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I/I o NBS TECHNICAL NOTE 1009

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

Laser Beam Profile Measurements

Using Spatial Sampling,

Fourier Optics,

and Holography
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act of Congress March 3, 1901. The

Bureau's overall goal is to strengthen and advance the Nation's science and technology and

facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau's technical work is performed by the National Measurement Laboratory, the National Engineering Laboratory,

and the Institute for Computer Sciences and Technology.

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THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts

research and provides scientific and technical services to aid Federal Agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and

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The National Bureau of Standards was reorganized, effective April 9, 1978.
Laser Beam Profile Measurements

Using Spatial Sampling,

Fourier Optics, and Holography

HatJonal Bareau of Standards MAR 2 3 1S7S

\xb'

Eric G. Johnson, Jr.

:iectromagnetic Technology Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

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.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

[NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued January 1979
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1009

Nat. Bur. Stand. (U.S.), Tech. Note 1009, 96 pages (Jan. 1979)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1979

For sale by the Superintendent of Documents, U.S. Government Printmg Office, Washington, D.C. 20402

Stock No. 003-003-02019-2 Price 2.75 (Add 25 percent additional for other than U.S. mailing)
CONTENTS

Page

1. INTRODUCTION 1 1.1 What this Introduction Contains 2 1.

2

Defining Beam Profile 2 1.3 Various Methods to Measure Beam Profile 3 1.3.1 The Methods Using Fewer Assumptions 3 1.3.2 The Focal Plane Method 4 1.3.3 The Mirror Scan Method Using Two Pinholes 4 1.3.4 The Hartmann Plate Method 5 1.3.5 The Holographic Method 5 1.4 The Purpose of this Paper 6

2. AN OVERVIEW OF THE BASICS USING THE HOLOGRAPHIC METHOD 7 2 .

1

What this Overview Contains 7 2.2 The Main Features of the Diffraction Process by a Single Aperture 7 2.3 Looking at the Interference Pattern in the Far Field and Fourier Plane

for Two Apertures 8 2.4 Important Properties of Fourier Optics as Applied to the Holographic

Method 10

2 . 5 Important Properties of the Holographic Process 10

2 . 6 Important Properties of the Cross-Correlation Process 11 2.7 Assembly of the Above Concepts to Define the Holographic Method 12

2.

8

Summary of this Overview 12

3. HOLOGRAM CONSTRUCTION APPARATUS 13

3.

1

What this Section Presents 13 3.2 A Discussion of the Prototype Apparatus Used to Generate the Hologram 13 3.3 The Mathematical Model Used to Understand the Hologram 13 3.4 Detailed Instructions for Assembling and Aligning the Apparatus in the

Construction of the Hologram 22

3 .

5

Results from Construction of the Hologram 27

4

.

THE CROSS-CORRELATION APPARATUS 28

4.

1

What this Section Presents 28

4.2 A Discussion of the Prototype Apparatus Used to Generate the

Cross-Correlation Pattern 28 4.3 The Mathematical Model Used to Understand the Cross-Correlation Pattern 28

4.3.1 The Model that Contains the Spatial Details of the Irradiance 28

4.3.2 The Simplified Model that Ignores Details of the Irradiance—the

Calibrated Model 31 4.4 Instructions for Assemblying and Aligning this Apparatus to Generate the

Cross-Correlation Patterns 34

4 . 5 Calibration of this Prototype Apparatus 36

4. 6 The Cross-Correlation Patterns 37

5. MODIFYING THE PROTOTYPE APPARATUS TO GENERATE A MORE EFFECTIVE UNIT 56

5.

1

What this Section Presents 56

5.2 A Scale Change between Original Hologram and a New Sampling Plate

Makes a Better Instrument 56

5.3 Thoughts about a Cross-Correlation System that Can' Make Beam Profile

Measurements that Are Orthogonal 57 5.4 Features of the Proposed Reflection System 59

5.5 A Beam Profile System that Can Be Mass Produced 66

6. CONCLUSIONS DRAWN FROM THE PROTOTYPE APPARATUS 68

7

.

ACKNOWLEDGMENTS 70

8

.

REFERENCES 71
APPENDICES

Page

IMPORTANT FORMULAS FOR FOURIER OPTICS 73

A. 1 The Fourier Transform 73

A. 2 The Imaging Formula 75

A. 3 The Displacement Formulas 75

THE DIFFRACTION DETAILS OF A SQUARE HOLE 76

B.l What this Appendix Presents 77

B. 2 The Far-Field Diffraction Pattern 77 B.3 The Fourier Transform Pattern 80

THE DIFFRACTION DETAILS OF A CIRCULAR HOLE 82

C.l What this Appendix Presents 82

C. 2 The Far-Field Diffraction Pattern 82

C. 3 The Fourier Transform Pattern 84

DEPTH OF FIELD ANALYSIS FOR FOURIER OPTICS 85

XV
LIST OF FIGURES

Page

Figure 1. The hologram generating apparatus - block diagram 14 Figure 2. The cross-correlation generating apparatus - block diagram 15 Figure 3. The pinhole sampling unit for the prefilter plane 16 Figure 4. The coordinates for the prefilter beams 19 Figure 5. The coordinates and structure of the spots at the cross-correlation

plane 19 Figure 6. Two examples of output at the cross-correlation plane - one exit

beam at prefilter 38

Figure 7. Two examples of output at the cross-correlation plane - two exit

beams at prefilter 39

Figure 8. Television picture of the cross correlation plane - two exit

beams with i, = -1, +1 and Jl„ = at prefilter with approximately

equal phase and amplitudes 40

Figure 9. Same picture as figure 8 except beams are approximately 180° out

of phase 4

1

Figure 10. Television picture with i, = +1 and St^ = ±1 at prefilter with

approximately equal phase and amplitudes 42

Figure 11. Same picture as figure 10 except beams are approximately 180° out

of phase 43

Figure 12. Television picture with £. = and S-« = ±1 at prefilter. Here

beams are in phase 44

Figure 13. Television picture with l^ = -1 and i„ = ±1 at prefilter. Here

beams are out of phase 45

Figure 14. Television picture with £, = +1, iy = -1 and i, =0, l^-o = 46

Figure 15. Television picture with I = -1, i^ = -1 and i =0, I = 47 Figure 16. Final examples of output at the cross-correlation plane 48

Figure 17. Television picture with all nine holes illuminated and the cross correlation output not saturating the vidicon-TV unit 51 Figure 18. Television picture showing saturation of vidicon-TV unit 51 Figure 19. Television picture showing the cross correlation not properly

aligned 52

Figure 20. A partial view of the beam profile apparatus 53

Figure 21. The remaining view of the beam profile apparatus 54 Figure 22. A picture of the object and calibration beam 58

Figure 23. An improved beam profile measurement scheme 58

Figure 24. The basic beam profile measurement process 60

Figure 25. Beam profile measuring apparatus using reflection optics

(not to scale) 61 Figure 26. Beam sampling pattern in the prefilter plane 62 Figure 27. Resulting spot pattern in the cross-correlation plane 64 Figure A.l. Relationships between surface "1," "2," and "3" for a simple lens 74 v

LASER BEAM PROFILE MEASUREMENTS USING

SPATIAL SAMPLING, FOURIER OPTICS, AND HOLOGRAPHY*

Eric G. Johnson, Jr.

Using appropriate holography, lenses, mirrors, and a two-dimensional

array of small holes to sample the electric field of a laser pulse, we demonstrate a technique for beam profile measurements at a preselected

observation plane. This method can measure the amplitude and phase of a laser pulse at each preselected sampling point on a transverse observation plane. Subject to constraints, these measurements can provide

separated visual images of the temporal pulse shapes for the amplitude

and phase at each of the sampled electric fields. This paper describes

the basic concepts for beam profile measurements via holography and uses

pictures generated by a prototype system for illustration.

This unit may be calibrated by using a cw laser beam of the same wavelength as the carrier or a Q-switched laser pulse. Therefore, a user

can make beam profile measurements of a laser pulse in real time. This

technique requires appropriate scaling of the lenses and mirrors, and uses a different hologram to allow the unit to work optimally for each

wavelength.

We define the potential limits and virtues of the holographic

method. In addition, we define limits and virtues of other beam profile

techniques such as the two-pinhole scanning method and the Hartmann plate

method.

Finally, we indicate possible directions for eventual commercial and

scientific exploitation of the holographic method. The technique can be

used for near- and far-infrared wavelengths and, with appropriate changes

in the optics, for visible wavelengths.

Key Words: Amplitude; beam profile; cross correlation; digital

sampling; fourier optics; holography; laser; laser parameters; laser

pulse; optical processing; phase; precision measurements; wave front.

1. INTRODUCTION

Using lasers for communication, material processing, and precision measurements is now

a significant activity and is expected to become a major element in the national economy.

This trend arises from the fact that a laser beam can be highly directional and can permit

efficient placement of high-power densities (irradiance) at well-defined locations. A consequence of these two properties is the use of lasers to communicate via optical fibers

as well as to weld and evaporate various materials.

Timely development and incorporation of lasers into our society require tools that measure what happens. The prime applications of lasers require knowledge of each beam

including the irradiance at every point in space and at every instant of time. Because

exact measurements are impossible to attain, we must use mathematical models and many

assumptions to define a minimum but sufficient group of measurements so we can infer the

values for key parameters of these laser beams. Some parameters of current interest are

laser power, energy, pulse shape, and beam profile (wavefront)

.

Work at the National Bureau of Standards (NBS) has been to generate and use methods to measure accurately power and energy in a continuous wave (cw) or a pulsed laser beam.

Existing calorimeters and power meters can measure selected ranges of power and energy in a

* Funded in part by the Combined Calibration Group (CCG) of the Department

of Defense under project number 77-109 and 78-109.

I
a

laser with an absolute uncertainty of less than one to five percent at discrete wavelengths

such as 0.6328, 1.06, and 10.6 ym. NBS's efforts provide a basis for controlling the quality

and safety of various lasers during their use in commercial and scientific equipment.

NBS is currently developing measurement systems that can document the time behavior of

the power in a laser pulse at a sample rate of one nanosecond or slower for the 10.6 and

1.06 ym wavelengths. This activity is a step toward real-time control of a laser beam.

The development of a beam profile measurement system for pulsed lasers is the subject

of this technical note. NBS's prime goal is to generate techniques that can permit simultaneous measurement of the beam profile, both irradiance and phase, in real time. Once we know how a pulse evolves in both time and space, we can use feedback techniques with

adaptive optics to improve the form of laser pulses for purposes such as launching them

into optical fibers, directing them into fusion devices, controlling their profile for

welding, drilling, and blasting on surfaces of microcircuits; and for communicating through

a highly scattering medium such as fog, smoke, or rain. Significant development remains to optimize the systems using lasers.

1.1 What this Introduction Contains

Because people in the laser community have asked me why beam profile measurements are

of interest, I include a brief discussion about those measurements in subsection 1.2. I then discuss the methods for beam profile measurements and indicate their virtues and

limits in subsection 1.3. The technical details of each method are abbreviated because key

features appear later in this technical note or are covered adequately in the literature.

I report the essential conceptual point about each method without requiring the reader to understand mathematical and experimental details.

My principal intent is to explain and document the holographic method, discussed

briefly in subsection 1.3.5. Other methods are explained in subsections 1.3.2 to 1.3.4.

Subsection 1.4 details this paper's purpose because, as I have found in numerous

conversations, there is a need for preliminary discussion to familiarize the reader with

the vagaries of beam profile measurements.

1.2 Defining Beam Profile

Laser beams are electromagnetic fields. Current evidence indicates they propagate

according to Maxwell's equations. Therefore, to completely prescribe the electromagnetic

field, the reader needs to know the medium of propagation, usually defined by an index of

refraction which can be a function of time and space, a state of polarization, or the amplitude and phase of the electric field as a function of time at a selected two-dimensional

surface. Because the transverse dimensions of these beams are large compared with their

wavelength, the vector nature of the beam usually can be approximately described by scalar

concepts. At some stage in design of a system, polarization must be considered. Since the

wavelength is fairly sharply defined, the time dependence of the carrier is usually treated

as harmonic. The spatial dependence of the index of refraction usually is assumed to be

known for a given apparatus. These approximations mean only the amplitude and phase of the

electric field along a selected transverse surface to the beam need be measured to predict

all subsequent space and time behavior of the beam.

Even with the above approximations, the complete measurement requires both the amplitude

and phase at all points on the chosen transverse surface.* From this information all other

amplitudes and phases on other transverse surfaces can be deduced using appropriate solutions to the Helmholtz equation. Because this set of measurements implies a four-fold

infinity* of numbers, many more assumptions about the structure of a laser beam in a given

apparatus are needed to permit restricted measurements to prescribe the beam.

*If a beam has strong directional character, then the number of measurements

can be significantly reduced to some countable or finite number.

-2-
This requirement for many (not necessarily unique) assumptions to get useful data can cause ambiguity in the definition of beam profile measurements. Briefly, the above discussion shows there can be no complete set of beam profile measurements; there can only be

approximate subsets. This means that the appropriate beam profile measurements are strongly

dependent on the lasers being used in a given system and on the goals of that system. For example, if your beams are purely Gaussian everywhere, cylindrically symmetric, and steady

in time, only the following five parameters on a given transverse surface need to be measured: (1) the two coordinates for the center of peak irradiance of the profile; (2) the

distance from the center to the half -peak irradiance value; (3) the total power in the beam;

and (4) the radius of curvature of the phase front.

In conclusion, I define beam profile measurements as amplitude and/or phase measurements

on a transverse plane of a beam that collect sufficient information to specify it. For

example, the method in subsection 1.3.2 only measures the amplitude. This method can still measure the beam profile provided certain facts about the beam are known. If these facts are not available, then a simple measurement of amplitude will not be sufficient. In this

case, a different method for beam profile measurement should be used.

1.3 Various Methods to Measure Beam Profile

There are numerous methods for performing beam profile measurements which can be grouped

into three classes. I make no attempt to present all possible methods. All methods can be

used for far- and near-infrared visible wavelengths. Each wavelength requires appropriate

detectors and optics. For example, mirrors can be used at 10.5 ym.

The first class assumes the beam profile has a known shape such as Gaussian; all that

is required is an irradiance scan to deduce the unknown width of the beam. One method of

this class uses an aperture of 10 to 100 ym diameter in front of a linear detector to intercept and scan the beam systematically. A second method uses a fine wire whose diameter is

comparable to the half-power width of the beam [1]. The detector captures the entire beam,

and the wire moves at constant velocity to generate a shadow of the wire on the detector.

By using appropriate assumptions about the beam profile and by studying the current pulse

from the detector, the width of the beam is deduced. Both methods are quantitative.

However, there is no information about the phase front of the beam in the plane of these

measurements. That fact requires additional measurements of irradiance at new planes along

the optical axis of the laser beam and the application of solutions to the Helmholtz

equation. This procedure only works for stable beams.

The second class of measurement methods uses an optical flat or some other beam dividing

device to generate an interference pattern across the beam profile [2] . These methods are quantitative if the original beam irradiance is uniform; hence, the output pattern is

determined by the phase front. If the irradiance of the original beam has nonunif ormities,

then the interference pattern becomes modified. To get quantitive results, it is necessary

to make additional assumptions about or measurements of the structure of the irradiance to determine its effect on the interference pattern. Using appropriate scan methods, making

measurements at two surfaces, and applying the appropriate solution to the Helmholtz equation

can permit this technique to become quantitative even if the irradiance is nonuniform.

The third class of methods is discussed in more detail in subsections 1.4.1 to 1.4.5

because it is less dependent on specific features of a beam profile. This means these

methods can be part of a general measurement instrument. Because these methods have more

physical and conceptual complexity than those in the other classes, the previous methods

should be used when appropriate.

1.3.1 The Methods Using Fewer Assumptions

The four methods described in subsections 1.4.2 to 1.4.5 make fewer simplifying

assumptions about the actual phase front and the irradiance than do the previous methods.

Each has a different limit for valid use. Those limits are described in the following

subsections. In all cases, the appropriate method to use with a given laser system depends

on desired results. Further, there is a brief discussion of calibration techniques for each

method.

II

-3-
1.3.2 The Focal Plane Method

This method accepts that only the irradiance distribution at the far field of a laser

beam has importance [3] . Consequently, the beam passes through a lens or reflects off a mirror with appropriate radii of curvature to generate the far-field pattern at the focal

plane. To measure the irradiance pattern, the detectors in the focal plane can be a vidicon with TV scan speed, a two-dimensional array of detectors, or a single detector with

appropriate moving mirrors and pinholes.

This method's limits are set by the linearity of the vidicon unit, the speed of the

scanning processes, the spatial resolution of the detector array, and the selection of the

appropriate lens or mirror to match key spatial details in the far-field irradiance pattern

with the spatial resolution of the detector system.

A direct calibration of a system using the focal plane method is complex, which is one very important constraint of the method. If we could have a uniform, high irradiance beam

of significant cross section with a known plane wavefront, then a straightforward scan technique could be developed using this beam to calibrate the focal plane system. However,

generation of such large area plane-waves at an adequate irradiance level can be difficult.

In principle, a known source could be used to develop the calibration of the system. One

likely possibility is a point source. To be sure the calibration of the system is adequate

requires a detailed two-dimensional scan by both a single point source and a pair of closely

spaced point sources that have their far fields well developed before their beams enter the

system. Scans by both source configurations should be done at different optical distances

from the focal plane to calibrate the curvatures of their phase fronts.

The above calibration works as long as the detectors are linear over the entire range.

If they are not, a beam splitter can be used to scale the unknown beam to the irradiance

levels of the calibration system, so that the focal-plane system can work as a calibrated

unit.

The principal value of the focal plane method is its measurement capability for the

details of the far-field irradiance pattern. As discussed in reference [3], any residual

radius of curvature in the near-field phase front can cause an apparent distortion of the

irradiance pattern in the focal plane. To minimize this problem for detectors in this plane

and to be sure the irradiance pattern is not significantly modified by the residual radius

of curvature, we require the focal length of the lens to be such that the depth of field is

large compared to the dimensions of the detectors. Under these conditions, an accidental

curvature causes no significant errors in measurement of the irradiance pattern.

1.3.3 The Mirror Scan Method Using Two Pinholes

This method uses a moving mirror to direct the beam in a systematic two-dimensional

scan across a fixed pair of pinholes followed by a beam splitter [4,5]. The device has two detectors. One measures the total irradiance through the two pinholes in one beam of the

beam splitter. A moving spatial filter intercepts the second beam's interference pattern,

and the filtered radiation is focused either by a lens or by a spherical mirror into a second detector. The time dependence of the output from the two detectors, coordinated

with the motions of the mirror and the filter, generates two electrical signals containing

details of the amplitude and phase of the beam profile.

The scan speed of the mirror limits the rate of change of the beam profile that can be

followed. The usual speed is about 0.01 second per frame, i.e., about the TV scan rate. As long as the beam changes at a rate that is slower than this scan, the measurement made

by the device is well defined and has high precision. If the beam profile changes at a speed comparable to or faster than this scan, then the output fails to map the input.

An apparent limitation arises from the spatial resolution of a TV scan. A telescope

can overcome it by changing the scale of the beam before the scan takes place, thereby

enabling the TV output to contain the details of interest.

-4-
This unit can be calibrated by using the one- and two-point sources described in the

previous section. With this method a few discrete positioning changes of the sources are

used rather than the detailed two-dimensional scan prescribed for the focal plane method.

The two-pinhole method has potential for high precision. Its limitations are the

extreme expense of the instrument, the scan speed, and the apparent lack of stability of

the calibration.

