Guide to Program
DIFFRACT (8.4)
Optical System Simulation Software
from
MM Research, Inc.

January 2007

Copyright © 1987–2007 by MM Research, Inc. All rights reserved.

Information in this document is subject to change without notice and does not represent a commitment on thepart of MM Research. The software described in this document is furnished under a license agreement or
nondisclosure agreement. The software may be used or copied only in accordance with the terms of theagreement. It is against the law to copy the software on any medium except as specifically allowed in the license
or nondisclosure agreement. The purchaser may make one copy of the software for backup purposes. No part ofthis manual may be reproduced or transmitted in any form or by any means, electronic or mechanical, including
photocopying, recording, or information recording and retrieval systems, for any purpose other than thepurchaser’s personal use, without the express written permission of MM Research.

© 1987-2007 by MM Research, Inc. All rights reserved.
Graphics by Nonie Veccia
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ii

Guide to Program
DIFFRACT (8.4)

For Windows 95, 98, 2000, NT and XP

Table of Contents
Contents................................................................................................................................. iii
Installing DIFFRACT to Hard Disk.......................................................................................vi
New Features in Version 8.4 ...................................................................................................x
New Features in Version 8.2 ............................................................................................... xiii
New Features in Version 7.9 ............................................................................................. xviii
1.

Introduction .............................................................................................................................1
1.1. Home Directory and Working Subdirectories..............................................................1
1.2. Coordinate System and the Mesh.................................................................................2
1.3. Save and Read ..............................................................................................................5
1.4. The Plot Option ............................................................................................................5
1.5. Internal Arrays and Memory Requirements .................................................................7
1.6. Data Files Created by DIFFRACT...............................................................................92.

Interactive and Non-interactive Sessions ..............................................................................212.1. Structure of the COMMAND.DAT File ....................................................................212.2. The $ Commands .......................................................................................................222.3. Running DIFFRACT in the Non-interactive Mode ...................................................312.4. Running DIFFRACT from the Command Line. ........................................................31
2.5. Building COMMAND.DAT Files with the Help of a Template ..................................32
2.6. Extracting Specific Signals from the SIGNALS.DAT File........................................33

3.

File Management and Graphics.............................................................................................35
3.1. Commands and their Actions .....................................................................................35

4.

General Features....................................................................................................................39
4.1. Incident Amplitude Distribution ................................................................................424.2. Propagation Regimes..................................................................................................48
4.3. Scalar and Quasi-Vector Treatments of Polarization .................................................49
4.4. Store Distribution .......................................................................................................50
4.5. Retrieve Distribution ..................................................................................................50
4.6. Phase /Amplitude Object ............................................................................................51
4.7. Interferometric Beam Combination............................................................................54
4.8. Launching Guided Mode............................................................................................56
4.9. Phase /Amplitude Extraction ......................................................................................57
4.10. Peak Intensity and FWHM Measurement ................................................................57
4.11. Wavefront Analysis..................................................................................................58
4.12. Spatial Filter ............................................................................................................67
4.13. FDTD Interface ........................................................................................................70

5.

Modifying the Coordinate System and the Mesh Parameters ...............................................73
5.1. Changing the Coordinate System ...............................................................................73
5.2. Rescaling the Mesh ....................................................................................................73

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6.

Detectors and Signal Processing ...........................................................................................74
6.1. Quad Detector ............................................................................................................74
6.2. Phi Detector................................................................................................................76
6.3. Split Detector..............................................................................................................77
6.4. Differential Detector...................................................................................................77
6.5. Signal Processing .......................................................................................................78

7.

Simple Phase/Amplitude Masks and Apertures ....................................................................80

8.

Polarization-sensitive Element ..............................................................................................819.

Flat Mirror.............................................................................................................................83

10.

Weak Biprism........................................................................................................................92

11.

The Lens Action ....................................................................................................................96
11.1. Positive Focusing Lens.............................................................................................97
11.2. Collimator.................................................................................................................98
11.3. General-purpose Lens ..............................................................................................99
11.4. Positive Cylindrical Lens .......................................................................................100
11.5. Astigmatic Lens......................................................................................................101
11.6. Toric_ring Lens ......................................................................................................102
11.7. Split Lens ...............................................................................................................103

12.

Diffraction Grating..............................................................................................................106
12.1. Analysis of Grating in the Scalar Approximation ..................................................107
12.2. Entering and Exiting the Grating Neighborhood ...................................................109
12.3. Importing Reflection/Transmission Coefficients from a Data-file.........................118
12.4. Format of the Data-file ...........................................................................................119

13.

The Multilayer.....................................................................................................................123
13.1. Orientation of the Multilayer..................................................................................123
13.2. Entering and Exiting the Multilayer.......................................................................125
13.3. Saving Multilayer Stack Parameters in Hard Disk Files ........................................127
13.4 Building XXXX.STK Files with the Help of a Template .........................................127

14.

Optical Disk Surface ...........................................................................................................136
14.1. Pregrooves..............................................................................................................137
14.2. Pregroove Wobble..................................................................................................138
14.3. Bumps and Pits.......................................................................................................144
14.4. Magnetic Domains .................................................................................................151
14.5. Multilayer Dye-Polymer Media .............................................................................155
14.6. Inspecting the Surface Pattern of an Optical Disk..................................................171

15.

Conversion to Rays .............................................................................................................172

16.

Geometric-optical System: Definition, Inspection, and Ray_Tracing ................................176
16.1. Surface Coordinates ...............................................................................................176
16.2. Arrangement and Succession of Surfaces ..............................................................178
16.3. Surface Types and Shapes......................................................................................179
16.4. Identifying an Optical System to the Program .......................................................182
16.5. Geometric-optical System Inspection by Selective Ray_Tracing ..........................184iv

16.6. Incident and Emergent Rays during Ray_Tracing .................................................189
16.7. Searching for Curvature/Tilt Phase-factor .............................................................190
16.8. Phase-factor Stripping and Wavefront Reconstruction ..........................................190
16.9. Examining a Geometric-optical System.................................................................191
16.10 Building XXXX.SYS Files with the Help of a Template .......................................203
17.

Diffractive Optical Elements...............................................................................................20417.1. Transmissive DOE on flat substrate .......................................................................206
17.2. Reflective DOE on flat substrate............................................................................210
17.3. Reflective DOE on a parabolic mirror....................................................................214
17.4. Transmissive DOE on an aspheric glass lens .........................................................219
17.5. Transmissive DOE on a cylindrical GRIN lens .....................................................224
17.6. Reflective DOE on a conical mirror surface ..........................................................228
17.7. Transmissive DOE on an axicon ............................................................................232
17.8. Transmissive DOE sandwiched between two flat substrates .................................236
Appendix A. Frequently asked questions about DIFFRACT ...............................................243
Appendix B. Relationships among various Menu Options .................................................251
Appendix C. Coordinate System(s) of DIFFRACT ..............................................................253
Appendix D. Command blocks as they appear in COMMAND.DAT ..................................261
Appendix E. Selected Test Runs .........................................................................................291
Appendix F. Test Runs available on DIFFRACT’s distribution CD-ROM ...........................407
Appendix G. OPN articles (and their resource files) available on DIFFRACT’s CD-ROM ..409
Appendix H. Reprint of an Article ......................................................................................411
Appendix I. Reprint of an Article........................................................................................435
Appendix J. Abbe’s Sine Condition....................................................................................447
Appendix K. Theory of Diffractive Optical Elements ........................................................455
References ...........................................................................................................................465v

Then the Lord said∇⋅D = ρ∇ × H = J + ∂D/∂t∇ × E = − ∂B/∂t∇⋅B = 0... and there was lightHard_key Installation and Setting up DIFFRACT on the Hard Disk
Your DIFFRACT package contains an installation CD-ROM and a Rainbow
Sentinel SuperPro™ hard_key, together with the necessary software driver tooperate the hard_key. The hard_key itself must be securely fastened to a parallel
port (or a USB port) on your computer. (Typically, the parallel port is the portthat connects to a printer.) To activate the hard_key on your system, the
INSTALL.EXE routine transfers the necessary software driver(s) to your harddisk, then runs SETUP.EXE from the <RAINBOW> subdirectory located within theworking directory.
To install the parallel-port key you must disconnect the printer cable (if any)
from the back of your computer, mount the hard_key directly on the parallel
port, tightly fasten the screws, then re-attach the printer cable (if any) to the backof the hard_key. (If you have received a USB key instead of a parallel-port key,
simply insert the key into an available USB port.) The driver software is
automatically loaded and activated by the installation routine.
To run DIFFRACT under Windows® 95, 98, 2000, NT, or XP you must first
install the program on your hard disk. This is done by following four simple
steps:
1. Insert the distribution CD in a CD-ROM drive, then switch to the CD’s
directory.
2. Double-click on INSTALL.EXE. The installation routine prompts you for thename of a home directory on the hard disk. (The default home directory is
C:\DIFFRACT.) Many of the files and directories on the CD-ROM are
subsequently transferred to the specified home directory, which will
eventually contain the subdirectories <RAINBOW>, <SYSTEMS>, <STACKS>,<TEMPLATES>, and the files LOGO.BMP, DIFFRACT.EXE, EXTRACT.EXE,
SETTINGS.EXE, SETTINGS.LOG, README.DOC.
3. The installation routine automatically installs the hard_key driver by running
SETUP.EXE from the <RAINBOW> subdirectory. At this point you may have torestart the computer to activate the hard_key driver.

vi

4. Once the program and its hard_key are properly installed, switch to
DIFFRACT’s home directory, then double-click on the program’s icon. This is
all you need to do to get the program running.
DIFFRACT opens three windows on the monitor: The Dialogue Window iswhere the program asks questions and expects to receive your answers. Help is
automatically displayed in the Guiding Information window; this window
stays in the background and you must switch to it manually if you need to lookup its contents. The Graphics Window becomes active when you invoke theplotting routine. Use the mouse-click or press Ctrl + Tab to switch from one
window to another.
Files and subdirectories: During installation several files will be copied to
DIFFRACT’s home directory. The main file is DIFFRACT.EXE which you invoke
every time you run the program. There is also SETTINGS.EXE and the text file
SETTINGS.LOG which you may use to set upper bounds on some of the fixedparameters of the program. The file EXTRACT.EXE should be run whenever you
need to extract one or more signals from the data file SIGNALS.DAT, an ASCII
file generated by DIFFRACT. The bitmap file LOGO.BMP contains the program’slogo, which appears in the beginning of each session. The logo may be disabled
by renaming this file or removing it from the home directory. In addition tothese files there are four subdirectories: <RAINBOW>, <STACKS>, <SYSTEMS>,
and <TEMPLATES>. The first subdirectory contains the driver for the hard_key.
The second and third subdirectories contain, respectively, examples ofmultilayer stacks and geometric-optical systems which you might find useful ingetting acquainted with some of the features of the program. Templates forbuilding various files for later importation to DIFFRACT are contained in the<TEMPLATES> subdirectory.
Test Runs: The distribution CD-ROM for DIFFRACT™ comes with a set of test
runs included in the subdirectory <TESTRUNS>. These test runs are NOT
automatically transferred to the hard disk during installation, but the user may
transfer them at a later time. The <TESTRUNS> subdirectory could be dragged
and dropped into DIFFRACT’s working directory, or into any other directory on
the hard disk. Alternatively, the test runs could be opened from the CD-ROM andviewed on the monitor or printed out. The test runs contain numerous
application examples, and the accompanying COMMAND.DAT files can be usedas templates for creating new simulations. Because the test runs have beencollected over a period of time with different versions of the program, there existvii

minor differences between the sequence of commands in the supplied
COMMAND.DAT files and the sequence of commands that the current version of
DIFFRACT actually accepts. Do not attempt to run DIFFRACT with the exact
COMMAND.DAT files supplied; instead, use the various test runs as a guide tosetting up and running the program.
OPN Directory: The distribution CD-ROM also comes with a set of resource
files relating to articles – based on DIFFRACT simulations – that were originallypublished in Optics & Photonics News (OPN). The < OPN > directory is NOT
automatically transferred to the hard disk during installation, but the user may
transfer it (or parts of it) at a later time. The < OPN > directory may be dragged
and dropped into DIFFRACT’s working directory, or into any other directory on
the hard disk. Alternatively, the various resource files in this directory can be
opened from the CD-ROM and viewed on the monitor or printed out. Since the
OPN articles have been written over a period of several years using different
versions of the program, there exist minor differences between the sequence ofcommands in the supplied COMMAND.DAT files and the sequence of commands
that the current version of DIFFRACT actually accepts. Do not attempt to run
DIFFRACT with the exact COMMAND.DAT files supplied; instead, use the various
resource files as a guide to setting up and running the program.
Graphic Capabilities: DIFFRACT’s graphics routines generate two-dimensional
color as well as black-and-white plots for viewing in the Graphics Window.
These plots are subsequently saved in the working directory as bitmap files, TIFF
files, or ASCII files. The bitmap and TIFF files can be read by Windows® utility
programs, such as Word® and Paint®, modified when necessary, andincorporated into documents. For other types of graphics, you may port the
ASCII data-files created by DIFFRACT to other commercially available software.
These data-files are compatible with most graphics packages such as Matlab®,
Origin®, and Mathematica®.
Install Routine Failure. In some instances the operating system gives thewrong information concerning the available disk space to DIFFRACT's installroutine. When this happens, the install routine, believing the available disk space
inadequate, refuses to install the program. What you should do here is create a
directory on your hard disk (call it Diffract, if you wish), then transfer all thefiles and subdirectories from the installation CD-ROM to this directory on yourhard disk. The transfer is done simply by selecting everything (except
Install.exe) on the CD-ROM and dragging them to the newly created directory on
the hard disk. Once everything is on the hard disk, open the <RAINBOW>
viii

subdirectory and double click on SETUP.EXE. This will install the Hard_key
driver software. (At this point the Hard_key itself must be attached to theparallel port in the back of the computer.) In some systems you will have torestart the computer to get the Hard_key driver initialized; in others there is noneed to do so. Once the above steps are completed, your software package will
be ready to run from the hard disk. Simply go to the DIFFRACT directory you
have set up on the hard disk, and double click on DIFFRACT.EXE.
Problem with Blank Windows. In some cases, when DIFFRACT™ is installed
under Windows 95, 98, 2000, or NT, the Graphics and Guiding Information
Windows appear blank. This is caused by a setting of the desktop colors. To
solve the problem, follow these instructions: Click on the Start button. Click on
Settings, then on Control Panel. Double-click the Display icon, which will open
the Display Properties window, then click on Settings. There, change the settingfrom True Color to something else, such as High Color 16 bits. (Any setting
other than True Color should work.) Then click the Apply button, which will
reset your computer. You should be able to use the Graphics and Guiding
Information Windows now with the new settings.
Problems with Hard_key: The driver software is automatically loaded andactivated by the installation routine. However, in case of difficulty with the
hard_key, try the following steps:
1. Insert the installation disk in the CD-ROM drive. Switch to this drive and openthe <RAINBOW> directory. Double click on SETUP.EXE and follow the on-line
instructions. (Alternatively, you may transfer the <RAINBOW> directory to
your hard drive, then run SETUP.EXE from the hard drive. In this way, in theunlikely event that you will need to reinstall the hard_key driver in the future,
the driver software will already reside on the hard disk, and the installation
CD-ROM will no longer be needed.)
2. Remove the Installation CD-ROM from its drive and keep it in a safe place.
You might need this disk again in the future to reinstall the program in caseof a system failure, or for installing the program on a different computer.
3. You may have to restart your computer to activate the hard_key driver.
The hard_key is now activated on your system, and you may proceed to run yourprotected application. If difficulties persist, please contact MM Research, Inc.

ix

New Features in Version 8.4
1. Diffractive Optical Elements. The various surfaces within the GeometricOptical_System (GEOP) option of DIFFRACT™ may now be coated withDiffractive Optical Elements (DOEs). In previous versions, a Glass surface
could be specified as bare or antireflection coated. In the new version, a Glasssurface, be it Flat, Spherical, Conical, Aspherical, Cylindrical, or Axiconal,
may be specified as Bare, Antireflection-coated, or Coated with DOE. The newcommand line in the command.dat file is Coating (AR/DOE/BARE):,
which replaces the old command line AR Coating (Y/N):.
Mirror surfaces can now be coated with Diffractive Optical Elements as well. In
previous versions, a Mirror surface, whether Flat, Spherical, Conical,
Aspherical, Cylindrical, or Axiconal, was characterized by its complex
refractive index (n, k), which determined the Fresnel reflection coefficients ofthe surface. If both n and k were set to zero, the Mirror was understood to be aperfect reflector. In the new version, a Mirror surface can be Bare, Perfect
Reflector (PR), or Coated with DOE. The following (new) command line
appears in the command.dat file: Coating (PR/DOE/BARE):. If the user
selects the perfect reflector (PR) option, no further questions will be asked, but
if the chosen option is BARE, the user will be prompted for the complex
refractive index (n, k) of the Mirror material.
When the DOE coating option is chosen for Glass and Mirror surfaces, the user
is prompted for the various characteristics of the DOE. These include theconstruction wavelength λc, the desired diffracted order m, and the polynomial
coefficients of the phase function encoded by the DOE. If the operating
wavelength λo happens to be identical with the construction wavelength λc, the+1st diffracted order (i.e., m = +1) will have 100% efficiency while all otherorders vanish. For λo ≠ λc, however, the user may choose values of m other than
+1. In general, the DOE phase profile is centered at the vertex of thecorresponding surface and is specified in terms of a circularly symmetricpolynomial F (r), where r is the radial distance from the center in the plane thatis tangent to the surface at its vertex. In the special case of a Flat surface, theuser is given the option to specify the phase polynomial as F (x, y), where x andy are Cartesian coordinates in the plane of the Flat surface. The orientation of thexy-plane of the Flat surface relative to the xyz-coordinates of incidence may also
be specified by the user.
x2. Higher-order Aspheric Coefficients. In previous versions of DIFFRACT™,
the sag polynomials for Aspheric Glass and Mirror surfaces within theGeometric-Optical_System (GEOP) option could be specified with coefficients
up to the 12th order. In version 8.4 these polynomials are specified with 21
coefficients. In the command.dat file, therefore, after the line A10,A11,A12:,
the following three lines are added for the new aspheric coefficients:
A13,A14,A15:
A16,A17,A18:
A19,A20,A21:
3. Surface Arrangement Analysis within the GEOP Option. DIFFRACT™
8.4 conducts an automatic analysis of the Geometric-Optical Systems defined
under the GEOP option. Since each surface, depending on whether it is hit ormissed by incoming rays, can lead to two other surfaces, the number of possible
paths through the system can be enormous. DIFFRACT™ conducts a complete
search through the configuration space, finds all admissible paths, anddetermines their logical consistency. The user is then provided with a list ofadmissible paths and informed of any logical inconsistencies that might exist.
4. Power content of the beam. In previous versions of DIFFRACT™ the total
optical power of the beam was initially set to unity. Passage of this beamthrough an optical system would then cause its power to decline as the beam
went through various lossy elements. The user could keep track of the total
optical power by monitoring the values of XPOWER and YPOWER(corresponding to the x- and y-components of polarization), regularly displayedin the table above the main menu. Since the optical power was dimensionless,the beam intensity was similarly normalized and displayed in units of 1/λ2,
where λ = λo / NVIRON was the wavelength of the light beam within its
environment.
In Version 8.4, the user specifies at the outset the power of the beam in units ofmilliwatts. The values of XPOWER and YPOWER (as well as other similarparameters, e.g., ZPOWER) are now displayed in units of milliwatts. Moreover,the intensity (or irradiance) of the beam, signifying the time rate of flow of
optical energy per unit cross-sectional area, is now displayed in units ofWatts/m2. The only change in the command.dat file is within the BEAM block,where the line containing the beam’s center coordinates, BCX,BCY: is replacedwith BCX,BCY,POWER(mW):.

xi

5. Dye-polymer optical disk containing multiple sets of grooves and marks.
In DIFFRACT™ 8.4 the multilayer stack of a dye-polymer optical disk may
support up to four different sets of grooves and also, independently, up to four
different sets of marks (i.e., bumps/pits). For the first groove-set the user
specifies two layers whose thicknesses are affected by the presence of thegrooves; these grooves may be with or without wobble (both Sine and Blip
wobble are allowed). The user then specifies another pair of layers whose
thicknesses are modified by the second set of grooves, defines the characteristicsof the grooves (including wobble) independently of the preceding groove-set,
and continues in the same vein until all groove-sets are specified. In a similar
fashion, the user specifies the number of different mark-sets that must be
embedded in the multilayer stack. Each set of marks affects the thickness of twolayers, independently of all the groove-sets and of the other mark-sets. Withineach mark-set an arbitrary number of marks may be specified, each with its own
location, orientation, shape, and dimensions.
6. Transmission mode for dye-polymer optical disk. Whereas in previous
versions of DIFFRACT™ the dye-polymer optical disk was limited to thereflective type of media, in version 8.4 both reflective and transmissive types areallowed. In addition, each multilayer stack may have a (transparent) protective
cover-layer on either or both sides of the stack. The user can thus specify amedium of incidence and an emergence medium for the dye-polymer disk, while
defining multiple groove-sets as well as marks (i.e., bumps/pits) that can beplaced at various depths within the multilayer structure.

xii

New Features in Version 8.2
Length Units. In previous versions of DIFFRACT™ all lengths were entered anddisplayed in normalized form. The normalization was with respect to thewavelength of the beam in its current environment. Thus, if λo denotes thevacuum wavelength and NVIRON the refractive index of the environment of thebeam, all lengths would be divided by λo/NVIRON. This is the way DIFFRACT™
treats the lengths internally, but for input/output purposes the normalization is
tedious and occasionally misleading. (The confusion usually arises when the
beam enters a medium of different refractive index, which results in a change oflengths by the ratio of the refractive indices, even though the physical lengths
remain the same.)
DIFFRACT 8.2 eliminates the above problem by requiring that the user specify thevacuum wavelength λo in units of cm, mm, um, or nm. (In the past, the user did
not have to specify λo because, in linear optics, once the lengths are properly
normalized by wavelength within the environment, λo disappears from all
equations.) As in previous versions, the user of DIFFRACT 8.2 is asked to specify
at the outset the refractive index NVIRON of the initial environment of the beam.
The user is then prompted for Length_Units, which may be cm, mm, um, or nm.
(To maintain some degree of compatibility with older versions of DIFFRACT, theuser may, in response to the Length_Units question, simply press the ENTER
key, in which case the lengths will be assumed to be dimensionless, i.e.,
normalized by the vacuum wavelength λo).
Note that there is a difference between the units of λo and the Length_Units used
for all other lengths in DIFFRACT™. The units for λo are specified at the outset
and remain the same throughout the simulation, whereas the Length_Units forother length parameters can change as the beam evolves. For each command that
requires the specification of one or more length parameters, the user is prompted
for the corresponding Length_Units. Thus, in a given simulation, the BEAM
command may operate with millimeter Length_Units, the LENS command may
use centimeters, while the PLOT command may use microns. The important
point to recognize is that, as the beam propagates through an optical system, it
might expand or shrink in such a way that the Length_Units used in one part ofthe system may become inadequate for other parts. DIFFRACT thus allows theuser to specify the Length_Units for each command independently of all thexiiiother commands, and also independently of the units used to specify the
wavelength.
When displaying the simulation parameters (or results of a particular
calculation) DIFFRACT always shows the relevant Length_Units, either on themonitor or in prominent locations within the SESSION.DAT file, to ensure thatthe displayed lengths are properly interpreted. During interactive sessions,consult the box above the main menu for information about the vacuum
wavelength λo, refractive index of the current environment (NVIRON),
Length_Units, mesh parameters NMAX, NMAY, LMAX, LMAY, and otherimportant information about the current state of the beam.
Zernike Polynomials. The maximum allowed order of the Zernike polynomials
has increased from 8 to 12 in the current release of DIFFRACT. The BEAM andth
LENS options can now be specified with Zernike aberrations up to the 12 order.
Also, the MASK option, which in previous versions lacked any kind of
wavefront distortions, can now be specified with either Seidel or Zernike
aberrations (up to the 12th order). Of the four types of MASK presently available(Circular, Elliptical, Annular, Rectangular) only the Circular mask offers theoption of Seidel or Zernike aberrations.
Spatial Filter. This new feature in DIFFRACT 8.2 allows the user to modify thespatial frequency spectrum of the beam by a circular phase/amplitude mask
placed in the Fourier domain. The same operation could have been done inprevious versions by a combination of Positive-focusing_LENS, MASK, andCollimator_LENS, but the new Spatial Filter simplifies the operation. In addition,
the new feature allows a choice between Fast Fourier Transformation (FFT) andDirect Fourier Transformation (DFT). The FFT method is faster, but the DFT
method is more flexible, allowing the user to set the resolution in the frequency
domain independently of the sampling interval in the space domain. The DFT
method of spatial filtering (which can be painfully slow) does not have an
equivalent in previous releases of DIFFRACT.
FDTD Interface. In DIFFRACT 8.2 the user may call upon the FDTD_Interface toexport a cross-section of the beam to an external file, which file is subsequently
used by a Finite_Difference_Time_Domain (FDTD) program for near-field
simulations. In Import mode, the FDTD_Interface reads a beam cross-section
from an external file created by the FDTD program as the end result of near-field
simulations. In this way, DIFFRACT 8.2 handles beam propagation and otherxivmanipulations that are best suited for the classical diffraction theory, leaving
near-field calculations (which require full-vector solutions to Maxwell’sequations) to an external FDTD program.
A good example of the above approach is furnished by the problem of opticaldisk readout. Here the beam of light emanating from a semiconductor laser
diode goes through a collimator lens, a quarter-wave plate, a beam-splitter, and afocusing lens before arriving at the disk surface. DIFFRACT simulates the entire
optical path from the laser to the disk surface, but its treatment of the interaction
between the focused spot and the pits at the disk surface is only anapproximation based on treating the pits as phase objects. For more accurate
simulations, therefore, it is necessary to solve the full-vector Maxwell equations
within the volume of the focused spot in the presence of the pits. To this end,
one may call upon the FDTD option of DIFFRACT 8.2 to export the E-field and Bfield components of the focused spot to an external data file. The content of thisfile is then used as initial distribution for an FDTD program that proceeds tocompute the reflected field in a plane immediately above the disk surface. The
resulting complex-amplitude distribution, i.e., the reflection of the focused spotfrom the disk surface, is then saved to a data file. The FDTD option of
DIFFRACT 8.2 is subsequently called upon (in the Import mode) to read this data
file and to propagate the reflected beam back through the objective lens and to
the photodetectors.
Wobbled Grooves for Optical Disk Surface. The grooves in Writable,
Magneto-optical, and Dye-Polymer types of optical media can now be wobbled.
The two edges of each groove can be assigned different amplitude, period, andphase values for the wobble sinusoid. This enables the simulation of various
types of wobble including groove-width-modulation and groove radial-positionmodulation. Also, adjacent grooves can have in-phase wobbles or exhibit a fixed
relative phase. A new type of wobble consisting of a periodic sequence ofisolated wiggles (or blips) along the length of the grooves is also implemented in
DIFFRACT 8.2.
Geometric-optical_System. Several new features have been introduced in theGeometric-optical_System option of DIFFRACT 8.2. The various Glass andMirror surfaces can now have a circular hole at their center. This is achieved by
specifying both a minimum radius Rmin and a maximum radius Rmax for the clear
aperture of each such surface.

xv

The Aperture surfaces, which were limited to circular and annular shapes, cannow be elliptical or rectangular as well. Aperture dimensions and orientation in
3D space can be chosen by the user without restrictions.
Aspheric Glass and Mirror surfaces in previous versions were limited toaspheric coefficients A4, A6, A8, and A10. In DIFFRACT 8.2 all aspheric coefficients
up to the 12th order, namely, A1, A2, A3, A4, …, A10, A11, A12 are allowed.
In previous releases of DIFFRACT, the Gradient Index (GRIN) media wererestricted to Type 1. In the new release, Type 2 GRIN media are also allowed.
The index profiles for the two types of GRIN media are given by the following
equations:6n (r, z) = no {1 + Σ Qm (r/Ro)

12

2m

m=1
6

n (r, z) = no {1 + Σ Qm (r/Ro)
2

2

m=1

}{1 +mΣ= 1Wm (z/Zo) m }(Type 1)122m}{1 +mΣ= 1Wm (z/Zo) m }(Type 2)The surfaces of a Geometric-optical System are numbered sequentially from 1
up to a maximum allowed integer. Each system must have a first surface and alast surface, the latter being always the Destination plane. All remaining
surfaces must be numbered between 1 and the number associated with the last
surface. In previous versions of DIFFRACT, the incoming rays could only visit
the surfaces sequentially, entering the first surface and ending up at the
Destination plane. Perhaps the most powerful new feature in DIFFRACT 8.2 is the
specification of Next_Surfaces for each surface within the Geometricoptical_System. In the new scheme, if an incoming ray intersects a given
surface, it will be directed toward one surface, but if the incoming ray misses agiven surface it will be directed toward a different surface. The introduction of
Next_if_Hit and Next_if_Miss surfaces thus provides for far more complex
systems than have heretofore been possible.
For each Glass, Mirror, and Aperture the user specifies two other surfaces(within the same system) as alternative choices for the next surface; these arereferred to as Next_if_Hit and Next_if_Miss surfaces. If a ray hits a given
surface, it will seek the (user-specified) Next_if_Hit surface as a subsequent
target. However, if a ray misses a surface, its next target becomes the associated
Next_if_Miss surface. Each incident ray arriving at a Geometric-optical Systemthus starts by targeting surface 1 and follows a path through the system – asdictated by the specified sequence of next_surfaces – until it reaches thexviDestination plane. The Destination plane is the only surface that has no
associated next_surfaces. A next_surface specified by the numeral 0 acts as anabsorber. Any Glass, Mirror, or Aperture surface becomes opaque (i.e., anabsorber) if its Next_if_Hit surface is set to 0. Similarly, rays that miss a given
surface can be blocked (i.e., absorbed) by setting the corresponding
Next_if_Miss surface to 0. The above sequencing scheme provides a powerfultool for simulating complex Geometric-optical Systems.

