







n
  ip 2 2e   mn (cos  ) q  i  02  q 2
exp  ik ( qy0  pz0 )  cos  d 
1/ 20k  mn
pmTM, n  

,

n
2
2
n
ip
0
k 0  0  q 22
e 2   mn (cos  ) q  i  0  q
exp  ik ( qy0  pz0 )  cos  d 



2

(8.81)







n
  ip 2 2

2
2
e

(cos

)

q

i


q
exp  ik ( qy0  pz0 )  cos  d  
mn
0

1/ 2

0
k  mn
qmTM, n  
 
,
n
2
2
n
k 0  0  q  ip 2 022e   mn (cos  ) q  i  0  q
exp  ik ( qy0  pz0 )  cos  d  



2

(8.82)and the BSCs of Gaussian light-sheet are  ip 2 2 02 2 2 

(cos

)exp
k q  exp  ik (qy0  pz0 )  cos  d 
e

mn

0
 4


1/ 2 w0 
pmTE,n = ik mn,22   ip 2 0 0 2 2 e   mn (cos  )exp   4 k q  exp  ik (qy0  pz0 )  cos  d 
2





(8.83)
  ip 2 2 02 2 2 
e

(cos

)exp

k q  exp  ik ( qy0  pz0 )  cos  d 


mn

0
 4


1/ 2 w0 
qmTE,n = ik mn22   ip 2 0 0 2 2 e   mn (cos  )exp   4 k q  exp  ik ( qy0  pz0 )  cos  d 
2





(8.84)
  ip 2 2 02 2 2 
k q  exp  ik (qy0  pz0 )  cos  d e 0  mn (cos  )exp   4
1/ 2 w0 
pmTM,n = k mn,22   ip 2 0 0 2 2 e   mn (cos  )exp   4 k q  exp  ik (qy0  pz0 )  cos  d 
2





(8.85)

350

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)  ip 2 2 02 2 2 
e

(cos

)exp

k q  exp  ik ( qy0  pz0 )  cos  d 


mn

0
 4


1/ 2 w0 
qmTM,n = k mn22   ip 2 0 0 2 2 e   mn (cos  )exp   4 k q  exp  ik ( qy0  pz0 )  d 
2





(8.86)

8.3 Light Scattering of Laser Light-sheet by Dielectric Sphereof Arbitrary Size
8.3.1. Far-field Scattering Intensity
According to Eq. (8.44), the scattering field is expanded as


n

pol
(3)

Es (r , , ) =   iEmn  ampol,n N (3)mn ( kr )  bm , n M mn ( kr ) (8.87)

n =1 m =  n(3)(3)
The VSWFs M mn
and N mnare expressed as

(1)
im
M (3)
mn ( kr ) = i mn (cos  )e   mn (cos  )e  hn ( kr )e ,1 d rhn(1) (kr )  eim N (3)=  mn (cos  )e  i mn (cos  )e mn ( kr )kr dr h(1) (kr ) ime r n(n  1) P (cos  ) ne ,kr(8.88)(8.89)

mn

where hn(1) is the spherical Hankel function of the first kind. Considering the far-field limitof the spherical Hankel function:hn(1)  (i)n1 eikr / (kr ),(8.90)

Eqs. (8.88) and (8.89) become
M (3)mn ( kr ) =  mn (cos )e  i mn (cos  )e 1( i) n eikr eim ,kr(8.91)N (3)mn ( kr ) =  mn (cos  )e  i mn (cos  )e 1( i) n eikr eimkr

(8.92)

Substituting Eqs. (8.91) and (8.92) into Eq. (8.87):

Es = E e  E e ,(8.93)
351

Advances in Optics: Reviews. Book Series, Vol. 5E =i ikr  nie   Emn (i ) n  ampol,n mn (cos  )  bmpol,n mn (cos )  eim = eikr S 2 , (8.94)kr
kr
n =1 m =  nE =1 ikr  n
1e   Emn (i ) n  ampol,n mn (cos )  bmpol,n mn (cos )  eim = eikr S1 , (8.95)krkr
n =1 m =  n

where


n

S1 =   Emn (i) n  ampol, n mn (cos )  bmpol,n mn (cos )  eim ,(8.96)

n =1 m =  nnS2 =   Emn (i ) n  ampol,n mn (cos )  bmpol,n mn (cos )  eim(8.97)

n =1 m =  n

The dimensionless far-field scattering intensity can be evaluated as:
I ( r , ,  ) =1 
22
S1 ( , )  S2 ( ,  ) 
k 2r 2 (8.98)For convenience, the dimensionless normalized far-field scattering intensity can be
defined such that:I  ( , ) =  k 2 r 2  limI (r , , ) = S1 ( , )  S 2 ( , )
2

r 

2

(8.99)

8.3.2. Scattering, Extinction and Absorption Efficiencies
The extinction and scattering cross sections are Cext = Wext / I 0 , Csca = Wsca / I 0 , where
Wext and Wsca are extinguished and scattered energies by the dielectric sphere,
respectively, and I 0 = E02 / (2* Z0 ) is a characteristic intensity factor, where E0 is theamplitude of the incident electric field, and Z 0 is the wave impedance of the surrounding
medium. The dimensionless extinction and scattering efficiencies Qext = Cext / ( a2 ) andQsca = Csca / ( a2 ), where a is the radius of the dielectric sphere, are expressed as [43]:Qext =

n4
 pmpol,n ampol,n*  qmpol,n bmpol,n *  ,
R
e


2 2
k a
n =1 m =  n(8.100)

Qsca =

2
4  n  pol 2a bmpol,n  ,2 2    m,n

k a n =1 m =  n (8.101)

where the superscript symbol * denotes a complex conjugate. The absorption efficiency
is given by the Qext and Qsca that352

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)

Qext = Qabs  Qsca

(8.102)8.3.3. Radiation Force and TorqueIn the framework of GLMT, the integral of a Maxwell’s stress tensor Γ is a typicalcalculation method of optical radiation force [47]F =  nˆ  Γ dS ,(8.103)S

where differential surface element is represented by dS , the unit vector is denoted byn̂ , and the n̂ is on the surface that is enfolding the dielectric sphere. Γ is defined as
[47]

1 
1

Γ = Re  E  E*   H  Η*   E  E*   H  H* I  ,2 2(8.104)

where unit tensor is I , a tensor product is marked by  . Substituting the total of incident
and scattering EM fields ( E and H ) into Eq. (8.104) [48], the optical force components
can be obtained as follows,
Fx = Re  F1  , Fy = Im  F1  , Fz = Re  F2  ,(8.105)

where
F1

1

pol * polpol * poln2  (n  m)(n  m  1)  2  am ,n bm 1,n  bm ,n am 1,n 
2= 2 | E0 |     pol * polpol * pol 
p q

k
n(n  1)n =1 m =  n 
m , n m 1, n  qm , n pm 1, n 1

pol * pol
 n(n  2)(n  m  1)(n  m  2)  2  ampol,n am* pol1, n 1  bm , n bm 1, n 1   pol * pol2
pol * pol (n  1)  2n  1 2n  3    pm ,n pm 1,n 1  qm ,n qm 1,n 1 
1

pol
* pol

 n(n  2)(n  m)(n  m  1)  2  ampol,n 1am* pol1, n  bm , n 1bm 1, n,   pol * pol2pol* pol   (n  1)  2n  1 2n  3    pm ,n 1 pm 1, n  qm , n 1qm 1, n  


F2

=


n
 m
4
2
pol * pol|E| ampol,n bm* ,pol




0
n  pm , n qm , n
2
k
n
(
n

1)
n =1 m =  n 





1

pol * pol
 n(n  2)(n  m  1)(n  m  1)  2  ampol,n am* pol, n 1  bm , n bm , n 1 ,
   pol * pol2
pol * pol  (n  1)  2n  1 2n  3    pm,n pm,n 1  qm ,n qm ,n 1  353Advances in Optics: Reviews. Book Series, Vol. 5

and
1 pol pol 1 pol
pm , n , pm, n = pm , n ,2
2
1
1
bmpol, n = bmpol, n  qmpol, n , qmpol, n = qmpol, n22ampol, n = ampol, n (8.106)The integral of a time-averaged moment of the angular momentum flow tensor K is afeasible calculation method to calculate OST vector, that is [28]T =   nˆ  K dS ,(8.107)1 
1K = Re  E  E*   H  H*   E  E*   H  H*  I   r ,2 2(8.108)S

where [49-51]

where position vector is denoted by r. Considering the background medium is lossless,the OST components are [52, 53]
Tx = Re  N1  , Ty = Im  N1  , Tz = Re  N 2  ,(8.109)

1



n
2 0
2
[(n  m)(n  m  1)]2

N1 = 3 E0   ,kn =1 m =  n  a pol a* pol  b pol b * pol  p pol p* pol  q pol q* polm , n m 1, n
m , n m 1, n  m,n m 1,n m,n m 1,n



N2 = 


n
2 0
2
E
m | ampol, n |2  | bmpol, n |2  | pmpol, n |2  | qmpol,n |2


0
3
k
n =1 m =  n







(8.110)(8.111)8.4. Numerical Results and Discussions
Next, the simulation of dimensionless far-field scattering intensity, energy efficiencies, aswell as radiation force, torque on a sphere are shown. Both the three types of incident
light-sheet, such as Airy, Gaussian, Bessel pincer light-sheets, have been discussed.
Besides, the TE-polarized and TM-polarized incident light-sheets are studied. The
wavelength of the incident light is  =1.064  m, and the focused point of light sheets is( y0 , z0 ) . The center point of the scatters (relative permeability is  = 1) is ( y, z)  (0,0).
The amplitude of incident light is E0 = 1.0  10 6 V/m.
8.4.1. Airy light-sheet
The influence of the parameters of dimensionless kys ,  of the Airy incident field and theka (a is the radius of the sphere) of the sphere are considered.
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Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)

8.4.1.1. Far-field Scattering Intensity
Based on Eq. (8.99), the far-field scattering intensity of Airy light-sheet on a losslessdielectric sphere is studied as follows. Assuming the scatterer is a polystyrene sphere in
vacuum, and the corresponding refractive index are n1 = 1.59 and n0 = 1.0, respectively.
The scatterer is located at the main lobe of incident field.
Fig. 8.5 shows the effect of kys on I  with TE-polarized incident field, where kys takes
values as 1, 10, 50, respectively,  = 0.1, and ka =10. When kys changes from 1 to 10,
the amplitude of I  becomes bigger. But the amplitude of I  in kys = 50 is same as it inkys = 10. It means when kys < ka, I  increases as kys increases. But when kys > ka, I does not increase any more as kys increases. Fig. 8.6 is same as Fig. 8.5, but
TM-polarized Airy light-sheet is paid attentions. It shows that the state of polarization has
a minor effect on I .

Fig. 8.5. I  (Log) of a TE-polarized Airy light-sheet scattered by a dielectric sphere. ka = 10,

 = 0.1, kys varies as following list [1,10,50].

Fig. 8.6. The same as in Fig. 8.5, but the incident electromagnetic field is a TM-polarized.

In Figs. 8.7 and 8.8, the effect of  is analyzed, where  is set as 0.05, 0.1, 0.3,
respectively, and kys = 10, ka =10. Fig. 8.7 shows that all the maximum amplitude ofI  is in the direction of wave propagation (  = 0 ). The change of  can not affect theI  . And the polarization of Airy light-sheet has little effect on I .
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Advances in Optics: Reviews. Book Series, Vol. 5

Fig. 8.7. I  (Log) of a TE-polarized Airy light-sheet scattered by a dielectric sphere. ka = 10,

kys = 10,  varies as following list [0.05,0.1,0.3].

Fig. 8.8. The same as in Fig. 8.7, but the incident electromagnetic field is a TM-polarized.

Figs. 8.9 and 8.10 discuss the influence of size parameter ka, which are 2, 10, 20,
respectively, and kys = 10,  = 0.1. Compared with Figs. 8.9 and 8.10, the type ofpolarization have an impact on the distribution of I  only when ka = 2. As ka increases,
the maximum amplitude of I  increases. However, the distribution of I  has no changewith different  .

Fig. 8.9. I  (Log) of a TE-polarized Airy light-sheet scattered by a dielectric sphere.  = 0.1,

kys = 10, ka varies as following list [2, 10, 20].

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Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)

Fig. 8.10. The same as in Fig. 8.9, but the incident electromagnetic field is a TM-polarized.

8.4.1.2. Scattering, Extinction and Absorption Efficiencies
Next, the energy efficiencies have been computed base on the Eqs. (8.100)-(8.102). The
complex refractive index of the object (gold spherical particle) is n1 = 0.26  7.0i
suspended in water (n0 = 1.33). The scatterer is also located at the main lobe of Airy lightsheet. And ka transforms from 0.01 to 66.
Fig. 8.11 gives the calculated results for the scattering efficiency Qsca , investigating theeffect of the type of polarization and different kys taking values as 5, 10, 15, respectively,
where  = 0.1, ka varies in range (0,60). It is observed that Qsca of the three lines
increases rapidity in the range of 0 < ka <1 in Fig. 8.11, but when ka >1, increasing ka
makes Qsca decreases. And they are consistent in the range 0 < ka < 5. However, the plotof kys = 5 decrease quickly than the other two plots. And the two lines of kys being 10
and 15 are separated after ka >10. It means that Qsca is sensitive to the relative size ofka and kys . when ka > kys , Qsca increases as kys increases. The polarization of incident
light also shows a minor effect on the kys in this case.

Fig. 8.11. The Qsca of an Airy light-sheet scattered by a dielectric sphere. ka changes
in the range 0.01 < ka < 66. kys varies as following list [5,10,15], and  = 0.1.

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Advances in Optics: Reviews. Book Series, Vol. 5

Fig. 8.12 shows the extinction efficiency Qext , the parameters are same as Fig. 8.11. Since
Qext is the sum of Qsca and Qabs and Qabs is much smaller than the Qsca , the plots of Qext
and Qsca have the approximate trend and values.

Fig. 8.12. The Qext of an Airy light-sheet scattered by a dielectric sphere. ka changes
in the range 0.01 < ka < 66. kys varies as following list [5,10,15], and  = 0.1.

Fig. 8.13 are the same Figs. 8.11 and 8.12, but it calculate Qabs . Compared with Qsca andQext , the amplitude of Qabs is much smaller. Although the amplitude is different, the trendof Qabs and Qext , Qsca are similar. It increases when 0 < ka <1, and decrease in the range
ka >1. When the ka >10, increasing kys leads Qabs become bigger. In the panel (b),where the incident EM field is TM polarization, Qabs decreases more quickly than it inpanel (a), especially in ka >10.

Fig. 8.13. The Qabs of an Airy light-sheet scattered by a dielectric sphere. ka changes
in the range 0.01 < ka < 66. kys varies as following list [5,10,15], and  = 0.1.

Figs. 8.14, 8.15, and 8.16 discussed the influence of  , which varies in the list
[0.05, 0.1, 0.3]. And kys = 10, ka = 10. Similarly, the polarization of Airy light-sheet
affects Qsca and Qext lightly, but for Qabs TM-polarized incident filed makes Qabs
decreases more quickly. From Fig. 8.14, Qsca also increases as ka increases from 0 to 1.
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Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)

Then, Qsca deduces as ka increase. And the increasing of  cause Qsca decrease.
Moreover, the same case happens in Fig. 8.15, and 8.16.

Fig. 8.14. The Qsca of an Airy light-sheet scattered by a dielectric sphere. ka changes
in the range 0.01 < ka < 66.  varies as following list [0.05,0.1,0.3], and kys = 10.

Fig. 8.15. The Qext of an Airy light-sheet scattered by a dielectric sphere. ka changes
in the range 0.01 < ka < 66.  varies as following list [0.05,0.1,0.3], and kys = 10.

Fig. 8.16. The Qabs of an Airy light-sheet scattered by a dielectric sphere. ka changes
in the range 0.01 < ka < 66.  varies as following list [0.05,0.1,0.3], and kys = 10.

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Advances in Optics: Reviews. Book Series, Vol. 5

8.4.1.3. Radiation Force and Torque
Optical force on a spherical particle with refractive index n1 = 1.59 in vacuum (refractive
index n0 = 1.0) is studies as follows. An lossless Rayleigh spherical particle ( ka = 0.1) ispresented in Figs. 8.17-8.20. The optical force of different directions Fy and Fz is shown,with particularly emphasize the negative pulling force Fz < 0. The effect of kys ,  andthe type of polarization is considered, where kys takes values as 5, 10, and 
are 0.05, 0.1.

Fig. 8.17. The optical force on a lossless Rayleigh spherical particle with TE-polarized Airy light,
ka = 0.1,  = 0.1. kys varies as following list [5,10].

In Fig. 8.17, it is showing that different values of kys have an obvious influence on theamplitude and distribution. When kys increases from 5 to 10, the maximum amplitude of
all the panels becomes smaller, but the distribution scope of force in the yz -plane isbigger. It means that the distribution scope of negative pulling force Fz increases too,
which has important practical significance. Besides, in the case of the same parameters,
the amplitude of Fz is smaller than Fy . Fig. 8.18 is the same case as Fig. 8.17, but
incident light is TM-polarized. It is distinct that the maximum of radiation force whenkys = 5 in panels (a), (b), (c) of Fig. 8.18, are smaller than that in Fig. 8.17. However, thedistribution of optical force is as broad as it’s long. Fig. 8.19 displays the effect of  .
When  changes from 0.05 to 0.1, the maximum amplitude of optical force decrease in
all the Fy and Fz . And the distribution scope of Fz < 0 are smaller too. Similarly, when360

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet) = 0.05 the maximum amplitude in Fig. 8.20 are smaller than the corresponding panels
in Fig. 8.19. But the same situation do not happen in the case of  = 0.1.

Fig. 8.18. Being same as Fig. 8.17, but is the Airy light-sheet is TM-polarized.

Fig. 8.19. The optical force on a lossless Rayleigh spherical particle with TE-polarized Airy light,
kys = 10.  varies as following list [0.05,0.1].

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Advances in Optics: Reviews. Book Series, Vol. 5

Fig. 8.20. Being same as Fig. 8.19, but is the Airy light-sheet is TM-polarized.

The optical force on an lossless Mie spherical particle (ka = 10) is studied in
Figs. 8.21-8.24. The lossless Mie spherical particle is located in z = 1  m of theyz -plane, and y varies in the range -25 to 25 ( m). Fig. 8.21 gives the simulated plotsof Fy for different values of kys , where  = 0.1. It is shown that the maximum amplitude
of positive Fy occurs when kys = 10. And in the position of y = 0, the direction of Fy
is all towards the negative y -axis. Besides, the amplitude of these negative Fy increases
as kys decreases. The amplitude of Fy on Mie particles illuminated by the TM-polarized
incident field is slightly larger than it illuminated by the TE-polarized incident field.
Fig. 8.22 is the same as Fig. 8.21, but is Fz . It is observed that all the directions of Fz ispositive. As kys increases, Fz increases, which means the maximum amplitude of Fz isin the plot of kys = 15, different from Fy . Fig. 8.23 shows the effect of  on Fy , where changes as following list 0.05, 0.1, 0.3, and kys = 10. As  increase, the maximum
amplitude of Fy in the positive y -axis decreases. And Fy in panel (b) which is
TM-polarized incident Airy light-sheet are bigger than it in panel (a) of TE polarization.
Fig. 8.24 is same as Fig. 8.23, but is Fz . Similar to Fig. 8.23, increasing  makes Fz
decreases. And the type of polarization of incident field has a minor effect on Fz .
Then the optical torque on an absorptive spherical particle (n1 = 0.26  7i) in water(n0 =1.33) is calculated. A Rayleigh spherical particle (ka = 0.1) is first presented in
Figs. 8.25 and 8.26, particularly analyzing negative Tx . Since the optical torque only
occurs in the case of TM-polarization, and only Tx exists but Ty , Tz = 0 [28], we only362

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)analysis the influence of  and kys on Tx . In Figs. 8.25, as kys increases, the trajectoryof Tx become straight, and the amplitude of Tx in this calculated scope become smaller,
but the distribution scope of negative Tx in panels (d), (e), (f) become bigger. The main
lobe of Tx become wider too. Fig. 8.26 shows how  influence Tx . As  changes from0.05 to 0.1, the amplitude of Tx has just a minor difference. The distribution scope of Tx
becomes slightly smaller too. However, when  increases from 0.1 to 0.3, Tx decreases,
the distribution scope of Tx are much smaller including negative Tx . Fig. 8.27 shows the
optical torque on a Mie spherical particle (ka = 10) as 1D plots, where z = 1  m.
Considering the different values of kys (5, 10, 15) and  (0.01, 0.1, 0.3). In panel (a),where  = 0.1, the maximum amplitude of positive Tx occurs when kys = 15, however
the negative Tx occurs in kys = 10. In panel (b), where kys = 10, the maximum amplitude
positive Tx does not change in different  . However, the amplitude of negative Tx
located in about y = 0, increases as  decrease.

Fig. 8.21. Fy on a lossless Mie spherical particle of Airy light-sheet with TE and TM
polarizations. The values of kys are 5, 10, 15, respectively, and  = 0.1.

Fig. 8.22. Being same as Fig. 8.21, but the optical force is Fz .

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Advances in Optics: Reviews. Book Series, Vol. 5

Fig. 8.23. Fy on a lossless Mie spherical particle of Airy light-sheet with TE and TM
polarizations. The values of  are 0.05,0.1, 0.3, respectively, and kys = 10.

Fig. 8.24. Being same as Fig. 8.23, but the optical force is Fz .

Fig. 8.25. Tx on an absorptive Rayleigh spherical particle with TM-polarized Airy light,
where kys are 5, 10, 15, respectively, and  = 0.1.

364Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)

Fig. 8.26. Tx on an absorptive Rayleigh spherical particle with TM-polarized Airy light,
where  are 0.05, 0.1, 0.3, respectively, and kys = 10.

Fig. 8.27. Tx on an absorptive Mie spherical particle with TM-polarized Airy light. The valuesof kys are 5, 10, 15, respectively, and  = 0.1 in panel (a). But in panel (b)  takes values
as 0.05, 0.1, 0.3, and kys = 10.

8.4.2. Bessel Pincer Light Sheet
8.4.2.1. Far-field Scattering Intensity
Here, the I  on a dielectric sphere with a Bessel pincer light-sheet incident is analyzed.
Assuming the object is a polystyrene sphere (refractive index is n1 = 1.59) in vacuum(refractive index is n0 = 1.0). Note that the amplitude of incident electric field is
365

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E0 = 1.0  10 6 V/m. ka varies [2, 10, 20],  0 varies [0.2, 0.5, 1], and l varies[10, 45, 100].
Fig. 8.28 gives the far field scattering distribution ( I  ) under the TE-polarized Bessel
pincer light-sheet. Panels (a)-(c) gives the I  of  0 = 0.5, l = 10 with different ka. In
panel (a) (ka = 2), there are only two beam lobes (The left one is minor, and the right one
is big). The I  is symmetrical along 0-  -axis. The gain value magnitude of I  is equal
to 80. In panel (b) (ka = 10), there are a few beam lobes (like a small cat). Except for thethe symmetrical main lobes on the right side, there are many side lobes symmetricallyalong 0-  -axis. The gain value magnitude of I  is equal to 90. In panel (c) (ka = 20),
there are a few beam lobes (like a fried hairy cat). The gain value magnitude of I  isequal to 100.

Fig. 8.28. I  (Log) of a TE-polarized Bessel pincer light-sheet scattered by a dielectric sphere.
 0 = 0.5, l = 10, ka varies as following list [2,10, 20].

Fig. 8.29 gives the far field scattering distribution ( I  ) under the TM-polarized Bessel
pincer light-sheet. Panels (a)-(c) gives the I  of  0 = 0.5, l =10 with different ka. In
panel (a) (ka = 2 ), different from panel (a) of Fig. 8.28 there is no main beam lobes. The
I  is symmetrical along 0-  -axis. The gain value magnitude of I  is equal to 80. In
panel (b) (ka = 10) and panel (c) (ka = 20), the regularity of I  and the beam lobes aresame the TE-polarized mode case of panels (b) and (c) in Fig. 8.28.
Panels (a), (b) and (c) of Fig. 8.30 gives the far field scattering distribution ( I  ) of
ka = 20, l = 10 with different  0 (0.2, 0.5, 100) under the TE-polarized Bessel pincer
light-sheet. In panel (a) (0 = 0.2), there are a few beam lobes (like a small cat). The main
lobes on the right side have the gain value magnitude of I  (equal to 80), there are manyside lobes symmetrically along 0-  -axis. In panel (b) (0 = 0.5), there is only one pairof the main lobe distributing symmetrically, and less side lobes (comparing to panel (a)).
The gain value magnitude of I  is equal to 100. In panel (c) (0 = 1.0), there are plenty366

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)of main lobes with the same gain value magnitude (equal to 120). But there are less side
lobes comparing to panel (b).

Fig. 8.29. The same as Fig. 8.28, but the incident electromagnetic field is a TM-polarized.

Fig. 8.30. I  (Log) of a TE-polarized Bessel pincer light-sheet scattered by a dielectric sphere.

ka = 20, l = 10,  0 varies as following list [0.2,0.5,1.0].

Panels (a), (b) and (c) of Fig. 8.31 gives the far field scattering distribution ( I  ) of
ka = 20, l = 10 with different  0 (0.2, 0.5, 100) under the TM-polarized Bessel pincer
light-sheet. In panel (a) (0 = 0.2), there are a few beam lobes (like a small cat). The main
lobes on the right side have the gain value magnitude of I  (equal to 80), there are manyside lobes symmetrically along 0-  -axis. In panel (b), there is only one pair of the main
lobe distributing symmetrically, and less side lobes (comparing to panel (a)). The gain
value magnitude of I  is equal to 100. In panel (c) (0 = 0.5), there are plenty of main
lobes with the same gain value magnitude (equal to 120). But there are less side lobes
comparing to panel (b). The gain values and symmetrical characteristics is similar in two
kinds of polarizations.
Panels (a), (b) and (c) of Fig. 8.32 gives the far field scattering distribution ( I  ) of
ka = 20,  0 = 0.5 with different l (10, 45, 100) under the TE-polarized Bessel pincer
light-sheet. In panel (a) (l = 10), the one pair of the main lobe distribution. There are367

Advances in Optics: Reviews. Book Series, Vol. 5

many side lobes distributing symmetrically. The maximal gain value magnitude is equal
to 120. And there are a few lobe sides along 0-  -axis. In panel (b) (l = 45), the regionof main lobes shrinks. There are many side lobes distributing symmetrically. The maximal
gain value magnitude is equal to 110. Different in panel (a), there is little side lobes along0-  -axis. In panel (g) (l = 100), the region of main lobes shrinks. There are many side
lobes distributing symmetrically. The maximal gain value magnitude is equal to 120.
Different in panel (b), there are shorter side lobes than that in panel (c).

Fig. 8.31. The same as Fig. 8.30, but the incident electromagnetic field is a TM-polarized.

Fig. 8.32. I  (Log) of a TE-polarized Bessel pincer light-sheet scattered by a dielectric sphere.

 0 = 0.5, ka = 20, l varies as following list [10, 45,100].

Panels (a), (b) and (c) of Fig. 8.33 gives the far field scattering distribution ( I  ) of
ka = 20,  0 = 0.5 with different l (10, 45, 100) under the TM-polarized Bessel pincer
light-sheet. The cases of beam lobes, gain values and symmetrical characteristics is nearly
to the situations in Fig. 8.32.
In summary, 0 , l , and ka are sensitive to the I . Firstly, the gain value of lobe width
remains same, but the lobe distribution shape becomes complex with symmetricalcharacteristic, as the increasing of dimensionless parameter ka. Secondly, the larger thescaling parameter  0 is, the greater the gain value of lobe width is. Meanwhile, the shapeof main lobe distributes with more second lobes with the increasing of scaling parameter
368

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)0 . Lastly, the gain value of lobe width turns minor, and the main lobe appears little bylittle, while increasing the beam order l.

Fig. 8.33. The same as Fig. 8.32, but the incident electromagnetic field is a TM-polarized.

8.4.2.2. Scattering, Extinction and Absorption Cross Section
In this section, the scattering Qsca , extinction Qext and absorption efficiencies Qabs areanalyzed. The complex refractive index of the object (gold spherical particle) is
n1 = 0.26  7.0i suspended in water (n0 = 1.33). The ka of the sphere varies from0 to 100.
The panels (a) and (b) in Fig. 8.34 show that Qsca with Bessel pincer light-sheet incident,
versus ka, with different l (10, 45, 100) for the scaling parameter  0 = 0.8. The Qsca inpanel (a) with different l is under TE-polarized while panel (b) with TM-polarized. In
panel (a), the maximum magnitude of Qsca is about 3.2 103 for l = 10, 0.5 103 forl = 45 and 3.2 108 for l = 100. When the value of order l = 10, there is a little peakwith magnitude 0.25 103 at ka = 5. And then there is an abrupt increasing till maximum
magnitude of Qsca near at ka = 12. subsequently, after a lower peak, the second
maximum of Qsca appears about at ka = 19. And the third maximum magnitude of Qsca
emerges at ka = 20. After ka larger than 20, the value of Qsca goes down till thatconverges. But for l = 45, the value of Qabs shows an rising trend until ka reaches 40,
with a lower magnitude of Qsca . The trend towards convergence is relatively slow.
However, the value of Qsca for l = 100 is almost 0, comparing to the value of that forl = 10 and l = 45. Panel (b) is the distribution of TM-polarized, which is similar with the
distribution in panel (a). The difference of TM-polarization and TE-polarization is thelocation of the main peaks. The maximum magnitude of Qsca in panel (b) is further thanthat in panel (a), the same to the second maximum magnitude for l =10. The trends of
Qsca in panel (b) is near that in panel (a) for l = 45 and l = 100.
The panels (c) and (d) in Fig. 8.34 give the scattering efficiency Qsca versus ka, withdifferent  0 (0.2, 0.5, 1.0) for the beam order l = 10. The Qsca in panel (c) with different369

Advances in Optics: Reviews. Book Series, Vol. 5

 0 is under TE-polarized while panel (d) with TM-polarized. In panel (c), the maximum
magnitude of Qsca is about 3.7 103 for  0 = 0.2, 2.7 103 for  0 = 0.5 and 2.0 103
for  0 = 1.0. For  0 = 0.2, the maximum peak of Qsca is at ka = 10, with the second
maximum peak at ka =16. And the trend downward to convergence is relatively quick.
Comparing to  0 = 0.2, the location of maximal peak for  0 = 0.5 is at ka = 23, further
than that of particle center. The downward trend for  0 = 0.5 is relatively steep than thatof  0 = 0.2. For  0 = 1.0, the location of maximal peak with ka = 60, is the furthest
among the value of 0 . But, the downward trend of Qsca is same to the case of  0 = 0.5.
As for TM-polarized situation, the maximum magnitude of Qsca is lower than that of
TE-polarization. For  0 = 0.2, there is one more peaks with maximum magnitude of

Qsca = 3.1103. For  0 = 0.5, the maximum magnitude of Qsca is about 2.6 103. The
trend of Qsca in panel (d) is similar with the trend in panel (c). From the standpoint of
convergence, whatever situations in TE or TM-polarized, the rates of convergence is fastfor  0 = 1.0, medium for  0 = 0.5, low for  0 = 0.2. And the amplitude of the peaks is
close for the same scaling parameter  0 or beam order l.

Fig. 8.34. The Qsca of a Bessel pincer light-sheet scattered by a dielectric sphere, ka changes
in the range 0 < ka < 100.  0 = [0.2,0.5,0.8], ł = [10, 45,100].

The panels (a) and (b) in Fig. 8.35 show the scattering efficiency Qext versus ka, withdifferent l (10, 45, 100) for the scaling parameter  0 = 0.8. The Qext in panel (a) withdifferent l is under TE-polarized while panel (b) with TM-polarized. In panel (a), the370

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)maximum magnitude of Qext is about 3.2 103 for l = 10, 0.5 103 for l = 45 and3.2 108 for l = 100. And when the value of l is 10, there is a little peak with magnitude
0.25 103 at ka = 5. And then there is an abrupt increasing till maximum magnitude of
Qext near at ka = 12. Subsequently, after a lower peak, the second maximum of Qext
appears about at ka = 19. And the third maximum magnitude of Qext emerges at
ka = 20. After ka larger than 20, the value of Qext goes down till that converges. But forl = 45, the value of Qext shows an rising trend until ka reaches 40, with a lower
magnitude of Qext . The trend towards convergence is relatively slow. However, the valueof Qext for l =100 is about 0, comparing to the value of that for l = 10 and l = 45. Panel(b) is the distribution of TM-polarized, which is similar with the distribution in panel (a).
The difference of TM-polarization and TE-polarization is the location of the main peaks.

Fig. 8.35. The Qext of a Bessel pincer light-sheet scattered by a dielectric sphere, ka changes
in the range 0 < ka < 100.  0 = [0.2,0.5,0.8], ł = [10, 45,100].

The maximum magnitude of Qext in panel (b) is further than that in panel (a), the same tothe second maximum magnitude for l = 10. The trends of Qext in panel (b) is near that inpanel (a) for l = 45 and l =100. The panels (c) and (d) in Fig. 8.35 give the scattering
efficiency Qext versus ka, with different  0 (0.2, 0.5, 1.0) for l = 10. The Qext in panel(c) (different 0 ) is under TE-polarized while panel (d) with TM-polarized. In panel (c),the maximum magnitude of Qext is about 3.7 103 for  0 = 0.2, 2.6 103 for  0 = 0.5
and 2.0 103 for  0 = 1.0. For  0 = 0.2, the maximum peak of Qext is at ka = 10, withthe second maximum peak at ka = 16. And the trend downward to convergence is371

Advances in Optics: Reviews. Book Series, Vol. 5

relatively quick. Comparing to  0 = 0.2, the location of maximal peak for  0 = 0.5 is atka = 23, further than that of particle center. The downward trend for  0 = 0.5 is relatively
steeper than that of  0 = 0.2. For  0 = 1.0, the location of maximal peak with ka = 60,
is the furthest among the value of 0 . But, the downward trend of Qext is same to the caseof  0 = 0.5. As for TM-polarized situation, the maximum magnitude of Qext is lower thanthat of TE-polarized. For  0 = 0.2, there is one more peaks with maximum magnitude of

Qext = 3.1103. For  0 = 0.5, the maximum magnitude of Qext is about 2.6 103. The
trend of Qext in panel (d) is similar with the trend in panel (c). From the standpoint of
convergence, whatever situations in TE or TM-polarized, the rates of convergence is fastfor  0 = 1.0, medium for  0 = 0.5, low for  0 = 0.2. And the amplitude of the peaks is
close for the same  0 or l.
Panels (a)-(d) in Fig. 8.36 show Qabs (given by Qsca  Qext ) under different l(l = 10, 45, 100 in panels (a) and (b)) and  0 ( 0 = 0.2, 0.5, 1.0 in panels (c) and (d)). As
shown in panels (a) and (b) of Fig. 8.36, the value of Qabs with  0 = 0.8 but different l(10, 45, 100) is definitely different. For order l = 10 in panel (a), there is a obviously
steep change from ka = 0 to ka = 20 under TE-polarization, but relatively slower thanthat in panel (b) with which ka varies from 0 to 10. For the situation of Qabs for ka > 20
but ka < 58 in panel (a), the downward rate of Qabs is relative low step by step tillka > 58. But for TM-polarized case, after ka > 10, the trend of Qabs is downward lower
and lower. For l = 45 in TE-polarized case (panel (a)), there is a peak at ka = 58. And
then, the downwards for ka > 58, go to converge slowly. For l = 45 in TM-polarized case(panel (b)), there is a period of steady trends for ka between 45 and 58. And then, thedownwards is still continuous to converge. For l = 100, the value of Qabs is nearly tozero. Panels (c) and (d) of Fig. 8.36 gives the value of Qabs with l = 10 but different  0(0.2, 0.5, 1.0). The changes of Qabs with different location of ka, still have the great
effects. On the one hand, the maximal peak of Qabs is relevant different scaling parameter0 . In panel (c), for scaling parameter  0 = 0.2, the location of peak is ka = 18,
 0 = 0.5 for ka = 30, and  0 = 1.0 for ka = 58. In panel (d), for scaling parameter 0 = 0.2, the location of peak is ka = 15,  0 = 0.5 for ka = 19, and  0 = 1.0 for
ka = 42. On the other hand, the polarization characteristic is influential. For  0 = 0.2,
after ka = 20, Qabs in TE-polarized goes to a downward trend, relatively slower than thatin TM-polarized. The same to  0 (0.5 and 1.0), the rate to obtain the maximal peak or theconvergence in TM-polarized models quicker than that in TE-polarized mode.
Based on the above simulation and analysis, it is obtained that the location of main peak
and polarized mode is affected by  0 and l of Bessel pincer light-sheet. First, the bigger
the l is, the closer the location of main peak to the source of particle is, but the smaller
the amplitude of the main peak is. Secondly, the larger the scaling parameter  0 is, the372Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)closer the location of main peak to the source of particle is, but the smaller the amplitudeof the main peak is. Lastly, the convergence rate under TE-polarized mode is slower thanthat of under TM order.

Fig. 8.36. The Qabs of a Bessel pincer light-sheet scattered by a dielectric sphere, ka changes
in the range 0 < ka < 100.  0 = [0.2,0.5,0.8], ł = [10, 45,100].

8.4.2.3. Radiation Force and TorqueHere, the optical force (mainly focusing on Fy , Fz and Fz < 0) with different  0 and lof Bessel pincer light-sheet are discussed. The complex refractive index of the object is
n1 = 1.59 in vacuum ( n0 = 1.0). The dimensionless parameter ka is chosen as ka = 0.1
and ka = 10.
As can be gathered from the panels (a)-(l) of Fig. 8.37, Fy displaying positive values,
interferes on the object, incoming of a TE-polarized Bessel pincer light-sheet, withka = 0.1, for  0 = 0.2 and 0.5, l = 10 and 45. In panels (a) (0 = 0.2, l = 10) and (d)(0 = 0.2, l = 45), the different order l affects the distribution of Fy . It can be seen thatin panel (a), there are some beam clusters along z -axis, and the beam focusing spot isabout at z = 70 m. The magnitude of maximal force is 1020 N. It’s symmetric, and thewidth region of two bending arcs is  10,10  m along the y -axis. In panel (d), there islittle clusters along z -axis( some margin about y = 30 m) , and the beam focusing spot
doesn’t appear in our calculation region (the focusing spot is about z = 400 m in minor
figure). The characteristic of symmetric didn’t appear in our calculation region,373

Advances in Optics: Reviews. Book Series, Vol. 5either.(width region is about  50,50  m along y -axis in minor figure). Panels (g) and(j) give the distribution of Fy with  0 = 0.5, the beam order l = 10 and 45, respectively.
Identically, there are some clusters along z -axis, which the focusing spot is about at
z =18 m . The magnitude of maximal force is 1019 N. It’s symmetric, and the width
region of two bending arcs is  5,5 m along the y -axis. In panel (j), there is further

focusing spot at z = 60 m, and the width region of two bending arcs is  15,15 malong the y -axis, symmetrically. Comparing to the margin cluster distributing region after
the focusing spot in panels (g) and (j), we can see that the region is  30,30  m in panel(g), and the region is  12,12  m in panel (j). The distributing region after focusing inpanel (g) is wider than that in panel (j).

The panels (b), (e), (h) and (k) of Fig. 8.37 represents the plots for the longitudinalradiation force component Fz (the positive values), which interferes on the object,incoming of a TE-polarized Bessel pincer light-sheet, with ka = 0.1, for  0 = 0.2 and 0.5,
l 10 and 45. In panel (b) (0 = 0.2, l = 10) and (e) (0 = 0.2, l = 45), the different orderl effects the distribution of Fz . It can be seen that in panel (b), there are come beamclusters along z -axis, and the beam focusing spot is about at z = 70 m (the same withfocusing spot in panel (a)). The magnitude of maximal force is 1021 N (smaller than thatin panel (a)). It’s symmetric, and the width region of two bending arcs is  10,10  malong the y -axis (the same with distributing region in panel (a)). In panel (e), there is little
clusters along z -axis (some margin about y = 30 m) , and the beam focusing spot
doesn’t appear in our calculation region (the focusing spot is about z = 400 m in minor
figure) (the same to focusing spot in panel (d)). The characteristic of symmetric didn’t
appear in our calculation region, either.(width region is about  50,50  m alongy -axis in minor figure) (the same with distributing region in panel (d)). Panels (h) and (k)give the distribution of Fz with  0 = 0.5, the beam order l = 10, 45, respectively.
Identically, there are some clusters along z -axis, which the focusing spot is about at
z =18 m. The magnitude of maximal force is 1019 N. It’s symmetric, and the width
region of two bending arcs is  5,5 m along the y -axis. In panel (k), there is further
focusing spot at z = 60 m, and the width region of two bending arcs is  15,15 malong the y -axis, symmetrically. Comparing to the margin cluster distributing region after

the focusing spot in panels (h) and (k), we can see that the region is  30,30  m in panel(h), and the region is  12,12  m in panel (k). The distributing region after focusing inpanel (h) is wider than that in panel (k). It is interesting that there are additional beamcluster along the margin beam in panels (b), (h) and (k), comparing to thecases of Fy .
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Panels (c), (f), (i) and (l) of Fig. 8.37 indicates the plots for the longitudinal radiation force
component Fz (with negative values), which interferes on the object, incoming of a
TE-polarized Bessel pincer light-sheet, with ka = 0.1, for  0 = 0.2 and 0.5, l 10 and 45.
The object experiences a negative pulling longitudinal force; thus, the components affects
in the opposite direction comparing to the direction of wave propagation. The influenceof the scaling parameter  0 and the beam order l is similar to that in cases of Fz .

Fig. 8.37. The optical acting on a dielectric sphere illuminated by a TE-polarized Bessel pincer
light-sheet. ka = 0.1, alpha0 = [0.2,0.5], ł = [10, 45].

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Panels (a)-(l) of Fig. 8.38, gives the transverse radiation force component Fy , but theobject influenced by the incoming beam of a TM polarized Bessel pincer light sheet. In
panel (a) ( 0 = 0.2, l =10) and (d) ( 0 = 0.2, l = 45), the different order l effects the
distribution of Fy . Similar with the distribution of Fy under TE-polarized mode, thefocusing spot is nearly at z = 70 m for panel (a) ( 0 = 0.2, l = 10), and z = 400 m
for panel (d) ( 0 = 0.2, l = 45). The symmetrical characteristic still exists along y -axis.

Fig. 8.38. The same to Fig. 8.37, but TM-polarized.

The width region of two bending arcs is about 10,10m for panel (a)( 0 = 0.2, l = 10),  50,50  m for panel (b) ( 0 = 0.2, l = 45) along the y -axis. In the376Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)same way, the distribution of Fy with  0 = 0.5 and the beam order l =10 and 45, isshown as panels (g) and (j). The location of focusing spot and the width region of bendingarcs is the same as the cases in panels (g) and (j) of Fig. 8.37. The panels (b), (e), (h) and(k) of Fig. 8.38 illustrates the plots for the longitudinal radiation force component Fz(positive values), which interferes on the object, incoming of a TM-polarized Bessel
pincer light-sheet, with ka = 0.1, for  0 = 0.2 and 0.5, l 10 and 45. The regularity for thelocation of focusing spot and the width region of bending arcs, is familar with thesituations in panels (b), (e), (h)and (k) of Fig. 8.37. Panels (c), (f), (i) and (l) of Fig. 8.38
depicts the plots for the longitudinal radiation force component Fz (with negative values),
which interferes on the object, incoming of a TM-polarized Bessel pincer light-sheet, withka = 0.1, for  0 = 0.2 and 0.5, l = 10 and l = 45. In the same way, the components of
optical force act in opposite direction of wave propagation, with the influence of  0 andl , similar to that in cases of Fz .
Here, the transverse radiation force component Fy interfering on the object illuminatedby a Bessel pincer light-sheet are analysis, chiefly focusing on the influences of 1D (onedimension) with different  0 and l in panels (a)-(d) of Fig. 8.39. The dimensionless size
parameter is ka =10. The value of y is 4 m. Panel (a) gives the trend of the force
component Fy acting on the sphere illuminated by TE-polarized Bessel pincer light-sheet,as z changes under different  0 for l = 10. For  0 = 0.5 (the blue line), the amplitude
of Fy is equal to zero along z -axis. For  0 = 0.7 (the yellow line), there is a wave crestat z = 35 m with maximal magnitude of Fy = 6  1017 N. There is a trough about at

z = 50 m with minor magnitude of Fy = 11018 N. There is a period of second peakat the region 60,80 m along z -axis. After z > 85 m, Fy shows an upward trend.
For  0 = 0.9 (the green line), Fy shows a downward trend till z = 40 m (the firsttrough with magnitude Fy = 0). There is a second crest (same magnitude with the firstcrest) at region  40,70 m along z -axis. The second trough is about at z = 70 m.
After z > 80 m, there is negative and positive value of Fy , with some unstablechanges. The maximal magnitude of Fy is 5.9 1017 N, while the minimal with

5.9 1017 N. Panel (b) gives the trend of the force component Fy acting on the sphere
illuminated by TM-polarized Bessel pincer light-sheet, as z changes under different  0
for l =10. For  0 = 0.5 (the blue line), the amplitude of Fy is equal to zero alongz -axis. For  0 = 0.7 (the yellow line), there is a wave crest at z = 35m with maximal
magnitude of Fy = 6  1017. There is a trough about at z = 50 m with minor magnitudeof 11018 N. There is a period of second peak at the region 60,80 m along z -axis.
After z > 85 m, Fy shows an upward trend. For 0 = 9 (the green line), Fy shows a377

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downward trend till z = 40 m (the first trough with magnitude Fy = 0 N). There is asecond crest (same magnitude with the first crest) at region  40,70 m along z -axis.
The second trough is about at z = 70 m. After z > 80 m, there is negative andpositive value of Fy , with some unstable changes. The maximal magnitude of Fy is

5.9 1017 N, while the minimal with 5.9 1017 N. The differences between twopolarized modes (panels (a) and (b)) is that the trends of Fy is stable for TM afterz > 90 m. Panel (c) gives the trend of Fy acting on the sphere illuminated byTE-polarized Bessel pincer light-sheet, as z changes under different l for  0 = 0.5. For
different l (0, 5, 10), the value is positive, and the trend goes toward along z -axis. The
difference is that the l = 0 (the blue line) changes rapidly, l = 5 changing medium, andl =10 changing slow. Panel (d) gives the trend of the force component Fy acting on thesphere illuminated by TM-polarized Bessel pincer light-sheet, as z changes under
different l for  0 = 0.5. The trends of Fy is similar to the cases in panel (c).

Fig. 8.39. Optical force ( Fy ) with different  0 = [0.5,0.7,0.9] and l = [0,5,10]
under TE and TM mode.

Fig. 8.40 demonstrates the longitudinal radiation force component Fz acting on adielectric sphere illuminated by Bessel pincer light-sheets are analysis, primarily focusing
on the influences of 1D (one dimension) with different  0 and l. And the value of
y = 4 m. Panel (a) gives the trend of the force component Fz acting on the sphere
illuminated by TE-polarized Bessel pincer light-sheet, as z changes under different  0
for l = 10. For  0 = 0.5 (the blue line), the amplitude of Fz is equal to zero along378

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)z -axis. For  0 = 0.7 (the yellow line), there is a wave crest at z = 42 m with maximal
magnitude of Fz = 7 1017 N. There is a trough about at z = 60 m with minor
magnitude of 11018 N. There is a period of second peak at the region 60,80 malong z -axis. After z > 85 m, Fz shows an upward trend. For  0 = 0.9 (the greenline), Fz shows a downward trend till z = 60 m (the first trough with magnitude
Fz = 0 N ). There is a second crest (same magnitude with the first crest) at region

40,75 m along z -axis. The second trough is about at z = 60 m. The second troughis about at z = 70 m. After z > 75 m, there is some unstable value changes of Fz .
The maximal magnitude of Fz is 7.5 1017 N, while the minimal with 1.8 1017 N.

Fig. 8.40. Optical force ( Fz ) with different  0 = [0.5,0.7,0.9] and l = [0,5,10]
under TE and TM mode.

Panel (b) gives the trend of the force component Fz acting on the sphere illuminated by
TM-polarized Bessel pincer light-sheet, as z changes under different  0 for l =10. For
0 = 0.5 (the blue line), the amplitude of Fz is equal to zero along z -axis. For  0 = 0.7(the yellow line), there is a wave crest at z = 42 m with maximal magnitude of

Fz = 7 1017 N. There is a trough about at z = 60 m with minor magnitude of11018 N. There is a period of second peak at the region 60,80 m along z -axis.
After z > 85 m, Fy shows an upward trend. For  0 = 0.9 (the green line), Fz shows adownward trend till z = 60 m (the first trough with magnitude Fz = 0 N). There is asecond crest (same magnitude with the first crest) at region  40,75 m along z -axis.

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The second trough is about at z = 60 m . The second trough is about at z = 70 m.
After z > 75 m, there is some unstable value changes of Fz . The maximal magnitudeof Fz is 7.5 1017 N, while the minimal with 1.8 1017 N. Panel (c) gives the trend of
Fz acting on the sphere illuminated by TE-polarized Bessel pincer light-sheet, as zchanges under different l for  0 = 0.5. For different l (0, 5, 10), the value is positive,
and the trend goes toward along z -axis. The difference is that the l = 0 (the blue line)
changes rapidly, l = 5 changing medium, and 1 = 10 changing slow. Panel (d) gives thetrend of the force component Fz acting on the sphere illuminated by TM-polarized Bessel
pincer light-sheet, as z changes under different l for  0 = 0.5. The trends of Fz issimilar to the cases in panel (c).
The spin torque can cause the object rotate around its center of mass in Figs. 8.41, 8.42.
But only the non-vanishing axial spin torque component is Tx  0, The optical spin torque(mainly focusing on Tx , and Tx < 0) illuminated by a TM-polarized Bessel pincer
light-sheet with different scaling parameter  0 and beam order l are discussed. The
refractive index of the dielectric sphere is n1 = 0.26  7.0i, and that of surrounding
medium is n0 = 1.33. The dimensionless parameter ka is chosen as ka = 0.1
and ka = 10.

Fig. 8.41. Tx acting on a dielectric sphere illuminated by a TM-polarized Bessel pincer
light-sheet. ka = 0.1,  0 = 0.2, l = [10, 45,100].

Fig. 8.41 demonstrates the optical spin torque component Tx (and Tx < 0, displaying
positive values), interfering on the object, incoming of a TM-polarized Bessel pincer
light-sheet, with ka = 0.1, for  0 = 0.2 and l = 10, 45, 100. Panels (a)-(c) is the Tx
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distribution of different l. It can be seen that in panel (a) (l = 10), there are come beamclusters along z -axis, and the beam focusing spot is about at z = 70 m. The magnitudeof maximal torque is 1028 N.m. It’s symmetric, and the width region of two bending arcs
is  10,10  m along the y -axis. In panel (b) (l = 45), there is little clusters alongz -axis (some margin about y = 30 m), and the beam focusing spot doesn’t appear in
our calculation region (the focusing spot is about z = 400 m in minor figure). The
characteristic of symmetric didn’t appear in our calculation region, either. (The width
region of that is about  50,50  m along y -axis in minor figure). In panel (c) (l = 100),
there are a few horizontal lines, which means the zero value. From the minor figure, there
are some beam clusters along z -axis, and the beam focusing spot is about at
z = 800 m. The magnitude of maximal torque is 1028 N.m. It’s symmetric, and thewidth region of two bending arcs is  100,100 m along the y -axis. After the focusing

spot, the margin beam region is  30,30  m in panel (a) (l = 10),  100,100 m inpanel (b) (l = 45) and  30,30  m in panel (c) (l = 100). The negative component(panels (d)-(f)) nearly has the same regularities in cases (panels (a)-(c)) of Tx .

Fig. 8.42. Tx acting on a dielectric sphere illuminated by a TM-polarized Bessel pincer
light-sheet. ka = 0.1,  0 = [0.2,0.5,1.0], l = 10.

Fig. 8.42 depicts the optical spin torque component Tx (and Tx < 0, displaying positive
values), interfering on the object, incoming of a TM-polarized mode Bessel pincer
light-sheet, with ka = 0.1, for  0 = 0.2, 0.5, 1.0, and l = 10. Panels (a)-(c) is the Tx
distribution of different 0 . It can be seen that in panel (a) (0 = 0.2), there are comebeam clusters along z -axis, and the beam focusing spot is about at z = 70 m. The
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magnitude of maximal torque is 1028 N.m. It’s symmetric, and the width region of two
bending arcs is  10,10  m along the y -axis. In panel (b) (l = 0.5), there are someclusters along z -axis, which the focusing spot is about at z =18 m. The magnitude of
maximal torque is 1028 N.m. It’s symmetric, and the width region of two bending arcs is 5,5 m along the y -axis. In panel (c) (0 = 1.0), there are some clusters alongz -axis, which the focusing spot is about at z = 2 m. The magnitude of maximal torqueis 1028 N.m. It’s symmetric, and the width region of two bending arcs is  1,1 malong the y -axis. The negative component Tx (panels(d)-(f)) nearly has the same cases(panels(a)-(c)) of Tx .
Here, Tx acting on an absorptive object illuminated by Bessel pincer light-sheets areanalysis, basically focusing on the influences of 1D (one dimension) with different  0
and l in panels (a(d) of Fig. 8.43. The dimensionless parameter is chosen as ka = 10. And
the value of y is 4 m.

Fig. 8.43. Tx with different  0 = [0.5,0.7,0.9] and l = [0,5,10] under TE and TM mode.

Panel (a) of Fig. 8.43 gives the trend of the spin torque component Tx acting on the sphere
illuminated by TM-polarized Bessel pincer light-sheet, as z changes under different  0
for l = 10. For  0 = 0.5 (the blue line), the amplitude of Tx is equal to zero alongz -axis. For  0 = 0.7 (the yellow line), there is a wave crest at = 40 m with maximal
magnitude of Tx = 11022 N.m. There is a trough about at z = 53 m with minor
magnitude of 7 1023 N.m. There’s a region along the z -axis at region 60,90 mthat’s symmetrically distributed along z = 70 m. After z > 85 m, Tx shows an
upward trend, till the peak Tx = 8 1023 N.m. For  0 = 0.9 (the green line), Tx shows adownward trend till z = 28 m (the first trough with magnitude Tx = 11023 N.m).

There is a first crest at region  40,70 m along z -axis. The second trough is about at

z = 68 m. After z > 80 m, there is negative and positive value of Tx , with someunstable changes. The maximal magnitude of Tx is 1.5 1022 N.m, while the minimal
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Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)with 1.2 1017 N.m. There are few differences in two polarized mode. Panel (b) ofFig. 8.43 gives the trend of the torque component Tx acting on the sphere illuminated by
TM-polarized Bessel pincer light-sheet, as z changes under different l for  0 = 0.5. The
trends of Tx is decreasing as the increasing of l , and there is a few wave vibrating alongz -axis.
As has been noted, 0 > 0 represents the auto-focusing characteristic of the beam. ł (anyreal order) represents the bending characteristic. Whatever situation of optical torquecomponents ( Fy , Fz and Fz < 0) and optical torque (Tx , Tx < 0), the scaling parameter 0 and the beam order l is sensitive to the raising or reducing. When the  0 is the samevalue, the width region of two bending arcs before focusing is wider and wider as theincreasing of the beam order l. And the focusing spot location is further and further withthe rising of the beam order. While the value of beam order is invariable, the width regionof the two bending atcs before focusing is narrower and narrower, the focusing spot iscloser and closer, with the increasing of the scaling parameter 0 . There is symmetricalcharacteristic of the optical force and torque along the y -axis.
8.4.3. Gaussian Light-sheet
Gaussian light-sheet is the two-dimensional (2D) slice of a Gaussian beam which is atypical high-focused light. The optical scattering and mechanical effects when Gaussian
light-sheet illuminates a dielectric sphere are paid attention to in this section. The effectof dimensionless waist radius k0 of the Gaussian light-sheet and ka is considered. BothTE-polarized and TM-polarized Gaussian light-sheet are studied.
8.4.3.1. Far-field Scattering Intensity
Assuming the scatterer is located at the main lobe of incident light. Surrounding medium
is vacuum (n0 = 1.0) and refractive index of scatterer is n1 = 1.59. The impact of k0 ,ka and polarization type on I  is shown in Figs. 8.44-8.47.

Fig. 8.44. I  (Log) of a TE-polarized Gaussian light-sheet scattered by a dielectric sphere.

ka = 10. k0 varies as following list [6, 12, 18].

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Fig. 8.45. The same as Fig. 8.44, but a TM-polarized Gaussian light-sheet is assumed.

Fig. 8.46. The normalized far-field intensity of a TE-polarized Gaussian light-sheet scatteredby a dielectric sphere. k0 = 12. ka varies as following list [2, 10, 20].

Fig. 8.47. The same as Fig. 8.46, but a TM-polarized Gaussian light-sheet is assumed.

When a TE-polarized Gaussian light-sheet with varying k0 illuminates a dielectric
sphere, I  is revealed in Fig. 8.44. k0 takes values as [6, 12, 18]. As the dimensionlesswaist radius k0 increases, the intensity of the main and side lobes increases. No matterhow k0 varies, the distribution of I  is always symmetric about z -axis; the center andzone of the main and side lobes remain constant; the strongest intensity is located at = 0 . When the incident beam becomes TM-polarized (Fig. 8.45), the intensity of lobes
are weaker, and k0 affects amplitude of lobes more weakly.
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Fig. 8.46 (Fig. 8.47) exhibits the influence of ka on the far-field scattering intensitywith
TE-polarized (TM-polarized) incident light. ka takes values as [2, 10, 20]. The
distribution of I  still has perfect symmetry about z -axis. As ka increases, more side
lobes appear, the intensity of the main lobe becomes stronger, and the width of the main
lobe becomes smaller. When a TM-polarized Gaussian light-sheet illuminates, theamplitude of the main lobe becomes larger than the TE-polarized case.
8.4.3.2. Scattering, Extinction and Absorption Efficiencies
The energy efficiency reflects the scatter’s ability to transform the incident energy to thescattering, extinction or absorption energy. We discuss the scattering, extinction andabsorption efficiencies Qsca , Qext , Qabs in order. Suppose a Gaussian light-sheet shines ona gold particle (n1 = 0.26  7.0i) immersed in water (n0 =1.33). In this case,
Figs. 8.48-8.50 show the variety of energy efficiencies with k0 and ka. k0 takes
values as the list [6, 12, 18]. ka transforms from 0.01 to 60. The scatterer is also locatedat the main lobe of Airy light-sheet.

Fig. 8.48. The Qsca of a Gaussian light-sheet scattered by a dielectric sphere. ka changes
in the range 0.01 < ka < 60. k0 varies as following list [6, 12, 18].

Fig. 8.49. The Qext of a Gaussian light-sheet scattered by a dielectric sphere. ka changes
in the range 0.01 < ka < 60. k0 varies as following list [6, 12, 18].

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Fig. 8.50. The Qabs of a Gaussian light-sheet scattered by a dielectric sphere. ka changes in therange 0.01 < ka < 60. k0 varies as following list [6, 12, 18].

We discuss the scattering efficiency first. As shown in Fig. 8.48, whether the incident
beam is TE-polarized or TM-polarized, Qsca rapidly increases to the maximum value inthe range of 0.01< ka < 0.813 first; within the range of 0.813  ka  2.819, Qsca
oscillates and has some peaks at ka = 0.813,1.414,2.217,2.819; in the range of
2.819 < ka < 60, Qsca decreases gradually and the decreasing speed declines with ka
increases. Observing the three curves with different dimensionless waist radii, the largerk0 , the larger amplitudes of Qsca and peaks; with ka increases, the effect of k0 on
the growth of Qsca increases first and then decreases.
As for the impact of polarization, the curves of TE-polarization and TM-polarization are
similar with the same parameters, but its value of the scattering efficiency is not equal.
Within the range of ka < 57.994, the value of Qsca with TE-polarization is bigger thanthat with TM-polarization, however, the rule is contrary out of that boundary. As k0
increases, the point with the strongest effect of polarization on the scattering efficiency
corresponds to a larger ka.
Fig. 8.49 displays the impact of k0 and ka on the extinction efficiency Qext . The
tendency of three curves representing Qext owns a striking similarity to that of Qsca . Dueto the existence of absorption efficiency (the imaginary part of n1 is larger than zero), Qext
is larger than Qsca throughout the computational zone.
The absorption efficiency Qabs is also considered in Fig. 8.48. The trend of Qabs closely
resembles that of Qsca . However, there are some differences between them. For Qabs , theamplitude is much smaller than that of Qsca ; the amplitude of peaks are located at
ka = 0.612,1.414,2.016,3.421; the amplitude with TE-polarization is always larger thanthat with TM-polarization.
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8.4.3.3. Radiation Force and Torque
Optical force acting on a Rayleigh (Figs. 8.51, 8.52) or Mie (Fig. 8.53) particle shined by
a Gaussian light-sheet with different k0 is discussed. ka of the Rayleigh and Mie
dielectric spheres are 0.1 and 10, respectively. We assign k0 value as the list of[6, 12, 18]. Both transverse and longitudinal force components Fy and Fz are considered.
We pay more attention to negative longitudinal force Fz < 0. Assuming a Gaussian
light-sheet illuminates on a sphere (n1 =1.59) in the vacuum (n0 =1.0).

Fig. 8.51. The optical force acting on a dielectric sphere illuminated by a TE-polarized Gaussian
light-sheet. ka = 0.1. k0 varies as following list [6, 12, 18].

We consider the Rayleigh scatters first. The calculation region is 20 < y < 20 and
0 < z <100 ( m). Due to the interaction between particle and light, Fy has two lobes
that are symmetric about z -axis and have the opposite value, Fz has one negative main387

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lobe near z -axis and two positive symmetric lobes on both sides of the negative lobe. A
negative lobe of Fy is located in the positive y -axis. With k0 increasing, thedistribution of main or side lobes of Fy and Fz enlarges, but the numbers of that are
constant. When incident light-sheet is TM-polarized (Fig. 8.52), the same rule appears as
TE-polarization.

Fig. 8.52. The same as Fig. 8.51, but a TM-polarized Gaussian light-sheet is assumed.

Changing the size parameter of scatterer to 10, Fig. 8.53 displays Fy and Fz withdifferent k0 . The value of Fy and Fz with TE-polarization is equal to that with
TM-polarization. Fy and Fz are symmetric about the origin and the line of y = 0,
respectively. In panels (a,b), Fy has one negative lobe on the negative y -axis and one
positive lobe on positive y -axis; as k0 increases, the main lobe has a smaller amplitude
and larger zone, and the center of the main lobe move further away from the origin. Fz
ranges more largely with k0 growing.
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Fig. 8.53. The optical force acting on a dielectric sphere illuminated by a Gaussian light-sheet.
ka = 10. k0 varies as following list [6, 12, 18].

The radiation torque providing another mechanical degree of freedom is also studied asshown in Figs. 8.54, 8.55. Suppose n0 = 1.33 and n1 = 0.26  7.0i. We discuss Tx in thissection. In Fig. 8.54, scatterer is a Rayleigh sphere (ka = 0.1). Tx owns one negative lobe
( y > 0) and one positive lobe ( y < 0) near z -axis. The larger k0 , the smaller the
maximum value is, and the wider zone Tx takes.Finally, the case of a Mie sphere(ka = 10) is calculated (Fig. 8.55). Each curve of Tx is exactly symmetrical about region.
With increasing k0 , the center of main lobe of Tx move towards the larger ka, and theamplitude of main lobe decreases.

8.5. Conclusions
This chapter derives the equations for the light-sheet in the framework of GLMT. Thispart of the theory is suitable to any form of light-sheet and a spherical particle of arbitrary
size. Then we deduced the scattering and mechanical properties of arbitrary light-sheet
polarized by TE and TM illuminating a spherical particle of any size. When the incident
light is an Airy light-sheet, the change of ys and ka has a huge influence on the far-field
scattering intensity, the change of  has a little effect on it, and the polarization state has
almost no influence on it. Qext , Qabs , and Qsca have roughly the same fluctuation trend,
and the larger the ys , the smaller the ability of the particle to absorb and scatter the Airy
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light-sheet as ka increases.  has the opposite effect on this ability. The change of  andpolarization state has little effect on the force. Torque is very sensitive to the variation ofys and  . In the case of a Gaussian incident light-sheet, as the dimensionless k0
increases, the intensity of main and side lobes increases. The distribution of intensity has
perfect symmetry about the z -axis. As ka increases, more side lobes appear, the intensityof main lobe becomes stronger, and the width of main lobe decreases. Qext , Qabs , andQsca have roughly the same fluctuation trend and the larger the k0 . The smaller theability of the particle to absorb and scatter the Gaussian light-sheet as ka increases. Forceand torque are very sensitive to the variation of k0 , but they are not affected by thepolarization state. When a Bessel pincer light-sheet shines, the far-field scattering
intensity is susceptible to the scaling parameter  0 and beam order l , and thedimensionless parameter ka. Whatever situation of optical force components ( Fy , Fz and

Fz < 0) and optical torque (Tx , Tx < 0), their raising or reducing is susceptible to  0
and l. The research content of this chapter is expected to be applied to super-resolution
imaging and optical control of light-sheet.

Acknowledgements
This work was supported by the National Natural Science Foundation of China[61901324, 62001345], the China Postdoctoral Science foundation [2019M653548,
2019M663928XB], the Project (B17035) and the CETC Key Laboratory of Data Link
Technology (No: CLDL-20182418).

Fig. 8.54. Tx acting on a dielectric sphere illuminated by a TM-polarized Gaussian light-sheet.

ka = 0.1. k0 varies as following list [6, 12, 18].

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Fig. 8.55. Tx acting on a dielectric sphere illuminated by a Gaussian TM-polarized light-sheet.

ka = 10. k0 varies as following list [6, 12, 18].

References[1]. J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, E. H. K. Stelzer, Optical sectioning deep
inside live embryos by selective plane illumination microscopy, Science, Vol. 305,
Issue 5686, 2004, pp. 1007-1009.
[2]. R. Tomer, K. Khairy, F. Amat, P. Keller, Quantitative high-speed imaging of entire
developing embryos with simultaneous multiview light-sheet microscopy, Nature Methods,
Vol. 9, 2012, pp. 755-763.
[3]. G. Diemunsch, J. P. Prenel, A compact light sheet generator for flow visualizations, Optics& Laser Technology, Vol. 19, Issue 3, 1987, pp. 141-144.
[4]. G. Gréhan, K. F. Ren, G. Gouesbet, A. Naqwi, F. Durst, Evaluation of a particle sizing
technique based on laser sheets, Particle & Particle Systems Characterization, Vol. 11,
Issue 1, 1994, pp. 101-106.
[5]. H. Hama, H. Hioki, K. Namiki, T. Hoshida, H. Kurokawa, F. Ishidate, T. Kaneko, T. Akagi,
T. Saito, T. Saido, A. Miyawaki, Scales: an optical clearing palette for biological imaging,
Nature Neuroscience, Vol. 18, Issue 10, October 2015, pp. 1518-1529.
[6]. F. G Mitri, Nonparaxial scalar Airy light-sheets and their higher-order spatial derivatives,
Applied Physics Letters, Vol. 110, Issue 9, 2017, 091104.
[7]. Z. Zhao, W. Zang, J. Tian, Optical trapping and manipulation of mie particles with airy beam,
Journal of Optics, Vol. 18, Issue 2, 2016, 025607.
[8]. S. Pei, Q. Pan, F. Cui, S. Xu, Z. Cao, Electromagnetic scattering of an airy beam light sheet
by a multilayered sphere. Optik, Vol. 180, 2019, pp. 379-386.
[9]. S. Zhang, R. Li, B. Wei, J. Zhang, H. Sun, N. Song. Scattering of a non-paraxial Bessel
light-sheet by a sphere of arbitrary size. Journal of Quantitative Spectroscopy & Radiative
Transfer, Vol. 245, April 2020, 106869.
[10]. T. Meinert, A. Rohrbach, Light-sheet microscopy with length-adaptive bessel beams,
Biomedical Optics Express, Vol. 10, Issue 2, 2019, pp. 670-681.
[11]. C. Zhai, Z. Cao, Elastic light scattering of a single sphere in a Gaussian light sheet, Optik,
Vol. 194, 2019, 163083.
[12]. F. G. Mitri, Radiation force and torque of light-sheets, Journal of Optics, Vol. 19, Issue 6,
June 2017, 065403.
[13]. G. A. Siviloglou, J. Broky, A. Dogariu, D. N. Christodoulides, Observation of accelerating
airy beams, Phys. Rev. Lett., Vol. 99, Nov 2007, 213901.

391

Advances in Optics: Reviews. Book Series, Vol. 5[14]. Q. Zhang, X. M. Cheng, H. W. Chen, B. He, Z. Y. Ren, Y. Zhang, J. T. Bai, Diffraction-free,
self-reconstructing Bessel beam generation using thermal nonlinear optical effect, Applied
Physics Letters, Vol. 111, Issue 16, 2017, 161103.
[15]. J. Broky, G. A. Siviloglou, A. Dogariu, D. N Christodoulides, Self-healing properties of
optical Airy beams, Optics Express, Vol. 16, Issue 17, 2008, pp. 12880-12891.
[16]. X. Chu, W. Wen, Quantitative description of the self-healing ability of a beam, Opt. Express,
Vol. 22, Issue 6, Mar 2014, pp. 6899-6904.
[17]. F. G Mitri, Nonparaxial Bessel and Bessel-Gauss pincers light-sheets, Physics Letters A,
Vol. 381, Issue 3, 2017, pp. 171-175.
[18]. F. G Mitri, Nonparaxial fractional Bessel and Bessel-Gauss auto-focusing light-sheet pincers
and their higher-order spatial derivatives, Journal of Optics, Vol. 19, Issue 5, 2017, 055602.
[19]. Z. Cao, C. Zhai, Scattering of one-dimensional Airy beam light sheet with finite energy
by a sphere, Applied Optics, Vol. 56, Issue 12, 2017, pp. 3491-3496.
[20]. Q. Pan, S. Pei, F. Cui, S. Xu, Z. Cao, Scattering of an Airy light sheet by a chiral sphere,
Applied Optics, Vol. 58, Issue 26, 2019, 7151.
[21]. G. G. Gouesbet, J. A. Lock, G. Gréhan, Generalized Lorenz-Mie theories and description ofelectromagnetic arbitrary shaped beams: localized approximations and localized beammodels, A review, Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 112,
Issue 1, 2011, pp. 1-27.
[22]. H. Wang, X. Liu, Y. Li, B. Han, L. Lou, K. Wang, Optical technic of isolation a single
chromosome, Acta Biophysica Sinica, Vol. 20, Issue 1, 2004, pp. 50-56.
[23]. C. M. Ruiz, J. J. Simpson, W. Shao, J. L. Li, G. Ala, E. Francomano, M. F. Pantoja,
A. R. Bretones, S. Garcia, X. L. Travassos, et al., Cepstral analysis of photonic
nanojet-illuminated biological cells, Applied Computational Electromagnetics Society
Journal, Vol. 27, Issue 3, 2012, pp. 215-222.
[24]. T. Li, S. Kheifets, D. Medellin, M. G. Raizen, Measurement of the instantaneous velocity
of a brownian particle, Science, Vol. 328, Issue 5986, 2010, pp. 1673-1675.
[25]. F. G. Mitri, Airy acousticalвЂ“sheet spinner tweezers, Journal of Applied Physics, Vol. 120,
Issue 10, 2016, 104901.
[26]. F. G. Mitri, Interaction of Bessel pincers light-sheets with an absorptive subwavelength
sphere coated by a plasmonic layer, JOSA B, Vol. 34, Issue 7, 2017, pp. 1471-1477.
[27]. F. G. Mitri, Radiation force and torque on perfect electricallyвЂ“conducting (PEC)
corrugated circular and elliptical cylinders in TE or TM polarized plane progressive waves
with arbitrary incidence, Journal of Quantitative Spectroscopy & Radiative Transfer,Vol. 235, pp. 15-23, 2019.
[28]. N. Song, B. Wei, R. Li, H. Sun, B. Wei, S. Zhang, J. Zhang, F. G. Mitri, Optical torque on anabsorptive dielectric sphere of arbitrary size illuminated by a linearly-polarized Airylight-sheet, Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 256, 2020,
107327.
[29]. N. Song, R. Li, H. Sun, J. Zhang, B. Wei, S. Zhang, F. G. Mitri, Optical radiation force on adielectric sphere of arbitrary size illuminated by a linearly polarized Airy light-sheet, Journalof Quantitative Spectroscopy & Radiative Transfer, Vol. 245, 2020, 106853.
[30]. F. G. Mitri, Circularly-polarized Airy light-sheet spinner tweezers and particle transport,
Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 260, 2021, 107466.
[31]. R. Li, C. Ding, F. G. Mitri, Optical spin torque induced by vector Bessel (vortex) beams withselective polarizations on a light-absorptive sphere of arbitrary size, Journal of Quantitative
Spectroscopy & Radiative Transfer, Vol. 196, 2017, pp. 53-68.
[32]. R. Li, R. Yang, C. Ding, F. G. Mitri, Optical torque on a magneto-dielectric Rayleighabsorptive sphere by a vector Bessel (vortex) beam, Journal of Quantitative Spectroscopy &Radiative Transfer, Vol. 191, 2017, pp. 96-115.
[33]. L. Novotny, B. Hecht, Principles of Nano-Optics, Cambridge University Press, 2012.

392

Chapter 8. Light-sheet Scattering by a Dielectric Sphere of Arbitrary Size (GLMT for Light-sheet)[34]. J. Yang, L. W. Li, K. Yasumoto, C. H. Liang, Two-dimensional scattering of a Gaussian beam
by a periodic array of circular cylinders, IEEE Transactions on Geoscience & Remote
Sensing, Vol. 43, Issue 2, 2005, pp. 280-285.
[35]. W. H. Carter, Electromagnetic field of a Gaussian beam with an elliptical cross section,
Journal of the Optical Society of America, Vol. 62, Issue 10, 1972, pp. 1195-1201.
[36]. K. F. Ren, Diffusion des Faisceaux Feuille Laser par une Particule Sph érique et Applications
aux Ecoulements Diphasiques, PhD Thesis, Université de Rouen, 1995.
[37]. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, M. L. Schattenburg, Analyses of vector
Gaussian beam propagation and the validity of paraxial and spherical approximations,
Journal of the Optical Society of America A Optics Image Science & Vision, Vol. 19, Issue 2,
2002, pp. 404-412.
[38]. P. Liu, B. Lü, The vectorial angular-spectrum representation and Rayleigh-Sommerfeld
diffraction formulae, Optics & Laser Technology, Vol. 39, Issue 4, 2007, pp. 741-744.
[39]. G. A. Siviloglou, J. Broky, A. Dogariu, D. N. Christodoulides. Observation of accelerating
Airy beams. Physical Review Letters, Vol. 99, Issue 21, 213901, 2007.
[40]. G. A. Siviloglou, D. N. Christodoulides. Accelerating finite energy Airy beams, Optics
Letters, Vol. 32, Issue 8, 2007, pp. 979-981.
[41]. Y. Xu, G. Zhou, The far-field divergent properties of an Airy beam, Optics & Laser
Technology, Vol. 44, Issue 5, 2012, pp. 1318-1323.
[42]. S. N. Khonina, Specular and vortical Airy beams, Optics Communications, Vol. 284,
Issue 19, 2011, pp. 4263-4271.
[43]. G. Gouesbet, G Grehan, A generalized Lorenz-Mie theory, Journal of Optics, Vol. 13, 1982,
pp. 97-103.
[44]. G. Mie, Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen, Annalen der
Physik, Vol. 330, Issue 3, 1908, pp. 377-445.
[45]. W. Lu, H. Chen, S. Liu, Z. Lin, Rigorous full-wave calculation of optical forces on dielectricand metallic microparticles immersed in a vector Airy beam, Optics Express, Vol. 25,
Issue 19, 2017, pp. 23238-23253.
[46]. N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical
Physics, John Wiley and Sons, 1996.
[47]. J. D. Jackson, Classical Electrodynamics, 3rd Ed., John Wiley & Sons, New York, 1999.
[48]. H. Chen, N. Wang, W. Lu, S. Liu, Z. Lin, Tailoring azimuthal optical force on lossy chiralparticles in Bessel beams, Physical Review A, Vol. 90, Issue 4, 2014, 043850.
[49]. J. Schwinger, L. L. Deraad, K. A. Milton, W.-Y. Tsai, Classical Electrodynamics, Perseus
Books, 1998.
[50]. Y. Jiang, H. Chen, J. Chen, J. Ng, Z. Lin, Universal relationships between optical force/torqueand orbital versus spin momentum/angular momentum of light, arXiv:Optics, 2015.
[51]. Q. Ye, H. Lin, On deriving the maxwell stress tensor method for calculating the optical force
and torque on an object in harmonic electromagnetic fields, European Journal of Physics,
Vol. 38, 2017, 045202.
[52]. N. Ji, M. Liu, J. Zhou, Z. Lin, S. T. Chui, Radiation torque on a spherical birefringent particle
in the long wave length limit: analytical calculation, Optics Express, Vol. 13, Issue 14, 2005,
pp. 5192-204.
[53]. M. Liu, N. Ji, Z. Lin, S. T. Chui, Radiation torque on a birefringent sphere causedby an electromagnetic wave, Physical Review E, Vol. 72, Issue 5, 2005, 056610.