1.3.4 The Hartmann Plate Method

This method bypasses the TV scan limit suffered by the previous systems [6,7,8]. The

basic device is a plate with holes that intersect and diffract the laser beam. These holes

are placed in a pattern to make the desired spatial sampling of the beam profile. The

amplitude and phase at each sampled point are extracted by a system of detectors. The

detector stucture at each hole stands on the optical axis at the closest point where the

diffraction pattern is a well-developed Airy disk. Each disk has a center of irradiance

which depends on the phase gradient relative to the plane defined by the hole. Further,

the total power in each Airy disk is proportional to the irradiance at the sampling hole.

To find the center of irradiance requires either a two-dimensional array of detectors for

each hole or an X and Y position sensing (helipot) system [9, 10] . With appropriate electronics at each sampled hole, it is thus possible to measure both the phase and amplitude

of a laser beam.

The space required for each Airy pattern to prevent significant overlap limits the

spatial resolution. A diverging lens or mirror can improve this.

A second limitation is the time constant of the electronics. A detector array can have

a response time of one nanosecond. The helipot appears to have a response time slower than

one microsecond. This fact distorts the output relative to the input at high speeds.

The speed and accuracy with which the phase information can be extracted from the

electronics defines the final limit. To deduce the phase requires a detailed unfolding of

the time and spatial response structures of the detection system. As long as this unfolding

is not complex, the system can perform accurate phase measurements. The detector array

permits the best accuracy in phase measurement. The price is a system with numerous

channels of output that must be processed with parallel electronics or stored on a multichannel data system.

Unlike the previous methods, the Hartmann method does not need point sources for

calibration. The distances between sampling holes may make possible a calibration by a uniform irradiance and a uniform phase front at each pair of neighboring holes in the plate.

Unlike the point source technique described in the focal plane method, this calibration can

be done at power levels comparable to those in the laser beam of interest. This implies a less severe constraint on the actual linearity of the detectors.

1.3.5 The Holographic Method

This subsection presents the essential operations of the holographic method which uses

elements from all three previous methods. This method minimizes the limitation of the electronic response time; hence, laser pulses can be better measured in real time. Obviously,

there is always a final limitation to the response time for any system due to the time

constants in the detectors.

The holographic method makes a beam profile measurement in the following sequence.

First, a Hartmann plate with a periodic pattern of holes intercepts and samples the beam.

Second, a series of lenses or mirrors generates the Fourier transform of the pattern from

the plate at the focal plane. Third, the transformed pattern is passed through a hologram

located at the focal plane. The hologram was previously generated by a beam profile of

approximately known amplitude and phase characteristics. Fourth, the filtered beam exiting

from the hologram is transformed by a second lens or mirror system to get the cross correlation. Under proper conditions, using the appropriate hologram* and proper sperturing

of the sampled laser beams, the resulting pattern in this correlation plane now has spots

with intensities showing only amplitude information about each sampled hole and spots with
is generated at the speed of light and represents an optical processing sequence. All

electronics are delayed to the last step. This fact minimizes the unfolding effects of

finite response time in detectors. Since the hole pattern is stationary in this plane the

size of each detector can be small, which substantially decreases the detector's time

constant.

The physical complexity of this system is greater than the previous systems. However,

the calibration process has been simplified greatly, and it appears that the precision will

eventually be higher than for previous methods. The number of calibration beams can be

greatly reduced. Even fewer calibration values are necessary compared with the Hartmann

plate method. Details of the calibration process are presented in subsections 4.3.2 and

4.5.

The relative phase between neighboring

radians, where n is some arbitrary integer.

in the phase front between sampled holes mus profile data. For highly directional laser

divergence, the holographic method may have more appropriate. Usually rapid variations

that the spatial coherence properties are no directionality. This poor directional chara

could work as well or better.

beams is determined only to a modulus of 2 Trn This fact means the range of allowed variation

t be restricted to obtain unambiguous beam

beams, this is not critical. For beams of high

little value. Some other method is probably

in the phase front from a plane wave indicates

t important. The beam does not have strong

cter may imply a spatially incoherent source

The sampling pattern of the Hartmann plate limits the spatial resolution of the

measurement of the beam profile. Appropriate use of beam expanders in front of the Hartmann

plate can minimize the impact of this technical limit.

The ultimate limitations of the holographic method are set by the total number of

detectors used to measure the time behavior of the beam profile and by the physical size of

each sampling hole. Each sampling hole needs about 2.25 channels; thus, 100 sampling holes

imply at least 200 detectors.

The above completes the general discussion on the beam profile methods;

next subsection and make a detailed study of the holographic method.

we pass to the

1.4 The Purpose of this Paper

At this point the reader should have a sense of what beam profile measurements are as well as how and why they are done. The reader may not yet have detailed understanding of

how wave propagation works, what are Fourier optics, and how beam sampling permits phase

and amplitude measurements with high precision. To provide necessary background, appendix A

details the Fourier optics notions, and appendices B and C describe diffraction patterns of

square and circular holes in the Fourier optics approximation. These results are then

summarized and used in section 2 to show how ideally-shaped laser beams can be used for

precision measurements.

After we complete the technical overview in section 2, we describe in sections 3 and 4 the prototype system constructed during 1977 at NBS (Boulder, Colorado) to demonstrate the

holographic method. In section 3, we describe both the derivations of the mathematical

models for the method and the alignment process for the apparatus. Section 4 describes the

apparatus that generated the cross-correlation patterns. Because the alignment process

requires an understanding of the concept of depth of field, I include appendix D which

explains the idea in a form different from that usually found in the literature.

Because the original system is a prototype and the basic holographic method is new, many things were learned during this study. Section 5 describes what was learned and

suggests improvements to the basic system.

*This hologram can be generated so that it works for far-and near-infrared.

This paper is a demonstration of the concepts using visible light. Subsection 5.4

describes the reflection system that can be used for 10.6 ym. -6-
Finally, section 6 draws conclusions from the evaluation of the prototype unit and from

analysis of questions raised by various people in the laser community.

2. AN OVERVIEW OF THE BASICS USED IN THE HOLOGRAPHIC METHOD

Subsection 1.4.5 describes briefly the holographic method using terms defined in various

books and papers on holography and Fourier optics. Section 2 explains in more detail the

meaning of those terms and techniques so that subsequent sections can use freely those

notions to define the holographic method.

2.1 What this Overview Contains

To understand the holographic method requires a thorough comprehension of the diffraction of coherent light by a single aperture. If an aperture is small compared to the spatial

details in the original beam, then the subsequent structure of radiation out of this aperture

is known from the theory and previous experiments. To m.ake precision measurements of beam

profile, only the details of the Fourier plane and the near field of the diffraction pattern

are needed. The extremely complex intermediate field structure can be ignored. Subsection

2.2 gathers the relevant information about the single aperture that is necessary for the

precision measurements.

Because a single aperture generates insufficient information about an arbitrary laser

beam profile, we must use multiple apertures in a sampling plane and understand the resulting

Fourier plane pattern, and we need to know how to use the pattern to make precise and proper

measurements of an arbitrary beam profile. To help explain the use of multiple apertures,

we discuss in subsection 2.3 the diffraction by two apertures under far-field and Fourier

plane conditions.

The analysis of multiple beam sampling for the actual holographic method is discussed

in subsections 3.3 and 4.3.

Three concepts in the holographic method may be difficult to understand: Fourier optics,

holograms, and cross correlation. To be sure the reader has a clear picture of what is

important about each, subsection 2.4 addresses Fourier optics, 2.5 describes the holograms,

and 2.6 discusses cross correlation.

Subsection 2.7 assembles the above ideas to expand the discussion in subsection 1.3.5

about the properties of the holographic method.

In subsection 2.8, I summarize the holographic method and indicate what this paper

presents in the remaining sections using the terminology developed in section 2 and the

appendices A, B, C, and D.

2.2 The Main Features of the Diffraction Process by a Single Aperture

The key reason for using apertures to sample a laser beam profile is to gather precise

information about two important beam parameters, the average intensity and phase across each

aperture. The apertures suppress the effects of the gradients in the irradiance and the

phase across each aperture. By using the Fourier plane as a critical measurement surface, we can perform precise measurements on the average intensity and phase with a minimum

generation of error caused by phase and irradiance gradients and by variations in the exact

form of the apertures.

Because the beam is sampled just at selected coordinates, we have precise information

only on selected features of a laser beam. Therefore, to have meaningful beam profile data

requires that information is unambiguous and consistent with subsequent use of the beam.*

In short, the measurement technique for beam profile must be appropriate. For example, if

*The beam measured here would usually be one generated from the main laser

beam by a grating or a wedge-beam splitter.

-7-
only the intensity distribution of a laser beam's far field is important but the effects of

the propagation process are unimportant, then aperturing the laser beam for profile measurements can be a mistake. In this case, it makes more sense to generate the Fourier plane of

the laser beam without an aperture in front of the optics, and then to aperture the irradiance pattern in the Fourier plane for the selected beam profile information. However, to control or know this irradiance pattern for near, intermediate or far field in real time,

details on the phase front in the near field of the laser beam are needed. In this case, the far-field measurements are insufficient and not appropriate. There are circumstances

when it is not possible to measure the focal plane irradiance because the distribution is too small and the available space is limited; therefore, near-field phase and amplitude measurements are necessary to infer the far-field irradiance pattern.

Returning specifically to diffraction details by a selected aperture, we develop the

far-field and the Fourier plane patterns for two shapes, namely a square and circle, in

appendices B and C. In both, the information about the average irradiance and phase multiplies the field pattern generated by the unknown laser beam illuminating these apertures (see

equations B.6 and C.5). Note that the patterns are different for the square and circular

apertures. Further, to get accurate results, the variation in position off the optical axis

for the peak irradiance caused by the gradient in the phase (b^ and c. are not zero) must

be unimportant. To become insensitive to the shape of the aperture requires measurements

that correct for this influence. The holographic method reduces sensitivity by using a significant fraction of the field pattern in the Fourier plane and by comparing that pattern

with an appropriate pattern stored on a filter in that plane. By this process, the differences between the two patterns are minimized and their similarities are maximized. We see these details in sections 4.3, 5.4, and 5.5.

Shrinking an aperture to such a size that gradients in the phase are insignificant is

easy. Unfortunately, the amount of laser power through each aperture is so greatly reduced

that this can impose a stringent demand on the allowed detectors. One potential solution to this problem would be to use a laser amplifier at each aperture that would preserve the

details of the average irradiance and phase striking the aperture to eliminate completely

the gradient effects and yet generate sufficient gain to compensate for the small sampling

area. At present, there are no such amplifiers; therefore, we compensate by using larger

apertures and then restricting the allowed gradients across a given aperture. Of course,

in high power or energy lasers the size of the apertures can be smaller than those needed

for low power or energy lasers.

Incidentally, the constraint that the gradient of the phase across an aperture be less

than some value is opposite to the conditions for optimum operations of the Hartmann plate

method. In this case, it is necessary for the phase gradient be greater than some prescribed value so that variations in the phase front beyond a trivial plane wave can be

detected. I do not show constraints applied to the holographic method in the later subsections; they strongly depend on the actual apparatus. Sections 3.3 and 4.3 give the

correct mathematics for the square aperture case. In final analysis, when a precision

apparatus is built, the precise dynamic range of each parameter at each aperture is measured.

At this time we make approximations and computer simulations of ideal systems.

2.3 Looking at the Interference Pattern in the Far Field

and Fourier Plane for Two Apertures

The far-field pattern for two square apertures H^ and 2.„ can be directly written from

the results in eq. (B.6) as (see note below*):

*Note "*" means to multiply in this and subsequent expressions except when used

for complex conjugation. The difference is obvious in each expression.
^l^^'r^l^ = [A^ a} e^^^/{±\z)]

^^^l fik 2 9

f r(x2-g,i) ^

b^^ , 1 Az 2Tr sine

id.

+ ^—I a Az 2t:

^^2 ik 2 9 + e exp |- [(x2-g^2) + (^9-^0)]

1^2-§£2^ ^

^£2

Az 277 a

^ J

2 £2'

sine ^^2^2) ^

^£2

Az 2ir

To qualitatively understand this equation, we simplify it by assuming Az is large and the aperture diameter a is small enough, and we only eonsider X2, y2 ^ 0. In

ease, the above expression simplifies to:

(2.3.1)

enough

this

A 2 ikz A a e "2^^2'y2^ ^^-TXz

e " expf [(x-2g,i)'+ (y-2\i)']

^<^£2 fik 2 9 +^ exp \j-^ [(x-2g,2) + (y-2\2)

(2.3.2)

Next we presume the irradiances on the two apertures are identical; thus d„„ = d„,

Z£^ zi With these simplifications, the irradianee pattern in the far field for two apertures

becomes:

I[x2,y2] = 21^

1+eos ^d^,^-di,^+2i

^^-2S£l)^ - (^-28£2^^

^^^2-^1^^ - ^^l-'^ll^^

(2.3.3)

where I A^ 2

T— A exp(-2d„„ ). This 100% modulation pattern contains detailed ^ " I o 2£ ^ Az information about the relative phase between two apertures, namely d. ~d„ , as well as

the irradianee illuminating both apertures, namely the value fixed by d„ .

By appropriate study of the details of the far-field irradianee pattern without all the

above simplifications, you can get the relative phase between the two apertures as well as

the individual irradiances, namely those values fixed by d„ and d„ . Unfortunately

II

-9-
measurement of those details can be complex. That is why constraints are imposed on the

system so we can make accurate beam profile measurements.

The Fourier plane pattern U^(x„, Y„) , for two square apertures Z, and 2.^ can be

directly written from the results (B.24) as:

A a^

U£(x2,y2) = ^Xf~ ^""P [i(2kf+(n-l)tk)]

e sine

+e

^[^£2 - f^^2§£2+y2^2)l

sine

^^2 ^

^£1

xf

•" 27-1^

\f 2-n

(^2 ^^£1

Xf + 2^^

Xf 2tt

(2.3.4)

Again to illustrate the key features of this pattern, if a is small enough and f is

large enough, then the sine functions are equal to one. If we further assume the irradiance

at each aperture is the same, then the resulting irradiance in the Fourier plane becomes:

I(x2,y2) = 2I^|1 + cos (d^^^-d^^^^ - - [^2^g^^-g^2^ + y2(h,i-h,2) 1 (2.3.5)

)^

where

I^ = A

f o

2 ^2

a

Xf exp (-2d„ ) 1

We see that the irradiance pattern for both the far field and Fourier plane are similar.

The only difference is the phase shift constant,

'T~~[gn-] ~ Sjio + h^ - h „], which is present in the far-field expression.

2.4 Important Properties of Fourier Optics as Applied to the Holographic Method

Appendices A and D describe key features of Fourier optics [11,12]. Fourier optics is

the holographic method because you generate the Fourier transform of an arbitrary beam

profile. Further, this technique allows linear filtering of that transform for another

Fourier transform.

Why is the transform important? As already discussed, precision measurements require numerous apertures to extract needed irradiances and phases. As shown in subsection A. 3,

the Fourier plane has the position details for each aperture buried in the phase term. Thus,

the linear filter used to generate the required cross correlations acts uniformly and simultaneously on each electric field from all the apertures. This fact allows optical computing

rather than electronic computing; consequently, the measurement rate is faster and probably

more accurate.

2.5 Important Properties of the Holographic Process

Holography has two points of importance, both due to the use of a reference beam in

conjunction with an unknown object beam. First, using a plane wave as a reference allows

a photographic plate to record both phase and irradiance information of the object beam.

Second, because the reference beam has its phase front at an angle to the object beam, the

density recorded in the plate can generate spatially separated laser beams one of which

contains the desired cross-correlation information.

-10-
To understand the above points, we discuss briefly the holographic process. We symbolize

the incident radiation as:

ik„z ik X U(x,y) = e [0(x,y) + R(x,y)e ],

Ik^x

where 0(x,y) represents the electric field for the object beam; R(x,y)e represents the

reference beam; k labels the vector component along the x axis for the propagation of the

reference beam; and k„ labels the vector component along the z axis for the propagation of

both the reference and object beams. The latter lias no propagation along the x axis.

The photographic emulsion records an irradiance pattern, namely:

I(x,y) = U(x,y) U (x,y) . (See note below*)

If the irradiance of the reference beam is four or more times the irradiance of the object

beam [11, pages 45-60], then the recorded transmittance, t, on the film can be considered

to have the form:

t(x,y) = f[R(x,y)R (x,y)] + a^[R(x,y)0 (x,y)e ]

* -ik^x + a^ [R (x,y)0(x,y)e ] + neglected terms.

The function f is the average transmittance and the a is the derivative of this function

biased at the average transmittance. The neglected terms are higher derivatives, the biasing

effects of 00* and the nonlinearities of the holographic process. We presume the film has

a linear response region. This describes the first feature of the holographic process.

When the transmittance t(x,y) intercepts and filters a laser beam, the exit radiation

pattern is given as:

E(x,y) = S(x,y) t(x,y) .

S(x,y) is the incident electric field before it strikes the photographic emulsion, and E(x,y)

is the exiting electric field. Notice there are three beams exiting from this emulsion.

A First, the beam, a S(x,y)0 (x,y)R(x,y), diffracts to propagate with a vector component along

the X axis of +k . Second, the beam a S(x,y)0(x,y)R (x,y), diffracts to propagate with a vector component along the x axis of -k^ . Third, the beam S(x,y) [R(x,y)R (x,y)], contains

no diffraction or beam bending; therefore, this beam continues along the original direction

of the incident beam, S(x,y).

2.6 Important Properties of the Cross-Correlation Process

As already mentioned in subsections 2.4 and 2.5, the cross-correlation process

involves spatial filtering [13]. After the filtering, the appropriate beam,

*

C = S(x,y)0 (x,y)R(x,y)a , becomes transformed. This step converts the C into a series of

spots.

*Note: The U*(x,y) means complex conjugate of U(x,y)

-11-
These spots are distorted Images of the original illuminated apertures. The distortion

comes from three sources: the aperturing by the filter, the pattern structure of the filter,

and the aperturing by the final Fourier transform lens. The number of spots can increase

substantially over the original number of apertures used to sample the unknown laser beam.

Each spot can have contributions of both irradiance and phase from each sampled beam. The

subsequent sections describe filters and approaches to control the form of these contributions.

In conclusion, in the cross-correlation process the filter acts as a grating and beam

splitter to direct known fractions of radiation from one aperture to selected spots in the

cross-correlation plane. By this diffraction process, the phase and irradiance at each

aperture can be referenced in real time to the phases and irradiances of other apertures.

This situation permits application of a differential measurement technique comparable to that used in electronic bridges. Because these comparisons are made at localized points,

there is substantial reduction both in the response time of detectors and on their effect

on the precision of phase and irradiance measurements. This condition is a prime reason

for making the detection process in the cross-correlation plane rather than in the Fourier

or far field of the original apertures.

2.7 Assembly of the Above Concepts to Define the Holographic Method

I suggest the reader return to subsection 1.3.5 and reread the second paragraph using

the understanding generated by discussions in subsections 2.1 to 2.6. Once this is completed,

the reader will have a clear overall picture of the holographic method and will be ready to study the details of this method.

2.8 Summary of this Overview

We have seen that developing an understanding of propagating laser beams is important.

There is extensive interplay between the ideal understanding of the sampling process, as shown by the discussion of aperturing a laser beam, and the corresponding technical implications of Fourier transform, filtering, and holography [11,12,14]. This interplay

generates options for optical processing [13] . By avoiding conversion of the beam profile

information as carried by sampled beams, until the last possible moment, we increase our speed for precise measurements. This allows us to use feedback at speeds not currently

available and implies a significant technical advance.

^

-12-
3. HOLOGRAM CONSTRUCTION APPARATUS

This section has three goals. First, we describe the front part of the beam profile

apparatus built at NBS in 1977. Second, we construct the mathematical model simulating

this apparatus. Third, because we want to accelerate transfer of these concepts, we discuss

actual equipment.