xviiNew Features in Version 7.9
Incident Beam: In addition to Uniform, Simple Gaussian, and Laser Diodebeams, the new version allows the user to choose a General Gaussian, aLaguerre Gaussian, or a Hermite Gaussian beam from the menu. Perhaps more
importantly, the incident beam can now exist in a medium of refractive indexn ≥ 1, thus allowing propagation in an environment other than free space.
Propagate in Environment: In addition to propagation in an environment ofrefractive index n ≥ 1, the user now has the option of shifting the beam to its
centroid before propagating to the near field, intermediate field, or the far field.
When this option is chosen, the program shifts the origin of the coordinate
system to the beam’s centroid, performs the propagation calculations, thenreturns the beam to its original position. This is a more accurate method ofhandling beam propagation in the Fresnel and Fraunhofer regimes, especiallywhen the centroid is far from the origin of the coordinate system.
Launch Guided Mode: This new feature of the main menu enables the user tocreate a Laguerre-Gaussian or a Hermite-Gaussian beam of specific properties,
then compute its overlap integral with the beam residing in the mesh. The
overlap integral may be saved in the SIGNAL array and subsequently printed to
the SIGNALS.DAT file. This feature is useful when the coupling efficiency to afiber or waveguide is desired. In addition, the Laguerre-Gaussian or theHermite-Gaussian beam thus created replaces the existing beam in the mesh,assuming the amplitude and phase of the overlap integral. Thus, not only is theoverlap integral computed, but the excited mode is launched as well.
Wavefront Analysis: This new feature of the main menu allows the user todetermine the effective wavefront error of the beam at any given cross-section.
This effective wavefront error is related to the r.m.s. wavefront error and to theStrehl ratio of the beam, provided that the wavefront error is not too large.
Phase/Amplitude Object: In version 7.9, when an external object is imported to
DIFFRACT™, the user will be given the option of tiling the available aperture
with multiple copies of the object. If this option is chosen, several copies of the(rectangular) object will be placed side by side, creating a periodic structure
within the specified aperture.
xviii

Multilayer Structure: The beam transmitted through a multilayer stack may
now stay within the homogeneous and isotropic medium of its substrate, andcontinue to propagate in this new environment. The program automatically sets
the refractive index of the new environment to that of the substrate; allsubsequent length parameters will have to be normalized by the wavelength ofthe light in this new environment. Similarly, in the reflection mode, if themedium of incidence on the multilayer happens to differ from the environmentof the beam, the user will be given the option of returning the beam to its initialenvironment, or to keep it inside the medium of incidence, which then becomesthe new environment for the reflected beam. The options of Beam Compression
and Spectral Compression, which existed in previous versions of DIFFRACT andallowed for the entrance/exit of the beam through a flat or a spherical interface,are still available. These features, combined with the new possibility of keeping
the beam inside the medium of incidence (reflection mode) or inside thesubstrate (transmission mode) provide many degrees of freedom forinvestigating the interaction of the beam with a multilayer stack using various
entrance/exit strategies.
Diffraction Grating: The beam transmitted through a grating may now staywithin the homogeneous and isotropic medium of its substrate, and continue topropagate in this new environment. The program automatically sets therefractive index of the new environment to that of the substrate; all subsequent
length parameters will have to be normalized by the wavelength of the light inthis new environment. Similarly, in the reflection mode, if the medium of
incidence on the grating happens to differ from the environment of the beam, theuser will be given the option of returning the beam to its initial environment, or
to keep it inside the medium of incidence, which then becomes the newenvironment for the reflected beam. The options of Beam Compression andSpectral Compression, which existed in previous versions of DIFFRACT andallowed for the entrance/exit of the beam through a flat or a spherical interface,are still available. These features, combined with the new possibility of keeping
the beam inside the medium of incidence (reflection mode) or inside thesubstrate (transmission mode) provide many degrees of freedom forinvestigating the interaction of the beam with a grating using various
entrance/exit strategies.
Extract Phase/Amplitude: In addition to removing the phase of the beam oreliminating its amplitude non-uniformity, the user now has the option ofreversing the sign of the phase, thus creating the complex conjugate of the beam.
xix

This is similar to the action of an ideal phase-conjugate mirror (PCM) on an
incident beam.
Geometric-optical System: In addition to flat, spherical, conical, and asphericalsurfaces, the new version allows cylindrical, conic-cylindrical, and axiconicalsurfaces within the Geometric-optical_System option of the main menu. Also,
whereas in previous versions the Destination plane had to be in free space, inversion 7.9 the Destination plane can be placed inside a Glass medium as well.
While tracing rays through a geometric-optical system, if the rays enter a Glassmedium and the Destination plane is introduced before specifying an exit
surface, then the emergent rays will remain within the last Glass medium, whoserefractive index becomes the index of the new environment for the emergent
beam.
Within a geometric-optical system, the gradient-index (GRIN) media can haveboth radial and longitudinal gradients. Version 7.9 raises the order of thepolynomial that represents the longitudinal gradient from 6 to 12. This featureallows GRADIUM™ glasses to be simulated within DIFFRACT’s Geometricoptical_System environment. The combination of cylindrical surfaces and radial
GRIN media also enables the simulation of special lenses (such as Doric™) usedin laser diode collimating optics.
File Management/Graphics: A new option allows the contents of DIFFRACT’s
Graphics Window to be saved to a user-specified bitmap file XXXX.BMP. Also,
when adding two data files, the second one can now be multiplied by a userdefined constant. If, for example, the value of this constant is –1, then thesecond file will be subtracted from the first.
Phase/Amplitude Mask: In addition to circular, rectangular, and annular masks,
version 7.9 allows elliptical masks of arbitrary orientation to be placed in the
XY-plane. Also, the transition between adjacent regions of the various masks can
now be smoothed out through the specification of a new parameter, α. When α
is set to zero the transition will be sharp. However, when 0 < α ≤ 1, thephase/amplitude values of the mask in one region blend into the phase/amplitude
values of the neighboring region at a rate determined by the value of the αparameter.
Process Signals: The assignment operation (=) now allows the mesh parameters(NMAX, NMAY, LMAX, LMAY) to be assigned to the SIGNAL elements.
xxTypically, when Assignment (=) is requested, the user is prompted for thelocation ‘i’ within the SIGNAL array and the value C of the constant to be
assigned to that location. If, instead of entering a number, the user types NMAX,
NMAY, LMAX, or LMAY, the current value of the chosen parameter will beassigned to element ‘i’ of the SIGNAL.
$ Commands: A new command ($MAKE) is added to the set of $ commands thatare used when programming the COMMAND.DAT file for non-interactive
sessions. The syntax of the $MAKE command is$MAKE#ln,ITEMi~#lm,ITEM j:
The function of this command is to modify the value of an item on a given lineof COMMAND.DAT by an amount depending on the value of another item on thesame line or on a different line. The MAKE statement appears immediately after
the $ sign; 'ln' is a 2-digit line number in COMMAND.DAT, which specifies theline targeted for modification; 'i' is a 1-digit number which specifies thetargeted item within that line (i = 1, 2, 3); ‘~’ is one of the ten operators +, −, *,/, =, E, L, S, C, T, which stand for addition, subtraction, multiplication, division,
setting equal to, exponential function, natural logarithm, sine, cosine, andtangent. The 2-digit line number 'lm' and the 1-digit item number 'j' specify theitem within COMMAND.DAT that determines the amount of change in thetargeted item. In the process only the first item (i) is modified; the second item(j) remains intact.
Multiple Directories: At the start of a session, DIFFRACT asks for the name ofthe subdirectory in which all the needed files reside (and all new files that willbe created during the session will be stored). In Version 7.9, the user can specify
up to 20 subdirectories. When DIFFRACT operates in the non-interactive ZOOM
mode, the COMMAND.DAT files residing in the specified subdirectories will besequentially executed. The program comes to a halt only after the last
COMMAND.DAT has been executed. This feature allows multiple batch jobs to besubmitted all at once.
Templates: The new <TEMPLATES > subdirectory located in DIFFRACT’s home
directory contains templates for constructing your own COMMAND.DAT,XXXX.SYS, and XXXX.STK files. For example, you may open on your monitor
TEMPLATE.DAT and COMMAND.DAT files side by side, then copy desiredcommand blocks and paste them (in a logical order) onto the (initially empty)xxi

COMMAND.DAT file. Proceed to modify the various command parameters until

the elements of your system are properly represented within the newly-created
COMMAND.DAT. When all the elements of the system are in place, you may
want to copy one or more $commands to appropriate locations within thecompleted COMMAND.DAT file. These $commands enable the creation of loops(or other programming constructs) for the purpose of repeating a given
simulation with different sets of parameters in non-interactive sessions.
In similar fashion, you can construct new XXXX.SYS files for importation to theGeometric-optical_System feature of DIFFRACT, or build new XXXX.STK files
for use in conjunction with the Multilayer option. Copy GEOP.SYS or MLYR.STK
from <TEMPLATES> to a working subdirectory, then open this text file, andfollow the instructions given in the file itself.
More Examples of DIFFRACT Applications: The new manual contains many
more examples of DIFFRACT applications. Also, during the last several years, wehave written over 40 journal articles to describe the varied applications of
DIFFRACT. A bound collection of these articles – which were originallypublished in Optics & Photonics News – is included in the DIFFRACT package,along with the program’s manual and a distribution CD-ROM. The distribution
disk contains DIFFRACT’s executable code as well as an extensive set of test
runs that can be used as guides for setting up and running various
COMMAND.DAT files. The new CD-ROM also includes an <OPN> directory,
which contains many resource files extracted from the optical systems originallydescribed in the OPN articles.

xxii

Guide to DIFFRACT

1

1. Introduction
DIFFRACT™ is a Fortran-based program for calculating diffraction patterns at
various cross-sections of the optical path in a coherent optical system. Details of
the analytical and numerical techniques used in this program as well as several
application examples may be found in the references cited at the end of thismanual. References 3 and 5, in particular, contain the basic theory behind
DIFFRACT, and are included in this manual as Appendices H and I.
The user of DIFFRACT defines the optical path and decides the values of all the
relevant parameters. This is done by choosing from a list of options and byanswering questions posed by the program during an interactive session. The
questions asked by DIFFRACT are straightforward and easy to answer; in case ofdifficulty, however, the user may consult the accompanying examples and test
runs for guidance. In addition, on-line information about every aspect of the
program is available and can be accessed via the HELP command from the main
menu. The same guiding information also appears automatically in a backgroundscreen during interactive sessions. The following paragraphs contain some
introductory remarks and general information about DIFFRACT™.
1.1. Home Directory and Working Subdirectories. DIFFRACT’s home
directory, created during the program installation, contains the executable
DIFFRACT.EXE which you invoke every time that you run the program. Also
residing in this directory are the files EXTRACT.EXE, SETTINGS.EXE,
SETTINGS.LOG, README.DOC, and LOGO.BMP. In addition, there are four
subdirectories: <RAINBOW>, <STACKS>, <SYSTEMS>, and <TEMPLATES>. The
first subdirectory contains the driver for the hard_key. The second and third
subdirectories contain, respectively, examples of multilayer stacks andgeometric-optical systems which you might find useful in getting acquainted
with some of the features of the program. The last subdirectory contains
templates for creating various data files that DIFFRACT can read and interpret.
All data-files created during a session of DIFFRACT will be recorded within aworking subdirectory, and all data-files needed by the program must residewithin the same subdirectory. The user is prompted to specify the name of thisworking subdirectory at the outset. If the name is new, the program creates thesubdirectory and makes it home directory for the entire session. If, on the other
hand, the name belongs to an existing directory, the user is warned that some of
the files resident within that directory might be subsequently renamed or2Guide to DIFFRACT

deleted. The user can then accept this option or specify a different working
subdirectory. It is also possible to specify more than one subdirectory (separated
by blanks or commas) in the beginning of a session. The program then begins in
the first specified directory and, upon completing the run, moves to the second
directory, where it starts a new run. The process continues until all specified
directories are exhausted. This multiple-directory capability of the program isparticularly useful for multiple batch runs in the ZOOM mode, where each run
has its own COMMAND.DAT in a separate directory. (See section 2, Interactive
and Non-interactive Sessions.)
1.2. Coordinate System and the Mesh: A coherent, monochromatic amplitude
distribution is defined in the XY-plane at Z = 0. This distribution defines the
cross-section of a beam propagating along the Z-axis. The discrete meshrepresenting a two-dimensional sampling of this distribution on a rectangularlattice is shown in Fig. 1. There are NMAX pixels along X and NMAY pixels
along Y. The mesh is centered at (x, y) = (0, 0), and its physical dimensions along
X and Y are LMAX and LMAY, respectively.
Note: DIFFRACT allows the user to specify the lengths (including LMAX andLMAY) in units of centimeter cm, millimeter mm, micrometer um, nanometer
nm, or normalized by the beam’s vacuum wavelength λo. (For instance, whenλo = 0.8 µm, a length of 4 mm in normalized form is represented as 5000.)
Since the beam typically expands or shrinks as it propagates, the length units
that are most convenient at one location may become cumbersome at otherlocations. To maintain flexibility, DIFFRACT prompts the user for the desired
length units at the start of each command. In general, at any given location, amonochromatic beam having vacuum wavelength λo will be within ahomogeneous environment of refractive index NVIRON. Internally, DIFFRACT
normalizes all lengths by the wavelength λ within the current environment of
the beam (λ = λo/NVIRON), but for input/output purposes the lengths will bein user-specified units.
The X- and Y-polarization components of the distribution are contained in arrays
AX and AY, respectively. AX and AY are complex two-dimensional arrays, withidentical dimensions NMAX × NMAY. The maximum allowed value for both

Guide to DIFFRACT

3

NMAX and NMAY is set in the beginning of the program by the user through the
choice of the parameter NDIM.†

As the beam propagates along Z and encounters various optical elements (i.e.,
lenses, gratings, apertures, polarizers, retarders, mirrors, multilayer stacks,
optical disks, detectors, etc.), the contents of AX, AY are updated. In other words,
at any given time during the computations, AX and AY contain the amplitude
distribution at the latest cross-section of the system.

Figure 1. Discrete mesh in the XY plane. The beam propagates along the positive Z-axis,
where it encounters various optical elements. The number of pixels in the mesh,
NMAX × NMAY, remains constant at all cross-sections of the optical path. The physicaldimensions of the mesh, LMAX × LMAY, however, undergo automatic adjustments as thebeam propagates through the system.

†

AX and AY are actually NDIM × NDIM square arrays that reside in the random accessmemory (RAM) of the machine. The user is prompted at the outset to specify NDIM, so thatsufficient memory will be allocated for these arrays. NDIM must be greater than or equal tothe intended values of both NMAX and NMAY, which the user will subsequently specify.
The presently set upper bound on NDIM is 256 (Student Edition) and 2048 (General Edition);in the latter case the user can increase the upper bound by modifying the SETTINGS.LOG file.
The maximum useful value of NDIM is ultimately limited by the available memory of thecomputer.

4

Guide to DIFFRACT

Certain useful information about the mesh and its contents are frequently
updated and displayed during the session. Here is an example of the displayed
information.
Home: <Working Directory>
Vacuum wavelength: 0.633000E+03 nm
Index of local environment (NVIRON) = 1.00000
Length_Units: mm
LMAX = 30.00
LMAY = 30.00
NMAX = 512
NMAY = 512
Beam centroid: (X = -0.650E-03, Y = 0.203E-02)
DIAM = 0.5320E+01
XPOWER(mW) = 0.950 YPOWER(mW) = 0.500E-02 Curvature = 0.2000E-2
Generally, the beam is within an isotropic, homogeneous, transparent medium of
refractive index n ≥ 1, known as the “environment.” The index of theenvironment (NVIRON) always appears in the above table, along with thevacuum wavelength of the beam and the length units that are currently in effect.
Beam centroid is the center of mass, and DIAM is the r.m.s. diameter of theintensity distribution in the XY-plane. (Since the Z-component of polarization isusually not available to the program, Centroid and DIAM represent only the Xand Y-components of polarization.) XPOWER and YPOWER (in units of mW) are
the total integrated intensity of the X- and Y-components of polarization within
the mesh. If the beam happens to have a curvature that is carried by the program
outside the mesh,‡ the magnitude of this curvature will also appear in the abovetable. (Curvature, which is the inverse of the beam’s radius of curvature, ispositive for a diverging beam and negative for a converging beam.) Aside fromthe vacuum wavelength which has its own units, all lengths such as LMAX,
LMAY, centroid coordinates, DIAM, radius of curvature, etc., have theLength_Units specified in the table (millimeters in the above example). If theLength_Units are left blank, the various lengths will be dimensionless, that is,normalized by the vacuum wavelength.

‡

In certain circumstances the beam acquires a curvature, yet it remains feasible to separate thecorresponding phase-factor from the distribution and carry it outside the mesh. For example,
when a beam is propagated to the far field (either by the Fraunhofer formula or by theextended Fresnel method), or upon Ray_Tracing through a lens, the emerging beam isgenerally endowed with a curvature. In all such cases, the program keeps track of thecurvature outside the mesh, so that the corresponding phase-factor need not be multiplied into
the AX, AY arrays. The advantage of this approach is that, when the curvature is large, thenumber of pixels in the mesh does not have to become excessively large in order toaccommodate the rapid oscillations of the wavefront.

Guide to DIFFRACT

5

1.3. Save and Read: To store the current contents of the mesh before
proceeding further, use the SAVE option. With the aid of the READ option, astored array can be read back into the program, either later in the same run or
during a subsequent run, in which case it serves as the starting distribution. Also,using the INTERFERE option, a saved array may be used for interference withanother distribution. If you do not plan to use a given cross-section of the beam,
however, save time and disk space by avoiding the use of SAVE.
1.4. The Plot Option: To obtain plots of intensity, log_intensity, phase,polarization, or ray_distribution at any given cross-section of the system, use
the PLOT option from the main menu. The intensity distribution IX(x, y) for the
X-component of polarization is |AX(x, y)|2. Similarly, IY(x, y) = |AY(x, y)|2 is theintensity distribution for the Y-component. The phase distributions are PX(x, y)
and PY(x, y). The angle between the major axis of the ellipse of polarization and
the X-axis is RT(x, y) where RT stands for rotation angle; polarization ellipticity
is EL(x, y). The geometric-optical rays are specified in terms of their Cartesiancomponents RX(x, y), RY(x, y), and RZ(x, y).
When plotting log_intensity you will be prompted for SCALE, a parameter thatsets the cutoff value at the low-end of the logarithmic scale. The program
normalizes the intensity distribution, say, IX(x, y), by its peak value, Imax, thencomputes the logarithm (base 10) of the normalized intensity distribution. The
peak value of the logarithmic distribution is thus 0, while its lowest value is cut
off at −SCALE. Thus, for example, if SCALE is set to its default value of 4, thelogarithm of normalized intensity in the interval [0.0001, 1] is computed anddisplayed.
When plotting intensity, log_intensity, or phase, the user has the option to plot
the Z-component of the distribution (in addition to the usual X- and Ycomponents). Because the Z-component is usually not carried along in the mesh,its computation requires two FFT's, which is time-consuming. Do not use thisoption unless you really want to see the Z-component. When plotting isrequested, the Z-component is computed and placed in the DSPZ array. This
array may subsequently be committed to disk upon request by the user.
The plotting routine will not provide the option to plot the ray_distribution
corresponding to a given complex-amplitude distribution unless the rays havealready been generated. (To generate the rays you must choose the RAYS option
from the main menu.) Once the rays are computed, you may call the plotting
routine and produce a plot of their X-, Y-, and Z-components.

Guide to DIFFRACT

6

When the PLOT option is selected, the distributions in the interval [Xmin : Xmax,
Ymin : Ymax] are computed and stored in arrays DSPX, DSPY, DSPZ, which may
subsequently be written to disk as files IXnn.DAT, IYnn.DAT, IZnn.DAT, or
PXnn.DAT, PYnn.DAT, PZnn.DAT, or RTnn.DAT, ELnn.DAT, or RXnn.DAT, RYnn.DAT,
RZnn.DAT. The arrays DSPX, DSPY and DSPZ are square two-dimensional arrays,
each with NDSP rows and NDSP columns.§ Before committing DSPX, DSPY, and
DSPZ to disk, the requested plots are displayed on the monitor. The user is given
the option to visualize the data either in black-and-white or in color. The grayscale has 256 levels between black (minimum value of the function) and white(maximum value). Color displays also use 256 levels, starting with blue (for theminimum value) and progressing through colors of the rainbow to red(maximum value).
Multiple plots are allowed in DIFFRACT by assigning two-digit ID numbers to
the data files. For each set of plots, the user is prompted for an integer ID
between 0 and 99. The corresponding files are then tagged with this ID, forexample, IXnn.DAT or RTnn.DAT, where nn is the assigned ID (from 00 to 99). In
the interactive mode, if a file with the same name and ID happens to reside on
the disk, the user will be prompted for a different ID. In the non-interactivemodes (both the STEP mode and the ZOOM mode) an existing file having thesame name and ID as the requested file will be overwritten.
A typical data file such as IXnn.DAT is created with the following set of Fortran
instructions:
CHARACTER FNAME*8,FRMT*10
FNAME=‘IX .DAT’
WRITE (FNAME(3:4),‘(I2.2)’)ID
OPEN (UNIT=21, FILE=FNAME, STATUS='NEW')FRMT='(
E14.6)'WRITE (FRMT(2:4),'(I3)')NDSP
DO J=1,NDSP
WRITE (21,FRMT) (DSPX(I,J), I=1,NDSP)END DOCLOSE (UNIT=21, STATUS='KEEP')§The user will be prompted at the outset to specify NDSP, so that data-files created by the
PLOT option will have the user’s desired dimensions. NDSP is upper bounded by the linear
dimension, NDSP0, allocated within the program to DSPX, DSPY, DSPZ arrays. At present
NDSP0 is set to 150, but the user may change it within the SETTINGS.LOG file to any desired
value.

Guide to DIFFRACT

7

A similar set of instructions is used for creating IYnn.DAT, IZnn.DAT, PXnn.DAT,
PYnn.DAT, PZnn.DAT, RTnn.DAT, ELnn.DAT, RXnn.DAT, RYnn.DAT, and RZnn.DAT.
Aside from a header box (described below), these ASCII files each contain asquare array of real numbers, and are readily accessible by most graphics
packages. Once stripped of the header box and imported to a graphics
environment, these arrays can be displayed as 3D, contour, gray-scale, colorcoded, or any other type of plot that the user may desire.
A typical header box (written at the top of a PLOT data-file) is shown below:*********************************************************************** NVIRON= 1.50000
Vacuum wavelength=0.650000E+03 nm**** Number of Rows= 101
Number of Columns= 101**** IXmin= 0.2773E-03
IXmax= 0.1589E+03**** Xmin= -0.500000E+01
Xmax= 0.500000E+01
Log10_Scale: 4.00 **** Ymin= -0.500000E+01
Ymax= 0.500000E+01
Length_Units: mm***********************************************************************The meaning of most elements in the header box is self-evident. The exception
perhaps is the logarithmic scale-parameter Log10_Scale (always ≥ 0). If
SCALE = 0, the data-file is not logarithmic; otherwise it is, and its content covers
the range [−SCALE : 0]. Note that the min/max values shown in line 3 of theheader box correspond to the values of the data before the logarithm was taken.
The actual range of the data contained in the file is thus [−SCALE : 0], differingfrom the listed min/max values. The vacuum wavelength (λo) has its own unitsof cm, mm, um, nm (indicated on line 1), whereas the Length_Units, being cm,
mm, um, nm, or blank (indicating normalization by λo) apply to [Xmin : Xmax,
Ymin : Ymax]. The nature of the data is specified by the 2-letter labels associatedwith the min/max values: IX, IY, IZ represent intensity; PX, PY, PZ represent
phase; RT, EL represent polarization rotation angle and ellipticity; RX, RY, RZ
represent ray distributions, and SP is used for Optical_Disk surface pattern.
1.5. Internal Arrays and Memory Requirements: There are six large arrays in
DIFFRACT that account for the bulk of its memory requirements. These are
two-dimensional, complex-valued arrays AX, AY, WRKX, WRKY, BX, BY. Each ofthese arrays has NDIM × NDIM elements, thus occupying 8 × NDIM2 bytes ofrandom-access memory (RAM). For example, if you set NDIM = 512 in the
beginning of the program, you will be allocating 6 × 8 × 5122 bytes (or 12 MB)
of RAM to these arrays.

Guide to DIFFRACT8

The complex amplitude of the beam at a given cross-section of the system isgenerally stored in AX and AY (one array for each polarization component).**WRKX and WRKY are work-arrays that keep the intermediate results during acomputation. BX and BY are used only for ray-tracing; that is why they can beeliminated at the outset when the user declares his intention to not invoke any
ray-tracing options.

AX

AY

WRKX

DSPX

WRKY

DSPY

BX

BY

DSPZ

SIGNAL
Figure 2. Schematic diagram showing the internal arrays of DIFFRACT. The six arrays
shown in the top row are square, complex-valued, NDIM × NDIM arrays. The value of
NDIM is chosen by the user in the beginning of the program. (Arrays BX and BY may beeliminated if the user declares his intention to not use ray_tracing during a given session.) The
three arrays in the middle row are square, real-valued, NDSP0 × NDSP0 arrays. (The value of
NDSP0 is presently set to 150.) These arrays are mainly used for plotting and displaying thedata, but they are also needed as temporary storage by certain other routines. The SIGNAL
array shown at the bottom is a one-dimensional, real-valued array, capable of storing up to
100 data items. The outputs of the various detectors as well as certain characteristic
parameters of the beam are assigned to this array by the user. The contents of SIGNAL can be
manipulated by the Process_Signals option, and selected elements from this array may be
printed to the SIGNALS.DAT file.

There are also three smaller arrays, used for displays and plots as well as for avariety of intermediate calculations. These are two-dimensional, real-valued
arrays DSPX, DSPY, and DSPZ. Each of these arrays has NDSP0 × NDSP0
elements. Since NDSP0 is presently set to 150, the display arrays occupy a totalof 264 KB of RAM. Note that the user decides at the outset the array dimensions**

The Z-component of polarization is usually not carried in a separate array because, whennecessary, it can be computed, on the spot, from the X- and Y-components. Once computed,
however, the Z-component will reside in WRKX.

Guide to DIFFRACT

9

for displays and plots. However, the user-specified parameter for this purpose,
NDSP, does not affect the internal size of the display arrays, since the program
needs the full-sized (NDSP0 × NDSP0) arrays for other purposes. The
specification of NDSP by the user affects only the size of the files generated by
the PLOT option.
The outputs of the various detectors as well as certain characteristics of the beam
are assigned to an internal array called SIGNAL. This real-valued, onedimensional array has 100 elements, which are identified by a 2-digit ID numberthat runs from 00 to 99. The Process_Signals (PSIG) option of DIFFRACT™
allows access to this array, permitting the user to perform various operations(e.g., add, subtract, divide, multiply) on its contents before printing selected
items to an external file.
1.6. Data Files Created by DIFFRACT: During a session, DIFFRACT creates anumber of files and stores them in the working directory on the hard disk.
Figure 3 shows the complement of files that can be created, although not every
file in this diagram will be present at the end of a given session; the files that are
actually created depend on the options chosen by the user.
The input/output information during a given run is stored in the text file
SESSION.DAT. As an example, for the simple optical system whose schematic
diagram is shown in Fig. 4, a printout of SESSION.DAT appears in Fig. 5(a). You
may print SESSION.DAT at the end of each run in order to have a record of thesession.
Note: If a SESSION.DAT file happens to reside in the working directory, it will
be renamed SESSION.nnn at the start of a new run. The extension nnn will be thesmallest 3-digit number (in the range of 000 to 999) for which a corresponding
file does not exist within the working directory.
The sequence of key-strokes during a run is stored in COMMAND.DAT. This is atext file that can be edited by the user with the aid of a text editor. During
interactive sessions the user will be given the option to create COMMAND.DAT.
(For the system of Fig. 4 the corresponding file is shown in Fig. 5(b).) During
non-interactive sessions, DIFFRACT reads the COMMAND.DAT file from theworking directory and executes its commands as if they were entered from the
keyboard. (For more information about COMMAND.DAT and its various features
see Section 2.)

Guide to DIFFRACT10SESSION.DAT

SIGNALS.DATCOMMAND.DAT

XXXX.DST

IXnn.DATIYnn.DATIZnn.DAT

PXnn.DATPYnn.DAT

PZnn.DAT

RTnn.DAT

ELnn.DAT

RXnn.DATRYnn.DAT

RZnn.DAT

SPnn.DAT

ISnn.BMP

PSnn.BMP

QSnn.BMP

RSnn.BMP

DSnn.BMP

IXnn.TIF

IYnn.TIF

IZnn.TIF

PXnn.TIF

PYnn.TIF

PZnn.TIF

RTnn.TIF

ELnn.TIF

RXnn.TIF

RYnn.TIF

RZnn.TIF

XXXX.STK

SPnn.TIF

XXXX.SYS

Figure 3. Diagram showing the complement of files that DIFFRACT can create. These files
reside in the working directory, and may be overwritten in subsequent runs of the program,
especially during non-interactive runs. The user must rename or remove from the working
directory those files that he/she intends to keep.

Guide to DIFFRACT11

The file SIGNALS.DAT is created during non-interactive sessions only. It
contains those elements of the SIGNAL array that are specifically selected forprinting to this file (see Section 5.5 for more information on signal processing).
The data in SIGNALS.DAT is arranged into four columns. The first column shows
the location of the selected element within the SIGNAL array. The second column
is the actual value of the SIGNAL at the specified location. The third and fourthcolumns are the values of the tags that are in effect at the time of printing to thefile (see Section 2 for more information on tags). SIGNALS.DAT provides alisting of all output signals for the system under consideration, along with thosevalues of the system parameters that are associated with each output. From thisfile, the user may extract the individual signals and their associated tags forplotting purposes.
The entire distribution at a given cross-section of the system, namely, thecontents of the mesh, may be stored in the hard disk file XXXX.DST. (Thefilename extension DST stands for distribution.) The menu option SAVE allows
the user to select the filename XXXX. (In the interactive mode DIFFRACT does
not accept the name of an existing file and, in case of duplication, prompts theuser for an alternative filename. In the non-interactive mode, however, an
existing file will be overwritten without warning.) These binary DST files cannot
be modified by the user, nor can they be read into other programs. The files
created by the SAVE option are accessible only to DIFFRACT and should not be
used for other purposes (e.g., graphics).
Data for plotting various distributions will be stored in one or more of thefollowing files: IXnn.DAT, IYnn.DAT, IZnn.DAT, PXnn.DAT, PYnn.DAT, PZnn.DAT,
RTnn.DAT, ELnn.DAT, RXnn.DAT, RYnn.DAT, RZnn.DAT, and SPnn.DAT. IX is theintensity distribution (in the XY-plane) for the polarization component along X;
IY and IZ are intensity patterns for polarization along Y and Z; etc. If the beam
is linearly polarized along one of the coordinate axes, either IX or IY, but notboth, will be generated. IZ will be created if requested by the user, or if the Zcomponent of polarization happens to be available from a previous calculation.
The same considerations apply to PX, PY, and PZ. Also generated upon request isSPnn.DAT, which is the pattern of grooves, bumps, pits and magnetic domains on
an Optical_Disk_Surface (SP stands for surface pattern); for information on the
Optical_Disk_Surface option see Section 14. The files created for plotting
purposes have identical format. Instructions used for creating these files are thesame as those given for IXnn.DAT in Section 1.4 (see page 6).