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Chapter 9. Laser Characterizations for Optical Wireless Communications

Chapter 9

Laser Characterizations
for Optical Wireless Communications
Salam Nazhan1

9.1. Introduction to Optical Wireless Communications
At the present time, optical communication technology, represented by optical wirelesscommunication (OWC), which also known as free space optical (FSO) communication,or fibreless optical communications, that uses laser beam propagating through space, has
substantially becomes a major leading topic in research articles. Laser-based OWC has
basically given wide attention in research articles [1-3]. OWC is concerned with thetransmission of information through the atmosphere from one point to another by using
visual or laser beam to get optical communications, Fig. 9.1. A significant progress has
been made since 1960s in OWC with the discovery of optical sources, particularly laser
sources. Since then, the OWC has become one of the important fields in communications
and has attracted a large number of researchers at a global level. The OWC first stated in
the military application for covert communication because of inherent security comparedwith the radio frequency based technologies [4]. OWC technologies have a number of
attractive features such as narrow optical beam width, thus avoiding potential interferewith the other beam, low costs system requirements, no fibre optic cables to lay (so noneed for expensive rooftop connections equipment), very large bandwidth, excellent
frequency re-use capabilities and compatibility with the exiting optical fibre
communication networks [5, 6].
The optical sources that utilized for such systems are light emitted diodes or lasers. Bothlaser diode (LD) and light emitted diode are widely used in OWC systems. LDs are mainly
used for outdoor and medium to long range applications, nevertheless the latter is ideal
for high speed line of sight links, in both indoor and outdoor applications [7, 8].
Laser sources are one of the key components in the optical communication systems andnetwork to enable high-speed data communications, therefore recent applications for thelaser sources are highlighted in brief in the next sections.

Salam Nazhan
Department of Communication, College of Engineering, University of Diyala, Iraq

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Advances in Optics: Reviews. Book Series, Vol. 5

Laser sources offer advantages in bandwidth and speed for free space applications overtraditional systems which operated at shorter wavelength [2]. The laser beam increases
the wireless capacity of 5G services and beyond for the future applications [1, 3].
Recently, high-speed underwater is also given wide interest in applications of OWC
systems [9], as well as THz emitter based on laser chips [10]. The OWC technology is
influenced by propagation of the laser beam through the space. Over the last decade, theOWC technology has to use applications such as satellite-to-satellite cross links andamong mobile or stationary terminals to provide high bandwidth wireless communication
links. OWC will be a powerful technique and one of the most unique tools to address
issues of bandwidth limitation and security that have been created in high speed
communication [11-13]. In addition, OWC based interconnect is the most promising
scheme that could lead to increased speed, reduced size and compact packaging in future
integrated circuits [14]. Normally, the free space communication systems use thewavelength range between 750 and 1600 nm because of the optical energy that travels
through atmosphere have comparable properties at visible and near-IR wavelength [8].
However lower and higher wavelength are also being considered for specific applications.
The wavelengths between 780-850 nm are the most popular because of readily available
and inexpensive components, which have an attenuation of less than 0.2 dB/km [8]. The
majority of OWC systems designed to operate at a transmission window located at awavelength range of around 780-850 nm. Laser devices that emit 780 nm are available
and inexpensive, but have short lifespan so must be considered during system design.
While wavelength of 850 nm, high-performance transmitter, inexpensive and reliable,
therefore they are commonly used in network and transmission systems. Furthermore,
there are many devices that operate in this wavelength, such as avalanche photodiode(APD), highly sensitive silicon (Si) and vertical cavity surface emitting lasers (VCSELs).
The latter, are extremely inserted in wide applications in such modern communication
systems.
In vast majority of the semiconductor lasers (SLs) diode applications, such ascommunication systems employing a high pump source and on optical fibre, the edge
emitting lasers (EELs) are the dominant source and are widely used. However, EELs are
too costly requiring optical fibre coupling, which results in additional power loss.
Therefore, attention has been focused on VCSEL devices with potentially low
manufacturing cost for various applications including optical communication systems.
VCSEL is a SL, which has a resonant cavity that is vertically formed with the surfaces ofthe epitaxial layers. The lasing of VCSELs was first demonstrated in 1979 where a gallium
indium arsenide phosphide grown on an indium phosphide GaInAsP/InP material wasused for the active region, the emitted light had a wavelength of 1300 nm. VCSEL devices
with optical and electrical properties offer a number of advantages such as a low thresholdcurrent, which enables these devices to be directly modulated at high frequencies, small
beam divergence that allows good coupling efficiency with the optical fibre, symmetrical
output beam profile, small switching transients, circular light-output mode, highpackaging density, and very low power consumption compared to the conventional edgeemitting semiconductor laser devices [15]. VCSELs are the key and dominant source inlocal area networks thanks to their advantages such as wafer scale testing and ease of
fabrication [16].
396Chapter 9. Laser Characterizations for Optical Wireless Communications

Fig. 9.1. An alternative to traditional sources laser-based communications, or optic-based
technologies, are prominent as another appliance for OWC systems.
(AFCEA International, Airbus).

The FSO communication systems typically uses wavelength between 750 and 1600 nm
because of the optical energy that travels through atmosphere have similar properties at
visible and near-IR wavelength rang [8]. However lower and higher wavelength are also
being used for specific applications. The wavelengths between 780-850 nm are wide
common owing to readily available and inexpensive components, which also has anattenuation of less than 0.2 dB/km [8]. The majority of OWC systems designed to operate
at a transmission window located at 780–850 nm wavelengths. Lasers emit 780 nm areavailable and inexpensive, but have short lifespan so must be considered during designing
such systems. While devices that emit 850 nm have high-performance, inexpensive andreliable, therefore they are generally used in network and transmission systems. Moreover,there are many devices that operate in this wavelength, such as avalanche photodiode,highly sensitive silicon and VCSELs. High security transmission data over a network is
demand with the wide distributed of communication systems. OWC communications
technology based on chaos becomes one of the most promising technique to avoid hacking
information [11]. The message can be encoded with spectrum of the chaos and be sent
through a free space channel. In the past few years, the chaotic dynamics of VCSELs has
been the subject of great interest for researchers theoretically and experimentally [17, 18];
particularly, for secure optical communications.

9.2. Underwater Wireless Optical Communication
Underwater wireless optical communication (UWOC) faces today challenges ofabsorption, scattering, and channel turbulence which still need an active solution,
particularly for long distance wireless communication and high data rate transmission.
These are real problems which could face many applications of UWOC channel. Opticalwave offer high-data rate with power efficiency and short length data transmission
compared with acoustic and radio frequency (RF). Latter suffer from limitation of high397

Advances in Optics: Reviews. Book Series, Vol. 5

data transmission range owing to high water channel attenuation [19, 20]. Numbers oflight sources, such as edge emitting laser diodes, super-luminescent diodes and VCSELs
have been discussed in terms of modulation bandwidth and performance in the field of
UWOC. High-speed underwater data transmission rate of about 3.4 Gbps wasdemonstrated with mixing lasers, with corresponding BERs of 3.5 × 10−3 where presented
in [9]. An experimental results based on a green laser diode have achieved 34.5 m long
distance UWOC with data rate of 2.70 Gbps and light attenuation coefficient of~0.44 dB/m, which reported in [21]. For UWOC links the maximum attenuation length of
35.88 with symbol error rate of 6.31×10−4 at a distance of 120 m in clear ocean water at a
bandwidth of 13.7 MHz have experimentally demonstrated in [22]. Based on boththeoretical and experimental study, hundreds of meters long distance UWOC is quitepossible in the future through optimization parameters of the optical links such as, powerand beam divergence angle. Laser polarization modes, chaos and coherent beam areextremely attractive properties for the demand of high data rate transmission, power andsecurity.

9.3. Chaos Based Secure Optical Communications
Chaos-based optical communications using laser intensity have attracted intensive
research interest due to its encryption capabilities [23]. The information signal is used
intensity of modulated laser, which is set to operate in the chaotic oscillation of thepolarization mode, and can be recovered at the receiver upon synchronization with an
identical laser [17, 24]. Chaotic systems with a number of unique features including anoise-like shape with a broadband spectra, lower power implementations, andnonlinearity, have become appealing for modern secure communications applications. In
such systems, a message is encoded into a noise like signal generated by a laser sourcewith the chaotic behavior [25].
The chaotic communication system has three main parts, as depicted in Fig. 9.2,transmitter, channel, which including noise, and receiver [26]. The time delay between
the two chaotic synchronization waveforms is the key to distinguish massage from these
two signals of synchronization. It is important to match the corresponding parameters(internal and external) of transmitter and receiver to achieve a complete synchronization.
External parameters such as bias current, temperature and so on of both transmitter andreceiver can be controlled easily; however, the internal parameters such as optical loss,
gain and quantum efficiency are difficult to be accurately controlled. Under the additive
chaos modulation encryption scheme, the encoded messages can be successfully extracted
based on a good synchronization oscillation between the transmitter and receiver signal.
At the transmitter a chaotic carrier is employed to improve the overall security of thesystem. The modulated chaotic carrier signal is transmitted over the communication
channel. At the receiver detection is carried out using exactly the same chaotic oscillator
as in the transmitter to ensure successful recovery of the information signal.
Successful message encoding/decoding in an unpredictability-enhanced chaotic VCSELs
system is achieved numerically via the chaos-shift-keying technique. Using two VCSELs,
which are subject to OF, and polarization-preserved optical injection, are adopted.
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Chapter 9. Laser Characterizations for Optical Wireless Communications

Improved decoding performance is achieved by choosing a proper polarizer angle, and the
security is, to some extent, enhanced owing to the variple polarization OF [27]. Message
encoding and decoding using the chaotic of VCSELs devices has been experimentallydemonstrated in [28].

Signal + Noise

ChannelSynchronization
technique
Demodulator

Chaotic
signal

MassageTransmitter

ReceiverFig. 9.2. Block diagram of chaotic communication system.

9.4. Vertical-cavity Surface-emitting Lasers (VCSELs)
VCSELs are relatively recent type of semiconductor laser devices, which are attractive
for a number of applications from sensors to telecommunications. VCSEL is asemiconductor laser (SL), which has a resonant cavity that is vertically formed with thesurfaces of the epitaxial layers. Typically SLs devices consist of semiconductor layersgrown on top of each other on a substrate. Usually the growth processing is achieved in amolecular-beam-epitaxy (MBE) or metal-organic-chemical-vapor-deposition (MOCVD)
growth reactor. Fig. 9.3 shows the oxide-confined VCSEL top-emitting structure [26].
The construction of VCSEL is particularly different from other lasers, an active layer
sandwiched between highly reflectivity mirrors in range of about 99.5 to 99.9 %, whichis placed at the bottom and top of the device structure as shown in the Fig. 9.3. Thesemirrors consist of distributed Bragg reflectors (DBRs), which is made up of several
quarter-wavelength-thick layers of semiconductors. These layers have alternatively highand low refractive indexes. The VCSEL emission can be from the top or bottom of thedevice [29]. There are several categories of VCSELs based on the optical and electricalconfinement techniques, active layer design and wavelength emission. The mostimportant issue in fabrication such devices is the optical confinement factor, because of
the conversion efficiency of the electrical-to-optical signal.
VCSEL has become a very important source, which effectively displacing EELs forapplications such as high speed data communication and chaos for local area networks[30]. VCSELs consist of a small active volume (few μm) and these have a very low
threshold current (few μA) compared with EELs (~300 μm) [31]. This is one of theattractive features that give the higher reliability than edge-emitters devices.
399

Advances in Optics: Reviews. Book Series, Vol. 5n-contactLight
outputOxide
Active Aperture
RegionDBRsp-contact
Substrate

Fig. 9.3. VCSEL with a selectively-oxidized top-emitting structure [26].

9.4.1. Optical and Electrical Properties of VCSELs
As discussed above the structure of VCSEL is completely different from othersemiconductor structures devices that have been realised so far, such as distributed
feedback laser (DFB) or EELs [32]. What makes VCSEL structure unique is that theemitting light is perpendicular to the surface of the laser. Consequently, this dramaticdifference in the VCSEL structure makes a similar design technique impossible forconvention facet emitting lasers. Because of the main concern in these devices is toachieve high-longitudinal side-mode suppression, which is completely disregarded in the
VCSEL design [33].
VCSELs gained a reputation as a superior technology for applications such as
fibre-channel, Gigabit Ethernet and intra-systems, free space optical communications,
optical fibre communications and optical recording [34-36]. However, VCSELs have anumber of problems based on large frequency chirp and polarization insensibility, which
limit their performance in fibre-optic communication systems, as well as causing
limitations in transmission distances and speed [37]. Such drawbacks are related to thelaser noise properties and also depend on polarization mode fluctuation [38]. VCSELs arevery sensitive to the effects of optical feedback (OF) and optical injection because of theirhigh gain and very short cavity length [39].
9.4.1.1. Light-current (L-I) Curve Characteristics
VCSEL when operating near threshold current usually lases in a single polarization mode[40]. However, VCSEL can oscillate with the orthogonally polarized simultaneously,
owing to their circular symmetry structure [41]. VCSEL emit mainly linearly polarized
light. However, its orientation is not well distinguish because of the laser cavity and thegain medium are quasi isotropic in the active layer. In most sensing applications and data
communications, a polarization stability of VCSEL and how it can be controlled isessential [42]. Consequently, the polarization mode linearity of VCSELs when modulated
is of critical concern; particularly, in applications such as optical communications and400

Chapter 9. Laser Characterizations for Optical Wireless Communications

optical memory [43]. One of the essential parameter of a SL is light-current (L-I) curve,
the laser efficiency can be predicted from the L-I curve properties [44].
The VCSEL lases in a fundamental mode with two orthogonal polarization modes for theentire range of bias current 𝐼𝑏 used as shwing in an experimental Fig. 9.5 (b). Fig. 9.4
displays the experimental setup for the free running L-I curve measurements of VCSELs.
The laser is driven by laser diode driver (Newport, 505B) and is temperature controlled
by temperature controlled with a thermoelectric temperature controller (TED 200C) towithin 0.01 oC. The laser output is collimated using objective lens (Aspheric Lens,
f = 4.51 mm). A half wave plate (HWP) (Zero-Order Half-Wave Plate) and a polarization
beam splitter (PBS) (Cube, 620-1000 nm) are used to direct the orthogonal polarizationsof the VCSEL to the photodetectors (PD) (New Focus Nanosecond photo detector, model
No). 1621. The HWP and PBS are used only when measured the VCSEL polarizationresolved L-I curve using an optical power meter (Anritsu, ML9001A) and then removed
when measuring the total output power (IT). The L-I plots were obtained using LabVIEW
controlled by a personal computer.
Power meter

Temperature&
current controller

PD
HWP
Lens

PD

VCSEL

PBS

Fig. 9.4. Experimental setup to measure output power properties of VCSELs, half wave plates(HWP), polarization beam splitter (PBS) and photo detector (PD).

Fig. 9.5 displays the L-I characteristics of VCSEL, where in Fig. 9.5(a) the total output
power and in Fig. 9.5(b) the polarization-resolved output power of the VCSEL arepresented. The first lasing mode in Fig. 9.5(b) with a full square black line refers to the
X-polarization (XP) and the suppressed mode with a full dot red line corresponds to theY-polarization (YP) mode. Fig. 9.6 shows the total output power (a) and the polarization
resolved output power (b) of the standalone VCSEL combined with two polarization
switching (PS) without any external perturbation. PS occurs between the orthogonal
polarization modes of the VCSEL when 𝐼𝑏 increase from zero to 9 mA. The first type ofswitching (PSI) occurs at ~6.3 mA from the high-frequency mode (XP) to the mode withlow frequency (YP). While the second type of switching (PSII) is observed at 𝐼𝑏 of~7.6 mA corresponding to PS from the XP (low frequency mode) to the orthogonal mode
(YP) with the high frequency mode, as depicted in Fig. 9.6(b). Both types of switching(i.e., PSI and PSII) are defined and studied in detail in [45].
401Advances in Optics: Reviews. Book Series, Vol. 5
0.6

Total

Output power (mW)

Output power (mW)

0.6

0.4
(a)
0.2

0.0
0

2

4

6

XP-ModeYP-Mode

0.4
(b)
0.2

0.0
0

8

2

4

6

8

Bias Current (mA)

Bias Current (mA)

Fig. 9.5. Light-current characteristics of free-running VCSELs under study (a) the total output
power and (b) the polarization-resolved output power.

Output power (mW)

Output power (mW)

XP-ModeYP-Mode

0.6

Total

0.6
0.5
0.4

0.3 (a)
0.2
0.1
0.0

0.5
0.4
0.3 (b)
0.2
0.1
0.0

0

2

4

6

Bias Current (mA)

8

0

2

4

6

8

Bias Current (mA)

Fig. 9.6. The L-I characteristic of standalone VCSEL: (a) Total output power;(b) Polarization-resolved output power with two PS.

9.4.1.2. Polarization Switching
Most VCSELs devices typically emit linearly polarized light. In fact, instability is acommon features in VCSELs devices owing to weak material and cavity anisotropies [46],therefore polarization instability can happen without external perturbation. However,under some conditions - such as when the bias current is increased or an external OF is
introduced - the linearly polarized state switches to the orthogonal linearly polarized state.
This usually occurs due to changes in the gain and loss of the orthogonally polarizedmodes [47], and changes in the operating temperature as well as the magnitude anddirectionally of the bias current [48]. Based on the relevant studies in the literature,experimental and theoretical works on VCSELs have shown that increasing the injection
current and the OF level can lead to increased polarization inversion and mode
competition between the laser polarization modes. Mode competition is involving
multiple switching, which destabilizes the laser emission and dynamic solutions.
Interestingly, variable polarization based OF was recently proposed for external cavity
feedback as an effective tool to control the polarization instability of VCSEL [46]. The
PS features of SLs are of interest for a wide range of applications including opticalswitching, storage system and high capacity data processing. It has been observed that PS
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Chapter 9. Laser Characterizations for Optical Wireless Communications

can occur under the fixed bias current and OF conditions [49]. In VCSEL the position of
PS is mainly determine by three factors; the net gain of the two polarization modes, theelectric field and the injection current [50]. A critical issue in developing such laser
devices is how to determine and control their polarization instability. Controllable PS has
been investigated in [40, 51, 52] by considering the effects of a number of parameters
including the OF strength, optical injection and frequency detuning.
The first experimental work that demonstrated the PS in VCSELs was reported in 1993[53], using optical injected. Since that time the polarization bi-stability of VCSELs has
been the subject of extensive research from both theoretical and experimental perspective.
However, the majority of previous studies have focused on influences of conventional OFon the dynamics and polarization characteristics of VCSELs [54-56]. The polarization
properties of SLs subject to the OF with non-rotated [40, 41, 50-52, 54, 55, 57-61] andvariable rotated polarization angles have been studied experimentally and theoretically in[49, 62-66]. It has shown that a laser under the OF effect can emit in certain polarization
mode and increasing the level of dose lead to suppressed the polarization instability.
Controllable PS of VCSEL has also been investigated in [40, 51, 52], where the effects ofa number of parameters such as the OF strength, optical injection and frequency detuning
were considered. Polarization mode hopping is also observed, which is associated withimprovement of the antiphase synchronization dynamics [67].
Recently, numerical simulations results on the influence of polarization-rotated OF on thepolarization properties of VCSEL have been reported in [49]. It has been observed that
PS can be occur even for a fixed bias current and an OF level. Furthermore, PS propertiesare depending on the OF level and the polarization angle 𝜽𝒑 , where smaller 𝜽𝐩 is required
to implement the PS when the OF level increase. In VCSELs it is possible to achieve PS
by means of thermal effects [68], optical injection [69], strong enough OF [70] and rotatedpolarization angle of [65]. In the later, it was shown that by employing a polarization
controller to obtain the variable polarization angle, VCSEL exhibit dominant PS for a
fixed bias current and OF for the case of the selected polarization feedback, whereas incase of preserving the OF the dominant PS is not observed in the entire parameter space.

9.5. OWC Based on VCSELs
Since thirty seven years ago, the VCSELs idea which was established in Japan is growing
up rabidly worldwide [71]. After that VCSEL technology offers an exceptional advantage
to high speed OWC links. Thanks to their unique features of low costs, power efficiency
and low threshold current as well as desired output-power beam profile (circular and,single-longitudinal operation) with small divergence angle, VCSEL have introduced inwide ranges of applications including optical sensing, optical switching, short and longdata transmission and networks, optical storage, optical HDMI cables, intra-chipcommunications, network security etc. [72-76]. Recently, VCSELs have been used as apromising candidate for photonic neuron networks. A neuron-like response dynamic
based on polarization switching (PS) and dynamics nonlinearity, which have reported in
VCSEL devices with applying polarization dynamic injection [47, 77, 78]. Underwater
wireless communication (UWC) technologies have utilized VCSELs owing to desired403

Advances in Optics: Reviews. Book Series, Vol. 5

modulation bandwidth range which past GHz range [79-81]. OWC based on VCSELs is
becoming more common in wide variety of applications in optical communication
including FSO communication [82], UWC [80], Light Fidelity or LiFi transmission basedon VCSEL with injection technique, which reported in [82] and VCSELs’ chaos-based
data transmission security as demonstrated in [67, 83]. VCSEL also adaptive for future
metro applications, where it presented up to 50 Gb/s connections per flow over hundredsof km to address the needs of capacity and support multi-terabit connections
networks [84].
Among wide light sources, VCSELs are recommended to be best candidate for both
fiber-optics and wireless communication systems, in terms of their cost efficiency, higher
data transmission and low power consumption [85, 86]. VCSELs are also considered tobe ideal for gigabit Ethernet and optical interconnects [34, 87]. Lately, in [88, 89],high-speed and energy-efficient 850 nm VCSELs with direct modulation operated errorfree up to 57 Gbit/s. The VCSELs’ low power consumption minimizes the overall power
dissipation of the optical sources, when it is used in high-speed data transmission systems.
Conventional light sources, such as light-emitting diodes at a low-biasing current,
experience an efficiency droop, which is a direct consequence of non-radiative
recombination [90]. Both [91, 92] have shown that lowering the drive current of opticalsources can achieve lower power consumption, as well as higher modulation speed.
Indeed, reducing the lasing current below the threshold is essential, in order to decrease
the optical device’s heat dissipation and increase its modulation bandwidth. Certainly
VCSELs recorded very low threshold current of several micro ampere with high-efficient
output power and performance [32, 93].

9.6. Nonlinearity of VCSELs
The nonlinear behaviour of SL with modulation signal is a subject of great interest in
many researches [94] particularly for optical communication due to, for instance theireffects to limit the RF dynamic range of optical devices. The nonlinear behaviour of
optical devices such as EEL and VCSEL is a major limiting factor in analogue opticalcommunications [95] and has been widely investigated [96-99]. The nonlinear dynamics
of VCSELs as a result of optical injection and OF was investigated in [96]. Noise anddifferent physical mechanisms, such as spatial-hole burning, phase coupling, gain
anisotropy and birefringence effects can lead to rich nonlinearity properties in SL. In[100, 101] it was shown that nonlinear induced harmonic distortion can lead to decreased
power efficiency of VCSEL. Recently, special attention has been paid to light polarization
in VESCELs [96].
In the VCSELs under current modulation (CM) the nonlinear dynamics and chaos areeasily realized due to the mode competition or polarization mode switching [102].
Furthermore, in semiconductors laser the nonlinear distortion appears when the laser isbeing driven near threshold current [101]. High nonlinearity in the L-I curve of VCSEL
devices has been reported when the laser is biased near the threshold current at highmodulating frequency [97], which has led to harmonic distortions. Nonlinear gain
saturation of the lasing transition produce a nonlinear behaviour in the L-I characteristics
404

Chapter 9. Laser Characterizations for Optical Wireless Communicationsof VCSELs devices, which lead to increase the relaxation oscillation damping andeliminate the light intensity through the reduction of differential gain [103]. The result
obtained in [104] demonstrated that the nonlinearity of VCSELs can entirely suppressedwith orthogonal OF, and the 3-dB bandwidth of the devices is enhanced. As well as the
harmonics distortions are significantly improved and reduced to the noise floor. This
introduced OF as an effective tool to recover the nonlinearity behavior and improve theanalogue performance of VCSELs.

9.7. VCSELs-based Chaos Dynamics
In a number of applications such as communication systems, high pump sources andoptical fiber systems, a potential light source that could be adopted is the VCSEL withunique features including high data rates [105], low power consumption and lower
manufacturing costs [86]. VCSELs are sensitive to the OF and optical injection due totheir higher mirror reflectivity inside the laser cavity [106, 107]. However, OF does leadto a number interesting complex dynamic behaviors in VCSELs including chaotic,
time-period pulsing dynamics and PS [91, 108]. In the past few years, the chaotic
dynamics of VCSELs with OF has been the subject of interest by researchers theoreticallyand experimentally [17, 18] particularly for secure optical-based communication systems.
Recently, synchronization of the orthogonal polarization modes of VCSEL received wide
attention as a chaotic source in secure communication systems [109, 110]. The anti-phasecorrelation of a semiconductor laser was experimentally observed in [7] for the case where
the chaotic oscillation of the polarization modes was lower than the relaxation oscillation
frequency. In VCSELs, detailed characteristics of chaos synchronization outlined in [15]showed that the anti-phase chaotic synchronization can be achieved between orthogonal
polarization modes of the two mutually coupled VCSELs. An anti-phase synchronization
phenomenon is a common feature in semiconductor lasers with polarization rotated OF
technique. Furthermore, the time lag between the two synchronization signals can be zeroin complete chaos synchronization [111]. A solitary VCSEL can exhibits strong
anti-phase dynamics between orthogonal polarization modes [110]. Numerically andexperimentally study in [11] outlined that the two polarization modes of VCSEL exhibit
anticorrelated dynamics with OF. However, a study presented an anti-phase oscillation
between transverse electric (TE) and transverse magnetic (TM) modes in a semiconductor
laser with polarization-rotated OF [112]. Maximum values for the correlation coefficientof the synchronization modes of -0.68 and -0.99 obtained experimentally andtheoretically, respectively with a zero time lag between the orthogonal modes underappropriate conditions have been reported [112].
In [113] demonstrated experimentally high anti-phase cross-correlation value of (0.99)
between the polarization modes of a VCSEL with zero time delay between thesynchronized signals. It shows that VCSEL can produce high-quality anti-phasepolarization-resolved chaos synchronization between their polarisation modes underrotating polarization-preserved OF. Furthermore, its allow illustration that a different
chaotic pattern can be achieved by means of rotated-polarization OF, which indicates thecomplex dynamic behaviour associated with the rotated polarization angle [114].
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9.7.1. Chaos Synchronization in VCSELIn the recent years, chaos synchronization has become a hot topic due to their potential
applications in optical communication systems where security is paramount. One of theimportant applications of the synchronized chaotic system is for encrypted
communications, where the chaotic dynamic of an optical laser is controlled via
synchronization phenomena using OF. Chaotic synchronization based on semiconductor
lasers and their applications in secure communications have attracted considerable
attention. The first demonstration of chaos synchronization was by Pecora and Carroll[115]. Recently, chaos synchronization in optical communications systems employing SL
diodes has been a hot topic in applications where security is paramount. In such systems,
a message is encoded into a noise like signal generated by laser source with chaotic
behaviour [25]. In this context, chaos-based optical communications have attracted
intensive research interest due to its encryption capabilities.
A popular optical source adopted in such systems is the VCSELs because of their uniquefeatures especially when subjected to polarization OF. Furthermore, a different chaotic
pattern and complex dynamics can be achieved under rotated-polarization angle of [17,64, 114, 116]. Nowadays, we have seen a number of experimental and theoretical research
activities on chaos, chaos synchronization and communication characteristics of VCSELs.
Synchronization of the chaos is achieved experimentally in un-directionally coupledexternal-cavity vertical-cavity surface-emitting SLs operating in an open-loop regime.
The polarization of the injected beam was perpendicular to that of the free-runningreceiver (x polarization). The injected beam and the y-polarized component of the receivershow good synchronization [17, 27, 28, 117-123]. It has been shown that two VCSELs
can be synchronized under appropriate conditions. However, synchronization is lost at ahigher mirror reflectivity (i.e., 50 %).
Chaotic oscillations can be achieved with an external OF and optical injection[17, 94, 124], even more with a free running operation of semiconductor lasers [125].
Several theoretical and experimental studies have reported that the dynamics ofpolarization-rotated OF [126] in a semiconductor laser is quite different from theconventional OF [64, 114, 125]. Synchronization of chaos have attracted increasing
attention especially in coupled VCSELs with a polarization-rotated OF and opticalinjection [17, 107]. The rotated polarization OF has employed to suppress the polarization
nonlinearity [104] and polarization switching in VCSEL with the modulation signal [127].
In contrast, a higher external reflectivity leads to a more chaotic behaviour in the output
power of VCSELs. VCSEL’s dynamic can be driven into the chaotic regime by means ofan external OF. The synchronization has been observed over a range of values of thecoupling parameter such as the drive current, external reflectivity and coupling coefficient[117]. Solitary VCSEL can exhibits strong anti-phase dynamics between own orthogonal
polarization modes [110]. However, one study has presented an anti-phase oscillation
between transverse electric (TE) and transverse magnetic (TM) modes in a SL withpolarization-rotated OF. It has achieved a maximum value of correlation coefficient of thesynchronization modes of -0.68 (-0.99) experimentally (theoretically) with zero time lag
between the orthogonal modes under appropriate conditions [112]. The antiphase chaotic
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Chapter 9. Laser Characterizations for Optical Wireless Communications

synchronization is enhanced as the angle of orthogonal polarization of the OF is increased.
Polarization modes are oscillated entirely in the chaotic regime in antiphasesynchronization, with no time delay at low and high bias currents of 1.2 mA and 1.7 mA,
respectively [67].
However, the synchronization quality of the two modes completely deteriorates when thebias current is increased to 1.7 mA at a polarization angle of 70o, where power mode
differences are increased.

9.8. Theoretical Analysis of Lasers Dynamics
In order to assess semiconductor lasers performance theoretically, it is imperative to utilise
rate equations describing the time variation of the carrier and photon density.
9.8.1. Rate Equations Describing Carriers and Photons Density Dynamics
These equations are shown below describing the rate equations for the carrier decay N andthe photon decay S as a function of time t. These can be written as the subtracting carrier
recombination rate via laser emission G νg S, and loss mechanisms N/τc, from the carrier
generation rate ηi I/qV, by stimulated and spontaneous emissions [64, 97, 128]:
𝑑𝑁(𝑡)
𝜂𝑖 𝐼
𝑁(𝑡)
= 𝑞𝑉
− 𝜏 − 𝐺 (𝑡)𝜈𝑔 𝑆(𝑡)
𝑑𝑡
𝑐

(9.1)

The sources of photons are stimulated and spontaneous emissions. However, only a small
fraction of the spontaneous emission is coupled into a given laser mode and thiscomponent may often be neglected. Losses of photons are governed by a photon
lifetime τp , is given by:
dS(t)
S(t)
= G (t)νg S(t) + Γβsp R sp −
,
dt
τp

(9.2)

where I is the current density, ηi is the injection efficiency, V is the active volume, q is theunit charge, G is the gain coefficient, 𝜏𝑐 is the carrier lifetime, vg is the photon groupvelocity, Γ is the confinement factor, ßsp is the spontaneous emission factor, and R sp is
the spontaneous recombination rate. The gain coefficient is assumed to be a linear functionof carrier density and can be expressed as:
N(t)−N

G(t) = g ο 1+ϵS(t)ο

(9.3)

Here g ο is the linear gain coefficient, ϵ the gain saturation coefficient, and Nₒ is the carrier
number at transparency. At the study-state condition, the gain of a laser above thresholdshould always equal the threshold gain, otherwise optical amplitude continues to increase
which cannot happen in the steady-state. The same argument is valid for the carrier density
in the steady-state because of both G and N are combine in the cavity of the laser, so [129]:407

Advances in Optics: Reviews. Book Series, Vol. 5

G = Gth , (I > Ith ),(9.4)

N = Nth , (I > Ith )(9.5)

At the study state condition, gain = loss and can be assumed to be constant, therefore
we have:
1

Gth = τ ,

(9.6)

p

where Ith is the threshold current. The average modal gain can be written as:
1

1< G >th = ΓGth = <ai > + L ln ( R),(9.7)

where L is the total cavity length, R is the mean mirror reflectivity, for simplicity the111
mirror loss term ( L ln( R) ) is abbreviated as am and the photon loss ( τ ) is defined by:p

1
τp= νg ( < ai > +am )(9.8)

The internal cavity loss of the laser is ( ai + am ), ai is the photon losses, and am is themirror loss parameter.
ΓGth =1
τp νg= < ai > +am(9.9)

At the study state condition and from equation (9.1) the photon density above threshold
can be written as:
η (I−I )ith
S = qVG, (I > Ith )νth g(9.10)

To obtain the output power of the laser we have to extract the total optical energy inside
the cavity Ein , which is defined in terms of the energy loss rate through the mirrors(νg am ) as:
Ein = Shν Vc,(9.11)

Pout = Shν Vc νg am ,(9.12)

where Vc is the cavity volume, and hv is the photon energy. Substituting from Eqs. (9.10)
and (9.9) and using Γ = V/Vc in Eq. (9.12), now the laser out power can be written in acommon expression as [130]amPout = ηi a +aihν
(I − Ith )m q

The differential quantum efficiency is given by:
408(9.13)

Chapter 9. Laser Characterizations for Optical Wireless Communicationsamηd = ηi a +aim

(9.14)

We can simplify Eq. (9.13) to be as:hνPout (t) = ηd q (I(t) − Ith )(9.15)

When a laser is subject to direct modulation current, the total injection current isgiven by:I(t) = Idc + Im (t),(9.16)

Idc is the bias current (time-independent), and Im (t) is the modulating current (timevarying), which could be given in a sinusoidal form as:
Im (t) = Im eifmt ,(9.17)

where fm is the modulation frequency (Hz).

9.9. Conclusions
In this chapter, an overview to optical wireless communication (OWC), underwater
wireless optical communication (UWOC), Chaos-based optical communications andproperties of one of the most important and commonly used semiconductor devices incommunication systems (VCSEL), including of the optical chaos, optical and electricalproperties, L-I characteristics, polarization switching (PS), and nonlinearity behaviours
were presented. In addition, theoretical analyses for the dynamical properties of thesemiconductor lasers (SLs) using the rate equations were described. It was known thatOWC is a powerful and promising technique to increase the data transmission rat incommunication systems. Laser devices are strongly introduced as alternative sources for
OWC system due to their advantages in the bandwidth and speed properties overtraditional sources.
Although, properties of SLs such that VCSELs, for OWC have intensively been studied
however, they are still under active investigation due to its potential applications as insecure communication and nonlinear optical systems. It was referring that the SLs
properties are still under active study and not fully understood, especially when subjected
to an external pertraptions. The dynamical properties of SLs were describe in a theoreticalmodel and discussed in detail using rate equations to better understanding of the laser
behaviours. Most OWC systems use a wavelength range between 780 to 850 nm. For thiswavelength range, which has a low attenuation less than 0.2 dB/km and the optical devices
are available and inexpensive. VCSELs devices can operated in instability polarization
modes emitting owing to weak material and cavity anisotropies, therefore OF technics
utalized to control the polarization instability of such devices. VCSEL under the effect ofpolarization OF can be a worthy applicant as a chaotic source for secure communication
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Advances in Optics: Reviews. Book Series, Vol. 5

system. For UWOC, optical wave-based communication offer high-data rate with power
efficiency compared with RF and acoustic channels.

Acknowledgements
The author would like to acknowledge research assistance from College of engineering
University of Diyala.

References
[1].
[2].
[3].

[4].
[5].
[6].

[7].
[8].
[9].

[10].
[11].
[12].

[13].

[14].
[15].
[16].
[17].

410

M. Z. Chowdhury, et al., A comparative survey of optical wireless technologies:
Architectures and applications, IEEE Access, Vol. 6, 2018, pp. 9819-9840.
H. Kaushal, V. Jain, S. Kar, Free Space Optical Communication, Vol. 1, Springer, 2017.
G. Isoe, R. Karembera, T. Gibbon, Advanced VCSEL photonics: Multi-level PAM forspectral efficient 5G wireless transport network, Optics Communications, Vol. 461, 2020,125273.
W. Popoola, Subcarrier intensity modulated free-space optical communication systems,
Northumbria University, 2009.
S. G. Wilson, et al., Free-space optical MIMO transmission with Q-ary PPM, IEEE
Transactions on Communications, Vol. 53, Issue 8, 2005, pp. 1402-1412.
N. Cvijetic, S. G. Wilson, M. Brandt-Pearce, Performance bounds for free-space optical
MIMO systems with APD receivers in atmospheric turbulence, IEEE Journal on Selected
Areas in Communications, Vol. 26, Issue 3, 2008, pp. 3-12.
R. Paudel, et al., High speed train communications systems using free space optics,
Northumbria University, 2010.
S. Bloom, et al., Understanding the performance of free-space optics, Journal of Optical
Networking, Vol. 2, Issue 6, 2003, pp. 178-200.
X. Liu, et al., Laser-based white-light source for high-speed underwater wireless opticalcommunication and high-efficiency underwater solid-state lighting, Optics Express,
Vol. 26, Issue 15, 2018, pp. 19259-19274.
H.-J. Song, et al., Terahertz wireless communication link at 300 GHz, in Proceedings of theIEEE International Topical Meeting on Microwave Photonics (MWP’10), 2010, pp. 42-45.
A. K. Majumdar, Advanced Free Space Optics (FSO): A Systems Approach, Vol. 186,
Springer, 2014.
W. Jin-Yuan, et al., Outage analysis for relay-aided free-space optical communications overturbulence channels with nonzero boresight pointing errors, IEEE Photonics Journal,
Vol. 6, Issue 4, 2014, pp. 1-15.
A. M. Khalid, et al., 1-Gb/s transmission over a phosphorescent white LED by using
rate-adaptive discrete multitone modulation, IEEE Photonics Journal, Vol. 4, Issue 5, 2012,
pp. 1465-1473.
H. Soda, et al., GaInAsP/InP surface emitting injection lasers, Japanese Journal of Applied
Physics, Vol. 18, 1979, pp. 2329-2330.
R. S. G. a. L. A. Coldren, Sub-milliamp threshold vertical cavity laser diodes, Appl. Phys.
Lett., Vol. 16, Issue 57, 1991, pp. 1605-1607.
J. A. Tatum, et al., Commercialization of Honeywell's VCSEL technology, Proceedings of
SPIE, Vol. 3946, 2000.
J. Liu, Z.-M. Wu, G.-Q. Xia, Dual-channel chaos synchronization and communication basedon unidirectionally coupled VCSELs with polarization-rotated optical feedback andChapter 9. Laser Characterizations for Optical Wireless Communications

[18].

[19].
[20].
[21].

[22].

[23].
[24].

[25].

[26].
[27].
[28].

[29].
[30].

[31].
[32].
[33].
[34].
[35].

[36].

[37].

polarization-rotated optical injection, Optics Express, Vol. 17, Issue 15, 2009,
pp. 12619-12626.
Y. Hong, P. Spencer, K. Shore, Wideband chaos with time-delay concealment in verticalcavity surface-emitting lasers with optical feedback and injection, IEEE Journal of
Quantum Electronics, Vol. 4, Issue 50, 2014, pp. 236-242.
N. Saeed, et al., Underwater optical wireless communications, networking, and localization:
A survey, Ad Hoc Networks, Vol. 94, 2019, 101935.
J. Shi, S. Zhang, C.-J. Yang, High frequency RF based non-contact underwatercommunication, in Proceedings of the OCEANS-Yeosu Conference, 2012, pp. 1-6.
X. Liu, et al., 34.5 m underwater optical wireless communication with 2.70 Gbps data rate
based on a green laser diode with NRZ-OOK modulation, Optics Express, Vol. 25,
Issue 22, 2017, pp. 27937-27947.
S. Hu, et al., 35.88 attenuation lengths and 3.32 bits/photon underwater optical wirelesscommunication based on photon-counting receiver with 256-PPM, Optics Express,
Vol. 26, Issue 17, 2018, pp. 21685-21699.
M. Sciamanna, K. Shore, Physics and applications of laser diode chaos, Nature Photonics,
Vol. 9, Issue 3, 2015, pp. 151-162.
Y. Hong, Experimental study of time-delay signature of chaos in mutually coupled verticalcavity surface-emitting lasers subject to polarization optical injection, Optics Express,
Vol. 21, Issue 15, 2013, pp. 17894-17903.
N. Jiang, et al., Multiaccess optical chaos communication using mutually coupled
semiconductor lasers subjected to identical external injections, IEEE Photonics Technology
Letters, Vol. 22, Issue 10, 2010, pp. 676-678.
S. Al Zaidi, Optical and electrical characteristics of vertical-cavity surface-emitting lasers
for free space optical communications, Northumbria University, 2016.
S. Y. Xiang, et al., Message encoding/decoding using unpredictability-enhanced chaotic
VCSELs, IEEE Photonics Technology Letters, Vol. 24, Issue 15, 2012, pp. 1267-1269.
M. W. Lee, Y. Hong, K. A. Shore, Experimental demonstration of VCSEL-based chaotic
optical communications, IEEE Photonics Technology Letters, Vol. 16, Issue 10, 2004,
pp. 2392-2394.
F. H. Peters, et al., Vertical-cavity surface-emitting laser technology, Proceedings of SPIE,
Vol. 1851, 1993.
A. Valle, L. Pesquera, Theoretical calculation of relative intensity noise of multimodevertical-cavity surface-emitting lasers, IEEE Journal of Quantum Electronics, Vol. 40,
Issue 6, 2004, pp. 597-606.
A. Hayat, et al., Optical Injection-Locking of VCSELs, INTECH Open, France, 2010.
K. Iga, Surface-emitting laser-its birth and generation of new optoelectronics field, IEEE
Journal of Selected Topics in Quantum Electronics, Vol. 6, Issue 6, 2000, pp. 1201-1215.
S. F. Yu, Analysis and Design of Vertical Cavity Surface Emitting Lasers, Vol. 7, John
Wiley & Sons, 2003.
I. Kenichi, Vertical-cavity surface-emitting laser: Its conception and evolution, Jpn. J. Appl.
Phys., Vol. 47, Issue 1R, 2008, 1.
M. Ahmed, et al., Enhancing the modulation bandwidth of VCSELs to the millimeterwaveband using strong transverse slow-light feedback, Optics Express, Vol. 23, Issue 12,2015, pp. 15365-15371.
A. Boletti, et al., Performance analysis of communication links based on VCSEL and silicon
photonics technology for high-capacity data-intensive scenario, Optics Express, Vol. 23,
Issue 2, 2015, pp. 1806-1815.
D. Parekh, Optical injection locking of vertical cavity surface-emitting lasers: Digital andanalog applications, PhD Thesis, University of California, Berkeley, 2012.

411

Advances in Optics: Reviews. Book Series, Vol. 5[38]. Y. Hong, et al., The effects of polarization-resolved optical feedback on the relative
intensity noise and polarization stability of vertical-cavity surface-emitting lasers, Journalof Lightwave Technology, Vol. 24, Issue 8, 2006, pp. 3210-3216.
[39]. A. Uchida, Optical Communication with Chaotic Lasers: Applications of Nonlinear
Dynamics and Synchronization, John Wiley & Sons, Saitama, Japan, 2012.
[40]. Y. Hong, P. S. Spencer, K. A. Shore, Suppression of polarization switching invertical-cavity surface-emitting lasers by use of optical feedback, Optics Letters, Vol. 29,
Issue 18, 2004, pp. 2151-2153.
[41]. K. D. Choquette, et al., Gain-dependent polarization properties of vertical-cavity lasers,
IEEE Journal of Selected Topics in Quantum Electronics, Vol. 1, Issue 2, 1995,
pp. 661-666.
[42]. J. M. Ostermann, R. Michalzik, Polarization control of VCSELs, in VCSELs, Springer,
2013, pp. 147-179.
[43]. M. Wu, et al., Polarization induced enhancement of relative intensity noise and modulation
distortion of vertical cavity surface emitting lasers, in Guided-Wave Optoelectronics,
Springer, 1995, pp. 59-65.
[44]. J.-W. Shi, et al., High-power and high-speed Zn-diffusion single fundamental-mode
vertical-cavity surface-emitting lasers at 850-nm wavelength, IEEE Photonics Technology
Letters, Vol. 20, Issue 13, 2008, pp. 1121-1123.
[45]. A. Valle, et al., Transverse mode switching and locking in vertical-cavity surface-emitting
lasers subject to orthogonal optical injection, IEEE Journal of Quantum Electronics,
Vol. 43, Issue 4, 2007, pp. 322-333.
[46]. S. Y. Xiang, et al., Influence of variable-polarization optical feedback on polarization
switching properties of mutually coupled VCSELs, IEEE Journal of Selected Topics
in Quantum Electronics, Vol. 19, Issue 4, 2013, pp. 1700108-1700108.
[47]. J. Martin-Regalado, et al., Polarization properties of vertical-cavity surface-emitting lasers,
IEEE Journal of Quantum Electronics, Vol. 33, Issue 5, 1997, pp. 765-783.
[48]. K. Panajotov, et al., Polarization switching in VCSEL's due to thermal lensing, IEEE
Photonics Technology Letters, Vol. 10, Issue 1, 1998, pp. 6-8.
[49]. S. Xiang, et al., Polarization properties of vertical-cavity surface-emitting lasers subject tofeedback with variably rotated polarization angle, Applied Optics, Vol. 48, Issue 27, 2009,
pp. 5176-5183.
[50]. H. Yanhua, et al., Investigation of polarization bistability in vertical-cavity surface-emitting
lasers subjected to optical feedback, IEEE Journal of Quantum Electronics,
Vol. 41, Issue 5, 2005, pp. 619-624.
[51]. Y. Chung, Y. Lee, Spectral characteristics of vertical-cavity surface-emitting lasers withexternal optical feedback, IEEE Photonics Technology Letters, Vol. 3, Issue 7, 1991,
pp. 597-599.
[52]. J. Paul, et al., Impact of orthogonal optical feedback on the polarization switchingof vertical-cavity surface-emitting lasers, JOSA B, Vol. 24, Issue 8, 2007, pp. 1987-1994.
[53]. Z. G. Pan, et al., Optical injection induced polarization bistability in vertical‐cavity surface‐emitting lasers, Applied Physics Letters, Vol. 63, Issue 22, 1993, pp. 2999-3001.
[54]. P. Spencer, C. R. Mirasso, K. A. Shore, Effect of strong optical feedback on vertical-cavity
surface-emitting lasers, IEEE Photonics Technology Letters, Vol. 10, Issue 2, 1998,
pp. 191-193.
[55]. C. Masoller, N. Abraham, Polarization dynamics in vertical-cavity surface-emitting lasers
with optical feedback through a quarter-wave plate, Applied Physics Letters, Vol. 74,
Issue 8, 1999, pp. 1078-1080.
[56]. X. F. Li, et al., Static and dynamic characteristics of VCSELs with polarisation-selective
optical feedback, IEE Proceedings – Optoelectronics, Vol. 153, Issue 2, 2006, pp. 67-74.

412

Chapter 9. Laser Characterizations for Optical Wireless Communications[57]. D. V. Kuksenkov, H. Temkin, Polarization related properties of vertical-cavity
surface-emitting lasers, IEEE Journal of Selected Topics in Quantum Electronics, Vol. 3,
Issue 2, 1997, pp. 390-395.
[58]. A. Chavez‐Pirson, et al., Polarization properties of a vertical cavity surface emitting laserusing a fractional layer superlattice gain medium, Applied Physics Letters, Vol. 62,
Issue 24, 1993, pp. 3082-3084.
[59]. X. Li, et al., Static and dynamic characteristics of VCSELs with polarisation-selective
optical feedback, IEE Proceedings – Optoelectronics, Vol. 153, Issue 2, 2006, pp. 67-74.
[60]. M. Arteaga, et al., Study of polarization properties of VCSELs subject to optical feedback
from an extremely short external cavity, in Proceedings of the Conference on Lasers andElectro-Optics Europe (CLEO EUROPE’05), 2005, p. 148.
[61]. E. A. Viktorov, P. Mandel, Low frequency fluctuations in a multimode semiconductor laser
with optical feedback, Physical Review Letters, Vol. 85, Issue 15, 2000, 3157.
[62]. J. Martin-Regalado, S. Balle, M. San Miguel, Polarization and transverse-mode dynamics
of gain-guided vertical-cavity surface-emitting lasers, Optics Letters, Vol. 22, Issue 7, 1997,
pp. 460-462.
[63]. S. Nazhan, et al., Hysteresis properties induced by variable polarization angle in thepolarization switching of VCSELs, in Proceedings of the 9th International Symposium on
Communication Systems, Networks & Digital Signal Processing (CSNDSP’14), 2014,
pp. 325-329.
[64]. N. Shibasaki, et al., Characteristics of chaos synchronization in semiconductor lasers subject
to polarization-rotated optical feedback, IEEE Journal of Quantum Electronics, Vol. 42,
Issue 3, 2006, pp. 342-350.
[65]. S. Xiang, et al., Variable-polarization optical feedback induced hysteresis of the polarization
switching in vertical-cavity surface-emitting lasers, JOSA B, Vol. 27, Issue 12, 2010,
pp. 2512-2517.
[66]. S. Xiang, et al., Polarization degree of vertical-cavity surface-emitting lasers subject tooptical feedback with controllable polarization, JOSA B, Vol. 27, Issue 3, 2010,
pp. 476-483.
[67]. S. Nazhan, Z. Ghassemlooy, Antiphase chaotic synchronization enhancement in a verticalcavity surface emitting laser, Applied Optics, Vol. 58, Issue 35, 2019, pp. 9491-9497.
[68]. J. Martin-Regalado, et al., Polarization switching in vertical-cavity surface emitting lasers
observed at constant active region temperature, Applied Physics Letters, Vol. 70, Issue 25,
1997, pp. 3350-3352.
[69]. B. Ryvkin, et al., Optical-injection-induced polarization switching in polarization-bistablevertical-cavity surface-emitting lasers, Journal of Applied Physics, Vol. 96, Issue 11, 2004,
pp. 6002-6007.
[70]. M. A. Arteaga, et al., Investigation of polarization properties of VCSELs subject to opticalfeedback from an extremely short external cavity-Part I: Theoretical analysis, IEEE Journalof Quantum Electronics, Vol. 42, Issue 2, 2006, pp. 89-101.
[71]. F. Koyama, Advances and new functions of VCSEL photonics, Optical Review, Vol. 21,
Issue 6, 2014, pp. 893-904.
[72]. J. A. Tatum, et al., VCSEL-based interconnects for current and future data centers, Journalof Lightwave Technology, Vol. 33, Issue 4, 2015, pp. 727-732.
[73]. M. I. Olmedo, et al., Multiband carrierless amplitude phase modulation for high capacity
optical data links, Journal of Lightwave Technology, Vol. 32, Issue 4, 2014, pp. 798-804.
[74]. K. Szczerba, et al., 94-Gb/s 4-PAM using an 850-nm VCSEL, pre-emphasis, and receiver
equalization, IEEE Photonics Technology Letters, Vol. 28, Issue 22, 2016, pp. 2519-2521.
[75]. J. Yan, et al., An electrooptothermal-coupled circuit-level model for VCSELs under pulsed
condition, IEEE Transactions on Industrial Electronics, Vol. 66, Issue 2, 2019,
pp. 1315-1324.

413

Advances in Optics: Reviews. Book Series, Vol. 5[76]. C. Xie, et al., Generation and transmission of 100-Gb/s PDM 4-PAM using directly
modulated VCSELs and coherent detection, in Proceedings of the Optical Fiber
Communication Conference (OFC’14), San Francisco, California, 2014, pp. 1-3.
[77]. S. Xiang, et al., Photonic generation of neuron-like dynamics using VCSELs subject
to double polarized optical injection, Journal of Lightwave Technology, Vol. 36, Issue 19,
2018, pp. 4227-4234.
[78]. A. Hurtado, I. D. Henning, M. J. Adams, Optical neuron using polarisation switching in a
1550 nm-VCSEL, Optics Express, Vol. 18, Issue 24, 2010, pp. 25170-25176.
[79]. H. M. Oubei, et al., Light based underwater wireless communications, Japanese Journalof Applied Physics, Vol. 57, Issue 8S2, 2018, 08PA06.
[80]. C.-Y. Li, et al., A 5 m/25 Gbps underwater wireless optical communication system, IEEE
Photonics Journal, Vol. 10, Issue 3, 2018, pp. 1-9.
[81]. W. Hofmann, et al., 40 Gbit/s modulation of 1550 nm VCSEL, Electronics Letters, Vol. 47,
Issue 4, 2011, pp. 270-271.
[82]. H.-H. Lu, et al., 64 Gb/s PAM4 VCSEL-based FSO link, Optics Express, Vol. 25, Issue 5,2017, pp. 5749-5757.
[83]. N. Li, et al., Secure communication systems based on chaos in optically pumped
spin-VCSELs, Optics Letters, Vol. 42, Issue 17, 2017, pp. 3494-3497.
[84]. M. S. Moreolo, et al., VCSEL-based sliceable bandwidth/bitrate variable transceivers,
Proceedings of SPIE, Vol. 10946, 2019, 1094606.
[85]. J. W. Shi, et al., High-performance Zn-diffusion 850-nm vertical-cavity surface-emitting
lasers with strained InAlGaAs multiple quantum wells, IEEE Photonics Journal, Vol. 2,
Issue 6, 2010, pp. 960-966.
[86]. J.-W. Shi, et al., Single-mode, high-speed, and high-power vertical-cavity surface-emitting
lasers at 850 nm for short to medium reach (2 km) optical interconnects, Journalof Lightwave Technology, Vol. 31, Issue 24, 2013, pp. 4037-4044.
[87]. P. Westbergh, et al., High-speed oxide confined 850-nm VCSELs operating error-free at
40 Gb/s up to 85 ºC, IEEE Photonics Technology Letters, Vol. 25, Issue 8, 2013,
pp. 768-771.
[88]. P. Westbergh, et al., High-speed 850 nm VCSELs operating error free up to 57 Gbit/s,
Electronics Letters, Vol. 49, Issue 16, 2013, pp. 1021-1023.
[89]. P. Wolf, et al., Energy efficient 40 Gbit/s transmission with 850 nm VCSELs at 108 fJ/bit
dissipated heat, Electronics Letters, Vol. 49, Issue 10, 2013, pp. 666-667.
[90]. N. Li, et al., Ultra-low-power sub-photon-voltage high-efficiency light-emitting diodes,
Nature Photonics, Vol. 13, Issue 9, 2019, pp. 588-592.
[91]. R. Michalzik, VCSELs: Fundamentals, Technology and Applications of Vertical-Cavity
Surface-Emitting Lasers, Springer, Berlin, Heidelberg, 2013.
[92]. J.-W. Shi, et al., Oxide-relief vertical-cavity surface-emitting lasers with extremely highdata-rate/power-dissipation ratios, in Proceedings of the Optical Fiber Communication(OFC’11) Conference, 2011, pp. 1-3.
[93]. P. Dapkus, M. MacDougal, Ultralow threshold vertical cavity surface emitting lasers, IEEE
LEOS Newsletter, Vol. 9, 1995, pp. 3-5.
[94]. S. Yu, Nonlinear dynamics of vertical-cavity surface-emitting lasers, IEEE Journalof Quantum Electronics, Vol. 35, Issue 3, 1999, pp. 332-341.
[95]. C. H. Cox III, et al., Limits on the performance of RF-over-fiber links and their impact on
device design, IEEE Transactions on Microwave Theory and Techniques, Vol. 54, Issue 2,
2006, pp. 906-920.
[96]. K. Panajotov, et al., Nonlinear dynamics of vertical-cavity surface-emitting lasers,
Advances in Optical Technologies, Vol. 2011, 2011, pp. 1-16.

414

Chapter 9. Laser Characterizations for Optical Wireless Communications[97]. H.-F. Liu, W. F. Ngai, Nonlinear dynamics of a directly modulated 1.55 μm InGaAsP
distributed feedback semiconductor laser, IEEE Journal of Quantum Electronics, Vol. 29,
Issue 6, 1993, pp. 1668-1675.
[98]. S. Yu, et al., Theoretical analysis of modulation response and second-order harmonic
distortion in vertical-cavity surface-emitting lasers, IEEE Journal of Quantum Electronics,
Vol. 32, Issue 12, 1996, pp. 2139-2147.
[99]. A. Valle, M. Sciamanna, K. Panajotov, Irregular pulsating polarization dynamics
in gain-switched vertical-cavity surface-emitting lasers, IEEE Journal of Quantum
Electronics, Vol. 44, Issue 2, 2008, pp. 136-143.
[100]. G. Shtengel, et al., High-speed vertical-cavity surface emitting laser, IEEE Photonics
Technology Letters, Vol. 5, Issue 12, 1993, pp. 1359-1362.
[101]. G. S. Gordon, et al., High-order distortion in directly modulated semiconductor lasers inhigh-loss analog optical links with large RF dynamic range, Journal of Lightwave
Technology, Vol. 29, Issue 23, 2011, pp. 3577-3586.
[102]. B. Kuntsevich, A. Pisarchik, V. Kononenko, Nonlinear dynamics of a directly modulated
semiconductor laser with cavity detuning, Optical and Quantum Electronics, Vol. 37,
Issue 7, 2005, pp. 675-693.
[103]. M. Giudici, et al., Polarization dynamics in vertical-cavity surface-emitting lasers with
optical feedback: experiment and model, JOSA B, Vol. 16, Issue 11, 1999, pp. 2114-2123.
[104]. S. Nazhan, et al., Suppressing the nonlinearity of free running VCSEL using selectiveoptical feedback, IEEE Photonics Technology Letters, Vol. 28, Issue 2, 2016, pp. 185-188.
[105]. S. Blokhin, et al., Oxide-confined 850 nm VCSELs operating at bit rates up to 40 Gbit/s,
Electronics Letters, Vol. 45, Issue 10, 2009, pp. 501-503.
[106]. K. Panajotov, et al., Optical feedback in vertical-cavity surface-emitting lasers, IEEE
Journal of Selected Topics in Quantum Electronics, Vol. 19, Issue 4, 2013,
pp. 1700312-1700312.
[107]. N. Fujiwara, Y. Takiguchi, J. Ohtsubo, Observation of the synchronization of chaos
in mutually injected vertical-cavity surface-emitting semiconductor lasers, Optics Letters,Vol. 28, Issue 18, 2003, pp. 1677-1679.
[108]. S.-Y. Lin, et al., 10-Gbit/s direct modulation of a TO-56-can packed 600-μm long laser
diode with 2 % front-facet reflectance, Optics Express, Vol. 21, Issue 21, 2013,
pp. 25197-25209.
[109]. M. Ozaki, et al., Leader-laggard relationship of chaos synchronization in mutually coupled
vertical-cavity surface-emitting lasers with time delay, Physical Review E, Vol. 79, Issue 2,
2009, 026210.
[110]. A. Uchida, et al., Synchronization of chaos in mutually coupled vertical-cavity
surface-emitting lasers with time delay, in Proceedings of the IEEE/LEOS Winter Topicals
Meeting Series (LEOSWT’09), 2009, pp. 126-127.
[111]. J. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos, Vol. 111, Taylor &Francis, 2006.
[112]. Y. O. Junji Takeuchi, R. Shogenji, Chaotic dynamics in semiconductor lasers subjected
to polarization-rotated optical feedback, Applied Physics Letters, Vol. 93, Issue 18, 2008,
pp. 181105-181105.
[113]. S. Nazhan, Z. Ghassemlooy, K. Busawon, Chaos synchronization in vertical-cavity
surface-emitting laser based on rotated polarization-preserved optical feedback, Chaos:
An Interdisciplinary Journal of Nonlinear Science, Vol. 26, Issue 1, 2016, 013109.
[114]. T. Heil, et al., TE-TM dynamics in a semiconductor laser subject to polarization-rotatedoptical feedback. Physical Review A, Vol. 68, Issue 3, 2003, pp. 033811.
[115]. L. M. Pecora, T. L. Carroll, Synchronization in chaotic systems, Physical Review Letters,Vol. 64, Issue 8, 1990, pp. 821-824.

415

Advances in Optics: Reviews. Book Series, Vol. 5[116]. D. W. Sukow, et al., Experimental synchronization of chaos in diode lasers withpolarization-rotated feedback and injection, Optics Letters, Vol. 29, Issue 20, 2004,
pp. 2393-2395.
[117]. P. Spencer, et al., Modeling of optical synchronization of chaotic external-cavity VCSELs,
IEEE Journal of Quantum Electronics, Vol. 34, Issue 9, 1998, pp. 1673-1679.
[118]. A. Scire, et al., Polarization message encoding through vectorial chaos synchronizationin vertical-cavity surface-emitting lasers, Physical Review Letters, Vol. 90, Issue 11, 2003,
113901.
[119]. Z.-L. Zhang, G.-Q. X, X.-D. Lin, Z.-M. Wu, Comparison of closed-loop and open-loop
in incoherent optical feedback chaos synchronization system, Optoelectronics andAdvanced Materials, Vol. 3, Issue 5, 2009, 5.
[120]. S. Y. Xiang, et al., Quantifying chaotic unpredictability of vertical-cavity surface-emitting
lasers with polarized optical feedback via permutation entropy, IEEE Journal of Selected
Topics in Quantum Electronics, Vol. 17, Issue 5, 2011, pp. 1212-1219.
[121]. T. Kingni, S. J. Hervé Talla Mbé, P. Woafo, Semiconductor lasers driven by self-sustained
chaotic electronic oscillators and applications to optical chaos cryptography, Chaos:
An Interdisciplinary Journal of Nonlinear Science, Vol. 22, Issue 3, 2012, 033108.
[122]. Y. Hong, et al., Synchronization of chaos in unidirectionally coupled vertical-cavity
surface-emitting semiconductor lasers, Optics Letters, Vol. 29, Issue 11, 2004,
pp. 1215-1217.
[123]. S. Priyadarshi, et al., Experimental investigations of time-delay signature concealment inchaotic external cavity VCSELs subject to variable optical polarization-angle of feedback,IEEE Journal of Selected Topics in Quantum Electronics, Vol. 19, Issue 4, 2013, 1700707.
[124]. J. Ohtsubo, Chaos synchronization and chaotic signal masking in semiconductor lasers with
optical feedback, IEEE Journal of Quantum Electronics, Vol. 38, Issue 9, 2002,
pp. 1141-1154.
[125]. M. Virte, et al., Deterministic polarization chaos from a laser diode, Nature Photonics,
Vol. 7, Issue 1, 2013, pp. 60-65.
[126]. Y. Takeuchi, R. Shogenji, J. Ohtsubo, Chaos synchronization in semiconductor lasers withpolarization-rotated optical feedback, Optical Review, Vol. 17, Issue 5, 2010, pp. 467-475.
[127]. S. Nazhan, et al., Investigation of polarization switching of VCSEL subject to intensity
modulated and optical feedback, Optics & Laser Technology, Vol. 75, 2015, pp. 240-245.
[128]. L. A. Coldren, S. W. Corzine, M. L. Mashanovitch, Diode Lasers and Photonic Integrated
Circuits, Vol. 218, Wiley, 2012.
[129]. L. A. Coldren, S. W. Corzine, Diode lasers and photonic integrated circuits, OpticalEngineering, Vol. 36, Issue 2, 1997, pp. 616-617.
[130]. S. Li, UWB Radio-over-Fiber System Using Direct Modulated VCSEL, UWSpace, 2007.

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Chapter 10. Compact Solar-pumped Lasers

Chapter 10

Compact Solar-pumped Lasers
Hiroshi Ito, Kazuo Hasegawa, Shintaro Mizuno,
Yasuhiko Takeda and Tomoyoshi Motohiro1

10.1. Introduction
10.1.1. Solar-pumped Lasers and Their Possible Applications
A solar-pumped (or sun-pumped) laser (SPL) can be deemed as a variant of a conventional
flash-lamp pumped laser. In SPLs, natural sunlight is used instead of flash-lamps forpumping the laser medium. Getting rid of electricity-consuming flash-lamps, SPLs are
essentially renewable energy devices that convert incoherent sunlight with a wide spreadof spectrum and low areal density into a monochromatic coherent laser beam withextremely high energy flux density. SPLs can be used to transmit the collected solar
energy as laser beams across large distances wirelessly or via optical fibers. Some possible
examples include power transmission between an orbiting space solar power station andthe ground [1-5], power transmission between a lunar lander and a lunar rover exploring
deep into permanently shadowed craters for frozen water [6, 7], or powering mobile
objects such as drones, robots, and electric vehicles [4, 8, 9]. The transmitted laser beams
can be received and converted into electricity by solar cells specially designed for the
monochromatic light of the laser beams [8, 10-18]. SPLs have also been investigated toheat up materials to a higher temperature than the temperature that can be attained bysimple solar concentration. A typical example includes a magnesium-based energy cycle
in which solar energy is eventually stored and utilized by reduction of magnesium oxideusing SPL irradiation and hydrogen production by reduction of water withmagnesium [19, 20].
10.1.2. Brief History of SPLs
SPLs were first reported in the 1960s [21-23], soon after the discovery of laser, that is,
stimulated optical radiation in ruby [24]. Since then, more than 600 papers publicly

Tomoyoshi Motohiro
Institute of Materials Innovation, Institutes of Innovation for Future Society, Nagoya University, Japan

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Advances in Optics: Reviews. Book Series, Vol. 5

available papers have been published on SPLs. Iodine or iodides were the major laser
mediums (LMs) intensively studied by research groups in USA (NASA and University of
Florida, etc.) [25, 26], and Russia (Institute of Laser Physics, etc.) [27]. SPLs based on
gas-phase dimer molecules have been actively studied by an Israeli research group(Weizmann Institute of Science) [28]. Various solid-state mediums, such as phosphateglasses [29], tellurite glass, silicate glass, fluoride glasses [22, 30-35], glass ceramics [36],borate crystals [37, 38], GdScGa-garnet crystals [39, 40], and YAlO3 [41], have also beenstudied for application as LMs. In one of the above-mentioned initial works on SPLs in
the 1960s [23], Nd-doped yttrium aluminum garnet (Nd:YAG) single crystals were
employed as LMs. Since then, Nd:YAG crystals have been the primary option for LMs[42-44]. Then, transparent Nd:YAG ceramic LMs were successfully fabricated and usedwidely in LMs [45, 46]. Further, transparent Cr-codoped Nd:YAG (Cr, Nd: YAG) ceramic
LMs have been intensively studied by Institute for Laser Technology, Osaka and Instituteof Laser Engineering, Osaka University, Japan [47]. Significant contributions have been
made by Yogev et al. of Weizmann Institute of Science, Israel [48-55]. Active research
on SPLs has also been ongoing in Academy of Sciences of Uzbekistan [56] and in NOVA
University of Lisbon, Portugal [57-60].
10.1.3. Stance and Orientation of This Work
Typical sizes or diameters of solar concentrators in conventional SPL studies are in therange of 1-2 m [61, 62], such as parabolic mirrors [22, 23] and Fresnel lenses [61-64].
Some studies have employed larger solar concentrators, such as solar furnaces (mirrors)of 10 m aperture [42] and 54×48 m rectangle [65]. This trend is quite reasonable because
it is necessary to harvest substantial amount of solar energy for practical solar energy
utilization owing to the low areal density of sunlight (in the order of sub-kW/m2).
However, for the next generation of SPLs that are more likely to be widely used for solar
energy utilization (e.g., present solar cells), cost-effective and much smaller SPLs might
be advantageous because they are suitable for mass production. Therefore, we exploredthe merits of compact SPLs and have developed prototype compact SPLs employing⌀50.8 mm and ⌀76.2 mm off-axis parabolic mirrors (OAPs) for solar concentrators. Aninterim progress report of this research activity is provided in this chapter.

10.2. Background of the Concept Design
10.2.1. Preferable Oscillation Wavelength of SPLs
Fig. 10.1(a) schematically shows the conversion of incoherent sunlight with a wide spreadof spectrum into a monochromatic coherent laser beam by an SPL. Stimulated emissionof photons from an SPL at an oscillation wavelength of λL is caused by the energy of theincident sunlight of wavelength λi shorter than λL unless a means of upconversion to utilize
sunlight of wavelength longer than λL is employed [66-75]. If all the photon energies of
the incident sunlight of λi < λL contribute to the laser-beam output of the SPL ideally, theconversion efficiency from solar energy to laser energy η, that is, the ratio of the output
photon energy to the total incident photon energy is given by the following expression:
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Chapter 10. Compact Solar-pumped Lasers
2



i

   I    d 
i

1

L

2



I  i  d ii,

(10.1)

1

where I(λi) stands for the areal power density of the incident spectral sunlight[J/(sec‧μm‧m2)]. The wavelengths λ1 and λ2 defining integral range are ideally 0 and ∞,respectively; however, in practice, they are 300 nm and 4000 nm if the main part of theintensity distribution of the solar spectrum is taken into consideration. The factor λi / λL in
the integrand in the numerator in Eq. (10.1) comes from the ratio of the photon energies,(h‧c/λL)/ (h‧c/λi), representing the quantum defect caused by down-conversion. Variationof η as a function of λL is plotted in Fig. 10.1(b) together with the solar spectrum.
Fig. 10.1(b) shows that η takes the optimal value in the range 1100 nm < λL < 1300 nm.
Fig. 10.1(b) also shows that η of this single-wavelength emission type SPL cannot exceed
0.5 even in the ideal condition [76]. If an SPL has two emission wavelengths λL1 and λL2(λL1 < λL2) employing two LMs, as shown in Fig. 10.2(a), η can be increased over 0.6 in
the wavelength regions around 864 nm for λL1 and 1775 nm for λL2, as shownin Fig. 10.2(b).

Fig. 10.1. (a) Function of a solar-pumped laser, (b) Variation of η as a function of the SPL
emission wavelength λL displayed with the solar spectrum.

Fig. 10.2. (a) Function of a two-wavelength emission type SPL, and (b) Variation of η as afunction of SPL emission wavelengths λL1 and λL2.

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Solar-pumped lasers utilizing emissions from Nd ions such as a Nd-doped Y3Al5O12(Nd:YAG) crystal, described in Section 10.1, are preferable practical systems in the caseof single-wavelength emission type SPL because their emission wavelength of around1.06 μm is near the optimal wavelength region between 1100 nm and 1300 nm shownin Fig. 10.1(b). In this case, η at a wavelength of 1.06 μm is obtained as 0.48 via Eq. (10.1).

10.2.2. Incident Power Density Required to Oscillate SPLs
Here, we consider a case of an LM working on a four-level scheme, like the Nd:YAG
transparent ceramic LMs shown in Fig. 10.3. When a four-level laser provides a constant
output power under stable oscillation, the stable photon density in the LM Np [1/m3] isgiven by the following expression:

N P   C  RP 1  L   c ,

(10.2)

where Rp [1/(sec•m3)] stands for the pumping efficiency, that is, the number of atoms
excited per second per cubic meter; σ(λL) stands for the stimulated emission cross-section
at the oscillation wavelength λL; c stands for the velocity of light; τ stands for the lifetimeof spontaneous emission, which is equal to the reciprocal of the Einstein A coefficient (therate of spontaneous emission, A32); and τc [sec] stands for the cavity photon lifetime,
which is expressed asC 

2  L,c    ln  R1  R2 (10.3)

where L stands for the cavity length which is almost equal to the length of the LM, δ stands
for the round-trip resonator loss, and R1 and R2 are the reflectivity of the cavity mirrors.
Eq. (10.2) is schematically visualized in Fig. 10.4.

Fig. 10.3. Four level laser scheme.

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Fig. 10.4. Stable photon density in the LM as a function of pumping efficiency.

As elucidated in Fig. 10.4, the value of Rp when Np decreases to zero, Rpth is the minimum
pumping efficiency, that is, the threshold pumping efficiency to cause laser oscillation.
Using Eqs. (10.2) and (10.3), Rpth is expressed asRPth    ln  R1  R2 
2  L    L(10.4)

Here, we consider a rod type LM of diameter 2r and length L pumped by the light Ie[W/m2] of frequency νe from the left mainly on the left end facet and partly on the sidesurface, as schematically illustrated in Fig. 10.5. The total number of photons that
contribute to pumping the rod is given byIe  r 2  2  r   L  ,h  e(10.5)

where h stands for the Planck constant = 6.626×10-34 J‧sec. Here, the factor α(0 ≦ α ≦ 1) is used to express effective length αL of the side surface that receives photons.
Dividing Eq. (10.5) by the volume of the rod, π‧r2‧L, the number of excited ions per second
per volume Re is expressed by I   1 2  Ie  1
Re   e       1  2  A  ,
    h  e   L r  h  e  L(10.6)

where A ( = L / r) is the aspect ratio of the rod, and η is the total pumping efficiency; it isexpressed as follows:  QD QE abs m ,(10.7)

where ηQD stands for quantum defect representing the ratio of photon energy of the emitted
laser to that of the pumping light, ηQE stands for quantum efficiency representing the ratio
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Advances in Optics: Reviews. Book Series, Vol. 5of number of photons contributing the excitation of ions to the number of absorbed
photons, ηabs represents the ratio of the number of absorbed photons to the number ofincident photons to the rod reflected by the degree of matching of the absorption spectrumof the rod material with the sunlight spectrum, ηm stands for mode-matching efficiency
representing the ratio of number of exited ions distributing in the laser oscillation modeand contributing to the laser oscillation to the total excited ions in the rod. Laser oscillation
takes place when Re in Eq. (10.6) exceeds RPth in Eq. (10.4). Combining Eq. (10.4) with
Eq. (10.6), the threshold of the incident power density to start laser oscillation Ieth [W/m2]
is expressed as follows: h  e     ln  R1  R2   11
 I eth      
2
  1  2 A   QD QE  abs  m    L    (10.8)

Fig. 10.5. A rod type LM pumped by light mainly on the left end facet and partly
on the side surface.

Eq. (10.8) shows that to decrease Ieth for ease of laser oscillation, lower νe, in other words,
larger λe under the constraint of λe < λL (the lower quantum defect); larger σ(λL) ‧τ; lower
cavity loss; higher reflectivity of the cavity mirrors; larger effective aspect ratio αA; andlarger ηQD, ηQE, ηabs, and ηm are preferable. It should be noted that L is not related to Ieth in
Eq. (10.8) explicitly because both Eqs. (10.4) and (10.6) contain L in their denominators.
In a typical case of Nd:YAG laser with λL of 1064 nm, λe of 808 nm, σ(λL) of
2.8×10-19 cm2, τ of 230 μsec, R1 of 0.99, R2 of 0.95, ηQD of 0.61 ( = 650 nm / 1064 nm),ηQE of 1, ηabs of 0.3, ηabs of 0.29, A of 20 ( = 10 mm / 0.5 mm), and α of 0.03, Ieth was
calculated to be 5427 kW/m2. Therefore, a solar concentration ratio of more than 5500 is
required for 1 Sun solar insolation of 1 kW/m2 to start oscillation in this laser rod.
10.2.3. Preferable Shape of LMs for SPLs
The volume V and the total surface area S of the typical LM shown in Fig. 10.5
are given by422

V    r 2  L,

(10.9)

S  2  r  L  2  r 2 ,(10.10)Chapter 10. Compact Solar-pumped Lasers

respectively. Using Eq. (10.9), A = L / r, and normalizing S by the values of S when A is2, SA = 2, under the fixed V, Eq. (10.10) can be modified as2

23 1  1 
  A3  1  
S A 2 3
 A
S(10.11)

Fig. 10.6 shows the variation of S/SA = 2 as a function of A. Fig. 10.6 shows that the total
surface area of the LM takes high values under fixed V in the left-hand side and the righthand side of the graph. Therefore, the total number of pumping photons expressed in
Eq. (10.5) also takes a large value in the small A range and in the large A range in the fixedvolume. This means that thin disk-type LMs, and thin fiber-type LMs are preferable for
SPL because of the low areal energy density of sunlight. The high A values in the thindisk-type LMs and thin fiber-type LMs are also advantageous for thermal dissipation, as
intuitively shown by the pink arrows in Fig. 10.6. In the case of a Nd:YAG crystal, forexample, the theoretical upper limit of solar energy conversion efficiency into
monochromatic laser at a wavelength of 1.06 μm is calculated to be 0.48, as described in
Subsection 10.2.1. However, actual feasible conversion efficiency will be much lower[76]. Since most of the concentrated solar energy is converted to heat mainly in the LM,
without rapid thermal dissipation, σ(λL) ‧ τ in Eq. (10.8) decreases and an inhomogeneoustemperature increase in the LM can take place. This causes degradation of the laser-beam
quality because of temperature dependent refractive index, including thermal lens effect,loss of the energy conversion efficiency, and even the breakage of LMs. Therefore, theadvantage of thermal dissipation in thin disk-type, and thin fiber-type LMs is of key
importance. Here, it is suggested that a glass LM is promising for SPLs because it can beeasily molded into thin disks and thin fibers. High solubility of rare earth elements suchas Nd is another attractive property of glass LMs. Further, the suitability of glass LMs formass production ensures low production costs, which is a key factor for wide adoption of
SPLs in the society.

Fig. 10.6. Variation of S/SA = 2 as a function of A.

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Advances in Optics: Reviews. Book Series, Vol. 5

10.2.4. Theoretical Limit of Solar Concentration
Suppose the Sun is a perfect blackbody of radius RS (m) and surface temperature TS (K),the total radiant flux power from the Sun, PO (W), is given by

P0  4  RS2    TS4 ,(10.12)where σ stands for the Stefan-Boltzmann constant = 5.67×10-8 W/(m2K4), and σ‧Ts4
represents the radiant flux power density at the surface of the Sun based on the StefanBoltzmann law. At the average radius of Earth’s orbit around the Sun, that is, one
astronomical unit Au (m), the radiant flux power density IO (W/m2) is called the solar
constant and is given by
IO 

PO
RS2

   TS42
2
4    Au Au(10.13)

IO is calculated to be 1367 W/m2. The solar concentration ratio C is defined as the ratio of
the radiant flux power density at the most concentrated location near the focal point of thesolar concentrator to the radiant flux power density of sunlight at the entrance of theconcentrator. Thus, the radiant flux power density at the most concentrated location near
the focal point of the solar concentrator I’ (W/m2) is given by

I'C

RS2
   TS4    T '4
2
Au

(10.14)

Suppose the most concentrated location near the focal point of the solar concentrator is ona perfect blackbody surface, the surface temperature T K of this blackbody is required to
be lower than TS based on the principles of thermodynamics. Then the following inequality
expression can be deduced.

Au 2
1C 2 ,RS (sin  S )2

(10.15)where θS is the apparent Sun semidiameter observed from the average Earth’s orbit aroundthe Sun, as illustrated in Fig. 10.7.

Fig. 10.7. Schematic geometry of the Sun, a solar concentrator on the average Earth’s orbit
around the Sun and a blackbody absorber near its focal point.