3.1 What this Section Presents

There are four subsections in this section. Subsection 3.2 defines the structure of

the prototype apparatus that generated the hologram used for the cross-correlation process.

Subsection 3.3 uses this prototype to fix the necessary mathematics for the simulation

describing how the apparatus generates the hologram and how the front end of the cross- correlation process works. Subsection 3.4 specifies the assembly and alignment of the

prototype apparatus, and subsection 3.5 lists conclusions relevant to the unit.

3.2 A Discussion of the Prototype Apparatus Used to Generate the Hologram

Using concepts of holography [1] and spatial filtering [13,14,15,16,17], we define the

prototype apparatus using the block diagrams in figures 1 and 2 to identify each basic

optical component. Each component is discussed in the order in which the laser beam goes

through that component.

We generate the reference and object beams by using a beamsplitter to create two beams

from the single beam exiting the laser. Because it is convenient for prototyping to study

beam profiles with cross sections near 10 mm, we use a beam expander to expand original

laser beam from its nominal 1 mm to a 10 mm diameter.

Following the reference beam, a beam expander creates a slowly diverging beam with an approximately Gaussian profile. The beam diameter at the half-power point is 25 mm when

the reference beam intersects the film within the liquid gate.

Following the object beam, the attenuator in the path of this beam reduces its power

to a convenient level for the hologram. The beam expander generates a Gaussian-like beam

with a beam diameter near 10 mm and a plane phase front. This beam is now intersected by

the prefilter which samples with nine apertures (see fig. 3). The exiting beams from the

prefilter pass through a Fourier transform lens and an image magnification lens to intersect simultaneously the reference beam and the film.

We construct the hologram by exposing the film to the two beams and by developing the

exposed film.

3.3 The Mathematical Model Used to Understand the Hologram

We examine each beam in figure 1 with the mathematics defined In the appendices to derive the resulting irradiance pattern recorded by the film. During this process, necessary notation is defined and symbolic abbreviations of the complex details are generated

to clarify the essential structure of the process for generating the hologram.

We begin with the reference beam at a position one focal length in front of the first

lens of the beam expander. At first, we assume the beam profile is arbitrary, namely

U^(x^,y^,z^) , (3.3.1)

where z is the distance between the above focal point and the position where the laser

beam is split to generate the reference beam.

NOTE: The optical axis is defined operationally as the center of intensity

for a cylindrically symmetric laser beam passing through the various optics.

-13-
Reference Beam

APrefilter/h

\ ->

BeamExpander

Fourier

Transform

Lens \

\

I

Attenuator

BeamExpander

/

Object Beam

^

Beamsplitter

I^Liquid Gate

(film)

Laser

Figure 1. The hologram generating apparatus - block diagram.

-14-

I
The Plane Where the

Beam Profile Is Measured

t

Pref liter

Calibration

or Test Beam

Laser

-ourier

Transform

Lens

^

i

Test or

Calibration

Beam

Hologram

Fourier Transform

Lens for Cross

Correlation

Cross-Correlation Plane

Figure 2. The cross-correlation generating apparatus - block

diagram.

-15-
-16-
The first lens has an equivalent focal length, f„. The field at the Fourier plane of

this lens as shown by figure A.l and eqs. (A. 4) and (A. 5) is:

A„e

V^(x^,Y^;z^) = -^— / dx-^dy-^ U^(x-^,y^;z^) h^(l,2;f2) , (3.3.2)

where the phase shift is (j)„ = k[2f„ + (n„-l)t„]; the distance along the optic axis (see

note below) is z„ = z + 2f„; the attenuation factor and additional phase shift are contained

in A„; and the transform function is h (l,2;f„) = exp[-i(u„x + v„y ) 2Tr] . The spatial

frequencies are defined as u^ = x„/Xf„ and v„ = y„/Xf„. The apparent thickness of the

lens is t„, and the effective index of refraction is n„. The spatial filter at z„ changes

the exit beam U. to:

\]2(x^,y2;z^) = \]^(x2,y2''Z2^ circ(r2/a^). (3.3.3)

2 2 1/2

The cylindrical coordinates center around the optical axis as r = (x + y ) , and a is

the diameter of the filter. The circ function is defined in appendix C.

The second lens has a focal length, f^. It is placed so that a magnified image of the

filtered field U^ is formed at the film. Using eq. (A. 7), the reference beam at the film

is given as:

1 a ""3 ^3

^i^^3y3'"3) = M^ h[x3,y3;s3] U^ (- — , —; z^) (3.3.4)

* e A3

where the magnification is M„ = d,/d^; the effective radius of curvature is s_ = d ,

/

° 2 4 3 3 4

(1 + 1/M ) ; the phase shift is d_ = k[d_+d,+(n -1) t ]

; the optical distance from this

second lens to the focal plane of the first lens is d ; and the optical distance from the

second lens to the film is d . . The imaging condition is satisfied, namely 1/d + 1/d. =

1/f-. The additional attenuation and phase shifts due to mirrors and the second lens are contained in A . The optical path length is z = d, + d + z„.

We now examine the evolution of the object beam. There are two stages showing its

development in front of and behind the prefilter.

Because the actions of the attenuator and beam expander are arbitrary, we address the

development of the object beam behind the prefilter and delay the discussion of the object

beam in front of the prefilter until the end of this subsection.

As already assumed in Appendix B, we presume that the beam shape across each square

aperture in the prefilter can be adequately represented as:

U^(x^,y^;z^) = n[(x^-£^b)/a]n[(y^-£2b)/a]*A^

* exp

j i[bj^(x^-^^b) + c^iy^-l^^) + d^] , (3.3.5)

-17-
where z, is the optical path distance from the beam splitter; x, ,y, are the transverse DO

coordinates at the prefilter; b is the distance between the apertures in either x, or y,

D o directions; a is the edge dimension of each aperture; g„ = £ b labels the x, coordinate of

the £ aperture (see figure 4); h„ = £„b labels the y, coordinate of the £ aperture; and

b„, c., d. are the parameters for the beam profile across the £ aperture. We abbreviate

the (£^,£„) label as "£". Only when it becomes necessary to give a detailed label will we

expand the notation.

The beam form exiting the prefilter is a simple superposition of the terms in eq. (3.3.5); thus

"p = I h^^e^ye'^6^ - (3.3.6)

The prototype in figure 3 has nine such apertures.

After exiting the prefilter, the object beam goes through a Fourier transform lens

arrangement. We can write out the beam profile at the Fourier plane as:

a A e 7 b. c- Ug[x^,y^;z^] = —rf^ B^ I sine [ (u^ + ^)a]sinc[(v^ + j^)a]

* exp \ i[d^ 2u:u^gj^ + v^h^)] (• (3.3.7)

The phase shift is $ = k[2f + (n -l)t 1; the focal length of the Fourier transform lens

is f ; the index of refraction is n ; and the thickness of the lens is t^. The optical

path distance at the Fourier plane is z x z, + 2f . The spatial frequencies are

u = x /Xf and v = y /Xf . The additional attenuation and phase shift due to the mirrors

and lens are contained in B .

Because the dimensions of the Fourier transform are too small for effective use of the

film, we have an image lens which magnifies the pattern [use eq. (A. 7)]. Thus, at the film

where the hologram is generated, the object beam is now:

i$ Bge 8 Xg yg

Uy(xg,yg;zg) =— h[xg,yg;sg] Ug(- —, ^ ; ^j) (3.3.8)

The magnification is M = do/d-,; the next optical path location is z^ = z., + d^ + d-,; the

o o / o / o / radius of curvature is s^ = [do/(l + I/Mq)]; and the extra phase shift is $„ = k[do + d-, +

o o o o o / (n -1)z ]. The additional attenuation and phase shift corrections due to the lens and

mmirrors are contained in B . This factor also includes the reflections at the liquid

o gate.

Before we explain the details of the beam at the film plane, we discuss the conditions

that allow us to ignore the b„,c„ terms in eq. (3.3.8). Essentially, a convenient condition

for negligible b„,c„ is that the modification of the sine function around the first zero

-18-
Prefilter AA ' -1 +1

•^2

-1 D D D

D D D

n D D D

l. ^J.2 labels the spatial position of the laser beams.

Figure 4. The coordinates for the prefilter beams,

Cross Correlation Plane can show a 5 by 5 pattern of spots

k 2V^ 2 1 0-1-2

-1

D D D D D

D D D D D

D D D D D

D D D D D

D D D D D

k^ , kg labels the spatial position of spots in the Cross-Correlation Plane.

Figure 5. The coordinates and structure of the spots at the

cross-correlation plane.

-19-
be unimportant. This means the magnitude of the gradients, namely |b.|, |c.|, must be such

that either is less than

(.01)

27T (3.3.9)

Thus, the smaller the edge of the sampling aperture is, namely a, the greater the allowed

gradients are. This situation is equivalent to having a large depth of field in the beam

profile instrument. For our subsequent discussion, we presume this condition is satisfied

and drop the b, and c„ terms in the sine functions.

Clearly, these constraints finally fix. the allowed rate of change of the beam profile

between different apertures and show up as a limit to the range of the d„ terms in the

final model used to represent the beam.

I

The next approximation is to ignore the angle between the reference and object beams

(see figure 1). The x ,y and x ,y define planes different from the film plane and each

o o i i other. A rigorous theory generates numerous corrections to deduce the true beam profile

recorded in the film plane. Under the conditions that both beams have significant depth of

field, then most corrections are insensitive to this angle. Only the plane wave phase term

shows any importance and any distortions in the theory will be unimportant.

We show this by extracting temporarily this phase term and writing the reference and

object terms as: ikz^

R{-Kyy^;z^) = e R and

ikZg

0(xg,yg,zg) = e 0^-

where R and are the rest of the details in each beam. Here y„ = y„, if we assume that

a a 3 8 the various x coordinates are in the plane defined in figure 1. Then x = x„ cos 8 and

x„ = x„ COS 9, if we assume the normal to the film bisects the reference and object beams.

8 9 The z JOand z become modified to z 39= z cos 9 - x sin 9 and z = z cos 9 + Xq sin 9. We

9 o9 9 absorb the z_ cos 9 factor in R and , and note that cos 9 = .995 or is about one when

9 a a the angle 9 is less than 0.1 radian. Thus, we write:

and

R^ = p R^ and 0^ = pO^ (3.3.10)

where

p = exp[ikx sin (9)]

9 (3.3.11)

The irradiance recorded on the film for a cw laser with no incoherence between the two beams is broken into terms

* 2 * 2

I = I^ + I^(p ) + I^(p) (3.3.12)

where the biasing term is I = R-yR,, + OuOu; the cross-correlation term is I^ = RuOu' ^'^'^ * 2 the virtual image term is I . The terms p and , ''=.2 cause a sine wave modulation of the

intensity pattern and act as a diffraction grating when the hologram is irradiated.

-20-
The film exposure has the transmittance as t(x,y) = t[I] just as is shown in equation

(2.3.3)- (Note there is a time of exposure that has been suppressed in this discussion.)

For convenience, we note the form is:

t(I) = f(I^) + a^(I^(p*)^ + lj(p)^) . (3.3.13)

We have generated the hologram and are now ready to discuss the evolution of the object

beam in front of the prefilter.

For the hologram generated to demonstrate the method, we choose an approximately

uniform irradiance pattern with zero phase difference between the apertures (other patterns

may be used) . Since our purpose requires the cross-correlation structure give equal weight

to each aperture, we use the plane wave with uniform irradiance.

To generate the desired beam profile with appropriate mathematical terms, we model the

attenuator and the beam expander for the object beam.

First, assume the beam profile at the beam splitter for the object beam is:

U(x,y;o) . (3.3.14)

If we accept that the attenuator is at z , and we use eq. (A.l) to deduce the profile at

z , then the profile is:

^ 4 \<^^4'y4'^4) " ~iXz— / dxdy U(x,y;o)h(x^-x,y^-y;z^) . (3.3.15)

4

Here the phase shift is $, = kz,, and the effects of the attenuator are contained in coefficient B, . Technically, the attenuator also can produce beam distortion. For this discussion, we presume there is none.

Now we assume the position, z. , is located at one focal length in front of the first

lens of the beam expander. This implies that a beam profile at the focal plane after the

object beam exits the first lens is the Fourier transform of eq. (3.3.15) as prescribed in

eqs. (A. 4) and (A. 5) . To make this beam profile as close to a known profile as possible,

we insert a pinhole filter in this plane, namely at z^ = z, +2f ,

.

5 4 4 The f , is the equivalent focal length of the first lens. If the first lens has a small focal length and we choose arbitrarily a circular filter with a diameter equal to ten

wavelengths, then when w = d„ or d = f in eq. (A. 3), the exiting beam profile from the

second lens system at the prefilter becomes:

^^^e'^e'^e^ = TTf; ^ '^^5^y5 U(x3,y3^Z3)h[x3,y3:s3]

r iK r . M . '*6 (3.3.16)

The phase shift (\) E k(f + d + (n - l)t ); the radius of curvatures are s = fi-/(l - d„/

S3 = f 1-/(1 - d /f ); the focal length of the second lens is f ; the index of refraction is

n ; and the equivalent thickness of the lens is t . The distance between the second lens

and the prefilter is d„.

-21-
Applying the conditions that the filter at plane x , y^ has all x^-, y^. within the 5X

and that the distance d„ = mf_, thus s_ = f_/(l-in), we find the worst phase error in

h[x^,y^;s^] is:

k a^

2 s.

25A7T(l-in) (3.3.17)

If f^ = 30 mm, A = 1 ym and m is less than 11, then this phase error is less than

75 * 10

30 10

-6

* 10 or 0.02 radians. In this case, we set h = 1 and assume that the beam profile

at prefilter is a Fourier transform of the pattern at the pinhole. If we now assume that

the latter pattern is uniform and constant over the aperture, then the prefilter pattern is an Airy disk pattern with a plane wave front. The amplitude variation Is deduced from eq. (C.5) and is

A 5 A

2 ral r^

1 ^ ^ ^(d^ (3.3.18)

near the center of the beam. Given a- = 5A and d^ = 10 f^, this amplitude becomes

^ = *„ii ^

^h

25

100

2~l 2^

] . If we require less than 1% amplitude error, then

< . 04 or r < 6 mm for the f_ = 30 mm.

We now see that the sampling of the object beam in the prefilter plane in a circle of

6 mm radius will show less than 1% error in the uniformity of amplitude and less than 2% error for the Irradlance.

Concluding this subsection, we note three points. First, by appropriate choice of beam

expander, pinhole, and distance between expander and prefilter, we can make arbitrarily

accurate plane waves over selected cross sections with limited ranges of irradlance for the

object beam. Second, by similar use of a beam expander and its filter, we make the reference beam have a uniform Irradlance and a spherical wave front over a controlled cross section of the film. This means we can construct a high accuracy cross-correlation pattern

in the hologram. Third, there is a technical limit to this procedure. Thus, the larger the

desired sampling region is, the harder It is to get the precision hologram. However, the

computer-generated structures alluded to in subsections 5.2 or 5.4 allow us to generate an appropriate substitute for the hologram constructed by laser beams.

3.4 Detailed Instructions for Assembling and Aligning the Apparatus

in the Construction of the Desired Hologram

In our prototype apparatus we used an HeNe laser with a 7 mW output at 0.6328 mm and

with a linear polarization. This laser was chosen because we wanted to have sufficient

power for the cross-correlation process and because we wanted to do this evaluation of

concepts at a visible laser wavelength.

The choice of a fixed electric-field direction eliminates the need to consider polarization effects. The electric field remains fixed throughout the apparatus if the field

points only along either the parallel or the perpendicular direction to a plane defined by

the various laser beams within the apparatus. Our apparatus used the perpendicular direction for the electric field.

-22-
We used a large metal table with a flat surface to hold various optical components and

to fix their relative positions against vibrations caused by moving, rotating, and vibrating

equipment; moving people; minor earth tremors; etc. In this prototype, the metal table

weighs about 250 kg and has a machined flat surface 0.9 by 1.4 m. Because the room is at ground level, this table has no special shock mounting. All the components shown in figure

1 are on the surface of the table except the laser. It was mounted on a secondary table

because there was insufficient space on the main table. It would be better to mount the

laser on the main table; however, as long as the reference and object beams are formed and

recombined on this table and the time scale for generating the hologram is short enough, the

vibrations do not affect significantly the quality of the hologram. This fact is proven by

the quality of the resulting cross correlations.

Figure 1 shows numerous plane mirrors used to deflect both beams around the table.

These mirrors keep the reference and object beams confined within the table boundaries and

allow the beams to cross at the liquid gate.

During construction of a hologram, we split the laser beam into a reference and object

beam and kept their electric fields perpendicular to the surface of the table. The beam

splitter causes about 50% of the laser power to go into each beam. This beam splitter uses

a dielectric film to create the two beams. We need comparable power in each beam to get a proper hologram. ^

In this prototype apparatus there are no antiref lection coatings on the various optical

components. The many reflected beams from the various optical surfaces have significant

power and can cause serious measurement problems. One way to avoid these beams is to use reflection optics. Unfortunately, this step is not a cure all; there are measurement

problems in both types of optics because each interaction of a laser beam with an optical

component can change its profile in an unknown way. For example, a plane mirror is not

truly plane; therefore, the wavefront is distorted by the irregularities of the mirror.

Careful scaling and choice of optical components can reduce these problems to desired tolerances. The other way to avoid these beams is to use antiref lection coatings which make the

optics expensive.

Continuing to follow the primary laser beam, we now follow the evolution of the reference beam from the splitter to the liquid gate. The reference beam is about 1 mm in diameter at the half irradiance points after the beam exits from the splitter. To avoid resolution problems in the photographic emulsions at the hologram plane and to generate an almost plane wave at this plane for this reference beam, we insert a pair of lenses and a pinhole to make a beam expander.

The initial lens in the beam expander through which the reference beam passes is a "20x" (8 mm focal length) microscope objective. Bubbles and other imperfections in this

lens create interference rings in a coherent beam. Precision work requires that these rings

be eliminated. Most of the power in the reference beam is concentrated at a plane near the

focal plane of the microscope objective. When we passed the beam through an 18 ym pinhole

located in this plane, the prototype apparatus removed the rings and also many non-TEM^^^

modes present in the original laser beam and thereby left mainly a TEM^^ mode (see reference

[18] for discussion of laser modes). The filter process is not perfect; consequently, the

recorded reference beam introduces error in the hologram. Fortunately, the sampling process

at the prefilter plane makes it unnecessary for the structure of the beam used to make the

hologram a pure TEM__ mode. Subsection 3.3 assumes a different beam instead of the TEM__

mode. The exact form of the beam depends on the various components in an apparatus.

Finally, we complete the beam expander using a lens with a focal length of 72 mm and a diameter of 60 mm which changes the curvature of the reference beam so that it exits from

the beam expander with a half-power diameter of about 9 mm and shows a slowly diverging

wavefront.

At the liquid gate, the reference beam is about 25 mm in diameter at the halfirradiance points. The optical path distance between the beam expander and the liquid gate

-23-
is about 2.5 m. This distance insures that at the liquid gate the depth of field, as defined in appendix B, for the reference beam is substantial; therefore, there is no sensitive adjustment necessary in the beam expander or in the mirrors except to make sure the

TEM-^ mode is present and both the reference and object beams overlap at the film plane.

The choice of lenses in the beam expander was determined by readily available optics which

had minimal defects.

The preceding discussion defines the reference beam. We now turn to the evolution of

the object beam.

At first we need to consider what happens to the unwanted reflected beams from the

many optical components in the object beam because they could induce measurement problems

and be a safety hazard. Briefly, each spurious beam should be stopped and absorbed. In the

interest of brevity, this paper does not describe where each of these beams is located. If

a reader wishes to construct some version of this apparatus, he should test his apparatus

with visible laser beams to confirm that the spurious beams cannot cause problems or hazards.