Guide to DIFFRACT

12

Figure 4(a). Schematic diagram of a simple optical system. A truncated Gaussian beam isincident on an objective lens, which brings the beam to focus on the grooved surface of an
optical disk. The reflected beam returns through the same lens and is directed by a beamsplitter towards a four-quadrant detector.

+4

y/λ

−4

−4

x/λ

+4

−4

x/λ

+4

−4

x/λ

+4

Figure 4(b). Logarithmic plots of intensity distribution at the focal plane of the objective
lens. The incident beam is linearly polarized at 45° to the X-axis. The comatic shape of the
focused spot is due to a quarter-wave of third order coma placed on the lens. From left toright: log_intensity for the X-, Y-, and Z-components of polarization. The peak intensities arein the ratio of 1: 1: 0.07.

Guide to DIFFRACT

+5

y/λ

−5

13

3200

y/λ

−3200

x/λ
+5
−3200
x/λ
3200
−5
Figure 4(c). Grooved pattern of the disk surface (left), and the computed intensity distribution
at the plane of the detector (right).

Upon selecting the PLOT option, the requested distributions are displayed on themonitor, both during interactive sessions as well as during non-interactive runs
in which the user chooses to STEP through COMMAND.DAT. The displayed
screen may be saved as a bitmap file within the working directory. Whenever theuser requests saving of the data file(s), the screen is automatically dumped to acorresponding bitmap file. Screens displaying intensity and log_intensity
distributions, for example, will be dumped to ISnn.BMP, where I stands for
Intensity, S for Screen, nn is the 2-digit user-defined ID number given to thecorresponding data files, and BMP is the standard filename extension for bitmap
files. In similar fashion, PSnn.BMP is the screen dump for phase distributions,
QSnn.BMP corresponds to polarization patterns, RSnn.BMP corresponds to ray
distributions, and DSnn.BMP is the screen dump for Optical_Disk_Surface
patterns.
The File_Management/Graphics option of DIFFRACT allows the user to load anyof the graphic data-files (i.e., those having a .DAT extension) into a temporary
array, convert them to TIFF format, and store the resulting files on the hard disk
as XXXX.TIF. These TIFF files may later be imported into other programs (suchas the Microsoft Word®) for incorporation into documents.
If you call the Multilayer option to place a multilayer stack in the beam path,
you will be given the option to store the stack parameters on hard disk in a file
named XXXX.STK. Similarly, the Geometric-optical_System option allows theuser to store in XXXX.SYS a complete description of the optical system throughwhich rays are traced. In both cases, the choice of the filename (XXXX) is withthe user. Both the .STK and .SYS files are text files, which the user can modify

Guide to DIFFRACT

14

using a text editor. These files can be readily imported into DIFFRACT during
interactive sessions, thus eliminating the tedious task of entering the stackparameters or the optical system parameters from the keyboard. As for noninteractive sessions, the .STK and .SYS files can be inserted into the
COMMAND.DAT file once their headers have been removed.
Figure 5(a). SESSION.DAT created by DIFFRACT during an interactive session in which
the system of Fig. 4(a) was simulated. The focused spot was stored in a file named SPOT.DST
for later use. As far as the present simulation is concerned, there was no need to retrieve
SPOT.DST immediately after storing it, since SAVE does not affect the contents of the mesh.
The retrieval was done, however, in preparation for the recursive, non-interactive run
described in Section 2.
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 8.4
File SESSION.DAT $$$$ Date: 06/13/2005
Time: 13:15 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$_______________________________________________________DIMENSIONLESS LENGTHS ARE IN UNITS OF VACUUM WAVELENGTH
ALL ANGLES ARE IN DEGREES_______________________________________________________
Array dimensions: NDIM= 512, NDSP= 149**************************************
Designated Working Subdirectory: <TEST01>*****************************************_________________________________________________________________
Vacuum lambda (um)= 1.0000
NVIRON= 1.0000 Length_Units: um
LMAX=
10000.0
LMAY=
10000.0
NMAX= 512
NMAY= 512_________________________________________________________________
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Length Units: None (normalized by vacuum wavelength)____________________________________________________
Simple Gaussian: Center coordinates: BCX=
0.00000
BCY=
0.00000
Power(mW)=
1.00000
RA=
3000.00
THETA= 0.00
RMIN=
4000.00
RMAX=
4000.00
Aberrations: None
Polarization state: Rho= 45.0000
Eta= 0.0000_________________________________________________________________
Vacuum lambda (nm)= 780.00
NVIRON= 1.0000 Length_Units:
LMAX=
40000.0
LMAY=
40000.0
NMAX= 512
NMAY= 512
Beam centroid:(X= 0.49573E-06, Y=-0.65586E-05) DIAM=0.542316E+04
XPOWER(mw)= 0.50000
YPOWER(mw)= 0.50000 Curvature= 0.00000E+00_________________________________________________________________LENS ACTION***********
Length Units: None (normalized by vacuum wavelength)____________________________________________________
TYPE: Positive_Focusing(LCX,LCY)=( 0.0000, 0.0000)
Aberrations: Seidel
Spherical: C40= 0.000000
Coma: C31= 0.250000
Phi31= 45.00
Astigmatism: C22= 0.000000
Phi22=

NA=0.500000

0.00

FL=

6000.00

Guide to DIFFRACT

15

Curvature:
C20= 0.000000
Distortion: C11= 0.000000
Phi11=
0.00
Quasi-vector treatment
Propagation distance=
6000.00__________________________________________________APRX method: NGUIDE=
100 LGUIDE= 0.30000E+05___________________________________________________________________________________________________________________
Vacuum lambda (nm)= 780.00
NVIRON= 1.0000 Length_Units:
LMAX=
76.8000
LMAY=
76.8000
NMAX= 512
NMAY= 512
Beam centroid:(X= 0.27886E+00, Y= 0.27884E+00) DIAM=0.790450E+01
XPOWER(mw)= 0.50082 YPOWER(mw)= 0.50082
Curvature= 0.00000E+00_________________________________________________________________
PLOTTING ROUTINE****************
Length Units: None (normalized by vacuum wavelength)____________________________________________________
Base 10 logarithm of normalized intensity distribution(Range: from 0 at peak_intensity to -4.0000 at cutoff)___________________________________________
Xmin= -0.400000E+01
Xmax= 0.400000E+01
Ymin= -0.400000E+01
Ymax= 0.400000E+01___________________________________________
IXmin=
IYmin=
IZmin=

0.8588E-06
0.8588E-06
0.2008E-08

IXmax=
IYmax=
IZmax=

0.3626E+00
0.3626E+00
0.2586E-01

Latest screen (intensity pattern) dumped in IS00.BMP
Intensity pattern for X-polrz stored in IX00.DAT
Intensity pattern for Y-polrz stored in IY00.DAT
Intensity pattern for Z-polrz stored in IZ00.DAT
Peak intensity=0.727E+00 XPEAK= 0.150000
YPEAK= 0.150000(Total energy flux=0.100E+01
ZPOWER=0.555E-01)
STORE DISTRIBUTION******************
Distribution stored in SPOT.DSTRETRIEVE DISTRIBUTION*********************
Distribution retrieved from SPOT.DST_________________________________________________________________
Vacuum lambda (nm)= 780.00
NVIRON= 1.0000 Length_Units:
LMAX=
76.8000
LMAY=
76.8000
NMAX= 512
NMAY= 512
Beam centroid:(X= 0.27886E+00, Y= 0.27884E+00) DIAM=0.790450E+01
XPOWER(mw)= 0.50082 YPOWER(mw)= 0.50082
Curvature= 0.00000E+00_________________________________________________________________OPTICAL DISK SURFACE********************
Length Units: None (normalized by vacuum wavelength)____________________________________________________
Disk type: WRITABLE
Aperture: SCX= 0.0000
SCY= 0.0000
Air-incidence configuration (Ninc= 1.000000)

PREGROOVES__________

RA=

20.0000

Guide to DIFFRACT

16
A= 0.1500E+00
ZETA= 0.1250E+00

B= 0.5000E+00
THETA=
45.00

C= 0.6500E+00
GCX= 0.0000E+00

D= 0.2000E+01
GCY= 0.0000E+00

Amplitude reflection coefficient on land center (RL): 0.8000
Amplitude reflection coefficient on groove center (RG): 0.8000
Central groove-edge's wobble Amplitude, Period, Phase Step:
A0= 0.0000E+00, P0= 0.0000E+00, Q0=
0.00
Adjacent groove-edge's wobble Amplitude, Period, Relative Phase:A1= 0.0000E+00, P1= 0.0000E+00, Q1=
0.00
Latest screen (disk_surface_pattern) dumped in DS00.BMP_____________________________________
Surface pattern saved in SP00.DAT
Xmin=-0.50000E+01
Xmax= 0.50000E+01
Ymin=-0.50000E+01
Ymax= 0.50000E+01_____________________________________
Propagation distance=
6000.00
Propagation regime: Fraunhofer_______________________________________________________NGUIDE=
999 LGUIDE= 0.200000E+03 SMAX= 0.500000________________________________________________________________________________________________________________________
Vacuum lambda (nm)= 780.00
NVIRON= 1.0000 Length_Units:
LMAX=
40000.0
LMAY=
40000.0
NMAX= 512
NMAY= 512
Beam centroid:(X= 0.43066E+03, Y=-0.42816E+03) DIAM=0.130782E+05
XPOWER(mw)= 0.27875
YPOWER(mw)= 0.27887 Curvature= 0.16667E-03_________________________________________________________________LENS ACTION***********
Length Units: None (normalized by vacuum wavelength)____________________________________________________
TYPE: Collimator(LCX,LCY)=(0.0000, 0.0000)
NA=0.500000
FL=
6000.00
Aberrations: Seidel
Spherical: C40=
0.000000
Coma: C31=
0.250000
Phi31= 45.00
Astigmatism: C22= 0.000000
Phi22=
0.00
Curvature: C20=
0.000000
Distortion: C11= 0.000000
Phi11=
0.00
Quasi-vector treatment_________________________________________________________________
Vacuum lambda (nm)= 780.00
NVIRON= 1.0000 Length_Units:
LMAX=
40000.0
LMAY=
40000.0
NMAX= 512
NMAY= 512
Beam centroid:(X= 0.47404E+03, Y=-0.47875E+03) DIAM=0.507563E+04
XPOWER(mw)= 0.21130
YPOWER(mw)= 0.21133 Curvature= 0.00000E+00_________________________________________________________________PROPAGATION IN ENVIRONMENT**************************
Length Units: None (normalized by vacuum wavelength)____________________________________________________
Propagation distance=
20000.00(Forward)
Beam is NOT repositioned
Propagation regime: Direct Fresnel_______________________________________________________NGUIDE=84 LGUIDE= 0.253782E+05 SMAX= 0.000167_______________________________________________________
Quasi-vector treatment_________________________________________________________________
Vacuum lambda (nm)= 780.00
NVIRON= 1.0000 Length_Units:Guide to DIFFRACT

17

LMAX=
40000.0
LMAY=
40000.0
NMAX= 512
NMAY= 512
Beam centroid:(X= 0.47389E+03, Y=-0.47857E+03) DIAM=0.507552E+04
XPOWER(mw)= 0.21130
YPOWER(mw)= 0.21133 Curvature= 0.00000E+00_________________________________________________________________
PLOTTING ROUTINE****************
Length Units: None (normalized by vacuum wavelength)____________________________________________________
Intensity distribution___________________________________________
Xmin= -0.320000E+04
Xmax= 0.320000E+04
Ymin= -0.320000E+04
Ymax= 0.320000E+04___________________________________________
IXmin=
IYmin=

0.9803E-20
0.5343E-20

IXmax=
IYmax=

0.1828E-07
0.1831E-07

Latest screen (intensity pattern) dumped in IS01.BMP
Intensity pattern for X-polrz stored in IX01.DAT
Intensity pattern for Y-polrz stored in IY01.DAT
PHOTODETECTOR*************
Length Units: None (normalized by vacuum wavelength)____________________________________________________
Detector type: Quad
DCX=
LX=

0.00000
5000.00

DCY=
LY=

0.00000
5000.00

Theta= 45.00
GW=
0.00000

Signals S1,S2,S3,S4 saved in SIGNAL array as S(1), S(2), S(3), S(4)S1=0.571E-01
SUM=0.423E+00

S2=0.572E-01
P12M34= -0.45907

S3=0.155E+00
P13M24= 0.00252

SIGNAL PROCESSING*****************
Printing:
Printing:
Printing:
Printing:

S( 1) =
S( 2) =
S( 3) =
S( 4) =

0.571278E-01
0.571815E-01
0.154723E+00
0.153603E+00

End of session: DIFFRACT terminated normally********************************************

S4=0.154E+00
P14M23= -0.00278

18

Guide to DIFFRACT

Figure 5(b). COMMAND.DAT created by DIFFRACT during the interactive session in which
the system of Fig. 4(a) was simulated. All the key-strokes entered by the user appear in
COMMAND.DAT, in exactly the same order in which they were entered. (The SESSION.DAT
file corresponding to this run appears in Fig.5(a).) The first 24 columns in COMMAND.DAT
contain guiding information for the benefit of the user only; they are not used by DIFFRACT
in non-interactive sessions. The colon (:) must always appear in column 25, since it signals tothe program the beginning of a line of data. The # sign must always appear in column 75,
since this is where the program expects to find the line number (if one has been assigned).
The title box (generally ignored by DIFFRACT) contains templates for the $ commands.
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/13/2005
Time: 13:15 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: ) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
1/e radii Rmin,Rmax:
4000.0000
4000.0000#
RA,THETA:
3000.0000
0.0000000#
Aberrations: None#
Polarization RHO,ETA:
45.000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
40000.000
40000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: ) #Type: PFOC#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.5000000
6000.0000#
Aberrations: Seidel#
Spherical C40:
0.0000000#
Coma C31,Phi31:0.2500000
45.000000#
Astigmatism C22,Phi22:0.0000000
0.0000000#
Curvature C20:
0.0000000#
Distortion C11,Phi11:0.0000000
0.0000000#
Scalar/Quasi-vector: QV#
Calculation method: APRX#Guide to DIFFRACT
Propagation distance:
6000.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: ) #Type: Log(intensity)#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-4.0000000
4.0000000#
Ymin,Ymax:-4.0000000
4.0000000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Store distribution: SAVE#
File name: SPOT.DST#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Retrieve a distribution: READ#
File name: SPOT.DST#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units: ) #Type of medium: WR#
SCX,SCY,RA:
0.0000000
000000000
20.000000#(Incidence medium) Ninc:
1.0000000#
Pregrooves: Y#
A,B,C:
0.1500000
0.5000000
0.6500000#
D,ZETA:
2.0000000
0.1250000#
Reflection coeffs. RL,RG:
0.8000000
0.8000000#
THETA,GCX,GCY:45.000000
0.0000000
0.0000000#
Wobble (Sine,Blip,None): None#
Marks (bumps/pits): N#
Inspect surface pattern: Y#
Xmin,Xmax:-5.0000000
5.0000000#
Ymin,Ymax:-5.0000000
5.0000000#
Color or Gray-scale: G#
Save data file (Y/N): Y#
File identifier: 0#
Propagation distance:
6000.0000#
Propagation regime: FRNHF#
Smax:.50000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: ) #Type: COLL#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.5000000
6000.0000#
Aberrations: Seidel#
Spherical C40:
0.0000000#
Coma C31,Phi31:0.2500000
45.000000#
Astigmatism C22,Phi22:0.0000000
0.0000000
#

19

20

Guide to DIFFRACT
Curvature C20:
0.0000000#
Distortion C11,Phi11:0.0000000
0.0000000#
Scalar/Quasi-vector: QV#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Propagate in environment: PROP
(Length_Units: ) #Propagation distance:
20000.000#
Multiply curvature (Y/N): N#
Reposition beam (Y/N): N#
Propagation regime: DFRNL#
Smax:
0.10000000#
Scalar/Quasi-vector: QV#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: ) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-3200.0000
3200.0000#
Ymin,Ymax:-3200.0000
3200.0000#
Color or Gray-scale: C#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Detector: DTCT
(Length_Units: ) #Quad/Split/Phi/Diff: Q#
DCX,DCY,THETA:
0.0000000
0.0000000
45.000000#
LX,LY,GW:
5000.0000
5000.0000
0.0000000#
ID numbers for S1,S2:
1
2#
ID numbers for S3,S4:
34#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Process signals: PSIG#
Action: P#
Print S(i)i:1#Action: P#
Print S(i)i:2#Action: P#
Print S(i)i:3#Action: P#
Print S(i)i:4#Action: E#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Guide to DIFFRACT

21

2. Interactive and Non-interactive Sessions
When you run DIFFRACT, it asks in the beginning whether or not the session isgoing to be interactive. If your answer is positive, all subsequent commands andparameters must be entered from the keyboard. If your answer is negative,
DIFFRACT assumes that all commands and parameters are stored in a file called
COMMAND.DAT; it then proceeds to read that file and execute its commands.
Note: In the non-interactive mode you may choose to STEP throughCOMMAND.DAT or let the program run its course automatically (ZOOM). With
the STEP option the program will pause at the end of each screen and sound atone. You must then press ENTER in order to resume program execution.
2.1. Structure of the COMMAND.DAT File. In the interactive mode you will begiven the option to create COMMAND.DAT, a file which contains all the keystrokes entered during a session. An example of COMMAND.DAT is shown in
Fig. 5(b). The title box of this file contains the syntax of the $ commands, which
are described in the next subsection. After the title box, there appear in columns
26-74 all the key-strokes in the same sequential order in which they have beenentered. The guiding information appearing in columns 1-24 is intended to help
the user, but is not interpreted by DIFFRACT. The colon (:) in column 25
separates the guiding information from the actual key-strokes.
A number sign (#) is printed at the end of each line of COMMAND.DAT. You canassign any 2-digit number (00 - 99) to a given line by placing the number
immediately after # at the end of the line. These numbers are to be used inconjunction with the $ commands that you may choose to incorporate into
COMMAND.DAT. You can change the commands and/or the parameter values
appearing in COMMAND.DAT, so long as the new commands and/or parameters
are acceptable to DIFFRACT. For example, you can change the value of 512
assigned to NMAX in Fig. 5(b) to 256 or to any other acceptable value.
Each command block within COMMAND.DAT has a header box that contains thevalue of vacuum wavelength λo and the index of the environment NVIRON.
There is also space within the header box for the user to write his/her remarks
about the command block that follows. The header box is bounded by two solidlines: the top line begins with a $ sign and both lines end with a # sign. For the
BEAM command, the values of vacuum wavelength λo and NVIRON listed in theheader box are essential specifications. For other commands, however, these are

22

Guide to DIFFRACT

non-essential parameters which are provided as information for the user, andmay be removed (or modified) without any consequences.
Typically, λo remains constant throughout a simulation, although introducing a
new BEAM or reading the distribution from an external file (e.g., via READ or
FDTD commands) can cause a change of the wavelength. DIFFRACT keeps trackof the wavelength, and if the value listed in a header box is found to beinconsistent with the current λo, DIFFRACT issues a warning and proceeds toignore the value listed in the header box. If, however, a new BEAM with adifferent λo is introduced, if λo of the BEAM is changed by a $ command, or if a
new distribution having a different wavelength is imported (via READ or FDTD
commands), DIFFRACT accepts the new value and discards the old one.
The value of NVIRON is either specified in the header box of a BEAM, orimported from an external file along with other parameters of a distribution. As
the beam moves from one environment to another during a simulation, the valueof NVIRON can change accordingly. DIFFRACT keeps track of these changes, andif the value specified in a header box is found to be inconsistent with the current
environment’s refractive index, DIFFRACT issues a warning and proceeds toignore the value listed in the header box.
When a COMMAND.DAT is being created during an interactive session,
DIFFRACT ensures that the values of λo and NVIRON listed in each header box
represent the actual wavelength and environment for the command block that
follows. However, if the user creates a COMMAND.DAT by cutting, pasting, andrearranging various command blocks, the information in the header boxes may
become invalid. By tracking λo and NVIRON during non-interactive sessions,
DIFFRACT prevents such inadvertent errors from corrupting the simulation.
2.2. The $ Commands: $ commands control the execution of DIFFRACT in the
non-interactive mode. The syntax of all available $ commands appears in thetitle box of COMMAND.DAT. These commands enable DIFFRACT to operate in aloop by allowing the user to modify specific items within COMMAND.DAT, andalso by allowing conditional jumps to marked positions within COMMAND.DAT.
Some of the lines appearing in COMMAND.DAT begin with a $ sign and end witha # sign, but contain no information otherwise; these lines identify the end of
one command block and the beginning of the next. It is at such locations,
namely, just before the beginning of a command block, that $ commands may beinserted. One may insert as many successive $ commands as is necessary at suchlocations. The $ commands have a rigid format; the best way to use them is tocopy them from the templates provided in the title box, and then substituteGuide to DIFFRACT

23

appropriate numerical values for the question marks (?) appearing within thetemplate.
The $ commands generally target specific lines of COMMAND.DAT, or specificitems within these lines. A targeted line must be identified by a unique 2-digit
number placed immediately after the # sign at the end of the line. Within eachline there can be at most three items, identified as ITEM1, ITEM2 and ITEM3. A
detailed description of the $ commands follows.
i) The syntax of the INCREMENT command is$INCREMENT#??,ITEM?...BY: ??
Once a line in COMMAND.DAT has been labeled with a 2-digit number, it can beaddressed by the INCREMENT command. The ?? appearing after # in thecommand syntax is the line number which contains the item to be incremented.
Each targeted line could contain up to three items. The targeted item number (1,2 or 3) must replace ? immediately after ITEM in the command syntax. The
amount by which the targeted item is to be incremented replaces ?? after thecolon. The increment amount can be any arbitrary positive or negative real
number.
ii) The syntax of the MULTIPLY command is$MULTIPLY#??,ITEM?....BY: ??
This command is similar to the INCREMENT command, except that the targeted
item is multiplied (rather than incremented) by the specified constant. Any line
labeled with a 2-digit number within COMMAND.DAT can be addressed by the
MULTIPLY command. The ?? appearing after # in the command syntax is the line
number which contains the item to be modified by multiplication. Each targetedline could contain up to three items. The targeted item number (1, 2 or 3) must
replace ? immediately after ITEM in the command syntax. The real-valued
constant by which the targeted item is to be multiplied replaces ?? after thecolon; this constant can be any arbitrary positive or negative number.
iii) The syntax of the MAKE command is$MAKE # ln,ITEM i ~ # lm,ITEM j:
The function of this command is to modify the value of an item on a given lineof COMMAND.DAT by an amount depending on the value of another item on thesame line or on a different line. The MAKE statement appears immediately afterGuide to DIFFRACT

24

the $ sign; 'ln' is a 2-digit line number in COMMAND.DAT which specifies theline targeted for modification; 'i' is a 1-digit number which specifies the targeted
item within that line (i = 1, 2, 3); ‘~’ is one of the ten operators +, −, *, /, =, E, L,
S, C, T, which stand for addition, subtraction, multiplication, division, setting
equal to, exponential function, natural logarithm, sine, cosine, tangent. The 2digit line number 'lm' and the 1-digit item number 'j' specify the item within
COMMAND.DAT that determines the amount of change in the targeted item. In
the process only the first item (i) is modified; the second item (j) remains intact.
+
−
*
/
=
E
L
S
C
T

adds item j to item i.
subtracts item j from item i.
multiplies item j into item i.
divides item i by item j.
sets item i equal to item j.
replaces item i with the exponential of item j.
replaces item i with the natural logarithm of item j.
replaces item i with the sine of item j (angle in degrees).
replaces item i with the cosine of item j (angle in degrees).
replaces item i with the tangent of item j (angle in degrees).

iv) The syntax of the JUMP command is$JUMP#??_IF#??,ITEM?.GT.: ??
Upon reading the JUMP command, DIFFRACT examines the current value of a
specified item within a specified line; if the item is greater than (or less than) a
certain amount, control will be transferred to a line also specified in the JUMP
command line; otherwise JUMP is ignored and the normal execution of the
program continues.
The destination address (i.e., the address to which control will be transferred)appears immediately after the JUMP command. The destination must be either a$ command or the beginning of a command block (i.e., a line which begins witha $ sign, ends with a # sign followed by a 2-digit number, and contains noinformation otherwise). These are the only types of line to which a JUMP may bedirected; in other words, DIFFRACT cannot be instructed to JUMP to the middle
of a command block.
The line-number and the item within that line that serves as the condition
parameter for JUMP appear after IF in the command syntax. (Usually, this same
item is also the target of an INCREMENT, MULTIPLY, or MAKE command,changing value with each iteration.) The condition parameter is compared with aterminal value appearing after the colon (:) in the JUMP command. The

Guide to DIFFRACT

25

comparator .GT. stands for Greater Than. Other comparators such as .LT. (LessThan), .GE. (Greater than or Equal to), and .LE. (Less than or Equal to) may
also be used in this position.
Use extreme caution in choosing the condition parameter, the comparator, andthe terminal value for the JUMP; a mistake in this command may either cause thejump to be ignored, or it may result in an infinite number of jumps. Also makesure that the jump destination is a valid starting point for a loop. Note inparticular that the contents of AX, AY after the jump are the same as those before
the jump; therefore, if you plan to repeat a computation with a certain initial
amplitude_distribution, you must read the desired distribution into AX, AY using
the READ option at the beginning of each iteration.
v) The syntax of the TAGS command is$TAGS#??,ITEM?_#??,ITEM?:
Whenever this command is executed, the two targeted items are associated with
two internal variables, Tag1 and Tag2. Later, when the Process_Signals option
is invoked to record a signal to the SIGNALS.DAT file, the current values of Tag1
and Tag2 will appear next to the recorded signal. (Ideally, items used as tags
should also be targets of INCREMENT, MULTIPLY, or MAKE commands, whichassume different values in different iterations.)
The TAGS command must specify exactly two items (no more and no less)within COMMAND.DAT for association with Tag1 and Tag2. These items,
however, may have the same address or they may have different addresses. Tag1
and Tag2 are dynamically associated with the targeted items, their values being
updated whenever an INCREMENT, MULTIPLY, or MAKE command changes thecorresponding items.
The user may insert the TAGS command in more than one place within
COMMAND.DAT, thus changing the associations of Tag1 and Tag2 during
program execution. At the time when a particular output is being written to
SIGNALS.DAT, however, the tags in effect will be the last ones invoked.
Example: Figure 6(a) shows a modified version of the COMMAND.DAT filedescribed earlier in conjunction with the system of Fig. 4. Comparing with theoriginal COMMAND.DAT of Fig. 5(b), we observe the following features of the
new file:• Mesh parameters NMAX, NMAY, LMAX, LMAY have been modified.

26

Guide to DIFFRACT• Three of the lines have been marked as #10, #20, and #30. These are the lines
targeted by the $ commands described below.
• Towards the end of the file we have inserted five $ commands. The firstcommand targets line #20, ITEM2 (namely, the GCX parameter) for
incrementing in the amount of 0.1. The second command decrements ITEM3of the same line (namely, GCY) by 0.1. The third command targets line #30
and increments ITEM1 (namely, the file identifier for saving the plots) by 1.
The fourth command assigns GCX to Tag1 and GCY to Tag2. The fifthcommand transfers control of DIFFRACT to line #10 whenever GCX happens
to be ≤ 0.5.
Note that line #10 is a proper jump destination, since it begins with a $ sign,
ends with a # sign, and is empty otherwise. Also observe that at the beginning ofeach iteration the distribution is loaded from the (previously stored) file
SPOT.DST. We cannot overemphasize the importance of loading the proper
distribution for each iteration since, in the absence of this crucial step, iterations
beyond the first one will start with the distribution that is already in the meshprior to the JUMP command. (In the present example this is the distribution at thedetector, which, clearly, is not the correct distribution for the focused spot at the
Optical_Disk_Surface.)
Figure 6(b) shows the contents of SIGNALS.DAT, a data file created in the noninteractive run conducted by COMMAND.DAT of Fig. 6(a). Each line of this filecontains a specific detector output (S1, S2, S3, S4) as well as the values of Tag1
and Tag2 corresponding to each iteration. The tags, of course, represent thevalues of (GCX, GCY), which are iteration variables.