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Chapter 10. Compact Solar-pumped Lasers

This is the theoretical limit of the solar concentration ratio CO deduced from the laws ofthermodynamics. The same results can be obtained from the Lagrange-Helmholtz
invariant or Luminance conservation law, naturally. By substituting the values
RS = 6.96 ×108 m and Au = 1.496×1011 m into Eq. (10.15), we obtainC  CO  46200(10.16)10.2.5. Size of Images of the Sun and Concentration Ratios by Solar Concentrators
10.2.5.1. Convex Lens
Fig. 10.8 illustrates the geometry of image formation by a convex lens comprised of an
object, a lens, and an image of the object located graphically by means of geometrical
optics [77]. The white arrow on the left side of the lens represents an object of length RS.
The distance between the foot of this white arrow and the center of the lens O is supposed
to be Au. Correspondingly, the smaller upside-down white arrow of length rS appears on
the right side of the lens as an image of the white arrow on the left side of the lens. The
distance between the foot of the image and O is expressed as f + x, where f stands for the
focal length of the lens. A black arrow of length D, which is equal to the radius of the lensis marked on the white arrow on the left of the lens. Correspondingly, a black arrow oflength d also appears upside-down on the image of the white arrow on the right side ofthe lens. From the two similar triangles with a common vertex F, the following Eq. (10.17)
can be obtained:
d
D

x
f

(10.17)

Fig. 10.8. Illustration of image formation by a convex lens tracing several characteristic rays.

From the other two similar triangles with common vertex O, the following Eq. (10.18)
can be obtained:dDf  x Au

(10.18)From Eqs. (10.17) and (10.18), rS is expressed as follows:425Advances in Optics: Reviews. Book Series, Vol. 5

rs  d R S RS
R
f


 Sf
D
Au 1  f
Au
Au(10.19)In Eq. (10.19), the denominator was partly approximated using the relation f /Au≪1. In a
specific case of f = 50.8 mm, the radius of the image of the Sun is obtained to be
0.236 mm. The concentration ratio can be expressed as follows:2 A  D D2D2  u   2
2
 rS 
  RS   f 
R
f 
 S

 Au 1  f 

Au 


2

(10.20)As indicated in Eqs. (10.15) and (10.16), (Au/RS)2 is the theoretical upper limit of solar
concentration ratio. In a typical case of f = 50.8 mm and 2D = 50.8 mm, the concentration
ratio is 25 % of the theoretical upper limit. In practical aplanatic lenses that are nearly freeof coma and spherical aberration, D/f does not exceed 0.7, usually. Therefore, solar
concentration ratios are usually less than 50 % of the theoretical upper limit. In practice,a lens transmittance loss of between 0.25 and 0.9 ensures that solar concentration ratios
are between 12.5 % and 45 % of the theoretical upper limit.
10.2.5.2. Parabolic Concave Mirror
Fig. 10.9 illustrates the geometry of image formation by a parabolic concave mirror in thesame manner as Fig. 10.8. Through similar geometrical consideration, the same relationas Eq. (10.18) is obtained. The equation corresponding to Eq. (10.17) can be obtained as
Eq. (10.21).

d

x

D
D2
f
4f

Fig. 10.9. Illustration of image formation by a parabolic concave mirror tracing
several characteristic rays.

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Chapter 10. Compact Solar-pumped Lasers

Here, the factor D2/4f in the right-hand denominator corresponds to the displacement of
the mirror surface to the left at its edge. From Eqs. (10.18) and (10.21), rS is expressed asfollows:

rS 

RS
R
d  S 
D
Au

R
f
 Sf
2
1 
D  Au1  f 

Au 
4f 

(10.22)In Eq. (10.22), the denominator was partly approximated using the relation f /Au≪1. In atypical case of f = 50.8 mm, the radius of the image of the Sun is obtained to be 0.236 mm.
The concentration ratio can be eventually expressed in the same manner as Eq. (10.20):
22

 A  D
 D 2  Au D2  u   22 rs  Rs    Rs   f 


f



2 
1  1   f  D   Au 4 f  2

(10.23)Thus, for a typical case of f = 50.8 mm, and 2D = 50.8 mm, the concentration ratio is 25 %of the theoretical upper limit.
10.2.6. Merits of Miniaturization of SPLs
Fig. 10.10 shows photos of Odeillo solar furnace at the Process, Materials and Solar
Energy (PROMES)-CNRS laboratory. It is one of the largest solar furnaces in the world,
situated in the Pyrenees mountain range in the south of France, and has been used as asolar concentrator for an SPL [65]. Its height, width, and focal length are 54 m, 48 m, and
18 m, respectively. With the aid of 63 flat plate heliostats, it attains a temperature of3500 ºC and concentrates 1 MW at the focus. It should be noted that the expressions forthe solar concentration ratio of a convex lens (Eq. (10.20)) and of a parabolic mirror(Eq. (10.23)) do not contain the absolute sizes of the solar concentrator, such as the focal
length f and the diameter 2D, but contain the ratio D/f. Therefore, the same radiant flux
power density as the Odeillo solar furnace can be attained by a much more compact solar
concentrator, for example, of height, width, and focal length = 5.4 cm, 4.8 cm, and
1.8 cm, respectively, although the total amount of the concentrated solar power would bemuch less than 1 MW.
As a typical example of a conventional SPL, a 1.4×1.05 m Fresnel lens with a focal lengthof 1.2 m and a Cr-codoped Nd:YAG ceramic LM was used for an SPL system and18.7 W laser output was obtained [64]. The corresponding NA was 0.4-0.5. For simplicity,
let us consider an SPL system composed of a ⌀1.37 m lens with a focal length of 1.2 m(NA = 0.5) and a ⌀10×100 mm cylindrical Cr-codoped Nd:YAG LM, as shown in
Fig. 10.11(a). The sunlight-receiving area (1.37/2)2 × π is equal to 1.4×1.05 m2. Most of427

Advances in Optics: Reviews. Book Series, Vol. 5

the focused sunlight is introduced into the rod because the size of the image of the Sun iscalculated to be approximately ⌀8.2 mm according to Eq. (10.19). It is duly conjectured
that this SPL can also oscillate and yield a similar amount of laser output as in reference[64]. Then, let us consider a miniaturized SPL system composed of ⌀76.2 mm with a focal
length of 66 mm (NA = 0.5) and a cylindrical LM of ⌀1 × 10 mm as shown in
Fig. 10.11(b). Most of the focused sunlight is introduced into the rod because the size ofthe image of the Sun is calculated to be approximately ⌀0.62 mm according to Eq. (10.19).
This miniaturized SPL will be able to oscillate in natural sunlight because the focusedradiant flux power densities are the same between the ⌀1.37 m lens and the miniaturized⌀76.2 mm system because Eqs. (10.20) and (10.23) do not relate to the absolute size D or
f but only D/f. However, the absolute value of the laser output power is approximately
1/323 in proportion to the ratio of the sunlight-receiving areas. Therefore, an array of
324 miniaturized SPLs will make more laser output power than the system of ⌀1.37 mlens. Here, the total volume of the ceramics LMs in this array is 810π mm3(2.5π mm3 × 324), which is much less than the 2500π mm3 (25π × 100) in the ⌀1.37 m
lens system. In addition to the advantage of having less volume, the ⌀1 × 10 mm
Cr-codoped Nd:YAG LMs are easier to fabricate and more suitable for mass production
compared to the ⌀10 ×100 mm LM. The ⌀1 × 10 mm micro laser rod is also advantageousin comparison with the conventional size laser rod, such as the one of ⌀10 ×100 mm, fromthe view point of rapid thermal dissipation owing to reasons explained earlier. As for thesolar concentrator, compact solar concentrators together with the corresponding
solar-tracking systems are cost-effective because they have already been mass produced
and used in various commercial applications, such as the solar lighting system“Himawari” with a solar-tracking system by La Foret Engineering Company (Japan) [78].
This is in contrast to the large systems, such as ⌀1.37 m lens, which are still in the
made-to-order stage.

Fig. 10.10. Odeillo solar furnace of the Process, Materials and Solar Energy (PROMES)-CNRS
laboratory in France (photograph taken on October 9th, 2018).

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Chapter 10. Compact Solar-pumped Lasers

Fig. 10.11. Concept of miniaturization of SPLs keeping laser output power unchanged.

10.2.7. Direct Solar Radiation and Diffuse Solar Radiation
Fig. 10.12 shows the daytime variations in direct solar radiation and diffuse solar radiationon the horizontal ground plane simulated using the “SPECTRAL2” solar spectrum modelby Bird and Riordan [79]. Detailed simulated conditions are as follows:
longitude = 137°3’E and latitude = 35°10’N, August 1st, AM1.5, total column
ozone = 0.34 cm, total precipitable water vapor = 1.34 cm, and aerosol optical depth at
the wavelength of 500 nm = 0.27. It can be observed in the figure that the shape of thespectrum of the diffuse solar radiation does not vary but its intensity varies with time. In
contrast to this, the spectra of the direct solar radiation shows a significant red shift withtime in the afternoon. The effects of this spectrum shift must be seriously considered in
the case of SPLs because they mainly work with direct solar radiation. If an aplanatic lens
is used for solar concentration, the spectrum shift in the afternoon may affect the SPL
oscillation in connection with the chromatic aberration of the lens. The spectrum shift may
also affect the SPL oscillation in connection with the absorption and excitation spectra of
the LM. The total radiant flux power density between wavelengths of 300 nm and 4 μm
can be calculated as follows: Global solar radiation: 1008 W/m2 (100 %); Direct solar radiation: 787 W/m2 (78 %); Diffuse solar radiation: 221 W/m2 (22 %).
In a typical case in which a Cr,Nd:YAG transparent ceramic LM is employed, the total
radiant flux density is considerably reduced because the wavelength range of theabsorption spectrum is reduced to being between 300 nm and 900 nm, as shown below:
 Global solar radiation: 696 W/m2 (100 %); Direct solar radiation: 508 W/m2 (73 %); Diffuse solar radiation: 188 W/m2 (27 %).
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Advances in Optics: Reviews. Book Series, Vol. 5

Fig. 10.12. Daytime variations of solar spectrum for direct solar and diffuse solar radiations
on the horizontal ground plane simulated using the “SPECTRAL2” solar spectrum model by Bird
and Riordan [79] for longitude = 137°3’E and latitude = 35°10’N, August 1st, AM1.5, equivalentozone depth = 0.34 cm, total precipitable water vapor = 1.34 cm, and aerosol opticaldepth = 0.27@500 nm.

On an inclined plane or on a plane with its surface normal tracking the Sun, the total
radiant flux density can be improved. Fig. 10.13 shows daytime variations of global solar
radiation power density (1) on a horizontal plane, (2) on a plane inclined south at 37° tothe horizontal plane, (3) on a plane with its surface normal tracking the Sun, and (4) direct
solar radiation on a plane with its surface normal tracking the Sun, calculated using the“SPECTRAL2” solar spectrum model by Bird and Riordan [79] for longitude = 137°3’E
and latitude = 35°10’N, under typical clear skies, on a Spring/Autumn day, with diffuse
fraction = 25 %, and transmittance of the atmospheric layer = 75 %. Fig. 10.13 shows thatsun-tracking can gain larger solar radiation both in the morning and in the afternoon. The
following integrated solar radiation power densities throughout the day also show theadvantages of sun-tracking: Global, on the horizontal plane: 6462 Wh/m2;
 Global, on the plane inclined south at 37° to the horizontal plane: 6530 Wh/m2;
 Global, tracking the Sun: 8572 Wh/m2;
 Direct, tracking the Sun: 6940 Wh/m2.
In Subsection 10.2.2, a typical threshold power for oscillation of Nd:YAG laser was
calculated to be approximately 5500 kW/m2. As described in Subsection 10.2.4, in actual
practice, solar concentration ratios are between 12.5 % and 45 % of the theoretical upper
limit of 46200. Therefore, the radiant flux power density is 5775-20790 kW/m2 when the
direct solar radiation was 1 kW/m2. Therefore, the direct solar radiation utilized by Cr,
Nd:YAG at noon (508 W/m2) can make the practical concentrated radiant flux power
density = 2994-10561 kW/m2. This shows that it is important to use lenses or mirrors oflarger D/f and higher transmittance or reflectivity for stable oscillation in the daytime –almost from sunrise to sunset – tracking the Sun.
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Chapter 10. Compact Solar-pumped Lasers

Fig. 10.13. Daytime variations of global solar radiation power density (1) on a horizontal plane(a black line), (2) on a plane inclined south at 37° to the horizontal plane (a green line),(3) on a plane with its surface normal tracking the Sun (a pink line) and (4) direct solar radiation
on a plane with its surface normal tracking the Sun (a red line). Simulations were performed using
the solar spectrum model “SPECTRAL2” by Bird and Riordan [79] for longitude = 137°3’E
and latitude = 35°10’N, under typical clear skies, on a Spring/Autumn day, with the diffuse
fraction = 25 %, and the transmittance of the atmospheric layer = 75 %.

We also measured the direct solar radiation on Wednesday, August 9th, 2017 on the groundat 136°58’E longitude and 35°10’N latitude (Nagoya, Japan) using a pyrheliometer for the
direct solar radiation of ⌀18 mm caliber mounted on an equatorial mounting that tracked
the Sun. The power of the direct solar radiation in the view angle of 5° and in the
wavelength range of 190 to 25000 nm captured with an aperture and a pinhole wasmeasured with a THOLABS S302C thermal power meter sensor attached to thepyrheliometer, as shown in Fig. 10.14. The THOLABS S302C power meter sensor had
been calibrated with an EKO INSTRUMENTS MS-53 (ISO 9060 First Class
Pyrheliometer), which provided the value of the direct solar radiation in terms of W/m2.
The measured value was 905 W/m2 for similar conditions in which 787 W/m2 was
obtained via SPECTRAL2.

Fig. 10.14. A THOLABS S302C thermal power meter sensor with a pinhole calibrated
and attached to a pyrheliometer.

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Advances in Optics: Reviews. Book Series, Vol. 5

We also monitored the daytime variations of the solar spectrum change using an Ocean
Optics MAYA2000 solid-state optical spectrum analyzer. Fig. 10.15 shows variation in
the measured solar spectrum as a function of the clock time. The shapes of the spectra
qualitatively show a gradual red shift of the peak of the solar spectrum as the afternoon
grew late, similar to the results obtained from the “SPECTRAL2” solar spectrum model
shown in Fig. 10.12. Therefore, chromatic aberration must also be reduced in an aplanaticlens for solar concentration because of the wide-band range of the solar spectrum and its
red shift with elapse of the time of the day. However, achromatic aplanatic lenses are notcost-effective. Therefore, parabolic mirrors are preferable as solar concentrators for SPLs.
However, large-scale parabolic mirrors are also not cost-effective. Therefore, thepreferable choice for the solar concentrator will be a small parabolic mirror.

Fig. 10.15. Variation in the measured solar spectrum as a function of the clock time.

10.3. Compact Solar-Pumped Fiber Laser (SPFL) System
10.3.1. Choice of Laser Mediums (LMs)
10.3.1.1. Shape of LMs
As described in Subsection 10.2.3, either disk-like and fiber-like shapes are advantageousfor LMs from the point of efficient excitation and efficient thermal dissipation. As
described in Subsection 10.2.1, solar-pumped lasers utilizing emissions from Nd ions are
preferable practical systems in the case of a single wavelength emission type because theiremission wavelength λL (approximately1.06 μm) is near the optimal wavelength region
situated between 1100 and 1300 nm, as shown in Fig. 10.1(b). So far as the LMs utilizing
emissions from Nd ions is concerned, the long optical path of a fiber-type LM enables
absorption of almost all the pumping light propagating along its axial direction. In
contrast, the short interaction length of a disk-type LM is insufficient for the same amount432

Chapter 10. Compact Solar-pumped Lasersof pumping sunlight to be absorbed. Therefore, a fiber-type LM was chosen to develop asolar-pumped fiber laser (SPFL).
10.3.1.2. Glass LMs
To realize a fiber-type LM as well as a disk-type LM, it was suggested in
Subsection 10.2.3 that glass LMs are promising for SPLs because of their high moldability
into thin disks/thin fibers, high solubility of rare-earth elements such as Nd, and suitability
for mass production.
Thus, TeO2-K2O (tellurite) glass, SiO2-B2O3-Na2O-Al2O3-CaO-ZrO2 (SBNACZ,
borosilicate) glass, and ZrF4-BaF2-LaF3-AlF3-NaF (ZBLAN, fluoride) glass were
examined as host glasses [80]. The content of Nd was changed in the range of 0-5.0 mol%. TeO2-K2O (tellurite) glass has the advantage of having the largest integrated absorption
cross-section of Nd3+ from 400 to 950 nm among the approximately 250 glass hosts. For
example, it is about 1.5 times higher than those in borosilicate glass and fluoride glass[81, 82]. The effective phonon energy in tellurite glass is usually less than 800 cm-1, which
is lower than that of the other oxides such as silicate (1000 cm-1) and phosphate(1200 cm-1) [83]. This enables efficient fluorescence from Nd3+ ions doped in the tellurite
glass. SBNACZ borosilicate glass has the advantage of having excellent rare-earthsolubility and chemical and thermal stability against crystallization because of its highglass transformation temperature. Crystallization behavior was not noticed for rare-earth
doping less than 10 wt. %. Therefore, it was initially suggested as a host medium toimmobilize high level nuclear waste [84]. SBNACZ borosilicate glass can hold up to
30 wt. % Nd2O3 without evidence of Nd clustering and has been proposed as an LM[81, 82]. ZBLAN fluoride glass, found in 1975 [85], has advantages in terms of its wide
transparent window between 0.22 and about 8 μm, low minimum loss as low as 1 dB/km[86], low effective phonon energy of less than 600 cm-1 [87], and high allowable doping
levels (up to 10 mol %) of rare-earth ions [34].
The internal quantum efficiencies under sunlight tracking the Sun using an altazimuth andTi:sapphire laser excitation (at 807 nm for tellurite and borosilicate glasses and at 794 nm
for fluoride glass) were measured directly using an integrating sphere method(cf. Fig. 10.22 (a)). The radiative quantum efficiency obtained by Judd-Ofelt analysis wasalmost 100 % for the tellurite and ZBLAN fluoride glasses and 50 % for the SBNACS
borosilicate glasses. The quantum efficiency under laser excitation was respectively foundto be 86 %, 34 %, and 88 % for the tellurite, SBNACS borosilicate, and ZBLAN fluoride
glasses at a low Nd3+ content. Parasitic absorption in the SBNACZ borosilicate glass might
have caused the low quantum efficiency. The quantum efficiency under sunlight
excitation was up to 33 %, 21 %, and 70 % for the tellurite, SBNACS borosilicate, and
ZBLAN fluoride glasses, respectively. Therefore, ZBLAN fluoride glass was chosen for
the glass LM. The stimulated emission cross-section σ(λL) in the transition 4F3/2 to 4I11/2
corresponding to the 1064 nm emission for 0.5 mol. % Nd3+ doped in ZBLAN glass wascalculated to be 2.95 × 10−20 cm2. With a fluorescence lifetime = 496 μsec for glass, σ(λL)‧ τ was obtained to be 1.46 × 10−27 m2s, which is higher than 0.75 × 10−27 m2sec for
Nd-doped silica glass codoped with Al [88].
433Advances in Optics: Reviews. Book Series, Vol. 5

10.3.2. Solar Concentrator
An off-axis parabolic mirror (OAP) was chosen as the solar concentrator because it is freefrom chromatic aberration and because products of relatively short focal length, incomparison with the caliber diameter, can be commercially available. Specifically, a⌀50.8 mm caliber aluminum OAP (Edmund Optics) with an aluminum coating layer(reflectance = 92 %) was employed for concentration of sunlight. Fig. 10.16 shows
schematic drawings of the OAP, including typical traces of light rays emitted from certain
points of an object and reflected at OAP.

Fig. 10.16. Schematic drawings of an OAP of caliber diameter 2D = 50.8 mm with typical tracesof light rays emitted from the head of the white arrow-shaped object of length RS and the headof a black arrow drawn on the white arrow-shaped object of length D. (a) A side view of a casein which the arrow-shaped object is pointing right in a side view of the system, (b) A case in which
the arrow-shaped object is pointing left in a side view of the system, (c) A case in which
the arrow-shaped object is pointing perpendicular to the plane of paper. A front view is also
provided to facilitate understanding of the geometry.

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Chapter 10. Compact Solar-pumped Lasers

The offset angle of this OAP was 90º. The focal point F is located on the horizontal line,
which penetrates the center point O of the OAP. The distance between F and O was also
designed to be 50.8 mm. Among the commercially available products of ⌀50.8 mm caliber
OAP, this was of the shortest FO length. Based on the geometrical consideration in
Fig. 10.16 (a), the length of the inverted image of the black arrow d can be given asD2Dd   
 Au  1  12  D
5 Au(10.24)

Supposing the ratio of the length of the image of the white arrow-shaped object rS to d isthe same as the ratio RS to D, rS can be given asrS  d R RS  RS 
2D  
  S   2D
D  Au  1  12  D  Au 
5 Au

(10.25)Here, D/Au ≪1 is used for the approximation. The lateral location of the image is definedby the distance from F, x, which can be expressed asx

12  D     2D
5  Au (10.26)

Because D/Au ≪1, the image is located at x ≈ 0, that is, on the focal point F. Similarly,
based on geometrical consideration in Fig. 10.16 (b), the length of the inverted image ofthe black arrow d can be expressed similar to Eq. (10.24). As for the lateral location of theimage, x is obtained as follows:
4 Dx      2D
3  Au (10.27)

Because D/Au ≪1, both the images in Figs. 10.16 (a) and (b) are located at x ≈ 0, that is,on the focal point F, as indicated in Eqs. (10.26) and (10.27). Figs. 10.16 (c) and (d) show
the side view and front view for cases in which the arrow-shaped objects are pointing
perpendicular to the plane of paper, respectively. Traces of two typical light rays emitted
from the head of the black arrow drawn on the white arrow-shaped object are indicatedwith blue and red lines. It is shown that the two lines, blue and red, do not intersect witheach other at a point three-dimensionally. More specifically, the red ray penetrates the
focal point F but the blue ray does not. This is because the present OAP is not precisely
an imaging system. However, the blue ray passes near the focal point. At the point of thenearest approach, the distance from F is (D/Au)‧2D. As for the light ray emitted from thehead of the white arrow-shaped object corresponding to the blue line, the distance between
the point of the nearest approach and F can be calculated as (RS/Au)‧2D. This is the sameas Eq. (10.25). Therefore, all incident sunlight at the entrance of the OAP crosses inside435Advances in Optics: Reviews. Book Series, Vol. 5

the circle of radius rS at the focal point F. Therefore, 2 rS is not a diameter of the image of
the Sun. It should be called the diameter of the pseudo image of the Sun. The solar
concentration ratio can be expressed as follows:2A  1
 D2
D2
 u  22
 rs  R 
 Rs  42S   4D
 Au 

(10.28)Using 50.8 mm for 2D, 6.96 ×108 m for RS, and 1.496×1011 m for Au into Eqs. (10.25) and(10.28), the diameter of the pseudo image of the Sun and corresponding solar
concentration ratio can be obtained as follows:
2rs  0.473 mm,(10.29)

 D2
 11550
 rs2(10.30)10.3.3. SPL Resonator Based on a Double Cladding Optical Fiber
Using the OAP in Fig. 10.16, the diameter of the image of the Sun near the focal point is
0.473 mm as shown in Eq. (10.29). A double-clad fiber (DCF) structure composed of acore (⌀5 μm), an inner cladding (⌀125 μm), and an outer cladding (⌀200 μm) was
employed. Both end facets were coated with thin film mirrors having reflectivity of 98 %at 1050 mm [30-32]. A substantial part of the focused sunlight that hits out of the core canenter the inner cladding and propagates in it, after which it eventually enters the core to
be absorbed by ions in the core as schematically shown in Fig. 10.17(a). This 10 m long
Nd 0.5 at% doped ZBLAN DCF was spooled on a kidney-shaped spool to increase thechance of sunlight propagating in the inner cladding to enter the core. As a consequence,
the mode-matching efficiency is as high as 80 %. The absorption spectrum, in comparison
with the broadband solar spectrum, gives the spectrum matching efficiency of absorptionof 13 %. Fig. 10.17 (b) shows a schematic diagram of this SPFL system. Fig. 10.18 shows
photographs of this SPFL system mounted on an altazimuth-type solar-tracking system(Vixen SKYPOD).
In this system, the factor in the first parenthesis in Eq. (10.8), h‧νe / (σ(λL) ‧τ), is
1.69×105 kW/m2. Supposing 0.8 for δ, and using 0.98 for R1 and R2, the factor in thesecond parenthesis in Eq. (10.31), (δ – ln (R1‧R2)/2, is 0.42. Supposing 0.62(= 650 nm / 1050 nm) for ηQD, 1 for ηQE, 0.13 for ηabs, 0.8 for ηm, 4×106 ( = 10 m /2.5 μm)
for A, and 2×10-4 ( = 2 mm / 10 m) for α (assuming 2 mm of cladding from the end facet
functions effectively), the threshold power density is calculated to be 686 kW/m2.
Supposing the incident direct solar radiation is 508 W/m2 as described in
Subsection 10.2.7, and geometrical and reflectivity loss of the solar concentrator is 30 %,the required concentration ratio is 1930 (686 kW/m2 / (0.508 kW/m2×0.7)). Therefore, it
was conjectured that the system shown in Figs. 10.17 and 10.18 has a good chance torealize oscillation.
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Fig. 10.17. Schematic diagrams of (a) double clad optical fiber (DCF), and (b) a prototype SPFL
system composed of OAP and Nd 0.5 at% doped ZBLAN DCF laser medium.

Fig. 10.18. Photographs of the prototype SPFL system comprising an OAP, a kidney-shaped
spool for a 10 m long Nd 0.5 at% doped ZBLAN DCF, and an optical fiber for the SPFL output.
The system is mounted on an altazimuth type solar-tracking system.

10.3.4. Outdoor Oscillation Performance of the Compact SPFL
Lasing experiments using sunlight were performed at longitude 137°3’E and latitude
35°10’N under clear skies with occasional cumuli. Fig. 10.19 (a) shows the direct sunlight
spectrum and the measured spectrum after passing through the Nd 0.5 at% -doped ZBLAN
fluoride glass fiber together with the lasing spectrum. The direct sunlight spectrum wascalculated according to the model by Bird and Riordan [79] under the following
conditions: elevation angle = 37°, aerosol optical depth at 500 nm wavelength = 0.27, total
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Advances in Optics: Reviews. Book Series, Vol. 5

column ozone = 0.34 cm, surface albedo = 0.20, total precipitable water vapor = 1.42 cm,
longitude = 137° 3′ E, and latitude = 35° 10′ N. Absorption bands observed were indicatedwith the identified related excited electronic levels of Nd3+ ion. Sunlight was completely
absorbed around the bands at wavelengths of 520 nm, 575 nm, 740 nm, 795 nm, and867 nm. However, strong intensity transmissions, such as blue light with wavelengths
between 410 nm and 510 nm and red light with wavelengths between 590 nm and725 nm, still remain. The detailed lasing spectra around 1053.7 nm obtained at different
times are shown in Fig. 10.19(b). Many peaks were observed between 1052 nm and1054 nm. Typically, the full width of the half maximum of the peak at 1053.7 nm is
approximately 0.01 nm. The spectra showed a complex and congested nature andfluctuated quickly. This was caused partly because of inhomogeneous broadening from
Nd ions in glass and longitudinal-mode hopping.

Fig. 10.19. (a) The direct sunlight spectrum (in red) and the measured spectrum after passing
through the Nd 0.5 at% -doped ZBLAN fluoride glass fiber together with the lasing spectrum(in blue open circles). The locations of absorption bands are indicated with the identified related
excited electronic levels of Nd3+ ion. (b) Detailed lasing spectra around 1053.7 nm obtainedat different times.

Fig. 10.20 shows the plots of the SPFL output power as a function of the input sunlight
power captured in the inner cladding of the fiber. The input sunlight power was controlled
by changing the flare angle of the bow-tie aperture placed at the sleeve for the parabolic
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mirror (cf. Fig. 10.26). A clear lasing threshold at 49.1 mW is confirmed by natural
sunlight excitation. The slope efficiency and the total efficiency were 3.3 % and 0.88 %,respectively. The maximum laser output power was 0.57 mW. Because the reflectivity ofboth ends of the fiber are the same, adding the output powers from both ends, the slope
efficiency and the total efficiency are doubled to be 6.6 % and 1.76 %, respectively [34].
This is “the first solar-pumped fibre laser” as picked up by Nature Materials [89].
The threshold power was derived to be 49.1 mW. Dividing the cross-sectional area within⌀125 μm of the inner cladding and the core of the ZBLAN optical fiber, the threshold
power density Ieth at the end facet of the fiber to start laser oscillation is approximately
4 MW/m2. Passing through the preposing optical system including a UV filter with atransmittance of 95 % and OAP with a reflectance of 92 %, the total transmittance of the
preposing optical system was 71.5 %. Taking the solar concentration ratio (11550) of thisOAP into consideration, the power density before the concentration is 484 W/m2.
Similarly, the maximum sunlight power density in this measurement was obtained as
651 W/m2. In the example of a numerical estimate based on the Eq. (10.8) in
Subsection 10.2.2, Ieth was estimated to be about 686 kW/m2, which is much less than theexperimentally obtained value of 4 MW/m2. It is conjectured that a larger value of δ andsmaller value of ηQD, ηQE, ηabs, ηm and α in practice may have caused this result.

Fig. 10.20. Output laser power of SPFL as a function of input solar power.

10.4. Improvement of Spectral Matching Efficiency by Cr Codoping
The spectral absorption range of the LM should cover a wide spectral range of sunlight –from UV to IR. This is referred to as “spectral matching”. It has been considered that an
SPL using a single rare-earth element such as Nd cannot be efficiently driven because of
the extremely narrow absorption bands of rare-earth elements [23]. A promising means
for the improvement of spectrum matching is to introduce a sensitizer of another elementthat extends the absorption range considerably [90]. Particularly, it has been found thatthe transition metal Cr3+ is a promising sensitizer, which has broadband absorption
characteristics and transfers the absorbed energy to Nd3+ [20, 42, 58, 63, 91-97]. In our
ZBLAN fiber, the mode-matching efficiency ηm is as high as 80 % while the spectral439

Advances in Optics: Reviews. Book Series, Vol. 5matching efficiency ηabs is 13 %. Therefore, it is important to increase the spectral
matching efficiency by Cr codoping in addition to Nd. Although Cr3+ codoping is apossible tactic to improve the performance of SPFL, it was eventually found difficult tocodope Cr into Nd-doped ZBLAN fibers. Therefore, instead of taking advantage of thelarge aspect ratio A of fiber LMs shown in Fig. 10.6, we used Cr-codoped Nd:YAG
transparent ceramic micro rods to make use of the advantages described in
Subsection 10.2.6 and Fig. 10.11. Fig. 10.21 shows the measured spectral absorption
coefficients of transparent ceramic mediums of (1) Nd (1.0 at%):YAG, (2) Cr (0.1 at%),Nd (1.0 at%):YAG, and (3) Cr (0.4 at%), Nd (1.0 at%): YAG. In contrast with the narrowadsorption peaks of (1) Nd (1.0 at%):YAG, (2) Cr (0.1 at%), Nd (1.0 at%):YAG and(3) Cr (0.4 at%), Nd (1.0 at%):YAG showed broad absorption peaks. It was also shownthat the increase in Cr content from 0.1 at% to 0.4 at% caused an approximate 4 times
increase in the broad absorption peaks. It is expected that spectral matching efficiency
improves from 13 % to 35 % by the codoping of Cr ions. Then, introducing sunlight into
an integrating sphere, spectral absorption, and emission properties of Nd:YAG and
Cr-codoped Nd:YAG were examined. Fig. 10.22 (a) schematically shows a system tomeasure the spectral absorption and emission from the sample in an integrating sphere in
the wavelength range of 300 nm to 1600 nm under natural sunlight irradiation (LUCIRFQEM-L1). Fig. 10.22 (b) shows the observed spectra with and without a sample.
Subtracting the spectrum without a sample from the spectrum with a sample, a differential
spectrum could be obtained that showed absorption of the solar spectrum and thespontaneous emission from the sample. Fig. 10.22 (c) shows a differential spectrum
measured for the Nd (1 at%):YAG ceramic sample, which shows absorption of the solar
spectrum and emission (1064 nm) from Nd3+ ions. Fig. 10.22 (d) shows a similardifferential spectrum for Cr (0.5 at%)-codoped Nd (1 at%):YAG ceramics sample. In
comparison with the spectrum in Fig. 10.22 (c), the absorption of the solar spectrum wasenhanced, and the emission (1064 nm) increased 1.5 times in Fig. 10.22 (d). This indicatesthe photon energies absorbed by Cr3+ ions were transferred to Nd3+ ions and contributed
the spontaneous photoemission. Thus, Cr doping to Nd:YAG transparent ceramic rod wasexperimentally reconfirmed to be a promising tactic to improve SPLs.

Fig. 10.21. Measured spectral absorption coefficients of transparent ceramic mediums of (1) Nd(1.0 at%): YAG; (2) Cr (0.1 at%), Nd (1.0 at%): YAG; and (3) Cr (0.4 at%), Nd (1.0 at%): YAG.

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Fig. 10.22. Measurement of differential spectra showing absorption of sunlight and spontaneousemission from Nd:YAG ceramic samples and Cr-codoped Nd:YAG ceramic samples by guiding
the natural sunlight into an integrating sphere using a solar-tracking system and optical fibers.

10.5. Compact Solar-Pumped Micro-Rod Laser (μSPL) System
We replaced the Nd-doped ZBLAN glass fiber with a 1 × 1 × 5 mm quadrangular-prismshaped, Cr 0.1 at% codoped, Nd 1 at%:YAG transparent ceramic micro laser rod supplied
by World Lab. Co. Ltd, Japan as shown in Fig. 10.23.

Fig. 10.23. 1×1×5 mm, Cr 0.1 at% codoped, Nd 1 at%:YAG transparent ceramic
micro laser rod.

The solar concentrator was changed to a ⌀76.2 mm caliber OAP (Edmund Optics) keeping
the distance between the focus F and the center of OAP, O, unchanged from that of the⌀50.8 mm caliber OAP used in the SPFL described in Section 10.3, that is, 50.8 mm.
Among the commercially available products of ⌀76.2 mm caliber OAP, this is of theshortest OF length. Expressing the caliber diameter 76.2 mm as 2D in this section, thedistance between F and O is 4D/3 as shown in the schematic drawing Fig. 10.24, which is441

Advances in Optics: Reviews. Book Series, Vol. 5

drawn in the same manner as Fig. 10.16. Based on the geometrical consideration in
Fig. 10.24(a), the length of the inverted image of the white arrow-shaped object rS is
obtained using a similar process as the derivation of Eq. (10.25):

R  4rS   S   D
 Au  3(10.31)The same result is also obtained from Fig. 10.24(b). Then, it can be shown that all theincident sunlight at the entrance of the OAP crosses inside the circle of radius rS at the
focal point F in the same manner as shown in Subsection 10.3.2. The solar concentration
ratio can be expressed as follows:2 Au  9
 D2
D2


  
2
2
2
 rS
 RS   4 
 RS  162D   
 Au   3 (10.32)Using values 36.1 mm for D, 6.96 ×108 m for RS, and 1.496×1011 m for Au into Eqs. (10.31)
and (10.32):
2rs  0.448 mm,(10.33)

 D2
 25990
 rS 2(10.34)

Fig. 10.24. Schematic drawings of an OAP of a caliber diameter 2D of 76.2 mm with typical tracesof light rays emitted from the head of a white arrow-shaped object of a length Rs and the headof a black arrow drawn on the white arrow-shaped object of a length D. (a) A case in which
the arrow-shaped object is pointing right in a side view of the system, (b) A case in which
the arrow-shaped object is pointing left in a side view of the system.

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These results are more favorable than the results for the ⌀50.8 mm caliber OAP shown in
Eqs. (10.29) and (10.30). Sunlight can be focused within the end facet of the ⌀1 × 10 mmrod. Fig. 10.25 schematically shows this μSPL system. Output coupler (OC) of different
curvature radii were tested. In a typical case of Nd:YAG laser where λL was 1064 nm, λe
was 808 nm, σ(λL) was 2.8×10-19 cm2, τ was 230 μsec [98], δ was 0.02, R1 was 0.99, R2
was 0.9995, ηQD was 0.61 ( = 650 nm / 1064 nm), ηQE was 1, ηabs was 0.35, ηm was 0.29,
A was 10 (5 mm /0.5 mm) and α was 0.03, Ieth was calculated to be 5.9 MW/m2. Therefore,
solar concentration ratio of more than 16244 is required for the direct solar radiation of
508 W/m2 to start oscillation of this laser rod, taking the transmittance of the preposing
optical system to 71.5 % including a UV filter with a transmittance of 95 %, and OAP
with a reflectance of 92 %. Fig. 10.26 shows a photograph of the nucleus of a μSPL.
Bow-tie apertures of different flare angles were used to control the incident sunlight power
to measure the input solar power dependence of the output laser power. Fig. 10.27 shows
the μSPL system mounted on an altazimuth-type solar-tracking system (Vixen SKYPOD)
in an outdoor oscillation experiment tracking the Sun. The μSPL is compact enough touse an altazimuth fabricated to mount an astronomical telescope for observation ofcelestial objects (Vixen SKYPOD) as a solar-tracking system.

Fig. 10.25. Schematic diagrams of a prototype μSPL system comprised of an OAP and a Cr
0.1 at%, Nd 1 at% doped transparent YAG ceramic rod laser medium in an Al heat sink, an OC,
and an optical fiber for laser output.

Fig. 10.28 (a) shows the measured output laser power as a function of the input solar
power into the rod shown in the abscissa at the bottom. An OC of a curvature radius = 100
mm was used. The input solar power into the OAP is also indicated in the abscissa at the
top. The maximum input solar power into the OAP was 4.28 W, which corresponds to a
direct solar radiation of 939 W/m2. Passing through the preposing optical system,
including a UV filter with a transmittance of 95 % and OAP with a reflectance of 92 %,the maximum input solar power into the rod was 3.06 W. The total transmittance of the
preposing optical system was 71.5 %. The threshold power at the end facet of the rod was
1.8 W. Dividing it by the area of the focused size, π‧rS2 of 1.58×10-7 m2, the threshold443

Advances in Optics: Reviews. Book Series, Vol. 5

power density of 11.4 MW/m2 could be obtained. The slope efficiency against the input
solar power at the rod and the total efficiency against the input solar power at the rod were
2.1 % and 0.9 %, respectively. The external slope efficiency and the external total solar
conversion efficiency against the input solar power into the OAP were 1.5 % and 0.7 %,respectively. Fig. 10.28(b) shows optical transmittance spectra of sunlight before and after
passing through the laser rod. The difference between the two spectra corresponds to theabsorbed light by the rod. Fig. 10.28 (c) shows the laser emission spectrum peaking at
1064 nm.

Fig. 10.26. Nucleus of the μSPL system composed of an OAP and a resonator comprisinga Cr-codoped, Nd:YAG transparent ceramic micro laser rod and an optical coupler. Bow-tie
apertures of different flare angles are used for controlling the incident sunlight power to measure
the input solar power dependence of the output laser power.

Fig. 10.27. Outdoor oscillation experiment of a μSPL on an altazimuth tracking the Sun.

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Fig. 10.28. (a) Output laser power of a 1×1×5 mm Cr (0.1 at%), Nd (1 %):YAG transparent ceramic
rod type μSPL as a function of the input solar power; (b) Optical transmittance spectra of sunlight
before and after passing through the laser rod; (c) Spectrum of the output laser. The inset figure
shows a detailed peak profile between 1060 nm and 1068 nm.

Fig. 10.29 shows the measured output laser power as a function of the input solar power
into the ⌀1 × 10 mm, Cr (0.1 at%) codoped Nd (1.0 at%):YAG transparent ceramic rod(Konoshima Chemical Co. Ltd.) shown in the abscissa at the bottom using the same μSPL
system shown in Fig. 10.25. An OC of a curvature radius = 100 mm was used. The input
solar power into the OAP is also indicated in the abscissa at the top. The maximum input
solar power into the OAP was 3.61 W, which corresponds to direct solar radiation of
792 W/m2. Passing through the preposing optical system of the total transmittance of
71.5 %, the maximum input solar power into the rod was 2.59 W. The threshold power atthe end facet of the rod was 1.56 W, which was lower than that in Fig. 10.28. Dividing
the value by the area of the focused size, π‧rS2 of 1.58×10-7 m2, threshold power densityof 9.87 MW/m2 could be obtained. The slope efficiency and the total efficiency against
the input solar power into the rod were 2.6 % and 1.0 %, respectively. The external slope
efficiency and the external solar energy conversion efficiency against the input solar
power at the OAP were 1.8 % and 0.73 %, respectively.
The effects of different OC curvature radii, 500 and 1000 mm, were also examined.
Further, two quadrangular-prism-shaped transparent ceramic rods of 1 ×1× 10 mm of Cr
0.1 at% codoped Nd 1 % doped YAG and Cr 0.4 at% codoped Nd 2 % doped YAG
supplied by World Lab. Co. Ltd, were also tested in the same μSPL system shown in
Fig. 10.25. Table 10.1 summarizes the results. Experiment (a) shown in Fig. 10.29 is thebest data among the five results.
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Fig. 10.29. Output laser power of the μSPL as a function of the input solar power into the rod.
The input solar powers into the OAP are indicated with italic letters to show the realsoler energy conversion efficiency.
Table 10.1. Slope efficiencies and threshold powers in the different experimental setups.
Exp.
No.
(a)
(b)
(c)
(d)
(e)

Size
(mm)
⌀1 × 10⌀1 × 10⌀1 × 101 × 1 × 10
1 × 1 × 10

Dopants
Cr0.1 %, Nd1 %Cr0.1 %, Nd1 %Cr0.1 %, Nd1 %Cr0.1 %, Nd1 %Cr0.4 %, Nd2 %Radius of OC
(mm)
100
500
1000
100
100

Slope efficiency
( %)
2.60
1.68
1.73
0.78
1.40

Thresholdpower (W)
1.56
1.26
1.38
2.26
1.20

10.6. Continuous Oscillation of μSPL for Over 6.5 Hours Tracking the Sun
Using the same condition as Experiment (a) in Table 10.1, continuous oscillation was tried
outdoors tracking the Sun. Instead of the altazimuth (Vixen SKYPOD) in Fig. 10.27, acommercially available automatic equatorial mounting (VIXEN SXD2) with a tripod was
employed seeking perfect solar tracking. The system was set at the roof terrace of Building
3 of Faculty of Engineering, Nagoya University located at latitude 35°10’N north andlongitude 136°58’E, Nagoya, Japan. A continuous oscillation experiment was performed
on a clear weather day, Wednesday, August 9th, 2017. The μSPL experimental rig is
pictured in Fig. 10.30. Output power of the μSPL was measured with a power meter sensor
every 0.3 seconds and the measured value was stored in the memory, coinstantaneously.
A pyrheliometer shown in Fig. 10.14 and a solar telescope were fabricated and mounted
on the same equatorial mounting with the μSPL and tracked the Sun simultaneously. The
power of the direct solar radiation in 5 degrees in view angle and in the wavelength rangeof 190 nm to 25 μm captured with an aperture and a pinhole was measured with a
THOLABS S302C thermal power meter sensor attached to the pyrheliometer. The
THOLABS S302C power meter sensor had been calibrated with an EKO
INSTRUMENTS MS-53 (ISO 9060 First Class Pyrheliometer) that provided the value ofthe direct solar radiation in terms of W/m2. These data were measured as well as recorded
in memory every 0.3 seconds. The inset at the lower left in Fig. 10.30 is an image of the446Chapter 10. Compact Solar-pumped Lasers

Sun on a computer display monitored by the solar telescope using a video-capturingattachment. As reference, global solar irradiance was measured with a digital
illuminometer. After adjustment of the direction of the μSPL roughly using the image of
the Sun on the PC display monitored by the solar telescope, precise adjustment of screws
fixing the μSPL onto the mounting provided a sudden rise in the output power of the μSPL
from several μW to several mW at 10:50, indicating that laser oscillation was taking place.
After start of the μSPL oscillation, the solar telescope was replaced by the pyrheliometer,
and the measurement of the direct solar radiation was performed in parallel with themeasurement of the output power of the μSPL.

Fig. 10.30. μSPL system tracking the Sun automatically on an equatorial mounting (VIXEN
SXD2). The inset figure in the lower left is an image of the Sun on a PC display monitoredby the solar telescope using a video-capturing attachment for a rough adjustment of directionof μSPL toward the Sun.

Fig. 10.31 shows the variation in μSPL output power as a function of time in the lower
blue line. The variation of the direct solar radiation in terms of kW/m2 is also shown in
the upper red line. For reference, the global solar irradiances at three different clock times
indicated on the illuminometer—109000 lx at 11:08, 114000 lx at 11:59, and 43600 lx –at 16:48 are also shown at the upper side of the figure. Simultaneous instantaneous falls
both in the μSPL output and in the direct solar radiation around 13:22-13:36 and 17:17
were caused by passing clouds. With occasional manual readjustments of the equatorial
mounting such as at 11:10, 11:23, 11:33, 11:43, 11:52, 12:00, 12:25, 12:55, 13:48, 14:46,
15:16, 15:37, 16:11, 16:29, and 17:00, the output power recovered 2-3 mW. Especiallythe readjustment at 12:00 provided an approximately 4.5 mW increase in the μSPL output.
The maximum direct solar radiation over 898 W/m2 took place between 12:00 and 13:00
in which the μSPL output also reached its maximum of around 12 mW. The μSPL output
kept going over 10 mW until 14:30. After continuous decrease, the output power of theμSPL suddenly fell from 0.2 mW to 0.0017 mW between 17:34 and 17:35, althoughinstantaneous oscillations were observed several times beyond 17:35. After 17:35 until
17:51, almost all the measured output powers were around 0.0017 mW. Therefore, it isconsidered that the μSPL stopped continuous oscillation between 17:34 and 17:35. It was
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positive outcome that the μSPL kept continuous oscillation even if the direct solar
radiation was lower than 550 W/m2 around 17:30, that is, 61 % of the maximum value.
This indicates the possibility of continuous oscillation for 11 hours from 6:30-17:30,
which is considerably longer than our expectation and favorable forsolar energy utilization. Fig. 10.32 shows a plot of these synchronous 47012 data sets, that
is, a plot of μSPL output against the incident direct sunlight power at the entrance of theOAP. The distribution of the data clearly shows the existence of a threshold incident
direct sunlight power around 1.47 W (2.05 W) for laser oscillation and a linear increasein μSPL output power with incident direct sunlight power larger than 1.47 W (2.05 W).
This plot was realized because the measurement and the data acquisition of μSPL output
power were continued beyond 17:30 after the oscillation stopped because of shadowing
of sunlight by clouds spreading flat to the western horizon. A more detailed inspection ofFig. 10.32 reveals existence of two straight lines: one is in the region under 2.7 W of thetotal incident power (under 5 mW of the μSPL output), and the other is in the region over
2.7 W of that (over 5 mW of the μSPL output), with an inflection point around 2.7 W.
Since the μSPL output was over 5 mW in the major day time before 16:30 in thisexperiment as shown in Fig. 10.31, we estimated the slope efficiency (external slope
efficiency) to be 1.15 % (0.82 %) from the straight line over 2.7 W. The slope efficiency
based on the straight line below 2.7 W is discussed next using the inset figure
in Fig. 10.32.

Fig. 10.31. Variation of μSPL output power as a function of time with an occasional manual
readjustment of equatorial mount during automatic solar tracking. Variation of the direct solar
radiation in terms of kW/m2 measured by a pyrheliometer is also shown. For reference,
the measured global solar irradiance in terms of lx at three different clock times are shownat the upper side.

To understand whether the red shift as the afternoon grew late (as displayed in Figs. 10.12
and 10.15) influenced the oscillation properties of the μSPL or not, a replot is shown in
the inset figure in Fig. 10.32 in which the 8000 data points measured in the last 40 minutes
after 16:50 are indicated using red markers while the others are indicated using graymarkers. Those red-marked data were measured under insolation with the spectrum almost
equal to or more redshifted than that at 16:50 indicated in Fig. 10.15. The distribution of448

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the red markers is consistent with the second straight line below 2.7 W of the total incident
power described above. The slope efficiency deduced from this second straight line isevidently lower than that deduced from the total data in the main figure. To compare theslope efficiency quantitatively, the slope efficiency (external slope efficiency) of 1.08 %(0.77 %) was deduced from the second straight line while that deduced from the first line
was 1.15 % (0.82 %). The two straight lines do not positively indicate the existence of
two different threshold incident direct sunlight powers for laser oscillation but rather thesame one around 1.47 W (2.05 W).

Fig. 10.32. Plots of μSPL output power against incident direct sunlight power at the entranceof the OAP using the 47012 data sets from those shown in Fig. 10.31. The inset at the upper left isa replot emphasizing the 8000 data points that correspond to the last 40 minutes after 16:50
indicated using red markers while others are marked using gray markers.

The results in Figs. 10.29 and 10.32 were obtained employing the same laser rod together
with the same solar concentrator of ⌀76.2 mm caliber OAP. Although there is noremarkable difference in the threshold power between the two experiments, the slope
efficiency and the total efficiency are more than halved in the result in Fig. 10.32 incomparison with that in Fig. 10.29. Almost two years had passed when the experiment in
Fig. 10.32 was performed since the experiment in Fig. 10.29. Between the two
experiments, repeated test experiments were done using the same laser rod. It is known
that there is a degradation of Nd:YAG transparent ceramics, known as solarization, whichis thought to be caused by sunlight exposure. However, the origin of solarization has notbeen clarified yet because the degradation is difficult to detect via instrumental analyses.
Although a UV filter (Kenko Tokina Zeta UV L41) was always set at the entrance of the
OAP in our experiments, degradation of the laser rod is a possible cause of the decreasein the efficiencies. The prevention of degradation of the laser rod will be one of key
challenges before commercial adoption of SPLs.
In previous study [23], a water-cooled Nd:YAG crystal rod was pumped with a ⌀61 cm,
solar tracking, equatorial mount solar collector to give 1 W of continuous wave (CW)
laser output. Although the study claims that “Operation over many hours was obtainedwith no evidence of reduction of output,” no evidential data of continuous oscillation wasprovided. An evidential record of continuous oscillation for 11 min was reported
employing a water-cooled Nd:YAG rod and a 10 m aperture solar collector [42], in which
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tracking error was corrected manually 8-10 times during this 11 min continuous operation.
A record of 60 sec stable continuous oscillation was also reported employing a watercooled Nd:YAG ceramic rod and 2 m × 2 m Fresnel lens to generate 110 W CW laser
output [63]. More recently, records of continuous oscillation of solar-pumped 1064 nm
laser emission with a simple Gaussian intensity distribution around the center axis of thebeam (TEM00-mode) for 4 minutes have been reported [99]. Therefore, the result shownin Fig. 10.31 is a record long continuous laser oscillation of SPL tracking the Sun [100].
This is an indispensable experimental step if SPLs are to be applied to terrestrial solar
energy utilization.

10.7. Effect of Cr Content in a Cr-Codoped Nd:YAG Transparent Ceramic
Laser Rod for μSPLs
Direct experimental comparison of the effect of Cr, Nd:YAG ceramic LMs with different
Cr contents on the SPL output was performed. Quadrangular-prism-shaped LMs oftransparent Cr-codoped Nd:YAG ceramics (1 × 1 × 10 mm, and 1 × 1 ×20 mm) wereprepared with Cr contents of 0.0, 0.4, 0.7, and 1.0 at% with a fixed Nd content of 1.0 at%.
These LMs were fabricated by World Lab. Co. Ltd., Atsuta, Nagoya, Japan [45, 46]. The
μSPL system is the same as shown in Figs. 10.25 and 10.26. Fig. 10.33 shows aphotograph of the prepared LMs; the greenish color is attributed to Cr3+. The end facet of
an LM was coated with a high-reflectivity (HR) coating consisting of a multilayered thinfilm optical filter with 95 % transmittance for focused sunlight coming from the OAP anda reflectivity (RHR) of 99.95 % for the 1064 nm monochromatic light emitted by Nd3+ inthe LM. The other side of the LM was coated with a broadband antireflection (BBAR)
coating consisting of a multilayered thin film optical filter with more than 95 %transmittance of sunlight with wavelengths between 780 and 1064 nm and almost 100 %transmittance of 1064 nm monochromatic light. The laser cavity contained the HR coatingon the end facet of the LM and the concave mirror of OC with a curvature radius r of
100 mm and a diameter of 10 mm.

Fig. 10.33. Photograph of the prepared Cr-doped Nd (1.0 at%): YAG rods. Cr content is indicatedon the left. Length of the rod is indicated at the top. The black points near the left ends of the rods
indicate that the left end facets of the rods were coated with HR coatings, whereas the right end
facets were coated with BBAR coatings.

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An outdoor μSPL oscillation experiment was performed on the same location as shown in
Figs. 10.27 and 10.30. Figs. 10.34(a) and (b) show plots of measured μSPL outputs against
the incident direct sunlight power into the OAP (IOAP) for the 10 mm and 20 mm LMs,
respectively [101]. The incident sunlight power at the edge of LM, ILM, is 71.5 % of IOAP
in this light collection system employing the OAP, as was reported in Section 10.3. Here,the OC had a reflectively Rc of 99 % for laser emission (1064 nm). For both the LMs, theoutput power of the μSPL took the maximum at a Cr content of 0.4 at%. The output power
at a Cr content of 0.4 at% was approximately 8.5 times that at a Cr content of 0.0 at%.
The output power decreased as the Cr content increased beyond 0.4 at%. At a Cr content
of 1.0 at% for the 20 mm LM, the output power was even lower than that for the LM
without Cr doping. It is not easy to define the portion of linear increase of μSPL output
with IOAP. However, we can compare the threshold input powers and slope efficiencies of
the LMs using the values of the IOAP intercept Ieth with the line connecting the point of thehighest μSPL output and the point of the second highest μSPL output obtained in thisexperiment for each Cr content and the values of the corresponding slope ηslope. Table 10.2
lists the obtained values of Ieth and ηslope.

Fig. 10.34. Plots of measured μSPL output power against the incident direct sunlight power
into the OAP, IOAP for (a) 10 mm and (b) 20 mm transparent Cr-doped Nd (1.0 at%) YAG LMs
with different Cr contents.

In addition to the outdoor laser oscillation experiments pumped by direct sunlight, laser
oscillation experiments in which Nd3+ was pumped directly using an 808 nm LD were
performed in an indoor laboratory. To couple the 808 nm excitation beam, along-working-distance objective lens (focal length = 10 mm) was used. Here, OCs of
r = 100 mm, diameter = 20 mm, and reflectivity Rc = 90 %, 95 %, and 99 % for laser
emission (1064 nm) were employed. Fig. 10.35 (a) shows a typical oscillation property of
the 10 mm LM of Cr (0.4 at%), Nd (1.0 at%):YAG ceramic obtained in the laser
oscillation experiments by pumping Nd3+ directly using an 808 nm LD in an indoor
laboratory. Here, an OC with a reflectivity Rc of 95 % for the laser emission (1064 nm)
was employed. To evaluate the round-trip loss δ808 in the laser cavity, the oscillation
properties were also obtained using OCs with Rc of 90 % and 99 %. Fig. 10.35 (b) shows
a plot of Ieth (808) for the three measurements at different values of Rc against loge Rc. The
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extrapolation of a line determined by least-square fitting of the three data points intercepts
the horizontal loge Rc axis at -0.026. Therefore, δ808 was found to be 0.026 [102]. Similarly,
the δ808 values of the eight LMs were obtained and are listed in Table 10.2, together withthe ηslope (808) and Ieth (808) values obtained in these indoor experiments.
The results in Table 10.2 show that δ808 increased with increasing Cr content. The valueof ηslope (808) did not increase as dramatically as ηslope but did show a slight increase between
0.0 at% Cr and 0.4 at% Cr. At Cr contents exceeding 0.4 at%, ηslope (808) decreased
gradually in both the 10 mm and 20 mm LMs. The increase in Ieth (808) was like that of δ808
for both the 10 mm and 20 mm LMs. These results clearly show that the presence of Cr3+
does not affect the oscillation process because of direct excitation of Nd3+ by the LD. In
contrast, for both the 10 and 20 mm LMs, Ieth showed a steep decrease between 0.0 at%
Cr and 0.4 at% Cr, reaching a minimum between 0.4 at% Cr and 0.7 at% Cr and increased
again toward 1.0 at% Cr. According to Eq. (10.8), the increase in Cr content is thought tocontribute to the increase in Ieth via the increase in δ. However, the increase in the Crcontent also contribute to the decrease in Ieth via the increase in ηabs in Eq. (10.8). The
minimum value of Ieth at a Cr content of 0.4 at% is attributed to these two conflicting
contributions. Because the increase in Ieth was observed even at low Cr contents between
0.4 % and 0.7 %, the possible effect of concentration quenching is thought to have little
effect on the increase in Ieth in this experiment. Recall that ηslope increased dramaticallybetween 0.0 at% Cr and 0.4 at% Cr and decreased gradually beyond 0.4 at% Cr. The
maximum ηslope at 0.4 at% Cr was 2.43 % for the 10 mm LM and 3.19 % for the 20 mm
LM. These values are converted to 3.40 % and 4.46 %, respectively, if the μSPL output isplotted against not IOAP but ILM. These values are competitive with the current
state-of-the-art values [60].
Table 10.2. Measured values: Ieth and ηslope. of the eight LMs obtained outdoors under sunlight,
and the round-trip loss L808, Ieth (808) and ηslope (808) obtained indoors under 808 nm LD pumping.
10 mm LMCr3+
I e th η slope δ 808 η slope(808) I e th(808)(at%)(W) (%) (%)
(%)
(mW)
0
2.98 0.36 2.5
47.2
45.5
0.4 2.41 2.43 2.6
49.2
50.6
0.7 2.49 2.00 3.0
37.4
70.9
1
2.52 1.15 4.5
26.6
85.3

I e th
(W)
3.03
2.64
2.52
3.26

20 mm LMη slope δ 808 η slope(808) I e th(808)(%) (%)(%)(mW)
0.67 2.8
37.9
58.6
3.19 2.6
43.5
59.7
1.44 3.8
32.3
79.3
0.34 5.8
18.1
93.8

These results indicate that Cr doping is effective for decreasing Ith and increasing ηslope
because of increased optical absorption by Cr3+. Although it was demonstrated that Cr
doping increases δ808, it was also found that the increase in δ808 between 0.0 at% Cr and
0.4 at% Cr was small compared with the increase between 0.4 at% Cr and 1.0 at% Cr.
Therefore, the μSPL output showed an outstanding maximum at 0.4 at% Cr, because it isdetermined by competition between the positive effect of increased energy transfer from
Cr3+ to Nd3+ and the negative effect of increased δ with increase in Cr content. It appears
that these conditions were the same for both 10 and 20 mm LMs.
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Fig. 10.35. (a) Plots of μSPL output against the incident 808 nm laser power I(808) into the 10 mm
LM of Cr (0.4 at%),Nd (1.0 at%):YAG, where an OC with r of 100 mm, a diameter of 20 mm,
and a reflectivity Rc of 95 % at 1064 nm was employed. (b) Plot of Ieth (808) for Rc of 90 %, 95 %,and 99 % against loge Rc.

10.8. For Improvement of Mode-matching Efficiency
In this work, it has been established that sunlight can be concentrated to exceed Ieth at theend facet of LMs by using OAPs. However, this raises a serious issue, known as the“mode-matching problem,” which means that there is sufficient overlap between the lasermode profile and the propagation mode of the pumping light is difficult to achieve.
Fig. 10.36 schematically shows how the concentrated direct sunlight propagates in an LM.
The thin, red cylindrical region stands for a simplified laser mode profile. The overlap ismaximum when the length of the LM is small, that is, in the case of disk-shaped LM. With
increase in the length of LM, the overlap decreases toward the constant value because theconverged sunlight at the end facet of the LM diverges again in the LM. Pumping theregion not overlapping with the laser mode does not contribute to the laser output but only
leads to energy loss. In Figs. 10.16 and 10.24, several typical traces of light rays areindicated to show the image formations. Here, more detailed tracing of sunlight was
performed using a commercially available simulation software ZEMAX. Fig. 10.37(a)shows the result for traces of sunlight reflected by an OAP and converged on an end facetof ⌀1 × 10 mm LM. Fig. 10.37(b) shows a distribution of the absorbed power of thepumping light propagating in the LM. The distribution reflects asymmetric nature of theconverged light by OAP in the first 1 mm and extends widely in the LM, implying
significant loss in the peripheral region out of the laser mode. Combining a laser cavity
analysis software LASCAD, the output laser powers were simulated as a function of the
input solar power into the rod assuming 2.8×10-19 cm2 for σ(λL), 230 μsec for τ [98], 0.02
for δ, and 0.8 for ηQE. The results are shown in Fig. 10.38. The simulated results give athreshold power of 1.44 W and a slope efficiency of 2.8 % while the corresponding
experimental values are 1.56 W and 2.6 %. Although the simulated results approximately
reproduced the experimental results in Fig. 10.29, the slightly better threshold power andthe slope efficiency imply ηQE lower than 0.8, thereby reflecting poor energy transfer
efficiency from Cr3+ to Nd3+. If the doping region in ⌀1 × 10 mm rod could be reduced,453

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for example, within ⌀0.4 mm, 0.6 mm, or 0.8 mm as shown in Fig. 10.39(a), the absorptionof the light would take place only within the doped region as shown in a simulated resultin Fig. 10.39(b). Here, we call this structured rod shown in Fig. 10.39(a) as a composite
rod. Simulated output laser powers plotted against the input solar power into the rods by
ZEMAX and LASCAD shown in Fig. 10.39 (c) indicate a decrease in the threshold powerwith decrease in the diameters of the doped regions. This is possibly caused by the increasein mode-matching, that is, the decrease of the absorption out of the laser mode that does
not contribute the laser output. However, reduction of the doped region less than ⌀0.8 mm
led to decrease in the slope efficiency, which is possibly caused by the decrease in theabsorbed light in the laser mode region because of the excessive reduction in the
doped region.

Fig. 10.36. Schematic representation of a laser mode profile and pumping light propagation
in a laser medium.

Fig. 10.37. (a) Ray tracing of the sunlight concentration onto ⌀1 × 10 mm, Cr (0.1 at%) codoped
Nd (1.0 at%):YAG transparent ceramic rod by an OAP; (b) Absorbed power distribution
in the laser medium (optical simulation code: ZEMAX).

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Fig. 10.38. Output laser power of the μSPL as a function of the input solar power into the rod
simulated combining ZEMAX and LASCAD. The corresponding experimental data shownin Fig. 10.29 were also replotted for reference.

Fig. 10.39. Effects of reduction of the doped region in a YAG micro rod. (a) A schematic composite
rod structure; (b) Absorbed power distribution in the composite rod simulated using ZEMAX;(c) Output laser power of the composite rod SPL as a function of the input solar power into the rod
simulated combining ZEMAX and LASCAD.

Although the reduction in the doped region to form a composite rod is effective for
improving mode-matching, the difference in the refractive indices between the core andthe cladding causes another loss, that is, diffraction loss between the exit end facet of therod and a resonator mirror as shown in Fig. 10.40 (a). By doping 5 at% non-absorbing Gd,
the refractive index of the cladding was matched with that of core (Fig. 10.40 (c)), and thediffraction loss could be eliminated as shown in Fig. 10.40(b). For ease of production,
quadrangular-prism-shaped composite rods in place of concentric cylinders, as shown in
Fig. 10.39 (a), were chosen. Fig. 10.41 shows photographs of (a) a colorless Gd 5 at%doped YAG transparent ceramic LM, (b) a greenish Cr 0.4 at%, Nd 1 at% doped
transparent ceramic LM, (c) a cross-sectional view of 2D composite LM, (d) a455

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cross-sectional view and part side
three-dimensional (3D) composite LM.

view

of

a

quadrangular-prism-shaped

Fig. 10.40. Refractive index matching between the Cr, Nd: YAG core and the Cr, Nd non-doped
YAG cladding by non-absorbing Gd in the cladding to reduce the diffraction loss between
the exit end facet of the rod and a resonator mirror.

Fig. 10.41. Photographs of (a) a colorless Gd 5 at% doped YAG transparent ceramic LM,(b) a greenish Cr 0.4 at%, Nd 1 at% doped transparent ceramic LM, (c) a cross-sectional viewof a quadrangular-prism-shaped two-dimensional(2D) composite LM, (d) a cross-sectional view,
and a part of a side view of a quadrangular-prism-shaped three-dimensional(3D) composite LM.

Using a convergent monochromatic light of NA = 0.4 formed from an 808 nm diode laserthat directly excites Nd3+ ions (Fig. 10.42 (a)), preliminary evaluations of composites rods
shown in Fig. 10.41 (c) and (d) were performed. The results shown in Fig. 10.42 (b)elucidate a lower threshold power and a higher slope efficiency for 2D composite rod incomparison with those for a conventional homogeneous rod, and even better results for3D composite rod than that for 2D composite rod [103]. Although an experimental proofof the advantages of 3D composite rods by outdoor oscillation tracking the Sun remains achallenge for the future, the slope efficiency in a 3D composite rod with a 0.4×0.4 mmcore in a 1×1 ×10 mm LM under white light pumping is simulated to be 18 %, as shownin Fig. 10.43 whereas that in a 2D composite rod is 3.1 % and that in a homogeneous(conventional) rod is 1.3 %. Therefore, more precise adjustment of core size may bringbetter mode-matching and better efficiencies.
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Fig. 10.42. (a) Monochromatic convergent beam optics for preliminary evaluation of 2D and 3D
rods; (b) Results of comparative evaluations of a conventional homogeneous rod, 2D composite
rod, and 3D composite rod.

Fig. 10.43. Simulated oscillation property of a 3D composite rod under white light pumping.

10.9. Evaluation of Energy Transfer Efficiency from Cr3+ to Nd3+ in μSPL
in Outdoor Operation
As shown in Fig. 10.34, Cr codoping is effective in increasing the output laser power at
0.4 at% in comparison with no codoping. However, in the cases of codoping of 0.7 at%
Cr and 1 at% Cr, the demerit of the increase in the round-trip loss δ offset the merit andthe output laser power decreased. There is a previous study that reports that theperformance of Cr, Nd:YAG ceramics was inferior to that of undoped Nd:YAG crystals
in an SPL where a 3 × 9 × 100 mm transparent Cr, Nd:YAG ceramic LM was tested in a
specially designed SPL employing end-pumping and side-pumping simultaneously using
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a liquid light-guide lens [91]. The reason suggested was as follows: (1) Cr codoping
increased the scattering coefficient of transparent Nd:YAG ceramic rods, (2) the increasein the scattering coefficient of the rods leads to an increase in the round-trip loss δ in thelaser cavity, (3) the increase in δ leads to a decrease in ηslope, (4) the decrease in ηslope
offsets the advantage of the decrease in the saturation gain caused by increased opticalabsorption by Cr3+ followed by energy transfer from Cr3+ to Nd3+. This agrees with theresults observed in Fig. 10.34. Since then, undoped transparent Nd:YAG LMs have beenused in SPLs [58, 59]. However, using ⌀4.5 mm × 35 mm Cr (0.1 at%)-doped Nd(1.0 at%):YAG ceramic rod, the same authors recently reported a slope efficiency
1.28 times their own previous record [58, 60]. Therefore, the advantage of usingtransparent Cr, Nd:YAG ceramic LMs instead of Nd:YAG single-crystal LMs in SPLs is
not conclusive.
It was also suggested from the simulated result in Fig. 10.38 that ηQE might be less than0.8, reflecting poor energy transfer efficiency from Cr3+ to Nd3+. Fig. 10.44 schematicallyshows the energy diagram of Cr3+ and Nd3+ in YAG crystal including energy transfer
process from Cr3+ to Nd3+. Cr3+ ions excited by sunlight relax to 4T2 state or 2E state.
Through the transition of Cr3+ ions from these excited states (4T2 or 2E) to the ground state(4A2), it is considered that energy is transferred from Cr3+ to Nd3+ ions to put Nd3+ ions
eventually into the metastable excited state 4F3/2. It is considered that this process increases
the pumping efficiency and decreases the threshold power for laser oscillation.
However, the direct excitation of Nd3+ ions by sunlight may increase the number of Nd3+
ions at the metastable excited state 4F3/2 so much that Cr3+ ions excited by sunlight cannot
find enough Nd3+ ions at the ground state to which energy is transferred. This may cause
poor energy transfer efficiency from Cr3+ to Nd3+. To simulate this situation
experimentally, laser oscillation experiments with simultaneous direct excitation of Nd 3+
by an 808 nm laser and the direct excitation of Cr3+ by 561 nm that does not excite Nd3+
directly were performed as shown in Fig. 10.45 (a). Fig. 10.45 (b) shows output laser
powers as a function of 808 nm pumping power in cases with and without 50 mW pumping
at 561 nm. With 50 mW pumping at 561 nm, the output laser power increased incomparison with the case without 50 mW pumping at 561 nm. The increment can be
attributed to the Cr to Nd energy transfer. Fig. 10.45(c) shows the decrease in the threshold
power with increase in the input 561 nm power. From these results, the energy transfer
efficiency was estimated to be 65.3 % [97].
A more realistic approach to access the Cr to Nd energy transfer efficiency was also tried
using natural sunlight pumping outdoors. For the samples, Cr-codoped Nd (1.0 at%):YAG
transparent ceramic LMs of 1 × 1 × 10 mm in size with Cr contents of 0.0 at%, 0.4 at%,
0.7 at%, and 1.0 at% shown in Fig. 10.33 were used again. Apart from the measurementof the stimulated emission, as shown in Fig. 10.25, the spontaneous emission spectra by
sunlight excitation were also measured in a similar geometry as shown in Fig. 10.46 (a).
Figs. 10.46 (b) and (c) show measured spontaneous emission spectra in the cases ofsunlight intensity of 1.12 W and 3.07 W, respectively. Using the decrease in the threshold
power caused by the energy transfer from Cr3+ ions to Nd3+ ions, the values of Cr3+ to Nd3+
energy transfer efficiency ηCr→Nd were estimated as shown in Fig. 10.47 [104]. In the458Chapter 10. Compact Solar-pumped Lasers

figure, the values estimated are not for ηCr→Nd but for (ηCr/ ηNd) ⋅ηCr→Nd. Here, ηCr is thequantum efficiency of the formation of the excited Cr3+ ions in the energy level 2E and 4T2
from the number of absorbed photons by Cr3+ ions, and ηNd is the quantum efficiency of
the formation of the excited Nd3+ ions in the metastable energy level of 4F3/2 from thenumber of absorbed photons by Nd3+ ions. In other words, ηCr stands for the efficiency ofrelaxation of the excited electrons in higher energy levels of Cr3+ to their 4T2 or 2E levels,and ηNd stands for the efficiency of relaxation of the excited electrons in higher energy
levels of Nd3+ to the metastable 4F3/2 levels. The ratio γ ( = ηCr/ηNd) is not known. For the
first-order approximation, γ can be assumed to be unity because both ηCr and ηNd represent
similar properties of the isolated metal ions doped in YAG ceramics. For the second-orderapproximation, γ can be assumed to be a value slightly less than unity, γ ≾1. Because theenergy levels of rare-earth ions such as Nd3+ are not significantly influenced by thesurrounding electronic structure of YAG matrix atoms, ηNd is considered to be higher thanηCr of a transition metal ion, Cr3+. Since γ, which is nearly equal to unity, does not majorly
influence the present discussion, the values listed in Fig. 10.47 are deemed to represent
ηCr→Nd. When IOAP was increased to 3.07 W, ηCr→Nd in spontaneous emissions decreased to
0.675γ (0.4 at%), 0.623γ (0.7 at%), and 0.625γ (1.0 at%). In stimulated emissions, ηCr→Ndfurther decreased to 0.448γ (0.4 at%), 0.213γ (0.7 at%), and 0.166γ (1.0 at%). The decreasein ηCr→Nd in the increased IOAP in spontaneous emissions, and further decrease in ηCr→Nd instimulated emissions may be attributed to the fact that the energy of the Cr3+ ions excitedby sunlight cannot transfer to Nd3+ ions because the number of Nd3+ ions directly excitedby sunlight and already occupying the metastable excited state, 4F3/2, increased withincreasing IOAP in spontaneous emissions, and more seriously so in the population
inversion in stimulated emissions. As for the dependence on the Cr content, the highest
values of ηCr→Nd were obtained at 0.4 at% in line with the Cr content dependence of output
powers of SPL shown in Fig. 10.34.

Fig. 10.44. Energy diagrams of Cr3+ and Nd3+ in YAG crystal under the sunlight excitation.

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Fig. 10.45. (a) Experimental set-up of the Nd/Cr: YAG pumped by 808 nm and 561 nm lasers.
(b) Laser output power property by two-wavelength pumping. (c) Laser oscillation thresholdvs. the 561 nm pumping power.

Fig. 10.46. (a) Geometry to monitor solar-pumped spontaneous emission spectra. The opticalfilter transmits 70 % for λ of 1064±2 nm; (b) and (c) Measured spontaneous emission spectrain 1060 nm<λ<1070 nm at two different sunlight intensities of 1.12 W and 3.07 W, respectively.

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Fig. 10.47. Results of estimated ηCr→Nd·γ displayed in bar charts to show (1) the dependence
on the sunlight power in spontaneous emissions, (2) the difference between spontaneous emissions
and stimulated emissions and (3) dependence on the Cr content. The Cr content dependenceof the output power of μSPL at the pumping power of sunlight = 3.07 W is also attached. γ isthe ratio ηCr/ηNd ≾ 1, ηCr is the quantum efficiency of the formation of the excited Cr 3+ ionsin the energy level 2E and 4T2 from the number of absorbed photons by Cr3+ ions, and ηNd is
the quantum efficiency of the formation of the excited Nd 3+ ions in the metastable energy levelof 4F3/2 from the number of absorbed photons by Nd 3+ ions.

Deeming γ to be unity, the values of ηCr→Nd obtained here in stimulated emissions are alsomuch less than the values of ηCr→Nd in spontaneous emission. This indicates that
Cr-codoped Nd:YAG is not an ideal LM. Improvement of ηCr→Nd is awaited for SPL. Thisis consistent with the motivation to develop novel LMs such as in Cr, Nd-doped garnet
crystal families for SPLs in place of Cr, Nd:YAG [105, 106]. However, the advantages of
Cr, Nd:YAG ceramics over novel LMs, such as grain sizes of a few micrometers resulting
in suppressed scattering on the grain boundary, controllability of Cr and Ndconcentrations, and thermal conductivity, have been still unchallenged. As briefly
explained earlier, the problem of degradation due to solarization also needs to be solvedin Cr-codoped Nd:YAG transparent ceramic LMs. To improve the spectral matchingefficiency, we believe that it is promising to employ a vertical cavity surface emitting
semiconductor laser (VECSEL) that absorbs all the solar photons with their energy larger
than the bandgap of a semiconductor as a future challenge [107-109]. A VECSEL isessentially a disk-type LMs shown as an ideal form of LM in the top left corner in Fig.
10.6, in contrast to the current Nd:YAG rod-like micro LMs in our μSPL.

10.10. Coordinated Solar Tracking of an Array of μSPLs to Harvest Larger
Amount of Solar Energy
In contrast to a conventional SPL employing a large Fresnel lens or a large converging
mirror with typical sizes or diameters in the range of 1-2 m and a water-cooled thick LM
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Advances in Optics: Reviews. Book Series, Vol. 5of a typical size of ⌀10×100 mm [63-65], a μSPL can be mounted on a commerciallyavailable solar-tracking system for a telescope for amateur astronomers, as shown in
Fig. 10.30, for all-day continuous lasing tracking the Sun only with natural air convection.
This is enabled by rapid thermal dissipation from a thin micro-LM of a typical size of⌀1 ×10 mm. However, a ⌀76.2 mm OAP can harvest only 4.1 W solar energy even in therelatively high direct solar radiation of 905 W/m2 we observed, using the apparatus shownin Fig. 10.14. The improved sunlight-to-laser energy conversion efficiency of 14 % shownin Fig. 10.43, and the thermodynamical limit deduced from a detailed valance theory
31 % [76] gives the output laser power of only 0.6 W and 1.3 W, respectively. For
meaningful amount of solar energy utilization, arrays of large numbers of μSPLs should
be used.
Mass production and cost-reduction of μSPLs is indeed feasible, as demonstrated by thecommercially available “Himawari” solar lighting system composed of compact
concentrators and a solar-tracking system [78]. Large-sized solar concentrators with asolar-tracking system and solar-tracking type solar panels cannot be operated on windydays because they suffer from wind forces. Taking this fact into account, a coordinatedsolar tracking system of an array of compact solar-pumped lasers was designed, as shownin Fig. 10.48 (a). In Fig. 10.48 (a), parallel sliding motion of an upper white paneltransmits motions of a single solar-tracking engine to 25 μSPLs to realize coordinatedsolar tracking. The upper white panel is always kept horizontal during its motion to keep
the wind pressure minimum. The distances between neighboring μSPLs are determined in
the array to avoid spatial interventions between them during the coordinated solar
tracking. In the outdoor performance test shown in Fig. 10.48(b), thermal expansion of
the upper panel by solar irradiation could disturb precise coordinated tracking.
Application of super Invar alloy, whose thermal expansion coefficient between 20 and
90 ℃ is lower than 0.1×10-6 K-1 (which is as low as 1/100 of that of the popular stainlesssteel), could overcome this problem. However, application of more cost-effective
materials or structures to realize low thermal expansion remains a future challenge [110].

Fig. 10.48. (a) Schematic rendering of a coordinated solar tracking system of 5×5 compact SPLs
and (b) a photograph of outdoor solar tracking operation test.

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10.11. SPL-PV Combined System and Its Application to Optical WirelessPower Transmission
From the spectral quantum efficiency measured on a Si solar cell of energy conversion
efficiency η = 24.0 % [111], the spectral energy conversion efficiency η(λ) can becalculated as shown in Fig. 10.49, where λ stands for wavelength [8]. η(λ) takes amaximum value of over 40 % at λ just below the optical absorption band edge of
Si = 1117 nm. The solar spectrum, which is also illustrated with its intensity in arbitrary
unit for comparison, has a maximum at around 500 nm. Obviously, the solar spectrum
does not make a good fit with η(λ). On average in the range of the solar spectrum below
1117 nm, η for crystalline Si solar cells decrease to a level as low as 25-30 % as those in
the market.