Continuing with the object beam out of the beam splitter, we see in figure 1 that this

beam goes through an attenuator. This unit contains a half -wave plate and a polarizer. The

former is inclined at the brewster angle to eliminate or reduce, substantially, interference

between the various transmitted beams. This plate rotates the polarization plane of the

object beam by a controlled amount which depends on the orientation of the plate [19] . The

polarizer is oriented to pass radiation (1000 to 1) with the electric field perpendicular to the metal table. We use the attenuator to adjust the peak power density of the object beam

to be one fourth the irradiance of the reference beam at the film plane. The literature

suggests that this ratio of irradiance makes the cross-correlation process efficient and

minimizes potential distortions in the hologram [20] . Because this prototype only demonstrates and checks key elements in the cross-correlation process, it is premature to design

for optimum operation; therefore, we accepted suggestions in the literature and based many

decisions on incomplete information. The section discussing problems and conclusions suggests how this prototype may be improved and what should be considered to make an optimum

design.

Continuing with the object beam, we employ a beam expander like the one used for the

reference beam. In this prototype apparatus, the translation mounts in this expander do not allow as precise adjustment and mechanical stability as in the beam expander for the reference beam. As a consequence, the pinhole here is 25 ym in diameter rather than 18 ym in

diameter. Again, subsection 3.3 considers a different strategy for the filter in the beam

expander. For this prototype, the discussion in this subsection is sufficient. For more

precise details, consider the issues discussed in subsection 3.3.

The best translation equipment used within this apparatus has 10 ym resolution. An improved version of this apparatus should use translation equipment which can be adjusted

and remain stable to better than 1 ym; otherwise, the apparatus will not remain calibrated

for significant periods.

During the initial construction phase, it is necessary to adjust the optical path

lengths of both the reference and object beams between the beam splitter and liquid gate so that their lengths are equal to ± 1 cm accuracy. One convenient method having this accuracy

for a three-meter path length uses a Ronchi ruling in front of the beam splitter, temporarily removes the beam expanders, and observes the diffraction pattern in the film plane.

When the spatial separation between the two first-order diffraction maxima from each beam

are equal then their path lengths are equal. After the paths are equalized, the beam

expanders are reinserted and the ruling is removed. This adjustment is not critical; it is

only necessary to be within the coherence length of the two beams. The next step verifies

the adequacy of the adjustments.

Also during the initial construction phase of this apparatus, we confirmed that the

metal table is stable. One method uses a magnified image of the interference pattern between

the object and the reference beam at the film plane. One way to obtain a magnified image is

to use a lens behind the liquid gate near the cross-correlation position (fig. 2) to form an

-24-
image of the film plane on a distant, painted metal surface of the wall of the laboratory.

An image of the interference pattern of the two beams can be seen if the apparatus is stable.

The metal surface creates a flat image plane; a paper surface will not work. Because the

spatial frequency of the pattern is high, a magnifying lens is used to see the interference

pattern. If no pattern is visible, then the apparatus has insufficient stability for making

holograms^

In our prototype apparatus, we had a very quiet laboratory on the ground floor and

used a very heavy table. Therefore, the pattern was sharp even though the path length

between beam splitter and film plane was about 3 meters.

Continuing the placement of optics into the object beam, we note the following

conditions on the liquid gate:

(1) The angle between the reference and object beams should be near 5° to allow

maximum resolution [21,22] in the photographic plate and to permit a reasonable separation

between the cross correlation [11, page 194] and image beam leaving from the film (see

fig. 2). This angle is approximate and not critical.

(2) It is desirable to make the normal to the film plane bisect the angle between the

two- beams. This adjustment maximizes the depth of field for the resulting hologram and

permits its efficient use as either a cross correlation or an image beam generator. Again

this adjustment is not critical.

(3) The liquid gate is filled with water during all adjustments. (To prevent algae

from growing in the water, add 10-20 ppm of CuSO,.) IR and UV wavelengths may require

different liquid gates and liquids.

The placement of the optical components for the object beam is almost complete. We

have two remaining components, the Fourier transform lens and the prefilter. Both units

require critical adjustment and clear understanding of their functions, which requires a detailed study of Fourier optics [11,13,14,15,16]. (Appendices A and D of this paper

present selected aspects of Fourier optics.)

In this prototype apparatus, we use a single element convex lens with a focal length of

305 mm and a diameter of 89 mm to perform the Fourier transform process. Because the trans- verse dimensions of the Fourier transform are not large enough for efficient use of the

film, we use a second lens with a focal length of 52 mm and a diameter of 35 mm to magnify

the Fourier transform and to place that image at the film plane. To be efficient, we make

the spatial frequencies for the Fourier transform of the prefilter plane several orders of

magnitude lower than the cutoff of the film. This process will keep the potential accuracy

for a phase measurement as high as possible. Detailed analysis is necessary to establish

the optimum level of magnification. That has not been done here. To align this Fourier

transform unit, the following steps are taken:

(1) First, select the location of the prefilter plane, then locate the Fourier

transform plane two focal lengths (610 mm) further along the object beam using a variable

aperture at the transform plane. This aperture, which must be capable of ranging from 1 mm

to 10 mm in diameter, will be used to fix precisely the Fourier plane at the film plane.

(2) To center the variable aperture, remove the beam expander for the object beam and

stop the aperture at the Fourier transform plane to 1 mm in diameter and center on the

unexpanded laser beam. Note where the center of this beam strikes the film plane and adjust

the appropriate mirror to make this position also the center of the expanded reference beam.

(3) Replace the beam expander into the object beam and adjust the expander so that the

beam that passes through the aperture at the Fourier transform plane centers at the previous

position on the film plane.

(4) Insert the magnifying lens behind the Fourier transform plane to construct as sharp an image as possible of the aperture on the film plane. To make the spatial

II

*NOTE: A choice of convenience. There may be better liquids,

-25-
adjustment of this lens as precise as possible, open up the aperture to the full 10 mm

diameter which decreases the depth of field at the film plane (see appendix D) . (This lens

is not shown in figures 1 or 2)

This report does not discuss the technique for placement of the various translation and

rotation mounts because these details are dependent on the type of mount. It is more con- venient if a mount adjusts independently of other mounts. Assembly of an optical component

requires careful thought about the form of the necessary translations and rotations, which

simplify adjustment of the beam profile system. Ideally, the most critical adjustment

should be independent of the other adjustments.

(5) Stop down the aperture to 1 mm in diameter at the Fourier plane and insert the

Fourier transform lens. Adjust this lens until the spot is centered in the aperture and is as sharp as possible on the film plane. There is substantial depth of field for the adjustment of the object beam along its axis. Adjust the mount so that the translation along

the optical axis keeps the spot centered in the aperture over the entire range of adjustments of that mount.

(6) The final adjustment is the accurate positioning of the Fourier transform lens

along the optical axis. Temporarily place a mirror in the prefilter mount to reflect the

object beam back into the beam expander. Adjust this prefilter so that all adjustments keep

the optical axis of the reflected beam parallel to the optical axis of the incident beam.

This adjustment can be made accurately by noticing the lack of motion by the reflected spot

at an aperture just in front of the laser's exit port. (Here the 25 ym filter is not

present in the beam expander.)

When the prefilter unit is adjusted properly, we can make a precise transverse motion

of a filter made of two 100 ym pinholes separated by 1 mm mounted in the prefilter unit. By

temporarily blocking the reference beam and looking at the image of the film plane for the

object beam as formed on the wall of the laboratory, we see an Airy disk superimposed on a series of parallel dark and bright interference lines. When the prefilter unit is moved

transversely to the optical axis, the interference fringes will shrink or stretch unless the

Fourier transform plane is coincident to the film plane. This condition can be attained by

alternately adjusting the position of the Fourier transform lens along the optical axis and

testing whether or not the interference pattern remains constant.

Following the above procedures guarantees an accurately placed Fourier transform unit.

The following sequence describes how to place the prefilter unit.

(1) Replace the two-pinhole filter with one-pinhole filter. Make sure both filters

lie in the same transverse plane. Add a mirror behind the Fourier transform lens to reflect

the laser beam back to the prefilter plane. Adjust the position of the pinhole along the

optical axis so the speckles reflected from the prefilter plane are as large as possible

[23].

(2) Remove the mirror and replace the pinhole filter with the prefilter pattern shox^m in figure 3, being very careful to locate the new prefilter in the same plane as the previous filters. The prefilter unit is now adjusted properly along the optical axis.

(3) The transverse positions are determined by the part of the Incident object beam to be recorded on the hologram. In the prototype apparatus, the prefilter samples the center

of the object beam, where the beam is more nearly a pure TEM mode than at the edges.

(4) Adjust the beam expander to provide a flat wavefront at the prefilter plane. For example, use a mirror placed just in front of the prefilter plane to generate a reflected

beam and adjust the beam expander to get the largest speckle reflected from the pinhole

within the beam expander.

This discussion completes the adjustments of the object and reference beams. We

complete construction of the hologram. Using a silicon power meter, we measured 3.6 yW at the film plane for the reference beam and 0.8 yW for the object beam. These values imply an approximate peak irradiance of 400 yW/cm^. To get efficient phase holograms requires an

-26-
optical density of 2 which corresponds to a total energy density of 80 yJ/cm . We therefore

used a shutter speed of 0.2 s.

3.5 Results from Construction of the Hologram

The original intent in this evaluation was to generate a phase transmission hologram

because it is more efficient for cross-correlation measurements [11, page 148] ; therefore,

all holograms produced by this apparatus used the 0.2 s exposure. Technical problems developed in making adequate phase holograms. Since the prime goal of the prototype was to demonstrate the concepts of correlation process, the final hologram was not bleached. The

cross correlations shown in subsection 4.6 are from the resulting amplitude hologram. The

procedures for developing the plate followed the instructions in references [21,24]. The

entire development process left the plate in place. This is the prime reason for using a liquid gate.

After we constructed the hologram and understood the subsequent production of cross correlations, we concluded that the liquid gate is unnecessary and even undesirable. It is

necessary only to replace the film plate within 0.1 mm, and to steer the object beam with

precision so that the resulting hologram and the object beam used in the cross-correlation

process are properly matched. In addition, all potential phase error effects in the hologram

used without a liquid gate, such as shrinkage, can be minimized by using an appropriate

calibration procedure provided there are no significant humidity effects. In this case, the

emulsion must be shielded and sealed by a glass cover.

Information about photographic chemical development is not included because we did not determine the best approach. The reader must get those details from the literature. When

it becomes important to increase the accuracy of the beam profile measuring apparatus, then

extensive evaluation for appropriate holograms becomes useful. At present, the best choice

is an educated guess.

-27-
4. THE CROSS-CORRELATION APPARATUS

This section has three goals similar to those in section 3. First, we describe the

cross-correlation part of the beam profile apparatus built at NBS in 1977. Second, we

construct the two mathematical models simulating this apparatus. Third, because I want to accelerate transfer of these concepts, I discuss actual equipment.

4.1 What this Section Presents

There are five subsections in this section. First, subsection 4.2 defines the structure

of the prototype apparatus that generates cross correlations between the illuminated ninehole prefilter and its hologram. Second, subsection 4.3 uses the unit to fix the mathematics

for simulating the cross-correlation process. Third, subsection 4.4 specifies assembly and

alignment of the prototype apparatus for optimum correlation. Fourth, subsection 4.5 describes how to calibrate this apparatus. Fifth, to aid intuitive understanding of cross correlation, subsection 4.6 shows both TV pictures and conceptual sketches of observed

correlations. The conclusions drawn from this section are discussed in section 6.

4.2 A Discussion of the Prototype Apparatus Used to Generate

the Cross-Correlation Pattern

Presuming the reader is now aware of holographic aspects of this apparatus, we define

the cross-correlation features using block diagrams in figure 2 to identify each basic

component. This subsection explains the function and action of each component. Each

component is selected by the order in which the laser beam goes through.

We direct the test or calibration laser beam through the prefilter. In our test apparatus, we use this same beam to generate the hologram, except that we remove the beam

splitter. In a beam profile measuring apparatus, it is not necessary to put a beam expander

in front of the prefilter. We ignore that operation in the discussion of the prototype

unit. For this discussion we presume the input beam has at least a 10 mm diameter.

The illuminated apertures in the prefilter provide the required radiation sources for

the Fourier transform lens system. To make the cross correlation work, we use exactly the same setup as we used to generate the hologram. Thus, the position of the apparatus from

the prefilter through the hologram is unchanged. We add the Fourier transform lens behind

the hologram to form images in the cross-correlation plane from the beams generated by

radiation passing through the pref ilter-hologram sequence.

To make these images in this plane easy to see and understand, a magnifying lens is

used in conjunction with a TV camera and video receiver to show the different irradiance

patterns in that plane on a CRT.

4.3 The Mathematical Model Used to Understand the Cross-Correlation Pattern

In subsection 3.3 we generated the first level of mathematics for the prototype system.

Here we proceed to the final levels.

In subsection 4.3.1 we derive the expressions that show the expected spot pattern at the cross-correlation plane. To show that the resulting cross correlation can be made

insensitive to the details of the spot shape requires numerous assumptions and approximations which are shown here.

A convenient model for operational use of the prototype is the prime goal of subsection

4.3.2 and is therefore generated here. In addition, this subsection shows the analysis of a simple beam profile measuring unit with two apertures at the prefilter. This discussion

covers the entire cycle of measurements—namely the calibration of the unit and also the use

of the unit for real time beam profile measurements.

4.3.1 The Model that Contains the Spatial Details of the Irradiance

Here we want the form of the test beam at the hologram. The test or calibration beam

exiting from the prefilter is represented in eq. (3.3.5). Only the bp, c^, and d. change to

-28-
new values, say Q., Cnj and ri. respectively. Following the arguments in subsection 3.3,

after eq. (3.3.5) through eq. (3.3.9), we find that the test beam can be symbolized as: 1*8

h(x8,y8:s8) \ (-

^ ' "C ' ^7^' (^-^-D

1 o o \^^8'y8'"8) M.

where

2 ^7

^ ^o^ V ^i ^l

U^^(x^,y^;z^) = iXf^ By

\

^^^^[(^7 + 2^^^^ "^'"^ ^ ^^ + 2^^^^

* exp^i[nj^2TT(u-,gjj^+v^h^)] } . (4.3.2)

This U beam can be restructured according to the discussion near eqs. (3.3.10) and (3.3.11)

as:

"t = p "t ' (4.3.3)

to reflect the fact that the test beam is at an angle, 9, to the hologram's normal.

Next, using the transmission in eq. (3.3.13), we denote the exit beams from the

hologram after Illumination by the test beam U as:

U^ ^^(X8,y8;zg) = p U^ f(I^) + a^ p I^ U^ + a^(p)^I^ U^ (4.3.4)

As indicated by the presence of p, the first term in this expression shows the beam propagates along the direction of the original test beam. In contrast, the second term shows a beam propagating coaxially to the original reference beam. The final term shows that this

beam deflects from the mean normal of the test beam by an angle 26. The first and last

beams are ignored in the cross-correlation process; therefore, we drop those beams and

reduce the eq. (4.3.4) to:

u.Cxg^ygS-g) = «i RH^H^t

(4.3.5)

We have identified the proper beam. Also we have absorbed the p term into R^ and recovered

the form shown in eq. (3.3.4).

To complete the cross-correlation process, we use the lens in figure 2 to Fourier

transform U into:

c

UQ(^Q.yQ;zQ) = Iy^ / dx_dy. U (x„,y„) 9^ r'r 9' iXf

* exp [i2TT(ugX8+Vgy8)] (4.3.6)

-29-
This expression can be rewritten with extraneous details suppressed; thus Uq is:

i(n -d )

^ 2

(4.3.7)

where

/ o

Xf^M- 27r 7 o

sine

* N

* ^

Xf-,M_ 27T

/ o

Xf^M_ 2tt

/ o ^A^^S'^S^

* exp |-2Tri[(g^g ) ^8^f7 + (h„hj + u„x_+v_y_] , o /

(4.3.8)

and where

A E Mg(Xf^)^iXfg

(4.3.9)

Using the discussion in subsection 3.3 near eq. (3.3.9), we assume Q,., h^, I, , c„ can all

be neglected. In this case, we can define s^ = a/Xf-,M , o / o

_ 2 2 H(xg,yg) - a^R^ sine [XgSg] sine [YgSg] (4.3.10)

and its Fourier transform as:

K(x ,y ) = 9''9 / dx„dy H(x ,y )exp -2TTi(u x^+v^y ) '9 8 9-' (4.3.11)

We now rewrite H. . as:

^£,j = Wf^9"S9<^^i~ii)' y9-S9(^2~^2-*^ ' (4.3.12)

where we have defined s = f b/(f M^), and made explicit the individual components of the ^ 9 9 7 8 vectors, it and j . With appropriate choice of a Kronecker delta function we simulate the

positions of the spots in the cross-correlation plane (see figure 5) . We rewrite the

electric field pattern as: ^ i(n^-d ) UQ(xQ,yQ) = A J e

-J W [Xo-s„(£,-j,), y„-s„(£,-j^)] 9' g'-'g^

k 1 2 ^:

9 9' 1 -'r ' '9 9' 2

-'2'

h ^2

A I W[x-s k] I 5l_^e

k 1,2 i-2

i(n,-d.) (4.3.13)

-30-
The vectors are defined as x = (xQ,yQ); k = (k^,k^); i = (I ,i„)- and j = (3^,3-;,)' We

define the detected electric field at the k spot as:

'l'-J2^

D (x - k Sq) = a B W(x - k Sg) (4.3.14)

where the B, is defined in component form as:

!l^,3^ «-2'J2 ^ ^ ^ ^

(4.3.15)

The power captured by the detector is;

^k " ^k^k^k

(4.3.16)

where

G^ E / dxgdyg A A* W W*

(Detector)

Note that G depends on a [see eq. (4.3.9)]; therefore, an increase of 10% in a implies an increase of approximately 100% in power received. Further, the detectors have a conversion

term, G„ , that gives the information in volts. We simulate this fact by:

\ = Sk ^k ' (4.3.17)

where we ignore the problems of finite response time.

This completes the generalized discussion of subsection 4.3.1. Subsection 4.6 shows

many possible patterns that reflect the approximations pertinent to eq. (4.3.17).

4.3.2 The Simplified Model that Ignores Details in the Irradiance

the Calibrated Model

We restructure eq. (4.3.15) through (4.3.17) to a more convenient form for the

calibration model. Let:

(4.3.18)

I'O -LX^^)X/rt (4.3.19)

E„

J = e .and

^1, 2 (4.3.20)

1 ' 9

-La/- y K/n (4.3.21)

i

-31-
The voltage at the kth detector is:

where the effective electric field on the kth detector at the cross-correlation plane is:

\= ^- h -k I -k h Z

^- • (4.3.23)

£ £ 1 1' 2

"^2 1' 2

Here A, „ = when either £ or £„ exceed the range of apertures in the prefilter.

Subsection 4.6 shows that this happens for the prototype unit when either |£ | > 2 or

I £„ I ^2. Incidentally, the range of the cross-correlation index has:

|k I < 2 and |k | < 2

for the prototype unit.

To illustrate this calibration model, we assume that instead of a three-by-three

pattern, we have only two apertures in the prefilter. Therefore, we drop the £„, k„ label

and let £ = £ and k = k.. . To obtain a correct count in the notation, we set £ = ± 1 and

k = ± 2, 0. The prefilter has the pattern:

£ = + 1 -1

The cross-correlation pattern is:

k = -2 +2 .