Guide to DIFFRACT

27

Figure 6(a). A modified version of the COMMAND.DAT file shown in Fig.5(b). When used in
the non-interactive mode, this COMMAND.DAT puts DIFFRACT through a loop that covers a
range of positions of the Optical_Disk center (GCX, GCY).
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/13/2005
Time: 13:15 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: ) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
1/e radii Rmin,Rmax:
4000.0000
4000.0000#
RA,THETA:
3000.0000
0.0000000#
Aberrations: None#
Polarization RHO,ETA:
45.000000
0.0000000#
NMAX,NMAY:
256
256
#
LMAX,LMAY:
50000.000
50000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: ) #Type: PFOC#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.5000000
6000.0000#
Aberrations: Seidel#
Spherical C40:
0.0000000#
Coma C31,Phi31:0.2500000
45.000000#
Astigmatism C22,Phi22:0.0000000
0.0000000#
Curvature C20:
0.0000000#
Distortion C11,Phi11:0.0000000
0.0000000#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance:
6000.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: ) #Type: Log(intensity)#
Logarithmic SCALE:
4.0000000#

28

Guide to DIFFRACT
Xmin,Xmax:-4.0000000
4.0000000#
Ymin,Ymax:-4.0000000
4.0000000#
Color or Gray-scale: C#
Z-component (Y/N): N#
Save data files (Y/N): N#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Store distribution: SAVE#
File name: SPOT.DST#$________________________________________________________________________ #10
Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Retrieve a distribution: READ#
File name: SPOT.DST#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units: ) #Type of medium: WR#
SCX,SCY,RA:
0.0000000
000000000
20.000000#(Incidence medium) Ninc:
1.0000000#
Pregrooves: Y#
A,B,C:
0.1500000
0.5000000
0.6500000#
D,ZETA:
2.0000000
0.1250000#
Reflection coeffs. RL,RG:
0.8000000
0.8000000#
THETA,GCX,GCY:45.000000
0.0000000
0.0000000#20
Wobble (Sine,Blip,None): None#
Marks (bumps/pits): N#
Inspect surface pattern: Y#
Xmin,Xmax:-5.0000000
5.0000000#
Ymin,Ymax:-5.0000000
5.0000000#
Color or Gray-scale: G#
Save data file (Y/N): N#
File identifier: 0#
Propagation distance:
6000.0000#
Propagation regime: FRNHF#
Smax:.50000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: ) #Type: COLL#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.5000000
6000.0000#
Aberrations: Seidel#
Spherical C40:
0.0000000#
Coma C31,Phi31:0.2500000
45.000000#
Astigmatism C22,Phi22:0.0000000
0.0000000#
Curvature C20:
0.0000000#
Distortion C11,Phi11:0.0000000
0.0000000#
Scalar/Quasi-vector: QV#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Propagate in environment: PROP
(Length_Units: ) #Guide to DIFFRACT
Propagation distance:
20000.000#
Multiply curvature (Y/N): N#
Reposition beam (Y/N): N#
Propagation regime: DFRNL#
Smax:
0.10000000#
Scalar/Quasi-vector: QV#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: ) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-3200.0000
3200.0000#
Ymin,Ymax:-3200.0000
3200.0000#
Color or Gray-scale: C#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#30$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Detector: DTCT
(Length_Units: ) #Quad/Split/Phi/Diff: Q#
DCX,DCY,THETA:
0.0000000
0.0000000
45.000000#
LX,LY,GW:
5000.0000
5000.0000
0.0000000#
ID numbers for S1,S2:
1
2#
ID numbers for S3,S4:
34#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Process signals: PSIG#
Action: P#
Print S(i)i:1#Action: P#
Print S(i)i:2#Action: P#
Print S(i)i:3#Action: P#
Print S(i)i:4#Action: E#$#$INCREMENT#20,ITEM2...BY: 0.1#$INCREMENT#20,ITEM3...BY: -0.1#
$INCREMENT#30,ITEM1...BY: 1#$TAGS#20,ITEM2_#20,ITEM3:#$JUMP#10_IF#20,ITEM2.LE.: 0.5#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

29

Guide to DIFFRACT

30

Figure 6(b). A printout of SIGNALS.DAT created by DIFFRACT in the non-interactive mode
when the COMMAND.DAT of Fig. 6(a) guided the execution of the program. The various
signals recorded in this file are those elements of the SIGNAL array that were explicitly
printed out using the print command within the Process_Signals option. The first four
elements of the SIGNAL array in this case were assigned the outputs of the quad detector S1,
S2, S3, S4. Each output signal is tagged with two user-defined parameters (Tag1 and Tag2),
which are the values of GCX and GCY in the present example.
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 8.4
File SIGNALS.DAT $$$$ Date: 06/13/2005
Time: 13:15 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
ID
****
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4

SIGNAL****************
0.54614019E-01
0.54914065E-01
0.15300818
0.15163633
0.64985648E-01
0.66011369E-01
0.12760714
0.12595399
0.86366005E-01
0.87938532E-01
0.10323367
0.10142922
0.10961167
0.11130949
0.77218354E-01
0.75864837E-01
0.13733993
0.13893650
0.59626676E-01
0.58892675E-01
0.16181727
0.16333540
0.55273868E-01
0.55500906E-01

TAG1****************
0.0000000
0.0000000
0.0000000
0.0000000
0.10000000
0.10000000
0.10000000
0.10000000
0.20000000
0.20000000
0.20000000
0.20000000
0.30000001
0.30000001
0.30000001
0.30000001
0.40000001
0.40000001
0.40000001
0.40000001
0.50000000
0.50000000
0.50000000
0.50000000

TAG2****************
0.0000000
0.0000000
0.0000000
0.0000000-0.10000000-0.10000000-0.10000000-0.10000000-0.20000000-0.20000000-0.20000000-0.20000000-0.30000001-0.30000001-0.30000001-0.30000001-0.40000001-0.40000001-0.40000001-0.40000001-0.50000000-0.50000000-0.50000000-0.50000000

Guide to DIFFRACT

31

2.3. Running DIFFRACT in the Non-interactive Mode. A COMMAND.DAT fileis usually created interactively, then opened with a text-editor (e.g., NotePad or
WordPad) for modification by the user. The modifications can be as simple aschanging a few of the numbers, or as extensive as cutting and pasting command
blocks from different files into one, adding $ commands, creating nested loops,
etc. Once a COMMAND.DAT file is ready, it should be placed within a
subdirectory of the home directory. The user then starts the program by doubleclicking on DIFFRACT.EXE from the home directory, identifying as working
directory the subdirectory within which the desired COMMAND.DAT file resides,
and instructing DIFFRACT to operate either in the STEP mode or in the ZOOM
mode. The program then moves automatically to the specified subdirectory,reads the COMMAND.DAT file, and executes its commands sequentially.
It is also possible to specify more than one subdirectory (separated by blanks orcommas) in the beginning of a session. The program begins in the first specified
subdirectory and, upon completing the run, moves to the second subdirectory,
where it starts a new run. The process continues until all specified subdirectories
are exhausted. This multiple-directory capability of the program is particularly
useful for multiple batch runs in the ZOOM mode, where each run has its own
COMMAND.DAT file in a separate subdirectory of the home directory.
2.4. Running DIFFRACT from the Command Line. When DIFFRACT.EXE isinvoked from a command line, it attempts to pick up its initial parameter values
from the same command line. These include the name(s) of one or more working
subdirectories, the desired mode of operation (i.e., STEP, ZOOM, or Interactive),
the values of NDIM and NDSP, and a parameter that indicates whether or not todisable the ray-tracing features of the program. If DIFFRACT is invoked simply
by double-clicking on its icon, or if the command line contains no parameters,
an insufficient number of parameters, or incorrect parameters, control willtransfer to the Dialogue Window where DIFFRACT interrogates the user for its
initial parameter values. DIFFRACT’s command line has the following format:
Run Drive:\Directory\Diffract.exe Subdirectory Mode NDIM NDSP NoRays
The command-line interpreter is case-insensitive, and the various command line
parameters must be separated from each other by at least one blank space.
A working subdirectory (specified as the first parameter of the command line)
may exist within the home directory prior to launching DIFFRACT; otherwise thesubdirectory will be created. In either case the subdirectory becomes theworking directory during the session, where all data files needed by the program32

Guide to DIFFRACT

must reside, and where all files created by the program will be stored. More than
one subdirectory may also be specified on the command line if the adjacentnames of various subdirectories are separated by commas, colons, semicolons,backslashes or foreslashes. (Note that blanks are not allowed anywhere within
the list of subdirectories.) When a specified subdirectory happens to beunacceptable, the user will be prompted for one or more replacements.
Mode may be either STEP or ZOOM; anything else (even a blank) will launch
DIFFRACT in the Interactive Mode.
NDIM is the maximum linear dimension of DIFFRACT’s main mesh, while NDSP

is the linear dimension of the display mesh. The upper bounds on theseparameter values are set in the settings.log file located in DIFFRACT’s home
directory. When the specified NDIM and/or NDSP are out of bounds, the user will
be prompted for acceptable values of these parameters.
The presence of the six-letter word “NoRays” at the end of the command line
disables all ray-tracing features of the program; anything else (even a blank) inplace of this command-line parameter will enable the ray-tracing features.
Examples of valid command lines are:
Run C:\Diffract\Diffract.exe
Run D:\Reports\Work\Diffract.exe Opt123 ZOOM 512 129 NORAYSRun E:\Projects\Diffract.exe JohnDoe STEP 256 109 NoRays
Run F:\Diffract\Testruns\Diffract.exe Test01,Test02,Test03 1024 99
By setting the command-line parameters to their proper values before starting
the program, the user can launch DIFFRACT in the ZOOM mode, without having
to click any buttons or enter any parameter values from the keyboard during thesession.
2.5. Building COMMAND.DAT Files with the Help of a Template. If you arealready familiar with the basic structure of DIFFRACT, use the <TEMPLATES >sub-directory (located in DIFFRACT’s home directory) to create your own
COMMAND.DAT files. To avoid accidental deletion or modification of the files
contained in <TEMPLATES >, copy the text files TEMPLATE.DAT and
COMMAND.DAT to another (working) sub-directory, then open both files inside
your working sub-directory and place them side by side on the monitor. Copy
desired command blocks from TEMPLATE.DAT and paste them (in a logicalorder) onto the empty COMMAND.DAT file. Proceed to modify the various

Guide to DIFFRACT

33

command parameters until the elements of your system are properly represented.
When all the elements of the system are in place, you may want to copy one ormore $ commands to appropriate locations within the completed
COMMAND.DAT file. These $ commands enable the creation of loops (or otherprogramming constructs) for the purpose of repeating a given simulation withdifferent sets of parameters.
The vacuum wavelength λo and the refractive index NVIRON of the beam’s
environment must be specified in the header box of the BEAM command; theunits of wavelength (cm, mm, um, nm) must also be specified in the header box.
For all other commands, header boxes are optional, but it is generally a goodpractice to insert these boxes before each command block, and to write
comments and supplemental information in the Remarks: area of the box for
future reference. (The values of λo and NVIRON written in these header boxes aswell as any remarks will be ignored by DIFFRACT. The program issues awarning, however, if the header box values of λo and NVIRON are found to beinconsistent with the current values during a simulation.)
2.6. Extracting Specific Signals from the SIGNALS.DAT File: During
installation, the routine EXTRACT.EXE is copied to the home directory of
DIFFRACT on your hard disk. If you invoke this routine (by double-clicking on
its name), it will scan the file SIGNALS.DAT and extract from it the signal(s) thatyou specify. Also extracted for each signal are the corresponding values of Tag1
and Tag2. The user is then prompted for the names of files in which theextracted data are to be saved.
If a SIGNALS.XXX file does not exist in the working directory, you will receive amessage to this effect, and EXTRACT will terminate. Otherwise, you should
specify the filename extension XXX either as DAT or as any 3-digit number 000
to 999. When a SIGNALS file with the specified extension is located within theworking directory, you will be prompted to enter the ID of the signal item
(0 : 99) that you wish to extract from SIGNALS.XXX. If this item exists in thefile, you will be informed of its successful extraction, the number of extracted
elements, and the minimum and maximum values of the extracted signal. You
will then be prompted for filenames XXXX.DAT, in which extracted data will besaved. (The one-dimensional ASCII files thus created can be ported to any
commercially available plotting routine for graphical rendering.) You will beallowed to extract as many elements of the SIGNALS.DAT as you wish, before
terminating EXTRACT.

34

Guide to DIFFRACT

In case you do not have access to graphics software, you may use EXTRACT’s
primitive graphic routine. The routine enables you to draw one or more curves
on a set of XY-coordinates within a square frame. Input data-files to the graphic
routine are one-dimensional ASCII files XFILE.DAT and YFILE.DAT (filenames
are supplied by the user and may consist of up to 10 characters). The first plot is
automatically scaled, although the user is allowed to rescale the frame. Oncesatisfied with the scale, the user will be given the option of overlaying morecurves on the same plot. For each new curve the user must specify filenames
which contain the X- and Y-coordinates of the curve. There is no limit to thenumber of curves that can be simultaneously displayed, but the scale of theframe, once set for the first curve, cannot be changed.
The user selects the line-style and color of each curve by specifying a four-digit
number SRGB in the range [0000 : 9999]. ‘S’ is departure from solidity; thus
S = 0 for solid, S = 1 for long dash, S = 2 for shorter dash, and so on. ‘R’signifies the strength of the color Red in the curve; the color is absent whenR = 0 and fully present when R = 9. Similarly, ‘G’ and ‘B’ represent thestrengths of Green, and Blue in the plot. The default value of SRGB is 0000,
yielding a solid black curve.
At the end, the user will be given the opportunity to save the screen in a bitmap
file, FILE.BMP (the filename is user-selectable). Bitmap files may be readily
imported to various Microsoft utility programs (such as Paint® and Word®) for
incorporation into documents.

Guide to DIFFRACT

35

3. File Management and Graphics
This option, available through the main menu of DIFFRACT™, allows the user tomanage his/her files by providing access to the MS-DOS environment. In thismode of operation, the user can move freely between drives and directories inorder to delete, copy, transfer, rename, and otherwise manipulate various files.
The EXIT command will return control to DIFFRACT, where the user will findhimself once again in the home directory, within the File_Management/
Graphics environment. Once in this environment, the user can import graphic
data-files from the home directory and load them into a temporary array called
TEMP. (Graphic files are those with a .DAT extension, created by DIFFRACT'splotting routines.)
Using TEMP as a work-array, the user can import and add (pixel-by-pixel) two or
more graphic data-files together. Also, the contents of TEMP may be normalized
by the peak value of the array, then converted to base-10 logarithm. At any
given stage the contents of TEMP can be displayed on the monitor (in color or ingray-scale), or saved within the home directory either as a graphic data-file or as
a TIFF file. (TIFF files obey a standard format that enables their importation by
many commercial software, such as Microsoft Word®, Corel Draw®, etc. TIFF
files can be readily inserted into text and graphic documents.)
3.1. Commands and their Actions: The various options available within theFile Management and Graphics environment are listed below.
S) Switch to DOS environment to observe/manage graphic files
L) Clear temporary array TEMP, then load a data-file into itA) Load-and-Add (pixel by pixel) a data-file into TEMP arrayB) Save contents of Graphics Window to bitmap file XXXX.BMPC) Compute logarithm (base = 10) of the normalized TEMP arrayD) Display the current contents of TEMP array on the monitor
F) Save TEMP array in a hard disk data-file called XXXX.DATT) Save TIFF version of TEMP array in a file called XXXX.TIF
Q) Return to DIFFRACT ′s main menu after clearing TEMP array
Switching to DOS (via the S command) is allowed in the interactive mode of the
program, as well as in the non-interactive mode when the user chooses to STEP
through the file COMMAND.DAT. (In the non-interactive mode with ZOOM
option, requests for switching to DOS will be ignored.) Upon switching to DOS,
the following message will appear:36

Guide to DIFFRACT

You are now in the MS-DOS environment. Here you may issue as many DOS
commands as you wish. In particular, use the DIR command to take note ofgraphic data-files created by plotting routines; these are files with a .DAT
extension. Upon returning to DIFFRACT you will be given several options
for viewing these files, combining them, and converting them to TIFF files.
When ready to quit the DOS environment simply type EXIT, and you will be
returned to DIFFRACT.
Loading a data-file to the TEMP array (via the L command) will cause TEMP's
previous content, if any, to be erased. In the interactive mode, if the user
attempts to do so, the following message will appear:
Warning: The TEMP array is not empty. It will be overwritten if you proceedwith loading the file.
The user will thus be given one last opportunity to retract his request. (In the
non-interactive mode, there are no warnings and TEMP is automaticallyoverwritten.)
The pixel-by-pixel addition of a data-file to TEMP (via the A command)requires that another data-file be previously resident in TEMP. In the interactivemode, if the user tries to Load-and-Add a file to an empty TEMP, the following
message will appear:
Warning: TEMP array is empty. You may load a data-file into it, but there isnothing there to be added to the data-file.
In the non-interactive mode, there is no such warning and the file will be loaded
into an empty TEMP, without being augmented by a previously loaded array.
Thus, when TEMP happens to be empty, the A command is similar to the Lcommand.
The content of the specified data-file in the Load-and-Add option may be
multiplied by a user-defined constant (or weight-factor) before being added to
the TEMP array. The weight-factor may be positive, zero, or negative.

Guide to DIFFRACT

37

Note: TEMP is a square, NDSP0 × NDSP0 array; as such, it accepts any graphic
data-files whose linear dimensions are ≤ NDSP0. However, two arrays can be
added (via the A command) only if their dimensions are identical. The program
rejects any requests involving arrays with dimensional incompatibility.
The Load_and_Add option allows the content of any data-file created byplotting routines to be added to the TEMP array, so long as the chosen file has
the same number of horizontal as well as vertical pixels as TEMP. (Note:dimensions of TEMP are determined by the file initially loaded into it; thenumber of pixels in each dimension, of course, cannot exceed NDSP0.) If theselected file has, in each dimension, the same number of pixels as the current
content of the TEMP array, its content will be added, pixel by pixel, to thecontent of TEMP. It is the user’s responsibility to ensure that such an operation ismeaningful; for example, the physical dimensions [Xmin : Xmax, Ymin : Ymax] of
the arrays being added together should also be the same. Ideally, the files being
added should be created in similar environments, with identical λo (vacuum
wavelength) and NVIRON, and should be of the same general type. For instance,
Intensity plots may be added together, but there is no point in combining
Log_intensity plots using the Load_and_Add option, as this is tantamount tomultiplying the corresponding Intensity distributions. Similarly, adding a Phase
plot to an Intensity plot is pointless. Nevertheless, DIFFRACT™ does not checkfor this kind of compatibility; it simply proceeds to add the specified files pixelby pixel. The combined file acquires (by default) all the properties of the last
added file; these include the environment parameters (λo and NVIRON) as well as
the physical dimensions [Xmin : Xmax, Ymin : Ymax].
The B command may be invoked whenever the Graphics Window contains one
or more plots. The user is prompted for the filename XXXX.BMP to which thegraphic screen will be dumped.
When the contents of TEMP are non-negative, their base-10 logarithm may becomputed by invoking the C command. DIFFRACT first finds the largest
element of the array (i.e., its peak value), then normalizes each element of thearray by this peak value before evaluating its logarithm. In the process thecontents of the array are replaced by the base-10 logarithm of their normalized
values. Also, the user is prompted for the value of the lower-cutoff point, anegative parameter called SCALE; any array element whose log10(normalized
value) is below the cutoff will be truncated and reset to SCALE.

38

Guide to DIFFRACT

The user may, at any time, observe the contents of TEMP by choosing thedisplay option (the D command). Either color-coded or gray-scale display of thearray may be requested.
The content of TEMP may be committed to disk as an ASCII file, by invoking the
F command (or as a TIFF file, by invoking the T command). The user will beprompted for a filename, XXXX, and TEMP will be subsequently written to
XXXX.DAT (or XXXX.TIF).
Note: In the interactive mode, if XXXX.DAT or XXXX.TIF or XXXX.BMP happen
to exist within the working directory, the program will refuse to overwrite them.
In the non-interactive mode, however, SAVE requests take precedence, and theexisting files are overwritten.

39

Guide to DIFFRACT

4. General Features
DIFFRACT calculates the distribution of light at various cross-sections of an

optical system. The incident amplitude distribution may be specified as uniform,
simple Gaussian, Laguerre-Gaussian, or Hermite-Gaussian, it may be specified
as the output of a semiconductor laser diode, or it may be read from a disk file.
There can be a number of optical elements in the system including various
apertures, lenses, gratings, polarization elements, detectors, grooved surfaces
with bumps and pits, etc. DIFFRACT computes the distribution of amplitude onplanes perpendicular to the optical axis and, upon request, stores the resulting
arrays on disk files. The various elements of the optical system are treatedsequentially, but otherwise independently. The output distribution from oneelement generally serves as input to the next element.
The beam that goes through various optical elements in DIFFRACT is a coherent,
monochromatic beam of vacuum wavelength λo. The units of λo (cm, mm, um,nm) are specified by the user at the outset. At any given point along the opticalpath, the beam is in a homogeneous environment of refractive index NVIRON;
the value of NVIRON, however, may change during a simulation as the beam
passes from one environment to another.
For each optical element, the user is prompted to specify the Length_Units
appropriate for that element. The Length_Units may be cm, mm, um, or nm.
(Alternatively, the user may choose to represent lengths as dimensionlessquantities, i.e., normalized by the vacuum wavelength λo; this is done simply bypressing ENTER in response to the Length_Units question.) Note that, although
the units of λo remain the same throughout a simulation, the Length_Units could
vary from one element to the next. This is desirable, considering that the beam
could expand or shrink (as it propagates through the system) in such as way that
Length_Units that are suitable in one location may become inadequate at otherlocations.
All dimensionless lengths are normalized by the vacuum wavelength λo.
All angles are in degrees.
The environment of the beam is a homogeneous, isotropic, transparent mediumof refractive index NVIRON ≥ 1. The user specifies the parameters of the beam at
the outset, including the vacuum wavelength λo and the value of NVIRON. The
wavelength of the light in this environment is λ = λo/NVIRON. The beamsubsequently propagates in its environment, encountering various optical

40

Guide to DIFFRACT

elements embedded within the same medium of refractive index NVIRON. Some
optical elements such as Multilayer stacks (in transmission mode) andGeometric-optical Systems can bring the emergent beam into a different
environment. The program recognizes the corresponding change of NVIRON, andautomatically adjusts the beam amplitude to account for this change. The current
value of NVIRON is always displayed in the table that appears immediately
above the main menu of DIFFRACT.
The linear dimensions of the main array, NDIM, and of the display array, NDSP,
must be set in the beginning of each session. Upper bounds on these parameters
are presently set at NDIM = 256 (Student Edition), NDIM = 2048 (General
Edition), and NDSP = 150. The first set of parameters that you specify, therefore,
are the desired values of NDIM and NDSP. Also, disabling the ray_trace featuresof DIFFRACT will help reduce allocated memory by about 30%. If you intend touse any ray_tracing options during a given session, check the appropriate box in
the beginning, otherwise you can save memory by unchecking the box. (Thearrays used in ray-tracing are also needed for the FDTD option, Export mode, tokeep track of the magnetic field. Make sure the ray-tracing box is checked in the
beginning of the program if you intend to call upon the FDTD option in the
Export mode.)
The following options are available from the main menu of DIFFRACT:
Incident Beam
Propagate in Environment
Measure Peak and FWHM
Polarization Element
Multilayer Structure
Diffraction Grating
Change Coordinate System
Geometric-optical System
Extract Phase /Amplitude
Phase /Amplitude Object
Store Distribution
Wavefront Analysis
FDTD Interface
File Management /Graphics
Provide Guidance(BEAM)(PROP)(MEAS)(POLE)(MLYR)(GRAT)(CSYS)(GEOP)(XTRC)(XOBJ)(SAVE)(WAVE)(FDTD)(FMAN)(HELP)

Retrieve Distribution
Lens Action
Phase /Amplitude Mask
Flat Mirror
Optical Disk Surface
Launch Guided Mode
Weak Biprism
Convert to Rays
Photodetector
Interfere
Process Signals
Rescale the Mesh
Spatial Filter
Plot Distribution
Quit Program(READ)(LENS)(MASK)(MIRR)(DISK)(MODE)(BIPR)(RAYS)(DTCT)(INTF)(PSIG)(RSCL)(FLTR)(PLOT)(QUIT)

To define an initial distribution and to establish the proper mesh size, the first
item chosen from the menu must be either BEAM or READ. Selecting any otherGuide to DIFFRACT

41

option (except GEOP, PSIG, FMAN, HELP, and QUIT) in the beginning of asession will result in the following message and a return to the main menu.
Warning: An initial distribution has not yet been defined. You must returnto the main menu and choose either the BEAM or the READ option toestablish an initial distribution before you can proceed further.
In general, a rectangular mesh is used for a discrete representation of amplitude
distribution function(s) in the XY-plane; see Fig. 1. The mesh is centered at(x, y) = (0, 0) and has NMAX pixels along X and NMAY pixels along Y. The
physical length of the mesh along X is LMAX and that along Y is LMAY. To
ensure proper sampling in the Fourier domain, choose LMAX and LMAY to be
several times greater than the diameter of the initial beam.†
Note: During a typical run of DIFFRACT the mesh is Fourier transformed one ormore times using the fast Fourier transform (FFT) algorithm. Since FFT’s aredone most efficiently when the mesh dimensions are powers of 2, it is advisable
to choose the numerical values of NMAX and NMAY as 128, 256, 512, 1024, or
2048. Smaller values for these parameters are also allowed but they usually
result in under-sampling of the beam’s cross-section. In principle, larger values
for NMAX, NMAY can be selected, although this would require a computer
whose memory is large enough to maintain the necessary arrays in RAM at all
times. NMAX and NMAY are upper-bounded by the parameter NDIM that the user
specifies in the beginning of each run. The hard limit presently set on the
maximum allowed value of NDIM is 256 (Student Edition) and 2048 (General
Edition).
As the beam goes through the system, NMAX and NMAY remain constant but
LMAX and LMAY undergo automatic adjustments. To keep the user informed,
the new values of LMAX and LMAY are regularly displayed. The choice of
NMAX, NMAY, LMAX, LMAY must be made judiciously and with the entire
optical path in mind. If need be, however, mesh parameters may be adjusted at alater stage using the Rescale_the_Mesh (RSCL) option.
†

Default values of the mesh parameters are λo = 1µm, NVIRON = 1.0, Length_Units: µm,
NMAX = NDIM, NMAY = NDIM, LMAX = LMAY = 10000 µm. However, when the BEAM
option is chosen from the main menu, the user will be given the opportunity to set the meshparameters as desired.

42

Guide to DIFFRACT

4.1. Incident Amplitude Distribution: The incident distribution in the XYplane may be read from a previously stored file using the READ option of themain menu. Alternatively, it may be specified (using the BEAM option) as
Uniform Beam, Simple Gaussian, General Gaussian, Laguerre-Gaussian,Hermite-Gaussian, or emerging from a semiconductor Laser_Diode. A beamspecified through the BEAM option of DIFFRACT has vacuum wavelength λo,exists in a medium of refractive index NVIRON (a real number greater than orequal to 1.0), is centered at (x, y) = (BCX, BCY), is confined within a circular
aperture of radius RA, and its total optical power (i.e., integrated intensity)
within this aperture is Po (in milliwatts).
The Uniform Beam is specified simply by its center coordinates (BCX, BCY)
and by its radius of aperture RA.

a

b

c

d

−20

x/λ

20 −20x/λ20

Figure 7. Distributions of intensity (left) and phase (right) in the cross-sections of two
General Gaussian beams having different a, b, c parameters. The phase plots are encoded ingray-scale, with black representing –180° and white corresponding to +180°.
Top: a = 0.009 – i0.023, b = −0.006 – i0.002, c = 0.012 – i0.016.
Bottom: a = 0.011 – i0.023, b = −0.01 – i0.003, c = 0.016 + i0.012.

43

Guide to DIFFRACTThe Simple Gaussian beam is specified by its e−1 (amplitude) radii RMIN,
RMAX, by the angle Θ (0° : 180°) which the major axis of its elliptical crosssection makes with the X-axis, and by the aperture radius RA at which the beamis truncated. The beam is centered at (x, y) = (BCX, BCY).
The General Gaussian beam's complex amplitude distribution in the XY-plane
is proportional to exp[-π(ax2 + 2bxy + cy2)], where a = a1 + ia2, b = b1 + ib2, andc = c1 + ic2 are complex constants. The only restrictions on these parameters are:a1 > 0, c1 > 0, and a1c1 ≥ b12. The beam is centered at (x, y) = (BCX, BCY) and istruncated by a circular aperture of radius RA. Figure 7 shows the intensity andphase profiles of this type of beam for two sets of the a, b, c parameters.
The Laguerre-Gaussian beam is specified by the two mode-indices Nr, Na,which are the radial and azimuthal order-parameters of its generalized Laguerre
polynomial, by the e−1 (amplitude) radius Ro of its Gaussian envelope, and by theangle Θ (0 : 360°), which is the reference point from which the azimuthal
coordinate is measured. Both Nr and Na are integers, with the only restriction
being that Nr is non-negative. The beam is centered at (x, y) = (BCX, BCY) and istruncated by a circular aperture of radius RA.
a-10500bx/λ10500 -10500

cx/λ

10500 -10500

x/λ

10500

Figure 8. Plots of (a) Intensity, (b) Logarithm of intensity (scale = 3), and (c) Phase
distribution in the cross sectional plane of a Laguerre-Gaussian beam. (BCX, BCY) = (0, 0),
Nr = 4, Na = 6, Ro = 2500, Θ = 23°, RA = 10000. The phase plot is encoded in gray-scale, withblack representing –180° and white corresponding to +180°.

The Hermite-Gaussian beam is specified by the two mode-indices Nx, Ny,
which are the order-parameters of its Hermite polynomials, by the e−1 amplitude
radii RX0, RY0 of its Gaussian envelope, and by its angle of orientation
Θ (0 : 360°) relative to the X-axis of the incident coordinate system. Both Nx and44

Guide to DIFFRACT

Ny are non-negative integers. The beam is centered at (x, y) = (BCX, BCY) and istruncated by a circular aperture of radius RA.
a-10500bx/λ10500 -10500

cx/λ

10500 -10500

x/λ

10500

Figure 9. Plots of (a) Intensity, (b) Logarithm of intensity (scale = 3), and (c) Phase
distribution in the cross sectional plane of a Hermite-Gaussian beam. (BCX, BCY) = (0, 0),
Nx = 4, Ny = 5, Rx = 2800, Ry = 3300, Θ = 35°, RA = 10000. The phase plot is encoded ingray-scale, with black representing 0° and white corresponding to 180°.

For the Uniform Beam as well as all the various Gaussian distributions (i.e.,
Simple, General, Laguerre, Hermite), the state of polarization is uniform acrossthe aperture, but it may otherwise be chosen arbitrarily (see below). Also
primary (or Seidel) as well as Zernike aberrations are allowed for these
distributions. Defining ρ = R/RA as the normalized radial coordinate within thebeam’s aperture, and allowing φ to denote the azimuthal coordinate, the primaryaberrations of the wavefront are listed in Table 1.
Table 1. Primary (Seidel) aberrations. Dimensionless coefficients Cmn are normalized by λo.

Spherical:
W40 (ρ, φ) = C40 ρ4Coma:
W31 (ρ, φ) = C31 ρ3 cos(φ - φ31)
Astigmatism: W22 (ρ, φ) = C22 ρ2 cos2(φ - φ22)
Curvature: W20 (ρ, φ) = C20 ρ2
Distortion: W11 (ρ, φ) = C11ρ cos(φ - φ11)

(User identifies C40)
(User identifies C31‚ φ31)
(User identifies C3, φ22)
(User identifies C20)
(User identifies C11, φ11)

Similarly, Zernike’s circle polynomials may be used to define wavefront
aberrations. DIFFRACT allows Zernike polynomials up to the 12th order to be
specified. The user is prompted for the coefficients Cmn and the orientation angle
φmn of the various Zernike functions. Listed in Table 2 are the functional formsof all such aberrations up to the 12th order.

Guide to DIFFRACT

Table 2. Zernike’s wavefront aberrations up to and including the 12th order.
Dimensionless coefficients Cmn are normalized by λo.