Fig. 10.49. Spectral conversion efficiency of a crystalline Si solar cell under monochromatic
illumination of around 50 mW/cm2. The solar spectrum is also illustrated. The lasing wavelengthof the Nd:glass laser is indicated by an arrow.

The residual 70-75 % of the solar energy contributes only to the temperature rise of thecell. In contrast, the lasing wavelength of a Nd:glass laser is located just below 1117 nm
and closely fits the peak of η(λ). This means that the photon energy can be converted toelectricity with minimal heat loss. Therefore, it will be reasonable to convert the output
of μSPL into electricity via specially designed Si photovoltaic (PV) cells [10-17].
Optical wireless power transmission [109] to mobile objects may find applications in near
future such as power feeding to drones and electric vehicles [4-9]. The extraterrestrial
AM0 solar radiation on the Earth’s orbit is 1.36 W/m2, which is 36 % larger than theterrestrial AM1.5 solar radiation. Owing to a lack of interrupters such as clouds, theconcept of space solar power stations, on which electricity is generated using solar cells
and transmitted to the earth via microwaves or laser beams, has been proposed [1-5]. Morerecently, electricity generated by solar cells at a lunar lander has been planned fortransmission via laser beam to a lunar rover exploring deep into permanently shadowed
craters for frozen water [6, 7]. Because all the 1.36 W/m2 solar radiation in space is direct
solar radiation, SPLs can be also applied to generate laser beam for power feeding. The
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combined loss of solar energy via energy conversion from sunlight to electricity by solar
cells and that from photovoltaic electricity to laser beam by of a laser diode can be higher
than the single loss due to conversion from sunlight to laser beam by an SPL [8].
In the terrestrial application of SPLs to laser-beam power feeding, the high concentration
ratios of OAPs such as 11550 in Eq. (10.30) and 25990 in Eq. (10.34) exclude the chance
to utilize diffuse solar radiation. The reason why the high concentration is requested is torealize high radiation flux density that surpasses Ieth in Eq. (10.8). According to Eq. (10.8),
Ieth can be lowered if the aspect ratio A ( = L/r) of an LM is large, and the factor α, theeffective utilization rate of the length L of the LM is large. Disk-shaped LMs such as
VECSELs discussed in Section 10.9 and shown as an ideal form of an LM in the top left
corner of Fig. 10.6, show good promise in the future. As is also shown in the top rightcorner of Fig. 10.6, a fiber-type LM is another approach to attain large A and a large α byside-pumping in Eq. (10.8). As a promising approach, transverse excitation has been
introduced into a solar pumped fiber laser system to decrease the threshold excitation
power for oscillation using a low sunlight concentration at the ratio of 1/15 [112-115].

10.12. Conclusion
We have developed compact solar-pumped lasers (SPFL and μSPL) employing an
off-axis parabolic mirror with an aperture of 50.8 mm or 76.2 mm diameter and a
Nd-doped ZBLAN fiber or a Cr-doped Nd (0.1 at%):YAG transparent ceramic rod of⌀1 × 10 mm as the LM. Here, the stimulated emission takes place via the transition of
Nd3+ in YAG from its excited state to lower energy states. Cr was doped to absorb sunlightin broad wavelength ranges and to transfer the absorbed solar energy to Nd3+ because Nd3+
absorbs sunlight only in the limited narrow wavelength ranges. The laser oscillation
wavelength of 1.05-1.06 μm, just below the optical absorption edge of Si solar cells, issuitable for photoelectric conversion with minimal thermal loss after optical wireless(laser) power transmission to distant places. The small LMs and solar concentrators realize
more stable oscillation by rapid natural air convection cooling and increased mechanicalstability during wind exposure in contrast to the conventional large SPLs typically
employing a 2 m size solar concentrator. Outdoor operation tracking the Sun yielded
continuous oscillation exceeding 6.5 h, considerably improving upon the previously
reported 11 min. This shows the applicability of SPLs to whole-day operation andterrestrial and extraterrestrial solar energy utilization.
The measured incident sunlight-to-laser output total energy conversion efficiencies are
0.700.9 %. (slope efficiencies against the input solar power to LMs are 1.15-3.3 %.).
However, several promising data have been obtained for future improvements. The SPL
output increased more than eightfold between an LM with Cr content of 0.0 and that of
0.4 at%. The possibility of 14 % total energy conversion efficiency or more was shown inpreliminary experimental and simulated results by the improvement of mode-matchingefficiency introducing a refractive index matched core-cladding 3D composite structure
into an LM. In contrast, there was evidence of degradation of Cr-codoped Nd:YAG
transparent ceramic LMs, possibly by the accumulated sunlight irradiations, and relatively
low Cr3+ to Nd3+ energy transfer efficiency under broadband sunlight pumping. To
464

Chapter 10. Compact Solar-pumped Lasers

overcome these problems, future studies need to explore novel LMs or to investigate
alternative choices of LMs including side-pumped fiber LMs and VECSELs of highspectral and mode-matching efficiency.

Acknowledgments
This chapter contains research results supported by Advanced Low Carbon Technology
Research and Development Program (ALCA), Japan Science and Technology Agency,
and Toyota central R&D Laboratories Inc. The authors acknowledge Dr. A. Ikesue for his
essential contribution in fabrication of laser mediums. The authors also acknowledge Dr.
K. Higuchi, Dr. T. Ito, Dr. T. Kajino, Dr. A. Ichiki, Dr. H. N. Luitel, Dr. D. Inoue, Mr. T.
Ichikawa, Mr. Y. Suzuki, Mr. H. Terazawa, Mr. T. Kato, Ms. N. Hara, Mr. K. Watanabe,
Mr. S. Takimoto, Mr. D. Kano, Ms. L. T. A. Phuc, Dr. H. Iizuka, Dr. T. Tani, and Dr. M.
Umehara for their helpful contributions in this work. The authors also acknowledge Prof.
M. Yamaga (Gifu University), and Prof. Y. Ohishi and Assoc. Prof. T. Suzuki (Toyota
Technological Institute) for their suggestions and valuable discussions regarding thiswork. The authors would like to thank Editage (www.editage.com) for English language
editing. This work was started when all the authors were working for Toyota Central R&D
Labs., Inc., Nagakute, Aichi, Japan and was continued after H. Ito and T. Motohirotransferred to Nagoya University, Nagoya, Japan as a collaborative project. K. Hasegawa
is now continuing a related research in The Graduate School for the Creation of New
Photonics Industries (GPI), Hamamatsu, Shizuoka, Japan.

References
[1].

[2].
[3].
[4].
[5].

[6].

[7].

[8].

M. Mori, H. Nagayama, Y. Saito, H. Matsumoto, Summary of studies on space solar powersystems of the National Space Development Agency of Japan, Acta Astronaut., Vol. 54,
2004, pp. 337-345.
I. Cash, CASSIOPeiA – A new paradigm for space solar power, Acta Astronaut., Vol. 159,
2019, pp. 170-178.
X. Li, G. Fan, Y. Zhang, X. Ji, M. Li, A Fresnel concentrator with fiber-optic bundle-based
space solar power satellite design, Acta Astronaut., Vol. 153, 2018, pp. 122-129.
R. J. De Young, Beam profile measurement of a solar-pumped iodine laser, Appl. Opt.,
Vol. 25, Issue 21, 1986, pp. 3850-3854.
S. H. Choi, J. H. Lee, W. E. Meador, E. J. Conway, A 50-kW module power station of
directly solar-pumped iodine laser, J. Sol. Energy Eng. – Transactions of the ASME,
Vol. 119, Issue 4, pp. 304-311.
European Space Agency (ESA) Laser-powered Rover to Explore Moon’s Dark Shadows,
http://www.esa.int/Enabling_Support/Space_Engineering_Technology/Laser-poweredrover to explore Moons’ dark shadows
W. M. Farrell, T. J. Stubbs, J. S. Halckas, R. M. Killen, G. T. Delory, M. R. Collier,
R. R. Vondrak, Anticipated electrical environment within permanently shadowed lunar
craters, J. Geophys. Res., Vol. 115, 2010, E03004.
T. Motohiro, Y. Takeda, H. Ito, K. Hasegawa, A. Ikesue, T. Ichikawa, K. Higuchi, A. Ichiki,
S. Mizuno, T. Ito, N. Yamada, H. N. Luitel, T. Kajino, H. Terazawa, S. Takimoto,
K. Watanabe, Concept of the solar-pumped laser-photovoltaics combined system and its

465

Advances in Optics: Reviews. Book Series, Vol. 5

[9].

[10].
[11].

[12].

[13].

[14].

[15].

[16].
[17].

[18].

[19].

[20].

[21].
[22].
[23].
[24].
[25].
[26].

466

application to laser beam power feeding to electric vehicles., Jpn. J. Appl. Phys., Vol. 56,
2017, 08MA07.
A. Mohammadnia, B. M. Ziapour, H. Ghaebi, M. H. Khooban, Feasibility assessment of
next-generation drones powering by laser-based wireless power transfer, Opt. Laser
Technol., Vol. 143, 2021, 107283.
Y. Takeda, T. Ito, Light trapping for silicon photovoltaic cells illuminated by solar-pumped
lasers, Jpn. J. of Appl. Phys., Vol. 54, 2015, 08KD13.
Y. Takeda, H. Iizuka, S. Mizuno, K. Hasegawa, T. Ichikawa, H. Ito, T. Kajino, T. Motohiro,
Silicon photovoltaic cells coupled with solar-pumped fiber lasers emitting at 1064nm,
J. Appl. Phys., Vol. 116, 2014, 014501.
Y. Takeda, H. Iizuka, S. Mizuno, K. Hasegawa, T. Ichikawa, H. Ito, T. Kajino, T. Motohiro,
Light trapping for emission from a photovoltaic cell under normally incident
monochromatic illumination, J. Appl. Phys., Vol. 116, 2014, 124506.
Y. Takeda, T. Ito, N. Yamada, K. Hasegawa, S. Mizuno, T. Ichikawa, H. N. Luitel,
H. Iizuka, K. Higuchi, H. Ito, A. Ichiki, T. Motohiro, Crystalline silicon photovoltaic cells
used for power transmission from solar-pumped lasers: I. Light trapping concepts, Jpn. J.
Appl. Phys., Vol. 57, 2018, 08RF05.
Y. Takeda, N. Yamada, T. Ito, H. Ito, T. Motohiro, Crystalline silicon photovoltaic cells
used for power transmission from solar-pumped lasers: II. Practical implementation, Jpn. J.
Appl. Phys., Vol. 57, 2018, 08RF06.
N. Yamada, T. Ito, H. Ito, T. Motohiro, Y. Takeda, Crystalline silicon photovoltaic cells
used for power transmission from solar-pumped lasers: III. Prototype for proof of the lighttrapping concept, Jpn. J. Appl. Phys., Vol. 57, 2018, 08RF07.
Y. Takeda, Light trapping for photovoltaic cells used for optical power transmission, Appl.
Phys. Express, Vol. 13, 2020, 054001.
Y. Takeda, H. Iizuka, N. Yamada, T. Ito, Light trapping for photovoltaic cells using
polarization-insensitive angle-selective filters under monochromatic illumination, Appl.
Opt., Vol. 56, Issue 20, 2017, pp. 5761-5767.
R. Ishikawa, T. Kato, R. Anzo, M. Nagatake, T. Nishimura, N. Tsuboi, S. Miyajima,
Widegap CH3NH3PbBr3 solar cells for optical wireless power transmission application,
Appl. Phys. Lett., Vol. 117, 2020, 013902.
T. Ohkubo, T. Yabe, K. Yoshida, S. Uchida, T. Funatsu, B. Bagheri, T. Oishi, K. Daito,
M. Ishioka, Y. Nakayama, N. Yasunaga, K. Kido, Y. Sato, C. Baasandash, K. Kato,
T. Yanagitani, Y. Okamoto, Solar-pumped 80 W laser irradiated by a Fresnel lens, Opt.
Lett., Vol. 34, Issue 2, 2009, pp. 175-177.
T. Yabe, B. Bagheri, T. Ohkubo, S. Uchida, K. Yoshida, T. Funatsu, T. Oishi, K. Daito,
M. Ishioka, N. Yasunaga, Y. Sato, C. Baasandash, Y. Okamoto, K. Yanagitani, 100 W-classsolar pumped laser for sustainable magnesium-hydrogen energy cycle, Jpn. J. Appl. Phys.,Vol. 104, 2008, 083104.
Z. J. Kiss, H. R. Lewis, R. C. Duncan, Sun pumped continuous optical maser, Appl. Phys.
Lett., Vol. 2, Issue 5, 1963, pp. 93-94.
G. R. Simpson, Continuous sun-pumped room temperature glass laser operation, Appl. Opt.,
Vol. 3, Issue 6, 1964, pp.783-784.
C. G. Young, A sun-pumped CW one-watt laser, Appl. Opt., Vol. 5, 1966, pp. 993-997.
T. H. Maiman, Stimulated optical radiation in ruby, Nature, Vol. 187, Issue 4736, 1960,
pp. 493-494.
J. H. Lee, W. R. Weaver, A solar simulator-pumped atomic iodine laser, Appl. Phys. Lett.,Vol. 39, 1981, pp. 137-139.
C. K. Terry, J. E. Peterson, D. Y. Goswami, Terrestrial solar-pumped iodine gas laser withminimum threshold concentration requirements, J. Thermophys. Heat Tran., Vol. 10, 1996,
pp. 54-59.

Chapter 10. Compact Solar-pumped Lasers[27]. I. M. Belousova, O. B. Danilov, A. A. Mak, V. P. Belousov, V. Y. Zalesslii, V. A. Grigor’ev,
A. V. Kris’ko, E. N. Sosnov, Feasibility of the solar-pumped fullerene-oxygen-iodine laser(Sun-Light FOIL), Optic Spectrosc., Vol. 90, 2001, pp. 773-777.
[28]. I. Pe’er, I. Vishnevitsky, N. Naftali, A. Yogev, Broadband laser amplifier based on
gas-phase dimer molecules pumped by the Sun, Opt. Lett., Vol. 26, 2001 pp. 1332-1334.
[29]. T. Härig, G. Huber, I. A. Schcherbakov, Cr3+ sensitized Nd3+:Li-La-phosphate glass laser,
J. Appl. Phys., Vol. 52, 1981, pp. 4450-4452.
[30]. T. Suzuki, H. Kawai, H. Nasu, M. Hughes, S. Mizuno, K. Hasegawa, H. Ito, Y. Ohishi,
Excitation wavelength dependence of quantum efficiencies of Nd-doped glasses for solar
pumped fiber lasers, Proceedings of SPIE, Vol. 7598, 2010, 75981L.
[31]. T. Suzuki, H. Kawai, H. Nasu, M. Hughes, S. Mizuno, K. Hasegawa, H. Ito, Y. Ohishi,
Quantum efficiency measurements on Nd-doped glasses for solar pumped lasers,
J. Non-Cryst. Solids, Vol. 356, 2010, pp. 2344-2349.
[32]. T. Suzuki, H. Nasu, S. Mizuno, H. Ito, K. Hasegawa, Y. Ohishi, Spectroscopic investigationof Nd3+-doped ZBLAN glass for solar-pumped lasers, J. Opt. Soc. Am. B, Vol. 28, 2011,
pp. 2001-2006.
[33]. T. Suzuki, H. Kawai, H. Nasu, M. Huges, Y. Ohishi, S. Mizuno, H. Ito, K. Hasegawa,
Quantum efficiency of Nd3+-doped glasses under sunlight excitation, Opt. Mater., Vol. 33,
2011, pp. 1952-1957.
[34]. S. Mizuno, H. Ito, K. Hasegawa, T. Suzuki, Y. Ohishi, Laser emission from a solar-pumped
fiber, Opt. Express, Vol. 20, Issue 6, 2012, pp. 5891-5895.
[35]. Y. Iwata, T. Suzuki, Y. Ohishi, Optical properties of Er 3+-doped glasses for solar-pumped
lasers, Phys. Status Solidi C, Vol. 9, Issue. 12, 2012, pp. 2344-2347.
[36]. R. Reisfeld, Spectroscopic properties of Cr (III) in transparent glass-ceramics and energy
transfer from Cr (III) to Nd (III), in B. Jezowska-Trzebiatowska, J. Legendziewicz,
W. Stręk (Eds.), in Proceedings of the 1st International School on Excited Stated ofTransition Elements, World Scientific Publ. Co. Pte Ltd. Singapore, 1989, pp. 440-465.
[37]. H.-D. Hattendorff, G. Huber, H. G. Danielmeyer, Efficient cross pumping of Nd3+ by Cr3+
in Nd (Al, Cr)3(BO3)4 lasers, J. Phys. C Solid State Phys., Vol. 11, 1978, pp. 2399-2404.
[38]. H.-D. Hattendorff, G. Huber, F. Lutz, CW laser action in Nd (Al, Cr) 3 (BO3)4 laser, Appl.
Phys. Lett., Vol. 34, 1979, pp. 437-439.
[39]. D. Pruss, G. Huber, A. Beimowski, V. V. Laptev, I. A. Scherbakov, Y. V. Zharikov,
Efficient Cr3+ sensitized Nd3+: GdScGa-Garnet laser at 1.06 μm, Appl. Phys. B, Vol. 28,
1982, pp. 355-358.
[40]. G. A. Thompson, V. Krupkin, A. Yogev, M. Oron, Solar-pumped Nd: Cr: GSGG parallel
array laser, Opt. Eng., Vol. 31, 1992, pp. 2644-2646.
[41]. M. Bass, M. J. Weber, Nd, Cr:YAlO3 laser tailored for high-efficiency Q-switched
operation, Appl. Phys. Lett., Vol. 17, 1970, pp. 395-398.
[42]. H. Arashi, Y. Oka, N. Sasahara, A. Kaimai, M. Ishigame, A Solar-pumped CW 18 W
Nd:YAG Laser, Jpn. J. Appl. Phys., Vol. 23, 1984, pp. 1051-1053.
[43]. M. Weksler, J. Shwartz, Solar-pumped solid-state lasers, IEEE J. Quant. Electron., Vol. 24,
1988, pp. 1222-1228.
[44]. D. Cooke, Sun-pumped lasers: revisiting an old problem with nonimaging optics, Appl.
Opt., Vol. 31, Issue 36, 1992, pp. 7541-7546.
[45]. A. Ikesue, Y. L. Aung, Ceramic laser materials, Nat. Photonics, Vol. 2, 2008, 721.
[46]. A. Ikesue, Y. L. Aung, V. Lupei, Ceramic Lasers, Cambridge University Press, 2013.
[47]. T. Saiki, S. Motokoshi, K. Imasaki, K. Fujioka, H. Fujita, M. Nakatsuka, Y. Izawa,
C. Yamanaka, Effective fluorescence lifetime and stimulated emission cross section of
Nd/Cr:YAG ceramics under CW lamplight pumping, Jpn. J. Appl. Phys., Vol. 47, Issue 10,2008, pp. 7896-7902.

467

Advances in Optics: Reviews. Book Series, Vol. 5[48]. R. M. J. Benmair, J. Kagan, Y. Kalisky, Y. Noter, M. Oron, Y. Shimony, A. Yogev,
Solar-pumped Er, Tm, Ho:YAG laser, Opt. Lett., Vol. 15, 1990, pp. 36-38.
[49]. G. A. Thompson, A. Yogev, A. Reich, M. Oron, Effects of coolant temperature and pump
power on the power output of solar-pumped solid-state lasers, Opt. Eng., Vol. 31, 1992,
pp. 2488-2491.
[50]. I. L. Pe’er, N. Naftali, D. Abramovic, A. Yogev, M. Lando, Y. Shimony, Passive
Q-switching of high-power solar-pumped Nd:YAG laser, Proceeding of SPIE, Vol. 3092,
1997, pp. 273-276.
[51]. I. Pe’er, N. Naftali, A. Yogev, High power, solar pumped, Nd:YAG laser amplifier for freespace laser communication, Proceeding of SPIE, Vol. 3139, 1997, pp. 194-204.
[52]. N. Naftali, I. Pe’er, A. Yogev, Power transmission using solar pumped laser, Proceeding of
SPIE, Vol. 3139, 1997, pp. 259-265.
[53]. M. Lando, Y. Shimony, R. M. J. Benmair, D. Abramovich, V. Krupkin, A. Yogev, Visible
solar-pumped lasers, Opt. Mater., Vol. 13, 1999, pp. 111-115.
[54]. M. Lando, Y. Shimony, Y. Noter, R. M. J. Benmair, A. Yogev, Passive Q switching of asolar-pumped Nd:YAG laser, Appl. Opt., Vol. 39, 2000, pp. 1962-1965.
[55]. M. Lando, J. Kagan, B. Linyokin, V. Dobrusin, A solar-pumped Nd:YAG laser in the highcollection efficiency regime, Optic Commun., Vol. 222, 2003, pp. 371-381.
[56]. S. Payziyev, K. Makhmudov, Solar pumped Nd:YAG laser efficiency enhancement using
Cr: LiCAF frequency down-shifter, Optic Commun., Vol. 380, 2016, pp. 57-60.
[57]. J. Almeida, D. Liang, C. R. Vistas, R. Bouadjemine, E. Guillot, 5.5 W continuous wave
TEM00-mode Nd:YAG solar laser by a light-guide/2 V shaped pump cavity, Appl. Phys. B
Laser Optic., Vol. 121, 2015, pp. 473-482.
[58]. D. Liang, J. Almeida, C. R. Vistas, E. Guillot, Solar-pumped Nd:YAG laser with 31.5 W/m2
multimode and 7.9 W/m2 TEM00-mode collection efficiencies, Sol. Energy Mater. Sol.
Cells, Vol. 159, 2017, pp. 435-439.
[59]. D. Liang, C. R. Vistas, J. Almeida, B. D. Tibúrcio, D. Garcia, Side-pumped continuouswave Nd:YAG solar laser with 5.4 % slope efficiency, Sol. Energy Mater. Sol. Cells,
Vol. 192, 2019, pp. 147-153.
[60]. D. Liang, C. R. Vistas, B. D. Tibúrcio, J. Almeida, Solar-pumped Cr:Nd:YAG ceramic laser
with 6.7 % slope efficiency, Sol. Energy Mater. Sol. Cells, Vol. 185, 2018, pp. 75-79.
[61]. P. Xu, S. Yang, C. Zhao, Z. Guan, H. Wang, Y. Zhang, H. Zhang, T. He, High efficiency
solar-pumped laser with a grooved Nd:YAG rod, Appl. Opt., Vol. 53, Issue 18, 2014,
pp. 3941-3944.
[62]. T. H. Dinh, T. Ohkubo, T. Yabe, Development of solar concentrators for high-power
solar-pumped lasers, Appl. Opt., Vol. 53, 2014, pp. 2711-2719.
[63]. D. Liang, J. Almeida, Solar-pumped TEM00 mode Nd:YAG laser, Opt. Express, Vol. 21,
Issue 21, 2013, pp. 25107-25112.
[64]. T. Yabe, T. Ohkubo, S. Uchida, K. Yoshida, M. Nakatsuka, T. Funatsu, A. Mabuti,
A. Oyama, K. Nakagawa, T. Oishi, K. Daito, B. Behgol, Y. Nakayama, M. Yoshida,
S. Motokoshi, Y. Sato, C. Baasandash, High-efficiency and economical solar-energypumped laser with Fresnel lens and chromium codoped laser medium, Appl. Phys. Lett.,Vol. 90, Issue 26, 2007, 261120.
[65]. D. Liang, J. Almeida, C. R. Vistas, 25 W/m2 collection efficiency solar-pumped Nd:YAG
laser by a heliostat-parabolic mirror system, Appl. Opt., Vol. 55, Issue 27, 2016,
pp. 7712-7717.
[66]. H. N. Luitel, S. Mizuno, T. Tani, Y. Takeda, Broadband-sensitized upconversion of
ATiO3:Er,Ni (A = Mg, Ca, Sr, Ba), J. Ceram. Soc. Jpn, Vol. 125, Issue 11, 2017,
pp. 821-828.

468

Chapter 10. Compact Solar-pumped Lasers[67]. Y. Takeda, H. N. Luitel, S. Mizuno, Competing effects of sensitization and energy
dissipation by Ni2+ incorporation in La (Ga0.5Sc0.5) O3: Er, Ni, Nb upconverters, J. Lumin.,Vol. 194, 2018, pp. 778-784.
[68]. H. N. Luitel, S. Mizuno, T. Nonaka, T. Tani, Y. Takeda, Effect of Ti compositions for
efficiency enhancement of CaTiO3:Er3+, Ni2+ broadband-sensitive upconverter, RSC Adv.,
Vol. 7, Issue 66, 2017, pp. 41311-41320.
[69]. H. N. Luitel, S. Mizuno, Y. Takeda, CaTiO3:Er3+, Ni2+ broadband-sensitive upconverter:
An effective way to harvest unused NIR solar irradiation for crystalline silicon solar cells,
Phys. Status Solidi, Vol. 214, Issue 8, 2017, 1600899.
[70]. Y. Takeda, S. Mizuno, H. N. Luitel, T. Tani, Broadband-sensitive upconverters co-dopedwith Er3+ and Ni2+ for crystalline silicon solar cells, Proceeding of SPIE, Vol. 9937, 2016,
99370E.
[71]. H. N. Luitel, S. Mizuno, Y. Takeda, Effect of A-sites cations on the broadband sensitive
upconversion of AZrO3: Er, Ni (A = Mg, Ca, Sr, Ba) phosphors, Opt. Mater., Vol. 64, 2017,
pp. 314-322.
[72]. Y. Takeda, S. Mizuno, H. N. Luitel, K. Yamanaka, Broadband-sensitive cooperative
upconversion emission of La (Ga0.5Sc0.5) O3: Er, Ni, Nb, Appl. Phys. Express, Vol. 9,
Issue 11, 2016, 112402.
[73]. Y. Takeda, S. Mizuno, H. N. Luitel, K. Yamanaka, T. Tani, Energy transfer between Ni 2+
sensitizers and Er3+ emitters in broadband-sensitive upconverters La (Ga, Sc, In) O3: Er, Ni,
Nb, J. Appl. Phys. Vol. 120, Issue 7, 2016, 073102.
[74]. H. N. Luitel, S. Mizuno, T. Tani, Y. Takeda, Broadband sensitive Ni2+-Er3+ based
upconverters for crystalline silicon solar cells, RSC Adv., Vol. 6, Issue 60, 2016,
pp. 55499-55506.
[75]. Y. Takeda, S. Mizuno, H. N. Luitel, T. Tani, A broadband-sensitive upconverter
La(Ga0.5Sc0.5) O3: Er, Ni, Nb for crystalline silicon solar cells, Appl. Phys. Lett., Vol. 108,
2016, 043901.
[76]. S. Nechayev, C. Rotschild, Detailed balance limit of efficiency of broadband-pumped
lasers, Sci. Rep., Vol. 7, 2017, 11497.
[77]. F. A. Jenkins, H. E. White, Fundamentals of Optics, 3rd Ed., McGraw-Hill Kogakusha Ltd.,
1957.
[78]. Himawari Solar Lighting System La Foret Engineering Co. Ltd., https://www.himawarinet.co.jp/e_page-index01.html[79]. R. E. Bird, C. Riordan, Simple solar spectral model for direct and diffuse irradiance on
horizontal and tilted planes at the earth's surface for cloudless atmospheres, J. Appl.
Meteorol. Climatol., Vol. 25, Issue 1, 1986, pp. 87-97.
[80]. T. Suzuki, H. Nasu, M. Hughes, S. Mizuno, K. Hasegawa, H. Ito, Y. Ohishi, Quantum
efficiency measurements on Nd-doped glasses for solar pumped lasers, J. Non-Cryst. Solids,
Vol. 356, Issues 44-49, 2010, pp. 2344-2349.
[81]. S. E. Stokowski, R. A. Saroyan, M. J. Weber, Nd-doped Laser Glass Spectroscopic andPhysical Properties, Tech. Rep. 1, Lawrence Livermore National Laboratory, 1981.
[82]. S. E. Stokowski, R. A. Saroyan, M. J. Weber, Nd-doped Laser Glass Spectroscopic andPhysical Properties, Tech. Rep. 2, Lawrence Livermore National Laboratory, 1981.
[83]. H. Toratani, T. Izumitani, H. Kuroda, Compositional dependence of nonradiative decay rate
in Nd laser glasses, J. Non-Cryst. Solids, Vol. 52, Issues 1-3, 1982, pp. 303-313.
[84]. I. Bardez, D. Caurant, J. L. Dussossoy, P. Loiseau, C. Gervais, F. Ribot, D. R. Neuville,
N. Baffier, C. Fillett, Development and characterization of rare earth-rich glassy matrices
envisaged for the immobilization of concentrated nuclear waste solutions, Nucl. Sci. Eng.,
Vol. 153, Issue 3, 2006, pp. 272-284.
[85]. M. Poulain, M. Poulain, J. Lucas, Verres fluores au tetrafluorure de zirconium proprietes
optiques d'un verre dope au Nd3+, Mat. Res. Bull., Vol. 10, Issue 4, 1975, pp. 243-246.

469

Advances in Optics: Reviews. Book Series, Vol. 5[86]. T. Kanamori, S. Sakaguchi, Preparation of elevated NA fluoride optical fibers, Jpn. J. Appl.
Phys., Vol. 25, Issue 6A, 1986, pp. L468-L470.
[87]. F. Gan, Optical properties of fluoride glasses: A review, J. Non-Cryst. Solids., Vol. 184,
1995, pp. 9-20.
[88]. K. Arai, H. Namikawa, K. Kumata, T. Honda, Y. Ishii, T. Hanada, Aluminum or phosphorus
co-doping effects on the fluorescence and structural properties of neodymium-doped silica
glass, J. Appl. Phys., Vol. 59, Issue 10, 1986, pp. 3430-3436.
[89]. J. Heber, Solar fibre lasers, Nature Materials, Vol. 11, 2012, 266.
[90]. Z. J. Kiss, R. C. Duncan, Cross-pumped Cr3+-Nd3+: YAG laser system, Appl. Phys. Lett.,Vol. 5, Issue 10, 1964, pp. 200-201.
[91]. T. H. Dinh, T. Ohkubo, T. Yabe, H. Kuboyama, 120-watt continuous wave solar-pumped
laser with a liquid light-guide lens and an Nd: YAG rod, Opt. Lett., Vol. 37, Issue 3, 2012,
pp. 2670-2672.
[92]. T. Saiki, N. Fujiwara, N. Matsuoka, M. Nakatuka, K. Fujioka, Y. Iiida, Amplification
properties of KW Nd/Cr:YAG ceramic multi-stage active-mirror laser using white-lightpump source at high temperatures, Opt. Commun., Vol. 387, 2017, pp. 316-321.
[93]. M. Endo, Feasibility study of a conical-toroidal mirror resonator for solar-pumped thin-disklasers, Opt. Express, Vol. 15, Issue 9, 2007, pp. 5482-5493.
[94]. T. Ogawa, S. Wada, M. Higuchi, Development of Nd, Cr co-doped laser materials for solarpumped lasers, Jpn. J. Appl. Phys., Vol. 53, 2014, 08MG03.
[95]. Y. Honda, S. Motokoshi, T. Jitsuno, N. Miyanaga, K. Fujioka, M. Nakatsuka, M. Yoshida,
Temperature dependence of optical properties in Nd/Cr: YAG materials, J. Lumin.,Vol. 148, 2014, pp. 342-346.
[96]. M. Yamaga, Y. Oda, H. Uno, K. Hasegawa, H. Ito, S. Mizuno, Formation probability of
Cr-Nd pair and energy transfer from Cr to Nd in Y3Al5O12 ceramics codoped with Nd and
Cr, J. Appl. Phys., Vol. 112, 2012, 063508.
[97]. K. Hasegawa, T. Ichikawa, S. Mizuno, Y. Takeda, H. Ito, A. Ikesue, T. Motohiro,
M. Yamaga, Energy transfer efficiency from Cr3+ to Nd3+ in solar-pumped laser usingtransparent Nd/Cr: Y3Al5O12 ceramics, Opt. Express, Vol. 23, Issue 11, 2015,
pp. A519-A524.
[98]. W. Koechner, Solid-State Laser Engineering, 5th Ed., Springer-Verlag, Berlin, 2003.
[99]. S. Mehellou, D. Liang, J. Almeida, R. Bouadjemine, C. R. Vistas, E. Guillot, F. Rehouma,
Stable solar-pumped TEM00-mode 1064 nm laser emission by a monolithic fused silica
twisted light guide, Sol. Energy, Vol. 155, 2017, pp. 1059-1071.
[100]. Y. Suzuki, H. Ito, T. Kato, L.T.A. Phuc, K. Watanabe, H. Terazawa, K. Hasegawa,T. Ichikawa, S. Mizuno, A. Ichiki, S. Takimoto, A. Ikesue, Y. Takeda, T. Motohiro,
Continuous oscillation of a compact solar-pumped Cr, Nd-doped YAG ceramic rod laser
for more than 6.5 h tracking the sun, Sol. Energy, Vol. 177, 2019, pp. 440-447.
[101]. T. Kato, H. Ito, K. Hasegawa, T. Ichikawa, A. Ikesue, S. Mizuno, Y. Takeda, A. Ichiki,
T. Motohiro, Effect of Cr content on the output of a solar-pumped laser employing a
Cr-doped Nd: YAG ceramic laser medium operating in sunlight, Jpn. J. Appl. Phys.,Vol. 58, 2019, 062007.
[102]. D. Findlay, R. Clay, The measurement of internal losses in 4-level lasers, Phys. Lett.,Vol. 20, Issue 3, 1966, pp. 277-278.
[103]. K. Hasegawa, T. Ichikawa, Y. Takeda, A. Ikesue, H. Ito, T. Motohiro, Lasing characteristics
of refractive-index-matched composite Y3Al5O12 rods employing transparent ceramics forsolar-pumped lasers, Jpn. J. Appl. Phys., Vol. 57, 2018, 042701.
[104]. T. Kato, H. Ito, K. Hasegawa, T. Ichikawa, A. Ikesue, S. Mizuno, Y. Takeda, A. Ichiki,
T. Motohiro, Energy transfer efficiency from Cr3+ to Nd3+ in Cr, Nd YAG ceramics laser
media in a solar-pumped laser in operation outdoors, Opt. Mater., Vol. 110, 2020, 110481.

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Chapter 10. Compact Solar-pumped Lasers[105]. K. Takagi, M. Yamaga, E. G. Villora, K. Shimamura, K. Hasegawa, H. Ito, S. Mizuno,
Y. Takeda, T. P. J. Han, Energy transfer from Cr to Nd in substitutional crystal
Y3GaxAl5-xO12 codoped with Nd and Cr, J. Lumin., Vol. 169, 2016, pp. 65-71.
[106]. M. Yamaga, T. P. J. Han, O. Sawada, E.G. Villorac, K. Shimamura, K. Hasegawa,
S. Mizuno, Y. Takeda, Laser oscillation of Nd via energy transfer process from Cr to Ndin substitutional disordered garnet crystals codoped with Nd and Cr, J. Lumin., Vol. 202,2018, pp. 393-402.
[107]. A. H. Quarterman, K. G. Wilcox, Design of a solar-pumped semiconductor laser, Optica,
Vol. 2, Issue 1, 2015, pp. 56-61.
[108]. S. Unnikrishnan, N. G. Anderson, Quantum-well lasers for direct solar photo pumping,
J. Appl. Phys., Vol. 74, Issue 6, 1993, pp. 4221-4228.
[109]. T. Miyamoto, Optical wireless power transmission using VECSELs, Proceedings of SPIE,
Vol. 10682, 2018, 1068204.
[110]. T. Motohiro, A. Ichiki, T. Ichikawa, H. Ito, K. Hasegawa, S. Mizuno, T. Ito, T. Kajino,
Y. Takeda, K. Higuchi, Consideration of coordinated solar tracking of an array of compact
solar-pumped lasers combined with photovoltaic cells for electricity generation, Jpn. J.
Appl. Phys., Vol. 54, 2015, 08KE04.
[111]. J. Zhao, A. Wang, P. P. Altermatt, S. R. Wenham, M. A. Green, 24 % efficient perl siliconsolar cell: Recent improvements in high efficiency silicon cell research, Sol. Energy Mater.
Sol. Cells, Vols. 41-42, 1996, pp. 87-99.
[112]. T. Masuda, M. Iyoda, Y. Yasumatsu, M. Endo, Low-concentrated solar pumped laser via
transverse excitation fiber-laser geometry, Opt. Lett., Vol. 42, Issue 17, 2017,
pp. 3427-3430.
[113]. M. Endo, J.-F. Bisson, T. Masuda, Monte Carlo simulation of a transversely excitedsolar-pumped fiber laser, Jpn. J. Appl. Phys., Vol. 58, 2019, 112006.
[114]. T. Masuda, Y. Zhang, C. Ding, F. Liu, K. Sasaki, Q. Shen, M. Endo, All-inorganic cesium
lead halide perovskite nanocrystals for solar-pumped laser application, J. Appl. Phys.,Vol. 127, 2020, 243104.
[115]. T. Masuda, M. Iyoda, Y. Yasumatsu, S. Dottermusch, I. A. Howard, B. S. Richards,
J.-F. Bisson, M. Endo, A fully planar solar pumped laser based on a luminescent solar
collector, Commun. Phys., Vol. 3, 2020, pp.1-6.

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Chapter 11. Sub-micron Direct Silicon Processing by Microsphere Focused Femtosecond Infrared Laser

Chapter 11

Sub-micron Direct Silicon Processing
by Microsphere Focused Femtosecond
Infrared Laser
Fırat İdikut, Burcu Karagöz and Alpan Bek1

11.1. Introduction
The outstanding properties of the silicon attract the industry and the research centers tokeep making the investment on Si based applications. Si is transparent intelecommunication wavelength (1550 nm). It has high index of refraction in mid-infrared
range which makes it suitable for creating photonic crystal structures with complete
bandgaps. It is also resistant to the radiation damage. The most significantly, it has beenpracticed for decades in electronics and nano structuring. Therefore, there is a solidbackground in Si research [1].
Surface and subsurface modification of the silicon in electronics and photonics
applications have become one of the most popular research areas during the last decades[2-5]. Optical switches [6], waveguides [7, 8], photonic crystals [6, 9], modulators [10],and periodic structures [11-13] have been reported during the last two decades. Processingof Si surfaces in micron and submicron level is an important element to enhance theperformance of photonic devices. Lithography techniques; electron ion-beam [14] andfocused ion-beam [15, 16]; are known methods, which are used to pattern Si in theselevels. However, these methods are very time consuming and not cost effective.
Processing large scale of samples (> 1 cm) is the main difficulty on these methods. To
solve this, the direct writing method is an alternative approach to processing ofsemiconductors.
The use of pulsed lasers in order to form structures having photonic properties on thesurfaces of semiconductor crystal materials is quite common. The high peak power andthe controllable pulse duration within the designed limits open the gates of new design
techniques and the future technology, and enable the past ideas to put into practice inFırat İdikut, Burcu Karagöz
Department of Physics, Middle East Technical University, Ankara, Turkey

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Advances in Optics: Reviews. Book Series, Vol. 5

semiconductor fields. In last decades, the ultrafast lasers (femtosecond, picosecond andnanosecond) have started to be used for processing the material surfaces.
Surface and subsurface modification of semiconductors such as silicon became possible
with ultrafast lasers [8, 17]. The focal sizes formed in far-field processing techniques
remain 10 µm and above due to the limited numerical aperture (NA) [18]. In silicon, thespot size is enlarged due to high index of refraction of the silicon [17]. This causes the
wavelength bands in which the photonic structures that can be produced can work
effectively to be limited to mid- and far-infrared wavelengths. Therefore, in order to reachthe sub-micron processing alternative methods are needed. Reducing the processing pixel
sizes to 1 µm and below will be able to expand the potential photonic application on Si.
Femtosecond laser pulses have been used for precise processing and modification of arange of materials including silicon at micrometer scales [19] through the ultrashort pulse
duration and high intensity of these ultrafast pulses. Therefore, two-photon absorption(TPA) is higher in femtosecond lasers for low pulse energy. Ablation precision of
femtosecond laser pulses is also high through reduced heat affected zone and small
ablation threshold. Taking advantages of these characteristics, well-defined micro
structuring is possible with femtosecond pulses.
The smallest size of features that can be achieved by laser irradiation is limited due todiffraction of laser beams. The far-field resolution is therefore limited to approximately
half of the wavelength of illuminating beam. Direct laser writing cannot be used fornanoscale structuring due to the diffraction limitation of far-field patterning [20]. To
overcome this limitation in patterning of substrate surfaces, contact particle lens array(CLPA) technique where these particles serve as near-field focusing elements has been
extensively employed [21-25]. In this method, self-assembled monolayer of micrometer
sized spheres on a substrate subjected to a laser irradiation acts as a micro lens array that
focuses the laser beam to substrate. By scanning the laser beam through this microspherelattice, surface patterning beyond the diffraction limit can be observed on substrates [26].
The mechanism of this patterning technique is based on the intensity enhancement at
contact regions between spheres and substrate [27]. This enhanced field causes ablation
on substrate regions in contact with spheres and a patterned surface is obtained upon theremoval of ablated material.
In this chapter, the uniformly distributed microspheres on silicon (Si) substrate which isfunctioned as focusing micro lenses having capability to achieve the ablation of Si under
the two-photon absorption regime with 1.55 µm ultrafast laser will be discussed. In
addition to the direct writing with microspheres, periodic structuring inside the processedregion is reported.

11.2. Laser-material Interactions
11.2.1. Gaussian Beam Propagation
The beam shape is outcome of combination of the cavity modes generated in the laser. In
Fig. 11.1, several modes and their intensity profiles are represented. Depending on the474Chapter 11. Sub-micron Direct Silicon Processing by Microsphere Focused Femtosecond Infrared Laser

laser design for the usage of its application field, single mode or multimode beam profiles
are generated. In our case, TEM00 mode is considered as the output beam shape. This mode
is generally known as Gaussian beam [28].

Fig. 11.1. Representation of the intensity distribution of the transverse electromagnetic
modes (TEMij).

The Gaussian beam has an exponential decaying intensity distribution profile along x-yplane which is perpendicular to the propagation direction z. In equation (11.1), theintensity distribution of the Gaussian mode is indicated in cylindrical coordinates. The
distribution has dependencies along radial r and z. w(z) is the beam waist at themeasurement distance where the irradiance drops to 1/e2 factor, which is calculated withrespect to the beam waist (w0) as referred in equation (11.2).
𝑤0 2
−2𝑟 2
) 𝑒𝑥𝑝 (
),
𝑤(𝑧)
𝑤(𝑧)2

𝐼(𝑟, 𝑧) = 𝐼0 (2𝜆𝑧𝑤(𝑧) = 𝑤0 √1 + (𝜋𝑤 2 ) ,(11.1)(11.2)

0The equation (11.2) can be rearranged in terms of the Rayleigh range (ZR) and the axis
distance from the center. The Rayleigh range is the length at which the spot size increases
to 1.414w0.
𝑧𝑅 =

𝜋𝑤02
𝜆

(11.3)

Combining the equations (11.2) and (11.3), the beam waist is described by theparameter, z.
𝑧2𝑤(𝑧) = 𝑤0 √1 + (𝑧 )𝑅

(11.4)The beam width with varying z- distance is represented in Fig. 11.2.
The equation (11.4) and Fig. 11.2 are the general picture of the Gaussian beam spreading.
The spot sizes in specific distances (z- values) are𝑤(𝑧 = 0) = 𝑤0 ,(11.5)
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Advances in Optics: Reviews. Book Series, Vol. 5𝑤(𝑧 = 𝑧𝑅 ) = √2𝑤0 ,𝑤(𝑧 ≫ 𝑧0 ) =𝑤0 𝑧
𝜆= 𝜋𝑤 𝑧𝑧𝑅0(11.6)(11.7)The equation (11.5) is the spot size at the beam waist. At z = zR, the spot size is definedby the equation (11.6). ZR is Rayleigh distance, which is related to the focal depth of theGaussian beam focusing. zR is also taken as a reliable distance for measuring the focal
depth in near-field zone. In far-field approximation (z >> zR), the beam waist can becalculated by the equation (11.7) [29].

Fig. 11.2. The illustration of Gaussian beam.

The ideal Gaussian beam intensity distribution and the beam shape are defined by theequation provided above. However, in reality, the output beam shape may not match with
the ideal beam shape. It has some defects due to the quality of the equipment andenvironmental conditions. For this reason, the beam quality factor (M2) is introduced. It
provides the value that implies how close the ideal beam shape is obtained. The ideal valueis 1 for the ideal case. If the M value is introduced to equations (11.5), (11.6) and (11.7),
the spot sizes can be modified to the equations (11.8), (11.9) and (11.10),
respectively [28].
𝑤0,𝑒𝑓𝑓 = 𝑀𝑤0 ,(11.8)

𝑤𝑒𝑓𝑓 = 𝑀𝑤(𝑧),(11.9)

𝑤𝑒𝑓𝑓 ≅ 𝑀2 𝜋𝑤

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𝜆
0,𝑒𝑓𝑓

𝑧

(11.10)Chapter 11. Sub-micron Direct Silicon Processing by Microsphere Focused Femtosecond Infrared Laser

11.2.2. Ablation Mechanisms
In interaction of laser pulses with solids, there are different kinds of processes dependingon the material type and laser properties. These processes can be thermal or nonthermal
according to laser wavelength. In a thermal process, laser light energy absorbed byelectrons in materials is converted into heat and this transference of energy results inmelting and vaporization of the material [30]. For cases of photon energies sufficient tobreak atomic lattice, ablation of the material can occur without traditional heatingprocesses (nonthermal process) [31].
Laser pulse duration is also a crucial parameter in ablation process. Vapor pressure andrecoil pressure of laser light induce removal of materials through melting when lasers inlong pulse regime are used [32]. Ablation governed by a continuous wave (CW) and ananosecond wave is a thermal process. Because pulse durations of these ranges are longer
than thermalization time of materials, much of the incident beam energy is conducted toand deposited in exposed regions. Heat affected zone for ns pulses is smaller than thoseof CW lasers [33]. Material removal using lasers emitting ultrashort pulses can be thermal
or non-thermal [34]. Pulse durations of femtosecond and picosecond laser pulses areshorter than the necessary time for energy transference between the electrons and lattice.
This mechanism results in very small heat affected zones and low thermal effect enhances
micromachining quality of materials [19, 35].
Material ablation takes place when necessary energy is accumulated in processing region.
When a beam with intensity below ablation threshold illuminates the material, thetemperature of illuminated region will not increase sufficiently for ablation of the material.
The ablation threshold of a material depends on different parameters including pulse
duration. Ablation threshold for ultrashort pulses is smaller due to rapid pulse energy
deposition in laser spot.
When a laser beam interacts with a material, the energy of irradiating pulse may be
absorbed. Absorption mechanisms are varied depending on the material type and incident
photon energy. In semiconductors, incident light is strongly absorbed provided that totalenergy of incident photons exceeds the energy gap of the material. When photons energy
is smaller than the bandgap energy, light passes through the semiconductor as if it istransparent and the interaction of light with the material is weak. At high laser intensities,
nonlinear absorption becomes apparent.
The bandgap energy of crystalline silicon is 1.11 eV at 300 K [36] corresponding to awavelength value of 1.1 µm in vacuum. Absorption mechanism in silicon for photons ofwavelength longer than 1.1 µm is multiphoton absorption which is a nonlinear process.
The simplest form of multiphoton absorption is two-photon absorption (TPA) where
simultaneous absorption of two photons occurs. In a two photon absorbing material, lightintensity change along the propagation direction is𝐼

0𝐼(𝑧) = 1+𝛽𝑧𝐼,
0(11.11)477

Advances in Optics: Reviews. Book Series, Vol. 5

where 𝑧 is the propagation direction, 𝐼0 is the incident light intensity and 𝛽 is the TPA
coefficient of material.
The two-photon absorption is defined by the equation𝑑𝐼− 𝑑𝑧 = 𝛽𝐼 2

(11.12

Because absorption probability is proportional to the square of intensity, TPA is higher infemtosecond lasers for low pulse energy.
Silicon is also transparent to wavelengths longer than 1.1 µm. This characteristic enables
the use of lasers operating above this wavelength value in processing of both the surface
and bulk of the silicon.
For a pulsed laser, optical intensity is𝐸

𝐼 = 𝛥𝑡×𝐴,(11.13)where E is the energy per pulse, ∆t is the pulse duration and A is the beam spot area. A
higher energy value is provided as laser pulse duration gets shorter at a specific pulse
energy. Required pulse energy for ablation is therefore lower in femtosecond regime ascompared to nanosecond and picosecond regimes. The strong intensity dependence ofabsorption increases applicability of lasers operating in femtosecond regime inmultiphoton processing.
11.2.3. Simulations
The simulation programs have been started to use during the last decade in order to predictthe possible outcomes of the ideas before performing the experiments. In this part, thelight intensity distribution inside and vicinity of two different diameter spheres, which areplaced on smooth Si surface, under the laser exposure will be discussed. Two software(MNPBEM and Lumerical-FDTD) were used for our purpose.
The first one is MNPBEM. It is a MATLAB toolbox, which was developed by F. J. Garcia
de Abajo and A. Howie [37, 38]. By using the approach of the boundary element method(BEM), it solves Maxwell`s equations for arbitrary shaped (homogeneous orinhomogeneous) dielectric mediums which are expressed as surface integral equations.
A second alternative approach to calculate the intensity distribution of the model is the
Finite Difference Time Domain (FDTD) method. The method was introduced by Kane
Yee [39] in 1966. Similar to MNPBEM, Lumerical-FDTD also uses Maxwell’s equations
in order to calculate electric field distribution in and underlying region of models by
integrating over lines, surfaces or volumes. In this technique, field distributions can bemodeled in near- and far-field regions as well [40-42].

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11.3. Experimental Details
Our previous direct writing study on Si by using microspheres has been reported in
OPAL`2019 [43]. The custom design femtosecond laser having 1.55 µm central
wavelength, and 1 W average power at 1.0 MHz repetition rate was used in theexperiment.
The experimental setup was quite similar to the updated version which will be covered indetail. In first trials, the initial step was to determine the limits of the processing
parameters by observing the hot spot under the surface of the silicon in order to eliminate
surface damaging. To do so, a piece of c-Si (p-type) double side polished having 1 mm
thickness, 2 mm wide (mechanically diced and polished) and 3 cm length was prepared.
In Fig. 11.3a, the beam was focused by a lens having f = 8 mm and NA = 0.5. To monitor
the heated zone, which is created by the focused beam, a modified Si based camera wasused as an artificial IR camera (IR filter removed). The thermal radiation of the heatedzone at the focus generates photons having enough energy (>1.11 eV) for the detection by
Si based chips. The second step was to place the microspheres on the surface. In
Fig. 11.3b, scanning electron microscope (SEM) image shows that the spheres were
distributed by drop casting method and the single layer microsphere clusters containing
varying number of spheres was observed on the surface. In Fig. 11.3c (region 1), theablation threshold of c-Si with TPA was reached by using 4.5 µm diameter spheres. In
Fig. 11.3c (region 2), the misaligned sample with an amount of Rayleigh length along theoptical axis also created the linear pattern on c-Si surface. According to the first results in
Fig. 11.3, it was shown that the microspheres can be used as a micro lens array forprocessing c-Si with femtosecond laser at 1.55 µm wavelength by two photon absorption.

Fig. 11.3. (a) The plasma tail is observed by the camera; (b) SEM image of the distributed
microspheres having 4.5 µm in diameter; (c) In region 1, the ablation under the spheres wasobserved. In region 2, the ablation due to the direct writing was observed.

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Advances in Optics: Reviews. Book Series, Vol. 5

In order to process the large array structures by focusing the beam with microspheres indifferent diameters, it is needed to run the simulation for estimating the intensity
distribution on the surface by microlensing and to enlarge the area of c-Si which is coveredby the sphere clusters, also required to improve the experiment setup for pattering larger
areas (>1 cm). To do so, we first run some simulations for estimating intensity distributionof the focused beam on the surface by MNPBEM and Lumerical-FDTD. The spheres,
which are 1.5 µm and 4.5 µm in diameters, have an index of refraction 1.565 at 1.55 µm.
Si substrate has an index of refraction 3.5 at the same wavelength. A microsphere wasplaced on Si substrate and illuminated by a linearly polarized plane wave. Under these
conditions, the field distribution around the sphere is shown in Fig. 11.4. The outcomesof simulations indicate that both spheres in different diameters focus the incoming plane
wave to spot, which is close to the interface. Both simulation approaches gave thesimilar results.

Fig. 11.4. (a, b) The intensity distribution of focused laser beam with 1.5 µm and 4.5 µm
diameter spheres by MNP- BEM, respectively. (c, d) The intensity distribution of focused laser
beam with 1.5 µm and 4.5 µm diameter spheres by Lumerical- FDTD, respectively.

Secondly, the spin coating method was used for coating microspheres having 4.5 µm and1.5 µm in diameter. Certain amounts of aqueous suspension of these microspheres were
deposited by spin coating on silicon wafers which were made hydrophilic with an oxygen
plasma treatment before the deposition process. After a 24 h drying at room temperature,a self-assembled microsphere layer was generated on silicon substrate. In Fig. 11.7, theoptical microscope image of the microspheres (1.5 µm in diameter) layers is shown.
Thirdly, the optical path was changed to the periscopic configuration via integrating the480Chapter 11. Sub-micron Direct Silicon Processing by Microsphere Focused Femtosecond Infrared Laser

new 3-axis stage in Fig. 11.5a. The stage has resolution ~20 nm in z-axis and ~10 nm inhorizontal axes. The maximum scanning speeds are 20 mm/s in z-axis and 10 mm/s inhorizontal axes. Fig. 11.5b illustrates processing of c-Si by means of microspheres.
Further details about the experiment including spheres, processing and scanningparameters are listed in Table 11.1.

Fig. 11.5. (a) Position of the laser spot is controlled with a 3-axis stage. The polarization
direction of the beam on sample is shown. (b) The laser beam is focused on multiple areas
by microspheres.

Table 11.1. Experimental parameters of Fig. 11.6 and Fig. 11.7, respectively.
Spheres
4.5 µmLaser
Average Output Power
300 mW
Repetition Frequency
1 MHz
Pulse Duration
480 fsStage Parameters
Step Distance
50 µm
Scan Speed
1.5 mm/s
Diameter

1.5 µm
470 mW
1 MHz
480 fs50 µm
2.5 mm/s

In Fig. 11.6, optical microscope and SEM images obtained using 4.5 µm spheres areshown. After processing c-Si, the microspheres were cleaned and the patterned structure
was revealed. In Fig. 11.6 (c), SEM image shows that the line thickness of the processedzone is around 13.6 µm and the damaged zones were only observed in that region.
Therefore, the diameter of the spot on the surface can be taken the same as the line
thickness (13.6 µm).
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Advances in Optics: Reviews. Book Series, Vol. 5

Fig. 11.6. (a) The optical microscope image of the processed area with 50 µm step distance. It isshown that 4.5 µm diameter spheres were burned under the 1.55 µm laser pulses. The spheres were
evaporated in an instant of time under the laser exposure. (b) After the acetone cleaning
and chemical etching of the sample, the modified area was observed by optical microscope.
(c) The SEM image of the region is shown in (b).

Three regions (A, B, C) are indicated in Fig. 11.7. These are corresponding to multilayer,monolayer and uncoated regions on c-Si, respectively. In region A, multilayer of spheres(1.5 µm in diameter) causes the beam to defocus and the ablation threshold of c-Si cannot
be reached. The borders of the regions show that the direct writing is mostly functionalon monolayer area (region B). Intersection of the regions B and C is an evidence for TPA
processing of c-Si triggered by the microsphere focusing. Between the region A and B,
the surface pattering is also stopped because of the defocusing.
Different from SEM image given in Fig. 11.6c, a periodic structuring, which was createdby 1.5 µm spheres, is observed with SEM image in Fig. 11.8. The periodicity of thestructures inside the processed region is approximately ~300 nm and the simulation could
not predict this outcome.

11.5. Conclusions
In literature, microlensing with microspheres has been already demonstrated for thesurface structuring of c-Si via ultrafast lasers having central wavelength which has enoughphoton energy for direct absorption. However, in this work, we showed that submicron
processing of c-Si in TPA regime was achieved by focusing the femtosecond laser at
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Chapter 11. Sub-micron Direct Silicon Processing by Microsphere Focused Femtosecond Infrared Laser

1.55 µm wavelength with microspheres. c-Si was successfully processed by using two
types of microspheres (1.5 µm and 4.5 µm in diameters). The localized spot by the
microspheres overcame the diffraction limits of the optical system. The focal depth of thesystem is limited by Rayleigh length, which covers 1.5 µm microspheres as well. It isobserved that the spot was defocused by multiple scatterings inside of the spheres in
multilayer region and the monolayer region was the only processed area among 3-regions.
An unexpected outcome of this work was to observe the periodic structuring on Si under
the damaged spots by focusing the beam with 1.5 µm microspheres. The spatial period of
the structures was calculated around ~300 nm (~λ/5). The orientation of the periodic
structures depends on the polarization direction of the electric field. The structures aligned
perpendicular to the electric field polarization in our results. In the future, the triggering
mechanism of the periodic structuring under the ablated regions by TPA will be
investigated in detail.

Fig. 11.7. Three regions which are shown as A, B and C are corresponding microsphere (1.5 µm
in diameter) layers on Si surface. Region A includes multilayer microsphere coated area, B is
the monolayer coated region and C is Si substrate.

Fig. 11.8. SEM images of submicron periodic structure patterning with 1.5 µm microspheresshown in Fig. 11.7.

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Acknowledgements
The authors acknowledge the support and contributions of Mona Zolfaghari Borra andNasim Seyedpour Esmaeilzad at Micro and Nano Technology (MNT) program of the
Middle East Technical University in the initial phase of this study. This work is supportedby TÜBİTAK under grant nr. 118F132 and the Middle East Technical University under
grant nr. BAP-105-2018-2798. We thank Zafer Artvin for the simulations.

References[1]. B. Cowan, Optical damage threshold of silicon for ultrafast infrared pulses, AIP Conference
Proceedings, Vol. 877, 2006, 837.
[2]. F. Priolo, T. Gregorkiewicz, M. Galli, T. F. Krauss, Silicon nanostructures for photonics andphotovoltaics, Nature Nanotechnology, Vol. 9, 2014, pp. 19-32.
[3]. A. E. Lim, J. Song, Q. Fang, C. Li, X. Tu, N. Duan, K. K. Chen, R. P. Tern, T. Liow, Review
of silicon photonics foundry efforts, IEEE Journal of Selected Topics in Quantum Electronics,
Vol. 20, Issue 4, 2014, pp. 405-416.
[4]. G. T. Reed, G. Mashanovich, F. Y. Gardes, D. J. Thomson, Silicon optical modulators, Nature
Photonics, Vol. 4, 2010, pp. 518-526.
[5]. S. Y. Siew, B. Li, F. Gao, H. Y. Zheng, W. Zhang, P. Guo, S. W. Xie, A. Song, B. Dong,
L. W. Luo, C. Li, X. Luo, P. G. Lo, Review of silicon photonics technology and platform
development, Journal of Lightwave Technology, Vol. 38, Issue 13, 2021, 4374-4389.
[6]. T. G. Euser, L. W. Vos, Spatial homogeneity of optically switched semiconductor photonic
crystals and of bulk semiconductors, Journal of Applied Physics, Vol. 97, 2005, 043102.
[7]. T. K. Liang, H. K. Tsang, Silicon waveguide two-photon absorption detector at 1.5 μm
wavelength for autocorrelation measurements, Appl. Phys. Lett., Vol. 81, 2002, 1323.
[8]. O. Tokel, A. Turnalı, G. Makey, P. Elani, T. Çolakoğlu, E. Ergeçen, Ö. Yavuz, R. Hübner,
M. Z. Borra, I. Pavlov, A. Bek, R. Turan, D. K. Kesim, S. Tozburun, S. Ilday, F. O. Ilday,
In-chip microstructures and photonic devices fabricated by nonlinear laser lithography deep
inside silicon, Nature Photonics, Vol. 11, 2017, pp. 639-645.
[9]. A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez,
F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, H. M. van Driel, Large-scale
synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near
1.5 micrometres, Nature, Vol. 405, 2000, pp. 437-440.
[10]. R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen,
M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, A. O. Brajklev,
Strained silicon as a new electro-optic material, Nature, Vol. 441, 2006, pp. 199-202.
[11]. J. F. Young, J. S. Preston, H. M. van Driel, J. E. Sipe, Laser-induced periodic surface
structure. II. Experiments on Ge, Si, Al, and brass, Physical Review B, Vol. 27, 1983, 1155.
[12]. Y. F. Lu, W. K. Choi, Controllable laser‐induced periodic structures at silicon-dioxide/siliconinterface by excimer laser irradiation, Journal of Applied Physics, Vol. 80, 1996, 7052.
[13]. J. Bonse, J. Krüger, Femtosecond laser-induced periodic surface structures, Journal of Laser
Applications, Vol. 24, Issue 4, 2012, 042006.
[14]. P. W. Doll, A. Al-Ahmad, A. Bacher, A. Muslija, R. Thelen, L. Hahn, R. Ahrens, B. Spindler,
A. E. Guber, Fabrication of silicon nanopillar arrays by electron beam lithography andreactive ion etching for advanced bacterial adhesion studies, Materials Research Express,
Vol. 6, 2019, 065402.
[15]. M. Erdmanis, I. Tittonen, Focused ion beam high resolution grayscale lithography for siliconbased nanostructures, Appl. Phys. Lett., Vol. 104, Issue 7, 2014, 073118.

484

Chapter 11. Sub-micron Direct Silicon Processing by Microsphere Focused Femtosecond Infrared Laser[16]. W. R. W. Razali, N. N. M. Rashid, S. R. Hosman, N. K. Aid, Surface patterning of silicon
and germanium using focused ion beam for the development of FinFET structure,
in Proceedings of the IEEE Regional Symposium on Micro and Nanoelectronics (RSM’19),
2019, pp. 116-118.
[17]. D. Grojo, A. Mouskeftaras, P. Delaporte, S. Lei, Limitations to laser machining of siliconusing femtosecond micro-Bessel beams in the infrared, Journal of Applied Physics, Vol. 117,
2015, 153105.
[18]. R. Gattass, R. Mazur, Femtosecond laser micromachining in transparent materials, Nature
Photonics, Vol. 2, 2008, pp. 219-225.
[19]. X. Liu, D. Du, G. Mourou, Laser ablation and micromachining with ultrashort laser pulses,
IEEE Journal of Quantum Electronics, Vol. 33, Issue 10, 1997, pp. 1706-1716.
[20]. L. Li, M. Hong, M. Schmidt, M. Zhong, A. Malshe, B. l. Huis, V. Kovalenko, Laser
nano-manufacturing – State of the art and challenges, CIRP Annals – Manufacturing
Technology, Vol. 60, Issue 2, 2011, pp. 735-755.
[21]. Y. F. Lu, L. Zhang, W. D. Song, Y. W. Zheng, B. S. Luk’yanchuk, Laser writing of a
subwavelength structure on silicon (100) surfaces with particle-enhanced optical irradiation,
Journal of Experimental and Theoretical Physics Letters, Vol. 72, 2000, pp. 457-459.
[22]. H. Münzer, M. Mosbacher, M. Bertsch, J. Zimmermann, P. Leiderer, J. Boneberg, Local fieldenhancement effects for nanostructuring of surfaces, Journal of Microscopy, Vol. 202,
Issue 1, 2001, pp. 129-135.
[23]. G. Langer, D. Brodoceanu, D. Bäuerle, Femtosecond laser fabrication of apertures on
two-dimensional microlens arrays, Appl. Phys. Lett., Vol. 89, 2006, 261104.
[24]. A. Pena, Z. Wang, D. Whitehead, L. Li, Direct writing of micro/nano-scale patterns by meansof particle lens arrays scanned by a focused diode pumped Nd:YVO4 laser, Applied Physics
A, Vol. 101, 2010, pp. 287-295.
[25]. N. Bityurin, A. Afanasiev, V. Bredikhin, A. Alexandrov, N. Agareva, A. Pikulin, I. Ilyakov,
B. Shishkin, R. Akhmedzhanov, Colloidal particle lens arrays-assisted nano-patterning byharmonics of a femtosecond laser, Opt. Express, Vol. 21, Issue 18, 2013, pp. 21485-21490.
[26]. M. H. Hong, S. M. Huang, B. S. Luk`yanchuk, T. C. Chong, Laser assisted surface
nanopatterning, Sensors and Actuators A: Physical, Vol. 108, Issue 1-3, 2003, pp. 69-74.
[27]. J. R. Arias-González, M. Nieto-Vesperinas, Near-field distributions of resonant modes insmall dielectric objects on flat surfaces, Optics Letters, Vol. 25, Issue 11, 2000, pp. 782-784.
[28]. R. S. Quimby, Photonics and Lasers: An Introduction, Wiley-Interscience, 2006.
[29]. F. L. Pedrotti, L. M. Pedrotti, L. S. Pedrotti, Introduction to Optics, Addison Wesley, 2006.
[30]. R. E. Russo, X. Mao, H. Liu, J. Gonzalez, S. S. Mao, Laser ablation in analytical chemistry –A review, Talanta, Vol. 57, 2002, pp. 425-451.
[31]. S. S. Mao, X. Mao, R. Greif, R. E. Russo, Initiation of an early-stage plasma duringpicosecond laser ablation of solids, Appl. Phys. Lett., Vol. 77, 2000, 2464.
[32]. C. Earl, J. R. C.-P. P. A. Hilton, W. O’Neill, The dynamics of laser surface modification,
J. Manuf. Process., Vol. 21, 2016, pp. 214-223.
[33]. L. Lucas, J. Zhang, Industrial-lasers, https://www.industrial-lasers.com/cutting/article/
16488567/femtosecond-laser-micromachining-a-backtobasics-primer[34]. B. Rethfeld, D. S. Ivanov, M. E. Garcia, S. I. Anisimov, Modelling ultrafast laser ablation,
Journal of Physics D: Applied Physics, Vol. 50, Issue 19, 2017, 193001.
[35]. R. L. Harzic, N. Huot, E. Audouard, C. Jonin, P. Laporte, Comparison of heat-affected zones
due to nanosecond and femtosecond laser pulses using transmission electronic microscopy,
Appl. Phys. Lett., Vol. 80, 2001, 3886.
[36]. C. Kittel, Introduction to Solid State Physics, 8th Ed., John Wiley & Sons, 2005.
[37]. F. J. Garcia de Abajo, A. Howie, Retarded field calculation of electron energy loss ininhomogeneous dielectrics, Phys. Rev. B, Vol. 65, Issue 11, 2002, 115418.

485

Advances in Optics: Reviews. Book Series, Vol. 5[38]. U. Hohenester, A. Trügler, MNPBEM – A MATLAB toolbox for the simulation of plasmonic
nanoparticles, Comp. Phys. Commun., Vol. 183, 2012, 370.
[39]. K. Yee, Numerical solution of initial boundary value problems involving Maxwell's equationsin isotropic media, IEEE Transactions on Antennas and Propagation, Vol. 14, Issue 3, 1966,
pp. 302-307.
[40]. D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method, IEEE Press Series,
New York, 2000.
[41]. A. Taflove, Computational Electromagnetics: The Finite-Difference Time-Domain Method,
Artech House, Boston, 2005.
[42]. S. D. Gedney, Introduction to the Finite-Difference Time-Domain (FDTD) Method forElectromagnetics, Morgan & Claypool Publishers, 2011.
[43]. F. İdikut, M. Borra, N. Esmaeilzad, A. Bek, Subsurface silicon processing by microspherefocusing of ultrafast infrared laser, in Proceedings of the 2nd International Conference on
Optics, Photonics and Lasers (OPAL’19), Amsterdam, Netherland, 24-26 April 2019,
pp. 69-70.
[44]. A. Dhiman, Silicon photonics: A review, IOSR Journal of Applied Physics (IOSR-JAP),
Vol. 3, Issue 5, 2013, pp. 67-79.
[45]. X. Liu, D. Du, G. Mourou, Laser ablation and micromachining with ultrashort laser pulses,
IEEE J. Quantum Electron., Vol. 33, Issue 10, 1997, pp. 1706-1716.

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Chapter 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy
at the Grazing Angle Incidence

Chapter 12

Mid Infrared Quantum Cascade Laser
Reflection-absorption Spectroscopy
at the Grazing Angle Incidence
Annette M. Colón-Mercado, Karla M. Vázquez-Vélez,Francheska M. Colón-González, John R. Castro-Suárez,Nataly J. Galán-Freyle, Vladimir Villanueva-López,Edwin Caballero-Agosto, Leonardo C. Pacheco-Londoño,
and Samuel P. Hernandez-Rivera1

12.1. Introduction
Quantum cascade lasers (QCLs) have emerged as a promising technology for developing
applications in defense, security, and pharmaceutical processes. QCLs possess clear
advantages over conventional techniques due to their sensitivity, high signal-to-noise
ratios (S/N), rapid response, and remote detection capabilities. The high output power ofthese mid-infrared (MIR) sources allows the detection of analytes in the nanogram to the
picogram mass regime. These fast-scanning systems can achieve responses in less than
five seconds, offer portable detection at room temperature, and are commerciallyavailable. QCLs are semiconductor lasers that undergo sub-interband transitions that are
precisely engineered in the solid-state device. Modifying the heterostructure in the lasing
device allows multiple quantum well transitions. There is a constant effort to continue
improving techniques to detect analytes based on taking advantage of the QCLs'
capabilities. Most previous research has focused on detection using infrared spectroscopy(IRS) primary modalities: absorption, transmission, emission, reflection, and attenuatedtotal reflection. In this contribution, we describe a novel optical setup that couples a QCL
source to a compact grazing angle (GA) probe mount that enables a double-pass beam
trajectory that allows a higher sensitivity. At the same time, measurements at the GA
warrants a higher coverage area for detection and enhances the S/N, particularly when

Annette M. Colón-Mercado
Department of Chemistry, University of Puerto Rico, Mayagüez

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using highly reflective surfaces as sample substrates. By adapting the QCL’s vibrational
response to multivariate analysis (MVA) and artificial intelligence (AI) algorithms, abetter representation of the data is obtained. Therefore, the QCL-GAP represents apromising new path for detecting analytes in a wide range of applications.

12.2. Quantum Cascade Lasers
The invention of QCLs originated in 1994 at Bell Labs by Faist, Capasso, andcollaborators [1]. QCLs are semiconductor injection lasers that differ from ordinary diodelasers in the transition band process [2]. For the lasing process, diode lasers depend on a
direct transition between the valence band and the conduction band, and the population
inversion is achieved by optical or electrical pumping. On the other hand, QCLs the lasing
process is based on interband transitions in which the population inversion is distributed
over several overlapping bands, resulting in a relatively broad spectrum. One of thesignificant advances of QCLs is that modifications to the quantum confinements allow
tailoring the emission wavelength. Therefore, a wide spectral range from MIR to thesubmillimeter-wave region of the electromagnetic (EM) spectrum can be achieved. In
addition, wide bandgap materials can be used without the necessity of incrementing thetemperature.
In a QCL, the conduction band acquires a staircase shape where electrons stream down,
and photons are emitted at each step of the process. Population inversion occurs in eachstep where a discrete transition is achieved by electron tunneling. The resulting emission
has a higher density, thus enhancing the laser power. Fig. 12.1 shows the conduction band
diagram, where a discrete transition between a higher energy level n = 3 to n = 2 occurs.
Consequently, n = 2 and n = 1. In each transition, a photon is emitted while the tunneling
escape time decreases. A final transition between this first active region to the following
occurs from n = 1 to E3, where the photon is injected to the next level.

Fig. 12.1. Schematic of the conduction band structure of a QCL laser.

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12.3. Grazing Angle Reflection-Absorption Infrared Spectroscopy (RAIRS)
12.3.1. Design of the Optical Setup
One of the most sensitive techniques of infrared spectroscopy (IRS) is MIR spectroscopy
operating at the GA since this arrangement enables the optical effect known as reflectionabsorption infrared spectroscopy (RAIRS). The technique requires coupling a MIR
excitation source to an optical probe mount at an incident angle 80 with respect to thesurface normal). In the present case, a QCL based pre-dispersive spectrometer (model
LaserScan, Block Engineering, Southborough, MA) was coupled to a compact GAprobe (GAP) consisting of two compact gold-coated plane mirrors fixed near the grazing
angle (~82) as illustrated in Fig. 12.2). The LaserScan contains a thermoelectricallycooled mercury-cadmium-telluride (MCT) detector and a 3-in (diam) zinc selenide (ZnSe)lens that focuses the laser beam onto the right-hand mirror. Reflecting the laser beam
expands the laser beam. Gold (Au) coated mirrors were positioned as illustrated inFig. 12.2. The first gold-coated mirror was placed at 49 with respect to the surface
normal, deflecting the light at an angle of 8 with respect to the surface, hence 82 withrespect to the surface normal as shown in Fig. 12.2 [3].

Fig. 12.2. Setup for QCL-GAP for enabling RAIRS.

The LaserScan system has an output beam size of approximately 2×4 mm2. The GAP
optical setup expands the laser spot on the sample plane, forming an elliptical illuminatedimage for a wider sampling area. Thermal measurements show that the first pass of thelaser beam creates an ellipse of 40×4 mm2, and the returning beam produces an even larger
spot on the sample of 50×5 mm2. A comparison between a directly reflected beam from a
metal substrate and the GA probe setup is shown in Fig. 12.3.
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Fig. 12.3. Thermal images of the laser beam incident on an aluminum substrate.
a) QCL directly reflected beam b) QCL GAP reflected beam.

12.3.2. Advantages over Conventional Methods
Developing techniques for detecting emerging defense and security threats and organic
residues after cleaning sessions in pharma industries has been a top priority of thisresearch group for several years. Various methods for detecting and characterizing active
pharmaceutical ingredients (APIs) and high explosives (HEs) traces have been devised[4]. Chromatographic techniques have been the leading choice for analysis due to theirrobustness and low detection limits (DL) [5]. In spectroscopic applications in these criticaltasks, Raman scattering (RS) and IRS have been the two main branches used tocharacterize the vibrational signatures of targets. Modalities based on these techniques are
non-invasive, non-destructive, and require minimal sample preparation, making them
ideal for rapid characterization. To improve the performance, investigators have started tocouple existing techniques. In the MIR region, fiber optic probes have been coupled to
Fourier transform infrared (FT-IR) technology, achieving a higher sensitivity when
operating at the grazing angle [6-8].
The mentioned techniques still lack essential features that are needed when detecting
traces. High-performance liquid chromatography (HPCL) and mass spectrometry coupled
to gas chromatography (GC-MS) are time-consuming and require sample preparation.
Laser excitation of RS in the visible region of the EM spectrum can be plagued withfluorescence from the target analytes or impurities present in the samples. Anotherpossibility is sample thermal decomposition due to the high laser powers required in thenear-infrared region arising from the ~4 dependence of the inelastic scattering in a Raman
event [9]. IRS also presents limitations due to the radiation density of a thermal source
which results in low absorptions/reflectances [10]. The necessity of a fast and reliable
technology without sample preparation to detect residues at trace level was evident.
QCLs tunable lasers have emerged as a promising new source of MIR radiation capableof overcoming the challenges discussed. A laser source meets portability requirements
and adds stability, high absorptions/reflectances, low spectral noise. It is a high brightness
source that can be adapted to any optical setting to enhance the S/N. QCLs have theadvantage of working under ambient conditions, having minimal interference fromexternal sources [11, 12]. Furthermore, the high output power makes the system an
excellent choice for the standoff detection of threats [13]. Our approach using the
QCL-GAP optical setting goes one step further: coupling data acquisition with powerful490

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statistical models for data analysis to demonstrate how the technique surpasses common
challenges by understanding the behavior of the vibrational patterns in the spectra
obtained. This approach is critical when analyzing RAIRS data.

12.4. Sample Preparation for Trace Detection Analyses
12.4.1. Deposition Techniques
Quality assurance of an analyte on a sample surface is directly related to the deposition
technique used for the analysis. Sample homogeneity and the time of residence of theanalyte on the substrate are two factors that significantly affect the detection hence, thevibrational response of the instrument. There are complex deposition techniques used in
the field of materials science and engineering for manufacturing devices, particularly inthe areas of computer chips and electronic devices. Chemical vapor deposition (CVD),
molecular beam epitaxy (MBE), and ion beam deposition (IBD) have distinguishedthroughout the years. The need for practical analyte deposition methods for preparing
samples and standards has led to new alternate deposition techniques such as spin coating,inkjet printing, air spray, sample immersion (partial or total) meniscus-guided coating as
shown in Fig. 12.4. Spin coating deposition uses the centrifugal force caused by therotation of the surface to spread the liquid into a film. The thickness of the uniform film
ranges from 1-10 μm [14].. Regarding printing methods used by our research group,
thermal inkjet printing deposits samples on surfaces with high homogeneity, providing athin film deposition. The selection of the proper parameters ensures a deposition thatsimulates real-world samples. In the meniscus-guided coating deposition, the smearing
method uses a Teflon stub to spread the analyte on the surface of the substrate with theadvantage of minimal loss of the product. The properties of Teflon assist in minimizing
analyte aggregation; therefore, the sample is deposited on the surface nearlyhomogeneously. When using absorbent materials as substrates, a dip-coating approach isrecommended. Although, dip coating presents the challenge of not easily being
quantifiable. Sample sieving is also of great interest.

Fig. 12.4. Deposition methods: (a) Smearing; (b) Inkjet printer; (c) Smearing using micropipette);(d) Spin coating.