In this case, the electric field at the cross correlation has

e_|_2 = \^_i e ; k = +2 ;

£q = A^E^ e + A_^E_^e ; k = ;

and

i(ct)_.-e ) e_2 = A_^E^e ; k = -2 . (4.3.24)

-32-
The output voltage signal for each detector is:

2 ^2 = S,2 (^^-l)' k = +2 ;

V_2 = G^ _2 (A_-^E^) ; k = -2 ; and

^0 = S,0

(A^E^)^ + (A_^E_^)^

+ 2[A A_ E E_ ] cos Q^

; k = ; (4.3.25)

where the phase term is defined as:

1 ^-1 -1 1 (4.3.26)

To make the calibration process as simple as possible, we define:

(E_^) , F^E (A^) G2^2' h ^S,0/S, 2

'

_ 2 _ 2 S2 = (E^) , F2 = (A_^) G2^_2, T^ = ^2,0'^!, -2 '

A E e - e_ , and 6 = 4)^ - (1)_^. (4.3.27)

With this notation, the voltage equations become:

F,S^ ,

V_2 = F2S2 , and

Vq = F^S2r^ + F2S^r2 + 2 cos (6-9) (V2V_2r^r2)

1/2 (4.3.28)

The S, and S„ are the total power out of each aperture at the prefilter. The above

equations assume that the three detectors respond linearly to the received power in the

cross-correlation plane.

To perform the calibration process, we use a uniform irradiance beam at the prefilter

with a plane wave front. We must vary the relative phase between the two apertures. See

subsection 4.5 for a discussion of how this can be done. Also, we must control the total

power illuminating the prefilter. We measure the power directly at the prefilter with a calibrated power meter. The calibration process has three steps.

1. Block the second aperture and measure the exit power of the first

aperture as well as the response of the three detectors at the

cross-correlation plane. The method requires either a substitution

technique or a beam splitter with monitor in front of the prefilter

and beam steering unit. Results expected are a test of linearity,

dynamic range allowed by linearity, and the deduction of F, and r„F„.

-33-
Block the first aperture and unblock the second. Repeat the above sequence.

Results expected are a test of linearity, dynamic range allowed by linearity,

and the deduction of F„ and F r . 11

3. Unblock both apertures and set the power at midrange first. Then swing

the phase front over a range of phase values. The results here are the

allowed dynamic range of 6 in calibration model, and the deduction of 6. Repeat this process for the lowest and largest values of input power.

Again obtain the dynamic range of 6 in the calibration model and also

deduce the 6.

Given F , F„, T^, r„, and 6, we now can measure unknown beams. The three detectors

give V , V„, and V as a function of time. From this data we deduce the power at each

aperture, S and S„, as well as the relative phase as a function of time. The relationships are:

h = \/h '

^2 " ^2''"^2 ' ^^'^ cosO=V-VFr/F-VFr/F COS ^

^ v^ ^2 1 l' 2 1 2 2' 1 2[V2V2]

(4.3.29)

1/2

where the positive number Q relates to the relative phase by a many-to-one formula. The

sign of Q and the integer n are both unknown and, hence, arbitrary. The formula is:

6 ± Q + 2n7T (4.3.30)

If we scale the three detectors so that F = F^ F and r r„ = 1, then we have a simpler form for the three expressions; namely, S = V /F, S. = V /F, and cos Q = 1/2

(V -V -V )/2(V V~) . We see that phase measurements are most accurate when V^ is near

V2*

4.4 Instructions for Assemblying and Aligning this Apparatus

to Generate the Cross-Correlation Patterns

In this subsection, we presume that an appropriate hologram has been constructed by

the method discussed in subsection 3.4. We now follow the block diagram structure of

figure 2. All units in the object beam behind the beam splitter are the same as in

figure 1. On the other hand, the optics and mirrors that manipulate the reference beam can

be completely removed if new holograms are not needed. In this prototype apparatus, we simply remove the beam splitter to allow the full 7 mW of laser power to strike the

prefilter. The same prefilter, discussed earlier and 'shown in figure 3, acts on this

calibration beam.

To complete the cross-correlation process, we add two lenses in the cross-correlation

plane. Unlike the process for constructing a hologram, the placement of lenses for the cross correlation behind the hologram is not critical. We used a piano convex lens, which

has a 120 mm focal length and a 100 mm diameter, one focal length of optical path length

behind the hologram. The water and glass shorten the apparent distance of travel of the

laser beam, due to the larger indices of refraction. This scale change must be approximately accounted for.

-34-
I

A comment on the placement of the simple convex lenses in the prototype apparatus is

in order. Always place the (most flat) surface (longest radius of curvature) of a lens

facing the wave with the least curvature and make the optical axis go through the center of

the lens. This procedure will minimize the various aberrations of the lens both in the

production of a hologram and in the cross-correlation process. Place the side of the

microscope objectives without the screw thread toward the wavefront with the greatest

curvature—i.e., the apparent point source of focus. This placement minimizes aberrations

of these lenses.'

We complete construction of the cross-correlation unit by placing a 20x microscope

objective with x, y, and z translation mounts so that the lens magnifies and forms a real

image of the cross-correlation plane seen behind the 120 mm lens on the surface of the

detection unit. In this prototype case, a vidicop.-TV detector unit was used because it was

the only system conveniently available within NBS, Boulder, that could detect an array of

spots with laser power less than 10 nW. Subsection 4.6 of this paper shows examples of the

various cross correlations and illustrates the consequences of the cross-correlation process.

The final discussion in this subsection describes alignment of the apparatus when it

is in the cross-correlation configuration.

First, it is unnecessary for the cross-correlation apparatus to have been aligned

previously during construction of a hologram. If a transmission hologram that is computer

generated is used instead of one generated by the method described in subsection 3.4, then

the alignment described previously has not been done. The alignment procedure in subsection 4.4 assumes that the apparatus has no previous alignment history. In the interest

of brevity, those alignments within this section which are already described in subsection 3.4 are simply cross-referenced to that description.

Second, to make the cross-correlation process work with maximum accuracy and

sensitivity, it is necessary (1) to match the transverse dimensions of the Fourier transform

of the prefilter to the transverse dimension of the hologram and (2) to match the spherical

phase structure of the calibration beam to that within the hologram. The latter requirement means that either the hologram must contain the same spherical phase structure or additional lenses must be added to remove any spherical phase-front discrepancy between the

hologram and the Fourier transform unit. Properly matching these items maximizes the

potential dynamic range for the cross-correlation process. How rapidly this range changes

under mismatched conditions is a complex function of the hologram and of the laser beam

being compared with the hologram [see eq (4.3.8)].

Third, we assume the placement of the lens for the Fourier transform for the proper

scaling of the image and proper placement of the prefilter is accomplished, as discussed in

subsection 3.4.

Fourth, we assume the approximate placement of the cross-correlation unit and vidiconTV unit has been done according to subsection 3.4.

Fifth, we assume all subsequent adjustments fine tune the positions of the various

units in the basic cross-correlation apparatus, as shown in figure 2. All the remaining

adjustments make the cross correlation exhibit as strong a response as possible. To accomplish this end, the calibration beam is similar to the object beam recorded in the

hologram. In our apparatus we made a plane wave front with a nearly TEM intensity

profile at the prefilter plane by using the technique discussed in subsection 3.4. Our calibration beam is similar to the object beam.

Given the above beam, each translation of a lens, filter, or hologram is moved

individually to make the strength of the cross-correlation pattern as intense and compact

as possible (see subsection 4.6 for the illustration of the structure of the pattern).

Once all adjustments have been performed, the apparatus is aligned properly and can be

calibrated with a linear detector unit and a sufficiently stable cross-correlation apparatus. In this prototype, the vidicon-TV unit had a nonlinear response to irradiance, and

-35-
the beam expander for the object and calibration beams was not sufficiently stable mechanically to allow accurate calibration. Furthermore, as discussed in subsection 5.2, the

present spacing of sampling holes of the prefilter relative to the spacing of the sampling

holes in the hologram is not optimum for quantitative measurements; therefore, this paper

does not attempt to discuss in any detail a calibration procedure of the prototype. That

process will be discussed in a subsequent paper when an improved apparatus has been con- structed to have the necessary mechanical stability.

4.5 Calibration of this Prototype Apparatus

Although we report no numerical details here, the reader may wish to construct and use an apparatus similar to this prototype. Therefore, this subsection indicates a possible calibration process.

We must calibrate because:

(a) Someone can move a lens, mirror, etc.

(b) The laser source can change or be a different wavelength.

(c) Algae may be growing in the liquid gate.

(d) The theory is not complete.

(e) The various dimensions of the parts are not accurately known, etc.

To calibrate requires a mathematical model showing expected behavior. Try the simple

model defined in subsection 4.3.2. A basic calibration process could be:

(a) Use a cw laser of the same wavelength that you intend to use for your pulse

systems.

(b) Make sure the beam has almost uniform intensity and has its wavefront normal to the prefilter plane over at least two neighboring apertures in the prefilter.

(c) Have some means to translate the above beam so that all neighboring pairs of

apertures can be sequentially illuminated with uniform irradiance and zero relative phase.

(d) Have some means to tilt the above beam of-the normal so that the relative phases

can be changed in a controlled way. Appropriate transverse displacement of the lenses and

filter in the beam expander for the calibration beam can accomplish both the beam tilt and

its transverse translations. For larger dynamical ranges it may be necessary to insert

beam steering devices.

(e) Allow one aperture at a time to radiate. By varying the intensity of the input

and measuring the output at all 25 spots, you can develop the data to fix the A

1' 2 set. Use a small angle wedge (beam splitter) and a calibrated meter to monitor the absolute

power in the incident calibration beam. Use a second calibrated power meter just behind

the prefilter to determine the transfer ratio of power striking the monitor meter and

exiting from each aperture in the prefilter. Confirm that the process is linear under all

the conditions defined in (e) and (f). Capture the entire reflected and transmitted beam

in these measurements of laser power.

(f) Allow each pair of apertures to radiate in turn. By varying both intensities

and relative phases and measuring the corresponding output at all 25 cross-correlation

values, you can develop the necessary data to fix the values and to determine the allowed

dynamic ranges of of A

1' 2

£ £ 1' 2 for the simple model defined in subsection 4.3.2.

(g) Given the above calibration values, the apparatus can now be used to generate 25

time series signals of an incoming pulse. From that data the nine E. . and

1' 2 can be deduced.

£ £ 1' 2

(h) We make two final comments on the calibration process. First, only eight

!>-^,^2 ^^'^ eight 9^ ^ can be fixed; the remaining phases are arbitrarily fixed by some

-36-
convention. Second, the ultimate calibration accuracy depends on knowledge of both the

relative phase and the irradiance applied to a given pair of apertures in the prefilter.

Appropriate design of the control for beam steering can provide the needed information on relative phase. By using least squares and self -consistency in the results, you can deduce the appropriate calibrations and develop absolute ratio measurements.

4.6 The Cross-Correlation Patterns

This subsection contains many figures which show qualitatively the output of the

prototype device as defined in subsection 4.2. Because the beam expander for the calibration beam was dimensionally unstable and the responsivity of the vidicon-TV unit was nonlinear, a quantitative evaluation of the cross-cotrelation process was not useful.

Once the problems discussed in subsections 4.5 and 4.6 are solved during the

construction of a high precision and stable unit, the quantitative evaluation of the

holographic technique becomes appropriate. Even though this paper is qualitative, the

reader can estimate the promise of the holographic technique.

Figure 4 sketches the defined coordinates for the prefilter unit which is a 5 x 5 cm

glass plate with a photographic emulsion. Each hole is square with edge dimensions of

0.539 mm; the distance between holes is 1.001 mm. The accuracy of these dimensions is

± 0.001 mm. These dimensions are chosen to get a maximum signal out at the cross- correlation plane with insignificant overlap of the spots in that plane. Subsequent

figures show that this choice is qualitatively proper. For high precision quantitative

results, this ratio may not be adequate, and the size of the holes may need to be reduced.

Figure 5 sketches the cross-correlation pattern when all nine holes are illuminated

at the prefilter. Figures 17, 18, and 19 show the TV pictures of that pattern. These

figures are discussed later in this subsection. Figure 6 sketches two cases when a single hole in the prefilter is allowed to pass the laser radiation. Figure 7 sketches

what happens when two holes pass the laser radiation.

Given the above sketches, we proceed to actual pictures of the cross-correlation

plane. Figures 8 through 15 show cross correlations for various intensity and phase

relationships. Their captions should be adequate to explain qualitatively what happens.

The fine structure in these spots is a consequence of 'spatial filtering by the hologram

which passed only the center order of the Fourier transfomn for a square hole [11] . If the beam profile apparatus passes all orders accurately and the holes are small enough,

then the intensity profile for each spot in the cross-correlation plane is triangular for

a line scan through the center of a spot. Each TV picture in this report has a bright

line parallel to the vertical axis passing through some of the spots. A graph on the

side of the TV scan shows the intensity profile of those spots along that line scan. The

reader can see that this profile is not triangular. This result is expected, given the

size of the initial hole and the aperturing by the hologram. These results show how

insensitive this apparatus is to defects in the optics and indicate its promise as a precision system for beam profile measurements.

Figure 16 sketches what can happen when all nine beams pass the prefilter with equal

intensity.

-37-
Case a : Only hole at prefilter with i, =-l, JL^^ -\ has laser power

i.1 2 1 0-1-2

D D D

D D D

D D D

Caseb-- Only hole at prefilter with ^^=+1, Jl.^^ ^^ has laser power

2 1 0-1-2

nan

D D D

D n D

Figure 6. Two examples of output at the cross-correlation

plane - one exit beam at prefilter.

-38-
Case c : Two beams pass prefilter, namely ^p " "^ ' -^i ~ "^ °^^ """^

1 -1 -2

-1

-2

D D E D D

D D m D D

D D m D D

Spots at k^=0 has relative phase Information between the two beams

from the prefilter.

Case d Two beams pass prefilter, namely ji^ =-1, ^2 = ~1 j ^i = 0, j^ 2' ^

^'ll 2 1 0-1-2

D D n

D m EB D

D ffl a D

n n D

-2

Here four spots which have k^ = 1 , ; k2=1,0 give relative phase

information between the two beams from the prefilter.

Figure 7. Two examples of output at the cross- correlation

plane - two exit beams at prefilter.

-39-
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M

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1 4 9 4 1

4 16 36 16 4

9 36 81 36 9

4 16 36 16 4

1 4 9 4 1

Figure 16. Final examples of output at the cross-correlation plane.

-48-
Figures 17, 18, and 19 show the TV pictures of the cross-correlation plane under

three conditions. These pictures demonstrate the effects of beam misalignment within the

apparatus and of saturation at the vidicon-TV unit.

Figures 20 and 21 show several views of the apparatus. These pictures give scale to the various parts in the apparatus.

Figure 22 shows the intensity distribution of the object and calibration beams at the prefilter. The ring structure arises from the transverse modes in the laser which

are not filtered by the pinhole in the beam expander. Figure 4 shows where this laser

beam is sampled. The filter was about 3 mm by 3 mm, and the width of the beam was nearly

10 mm in diameter at the half- power points.

-49-
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-55-
5. MODIFYING THE PROTOTYPE APPARATUS TO GENERATE A MORE EFFECTIVE UNIT

This section makes numerous inferences . I discuss new possibilities for beam profile

measurements using holography. Each idea addresses a feature that proved to be a problem

in the prototype apparatus.

5.1 What this Section Presents

Subsection 5.2 notes that doubling the distance between the apertures in the prefilter

permits much simpler measurements. Unlike the prototype where up to nine beams interfere

for a single spot, there are cross-correlation spots where only two beams interact;

therefore, simple real-time and precision measurements become possible.

Previous discussions on the holographic method have not exhausted its capabilities

for beam profile measurements in real time; therefore, subsection 5.3 addresses beam

profile analysis by orthogonal cross-correlation structures. We indicate the concept,

athough an exact technical realization is unknown.

The remaining subsections address briefly two new units, each using the information

generated from lessons learned during development of the prototype.

Subsection 5.4 briefly describes the beam profile measuring apparatus [25] that will

use reflection concepts. Because design of this system is incomplete, this description

addresses only the gross features.

Finally, subsection 5.5 describes how the holographic method can be part of an Instrument that makes selected beam profile measurements. This unit, which can be made

cheaply [26] , should provide the economic means for real-time control of an arbitrary but

unknown laser beam.

5.2 A Scale Change between the Original Hologram and

a New Sampling Plate Makes a Better Instrument

Figure 16 shows that some spots in the cross-correlation plane have contributions

from more than two beams. In fact, the k = k„ = spot involves contributions from all

nine beams of the prefilter. This number of beams causes extensive mathematical complexity

when unfolding the phases and amplitudes in a test beam.

Figure 23(a) shows separating the sample holes to a new 3x3 beam pattern at the

prefilter as indicated by the unblocked beams in the 5x5 prefilter array 23(b) so that one cross-correlation spot has its intensity proportional to the intensity at the prefilter

hole (see spot k = 2 and k„ = 3). A different spot has information about the relative

phase between nearest neighboring holes in the prefilter plane (see spot k = 1 and

k„ = 3). Briefly, the distance between holes in the new 3x3 prefilter is twice the

distance between the holes defining the 3x3 pattern recorded in the hologram. Even

this relationship between the prefilter and hologram fails to eliminate all cross-correlation

spots with more than two beams contributing. Figure 23(c) shows there are spots with

four beams contributing (see spot k = k = 1)

Figure 23(a) shows a very limited array at the prefilter. If that array is extended

to a large array, say 20 x 20, then the cross-correlation plane would expand to 43 x 43

spots with a change in the pattern shown in figure 23(c). For example, location k^ = 3,

k„ = 1 would now have the contribution of two beams, and k, =3, k^ = would now have

the contribution of two beams, and k =4, k„ = would be a spot where only one beam

contributes. Note that the expansion to larger arrays need not be square or rectangular.

The hologram pattern could be many other shapes such as crosses or ribbon structures. A

cross would eliminate the four-beam contribution in the 7x7 array of the cross-correlation

plane.

-56-
5.3 Thoughts about a Cross-correlation System

that Can Make Beam Profile Measurements that Are Orthogonal

The original plan in this evaluation program of the holographic technique was to use

some form of orthogonal mode analysis using an appropriate hologram to measure beam

profile. Unfortunately, there remain extensive technical problems before such holograms

are realized. Consequently, we shifted to the digital sampling technique discussed in

this paper. We used the resulting hologram to demonstrate feasibility of the beam profile

measurements with a hologram as the basic optical processing instrument. In the event

the appropriate hologram eventually can be generated either by computer or by known laser

beam profiles, an orthogonal mode technique can be used. The literature [15] describes a way to realize orthogonal modes using random phase filters. Unfortunately, this method

will produce extreme signal-to-noise problems because too much of the laser power is

scattered by the random phase filter. Consequently, it is our opinion that this method

is unlikely to be used in a measurement system. To attain an orthogonal mode technique

that avoids the unnecessary scattering of the laser power requires a careful restriction

of allowed beam shape and a careful choice of the holograms. This discussion indicates

very briefly what must be done.

The cross-correlation process is written as:

"5 ^^5' ^5^

" du dv B(u,v) H*(u,v) exp[-i2TT(uS + VS2)] , (5.3.1)

where we continue to ignore the reference beam in the hologram and where H*(u,v) represents

the spatial structure in the hologram that will generate the desired cross correlations

with the incident laser beam, b(x , y, ) • B(u,v) is the Fourier transform of the incident

laser beam. Here u, v, S , S„, x , y , etc., are defined in subsection 3.3 and appendix A.

The concept of orthogonal cross correlation can be approached by noting the

following:

(1) First, we assume the hologram profile is represented by:

N _ H*(u,v) =

I D* G* (u,v)e ^^^^^"

, (5.3.2)

£=0

where G (u,v) is a set of complex spatial functions which represent the beam profile; the

la factor causes the cross correlation for the I value to be displaced in the x^ direction

by the distance £a. The D * is a measure of the strength to which the i mode is present

in the hologram. N is the total number of such modes used to represent the beam profile.