W11(ρ, φ) = C11 ρ cos(φ - φ11)
W20(ρ, φ) = C20 (2ρ2 - 1)
W22(ρ, φ) = C22 ρ2 cos[2(φ - φ22)]
W31(ρ, φ) = C31 (3ρ3 - 2ρ) cos(φ - φ31)
W33(ρ, φ) = C33 ρ3 cos[3(φ - φ33)]
W40(ρ, φ) = C40 (6ρ4 - 6ρ2 + 1)
W42(ρ, φ) = C42 (4ρ4 - 3ρ2 ) cos[2(φ - φ42)]
W44(ρ, φ) = C44 ρ4 cos[4(φ - φ 44)]
W51(ρ, φ) = C51 (10ρ5 - 12ρ3 + 3ρ) cos(φ - φ51)
W53(ρ, φ) = C53 (5ρ5 - 4ρ3) cos[3(φ - φ53)]
W55(ρ, φ) = C55 ρ5 cos[5(φ - φ55)]
W60(ρ, φ) = C60 (20ρ6 -30ρ4 + 12ρ2 - 1)
W62(ρ, φ) = C62 (15ρ6 - 20ρ4 + 6ρ2 ) cos[2(φ - φ62)]
W64(ρ, φ) = C64 (6ρ6 - 5ρ4 ) cos[4(φ - φ64)]
W66(ρ, φ) = C66 ρ6 cos[6(φ - φ66)]
W71(ρ, φ) = C71 (35ρ7 - 60ρ5 + 30ρ3 - 4ρ) cos(φ - φ71)
W73(ρ, φ) = C73 (21ρ7 - 30ρ5 + 10ρ3) cos[3(φ - φ73)]
W75(ρ, φ) = C75 (7ρ7 - 6ρ5) cos[5(φ - φ75)]
W77(ρ, φ) = C77 ρ7 cos[7(φ - φ77)]
W80(ρ, φ) = C80 (70ρ8 - 140ρ6 + 90ρ4 - 20ρ2 + 1)
W82(ρ, φ) = C82 (56ρ8 - 105ρ6 + 60ρ4 - 10ρ2 ) cos[2(φ - φ82)]
W84(ρ, φ) = C84 (28ρ8 - 42ρ6 + 15ρ4 ) cos[4(φ - φ84)]
W86(ρ, φ) = C86 (8ρ8 - 7ρ6 ) cos[6(φ - φ86)]
W88(ρ, φ) = C88 ρ8 cos[8(φ - φ88)]
W91(ρ, φ) = C91 (126ρ9 - 280ρ7 + 210ρ5 - 60ρ3 + 5ρ) cos(φ - φ91)
W93(ρ, φ) = C93 (84ρ9 - 168ρ7 + 105ρ5 - 20ρ3) cos[3(φ - φ93)]
W95(ρ, φ) = C95 (36ρ9 - 56ρ7 + 21ρ5) cos[5(φ - φ95)]
W97(ρ, φ) = C97 (9ρ9 - 8ρ7) cos[7(φ - φ97)]
W99(ρ, φ) = C99 ρ9 cos[9(φ - φ99)]

45

46

Guide to DIFFRACT

W10,0(ρ, φ) = C10,0 (252ρ10 - 630ρ8 + 560ρ6 - 210ρ4 + 30ρ2 - 1)
W10,2(ρ, φ) = C10,2 (210ρ10 - 504ρ8 + 420ρ6 - 140ρ4 + 15ρ2 ) cos[2(φ - φ10,2)]
W10,4(ρ, φ) = C10,4 (120ρ10 - 252ρ8 + 168ρ6 - 35ρ4) cos[4(φ - φ10,4)]
W10,6(ρ, φ) = C10,6 (45ρ10 - 72ρ8 + 28ρ6 ) cos[6(φ - φ10,6)]
W10,8(ρ, φ) = C10,8 (10ρ10 - 9ρ8) cos[8(φ - φ10,8)]
W10,10 (ρ, φ) = C10,10 ρ10 cos[10(φ - φ10,10)]
W11,1 (ρ, φ) = C11,1 (462ρ11 - 1260ρ9 + 1260ρ7 - 560ρ5 + 105ρ3 - 6ρ)× cos(φ - φ11,1)
W11,3(ρ, φ) = C11,3 (330ρ11 - 840ρ9 +756ρ7 - 280ρ5 + 35ρ3) cos[3(φ - φ11,3)]
W11,5(ρ, φ) = C11,5 (165ρ11 -360ρ9 + 252ρ7 - 56ρ5) cos[5(φ - φ11,5)]
W11,7(ρ, φ) = C11,7 (55ρ11 - 90ρ9 + 36ρ7 ) cos[7(φ - φ11,7)]
W11,9(ρ, φ) = C11,9 (11ρ11 - 10ρ9) cos[9(φ - φ11,9)]
W11,11(ρ, φ) = C11,11 ρ11 cos[11(φ - φ11,11)]
W12,0 (ρ, φ) = C12,0 (924ρ12 - 2772ρ10 + 3150ρ8 - 1680ρ6 + 420ρ4 - 42ρ2 + 1)
W12,2 (ρ, φ) = C12,2 (792ρ12 - 2310ρ10 + 2520ρ8 - 1260ρ6 + 280ρ4 - 21ρ2)× cos[2(φ - φ12,2)]
W12,4 (ρ, φ) = C12,4 (495ρ12 - 1320ρ10 + 1260ρ8 - 504ρ6 + 70ρ4)× cos[4(φ - φ12,4)]
W12,6 (ρ, φ) = C12,6 (220ρ12 - 495ρ10 + 360ρ8 - 84ρ6) cos[6(φ - φ12,6)]
W12,8 (ρ, φ) = C12,8 (66ρ12 - 110ρ10 + 45ρ8) cos[8(φ - φ12,8)]
W12,10 (ρ, φ) = C12,10 (12ρ12 - 11ρ10 ) cos[10(φ - φ12,10)]
W12,12 (ρ, φ) = C12,12 ρ12 cos[12(φ - φ12,12)]The incident beam, whether Uniform or Gaussian, may have an arbitrary state ofpolarization. The ellipse of polarization in the XY-plane is described by twoangles: ρ (0°: 360°) is the angle between X and the major axis of the ellipse;
η (- 45°: 45°) defines the beam's ellipticity; see Fig. 10. For linear polarization
set η = 0°; for right circular polarization (rcp) set η = 45°; for left circularpolarization (lcp) set η = -45°. (In cases of circular polarization the numericalvalue of ρ is inconsequential.)
The beam from the Laser_Diode (restricted to propagating along the Z-axis) is
specified in terms of its FWHM divergence angles Φ and Ψ, its astigmatic
distance ∆ (i.e., the separation in air between the two waists of the beam
perpendicular and parallel to the plane of the junction), the orientation angle Θ47

Guide to DIFFRACT(0°: 180°) of the junction relative to the X-axis, and the distance z between the
mid-point of the two waists and the XY-plane in which the distribution ispresently desired. The polarization state of the laser beam is linear and parallelto the plane of the junction.
Ey

ρ
η

Ex

Figure 10. The ellipse of polarization is uniquely specified by Ex and Ey, the complex-valued
electric field (E-field) components along the X- and Y-axes. The major axis of the ellipse
makes an angle ρ with the X-direction; the angle η facing the minor axis of the ellipse
represents polarization ellipticity.

b

a

−7200

x/λ

7200 −7200

x/λ

7200

Figure 11. Plots of (a) logarithm of intensity (Scale = 9) and (b) phase distribution for aLaser Diode beam. (BCX, BCY) = (0, 0), FWHM divergence angles (Φ, Ψ) = (10°, 25°),
astigmatic distance ∆ = 12, orientation angle Θ = 0°, distance from the mid-point of the waists
to the observation plane z = 5000. The phase plot is encoded in gray-scale, with black
representing –180° and white corresponding to +180°; the curvature C = 1/z has been
subtracted from the phase profile and carried by DIFFRACT outside the mesh.

48

Guide to DIFFRACT

4.2. Propagation Regimes: The three possible regimes of propagation in theenvironment (NVIRON) in DIFFRACT are: far field (Fraunhofer) regime for long
distances, near field (Direct Fresnel) regime for short distances, and intermediate(Extended Fresnel)‡ regime for distances in between the other two. Proper
selection of the propagation regime is ultimately the user’s responsibility, since
DIFFRACT has no internal rule for choosing one regime over another. The size ofthe beam, its wavelength, its spatial frequency content, and the propagation
distance are factors that determine the propagation regime. In general, there is no
simple rule for deciding the proper regime of propagation, and one must be
guided by experience. When a choice is not obvious, the user is encouraged totry different options, and to make a final decision based on the results of thesetrials.
Note: A Table displayed in the interactive mode provides guidelines for proper
selection of NMAX and LMAX (ditto NMAY and LMAY) for each regime of
propagation. These values, however, are only estimates and the user is not boundby them. The program also suggests a propagation regime, but it is the user who
decides whether or not to accept this suggestion.
The beam in DIFFRACT usually propagates in the +Z direction; this is referred to
as forward propagation. Since the optical system under consideration is
unfolded along +Z, DIFFRACT predominantly deals with forward-propagating
beams. There are circumstances, however, when it becomes necessary to find the
distribution at a cross-section on the negative side of Z with respect to thecurrent cross-section. This is when the desired propagation distance becomes
negative; the case is referred to as backward propagation.
When you choose the Propagate_in_Environment (PROP) option from the menu,
you may choose to propagate forward or backward. If the beam happens to carrya curvature outside the mesh, you will be given the option to multiply thecurvature phase-factor into the AX, AY arrays. Only when the curvature is mild,
however, should such multiplication be authorized. When the curvature is notmild one of the following situations may arise:(i)

Forward-propagating beam has a positive (divergent) curvature

All significant spatial frequencies of a given distribution are within a circle of radius Smax in
the Fourier domain. The value of Smax (0 : 1) is used to optimize computations in the Extended
Fresnel regime. The program has an estimate of Smax, which it uses by default unless
instructed otherwise by the user.
‡

Guide to DIFFRACT

49(ii) Backward-propagating beam has a negative (convergent) curvature(iii) Forward-propagating beam has a negative (convergent) curvature(iv) Backward-propagating beam has a positive (divergent) curvature
In cases (i) and (ii), DIFFRACT treats the beam as a smooth, diverging wavefront which maintains its shape while growing in size during propagation. In
cases (iii) and (iv), the beam is treated as if it had emerged from the exit pupil ofan appropriate positive or negative lens. In the latter two cases, the results ofcomputations will be accurate only when the chosen propagation distance brings
the beam to the vicinity of the focal plane of this imagined lens.
Frequently, when the beam centroid is not at the origin of the XY-plane, i.e.,(Xc, Yc) ≠ (0, 0), computations will be more accurate if, prior to beam
propagation, the origin of the coordinate system is laterally shifted to coincidewith the beam centroid, then returned to its initial location afterwards. The user
decides whether or not to perform this repositioning. For propagation in the
Fraunhofer regime (far-field) repositioning is generally recommended. In the
Fresnel regime (near-field) as well as in the Extended Fresnel regime(intermediate between near-field and far-field), repositioning is recommended
only when the beam centroid is far from the origin. (Note that the coordinate
shift in DIFFRACT is accompanied by a truncation of the initial distribution at a
certain large distance from the centroid.) Ideally, when the mesh is fine,
repositioning should improve the accuracy of the propagation calculations.
However, in rare occasions, the interpolation involved in shifting the beam backand forth over the mesh alters the complex amplitude of the beam in subtle
ways, causing small numerical errors.
4.3. Scalar and Quasi-Vector Treatments of Polarization: Propagation of thepolarization state in DIFFRACT is treated in either of two modes: Scalar andQuasi-Vector. In the scalar mode, the X- and Y-components of polarization aretreated independently of each other and according to the scalar theory ofdiffraction. The quasi-vector mode treats vector diffraction in an approximate
way, bending and mixing the components of polarization to a certain extent (seeRefs. 1-3 for a mathematical analysis of this subject).
The approximations involved in the Quasi-Vector treatment of polarization
propagation are valid only under certain circumstances. For example, when theincident beam is collimated or nearly so (i.e., when the spatial frequencies of theincident beam are relatively small), it is possible to infer the state of polarizationof each diffracted plane-wave from its propagation direction. Similarly, when a50

Guide to DIFFRACT

diverging beam is collimated (or nearly collimated) by a lens, it is possible toestimate the bending of the various polarization vectors from a geometric-opticalanalysis of the lens action. In these and similar situations, the Quasi-Vector
treatment is quite appropriate. On the other hand, when a highly focused beam(i.e., one that contains large spatial frequencies) is transmitted through a flat
surface or a surface having mild corrugations, the bending of the rays is rather
insignificant. In such a situation the Scalar treatment gives a better
approximation to reality.
When the choice regarding the method of polarization propagation isunambiguous, DIFFRACT makes the decision. In all other cases, the user is asked
to choose between the Scalar and Quasi-Vector methods of treatment. The
decision during each stage of propagation must be based on the circumstancesthat prevail during that particular stage, independent of all previous choices
made (or future choices to be made) regarding polarization.
4.4. Store Distribution: The Store_Distribution (SAVE) option of the main
menu enables the user to create within the working directory an unformatted(i.e., binary) data file called XXXX.DST, and to dump the current cross-section of
the beam into that file. (The filename XXXX is chosen by the user.) Typically,
the contents of the arrays AX, AY are saved, but if the Z-component ofpolarization, or the ray_distribution, happen to be in the mesh, the corresponding
arrays will also be saved. (In all cases, the vacuum wavelength λo, the index of
the environment NVIRON, and the mesh parameters NMAX, NMAY, LMAX,
LMAY are saved automatically.)
If, through the use of the Retrieve_Distribution (READ) option, a saved file isread back into the program at some future time, the stored distribution will
replace the contents of the mesh, thus becoming the current cross-section of thebeam. Stored distributions are also used in conjunction with the Interfere option,
when a previously generated (and saved) cross-section of the beam is calledback in order to be superimposed on the current cross-section.
4.5. Retrieve Distribution: The Retrieve_Distribution (READ) option enables
the user to read a previously saved distribution from the file XXXX.DST, and toplace it in the mesh. In this process the current contents of the mesh (if any) will
be discarded. When the mesh is not empty and you request the READ option, thefollowing warning will appear:Guide to DIFFRACT

51

Warning: You are about to choose a new distribution for the beam's cross-section.
Upon completion of this operation, the distribution presently residing in the mesh will
be discarded. You may want to return to the main menu and call the SAVE option
before proceeding.
DIFFRACT saves all length parameters (e.g., LMAX, LMAY) to XXXX.DST in

normalized form, i.e., divided by the vacuum wavelength λo. Upon readout,
since Length_Units are not known, all length parameters will be displayed inthis normalized form. The only exception to this rule is λo, which is always
saved together with its units, and is thus retrieved with actual dimensions.
4.6. Phase/Amplitude Object (XOBJ): This feature of DIFFRACT allows
operation in one of two modes. The IMPORT mode reads an external
phase/amplitude object from a data-file and places it in the beam's path. The
CREATE mode allows a photographic plate to be exposed to the beam's intensity
profile, and proceeds to save the developed phase/amplitude object in a data-file.
The files thus created can be imported later during the same run (or insubsequent runs) of the program as external phase/amplitude objects.
IMPORT Mode: The IMPORT option allows the user to place in the optical patha planar object, having an arbitrary, complex-amplitude transmissivity (orreflectivity) function F(x, y). The function F(x, y), defined on a discrete meshwith NX × NY pixels, must be created in advance and made available to theprogram as a hard disk file in the working directory. The physical dimensions of
the mesh that contains F(x, y) are LX × LY (in units of vacuum wavelength λo).
DIFFRACT treats F(x, y) as the transmission (or reflection) function of a planar
object that has been placed perpendicular to the optical axis. F(x, y) is read from§
FILE.DAT, which may be either an ASCII or a binary file. Also placed in
FILE.DAT must be an estimate of FRQMAX, which is the radius of the circle in
the Fourier domain that contains all significant spatial frequencies of F(x, y).**§ASCII files serving as external_object files for DIFFRACT may be created using a number ofcommercially available compilers, spreadsheets, etc. The important thing to remember whencreating such files is that the complex number F(I, J) must be written to the file as(Real, Imaginary), complete with its parentheses and a comma between the real andimaginary parts. Binary files are faster to write to and read from the disk, but, to assure format
compatibility, they must be created in a Fortran environment.
**

FRQMAX, a real number in the interval (0 : 1), is used by DIFFRACT for the sole purpose ofproviding the user with an estimate of the mesh parameters NGUIDE and LGUIDE. The valueof FRQMAX (supplied through FILE.DAT) does not affect the computations in any other way.

52

Guide to DIFFRACT

FILE.DAT must have the following format: NX, NY, LX, LY written on the firstline, FRQMAX written on the second line, and a single row of the array F(I, J)
written on each succeeding line. For a given value J0 of J (between 1 and NY) theline recorded in FILE.DAT contains the F(I, J) values for I from 1 to NX.

Once the file is read into DIFFRACT, the user is prompted for the center
coordinates of the object (OCX, OCY), and for its radius of aperture RA(measured from this center). The user will also be given the option to place
within the defined aperture a single or multiple copies of the external object(multiple copies are periodically repeated along X- and Y-axes). Since LX, LY
values retrieved from FILE.DAT are normalized by the vacuum wavelength λo,
DIFFRACT automatically converts these values to Length_Units to ensurecompatibility with other length parameters. Finally, interpolation is used to place
F(x, y) on the original mesh (that which contains the beam). DIFFRACT proceeds
to compute the transmitted (or reflected) distribution.
Y(OCX, OCY)

Figure 12. The external object(XOBJ) of length LX and width LY
is centered at (OCX, OCY) andreplicated in both X- and Ydirections throughout the mesh.
Each copy of XOBJ has NX × NY
pixels, while the total number ofpixels in the mesh is NMAX ×
NMAY. Only that part of thereplicated pattern which is within acircular aperture of radius RA
interacts with the incident beam.

RA

L
M
A
Y

X

LY
LX

LMAX

The external object has physical dimensions LX and LY along the X- and Y-axes,
and is centered at (OCX, OCY) within the main mesh, whose dimensions are
LMAX × LMAY. The center of the mesh itself is at (0, 0), where the X- and Yaxes cross; see Fig. 12. In some applications, it is desirable to have a periodic
array of the external object, so DIFFRACT automatically places copies of XOBJ in
the mesh in a periodic fashion. The user can always say “No” when prompted
for periodic repetition of the external object. Alternatively, the user may choosea sufficiently small radius of aperture RA to block all the replica, and expose

Guide to DIFFRACT

53

only a single copy of XOBJ to the incident beam. By asking for periodic
repetition and choosing a large RA, a periodic array of the same external object
will be exposed to the incident beam.
Note: The option to import a user-defined phase/amplitude_object provides ameans of introducing the cross-section of an arbitrary beam into the program.
The complex amplitude of the beam is represented by the function F(x, y), which
may be assumed to be the transmission function of a mask placed in the path of a
uniform incident beam. If the polarization state of the beam happens to beuniform over its entire cross-section, then a single mask will suffice; otherwise,
one must create two different masks for the x- and y-components of polarization,
Fx(x, y) and Fy(x, y), then superimpose these two distributions using the
INTERFERE option (see Sec. 4.7 below).
CREATE Mode: The CREATE option allows a photographic plate to be exposedto the current beam intensity profile. The resulting phase/amplitude object isthen saved to a standard data-file. The complex transmissivity τ exp (iφ) of theplate at any given point on its surface is a decreasing exponential function of theintensity I incident at that point. When exposing the plate, the user is given theoption to either include or exclude the Z-component of polarization from theintensity profile. Also, the user may choose to create a double-exposure plate:instead of recording the intensity pattern on a blank film, double-exposure
involves the use of an existing transparency. The previously-exposed plate must
reside in an existing data-file, and the two transmission functions will be
multiplied together. The product will be a true double-exposure if an exponentialfunction having the same set of characteristic parameters is used in both cases.
The user may choose to develop the plate into a positive or negativetransparency. At last the user specifies a filename, FILE.DAT, to which thesingle- or double-exposure transparency will be saved.
The photographic plate is an LX × LY rectangle, centered at (PCX, PCY) in the
XY-plane, and exposed to the incident beam through a circular aperture of radius
RA; the aperture is also centered at (PCX, PCY); see Fig. 13. The number ofpixels (NX, NY) used to digitize the transmission function of the plate is userselected, but may not exceed the upper bound NDIM placed on mesh size. The
exposed plate's complex transmission factor (τ, φ) varies exponentially with theintensity I, its value being (1, 0) at I = 0 and (τ0, φ0) at I = I0. The user mustspecify the values of I0 as strictly positive, τ0 in the interval (0, 1], and the phaseangle φ0 in degrees. One might either expose a blank film or a previously-

54

Guide to DIFFRACT

recorded plate; in the latter case the user will be prompted for a filename. The
exposed plate can be developed as is (i.e., a negative) or converted to a positivetransparency. The transmission factor of a negative is the same as the exposed
film’s, while that of the positive is (1 - τ, - φ).
Y

τ

1

τo
RA

LY

Io

×

φ

(PCX, PCY)

φo
LX

X

Io

Intensity

Figure 13. Photographic plate of length LX and width LY is exposed to incident intensity
distribution I (x, y) through a circular aperture of radius RA. The plate center (PCX, PCY)
coincides with the center of the aperture. The exposed negative acquires a complex amplitudetransmissivity τ exp (iφ), where τ (x, y) depends exponentially, and φ (x, y) linearly, on thelocal value of incident intensity I (x, y).

Exposure of a photographic plate may take place in free-space or within a
medium of refractive index NVIRON > 1. However, when storing thetransparency, neither NVIRON nor the vacuum wavelength λo are saved to
FILE.DAT. Moreover, the transparency’s physical dimensions LX, LY are saved
to FILE.DAT in dimensionless form (i.e., normalized by λo). The physicaldimensions of the transparency thus remain unchanged, so long as the vacuum
wavelength in the Import mode remains the same as that in the Create mode
(irrespective of any changes of NVIRON).
4.7. Interferometric Beam Combination: The beam combination option of
DIFFRACT, available through the main menu as the option INTERFERE, enables
the current distribution to be combined (interferometrically) with another
distribution residing in the working directory in the form of a XXXX.DST file.
This file must have been created using the SAVE option, either earlier in thesame run or during a previous run of DIFFRACT.

55

Guide to DIFFRACT

The input distribution is multiplied by a user-defined, complex-valued scalefactor α = A exp(i φ), where A is a positive constant, and φ is a phase angle in theinterval (0° : 360°). The scaled distribution is then added to the current
distribution. For multiple beam combinations, the user may invoke the
INTERFERE option as many times as is desired.
The input mesh must have the same number of pixels (NMAX, NMAY) as thecurrent mesh, but its physical dimensions (LMAX, LMAY) will be adjusted toaccommodate the current mesh, if necessary. Ideally, the input beam shouldhave the same NVIRON and the same vacuum wavelength λo as the current beam.
However, DIFFRACT issues a warning and proceeds to ignore the input beam’s
λo and NVIRON, if they happen to differ from current values. Since all lengthparameters in XXXX.DST are in dimensionless form (i.e., normalized by thevacuum wavelength), DIFFRACT interprets the normalization to be relative to thecurrent value of λo.
Note: When using the INTERFERE option, If the two beams do not overlap, or if
their polarization vectors happen to be orthogonal to each other, there will be nointerference, of course, but the combination will still be a valid superposition of
the two (complex) amplitude distributions.
a

−1050

c

b

x/λ

1050 −1050

x/λ1050 −1050

x/λ

1050

Figure 14. Interferometric combination of two beams of uniform amplitude and non-uniform
phase. (a) Phase pattern of a beam of radius RA = 1000, having a few waves of Seidel
aberrations. (b) Phase pattern of a beam of radius RA = 850, having a few waves of Zernike
aberrations. (c) Intensity profile of the superposition of the two beams depicted in (a) and (b).
The scale-factor α is 1.0, that is, A = 1.0 and φ = 0°.

56

Guide to DIFFRACT

4.8. Launching Guided Mode. A guided mode can be launched after evaluating
the overlap integral between the current distribution and a specific mode of awaveguide. The user may specify a Laguerre-Gaussian or a HermiteGaussian mode of arbitrary order to be launched by the incident beam (namely,
the distribution presently residing in the mesh). The launched mode will replace
the current distribution in the mesh; its various attributes will be those definedby the user, except for the mode's magnitude and phase, which will be
determined by the overlap integral between the incident beam and the specified
mode. The launched mode is centered at (MCX, MCY) in the XY-plane, and isconfined within a circular aperture of radius RA. Ideally, RA should be large
enough to contain the mode, otherwise orthogonality of the Laguerre-Gaussian(or Hermite-Gaussian) modes of differing orders cannot be guaranteed.
The Laguerre-Gaussian mode is specified by the two mode-indices, Nr, Na,which are the radial and azimuthal order-parameters of its generalized Laguerre
polynomial, by the e−1 (amplitude) radius Ro of its Gaussian envelope, and by theangle Θ (0° : 360°), which is the reference point from which the azimuthal
coordinate is measured. Both Nr and Na are integers, with the only restriction
being that Nr ≥ 0.
The Hermite-Gaussian mode is specified by the two mode-indices Nx, Ny,
which are the order-parameters of its Hermite polynomials, by the 1/e amplitude
radii Rxo, Ryo of its Gaussian envelope, and by its angle of orientation Θ(0° : 360°) relative to the X-axis of the incident coordinate system. Both Nx and
Ny are non-negative integers.
When the incident beam happens to have a curvature that cannot be directlytransferred to the launched mode, the following warning will be issued:
WARNING: Distance between the center of curvature of the incident beamand the launched mode's center is too large for the current mesh. You must
reduce this distance and/or reduce the coarseness of the mesh in order toobtain reliable values for the overlap integral(s).
The mode's Intrinsic Power (i.e., integrated intensity over the entire crosssectional area) should ideally be equal to 1.0 within the numerical accuracy ofthe program. When it substantially differs from 1.0, either the mesh is too coarse
for the chosen mode, or the selected aperture radius RA is too small to containthe mode. When this occurs, you will be given the option to ignore this problem

Guide to DIFFRACT

57

and continue with the selected mode parameters, or to cancel the MODE option
and return to the main menu.
4.9. Phase/Amplitude Extraction: This menu item provides the user with thefollowing options:(i) Eliminate the phase of the current distribution but maintain its magnitude
everywhere.
(ii) Preserve the phase but eliminate the non-uniformity of the amplitude.
(iii) Replace the current distribution with its own complex conjugate.
In all cases power and polarization components of the beam are preserved.
4.10. Peak Intensity and FWHM Measurement: This option allows the user todetermine the peak and the full-width at half-maximum (FWHM) intensity of thebeam. The intensity may be taken as that of the total E-field, encompassing thethree components (Ex, Ey, Ez) of polarization, or it may include just the in-planecomponents (Ex, Ey).
Note: DIFFRACT may yield erroneous values for the FWHM if the intensity
happens to have more than one major peak.
At the time the MEASURE option is invoked, the Z-component of polarization
may or may not be present in the mesh. Accordingly, one of the following
messages will appear:
Z-component of polarization is presently available.
Z-component of polarization unavailable but may be computed if necessary.
The user will then be given the option to include the Z-component in thecalculations. If included, the total intensity will become |Ex|2 + |Ey|2 + |Ez|2,
otherwise it will be |Ex|2 + |Ey|2. Either way, the program proceeds to find themagnitude and the location of the intensity’s peak within the XY-plane.
The peak's location in the XY-plane will be denoted by (XPEAK, YPEAK). The
width of the beam is measured along one or more straight lines in the XY-plane,
each going through (XPEAK, YPEAK) and making an angle θ (0°: 180°) with theX-axis. The user will be prompted for the desired value(s) of θ. Computation ofpeak intensity and beam-width does not alter the light amplitude distribution in58

Guide to DIFFRACT

any way; the user may thus continue to propagate (or otherwise process) the
beam upon returning to the main menu.
The values of XPEAK, YPEAK, Peak_Intensity, and FWHM are assigned to
specific elements within the SIGNAL array, as specified by the user. This data
can be processed later (using the Process_Signals option), and printed out to the
SIGNALS.DAT file in the non-interactive mode. (When different values of θ areexpected to yield different values for the FWHM, it is advisable to use θ as one of
the tags in COMMAND.DAT.)
4.11. Wavefront Analysis. The beam’s cross-section in DIFFRACT is fully
specified by the complex amplitude distributions AX(x, y), AY(x, y) and, whenavailable, AZ(x, y). For each component of polarization an effective measure of
wavefront distortion may be obtained by first computing the integral of A(x, y)over the beam's cross-section, normalized by the integral of |A(x, y)| over thesame region. This normalized integral, An, is a complex number whose
magnitude | An | is in the interval [0, 1]. The effective wavefront error, WFE, may
subsequently be defined as follows:
WFE = √ 1 − | An |2 /(2π).
In the limit of small aberrations, the above WFE approaches the well-known
r.m.s. wavefront error (in units of the wavelength λ). If the beam is brought tofocus by an aberration-free lens, the Strehl Ratio (defined as the on-axis
intensity of the focused spot normalized by the same quantity in the absence ofphase distortions) will be given by
Strehl Ratio = 1 − (2π WFE)2.
If the beam happens to carry a curvature phase factor outside the mesh, thecurvature will be excluded from the effective_wavefront_error calculation.
When AZ(x, y) is unavailable, the corresponding wavefront error will be set tozero. To obtain the correct value of WFEZ, you must return to the main menu,
call the PLOT option and ask for a plot of the Z-component. Once plotting iscompleted, you should call the Wavefront_Analysis option again.
The values of WFEX, WFEY, and WFEZ are assigned to three elements within the
SIGNAL array, as specified by the user. This data can be processed later (using
the Process_Signals option), and printed out to the SIGNALS.DAT file in the noninteractive mode.

59

Guide to DIFFRACT

Example. Figure 15 shows plots of Log_intensity and Phase for the threecomponents of polarization at the focal plane of a 0.5NA focusing lens. The
incident beam is a truncated Gaussian with small amounts of various Seidel
aberrations. The COMMAND.DAT and SESSION.DAT files for this simulation are
shown in Fig. 16. The effective WFE of the incident beam is 0.0958; thus theestimated Strehl Ratio of the focused spot is 0.638. In the simulation, thefocused spot’s peak intensity was obtained both with and without aberrations,yielding a Strehl Ratio of 0.714, which is reasonably close to the value obtained
from the incident WFE.

−8

x/λ

8 −8

x/λ

8 −8

x/λ

8

Figure 15. Plots of log_intensity (top) and phase (bottom) at the focal plane of a 0.5NA lens.
From left to right, the three panels in each row correspond to the X-, Y-, and Z-components of
the optical E-field. The incident beam is a truncated Gaussian with a small amount of various
Seidel aberrations. The focused spot’s Strehl Ratio is 0.714.