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12.4.2. Effect of the Substrate: Specular and Diffuse Reflectance Measurements
When analyzing the influence of the substrate, its chemical and physical characteristics
should be considered. Understanding the substrate role in the deposition process and thefinal product obtained helps determine the results obtained in the measurements
conducted. This is when analyzing the reflectance signals of the analytes and thesubstrates (background) on the spectra collected. The roughness and reflectivity of thesubstrate affect the laser beam trajectory. Hence, two different optical effects are present:
specular reflectance and diffuse reflectance. Specular reflectance results in the reflectionof light occurring at the same angle as the incident angle (with respect to the surface
normal) for highly reflective smooth substrates [15]. In diffuse reflectance, the light isreflected by rough surfaces in all directions. Most of the light reflected from roughmaterials comes from the surface, but some come from internal reflections. Diffuse
reflectance is mainly observed in non-reflective and opaque substrates.
12.4.3. Sample Preparation of HEs and APIs
We have recently established sample deposition protocols that improve the validation of
the detection process of the sensors developed. We have worked with all the mentioned
deposition methods. Here, we present the validation of the QCL-GAP system by usingreflective and non-reflective standards obtained from external laboratories (forcross-validation procedures) or prepared in our laboratories.
HE standards were obtained from the Navy Research Lab at Washington, DC. These wereprepared as part of a Department of Homeland Security (DHS) program that focuses on
standoff explosives detection on vehicles (SED-V) and also as part of DHS research
program entitled Methods for Optical Detection of Explosives (MoDEx). Sample coupons
were prepared by sieving deposition of HEs on acrylonitrile butadiene styrene (ABS)plastic substrates and aluminum (Al) substrates [5]. The HE included as primary standards
were RDX, Tetryl, and PETN. These 1 in. × 1 in. substrates (coupons) were dosed withwell-known surface concentrations of the HEs and used for validating our QCL-GAP
methodology. Fig. 12.5 shows the white light micrographs of the coupons.
In addition, the Applied Physics Laboratory of the Johns Hopkins University (Baltimore,MD) also participated in the MoDEx program, also prepared and sent primary standards
in the form of brushed AL coupons loaded with HEs. These were used to characterize andcompare two QCL acquisition modes: direct back-reflectance and GAP diffuse
reflectance. Thermal inkjet printing was used to deposit the HEs. The HE selected for the
study was C-4, which is a formulation of the nitramine RDX (91.0 %), dioctyl sebacate(5.3 %), polyisobutylene (2.1 %), and mineral/motor oil (1.6 %) [16] HPLC analysis
confirmed the mass deposited on the substrates, this being in a 0.1-10.0 g/cm2 range.
Fig. 12.6 shows white light micrographs of the coupons.
Besides HE detection experiments at near trace level, experiments designed forpharmaceutical cleaning validation applications were conducted to expand theapplicability of the methodology based on QCL-GAP. Sample preparation was carried
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using direct smearing deposition with a micropipette tip, minimizing the contact with thesurface. In this case, the Teflon stub was not required. We analyzed stainless steel plates
loaded with an API to look for similar substrates that represent surfaces inside a typicalpharmaceutical reactor contaminated with organic residues. The analytes under study
were [2-n-butyl-3-((2'-(1H-tetrazol-5-yl) biphenyl-4-yl) methyl)-1,3-diazaspiro (4,4)
non-1-en-4-one; the API] commonly known as irbesartan (IRBS) and RDX. Our goal wasto develop and analyze standards that represent real-life surfaces to generate a robust
technology capable of detecting trace amounts of analytes.

Fig. 12.5. White-light micrographs of ABS and Al substrates loaded with HEs used as standardsfor QCL-GAP validation. (a) Set of ABS substrates with HEs at various surface concentrations;(b) the second set of Al substrates loaded with the HEs.

Fig. 12.6. White light micrographs of JHU standards. Al brushed substrates are loadedwith C-4 HE in a 0.1 - 10.0 ug/cm2 range.

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12.5. Trace Detection Analyses Using QCL-GAP
12.5.1. Detection of HEs
When detecting HEs and APIs, the vibrational response from the QCL-GAP is used toachieve a functional group characterization of the target molecules. There are several steps
to follow: instrument performance check, spectra acquisition of the substrate, and spectra
acquisition of the analyte. A spectrum of solid potassium bromide (KBr) is performed tocheck the system stability determining the variation in the noise across the entire
spectrum. This process is denoted as acquiring the blank or the 100 % line (100 PC). To
obtain the substrate spectrum without the analytes, a KBr spectrum was used as areference. Fig. 12.7 shows typical spectra of the substrates without HE.

Fig. 12.7. 100 % line and reflectance spectra for Al and ABS substrates without the HE.

A total of 24 spectra were acquired for each sample. The obtained spectra werecharacterized by identifying the vibrational signals of the HEs by comparing them withthe ab initio computed MIR spectrum (in the case of Tetryl) or with the measured MIR
spectra for the neat substances (RDX and PETN). The IR spectrum of Tetryl was
simulated using OPT+FREQ-job setup in Gaussian-09 (Gaussian, Inc., Wallingford,
Connecticut, USA). The spectrum was generated using Density Functional Theory
6-311+G(2d,p) basis set and B3LYP hybrid functional. The aliphatic explosives RDX andPETN were synthesized in the lab, and the spectra were acquired using a small sample tocompare with the primary standards from the MoDEx program. Fig. 12.8 shows anaverage of the QCL-GAP reflectance spectra obtained for each of the HEs deposited on
ABS and Al. As mentioned before, signals can be affected by substrate roughness (ABS),but this must be taken into account. Also, RDX and PETN were synthesized in ourfacilities impurities could be present, affecting the results. Tables 12.1 and 12.2 show theidentification of the characteristic signals for each HEs.
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Fig. 12.8. Normalized MIR reflectance spectra for (a) RDX (b) Tetryl and (c) PETN.
Table 12.1. QCL reflectance signals for the HEs deposited on ABS substrates.
HE/ABS
TetrylRDX

PETN

Experimental band position(cm-1)
954
1312
1348
1149
1359
1080
942
1370
1171

Reference band
position (QCL, cm-1)
954
1287
1342
1148
1349
1093
954
1369
1176

Vibrational mode
Sym. ring str.
N-NO2 str.
N-NO2 sym. str.
CNstr./CH3 bend
NO2 sym. str.
CN str./ring str.
CH2 + CCC def.
CH2 wag + C5 sk. +NO2 rock

The study was extended by including the APL-JHU standards to compare two QCL
acquisition modes. The first is denoted as point detection. The second approach was basedon the QCL-GAP arrangement, where the optical setup allowed a larger sampling area
and a faster and more accurate analysis. Fig. 12.9 shows the two modes studied.
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Table 12.2. QCL reflectance signals for the HEs deposited on Al substrates.
HE/Al
TetrylRDX

PETN

Experimental band position(cm-1)
1144
1337
1075
954
1310
1351
955
1372
1175

Reference band
position (QCL, cm-1)
1148
1349
1093
954
1287
1342
954
1369
1176

Vibrationalmode
CN str./CH3 bend
NO2 sym. Str.
CN str./ring str.
Sym. ring str.
N-NO2 str.
N-NO2 sym. str.
CH2 + CCC def.
CH2 wag + C5 +NO2 rock.

Fig. 12.9. QCL acquisition modes: (A) Single-path optical mount with Mini-QCLTM system,(B) QCL-GAP double-path optical mount.

The sample coupons from JHU were loaded on the center of the substrates with C-4. The
averaged normalized spectra from the samples using both systems are shown in Fig. 12.10.
To identify C-4, it was necessary to use RDX QCL reflectance bands as a reference, which
are located at: 1220 cm-1 (-C-N stretch), 1261 cm-1 (-NO2 stretch), 1310 cm-1 (-N-N
stretch), and 1370 cm-1 (-NO2 stretch). In the analysis, the most prominent signal of RDX,
-NO2, located at 1261 cm-1, was used as evidence of the presence of the explosive in C-4.
The low surface loadings on the JHU samples required a confirmatory technique tounderstand how C-4 crystals were distributed. Raman microspectroscopy (RmS) allowed496

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at the Grazing Angle Incidence

the identification of the HE by focusing the particles at the sub-micron scale. Micrographs
and spectra confirmed the presence of the crystals on the Al substrate for the 0.1 g/cm2
and 10 g/cm2 surface concentrations as shown in Fig. 12.11. The identification of the NO2 vibrational signatures were found as (vs) symmetric and (vas) antisymmetric modes
in 1309 cm-1 for C-4, RDX at 1306, and 1304 cm-1. An excellent correlation between thepeak location for vs-NO2 of RDX and C-4 can be observed when comparing the RmS
spectra of RDX and C-4 in Fig. 12.12. Antisymmetric vibrations of -NO2 are shown at
1536 cm-1 for RDX and 1527 cm-1 for C-4. RmS confirmed the presence of this analyte on
the surface, which was previously detected using both QCL setups.

Fig. 12.10. Normalized spectra of C4 deposited on Al substrate. (A) System 1; (B) System 2.

Fig. 12.11. Micrographs of C-4 on Al: (a) 0.107 μg/cm2; (b) 1.074 μg/cm2 and (c) 11.096 μg/cm2.

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Fig. 12.12. Raman spectra of C-4 deposited on Al substrates. RDX and substrate.

12.5.2. Detection of API
The QCL-GAP back reflection setup was also used to detect irbesartan (IRBS, API,deposited on stainless steel substrates. RDX was also detected on the same substrate tocompare the methodology's performance, but our primary focus in this section will be
IRBS. As part of the data analysis, six (6) vertical regions were used following theelliptical illumination image formed by the GAP on the sample plane. Figs. 12.13a and12.13b illustrate the diffuse reflectance spectra for the concentration loadings of bothanalytes.
Band assignment for functional group characterization was achieved by reducing thespectral range to 1200 to 1428 cm-1. IRBS showed significant signals at 1236 cm-1
tentatively assigned to δNNH + νNN + νC-C bridge bond in biphenyl and at
1337.11 cm-1 corresponding to CH2 wagging were observed. Interference patterns present
in this region are shown in the clean substrates and on the samples. These patterns will befurther discussed in Section 12.6.3.

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Fig. 12.13. (a) Comparison of average spectra of clean SS substrate RDX/SS (~16 ng/cm2),
IRBS/SS (~60 ng/cm2) and their reference QCL spectra; (b) magnification of spectra(1200-1428 cm-1).

12.6. Improving Detection Performance
12.6.1. Development of Spectral Library
Instrumentation for vibrational spectroscopy, such as infrared and Raman, has evolved inrecent years. IRS has been used in academic, research, and industrial applications and has
become an alternative analytical technique in fieldwork. These developments are due totechnological advances and the miniaturization of equipment. The latter has led to smaller,easier to use, smarter devices, and faster data processing. A good example of these are the
QCL based portable and handled spectrometers. However, to achieve quick and timely
decisions during field analyses, a suitable data processing analysis of the acquired spectra
is necessary. To answer some of the following questions: is a target analyte present?
(i.e. detection) Is this material what I think it is ? (i.e. confirmation) What is this?
(i.e. identification) To which group does it belong? (i.e. classification) How much of thisis present? (i.e. quantification) [17]. Various algorithms that allow detection,
confirmation, classification, identification, and quantitation of a target are used to carryout this task. All these algorithms perform a comparison between the target spectra (test)499

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and the spectra of known compounds stored in a spectroscopic library. The simplest
spectral comparison is based on calculating the hit quality index (HQI) values [18]. The
HQI is a numerical quantity that indicates the correlation between two spectra and has
been widely used in spectroscopy to indicate the degree of spectral matching in library
searches [19-24]. In the process, a spectrum of an unknown compound is compared withall spectra of known compounds in a library, and the best match is determined based on
the calculated HQI values. HQI values are calculated using different algorithms, but themost commonly used are spectral correlation algorithms. A spectral correlation algorithm
utilizes the Pearson product-moment correlation coefficient, rxy, which is a measure of thestrength and direction (Euclidean distance) of the linear relationship between two
variables. This parameter is defined as the covariance of the variables divided by the
product of their standard deviations. In this case, the spectral correlation algorithm isapplied between two spectra: a reference (spectra form library) and an unknown spectrum
to be identified [20-23]. The rxy values are calculated using Eq. (12.1):
𝑟𝑥𝑦 =

1
𝑛

∑ 𝑥𝑖 𝑦𝑖 − (∑ 𝑥𝑖 )(∑ 𝑦𝑖 )
1
𝑛

2

1
𝑛

2

1⁄2

[{∑ 𝑥𝑖2 − (∑ 𝑥𝑖 ) }{∑ 𝑦𝑖2 − (∑ 𝑦𝑖 ) } ],(12.1)

where 𝑥 and 𝑦 represent the spectral responses of the reference spectrum and an unknown
spectrum, respectively, measured at the ith wavenumber for a set of n corresponding
wavenumber points; in this case, the HQI assumes values between +1 and -1. An HQI
value of +1 is obtained when the spectral similarities are maximum (the unknown
spectrum is identical to a library spectrum), -1 when the spectral similarities are inverted(the unknown spectrum is identical to a library spectrum but with peaks inverted withrespect to the library spectrum), for example, when the transmittance spectrum iscompared to an absorbance spectrum, and 0 when there is no spectral similarity. Often,
the values are rescaled to values between 0 and 1, taking the square of r (r2) [22, 23] toyield positive values between 0 and 1. The spectral correlation method can be used inalgorithms for identification or verification. The identification mode compares a test
spectrum to all of the spectra in a reference library and ranks the library spectra using an
HQI value. In identification mode, the library spectrum that gives the highest HQI value
identifies the most probable composition of the test material. The verification mode ismore common in the pharmaceutical industry (also in national defense and forensic
science applications). It is widely used to test incoming pharmaceutical raw materials andexcipient to verify if a compound is an explosive or a controlled substance. The
identification mode compares each sample spectrum acquired to all of the library spectra.
The verification mode operates by comparing the spectrum of each sample to apredetermined library reference spectrum. Verification tests use a minimum HQI
threshold to assign a “Pass” or “Fail” determination to the sample under study. A typicalminimum HQI threshold for a “Pass” is 0.85 [22].
This section illustrates the usefulness of the spectral correlation method, calculating the
HQI value for the identification of HE deposited on metallic surfaces. The explosives were
deposited on an Al plate, using the smearing technique (already described). Then, the QCL
reflectance spectra were recorded using back-reflectance (point detection setup). The
spectra were divided into two groups: calibration spectra and test spectra. The respective
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HQI value for each explosive tested was calculated. Ten explosives were analyzed, as
shown in Table 12.3. The analyzed compounds correspond to aromatic explosives (TNT,
2,4-DNT, p-nitrotoluene (p-NT), 3,4-dinitrotoluene (3,4-DNT), and 2,4-dinitrophenol(2,4-DNP)), aliphatic explosives (PETN and RDX), and peroxide explosives (TMDD,
HMTD, and TATP), Fig. 12.14 shows the spectra of HEs used.
Table 12.3 shows the spectral correlation coefficients for unknown spectra of HE
deposited on Al substrates. Before calculating the r values, a spectral normalization was
performed on the entire spectral range of 1000-1600 cm-1 for both the unknown(measured) and library spectra (measured on Al substrates). Normalization of a spectrum
for library searching is a two-step process, as recommended by ASTM E2310-04 (2009):“Standard Guide for Use of Spectral Searching by Curve Matching Algorithms with Data
Recorded Using MIR Spectroscopy.” [18] First, the minimum spectral response value inthe selected spectral range is subtracted from the complete spectral response in the samerange. The resulting values are then scaled by dividing by the maximum value in that
range. The net result is a spectrum in which the minimum intensity value is zero (0) andthe maximum value is one (1). It is generally accepted to consider a spectrum of an
unknown compound similar to a spectrum stored in the library when the spectral
correlation coefficient is larger than ~0.85. However, the spectral resolution of IR spectra(reference and target), the shift of baseline, and the sample preparation method's
difference may be considered factors affecting correlation coefficients. In these cases, an
HQI equal to 0.8 may be sufficient to confirm a questioned spectrum. As is shown in
Table 12.3, spectra were correctly identified when QCL spectroscopy coupled with
spectral correlation algorithms were used. In general, all the calculated HQIs presented
high values, allowing to confirm that the questioned spectrum belongs to an HE.
Table 12.3. Values of spectral correlation coefficients (HQI) for spectra of HEs using QCL.

LibraryspectraTNT
2,4 DNT
pNT
3,4 DNT
2,4 DNP
PETN
RDX
TMDD
HMTD
TATP

TNT
1.0
0.9
0.7
0.6
0.4
0.0
0.1
0.0
0.7
-0.2

2,4
DNT
0.9
1.0
0.6
0.6
0.4
0.0
0.0
0.1
0.6
-0.2

pNT
0.7
0.7
0.9
0.3
0.4
0.2
0.4
0.1
0.6
0.1

HQI/ Spectral Correlation
Spectra Questioned
3,4
2,4
PETN RDX TMDDDNT DNP
0.8
0.6
0.0
0.1
0.2
0.8
0.6
0.0
0.0
0.2
0.5
0.6
0.1
0.3
0.5
0.8
0.2
-0.1
-0.2
0.1
0.2
0.8
-0.1
0.1
0.3
-0.1
0.0
1.0
0.4
0.4
-0.1
0.2
0.5
1.0
0.6
0.0
0.1
0.3
0.3
0.8
0.7
0.5
0.0
0.2
0.3
-0.5
0.3
-0.2
0.2
0.2

HMTD

TATP

0.6
0.5
0.6
0.1
0.2
-0.1
0.2
0.1
0.8
0.2

-0.1
0.0
0.4
-0.3
0.2
0.9
0.2
0.1
0.1
0.9

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Fig. 12.14. IR Spectra of HEs tested for calculating HQI values using spectral correlation.

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12.6.2. MVA Routines
MVA offers new possibilities for data treatment in many fields of science, engineering,
agriculture, or even in the arts for understanding and helping to express the dependence
and independence of the variables in a data set. When conducting a study, 30 % effort isdirected to mathematical and statistical analysis to evaluate the data. How MVA plays avital role in detecting chemical threats and cleaning validation of vessels inpharmaceutical industries? In the detection arena, the research has migrated from a
univariate analysis to consider all the obtained data to decode a pattern fully. In cleaningvalidation, it is essential to have a panorama of the possible impurities that may appear
when conducting this procedure and MVA permits to control it. In both situations, MVA
can be used as a qualitative analysis tool or for quantitative analysis, which are different
processes of treating the data depending on the need in the study. Applying MVA withdata obtained from QCL helps us elucidate behaviors that we could not observe with the
naked eye. This analysis provides more relevant chemical information from a data set, andit can help us reduce the dimensions in a data set to identify patterns in the data. Today
several statistical programs allow better interpretation of the data on a larger scale, making
the process more user-friendly. This approach provides extensive research of vibrational
peak analysis of molecules in the MIR region, seeing where there is more significant
variation, and identifying patterns that discriminate the data.

12.6.2.1. Data Preprocessing
How can we be sure that we are analyzing the chemical effects of the data? It is crucial
that when the vibrational data is obtained, before performing the MVA, some
pretreatments (scatter corrections) are applied. These must be performed to ensure that thevariations observed in the mathematical models are from the chemical effects of theanalyte and not from a physical influence that may appear. Baseline distortion, dispersion,and noise are typical physical effects that can affect spectral data. These effects are particle
size, shape, surface roughness, and others may mask the samples' vibrations' chemical
aspects [25]. These effects can be dominant in the spectral data, which can invalidate
subsequent steps of the MVA models. For this reason, one should consider using some of
the following preprocessing steps to reduce these effects:a) Normalization
Normalization helps us reduce the baseline effect in the spectra, enabling all data sets to
be on the same scale. It allows seeing the interactions that would lead to spectral variations
not related to the composition of the samples. This pretreatment places the data on a scaleof 0 to 1, according to their respective intensities. If one desires to take the data to aquantitative MVA analysis, special care must be taken since the proportions in data canchange. This can lead to having concentration values different from what they are.

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Advances in Optics: Reviews. Book Series, Vol. 5b) Savitzky-Golay Algorithm for Data Smoothing
Noise can be a factor that interferes with the variations that we need to observe in thevibrational data. This algorithm allows estimating the spectrum's shape using apolynomial fit of a specific rank and degree.
c) Multiplicative Scatter Correction (MSC) and Standard Normal Variate (SNV)
These two also help with the baseline distortion and data dispersion that appear in someof the spectra. MSC can estimate the coefficient by regressing the spectrum to correct witha reference. SNV is a scaling algorithm, where the average of a data set is subtracted, andeach spectrum is divided by the standard deviation of the data set. It should be consideredthat these should only be applied to spectra that present physical information influenced
by these multiplicative effects. If misused, they can provide erroneous results or when
performing predictive MVA models [25].
12.6.2.2. Data Exploration
Once the spectral data is prepared, what would be the next step? When conducting an
MVA, it is essential to visualize the behavior of the spectral data to obtain a correctinterpretation of the variation that the study is trying to explain. With the help of what is
known as principal component analysis (PCA), an exploratory data analysis (EDA) canbe done. EPA is usually used for qualitative data analysis. PCA models provide
information based on the score and the loadings plots. The variation of the data isdecomposed in the form of principal components. The selection of the principal
components falls into the direction that has the most variation, and this is the principal
component 1 (PC 1). PC 2 is selected orthogonal or perpendicular to PC 1. This procedure
is carried out for all the PCs required to explain all the variations in the data. This isrepresented on a score plot as the distance from the mean along a PC for a sample. The
loadings plot provides information on the variables of the data. In spectral data, thevariables will be the wavenumber (cm-1) to see how much they contribute to the PC bycorrelation.
Before applying a PCA model, the data must be subjected to other treatments that help
visualize them, as seen in Table 12.4. This is done to determine how the spectral data
behaves and see which preprocessing step provides more information to interpret. The
best pretreatments of the spectral data are usually the first (1SG) and second (2SG)
Savizky-Golay derivatives. Derivatives are the most common treatments applied tospectral data to resolve peak overlapping (thus enhancing spectral resolution) and toeliminate constant and linear baseline drifts between samples. One disadvantage is thatapplication of derivatives could increase noise, but this can be evaluated using theloadings plots to determine which variables affect the variation on the data.
How do we know which of the pretreatments is ideal for my data? Selection of the
optimum preprocessing step (or combination of them) can be based on the patterns
observed in the data when PCA is performed with that pretreatment (s). The objective is504

Chapter 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy
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to place the data in clusters with similarities but differences compared to other clusters.
There are variations in the clusters’ characteristics. This helps to find an ideal model that
explains the topic of interest in the research. One also constantly seeks to obtain the mostsignificant interpretation of the data with the least number of PCs.
Table 12.4. Set of pretreatments that can be used in chemometric routines.

SNV
MSC
1SG
2SG

Pretreatments
SNV + 1SG
SNV + 2SG
MSC + 1SGMSC + 2SG
1SG + SNV
1SG +MSC
2SG + SNV
2SG + MSC

How do the PCs correlate with spectral data? Once the characterization of the data iscarried out and the ideal preprocessing step for the PCA is identified, one can further
explore what is happening with this data. An example of this can be seen in the PCAs
shown in Fig. 12.15, where the treatment selected was 1SG + SNV. In these PCAs, theobjective was to differentiate between the HEs deposited on ABS and Al, where the main
components PC 1 and PC 2 showed a higher percentage of variation for both substrates.
As seen in both PCAs, the clusters are from the differences between the vibrationalsignatures of PETN, RDX, and Tetryl. Another way to understand where the mostremarkable variation occurs in the spectral data is by viewing the loadings plots. The
loadings are the map of the variables; in this case, the wavenumber (cm-1). The vibrationalsignals from each HE can be evaluated on the loading plot to fully corroborate that thevariation analyzed is the one from each one of the vibrations of the nitro groups. The
correlations are shown in Fig. 12.16.

Fig. 12.15. Scores plot for PETN, RDX, and Tetryl on (A) ABS and (B) Al substrate in termsof PC-2 versus PC-1. The 95 % confidence level used in the analysis is also shown (dotted trace).

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Fig. 12.16. Scores plots for PC-1 and PC-2 for HEs/ABS performed for data collectedin the range of 833 to 1428 cm−1.

12.6.2.3. Data Discrimination
QCL spectral data can be further investigated by using MVA prediction and classification
models. Prediction models are known as Soft independent modeling by class (SIMCA) or
PCA projections. SIMCA allows objects to display intrinsic individualities as well as
common pattern characteristics. However, only the common properties of the classes aremodeled by PCA [26]. This model takes advantage of similarities among members of thesame class where a PCA is built for each class where samples well described by the modelare "accepted," and others are "rejected" or not classified. The model generates aclassification table where the samples from each of the model's classes are listed.
Can a quantitative analysis be done using spectral data from QCL? Partial Least Squares(PLS) is an excellent tool to analyzed quantitative data. As seen in Fig. 12.17, models likethis can detect explosives at low concentrations when using these multivariate routines.
For this reason, regardless of how the sample is created, the combination of PLS models
with data obtained from QCL helps the detection of explosives, making it have much
lower detection limits. Likewise, there are other mathematical models such as support
vector machines (SVM or VM), k-nearest neighbors (KNN), partial least-squarediscriminant analysis (PLS-DA), linear discrimination analysis (LDA), among others thatcan be applied to generate classification models to assist in the detection analysis.
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at the Grazing Angle Incidence

Fig. 12.17. PLS regression plots of predicted vs. measured surface concentrations for C-4
deposited on Al substrate: (A) System 1. (B) System 2. Spectral range, 950 -1374 cm-1.

12.6.3. Fourier Transform Data Preprocessing Algorithm
When the QCL-GAP system is well aligned, the interaction of the MIR light with thesample can produce interference patterns. At first, these patterns were considered a
problem in the acquired vibrational response, but it was noticed that they displayedchanges in the patterns observed when the analyte was deposited on the surface. The
preprocessing step to determine if these patterns could be used for identification andquantification analysis using chemometric methods is fast Fourier transform (FFT). FFT
is a tool used to extract robust and accurate information from reflectance spectra. Thisalgorithm transforms a function from the time domain to the frequency domain, andconversely. The result of applying the FFT is a complex function that consists of animaginary part [Im(n)] and a real part [Re(n)]. Its magnitude is expressed as the absolutevalue of lz(n)l, and “n” represents the number of points used. This information was used
for discrimination and quantification using PLS-DA and PLS. Figs. 12.18A and 12.18B
show the FFT preprocessing steps applied to the clean SS substrate when the GAP mirrors
are aligned and misaligned.

Fig. 12.18. (a) Representation of interference patterns; (b) applying FFT preprocessing,and representative spectral patterns obtained after applying FFT preprocessing.

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12.7. Conclusions
A QCL-GAP system was presented in this chapter as a novel technique for detecting APIsand HEs. Our approach demonstrated the system's feasibility in detecting HEs for national
defense and security applications and cleaning validation in pharmaceutical batch reactors
applications. The GAP setup presents many advantages that surpass common challenges
posed by conventional point detection systems: a double-pass (double reflectance) systemand the RAIRS advantage, known as the most sensitive MIR technique. Detection quality
can be affected by the deposition technique selected and the substrate properties.
Deposition techniques can affect the vibrational response in the analysis, mainly due tothe lack of homogeneity and the time of residence of the analyte. In terms of the substrate,
non-reflective and opaque surfaces produce a diffuse reflectance signal, while reflective
substrates with high homogeneity produce specular reflectance. Coupling chemometricroutines (MVA) to the QCL-GAP detection system improves the performance of theanalysis. FFT-derived functions using MATLAB were tested to evaluate the complete
spectral range, including the regions where interference patterns were present. These
patterns determined the alignment of the system. The QCL-GAP described herein will
have a far-reaching impact due to its high specificity and high sensitivity results whenanalyzing low concentrations of analytes.

Acknowledgments
Support from the US Department of Homeland Security under Award Number 2013-ST061-ED0001 is acknowledged. However, this document's views and conclusions are thoseof the authors. They should not be considered a representation of the official policies,
either expressed or implied, of the US Department of Homeland Security. The US Naval
Research Lab prepared the sieved sample coupons mentioned in Section 12.4.3 underDepartment of Homeland Security Inter-Agency Agreement HSHQPM-15-X-00166.

References[1]. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, A. Y. Cho, Quantum cascadelaser, Science, Vol. 264, Issue 5158, 1994, pp. 553-556.
[2]. Y. Yao, A. J. Hoffman, C. F. Gmachl, Mid-infrared quantum cascade lasers, Nature
Photonics, Vol. 6, Issue 7, Jul. 28, 2012, pp. 432-439.
[3]. S. P. Hernández-Rivera, L. C. Pacheco, Grazing Angle Probe Mount for Quantum Cascade
Lasers, Patent, Ser. #62/587,557, 2017, submitted.
[4]. W. Zhou, D. Wu, R. McClintock, S. Slivken, M. Razeghi, High performance monolithic,broadly tunable mid-infrared quantum cascade lasers, Optica, Vol. 4, Issue 10, Oct. 2017,
pp. 1228-1231.
[5]. A. M. Colón-Mercado, K. M. Vázquez-Vélez, E. Caballero-Agosto, V. Villanueva-López,R. Infante-Castillo, S. P. Hernández-Rivera, Detection of high explosives samples depositedon reflective and matte substrates using mid-infrared laser spectroscopy at the grazing angle
of incidence assisted by multivariate analysis, Opt. Eng., Vol. 59, Issue 9, Aug. 2020, 092011.
[6]. O. M. Primera-Pedrozo, Y. M. Soto-Feliciano, L. C. Pacheco-Londoño, S. P. HernándezRivera, Detection of high explosives using reflection absorption infrared spectroscopy withfiber coupled grazing angle probe/FTIR, Sens. Imaging, Vol. 10, Issues 1-2, 2009, pp. 1-13.

508

Chapter 12. Mid Infrared Quantum Cascade Laser Reflection-absorption Spectroscopy
at the Grazing Angle Incidence[7]. O. M. Primera-Pedrozo, L. C. Pacheco-Londono, L. F. De la Torre-Quintana,
S. P. Hernandez-Rivera, R. T. Chamberlain, R. T. Lareau, Use of fiber optic coupled FT-IR
in detection of explosives on surfaces, Proceedings of SPIE, Vol. 5403, 2004.
[8]. M. L. Ramírez-Cedeño, N. Gaensbauer, H. Félix-Rivera, W. Ortiz-Rivera, L. PachecoLondoño, S. P. Hernández-Rivera, Fiber optic coupled raman based detection of hazardous
liquids concealed in commercial products, Int. J. Spectrosc., Vol. 2012, 2012, 463731.
[9]. T. Vankeirsbilck, et al., Applications of Raman spectroscopy in pharmaceutical analysis,
TrAC – Trends Anal. Chem., Vol. 21, Issue 12, 2002, pp. 869-877.
[10]. D. T. D. Childs, R. A. Hogg, D. G. Revin, I. U. Rehman, J. W. Cockburn, S. J. Matcher,Sensitivity advantage of QCL tunable-laser mid-infrared spectroscopy over FTIR
spectroscopy, Appl. Spectrosc. Rev., Vol. 50, Issue 10, 2015, pp. 822-839.
[11]. C. Gmachl, F. Capasso, D. L. Sivco, A. Y. Cho, Recent progress in quantum cascade lasers
and applications, Reports Prog. Phys., Vol. 64, Issue 11, Oct. 2001, pp. 1533-1601.
[12]. J. Faist, et al., High power mid-infrared (λ∼5 μm) quantum cascade lasers operating above
room temperature, Appl. Phys. Lett., Vol. 68, Issue 26, Jun. 1996, pp. 3680-3682.
[13]. F. Capasso et al., New frontiers in quantum cascade lasers and applications, IEEE J. Sel. Top.
Quantum Electron., Vol. 6, Issue 6, 2000, pp. 931-946.
[14]. R. Smith, C. Peters, H. Inomata, Historical background and applications, in Supercritical
Fluid Science and Technology, Vol. 4, Elsevier B. V., 2013, pp. 175-273.
[15]. A. Kumar, Object appearance and colour, Book Chapter in Principles of Colour andAppearance Measurement Object Appearance, Colour Perception and Instrumental
Measurement, Elsevier, 2014, pp.53-102.
[16]. S. Chong, B. Long, J. K. Maddry, V. S. Bebarta, P. C. Ng, Acute C4 ingestion and toxicity:
Presentation and management, Cureus, Vol. 12, Issue 3, Mar. 2020, e7294.
[17]. C. Gardner, R. L. Green, Identification and confirmation algorithms for handheld
spectrometers, in Encyclopedia of Analytical Chemistry, John Wiley & Sons Ltd, 2000,
pp. 1-18.
[18]. Standard Guide for Use of Spectral Searching by Curve Matching Algorithms with Data
Recorded Using Mid-Infrared Spectroscopy, https://www.astm.org/DATABASE.CART/HISTORICAL/E2310-04R09.htm[19]. K. Banas, et al., Multivariate analysis techniques in the forensics investigation of the postblast
residues by means of Fourier transform-infrared spectroscopy, Anal. Chem., Vol. 82, Issue 7,
Apr. 2010, pp. 3038-3044.
[20]. E. C. Wong, J. C. Reid, Data-reduction and -search system for digital absorbance spectra,
Appl. Spectrosc., Vol. 20, Issue 5, Sep. 1966, pp. 320-325.
[21]. K. Tanabe, S. Saëki, Computer retrieval of infrared spectra by a correlation coefficientmethod, Anal. Chem., Vol. 47, Issue 1, Jan. 1975, pp. 118-122.
[22]. J. D. Rodriguez, B. J. Westenberger, L. F. Buhse, J. F. Kauffman, Quantitative evaluation of
the sensitivity of library-based Raman spectral correlation methods, Anal. Chem., Vol. 83,
Issue 11, Jun. 2011, pp. 4061-4067.
[23]. S. Lee, H. Lee, H. Chung, New discrimination method combining hit quality index based
spectral matching and voting, Anal. Chim. Acta, Vol. 758, Jan. 2013, pp. 58-65.
[24]. FT-IR Search Algorithm – Assessing the Quality of a Match,https://www.spectroscopyonline.com/view/ft-ir-search-algorithm-assessing-quality-match[25]. J. W. Jin, et al., Quantitative spectroscopic analysis of heterogeneous mixtures: The
correction of multiplicative effects caused by variations in physical properties of samples,
Anal. Chem., Vol. 84, Issue 1, Jan. 2012, pp. 320-326.
[26]. K. H. Esbensen, An introduction to Multivariate Data Analysis, including Process Analytical
Technology (PAT) and Quality by Design (QbD), Camo, Oslo, 2018.

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Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization,
Wavelength Switching, and Wavelength Tuning of Lasers

Chapter 13

Advances in Linewidth Compression,
Linewidth Measurement, Noise
Characterization, Wavelength Switching,and Wavelength Tuning of Lasers
Paul Ikechukwu Iroegbu, Min Liu and Tao Zhu1

13.1. Introduction
Light is vital to the human race and our persistent attempt to understand the world aroundus. Our earliest efforts to comprehend light dates far back to the dawn of civilization. If
we characterize the existence and evolution of light ranging from antiquity until the
middle of the 19th century, its classification is appropriately recognized as incoherent lightsource. Such a phrase in classification is adopted because it consists of the sun, candlelight, sodium lamp, or light bulb. In the middle of the 19th century, a new light source
called Laser (Light Amplification by Stimulated Emission of Radiation), which is acoherent light source, was invented and ultimately ended up as one of the greatest
inventions of the 20th century [1]. The concept of laser acting as a light source, as
introduced in 1958 by Townes and Schawlow, has revolutionized the world we live in by
actively participating in a vast range of modern applications. These applications are ultrahigh geographical monitoring and surveying with detection ranges as high as pico-meters,
highly delicate eye surgery, security, and surveillance devices capable of measuring
vibrations. This chapter presents the fundamental dynamics of lasers, their classification,and their applications to optical fiber communication and optical fiber sensing systems.
Furthermore, we present the advances done on lasers regarding linewidth compression,
linewidth measurements, noise characterization, wavelength switching, and wavelengthtuning. Lastly, we present our predictions on the future directions and development oflasers.

Paul Ikechukwu Iroegbu
School of Microelectronics and Communication Engineering, Chongqing University, Chongqing, China

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13.2. Laser Dynamics
The term laser is an acronym that stands for (Light Amplification by Stimulated Emissionof Radiation). This is because the output light signal from a laser is amplified in a cavity
regarded as the gain medium, which is fundamentally saturated by atoms or molecules.
The amplification of light signal in the gain medium is referred to as excitation, whereas
the reverse process, referred to as de-excitation, leads to photons' release. A propagational
encounter between these released photons and other excited atoms inevitably stimulates
those excited atoms/molecules sufficient enough to release their stored energy in a chain
reaction manner that creates an avalanche emission of radiation in the form of photons
with equivalent wavelength. In the knowledge that these released photons are ofequivalent wavelength and monochromatic (i.e., stimulated by the same photons
propagating nearby), it is further understood that these emitted photons are as well inphase with each other. This phenomenon establishes what is globally regarded andaccepted as a coherent light source and consistent across all laser types. Lasers generally
consist of three core elements; pump, gain medium, and resonant cavity. The difference
in types of lasers is mainly associated with the difference in the gain medium. Since lightsources are classified into coherent light sources and incoherent light source, candle lights
and sunlight are practical examples of incoherent light source [2].

13.3. Classification of Lasers
As a result of Townes and Schawlow's laser concept, various lasers have sprouted out over
the years with a broad range of physical and operating parameters. Provided lasers areclassified according to their gain medium's physical state, the description of lasers into
various classes such as solid-state lasers, liquid state lasers, and gas lasers is adopted.
However, besides these renowned classifications of lasers, other rather special cases exist
to classify lasers, such as free-electron lasers whose gain medium consists of free electrons
propagating past a periodic magnetic field relativistic velocity. In addition, more specialclassifications of lasers are infrared lasers, visible lasers, UV (Ultra-Violet) lasers, and
X-ray lasers which are resultants of cases where lasers are characterized by the wavelengthof their emitted radiation [3]. Furthermore, a semiconductor laser and fiber laser are bothclassified as independent classes of their own. An excellent example of a semiconductor
laser is a Distributed Feedback (DFB) laser. On the other hand, a perfect example of arare-earth-doped fiber laser is an EDF (erbium-doped fiber).

13.4. Lasers and Systems
In this section, we present semiconductor laser and fiber laser. We also present how thelaser parameters of semiconductor lasers and fiber lasers influence their applications incommunication and sensing systems. Various types of lasers are used as a light source ina large array of modern-day system applications. However, we will emphasize
semiconductor lasers and fiber lasers and their advances and applications in opticalcommunication and optical sensing systems. Our distinct choice and focus on theadvances relating to semiconductor lasers, and fiber lasers are centered on the fact that512

Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization,
Wavelength Switching, and Wavelength Tuning of Lasers

there has ever remained an essential device in communication and sensing systems. Thisis because lasers lasing at single longitudinal mode (SLM), whose laser linewidth iscompressed, low phase or frequency noise, low relative intensity noise (RIN), andwavelength-tunable or switchable, are critical to the performance level of these systems.
For example, the quality of the light source employed in terms of its compressed spectral
linewidth, phase or frequency noise, RIN, and wavelength-tunable range or switching has
significant impacts on an optical fiber communication system, such as its bandwidth (i.e.,channel capacity), signal-to-noise ratio (SNR), and sensitivity. Likewise, it also affects
the performance of optical fiber sensing systems, such as their detection range, sensitivity,
and accuracy. Hence, the development of these systems is tremendously accelerated by
the advances in these lasers.
13.4.1. Semiconductor Lasers
DFB lasers, which were simultaneously invented at three different industrial labs, MIT
Lincoln Laboratory, IBM, and General Electric, were inspired by the desire to investigate
a relatively new laser gain medium. These lasers have been highly desirable due to theirsmall size, low cost, stable single-longitudinal mode operation, and easy compatibility
with other devices [4]. Semiconductor optical amplifier (SOA), on the other hand, wasdiscovered around the early 1960s, wherein a gallium arsenide laser diode was found to
be a suitable laser diode. It was also discovered to be a good amplifier of infrared light,
provided its reflectivity at the facets is decreased with an anti-reflection coating material.
These lasers have been highly desirable due to their fast gain dynamics and simultaneous
wavelength operation. The gain medium commonly used in an SOA and a DFB are
indium, gallium, arsenide, phosphide (InGaAsP). SOA is often employed to amplifytransmitted optical light signals by stimulated emission in a fiber waveguide, which
attenuates long fiber length spans. Depending on the need, an SOA can be adopted as a
post-amplifier, in-line amplifier, pre-amplifier, or a wavelength converter [4-6].
13.4.2. Fiber Lasers
The emergence and continuous advancement of fiber lasers whose active gain medium isa piece of optical fiber that is doped with a rare-earth element such as thulium, ytterbium,
erbium, etc., are associated with the progressive development of various manufacturing
technologies. For example, low-loss erbium-doped fiber (EDF) with a commendable
monochromaticity was successfully prepared by the SB Poole team at the University ofSouthampton, UK, in the 1980s [7]. Since then, it has received wide-spread applications
due to its overall merits, such as high conversion efficiency, low insertion loss, hightemperature resistance, high output power, and strong anti-electromagnetic interference
capabilities.
13.4.3. Lasers in Optical Fiber Communication Systems
A typical optical fiber communication system consists of a transmitter (laser light source),
medium (optical fiber), and a receiver (optical detectors). Semiconductor lasers and fiber513

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lasers are good examples of light sources in optical fiber communication systems. A laser's
linewidth, wavelength, phase or frequency noise, and relative intensity noise, have often
been of particular interest and high demand in academia and industry for opticalcommunication systems. This is because the laser light source's linewidth often affects the
bandwidth of the fiber. In addition, the wavelength determines the number of accessible
channels. The phase or frequency noise determines the long-term stability of the laser andthe detection accuracy of local oscillators (LO) in an optical receiver. Likewise, therelative intensity noise influences the long-term stability of the laser and the sensitivity ofthe optical receiver. Hence, stringent requirements on the parameters of the laser lightsource employed have ever remained increasingly dominant over the years. These
requirements are inspired by the advent of emerging and promising opticalcommunication technologies that strive to resolve the growing capacity demands of
optical networks in long-haul, metro, and local area networks. Some of these technologies
are wavelength division multiplexing (WDM), dense wavelength division multiplexing(DWDM), digital coherent technology, multi-level modulation formats, and spatial
division multiplexing (SDM).
13.4.3.1. Lasers and Optical Communication System Technological Evolution
Over the years, considering the growing demand for large transmission capacity, academia
and industry are actively seeking to develop a system whose transmission capacity spans
beyond 10-Gbits/s. These same systems will be capable of adopting advanced modulation
formats such as differential phase-shift keying (DPSK), differential quadrature phase-shift
keying (DQPSK) as opposed to an intensity modulation (IM) format. In order to meet fastgrowing internet traffic, a combination of advanced multi-level modulation formats in a
coherent WDM/DWDM system that utilizes a wavelength-tunable or switchable laser isrequired. For example, a wavelength tunable range of 100 nm, can guarantee a high-speed
fiber optic transmission system with bitrates that span over 100 Gbits/s in today's optical
networks [8]. Regardless of the emergence of various technologies to resolve issues
relating to high capacity demand, it is essential to note that a broad array, if not all of
today's optical networks, adopts a combination of these technologies to resolve the sameproblem. This is practically evident in today's 100 Gb/s optical transmission systems that
adopt a combination of advanced modulation formats such as quadrature phase-shift
keying (QPSK), which work together with polarization division multiplexing (PDM) [9].
Like optical transmitters, the most current focus is to actualize digital coherent receivers
that can handle bit rates as high as 400 Gb/s in the nearest future. The trade-off inrequirements between an optical transmitter and an optical receiver is this. In opticalreceivers, phase signals are detected or identified with a local oscillator, which practicallysuggests that a local oscillator's detection accuracy in an optical receiver is muchdependent on the phase noise of the laser. Furthermore, the feasibility of attaining
high transmission capacity typically requires a laser of low relative intensity noise.
This is because the RIN noise level of the laser affects the SNR of the system. Hence, theneed to obtain a tunable laser with a low phase or frequency noise and low RIN isparamount [10].
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Wavelength Switching, and Wavelength Tuning of Lasers

In regards to the latest advancement towards attaining massive data rate increase andbreaking the Shannon capacity limit, SDM is today's world-leading technology to achieve
this tremendous progress in a submarine network system. This is because SDM is the lastunexplored physical dimension in optical communication. It is merely the act of adopting
multi-core fiber (MCF) or a few-mode fiber (FMF) to exceed the expected capacity
crunch. However, little attention was paid to this promising novel technology over theyears due to the effectiveness of other emerging technologies such as WDM, multi-level
modulation formats, and digital coherent technology. These technologies collectively andindividually played a critical role in resolving issues relating to capacity demand. Moreso, as the increase in capacity demand continued to rise exponentially in the past, more
technologies to mitigate this issue were developed and employed, such as Erbium-doped
fiber amplifier (EDFA) [11] and PDM [12]. This ultimately limited the need to adopt
SDM.
More importantly, it is anticipated that the massive increase in demand for higher data
rates will directly supersede today's optical network's bandwidth capacity, which will
create that which is globally coined as 'capacity crunch' [13]. In the era of capacity crunch(i.e., around the year 2018), global commercial systems are expected to deliver bandwidth
capacity as high as 80 Tbit/s, which is way beyond the limit of today's single-mode fiber(SMF) based optical networks [14]. Consequentially, SDM, as an unexplored opticaldomain, is regarded as a promising technology and a monumental leap forward towards
resolving these capacity demands.
In retrospect to the previous discussion on modulation formats that modulates phase andthe amplitude to increase data rates significantly in an optical communication system that
adopts a wavelength-tunable or switchable laser, another spectral characteristic of thelaser in terms of a narrow linewidth is likewise essential. The need to modulate the phaseof the light source directly places stringent requirements on the laser linewidth. For
example, a communication system that adopts a multi-level advanced modulation format
such as a 16-state quadrature amplitude modulation (16-QAM) requires a laser linewidth
below 100 kHz. However, recent research proves that higher modulation formats incontrast to 16-QAM, such as 32-QAM and 64-QAM, need the MHz laser linewidth of alaser to be deeply compressed down to the single kHz level [15]. This is because narrowlinewidth lasers with suitable phase or frequency noise are vital in optical communication
systems. Such systems seek a light source of long coherence length or superior phasesensitivity to expand the capacity of long-haul, metro, and short-distance optical networks.
13.4.4. Lasers in Optical Fiber Sensing Systems
A typical optical fiber sensing system consists of a laser, transmission fiber, detectors, andother parts. Once again, semiconductor lasers and fiber lasers are good examples of lightsources in optical fiber sensing systems. The quality of the laser employed has often beenof particular interest and high demand in academia and industry for sensing systems. Thisis because the laser light source's linewidth often determines the measurement distance.
In addition, the wavelength determines the accuracy and precision of the detection
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process. The phase or frequency noise determines the stability and sensitivity of thesystem. Likewise, the relative intensity noise determines the accuracy of the sensing
system towards detecting and demodulating small signals.
Hence, stringent requirements on the quality of the laser light source employed have everremained increasingly dominant over the years. These requirements often influence theperformance of various emerging and promising optical sensing technologies such as
distributed acoustic sensing (DAS), distributed vibration sensing (DVS), distributed
temperature sensing (DTS), and distributed strain sensing (DSS). These technologies findversatile applications in the oil and gas industries, roads, bridges, borders, perimetersecurity, etc. In order to realize the sensitivity, resolution, accuracy, and sensing range ofthese technologies used in a distributed optical fiber sensing system, these technologies,
such as DAS, DVS, DTS, and DSS, functions alongside various optical techniques. Someof these techniques are phase-sensitive optical time-domain reflectometry (φ-OTDR),
optical frequency domain reflectometry (OFDR), Brillouin optical time-domainreflectometry (BOTDR), Brillouin optical time-domain analysis (BOTDA), and Raman
optical time-domain reflectometry (ROTDR). Furthermore, other sensing technologies
such as light detection and ranging (LIDAR), which is also regarded as a sub-field (i.e.,
optical remote sensing) within the broader field of optical sensing, also places stringent
requirements on the quality of the laser light source employed for various applications.
Some of these applications are found in unmanned driving, law enforcement,
environmental monitoring, etc.
13.4.4.1. Lasers and Optical Sensing System Technological Evolution
LIDAR supports autonomous (self-driving) cars that adopt automotive coherent LIDAR
systems during the cruise as a technology for object recognition such as pedestrians andother vehicles. Object recognition is feasible when the receiver of the LIDAR systemcharacterizes the intensity or frequency information corresponding to the reflected laser
signal [16, 17]. Narrow-linewidth wavelength-tunable lasers are essential to extend the
dynamic sensing range and sensitivity of coherent LIDAR systems in autonomousvehicles to over 400 m. Linewidths of 100 kHz are sufficient for LIDAR applications [18].
When a tunable laser is used in LIDAR, its wavelength needs to be continuously linearlyscanned to obtain beat frequency signals related to the state to be measured. The linearityof laser wavelength scanning affects the positioning accuracy of the object to be measured.
Suppose we categorize the desired sensing parameters that distributed sensing systems
intend to detect. In that case, we can appropriately obtain four categories of optical sensing
technologies corresponding to each desired parameter. These categories of sensing
technology include DAS [19], DVS [20], DTS [21], and DSS [22]. These sensing
technologies of DAS, DVS, DTS, and DSS corresponds to the desired sensing parameterof acoustic, vibration, temperature, and strain, respectively. In order to realize thesedesired sensing parameters, we need to utilize at least either one of the three types ofscatterings in the optical fiber of the distributed sensing system, namely Rayleigh,Brillouin, or Raman scattering.
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The Rayleigh scattering effect in an optical fiber of a distributed sensing system is mainly
associated with two techniques. These techniques are φ-OTDR [23-26] and OFDR
[27-29]. Consequently, the φ-OTDR is used to detect the dynamic change invibration and acoustic sensing. In contrast, OFDR detects the slow change in strain andtemperature. The Brillouin scattering effect is mainly associated with two techniques,
namely BOTDR [30-32] and BOTDA [33-35]. Consequently, the BOTDR and BOTDA
are used for detecting strain and temperature. The Raman scattering effect is mainly
associated with the ROTDR [36-37], which is used for detecting temperature. Therefore,a distributed sensing system's principle of operation or technique mainly includes theabove-listed φ-OTDR, OFDR, BOTDR, BOTDA, and ROTDR. These techniques areused to realize the various sensing technologies listed above by monitoring the physicalparameter that changes with time and space within an optical fiber. In fiber-based sensing
systems, the fundamental characteristics of an optical signal launched from a laser
and utilized to conduct sensing during transmission within the fiber are subjected tochanges induced by various parameters. These parameters. are pressure, temperature,polarization, rotation, displacement, frequency, strain, bending, and vibration, which
affect its light path.
Hence, academia and industry have stringent requirements on the quality of laser sources
used in distributed sensing systems. For example, in OFDR based sensing systems, theserequirements now demand laser linewidth less than 100 Hz to be adopted in scenarios
where the sensing fiber length is 80 km [38]. In addition, the coherence of the laser
linewidth directly characterizes the impact of phase or frequency noise of these systems,
which hampers the linearity of frequency sweeps. Therefore, low phase or frequency noise
lasers are desirable for such systems [21]. OFDR based sensing systems also strive to use
a tunable light source to tune the frequency wherein aspects relating to frequency tuning
range, frequency tuning accuracy, and frequency tuning time of the tunable light source
are considered. Therefore, to achieve high accuracy coherent detection and demodulationof the optical signals in light frequency domain-based systems, it is essential to ensurethat the laser possesses fast-periodic sweep frequency attributes. This suggests that themodulation rate of the laser should be higher, with a sweep speed of approximately
10 pm/μs [39]. Similar requirements are demanded as well in BOTDR based sensing
systems [40].
Furthermore, we generally require a low RIN laser when demodulating small signals inthe φ-OTDR based sensing system [20]. Likewise, suppose we intend to significantly
increase the sensing range of a BOTDA based sensing system by employing Raman
amplification. In that case, a low RIN laser is desirable [41]. The ROTDR technique can
comfortably and independently conduct distributed temperature sensing. However, forcase scenarios where a multi-parameter hybrid-based sensing system that can conductsimultaneous temperature and acoustic sensing with the ROTDR technique andthe φ-OTDR technique, respectively, a laser linewidth less than 1 kHz is often required.
Such a hybrid sensing system utilizes an MCF fiber that contains SDM reflectometers forsensing [42].

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13.5. Linewidth Compression
This section presents laser linewidth and various methods used to compress the linewidthof semiconductor lasers and fiber lasers. The laser linewidth is a measurement of thespectral qualities of a laser light source which is predominantly related to the temporal
coherence often characterized by a coherence length and time. In other words, it is the full
width at half maximum (FWHM) of its optical spectral. In most situations, the linewidthof a laser results from quantum fluctuations generated from the excited spontaneousemission of ions and atoms that propagates without a specific direction and phase. Thus,
the photons of each spontaneous emission event randomly propagate in a manner that
ensures that an addition to the optical field in terms of a random phase is experienced.
Sometimes, the effect of the spontaneous emission ensures that some radiation is
inseparable from the stimulated emission, which is centered on the fact that they propagate
along an approximate direction. Thus, the core aspect of spontaneous emission noise isnone other than the desire to ensure that the laser emits a finite spectral width [43].
In typical practice, the value of a spectral linewidth often varies across different orders ofmagnitude and is predominantly consistent with the type of laser. The Schawlow-Townes
expression, which contributes to the linewidth as a result of phase changes initiated byspontaneous emission, is an expression that is globally accepted and adopted as afundamental value to characterize a laser’s spectral linewidth. It also suggests that thewidth of the Lorentzian is inversely proportional to the laser’s output power. The
Schawlow-Townes equation aimed to calculate the quantum limit of a laser’s linewidth
was put forward even before the emergence of the first laser. This Lorentzian spectral line
shape equation can be expressed as:

 laser   h   c  / Pout ,2(13.1)

where hv is the photon energy, vc is the bandwidth of the cavity (FWHM), and Pout isthe output power of the laser. The Schawlow-Townes linewidth is proportional to thesquare of the cavity bandwidth divided by the output power.
It can also be written as:


P2
hw 2
wST 
[] c ,
P2  P1 Pout

(13.2)

where P2 and P1 represents the steady-state lower-level and upper-level population,
respectively.  c  (c / 2d )ln( R1R2 ) is the power damping rate of the laser cavity. crepresents the speed of light in a vacuum, and d is the laser’s resonator length. R1 and R2
are used to represent the reflectivity of the mirror [44].

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Wavelength Switching, and Wavelength Tuning of Lasers

13.5.1. Whispering Gallery Mode (WGM)
Whispering gallery mode resonator with high quality (Q) factor in the order of 1010,
mainly found in calcium fluoride (CaF2) crystal-line disk and characterized by theirrelatively smaller structure size, is a rather promising candidate for filtering and stability
in a laser cavity. For example, Sprenger et al. [45] demonstrated a ring laser linewidthcompression system that adopts a calcium fluoride (CaF2) with a diameter of 5 mm, thepassive linewidth of 15 MHz, and Q factor of 107 acting as a whispering gallery mode
resonator for linewidth compression of an SOA as shown in Fig. 13.1 (a).
Compared to other research, the authors further enhanced the stability of their system byadopting a dual prism structure attached to both sides of the CaF2, which ensures that lightis coupled to and from the SOA in numerous cyclic roundtrips. Even more, a single
longitudinal mode operation is guaranteed by mainly adjusting the coupling on either sideof the resonator. Furthermore, the authors characterized their detected compressed laser
linewidth by adopting the three-cornered-hat method to ensure absolute accuracy, as
shown in Fig. 13.1 (b). To achieve this, in the linewidth detection method, the authors
utilized two other lasers alongside the ongoing investigated CaF2 disk laser to beat theoutput of the lasers and detect them by three photodetectors whose results are shown in
Fig. 13.1 (c). These two other lasers are MP 100-Littro configuration grating stabilized
laser and Toptica DL Pro laser.
According to the authors, by utilizing the renowned Allan deviation, it appears to be thatthe spectral linewidth of the MP 100-Littro configuration grating stabilized laser isrelatively larger in comparison to the other two corresponding lasers, as shown in
Fig. 13.1 (d). In other words, after numerous averaging times, the spectral linewidth of
the CaF2 disk laser is enhanced by a factor of over 103 order of magnitude. The authors
verified their claims by further characterizing the detected spectral linewidth with a
traditional self-heterodyne method which adopted a long delay fiber of 45 km, whose
results are shown in Fig. 13.1 (e). Remarkably, the authors compressed the passive 15
MHz linewidths of the CaF2 disk laser to 13 kHz for a Q-factor of 103. This linewidthcompression method appears to be effective towards deep linewidth compression andensuring stability to an extent.
Utilizing a WGMR (Whispering gallery mode resonator) is arguably an effective approachto compress the spectral linewidth of any commercial semiconductor laser while still
maintaining its well-renowned desirable fundamental qualities and further guarantee
stability. The reason being, it is an alternative approach to circumvent the drawbacks
associated with other linewidth compression methods tailored for semiconductor lasers,
which, when adopted, undesirably diminishes the fundamental qualities of semiconductor
lasers (i.e., size, weight, and power).
Typically, the host material of any WGMR usually contains, for example, some crystallike material. This is because such crystal-like materials are highly transparent and canguarantee that material absorption of the laser light is mitigated. More so, crystals arearguably the most transparent material in the entire electromagnetic spectral.
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(a)

(c)

(a)

(c)

(b)

(b)
(d)

(d)

(e)

(e)

Fig. 13.1. (a) Stabilized laser setup. Gain from a SOA passes through polarization control (PC)and is coupled through a CaF2 disc, using two prisms and GRIN lenses. 1 % of the 1550 nm
emission is coupled out, and unidirectional lasing is ensured by an isolator. (b) Schematicof the frequency mixing for the three-cornered-hat measurement. (c) Allan deviationsof simultaneous beat signals between a Toptica DL Pro, a home-built grating-stabilized diode laser(MP 100), and the CaF2 disk stabilized laser. A common 1:5 kHz harmonic due to vibration isremoved from the beats with the MP 100 before the Allan deviation calculation. Inset, frequenciesof the beats as a function of time (10 ms). MP 100 and CaF2, DL Pro and CaF2, then DL Pro
and MP 100 from top to bottom. (d) Allan deviations of the three individual lasers, calculated using
the three-cornered-hat method. (e) Self-heterodyne technique measurements. Beats are at the AOM
frequency of 100 MHz. From left to right, recording spans of the MP 100, DL Pro, and CaF 2
stabilized laser are 5 MHz, 1 MHz, and 100 kHz, respectively. 3 dB linewidths are 550 kHz,60 kHz, and 18 kHz. The delay is 45 km [45].

In 2015, Liang et al. [46] demonstrated a diode laser that is self-injection locked to a
WGM micro-resonator that consists of a crystalline host material which guaranteed thestability of the heterogeneously integrated laser. Furthermore, the authors ensured thatissues relating to size are resolved by meticulously placing the entire component (laser,
optics, and resonator) into a single package, as shown in Figs. 13.2 (a), (b), (c). Throughthe self-injection locking technique, Rayleigh scattering, an effect of surface and/or
volumetric inhomogeneities, is backscattered into the laser when the frequency of theemitted propagating light signal interferes with the frequency of the selected resonatormode. Hence, the resultant linewidth is gradually compressed until it achieves dynamic
equilibrium. In other words, the 2 MHz free-running laser linewidth was compressed toseven orders of magnitude, as shown in Figs. 13.3 (a), (b).
Such a deep linewidth compression was feasible simply because, with the aid of a prism,
the authors coupled the light signal from a diode laser into a WGMR whose host materialis MgF2 (Magnesium fluoride). The loaded quality factor is ~6 ×108, and its unloaded
resonator Q-factor is ~6 ×109. The authors reported that the frequency noise of their laser
is on the order of 0.3 Hz Hz-1/2 above 10 kHz, as shown in Fig. 13.4 (a), (b). The
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uniqueness of the self-injection locking technique also improves stability by reducing
ambient acoustic noise which affects the laser frequency.

(a)

(b)

(c)
Fig. 13.2. Schematic of the experimental set-up. Light from the pump laser (a semiconductor DFB
laser) enters the whispering gallery mode resonator (WGMR) through the prism. Part of light isreflected back to the laser because of Rayleigh scattering in the resonator. The light exiting
the prism is collimated and used for processing. Insets show power distribution in the resonatormode (a), a picture of the actual resonator (b) as well as a picture of the packaged laser (c) [46].

(a)

(b)

Fig. 13.3. Power spectral of the RF signal generated by beating two self-injection locked DFB
lasers on a fast photodiode. RF carrier frequency is kept at 8.7 GHz, resolution bandwidth is
300 kHz and video bandwidth is 3 kHz. The noise floor is determined by Johnson–Nyquist noiseof the photodiode. Inset (a): linewidth measurement performed with 30-Hz resolution bandwidth.
Points stand for the experimental data. Continuous red line is a 60-Hz Lorentzian fit of the data.
Inset (b): comparison of the RF spectra generated by beating of two self-injection locked lasers(curve (2)) and one injection locked and one free running lasers (curve (1)) [46].

521Advances in Optics: Reviews. Book Series, Vol. 5

(a)

(b)

Fig. 13.4. Spectral purity and stability characteristics of the laser. (a) Linear frequency noiseof the RF beat note of the lasers, (1) compared with the noise determined by conversion of the laser
power fluctuations to the frequency fluctuations, (2) as well as fundamental thermorefractive,(3) and thermoexpansive (4) noise. (b) Allan deviation interpolated using the frequency noise data
(1) and actual measurement of the Allan deviation (2). The 500-s peak in curve (2) resultsfrom the air conditioner cycle in the laboratory. This systematic frequency shift exceeds the internallaser noise rather significantly [46].

13.5.2. Parallel Feedback Mechanism
Wang et al. [47] demonstrated a linewidth compression mechanism based on an external
parallel feedback cavity. This structure was established by mainly inserting several piecesof MMF (Multimode fiber) whose core diameter is 62.5 µm, length is 15 m, into a linear
single-mode fiber cavity as shown in Fig. 13.5 (a). The authors coined this external
parallel feedback cavity as a single-mode-multimode-single-mode fiber (SMS) structure
which was used in the experimental scheme shown in Fig. 13.5 (b), (c). According to theauthors, the parallel feedback mechanism generated by this external parallel feedback
cavity is similar to the Weiner effect, alternatively known as the cursor effect.
In principle, compared to the propagation of light signals only within the main mode asconsistent with an SMF, the propagation of the light signal in an MMF is very different,
resulting in a parallel traverse light signal with different propagation constant. Such aphenomenon is present within the authors' external parallel feedback cavity structure,
which renders a compression in laser linewidth since the internal cavity of the DFB laser
and that of the external structure correspond to a compound resonator cavity. In otherwords, the propagating light signal within such a compound cavity must conform to every
restriction placed by various light paths within the cavity, which ultimately leads tocompression in spectral linewidth.
The authors compressed the free-running 200 kHz spectral linewidth of a normatively
designed DFB laser without an internal isolator down to 1.25 kHz and 430 kHz, as shownin Fig. 13.6 (a), (b), and in Fig. 13.7 (a), (b). This was feasible by optimizing their laser
cavity wherein an EDFA is absent and present, respectively, for the same structure atdifferent compression scenarios. Firstly, the reported value of the compressed spectral
linewidth (i.e., 1.25 kHz) was characterized by a traditional delayed self-heterodyne
522Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization,
Wavelength Switching, and Wavelength Tuning of Lasersinterferometer (DSHI) system were at different detection scenarios, a delay fiber of 100
m and a single acoustic optic modulator (AOM) of 70 MHz was used in the detection
structure at a particular time. Likewise, two different AOM of 70 MHz and 75 MHz
frequency shifts alongside the delay fiber of 100 m were used in the interferometer
structure at another particular time.
(a)

(b)

(c)

Fig. 13.5. (a) The single-mode–multimode–single-mode fiber (SMS) Structure; (b) Experiment
setup for compressing the line-width of a DFB Laser with a passive External Parallel Feedback
Cavity, and (c) Experiment setup for compressing the line-width of a DFB Laser with an active
External Parallel Feedback Cavity [47].

(a)

(b)

Fig. 13.6. Measurement of laser line-width compressed by the external cavity: (a) The whole spanview, and (b) An image zoomed in [47].

523Advances in Optics: Reviews. Book Series, Vol. 5

(a)

(b)

Fig. 13.7. Measurement of laser line-width compressed by the external cavity with an EDFA:(a) is the whole span view, and (b) is an image zoomed in [47].

Secondly, the authors argued that provided the narrow linewidth laser's coherence length
is longer than the length of the delay fiber, accuracy is questionable. Therefore, theexternal parallel feedback cavity, whose parallel feedback mechanism was optimized byattaching an EDFA within the compound structure for loss compensation, had its spectral
linewidth rather characterized by a loss-compensated recirculating delayed selfheterodyne interferometer (LC-RDHSI). This interferometer structure consisted of an
adjustable gain EDFA, an AOM of 500 kHz frequency shift, and a delay fiber of 100 m.
Through the LC-RDHSI, the authors reported a compressed spectral linewidth of 430 Hz.
Based on the authors findings, this is indeed in many ways, a commendable approach toattain linewidth compression of such magnitude.
13.5.3. Optical Self-Injection Feedback
In 2018, Kasal et al. [48] demonstrated a 1.5 µm distributed feedback-laser diode(DFB-LD) array system with the characteristics of a highly compressed, high output
power, and wavelength-tunable merits by adopting two different types of self-opticalfeedback circuits. According to their findings, in the linewidth compression structure that
adopts a monolithically integrated 12 DFB-LD and an optical circulator-based feedback
circuit as shown in Fig. 13.8 (a), the authors reported a compressed spectral linewidth of
8 kHz as shown in Fig. 13.8 (b). Similarly, in the linewidth compression structure that
adopts a monolithically integrated 12 DFB-LD, a Multimode interferometer (MMI), an
SOA, and a partial reflection mirror as shown in Fig. 13.8(c), the authors resolved issues
relating to compactness and obtain a spectral linewidth of 11 kHz as shown in
Fig. 13.8 (d). In other words, both systems generally adopted the well renowned opticalself-injection feedback method for such a large laser diode array with a wavelength
oscillation separation of 3.2 nm and lasing at the 1.5 µm wavelength.
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(a)

(c)

(b)

(d)

Fig. 13.8. (a) Configuration of tunable DFB LD array with circulator-based self-optical feedback
circuit; (b) Delayed self-heterodyne spectrum of LD 9; (c) Configuration of tunable DFB LD arraywith partial reflection mirror-based self-optical feedback circuit; (d) Delayed self-heterodyne
spectrum of LD 9 [48].

A minor drawback to this laser structure that adopts this compression method is theinability to operate all twelve LD simultaneously. Therefore, the performance test of one
single LD lasing at 1538.8 nm (i.e., hereinafter referred to as LD 9) was driven at an LD
current of 250 mA, whereas the SOA was driven at 400 mA. The stability of the laser
structure is enhanced by packaging the LD array, MMI, SOA, and two lenses in atemperature-controlled butterfly-type pigtail module. Even more, the authors packaged
the self-optical feedback circuit into a metal structure of reasonable dimensions. The
authors ensured that the intrinsic loss of the MMI coupler situated in between the LD arrayand the SOA is compensated for by the SOA.
As a typical self-optical feedback circuit, light signals are fed back into the DFB-LD array
after crossing through the polarization maintaining (PM) circulator, whose short loop
length of 1.4 m has its propagating optical feedback power finely controlled by a variable
optical attenuator attached within the loop. The length of the short loop plays a decisiverole in the linewidth compression effect. This is because the Q-factor of the cavity
increases provided the length of the loop is meticulously increased, thereby resulting inspectral linewidth compression. The authors adopted the delayed self-heterodyne
interferometer (DSHI) method with a delay fiber of 25 km to characterize the compressed
spectral linewidth of 8 kHz. Such attention and design is focused on the laser structurethat adopts the optical circulator-based feedback circuit.
On the other hand, for the same DFB LD array module packaged with its previously
mentioned components, the authors replaced the previously used self-optical feedback525Advances in Optics: Reviews. Book Series, Vol. 5

circuit with a partial reflection mirror acting as the self-optical feedback mechanism. Such
optimization enabled the authors to utilize a PM partial reflection mirror in a linear cavity
length of 30 cm to feedback 10 % of optical light signals into the DFB-LD array. In thisconfiguration, the authors studied the linewidth compression behavior of their laser
system by fixing the drive current of the LD array to 250 mA and gradually alternating
the SOA drive current where they discovered an increase in SOA drive current leads to adecrement in linewidth. In other words, the optical feedback power also increases
sufficient to obtain a compressed linewidth that is less than 11 kHz at an optical feedback
power above 1.5 mW. For this optimized system, the reported output power was mainly
as high as 18 mW. Thus, it demonstrates a slight difference in spectral linewidth andoutput power compared to the first linewidth compression structure (i.e., the opticalcirculator-based feedback circuit). The authors claimed that such a difference ininvestigated performance and reported results for the same LD 9 was induced by the shortfeedback length of the partial reflection mirror-based linewidth compression structure.
This linewidth compression method is a significant improvement in laser stability.
13.5.4. Microfibers
Fan, et al. [49] demonstrated a microfiber ring laser whose spectral linewidth wascompressed to 2 kHz, and the side mode suppression ratio (SMSR) is as high as 38 dB
with an output power higher than 0.95 µW. Structurally wise, the microfiber employed as
the gain medium in this laser configuration lasing at the 1536.1 nm wavelength is madeup of a 125 µm Er3+/Yb3+ co-doped phosphate glass fiber which is 1.88 µm in diameter.
The authors ascertain SLM operation in their laser structure by spooling two fiber knot
with the diameter of 206 and 351 µm, thereby creating a high-Q double-knot resonator
whose stability was further enhanced by placing the resonator on an MgF2 crystal substrate
as shown in Fig. 13.9 (a), (b), (c). The characteristics of a non-flat gain profile of thegain medium, which is ideal for an easier single-mode generation in this laser
configuration and its corresponding validation of a satisfactory single longitudinal mode
operation without mode hopping or mode competition, are shown in Fig. 13.10 (a), andin Fig. 13.10 (b), respectively.
The authors paid particular attention to the joining point between the double-knot
resonator and the taper-drawn fiber to ensure that light is evanescently coupled to andfrom the resonator. This joint point was established by electrostatic and van der Waals
force. As a result, the traversing optical light signal within the resonator tends to propagate
in an anti-clockwise direction of the double knot. Furthermore, the authors claimed thatthe inter-twisted part of the double knot ensures that a part of the light signal is recoupledback into the double knot using the evanescent coupling effect. In other words, there is asomewhat interference between the injected pump light signal from the semiconductor
LD and the luminescent recirculated light signal, thereby rendering the resonator cavity
into a lasing operation with the merits of a 2 kHz compressed laser linewidth as shown in
Fig. 13.10 (c). This linewidth compression scheme and method appears to be a costeffective and portable method.
526

Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization,
Wavelength Switching, and Wavelength Tuning of Lasers

(a)

(b)

(c)

Fig. 13.9. Configuration of the microfiber laser: (a) Schematic diagram; (b) Optical microscope
image of the double-knot resonator with green upconverted photoluminescence,and (c) The scanning electron microscope image of the coupling region of the double-knot
structure [49].

(a)

(b)

(a)

(b)

(c)

(c)

Fig. 13.10. (a) Gain characteristics of the Er3+∕Yb3+ co-doped phosphate glass fiber; (b) Single
frequency characteristics of the microfiber laser investigated by the scanning Fabry-Perot
interferometer; (c) Line shape of the homodyne signal measured with 48.8 km fiber delay [49].

13.5.5. Electrical Feedback Control
In principle, spectral linewidth compression by an electrical feedback method is regarded
as converting frequency fluctuations into intensity variations through a frequency
discriminator that depends on a transfer function that creates a situation where a fasttransmission variation is plotted against its corresponding optical frequency. The transfer
527

Advances in Optics: Reviews. Book Series, Vol. 5function is created due to either an interference effect of optical wave(s) or the effect ofabsorption at a specific wavelength. Such a frequency discriminator could be either
a π-phase-shifted FBG or a high finesse FP resonator.
To that end, Poulin et al. [50] put forward the design choices and the electrical feedback
method adopted by TeraXion for laser linewidth compression, as shown in Fig. 13.11 (a).
As a linewidth compression method, a typical electrical feedback method adopts areference photodetector to limit the impact of the RIN of a laser pose towards the error
signal. The efficiency of the frequency discriminator is dependent on its isolation against
vibrations. Regarding utilizing a feedback loop to correct fast frequency fluctuations thatcan further reduce frequency noise at high frequency, the authors claimed that TeraXion
is currently conducting technological advances to attain it. The findings of their numericalanalysis are shown in Fig. 13.11 (b). According to the authors, TeraXion reported using a
34 mm long π-phase-shifted fiber Bragg grating (FBG) of 15 MHz widths serving as afrequency discriminator in their laser configuration whose transmission and reflection
spectra are shown in Fig. 13.11 (c), and Fig. 13.11 (d). In order to resolve issues relating
to the compactness of a laser module adopting this suppression method, the authors put
forward the technological prospects by TeraXion, which aims to reduce the size of theemployed FBG for a superior withstand against vibration effects. For any laser that adopts
this compression method, the spectral linewidth will be compressed by a servo feedback
control system that is proportional to the error signal generated by the intensity variations.
(a)

(b)

(c)

(d)

Fig. 13.11. (a) Schematic illustration of the electrical feedback method. (b) Simulations showing
the power spectral density of frequency noise (PSDFN) for a free-running semiconductor laser(dark blue), for the same laser with electrical feedback for linewidth reduction (light blue)
and width an additional fast loop (red). (c), (d) Theoretical reflection (red) and transmission (blue)
spectrum of a 34 mm long π-phase shifted FBG providing a 15 MHz wide peak [50].

528Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization,
Wavelength Switching, and Wavelength Tuning of Lasers

As shown in Fig. 13.12 (a) and Fig. 13.12 (b), the authors reported a linewidthcompression of less than 5 kHz (approx. 4.4 kHz) of a 365 kHz free-runningsemiconductor DFB laser that adopts this compression method in a compact laser module.
This laser module consists of a temperature controller, current driver, and other opticaland electronic components are integrated into a 10×15×2.5 cm3 sized module. For casescenarios that will adopt the well-renowned heterodyne configuration, in this work, theauthors emphasized that an additional LD with the ability to serve as an offset-frequency
will be phased locked to the laser under test (LUT). This is done to synthesize pure RF
signals attached along the optical parts created to induce the electrical feedback effect.
With this electrical feedback method, linewidth compression from a higher order ofmagnitude to lower orders of magnitude is very much feasible. However, to some extent,
the structure of an electrical feedback control method appears to be a somewhat complex
electrical system.

(a)

(b)

(b)

Fig. 13.12. (a), (b) Measured power spectral density of frequency noise (left) and calculated optical
spectrum (right) for the Narrow Linewidth Laser (red) and for the free-running laser (blue) [50].

13.5.6. Rayleigh Backscattering
Our research group put forward a new linewidth compression mechanism centered onRayleigh backscattering (RBS) with a great potential to be utilized in any waveguidestructure operating at any wavebands. For the first time, Zhu et al. [51] simulated andexperimentally demonstrated a Rayleigh backscattering linewidth compression model
529

Advances in Optics: Reviews. Book Series, Vol. 5(RBSLCM). In principle, Rayleigh backscattering has a strong compression effect on the
linewidth of a laser. This is because the linewidth of a propagating laser signal after
scattering in a high Rayleigh scattering structure is narrower than the linewidth of theincident laser light signal. The authors claimed that when an incident light signal isinjected into an optical fiber, the RBS signal centered at the central wavelength of thescattering intensity profile decreases faster only at a region beyond the central wavelength.
This is feasible since RBS is regarded to be inversely proportional to the fourth power ofthe incident light signal.
This same RBS signal undergoes a process of amplification, thereby generating an RBS
signal but with a relatively higher Rayleigh coefficient when it encounters an active fiberor gain medium such as the EDF employed by the authors. In principle, the light signal isusually scattered at every scattering point during propagation which then serves as an
input for the next scattering point. In other words, the amplified light signal recreates the
RBS signal with a spectral linewidth that is comparably narrower than the incident lightsignal. Consequently, the spectral linewidth of a laser will be compressed for every
excitation and amplification of cyclic roundtrips within the laser cavity. The numericalanalysis of the author's RBSLCM model utilized an incident Lorentzian linewidthof 5.03 × 10-5 nm at a fixed number of 5000 iterations in a simulated waveguide of
100 scattering points, as shown in Fig. 13.13 (a), (b), (c). Upon successful 5000 iterations,the authors discovered that the laser linewidth was compressed by ~ 99.89 %.
Furthermore, the authors discovered that the scattering medium's length and damping rate
play a decisive role in the linewidth compression.
(b)

(a)

(c)

Fig. 13.13. Simulation of RBS-based linewidth compression mechanism: (a) Rayleighbackscattering spectrum; (b) Power spectra with different iteration number; (c) Power spectra
under different scattering factors after 5,000 iterations with one scattering point [51].

530Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization,
Wavelength Switching, and Wavelength Tuning of Lasers

To further validate this novel mechanism, the authors utilized and investigated thiscompression mechanism in a fiber ring laser system which consists of a 12 m long EDF
serving as the gain medium that is a fusion spliced and connected to an array of otheroptical components, as shown in Fig. 13.14 (a). The authors ensured that the two points(point A and point B) presented a laser structure wherein the RBS effect can be applied toor isolated from the ring laser system at any given time. Therefore, another part of theirlaser structure was established by connecting another optical circulator (OC) to a highRayleigh backscattering structure (HRBSS). The far end of this structure wasaccompanied by a variable optical attenuator (VOA) and faraday rotation mirror (FRM).
Such a configuration was made for the solemn purpose of loss and gain balancing in thecavity. This HRBSS serving as a unique material to induce the desired RBS signal of highRBS coefficient, low optical attenuation, strong optical confinement, and high stimulated
Brillouin scattering (SBS) threshold, was obtained from a slightly tapered 110 m long
SMF with characteristics of ~112 m waist diameter, 2 cm tapering length, and a total
number of 21 tapers.
(a)
(a)

(b)
(b)

(d)
(d)

(c)
(c)

(e)
(e)

Fig. 13.14. (a) Schematic diagram of fiber ring laser combing RBS and self-injection feedback.
HRBSS high RBS structure. The coupling ratios are C1: 80/20; C2: 50/50; C3: 50/50; C4: 90/10,(b) The longitudinal mode characteristics of the output laser without and with the self-injection
feedback based on RBS, respectively, (c) The output spectra measured by the self-heterodyne
method with the change in feedback power, (d) The relationship between the laser linewidth
and the feedback power under the condition of pump power of ~165 mW, (e) Lorentz fitting
linewidth for the narrowest laser linewidth [51].