(2) Second, we assume the beam profile at the hologram is also represented by:

N

B(u,v) = ^ A G.(u,v) , (5.3.3)

£=0

where A gives the amount of each mode.

(3) The cross-correlation process thus is understood as the consequence of the

interaction between two modes:

^' £ (^' ^2)

-i2TT[u(S + (£'-£) a) + vS^]

du dv G*, (u,v) Gj^(u,v)e . (5.3.4)

-57-
(a) The Fourier transform of the pattern in the hologram filter.

-1 ^t- 1 D D D

D D D

D D D

(b) The 5x5 array in the prefilter with selected transmission pattern

J, /^ -2 -1 1 @ D @ D @

D D D D D

© D @ D @

n D D D D

@ D @ D ©

D Blocked beams

(3) Unblocked beams

(c) The resulting pattern at the cross-correlation plane.

.k. k2V 3

3

2

1

-1

-2

-3

1 -1 D D

D D

D D (§) D (§) D D

D D

D D ® D (D D n

D D

D D

One beam contributes here

D Two beams contribute here

(g) Four beams contribute here

Figure 23. An Improved beam profile measurement scheme.

-58-
For orthogonal decomposition to work properly, the terms in eq (5.3.4) should correspond

to a series of spots in the cross-correlation plane: if £' 4" ^, there is no well-defined

spot; and if £' = I, there is a sharp high intensity spot. Under these circumstances, we

have a method to get orthogonal decomposition of the beam profile. To realize the above

concept requires construction of the appropriate hologram and generation of appropriate

calibration beams to confirm that the decomposition via the hologram gives a set of spots

with the desired orthogonal decomposition.

5.4 Features of the Proposed Reflection System

Reference [25] will detail the mathematics and many properties of this reflection

system. This subsection just briefly indicates the structure of and the expectations for

the unit.

A convenient visual summary of the basic holographic technique for a transmission

system is shown in figure 24. The two lens elements are symbolic summaries of the more

complex optics used in the prototype device and in the proposed reflection system.

Figure 25 shows a block diagram of the reflection system. The system uses reflection

optics so that it can operate at various wavelengths such as 10.6 and 1.06 ym. This unit

has eleven surfaces of interest; each represents a definite stage in Fourier optics.

There is one significant correction to the theory defined in appendix A because the beam

for each optical component is off axis. This correction implies a corresponding modification of the actual optical components and thereby removes the first-order astigmatism.

The low-order aberrations are minimized by using concave mirrors with different, but

constant, radii of curvature for the two transverse coordinates. Variable curvature

mirrors that could further reduce the aberrations prove to be unnecessary in this unit.

This is fortunate because these units can be expensive if we require large aperture optics.

We define each surface and the transfer between surface pairs in the following list.

Surface // Function

1 Here is the basic prefilter as shown in figure 26.

2 Here is the first Fourier optic transform mirror with both radii of

curvature near 2 m. It is used for all wavelengths of interest.

3 Here is the Fourier transform surface of the prefilter plane. It

is placed one focal length, f ,

, from the mirror which is one focal

length from the prefilter. Trie radius of curvature is twice f ^ .

4 Here is the first mirror of a magnifying telescope. This mirror is

changed to reflect the change in the carrier frequency of the laser

pulse. Thus, the 10.6 ym wavelength needs a radius of curvature near

2 m and the 1.06 pm wavelength needs one near 0.2 m. This mirror

change permits a proper match of transverse dimension at the Fourier

plane of surface 3 to the "hologram" of surface 7.

5 This is a mathematical surface placed one focal length, f„, from

surface 4 which is one focal length, f„, from surface 3. For 1.06 ym, the focal length is about 0.1 m.

This mirror, which completes the telescope, has a radius of curvature

near 2 m and is unchanged for all wavelengths of interest, namely 1 to 20 pm. This element is placed one focal length, f„, from surface f

This Fourier plane is similar to the mathematical surface 3. Here,

a reflection hologram causes the beam from a hole in the prefilter

to be difracted into nine diverging beams. The hologram is

structured to generate only these nine beams for each hole. In addition, the hologram causes the beams to generate the desired

pattern shown in figure 27. This element is one focal length,

f_, from surface 6. -59-
0)o - 3

CO

toc

o- i 0) 3- 2^0. CO Q. 1 w.<UMO

_J o

4J

09 mouaB

nJ

(Uo

•H

COcd u

•H

-60-
Surface* 1

Prefilter

Plane

Surface** 2

Surface**4

First

Mirror

of

Magnifying

Telescope

Surface*?

Optical

Computer

"Hologram'

Surface "^lO

Magnification

Mirror

.a^oV^^°^ Fourier

Transform

Mirror

Second

Mirror

of

Magnifying

Telescope

Fourier

Transform

Mirror

Cross

Correlation

Plane

Fif^ure 25. Beam profile measuring apparatus usinc; reflection optics (not to scale)

-61-
o o o o o

o o o o o

o o o o o

o o o o o

5mnn

J-O o o o o

|-*- 5mm —

^1 mm

Figure 26. Beam sampling pattern in the prefilter plane,

-62-
8 This mirror creates the Fourier transform of the beams generated

by the hologram pattern at surface 7. It is spaced one focal

length, f ,, from that surface. The radius of curvature of this

mirror is about 2 m and is unchanged for those wavelengths of

interest.

9 This surface is a cross-correlation plane and is located one focal length, f , , from the mirror at surface 8.

10 Here we have a magnification mirror placed a distance, p, from

surface 9. Its radius of curvature is 2 m for 10.6 ym and 0.2 m

for 1.06 pm. This mirror allows a match of the pattern in

surface 9 to the distribution of the detectors in surface 11. This distance has p=2 m for 10.6 ym and 0.11 for 1.06 ym.

11 The detector pattern is shown in figure 27. We place detectors

at each spot where only one or two beams contribute. The fourbeam spots are not used. The distance from the magnification

mirror to this array is q. For 1.06 pm, q=l.l m, while for

10. 6 ym, q=2 m.

We have seen, briefly, the structure of the reflection unit; therefore, we now can discuss what is expected. There are four key points to note about this device: (1) the

peak irradiance for this device for cw use; (2) the range of phase difference between a pair of sampling holes; (3) the particular dimensions (curvatures, focal lengths, etc.);

and (4) the dynamic range for the carrier frequency permitted by a given physical setup.

Peak irradiance accepted by this reflection unit is fixed by the allowed peak

irradiance for a solid copper surface that is not water cooled and that is required to

2 show no significant change as a beam sampler. This irradiance is less than 100 kW/cm for

a cw signal of arbitrary time period. We assume the inout beam is a plane wave with

uniform irradiance at the prefilter and set the irradiance at each sampling hole at I^ 2 200 W/cm . The peak irradiance at surface 7 for that single hole is related to the

incident irradiance as:

o

f o 'h\

(5.4.1)

where f is the focal length of mirror 1, and M^ = t lf.„ is the magnification of the

telescope. For the nominal values given previously, the peak irradiance is I_ = 0.0055 I =1.1 W/cm for the two setups at either 10.6 or 1.06 ym. If we assume the acceptance

diameter for radiation at the prefilter is 10 cm, we find about 300 sampling holes within

this diameter. For a plane wave, the peak irradiance at surface 7 can be increased as the

2 2 square of the number of irradiated holes. Thus we can get I = N I, = (300) 1.1 =

2 P f 99kW/cm , which is near the allowed peak irradiance.

If we wish to look at cw power beyond this range, three adjustments to the setup are

possible. Make the sampling hole size, a, much smaller; make the effective focal length,

Mf , much larger; and make I smaller. The first change is controlled by the practical

size of the apparatus. The last change is accomplished by using beam splitters in front

of and/or behind the prefilter. These beam splitters should be gratings [27], if we want

reflection optics. Using a grating between the prefilter and the first mirror allows us

to use one of the higher diffraction orders as the collection of sampled beams. The

nonideal features of these gratings are accounted for by appropriate calibration as suggested in subsection 4.4. The remaining issues for extremely high power are discussed in

reference [25] .

-63-
o @ o o @ o

OXOXOXOXO

O @ O @ O O

OXOXOXOXO

O @ O @ O @ O

OXOXOXOXO

O @ O O © O —

2.5

^O

mm

XOXOXOXO

_Jl_ O @ O

^-—2.5mm

O @ O

X Four Beams contribute to ttiis intensity

Q Two Beanns contribute to this intensity

pattern. ( ptiase tiere ) One Beam produces this intensity.

igure 27. Resulting spot pattern in the cross-correlation plane,

-64-
Phase Shift

The allowed dynamic range of phase shift between neighboring holes is controlled by

the desired accuracy of the phase shift measurement. If we accept the nominal restriction

of ± 0.1 wavelengths as chosen in this paper, then we have the following estimates for the

dynamic range of the phase shift. With a designated spatial frequency as b/(27T), the

phase change across a single hole equals ba. The inferred phase shift between holes,

assuming a constant gradient, equals cb. Since the hole size is a = 1 mm, and the spacing

distance is c = 5 mm, the latter phase term is 5 times the former. This means a requirement of ± 0.1 wavelengths or less implies a change of ± 0.5 wavelengths or less between

neighboring sampling holes. If we accept that the phase front need not exhibit a constant

gradient, then the allowed phase change between neighboring holes can be many wavelengths.

There is one penalty when this latter condition is present: the actual phase front is

undetermined by the 2tt modulus of the phase (see the calibration discussion in subsection 4.4

on this point)

.

Dimensions

The dimensions of this proposed reflection apparatus are somewhat arbitrary. A beam

diameter of 10 cm was chosen because it represents a common beam size for many optical

setups using laser beams of high power. The sampling frequency represents a compromise

between the aperture of a detector versus that in the prefilter. Using a one-meter focal

length means that the apparatus can be contained on a 2 x 4 m holographic table. This

scale is consistent with commercial equipment. The remaining dimensions, not detailed

here, minimize the effects of various apertures and allow the mirrors and other units to be as small as possible so that their costs are not extravagent. (For example, a 20 cm (8

inch) mirror with one-meter focal length can cost $1,000. We need five of these plus

other equipment.) Reference [25] details the costs for this proposed system.

Carrier Frequency Range

To show briefly the allowed dynamic range of the carrier frequency (wavelength) for

a given physical setup requires some mathematics and skipping of many details. The theory

in reference [25] that accounts for the astigmatism and other such details shows the key

derivations. We show the final results without the astigmatism using Gaussian beams.

The output electric field at the cross-correlation plane for a single beam from the

prefilter and a selected diffraction beam from the hologram is:

"(S'^2'VV =-Sr^°(VV Mt (xV)(h^+ 1)

IX I \

[x + Mc(£^ - 2^ )] + [y + Mc(!l2 " 2a~)

1

'' o o exp ^-^r^ (5.4.2)

The total magnification factor is M = f,f„q/(f f p); the imaging magnification factor is M = q/p; the x coordinate for the fc th row of holes in the prefilter is )i,c; the y coordinate for the Jl„th row of holes in the prefilter is £ c; and A (£ ,2. ) is the electric

field strength at the {I ,9 ) hole. The order of the diffraction process at the hologram

is f, = ± 1,0 for the x coordinate and I = ± 1,0 for the y coordinate. The half-width of

X y the electric field at the prefilter is o ; the distance between the sampling holes is c; and A is the nominal design wavelength for the hologram. The remaining symbols are shown

in figure 25. Let X = A + A in the subsequent discussion on the dynamics of the allowed

range of carrier frequency.

-65-
A detector in the cross-correlation plane at a point where 1=1=0 has only one beam contributing to its irradiance. In this case, the irradiance is:

M

exp

M a -

(x+Mc«, )^ + (y+Mci,)^ '• '2^ /

(5.4.3)

If we presume the detector captures all the above irradiance, the power at this detector

is:

^^-^,^2^ ^\A ^^Y^^Y'^l) D(0,0)][A*(£^,£2) D*(0,0)] . (5.4.4)

A detector at a point where £„ = fc =0 and l^ = £ + 1 with £ = -1 has two beams ^

2 y 1 X contributing to make up the received power as:

P(£ + ^.0) = P(^,0) + P(£+1,0) + 2e ^(P(«.,0) * P(£+1,0) )"'

^ cos Q (5.4.5)

Here Q contains the relative phase. The mismatch from a designed wavelength X is shown

by the F term as:

2 2 McA

v2A ,

cA

a, A 1 o (5.4.6)

If we presume F <_ 0.1 for the carrier frequency shift to be insignificant for beam profile

measurements of moderate accuracy, we find, given c = \ fractional dynamic range for the carrier frequency is:measurements of moderate accuracy, we find, given c = 5 mm and a = 1 mm, that the allowed

< 0.009 (5.4.7)

o'

The above illustrates how a carrier frequency shift affects a single pair of beams. The

effect is similar for other pairs of beams. Clearly, for precise measurements, the carrier

frequency should be measured simultaneously with the output from all detectors and used to unfold the time dynamics of the relative phase between various sampled beams. The above

applies to both the transmission and reflection systems.

5.5 A Beam Profile System that Can Be Mass Produced

Reference [26] will detail this instrument which could be used in a feedback system

for real-time control of a laser beam profile. Possible uses of this unit are in conjunction with laser sources in communication, welding, and high energy laser beam units. This

subsection indicated the key concepts without the detailed mathematics. To make this unit

useful, simplifications of the basic cross-correlation process are made. These changes

make this unit both cheaper and more compact than the reflection system described in

subsection 5.4 and the transmission unit defined by this paper. Of course, these changes

make this unit less accurate and more restrictive in dynamic range for understandable

output from the array of detectors at the crosscorrelation plane.

-66-
In figure 24, the essential holographic method shows five elements: the prefilter,

the first lens or mirror for a Fourier transform, the hologram or filter for cross correlation, the second lens or mirror for Fourier transform, and the array of detectors in the

cross-correlation plane. The holographic devices described previously in this paper

require a total path length greater than 2 f + 2 f or about 4 meters with a probable

design set where f = f = 1 m. Clearly, this length is too large for convenient use in

field conditions. Also, the number of distinct elements is significant; therefore, it is

important to eliminate those that are unnecessary for the desired field instrument. If we

degrade the general flexibility in beam profile measurement, then we can make significant

simplifications to the unit shown in figure 24.

First, we eliminate the prefilter and the first lens. We directly illuminate the

hologram with the beam. If the beam profile has some appropriate constraints, such as that

the center of its peak should overlap the center of the hologram or the gradient of phase

and intensity is small compared to the sampling frequency of the hologram, then the output

of the detector at the cross-correlation plane may be used for real-time control of the

beam profile. The elimination of the first lens reduces the unit's length by two focal

lengths or two meters if f, = 1 m.

Second, we design an appropriate hologram and detector unit to accomodate beam inputs

that do not have precise beam centering capability. The details of this hologram are technically complex. I indicate only the concepts. (They may change when the design is

developed.)

(1) We subdivide the hologram into squares about 1 mm by 1 mm in size. We presume

that the change in the irradiance and phase from square to neighboring squares is less than

5% for irradiance and less than 0.1 rad for the phase.

(2) We make a picket fence grating with 25% modulation and with the base parallel to the X axis in each square. This grating acts as a sampler and multiple beam generator.

-2 -1

Each pattern for all squares has the same spatial frequency, perhaps 10 (mm) . Each

square has a different position in the y direction where an edge of a modulation peak

crosses the square. This shift of location induces an additional phase shift in the beams

generated by this grating. Having the same spatial frequency for each square ensures that

all beams of a given diffraction order exit parallel to those in the other squares. These

beams are deflected in the ± y direction. This process allows phase analysis of the beam

profile.

(3) To get amplitude analysis of the beam profile, we need further filtering of the

incident laser beam. To make the construction of a hologram cheap and simple, we may use

either a second picket fence or a variable width slit. This pattern is parallel to the y axis and perpendicular to the x axis. Since we are interested only in the zero-order beam

of deflection from this grating structure, we can change the spatial frequency, the

modulation level, or the slit width of this pattern for each square. These changes

accurately control the amount of power in the nondif f racted beam. The diffracted beams are

deflected in the ± x direction.

Third, to minimize the size of the device, we mount the hologram either on a concave

mirror for a reflection system or on the front flat surface of a plano-convex lens for the

transmission system. This arrangement means that the first focal length distance is

eliminated and only the second focal length remains. The choice of appropriate focal

length depends critically on the beam profiles being measured and controlled by this unit;

therefore, we have a fundamental length limit of f^. The lens or mirror causes the beams

of a given diffraction order to be focused on a unique spot. This spot has both relative

phase and amplitude information on the entire beam profile cross correlated with the

hologram. Each spot has a different cross correlation; therefore, it is possible to realize some version of the orthogonal analysis mentioned in subsection 5.2.

Fourth, to complete the unit, a fixed detector at each spot with an appropriate

capture cross section generates a voltage or current signal proportional to the laser power

-67-
received in the spot. The number of these detectors would be about 10 or less. The object

here is to have a minimum number of orthogonal modes in the basic beam profile.

I conclude this subsection by noting that the hologram can be tailored to selected

classes of beam profiles and clearly can be generated by using a step and repeat camera. The mass production feature is possible by replicating a master hologram with the above

information stored appropriately.

6. CONCLUSIONS DRAWN FROM THE PROTOTYPE APPARATUS

As can be expected in the construction of a prototype device, problems arise such as:

(1) Due to the high absorption in the hologram, the diffraction efficiency as measured

by the ratio of irradiance between the incident sampled laser beam and that at the final

correlation plane, equals 0.1%.

(2) The mechanical instability of the translation mounts in the beam expander for the

object beam destroys calibration stability.

(3) The nonlinearity in response to the irradiance in the vidicon unit even when it

is not saturated, destroys linearity.

(4) The fact that the apparatus must be restructured for use at each new laser

wavelength implies limited dynamic range of carrier wavelength.

(5) The "spurious" beams from each reflection and refraction reduce the available

energy for observing the cross-correlation process.

(6) The optimum hole size in the prefilter and the spacing of these holes are unknown.

This section addresses each problem as numbered and suggests a solution. To prove

that each suggestion is adequate requires construction of an appropriate unit incorporating the suggested solution.

(1) This 0.1% conversion efficiency is probably the most serious problem preventing

simple application of this hologram technique. To increase this efficiency to 10% and

better and to permit the unit to be used at 10.6 pm require construction of reflection

holograms. There are three types that can be generated, namely kinoform [11], blazed [11],

and binary [28] holograms. The former two have potential of 80-90% efficiencies; however,

they require many stages of fabrication to obtain the proper etching of the reflection

surface. The binary hologram requires only two stages of fabrication and achieves 10%

efficiency. The technical trade-offs here are the number of fabrication steps, the

complexity of the computer programs, and the resulting efficiency of the reflection

hologram. The description of the reflection unit [25] details this trade-off. If the

wavelength of the laser is in the near IR and visible, then phase holograms can permit

transmission holographic techniques near 10% efficiency.

(2) The instability of the beam expander can easily be cured by using translation

equipment that does not creep and that can be accurately reset; therefore, no further

discussion is necessary.

(3) The nonlinearity of the vidicon unit can be cured only by replacing this unit

with a different set of detectors. Four types of detectors exist which have sufficient

sensitivity and signal-to-noise ratios. These detectors can respond linearly at 1 pW to

1 nW power levels with electronic noise levels less than 1% of the input signal and can be

constructed in an array, one detector for each spot. The output signal of each detector

filters the spatial dependence within each spot of the cross-correlation plane to give a single output. For visible wavelengths, silicon detectors have sufficient linearity and

responsivity. Commercial literature suggests that, for the near infrared (1-3 ym) and for

far infrared (3-30 pm) , the Lead-Tin-Telluride detectors have sufficient linearity and

responsivity. The pyroelectric detector can be used for all laser wavelengths provided

that the acoustical noise in the background can be eliminated or made unimportant.