60

Guide to DIFFRACT

Figure 16(a). Printout of the COMMAND.DAT file used in computing the wavefront error of a
Gaussian beam containing small amounts of various Seidel aberrations. The beam issubsequently brought to a focus by a 0.5NA positive_focusing lens. The Strehl Ratio of the
focused spot is found as the ratio of the peak intensities with and without aberrations.
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 07/16/2006
Time: 15:10 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.0000000#
1/e radii Rmin,Rmax:
3000.0000
4000.0000#
RA,THETA:
5000.0000
45.000000#
Aberrations: Seidel#
Spherical C40:
0.1500000#
Coma C31,Phi31:
0.1200000
30.000000#
Astigmatism C22,Phi22:0.2500000
120.00000#
Curvature C20:
0.1000000#
Distortion C11,Phi11:0.1700000
135.00000#
Polarization RHO,ETA:
37.000000
19.000000#
NMAX,NMAY:
1024
1024
#
LMAX,LMAY:
70000.000
70000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Phase#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-5200.0000
5200.0000#
Ymin,Ymax:-5200.0000
5200.0000#
Color or Gray-scale: G#
Z-component (Y/N): Y#
Save data files (Y/N): N#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Guide to DIFFRACT
Wavefront analysis: WAVE
(Length_Units:) #SIGNAL ID for WFE X,Y,Z:
1
23#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: PFOC#
LCX,LCY:
0.00000000
0.0000000#
NA,FL:
0.50000000
10000.000#
Aberrations: None#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance:
10000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Log(intensity)#
Logarithmic SCALE:
6.0000000#
Xmin,Xmax:-8.0000000
8.0000000#
Ymin,Ymax:-8.0000000
8.0000000#
Color or Gray-scale: G#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Phase#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-8.0000000
8.0000000#
Ymin,Ymax:-8.0000000
8.0000000#
Color or Gray-scale: G#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Measure peak and FWHM: MEAS
(Length_Units:) #Include Z-component: Y#
ID numbers of PEAK I,X,Y:
4
5
6#
Compute/Return: C#
Theta:
0.0000000#
ID number for FWHM:
7#
Compute/Return: C#
Theta:
45.000000#
ID number for FWHM:
8#
Compute/Return: C#
Theta:
90.000000#
ID number for FWHM:
9#
Compute/Return: R#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ #
#

61

62

Guide to DIFFRACT
Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.0000000#
1/e radii Rmin,Rmax:
3000.0000
4000.0000#
RA,THETA:
5000.0000
45.000000#
Aberrations: NONE#
Polarization RHO,ETA:
37.000000
19.000000#
NMAX,NMAY:
1024 1024#
LMAX,LMAY:
70000.000
70000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Wavefront analysis: WAVE
(Length_Units:) #SIGNAL ID for WFE X,Y,Z:
10
11
12
#
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: PFOC#
LCX,LCY:
0.00000000
0.0000000#
NA,FL:
0.50000000
10000.000#
Aberrations: None#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance:
10000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Measure peak and FWHM: MEAS
(Length_Units:) #Include Z-component: Y#
ID numbers of PEAK I,X,Y:
13
14
15
#
Compute/Return: C#
Theta:
0.0000000#
ID number for FWHM:
16#
Compute/Return: C#
Theta:
45.000000#
ID number for FWHM:
17#
Compute/Return: C#
Theta:
90.000000#
ID number for FWHM:
18#
Compute/Return: R#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

63

Guide to DIFFRACT

Figure 16(b). Printout of the SESSION.DAT corresponding to COMMAND.DAT of Fig. 16(a).
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 8.4
File SESSION.DAT $$$$ Date: 07/16/2006
Time: 15:10 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ALL LENGTHS ARE IN UNITS OF WAVELENGTH; ALL ANGLES ARE IN DEGREES_________________________________________________________________
Array dimensions: NDIM=1024, NDSP= 149**************************************
Designated Working Subdirectory: <TEST>****************************************
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Simple Gaussian: BCX=
0.0
THETA= 45.00

BCY=
0.0
RMIN= 3000.0

RA=
5000.0
RMAX= 4000.0

Aberrations: Seidel
Spherical: C40= 0.150000
Coma: C31= 0.120000
Phi31= 30.00
Astigmatism: C22= 0.250000
Phi22= 120.00
Curvature: C20= 0.100000
Distortion: C11= 0.170000
Phi11= 135.00
Polarization state: Rho= 37.0000
Eta= 19.0000_________________________________________________________
NMAX=1024
NMAY=1024
LMAX= 70000.00
LMAY= 70000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
6748.8
XPOWER=0.609E+00
YPOWER=0.391E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Phase distribution___________________________________________
Xmin=-5200.00
Xmax=
5200.00
Ymin=-5200.00
Ymax=
5200.00___________________________________________
PXmin= -179.989548
PYmin= -179.974075
PZmin= -179.967865

PXmax=
PYmax=
PZmax=

179.938751
179.992981
179.995453(degrees)(degrees)(degrees)

Latest screen (phase pattern) dumped in PS00.BMP
Phase pattern for X-polrz stored in PX00.DAT
Phase pattern for Y-polrz stored in PY00.DAT
Phase pattern for Z-polrz stored in PZ00.DAT
Peak intensity=0.542E-07 XPEAK=(Total energy flux=0.999E+00

0.00 YPEAK=ZPOWER=0.164E-07)

0.00

Wavefront Analysis******************
Effective Wavefront Errors for the X,Y,Z Components of Polarization:(WFEX,WFEY,WFEZ)=(0.095762,0.095762,0.159155)
WFEX,WFEY,WFEZ saved in SIGNAL array as S( 1), S( 2), S( 3)64Guide to DIFFRACT_________________________________________________________
NMAX=1024
NMAY=1024
LMAX= 70000.00
LMAY= 70000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
6748.8
XPOWER=0.609E+00
YPOWER=0.391E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________LENS ACTION***********
TYPE: Positive_Focusing
LCX=
0.000
LCY=
0.000
NA=0.500000
FL=
10000.00
Aberrations: None
Quasi-vector treatment
Propagation distance=
10000.00__________________________________________________APRX method: NGUIDE=
100 LGUIDE=
50000.0___________________________________________________________________________________________________________
NMAX=1024
NMAY=1024
LMAX=
146.29LMAY=
146.29Beam centroid:(X=-0.14,Y=
0.30)
DIAM=
4.8
XPOWER=0.606E+00
YPOWER=0.389E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Base 10 logarithm of normalized intensity distribution(Range: from 0 at peak_intensity to -6.0000 at cutoff)___________________________________________
Xmin=
-8.00
Xmax=
8.00
Ymin=
-8.00
Ymax=
8.00
___________________________________________
IXmin=
IYmin=
IZmin=

0.3294E-07
0.2321E-07
0.1243E-08

IXmax=
IYmax=
IZmax=

0.2454E+00
0.1584E+00
0.9263E-02

Latest screen (intensity pattern) dumped in IS01.BMP
Intensity pattern for X-polrz stored in IX01.DAT
Intensity pattern for Y-polrz stored in IY01.DAT
Intensity pattern for Z-polrz stored in IZ01.DATPeak intensity=0.409E+00 XPEAK=(Total energy flux=0.100E+01-0.14 YPEAK=ZPOWER=0.357E-01)

0.29

PLOTTING ROUTINE****************
Phase distribution___________________________________________
Xmin=
-8.00
Xmax=
8.00
Ymin=
-8.00
Ymax=
8.00
___________________________________________
PXmin= -179.950470
PYmin= -179.987595
PZmin= -179.993057

PXmax=
PYmax=
PZmax=

179.997849
179.998978
179.999634(degrees)(degrees)(degrees)

Latest screen (phase pattern) dumped in PS01.BMP
Phase pattern for X-polrz stored in PX01.DAT
Phase pattern for Y-polrz stored in PY01.DAT
Phase pattern for Z-polrz stored in PZ01.DATPeak intensity=0.409E+00 XPEAK=(Total energy flux=0.100E+01-0.14 YPEAK=ZPOWER=0.357E-01)

0.29

65

Guide to DIFFRACTPEAK INTENSITY AND FWHM***********************
Z-component of polarization included in PEAK & FWHM measurement
Peak Intensity=0.409E+00
XPEAK=-0.14
YPEAK=
0.29
Peak Intensity, XPEAK,YPEAK saved in SIGNAL array as S( 4), S( 5), S( 6)Theta= 0.00
FWHM=
1.36
FWHM saved in SIGNAL array as S( 7)Theta= 45.00
FWHM=
1.26
FWHM saved in SIGNAL array as S( 8)Theta= 90.00
FWHM=
1.49
FWHM saved in SIGNAL array as S( 9)INCIDENT AMPLITUDE DISTRIBUTION*******************************
Simple Gaussian: BCX=
0.0
BCY=
0.0
RA=
5000.0
THETA= 45.00
RMIN=
3000.0 RMAX=
4000.0Aberrations: None
Polarization state: Rho= 37.0000
Eta= 19.0000_________________________________________________________
NMAX=1024
NMAY=1024
LMAX= 70000.00
LMAY= 70000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
6748.8
XPOWER=0.609E+00
YPOWER=0.391E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
Wavefront Analysis******************
Effective Wavefront Errors for the X,Y,Z Components of Polarization:(WFEX,WFEY,WFEZ)=(0.000000,0.000000,0.000000)*********************************************************************
NOTE: Since the Z-component of polarization is presently unavailable,
the corresponding wavefront error has been set to zero. To obtain thecorrect value of WFEZ you must return to the main menu, call the PLOT
option, and select "Plot Z-component." Once plotting is completed you
should call the "Wavefront Analysis" option again.
*********************************************************************
WFEX,WFEY,WFEZ saved in SIGNAL array as S(10), S(11), S(12)_________________________________________________________
NMAX=1024
NMAY=1024
LMAX= 70000.00
LMAY= 70000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
6748.8
XPOWER=0.609E+00
YPOWER=0.391E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________LENS ACTION***********
TYPE: Positive_Focusing
LCX=
0.000
LCY=
0.000
NA=0.500000
FL=
10000.00
Aberrations: None
Quasi-vector treatment
Propagation distance=
10000.00__________________________________________________APRX method: NGUIDE=
100 LGUIDE=
50000.0__________________________________________________

66

Guide to DIFFRACT_________________________________________________________
NMAX=1024
NMAY=1024
LMAX=
146.29LMAY=
146.29Beam centroid:(X=0.00,Y=
0.00)
DIAM=
4.4
XPOWER=0.606E+00
YPOWER=0.389E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PEAK INTENSITY AND FWHM***********************
Z-component of polarization included in PEAK & FWHM measurement
Peak Intensity=0.573E+00
XPEAK=0.00
YPEAK=
0.00
Peak Intensity, XPEAK,YPEAK saved in SIGNAL array as S(13), S(14), S(15)Theta= 0.00
FWHM=
1.30
FWHM saved in SIGNAL array as S(16)Theta= 45.00
FWHM=
1.21
FWHM saved in SIGNAL array as S(17)Theta= 90.00
FWHM=
1.28
FWHM saved in SIGNAL array as S(18)End of session: DIFFRACT terminated normally********************************************

67

Guide to DIFFRACT

4.12. Spatial Filter. Spatial filtering involves a Fourier transformation of thebeam’s cross-section, multiplication by a filter function in the frequency domain(Sx, Sy), followed by an inverse Fourier transformation that returns the beam to
the X Y-plane. In most cases, the conventional Fast Fourier Transform (FFT)
algorithm is adequate for both forward and backward transforms. However,under certain circumstances (involving over-sampling in one domain and undersampling in the other) it may be necessary to abandon the forward FFT in favorof the (much slower) Discrete Fourier Transform (DFT) method. The user mustchoose between the two methods of computing the forward Fourier transform,
namely, between FFT and DFT.
Sy

λNmay /Lmay

So
Sx
1

λNmax /Lmax
Figure 17. The Fourier plane SxSy has dimensions λNmax /Lmax along Sx and λNmay /Lmay along
Sy. The number of pixels along Sx is Nmax, that along Sy is Nmay. The samples are uniformly
spaced at intervals of (∆Sx, ∆Sy) = (λ/Lmax, λ/Lmay). Here Nmax, Nmay, Lmax, Lmay are parametersof the final mesh in which the filtered beam resides; λ is wavelength within the current
environment. (With the FFT method, the final mesh is the same as the initial mesh, but with
DFT the final mesh may be chosen differently.) The filter multiplies the spatial frequency
spectrum of the beam by the complex constant Aoexp(iφo) inside the circle of radius So, and by
A1exp(iφ1) outside the circle. All spatial frequencies residing outside the circle of unit radius(dashed) are evanescent and are, therefore, automatically eliminated by the Spatial Filter.

The filter function is a circular mask in the (Sx, Sy) Fourier domain, as shown in
Fig. 17. Since the Fourier domain is the space of direction cosines, the unit
circle in the Sx Sy-plane is the boundary between propagating and evanescentwaves. The mask is centered at (CSX, CSY), has radius So, and its complex
amplitude (to be multiplied into the spatial frequency spectrum of the beam) is
Aoexp(iφo) inside and A1exp(iφ1) outside this circle of radius So. In addition to
being filtered, the beam also sheds its evanescent components located outside
the unit circle in the Fourier domain.

68

Guide to DIFFRACTWith the FFT method, the mesh parameters Nmax, Nmay, Lmax, Lmay remain thesame before and after filtering. With DFT, however, the user can choose a
different mesh for the filtered beam. In both cases the Fourier domain will haveNmax × Nmay pixels with uniform spacing (∆Sx, ∆Sy) = (λ/Lmax, λ/Lmay), where Nmax,
Nmay, Lmax, Lmay are the parameters of the final mesh, namely, that which contains
the filtered beam; λ is the wavelength within the current environment.
Example. Figure 18 shows the intensity and phase distributions in the crosssection of a beam of diameter 2λ; the phase plot reveals the presence of a small
amount of aberrations. This beam is filtered through a spatial filter centered at
the origin (CSX, CSY) = (0, 0) in the Fourier domain, having radius So = 0.9. The
filter transmits the Fourier components inside the circle and eliminates all spatial
frequencies outside the circle.
(a)

(b)

Figure 18. Plots of (a) intensity and (b) phase in the cross-section of a beam of diameter 2λ.
The mesh on which the beam is defined has Nmax = Nmay = 1024 pixels, its linear dimensions
being Lmax = Lmay = 20λ. The sharp cutoff at the edge of the beam and the subwavelength
phase variations across the beam’s cross-section give rise to high-frequency content as well as
evanescent components that can be removed using a spatial filter.

Both FFT and DFT methods of filtering can be applied to this distribution, withresults shown in Figs. 19 and 20, respectively. Although the final mesh size isGuide to DIFFRACT

69

substantially different for the two methods, the results are nearly identical, as
they should be. The mesh parameters for the FFT method are Nmax = Nmay = 1024,
Lmax = Lmay = 20λ for the initial as well as the final mesh. For the DFT method
the initial mesh is the same as above, but the final mesh has Nmax = Nmay = 256,
Lmax = Lmay = 16λ. Despite a smaller mesh, computation with the DFT method ismuch slower than that with the FFT method. Nevertheless, for applications
invloving very high-resolution initial distributions (such as those imported by
the FDTD option; see section 4.13), the DFT method may be the only practical
way to perform spatial filtering.

Figure 19. From left to right, plots of intensity, log_intensity, and phase obtained when the
distribution depicted in Fig. 18 is subjected to a filter that removes all spatial frequencies
beyond So = 0.9 (FFT method).

Figure 20. Same as Fig. 19 except for the computation method, which in the present case isbased on a Discrete Fourier Transformation (DFT method).

70

Guide to DIFFRACT

4.13. FDTD Interface. By allowing the export/import of ASCII (or Binary) data
files, this menu option provides the interface with an external Finite DifferenceTime Domain (FDTD) program. The Export mode creates a file containing theE-field and B-field of the distribution presently residing in the mesh. This fileserves as input to the FDTD program. In the Import mode the program reads theoutput of the FDTD program in the form of a file containing the Ex, Ey
components of the E-field in an XY-plane. The imported distribution then serves
as an initial beam for DIFFRACT, and may be treated like any other complexamplitude distribution defined on an XY-plane.
To compute the B-field components in the Export mode, DIFFRACT requires that
additional memory be allocated at the start of the session. This is done byenabling the ray_tracing features, which have similar memory requirements. If
you disable the ray-tracing features and then proceed to invoke the FDTDInterface in the Export mode, you will receive the following message:
Ray-tracing features, which have been disabled at the start of thissession, are needed to compute the B-field distribution for exporting to
the external FDTD program. To reinstate these features, quit the program,then restart it and select the "Enable Ray Tracing" option.
In the Export Mode the user specifies the desired number of pixels NX, NY
along the X and Y directions, the FDTD mesh size LX, LY, and the pixel-size ∆zof the FDTD mesh in the propagation direction Z. (∆z may be positive or
negative, depending on whether the various field components are to be evaluatedin a direction parallel or antiparallel to the beam propagation direction Z.) NX
and NY cannot exceed NDIM, where NDIM is the maximum allowed meshdimension allocated at the start of the session.
With respect to the present center of the mesh, (xc, yc, zc) = (0, 0, 0), the FDTD
mesh (having pixel-size ∆x = LX/NX along X, ∆y = LY/NY along Y, and ∆z alongZ ) may or may not be staggered (upon user request) by a half-pixel in various
directions. When Stagger is requested, Ex is centered at (½∆x, 0, 0), Ey at(0, ½∆y, 0), Ez at (0, 0, ½∆z), Bx at (0, ½∆y, ½∆z), By at (½∆x, 0, ½∆z), and Bz at(½∆x, ½∆y, 0), as shown in Fig. 21. Without Stagger, however, Ex, Ey, Bz will becentered at (0, 0, 0), while Bx, By, Ez will be centered at (0, 0, ½∆z).

71

Guide to DIFFRACT

Figure 21. The unit cell of the FDTD mesh hasdimensions ∆x × ∆y × ∆z. The various components of the
E and B fields are defined at different locations on thisunit cell, as shown. When Stagger is requested, theexported field components will be properly shifted by ahalf-pixel in various directions. Without Stagger,
however, Ex, Ey, Bz will be centered at the center of themesh, (0, 0, 0), while Bx, By, Ez will be centered at(0, 0, ½∆z).

Ez
Bx
By
∆z

Ey
Bz

Ex
∆y

∆x

The ASCII (or Binary) file XXXX.DAT, which records the exported field
distribution, has the following format: The first line contains the refractive index
NVIRON of the present environment, the vacuum wavelength λo, and the 2-letter
lower-case METRIC (cm, mm, um, nm) that represents the units of λo. The secondline contains the mesh parameters NX, NY, LX, LY, with LX, LY normalized by
λo. The field component Ex is recorded as ((EX(I, J), I = 1, NX), J = 1, NY), followedby the remaining E-field components Ey, Ez. The value of ∆z (normalized by λo)is recorded on the next line, followed by B-field components Bx, By, Bz in
precisely the same format as the E-field components. The Fortran code fragment
below describes the structure of the exported file.
INTEGER NX,NY
REAL NVIRON,LAMBDA,LX,LY,DeltaZ
COMPLEX EX(NDIM,NDIM),EY(NDIM,NDIM),EZ(NDIM,NDIM)COMPLEX BX(NDIM,NDIM),BY(NDIM,NDIM),BZ(NDIM,NDIM)
CHARACTER METRIC*2
C
OPEN(UNIT=21,FILE=’XXXX.DAT’,FORM='FORMATTED')WRITE(21,*)NVIRON,LAMBDA,METRICWRITE(21,*)NX,NY,LX/LAMBDA,LY/LAMBDAWRITE(21,*)((EX(I,J),I=1,NX),J=1,NY)
WRITE(21,*)((EY(I,J),I=1,NX),J=1,NY)
WRITE(21,*)((EZ(I,J),I=1,NX),J=1,NY)
WRITE(21,*)DeltaZ/LAMBDAWRITE(21,*)((BX(I,J),I=1,NX),J=1,NY)
WRITE(21,*)((BY(I,J),I=1,NX),J=1,NY)
WRITE(21,*)((BZ(I,J),I=1,NX),J=1,NY)
CLOSE(UNIT=21,STATUS='KEEP')

Upon completion of Export, DIFFRACT issues the following message:
Cross-section of the beam has been transferred to an external file.

72

Guide to DIFFRACT

Note: Since computation of the B-field requires additional memory, the
Ray_Tracing features of DIFFRACT must be enabled at the start of any sessionthat calls upon the FDTD Interface routine in the Export Mode.
In the Import Mode the user specifies the name of a file, XXXX.DAT, residing
within the working directory. This ASCII (or Binary) file must have thefollowing format: The first line contains the refractive index NVIRON of the
beam's environment, the vacuum wavelength λo, and a 2-letter lower-case
METRIC (cm, mm, um, nm) that represents the units of λo. The second line
contains the mesh parameters NMAX, NMAY, LMAX, LMAY, with the physicalmesh dimensions LMAX, LMAY in dimensionless form, i.e., normalized by λo.
Subsequent lines contain the distribution of the field component Ex in thefollowing order: ((EX(I, J), I = 1, NMAX), J = 1, NMAY). This must be followed by
the distribution of Ey in precisely the same format as Ex. The Fortran codefragment below describes the structure of the imported file.
INTEGER NMAX,NMAY
REAL NVIRON,LAMBDA,LMAX,LMAY
COMPLEX EX(NDIM,NDIM),EY(NDIM,NDIM)
CHARACTER METRIC*2
C
OPEN(UNIT=21,FILE=’XXXX.DAT’,FORM='FORMATTED')
READ(21,*)NVIRON,LAMBDA,METRICREAD(21,*)NMAX,NMAY,LMAX,LMAY
READ(21,*)((EX(I,J),I=1,NMAX),J=1,NMAY)
READ(21,*)((EY(I,J),I=1,NMAX),J=1,NMAY)
CLOSE(UNIT=21,STATUS='KEEP')

The imported field replaces the current content of the mesh and may be treatedas an initial distribution for further processing by DIFFRACT. Upon completionof the Import operation, the following message will be issued:
Cross-section of the beam has been imported
from external file and placed in the mesh.

Guide to DIFFRACT

73

5. Modifying the Coordinate System and the Mesh Parameters
Several options in DIFFRACT™ allow the user to shift, rotate, and tilt thecoordinate system within reasonable limits. There is also an option for rescaling
the mesh at appropriate points along the optical path. The following paragraphs
describe these options in some detail. (Appendix C provides a detailed overview
of the coordinate system(s) in DIFFRACT™.)
5.1. Changing the Coordinate System: The mesh is generally in the XY-plane,
centered at (x = 0, y = 0), and the optical axis of the system under considerationis the Z-axis. The user may choose to switch to a different Cartesian system of
coordinates by selecting one of the following options.
SHIFT translates the center of the mesh to a different position (xc, yc), in the
XY-plane. To translate the optical axis Z parallel to itself, shift the XY-plane to
the new origin (xc, yc). This new origin remains in effect until the next shifting of
the coordinate system.
ROTATE keeps the center and the Z-axis intact, but rotates the XY-planearound Z through a specified angle θ (0°: 360°). The new orientation remains ineffect until the next rotation or tilt of the coordinates.
TILT maintains the center at (x = 0, y = 0, z = 0), but allows a solid tilt of the
XYZ system. The orientation of the new coordinate system (X ' Y ' Z ') within theold system (XYZ) is determined by three angles, as shown in Fig. 22. The axis Z 'is identified by its spherical coordinates θ0 (0°: 90°) and φ0 (0°: 360°), while thethird angle ψ0 (0°: 360°) determines the orientation of X ' . When ψ0 = 0, the new
system obtains by a solid rotation of the old system, one that brings Z onto Z ' byincreasing θ from 0 to θ0, while keeping φ constant at φ0. When ψ0 ≠ 0, theabove rotation is preceded by a rotation of the XY-plane around the Z-axis
through the angle ψ0. The outcome of this process is the distribution in the
X ' Y ' -plane at Z ' = 0.
5.2. Rescaling the Mesh: The plane of the new mesh, X'Y', is centered at a new
point, (X = Xc, Y = Yc), and is rotated in the XY-plane through the angle θ(0°: 360°) relative to the X-axis. The user must also specify the new meshparameters NMAX, NMAY, LMAX, LMAY. Upon invoking the Rescale option,
DIFFRACT interpolates the current amplitude_distribution residing in AX, AY
arrays, and places the rescaled version on the new mesh.

74

Guide to DIFFRACT

Figure 22. Orientation of the new coordinate system, X'Y'Z', within the old system, XYZ, isdetermined by the angles (θ0, φ0, ψ0). The origins of the two systems may be assumed to becoincident. In the most general case, two solid rotations can bring XYZ onto X'Y'Z'.
(a) When ψ0 = 0, the new system obtains by a solid rotation of the old system, one that brings
Z onto Z' by increasing θ from 0 to θ0, while keeping φ constant at φ0.
(b) When ψ0 ≠ 0 the above operation is preceded by a rotation of the XY-plane around the Zaxis through the angle ψ0.

6. Detectors and Signal Processing
A number of detectors may be called in DIFFRACT to compute signals that
correspond to various patterns of intensity and/or polarization distribution.
Detectors do not modify the contents of the mesh; they merely compute theintegrated intensity on various segments of the XY plane. The beam may thus
continue to propagate undisturbed beyond the detector, once the desired signals
have been computed.
A special array, called SIGNAL, is used within DIFFRACT to keep track of all
detector outputs. Individual detector signals are assigned user-defined ID
numbers (from 0 to 99), and are retained within the SIGNAL array. This array isaccessible from the main menu via the Process_Signals (PSIG) option, and its
contents can be processed to yield different combinations of the various detector
outputs. Remember that the ID of a given signal is its location within the
SIGNAL array. Thus, if the same ID is assigned to two different signals, thesecond assignment will override the first.
6.1. Quad Detector: This photodetector is centered at (DCX, DCY) in the XYplane, and is split into four identical quadrants, as shown in Fig. 23. The two
splitting lines are orthogonal to each other and cross at the center point, (DCX,

75

Guide to DIFFRACT

DCY). One of the splitting lines makes an angle θ (-90°: 90°) with the X-axis, theother makes the same angle with the Y-axis. The gap between adjacent detectors
is a dead-zone of width GW, and each detector segment is an LX × LY rectangle.
When θ = 0°, LX and LY are parallel to X and Y, respectively.

When (DCX, DCY) = (0, 0) and θ = 0°, detector 1 is in the first quadrant of the
XY-plane (X > 0, Y > 0), detector 2 in the second quadrant (X < 0, Y > 0),
detector 3 in the third quadrant (X < 0, Y < 0), and detector 4 in the fourth
quadrant (X > 0, Y < 0). All detectors have unity gain, and the individual
detector outputs are denoted by S1, S2, S3, S4. Also computed are:
SUM = S1+S2+S3+S4
P12M34 = (S1+S2-S3-S4) /SUM
P13M24 = (S1+S3-S2-S4) /SUM
P14M23 = (S1+S4-S2-S3) /SUM

Figure 23. Quad_Detector in the XY-plane. Individual segments of this photodetector coveridentical LX × LY areas. The detector is centered at (DCX, DCY), and its orientation angle
relative to the X-axis is θ.

In order to store detector outputs in the SIGNAL array, the user is prompted toassign ID numbers to the four outputs of the Quad_Detector (namely, S1, S2, S3,
S4). The user must choose four integers within the range 0 − 99 and keep a recordof this assignment for later usage.

76

Guide to DIFFRACT

6.2. Phi Detector: This four-segment photodetector is centered at the point(DCX, DCY) in the XY-plane. As shown in Fig. 24, the Phi_Detector resembles
its namesake, the greek letter φ. Defining the various segments of the detector
are a circle of radius R and a straight line through the circle at angle θ (0°: 180°)
relative to the X-axis. The gap-width GW is the width of the dead-zones that
separate the various segments. The segments, which combine to cover the entire
XY-plane, have unity gain.
Segments 1 and 2 are inside the circle, below and above the line, respectively.
Segments 3 and 4 are outside the circle, below and above the line. The
corresponding signals of the four detectors are S1, S2, S3, S4. Also computed are
the following combinations:
SUM = S1 + S2 + S3 + S4
P12M34 = (S1 + S2 - S3 - S4) /SUM
P13M24 = (S1 + S3 - S2 - S4) /SUM

Figure 24. Phi_Detector in the XY-plane. The circle has radius R, and the dead-zones
between the various segments of the detector have equal widths GW. Within their respective
half-planes, segments 3 and 4, which cover the region outside the circle, extend to infinity.

In order to store detector outputs in the SIGNAL array, the user is prompted toassign ID numbers to the four output signals from the Phi_Detector (namely,
S1, S2‚ S3, S4). The user must choose four integers within the range 0-99 andkeep a record of this assignment for later usage.
Note: Phi Detector is not available in the Student Edition.

Guide to DIFFRACT

77

6.3. Split Detector: This photodetector, shown in Fig. 25, is centered at(X = DCX, Y = DCY) in the XY-plane, and split into two identical halves. The
splitting line goes through the center at an angle θ (-90°: 90°) relative to the Xaxis. The gap-width is GW, and each half-detector covers a rectangular LX × LY
area, with LX being parallel to the splitting line. Both detectors have unity gain.
When θ = 0°, detector 1 is on the negative side (and detector 2 on the positiveside) of the Y-axis. Output signals of detectors 1 and 2 are denoted by S1 and S2,
respectively. Also computed are SUM = S1 + S2 and DIFF = (S1 − S2) /(S1 + S2).
In order to store detector outputs in the SIGNAL array, the user is prompted toassign ID numbers to the two output signals from the Split_Detector (namely, S1,
S2). The user must choose two integers within the range 0-99 and keep a recordof this assignment for later usage.

Figure 25. Split_Ddetector in the XY-plane. The detector is centered at (DCX, DCY), and its
orientation angle relative to the X-axis is θ. Individual segments of this photodetector coveridentical LX × LY rectangular areas.

6.4. Differential Detector: This detector module consists of a polarizing beamsplitter (PBS) and two wide-area photodetectors, as shown in Fig. 26. The
transmission axis of the PBS may be oriented at an arbitrary angleθ (−180°: 180°) relative to X. The detectors are ideal, with infinite area andunity gain. Detector 1 collects all the incident power polarized along thetransmission axis of the PBS; its output signal is S1. Detector 2 collects theremaining optical power (polarized perpendicular to the transmission axis); its
output signal is S2. Also computed are SUM = S1 + S2‚ and DIFF = S1 − S2.

78

Guide to DIFFRACT

Figure 26. Schematic diagram of a Differential_Detection_Module. The module may beoriented by a rotation around the optical axis Z through the angle θ. The rotation angle ismeasured counterclockwise from the positive X-axis.

In order to store the detector outputs in the SIGNAL array, the user is prompted
to assign ID numbers to the two output signals from the Differential_Detector(namely, S1, S2). The user must choose two integers within the range 0-99 andkeep a record of this assignment for later usage.
Note: Differential Detector is not available in the Student Edition.
6.5. Signal Processing: The Process_Signals option from the main menu allows
the user to access and to manipulate the contents of the SIGNAL array. (SIGNAL
has 100 elements, which are numbered sequentially from S00 to S99.) Prior tocalling for signal processing, the user must have stored various detector outputs
in known locations within the SIGNAL array. Processing involves any
combination of the following operations on SIGNAL elements: addition,
subtraction, multiplication, and division of pairs of elements; assignment ofconstant values to selected elements; amplification (or attenuation) of elements
by arbitrary factors; and printing of selected elements.
While in the Process_Signals environment, the user may request any number ofoperations before returning to the main menu. Below is a listing of the available
signal processing operations.