531Advances in Optics: Reviews. Book Series, Vol. 5

Two couplers (C1 and C2) were employed in the fiber ring laser system to induce the effectof optical self-injection feedback, whose feedback power was intuitively controlled by the
VOA2. Thus, issues relating to enhanced linewidth and laser operational stability are
guaranteed. In a way, there is a trade-off between the desired compressed linewidth andthe obtained SMSR. This is because the authors were compelled to adjust the VOA1
attached to the optical pathway wherein the HRBSS was situated, which ultimately leadto an increase in reported SMSR and a detected linewidth compression of ~ 300 Hz at thefirst instance of injection into the unique material as shown in Fig. 13.14 (b).
The authors also discovered that an increase in feedback pump power at a controllable
range of 0.2 to 900 µm by mainly gradually tuning the VOA2 attached to the self-injection
feedback scheme ensures an SLM operation (characterized by a broadened FSR) is
guaranteed. It is apparent that the side modes of the output laser gradually disappear at atuned feedback injection power of 20 µm, as shown in Fig. 13.14 (c). In this way, thespectral linewidth was further compressed to 130 Hz at a reported controllable feedback
power of ~600 µm, as shown in Fig. 13.14 (d) and Fig. 13.14 (e). Utilizing this novel
compression mechanism, the authors obtained an ultra-narrow linewidth laser output of~130 Hz and SMSR as high as 75 dB. The output power of the laser is considerably high,and the stability of the laser structure in terms of its output power and spectral linewidth
fluctuation is commendable for such a laser structure.
Since it has been predicted that Rayleigh backscattering could be collected in any
waveguide medium, member(s) of our research group [52] investigated the characteristicsof this weak signal in various types of optical fibers. They employed a heterodyne
detection scheme to isolate the Brillion scattering within a fiber laser to achieve this. Thesefibers are in polarization-maintaining fiber (PMF), large effective area fiber (LEAF), andyet again in the well renowned single-mode fiber (SMF-28e). The stimulated Rayleighscattering (STRS) investigated characteristics consist of its linewidth, threshold, andfrequency shift. A unique experimental system that consisted of an array of opticaldevices, as shown in Fig. 13.15, was employed to investigate aspects relating to linewidth
and threshold.

Fig. 13.15. Measurement of Rayleigh scattering in optical fibers. Laser: Fiber laser; EDFA:
erbium-doped fiber amplifier; PC1 and PC2: polarization controller, PBS: polarization beam
splitter; PM Circulator: polarization maintaining circulator; AOM: acoustic-optic modulator;
PD: photon detector; ESA: electrical spectrum analyzer [52].

532Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization,
Wavelength Switching, and Wavelength Tuning of Lasers

The authors' findings on the three employed types of fibers suggest that the amount ofapplied input power plays a decisive role in the level from which STRS occurs alone andthat the SBS is initiated henceforth. Even more, when the power of the SMF-28e, LEAF,
and PMF is within a controllable range of 11 dBm to 17 dBm, 4.5 dBm to 11.6 dBm, and16.5 dBm to 26 dBm, respectively, it is believed by the authors that the bandwidth of the
STRS is somewhat little in fluctuation. In other words, the authors believed and claimed
that the effect of STRS is evident in the region of an approximate 6 dB, excluding that of
the PMF, which exists at a power range of 10 dB. The spectral linewidth whose
evolutionary state of the spectra, contrast, and bandwidth variation is shown in
Fig. 13.16(a)-(f). According to the authors, the reported bandwidth of the three types offibers under test are 9 kHz, 10 kHz, and 11 kHz for the SMF-28e, LEAF, and PMF,
respectively.

Fig. 13.16. Evolution of spectra, 3 dB bandwidth changing, and contrast of Rayleigh scattering
signal for 2 km SMF-28e, 7 km LEAF, and 100 m PMF. (a), (c) and (e): Rayleigh spectra evolution
for SMF-28e, LEAF and PMF, respectively. (b), (d) and (f): 3 dB bandwidth and contrastof Rayleigh scattering for SMF-28e, LEAF and PMF, respectively [52].

The preceding (i.e., minimum) power value in those mentioned above 'controllable range'
represents the threshold power values of the STRS in each of the three different types offibers. Hence, any slight increase beyond the maximum power values of each of thosementioned above 'controllable range' for the different fibers under test will instead inject
power into the SBS process since the gain factor of SBS is more significant than that of
SBS the STRS. In addition, the authors discovered that the linewidth of STRS in the three533

Advances in Optics: Reviews. Book Series, Vol. 5

different fibers under test was compressed in comparison to the 11.2 kHz free-running
linewidth of the laser. What is more, a measured compressed linewidth of ~9 kHz for the
2 km SMF was the most compressed of them all.
In regards to the reported threshold power values of 10.2 dBm, 3.4 dBm, and 9.4 dBm forthe 2 km SMF, 7 km LEAF, and 100 m PMF, respectively, as shown in Fig. 13.16(a)-(f),the authors put forward two core reasons surrounding the difference in the measured
values. Firstly, these reasons are the dependency of the threshold power on thepolarization, fiber length, and effective core area of the three different types of fibers under
test. Secondly, the efficiency of STRS in a PMF fiber is arguably higher than that of
traditional optical fibers such as a LEAF and an SMF-28se. Hence, the approximate
corresponding Rayleigh gain coefficient of the three employed fibers are 2×10 -13 m/W,
2.8×10-13 m/W, and 1.0×10-12 m/W, for the 2 km SMF, 7 km LEAF, and 100 m PMF,
respectively.
Likewise, investigating the characteristics of frequency shift, the authors adopted the same
experimental structure in a modified manner, as shown in Fig. 13.17 (a). Regarding thefrequency shift, the authors claimed that the fundamental reason surrounding the reported
findings, as shown in Fig. 13.17 (b), is centered on the fact that the electrostriction andabsorption process possesses an equivalent magnitude when the applied input power islow. In other words, the input power plays a decisive role in observing a frequency shift
wherein a substantial increment in its value ensures that the electrostriction effect becomes
rather dominant. The authors further reported that when the input power is increased
beyond 25 dBm, the SBS effect is initiated within the 100 m PMF since the gain
coefficient of STRS is two orders of magnitude lower than that of the SBS. In light of this,a frequency shift measurement is arguably impossible in a fiber of longer lengths such as
the 2 km SMF and the 7 km LEAF, except if only the threshold values of SBS aresuppressed within such fibers in general.

Fig. 13.17. (a) Measurement system of frequency shift of Rayleigh signal in optical fibers, Laser:
fiber laser; EDFA: erbium-doped fiber amplifier; PM Circulator: polarization maintaining
circulator; PC: polarization controller, PBS: polarization beam splitter; AOM: acoustic-optic
modulator; PD: photon detector; ESA: electrical spectrum analyzer. (b) Frequency shift
experimental results [52].

Besides the classical discoveries of STRS in liquids and gases during its earliest days, themonumental findings of this work happen to be a significant addition to the literature of
scattering effects. The reason being, over the years, researches have been particularly534

Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization,
Wavelength Switching, and Wavelength Tuning of Lasers

tailored to discoveries on stimulated Brillion scattering (SBS) and stimulated Raman
scattering (SRS), which are renowned members of the non-linear scattering group. Thus,
the findings of this work on STRS as a linear scattering effect that focused on observing
the dynamic change from spontaneous to stimulated Rayleigh scattering by characterizing
its frequency shift, threshold, and linewidth, is an improvement on the literature.
Furthermore, the findings of this work distinctively proved that STRS, characterized byits low threshold power and narrow bandwidth, is indeed a weak effect in fibers and canbe employed for spectral linewidth compression purposes.
Member(s) of our research group [53] demonstrated ultra-narrow linewidth compression
in an erbium-doped fiber ring laser system which consists of a tapered optical fiber. The
linewidth suppression of this work was induced by the RBS effect from within a ~110 m
short tapered optical fiber. The SBS effect is suppressed because the shape of the alteredfiber core and cladding (i.e., the tapered fiber) influences the propagation of acoustic
waves (photons and phonons). The fabrication process of the tapered fiber (SMF-28,
Corning, Inc.) was done on a translation stage where an oxyhydrogen flame is exposed
onto the fiber's jacked-off sections to induce heat into the fiber to create the uniform
tapering process, as shown in Fig. 13.18 (a), and Fig. 13.18 (b). Consequently, the authors
emerged with a total number of 21 tapered zones along an ~110 m tapered fiber, thereby
increasing the SBS threshold value by approximately 6 dB.
(a)

(b)

Fig. 13.18. (a) Schematic of the fiber taper manufacture; (b) Schematic illustration of the tapered
fiber with two tapers. The left inset shows a side view of the waist and the right inset shows
the waist of the tapered fiber cross-section. The diameter of the waist is ∼112.09 µm, which isslightly different from single-mode fiber (D = 125 µm) [53].

The RBS effect in this laser configuration is evident within the tapered fiber, wheremultiple scattering occurs to present multiple reflections similar to a distributed mirror.
The most potent potion of these reflections, which gradually builds up with Rayleighscattering of other nearby scattering points, is typically situated at the front end of thetapered fiber. To further facilitate the feasibility of linewidth compression by thecirculating RBS propagating in nanoseconds, the employed EDF with characteristics of amillisecond-scale fluorescence relaxation time permits the RBS to undergo numerous
cyclic round trips within the ring laser. Thus, leading to laser performance modification(characterized by SLM operation).
Spectral linewidth compression occurs in the laser system as shown in Fig. 13.19 (a),when the laser system's cavity loss is increased by adjusting the VOA situated in between
535Advances in Optics: Reviews. Book Series, Vol. 5

an FRM and the spool of tapered fiber. When there is a significant cavity loss, the RBS
effect situated at the far end of the ~110 m tapered fiber plays a dominant role. It appears
to be relatively easier in amplification as compared to the reflected signal originating fromthe FRM. Hence, a cyclic round trip of numerous times inevitably creates a high gain,
thereby compressing the spectral linewidth of the laser to an ultra-narrow level of ~200
Hz, as shown in Fig. 13.19 (b), Moreover, obtaining an SMSR as high as 50 dB. The
authors further demonstrated the sensitivity of the laser linewidth evolution to acontinuously varied power by gradually adjusting the VOA, which directly increases thecavity loss of the laser, as shown in Fig. 13.19 (c). In so doing, the minimum linewidth of~200 Hz was discovered because by adjusting the VOA, the RBS signal dominates thefront end position of the fiber and can be easily amplified for deep linewidth compression
compared to the reflected pump signal. The authors also ensured that the growing demand
for ultra-narrow seed laser lasing at various wavelengths is met in their work. They
achieved this by ensuring that their system can be used at any desired wavelength due tothe wavelength selective filter (FBG) impact upon which the wavelength selection isdependent.
(a)

(b)

(a)

(b)

(c)

(c)

Fig. 13.19. (a) The configuration of the experimental setup. WDM: wavelength divisionmultiplexing device; EDF: erbium-doped fiber; FBG: fiber Bragg grating; OC1, OC2: opticalcirculators; C1, C2, C3: 50:50 couplers; PC: polarization controller; VOA: variable opticalattenuator; FRM: Faraday rotating mirror; AOM: acoustic optic modulator; PD: photodetector;
ESA: electrical spectrum analyzer. (b) The photocurrent power spectrum of the delayedself-heterodyne signal of the laser output and the Lorentz fitting results. (c) The minimum linewidth
the laser can achieve: its mean linewidth and its standard deviation [53].

In some situations, it appears to be that SBS and RBS could be simultaneously utilized ina laser system for spectral linewidth compression, provided the threshold value of SBS is
reduced intuitively. To that end, members of our group research; Huang et al. [54],536Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization,
Wavelength Switching, and Wavelength Tuning of Lasers

established a laser system, as shown in Fig. 13.20 (a) that utilizes multi-linewidthcompression methods to compress the spectral linewidth of their free-running laser to an
ultra-narrow level. These mechanisms include a dual cavity feedback structure (DCFC),
the SBS effect, and the RBS effect. Firstly, the authors utilized a DCFS whose principleof operation has been explained in their previous work [55] to compress the original MHz
linewidth of a commercial butterfly DFB laser to ~3 kHz to serve as the pump laser toinduce the SBS effect. Two spools of SMF fiber of 25 m and 17 m length were coupledalongside two couplers and two optical circulators to establish the DCFS. This was done
simply because the theoretical analysis of the authors predicted that a narrow linewidth ofthe pump laser in an SBS configuration would obtain a relatively narrower bandwidth of
the SBS spectral for situations where the spectral linewidth of the pump laser is less than
10 kHz.
(a)

(b)

(c)

(d)

Fig. 13.20. (a) Laser linewidth compression system based on SBS and RBS systems. (b) Frequency
domain characteristics of the final output laser by SCELD with a 50 km delaying fiber. (c) Different
frequency noise under different conditions: free-running DFB laser (blue curve), DFB laser
with a DCFS (red curve), and the final output laser after a DCFS, SBS, and RBS (purple curve).
(d) RIN of the final output laser [54].

Hence, the SBS effect further compresses the linewidth to ~200 Hz as the second
compression method. Reason being that a low loss fiber with a high gain nonlinear
scattering factor (i.e., the employed ~20 m HNLF) was coupled in a ring cavity
configuration with a polarization controller (PC), and an optical circulator. Thus, thisconfiguration was used to sufficiently accumulate the SBS effect and further guarantee
SLM operation. A C-band EDFA firstly amplified the pre-compressed 3 kHz linewidth
before being injected into the SBS ring resonator. To further validate their notion on thepump laser linewidth and SBS effect, the authors investigated a series of situations (output
characteristics of the SBS ring resonator) wherein the pump laser linewidth of the537

Advances in Optics: Reviews. Book Series, Vol. 5

resonator was changed. Hence, it appears to be that the gradual decrement in pump laser
linewidth (from 1.5 MHz to 150 Hz) leads to a gradual increment in SMSR (from 25 dB
to 62 dB), and likewise, the linewidth of the SBS ring resonator (from ~9 kHz
to ~200 Hz).
Finally, an ultra-narrow compressed linewidth of ~75 Hz with an SMSR as high as 70 dB
was achieved by an RBS configuration which consists of two pairs of FBG whosereflectivity are ~90 % and ~50 % each, with a spool of 50 m SMF situated in between
them as shown in Fig. 13.20 (b). The authors deliberately choose a mismatch in thepercentage of grating reflection (particularly the second FBG) to ensure that the SLM
operation of the laser achieved by the SBS effect is not altered. Similarly, this was done
to ensure that the randomly distributed Rayleighscattering gain is established by the SMF
and the second FBG precisely. Consequently, RBS is sufficiently accumulated in thischamber whose linewidth is narrower than the incident laser (SBS ring resonator). Hence,
this signal originating from the RBS structure is then feedback into the SBS chamberwherein numerous cyclic round trips and amplification occur to further compress the
linewidth to ~75 Hz. Furthermore, the parameter of the output laser in terms of its
frequency noise which is another representation of its compressed linewidth, is shown in
Fig. 13.20 (c). Likewise, the RIN was found to be −120 dB/Hz for frequency ranges
beyond 1 MHz, as shown in Fig. 13.20(d). The authors demonstrated a commendable laser
system with the merits of an ultra-narrow linewidth and stable frequency noise that seeks
wide-spread applications in communication and sensing systems.

13.6. Linewidth Measurement
This section present various linewidth measurement methods. The fundamental linewidthof a laser tends to be arduous whenever the need to measure it arises. This difficulty is
associated with the fact that all methods developed to determine the ideal linewidth of alaser are often subjected to a duration of time coined by academia and industry as a ‘finite
measurement time.’ In this instance, the compressed linewidth of a laser light source is
impacted by frequency jitter arising from multiple aspects such as vibrations, pump laser
noise, acoustic noise, etc. Therefore, for a compressed laser linewidth, the value of thedetected linewidth can be regarded as an integrated frequency jitter resulting fromtechnical noise originating across the integration duration of the linewidth detection
system [56].
13.6.1. Heterodyne Detection
Besides the fundamental reason for obtaining data from a laser linewidth measurement,
this particular method transcends its primary detection purpose and into a feasible method
to provide data of an optical power spectral. Likewise, it is feasible enough to characterizea non-symmetrical linewidth line shape of a laser. This method's effectiveness depends on
the quality of the reference laser light source employed in the detection scheme in termsof its narrow linewidth value and laser operational stability. A traditional heterodyne
detection scheme typically requires two laser light sources where one serves as the signal
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Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization,
Wavelength Switching, and Wavelength Tuning of Lasers

laser. In contrast, the corresponding laser serves as the reference laser, often referred to as
an LO, as shown in Fig. 13.21. Propagating light signals of the signal laser and the LO
laser are combined and mixed by a 3 dB optical coupler (OC). This mixed optical signal
is further connected to a photodetector, where the interference beat note is detected andconverted into an electrical signal. The measurement accuracy of this method is dependent
on the fact that two laser light sources are required in this configuration and that the
linewidth of the LO needs to be narrower or approximately comparable to the signal laser[57, 58].

Fig. 13.21. Schematic of the setup for optical heterodyne detection [58].

In typical practice, in order for the two lasers to undergo a signal mixture or impairment
and fall within the bandwidth of the photodetector, the central frequency of the reference
signal needs to be finely tuned to nearly coincide with the central frequency of the signal
laser. In this way, the photocurrent of the mixed frequency signal that is detected by thephotodetector can be expressed as [58]:i  t   R  Ps  t   PLO  t   2 P2  t  PLO cos  2  vs  vLO  t   (t )   ,

(13.3)

where Ps(t) and PLO(t) represents the power of the laser under test and the reference laser,
respectively. vs and vLO represents the central frequencies of the two laser signals.
Furthermore, Δϕ(t) and R represents the phase difference of the two lasers andresponsivity of the photodetector employed, respectively.
13.6.2. Delayed Self-Heterodyne Detection
In contrast to a traditional heterodyne detection method, the DSHI method is a relatively
more straightforward method to measure the spectral linewidth [57, 58]. This is because
it practically requires a single laser light source (signal laser) in its configuration insteadof having two laser light sources (i.e., the usual LO). It adopts a sizeable optical delay
fiber situated on one arm of the laser, which induces a delay of the optical signal, and an
acoustic optic modulator (AOM) situated on the second arm of the MZI, which induces afrequency shift of the optical signal. By adopting a directional optical coupler in thistraditional detection method, the optical light signal injected into the MZI structure andpropagating across, is split into two equal paths wherein one potion is delayed, and thecorresponding signal is frequency shifted as shown in Fig. 13.22.
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Fig. 13.22. Schematic setup for optical delayed self-heterodyne detection [58].

In this way, the two optical signals from the signal laser are mixed with themselves byanother directional optical coupler situated at another end of the MZI leading up to thephotodetector, where the interference beat note is detected and converted into an electricalsignal. Hence, the electric field that is detected by the photodetector can be expressed as[58, 59]:

ET  t   P  cos  2 f 0t    t   
 P0  cos  2  f 0  f1    t   d     t   d   ,(13.4)

where P and P0 represents the power of the upper and lower arms, respectively. f0 and f1
represents the frequency shift of the laser and AOM, respectively. τd represents the delaydifference induced by the optical path imbalance of the interferometer. Therefore, therelationship of the photocurrent I(t) and the light intensity at the PD can be expressed as:I  t   ET  t  ET  t   2  exp i   t     t   d  exp  it   c.c,(13.5)

where c.c represents the complex conjugate of the output light field. More so, it isunderstood that the photocurrent characterized by a constant amplitude that is equivalent
to the phase jump of the laser light field, has a quasi-monochromatic signal at an angularfrequency of Ω = 2πf1.
13.6.3. Delayed Self-Homodyne Detection
In a typical coherent detection method, the frequency of the optical light signal injected
into the measurement system is converted into the RF (Radio-Frequency) domain by using
a signal mixture or impairment with a LO (i.e., a reference laser signal). However, in a
traditional delayed self-homodyne detection method [57], this is not the case. The reason
being that a delayed self-homodyne detection method does not require an LO. In a delayedself-homodyne detection method, the optical light signal from the laser light source
practically mixes with what is rather referred to as a delayed version of itself originatingfrom the same laser after propagating across an extended length of delayed optical fiber.
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This delay fiber is usually situated on one arm, which causes an optical path differencecompared with its corresponding second arm without a delay fiber, as shown in Fig. 13.23.

Fig. 13.23. Typical experimental set-up for linewidth measurement
using self-homodyne method [57].

It is understandable that since the two mixed signals possess an equal mean opticalfrequency, the signal to be detected is mainly dependent on the precise length of the optical
path imbalance in the interferometer. Thus, in principle, by employing this MZI structure
to measure the spectral linewidth of the LUT, the overall concept is fixated on the factthat we intend to convert the laser’s optical phase or frequency fluctuations into variationsof intensity.
13.6.4. Amplitude Difference Comparison of Coherent Envelope
The desire to obtain laser linewidth measurement, which is less than 1 kHz, has propelled
the numerical analysis and experimental demonstration of a new linewidth measurement
method. This new method was coined as amplitude difference comparison of coherent
envelope (ADCCE) by members of our research group [60]. This linewidth measurement
method reveals the actual value of a signal laser less than 1 kHz by comparing the contrast
difference between the second peak and the second trough (CDSPST) of the coherent
envelope of the power spectral. Technically, this method follows the traditional DSHI
linewidth detection method. However, compared to a long delay optical path imbalance
adopted in a DSHI scheme, the ADCCE method utilizes a short delay fiber. In otherwords, it is somewhat analogous to a short delayed-self heterodyne interferometer(SDSHI).
This method is not only new but has proved to be stable and effective in characterizing
the spectral linewidth of a signal laser less than 1 kHz (i.e., 100 Hz ~ 1 kHz), thereby
rendering it a practical approach for narrow or ultra-narrow linewidth detection. This iscentered on the fact that the typical structure of this method, based on an SDSHI, ensuresthat the impact of the broadening spectral created by the 1/f frequency noise isapproximately ignored. Simply put, a short delay fiber is arguably promising to mitigate
this drawback. In principle, provided the linewidth is further compressed, and theLorentzian line shape is used to fit the power spectral in the DSHI, a delayed fiber ofhundreds of kilometers is required to be utilized to measure the spectral laser linewidth of541

Advances in Optics: Reviews. Book Series, Vol. 5

sub-kilohertz level. A typical structure of the SDSHI scheme is shown in Fig. 13.24,
consists of a signal laser, VOA, isolator (ISO), 1×2 optical couplers, AOM, delay fiber,PD, and an ESA.

Fig. 13.24. Schematic of the delayed self-heterodyne interferometer
to measure laser linewidth [60].

The propagation of the optical signal within this detection scheme can be explained asfollows. Firstly, a signal laser injects a single longitudinal mode beam via the VOA andISO, arriving at a typical MZI. The MZI consists of two optical couplers that ensure thatthe light signal is split into two paths (i.e., an equal path where a delay is induced andanother equal path where the signal is frequency shifted). Secondly, at the output arm of
the MZI, which holds another optical coupler, the light signal is therein recombined andpropagated towards the PD and ESA for power spectral light detection and analysis,
respectively. In this way, the output power spectrum of the SDSHI can be expressed as[60]:

S  f , f   S1S2  S3

(13.6)In the above equation, S1, S2 and S3 are respectively expressed as,
S1 

P02 f 2,
4 f 2  f 2

sin  2  f  f1  d  
S2  1  exp  2f  d  cos  2  f  f1  d   f
,f  f1


S3 

 P02
2

(13.7)

exp  2f  d    f  f1  ,where f represents the frequency, f1 represents the frequency shift of the AOM, the itemτd (τd = L/c) which refers to the time delay of an optical path with respect to the another
path, consists of L which represents the length of the delayed fiber; c is the speed of light,
and Δf is the FWHM of the power spectral.
The shorter the length of the delayed fiber, the narrower the laser linewidth, a more distinct
coherent envelope of the power spectral is bound to appear. Ultimately, the accuracy of542

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Wavelength Switching, and Wavelength Tuning of Lasers

this measurement method is directly dependent on the length of the delay fiber, theresolution of the ESA, and the narrow spectral linewidth of the signal laser under test.
Theoretically, the ADCCE remains unchanged once the length of the delayed fiber andspectral linewidth is determined. This ensures that the laser linewidth is revealed from thepower spectral of the coherent envelope by estimating the value of contrast difference
between the second peak and the second trough (CDPT). The second frequency is chosen
to evaluate the contrast difference between the peak and trough because it holds a more
distinct advantage in minor error detection and higher stability than the center frequency.
Furthermore, members of our research group established an equation as shown below to
express the value of the CDSPST (ΔS) in their subsequent work [61]:S  f   10log10 S peak  10log10 Strough  10log10

S peakSthough

2l  1 c
S  f1 , f 
2 nL

 10log10 cS  f1  m , f nL  2c 2  nfL  1    1  exp  2
c  
  nfL   
 10log10(l  2, m  2)
  3 c 2  nfL  1    1  exp  -2
c  
  2 nfL   

(13.8)

Similarly, f represents the frequency, f1 represents the frequency shift of the AOM, and n
is the refractive index of fiber. L represents the length of the delayed fiber, c is the speedof light, and Δf is the FWHM of the power spectral (Lorentzian linewidth). The positionof peaks and troughs can be represented by the item l and m whose values are within therange of 2,3,4, and 1,2,3, respectively.
In principle, the Lorentzian linewidth is induced by the white frequency noise, while theGaussian broadening is induced by the 1/f frequency noise of a long delay fiber. Voight
fitting is often adopted to separate these two effects. However, it is difficult to determine
the true spectral linewidth since the Lorentzian linewidth is enveloped by the 1/f frequency
noise of the long delay fiber. With the desire to resolve this drawback and ensure that an
ultra-narrow linewidth laser is accurately determined (i.e., Lorentzian linewidth), theGaussian broadening effect needs to be eliminated. Hence, the amplitude differencecomparison of the coherent envelope method our research group puts forward by mainly
estimating the contrast difference between the second peak and second trough of thesecond frequency in an SDSHI scheme is feasible and effective. Since the accuracy of thismethod depends on the spectral linewidth, ESA resolution, and the length of the delay
fiber, it is essential to know an exact choice of short delay fiber length for the SDSHI.
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Such emphasis is made because a suitable length can effectively filter out the
1/f frequency noise and is capable enough to characterize the actual value (Lorentzian
linewidth) of an ultra-narrow linewidth signal laser of different spectral magnitude. Sucha length decision can be determined by combining the estimated spectral linewidth and a
specific value of CDSPST.

13.7. Noise Characterization
This section present various phase or frequency noise, and relative intensity noisecharacterization methods. Phase noise is regarded as the phase fluctuations occurring
within a signal which are random, rapid, and short duration. Similarly, frequency noise isconsidered to be the fluctuations of the instantaneous frequency of a signal. The phase andfrequency noise of a laser are related because they are parts of the same noise and aremainly different ways of characterizing the same phenomenon. In fact, the measurement
method adopts the same setup. Some authors often prefer to present the power spectral
density (PSD) of the noise carrier in frequency due to the advantageous technicalities indemodulation. A typical example of a frequency noise spectrum which is easily identified
as series of jitter that equally spread across either side of a signal and manifesting itself inthe form of sidebands across the signal, is shown and discussed in some parts of thepreceding section (see Fig. 13.20. (c)).
Relative intensity noise is regarded as the fluctuations in the optical output power of alaser. The RIN noise of a laser can be induced from aspects relating to the fluctuations of
the laser gain medium, vibrations of the external cavity, or the intensity noise effectoriginating from a laser light source. An ideal level of RIN for high-performance lasers tomeet communication or sensing system application demands is lower than −155 dB/Hz[62]. A typical example of a relative intensity noise spectrum is shown and discussed insome parts of the preceding section (see Fig. 13.20. (d)).
13.7.1. Digital Cross Correlation
The phase noise or frequency noise characterization of a narrow linewidth laser is vital toevaluate the stability of a laser. In order to actualize this method, a digital cross-correlation
scheme can be established where one laser is the signal laser. In contrast, thecorresponding two lasers whose linewidths are relatively lower than the signal laser under
test act as the reference laser [63]. A typical structure of this characterization method
consists of chambers, as shown in Fig. 13.25.
To be precise, the three-laser light source is required to be in a chamber of theirs while its
light signal is transported over tens of meter fiber lengths to the detection chamber. The
detection chamber consists of two photodiodes. In addition, an optical amplifier is often
attached immediately after the PD to amplify the beat note signal, whose signal is first
amplified and then frequency down-converted by a frequency synthesizer. Hence, thesignal launched into the digital cross-correlator is power amplified, and a low pass filteredsignal. The first procedure undergone by this correlator is analog to digital conversion
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Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization,
Wavelength Switching, and Wavelength Tuning of Lasers(ADC) before analysis. A cascade of these three lasers ensures that the two reference
lasers beat with the signal laser to generate two RF electrical signals after photo-detection.
The photodiode requires that the frequency difference between the LUT and the reference
laser needs to be within the bandwidth of the PD in order for proper conversion from an
optical signal to an electrical signal. A digital cross correlator can analyze the detectedsignal. The noise features of the laser under test can be revealed by averaging the statisticalestimator of the cross PSD of the phases of these two beat-note signals. Such an approachis feasible since the noise level from the two other light sources is independent and does
not contribute to the noise level in the signal laser. Furthermore, this method does not
require the linewidth of the reference signal lasers to have a superior performance level tothe linewidth of the laser under test.

Fig. 13.25. Experimental setup of the laser phase noise characterization. Three separate ultra-stable
1542 nm lasers are used in this scheme. Laser B is beat with the reference Laser A and Laser C
to get two electrical signals. These two electrical signals are first mixed down to a lower frequency
and then analyzed by a home-designed digital cross correlator to reveal the phase noise PSD
of Laser B [63].

13.7.2. Michelson Interferometer
The emergence of the Michelson Interferometer (MI) method for phase or frequency noise
measurement was inspired by the desire to resolve drawbacks associated withmeasurement methods such as the digital cross-correlation method, which requires areference laser source of high coherence. The Michelson Interferometer method is anefficient and promising method that conveniently characterizes the phase and frequency
noise conveniently. This is feasible by demodulating an accumulated laser’s differential
phase in a delay time and then employing fundamental mathematical relations, which
involve the laser’s phase noise and differential phase. Such mathematical relations ensurethat the actual value of a laser’s phase and frequency noise is revealed [64]. Noise
measurement by this method involves a typical structure that consists of a signal laser,OC, 3×3 optical coupler, short delay fiber, etc., as shown in Fig. 13.26.
The signal laser generates the optical signal injected and traversing across the opticalcirculator and optical coupler before being reflected by the FRM. This reflection ensures
the optical signal propagates backward to be firstly mixed within the OC with different
545Advances in Optics: Reviews. Book Series, Vol. 5

delay times. Furthermore, it is detected by the photodetectors before being analyzed anddemodulated. The 3×3 optical coupler with two FRM attached at the far end of only two
coupler ports guarantees the restriction of polarization fading of the interferometer arisingfrom surrounding noise. Such a configuration is adopted in this unbalanced MI simply
because of its unique ability to accommodate a laser injected propagating light signal anddemodulate its corresponding differential phase.

Fig. 13.26. Experimental setup used to measure the laser phase and frequency noise, and the output
interference fringe of the PD1, PD2, PD3 (inset). LUT: laser under test, C: circulator, OC: opticalfiber coupler, FRM: Faraday rotation mirror, PD: photodetector, DAC: data acquisition board [64].

Even more, the 3×3 optical coupler holds superior performance simply because it does
not require any form of an active controlling operation. Thus, the unbalanced MI structure
does not suffer from the impact of polarization, nor does it require any adjustment. In thisway, the signal laser’s differential phase fluctuation generated by the MI is demodulated,
thereby paving the way to reveal the noise features of a compressed laser linewidth
utilizing PSD estimation and calculation of the differential phase.
Although the previously discussed delayed self-heterodyne detection method and thedelayed self-homodyne detection method can characterize the phase or frequency noise[65], most authors prefer to adopt the beat-note cross correlation method and theMichelson interferometer detection method as discussed above. In addition, the delayedself-heterodyne detection method and the delayed self-homodyne detection methodadhere to stringent requirements when employing these methods to characterize the phaseor frequency noise. These rigorous requirements seek a long delay fiber and the need tocontrol quadrature point using special, well-calibrated active feedback methods [63, 64].
Hence, the beat-note cross correlation and Michelson interferometer detection methods
are robust methods to characterize the phase or frequency noise accurately.
13.7.3. Cross-Spectrum Method
The cross-spectrum method is used to characterize the PSD of the relative intensity noiseof a laser [66]. The system configuration for this method possesses an upper and lower
arm that consists of a high-speed photodetector of low noise qualities used to detect thevarying propagating optical light signal. In some cases, a compromise can be made on the
choice of amplifier attached to this detection scheme, such as a PD with an internally546

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Wavelength Switching, and Wavelength Tuning of Lasers

packaged amplifier or a typical external stand-alone amplifier connected to the electricalpath. Considering a possible large flicker typically induced by a PD with an internally
packaged voltage amplifier, a low noise non-inverting configuration-placed amplifier ispreferred to be attached in front of the PD to amplify the detected signal in this scheme
shown in Fig. 13.27. In addition, a dual-channel Fourier frequency transform (FFT)
analyzer is utilized to measure and analyze the photo-detected and amplified signal by
conducting FFT on these two independent signals, which will then reveal the calculated
cross-spectrum of the RIN in the frequency domain.
It is assumed that for this detection configuration, the optical coupler is reasonably stable,
Hanbury-Brown Twiss effect free, and does not induce any partition noise on either arm
while operating with the intensity. Furthermore, it is also assumed that any environmental
impact does not influence the two channels. This cross-spectrum detection scheme tocharacterize the RIN seems to be of significant advantage compared to a conventional
direct detection scheme. A conventional direct detection scheme consisting of one channel
is often limited by the noise within the single channel consisting of a single PD and asingle external amplifier.

Fig. 13.27. Basic RIN measurement method [66].

13.8. Wavelength Switching and Wavelength Tuning
This section presents various methods for wavelength switching or tuning insemiconductor and fiber laser, either from the perspective of a single-wavelength
generation, dual-wavelength generation, or multi-wavelength generation. Wavelengthswitching is simply the act of reproducing the peak spectra of a particular wavelengthband into a different position within the wavelength band. In contrast, wavelength tuningis the act of gradually shifting the peak spectra of a specific wavelength band over aconsiderable range of the wavelength band. Furthermore, wavelength converted lasers
aimed to avoid wavelength blocking is another aspect of a laser wavelength that has
witnessed a sizable amount of widespread research.
13.8.1. Mechanical Switching/Tuning
In typical practice, various types of gain mediums such as ytterbium and erbium used
independently or simultaneously with each other can be exploited when it comes to547

Advances in Optics: Reviews. Book Series, Vol. 5

establishing a fiber-based laser system with attributes of wavelength switching or tuning.
However, for erbium-doped fiber lasers, the fundamental step towards attaining an
extended wavelength laser is to primarily adopt the approach of suppressing the amplified
spontaneous emission (ASE) and reducing the reabsorbing effect within the EDF. The
concept of ASE suppression is achievable by either utilizing an EDF with the parametricfiber structure of a depressed cladding or by basically employing a multistage scheme thatconsists of filters situated between each of the two stages. On the other hand, thereabsorbing effect can be eliminated by ensuring that a fiber of a shorter length isemployed to accommodate sufficient pumping that can guarantee a relatively highinversion level.
In light of this, Chen et al. [67] demonstrated an Erbium-doped fiber ring laser with awavelength switching phenomenon. The authors obtained a short wavelength band andwavelength switchable dual laser by the principle of bending-dependent loss. This was
feasible because the authors adopted a 12 m length of EDF with a depressed cladding
scheme. By spooling the EDF to a coil with a diameter of 6.25 cm, the authors were able
to induce the bending dependent loss effect on the EDF by basically pressing the fiberfrom either direction at the same time (i.e., from the top or the bottom) as shown in Fig.
13.28(a). At a fixed diameter of 6.25 cm, it appears to be that the laser system is somewhat
susceptible to the increment in pump power. The authors varied the pump power of theirlaser from 25 mW to 100 mW and discovered that the peak wavelength of their grating(i.e., 1488.9 nm) is less susceptible to the change in pump power. Even more, a constantincrease in pump power instead leads to the emergence of a new and unexpected peak that
progressively grows with the increment of pump power, as shown in Fig. 13.28(b).
In this work, the authors claimed that the dual-wavelength peaks' emergences are mainly
due to the FBG central wavelength (i.e., 1488.9 nm) and the strong reabsorbing gain effectof the EDF, which is believed to correspond to the peak wavelength of 1533.9 nm
according to the authors on an approximate explanation. When the fiber coil is reduced
from its initial diameter of 6.25 cm by a factor of 0.75 cm from the bottom up as inducedby the bending effect, the wavelength peak of 1533.9 nm completely switches to a rather
dominant wavelength peak of 1488.9 nm, as shown in Fig. 13.28 (c). However, the peakwavelength at 1533.9 nm is rather dominant than at 1488.9 nm when the diameter was at
its initial length. As the diameter decreases, this gradually switching phenomenon
progresses before reaching its lowest length of 5.50 cm, whose spectra phenomena werediscussed above. The authors emphasized that any further decrement of D beyond
5.50 cm will lead to a fall in power of the emitting peak. Also, the authors emphasizedthat the efficiency of the laser is influenced by the impurities incurred from the doped
EDF. Even more, a dual FBG should be considered as an effort to control the lasing power.
On the brighter side, this method is a cost-effective and portable approach to attainwavelength switching.
In 2007, Chen, et al. [68], with a length of 6.4 m EDF, proposed a ring cavity laser that iswavelength switchable and tunable, as shown in Fig. 13.29 (a). Compared to the consistentusage of wavelength-selective components in laser systems for wavelength switching or
tuning, such as the researches mentioned above, this structure, for the first time, instead548

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adopts a birefringent and high nonlinear Photonic crystal fiber (PCF). With this PCF,
single-wavelength lasing and dual-wavelength lasing are achieved. The dual-wavelength
peak of the laser is generated by the inclusion of the PCF, which directly creates twocavities of different lengths whose polarization state is different for each cavity. In thiswork, it is understandable that the operational mode of single and dual lasing of the laseris switched by mainly adjusting the PC attached behind the PCF, as shown in Fig. 13.29(b) and (c), respectively. This laser configuration's wavelength tunability and range
depend on the birefringence of the nonlinear PCF. The nonlinear coefficient of the PCF
was reported to be as high as 11 W-1 km-1 for the fiber length of 25 m.

(a)

(b)

(c)

Fig. 13.28. (a) Experimental setup of the switchable dual-wavelength erbium-doped fiber ringlaser; (b) Output spectrum variation with pump power; (c) Output spectrum variation with D.
Inset schematically shows the shape of the EDF loop and the definition of D [67].

Suppose we leverage the already established theory surrounding the feasibility to extend
the wavelength range of an erbium-doped fiber laser. In that case, it is not far-fetched tosuggest that the short length of the EDF will, in fact, alongside a highly birefringent PCF,
might attain a substantially wider tunable region beyond the range of 1563 nm to1569 nm, as shown in Fig. 13.30 (a). Another compelling reason surrounding theremarkable stability of the single lasing wavelength of the laser detected over some time,
as shown in Fig. 13.30 (b), is somewhat dependent on the fact that the gain medium isconsiderably short even for such a high pump power.
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(a)

(b)

(c)

(b)

(c)

Fig. 13.29. (a) Schematic configuration of the proposed fiber ring laser. LD: laser diode. WDM:
wavelength division multiplexer. OC: optical coupler. ISO: isolator. PC: polarization controller.
EDF: Erbium-doped fiber. PCF: photonic crystal fiber; (b) Output spectrum of the proposed fiber
laser with single-wavelength lasing; (c) Output spectrum of the proposed fiber laser
with dual-wavelength lasing [68].

(a)

(b)

Fig. 13.30. (a) Repeated scanning spectrum each 5 minutes. Inset shows the lasing wavelength canbe tuned by adjusting the polarization controller, and (b) Repeated scanning spectrum each
5 minutes [68].

Liu et al. [69] leveraged the multi-wavelength generation qualities of a PCF and thewavelength switching merits of a high birefringence (Hi-Bi) fiber grating to establish amulti-wavelength, stable channel spacing, and wavelength switchable EDF laser system
as shown in Fig. 13.31 (a). Quite similar to Chen, et al. [68], the authors adopted a high550

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Wavelength Switching, and Wavelength Tuning of Lasers

nonlinear coefficient PCF with a relatively long length of about 70 m. The PCF is inserted
into a loop to establish a loop mirror consisting of a rotated PC at an angle sufficient toattain multi-wavelength lasing with suitable channel spacing of 0.4 nm to 0.8 nm, as
shown in Fig. 13.31 (b) and (c). The stability of the multi-wavelength with an amplitude
variation of 0.2 dB, as shown in Fig. 13.31 (d), represents an equalization of the EDF
mode competition induced by the PCF loop mirror. By rotating the PC, this loop induces
an intensity-dependent cavity loss which is somewhat similar to that reported by Chen etal. [67]. However, slightly different since no bending effect was induced on the PCF but
instead intensity effect by the PC. The authors demonstrated and obtained a switchable
wavelength laser whose channel spacing functions at a three to four wavelength lasing
operation with the aid of the Hi-Bi and adjustment of the PC. It is understandable thatprovided the PC1 is fixed, and the PC2 is finely adjusted to an angle of 45 degrees, thechannel spacing grows even smaller (i.e., approx. 0.8 nm) with the lasing operation offour orders.
(a)
(b)

(a)

(b)

(c)

(d)

(c)

(d)

Fig. 13.31. (a) Schematic diagram of the proposed laser; (b), (c) Output spectra of the channelspacing and wavelength switchable multiwavelength EDF laser by adjusting the PC2; (d) Output
spectra of the stable multiwavelength lasing oscillation with 16 times repeated scans [69].

Indeed, a gain medium doped with a rare-earth element can provide high saturation power
and a significant gain that is also low polarization-dependent. Nevertheless, laser
structures that adopt gain mediums such as EDF lasing at room temperature are subjected
to drawbacks such as instability and intense mode competition. These drawbacks
primarily account for the homogeneous line broadening effect of the gain medium.
Provided we ignore a few aspects such as complexity and cost, it appears to be that quite
a few solutions have been put forward to mitigate these drawbacks, such as adopting a551

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four-wave mixing effect, utilizing a high-nonlinear fiber, and so on. Suppose we instead
leverage the effectiveness of the approaches mentioned above. In that case, it is regarded
that, to some extent, there have served to be of substantial impact in resolving the
drawback. More importantly, an ideal approach is to circumvent this drawback byadopting an SOA whose core attributes of inhomogeneous broadening can also facilitate
other aspects of a lasing laser’s wavelength, such as the generation and simultaneous
oscillations of multi-wavelengths.
In 2009, Zhang et al. [70] demonstrated an SOA-based fiber ring laser with multiwavelength generation and fine-tuning attributes. Based on the nonlinear polarization
rotation of an established polarizer consisting of two polarization controllers, apolarization-dependent isolator, and of course, an SOA, a multi-wavelength is easily
generated. The multi-wavelength generation is feasible by mainly adjusting the twopolarization controllers that function to adjust the polarization of the input light signal
traversing across the SOA and PDI. These two polarization controllers are placed in anonlinear direction to each other, as shown in Fig. 13.32 (a). If the polarization controllers
are appropriately adjusted, and of course in consideration of the phase difference between
the two modes (i.e., transverse electric and transverse magnetic) generated by the SOA,
the two adjusted PCs inevitably serve to guarantee that the output light signal traversing
across the PDI is decreased with respect to an increase of the light intensity. In otherwords, mode competition is avoided, leading to the ease generation and lasing of multiwavelength. Furthermore, as shown in Figs. 13.32 (b) and (c), the authors demonstrated amulti-wavelength comb laser and wavelength spacing as high as 126 and 0.08 nm,
respectively.
(a)

(b)

(c)

Fig. 13.32. (a) Experimental setup of our proposed multiwavelength SOA fiber laser. The partsurrounded by dashed line is a general configuration of nonlinear polarization rotation basedon SOA; (b) Multiwavelength output spectrum with wavelength spacing of 0.08 nm; (c) Zoom-inof the part surrounded by dashed lines in (b) [70].

Besides evaluating the stability of the multi-wavelength comb by changing thepolarization-dependent isolator to a polarization-insensitive isolator as shown in
Fig. 13.33 (a), the authors failed to demonstrate the stability of the multi-wavelength comb
over an extended duration of time. Even more, it appears to be that the length and fiber
characteristics of the polarization-maintaining fiber inserted in the Sagnac loop should
also play some decisive role alongside the polarization-dependent isolator towards thestability of the wavelength comb. More importantly, with this unique and commendable
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laser configuration, a fine wavelength tuning range as high as 20.2 nm has been achieved,
as shown in Figs. 13.33 (b)-(e).
(a)

(b)
(c)

(d)
(e)

Fig. 13.33. (a) Output spectrum from the SOA fiber laser with 0.08 nm Sagnac loop filter
when the PDI is replaced by a polarization-insensitive isolator. (b); (c); (d); (e); Tunablemultiwavelength generation through adjusting polarization [70].

In 2020, Hao, et al. [71] demonstrated a multi-wavelength tunable SOA fiber laser with aconsiderably ultra-narrow wavelength spacing of 0.081 nm and a wide tunable wavelength
range that spans from 168 nm to 250 nm. Similar to Zhang, et al. [70] the authors mitigatedthe mode competition effect induced by the SOA by mainly establishing a linear polarizer
consisting of two PCs, an SOA, and a polarizer, as shown in Fig. 13.34 (a). This linear
polarizer structure generates the intensity-dependent loss effect for the mode competition
suppression. It is also used to adjust the wavelength of the laser by mainly tuning the PC,
which directly controls the intensity of the independent-dependent loss. In principle, theorientation and components of the linear polarizer structure suggest that the direction of
propagating light signals ought to be in the forward direction. However, by adjusting thetwo PCs inserted in this structure, the authors could change the direction of the light signal.
Thus, creating a non-linear polarized light signal whose polarization state of the output
light signal is directly proportional and dependent upon the intensity of the input lightsignal. This change in orientation of the polarized light signal is only needed for ultranarrow wavelength spacing. Likewise, the corresponding tuning of the lasing lines isachieved by the induced effect of the intensity-dependent loss, whose intensity iscontrolled by adjusting the PCs to different settings.
Of course, the orientation of the propagating light signal, which ought to be a forward
light signal traversing across the linear polarizer, has its orientation changed with the aidof the polarizer attached within the structure. The multi-wavelength comb-like spectral of
the laser is generated by constructing an MZI consisting of two equally bidirectional
optical couplers with an SMF situated in between them. In this way, light traverses acrossthe interferometer in a double pass with a phase difference, creating a comb as shown in
Fig. 13.34 (b) and filtered by this same double pass MZI. The authors argued that thestability of their multi-wavelength laser, as shown in Fig. 13.34 (c), was mainly inducedby the presence of a Sagnac loop created and consisting of a considerably long fiber lengthof 600 m and a PC. Even more, the Sagnac loop also serves as a power equalizer. If we
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consider the length of the SMF the authors used in their Sagnac loop, compared to thatreported by Zhang et al. [70], the cost of a traditional telecommunication fiber is relatively
cheaper. In other words, a compromise between cost and stability needs to be considered.
The stability of this fiber laser is characterized by the progressive 3 min time intervalmeasurement over an entire duration or course of 15 min, which demonstrated little or noapparent power fluctuation, as shown in Fig. 13.34 (d).
(a)

(b)

(c)

(d)

Fig. 13.34. (a) Experimental setup of wavelength-number-tunable multi-wavelength SOA fiber
laser based on NPR effect; (b) Stable multiwavelength output with ultra-narrow wavelength
spacing and different number of laser lines; (c) Multiwavelength output spectrum without
Polarizer. (d) Repeated scans of the output optical spectra from1535 nm to1540 nm every 3 minover 15 min [71].

Yeh, et al. [72] demonstrated a hybrid fiber ring laser with a broad wavelength tuning thatspans from the C to L-band. This hybrid fiber ring laser was established by encompassinga C-band SOA and a C-band EDFA of 3 m EDF aligned serially (i.e., first stage and second
stage), as shown in Fig. 13.35 (a). This configuration was made to extend the effective
amplification range. Furthermore, the authors adopted an intracavity C-and L-bandtunable bandpass filter (TBF) with tuning ranges spanning from 1520 nm to 1560 nm and1560 nm to 1610 nm to lase and tune, respectively. In order to obtain a hybrid and wide
ASE source amplifier (i.e., consisting of both the first stage and second stage
amplification), the authors verified the threshold and maximum ASE state of the adopted
SOA. This verification was feasible by mainly varying the drive current of the first stage
SOA within the range of 80 to 200 mA. According to their findings, it appears to be thatthe wavelength of the SOA shifts to a lower wavelength region just as the current is
progressively increased. Thus, a hybrid amplifier (cascaded with the EDFA) is formed. In
other words, a bias current of 200 mA and a pump power of 25 mW on the SOA and EDF,
respectively, is sufficient for proper cascading, as shown in Fig. 13.35 (b).
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(a)

(b)

(c)

(d)

Fig. 13.35. (a) Experimental setup of CW tuning fiber ring laser scheme covering both Cand
L-bands; (b) Output ASE spectral of SOA at the driving current of 80, 140, and 200 mA
respectively. And red line is the ASE profile of proposed hybrid amplifier; (c) Output wavelength
spectra of the proposed fiber ring laser in the wavelengths of 1518.5–1610.0 nm, at the driving
current and pumping power of 200 mA and 25 mW, respectively; (d) Output power and SMSRof the proposed fiber ring laser under different lasing wavelength range of 1518.5–1610.0 nm [72].

With the aid of the TBF, a wavelength tuning range of 91.5 nm (i.e., from 1518.5 to1610.0 nm), as shown in Fig. 13.35 (c), is commendable for such a simple and costefficient system architecture. Furthermore, for such a hybrid fiber laser, the reported
maximum output power stability (characterized by its detected fluctuation) andwavelength variation in the order of 0.9 dB and 0.2 nm, respectively, is a massive leap inlaser system designs. However, although the authors demonstrated a laser system with thecharacteristics of a 62.5 dB high SMSR, there still exists room for further development in
the laser system to advance the reported 5.9 dBm in maximum output power as shown in
Fig. 13.35 (d).
Concisely and insightfully, Mao et al. [73] demonstrated an optical fiber short ring cavity
SOA laser system with a broad and continuous wavelength tuning range spanning from1498 nm to 1623 nm. The authors adopted a traveling wave SOA that acted as the gain
medium in the laser cavity, consisting of a fiber Fabry-Perot tunable filter (FFP-TF)
serving as the wavelength selective element and filter to attain a remarkable stablewideband tunable range. Compared to other authors, the authors mitigated the penalty of
ASE noise by brilliantly placing their FFP-TF with characteristics of 128 nm FSR andfinesse of 850 nm directly in front of the output coupler, as shown in Fig. 13.36 (a).
Moreover, for such a widely tuned wavelength range of 125 nm, as shown in
Fig. 13.36 (b), the laser exhibited a relatively stable output power with fluctuations
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accounted for approximately less than 2 dB and lasing at a single longitudinal mode
operational state. It appears to be that the presence of the polarization-independent isolator
functioning alongside the PC serves as a contributing factor towards profound stability
reported by the authors.

(a)

(c)

(b)

(d)

Fig. 13.36. (a) Configuration of the semiconductor fiber-ring laser; (b) The tunable semiconductor
fiber-ring laser spectrum over the 125 nm wavelength range; (c) The laser output power plottedagainst the laser emission wavelength for different driver currents. Driver currents: 250 mW,
200 mW, 150 mA, and 100 mA; (d) The measured laser output powers at different output ratios
of 0.5, 0.3, and 0.1 [73].

To further validate their laser system's effectiveness and performance level, the authors
investigated the drive current of their SOA and its corresponding output coupling ratio.
The authors discovered that at an SOA drive current of 250 mA (herein, we only
considered the maximum current investigated), the output power of their laser was foundto be 4.9 mW at an output coupling ratio of 30 %, shown in Fig. 13.36 (c). Besides theoutput coupler of 70:30 (characterized by an output coupling ratio of 0.3) used in theirlaser structure, the authors further investigated the feasibility of attaining and maintaining
such a wide tunable range. This was achieved by mainly leveraging various output
couplers such as 90:10, 50:50 which correspond to output coupling ratios of 0.1 (10 %)and 0.5 (50 %), respectively. According to the authors, it appears to be that a lower output
coupling ratio (i.e., either 0.1 or 0.3) is rather suitable if we intend to achieve an incrementin the tunable range. The reason is that cavity loss arising from a higher output coupling
ratio of 0.5 (i.e., a decrement in light signal injected back into the ring laser cavity) induces
substantial loss, which significantly impacts the cavity stability. On the other hand, a
relatively lower output coupling ratio of 0.1 demonstrated a more widely tunable range of556

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127 nm, as shown in Fig. 13.36 (d). For such a low-cost and compact ring laser, thereported output power is commendable. However, further optimization is needed if theauthors intend to meet the rapid and high demand for higher output power tunable lasers.
As much as the application needs of dual-wavelength lasers are in great demand, it isdifficult to achieve a dual-wavelength laser lasing at SLM operation because they both
adopt different methods to achieve the two output aspects of a laser simultaneously.
Therefore, various methods have been put forward to achieve these simultaneous aspectsof a laser. However, yet these methods suffer from drawbacks such as low output powerand somewhat operational complications associated with the laser's cavity. In 2017, Shenet al. [74] demonstrated a long ring cavity double SOA-based laser system with tunabledual-wavelength lasing qualities in SLM operation. A conventional SOA laser (i.e.,
SOA-S) is primarily utilized as the laser system's gain medium. In addition, for the sakeof attaining SLM operation based on spectral narrowing effect, the authors creatively
adopted a nonlinear semiconductor optical amplifier (NL-SOA) into their laser cavity to
induce the effect of inverse four-wave mixing (FWM), as shown in Fig. 13.37 (a).

(a)

(b)

(c)

(d)

Fig. 13.37. (a) Experimental setup of the proposed dual-wavelength SLM fiber laser; (b) Individualtransmission spectrum of the OCF (black solid line) and the OTF (red dotted line), and (c) overall
transmission spectrum of the OCF and OTF; (d) Output optical spectrum of the proposeddual-wavelength SLM fiber laser [74].

Naturally, such a laser structure is expected to generate a comb-like spectral similar to a
multi-wavelength laser. However, for mainly obtaining a dual-wavelength and tunable
laser, the authors endeavor to attach two key components (i.e., the TOF and the OCF) thatfunction basically in this manner. Firstly, the authors demonstrated a laser system with awavelength tuning range of approximately 56 nm by basically tuning the central
wavelength of the TOF, which in turn, filters out the various distinct number of channels
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stacked and adjacent to each other within the OCF, as shown in Fig. 13.37 (b). Secondly,
for the tuned wavelength, a channel spacing of 0.4 nm (in other words, wavelength
spacing) was achieved and dependent upon the wavelength separation of the tunable FSRof the OCF adopted in the laser system, as shown in Figs. 13.37 (c) and (d).
Ultimately, the previously reported tunable wavelength ranges at SLM operation and dualwavelength selection over a wide and narrow wavelength range, as shown in
Fig. 13.38 (a) and (b), was achieved by utilizing a tunable optical filter (TOF) and an
optical comb filter (OCF) within the laser cavity. The insertion loss and cost of this laser
system should be considered as we predict a compromise of these will not be inevitable.
Nevertheless, the authors demonstrated a relatively stable dual laser system that
tremendously mitigates the drawbacks of attaining dual-wavelength laser lasing at an
SLM through other methods. Furthermore, the reported tuned wavelength ranges andwavelength spacing for such a long ring cavity laser system are commendable.
(a)

(b)

Fig. 13.38. Wavelength tunability over (a) a wide wavelength range,and (b) a narrow wavelength range [74].

In 2012, Ummy, et al. [75] proposed and demonstrated an SOA-based dual-wavelengthswitchable and continuous wavelength-tunable fiber laser. As shown in Fig. 13.39 (a), theauthors established a relatively cost-effective laser system whose structure consists ofthree main components. These components include two Sagnac loop interferometersacting as a broadband reflection mirror, a C-band SOA acting as the gain medium whose
ASE propagates in both directions. It also consisted of wavelength selection components
such as an FBG (0.28 nm bandwidth) and a tunable filter (0.3 nm bandwidth) to switch
and tune the wavelength. Structure-wise, the second Sagnac loop is a lot similar to thefirst Sagnac loop. However, the slight difference in loop emerges from the absence of awavelength selection element and filter in the second loop whose reflectivity isdeliberately kept at 99.9 % in order to reflect, and adjust with a PC, a substantial amountof optical light signal towards the direction of the first loop.
The authors demonstrated a fiber laser with a single or dual-wavelength operation using
the two filters in the laser cavity. Thus, the wavelength reflected by the fiber gratings(reflection peak wavelength) and the wavelength selected by the thin film tunable filter(passband wavelength) establish the operation lasing modes. To be more precise, within
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the first Sagnac loop, a series of counterclockwise and clockwise propagation of lightsignals undergoes reflectance and transmittance. The ASE injected into the first Sagnac
loop encounters the FBG within the loop wherein the propagating light signal that falls
upon the resonance bandwidth of the grating (i.e., 1544.32 nm) is reflected in the same
direction of its initial propagation. In other words, the reflected signal becomes theselected wavelength as induced by the FBG. Acting as a standard mirror (i.e., the FBG),
the transmitted light signal effortlessly traverses across the grating in a propagatingmanner headed towards the thin film tunable filter.
(a)

(b)

Fig. 13.39. (a) Experimental setup of the SOA based dual loop mirror fiber laser; (b) Dualwavelength operation of the laser when the SOA is driven by a bias current of 100 mA [75].

In the counterclockwise direction, the first wavelength, also regarded as the selected
wavelength, survives a continuous circulation within the first Sagnac loop. However,since it is typically a reflected light signal (i.e., the wavelength signal reflected by the
FBG), it recombines with another reflected light signal induced by the almost totalreflection (i.e., 99.9 % reflected signal from the second Sagnac loop). This combination
produces a Michelson-like interference at the first output region of the laser cavity.
Regarding the second wavelength generation, it is essential to note that the ASE thatencounters the thin-film tunable filter mainly permits a wavelength signal(i.e., 1549.5 nm) that is selected by the tunable filter to propagate further. This selected
wavelength signal from the tunable filter effortlessly traverses across the FBG due to thedifference in resonance bandwidth of the fiber grating. A continuous circulation within
the loop ensures that either in the clockwise or counterclockwise direction, the selected
wavelength signal by the tunable filter is effortlessly recombined at the 3-dB splitter. The
PC within the first Sagnac loop is used to control the birefringence of the optical fiber,
which directly ensures that there is a control of the Sagnac loop’s reflectivity. In otherwords, a compromise in control of gain and cavity loss is feasible by adjusting the PC.
The authors were able to switch the wavelength operation of their laser system from asingle to a dual-wavelength and vice-versa by mainly adjusting the PC1 as shown in
Fig. 13.39 (b) and Figs. 13.40 (a), (b). In this way, the authors generated a dualwavelength laser system whose wavelength was induced due to the reflected wavelength
by the FBG and the selected wavelength by the tunable filter.
The wavelength tunability over the entire C-band was feasible by fixing the wavelengthof the FBG and finely tuning the band-pass filter's wavelength, which increases ordecreases the wavelength separation between the two adjacent generated wavelengths asshown in Figs. 13.41 (a)-(c). The separation between the dual adjacent wavelength is559

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dependent on the bandwidth of the filters. Provided a narrower bandwidth filter is adopted,
the minimum separation can be further reduced. In other words, a broader tunability isarguably feasible. The stability of the total output power of 6.9 dBm (characterized by avalue of 0. 33 dB) and wavelength stability (characterized by a value of 0.014 nm) is
commendable for such a distinguished laser configuration. This laser provides a rather
substantial cost-efficient and compact laser system with multiple functionalities.
(a)

(b)

Fig. 13.40. Single-wavelength operation of the fiber laser (a) wavelength selected by the FBG;(b) wavelength selected by the thin film based tunable filter [75].

(a)

(b)

(c)

Fig. 13.41. Variable wavelength difference Δλ obtained by tuning the tunable thin film filter.
(a) 5.3 nm, (b) 10.28 nm, (c) 15.28 nm [75].

13.8.2. Acousto-optic Switching
Delgado-Pinar et al. [76] developed a fiber laser system with the capacity of achieving
wavelength switching by adopting acoustic waves, as shown in Fig. 13.42. The authorscoined this method as AOSLM, an acronym that stands for acousto-optic superlattice
modulator. The wavelength is switched by controlling the relative peak reflectivity
between the sidebands originating from the AOSLM. Using this method, the authors
predict the feasibility of adjusting the wavelength spacing between adjacent lasing
wavelengths by mainly tuning the frequency of the RF signal. Undoubtedly, the acoustic
wave is based on a Bragg grating upon which the propagation occurs. The authors
employed a tapered boron codoped germanosilicate fiber. According to the authors, thefundamental essence of the tapering that is characterized by a 55 mm uniform waist, 100µm diameter, and a transition length of 12 mm, is to ensure that acousto-optic interaction
is enhanced in the FBG. The grating length, bandwidth, and reflectivity are 5 m, 0.1 nm,
and >30 dB, respectively.
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Fig. 13.42. Experimental setup [76].

In the fiber ring laser system, the authors endeavored to employ a piezoelectric disk driven
by an RF signal meticulously connected to the fiber grating through a silica horn togenerate a longitudinal acoustic wave that eventually propagates across the tapered FBG.
The authors argued that the acoustic reflection at the Bragg-based AOSLM system'stransitions is significantly reduced because the diameter of the tapered transitions
gradually decreases. Ultimately, sufficient acoustic power from the excitation system(characterized by the PZ-disk, silica fiber ‘Horn,’ and RF supply) effectively traverses
into the FBG.
The performance and findings of this acousto-optic superlattice modulator were
investigated firstly, as shown in Fig. 13.43 (a), (b), and (c). According to the authors, thereflection band of the FBG and fourth-order sidebands, as shown in Fig. 13.43 (a), is
reduced to 16 dB and detectable, respectively. The appropriate minimal voltage needed toensure reflectivity of the bands within this fiber laser configuration was explored, as
shown in Fig. 13.43 (b), to guarantee the laser's wavelength switching process. The
authors discovered that 7 V is suitable for the central band, from 7 to 17 V is suitable forthe first-order sidebands, and from 17 to 24 V is suitable for the second-order sideband.
What is more, the authors demonstrated the tunability of the first-order sidebands and thesecond-order sidebands, as shown in Fig. 13.43 (c). The linearity between the wavelength
shift from the central band and the RF frequency exists to reveal rates of 0.142 and0.285 nm/MHz for the first-order sideband and the second-order sideband, respectively.
With the change in applied voltage, the authors demonstrated a switchable wavelength
laser. A single wavelength peak corresponding to the central band exists for situations
wherein no voltage is applied, as shown in Fig. 13.44 (a). However, an increase in voltage
inevitably generates another wavelength peak regarded as the first-order right-handsideband, which creates a dual lasing wavelength laser in addition to the existing central
band, as shown in Fig. 13.44 (b). The authors claimed that the gain effect of the EDFA
ensured that the first order left-hand sideband does not exist since more gain is given to
the right-hand sideband only. Nevertheless, their laser system could lase a first-order lefthand sideband only when an additional FBG was cascaded into the ring laser cavity anddiminishes the right-hand sideband based on reflection and elimination by the circulator.
A switch between the first-order sideband, first and second-order sidebands, and a single
peak corresponding to the second-order sideband demonstrated by the authors is feasible
with the increase in voltage amplitude as shown in Fig. 13.44 (c), (d), and (e), respectively.
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Furthermore, the authors demonstrated the transient regime that occurs when a switchfrom the central band to the second-order sideband of their laser exists, as shown in theinset of Fig. 13.44.

(a)

(b)

(c)

Fig. 13.43. (a) Reflection spectra when an RF signal of frequency 1.017 MHz and amplitude 15 V
is applied to the piezoelectric (solid line). The reflection spectrum of the FBG is also shown (dashedline); (b) Peak reflectivity of the “central” band (triangles), and the first- (open circles)
and second-order (squares) sidebands against voltage applied to the piezoelectric, at a constantfrequency of 1.017 MHz; (c) Wavelength shift of the first- (open circles) and second-order(squares) sidebands as a function of frequency [76].

(a)
(b)
(c)
(d)
(e)

Fig. 13.44. Laser emission spectra for different voltages applied to the piezoelectric. Inset shows
the transient regime when the laser emission is switched from (a) to (e) [76].

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13.8.3. All Optical Tuning
Members of our research group, Li, et al. [77], established an all-fiber optical-controllable
laser assisted by graphene for wavelength tuning in a Q-switched mode-locking state laser.
The employed element for wavelength tuning in this work was mainly fabricated byattaching graphene onto the surface of a micro-FBG (MFBG) which was firstly acommercial Bragg grating whose reflectivity is 90 %, bandwidth is 0.2 nm, and lasing at
a wavelength of 1550 nm as shown in Fig. 13.45 (a). In order to avoid evanescent waves
from escaping the grating, a part of the cladding is etched in a hydrofluoric acid of 8 %concentration. In this way, with the aid of the graphene micro-FBG (GMFBG), thewavelength of the laser system, as shown in Fig. 13.45 (b), was easily tuned by adjusting
the intensity of the 1530 nm pump laser. In addition, the pump power was enhanced by a1530 nm EDFA to interact with the graphene appropriately.

(a)

(b)

(c)

(d)

Fig. 13.45. (a) Diagram schematic of the GMFBG, and the inset is the photograph of junction
region of the GMFB and MFBG; (b) Experimental schematic of the GMFBG-based wavelengthtunable passively QML fiber laser; (c) Output laser spectra under pump powers with a range
from 6.5 mW to 99.7 mW; (d) Peak wavelengths at different controlling pump powers with twotuning processes and the experimental data is linearly fitted by curve 1 and 2, respectively [77].

Furthermore, the authors discovered that the etching process altered the grating's typicalinternal structure (characterized by single-mode propagation), thereby rendering thegrating as a mix-waveguide (characterized by the propagation of single and multi-mode)and somewhat induced a loss of ~2 dB. Nevertheless, the authors validated the impact of
this fabricated element towards wavelength tuning by attaching it to a fiber ring laser
system mode-locked with a saturable absorber (SA) and a 1 m length of EDF serving as
the gain medium. Furthermore, the authors emphasized that envelopes of mode-locked
pulses are easily generated in their laser system. This was achieved with the aid of a563

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fabricated single-wall carbon nanotubes saturable absorber (SWCNT-SA) coupledalongside an OC, which modulated the laser system's net cavity loss.
In order to achieve the process of optical-controlled wavelength tuning, the authors
maintained the pump power of the 980 nm pump laser at 162 mW while tuning theintensity (pump power) of the 1530 nm pump laser from 6.5 to 99.7 mW. Hence, awavelength tuned range spanning from 1550.348 nm to 1552.520 nm was obtained asshown in Fig. 13.45 (c), with a relatively high wavelength tuning sensitivity of
23.5 pm/mW, which is dependent on the GMFBG's response time in the order of a few
milliseconds as shown in Fig. 13.45(d). The authors further mentioned that the filter
situated in front of the GMFBG is used to mainly tune the wavelength of the laser lasing
at a Q-switched mode-locking (QML) state. As a result, the stability of their laser system
had a fluctuation in the order of ∼±0.04 dB, which is rather commendable even for alab-based setting.
Li, et al. [78] employed graphene-coated micro FBG (GCMFBG) as shown in
Fig. 13.46 (a) to achieve a light-controlled wavelength-tunable fiber laser. An infraredcontinuous-wave light source continuously drives this fabricated device (GCMFBG). This
light source leverages the graphene wideband absorption property. In other words, theevanescent field of the controlling infrared light source which interacts with the grapheneensures that the central reflected wavelength is tuned when the photo-thermal effect ofgraphene is stimulated. This is mainly feasible by etching a standard FBG in hydrofluoric
acid to decrease the FBG's cladding, thereby creating a micro-FBG with a diameter of~15.6 µm, ensuring that the graphene is conveniently wrapped unto the surface of theetched FBG. Therefore, the evanescent field of the controlling light can adequatelyinteract with graphene, which is coated on a micro FBG, whose reflection spectrum andenergy distribution of the controlling light within the FBG are shown in Fig. 13.46 (b) and(c), respectively.
(a)

(b)

(c)

Fig. 13.46. (a) Experimental schematic of the fabricated GCMFBG and (b) its reflection spectrum;(c) Circular symmetrical energy distribution of the controlling light on the cross section [78].

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The fabricated GCMFBG in this work is attached to a fiber laser system similar to theexperimental scheme used by the same authors(s) in their previous work [77]. Comparedto their previous work, the modified experimental scheme of this present work, as shownin Fig. 13.47, employs an EDFA to amplify the laser and provide sufficient gain. It also
consists of a 20 m length of polarization maintaining fiber and two polarizationmaintaining devices (i.e., optical circulator 2 and optical coupler 4), forming a Brillouinlaser cavity for this laser configuration.

Fig. 13.47. Experimental setup of the light-controlled wavelength-tunable
narrow linewidth fiber laser [78].

The principle of operation in terms of wavelength tuning is somewhat similar, wherein
the controlling pump power is increased from 0 mW to 275.4 mW. In this way, thewavelength of the laser is optically tuned from 1549.936 nm to 1553.608 nm with wellpreserved linearity, as shown in Fig. 13.48 (a) and (b), respectively. This laser's
wavelength tuning speed depends on the switch time of graphene, which is ~ 10 ms.
Excluding the imperfections induced by EDFA and other environmental factors, thisfabricated device does not in any way alter the SLM operation and frequency stability of
the laser but rather enhances it, as shown in Fig. 13.48 (c) and (d).
13.8.4. Thermal Tuning
In 2017, Sun et al. [79] demonstrated a 1.5 µm narrow-linewidth fiber ring laser thatconsists of a π-phase-shifted fiber Bragg grating (π-PSFBG) which serves as a narrow
band filter, and a wavelength tuning component whose central wavelength is thermallytuned by a thermo-electric cooler (TEC). The fiber ring laser system presented in this work
consists of C-band active and passive fiber components, as shown in Fig. 13.49 (a). Likea traditional FBG, the π-PSFBG employed in this work possesses a typical periodic
refractive index modulation (i.e., grating period) along the fiber axis with a π-phase shift
whose length is odd times half of the grating period as shown in Fig. 13.49 (b).

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(a)

(b)

(c)

(d)

Fig. 13.48. (a) Tuned spectra at output 1 under different controlling pump powers;(b) Corresponding wavelength positions with different controlling pump powers; (c) Spectra
at output 1 and 2; (d) RF spectrum at output 2 with a frequency span of 1 GHz measured
by the delayed self-heterodyne system [78].

(a)

(b)

Fig. 13.49. (a) Experimental setup of the tunable narrow linewidth fiber laser. EDFA, erbiumdoped fiber amplifier; TEC, thermo-electric cooler; OSA, optical spectrum analyzer; PM, power
meter; HRFBG, highly reflective fiber Bragg grating; π-PSFBG, π-phase-shifted fiber Bragg
grating; (b) Schematic diagram of a π-PSFBG [79].

Such a grating structure creates an ultra-short resonant cavity along the fiber axis, which
induces an ultra-narrow transmission band in the central band of the π-PSFBG, whose
3-dB bandwidth is ~2.5 pm. According to the authors, the shape of this transmission band
spectrum is unchanged irrespective of the increase or decrease in temperature-induced on
it since the relationship between the length of the phase shift and grating period is constant.
Furthermore, the authors emphasized that the renowned and acceptable temperature
coefficient of the refractive index (α ≈1.1 × 10-5/°C) and the thermal expansion coefficientof silica fibers (β≈5.2 × 10-7/°C) are believed to be constant for the investigatedtemperature range in their laser configuration. In order words, the π-PSFBG employed inthis work is not damaged by the thermal effect, which presents the excellent durability ofthis laser system. However, when heat is induced on the grating with the aid of thethermoelectric cooler, which increases temperature, the shape of the transmission
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Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization,
Wavelength Switching, and Wavelength Tuning of Lasers

spectrum will undergo a red or blue shift along the fiber axis and inevitably result in ashift of the central wavelength. In essence, the central wavelength shift is feasible simply
because the grating period and the effective refractive index of the fiber core, which
determines the central wavelength, change in response to the induced thermal effect.
The authors employed two fiber gratings alongside an optical circulator to determine thelasing wavelength. The light signal is initially reflected and filtered by a high reflective
FBG (HR-FBG) with a bandwidth of ~0.18 nm. In contrast, the π-PSFBG is employed toserve as a narrow band filter and wavelength tuning component. The authors emphasizedthat a narrow linewidth laser whose wavelength is tunable is feasible only when thereflection band of the HR-FBG coincides with the narrow transmission band of the
π-PSFBG as shown in Fig. 13.50 (a) and (b).
(a)

(b)

Fig. 13.50. Spectra of (a) a π-PSFBG, where the blue solid line shows it measured with a tunable
laser with a resolution of 1 pm. Pink dashed line: measured with an OSA; (b) HR-FBG measured
with an OSA with a resolution of 0.02 nm [79].

Hence, these two gratings are separately placed on two refrigerator ceramic wafers,
somewhat like a substrate whose temperatures are controlled by two TECs. This renders
the simultaneous control of the central wavelength of these two gratings whose
wavelength shift demonstrated a linear bathochromic shift with a wavelength shiftcoefficient of ~12 pm/°C when the temperature values of the TECs are adjusted as shownin Fig. 13.51 (a) and (b).
The detected output spectral and wavelength tuning as a function of the temperature riseis shown in Fig. 13.52 (a) and (b) wherein the authors demonstrated that a gradualincrement in temperature of the π-PSFBG from 5 °C to 95 °C obtained a maximum tunable
range of 1.29 nm with a sensitivity of approximately 14.33 pm/°C. The authors further
emphasized that for the previously mentioned temperature tuned range, the laser system
required several seconds to obtain a stable laser output for each temperature changeof 10 °C.
The authors attached a second π-PSFBG with a transmission bandwidth of ~4 pm, as
shown in Fig. 13.53 (a) into their laser scheme, which serves as a reference narrow
bandpass filter. This reference grating was inserted in between the two couplers. Like the567

Advances in Optics: Reviews. Book Series, Vol. 5

other two initial gratings, it likewise has a TEC whose temperature value was adjusted toensure that the central wavelength coincides with the laser for each wavelength stability
observation process when the pump power is varied. In order words, the wavelengthstability can be characterized by observing the laser output power.

Fig. 13.51. Measured central wavelength of (a) a π-PSFBG
and (b) a HR-FBG with temperature [79].

(a)

(b)

Fig. 13.52. (a) Measured laser spectra as the temperature rises; (b) The laser wavelength as afunction of the temperature of the π-PSFBG [79].

As shown in Fig. 13.53 (b) and (c), it is observable that the maximum power fluctuation
is approximately 5%, suggesting that the maximum wavelength fluctuation isapproximately 0.2 pm (~25 MHz) and consistent with the transmission spectrum of thereference π-PSFBG. In this way, we can deduce that the wavelength of the laser system isvery stable. More work is to be done to expand the wavelength-tunable range. However,other commendable features of this laser are its high output power of approximately
1.1 W and low power fluctuation of < 3%, which is an impressive representation of thewatt-level output power stability even for such a low-cost laser configuration.
Furthermore, this laser tuning method is more straightforward and poses minor damage
when objectively compared with other wavelength tuning methods.

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Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization,
Wavelength Switching, and Wavelength Tuning of Lasers

(a)

(b)

(c)

Fig. 13.53. (a) Measured transmission spectrum of the reference π-PSFBG. The measured laser
power stabilities after passing through the reference π-PSFBG: (b) three power stages with same
wavelength, inset: zoom in near the power of 740 mW; (c) three different wavelengths with near
powers. All of the measurement durations are about 44 min [79].

13.9. Conclusion
We reviewed the advances on the laser parameters of semiconductor and fiber lasers byconsidering a few superior parameters such as linewidth, phase or frequency noise,
relative intensity noise, and wavelength. Improving a laser parameter is highly desirable
in modern-day systems such as optical communication and optical sensing systems, whose
performance level depends significantly on the parameters of the laser employed. Hence,various methods used to improve the performance of laser parameters for widespread
applications were presented in this work. Therefore, we predict that the future
development directions of lasers will focus on the persistent advances in material andstructural design, linewidth compression, high output power, low phase or frequency
noise, low relative intensity noise, wavelength switching, wavelength conversion, andwavelength tuning.

Acknowledgements
The first author will like to acknowledge his loving parents, Rev. Engr. Joseph Chidozie
Iroegbu and Bishop Agnes Benedicta Iroegbu for their loving support and unwavering
encouragement throughout this project. His siblings, Dr. Engr. Udochukwu Frank
Iroegbu, Chukwuemeka Daniel Iroegbu MD., Ph.D., and Pharm. Chioma Phoebe Iroegbu
was a tremendous source of inspiration and motivation to him.
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Advances in Optics: Reviews. Book Series, Vol. 5

This work was supported by the Ministry of Science and Technology(2016YFC0801202); Natural Science Foundation of China (61635004, 61825501,
61705024); Fundamental Research Funds for the Central Universities(106112017CDJZRPY0005).