Finally, over the UV and visible wavelengths, an array of photomultiplier or silicon

-68-
elements can be used for the detectors. Each wavelength range has detectors which do not

necessarily exist as arrays; therefore, technical problems must be solved before these

detectors are available for use in a hologram unit. Also, reviewing the available com- mercial units shows that the impulse response for these detectors varies radically from

group to group; hence, each must be chosen carefully to provide desired time response. In brief, the choice of detector is sensitive to wavelength, power level, and signal shape of

the laser pulse.

Note: The vidicon, as presently constructed, provides convenient qualitative results.

This unit allows critical alignment of the elements in the beam profile apparatus. If the

nonlinearity, apparently present in the response, could be removed by some appropriate

electronics or change in detectors and if the total spatial response at a given spot could

be integrated by appropriate electronics or computer analysis, then the vidicon could

economically be substituted for the detector array in a beam profile apparatus. If the

desired time response for the beam profile apparatus is faster than 1/60 second, then

current TV framing will not be adequate and will have to be substantially modified.

Parallel processing becomes necessary.

(4) There are two effects in the beam profile unit which modify the precision of the

unit as the wavelength of the laser beam changes.

If the beam profile apparatus has refractive optics, then there is a wavelength

dependence in the various indeces of refraction of the optics. This wavelength dependence

can be completely avoided by using reflective optics (see subsection 5.4 for an example).

An effect remains because the transverse dimension of the Fourier transform at the

hologram is wavelength dependent. One method to eliminate this effect requires changing

either the hologram or the Fourier transform lens to make the appropriate scale change for

each wavelength. This approach minimizes the number of reflective surfaces and simplifies

the number of optical components. However, a reflection hologram with appropriate hills

and valleys in the surface is not commercially available. Development of such a hologram

will be expensive. The change in the Fourier transform lens can preserve the convenient

dimensions of the apparatus.

A second method to compensate for wavelength dependence would keep the same hologram

for some wavelength ranges and would keep the apparatus a fixed size by using appropriate

telescopes in front of and behind the holograms to scale the Fourier transform and cross

correlation. The curvature of the phase front produced by the telescope in front of the

hologram must match the curvature of the phase front recorded in the hologram; otherwise,

there will be a significant Imprecision at the resulting spots of the cross-correlation

plane. The telescope behind the hologram need not match the curvature of the phase front

for all wavelengths. To realize these constraints requires appropriate design of the

telescopes which correspondingly requires appropriate use of the lens equation as discussed

in appendix A.

(5) To minimize the laser power lost from each surface requires using either antireflecting coatings on the optics or high quality reflective optics. The former increases

the wavelength sensitivity of the device; therefore, coatings should be used only if

refractive optics are the most economic for the apparatus and if a single wavelength is

appropriate.

(6) The hole size, number, and pattern in the prefilter determine the final precision,

the laser power density and heating of the hologram, and the cost of the beam profile

apparatus. The larger these holes are, the greater the received power is at the cross- correlation plane and, hence, the simpler the detector units can be. Unfortunately, with

increasing hole size the apparatus becomes more sensitive to additional details in the

relative position of each hole as well as to the intensity and phase variation across each

hole. For example, figure 18 shows the secondary spots generated because the hole size in

the prefilter is slightly too large. The more holes in the prefilter apparatus, the

greater is the resolution of the beam profile; however, this fact implies a corresponding

substantial increase in the size and cost of the detector array. The above effects can complicate the mathematical model needed to deduce the phase and amplitude in the beam

profile measurement. There are no simple criteria known that can fix the size, number,

-69-
and pattern of the holes; consequently, this report gives no solution to this problem.

However, if a measurement problem has a well-defined class of beam profiles, then the

choice of sampling size, etc., can be chosen appropriately. 3

If the problems found in this prototype apparatus can be considered as soluble, then

the concepts discussed in this report can be applied to measure, in real time, the amplitude and phase of a laser beam at a prescribed prefilter plane. For those interested in an integrated measurement of a laser pulse, see reference [29]

In conclusion, by proper construction of the hologram and by using appropriate types

of prefilters and optics, it appears possible to develop a device that can measure accu- rately changes in amplitude and phase of a laser beam profile for a particular wavelength

in times shorter than 1 ns. The detailed comparison of the relative value of the Hartmann

plate versus the holographic method for these speeds is discussed in [25]

7 . ACKNOWLEDGMENTS

The author wishes to thank the following persons who provided him with appropriate

advice: Mr. Aaron Sanders, Dr. Richard Smith, and Dr. Matt Young at NBS, Boulder. Further,

the course on "Coherent Optics" by Professor E. N. Leith and staff at the University of

Michigan helped the author develop "hands on" experience in Fourier optics and holography.

The author wishes to thank the following individuals for the fine work on typing and

setting up this paper: Sheila R. Aaker, Margaret L. Woolley, and Arlene Miller.

The author also wishes to thank Phyllis O'Rourke for her helpful editorial advice.

-70-
8. REFERENCES

[I] Yoshida, A., and Asakura, T., A simple technique for quickly measuring the spot size

of Gaussian laser beams. Optics and Laser Technology, 273 (Dec. 1976).

[2] Saunders, J. B. , J. Res. Nat. Bur. Stand. (U.S.) 65B, 239 (1961).

[3] Day, G. W. , and Stubenrauch, C. F., The focal plane transform technique for obtaining

far-field laser beam profiles, NBS Technical Report #1001 (1977).

[4] Angelbeck, A. W., and McClug, W. C, Laser wavefront analyzer using sliding reference

interferometry, NTIS Code #AD-922 980, Report (1974).

[5] Skolnick, M. L., and Gowrinathan, S., Laser wavefront analyzer using frequency offset

interferometry, NTIS Code #AD-922 980, Report (1974).

[6] Gillespie, R. W. , Wick, R. V., and Saxman, A. C, Laser wavefront profiling technique

for pulse radiation, NTIS Code #AD-A011 944/6ST Report (1975).

[7] Pfeffer, M. K. , and Loomis, J. S., Interf erometric measurements of windows on an environmental chamber. Third Conference on High Power Infrared Laser Window Material,

III : Surface, Coatings, AFCRL-TR-74-0085 (111) Special Report #174, p. 1191 (1973).

[8] Wick, R. v., Saxman, A. C, and Blankinship, E. A., Real time intensity profiling and

beam divergence measurement of a laser beam, Proc. Technical Program Electro-Optical

System Design Conference, p. 294, (1973).

[9] Block, W. , Nanosecond response time, room temperature infrared detection with thin

film metal. University of Illinois at Urbana, University Microfilms International, Ann

Arbor, Michigan (1973).

[10] Skipper, J. D., Goranson, R. W. , and Schaub, C. M., Application manual - Position

sensing detector for the far infrared, Dynalectron Corporation, P. 0. Box 18068,

Kirtland Air Force Base Station, Albuquerque, New Mexico 87115 (1975).

[II] Cathey, W. T. , Optical Information Processing and Holography (John Wiley & Sons, New

York, 1974).

[12] Young, M. , Optics and Lasers, An Engineering Physics Approach (Springer-Verlag, 1977).

A useful summary of various aspects of optics and lasers,

[13] Vander Lugt, Anthony, Coherent optical processing, Proc. IEEE 62^, 1300 (Oct. 1974).

[14] Goodman, J. W., Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

[15] Lee, Sing H. , Review of coherent optical processing. Applied Physics 1^, 203 (1976)

.

[16] Leith, Emmet N., Quasi-holographic techniques in the microwave region, Proc. IEEE 59 , 1305 (Sept. 1971).

[17] Gaskill, J. D., Optics for the electronics engineer, Electro-Optical System Design, 22 for part I (Feb. 1977), 28 for part II (June 1977), 43 for part III (June 1977).

[18] Koglnik, H. , and Li, T. , Laser beam and resonators, Proc. IEEE 54, 1312 (Oct. 1966).

[19] Ditchburn, R. W. , Light, Interscience (New York, pp. 371-372, 1958).

[20] Pennington, K. S., and Harper, J. S., Techniques for producing low-noise, improved

efficiency holograms. Applied Optics % 1643 (July 1970).

[21] Newport Research Corporation, Short Form Catalog, Film Selection Guide, 29. We

arbitrarily chose Holographic plate Type 120-02 by Kodak.

II

-71-
[22] Kodak Pamphlet No. P-311, Describes how to develop the Type 120 plate for transmission

holograms.

[23] Upatnieks, Juris, Coherent optics laboratory note. Short course at the University of

Michigan (1976).

[24] A private communication with Dr. Stephen Benton of the Polaroid Corporation helped me decide what photographic procedures to follow among the numerous methods in the

literature.

[25] Johnson, E. G. , Jr., Design of the reflection apparatus for beam profile measurements

(to be published as an NBS Technical Note)

[26] Johnson, E. G., Jr., Design of a field instrument for beam profile measurements (to be

published as a NBSIR)

[27] O'Neil, R. W. , Kleiman, H., Marquet, L. C, Kilcline, C. W. , and Northam, D., Beam

diagnostics for high energy pulsed C0„ laser, Applied Optics L3, 314 (Feb. 1974).

[28] Gallagher, N. , Jr., Angus, J. C, Cof field, F. E., Edwards, R. V., and Mann, J. A.,

Jr., Binary phase digital reflection holograms: Fabrication and potential applications. Applied Optics 1^, 413 (Feb. 1977).

[29] Miyamoto, T. and Yasmura, K., Measurement of the beam parameters of a laser beam and

its diffraction field, using a hologram. Applied Optics _10, 161 (Jan. 1971).

[30] Born, Max and Wolf, Emil, Principles of Optics (Pergamon Press Inc., New York, Third

ed., 1965).

[31] Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables

(Edited by Milton Abramowitz and Irene A. Stogen, National Bureau of Standards,

Applied Mathematics Series 55, pp. 229-233, for the si(z) and Ci(z) functions).

-72-
APPENDIX A. IMPORTANT FORMULAS FOR FOURIER OPTICS

We follow the conventions and definitions given in reference [11] to discuss three

formulas of interest: the Fourier transform, the imaging formula, and the effects of

displacement formula.

A. 1 The Fourier Transform

U^(x,y) is a scalar of the electric field where all polarization effects are ignored.

The fundamental formula using the approximations of Fourier optics for wave propagation

over a distance z is:

ikz

U2(x2,y2) = yIz ll'^^l '^^l ^l^^l' ^1-* h(x2-x^, y2-yi; 2) , (A.l)

where

(a) X is the wavelength of the radiation;

(b) x, , y, are the transverse coordinates of a plane for the initial

surface "1";

(c) x„, y„ are the transverse coordinates of a second plane for the

final surface "2" which is parallel to surface "1";

Ztt (d) k = T— is the wave number; and

A (e) h(x,y; z) = exp f (x^ + y^) is the Fourier optics approximation

for the transfer function between the surfaces "1" and "2." (A. 2)

The first formula of interest using a simple lens arises when surface "3" acts as the

Fourier transform of surface "1." A convex lens of focal length f, as shown in figure

A.l, generates this transform. The formula, which ignores aperturing by the lens and uses

the distances d^ and d^, as shown in figure A.l, is:

"3 ^^3' ^3^ = 1x1^ ^[^3' ^^3' d2/(l-w/d2)]

j dx^ dy-j^ U-|^(x^, y^^) h[x^, y^; d^/ (1-w/dj^) ]

*-P [ -^72 ^^1^3 + ^'l ^3)^ ' (^-3)

where — = — + — - — and where the exp [i(kz + ()>)] factor is suppressed as irrelevant.

The 1^ represents the phase shift due to the presence of a lens. Here z = d + d„.

The simplest Fourier transform case has d„ = f or w = d, as well as d^ = f. In this case:

\^^(x^,y^) = -Tj^ A^(u,v) , (A. 4)

-73-
(x.,yi) (Xp,y9) 2'J'2 (x3,yJ

ri

Figure A.l Relationships betxveen surface "1," "2 " and M oil ir ... -

' i tor a simple lens.

-74-

i
where the spatial frequencies u and v are given as:

^3 , _ ^3

u E —, and V = — ,

and where the Fourier transform is defined as:

r Aj^(u,v) - dx^ dy^ U^(x^,y^)exp[-i2TT(ux^+vy^)] . (A. 5)

A. 2 The Imaging Formula

The second lens formula uses a lens to form a scaled image of the radiation pattern

at surface "1." Here the distances between surface "3" and "1" satisfy:

— + — = —. (Figure A. 1 still applies for definitions.) (A. 6)

^1 "^2

The simplest image formulation assumes no aperturing effects by the lens; therefore,

an image law for Fourier optics is:

X y

U3(X3, y3) = Ih[x3, y3; d^/d+l/M)] U^( - ^, - ^)

where M = d„/d-. and again the exp [i(kz + 4))] phase factor has been suppressed. M is the

magnification of the image at surface "3" relative to surface "1."

Further, note that the radius of curvature of the phase front for the radiation at surface "3" is increased from the radius of curvature for the radiation at surface "1" by

the phase factor in h(x , y ; R) where R = d„/(l + 1/M) is an effective radius of curvature.

This phase curvature has very important consequences in the design of a beam profile

apparatus.

A. 3 The Displacement Formulas

To identify quantitatively the location of the correct Fourier plane requires knowledge

of the behavior of a Fourier transform under transverse displacement of the original

radiation pattern at surface "1." We show these effects for the Fourier transform in one

dimension, which is:

G(u) = dx g(x) exp [-i2T:xu] , (A. 8)

where g(x) represents the electric field distribution showing one spatial dimension exiting

from surface "1." If the radiation pattern moves by a distance a along the x-axis and the

phase changes by c, the electric field pattern g(x) becomes g(x-a) exp[icx]. This new

form for the field has the following Fourier transform at surface "3":

-75-
G[u-c/2Tr] exp [-iZirau + iac] . (A. 9)

This formula demonstrates that a rigid transverse displacement of a spatial field pattern

(for example, the two pinholes discussed in the text) shows no change in the position and

shape of the intensity distribution of the transform of that field pattern. However,

shifting the phase by c causes a corresponding shift in position of the Fourier transform

by c/27r. One can cause this change by bending the angle with which the incident beam

illuminates the original pattern, such as the true pinholes at surface "1."

APPENDIX B. THE DIFFRACTION DETAILS OF A SQUARE HOLE

Making proper beam profile measurements by either the mirror scan, the Hartmann plate,

or the holographic method depends on understanding and using appropriate diffraction

patterns from combinations of selected holes. There are two shapes that demonstrate the

necessary details and are easily realizable, namely, the square and circular holes [30]

Appendix C details the key features for the circular hole and this appendix details those

for the square hole.

To understand the diffraction pattern in detail for a single square hole, we represent

this hole as an electric field U (x-.,y ).

Here:

U^(x^,y^) = A^ * n[(x^-g^)/a]n[(y^-hp/a]

* exp 'li[b^(x^-g^) + c^(y^-h^) + d^] , (B.l)

where

A is a convenient reference field (it has the dimensions of

°

2 1/2 [watts/ (mm) ] ); g is the coordinate for the center of the £th hole along the x axis;

h is the coordinate for the center of the lith hole along the y axis;

Si is the label for the fcth hole;

b = b + ib„ is a complex number characterizing the average gradient

of the phase front and intensity profile along the x axis over the

ilth hole; (here b is the real part of b and b^ is the imaginary

part);

c = C-. + ic„ is a complex number characterizing the average gradient of

the phase front and intensity profile along the y.. axis over the

£th hole;

dj, = d

J, + ^^0 1 ^^ ^ complex number that characterizes the average phase and

intensity at the center of the £th hole; and finally, n[u]

is defined to be:

-76-
n[u] =1 if

= if

= 1/2 if

|u| < 1/2 ,

|u| > 1/2 , and

lul = 1/2.

(B.2)

B.l What this Appendix Presents

For efficient discussion in section 2, the subsection B.2 derives the conditions as well as the form for the far field of an arbitrary square aperture driven by the field

pattern in eq (B.l). In addition, the total power received by an adjustable aperture at the far field is detailed here to show the reader how large this aperture must be to capture a significant fraction of the laser power. Because this far-field pattern is used

differently by each method for beam profile measurements, corresponding changes in the

convenience, precision, and dynamic range for each technique result. Understanding this

far-field pattern provides the data base for discussion in section 2 for each of those

changes induced by each method.

To utilize correctly the Fourier optics and the aperturing process, the Fourier

transform of a given electric field must be known. Therefore, this subsection derives the

explicit formulas of this Fourier transform pattern for a square hole aperture with the

illumination represented by eq (B.l). The results are almost identical to those obtained

in the far-field case.

B.2 The Far-field Diffraction Pattern

Using the assumed form U in eq (B.l) for U , eq (A.l) can be rewritten as:

ikz a _1_ I dx. f + §£

dy^ e

exp

ik , 2 2 27 (^2"'y2) exp

ik , 2, 2,

27 (^1+^1^ (B.3)

If we define x^ =

'^i'''8£' ^i =

'^l''''^£' ^2 ~ ^2^^l' ^^"^ ^2 "" ^l^^i' ^^^^ ^^ ^^'^^ can be rewritten as:

ikz

"2^^2'^2^ = fxT -a

2

du.

a.2-a

2

dv, e A

1 o

exp

il^ / 2. 2> ik , ^ , 2l ^"2"^2) - ~ ("lU^+v^v^) exp

ik , 2, 2. 2i ("l+^l^

(B.4)

The far-field condition for the beam coming out of the ilth aperture occurs when the largest

1^1 I ~ l"^-| I ~ y » and the distance z is such that:

k_

2z (f]'+ (f)' 10

2.5 (B.5)

-77-
This approximation presumes we can neglect l/20th of wavelength changes in the phase

front. This is the far-field criterion for this entire paper. If this criterion is

satisfied, then eq (B.4) can have the second exp term in the bracket, { }, set equal to one, and the integral becomes:

l!^(u^,v^)

ikz

e exp

ik, 2 2 27^'^2+^2)

id

a A

Z

2-n sineXz 2tt (B.6)

TT *-u £ 4-- • / \ - sin(TTu) . - , Here, the function sinc(u) = —— = sinc(-u).

TTU :b.7)

If we define

^2-g£ \l

Xz 277

I^ E A

f o

Xz 2tt

2 -2d

Xz 21

n2 = M[2-n j

r^^i

[277 J a , and

(B.8)

then the intensity pattern for the far field can be written as:

I[?-L>n-|^] = Ij sinc[C-^+iC2]sinc[?^-iC2] * sine [n-j^+iTi2] sine [n-[^-in2] (B.9)

From this pattern we can determine the following: (1) The irradiance pattern is

sjmmietric about either inversion C-, ^ -C-, or n-, ^ -Ho or both. (2) The change in either

5^ -^ -t,„ or T\„ -+ -rip causes no change in the irradiance pattern; therefore, the far field

has no information about the sign of the gradient of irradiance at the square hole aperture.

(3) The coordinates for peak irradiance have l, = t] = 0. If ?„ and t] are small enough,

the peak irradiance is:

1(0,0) = I^[l + ^ r^\\^ fl + i" ^2^^ (B.IO)

And finally (4) the points of minimum irradiance have r, = n and ri = n„, where n^ and n are nonzero integers. These irradianees are approximately:

I(n^,n2)

2 2 If ?2 ^^2 2 2 2 2 [n^+C^] [n2+n2]

(B.U)

Note that if 5^ and n^ are both zero, these minimum irradianees are lines defined by

C-, = n with n-, arbitrary and by n-, = n with ? arbitrary. Again n and n„ are arbitrary

nonzero integers.