Guide to DIFFRACT

79(+) Addition:
S(i) = S(j) + S(k)(− ) Subtraction:
S(i) = S(j) - S(k)(×) Multiplication:
S(i) = S(j) × S(k)( /) Division:
S(i) = S(j) / S(k)(=) Assignment:
S(i) = C(A) Amplification:
S(i) = G × S(j)(P) Printing S(i) to monitor and to SIGNALS.DAT(E) End processing and return to main menu
Needless to say, repeated application of the above operations can yield manydifferent combinations of the various detector outputs.
The assignment operation (=) also allows the mesh parameters (NMAX, NMAY,
LMAX, LMAY) to be assigned to the SIGNAL elements. Typically, when
Assignment (=) is requested, the user is prompted for the location i of the
SIGNAL array and the value C of the constant to be assigned to that location. If,
instead of typing a number, the user types NMAX or NMAY or LMAX or LMAY,
the current value of the chosen parameter will be assigned to element i of the
SIGNAL array.
Note: In the non-interactive mode, printing of a SIGNAL element causes thatelement to be written not only to the SESSION.DAT file, but also to the
SIGNALS.DAT file. Although the SIGNAL array is accessible inside the program,its contents will be permanently lost once the program terminates. Therefore, it
is extremely important for the user to remember that before quitting the program
he must enter the Process_Signals environment and issue the Print command for
as many elements of the SIGNAL array as he intends to keep track of. In theabsence of this crucial step, the program will terminate and all trace of theoutput signals will be lost.

80

Guide to DIFFRACT

7. Simple Phase/Amplitude Masks and Apertures
The MASK option of the main menu allows the beam to be masked and/or
apertured by a number of simple phase/amplitude transparencies. The present
section describes this option and its corresponding parameters. More complex
apertures and masks may also be used in DIFFRACT. These, however, should be
defined by the user and imported to the program via the Phase/Amplitude Object(XOBJ) option described earlier in Section 4.6.
A beam transmitted through a MASK is multiplied in its different regions bydifferent complex coefficients, A exp (iφ). In each region the amplitude A is apositive real constant, and the phase angle φ is a constant (or variable) in theinterval [-180°: 180°]. The mask is parallel to the XY-plane and is centered at(MCX, MCY). Masks with circular, elliptical, annular, and rectangular patterns(as shown in Fig. 27) are presently available. In the case of circular masks, thephase inside the circle can vary across the aperture in accordance with Seidel or
Zernike polynomials.

Figure 27. Circular, Annular, and Rectangular masks/apertures. Also indicated are thecomplex-amplitude transmission coefficients, t, in various regions. A unique feature of thecircular mask is that it allows Seidel or Zernike phase aberrations inside the circle.

The amplitude/phase discontinuity at the borders between adjacent regions of
the mask may be smoothed out by choosing the transition parameter α (0 : 1).
For example, for a circular mask of radius Rc and parameter α, the radii between(1 − α) Rc and (1 + α) Rc fall in the transition region, where both the amplitude
and the phase of the mask vary continuously from their values inside the circle
to their corresponding values outside the circle.

81

Guide to DIFFRACT• A circular mask is specified by its radius Rc and its transition parameter α.
Also required are the (constant) values of amplitude and phase inside thecircle, (A1, φ1), as well as those outside the circle, (A2, φ2). A unique featureof circular masks is that they allow Seidel and Zernike phase aberrations
inside their circle of radius Rc. (See Tables 1 and 2 in section 4.1 for a
description of these aberrations.)• An elliptical mask is specified by the lengths of its short and long semi-axes(Rmin, Rmax), by the transition parameter α, and by the orientation angle θ(-90°: 90°) of its long axis relative to the X-axis. (When θ = 0° the major axisof the ellipse is parallel to X, and the minor axis is parallel to Y.) Also
required are the values of amplitude and phase inside the ellipse (A1, φ1), as
well as those outside the ellipse (A2, φ2). A circular mask is obtained in thespecial case when Rmin = Rmax.
• An annular mask is specified by the inner and outer radii of the annulus(RC1, RC2), by the transition parameters (α1, α2) at these radii, and by thevalues of amplitude and phase inside the inner circle (A1, φ1), in the region of
the annulus (A2, φ2), and outside the outer circle (A3, φ3).
• A rectangular mask is specified by its length, width, transition parameter α,
and the orientation angle θ (-90°: 90°) of its long side relative to the X-axis.
(When θ = 0° the long side of the rectangle is parallel to X, and the short sideis parallel to Y.) Also required are the values of amplitude and phase inside
the rectangle (A1, φ1) as well as those outside (A2, φ2).
8. Polarization-sensitive Element
This item, available from the main menu of DIFFRACT, simulates several
polarization-sensitive elements including linear polarizer/analyzer, leaky
polarizer/analyzer, quarter-wave and half-wave plates, and adjustable retarders(e.g., Soleil-Babinet compensator). The polarization element has a circular
aperture of radius RA, is centered at (x, y) = (ACX, ACY), and its mutuallyorthogonal axes A and B are both in the XY-plane (see Fig. 28). The first axis A,
whose angle with X is θ (0°: 90°), multiplies the projection of the incident
polarization along A with the complex coefficient α exp(iφa). The second axis B
multiplies its corresponding component of polarization with β exp(iφb). α and β
are real numbers in the interval [0,1], while φa and φb are phase angles in therange [−180°: 180°].

82

Guide to DIFFRACT

To simulate an ideal polarizer/analyzer set β = 0. To obtain a quarter-wave plate
set α = β = 1, φa = 0, φb = 90°. Other polarization-sensitive elements may besimulated in like manner.

Figure 28. Polarization-sensitive element in XY-plane. The orthogonal axes are A and B, thecenter of the element is C, and the orientation angle of A relative to the X-axis is θ.

b

a

−2250

x/λ

2250 −2250

x/λ

2250

Figure 29. (a) Polarization rotation angle ρ and (b) polarization ellipticity η in the crosssectional plane of a linearly-polarized beam with four different polarization-sensitiveelements in its path. The four elements have identical aperture radii RA = 1000. Placed in theupper-right quadrant is a polarizer with transmission axis at θ = 45° to the X-axis, (α, φa) =(1, 0°), (β, φb) = (0, 0°), yielding ρ = 45° and η = 0°. The element in the upper-left quadrant isa quarter-wave plate oriented at θ = 45° to the X-axis, (α, φa) = (1, 0°), (β, φb) = (1, 90°). This
yields a left circularly polarized beam with η = − 45°. In the lower-left corner there is a halfwave plate at θ = -45°, (α, φa) = (1, 0°), (β, φb) = (1, 180°), yielding a linearly polarized beamat ρ = -90°. The element in the lower-right quadrant is another quarter-wave plate oriented atθ = 45°, (α, φa) = (1, 90°), (β, φb) = (1, 0°), yielding a right-circularly polarized beam with
η = +45°.

83

Guide to DIFFRACT

9. Flat Mirror
The Flat_Mirror option allows the deflection of a beam in an arbitrary direction,
as shown in Fig. 30. The mirror material may have a complex refractive index(n, k). In the XYZ coordinate system of incidence, the mirror's surface normal isspecified by the spherical angles θ0 (90°: 180°] and φ0 [0°: 360°). The angle θ0 ismeasured from the positive Z-axis, while φ0 is measured counterclockwise fromthe positive X-axis in the XY-plane. DIFFRACT breaks up the incident beam intoa plane-wave spectrum (i.e., spatial_frequency decomposition), computes
Fresnel reflection coefficients for the various constituent plane-waves, andsuperimposes the reflected plane-waves to construct the emergent beam.
Z'

X'
X

Y'
N

φo

zo

θo
Z

Y
Figure 30. A beam of light, propagating within the XYZ coordinate system, is reflected off aflat, front-surface mirror. N, the mirror’s surface-normal, makes an angle θo with the Z-axis
and its projection in the XY-plane makes an angle φo with the X-axis. The emergent beam is
specified within the X ' Y ' Z ' system of coordinates. The propagation distance is zo .

The emergent (deflected) beam is attached to the X'Y'Z' coordinate system.
Orientation of the new (emergent) system within the old (incident) system isgiven by three angles: Z' is identified by its θ1 [0°: 180°] and φ1 [0°: 360°), whileψ1 [0°: 360°) determines the orientation of X' (see Fig. 22). When ψ1 = 0 (which
is usually the best choice) the new system is obtained by a solid rotation of theold system, a rotation that brings Z onto Z' by increasing θ from 0 to θ1, while
keeping φ constant at φ1. When ψ1 ≠ 0 the above operation is preceded by arotation of the XY-plane around the Z-axis through the angle ψ1. (Appendix C
provides a detailed description of DIFFRACT’s coordinate systems.) The output

84

Guide to DIFFRACT

from the mirror is the distribution in the X'Y'-plane at a distance of zo (along theZ'-axis) from the mirror.
Total internal reflection (TIR) at the flat interface between two homogeneous andisotropic dielectric media may also be simulated using the Flat_Mirror option of
DIFFRACT. It so happens that Fresnel’s reflection coefficients at such aninterface are precisely those of a mirror with a refractive index n = n1/n2 and anabsorption coefficient k = 0. (n1 and n2 are the indices of the rare and dense
media, respectively.) For example, if TIR occurs at the interface between the air(n1 = 1) and a glass of refractive index n2, then the equivalent index of the
Flat_Mirror will have to be (n, k) = (1/n2, 0). Note that the Flat_Mirror option
does not provide the beam with the means to enter and exit the dense medium; it
only allows computation of the reflected beam at the interface. Substitution of
Flat Mirror for a TIR prism, therefore, is admissible only insofar as the effects on
the beam of the entrance and exit facets of the prism can be ignored.
200

Z′

Y′′

y/λ
-200

-200 x/λ

200

Glass Prism

Y

Z′′

Y′
E
Incident
Beam

Aluminum
MirrorZ

Figure 31. A linearly polarized Gaussian beam propagating along the Z-axis is reflected froman aluminum mirror, enters a glass prism, and is reflected from the rear facet of the prism atslightly above the Brewster angle. The inset shows the intensity distribution in the XY-planeat Z = 0. The intensity and phase distributions of the beam after reflection from the mirror andbefore entering the prism are shown in Fig. 32. The profile of the beam emerging from theprism is shown in Fig. 33.

Example. Figure 31 is the diagram of an optical system in which a linearlypolarized Gaussian beam reflects off an aluminum mirror, propagates a shortdistance in free space, then enters a glass prism (n = 1.50). The beam issubsequently reflected from the rear facet of the prism at θ = 34°, slightly above

85

Guide to DIFFRACT

the Brewster angle of θB = tan−1(1/n) = 33.69°. The various parameters of thesystem are listed in the accompanying COMMAND.DAT file. The profile of thebeam after reflection from the aluminum mirror appears in Fig. 32. The bulk of
the reflected optical power (~81% of incident power) is contained in the Ycomponent of polarization. A small fraction of the incident power (~ 0.5 × 10−7),
however, emerges polarized along the X′ axis. Figure 33 shows the beam’s
intensity, phase, and polarization state as it emerges from the prism. Due to the
low reflectance of the prism for an essentially p-polarized beam near Brewster’sangle, the total emergent optical power is only ~ 0.48 × 10−4. More information
about the state of the beam throughout this simulation may be found in its
SESSION.DAT file.
a

b

c

d

−300

x/λ

300 −300x/λ

300Figure 32. Plots of (a, b) intensity and (c, d) phase distribution in the X′ Y′ Z′ coordinate
system for the beam reflected from the aluminum mirror (see Fig. 31). The component of theE-field along the X′-axis is depicted on the left-hand side (frames a, c), while that along Y′ isshown on the right-hand side (frames b, d). The integrated intensity of Ex is only 0.50 × 10−7,
while that of Ey is 0.81. The relative phase of the two lobes of Ex is 180°.

86

Guide to DIFFRACT

a

b

c

d

e

f

−300

x/λ

300

−300

x/λ

300

Figure 33. Plots of (a, b) intensity, (c, d), phase, and (e, f) polarization distribution in the
X′′ Y′′ Z′′ system emerging from the prism in the system of Fig. 31. The component of the Efield along the X′′-axis is depicted on the left-hand side (frames a, c), while that along Y′′ isshown on the right-hand side (frames b, d). The integrated intensity of Ex is 0.477 × 10−6,
while that of Ey is 0.480 × 10−4. The relative phase of the two lobes of Ex is 180°. The
polarization rotation angle ρ in (e) is either −90° (black) or +90° (white). The polarization
ellipticity η shown in (f) varies in the range (ηmin, ηmax) = (-12.85°, +8.53°).

Guide to DIFFRACT

Printout of the COMMAND.DAT file used in simulating the system of Fig. 31.
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/24/2005
Time: 11:07 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):550.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
1/e radii Rmin,Rmax:
100.00000
200.00000#
RA,THETA:
500.00000
45.000000#
Aberrations: None#
Polarization RHO,ETA:
90.000000
0.0000000#
NMAX,NMAY:
1024
1024
#
LMAX,LMAY:
3000.0000
3000.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):550.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-200.00000
200.00000#
Ymin,Ymax:-200.00000
200.00000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):550.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Flat mirror: MIRR
(Length_Units:) #Complex index (n,k):
2.0000000
7.0000000#
Normal Theta0,Phi0:
135.00000
90.000000#
Emergent Theta,Phi,Psi:
90.000000
90.000000
0.0000000#
Propagation distance:
10000.000#
Propagation regime: DFRNL#
Smax:
0.0100000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):550.0000
NVIRON: 1.000000 #_________________________________________________________________________ #

87

88

Guide to DIFFRACT#
Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-300.00000
300.00000#
Ymin,Ymax:-300.00000
300.00000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):550.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Phase#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-300.00000
300.00000#
Ymin,Ymax:-300.00000
300.00000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):550.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Flat mirror: MIRR
(Length_Units:) #Complex index (n,k):0.6667000
0.0000000#
Normal Theta0,Phi0:
146.00000
270.00000#
Emergent Theta,Phi,Psi:
112.00000
270.00000
0.0000000#
Propagation distance:
10000.000#
Propagation regime: DFRNL#
Smax:
0.0100000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):550.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-300.00000
300.00000#
Ymin,Ymax:-300.00000
300.00000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 2#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):550.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Phase#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-300.00000
300.00000#
Ymin,Ymax:-300.00000
300.00000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 2#$________________________________________________________________________ #89Guide to DIFFRACT
Remarks:#
Vacuum wavelength (nm):550.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Polarization#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-300.00000
300.00000#
Ymin,Ymax:-300.00000
300.00000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 2#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):550.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Printout of the SESSION.DAT file corresponding to the system depicted in Fig. 31.
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 8.4
File SESSION.DAT $$$$ Date: 06/24/2005
Time: 11:07 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ALL LENGTHS ARE IN UNITS OF WAVELENGTH; ALL ANGLES ARE IN DEGREES_________________________________________________________________
Array dimensions: NDIM=1024, NDSP= 149**************************************
Designated Working Subdirectory: <TEST>****************************************
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Simple Gaussian: BCX=
0.0
BCY=
0.0
RA=
500.0
THETA= 45.00
RMIN=
100.0 RMAX=
200.0
Aberrations: None
Polarization state: Rho= 90.0000
Eta= 0.0000_________________________________________________________
NMAX=1024
NMAY=1024
LMAX=
3000.00
LMAY=
3000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
316.2
XPOWER=0.000E+00
YPOWER=0.100E+01
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=-200.00
Xmax=
200.00Ymin=-200.00
Ymax=
200.00___________________________________________
IYmin= 0.7596E-20
IYmax= 0.3183E-04
Latest screen (intensity pattern) dumped in IS00.BMP
Intensity pattern for Y-polrz stored in IY00.DAT

90Guide to DIFFRACTFLAT MIRROR***********
Mirror's refractive index: (n,k)=( 2.0000, 7.0000)
Direction of mirror normal: Theta=135.0000
Phi= 90.0000
Emergent coordinates: Theta= 90.0000 Phi= 90.0000 Psi= 0.0000
Propagation distance=
10000.00
Propagation regime: Direct Fresnel_____________________________________________________NGUIDE=
49 LGUIDE=
6324.49 SMAX= 0.001581______________________________________________________________________________________________________________
NMAX=1024
NMAY=1024
LMAX=
3000.00
LMAY=
3000.00
Beam centroid:(X=0.00,Y=
0.06)
DIAM=
320.2
XPOWER=0.503E-07
YPOWER=0.811E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=-300.00
Xmax=
300.00Ymin=-300.00
Ymax=
300.00___________________________________________
IXmin=
IYmin=

0.1396E-20
0.2824E-15

IXmax=
IYmax=

0.1109E-11
0.2454E-04

Latest screen (intensity pattern) dumped in IS01.BMP
Intensity pattern for X-polrz stored in IX01.DAT
Intensity pattern for Y-polrz stored in IY01.DAT
PLOTTING ROUTINE****************
Phase distribution___________________________________________
Xmin=-300.00
Xmax=
300.00Ymin=-300.00
Ymax=
300.00___________________________________________
PXmin= -179.992508
PYmin=
0.000000

PXmax=
PYmax=

179.893936
150.430283(degrees)(degrees)

Latest screen (phase pattern) dumped in PS01.BMP
Phase pattern for X-polrz stored in PX01.DAT
Phase pattern for Y-polrz stored in PY01.DAT
FLAT MIRROR***********
Mirror's refractive index: (n,k)=( 0.6667, 0.0000)
Direction of mirror normal: Theta=146.0000
Phi=270.0000
Emergent coordinates: Theta=112.0000 Phi=270.0000 Psi= 0.0000
Propagation distance=
10000.00
Propagation regime: Direct Fresnel

91

Guide to DIFFRACT_____________________________________________________NGUIDE=
49 LGUIDE=
6404.03 SMAX= 0.001562______________________________________________________________________________________________________________
NMAX=1024
NMAY=1024
LMAX=
3000.00
LMAY=
3000.00
Beam centroid:(X=-6.78,Y=11.24)DIAM=
336.9
XPOWER=0.477E-06
YPOWER=0.480E-04
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=-300.00
Xmax=
300.00Ymin=-300.00
Ymax=
300.00___________________________________________
IXmin=
IYmin=

0.2729E-18
0.5767E-18

IXmax=
IYmax=

0.9094E-11
0.1241E-08

Latest screen (intensity pattern) dumped in IS02.BMP
Intensity pattern for X-polrz stored in IX02.DAT
Intensity pattern for Y-polrz stored in IY02.DAT
PLOTTING ROUTINE****************
Phase distribution___________________________________________
Xmin=-300.00
Xmax=
300.00Ymin=-300.00
Ymax=
300.00___________________________________________
PXmin= -179.976166
PYmin=
0.000000

PXmax=
PYmax=

179.681305
147.318558(degrees)(degrees)

Latest screen (phase pattern) dumped in PS02.BMP
Phase pattern for X-polrz stored in PX02.DAT
Phase pattern for Y-polrz stored in PY02.DAT
PLOTTING ROUTINE****************
Distributions of polarization rotation angle and ellipticity___________________________________________
Xmin=-300.00
Xmax=
300.00Ymin=-300.00
Ymax=
300.00___________________________________________
RTmin=
ELmin=-89.995468-12.852696

RTmax=
ELmax=

89.998917
8.529793(degrees)(degrees)

Latest screen (polarization pattern) dumped in QS02.BMP
Polarization rotation angle is stored in RT02.DAT
Polarization ellipticity is stored in EL02.DATEnd of session: DIFFRACT terminated normally********************************************

92

Guide to DIFFRACT

10. Weak Biprism
The Weak_Biprism is another menu option in DIFFRACT. Each one of its two
mild prisms in the XY-plane occupies a rectangular aperture of length L andwidth W, as shown in Fig. 34. The aperture center coordinates are (PCX, PCY),
and the aperture orientation is such that the longer side of the rectangle makes anangle ψ (-90°: 90°) with the X-axis. The two apertures corresponding to the two
prisms are fully independent of each other, but they must be non-overlapping;otherwise the program will issue a warning and will renew its request foraperture specifications.

Figure 34. Rectangular regions in the XY-plane, corresponding to apertures of individual
prisms of a generalized biprism. The center coordinates are C1 and C2, the lengths are L1 andL2, the widths are W1 and W2, and the orientation angles are ψ1 and ψ2 for the two prisms.
The dimensions and orientations of the prisms are independent of each other, so long as therectangles do not overlap. The inset shows the angular coordinates (θ, φ) of a beam exitingfrom a given prism, for an incident beam that travels along the Z-axis.

Each prism is characterized by the direction of a plane-wave (originallypropagating along +Z) after being refracted by that prism. For each prism, thepolar angle θ (0°: 90°) and the azimuthal angle φ (0°: 360°) specify the
emergent beam's direction. θ is measured from the +Z direction, while φ ismeasured counterclockwise from the X-axis (see the inset in Fig. 34). DIFFRACT
treats the Biprism as a mild phase-object, one that transmits the incident beam
while imparting a phase modulation to its wavefront.

93

Guide to DIFFRACT

Figure 35 shows a Fresnel Biprism followed by a Positive_Focusing Lens; thevarious parameters of this system are listed in the accompanying
COMMAND.DAT file. The incident beam is Gaussian, having a small positive or
negative curvature. Figure 36 shows the phase distribution immediately after the
biprism for three different amounts of incident beam curvature. The
corresponding intensity profiles at the focal plane appear in Fig. 37.
Lens

Z

Focal Plane

FresnelBiprism

Figure 35. A Gaussian beam having a small wavefront curvature is incident on a Fresnel
biprism, each prism of which imparts a 0.1° tilt to the incident beam. The 0.1NA lens has afocal length of 50000λ. Observations are made immediately after the biprism and at the focal
plane of the lens.

a

−5200

c

b

x/λ

5200 −5200

x/λ5200 −5200

x/λ

5200

Figure 36. Distribution of phase immediately after the biprism of Fig. 35. The incident
Gaussian beam has rxo = ryo = 3000λ and a small amount of Seidel curvature C20.
(a) C20 = -3.0, (b) C20 = 0.0, (c) C20 = +3.0.

94

Guide to DIFFRACT

a

−150

c

b

x/λ

150 −150x/λ

150 −150

x/λ

150Figure 37. Logarithmic plots of intensity distribution in the focal plane of the lens depicted inFig. 35 (scale = 3). The lens parameters are NA = 0.1, f = 50000λ. The incident Gaussian
beam (rxo = ryo = 3000λ) has a small amount of Seidel curvature. (a) C20 = -3.0, (b) C20 = 0.0,(c) C20 = +3.0.
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/21/2005
Time: 12:56 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #10
Remarks:#
Vacuum wavelength (um):
0.78000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.0000000#
1/e radii Rmin,Rmax:
3000.0000
3000.0000#
RA,THETA:
5000.0000
0.0000000#
Aberrations: Seidel#
Spherical C40:
0.0000000#
Coma C31,Phi31:
0.0000000
0.0000000#
Astigmatism C22,Phi22:0.0000000
0.0000000#
Curvature C20:
-3.0000000#20
Distortion C11,Phi11:0.0000000
0.0000000#
Polarization RHO,ETA:
0.0000000
45.000000#
NMAX,NMAY:
1024 1024#
LMAX,LMAY:
60000.000
60000.000#$________________________________________________________________________ #Guide to DIFFRACT
Remarks:#
Vacuum wavelength (um):
0.78000
NVIRON: 1.000000 #_________________________________________________________________________ ##Weak biprism: BIPR
(Length_Units:) #Prism #1:
PCX,PCY:
1768.0000
1768.0000#
L,W,PSI:
10000.000
5000.0000-45.000000#
THETA,PHI:
0.10000000
45.000000#
Prism #2:
PCX,PCY:-1768.0000-1768.0000#
L,W,PSI:
10000.000
5000.0000-45.000000#
THETA,PHI:
0.10000000
225.00000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.78000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Phase#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-5200.0000
5200.0000#
Ymin,Ymax:-5200.0000
5200.0000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 0#30$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.78000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: PFOC#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.10000000
50000.000#
Aberrations: None#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance:
50000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.78000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Log(intensity)#
Logarithmic SCALE:
3.0000000#
Xmin,Xmax:-150.00000
150.00000#
Ymin,Ymax:-150.00000
150.00000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 0#40$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.78000
NVIRON: 1.000000 #_________________________________________________________________________ ##$INCREMENT#20,ITEM1...BY: 3.0#$INCREMENT#30,ITEM1...BY: 1#$INCREMENT#40,ITEM1...BY: 1#$JUMP#10_IF#20,ITEM1.LT.: 3.1#$#
End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

95

96

Guide to DIFFRACT

11. The Lens Action
DIFFRACT allows a number of different lenses to be placed in the optical path.
The lens types presently available are: Positive_focusing (PFOC), Collimator(COLL), General_purpose (GENL), Cylindrical (CYLN), Astigmatic (ASTG),
Toric_ring (TRNG), and Split_lens (SPLN). Important features of these lenses

will be described in the following subsections. The most commonly used lenses
in practice, the Positive_focusing lens, the Collimator, and the General_purpose
lens, are discussed in sections 11.1 – 11.3. The other four lenses described in
sections 11.4 – 11.6 are less common, but, from time to time, they findapplication in various optical systems.
Note: General_purpose lens, Toric_ring lens, and Splitlens are not available in the Student Edition.
The Positive_focusing lens may be used to bring a collimated (or nearlycollimated) beam to a focus. The emergent beam from this type of lens may becomputed at or near the focal plane, but not too far from it. To collimate aconverging or diverging beam, one should call upon the Collimator. The
Collimator in DIFFRACT may be a positive or a negative lens, which is assumed
to be corrected for a conjugate ratio of infinity. The beam entering the
Collimator must have a radius of curvature close to the focal length of the lens,thus giving rise to an emerging beam that is (nearly) collimated. The
General_purpose lens can simulate both positive and negative lenses ofarbitrary conjugate ratios, so long as the computational method of stationaryphase approximation remains viable. The Cylindrical lens is generally used tobring a (nearly) collimated beam to a line focus. The Astigmatic lens option
should be invoked when the incident beam is collimated (or nearly so) and theastigmatism involved is fairly large.†† The Toric_ring lens is a special-purpose
lens, designed to bring a (nearly) collimated beam to focus on a circular ring.
The Split_lens is another special-purpose lens, made by cutting and then sliding
the two halves of a positive lens along a diameter of the lens. When placed in a(nearly) collimated beam, the two halves of the Split_lens give rise to two
focused spots in their common focal plane, separated along the line that splits
the original lens.
As alternatives to the Astigmatic lens, you may use the Positive_focusing lens, theCollimator, or the General_purpose lens, provided that the corresponding amount of
astigmatic aberration is added to these lenses. The Astigmatic lens option, however, gives
more accurate results when the astigmatism involved is relatively large.
††

Guide to DIFFRACT97

The lenses described in the present section will be defined strictly in terms of
their functionality, not their structure. For example, if a positive lens is one thatbrings a collimated beam to focus, then its full specification may entail thevalues of its numerical aperture, focal length, and aberration coefficients. From
this perspective, the number and type of glass surfaces used in the constructionof the lens will be inconsequential. In those instances where the detailed
structure of the lens needs to be considered for the analysis of the optical system,
the user must invoke the Ray_Tracing option of the program (see Section 16).
When a functional description of the lens is available, however, methods of thepresent section will suffice.
11.1. Positive_focusing Lens: This ordinary lens, generally used for bringing
the incident beam to a focus, may be constructed of any number of spherical,
conical, or aspherical surfaces. It is centered at (LCX, LCY) in the XY-plane,and has numerical aperture NA and focal length f l, as shown in Fig. 38. (BothNA and fl must be strictly positive.) The lens may be assumed to be corrected
for all aberrations, or else, its various aberration coefficients will have to be
specified. (See Section 4.1 for a description of the Seidel and Zernike
aberrations.) The Positive_focusing lens option of DIFFRACT, therefore, may be
used to simulate a lens as simple as a plano-convex singlet, or one as complex asa high-NA microscope objective. In general, the Positive_ focusing lens satisfies
Abbe’s sine condition. (For a detailed description of the sine condition see
Appendix G.)
For DIFFRACT to properly handle the Positive_focusing lens, the incident beam
must be collimated (or nearly so), and the output distribution must be sought inthe vicinity of the focal plane. This lens may therefore be considered to have a
conjugate ratio of infinity, or, equivalently, to be an imaging lens for objects
located at infinity (or sufficiently far away).
The severity of the above restrictions on the Positive_focusing lens (i.e., thedegree of collimation of the incident beam; the closeness of the destination planeto the focal plane, etc.) will depend on several factors, among them thenumerical aperture of the lens, the number of pixels in the mesh, and thephysical size of the mesh. In one form or another, all these restrictions arecaused by the curvature of the beam, which must either be of a type that can beextracted from the mesh and carried on the side, or else it must be incorporated
into a mesh of sufficient resolution to properly represent the rapid oscillations of
the corresponding phase-factor.

98

Guide to DIFFRACT

Figure 38. Schematic diagram showing a Positive_focusing lens.

The output distribution emerging from the Positive_focusing lens may becomputed using either the “exact,” multiple-FFT technique (EXCT), or theapproximate, stationary-phase method (APRX). (See Ref. [3] for a detailed
description of these methods.) The user should select the appropriate method
based on the parameter values of the lens, character of the incident distribution,and the available size of the mesh. The EXCT method should be used only when
the incident beam is highly nonuniform and/or aberrated. Application of the
EXCT method is limited by the fact that its required mesh size grows rapidly
with the increasing numerical aperture and/or focal length of the lens. For mostapplications, however, the approximate method (APRX) gives fairly accurate
results with a modest mesh size; consequently, it is the method of choice for
essentially all cases of practical interest.
Note: The EXCT method is not available in the Student Edition.
11.2. Collimator: This lens, which may be positive or negative, is placed with
its entrance and exit pupils perpendicular to the Z-axis, and is centered at(LCX, LCY) in the XY-plane, as shown in Fig. 39. The Collimator is defined by
its numerical aperture NA and focal length f l, and by its various aberration
coefficients. (The sign of f l determines whether the lens is positive or negative.)
For best results, the beam incident on the Collimator must be a converging ordiverging beam whose radius of curvature closely matches the focal length ofthe lens. (For converging beams the appropriate Collimator is negative, whereas
diverging beams require a positive lens for collimation.) Under such

Guide to DIFFRACT

99

circumstances, the beam emerging from the Collimator should have either asmall curvature or no curvature at all, in which case the emergent curvaturephase-factor can be safely multiplied into the mesh.
Note: A Collimator and a Positive_focusing lens may be used in tandem tosimulate any lens possessing a finite conjugate ratio. As an example, consider alens whose first principal plane is located a distance of z1 from the plane of an
object, and suppose that this lens
produces an image at a distance ofz2 beyond its second principal
plane. To simulate this system, youmust first propagate the distribution
contained within the object plane tothe first principal plane of the lens.
Next, place a Collimator having
f l = z1 in the beam’s path, andpropagateashort
distance(preferrably zero) beyond this lens.
Finally,
invokethePositive_focusing lens option,
select a focal length of z2, andpropagate the beam to the focal
plane of this second lens. In thisway you will obtain the image ofthe original object.
Figure 39. Diagrams showing a positive
and a negative Collimator.