References[1]. M. S. Zubairy, A Very Brief History of Light, Optics in Our Time (M. D. Al-Amiri,
M. M. El-Gomati, M. S. Zubairy, Eds.), Springer Nature, 2016.
[2]. S. Lee, Introduction to Laser Physics with Applications in Fiber Optics, Lulu.com.
[3]. O. Svelto, Principles of Lasers, 5th Ed., Springer Science + Business Media LLC, New York,2010.
[4]. G. P. Agarwal, Fiber Optic Communication Systems, 3 rd Ed., John Wiley & Sons Inc., 2003.
[5]. G. Kweon, Noise figure of optical amplifiers, Journal of the Korean Physical Society,Vol. 41, Issue 5, 2002, pp. 617-628.
[6]. B. Utreja, H. Singh, A review paper on comparison of optical amplifiers in opticalcommunication systems, Canadian Journal of Electrical and Electronics Engineering,Vol. 2, Issue 11, 2011, pp. 505-513.
[7]. S. Poole, D. Payne, R. Mears, M. Fermann, R. Laming, Fabrication and characterization of
low-loss optical fibers containing rare-earth ions, Journal of Lightwave Technology, Vol. 4,
Issue 7, 2003, pp. 870-876.
[8]. T. J. Xia, G. A. Wellbrock, A. Tanaka, M. Huang, E. Ip, D. Qian, Y. Huang, S. Zhang,
Y. Zhang, P. N. Ji, Y. Aono, S. Murakami, T. Tajima, High capacity field trials of 40.5 Tb/s
for LH distance of 1,822 km and 54.2 Tb/s for regional distance of 634 km, in Proceedingsof the Optical Fiber Communication Conference/National Fiber Optic Engineers
Conference, 2013, p. PDP5A.4.
[9]. K. Kikuchi, Digital coherent optical communication systems: Fundamentals and future
prospects, IEICE Electronics Express, Vol. 8, Issue 20, 2011, pp. 1642-1662.
[10]. T. Kimoto, G. Kobayashi, T. Kurobe, T. Mukaihara, S. Ralph, Narrow Linewidth tunable
DFB laser array for PDM-16QAM transmission, in Proceedings of the 18th Conference on
OptoElectronics and Communications, 2013, p. MK2_6.
[11]. E. Desurvire, J. R. Simpson, P. C. Becker, High-gain erbium-doped traveling-wave fiber
amplifier, Optics Letters, Vol. 12, Issue 11, 1987, pp. 888-890.
[12]. B. Glance, Polarization independent coherent optical receiver, Journal of Lightwave
Technology, Vol. 5, Issue 2, 1987, pp. 274-276.
[13]. A. Chraplyvy, The coming capacity crunch, in Proceedings of the 35th European Conference
on Optical Communication (ECOC’09), Vienna, Austria, 2009.
[14]. P. J. Winzer, Making spatial multiplexing a reality, Nature Photonics, Vol. 8, Issue 5, 2014,
pp. 345-348.
[15]. M. Seimetz, laser linewidth limitations for optical systems with high-order modulation
employing feed forward digital carrier phase estimation, in Proceedings of the Optical Fiber
Communication/National Fiber Optic Engineers Conference (OFC/NFOEC’08), San Diego,
California, United States, 24-28 February 2008, pp. 1-3.
[16]. S. Hancock, K. Anderson, K. J. Gaston, M. Disney, Measurement of fine-spatial-resolution
3D vegetation structure with airborne waveform lidar: Calibration and validation withvoxelised terrestrial lidar, Remote Sensing of Environment, Vol. 188, 2017, pp. 37-50.
[17]. M. Jaboyedoff, T. Oppikofer, A. Abellán, M. H. Derron, A. Loye, R. Metzger, A. Pedrazzini,
Use of LIDAR in landslide investigations: A review, Journal of the International Society for
the Prevention and Mitigation of Natural Hazards, Vol. 61, Issue 1, 2012, pp. 5-28.

570

Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization,
Wavelength Switching, and Wavelength Tuning of Lasers[18]. E. Dale, W. Liang, D. Eliyahu, A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, D. Seidel,
L. Maleki, Ultra-Narrow Line Tunable Semiconductor Lasers for Coherent LIDAR
Applications, in Proceedings of the Conference on Imaging and Applied Optics, 2014,
p. JTu2C.3.
[19]. D. Chen, Q. Liu, X. Fan, and Z. He, Distributed fiber-optic acoustic sensor with enhanced
response bandwidth and high signal-to-noise ratio, Journal of Lightwave Technology,
Vol. 35, Issue 10, 2017, pp. 2037–2043.
[20]. H. F. Martins, S. Martín-López, P. Corredera, M. L. Filograno, O. Frazão, and M. GonzalezHerráez, Phase-sensitive optical time domain reflectometer assisted by first-order Raman
amplification for distributed vibration sensing over 100 km, Journal of Lightwave
Technology, Vol. 32, Issue 8, 2014, pp. 1510–1518.
[21]. J. Song, W. Li, P. Lu, Y. Xu, L. Chen, and X. Bao, Long-range high spatial resolution
distributed temperature and strain sensing based on optical frequency-domain reflectometry,
IEEE Photonics Journal, Vol. 6, Issue 3, 2014, pp. 1–8.
[22]. Z. Zhang, X. Fan, and Z. He, Long-Range Distributed Static Strain Sensing With < 100 NanoStrain Resolution Realized Using OFDR, Journal of Lightwave Technology, Vol. 37,
Issue 18, 2019, pp. 4590-4596.
[23]. Z. Wang, L. Zhang, S. Wang, N. Xue, F. Peng, M. Fan, W. Sun, X. Qian, J. Rao, Y. Rao,
Coherent Φ-OTDR based on I/Q demodulation and homodyne detection, Optics Express,
Vol. 24, Issue 2, 2016, pp. 853-858.
[24]. F. Peng, H. Wu, X. Jia, Y. Rao, Z. Wang, Z. Peng, Ultra-long high-sensitivity Φ-OTDR forhigh spatial resolution intrusion detection of pipelines, Optics Express, Vol. 22, Issue 11,2014, pp. 13804-13810.
[25]. H. Wu, S. Xiao, X. Li, Z. Wang, J. Xu, Y. Rao, Separation and determination of the disturbing
signals in phase-sensitive optical time domain reflectometry (Φ-OTDR), Journal ofLightwave Technology, Vol. 33, Issue 15, 2015, pp. 3156-3162.
[26]. M. Chen, A. Masoudi, G. Brambilla, Performance analysis of distributed optical fiber acoustic
sensors based on φ-OTDR, Optics Express, Vol. 27, Issue 7, 2019, pp. 9684-9695.
[27]. R. Passy, N. Gisin, J. P. Von Der Weid, H. Gilgen, Experimental and theoreticalinvestigations of coherent OFDR with semiconductor laser sources, Journal of Lightwave
Technology, Vol. 12, Issue 9, 1994, pp. 1622-1630.
[28]. K. Takada, High-resolution OFDR with incorporated fiber-optic frequency encoder, IEEE
Photonics Technology Letters, Vol. 4, Issue 9, 1992, pp. 1069-1072.
[29]. S. Wang, X. Fan, Q. Liu, Z. He, Distributed fiber-optic vibration sensing based on phaseextraction from time-gated digital OFDR, Optics Express, Vol. 23, Issue 26, 2015,
pp. 33301-33309.
[30]. R. Linker, A. Klar, Detection of sinkhole formation by strain profile measurements using
BOTDR: simulation study, Journal of Engineering Mechanics, Vol. 143, Issue 3, 2017,
B4015002.
[31]. S. Z. Yan, L. S. Chyan, Performance enhancement of BOTDR fiber optic sensor for oil andgas pipeline monitoring, Optical Fiber Technology, Vol. 16, Issue 2, 2010, pp. 100-109.
[32]. F. Wang, C. Zhu, C. Cao, X. Zhang, Enhancing the performance of BOTDR based on the
combination of FFT technique and complementary coding, Optics Express, Vol. 25, Issue 4,
2017, pp. 3504-3513.
[33]. M. A. Soto, G. Bolognini, F. Di Pasquale, and Luc Thévenaz, Simplex-coded BOTDA fiber
sensor with 1 m spatial resolution over a 50 km range, Optics Letters, Vol. 35, Issue 2, 2010,
pp. 259–261.
[34]. M. A. Soto, G. Bolognini, and F. D. Pasquale, Long-range simplex-coded BOTDA sensor
over 120 km distance employing optical preamplification, Optics Letters, Vol. 36, Issue 2,
2011, pp. 232–234.

571

Advances in Optics: Reviews. Book Series, Vol. 5[35]. Y. Dong, L. Chen, and X. Bao, Extending the sensing range of Brillouin optical time-domain
analysis combining frequency-division multiplexing and in-line EDFAs, IEEE Journal ofLightwave Technology, Vol. 30, Issue 8, 2012, pp. 1161–1167.
[36]. A. Signorini, S. Faralli, M. A. Soto, G. Sacchi, F. Baronti, R. Barsacchi, A. Lazzeri,
R. Roncella, G. Bolognini, and F. Di Pasquale, 40 km Long-Range Raman-Based Distributed
Temperature Sensor with Meter-Scale Spatial Resolution, in Proceedings of the Conference
on Optical Fiber Communication, 2010, p. OWL2.
[37]. Y. Liu, L. Ma, C. Yang, W. Tong, and Z. He, Long-range Raman distributed temperature
sensor with high spatial and temperature resolution using graded-index few-mode fiber,
Optics Express, Vol. 26, Issue 16, 2018, pp. 20562–20571.
[38]. J. Geng, C. Spiegelberg, S. Jiang, Narrow linewidth fiber laser for 100-km optical frequency
domain reflectometry, IEEE Photonics Technology Letters, Vol. 17, Issue 9, 2005,
pp. 1827-1829.
[39]. Y. H. Gu, J. Z. Dai, Z. Y. Dai, Development and application of optical frequency domainreflectometer with high resolution, Infrared, Vol. 30, Issue 4, 2009, pp. 30-40.
[40]. X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, D. A. Jackson, Experimental and theoreticalstudies on a distributed temperature sensor based on Brillouin scattering, Journal ofLightwave Technology, Vol. 13, Issue 7, 1995, pp. 1340-1348.
[41]. S. Martin-Lopez, M. Alcon-Camas, F. Rodriguez, P. Corredera, J. D. Ania-Castañon,
L. Thévenaz, and M. Gonzalez-Herraez, Brillouin optical time-domain analysis
assisted by second-order Raman amplification, Optics Express, Vol. 18, Issue 18, 2010,
pp. 18769-18778.
[42]. Z. Zhao, Y. Dang, M. Tang, L. Wang, L. Gan, S. Fu, C. Yang, W. Tong, and C. Lu, Enabling
Simultaneous DAS and DTS Through Space-Division Multiplexing Based on Multicore
Fiber, Journal of Lightwave Technology, Vol. 36, Issue 24, 2018, pp. 5707-5713.
[43]. A. Yariv, Optical Electronics in Modern Communications, Oxford University Press, 1997.
[44]. P. Goldberg, P. W. Milonni, B. Sundaram, Theory of the fundamental laser linewidth,
Physical Review A, Vol. 44, Issue 3, 1991, pp.1969-1985.
[45]. B. Sprenger, H. G. L. Schwefel, Z. H. Lu, S. Svitlov, L. J. Wang, CaF 2 whispering-gallerymode-resonator stabilized-narrow-linewidth laser, Optics Letters, Vol. 35, Issue 17, 2010,
pp. 2870-2872.
[46]. W. Liang, V. S. Ilchenko, D. Eliyahu, A. A. Savchenkov, A. B. Matsko, D. Seidel, L. Maleki,
Ultralow noise miniature external cavity semiconductor laser, Nature Communications,Vol. 6, 2015, 7371.
[47]. Z. Wang, C. Wang, D. Wang, P. Lu, X. Yu, L. XU, Y. Yang, Y. Jiang, L. Zhu, Z. Li,
Line-width compression of the distributed feedback laser with an external parallel feedback
cavity, Proceedings of the SPIE, Vol. 8192, 2011, 81923L.
[48]. K. Kasai, M. Nakazawa, M. Ishikawa, H. Ishii, 8 kHz linewidth, 50 mW output, full C-band
wavelength tunable DFB LD array with self-optical feedback, Optics Express, Vol. 26,
Issue 5, 2018, pp. 5675-5685.
[49]. W. Fan, J. Gan, Z. Zhang, X. Wei, S. Xu, Z. Yang, Narrow linewidth single frequency
microfiber laser, Optics Letters, Vol. 37, Issue 20, 2012, pp. 4323-4325.
[50]. M. Poulin, S. Ayotte, C. Latrasse, Y. Painchaud, J. F. Cliche, A. Babin, M. Aubé,
M. Picard, F. Costin, M. Têtu, G. Lafrance, Compact narrow linewidth semiconductor laser
module, Proceedings of the SPIE, Vol. 7325, 2009, 73250O.
[51]. T. Zhu, S.H. Huang, L.L. Shi, W. Huang, M. Liu, K.S. Chiang, Rayleigh backscattering: amethod to highly compress laser linewidth, Chinese Science Bulletin, Vol. 59, 2014,
pp. 4631-4636.
[52]. T. Zhu, X. Bao, L. Chen, H. Liang, Y. Dong, Experimental study on stimulated Rayleighscattering in optical fibers, Optics Express, Vol. 18, Issue 22, 2010, pp. 22958-22963.

572

Chapter 13. Advances in Linewidth Compression, Linewidth Measurement, Noise Characterization,
Wavelength Switching, and Wavelength Tuning of Lasers[53]. T. Zhu, F. Y. Chen, S.H Huang, X. Y. Bao, An ultra-narrow linewidth fiber laser based onRayleigh backscattering in a tapered optical fiber, Laser Physics Letters, Vol. 10, Issue 5,2013, 055110.
[54]. S. Huang, T. Zhu, G. Yin, T. Lan, L. Huang, F. Li, Y. Bai, D. Qu, X. Huang, F. Qiu, Tens ofhertz narrow-linewidth laser based on stimulated Brillouin and Rayleigh scattering, Optics
Letters, Vol. 42, Issue 24, 2017, pp. 5286-5289.
[55]. S. H. Huang, T. Zhu, G. Yin, T. Lan, F. Li, L. Huang, M. Liu, Dual-cavity feedback assisted
DFB narrow linewidth laser, Scientific Reports, 2017, Vol. 7, 1185.
[56]. F. Riehle, Frequency Standards, Basics and Applications, Wiley, 2004.
[57]. A. H. Ali, S. N. Wahid, Analysis of self-homodyne and delayed self-heterodyne detections
for tunable laser source linewidth measurements, IOSR Journal of Engineering, Vol. 2,
Issue 10, 2012, pp. 01-06.
[58]. X. Chen, Ultra-Narrow Laser Linewidth Measurement, American State University, 2003.
[59]. D. Derickson, Fiber Optic Test and Measurement, Prentice Hall PTR, Upper Saddle River,1998.
[60]. S. H. Huang, T. Zhu, Z. Cao, M. Liu, M. Deng, J. Liu, X. Li, Laser linewidth measurement
based on amplitude difference comparison of coherent envelope, IEEE Photonics Technology
Letters, Vol. 28, Issue 7, 2016, pp. 759-762.
[61]. S. H. Huang, T. Zhu, M. Liu, W. Huang, Precise measurement of ultra-narrow laser linewidths
using the strong coherent envelope, Scientific Reports, Vol. 7, 2017, 41988.
[62]. R. Hui, M. O’Sullivan, Characterization of Optical Devices, Fiber Optic Measurement
Techniques (R. Hui, M. O’Sullivan,), Academic Press, 2009.
[63]. X. Xie, R. Bouchand, D. Nicolodi, M. Lours, C. Alexandre, Y. L. Coq, Phase noisecharacterization of sub-hertz linewidth lasers via digital cross correlation, Optics Letters,Vol. 42, Issue 7, 2017, pp. 1217-1220.
[64]. D. Xu, F. Yang, D. Chen, F. Wei, H. Cai, Z. Fang, R. Qu, Laser phase and frequency noise
measurement by Michelson interferometer composed of a 3×3 optical fiber coupler, Optics
Express, Vol. 23, Issue 17, 2015, pp. 22386-22393.
[65]. O. Llopis, P. H. Merrer, H. Brahimi, K. Saleh, and P. Lacroix, Phase noise measurement of anarrow linewidth CW laser using delay line approaches, Optics Letters, Vol. 36, Issue 14,
2011, pp. 2713–2715.
[66]. E. Rubiola, K. Volyanskiy and L. Larger, Measurement of the laser relative intensity noise,in Proceedings of the IEEE International Frequency Control Symposium Time Forum, 2009,
pp. 50-53.
[67]. S.-P. Chen, L. Ding, S.-M. Wang, Y.-G. Li, K.-C. Lu, S.-H. Zhou, Switchable
dual-wavelength erbium-doped fiber ring laser using bending dependent loss to switch theoperating wavelength, Laser Physics Letters, Vol. 3, 2006, 584.
[68]. D. Chen, L. Shen, Switchable and tunable Erbium-doped fiber ring laser incorporating abirefringent and highly nonlinear photonic crystal fiber, Laser Physics Letters, Vol. 4, 2007,
pp. 368-370.
[69]. Z.-Y. Liu, Y.-G. Liu, J.-B. Du, S.-Z. Yuan, X.-Y. Dong, Channel-spacing and wavelengthswitchable multiwavelength erbium-doped fiber laser using sampled Hi-Bi fiber grating andphotonic crystal fiber loop mirror, Laser Physics Letters, Vol. 5, Issue 2, 2007, 122.
[70]. Z. Zhang, J. Wu, K. Xu, X. Hong, J. Lin, Tunable multiwavelength SOA fiber laser withultra-narrow wavelength spacing based on nonlinear polarization rotation, Optics Express,
Vol. 17, Issue 19, 2009, pp. 17200-17205.
[71]. H. Hao, X. Zhou, M. Bi, M. Hu, G. Yang, Y. Lu, T. Wang, Wavelength-number-tunablemulti-wavelength SOA fiber laser based on NPR effect, Optical Fiber Technology, Vol. 58,
2020, 102296.

573

Advances in Optics: Reviews. Book Series, Vol. 5[72]. C. H. Yeh, C. W. Chow, J. H. Chen, K. H. Chen, S. S. Lu, Broadband C-plus L-band CW
wavelength-tunable fiber laser based on hybrid EDFA and SOA, Optical Fiber Technology,
Vol. 19, Issue 4, 2013, pp. 359-361.
[73]. X. Mao, S. Zhao, S. Yuan, X. Wang, P. Zheng, Continuously tunable wideband
semiconductor fiber-ring laser, Laser Physics, Vol. 27, Issue 8, 2017, 085102.
[74]. Z. Shen, L. Wang, X. Wang, Y. Cao, X. Feng, B. Guan, Tunable dual-wavelength
single-longitudinal-mode fiber laser based on spectral narrowing effect in a nonlinear
semiconductor optical amplifier, Optics & Laser Technology, Vol. 94, 2017, pp. 72-76.
[75]. M. A. Ummy, N. Madamopoulos, M. Razani, A. Hossain, R. Dorsinville, Switchable
dual-wavelength SOA-based fiber laser with continuous tunability over the C-band atroom-temperature, Optics Express, Vol. 20, Issue 21, 2012, pp. 23367-23373.
[76]. M. Delgado-Pinar, J. Mora, A. Diez, J. L. Cruz, M. V. Andres, Wavelength-switchable fiber
laser using acoustic waves, IEEE Photonics Technology Letters, Vol. 17, Issue 3, 2005,
pp. 552-554.
[77]. Y. Li, L. Gao, T. Zhu, Y. Cao, M. Liu, D. Qu, F. Qiu, X. Huang, Graphene-assisted all-fiber
optical-controllable laser, IEEE Journal of Selected Topics in Quantum Electronics, Vol. 24,
Issue 3, 2018, pp. 1-9.
[78]. Y. Li, L. Huang, S. Huang, L. Gao, G. Yin, T. Lan, Y. Cao, I. P. Ikechukwu, T. Zhu, Tunable
narrow-linewidth fiber laser based on light-controlled graphene, Journal of Lightwave
Technology, Vol. 37, Issue 4, 2019, pp. 1338-1344.
[79]. J. Sun, Z. Wang, M. Wang, Z. Zhou, N. Tang, J. Chen, and X. Gu, Watt-level tunable 1.5 μm
narrow linewidth fiber ring laser based on a temperature tuning π-phase-shifted fiber Bragg
grating, Applied Optics, Vol. 56, Issue 32, 2017, pp. 9114-9118.

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Chapter 14. Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked
Fiber Lasers

Chapter 14

Coherence Characterization of DSR-like
Square Pulses in Passively Mode-locked
Fiber Lasers
Meriem Kemel, Mohamed Salhi, Charles Ciret, Georges Semaan,
Ahmed Nady and François Sanchez1

14.1. Introduction
Over the past decades, high-energy lasers attract considerable attention in various fields
including medicine [1-4], security and defense [5, 6], and industry [7]. Until recently, thechirped pulse amplification (CPA) [8], which won the Nobel Prize of physics in 2018,
was the most used technique to generate ultrashort high-energy optical pulses inmode-locked oscillators. Despite the effectiveness of the CPA technique, its
implementation requires several amplifiers at the oscillator output.
To make a step forward, in 2004, Fernandez et al. has reported a new technique enabling
the generation of high-energy femtosecond pulses without having recourse to external
amplifiers [9]. The principle was to use a chirped multilayer mirrors in the oscillator to
produce a net positive dispersion over a broad spectral range. Then, the output chirped
pulses are compressed, using a dispersive delay-line at the output of the laser cavity. Since
then, several studies have emerged, offering a multitude of solutions for increasingly
compact, efficient, and high-energy lasers. In fiber lasers, Chong et al. studied the key
parameters (nonlinear phase shift, spectral filter bandwidth, and group-velocity
dispersion) to determine the behavior and properties of the laser [10], thus permitting abetter comprehension of the fiber laser functioning. Meanwhile, many investigations havebeen conducted in this regard and the concept of dissipative soliton resonance (DSR) was
introduced by Chang et al. in 2008. The authors have numerically studied a new regime
in which the energy of pulses increases indefinitely for particular values of the systemparameters of the cubic-quintic complex Ginzburg-Landau equation (CGLE) [11]. They

Meriem Kemel
Laboratoire de Photonique d’Angers, Université d’Angers, Angers, France575

Advances in Optics: Reviews. Book Series, Vol. 5

have also shown that this regime can occur in both normal and anomalous dispersion
regimes [12]. Based on these first reports, the generation of high-energy square pulses has
been extensively investigated numerically [13-18], and experimentally [19-29], in various
configurations of passively mode-locked fiber lasers.
In the experimental literature of the DSR regime, square pulses are generally assumed to
be a manifestation of the DSR when they reveal some features versus the pump power.
These features correspond to a linear increase of the pulse width, a linear increase of theenergy per pulse, constant peak power, flat-top pulse shape, and invariant optical
spectrum. This behavior occurs without experiencing any wave-breaking. Besides, to beconsidered as DSR, the square pulses must be temporally coherent, i.e., there have to be
no fine structures within the square envelope. Despite the importance to verify the
temporal coherence to affirm whether the square pulses are DSR or not, only a few authors
have investigated it by providing the autocorrelation traces [13, 15, 29]. It should be notedthat they mainly reported relatively short pulse durations (few hundreds of picoseconds)
with relatively high repetition rates. However, for higher energy pulses, the laser cavities
are adjusted to obtain much larger pulses. So, most of the reported square pulses in the
DSR regime are in the range of a few ns to few hundreds of ns of pulse width, and withlow repetition rates (in kHz). However, the usual autocorrelators do not always permit acorrect measurement when the laser's repetition rate is lower than a certain value. In ourcase, the autocorrelation trace is not accurate for repetition rates lower than 1 MHz.
Moreover, even when the repetition rate is compatible with the autocorrelator
characteristics, the resulting autocorrelation trace is not enough to conclude about thecoherence of the pulse.
In theory, if the pulse is not coherent, a peak appears in the autocorrelation trace whereas
it is absent in the case of a coherent pulse. The problem is that in the experiment, when apolarization controller (PC) is inserted at the input of the autocorrelator to optimize thesignal, the presence of a peak in the autocorrelation trace depends on the adjustment of
the PC. Therefore, it is impossible to conclude whether the square pulses are coherent or
not since the absence of a peak in autocorrelation trace can have several origins.
To circumvent these problems, we successfully implemented a new experimental setup to
investigate the coherence of large square-shaped pulses in passively mode-locked Er:Yb
co-doped double-clad fiber laser in anomalous dispersion regime. The setup consists of
two different methods of measurement: Mach-Zehnder interferometer (MZI) andDispersive Fourier Transformation (DFT). The MZI method is widely used for many
applications [30-35]. It is the same concerning the DFT method which is extensively usedin fundamental research, for example, to study pulse buildup dynamics [36-40], soliton
breathing phenomena [41-43], laser instabilities [44, 45], and multipulse regimes [46-48].
In this work, we use these two methods to study the coherence of large nanosecond squarepulses [49].
The chapter is organized as follows: Section 14.2 shows the experimental setup, including
the laser cavity and both MZI and DFT techniques for coherence characterization.
The experimental results of DSR-like square pulses are presented in Section 14.3. The
results of coherence measurements performed with the MZI and DFT techniques are
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Chapter 14. Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked
Fiber Laserspresented in Section 14.4 and Section 14.5, respectively. A general conclusion is made in
Section 14.6.

14.2. Experimental Setup
The experimental setup, implemented for the coherence characterization of DSR-likesquare pulses is illustrated in Fig. 14.1. It consists of three different parts: (a) the laser
cavity, (b) the MZI, and (c) the DFT line. The laser cavity is an all-fibered unidirectionalring (UR) enabling the generation of mode-locked square pulses using the nonlinear
polarization rotation (NPR) mechanism [50]. The cavity includes a C-band double-clad
co-doped Er:Yb 30 dBm fiber amplifier (EYDFA), a polarizer and two polarization
controllers (PC1 and PC2), a polarization-insensitive isolator (PI-ISO) to ensure theunidirectional propagation of the light and an additional coil of single-mode fiber(SMF 28). The total cavity length is about 290 m (which corresponds to a round-trip timeof 1.45 µs), including 5 m of double-clad fiber with second-order dispersion
β2 = −0.021 ps²m−1 and 285 m of SMF 28 with β2 = −0.022 ps²m−1. The net cavity
dispersion is about −6.375 ps². The formed signal is then extracted by a 1×2 optical
coupler (OC) from the output port with an extracting ratio of 20 %. The laser is adjusted
to generate square pulses in the nanosecond range (few tens of ns). Thus, the spatial
extension of the pulse is few meters, so it is not possible to scan it entirely by standard
interferometric methods. The coherence is then characterized by two different methods.

Fig. 14.1. Experimental setup. EYDFA, Erbium-Ytterbium co-doped fiber Amplifier; OC, optical
coupler; PC, polarization controller; SMF, single-mode fiber; PI-ISO, polarization-insensitive
isolator; DFT, dispersive Fourier transformation; PZT, Piezoelectric translator; MZI,
Mach-Zehnder interferometer; L, lens; TS, translation stage.

The first method is based on a Mach-Zehnder interferometer (MZI), as illustrated in part(b) of Fig. 14.1. A 50 % coupler splits the beam into two equivalent signals. One partpropagates through one arm, which includes two collimating lenses fixed on a translation
stage (TS). The second part of the signal propagates through a piezoelectric (PZT) fiber
stretcher which introduces a modulated delay δ(t) around the fixed path difference ∆. The
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total path difference is then ∆ + δ(t), as illustrated with the two pulses in the inset of part(b) of Fig. 14.1. The oval form PZT-based fiber stretcher consists of two U-shaped caps
made of polyethylene resin designed for winding the optical fiber around it. One of thecaps is fixed and made immobile while the other is mounted on a PZT translation stage
driven by a function generator with a triangular function at 200 Hz. The free-space
propagation made by the two lenses is crucial for a precise adjustment of the delay
between the two arms. It also permits to easily suppress the delay if needed, which istedious to achieve with all-fibered arms because of the eventual errors of the fiber length
introduced by the fusion splicing. Finally, both signals are recombined with a 50 % OC
and local temporal coherence of the square pulses is studied.
The second method is based on the dispersive Fourier transform (DFT) technique as
displayed in part (c) of Fig. 14.1. It consists of a simple delay-line of 30 km of SMF 28
through which the pulse propagates. DFT is a real-time measurement technique. In fiber
lasers, it aims either to measure the dispersion D (GVD per unit length) of optical fiber orthe spectrum waveform of a pulse [51]. This latter is performed by using the mapping
relation between time and wavelength, given by Δτ = L. D. Δλ, where L is the length of
the fiber, Δλ is the bandwidth of the laser, and Δ𝜏 is the time duration into which the laser
spectrum is mapped [52]. For ultrashort pulses of few tens of picoseconds or less, theeffect of DFT is to map the optical spectrum into a temporal waveform, i.e., the temporal
intensity trace of the pulses mimics the spectrum. In our work, we use the DFT techniquein a non-standard way to study the coherence of large nanosecond square pulses tocharacterize their coherence. These two methods (MZI and DFT) are completely
independent but each one permit to confirm results of the other thus concluding whether
the studied pulses are coherent or not.
The laser output characteristics and the coherence measurements are monitored by a
13-GHz oscilloscope, combined with two 12-GHz photo-detector, an optical spectrum
analyzer, and an electronic spectrum analyzer to investigate the temporal trace, the optical
spectrum, and the radio-frequency signal, respectively. At last, a high-power integrating
sphere is utilized to measure the laser average power.

14.3. DSR-like Square Pulses
In this section, the characteristics of square-shaped pulses are studied at the reference laser
cavity output (see the reference output in Fig. 14.1, part (a)). This allows verifying someof the DSR features of the pulses. The pulse temporal trace is given in Fig. 14.2 (a),obtained for the pump power of 2 W. There is one pulse per cavity round-trip, with a pulse
width of 35.4 ns, as shown in the inset of Fig. 14.2 (a). The corresponding optical spectrum
is centered at a wavelength of 1612 nm as shown in Fig. 14.2 (b). The spectral bandwidth
measured at -3 dB is equal to 9.2 nm, and the spectral waveform is typical for squarepulses in the DSR regime. Fig. 14.2 (c) represents the RF spectrum trace with a signal-tonoise ratio of 68 dB, indicating high signal stability. This is confirmed by the stability of
the regime for pump power ranging from 0.8 W to 4.02 W, without any polarization
controller (PC) readjustment. As illustrated in Fig. 14.3 (a), the temporal shape remains
perfectly square and there is one square pulse per cavity round-trip. So while increasing578

Chapter 14. Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked
Fiber Lasers

the pump power there is no wave-breaking occurred, which is an important feature of thedissipative soliton resonance (DSR) regime. The same observation is made concerning
the optical spectrum, which also keeps its initial shape, as shown in Fig. 14.3 (b), byholding the spectral bandwidth at -3 dB around the same value in function of the pump
power, as depicted in Fig. 14.3 (c).

Fig. 14.2. (a) Temporal trace of square pulses, (b) the corresponding optical spectrum,and (c) the radio-frequency trace.

Additional characteristics are required to affirm the eventual DSR nature of these squarepulses. As mentioned before, the most common characteristics of the DSR regime toverify are the linear increase of the pulse width and the pulse energy in the function of thepump power, and the peak power clamping effect, i.e., the peak power has to be constant
while the pulse width increases with the increase of the pump power. Fig. 14.4 (a)represents a linear increase of the pulse width (red stars fitted with a linear red curve), andthe blue dots represent a nearly constant peak power while increasing the pump power.
Pulse energy also follows the linear evolution with the pump power, as shown in
Fig. 14.4 (b). These three characteristics are as expected for a DSR pulse, and in many
publications, these criteria are enough to conclude that the obtained square pulses are in
the DSR regime. As specified before, the autocorrelation trace was provided in few studies
but for relatively short duration pulses. Herein, the study is completed by verifying the
temporal coherence of the obtained nanosecond pulses, and only then we can conclude
whether it is DSR pulses or not.
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Fig. 14.3. (a) Temporal trace of square pulses for pump power ranging from 0.8 W to 4.02 W;(b) The corresponding optical spectrum for the same range of pump power; (c) 3-dB bandwidthof the spectrum as a function of the pump power.

Fig. 14.4. Characteristics of square pulses. Evolution of the (a) pulse width (red stars linearly
fitted by red curve), the peak power (blue dots fitted by linear blue curve), and (b) the pulse
energy (black dots with a linear fit curve) with pump power ranging from 0.8 to 4.02 W.

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Fiber Lasers

14.4. Coherence Characterization of Square Pulses with MZI Technique
Square pulses in passively mode-locked fiber laser are a complex regime. Before studying
their coherence, the MZI is tested and validated with a continuous wave laser source. We
used a Distributed Feed-Back (DFB) laser source emitting at a wavelength of 1556.5 nm.
The optical spectrum of the DFB laser is represented in Fig. 14.5 (a). Fig. 14.5 (b) displays
the temporal trace of the signal at the output of the MZI.

Fig. 14.5. (a) Optical spectrum trace of the DFB laser source. (b) Temporal trace ofinterferometric fringes of the DFB laser at the output of the MZI.

The temporal trace exhibits the interferometric fringes with high contrast. The modulateddelay which is performed by the PZT fiber stretcher driven by a 200 Hz triangularfunction generator (see part (b) of Fig. 14.1), corresponds to the period T = 5 ms. One
can note that the curve is not a perfect sine as expected for a continuous phase variation,
but there exist phase jumps around 1 ms, 3.5 ms, and 6 ms. These phase jumps are locatedat every minimum and maximum of the triangular function which causes aninhomogeneous response of the fiber to the PZT fiber stretcher. The frequency of thetriangular function is chosen at 200 Hz, according to the characteristics of the PZT fiber
stretcher, the fiber response to the mechanical stretching, and also for better visualizationof the interferometric pattern on the oscilloscope. Finally, except for the phase jumps, the
MZI is operational and could be used to measure the coherence of more complex laser
regimes. From Fig. 14.5 (b) we deduce that there are about 33 fringes for one modulation
period of the PZT so that a 2𝜋-phase delay between the two arms of the interferometercorresponds to about 150 µs. The period of the interferometric fringes is about 100 times
greater than the round-trip time of the cavity (1.45 µs). Consequently, the interferometricfringes will not be visible along one square pulse but will modulate a large number ofpulses. Fig. 14.6 is a schematic representation of the expected temporal intensity evolutionof the square pulses at the MZI output.
In the case of coherent pulses, the intensity of interfering beams is expected to experiencea sinusoidal evolution corresponding to consecutive destructive and constructive
interference (as observed in the temporal trace of interferometric fringes of the DFB laser).
So, if we measure the intensity of the top of the center of each two interfering pulses (seered crosses in Fig. 14.6 (a)), the resulting curve would be sinusoidal, as shown in the same581

Advances in Optics: Reviews. Book Series, Vol. 5

figure (red curve representing the envelope versus time). However, in the case ofincoherent pulses, the intensity reached by the interfering part of each pulse remains
constant, i.e., as the pulses are not coherent, the resulting intensity is an incoherent
addition of intensities of the interfering pulses (see Fig. 14.6 (b)). The interpretation of
the interferometry results of the square pulses is made based on this reasoning.

Fig. 14.6. Temporal evolution of the pulse intensity, and the envelope, in case of (a) coherent
pulses, and (b) incoherent pulses.

Fig. 14.7 (a) represents the interferometric fringes of the square pulses with 26.5 ns pulse
width, obtained at a pump power of 1.67 W, for different fixed optical delays ∆ (~0, ~100,~200, and ~300 µm). For a fixed delay of ∆ ~0 µm, i.e., when the path difference is nearlyzero, the interferometric fringes are visible with relatively high contrast. When ∆increases to 100 µm and 200 µm the contrast of the fringes decreases and disappears for∆ ~300 µm. Fig. 14.7 (b) represents the corresponding peak intensity (envelope) traces foreach value of path difference, in which the evolution process of fringe visibility is muchclear. Taking into account the discontinuities introduced by the fiber response to thestretching, we can see that for ∆ ~0 µm, the contrast of the fringes is high and mimics thesine function as it is expected in the schematic of Fig. 14.6 (a). When ∆ increases, thecontrast of interferometric fringes decreases. For ∆ ~300 µm, the fringes completely
disappear, and we find the case of incoherent pulses of Fig. 14.6 (b), where the peakintensity is constant versus time.
The same measurements are conducted for different pulse durations. In Fig. 14.8, we
represent the results for square pulses with a duration of 69 ns, obtained at a pump powerof 4.02 W. As in the previous figure, one can observe a relatively high contrast
interferometric fringes for ∆ ~0 µm, but the coherence pattern disappears around the samevalue of the path difference as previously (∆ ~300 µm).
We conclude from Fig. 14.7 and Fig. 14.8, that when the path difference is equal or higher
than ∆ ~300 µm, corresponding to a time delay of 𝜏 = 1 𝑝𝑠, the coherence completely
disappears. This means that the square pulses are not coherent and could consist of many
shorter pulses within the square envelope. The duration of these pulses is estimated to beabout 1 𝑝𝑠.
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Fiber Lasers

Fig. 14.7. (a) Temporal evolution of the interferometry fringes of the 26.5 ns pulses at the pump
power of 1.67 W, for different optical delays ∆ (ranging from 0 µm – 300 µm), at the outputof MZI; (b) The corresponding peak intensity evolution.

Fig. 14.8. (a) Temporal evolution of the interferometry fringes of the 69 ns pulses at the pump
power of 4.02 W, for different optical delays ∆ (ranging from 0 µm – 300 µm), at the outputof MZI; (b) Intensity envelopes of the interferometry fringes.

In conclusion, by using an MZI, we prove that although the studied square pulses fulfill
almost all characteristics of the DSR pulses (such as linear increase of the pulse width andpulse energy, constant peak power, and invariant optical spectrum, versus the pump
power), they are not necessarily DSR but may be packets of very short pulses (which canbe difficult to detect by usual means).
These results are confirmed by other measurements conducted with the DFT technique.
Indeed, with the MZI, we can scan a short part of a square pulse so the coherence
measurement is local. It is possible to adjust the delay ∆ with the free-space collimatorsand scan different parts of the pulse as shown previously for different path differences ∆(~0, ~100, ~200, and ~300 µm). However, the studied square pulses have a spatial
extension of few meters and it is not possible to create such delays with collimators. The
DFT technique allows more global coherence characterization. By studying the shapeevolution of the whole pulse after propagating a long fiber coil, it is possible todiscriminate between coherent and incoherent square pulses.
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14.5. Coherence Characterization of Square Pulses with DFT TechniqueIn this section, the DFT technique is used to study the coherence of square pulses. As
explained in Section 14.2, the usual DFT technique allows performing real-timemeasurement of the spectral profile. This occurs if the fiber length 𝐿 satisfies the condition
𝐿 ≫ 𝐿𝐷 [53], where 𝐿𝐷 is the dispersion length. Experimentally, this condition can beeasily fulfilled with femtosecond or picosecond pulses while it is impossible in thenanosecond range. Considering a Gaussian pulse with a duration of 𝜏0 = 10 𝑛𝑠, therequired fiber length to perform an efficient DFT exceeds 109 𝑘𝑚. Despite this limitation
for large pulses, the DFT allows to easily discriminate between a coherent and incoherent
square pulse. Indeed, let us consider a square pulse in the nanosecond range and adispersive line of few tens of km of fiber. If the pulse is coherent, such fiber length is notenough to modify the pulse shape since the dispersion has no time to spread the pulse. In
contrast, if the square pulse experiences considerable modifications in the temporal profile
and imitates the spectrum waveform, we can conclude that the pulse is incoherent andassume that it is composed of subsequent fine structures. In our experiment, the studied
square pulses propagate through 30 km of SMF (see part (c) Fig. 14.1). Fig. 14.9 (a) shows
the temporal trace of the initial square pulses at the laser output (at the reference point inpart (a) of Fig. 14.1), while Fig. 14.9 (b) represents the temporal trace of the pulses at theoutput of the DFT line (part (c) of Fig. 14.1).

Fig. 14.9. Temporal traces of the square pulses for different pump powers (a) at the reference
output, and (b) at the output of the DFT-line. (c) The rise time of the initial pulses (red squares
fitted with the linear red curve) and after the DFT-line (blue dots fitted with the linear blue curve).

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Chapter 14. Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked
Fiber Lasers

Comparing the initial pulses and those after propagating through the DFT-line, the pulses
are no more square-shaped after 30 km of fiber. This shape-change is more pronounced
for the shortest pulses. We can see that for the pulses at the pump power of 0.46 W and
0.8 W, corresponding to initial duration of 6.5 ns and 12.2 ns, respectively, the temporal
trace of the pulses after propagation through the DFT-line are more likely to mimic the
spectral waveform as in standard DFT technique. For higher pump powers and larger
pulses, the DFT pulses remain flat at the center. However, the rise time of the initial squarepulses is around 0.1 ns while the rise time of these pulses after the DFT line is around10 ns as depicted in Fig. 14.9 (c). This means that if the available fiber is twice longer for
example (60 km instead of 30 km), but still much lower than the dispersion length, thepulse depicted in blue in Fig. 14.9 (a) would experience the same shape change as in thecase of the pulse depicted in red.
These observations permit us to conclude that the studied square pulses are not temporally
coherent but consist of many shorter pulses that allow the envelope to be stretched by only
30 km of fiber. Indeed, as explained above, if the square pulses were coherent with ananosecond range duration, they would not experience any change of the temporal profile
because of the considerable length of the fiber that would be necessary to observe the DFT
effect. For further investigation of the DFT-line effect on the pulses, additional numericalsimulations have been performed.
In the simulations, we have considered the optical fiber as a purely second-orderdispersive medium. Additionally, we consider the initial peak power of the pulse relatively
low so that the nonlinear effects can be neglected, which is the fundamental assumption
for DFT to be performed.
In the reference frame moving at the group velocity, the electric field envelope at theabscissa 𝑧 is related to the incident electric field envelope through the integral
formula [33]:1
1 (𝑡−𝑡′)2
𝐴(0, 𝑡′) exp (− 𝑖) 𝑑𝑡 ′ ,∫2𝑖𝜋𝛽2 𝑧2
𝛽2 𝑧

𝐴(𝑧, 𝑡) = √(14.1)

where 𝐴(0, 𝑡) is the initial pulse profile, and 𝐴(𝑧, 𝑡) is the pulse profile after propagation
through the fiber of length z. The coefficient 𝛽2 is the group velocity dispersion.
In these simulations, we consider a super-Gaussian pulse profile given by:
𝑡 𝑚

𝐴(0, 𝑡) = exp (−(1 + 2𝑖𝐶) ∙ (𝑡 ) )0(14.2)To start, the linear chirp parameter C is chosen randomly as 𝐶 = 15. The order of thesuper-Gaussian is 𝑚 = 100 to better fit the square shape of the experimental pulses. We
take 𝑡0 = 2.5 ns leading to an initial pulse duration of 5 ns, defined as the full width at
half maximum. The group velocity dispersion value of silica single-mode fiber at the
wavelength of 1.55 µm is 𝛽2 = −0.022 ps 2 m−1.
The results are shown in Fig. 14.10, where the initial square pulse and its spectrum are
represented in the blue curve in Figs. 14.10 (a) and (b), respectively. The red and black585

Advances in Optics: Reviews. Book Series, Vol. 5

curves in Fig. 14.10 (a) represent the pulse after propagating 30 km and 2.5 × 108 km offiber, respectively. One can note that the pulse shape after the 30 km of DFT line remains
almost square. The rise time difference is very low, increasing from 0.42 ns for the initial
pulse to 0.57 ns for the DFT pulse. However, after propagation in 2.5 × 108 km of fiber,
the temporal trace of the pulse is no more square but mimics the optical spectrum of thepulse. These observations demonstrate that to perform an efficient DFT on a coherent
pulse with an initial duration as large as 5 ns, the needed fiber length is at least equals to
2.5 × 108 km and the 30 km are far from being enough.

Fig. 14.10. (a) Temporal trace of the square pulses representing the initial pulse in blue, the pulse
propagation after 30 km in red, and the pulse propagated in 2.5 × 108 km in black;(b) The corresponding optical waveform for the initial square pulse.

Thus, in the case of the experimental results, as the 30 km of fiber is enough to strongly
modify the shape of the pulses and mimic the optical spectrum, the pulses are not coherent.
In the previous calculations, the chirp was fixed to C = 15, so further study of the effectof this parameter is required. Since DSR square pulses are highly linearly chirped [54],many values have been tested in the simulations considering two pulses with pulse widthsof 10 ps and 5 ns. The range of chirp values is estimated based on the duration and spectral
bandwidth of the pulse.
The 10 ps short pulse requires around 1 km of fiber to perform an observable DFT effect.
As shown in Fig. 14.11 (a), for an initial pulse width of 10 ps, the temporal trace after
1 km of fiber has mapped the optical spectrum shown in Fig. 14.11 (b). For the same 1
km propagation, different values of the chirp are tested such as C = 0.1, 1, 10, 20. The
curves are represented in a different color for each chirp value. One can note that for an
initial pulse with a duration as short as 10 ps, the chirp has no significant effect on the
DFT process even with the highest chirp C = 20. It is worth mentioning that 1 km of fiber
is a limit fiber length for an observable time-wavelength mapping, so if the chirp has any
influence on the DFT process, it would be visible in the simulations.
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Chapter 14. Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked
Fiber Lasers

Fig. 14.11. (a) Temporal trace of the square pulses representing the initial pulse in blue with a pulse
width of 10 ps, and the other curves represent the pulse after propagating 1 km of fiberwith different values of the initial chirp (C = 0.1, 1, 10, 20); (b) The corresponding optical
waveform for the initial square pulse.

Other chirp values (C = 15, 500, 5000, 50000) are applied for a pulse with an initial
duration of 5 ns. The result is shown in Fig. 14.12. We observe in Fig. 14.12 (a) that even
if a very high linear chirp is applied, the temporal trace doesn’t mimic the optical spectrum
shown in Fig. 14.12 (b).

Fig. 14.12. (a) The temporal trace of the square pulses representing the initial pulse in blue
with a duration of 5 ns, and the other curves represent the pulse after propagating 30 km of fiberwith different values of the initial chirp (C = 15, 500, 5000, 50000); (b) The corresponding optical
waveform for the initial square pulse.

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In conclusion, the initial chirp of the square pulses has no considerable impact on the DFT
process. So, the value of the chirp of experimental pulses, shown in Fig. 14.9 (b)(especially whose duration is about 5 ns), is not the reason why the pulses tend to mimic
the optical spectrum and perform an efficient DFT. As the DSR pulses are characterized
by a high chirp, these simulation results allow understanding the impact of the chirp while
the pulses are propagating in the fiber. Thus, both MZI and DFT methods confirm that thestudied square pulses are not temporally coherent. Therefore, the square pulses are not
DSR pulses.

14.6. Conclusion
In conclusion, using both MZI and DFT techniques, we studied the temporal coherenceof square pulses in a passively mode-locked fiber laser. In the case of MZI, we were able
to circumvent the limitations due to the repetition rate and the pulse’s duration, to visualize
in real-time the interferometric fringes on the oscilloscope. We observed that, as soon as
the interferometer’s time delay reaches 1 ps, the interferometric fringes disappear. This
proves that the square pulses contain shorter pulses within the square envelope. In
addition, through the DFT technique, we confirm that these pulses cannot be coherent as
the temporal shape of the pulses changes after only 30 km consisting of the DFT line. The
result of the DFT is more significant for the pulses with a duration of 6.5 ns and 12.2 ns
as they tend to mimic the optical spectrum. This would be impossible to occur in the caseof coherent pulses because of the length of the fiber needed to complete the DFT effecton these large duration pulses. The simulation of the DFT on a coherent square pulse of afew ns confirms that the 30 km available are far from being enough to produce any shapechange to mimic the optical spectrum. We also studied the effect of the linear chirp on the
DFT process, and confirm that its value doesn’t affect the shape-change of the pulses.
Finally, with both methods, we conclude that even if the square pulses verify most of thecharacteristics of DSR as a function of pump power (a linear increase of the energy andwidth of the pulses, clamped peak power, and invariant optical spectrum), they are notnecessarily DSR as long as they are not temporally coherent. We assume that these squarepulses may be noise-like pulses with fine structures that we are not able to detect with theusual acquisition system.

References[1]. A. A. Boni, D. I. Rosen, S. J. Davis, L. A. Popper, High-power laser applications to medicine,
J. Quant. Spectrosc. Radiat. Transfer, Vol. 40, Issue 3, 1988, pp. 449-467.
[2]. R. W. Waynant, I. K. Ilev, I. Gannot, Mid-infrared laser applications in medicine and biology,
Phil. Trans. R. Soc. Lond. A, Vol. 359, Issue 1780, 2001, pp. 635-644.
[3]. Q. Peng, A. Juzeniene, J. Chen, L. O. Svaasand, T. Warloe, K.-E. Giercksky, J. Moan, Lasers
in medicine, Rep. Prog. Phys., Vol. 71, Issue 5, 2008, 056701.
[4]. L. A. Dombrovsky, V. Timchenko, M. Jackson, G. H. Yeoh, A combined transient thermal
model for laser hyperthermia of tumors with embedded gold nanoshells, Int. J. Heat Mass
Transfer, Vol. 54, Issues 25-26, 2011, pp. 5459-5469.

588

Chapter 14. Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked
Fiber Lasers[5]. M. Eichhorn, Pulsed 2 µm fiber lasers for direct and pumping applications in defense andsecurity, Proceeding of SPIE, Vol. 7836, 2010, 78360B.
[6]. M. Silver, S. T. Lee, A. Borthwick, I. McRae, D. Jackson, W. Alexander, Compact,
diode-pumped, solid-state lasers for next generation defence and security sensors, J. Phys.:Conf. Ser., Vol. 619, 2015, 012022.
[7]. F. P. Gagliano, R. M. Lumley, L. S. Watkins, Lasers in industry, Proceedings of the IEEE,
Vol. 57, Issue 2, 1969, pp. 114-147.
[8]. D. Strickland, G. Mourou, Compression of amplified chirped optical pulses, Optics Comm.,
Vol. 56, Issue 3, 1985, pp. 219-221.
[9]. A. Fernandez, T. Fuji, A. Poppe, A. Fürbach, F. Krausz, A. Apolonski, Chirped-pulse
oscillators: a route to high-power femtosecond pulses without external amplification, Opt.
Lett., Vol. 29, Issue 12, 2004, pp. 1366-1368.
[10]. A. Chong, W. H. Renninger, F. W. Wise, Properties of normal dispersion femtosecond fiber
lasers, J. Opt. Soc. Am. B, Vol. 25, Issue 2, 2008, pp. 140-148.
[11]. W. Chang, A. Ankiewicz, J. M. Soto-Crespo, N. Akhmediev, Dissipative soliton resonances,
Phys. Rev. A, Vol. 78, Issue 2, 2008, 023830.
[12]. W. Chang, J. M. Soto-Crespo, A. Ankiewicz, N. Akhmediev, Dissipative soliton resonancesin the anomalous dispersion regime, Phys. Rev. A, Vol. 79, Issue 3, 2009, 033840.
[13]. X. Wu, D. Y. Tang, H. Zhang, L. M. Zhao, Dissipative soliton resonance in an all-normal
dispersion erbium-doped fiber laser, Opt. Express, Vol. 17, Issue 7, 2009, pp. 5580-5584.
[14]. X. Li, X. Liu, X. Hu, L. Wang, H. Lu, Y. Wang, W. Zhao, Long-cavity passively
mode-locked fiber ring laser with high-energy rectangular-shape pulses in anomalous
dispersion regime, Opt. Lett., Vol. 35, Issue 19, 2010, pp. 3249-3251.
[15]. X. Liu, Pulse evolution without wave breaking in a strongly-dissipative laser system, Phys.
Rev. A, Vol. 81, Issue 5, 2010, 053819.
[16]. P. Grelu, W. Chang, A. Ankiewicz, J. M. Soto-Crespo, N. Akhmediev, Dissipative soliton
resonance as a guideline for high-energy pulse laser oscillators, J. Opt. Soc. Am. B, Vol. 27,
Issue 11, 2010, pp. 2336-2341.
[17]. E. Ding, P. Grelu, N. Kutz, Dissipative soliton resonance in a passively mode-locked fiber
laser, Opt. Lett., Vol. 36, Issue 7, 2011, pp. 1146-1148.
[18]. F. Bahloul, K. Guesmi, M. Salhi, F. Sanchez, R. Attia, Control of the square pulse properties
in figure-of-eight microstructured fiber laser, Opt. Eng., Vol. 55, Issue 2, 2016, 026102.
[19]. A. Komarov, A. Dmitriev, K. Komarov, D. Meshcheriakov, F. Sanchez, Spectral-doublet
rectangular pulses in passive mode-locked fiber lasers with anomalous dispersion, Phys. Rev.
A, Vol. 94, Issue 4, 2016, 043827.
[20]. C. Liu, Q. Wu, Y. Yao, C. Wu, Y. Gan, Y. Fu, Y, Yang, J. Tian, K. Xu, Numerical researchon the dissipative-soliton-resonance peak power in normal-dispersion fiber lasers, Opt.
Commun., Vol. 474, 2020, 126163.
[21]. J. Yang, C. Guo, S. Ruan, D. Ouyang, H. Lin, Y. Wu, R. Wen, Observation of dissipative
soliton resonance in a net-normal dispersion figure-of-eight fiber laser, IEEE Photonics
Journal, Vol. 5, Issue 3, 2013, pp. 1500806-1500806.
[22]. S. Wang, Q. Ning, A. Luo, Z. Lin, Z. Luo, W. Xu, Dissipative soliton resonance in a passively
mode-locked figure-eight fiber laser, Opt. Express, Vol. 21, Issue 2, 2013, pp. 2402-2407.
[23]. N. Zhao, M. Liu, H. Liu, X. Zheng, Q. Ning, A. Luo, Z. Luo, W. Xu, Dual-wavelength
rectangular pulse Yb-doped fiber laser using a microfiber-based graphene saturable absorber,
Opt. Express, Vol. 22, Issue 9, 2014, pp. 10906-10913.
[24]. K. Krzempek, Dissipative soliton resonances in all-fiber Er-Yb double clad figure-8 laser,
Opt. Express, Vol. 23, Issue 24, 2015, pp. 30651-30656.
[25]. K. Krzempek, J. Sotor, K. Abramski, Compact all-fiber figure-9 dissipative soliton resonancemode-locked double-clad Er:Yb laser, Opt. Lett., Vol. 41, Issue 21, 2016, pp. 4995-4998.

589

Advances in Optics: Reviews. Book Series, Vol. 5[26]. G. Semaan, F. Ben Braham, J. Fourmont, M. Salhi, F. Bahloul, F. Sanchez, 10 µJ dissipative
soliton resonance square pulse in a dual amplifier figure of eight double-clad Er:Yb
mode-locked fiber laser, Opt. Lett., Vol. 41, Issue 20, 2016, pp. 4767-4770.
[27]. G. Semaan, A. Niang, M. Salhi, F. Sanchez, Harmonic dissipative soliton resonance squarepulses in an anomalous dispersion passively mode-locked fiber ring laser, Laser Phys. Lett.,Vol. 14, Issue 5, 2017, 055401.
[28]. R. Becheker, M. Tang, M. Touil, T. Robin, B. Cadier, M. Laroche, T. Godin, A. Hideur,Dissipative soliton resonance in a mode-locked Nd-fiber laser operating at 927 nm, Opt. Lett.,Vol. 44, Issue 22, 2019, pp. 5497-5500.
[29]. L. Duan, X. Liu, D. Mao, L. Wang, G. Wang, Experimental observation of dissipative soliton
resonance in an anomalous dispersion fiber laser, Opt. Express, Vol. 20, Issue 1, 2012,
pp. 265-270.
[30]. G. Yurtsever, B. Považay, A. Alex, B. Zabihian, W. Drexler, R. Baets, Photonic integrated
Mach-Zehnder interferometer with an on-chip reference arm for optical coherence
tomography, Biomed. Opt. Express, Vol. 5, Issue 4, 2014, pp. 1050-1061.
[31]. G. B. Hocker, Fiber-optic sensing of pressure and temperature, Appl. Opt., Vol. 18, Issue 9,1979, pp. 1445-1448.
[32]. F. Gao, H. Nan, R. Zhang, Q. Zhu, T. Chen, L. Wang, H. Chen, D. Hua, S. Stanič,
Quasi-monochromatic of laser echo signals on transmittance of Mach-Zehnder interferometerfor UV multi-longitudinal-mode high-spectral resolution lidar, J. Quant. Spectrosc. Radiat.
Transfer, Vol. 234, 2019, pp. 10-19.
[33]. P. Celliers, F. Weber, L. B. D. Silva, T. W. Barbee, R. Cauble, A. S. Wan, J. C. Moreno,
Fringe formation and coherence of a soft-x-ray laser beam illuminating a Mach-Zehnderinterferometer, Opt. Lett., Vol. 20, Issue 18, 1995, pp. 1907-1909.
[34]. W. Peng, P. Liu, Multiwavelength erbium-doped fiber laser based on a polarizationdependent in-line Mach-Zehnder interferometer, Opt. Quantum Electron., Vol. 51, Issue 9,2019, 300.
[35]. W. He, L. Zhu, M. Dong, X. Lou, F. Luo, Wavelength-switchable and stable-ring-cavity,
erbium-doped fiber laser based on Mach-Zehnder interferometer and tunable filter, Laser
Phys., Vol. 28, Issue 4, 2018, 045104.
[36]. J. Peng, M. Sorokina, S. Sugavanam, N. Tarasov, D. Churkin, S. Turitsyn, H. Zeng,
Real-time observation of dissipative soliton formation in nonlinear polarization rotation
mode-locked fibre lasers, Commun. Phys., Vol. 1, Issue 1, 2018, 20.
[37]. X. Liu, X. Yao, Y. Cui, Real-time observation of the buildup of soliton molecules, Phys. Rev.
Lett., Vol. 121, Issue 2, 2018, 023905.
[38]. X. Liu, M. Pang, Revealing the buildup dynamics of harmonic mode-locking states in ultrafast
lasers, Laser Photonics Rev., Vol. 13, Issue 9, 2019, 1800333.
[39]. X. Liu, D. Popa, N. Akhmediev, Revealing the transition dynamics from Q switching to mode
locking in a soliton laser, Phys. Rev. Lett., Vol 123, Issue 9, 2019, 093901.
[40]. I. Kudelin, S. Sugavanam, M. Chernysheva, Build-up dynamics in bidirectional soliton fiber
lasers, Photonics Res., Vol. 8, Issue 6, 2020, pp. 776-780.
[41]. Z. Wang, Z. Wang, Y. Liu, R. He, J. Zhao, G. Wang, G. Yang, Self-organized compound
pattern and pulsation of dissipative solitons in a passively mode-locked fiber laser, Opt. Lett.,Vol. 43, Issue 3, 2018, pp. 478-481.
[42]. Y. Du, Z. Xu, X. Shu, Spatio-spectral dynamics of the pulsating dissipative solitons in anormal-dispersion fiber laser, Opt. Lett., Vol. 43, Issue 1, 2018, pp. 3602-3605.
[43]. J. Peng, S. Boscolo, Z. Zhao, H. Zeng, Breathing dissipative solitons in mode-locked fiber
lasers, Sci. Adv., Vol. 5, Issue 11, 2019, eaax1110.
[44]. P. Ryczkowski, M. Nürhi, C. Billet, J.-M. Merolla, G. Genty, J. Dudley, Real-time full-fieldcharacterization of transient dissipative soliton dynamics in a mode-locked laser, Nat.
Photonics, Vol. 12, Issue 4, 2018, pp. 221-227.

590

Chapter 14. Coherence Characterization of DSR-like Square Pulses in Passively Mode-locked
Fiber Lasers[45]. S. D. Chowdhury, B. D. Gupta, S. Chatterjee, R. Sen, M. Pal, Rogue waves in a linear cavity
Yb-fiber laser through spectral filtering induced pulse instability, Opt. Lett., Vol. 44, Issue 9,2019, pp. 2161-2164.
[46]. Y. Luo, Y. Xiang, T. Liu, B. Liu, R. Xia, Z. Yan, X. Tang, D. Liu, Q. Sun, P. P. Shum,
Real-time access to the coexistence of soliton singlets and molecules in an all-fiber laser, Opt.
Lett., Vol. 44, Issue 17, 2019, pp. 4263-4266.
[47]. X. Wang, J. He, B. Mao, H. Guo, Z. Wang, Y. Yue, Y. Liu, Real-time observation ofdissociation dynamics within a pulsating soliton molecule, Opt. Express, Vol. 27, Issue 20,
2019, pp. 28214-28222.
[48]. C. Lapre, C. Billet, F. Meng, G. Genty, J. M. Dudley, Dispersive Fourier transformcharacterization of multipulse dissipative soliton complexes in a mode-locked
soliton-similariton laser, OSA Continuum, Vol. 3, Issue 2, 2020, pp. 275-285.
[49]. M. Kemel, M. Salhi, C. Ciret, G. Semaan, A. Nady, and F. Sanchez, Coherence
characterization of nanosecond rectangular pulses in passively mode-locked fiber lasers, OSA
Continuum, Vol. 4, Issue 2, 2021, pp. 279-289.
[50]. V. J. Matsas, D. J. Richardson, T. P. Newson, D. N. Payne, Characterization of a self-starting,passively mode-locked fiber ring laser that exploits nonlinear polarization evolution, Opt.
Lett., Vol. 18, Issue 5, 1993, pp. 358-360.
[51]. Y. C. Tong, L. Y. Chan, H. K. Tsang, Fibre dispersion or pulse spectrum measurement usinga sampling oscilloscope, Electronics Letters, Vol. 33, Issue 11, 1997, pp. 983-985.
[52]. K. Goda, B. Jalali, Dispersive Fourier transformation for fast continuous single-shotmeasurements, Nat. Photon., Vol. 7, 2013, pp. 102-112.
[53]. G. P. Agrawal, Nonlinear Fiber Optics, 2nd Ed., Academic, New York, 1995.
[54]. D. Li, L. Li, J. Zhou, L. Zhao, D. Tang, D. Shen, Characterization and compressionof dissipative-soliton-resonance pulses in fiber lasers, Sci. Rep., Vol. 6, 2016, 23631.

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Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers

Chapter 15

New Formulae for Obtaining Fourier Series,
Bernoulli Polynomials and Numbers
Do Tan Si1

15.1. Introduction
Every physicist and engineer know that a periodic function f (x) of period P may be
expanded into an infinite series of cosine, sine and exponential functions following theformulaef (x)    (n) cos(2 nx)
Pa  x  b  a  P,

(15.1)

f (x)    (n)sin(2 nx)
Pa  x  b  a  P,

(15.2)x)Pa  x  b  a  P,

(15.3)nZnZ

f (x)   c(n) exp(i 2 n  nnZThis revolutionary idea was introduced by Fourier in 1807 at the Academie Française
under the title “Treatise on the propagation of heat in solid bodies” [1].
Formula (15.3) means that if the variable x represents a time then “A periodic movementt  t0) with frequenciesof period P of an object creates a series of waves exp(i 2 n
Pn  2n / P each has relative amplitude equal to c(n) ”.

Do Tan Si
Ho Chi Minh-city Physical Association, Vietnam593

Advances in Optics: Reviews. Book Series, Vol. 5

For example the vibration of a guitar string generates octaves notes of music; rain water
creates wavelets on a pool, change of electron orbits creates light, etc.
Many works on Fourier series has been done afterward and may be found on the net for
examples in [4, 5]. Nevertheless looking at the integrations utilized until now to calculate
the coefficients  (n),  (n), c(n) we notice that these integrations are rather complicatedto perform, i.e. there is not yet a general formula easy to apply for obtaining them. This
work proposes such a formula where lengthy integrations are replaced with shortderivations. Moreover, from a simple property of the exponential function we obtain theequivalence between each term of a Fourier series of a function f ( x) with a polynomial.
Based on this property of the exponential function, we obtain also the recurrence formulafor calculating the Bernoulli polynomial Bm (x) from and only from Bm1 (x) which open
the way for obtaining the Lucas symbolic formula Bm (x) : ( B  x)m and a simple matrix
formula for calculating easily Bernoulli numbers Bm . Corollary the powers of pi, thevalues of positive even Riemann zeta functions may also be obtained. Details of
calculations are exposed in the hereafter paragraphs.

15.2. Fourier Series
15.2.1. Fourier Series Coefficients
Let f (x) be a periodic function defined on an interval a  x  b and has the periodP  b  a. We want to have the expansion of f (x) into a series of exponentials ef (x)   c(n)ei 2 nx
Pa  x  b  a  P,

i 2 nx
P(15.4)

nZ



b

a

e

 i 2 n 0x
Pxb   i 2 n0x
dx
Pf ( x)d ( )  c(0)  eaPP  c(n)(ei 2 ( n  n0 )

nZ
n 0

b
P

i 2 ( n  n0 )ei 2 (n  n 0 )aP)(15.5)

If n 0  0, the second term is equal to zero and we haveb f ( x)dx  c(0) Pa

If n 0  0, the first term is zero and we have594

(15.6)

Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers

b

 e

 i 2 n 0

x
P

a

x
e
f ( x)d ( )   c(n)P nZi  ( n  n0 )baPi  ( n  n0 )n  n0  c(n0 )ei  ( n  n0 )

nZ
n n0

b-a

P
(e
e
i 2 (n  n 0 ) i  ( n  n0 )

b-a
P

)



(15.7)baP

2i sin  ( n  n0 )
 0  c( n0 )i 2 (n  n0 )Combining (15.6) and (15.7) we get the formula for calculating the coefficients in a
Fourier seriesc ( n) x
 i 2 n
1 b
Pf ( x )edx
P a

(15.8)For continuous functions, in order to simplify the calculation in (15.8) we utilize themethod of integrations by partsx
 i 2 n
1 b
P
f
(
x
)
e
dx 
P a
b
a 
x
 i 2 n
 i 2 n
 i 2 n
i 
i b

P
P
P

 f(a)e
) 
f
'(
x
)
e
dx,
(f(b )ea2 n 
 2 nc ( n) (15.9)bxx
 i 2 n 
 i 2 n
i 
i b
P
P
c ( n) 
f(
x
)
e

f
'(
x
)
e
dx,


a
2 n 
 a 2 n(15.10)where we have introduced the notation
b



x
b
a
 i 2 n 
 i 2 n
 i 2 nPPP f(a)ef(x)e  f(b )ea(15.11)In the cases where the function has successive derivatives one may continue successively
integrations by parts beginning with



b

a

f '( x)e

 i 2 n

x
P

b

b

e  i 2 nx / P e i 2 nx / P
dx  P  f '( x)

P
f
"(
x
)
dx,

a
(i 2 n)  a( i 2 n)(15.12)in order to obtain an interesting new formula following our knowledge for calculating theFourier series coefficientsbx i 2 n 
1 
P k  ( k 1)Pc ( n)    () f( x )e ,n0P k 1 2i n 
a(15.13)595

Advances in Optics: Reviews. Book Series, Vol. 5

Remarks:o In actual literature about Fourier series it is considered integrations  f ( x)dx instead
b

a

b

of

 f ( x)dx. We think that this is not convenient for periodic functions f (x) whichaare discontinuous at b such as rectangular, Heaviside functions and so all;o The great advantage of Eq. (15.13) is that we don’t have to calculate integrations asin Eq. (15.9) but only derivatives.
15.2.2. Formula for Obtaining Fourier Series of Functions
From (15.11), (15.13) we get the formula for obtaining Fourier series of functions
b

x
m
b
 i 2 n 
1
1
P k i 2 n Px
Pf ( x)    f ( x)dx     f ( k 1) ( x)e
(
) e
,

a
P
P nZ k 1 
 a 2i n(15.14)n0

where m may be infinite. Remarking thate

 i 2 nbPe i 2 naP(15.15),

we may also writef ( x)  c(0) b
1 m
P k i 2 n x-aP
( k 1)


f
(
x
)
(

 a  2i n ) e
P k 1 nZ(15.16)n0

Eqs (15.14) and (15.16) may be written under the form of sum of sine and cosine Fourier
series, for example
m
 

b

x
 i 2 n 
1  2P 2 k 1x
P
f ( x)  c(0)    f (2 k ) ( x)e) 2i sin(2 n )  (P n 1 k 0 P
 a 2i n



m
2
 


1 
(  f (2 k 1) ( x)e
P n 1 k 1 x
 i 2 nP(15.17)

b


P 2k
x
) 2cos(2 n ) (P
 a 2i nbFrom (15.17) we see that a function f ( x) such that  f (2 k 1) ( x)   0, k is a pure Fourierasine series and such that  f (2 k ) ( x) 

596

b
a

 0, k a pure Fourier cosine series.

Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers

15.2.3. Fourier Series of Even and Odd Functions
We have for even functions f ( x)  f ( x),  a  x  a
a

a

x
x
 i 2 n 
 i 2 n  (2k)
P
P
f(
x
)
e

f
(
x
)
e



 0

a 
a

(15.18)

According to (15.18) we see that an even function has pure Fourier cosine series. Howevera pure Fourier cosine series may also came from functions such as Bernoulli polynomialsf ( x)  B2m ( x), a  0, b  1 as we can see in a hereinafter paragraph.
Similarly an odd function f ( x)   f ( x),  a  x  a
a

a

x
x
 i 2 n 
 i 2 n 
 (1) (2k 1)
P
P
f
(
x
)
e

f
(
x
)
e



 0

 a 
 a(15.19)is a pure Fourier sine series but functions which are not odd such as Bernoulli polynomialf ( x)  B2m1 ( x), a  0, b  1 has also pure Fourier sine series.
15.2.4. From Fourier Series to Fourier Transform
Consider a periodic functions f (x) defined on the interval a  x  b  a  P we havefollowing (15.4) and (15.8)
f (x) b
1eikx  f ( x)e  ikx dx

a
P nZ

Making the not forbidden extension a  , b   and putting k (15.20)2
n we get thePformulaf (x) 1eikx  f ( x)e  ikx dx


P kZ

(15.21)

which in a 3D space becomes1
f (r)   (  ei k.r f (r )dr ) ei k.r ,k R3 P 

(15.22)where P is determined within a multiplicative constant.
597

Advances in Optics: Reviews. Book Series, Vol. 5

We may conclude then that the Fourier transform is the limit when a  , b   of theFourier series and that the Fourier transform of a function, if it exists, is the sum of an
infinity of plane waves exp(i k.r) each has wavelength   2 / k and relative amplitudef (k )    ei k.r f (r ) dr ,(15.23)

where kr is dimensionless and f (k) is the Fourier transform of f (r).
It is not superfluous to inform readers that the formula (15.22) was found to be very
important because it alone may explain the principles in quantum mechanics and laws of
optics [2, 3].
For examples concerning laws of optics, we have deduced the theorem: “In the diffractionof a plane wave k0 by a 3D object represented by a function f D (r ) equal to unity forr  D and to zero for r  D under Fraunhofer conditions the diffracted wave iscomposed of plane waves k each having amplitude equal to the Fourier transform of theobject function calculated for the deviation k  k  k0 ”.
15.2.5. Obtaining Fourier Series of Current Functions
15.2.5.1. Periodic Function Built from Delta-like Function
 a ( x)   ( x),  a  x  a, P  2a
We simply havec ( n) 

1 a  i 2 n Px
1e
 ( x)dx  ,


a
P
2a

so that formallyx
2 i n
1
1
1 1 
n x
2a
 a ( x)    e  cos()
2a 2a nZ
2a a n 1an0This is not a convergent series.

598(15.24)

Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers

15.2.5.2. The Saw-tooth Function Built from f ( x)  x, 0  x  a
The saw-tooth function is described by the graph shown in Fig. 15.1.

Fig. 15.1. Graph of f ( x)  x, 0  x  a.

We havea xdx 

0

a

a

a 2   i 2 n P   a, 1.ei 2 n P   0,
 ,  xe



2 
0

0
x

x

so that from the formula (15.14) for calculating Fourier series we deduce thatxa  P i 2 n Px ()e
, 0  x  a, P  a,
2 nZ 2i n(15.25)

n0

i.e.


1

x

1

x

 n sin(2 n a )   ( 2  a ), 0  x  a, P  a(15.26)n 1Formula (15.26) permits to calculate the value of pi by the formula by putting 4x  a
1
3

1
5

1
7

  4(1     ....),(15.27)

which is the historical formula found by James Gregory (1638-1675) then by Euler in
1734 by a famous method very well explained by Ayoub in Ref. [6].
More generally we may prove that
b

b

b
a
a
 i 2 n
 i 2 n
 i 2 n i 2 n Px  i 2 n Px PPP xdx   b  a ,  xe

be

ae

Pe
,

1.e
  0,

a
2

a

a
b

2

2

599

Advances in Optics: Reviews. Book Series, Vol. 5

x

x-a

i 2 nabPP  ()e2nZ 2i n(15.28)

n0


P
x-a
  ( )sin(2 n) a  x  b, P  b  aPn 1  n

15.2.5.3. Periodic Function Defined from the Heaviside Function
Consider the function described by the graph shown in the following Fig. 15.2.
f ( x)  H a ( x)  0 if  a  x  0, H a (0) 1,
2(15.29)H a ( x)  1 if 0  x  a, P  2a

Fig. 15.2. Graph of H a ( x).

We havea H a ( x )   a,

a
a

0

0

a

x
1
x
x
 i 2 n 
 i 2 n 
 i 2 n 
 i 2 n 




2a
2a
2a
2a
H
(
x
)
e

H
(
x
)
e

H
(
x
)
e

H
(
x
)
e
 a

 a

 a

 a
 ,

a 
 a 
 0 
 0

so that by (15.14)

H a ( x) a
 i 2 n
1 1
1 1
2a i 2 n 2ax
2a   (0  0   (  )  (e 1))()e,
2 2a nZ
2 2
2i n
n0

600

Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers

H a ( x) 

1
2
n x

sin(
)
2 n 0  na(15.30)n odd

Formula (15.30) is conformed with the result given by Spiegel in Ref [4].
15.2.5.4. The Square Wave Function SW ( x)This is the function described by the graph in the following Fig. 15.3.

Fig. 15.3. Graph of SW ( x).

and related to the function H a ( x) by the relation
SW ( x)  2H a ( x)  1,

(15.31)so that according to (15.30)
4
n x
sin(
)
a
n 0  n

SW ( x)  (15.32)n odd

15.2.5.5. The Periodic Function Constructed from x
The function called repeating ramp is represented in Fig. 15.4 by the following graph.

601

Advances in Optics: Reviews. Book Series, Vol. 5

Fig. 15.4. Graph of the RR function x .

The considered function is not continuous so that we can’t utilize (15.14) for calculating
its Fourier series. Instead because its derivative function is identical to the square wave
function so that its Fourier series is obtainable by integration of (15.32) then calculate theaconstant of integration by putting x 2f ( x)  x  C  

n 0
n odd

4 a

 n n

cos(

n x
a
1
n x
)   4a  ( ) 2 cos()
a2an 0  nn odd

From (15.33) we may obtain the value of pi squared and many sums overFor examples by taking x  0 and x (15.33)1.
n2

a
we get
3

1 1
1
1
2
1 1
1
1
 2  2  2  2  .....,
 ( 2  2  2  2  ...)
8 1 3
5
7
12
1 3
5
7

2

(15.34)15.2.5.6. Fourier Series of a Function Corresponding to a Displaced Graph
Let f ( x) be defined in the interval a  x  b then g ( x)  f ( x   ) is defined in theinterval a    x  b  .
The graph of g ( x)  f ( x   ) is that of f ( x) displaced left-hand side a distanceequal to  .
Similarly the graph of g ( x)  f ( x)   ) is that of f ( x) displaced vertically down a
distance equal to  .
By displacing up-down and left-right a graph we may centralize it and get its
Fourier series.
602

Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers

For example the Fourier series of the function represented by the graph in Fig. 15.5 is
H a ( x   )     

12n ( x   )

sin(
)
2 n 0  na(15.35)n odd

Fig. 15.5. Graph of H a ( x   )   .

15.2.5.7. Fourier Series of a Function Corresponding to a Symmetrised Graph
Let f ( x) be defined in the interval a  x  b then g ( x)   f ( x) is defined in the sameinterval. The graph of g ( x)   f ( x) is symmetric with respect to the axis Ox of the graphof f ( x). Similarly the graph of the function g ( x)  2C  f ( x) is symmetric with respect
to the line parallel to Ox at y  C.
For example because the graph in the following Fig. 15.6 is obtainable from the graph off ( x)  x described in (15.33) by a symmetrization with respect to the axis Ox following
by a upward displacement of a distance a, we see that the Fourier series corresponding
to this graph isg ( x)   x  a 

a
1
n x
 4a  ( ) 2 cos()
2an0  n(15.36)n odd

Fig. 15.6. Graph of the triangular signal.

603

Advances in Optics: Reviews. Book Series, Vol. 5

15.2.5.8. The Periodic Function Built from f ( x)  x2 , 0  x  a, P  aThis function has f '( x)  2x and is described by the graph shown in Fig. 15.7.

Fig. 15.7. Graph of f ( x)  x2 , 0  x  a, P  a.

We have
a

a

a

3 
 i 2 n 
 i 2 n   i 2 n P 
2 PP x 2 dx   a ,  x 2 e

a
,
2
xe

2
a
,



 2e
  0,

0
3 
0

0

0

x

a

x

x

so that following (15.14) its Fourier series isx

i 2 n
a2a2a a 2 i 2 n axax   a (
)e
 (
) e
, 0  x  a,3
a nZ 2i nnZ 2i n
2

n0

x2 

n0

a2 a2  1x
a2  1
x

sin(2
n

)


 cos(2n a ), 0  x  a3
 n 1 na 2 n 1 n 2(15.37)

The formula (15.35) is given by Spiegel in Ref. [4] with the inconvenient remark that atx  0 and x   the series converges to 2 2 . We think that the difficulty results in thefact that the second term in this formula is undetermined for x  0. To bypass thisdifficulty we utilize the formula (15.25) to transform the Fourier series (15.37) into thesimpler one
x2 

a2
a
a2  1x a ( x  )  2  2 cos(2n ), 0  x  a32
 n 1 na(15.38)From (15.36) we get the equation
(

604


x2 x 1
1
x
2


)


cos(2n ), 0  x  a

2
2
a
a 6
a
n 1 n(15.39)

Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers

which gives by putting x  0 the values of pi squared and different sums over

1
such
n2

as the Riemann zeta function of second order


1

2

n  6(15.40)2n 1

Another way of calculating the Fourier series of f ( x)  x 2 is possible by remarking thatthe derivative of f ( x)  x 2 is f '( x)  2x so that by an integration of (15.24) we getx 2  ax 

a 2  1 i 2 n xaeC
2 nZ ( n) 2n0Integrating this function from x  0 to x  a and remarking that  eai 2 n

0

C

x
a

dx  0 we geta2
so that6a2
a 2  1 i 2 n xa
x   ax  e,0 xa6
2 nZ ( n) 2
2

(15.41)n0This result furnishes in turn the formula for calculating pi squared and sums related to


2
1
x
x 1
2 x
cos(2
n

)


(
  ), 0  x  a

2
2
a
a
a 6
n 1 nFor examples we get corresponding to1
n2(15.42)x
1 1
 0, ,a2 8

1 2 1 1 1 1
1
1 1
1 1
  2  2  2  2  ...,  2  2  2  2  2  ...,
6
1 2 3 4
12
1 2
3 4
11
2 1
1
1
1
2 
( 


 ...)
12.16
2 12 32 52 7 2Similarly for(15.43)x 1 1 1
 , , .
a 3 4 6605

Advances in Optics: Reviews. Book Series, Vol. 5

15.2.5.9. The Ramp Function Built from f ( x)  0,  a  x  0, f ( x)  x,0  x  a, 2a
The ramp function is represented by the graph in Fig. 15.8.