-78-
Knowing the power captured by a finite aperture detector at the far field defines the

allowed procedure for measurement of this irradiance pattern. Thus, the power captured by

a square aperture detector with edge at A/2 and centered at C = n-, = for the i aperture

is:

P. = du. dv2 l(c,^,ri-^) (B.12)

We define:

H(A^,?2)

^-A,

d?^ sine [c^+i^^] sine [C-|^-iC2] . (B.13)

with

, =Aa_

1

~ 2Xz

(B.14)

Thus, the power measured is:

?^ = I^ H[A^,?2] H[A^,C2] (B.15)

To get a convenient form for H in terms of basic functions that can be generated easily

on a computer, we manipulate H. First, we note that H[A,5] = H[A,-c] = H[A,]c|]. Hence,

with r,>0 for the remaining discussion:

H[A, k ] = A

dCn -A

[cosh(2TT?) - cos(2iT5 )]

2^2 [^2 + ^2^

1 . ,„ , 1 ^ -1 A —T- coshlZTrr) — tan — 1 A cos(2-rr^ ) dC. -A

1 2^2 ^1 "^ ?

(B.16)

H can be further changed by noting the integral in eq (B.16) equals — e - 2g(A,c).

Here:

g(A,C) =

•°° cos 2-nr,n

, ^^1 2 ^ 2

• A

?i + ^

(B.17)

By shifting the range of integration in g:

g(A,0 =

dx cos(2Trx)

dx[cos2TT(x+A)]

(x+A)^ + C^

cos(2ttA)

' o (x+A)''2,2+ C ~ sin(2iTA)

dx sin(2TTx)

(x+A)2 + C^ (B.18)

-79-
These two Integrals, which can be found in Fourier transform tables, imply:

Gnn'h/'97r7'^ fen ( 'y \A-ci-\ ( ^. g(A,C) =

2?

sinh(2TT5) [si(z)+si(z)]

- i cosh(2TT?)[Ci(z^) Ci(z^)] J , (B.19)

where:

z^ E (A+ic)2Tr and z = (A-i?)277 ,

and where si(z) and Ci(z ) are defined as sin and cos integrals in reference [31].

Therefore, H becomes:

^[^'^2^ =^ { cosh(2TrC2)tan-l [^)- f

e

+ T7TT^ 2c,

sinh(27r52) [si(z ) + si(z )] - i cosh(2^C2)[Ci(z^) - Ci(z*)] (B.20)

If we set 5„ = 0, H can be reduced to;

„ ^ (cos27tA -1)

H[A, ,0] = 1 + - si(2TTAJ + ^ 7 1 IT ^ 1^ TT^ A^ (B.21)

There are several values of A which are of interest in the design of an arbitrary

beam profile apparatus. The first minimum of irradiance has A Here H[1,0] = 0.90.

To make H = 0.98 requires A

the amount of power captured from the original power, P

5. Since H appears twice in the captured power eq (B.15),

transmitted through the iith aperture is P = 0.96 P for A 5 and 0.8 P„ for A = 1.

In addition to the above effects, there can be significant interference between the

far-field patterns from neighboring sampling apertures. For example, for n-, = and

C^ = ± 5.5, which locate the first peak just outside the far field apertures for A 5, we find this amplitude is 6% of the amplitude at peak irradiance, I, = t\ = 0. This fact means this power in the noncaptured part of the far field can interfere with the neighboring far-field pattern from the other apertures. To be sure of complete separation

between patterns so that no interference effects are greater than 1% requires | c^ | ^32.

This constraint can be a significant limit on the allowed spatial sampling frequency for

the Hartmann plate method.

B.3 The Fourier Transform Pattern

Equation (A. 4) shows that the Fourier transform pattern is:

12?: U2(x2,y2) = YXI

dx^dy^U^(x^,y^)e Af (x^x^+y^y^)

(B.22)

-80-
If U from eq (B.l) is used for each aperture, and the spherical phase front factor,

exp[ ~ {m\ + v\)] , (B.23)

is dropped from eq (B.4), then eq (B.20) has the same functional form as the far-field

form of eq (B.6) except that the focal distance, f, for the mirror or lens replaces the

distance, z, from the sampling plate. We write the Fourier transform U^(x„,y„) as: i[$+d^+9]

^o^ 2

""2 ^i ^2 ""l U2(x2,y2) = ^J^

a sinc[(y^ -

2:^)3] sine [ (-^ - -2^)3.] , (B.24)

-k

where 9 = -%- (x„g +y„h ), and <I> = 2kf+(n-l)tk. Here n is the index of refraction for the

lens or mirror and t is the thickness of that same lens. We see that most features of the

far-field pattern apply equally to the Fourier transform field except for the spherical wave front term (eq (B.2)), the plane wave phase shift defined by 6, and the fact that the

x„ and y„ are used instead of u„ and v„ in the sine functions (see subsection A. 3 about

this last point)

.

*Note: P^^ = I^ Xzl^ ,2 ^z I 2 = A e a

o

il-
APPENDIX C. THE DIFFRACTION DETAILS OF A CIRCULAR HOLE

Many of the details for diffraction pattern from a circular hole are discussed in

reference [29] . This appendix extracts these important features and parallels the development of appendix B. Many of the conventions and symbols are the same as in appendix B; therefore, this discussion only shows those new definitions and reduces substantially

the details for each derivation.

Here the U becomes:

s

U^(x^,y^) = A^ circ(r^/a)exp i[b^(x^-g^) + c^(y^-h^) + d^] (C.l)

where a is now the diameter of the aperture and circ is defined as:

2 2^/2^

circ(r/a) =1 if r^ e [(x^-g^^) + (y^^-h^) ] < y

= if r > a/2

= 1/2 if r^ = a/2. (C.2)

C.l What this Appendix Presents

Subsections C.2 and C.3 contain brief derivations of the far-field and Fourier trans- form patterns for the circular aperture and a discussion showing selected features of

those patterns.

C.2 The Far-field Diffraction Pattern

Equation (B.4) becomes changed to reflect the far-field condition:

z>f (2.5) . (C.3)

Using definitions:

1 u. k . 2 _^ 2,

!>^ = kz + 2j ("2 + V2),

C = C^ + ii;^ = a

zX 2tt

n = n, + lri2 = a

rv„ c '2 _ I zA 2tt

1/2

o , 2 , 2,1/2 , 2 , ^, „

, r, = (C-, + n,) , r = (c + n ) , tan e„ = n/? , and

tan 6 = v^ /u^ , (C.4)

-82-
we rewrite eq (B.4) to reflect the changes in aperture and notation. Further, we use reference [30, page 398] for Bessel function relationships to reduce the two integrals.

Thus, we get:

02(^2) = A e

o

i($, + d^)

iAz

J^(TTr2)a

(C.5)

By defining:

I. = A

f o Az4

(C.6)

fira^l 2 with the area, ~7~ ' ^°^ ^^^ circular aperture in place of the area, (a ), for the square, we can write the irradiance as:

1(^1, n^) = If

^ S (C.7)

The r„ can be written in terms of ? , ?„, etc., as:

2 2 1/2

(C.8)

From this irradiance, eq (C.7), we extract a number of facts: (1) Changing C-, "^ -?-, and

r\ -> -n, results in no change in the form of the irradiance. (2) Likewise, n^ "^ -n^

and 5„ -> -?„ imply no change in the form of the irradiance; therefore, we cannot deduce

the absolute sign of the gradient of the irradiance at the aperture. However, we can deduce the direction of this gradient. This situation means we have less symmetry than

with the square hole where we have two possible directions. (3) The coordinates for the

peak irradiance are unchanged, ^^ = n, = 0. If c,„ and rio are small enough, the peak

irradiance is:

1(0,0) =1^ tl + |- (?2 + ^2^^^ (C.9)

(4) Again for small f; , r\„, the rings of minimum irradiance in eq (C.7) are the values of

r where J (Trr°) is zero. The first three values are t° = 1.22, 2.23, and 3.24. (5) The

asymptotic form of J (Trr„) is:

/

2

TT r. cos(iTr„ ~ T ^)

-83-
The irradiance function for this region becomes:

I(Cl>n^) = ^k(^^ v*)

'i/'^ cos[i^r^ - ^]cos[^r2 - ^] (CIO)

Neglect 5„ and r]„, set r\ = 0, and look at local peak values of irradiance.

Then:

^1

(C.ll)

-4 For irradiance in eq (C.ll) to be 10 of eq (C.9), ? = 9.36 - 10. This distance

defined by ? for the circular aperture is about one-third of the same value for the

square aperture.

The capture of laser power for a given circular aperture with arbitrary values of

? and n^ cannot be determined in terms of known functions. A computer is necessary. For this discussion, we assume that there is no significant gradient at the aperture for the

irradiance; thus Co = Ho = 0. Under these conditions the measured power for a properly

centered detector is:

^ = ^0^1-^0^^^) - jJCwA^)] (C.12)

We set the diameter of the detector equal to A, and the equivalent scale of A = y

a_A

zX The total power, P

4 I,, is at the original aperture. The power in the first

O TT circle is only 0.84 P , in the second circle is 0.91 P , and in the third circle is 0. 94 P . ^ o o o Here A = 1.22, 2.23, and 3.24, respectively, for each of these cases.

C.3 The Fourier Transform Pattern

Equation (C.5) becomes:

- A e a J, (irrT) „ r fn _o 1 2 U2f^2] = "TXf f (C.13)

f _ 2 2 1/2 _ ^2 Z _ ^ A. where r^ = [(c^) + (n^) ] ' ^f = ^^xf " 27"'' ^^^ ^f ^ ^^Xf ~

Irf^

^2 ^i.

Please note the term -^ (x^g. + y^ff,) is exactly the same as for the square hole case

and again the Airy disk has no displacement due to g and h .

This concludes this discussion of the laser beam output produced by a circular

aperture in the Hartmann plate.

-84-
APPENDIX D. DEPTH OF FIELD ANALYSIS FOR FOURIER OPTICS

The proper assembly of lenses and mirrors to construct a correctly working cross- correlation system requires two distinct analyses for the depth of field of each type of

lens [30, page 441] . That is, we must fix the position of the lens used for Fourier

transform and the position of the lens for imaging. In both cases, the proper adjustments

of a lens require the appropriate h(x, y; R) function to be unity.

In the Fourier transform adjustment, let x = y = D //2 where D is the largest

radial dimension for interesting detail in surface "1." Further, we make the arbitrary

decision that the largest allowed phase shift in the depth of field adjustment which will

cause no significant error for design purposes has h = e 10 at this radius. This

10

value corresponds to ^^ of a wavelength fringe error. That is the level of error we can

see in a Michelson interferometer for moderate accuracy phase measurements. Design of a precision system should have this level of sensitivity. This phase shift implies a constraint on d„:

kD 2

d^ ^1

(D.l)

where R = d^ / (1-W/d, ) is the radius of curvature due to improper placement of surface "3"

(see figure A.l). Equation D.l can be rewritten as:

2D^2

1

< —

- 5 (D.2)

Thus, given D , d , A and f, eq (D.2) fixes the range through which d„ can be moved and

still make a proper Fourier transform of the pattern at surface "1." Equation (D.2) has

value for design work; the final adjustment within an actual system uses the insensitivity

to displacement discussed in appendix A, to fix accurately the Fourier plane.

Equation (D.2) can be simplified if d„ = f + A, where A is a small value. Here, A is

constrained to be:

- IQD, (D.3)

If we define an F number of the image on surface "1" as F = f/(2D ), we get the

convenient form for the depth of field at surface "3":

|a/a| < 0.4(Ff)2. (D.4)

-85-
In brief, and F less than one implies a very critical adjustment, and an F greater

than 10 allows ease in this adjustment. Unfortunately, the latter also implies long optical

distances. (Incidentally, this same large F^ limit tends to minimize spherical aberration

in the lens and permits using a single element lens to make beam profile measurements.)

The following formula fixes the depth of field for image plane placement:

kD 2

1 w ^5 (D.5)

The same arbitrary phase constraint on the h function is used here as was used in the Fourier]

transform depth of field analysis. If we set d„ = d^ + A in the lens formulas, we get ~ ~ / jON 2 > where the image equation -5—5- + 3— = t" applies. The A becomes:

A = ^

10

(D.6)

Here, we define a second number as F "3" reduces to:

// 2D^-

The depth of field for surface

Ia/x] £ 0.4(F2#)2 ,

which is the same form as eq (D.4). Again, long focal length optics and small

beam curvature at surface "3" imply large depths of field. On the other hand, short focal

length optics and large beam curvature at that same surface imply a small depth of field and

corresponding very critical mechanical adjustments. In practice, the adjustments of the

image plane are made by observing that the largest off-axis dimension of the image in

surface "3" is as visually sharp as possible. One procedure to specify this image plane

uses a variable aperture to block a plane wave at surface "1" (note the image of the

aperture at surface "3"). The sharpest image corresponds to the best adjustment of the

apparatus. The larger this aperture is, the more critical is the depth of field adjustment.

-86-
NBS-114A (REV. 9-78)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-10092.Gov't Accessiwj No. 5« fesCifHertt's AcCfi^sionWo,

4, TITLE AND SUBTITLE

Laser Beam Profile Measurements Using Spatial Sampling,

Fourier Optics, and Holography

5. Publication Date January 1979

6. Perft«iT«ng Or^mzalion t6de

7. AUTHOR(S)

Eric G. Johnson, Jr

,

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

le. Project/Task/Work Unit No.

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS rs(ree(. City, St^ite. ZIP)

Funded in part by:

Department of Defense

Combined Calibration Group (CCG)

13. Type of Report & Period Covered

14. ^nsor'tng Agaicy Code

15. SUPPLEMENTARY NOTES

I

I Document describes a computer program; SF-185, FIPS Software Summary, is ahached.

16. ABSTRACT (A 200-word or less factual summary of most significant infomnation. If document includes a significant bibliography or

literature survey, mention it here.) Using appropriate holography, lenses, mirrors, and a two-dimensional array of

small holes to sample the electric field of a laser pulse, we demonstrate a technique

for beam profile measurements at a preselected observation plane. This method can

measure the amplitude and phase of a laser pulse at each preselected sampling point

on a transverse observation plane. Subject to constraints, these measurements can

provide separated visual images of the temporal pulse shapes for the amplitude and

phase at each of the sampled electric fields. This paper describes the basic concepts

for beam profile measurements via holography and uses pictures generated by a prototype system for illustration.

This unit may be calibrated by using a cw laser beam of the same wavelength as

the carrier or a Q-switched laser pulse. Therefore, a user can make beam profile

measurements of a laser pulse in real time. This technique requires appropriate scaling of the lenses and mirrors, and uses a different hologram to allow the unit to

work optimally for each wavelength.

We define the potential limits and virtues of the holographic method. In addition, we define limits and virtues of other beam profile techniques such as the twopinhole scanning method and the Hartmann plate method.

Finally, we indicate possible directions for eventual commercial and scientific

exploitation of the holographic method. The technique can be used for near- and farj-nfrared wavelengths and, with appropriate changes in the optics, for visible wavelengths.

*'. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper na separated by semicolons) Amplitude; beam profile; cross correlation; digital sampling;

fourier optics; holography; laser; laser parameters; laser pulse; optical processing;

phase; precision measurements; wave front.

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citations to all recent NBS publications in NBS and non- NBS media. Issued six times a year. Annual subscription: domestic $17.00; foreign $21.25. Single copy, $3.00 domestic;

$3.75 foreign. Note: The Journal was formerly published in two sections: Section A "Physics and Chemistry" and Section B "Mathematical Sciences." DIMENSIONS/NBS (formerly Technical News Bulletin)—

This monthly magazine is published to inform scientists, engineers, businessmen, industry, teachers, students, and

consumers of the latest advances in science and technology,

Lwith primary emphasis on the work at NBS. The magazine

|highlights and reviews such issues as energy research, fire protection, building technology, metric conversion, pollution

ibatement, health and safety, and consumer product per- formance. In addition, it reports the results of Bureau pro- grams in measurement standards and techniques, properties

, of matter and materials, engineering standards and services, [instrumentation, and automatic data processing.

Annual subscription: Domestic, $11.00; Foreign $13.75

NONPERIODiCALS

Monographs—Major contributions to the technical liter- ature on various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and industrial practice (including safety codes) developed in cooperation with interested industries, professional organizations, and regulatory bodies.

iSpecial Publications—Include proceedings of conferences sponsored by NBS, NBS annual reports, and other special

publications appropriate to this grouping such as wall charts, pocket cards, and bibliographies.

Applied Mathematics Series—Mathematical tables, manuals, and studies of special interest to physicists, engineers,

chemists, biologists, mathematicians, computer programmers,

and others engaged in scientific and technical work.

^National Standard Reference Data Series—Provides quanti-

:ative data on the physical and chemical properties of

aterials, compiled from the world's literature and critically evaluated. Developed under a world-wide program co- ordinated by NBS. Program under authority of National (Standard Data Act (Public Law 90-396).

og.

NOTE: At present the principal publication outlet for these data is the Journal of Physical and Chemical Reference Data (JPCRD) published quarterly for NBS by the American Chemical Society (ACS) and the American Institute of

Physics (AIP). Subscriptions, reprints, and supplements

available from ACS, 1155 Sixteenth St. N.W., Wash., D.C.

20056.

Building Science Series—Disseminates technical information developed at the Bureau on building materials, components,

systems, and whole structures. The series presents research

results, test methods, and performance criteria related to the

structural and environmental functions and the durability and safety characteristics of building elements and systems. Technical Notes—Studies or reports which are complete in themselves but restrictive in their treatment of a subject.

Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often

serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures

published by the Department of Commerce in Part 10, Title 15, of the Code of Federal Regulations. The purpose

of the standards is to establish nationally recognized require- ments for products, and to provide all concerned interests with a basis for common understanding of the characteristics of the products. NBS administers this program as a supple- ment to the activities of the private sector standardizing

organizations.

Consumer Information Series—Practical information, based on NBS research and experience, covering areas of interest

to the consumer. Easily understandable language and

illustrations provide useful background knowledge for shopping in today's technological marketplace.

Order above NBS publications from: Superintendent of Documents, Government Printing Office, Washington, D.C.

20402. Order following NBS publications—NBSIR's and FIPS from

the National Technical Information Services, Springfield, Va. 22161.

Federal Information Processing Standards Publications (FIPS PUB)—Publications in this series collectively constitute the Federal Information Processing Standards Register.

Register serves as the official source of information in the Federal Government regarding standards issued by NBS

pursuant to the Federal Property and Administrative Services Act of 1949 as amended. Public Law 89-306 (79 Stat. 1127), and as implemented by Executive Order 11717

(38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal Regulations).

NBS Interagency Reports (NBSIR)—A special series of

interim or final reports on work performed by NBS for outside sponsors (both government and non-government).

In general, initial distribution is handled by the sponsor;

public distribution is by the National Technical Information

Services (Springfield, Va. 22161) in paper copy or microfiche form.

BIBLIOGRAPHIC SUBSCRIPTION SERVICES

e following current-awareness and literature-survey bibli- ographies are issued periodically by the Bureau:

Cryogenic Data Center Current Awareness Service. A litera- ture survey issued biweekly. Annual subscription: DomesItic,

$25.00; Foreign, $30.00.

Liquified Natural Gas. A literature survey issued quarterly.

Annual subscription: $20.00.

I

Superconducting Devices and Materials. A literature survey

issued quarterly. Annual subscription: $30.00. Send subscription orders and remittances for the preceding bibliographic

services to National Bureau of Standards, Cryogenic Data

Center (275.02) Boulder, Colorado 80302.
U.S. OEPARTMENT OF COMMERCE

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