Fig. 15.8. Graph of f ( x)  0,  a  x  0, f ( x)  x,0  x  a.

This function f ( x) is not continuous so that we can’t apply the formula (15.14) for
obtaining its Fourier series.
Instead by remarking that by derivation it becomes a Heaviside-like function given in(15.29) we integrate (15.29) and get

1
2
n x
x
x
 1 
f ( x)   (  sin())   2a   cos( n )  C
2 n0  na2an odd  n 
2

n 0

n odd

Putting x  a then x  0 and summing the results we get the constant of integrationa
C  and the Fourier series of the ramp function
4
2

x a
x
 1 
f ( x)    2a    cos( n ) , a  x  a2 4an odd  n 
n 015.2.5.10. The Parabolic Function f ( x)  x2 , a  x  a, P  2aThis function is described by the graph shown in Fig. 15.9.

606(15.44)

Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers

Fig. 15.9. Graph of f ( x)  x2 , a  x  a.

Becausea  i 2 x 
 x 2 dx   2a ,  x 2 e 2 a   a 2 e  i  a 2 ei  0,   a
3 
a
a

3

a

 i 2 2xa  ii 2 xe  2ae  2ae  4a,a
we obtain from (15.14) the Fourier series of f ( x)  x2 , a  x  a

x2 



a2
1 i 2 n x2aa a 21xa2
 2a 2 e4acos(2 n
)

2
2
3
3
a
nZ ( n)n 1 ( n)(15.45)n0Another way to get the Fourier series of f ( x)  x2 , a  x  a is possible by a suitable
integration of f '( x)  2x given by (15.28)

x



ab
P i 2 n x-aPP
x-a  ()e  ( )sin(2 n) a  x  b, P  b  a,2
PnZ 2i nn 1  n
n0



1
x-a
)sin( n),  a  x  a, P  2aan 1  nx  2a  (Indeed by integration from 0 to x we get
1 2
x a ) (cos  n
  cos(n )),  a  x  a, P  2aa n 1  n

x 2  4a 2  ((15.46)607

Advances in Optics: Reviews. Book Series, Vol. 5

Performing now once more integration from a to 0 and remarking thatx-a

 cos(n a )dx  0,0

(15.47)a

we get the following resultsa31 4a 3  ( ) 2 cos n ,
3

n
n 1

which leads to
2
12



  ( ) n
n 1

1
,
n2

(15.48)and the final form of (15.44)
x2 


a2
1
x-a 

 4a 2  ( ) 2 (cos  n,  a  x  a
3

n
a 

n 1

15.2.5.11. The Periodic Function Built from f ( x)  x3 , 0  x  a, P  aThis function is described by the graph shown in Fig. 15.10.

Fig. 15.10. Graph of x3 , 0  x  a, P  a.

We haveaax
 i 2 n 
 3 i 2 n Px 
a
3 
2
2
P

a
,
3
x
e
 x3d x   a ,  x e  3a ,

0
4 
0

0


608

4

(15.49)

Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbersaa i 2 n Px  i 2 n Px 
6
xe

6
a
,


6e
 0

0

0
so that its Fourier series following (15.14) isx
i 2 n
a3
aa 2 i 2 n ax
a 3 i 2 n ax
2ax   a ()e
 3a  () e
 6 ()e4nZ 2i nnZ 2i nnZ 2i n
3

n0

n0

(15.50)n0Thank to (15.25) and (15.39) we get a useful form of the Fourier series ofx3 , 0  x  a, P  ax3  

a2
3ax 2
a 3 i 2 n axx 6 ()e,22nZ 2i n(15.51)n0

and a beautiful formula for calculating pi tripled and sums containing(1x
sin(2n )3
na
x3 3x 2x 2 31x
 2 )
  3 sin(2n )
3
a
2a
2a 3
a
n 1 n(15.52)15.2.6. Polynomial Equivalent to an Individual Series in a Fourier Series
From the above experiences in getting the Fourier series of x, x2 , x3 we realize that
Lemma 1:
By successive integrations from 0 toa
A1 ( x)  x 2xof the Fourier series of the binomialx
i 2 naaA1 ( x)  x    ()e a , 0  x  a , P  a ,2
nZ 2i n(15.53)n0

one obtains a set of polynomials Am ( x)Am ( x)   Am 1 ( x) dx  Am (0),x(15.54)0each equal to an individual series as shown in the following formulae

609

Advances in Optics: Reviews. Book Series, Vol. 5
a m i 2 n xa
) e
, 0  x  a, P  anZ 2i n

Am ( x)   ((15.55)n0Moreover by remarking thata i 2 nexadx  0,(15.56)

 A ( x)dx 0,(15.57)0

we geta0m

and finally by (15.54) the useful formula
Am (0)  x
1 adx  Am 1 ( x)dx00a(15.58)

According to (15.58) we get the interesting formula for calculating Am ( x) from Am1 ( x)Am ( x)   Am 1 ( x)dx 
x

0

x
1 a
dx  Am 1 ( x)dx0
a 0

(15.59)For examplesx
a
1 a x 2 ax
A2 ( x)   ( x  )dx   (  ) dx 02
a 0 2 2x 2 ax 1 a x 2 ax
x 2 ax a 2
 (  )   (  ) dx  (  ) 2 2
a 0 2 22 2
12A3 ( x)  (x3 ax 2 a 2 x 1 a x 3 ax 2 a 2 x
x 3 ax 2 a 2 x)  ( )dx  3!
4
12
a 0 3!
4
12
3!
4
12

(15.60)and by putting Am (0)  Am

Am ( x) :xmx m 1
xm2
x mm
 A1
 A2
 ...  Amm!
(m  1)!
(m  2)!
(m  m)!

By construction Am ( x) has the properties

P m i 2 n Px) e,nZ 2i n

Am ( x)   (
n0

610

(15.61)

Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers(15.62)A 'm1 ( x)  Am ( x),


P 2m
) ,
2i n

(15.63)

P 2 m 1P 2 m 1) ())  0, m  0,
2i n
2i n

(15.64)

A2 m (0)   2(n 1

A 2 m 1 (0)   ((
n 1

a
a
 ,
2
2

(15.65)Am (1)  Am (0)   m1

(15.66)

A1 (0)  0 

15.2.7. Obtaining Bernoulli Polynomials as Fourier Series
Defining the Bernoulli polynomial Bm ( x) by an operator applying on the monomial xm
as followed [7]Bm ( x) 

we get the properties Bm ( x  1) 

x
xm ,
e 1

(15.67)x

x( x  1) me 1
x

(15.68)B 'm ( x)  mBm1 ( x),and the generating function Bm ( x)

m0

tm
t

e xt ,
m ! et  1(15.69)1
2(15.70)which by identification gives
B0 ( x)  1, B1 ( x)  x 

Now, from the Fourier series of B1 ( x) obtained from (15.14), (15.25)
B1 ( x)  x 

11
  ()ei 2 nx , 0  x  1
2
2
i

n
nZ

(15.71)n0

we get by integration from 0 to x

611

Advances in Optics: Reviews. Book Series, Vol. 51

 B ( x)dx  1! ( 2i n ) (e
x

0

1

2

i 2 nx(15.72)

 1)

nZ
n0

Taking another integration from 0 to 1 and utilizing the formula (15.57) we get1 dx  B ( x)dx  1! ( 2i n ) ,
1

0

x

1

0

(15.73)

2

nZ
n0

so that
1 2
1 2 i 2 nx)  1! () enZ 2i nnZ 2i n B1 ( x)dx  1! (
x

0

n0

Baptizing(15.74)n0

1
B2 ( x) the left hand side we get2

1 2 i 2 nx
) e
,
2
i
n
nZ

B2 ( x)  2! ((15.75)n0

and see that B2 ( x) has the properties of a Bernoulli polynomial of order 2 B '2 ( x)  2B1 ( x),B2 (0)  B2 (1). By recurrence we get

1 m 1 i 2 nx
) e
,
2
i
n
nZ

Bm 1 ( x)  (m  1)! ((15.76)n0

Bm 1 ( x)  (m  1)! Bm ( x)dx  (m  1)!  dx  Bm ( x)dx,
x

1

x

0

0

0



1 m 1)nZ 2i n

Bm 1  (m  1)! ((15.77)(15.78)n0

From (15.78) we obtain immediately the well-known properties B2m1  0 if m  0 andB2 m B2 m 2  0.

The formula (15.76) giving the equivalence between a Bernoulli polynomial with a
Fourier series was announced by Hurwitz without proof in 1890 as private communicationand may be found in Ref. [8].Formula (15.77) obtained from a property of ei 2 nx (15.57)
allows the calculation of Bm1 ( x) from Bm ( x). With the formula (15.77) we get the set of
Bernoulli polynomialsx

B0 ( x)  1, B1 ( x)  1! 1dx  1!
0

612

1

 xdx  x  2  B x  B ,
1

0

0

1

Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers

 x2
 x3
B
B
B x
x
x2 
B2 ( x)  2! B0  B1   2! 2 , B3 ( x)  3! B0  B1   3! 2  3! 3 ,
2
1
2!
2! 
2! 1!
3!

 3!
(…)

Bm ( x)B
B xm2x m B x m 1
 B0  1
 2
 ...  m ,m!
m! 1! (m  1)! 2! (m  2)!
m!
m
 m
Bm ( x)    Bk x m  k : ( B  x)mk 0  k (15.79)The formula (15.79) was established by Lucas in 1891 [9]. It is a symbolic formula where
undefined notations Bm are to be replaced with well defined Bm .
Combining (15.76) with (15.79) we finally getmm
1
Bm ( x)    Bk x m  k : ( B  x) m : m!e 2i  n x
m
k
(2
i

n
)
k 0 
nZ(15.80)n 0The above formula permits to obtain the Bernoulli polynomials Bm ( x) simultaneouslywith the Bernoulli number Bm .
The inconvenience is that it is fastidious for calculate Bm with high values of m.
Fortunately we have by operator calculus [7] a convenient formula due to Euler in 1738
[9] for calculating Bmm m(1  m) Bm    ()k Bk Bm  k : ( B  B) m , m  1,
k 0  k (15.81)

which leads because ()k  ()2m1k  0 immediately to the property(2m) B2 m1  ( B  B)2 m1  0(15.82)

From (15.81) we get for examples
3B2  2B1 B1 , 7 B6  30B2 B4  5 / 30,

(…)

613

Advances in Optics: Reviews. Book Series, Vol. 5

 20 
 20 
 20 
 20 
 20 
21B20  2   B2 B18  2   B4 B16  2   B6 B14  2   B8 B12  2   B10 B10
2
4
6
8
 10 
(15.83)
In Ref. [7] there is another method which is more convenient for calculating Bm .
It consists in utilizing the Lucas formula (15.79) corresponded to x  1

 m
 m
 m
Bm (1 )  B m    Bm 1    Bm  2  ...    B0 ,
1
2
 m

(15.84)B1 (1 )  B1 (0)  1(15.85)

 m
 m
 m
 m 
Bm (1)  Bm   m1    B0    B1  ...    Bk  ...   Bm 1 ,
0
1
k
 m  1(15.86)and the formulato get the recursion relation [7]Theorem:which may be written under the matrix form where the first line corresponds to m  0 as
shown in the following formula
1
1
1
1
...

...
...
3
...
6 4 ...
... ... ...

B0
1
B1
0
B2
0
=
B3
0
...
...
 m  1  m  1 m  10 
 ... ... ...  Bm
 0   1 
 m 
2
3
4
...

(15.87)This matrix equation may be resolved by doing linear combinations over lines from thesecond one in order to replace them with lines each containing only some non-zerorational numbers.
For instance for calculating successively B0 , B1 , B2 , B4 , B6 ,..., B18  we may utilize thefollowing matrix equation

614

Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers

1
1
0
0
0
0
0
0
0
0
0

2
1
0
0
0
0

3
1
5
1
0
7
9/5 0
0
5
0
0
61
3

0
0
105
35
0
0
0
240
0
17
0
2052
0
0 775

9
0 11
0

0 13

0
0
0

0
0
0

0 15
0 0 170 0 0 19

B0
1
B1
0
B2
0
B4
0
B6
0
B8 = 0
B10
0
B12
0
0
B14
0
B16
B18
0.

(15.88)We remark that the last line of this matrix has replaced the line19   , i  0,1, 2, 4,6,8,10,12,14,16,18
 i For example from (15.88) we have2052 B1  775B6  19 B18  0  B18  (1026 

775 1
)
42 19

This simply obtained value is to be compared with that given by Coen by anothermethod [10]
B18 

43867
798

15.2.8. Obtaining Powered of Pi and Even Positive Zeta Functions
From (15.71) we get the formulae for calculating the powers of pi
1 m i 2 nx) e, m  1,
2
i
n
nZ

Bm ( x)  m! (n0(2 ) 2 m  () m 1 (2m)!

21
cos(2 nx),
B2 m ( x) n 1 n 2 m(15.89)615

Advances in Optics: Reviews. Book Series, Vol. 5(2 ) 2 m 1  () m 1 (2m  1)!

1
B2 m 1



1

n
( x)
n 1

2m

(15.90)sin(2 nx),

1 1 1 1 1
where we may conveniently take x  0, , , , , .
8 6 4 3 2

For example(2 ) 2  2!


2  1
4
1
1 1 1
cos( n) () n 2  48( 2  2  2  ...)

2
1 n 1 n
1 1 1 n 1n
1 2 3
B2 ( )(   )
2
4 2 6

As for positive even Riemann zeta functions we find again from (15.77) the formulae firstproven by Euler as reported by Ayoub [6]1() m 1
 (2m)   2 m (2 ) 2 m B2 m(2m)!2n 1 n(15.91)1(2 ) 4 B448

(15.92)For examples
 (2)   2 B2\ ,  (4)  

15.2.9. Fourier Series of Polynomials
From (15.14) we may calculate the Fourier series of monomials from the matrix equation  f ( x )   f ( x)   f '( x)   f "( x)  ... ...  f
1f ( x)

0

1

1

1

1

0

0

0

( m 1 )



...
1/2

1

0

1

x



1

( x)

2



i 2  nx

/ (2i n )i 2  nx

/ (2i n )2i 2  nx

/ (2i n )

3

n0

1!


x

e

e
n0

1/3

1

2!

...



(15.93)

e
n0

x

3

...
x

m

1/4

1

3

3!

...

...

...

...

...

...

...

1 / ( m  1)

1

m

m(m-1)

...

m!

...


e

i 2  nx

/ (2i n )mn0

For example for a Hermite polynomial of fourth order in the interval 0  x  1 we have616

Chapter 15. New Formulae for Obtaining Fourier Series, Bernoulli Polynomials and Numbers

H 4 ( x)  16 x 4  48 x 2  12  


4
i
1
i
1
 32 (
6
 12
) ei 2 nx 23
5(2 n)(2 n)(2 n) 4n  0 (2 n)



4
1
16
48
96
 32( x  )   (

)
cos(2

nx
))

(
sin(2 nx)

2
4
3
5
2
( n)
n 1 ( n)n 1 ( n)(15.94)and may write
H 4 ( x)  



76
16
48
96 sin(2 nx) (15.95)
 32 x   (

)
cos(2

nx
))

2
x
(
,

2
4
2
5
( n)
2 nx
n 1 ( n)n 1 ( n)so that for x  0
H 4 (0) 

76 16 48
 
 12
5
6 90

(15.96)15.3. Remarks and Conclusion
The main remark of this work is that it clarifies for us the question why Fourier transformand Fourier series are so important for the studies of Optics and Acoustics. That is becausexboth utilize the exponential functions exp(2n ) that represent waves of frequenciesPn. They also are crucial in quantum mechanics where r and p represent theposition and momentum of a wave that we‘ve resumed in the formula  k  FT  r [2].
Apart from recognizing this remark, in this work is given firstly a new formula for
calculating the Fourier coefficients of continuous periodic functions defined on any
interval a  x  b as so as of functions having primitive or derivative continuous
currently known, including the Heaviside-like function and those derived from it byintegration, displacement or/and symmetrization with respect to an horizontal axis.
Secondly, it utilizes the fact that  e
P0i 2 n

xPdx  0, n  Z for obtaining the polynomial

P m i 2 n Px) epermitting to divide the Fourier series of anZ 2i nequivalent to the series  (n0function into sums of polynomials and series; to calculate positive even Riemann zetafunctions and many serial forms for each power of pi.
Thirdly concerning Bernoulli polynomials [8] this work proposes a concise method for
obtaining Bm ( x) from Bm1 ( x) which leads to the symbolic formula Bm ( x) : ( B  m)mthat was done in another way by Lucas in 1891 [9]. From this formula and the fact thatBm (1)  B m (0)   m1 we get an elegant matrix formula for calculating Bernoulli numbers
617

Advances in Optics: Reviews. Book Series, Vol. 5

which are omnipresent in many problems such as in heat equations, in powers sums onarithmetic and algebraic progressions [7].
We wish that the new formula for Fourier coefficients, the relation between Bm ( x) andBm1 ( x), the matrix formula for Bm should be judged as worth to be added to lessons on
Fourier series and Bernoulli numbers taught at universities.

Acknowledgements
The author thanks very much his adorable spouse and his responsible daughters Do Thi’sfor the healthcare they devote to him during the hard period of corona virus isolation.
He acknowledges Prof. Sergey Yurish for appreciations about his works now andin the past.

References[1]. J. B.-J. Fourier, Oeuvres de Fourier (D. Darboux, Ed.), Gauthier-Villars et Fils, Paris, Gallica,
1888, pp. 218-219.
[2]. D. T. Si, Principles of quantum mechanics and laws of wave optics from one mathematicalformula, Applied Mathematics, Vol. 10, 2019, pp. 892-906.
[3]. D. T. Si, The Fourier transform relation between Dirac Bras  k  FT  r and wave optics,
Advances in Optics: Reviews Book Series, Vol. 4, 2019, pp. 117-146.
[4]. Wikipedia, https://en.wikipedia.org/wiki/Fourier_series & M. R. Spiegel, Theory andProblems of Fourier Analysis, Schaum’s Outline Series, Mc Graw-Hill, 1974.
[5]. G. B. Folland, Table of Fourier Series, Fourier Analysis and Its Applications, Wadsworth &Brooks/Cole, 2014, pp. 1-4.
[6]. R. Ayoub, Euler and the Zeta function, The American Mathematical Monthly, Vol. 81,
Issue 10, 1974, pp. 1067-1086.
[7]. D. T. Si, Obtaining easily powers sums on arithmetic progressions and properties of Bernoulli
polynomials by operator calculus, in New Insights into Physical Science, Vol 3, Book
Publisher International, 2020, pp. 6-83.
[8]. F. Costabile, F. Dell’Accio, M. I. Gualtieri, A New Approach to Bernoulli Polynomials,
Rendiconti di Mathematica, Serie VII, Vol. 26, http://www1.mat.uniroma1.it/ricerca/
rendiconti/ARCHIVIO/2006(1)/1-12.pdf[9]. E. Lucas, Théorie des Nombres, Chapter 14, 1891 Topics: Number theory, Gauthier-Villars,
https://archive.org/details/thoriedesnombr01lucauoft/page/n7/mode/2up[10] L. E. S. Coen, Sums of powers and the Bernoulli numbers, MD Thesis, Eastern Illinois
University, 1996.
[11]. Group of authors, Bernoulli Numbers, https://vdocuments.site/bernoulli-number.html

618

Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculus

Chapter 16

Obtaining Fourier Transforms of Functions
by Differential Calculus
Do Tan Si1

16.1. Introduction
The notion of Fourier transforms was introduced 200 years ago, in 1822, by Joseph
Fourier in his famous work “Analytical Theory of Heat” [1]. Afterward tables of Fouriertransforms of functions, distributions are also available for examples in Refs. [2, 3].
Everything seems sufficient for researchers when dealing with the problems in
Heat theory.
However we now know that the Fourier transform is considered as a natural relation
between a spatial 3D object described by a function f (r ) and a sum of infinity of planewaves each with wave vector k , wavelength   2 / k , amplitude being the Fouriertransform f (k ) of f (r ).
From this consideration we have enlarged the fields of applications of Fourier transformto the domain of principles of quantum mechanics [4] and in optics in confirming that ina diffraction of a plane wave k 0 by the object f (r ) the component k of the diffracted
wave has amplitude proportional to f (k  k0 ) [5].
It has then for us a signification to re-actualize the study of the Fourier transform offunctions. To realize this aim, we firstly will introduce the notions of operator calculus [6]which leads to the differential representation of the Fourier transform. Afterward a method

Do Tan Si
Ho Chi Minh-city Physical Association, Vietnam, Université libre de Bruxelles, UEM, Belgium

619

Advances in Optics: Reviews. Book Series, Vol. 5

free of integrations for calculating simply the Fourier transforms of functions, a list of
current properties and a table of Fourier transforms are given.

16.2. Differential Operators Realizing Fourier Transform
16.2.1. Introduction to Operator Calculus
Let Dx be the derivative operator, X̂ the Eckaert operator “multiply by the variable x ”
and Iˆ the unity operator [7]Dx f ( x)  f '( x),(16.1)

Xˆ f ( x)  x f ( x),(16.2)

Iˆ f ( x)  f ( x),(16.3)

we get the identity between operatorsˆ  Iˆ,Dx Xˆ  XD
x

(16.4)

as we can verify by applying both members of it on any entire function f ( x)Dx xf ( x)  xf '( x)  f ( x)
From the identity (16.4) we may deduce by recurrence the followingˆ m  mD m1 , m  N
Dx m Xˆ  XD
x
x

(16.5)

In fact suppose that (16.5) is correct we haveˆ m  mD m  XD
ˆ m1  (m  1)D m1 QED
Dx m1 Xˆ  Dx Dx m Xˆ  Dx XDxxxxNow let f ( x) be an entire function and f ´( x) its derivative function we get from (16.5)
the identity f ( Dx ) Xˆ  Xˆ f ( Dx )  f '( Dx ) , i.e.

f (Dx ) Xˆ f 1 (Dx )  Xˆ  f '(Dx ) f 1 (Dx )(16.6)

From (16.6) we get for example

eaDx Xˆ eaDx  Xˆ  aIˆ,
620

(16.7)

Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculus
22e aD Xˆ e  aD  Xˆ  2aDx

(16.8)

As first application we get from the identity (16.7)

eaDx Xˆ meaDx  ( Xˆ  aIˆ)m ,(16.9)

then by applying the latter on a constant the formulae

eaDx xm  ( x  a)m ,(16.10)eaDx f ( x)  f ( x  a),(16.11)

which means that eaD realizes the translation of entire functions.
x

A extremely precious remark is that because (16.7), (16.8) are obtained from and only
from (16.4), by replacing the couple ( D , Xˆ ) with any other couples of operator ( Aˆ , Bˆ )xverifying the same commutation relation with the couple ( Dx , Xˆ ), saysˆ ˆ  BAˆ ˆ  Iˆ,
 Aˆ , Bˆ   AB


(16.12)

we will get from any identity between ( Dx , Xˆ ) exactly the same between ( Aˆ , Bˆ ).
aD 2 ˆ  aDx2
 Xˆ  2aDx ,For example from (16.7), (16.8) we get e x Xe

(16.13)ˆ2
ˆ  aAˆ 2  B
ˆ  2aBˆ
eaA Be(16.14)ˆˆ ˆ ˆ  ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
 XD
 x ,ln X   X  Dx ,ln X    X ,ln X  Dx  X ( Dx ln X  ln XDx )  I ,(16.15)More concretely because

we have the relations

eaXDx ln Xˆ e  aXDx  ln Xˆ  aIˆ,ˆˆ

(16.16)ˆˆ
eaXDx (ln Xˆ ) m e  aXDx  (ln Xˆ  aIˆ) m ,

ˆ
ˆ
ˆ ˆ
ˆ
eaXDx Xˆ e  aXDx  eln X  a I  e a eln X  e a Xˆ ,ˆwhich give us the dilatation operator e aXD

ˆ(16.17)x

ˆeaXDx x  e a eln x , e aXDx f ( x)  f (e a x),(16.18)621

Advances in Optics: Reviews. Book Series, Vol. 5that we will use hereinafter to calculate the Fourier transform of f (ea x).
For curiosity we here give some other examples that may be found in Ref. [6].
2x
  Xˆ 2 Dx , Xˆ 1   I  e  aXˆ Dx f ( x)  f (),

1  ax(16.19) Xˆ ln Xˆ Dx ,ln ln Xˆ   I  eln Xˆ ln Xˆ Dx f ( x)  f ( x a )(16.20)16.2.2. Operators Realizing the Fourier Transform
Consider the Fourier transform of a function f ( x) defined by the formula
1e  ikx f ( x)dx,2f (k )  FTf ( x) 

(16.21)

we have by direct calculations the properties1e  ikx xf ( x)dx 2 FTxf ( x) 1
iDk e  ikx f ( x )dx,

2 

i.e.

FTxf ( x)  (iDk ) f (k )(16.22)On the other side
FTf '( x) 

1
ik   ikx
e  ikx f ( x)  
 e f ( x)dx, i.e.

2
2 

FTf '( x)  (ik ) f (k )(16.23)

Thirdly
FT  ( x) 1e  ikx ( x)dx, i.e.
2FT  ( x) 12

(16.24)

Searching for a differential transform having the properties of (16.22), (16.23) we see
from the differential identities (16.8), (16.9) that


i ˆ2
X

e 2

622

i

(e 2

Dx 2



i ˆ2
X

(e 2

i ˆ2

i

i ˆ2

i ˆ2

i

i

i ˆ2

X
 Dx
X
 X
Dx
 Dx
X
Xˆ e 2 )e 2 )e 2  e 2 (e 2 Xˆ e 2 )e 2 222Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculus



i ˆ2
X

e 2

i ˆ2
X

( Xˆ  iDx )e 2

 Xˆ  Xˆ  iDx  iDx ,

i

i ˆ2
X

(16.25)and similarly


i ˆ2
X

e 2

(e 2

Dx 2



i ˆ2
X

(e 2



Dx e 2

i

Dx 2

)e 2

i ˆ2
X

 iXˆ

)e 2

(16.26)

From (16.25), (16.26) we arrive to define the operator FT


i ˆ2
X

FT   e 2

i

e2

i ˆ2
X

FT 1   1e 2

Dx 2



i ˆ2
X

e 2

,

i
 Dx 2

i ˆ2
X

e 2

e2

(16.27)(16.28),having the property that when applying on a function, ( Dx , Xˆ ) are changing into ( Dk , Kˆ )i
 Kˆ 2

FTf ( x)   e 2

i

e2

Dk 2

i
 Kˆ 2e 2(16.29)f (k )The operator FT has the properties

FTDx FT 1  iXˆ ,(16.30)FT Xˆ FT 1  iDx(16.31)At this state the crucial problem is the calculation of the factor  .
For this let us remark thatx2

i
( Dx  iXˆ )e 2  0,

FT ( Dx  iXˆ ) FT 1 FTe

i

x2
2

 (iKˆ  Dk ) FTe(16.32)
i

x2
2

0

(16.33)x2i
But ( Dx  iXˆ )e 2  0, so that
i

x2

i

k2

FTe 2  C e 2(16.34)

Similarly we have
i

x2

i

x2

FTe 2  C e 2(16.35)623

Advances in Optics: Reviews. Book Series, Vol. 5

Combination of (16.34) and (16.35) gives eCC  1

i

x2
2

 FTFTeix2
2

 C FTeik22 C C e

i

x2
2

, i.e.

TakeniC  e 4 ,(16.36)

we get finally the differential form of the Fourier transformii Kˆ 2i

FT  e 4 e 2 e 2

Dk 2i Kˆ 2(16.37)e 2

As first application of (16.37) we have
x
x

k

k
i
i
i
i
i
i
x2
k2 
2 FT cos( )  FT (e 2  e 2 )  (e 4 e 2  e 4 e 2 )  2cos(  ),
2
2
4
2

2

2

2

FT cos(

x2
2
k2
k2
)
(cos( )  sin( ))2222(16.38)FT sin(

x2
2
k2
k2
)
(cos( )  sin( ))2222(16.39)Similarly

Eqs. (16.38), (16.39) were proven by Easton [8].
Terminating beautifully this example let us remark from (16.21) that for k  0i
1   i x2
4
e
dx

C

e
,


2 
2

f (0) 1f ( x)dx, so that (16.34), (16.35) give

2 


i
1  i x2
4
e
dx

C

e
, and (16.38), (16.39) give


2 
2

2
1 
x2

cos(
)dx,

2
2
2 

(16.40)

2
1 
x2

sin( )dx

2
2
2 

(16.41)Another operator representing the Fourier transform may be obtained by taking theFourier transform of itself FT  FT FT FT
624

1

i



i

 e 4 e2

Dk 2



i ˆ2
K

e 2

i

e2

Dk 2

.

Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculus

For completeness we add that according to Wolf [9]
i



i



FT  e 4 e 4( Dk 2  Kˆ 2 )(16.42)

As conclusion we may state that the Fourier transform may be represented by any one of
the three differential operators (16.40), (16.41) or (16.42).
16.2.3. The Fourier Transform of f ( x)  1
Applying (16.37) on the Dirac delta distribution  ( x) we get by (16.24)i

i
 Kˆ 2
4
2

FT  ( x)  e e

e

i 2
i
Dk  Kˆ 2
2
2

e

1
,
2

 (k ) (16.43)i.e. because x ( x)  0i

i2 e 4 e 2

Dx 2

i

 ( x)  e 2

x2(16.44)

Replacing x with ix we get formallyii
 Dx 22 e 4 e 2i x2

 (ix)  e 2(16.45)1
 ( x),
a(16.46)1i(16.47)

From the property of Dirac delta (ax) 
we recommend that

 (ix)   ( x)With this recommendation we get
i

e2

Dx 2

i
 x2

e 2



i 1
i
 2 e 4  ( x)  2 e 4  ( x),i(16.48)and finallyii
 Kˆ 2

FT 1  e 4 e 2

i

e2

Dk 2

i
 Kˆ 2ii
 Kˆ 2e 2 1,  e 4 e 2i( 2 e 4 e

i


4

i ˆ2
K

e 2  ( x))  2 (k )(16.49)625

Advances in Optics: Reviews. Book Series, Vol. 5Classically this result must come from (16.21) FT 1 1e  ikx dx, but for us it is not2 

easy to prove it.
16.2.4. Geometric Signification of a Convolution ProductConsider 3D objects represented by functions f (r ) equal to zero if r is outside the object
and to unity if inside. The integral r R 3f (r ) g (r )dr represents then the volume of theintersection between two corresponding objects.
If we displace the object f (r ) with respect to the origin by a vector r0 then the volume of
the intersection between the objects described by f (r  r0 ) and g (r ) becomes



r R3

f (r  r0 ) g (r )dr(16.50)i.e. the value at r0 of the convolution product between f (r ) and g (r ), the all denotedby( f  g )(r0 )  r R3

f (r0  r ) g (r )dr(16.51)

As example let f (r ) represents the following square and g (r )  f (r )Fig. 16.1. Graph of Sq( x) .

When we right-displaced one of the squares an amount x with 0  x  a / 2 the new areaof the intersection is equal to (a  x)a1 so that the convolution product Sq( x)  Sq( x)corresponding to a  1 is the triangular function tri( x).

626

Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculus

Fig. 16.2. Area of Sq( x)Sq( x)  (a  x)a 2 .

Fig. 16.3. Graph of tri( x)  a  x .

As extension of the above method we think that it is feasible for calculating theconvolution products tri( x)  tri( x), Sq( x)  tri( x) etc. and the Fourier transforms ofthem as we can see hereafter.
16.2.5. Fourier Transform of a Convolution ProductConsider the convolution product ( f  g )( x)   f (x  x0 ) g ( x0 )dx0 .
We may write from the formula for translated functions (16.11)
FTf ( x  a)  FTeaDx f ( x)  eiak FTf ( x),

that








FT ( f  g )( x)   FTf (x  x0 ) g ( x0 )dx0   e  ix0 k FTf ( x ) g ( x0 )dx0  FTf ( x)  e  ix0 k g ( x0 )dx0  2 FTf ( x) FTg ( x),(16.52)

i.e. ”The Fourier transform defined by (16.21) of a convolution product of two functions
is 2 time the product of their Fourier transforms”.

627Advances in Optics: Reviews. Book Series, Vol. 5

The above formula is well-known for example in Ref. [10]. It may also be proven bydifferential calculus as we will see in the hereafter paragraph.

16.3. General Properties of the Fourier Transform
From the differential realization of the Fourier transform (16.30)

FTf ( x) 


i
i 2
i
i
 Kˆ 2
Dk  Kˆ 2
1   ikx
4
2
2
2
e
f
(
x
)
dx

e
e
e
e
f (k ),
2 (16.53)

we get simply the following general properties.
16.3.1. The Primordial Property
Thank to (16.49) we may write down the formulaf (k )  FTf ( x)  FTf ( Xˆ )1  f (iDk ) FT1  2 f (iDk ) (k ),(16.54)

which seemingly is a newly highlighted property of the Fourier transform. It may beutilized for obtaining in a simple manner the following formulae.
16.3.2. The Convolution ProductFTf ( x) g ( x)  f (iDk ) g (k )  f (iDk )   (k  k0 ) g (k0 )dk0 


1 
1

f (k  k0 ) g (k0 )dk0 
f (k )  g (k ),

2 
2

(16.55)i.e.

FT ( f ( x)  g ( x))  2 f (k ) g (k )(16.56)16.3.3. The double Fourier Transform
FT FTf ( x)  2 FTf (iDk ) (k )  2 f (i 2 Xˆ ) FT  (k )  f (i 2 x)  f ( x)(16.57)16.3.4. The Inverse TransformEq. (16.57) gives

FT f (k )  f ( x)628(16.58)

Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculus

16.3.5. Transform of Complex Conjugate

FTf * ( x)  f * (iDk ) (k )  ( f (iDk ) (k ))*  f * (k )(16.59)16.3.6. Transform of Translated Functions Following (16.11)

FTf ( x  a)  FTeaDx f ( x)  eiak f (k )(16.60)16.3.7. Transform of a Dilated Functionˆ
From the operator realizing a dilatation e a XD
(16.18) we getx(i) For f ( x) even, f ( x)  f ( x)ˆˆ

ˆ

FTf (e a x)  FTe aXDx f ( x)  e  a Dk K f (k )  e  a ( KDk 1) f (k )  e  a f (e  a k ),(16.61)

FTf (ea x)  ea f (ea k )(16.62)(ii) For f ( x) odd, f ( x)   f ( x)FTf (ea x)  ea f (ea k )(16.63)

Joint these two cases we get the well-known formula

FTf ( x) 

1 k
f( )
 

(16.64)16.3.8. Transform of a Sum of Two Functions

FT ( f ( x)  g ( x)  f (k )  g (k )(16.65)

It is evident that (16.65) is necessary for calculating the transforms of sinusoidal functions.

16.4. Table of Fourier Transform of Differential Operators
ˆOperator A

Dx

ˆ : FT A
ˆ FT 1
Fourier transform of A
i Kˆ

X̂

iDk

Dx Xˆ

 Kˆ Dk

629

Advances in Optics: Reviews. Book Series, Vol. 5
Xˆ Dx Dk Kˆ

Dx  Xˆ
D  Xˆ

i( Dk  Kˆ )
 i( D  Kˆ )Dx 2  Xˆ 2
f ( D , Xˆ )x

ˆ 2)
 ( Dk 2  K
ˆ , iD )f (i Kf ( Dx , Xˆ )f ( iKˆ ,iDk )

x

e

k

k

e

16.5. Table of Fourier Transform of Functions
f ( x)

1



e

 ikx

2 

1

2  (k ) ( x)(2 )1/ 2

f (  x)f ( k )f ( x)f ( k )Dx f ( x)

ik f (k )f ( x)dxf ( x)  f ( x)1

2 f (iDk ) (k )eiax f ( x)e aDk f (k )  f (k  a)eiax

2 (k  a)cos(ax)2( (k  a )   (k  a ))2

sin(ax)2( (k  a)   (k  a))2

H ( x)  Heaviside function,
H '( x)   ( x),Dx ( H ( x)  C )   ( x),Dx ( H ( x)  C )   ( x),H ( x)  H (  x)  1

H ( x)H ( x  a)

H ( x  a)  H ( x  a )630

f (k )  FTf ( x) ik ( H (k )  C 2 (k ))  ( 2 )1 ,
ik ( H (k )  C 2 (k ))  ( 2 )1 ,H (k )  H (k )  2C 2 (k ) 2 (k )  2C  1
1
1
2 (k )  ( 2 ) 1
2
ik
1 1 1
1 1

2 (k )) (   ( k ))
ik
2
2
2 ik
1 1
(   (k ))eiak
2 ik
1 2
sin(ak )2 k
 H (k ) Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculus

Rectan gular function
Sq ( x, a )  0,
a
Sq( x, a) x a/2

1

1 sin(ak / 2)
(ak / 2)2x  a / 2,1aa
H (( x  )  H ( x  ))
a
2
2
1

2
2 ik

sign( x)  H ( x)  H ( x)

1
ix



Dx sign( x)  2 ( x)2
sign(k ) Note : FT f (k )  f ( x)2
1 2
2
ikFTsign( x)  ik

2 ik
2

tri( x)  Sq(1, x)  Sq(1, x)

1 sin 2 ( k / 2)2 (k / 2) 2

( Dx 2  a 2 )e a x  2a ( x)(k 2  a 2 ) FTe a x  2aFT  ( x)

e

2a

1
2
2 (k  a )

a x

2

2
exp( a k )2a1x2  a2
1
Xˆ  1x

1x

iDk FT2
2
Dk ( H (k )  H (k )) Dk sign(k )222sign(k )  C
2i
1
Notification : C  0 because oddx1() n 1
1

Dx n 1n(n  1)!
xx

1Dx (ln x  C )  , x  0 ln x, x  0
x11
 e iaDx , a  0
x  iax
11
 eiaDx , a  0
x  iax1
 2  (k ) x() n 1
1
(ik ) n 1 FT(n  1)!
x

ikFT (ln x  C ) 2
sign(k ), k  02i

2sign(k )  C 2 (k )
2k


2 ak
e H (k )i2  ake H (k )i631

Advances in Optics: Reviews. Book Series, Vol. 5( x2  a2 )( Dk 2  a 2 ) FT

1
1
2
( x  a2 )1
 2 (k ),x  a22Note : ( Dk 2  a 2 )e a k  2a (k )1
( x2  a2 )1  a ke
a 22( Dx  Xˆ )e  x / 2  0,

2
i( Dk  Kˆ ) FTe k /2  0,e x / 2e k / 2
2

2



1

e

1

k2

(2a ) 2 e 4 a , a  0 or a  i ,

 ax2

Notification : eax is dilatation of e x /2
2

2

2
( Dx  iXˆ )e ix / 2  0

2( Dk  iKˆ ) FTeix /2  0( Dx  iXˆ )eix /2  0

2( Dk  iKˆ ) FTeix /2  0

2

i

i
 x2

C e 2

e 2



e

i

C e 2

i 2
x
2

x2

i

Note : e 2

x2

k2

i

 FTFTe 2ix2

 C C  1  C  e 4 , C  e
2

2

x
i
x2
1 ix
)  (e 2  e 2 )
2
2

sin(

x2
)
2

FT cos

i




4

i
k

i
x 2 1 i4 i k2
 (e ee 4e 2 ) 
2 2
2
k2
k2
(cos( )  sin( ))
2
2
2
2

cos(

i
 x2

 C FTe 2

2

2
k2
k2
(cos( )  sin( ))222((eiDk  eiDk )ik  (eiDk  eiDk )) f (k )  0 (ch( x) Dx  sh( x)) f ( x)  0

 (ik  ii  1) f (k  i)  (ik  ii  1) f (k  i)  0 
 f ( k  i )  f (k  i )  0

1f ( x) ch( x)f (k )  ch( k ) or ch 1 ( k ),22Note : ch( (k  i)  ch( (k  i)  0
2
2

1

1

ch(


2

x)


2

k)

eax  eaX 1,

FTeax  eiaDk 2  (k ),e ax

2  (k  ia)

ˆ

632

ch(

Chapter 16. Obtaining Fourier Transforms of Functions by Differential Calculusˆ  1 ) g ( x)  0
( XD
x
2

1
ˆ  1 ) g (k )  0( Dk Kˆ  ) g (k )  ( KD
k
2
2

1

g ( x) 1

g (k ) 

k

x
e

 x2
2

i

Note : ( Dx 2 Xˆ 2 )e  (2n  1)e

 x2
2



i



 e 4eH n ( x)i


4

(2 n 1)

e

e

2

4
1
1

,
4
3
1 x
k 1 4 xk ( x  xk )2

  x3 x2 

1  i2

H n (k ) H n (k )  e
i

1
 x2

1 i

e



 in


2

1
 k2

e 2 H n (k ),

i ( Dk 2  Kˆ 2 )Note : FT  e 4 e 4

H n ( x)e 2 H n ( x)

x1 

1
 k2
2

1
 k2
2

ˆ2

i (( Dk  K ))2e 4e 4

H n ( x),
 x2
2



 in


2

1
 k2

e 2 H n (k )sign(k )(ch(ak )sin(ak )  sh(ak ) cos( ak ) ch(ak ) cos(ak ),2
1
2  ak
Note : FTe H (k ), a  0
x  iai  x416.6. Remarks and Conclusion
Firstly this work introduces some knowledge about differential transforms inmathematical analysis based on the Eckaert operators Dx , Xˆ . From which we may factorizeˆˆ
aD 2
 aD 2sums of operators such as Dx  aIˆ  e  aX Dx e aX , 2aDx  Xˆ  e x Xˆ e x , then obtain the
differential forms of the Fourier transform sayii Kˆ 2i

FT  e 4 e 2 e 2

Dk 2i Kˆ 2

e 2

i



i
 Dk 2

 e 4e 2i ˆ2Ki Dk 2e2 e 2ii e 4e 4( Dk 2  Kˆ 2 )Secondly, from these differential forms we may deduce almost all properties of the Fouriertransform of functions without needs of integrations. Thank to this facility, we establish atable of Fourier transforms of operators and of many functions rather easily with concise
details, some may be added to the existing Tables of Fourier transforms we may find in
Refs [2], [3].
By the way we obtain the formula FT1  2 ( x), the geometric interpretation of
convolution product which happens to be very useful for calculations of Fouriertransforms.
As for Fourier transforms of functions in plane and space such as points, straight lines,
planes, pyramids, ellipses, cylinders, spheres, the readers may find in Refs. [5], whereapplications of Fourier transform and Dirac delta functions in optics are shown.
633

Advances in Optics: Reviews. Book Series, Vol. 5

Acknowledgments
The author acknowledges Prof. Sergey Y. Yurish at Barcelona for the appreciation andencouragement about his contributions in mathematics and optics. He thanks sincerely thereferee who justly demands him many precisions. He think that this work gives many joys
to his spouse and daughters who carefully take care of him in this hard period ofworldwide Covid disaster.

References[1]. J. Fourier, Analytical Theory of Heat, The Cambridge University Press, 1878.
[2]. A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Tables of Integral Transforms,
Vols. 1-2, McGraw-Hill, New York, 1954.
[3]. D. Kammler, A First Course in Fourier Analysis, Prentice Hall, 2000.
[4]. D. T. Si, Principles of quantum mechanics and laws of wave optics from one mathematicalformula, Applied Mathematics, Vol. 10, 2019, pp. 892-906.
[5]. D. T. Si, The Fourier transform relation between Dirac bras  k  FT  r and wave optics,
Chapter 5, in Advances in Optics, Vol. 4, International Frequency Sensor Association
Publishing, Barcelona, 2019, pp. 117-146.
[6]. D. T. Si, Operator Calculus, Edification and Utilization, LAP Lambert Academic Publishing,
Saarbrücken, Deutschland, 2016.
[7]. C. Eckart, Operator calculus and the solutions of the equations of quantum dynamics, Phys.
Rev., Vol. 28, 1926, pp. 711-726.
[8]. R. L. Easton Jr., Fourier Methods in Imaging, John Wiley & Sons, 2010.
[9]. K. B. Wolf, Hyperdifferential operators and integral transforms, Rev. Mexicana Fis. Math.,Vol. 25, Issue 1, 1976, pp. 55-60.
[10]. E. W.Weisstein, Convolution Theorem, https://mathworld.wolfram.com/
ConvolutionTheorem.html, 2010634

Index

Index
2
2.5D
interposer chip, 159
package, 135

3
3D
package, 135
torus network, 136Aa square-law detector, 27
ablation, 474, 477-479, 482
process, 477
threshold, 474, 477, 479, 482
absorption, 477, 478, 482, 519, 528, 534,
564
measurement, 301, 305–311, 319, 321,
327
mechanism, 477
spectral dependence, 301, 309–311
spectrum, 422, 429, 436
two-photon absorption, 474, 477-479
accelerating charge, 29, 30
accelerator processors, 115
Acceptance angle, 182, 185, 186, 192-194,
213, 214, 216, 220, 222, 226, 231
acoustic waves, 535, 560
acousto-optic superlattice modulator, 560,
561
adhesive technology, 151
AGN. See noise, Gaussian
AI processor, 115
Airy function, 338
Allan deviation, 519, 522
Altazimuth, 433, 436, 443, 446
Ampere’s law, 23, 29
amplitude difference comparison ofcoherent envelope (ADCCE), 541
amplitude division, 76, 79, 81

Amplitude
Point Spread Function (APSF), 66
Transfer Function (ATF), 66
Analog-to-digital converter, ADC, 318
Analyte deposition, 491
meniscus-guided coating, 491
spin coating, 491
thermal inkjet printing, 491
thin film, 491
analytical symbolic computation, 276
analyzer, 88
Angularmomentum flow tensor, 354
spectrum, 50, 54, 58, 333
decomposition method (ASDM), 334
of Airy light-sheet, 339
of Bessel pincer light-sheet, 339
of Gaussian light-sheet, 340
of initial electric field, 335
anti-reflection coating (AR), 154
Aperiodic
Sequence, 239
Zone Plates, 239
Aplanatic lens, 426, 429
achromatic, 432
Apparent Sun semidiameter, 424
Arbitrary size, of particles, 333
Arbitration, 146
architecture license, 119
ARM processor, 119assembly language, 110
Astronomical unit, 424
ATAC architecture, 143
Atoms, 512
autocorrelation, 45, 46, 68
avalanche emission, 512
axial image chromatic aberration, 275

Bback focal length, 272
Bandgap, 461
BBAR. See Broadband antireflection
coatingBeam635

Advances in Optics: Reviews. Book Series, Vol. 5

order, 335, 339
quality factor, 476
shape coefficients (BSCs), 334, 341
General expressions, 348
of Bessel pincer-sheet, 349
of Gaussian light-sheet, 350
of TE-polarized Airy, 349
of TM-polarized Airy light-sheet, 349
waist, 475, 476
bending-dependent loss, 548
Bernoulli
numbers, 594
polynomial, 593, 594, 610
big cores, 119bilinear function, 269, 270, 272, 285, 287,
288
birefringence, 89-91, 93
birefringent, 549
bit-level, 115
Blackbody, 424
BLIM
Bidirectional Lambertian Irradiation
Method, 188, 198-202
Bolosilicate glass, 433
SBNACZ, 433
boundary conditions, 35, 36, 38, 104, 107,108, 343
Bow-tie aperture, 438, 443, 444
branchinstruction, 116prediction, 116predictor unit (BPU), 129
Brewster’s angle, 87, 88
Brillouin
scattering, 517
optical time-domain
analysis (BOTDA), 516
reflectometry (BOTDR), 516
Broadband antireflection coating, 450
Built on ARM Cortex, 119butt coupling, 150C
cache
coherence, 116part, 111
caching/home agent (CHA), 131
calcium fluoride (CaF2), 519
Carriers and Photons Density Dynamics

636

Rate Equations, 407
Cavity loss. See Round-trip resonator loss
Cavitymirror, 420, 422
photon lifetime, 420
CCD, 180, 182, 187
camera, 164, 166-169, 172, 176
Center Point Strategy, 172, 173, 175, 176
Chirp, 585, 586, 587, 588
Chirped pulse amplification (CPA), 575
Chromatic aberration, 260, 429, 432, 434
circular polarization, 27, 97
Cladding, 436, 438, 455, 464
Clear Imaging, 252
coherence, 44-46, 48, 50, 68, 69, 82-85, 103
Spatial coherence, 48, 85
Temporal coherence, 46, 50
time, 46, 82
coherent
Coherence area, 48
detection, 517, 540
Illumination, 68
light source, 511, 512
volume, 48
Colourful Imaging, 257
Coma aberration, 426
complex instruction set computer (CISC),
110
Composite rod, 454, 455
2D, 456, 457
3D, 456, 457
compute node, 137
Concentration quenching, 452
Continuous wave, 449
contrast difference between the second peak
and the second trough (CDSPST), 541
converged mesh stop (CMS), 130
Conversion efficiency, 418, 423, 444, 462,
463
Convex lens, 425, 427
Convolution Product, 626
Core, 436, 439, 455, 464
Corona Architecture, 143
correlation
function, 45, 46
matrix, 102
cortex license, 119
Cortex-A78, 118
CPC, 180, 181, 184, 187, 192, 193, 202208, 210-215, 220, 222, 225, 231, 232

IndexCr-codoped Nd:YAG, 418, 427, 440, 450,
461
Cross sections, extinction and scattering,352
cross-correlation, 45
cursor effect, 522
CW. See Continuous wave
cycles per instructions (CPI), 115D
damping rate, 518, 530
data cache L1, 128
Data preprocessing
fast Fourier transform, 507
Multiplicative Scatter Correction, 504
Normalization, 501, 503
Savitzky-Golay Algorithm, 504
Soft independent modeling by class, 506
Standard Normal Variate, 504
data-level, 115
DCF. See Double-clad fiber
De-excitation, 512
Deformation theory, 312
delayed self-heterodyne interferometer(DSHI), 523, 525
dense wavelength division multiplexing(DWDM), 514
Detailed valance theory, 462
dextrorotatory, 94
diattenuation, 88
differential
phase-shift keying (DPSK), 514
quadrature phase-shift keying (DQPSK),
514
diffracted wave, 598
diffraction, 26, 37-44, 48, 50, 54, 56-66, 85,
105-107, 312, 320
limit, 474, 483
limited, 267, 294, 296
Diffuser, 179, 180, 200, 223, 224
digitalcoherent
receivers, 514
technology, 514, 515
cross-correlation, 544
signal processor (DSP), 115
thermal sensors (DTS), 134
DirectLambertianabsorptance, 189
Conductance, 190
Irradiation Method, 188
reflectance, 189
Disk-type LM, 423, 432, 433, 461
Displaced Graph, 602
displacement current, 23
Dissipative soliton resonance (DSR), 575,
576, 578, 579, 588
distributed
acoustic sensing (DAS), 516
Feedback (DFB) laser, 512
Feed-Back (DFB) laser, 581
strain sensing (DSS), 516
temperature sensing (DTS), 516
vibration sensing (DVS), 516
DLIM, 187-189, 195, 196, 199, 202-204,
207, 211
double precision, 116
Double-clad fiber, 436
Down-conversion, 419
DSR-like square pulses, 576, 577
dual cavity feedback structure (DCFC), 537
dual-wavelength, 547-550, 557-559
DynamIQ technology, 119

E
EDF (erbium-doped fiber), 512
Effective
aspect ratio, 422
focal length, 273, 274
Einstein A coefficient, 420
Electric field amplitude, 338
electrical feedback, 527-529
electromagnetic waves, 23, 29
electrostatic, 526
electrostriction, 534
EMesh, 143
Energyefficiency, 334
absorption, 352
extinction and scattering, 352
of Airy light-sheet, 357
of Bessel pincer light-sheet, 369
of Gaussian Light-sheet, 385
scattering, absorption, and extinction,333
transfer, 452, 453, 458, 464
entrance pupil, 64637

Advances in Optics: Reviews. Book Series, Vol. 5

equalization, 551
Equatorial mounting, 431, 446, 447
Erbium, 513, 547-549, 566
Erbium-doped fiber amplifier (EDFA), 515
étendue, 190
Ethernet switch, 138
evanescentcoupling effect, 526
wave, 31-33
Excitation, 512
spectra, 429
exit pupil, 64-68
external parallel feedback, 522, 524
Extinguished and scattered energies, 352
extraordinary- or e-ray, 90

FFaraday’s Law, 23
far-field
approximation, 476
patterning, 474
processing, 474
regions, 478
resolution, 474
scattering intensity, 333
dimensionless far-field scattering
intensity, 352
dimensionless normalized far-field
scattering intensity, 352
of Airy light-sheet, 355
of Bessel pincer light-sheet, 366
of Gaussian Light-sheet, 383
fat-tree network, 136
architecture, 141
femtosecond, 474, 477-479, 482
few-mode fiber (FMF), 515
fiber
Fabry-Perot tunable filter (FFP-TF), 555
laser, 512-515, 527, 534, 548-550, 552555, 557-561, 563-566
Fiber-type LM, 423, 432, 433, 464
Fibonacci Zone Plate, 249
finite measurement time, 538
Flash-lamp pumped laser, 417
floating point operations per second(FLOPS), 115
Fluorescence, 433
lifetime, 433
Fluoride glass, 418, 433, 437, 438

638

ZBLAN, 433, 436, 437, 441, 464
Focal point, 335
Force and torque, negative, 333
Fourierseries, 50, 51, 593, 594
transform, 25, 41-43, 47, 50, 52, 54-59,
61, 63, 66- 69, 82, 597, 619
Dispersive Fourier transform (DFT),576-578, 583-586, 588
double Fourier transform, 628
infrared spectroscopy, FTIR
spectroscopy, 301
fast Fourier transform, 41
frequency transform (FFT), 547
Spectroscopy, 68
Four-level scheme, 420
four-wave mixing, 552, 557
FPB. See full Poincaré beam
FPB Polarimetry, 164
Fractal Zone Plate, 243
Fraunhofer
conditions, 598
diffraction, 38, 42
free-electron lasers, 512
frequency
discriminator, 527, 528
jitter, 538
Fresnelbiprism, 78
diffraction, 38, 41, 42, 65
double-sided mirror, 78
integral formula, 241
lens, 418, 427, 450, 461
fringes, 37, 70, 71, 76-81, 83-85
front focal length, 272
FSO, 395, 397, 404
Fujitsu’s Fugaku (442 PFlops), 136
full
Poincaré beam, 164, 166, 168, 172
width at half maximum (FWHM), 518
fused multiply-add (FMA), 116Ggain medium, 512, 513, 526, 530, 531, 544,
547, 549, 551, 555, 557, 558, 563
gallium arsenide, 513
GaP prism, 152
gas lasers, 512
Gaussian, 474-476

Indexbeam, 315, 475, 476
broadening, 543
beam spreading, 475
Law, 23
mode, 475
Generalized Lorenz-Mie theory (GLMT),333
Glan-Foucault prism, 92
GLMT, 334, 341
Global solar irradiance, 447, 448
Goos-Hänchen Shift, 32
Gordon, 136
Graphene, 563-565
graphics processing unit (GPU), 115
grating coupling, 150
Green’s theorem, 38, 104, 105
group velocity, 74, 75
dispersion, 585

H
half-wave plate, 94, 99
Hanbury-Brown Twiss effect, 547
Heaviside Function, 600
Heliostat, 427
Helmholtz equation, 26, 38, 39, 52, 104,
105, 107
heterodyne detection, 538-540
high finesse FP resonator, 528
High-reflectivity coating, 450
HR coating, 450
hit quality index, 500
homogeneous line broadening, 551
host channel adapter (HCA), 137
Huygens-Fresnel’s principle, 37, 39hydrofluoric acid, 563, 564
hyper threads, 114

I
I/O node, 137
IBM's Summit (148.6 PFlops), 136
Ideal linear polarizers, 98
ILIM, 186-189, 195, 200, 202, 213-215,
230
Illuminometer, 447
image
processing unit (IPU), 115
signal processor (ISP), 115
imaging system, 64-68

impedance of the light field, 27
impulse response, 39, 64
Impurities
concentration measurement, 303–305,
309, 310, 322, 327–328
Cu, 306, 320, 327, 328
Fe, 305–308, 311, 327, 328
OH group, 301, 305, 308–311
Inclusions. See. Impurities
incoherent
illumination, 50, 68
ligh, 46, 64, 67, 103
light, 67
light source, 511, 512
index of refraction, 24, 25, 27, 30, 32, 33,
37, 87, 90
Indium, 513
Inductively Coupled Plasma Mass
Spectrometry, ICP-MS, 304
detection limit, 304, 306
Infiniband switch, 138
infraredlasers, 512
light, 513, 564
Initial electric field, 335
of Gaussian light-sheet, 340
of the Airy laser-sheet, 338
of the Bessel pincer light-sheet, 339
in-line amplifier, 513
Instruction cache L1, 128
instruction-level, 115
instructions per cycle (IPC), 115
Integrating sphere, 180, 199, 200, 225, 226,
433, 440, 441
intensity
distribution, 475, 476, 478, 480
modulation (IM), 514
Interference Phenomena of Partial Coherent
Light, 81
Interferometric fringes, 581, 582, 588
Internal and scattered field, 335, 342
Internal field coefficients, 342
Inverse Lambertianabsorptance, 189
Conductance, 190
Irradiation Method, 186, 188, 230
reflectance, 189

639

Advances in Optics: Reviews. Book Series, Vol. 5

J
Jones vector, 95, 96, 97, 99
Judd-Ofelt analysis, 433

K
Kidney-shaped spool, 436, 437
KX-6000, 126L
L2 cache, 128
Lambertianabsorbance, 183, 205-213
conductance, 183, 184, 190-192, 199,
204, 207, 208, 211, 212, 223, 225,
231
diffuser, 179
irradiation, 179, 188-190, 198, 199, 201,
202, 211, 230, 231
light, 180-182, 186-189, 193, 194, 196,
199, 202, 208, 213, 214, 225, 231
radiation, 179, 186, 187, 213-216, 221,
222, 231
reflectance, 183, 203-212
screen, 187
source, 181, 185, 187, 188, 195-197,
200
transmittance, 183, 184, 188, 190, 192,
195, 202-204, 206-208, 210, 211large effective area fiber (LEAF), 532
LASCAD, 453, 455
Laser, 179, 180
Ablation Inductively Coupled Plasma
Mass Spectrometry, LA-ICP-MS,
304, 305
detection limit, 304, 306–7, 327, 328
cavity, 450, 451, 453, 458
Characterizations, 395
medium, 417, 418, 437, 443, 454
oscillation mode, 422
rod, 422, 428, 441, 445, 449
LASER (Light Amplification by Stimulated
Emission of Radiation), 511
Lasing threshold, 439
latency, 117
lateral magnification, 273
law

640

of reflection, 35, 36
of refraction, 35
levorotatory, 94Lightcones, 181, 187, 192, 193, 207-210, 216,
220, 221, 231
detection and ranging (LIDAR), 516
intensity, 26, 27, 64, 75-77, 83, 88, 89,477, 478
Light-sheet, 333
Airy light-sheet, 333, 334, 338
auto-focusing Bessel pincer light-sheet,
334
Bessel pincer light-sheet, 333, 334
Bessel pincer-sheet, 340
circularly-polarized Airy light-sheet,
334
Gaussian light-sheet, 333, 334, 340
Nonparaxial Bessel pincer light-sheet,334
linear polarizer, 164, 168, 174, 175
Linewidth, 513-522, 524-539, 541-546,
565-567
compression, 511, 519, 520, 522, 524530, 532, 535-537, 569
measurements, 511
liquid state lasers, 512
little cores, 119
Lloyd’s mirror, 79
LM. See Laser medium
local oscillators (LO), 514
Lock-in amplifying, 302
logic part, 111
Longitudinal-mode hopping, 438
Loongson, 140
3B4000, 140
X-CPU, 140
loss-compensated recirculating delayedself-heterodyne interferometer (LCRDHSI), 524
low pressure chemical vapor deposition(LPCVD), 149Mmachine language, 110
Mach-Zehnder interferometer (MZI), 576,
577, 581, 583, 588
macro-operations (MOPs), 128
Made-to-order stage, 428

IndexMagnesium-based energy cycle, 417
magnetic field relativistic velocity, 512
main focus, 243
Malus’ law, 88, 89
Mass production, 418, 423, 428, 433, 462
matrix, 269-271
transformations, 268
Maxwell’s
equations, 23, 27, 31, 33
stress tensor, 353
Mechanical effect, 333
mesh interconnect architecture, 130
MESI, 117
MethodBLIM, 181, 187, 188, 189, 199-202,
204, 207, 211, 213-215
Parretta-Herrero, 187
PH-Method, 187
MgF2 (magnesium fluoride), 520
Michelson
Interferometer (MI), 545
Stellar Interferometer, 85
micro-FBG (MFBG), 563
Microfiber, 526, 527
microlenses, 151
microlensing, 480, 482
micro-operations (μOP), 128
microspheres, 474, 479-483
MMF (multimode fiber), 522
Mode-matching efficiency, 422, 436, 439,
464
Mode-matching Efficiency, 453
Modulation
parameter, 335, 338
transfer function, 68, 253
Molecules, 512
Monochromatic, 512, 540
coherent laser beam, 417, 418
wave, 25, 26
monochromaticity, 513
Mueller matrix, 163, 164, 166, 167, 172176
multi-conical surface, 292, 293
multi-core fiber (MCF), 515
multi-level modulation formats, 514, 515
multimode interferometer (MMI), 524
Multipole expansion, 335
Multivariate analysis
classification table, 506
exploratory data analysis (EDA), 504
loadings plots, 504, 505

Partial Least Squares (PLS), 506
principal component, 504
analysis (PCA), 504
score plot, 504
Multivariate analysis, MVA, 488
multi-wavelength, 547, 550, 552, 553, 554,
557

N
Natural air convection, 462, 464
Nd-doped yttrium aluminum garnet, 418
Nd
YAG, 418, 430, 440, 449, 457
Nd-doped Y3Al5O12, 420
Nearest Neighbor Strategy, 173, 175, 176
near-field
focusing, 474
zone, 476
regions, 478
negative uniaxial crystal, 91
neural processing unit (NPU), 115
noise, 165, 167, 168, 174-176, 305, 322–28,
324
accumulated, 168
characterization, 511, 544, 545
component of the measured signal, 305,
308, 317, 320
dark shot, 167
fixed pattern, 167, 168
Gaussian, 172, 174
photon shot, 167
read, 167
RMS deviation, 323–27
source, 323, 325
non-blocking network, 141
Non-diffraction, 333
nonlinear
polarization rotation (NPR) , 552, 577
scattering, 537
semiconductor optical amplifier (NLSOA), 557
nonmonochromatic, 25, 43, 95
Non-paraxial Airy “acoustical-sheet”, 334
Non-periodic Functions, 51
non-uniform memory access (NUMA), 136
north bridge, 130
numerical aperture, 292, 474

641

Advances in Optics: Reviews. Book Series, Vol. 5

OOdeillo solar furnace, 427, 428
Off-axis parabolic mirror, 418, 434, 464
OAP, 418, 434, 443, 451, 462
Offset angle, 435
on-axis spherical aberration, 277, 288, 291,
295, 296, 298
ONet, 143
on-package interconnect (OPI), 130
Operator Calculus, 620
Opticalabsorption edge, 464
apparatus, 180, 181
axis, 182, 186, 187, 197, 202
Bidirectional Lambertian Conductance,
179, 187, 190, 191, 198, 231
circulator-based feedback circuit, 524,
525, 526
characterization, 179, 180, 186, 187
code, 195, 200, 201
comb filter (OCF), 558
concentration ratio, 182, 185, 186
conductance, 181, 191, 224, 231
conduction, 191
configuration, 179
Direct Lambertian Irradiation Method,187
efficiency, 180, 181, 182, 185, 186, 187,
188, 213, 215, 216
element, 190, 191, 195, 196, 198, 199,
211, 213, 231
elements, 181, 182, 187, 191, 192, 198,
213, 222, 225, 231
force, 353
of Airy light-sheet, 360
of Bessel pincer light-sheet, 373
of Gaussian Light-sheet, 387
frequency domain reflectometry(OFDR), 516
Imaging, 252
Interferometry, 69, 75
Lambertian conductance(OLC) , 187,
190-192, 195, 231
Bidirectional LambertianConductance (OBLC), 187, 190,
198, 202
law, 181, 231
manipulation, 333
maps, 194642

network chip (ONC), 159
path, 204
difference, 577, 582, 583
properties, 179, 185, 187, 231
quantities, 181, 186, 187, 188, 189, 192,
203, 206, 211, 230
receiver, 514
reversibility, 181, 231
self-injection feedback, 524, 532
simulation, 196, 197
simulations, 182, 192, 194, 195, 202,
231
solar concentration, 186
spectrum analyzer, 432
spin torque, 334, 354
of Airy light-sheet, 362
of Bessel pincer light-sheet, 380
system, 181
systems, 186, 231
terminal, 179, 184, 185, 191
terminals, 191, 231
Transfer Function (OTF), 68
transmission efficiency, 192
transmittance, 202, 231
transmitters, 514
Tweezers, 243
Wireless Communications, 395
wireless power transmission, 463
ordinary- or o-ray, 90
oscillating dipole, 29, 30, 88
Oscillation wavelength, 418, 420, 464
out-of-order, 116
Output coupler (OC), 443, 445, 446, 450,451
OWC, 395-397, 403, 409

P
Package on Package (PoP), 135
Parabolic
Function, 606
mirror, 418, 427, 432, 439
parallelcomputing, 114
thin films, 79
paraxial approximation, 42, 55, 57
partially
coherent light, 46
polarized light, 100, 103
Particles, spherical, 333

Index

Passively mode-locked fiber laser, 576, 581,
588
PCH, 130
Periodic Functions, 50
Phase
jump, 581
or frequency noise, 513-517
shift, 37, 39, 52, 68, 93
Transfer Function, 68
velocity, 73, 74, 75
phase-sensitive optical time-domainreflectometry (φ-OTDR), 516
phase-shifted FBG, 528
Phosphate, 418, 433
Phosphide (InGaAsP), 513
Photoelasticity tensor, 313
Photonic crystal fiber (PCF), 549
Photons, 512
Photothermal Common-path Interferometry,
PCI, 302, 307–309, 311, 319
detection limit, 307–9, 311
photo-thermal effect, 564
pipelines, 114
plane of incidence, 33-35, 87
Poincaré sphere, 166, 167, 172, 173
polarimetric system, 164
polarimetry, 163, 164
polarization, 27, 163, 164, 166, 172, 173,
176
Elliptical polarization, 27, 96
input state, 166
linear polarization, 27, 96, 97
state, 163, 164, 166, 167, 172, 173, 176
polarization
division multiplexing (PDM), 514
state analyzer, 164, 168, 172, 176
state generator, 163, 172
polarization-maintaining fiber (PMF), 532
polarized light, 27-29, 86-89, 92-97, 100103
Completely Polarized Light, 95, 102,
103
polarizers, 86, 91, 95, 98, 100
Polluting chemical elements. See.
Impurities
Polynomial Equivalent, 609
Population inversion, 459
post-amplifier, 513
Powerfeeding, 463, 464
meter sensor, 431, 446

pre-amplifier, 513
principal section, 90, 91
PROMES-CNRS laboratory, 427, 428
PSA. See Polarization state analyzer
PSG. See Polarization state generator
pulse duration, 473, 474, 477, 478
Pump, 512, 521, 526, 531, 532, 536-538,
548, 549, 554, 555, 563-566, 568
Pumping efficiency, 420, 421, 458
Pyrheliometer, 431, 446, 448

Q
Q-switched mode locking (QML), 564
Quad Small Form-factor Pluggable (QSFP),
138
quadratureamplitude modulation (16-QAM), 515
phase-shift keying (QPSK), 514
Quantum cascade lasers, 487
conduction band, 488
Diffuse reflectance, 492
diode lasers, 488
discrete transition, 488
grazing angle, 487, 489, 490
incident angle, 489, 492
Population inversion, 488
tunable lasers, 490
Quantum
defect, 419, 421, 422
efficiency, 421, 433, 459, 461
fluctuations, 518
limit, 518
quarter wave phase plate, 164, 172, 174
quarter-wave plate, 93, 99
Quartz glass, ultrapure, UQG, 301, 303–
311, 316, 318–320, 321, 326-328
quick path interconnect (QPI), 132
QWP. See quarter wave plate

R
Radiance, 179-184, 186-190, 195-201, 208,
213-225, 231
Radiant flux power, 424, 427-430
Raman
amplification, 517
microspectroscopy, 496
optical time-domain reflectometry(ROTDR), 516

643

Advances in Optics: Reviews. Book Series, Vol. 5

scattering, 516, 517
Ramp Function, 605
rare-earth-doped fiber, 512
Rayleighbackscattering (RBS), 529
length, 479, 483
range, 475
scattering, 517, 520, 521, 530, 532, 533,
535, 538
Rayleigh-Sommerfeld formula, 38, 39, 40,
104, 108
reabsorbing effect, 548
Red shift, 429, 432, 448
reduced instruction set computer (RISC),
110
refractive index, 149, 543, 566
matching, 456
nonlinearity, 316, 317
register part, 111
relative
intensity noise (RIN), 513, 528
permeability, 24
permittivity, 24
Renewable energy, 417
ReOrder Buffer, 128
residualchromatic aberration, 291
spherical aberration, 296
resonant cavity, 512, 566
Riemann zeta functions, 594
rigorous analytical dependences, 268, 288,
298
ring bus, 127
Ronchi’s gratings, 59
Round-trip resonator loss, 420Ssagittal
radius, 278-280, 282, 292
radius-coefficient, 277-279, 285
Sagnac loop interferometers, 558
sample thermal decomposition, 490
saturable absorber (SA), 563
saturation power, 551
Scaling parameter, 335, 339
scanning electron microscope (SEM), 153
scattering, 30, 88-90, 333, 353
field coefficients, 342
Self-healing, 333

644

self-heterodyne, 519, 531, 540, 542, 566
self-homodyne detection, 540
self-injection locked, 520, 521
semiconductor
laser, 512, 515, 519, 528
optical amplifier (SOA), 513
Shannon capacity, 515
side mode suppression ratio (SMSR), 526
Sierra (94.64 PFlops), 136
Signal-to-noise ratio (SNR), 305, 306, 308,
316, 319–321, 325, 326, 513
silicon, 303, 304, 473, 474, 477-480
dioxide (SiO2), 148
nitride (Si3N4), 148
oxynitride (SiOxNy), 148
singlelongitudinal mode (SLM), 513
precision, 116Single-mode fiber (SMF) , 515, 577, 578,
584, 585
single-wall carbon nanotubes saturable
absorber (SWCNT-SA), 564
single-wavelength, 547, 549, 550
Size parameter, 335
Slope efficiency, 439, 444, 453, 456, 458
slope efficiency, external, 444, 445, 448,
449
Small Form-factor Pluggable (SFP), 138
Snell’s law, 31, 87
Solarangular dispersion, 186
cell, 185, 417, 418, 463, 464
concentration ratio, 422, 424, 430, 436,
442
concentrator, 179-182, 185, 187, 188,
190-193, 196, 202, 205, 214, 230,
231, 424, 427, 434, 449, 462
compound parabolic, 179
constant, 424
furnace, 418, 427
lighting system, 428, 462
radiation, 179, 185, 186, 223, 224, 430,
463
AM0, 463
AM1.5, 463
diffuse, 429, 464
direct, 429, 436, 443, 446, 462
global, 429, 430, 431
spectrum, 419, 432, 436, 440, 463
sunlight spectrum, 422, 437, 438

Indexmodel by Bird and Riordan, 429, 430
telescope, 446, 447
tracker, 181, 186
tunnel, 181, 187, 192, 194, 210-213,
221-223, 225, 231
Solarization, 449, 461
Solar-pumped laser, 417, 420, 432, 462
fiber laser, 433, 439, 464
SPFL, 432, 433, 436, 438, 464
SPL, 417, 423, 433, 449, 457
sun-pumped laser, 417
μSPL, 441, 445, 447, 450, 462
Solar-tracking system, 428, 436, 443, 462
coordinated, 462
solid-state lasers, 512
south bridge, 130
Space solar power station, 417, 463
spatialdivision multiplexing (SDM), 514, 515
light modulator, 243
Specific Optical
Bidirectional Lambertian Conductance(SOBLC) , 184, 191, 192, 204, 207,
208, 211, 212
Lambertian Conductance (SOLC), 191
spectralcorrelation algorithms, 500, 501
matching, 439, 440, 461
quantum efficiency, 463
SPECTRAL2, 429, 430, 431, 432
specular reflectance, 492, 508
Spherical
aberration, 426
Bessel function of first kind, 339
Spontaneous emission, 420, 440, 458, 459,
461, 518
spot size, 474-476
Square Wave Function, 600
SRAM, 112
Standing Waves, 72
Stiffness tensor, 314
stimulated
Brillouin scattering (SBS), 531
emission, 513, 518
emission, 418, 420, 433, 458, 461
Raman scattering (SRS), 535
Rayleigh scattering (STRS), 532
stochastic light fields, 86, 100
Stokesparameters, 163, 164, 166, 168, 172
polarimeter, 164

vector, 163, 164, 166, 172
submicron, 473, 482, 483
subsidiary foci, 243
SunwaySW26010 1.45 GHz, 136
TaihuLight (93.01 PFlops), 136
Super Invar alloy, 462
Super-Gaussian pulse profile, 585
Superposition of Waves, 70, 72
Super-resolution imaging, 333
superscalar processor, 114
Surface albedo, 438
Sylvester's matrix, 277
symbolic computation, 287
Symmetrised Graph, 603
Synthetic crystalline quartz, SCQ, 303–304,
306, 310, 316, 319–321, 328
system on chip (SoC), 115

Ttangential
radius, 278
radius-coefficient, 278
taperedboron codoped germanosilicate fiber,
560
optical fiber, 535
task-level, 115
TE and TM polarized, 334
TE-polarized laser sheet, 335
TM-polarized laser sheet, 335
Tellurite glass, 418, 433
TEM00-mode, 450
temperature
coefficient, 566
dependent refractive index, 423
Temporal coherence, 576-579, 581, 583,
584, 588
TE-polarized electromagnetic fields, 336
tetrahedron, 167, 172-175
thermaldesign power (TDP), 134dissipation, 423, 428, 432, 462
effect, 566
expansion, 462
expansion coefficient, 566
lens effect, 423
Thermodynamical limit, 462
thermo-electric cooler (TEC), 565

645

Advances in Optics: Reviews. Book Series, Vol. 5

Thermo-optical parameter, 313, 314, 319
thin film filter, 155
threads, 114
three-cornered-hat method, 519, 520
Thresholdpower, 430, 436, 443, 449, 456
pumping efficiency, 421
through silicon via (TSV), 135
Thue-Morse Zone Plate, 250
Thulium, 513
Ti sapphire laser, 433
Time dependence, 336
Time-resolved Photothermal Common-path
Interferometry, TPCI, 311–321, 327, 328
calibration, 313, 315, 320, 327, 328
detection limit, 319, 327, 328
TM-polarized electromagnetic fields, 337
Total
efficiency, 439, 444, 449
internal reflection, 31, 33
Townes and Schawlow, 511, 512
Traditional Mie scattering coefficients, 343
transfer function, 527
transmission loss, 149
transverse
electric, 552
magnetic, 552
scale, 335, 338
tunable bandpass filter (TBF), 554
tunable optical filter (TOF), 558
two equal-intensity foci, 258
two-photon absorption, 474, 477-479

Uultrafast lasers, 474, 482
ultra-path interconnect (UPI), 132
unpolarized light, 86, 89, 102
Upconversion, 418
UV (Ultra-Violet)
lasers, 512
filter, 439, 443, 449

Vvan Cittert-Zernike theorem, 48, 49, 50
van der Waals, 526
VCSEL, 133
Chaos Dynamics, 405
Chaos Synchronization, 406

646

L-I Curve Characteristics, 396, 399,
400, 401, 402, 403, 404, 405, 406,
409
Nonlinearity, 404
Optical and Electrical Properties, 400
Polarization Switching, 402
vectorinstruction, 115
processor, 115
spherical wave functions (VSWFs), 334
verification mode, 500
Vertical cavity surface emitting
semiconductor laser, 399, 461
VECSEL, 461, 464, 465
VIA, 126
visible lasers, 512
Voight fitting, 543
VSWFs, 341

W
Waist radius, 335
wave
aberration, 65
impedance, 352
speed, 24, 25
vector, 335
wavefront division, 76-78, 81
wavelength
blocking, 547
converted lasers, 547
converter, 513
division multiplexing (WDM), 514
switching, 511, 547, 548, 550, 560, 561,
569
tuning, 511, 547, 553-555, 557, 563565, 567-569
wavelength-tunable or switchable, 513, 514,
515
Wavenumber, 335
WDM, 155
weiner effect, 522
whispering gallery mode, 519
white frequency noise, 543
Wollaston prism, 92X
x86 instruction set, 110
Xeon Platinum 9282, 130

IndexX-ray lasers, 512

Z

Y

Zaoxin, 126
ZEMAX, 453, 454
Zeta Functions, 615

Young's experiment, 76, 77, 81, 84
Ytterbium, 513, 547

647