11.3. General_purpose Lens: This lens, which may be either a positive or anegative lens, is assumed to be corrected for given object and image distances,
and also to satisfy Abbe’s sine condition. (For a detailed description of the sinecondition see Appendix G.) The sign of the focal length fl determines whether
the lens is positive or negative. For an object source point, located on the opticalaxis at the correct distance from the lens, the cone of light captured by theentrance pupil has a numerical aperture NA1. The corresponding cone of light in
the image space (i.e., that leaving the exit pupil of the lens and converging
towards the conjugate image point) has a numerical aperture NA2, as shown in
Fig. 40. Depending on whether the source point and its image are real or virtual,
NA1 and NA2 may be positive or negative. The conjugate object and image

100

Guide to DIFFRACT

distances, Zobject and Zimage, which are the corresponding distances to the first andsecond principle planes, are related to f l, NA1, and NA2 as follows:
Zobject = f l (1 + NA2/NA1)

Zimage = f l (1 + NA1/NA2)

The only physical constraint on the parameter values of the lens is thatf l (NA1 + NA2) must be strictly positive. The General_purpose lens does not
require that the emergent beam be fully or nearly collimated. However, when thecurvature phase-factor of the emerging beam is large, it must remain outside themesh, that is, the user must instruct DIFFRACT to not multiply the curvaturephase-factor into the AX, AY arrays.
Note: The General_purpose lens is not available in the Student Edition.

Figure 40. Schematic diagram showing the General_purpose lens.

11.4. Positive Cylindrical Lens: This lens, whose entrance and exit pupils arecoincident with the XY-plane, is centered at (LCX, LCY), with focal length f l anda circular aperture of radius RA, as shown in Fig. 41. The cylinder axis makes anangle θ (-90°: 90°) with the X-axis.
Note: For DIFFRACT to be able to handle the Cylindrical lens, the incident beam
must be collimated (or nearly so) and the output distribution should be computedin the vicinity of the focal plane of the lens.
When the Cylindrical lens brings a collimated beam to a line focus, toaccommodate the long, thin focused line, the mesh shrinks along one of its

101

Guide to DIFFRACT

dimensions (i.e., either LMAX or LMAY becomes small). As a result, only small
deviations of the cylinder axis from the X-axis or from the Y-axis are tolerated.
In other words, θ must be either in the vicinity of 0°, or close to ±90°; otherwise,
the focused line will not fit into the highly elongated mesh.

Figure 41. (a) The cylindrical lens iscentered at C in the XY-plane; its aperture
radius is RA, and its axis makes an angle θwith the X-axis. (b) The cylindrical lensbrings a collimated incident beam into aline focus parallel to the cylinder axis.

Figure 42. (a) The clear aperture of theastigmat is elliptical, its semi-axes are RA1,
RA2, and the angle of its major axis with
the X-axis is θ. (b) Focusing of a collimatedbeam by an astigmat; the line foci appear atdifferent locations along the optical axis.

11.5. Astigmatic Lens: This lens, whose entrance and exit pupils are coincident
with the XY-plane, is centered at (LCX, LCY). The numerical aperture of the lensis NA, and its two focal lengths are f l1 and f l2 (see Fig. 42). The clear aperture ofthis astigmat is an ellipse whose major axis is oriented at an angle θ relative to
the X-axis. The semi-axes of this elliptical aperture, RA1 and RA2, are related toNA, f l1 and f l2 as follows:
RA1,2 = √ f l0 f l1,2 tan [asin(NA)]
Here f l0 = (2 f l1 f l2) / (f l1 + f l2) is a kind of average focal length for the lens. (SeeRef. [3] for more detail.)102

Guide to DIFFRACTFor DIFFRACT to be able to handle the Astigmatic lens, the incident beam mustbe (nearly) collimated and the output should be computed in the region between
the two focal planes of the lens, not too far away from their mid-plane. The
Astigmatic lens is treated in the stationary-phase approximation.
Note: A mild astigmat may be simulated either with the Astigmatic lens option
or with the Positive_focusing lens to which astigmatism has been added as anaberration. When the astigmat is not mild, however, the Astigmatic lens option
is superior, since its computations are based on a more versatile and somewhat
more accurate algorithm.
11.6. Toric_ring Lens: This lens, whose entrance and exit pupils are coincident
with the XY-plane, is centered at (LCX, LCY). As shown in Fig. 43 its radius ofaperture is RA, its focal length is f l, and the radius of its hollow central region is
R0. When the incident beam is collimated, the Toric_ring lens will produce a
sharp, bright circle of radius RING within its focal plane .
Note: For DIFFRACT to be able to handle the Toric_ring lens, the incident beam
must be collimated (or nearly so), and the output distribution should becomputed in the vicinity of the focal plane of the lens.
In general, the Toric_ring lens is treated as a phase-object, although two
methods are available for beam propagation beyond the lens. When the lensing
action is mild (i.e., when the effective numerical aperture of the lens is small),the beam may be readily propagated in an appropriate regime such as near-field(DFRNL), intermediate-field (XFRNL), or far-field (FRNHF). Under certain
circumstances, however, if the phase-factor that the lens imparts to the beam isfactorized, the constraint of small numerical aperture may be relaxed. These two
schemes of beam propagation are referred to as direct phase-object method(DPHOB) and extended phase-object method (XPHOB), respectively. The user
will be given the option to choose one or the other of these methods based on the
numerical values of lens parameters. (Reference [3] contains a comprehensive
discussion of the Toric_ring lens and of the two methods of computation.)
Note: The Toric_ring lens is not available in the Student Edition.

Guide to DIFFRACT

103

Figure 43. (a) Diagram showing the Toric_ring lens. The cross-sectional view of the lens on
the right-hand-side shows the aperture radius RA, the Ring radius, and the radius Ro of thehollow central region. (b) A sharp, diffraction-limited ring appears in the focal plane of thelens when the incident beam is collimated. For converging (diverging) incident beams thering broadens inward (outward).

11.7. Split Lens: This lens is composed of two halves of an ordinary, positive
lens, which halves have been slightly separated and/or shifted relative to eachother within the XY-plane (see Fig. 44). Before splitting, the lens has had focal
length f l and numerical aperture NA. The angle between the splitting line, whichgoes through the center of the lens, and the X-axis is θ (-90°: 90°). The positionof each half-lens in the XY-plane is specified by the coordinates of the point thatused to be the physical center of the full lens (before splitting), and which now
belongs to both halves.
Note: For DIFFRACT to be able to handle the Split_lens, the incident beam must
be collimated (or nearly so), and the output should be computed in the vicinityof the focal plane of the lens. In general, the Split_lens is treated in thestationary-phase approximation.

104

Guide to DIFFRACT

Figure 44. (a) The two halves of the split lens are centered at C1 and C2, and the splitting line
makes an angle θ with the X-axis. The diagram in (b) shows a collimated beam, brought totwo distinct foci by the split lens. A four-quadrant detector, placed at the focal plane of thelens, may be used to determine the curvature of the incident beam.
100

y/λ

−100
−100

x/λ

100

−100

x/λ

100

−100

x/λ

100

Figure 45. Logarithmic plots of intensity distribution at the focal plane of a Split_lens(NA = 0.1, f = 30000λ). The splitting line is at -45° to the X-axis, and the half-lenses are
shifted along the splitting line by ±50λ. From left to right: the incident beam has -2 waves,zero, and +2 waves of primary (Seidel) curvature.

Guide to DIFFRACT

105

An important application of the Split_lens is in detecting the curvature of anincident beam. When the incident beam is collimated, both focused spots will besymmetric with respect to the line that splits the lens. If the incident beam
happens to have positive or negative curvature, however, the two spots become
elongated in opposite directions, as shown in Fig. 45. A quad detector placed in
the focal plane of this lens thus produces four signals which, when properly
combined, will yield a bipolar signal in proportion to the incident beam’s
curvature.
Note: The Split_lens is not available in the Student Edition.

Guide to DIFFRACT

106

12. Diffraction Grating
A useful and highly versatile optical element available through the main menu
of DIFFRACT is the Diffraction_Grating. Gratings modulate the phase and/or theamplitude of an incident beam upon reflection from (or transmission through) a
periodic, one-dimensional, surface-relief structure. Both reflection-type andtransmission-type Gratings are allowed in DIFFRACT. Briefly, the method oftreatment used in DIFFRACT is as follows: At first, the incident beam isdecomposed into its plane-wave constituents. Each plane-wave is then allowed
to diffract from the Grating, giving rise to multiple diffraction orders. The
resulting orders are eventually superimposed to produce the distribution of theemergent beam.
The Grating is attached to the coordinate system X'Y'Z', with grooves parallel to
X' and surface-normal along Z'. Orientation of X'Y'Z' within the incident XYZ
system is as follows: Z' has coordinates θ0 [0° : 90°]† and φ0 [0° : 360°), whileorientation of X' is given by ψ0 [0° : 360°] (see Fig. 22). When ψ0 = 0, X'Y'Z' isobtained by a solid rotation of XYZ which brings Z onto Z' by increasing θ from0 to θ0, while keeping φ constant at φ0. When ψ0 ≠ 0, the above operation ispreceded by a rotation of the XY-plane around the Z-axis through the angle ψ0.
The freedom to choose the values of θ0, φ0, ψ0 means that the user can rotate
and/or tilt the Grating in 3-dimensional space without any constraints.
The emergent beam's coordinates are X"Y"Z". This system is identified with theangles θ1 [0 : 180°], φ1 [0 : 360°) and ψ1 [0° : 360°] relative to the XYZ
coordinates of incidence (again see Fig. 22 for a definition of the Euler angles
θ, φ, ψ). By allowing the user to choose the emergent coordinate system,
DIFFRACT keeps track of those portions of the reflected or transmitted beam thatare of particular interest to the user.
As for specification of the physical characteristics of the Grating, the following
two options are available:a) The user specifies the Grating shape and other relevant parameters. The
program then computes reflection (or transmission) coefficients for thevarious diffracted orders by assuming that the Grating is essentially a scalar
phase/amplitude object. This method largely ignores the physics of interaction†

The surface normal is assumed to point into the grating for both reflection-type andtransmission-type gratings. Since the incident beam generally propagates along the positive Zaxis, the grating’s surface normal is also assumed to have a positive projection on the Z-axis.

Guide to DIFFRACT

107

between the Grating and the beam of light by simply assuming that, inconformity with the relief pattern, the Grating imparts to the incident beam aphase/amplitude modulation. This assumption may provide a fair
representation of the physical reality in some cases, but it is anoversimplification in many other situations.
b) The user supplies a data-file containing reflection (or transmission)coefficients of the Grating for the admissible diffraction orders. DIFFRACT
proceeds to read this file, superimpose the various diffracted beams, andcreate the resultant distribution. This method should be employed when thecomplete response of the Grating to plane-wave illumination at all incidence
angles of interest is available, information which may be obtained by solving
Maxwell’s equations for the Grating of interest. DIFFRACT presently does nothave the capability to compute the exact diffraction coefficients, requiring
instead that the file containing these coefficients be generated externally. A
detailed description of the required file format is given in section 12.4 below.
Note: Option (b) is not available in the Student Edition.
12.1. Analysis of Grating in the Scalar Approximation: Under certain
conditions a Grating may be treated as a simple phase/amplitude object, towhich the approximate scalar diffraction theory applies. Shown in Fig. 46(a) is a
Grating in the X'Y' plane, with grooves parallel to X'. The Z'-axis crosses X'Y' at
the origin (X' = 0, Y' = 0). The trapezoidal cross-section of the grooves is
identified by corner coordinates in the Y'Z' plane at (Y', Z') = (A, 0), (B, ζ), (C, ζ),(D, 0). Thus ζ is the groove depth,‡ C – B is groove width at the bottom, and D – A
is groove width at the top. The period of the Grating is denoted by P. Amplitudereflectivity (or transmissivity) is AL on the land region, AG on the groove
bottom, and linearly varying between these two values on the groove sidewalls.
(Both AL and AG are real numbers between 0 and 1.)
Note: DIFFRACT treats such Gratings as phase/amplitude-objects, ignoring thecomplex, vectorial nature of the interaction between the beam of light and thecorrugated (or otherwise modulated) medium of the Grating. In other words,
DIFFRACT does not solve the full Maxwell's equations, but provides a solution‡

ζ is the effective (rather than physical) groove depth; it represents the relative phase shiftimparted to the beam upon reflection from (or transmission through) the land and the groove.
The phase shift factor is exp(-i4πζ) for the reflection-type grating and exp(i2πζ) for thetransmsission-type grating, irrespective of the refractive index of the medium of incidence orthat of the substrate.

Guide to DIFFRACT

108

(a)

(b)

Environment(c)Environment
Transparent
hemisphere(medium of
incidence)

Transparent
medium ofincidence
Grating
Substrate

Center of spherical surface

Spectral Compression

Beam Compression

Figure 46. (a) Grating in the X'Y'-plane; Z' is the Grating’s surface-normal. The X'Y'Z'
coordinates of the Grating may have an arbitrary orientation within XYZ, the coordinates of
the incident beam. P is the Grating period, C − B is the groove width at the bottom, D − A isthe groove width at the top, and ζ is the effective groove-depth. The distance of a groove edge
from the origin of the X'Y'-plane may be adjusted with the help of the parameter A. The
amplitude reflection (or transmission) coefficients for land and groove regions are AL and AG,
respectively. Although not shown here, the user must also specify the orientation of thecoordinate system X"Y"Z" within which the emergent beam travels. (b) Spectral compression
means that upon entering and leaving the medium of incidence (or the substrate), oblique rays
bend according to Snell's law. (c) Beam compression means that an index-matched hemispherical cap, placed at the interface between the environment of the incident beam and theGrating’s medium of incidence (or at the junction between the Grating substrate and theemergence environment), eliminates the refraction of oblique rays upon entering and/or
exiting the Grating’s neighborhood.

Guide to DIFFRACT

109

that is approximately valid under certain experimental conditions. For example,
shallow gratings, gratings with a large period (compared to the wavelength oflight), non-metallized gratings, etc., are more suitable for this type of analysis
compared to deep-groove, short-pitched, metallized gratings that require themore rigorous treatment based on Maxwell's equations. A noteworthy feature ofmetallized gratings, for instance, that requires the solution of Maxwell'sequations (and, therefore, cannot be predicted by the scalar approximation) is
the exhibition of one of Wood's anomalies involving the excitation of surface
plasmons.
Since the number of diffracted orders for each incident plane-wave is
proportional to the Grating period P, the computation time increases with theincreasing of P. In practice one can expect reasonable simulation times if P is
kept below about 50λ, where λ is the wavelength of light in the incidence
medium (for reflection Gratings) or in the substrate (transmission Gratings).
12.2. Entering and Exiting the Grating Neighborhood. In addition to Grating
parameters, the user specifies the refractive index of an incidence medium, ninc,through which light impinges upon the Grating (ninc ≥ NVIRON). Also, in thecase of transmission Gratings, the user is prompted for the refractive index of
the substrate, nsub, into which the light emerges after passage through theGrating. When ninc = NVIRON, the refractive index of the beam’s environment(also, when nsub = 1.0 in the case of a transmission Grating) no further questions
will be asked, and the program proceeds to compute the emergent distribution.
When you choose an incidence medium other than the current environment of
the beam (i.e., ninc > NVIRON), you are forcing a change in the nature of theincident distribution. DIFFRACT allows you to treat this problem in two different
ways:• Leave the incident distribution within the XY-plane intact. Here, as shown in
Fig. 46(b), the angular spectrum of the beam will be compressed toward theorigin (in Fourier domain) in order to preserve its spatial dimensions.
• Leave the incident angular spectrum intact. In this case, as Fig. 46(c)indicates, the beam itself will be compressed toward the origin in the XYplane in order to preserve the directions of its various rays.
The Grating may be used either in reflection mode or in transmission mode. In
reflection mode, the beam may stay within the incidence medium (n = ninc) or it
may emerge into its original environment (n = NVIRON) by exiting the incidence

110

Guide to DIFFRACT

medium the same way as it had entered this medium (i.e., by decompressing thebeam or its angular spectrum). The propagation distance after reflection ismeasured from the interface of the Grating with the medium of incidence.
In transmission mode, the beam will enter yet another transparent medium, thesubstrate, whose refractive index nsub ( ≥ 1) must be specified by the user. Whennsub = 1, the beam emerges directly into free-space and no further questions will
be asked. When nsub > 1, the beam may stay within the substrate, or it may exit
into another transparent, isotropic medium, the emergent environment, which
subsequently becomes the new environment for the beam. In the latter case theuser will be prompted to specify the shape of the surface through which the lightmust pass in order to exit the substrate (i.e., beam or angular-spectrum
decompression, corresponding to spherical or flat exit surfaces, respectively).
The propagation distance after transmission is measured from the interface
between the Grating and the substrate.
When the beam reflected from the Grating exits the medium of incidence toreturn to its pre-incidence environment, the compression performed on theincident beam (whether done on the beam itself or on its angular spectrum) will
be automatically reversed; that is, the reflected beam will properly trace theincidence path in the reverse direction. The beam transmitted through theGrating will also have the option of remaining within the substrate or exiting thesubstrate into a new, user-defined environment. If the refractive index of the
new environment happens to differ from nsub (that is, NVIRON < nsub) you may
choose to decompress the angular spectrum of the transmitted beam (inaccordance with Snell's law) in order to simulate the case of a beam exiting
through a flat substrate. Alternatively, you may leave the angular spectrum
intact and decompress the transmitted beam instead, as would be required, forinstance, when the beam exits through a hemispherical substrate.
Note: DIFFRACT assumes the optical system as a whole is designed to correctaberrations that result when the beam enters and exits the Grating through themedium of incidence and/or through the substrate; all such aberrations,
therefore, are ignored by the program. The propagation distance you will be
asked to specify when the beam is reflected from the Grating and returned to its
original environment (NVIRON < ninc), or when the beam is transmitted throughthe Grating and has emerged from the substrate into a different environment(NVIRON < nsub), is an effective distance which would have applied had the entire
propagation path been in the beam’s pre-incidence environment (reflection), or
post-incidence environment (transmission).

Guide to DIFFRACT111

The user may specify the top surface of the medium of incidence and the bottom
surface of the substrate as “bare” or “antireflection-coated.” When a surface isbare the various rays enter and exit through that surface with proper attenuationin accordance with Fresnel's transmission coefficients. When the surface isantireflection-coated the transmission coefficient will be assumed to be unity
regardless of the angle of incidence.
Example 1. A uniform beam (λo = 0.65µm, diameter = 20µm) is reflected froma grating at normal incidence. The trapezoidal grooves are 3 µm wide at the top,
1 µm wide at the bottom, and 0.1 µm deep; the grating period is 5 µm. The
amplitude reflectivity of the land regions is 1.0, while that of the grooves (at thebottom) is 0.8. Figure 47 shows plots of intensity and phase distribution
immediately after reflection from the grating surface. Simulation parameters arelisted in the accompanying command.dat and session.dat files.

Figure 47. (Left) Intensity and (right) phase profiles of a uniform beam immediately afterreflection from a diffraction grating. The trapezoidal grooves are 3 µm wide at the top, 1 µm
wide at the bottom, and 0.1 µm deep; the grating period is 5 µm. The amplitude reflectivity of
the land regions is 1.0, while that of the grooves (at the bottom) is 0.8.
Printout of COMMAND.DAT file pertaining to Example 1@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 05/11/2005
Time: 08:33 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#

112

Guide to DIFFRACT$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: um) #Type: UB/SG/GG/LG/HG/LD: UB#
BCX,BCY,POWER(mW):
0.000000
0.000000
10.00000#
Radius of aperture:
10.00000#
Aberrations: None#
Polarization RHO,ETA:
0.000000
0.000000#
NMAX,NMAY:
1024
1024
#
LMAX,LMAY:
140.0000
140.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Diffraction grating: GRAT
(Length_Units: um) #Reflection/Transmission: R#
Parameters: A,B,C:
1.000000
2.000000
3.000000#
D,Period,Zeta:
4.000000
5.000000
0.100000#
AL,AG:
1.000000
0.800000#
Nsub,Ninc:
1.000000
1.000000#
Compress incident: NONE#
Entrance surface: BARE#
Emergent environment's n:
1.000000#
Decompress emergent: NONE#
Exit surface: BARE#
Grating Theta,Phi,Psi:
0.000000
0.000000
45.00000#
Emergent Theta,Phi,Psi:
180.0000
0.000000
0.000000#
Propagation distance:
0.000000#
Propagation regime: DFRNL#
Smax:
0.100000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-12.00000
12.00000#
Ymin,Ymax:-12.00000
12.00000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #Guide to DIFFRACT

113_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Phase#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-12.00000
12.00000#
Ymin,Ymax:-12.00000
12.00000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Printout of SESSION.DAT file pertaining to Example 1$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 8.4
File SESSION.DAT $$$$ Date: 05/11/2005
Time: 08:49 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$_______________________________________________________DIMENSIONLESS LENGTHS ARE IN UNITS OF VACUUM WAVELENGTH
ALL ANGLES ARE IN DEGREES_______________________________________________________
Array dimensions: NDIM=1024, NDSP= 149**************************************
Designated Working Subdirectory: <EXAMPLE>******************************************_________________________________________________________________
Vacuum lambda (um)= 1.0000
NVIRON= 1.0000 Length_Units: um
LMAX=
10000.0
LMAY=
10000.0
NMAX=1024
NMAY=1024_________________________________________________________________
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Length Units: um________________
Uniform Beam: Center coordinates: BCX=Power(mW)=

0.00000
10.0000

BCY=
RA=

0.00000
10.0000

Aberrations: None
Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________________________
Vacuum lambda (nm)= 650.00
NVIRON= 1.0000
Length_Units: um
LMAX=
140.000
LMAY=
140.000
NMAX=1024
NMAY=1024
Beam centroid: (X= 0.15630E-09, Y= 0.10990E-06)
DIAM=0.200069E+02
XPOWER(mW)= 9.9991
YPOWER(mW)= 0.0000
Curvature= 0.00000E+00_________________________________________________________________________DIFFRACTION GRATING*******************
Length Units: um________________

Guide to DIFFRACT

114
Grating Type: Reflection
A= 0.10000E+01
D= 0.40000E+01
AL= 1.0000

B= 0.20000E+01
Period= 0.50000E+01
AG= 0.8000

C= 0.30000E+01
Zeta= 0.10000E+00

Refractive index of medium of incidence= 1.00000
Refractive index of emergent environment= 1.0000
Grating coordinates: Theta= 0.00
Emergent coordinates: Theta=180.00

Phi=
Phi=

0.00
0.00

Psi= 45.00
Psi= 0.00

Propagation distance=
0.00000_________________________________________________________________________
Vacuum lambda (nm)= 650.00
NVIRON= 1.0000
Length_Units: um
LMAX=
140.000
LMAY=
140.000
NMAX=1024
NMAY=1024
Beam centroid: (X=-0.11544E-04, Y=-0.12236E-04)
DIAM=0.199693E+02
XPOWER(mW)= 8.3598
YPOWER(mW)= 0.39118E-02
Curvature= 0.00000E+00_________________________________________________________________________
PLOTTING ROUTINE****************
Length Units: um________________
Intensity distribution___________________________________________
Xmin= -0.120000E+02
Xmax= 0.120000E+02
Ymin= -0.120000E+02
Ymax= 0.120000E+02___________________________________________
IXmin=
IYmin=

0.9736E+02
0.1864E+01

IXmax=
IYmax=

0.3978E+08
0.1904E+06

Latest screen (intensity pattern) dumped in IS00.BMP
Intensity pattern for X-polrz stored in IX00.DAT
Intensity pattern for Y-polrz stored in IY00.DAT
PLOTTING ROUTINE****************
Length Units: um________________
Phase distribution___________________________________________
Xmin= -0.120000E+02
Xmax= 0.120000E+02
Ymin= -0.120000E+02
Ymax= 0.120000E+02___________________________________________
PXmin= -179.989532
PYmin= -179.967316

PXmax=
PYmax=

179.994736
179.997498(degrees)(degrees)

Latest screen (phase pattern) dumped in PS00.BMP
Phase pattern for X-polrz stored in PX00.DAT
Phase pattern for Y-polrz stored in PY00.DAT
End of session: DIFFRACT terminated normally********************************************

Guide to DIFFRACT

115

Example 2. A circularly-polarized Gaussian beam (λo = 0.65µm) is focusedthrough a 0.6NA objective lens onto a grating (period = 1.5µm, groove depth= 125nm). The reflected beam is collected by the same objective, and theintensity and phase distributions at the exit pupil of the lens are shown in
Fig. 48. The various simulation parameters are listed in the accompanying
command.dat file. In the case of Fig. 48(a), the medium of incidence on thegrating is a flat cover layer of index ninc = 1.58, that is, in the GRATING
command block, Compress incident: SPEC and Decompress emergent: SPECare used. The parameters used in the simulation that lead to Fig. 48(b) wereidentical, except for the shape of the incidence medium on the grating, which
was hemi-spherical in this case (i.e., in the GRATING command block,
Compress incident: BEAM and Decompress emergent: BEAM were used).

Figure 48. Distributions of intensity (left) and phase (right) at the exit pupil of a 0.6NA
objective lens. (Top) The medium of incidence on the grating is a transparent, flat cover layer(ninc = 1.58). (Bottom) Incidence medium is a hemisphere of index ninc = 1.58. The position of
the groove-edge with respect to the focused spot is adjusted using the grating parameter A (solong as B,C and D are equally shifted); the focused spot is centered at the groove center.

116

Guide to DIFFRACT

Printout of COMMAND.DAT file pertaining to Example 2@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 05/10/2005
Time: 20:54 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: mm) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,POWER(mW):
0.000000
0.000000
5.000000#
1/e radii Rmin,Rmax:
5.000000
7.000000#
RA,Theta:
3.000000
0.000000#
Aberrations: None#
Polarization RHO,ETA:
0.000000
45.00000#
NMAX,NMAY:
1024
1024
#
LMAX,LMAY:
50.00000
50.00000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: mm) #Type: PFOC#
LCX,LCY:
0.000000
0.000000#
NA,FL:
0.600000
5.000000#
Aberrations: None#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance:
5.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Diffraction grating: GRAT
(Length_Units: um) #Reflection/Transmission: R#
Parameters: A,B,C:
1.000000
1.200000
1.800000#
D,Period,Zeta:
2.000000
1.500000
0.125000#
AL,AG:
0.7000000.700000#
Nsub,Ninc:
1.000000
1.580000#
Compress incident: SPEC#

Guide to DIFFRACT
Entrance surface: BARE#
Emergent environment's n:
1.000000#
Decompress emergent: SPEC#
Exit surface: BARE#
Grating Theta,Phi,Psi:
0.000000
0.000000
45.00000#
Emergent Theta,Phi,Psi:
180.0000
0.000000
0.000000#
Propagation distance:
5000.000#
Propagation regime: FRNHF#
Smax:
0.100000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: mm) #Type: COLL#
LCX,LCY:
0.000000
0.000000#
NA,FL:
0.600000
5.000000#
Aberrations: None#
Scalar/Quasi-vector: QV#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-3.200000
3.200000#
Ymin,Ymax:-3.200000
3.200000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Phase#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-3.200000
3.200000#
Ymin,Ymax:-3.200000
3.200000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

117

Guide to DIFFRACT

118

12.3. Importing Reflection/Transmission Coefficients from a Data-file.
Consider a plane-wave propagating along the direction of the unit-vector
S = (Sx, Sy, Sz). Since Sx2 + Sy2 + Sz2 = 1, the direction of propagation may be
identified by a point within the unit-circle in the SxSy-plane. Next, consider arectangular mesh centered at the origin of the SxSy-plane, having lattice constants∆Sx and ∆Sy, as shown in Fig. 49. Those mesh points located at (m∆Sx, n∆Sy)which fall within the unit-circle may be used to define a discrete set of planewaves in the incidence space.
Sy

∆Sy
Sx

Figure 49. Rectangular grid in the SxSy-plane.
The mesh points within the unit circle
correspond to a set of propagation directions
S = (Sx, Sy, Sz) for incident plane waves.

∆Sx

Upon reflection from (or transmission through) the Grating, each incident planewave will give rise to one or more diffracted orders. Let us assume that thegrooves of the Grating are parallel to the X-axis in some Cartesian frame ofreference, and that P denotes the period of the Grating (normalized by thevacuum wavelength λ0). Also for simplicity we assume that both the medium of
incidence and the substrate are free space, that is, ninc = nsub = 1. Then the k’th
diffracted order will propagate along the unit vector S’, where Sx’= Sx andSy’= Sy + k/P. Of course, only those values of k are acceptable that keep
(Sx’, Sy’) within the unit circle; therefore,− P (√ 1 − Sx2 + Sy ) ≤ k ≤ P (√ 1 − Sx2 − Sy)
For each incident plane-wave whose propagation direction is given by(Sx, Sy), and for each integer value of k satisfying the above inequality, there are
four complex coefficients Cxx, Cxy, Cyx, Cyy, which relate the amplitude of theincident plane-wave to that of the reflected (or transmitted) plane-wave.
Denoting the components of the electric field vector along the X- and Y-axes by
Ex and Ey, we have

Guide to DIFFRACT

119

Ex’ = Cxx Ex + Cxy Ey
Ey’ = Cyx Ex + Cyy Ey
where the primed E’s correspond to the diffracted field, and the unprimed E’s
represent the incident field. The set of complex coefficients (Cxx, Cxy, Cyx, Cyy)
for all incident plane-waves (Sx, Sy) and for all admissible values of thediffraction order k may be obtained by solving Maxwell’s equations for theGrating of interest. These coefficients should then be saved in a particular orderin an ASCII (or Binary) file for subsequent retrieval by DIFFRACT.

12.4. Format of the Data-file: The data-file containing the reflection ortransmission coefficients must have a four-letter name, followed by theextension .DAT (e.g., NAME.DAT). The contents of this file may be either ASCII
or Binary characters and numbers. Written on the first line of the file must be theletter R (for reflection) or T (for transmission), which signifies the type of theGrating. The second line must contain five real numbers: the refractive index
nsub of the substrate, the refractive index ninc of the medium of incidence, theGrating period PERIOD (normalized by the vacuum wave-length λo of the light),
and the sampling intervals ∆Sx and ∆Sy, which specify the mesh in the space ofincident plane-waves (see Fig. 49). Subsequent lines of the file contain the range
parameters Imin, Imax, Jmin, Jmax, Kmin, Kmax and the complex coefficients Cxx, Cxy,
Cyx, Cyy in the specific order shown in the Fortran code fragment of Fig. 50.
The character variable GTYPE specifies the type of the Grating as reflection (R)or transmission (T). The real number NSUB (≥ 1) is the refractive index of themedium into which the beam of light enters immediately after transmission
through the Grating. This parameter is not used by the program when theGrating is of the reflection-type, but it is included here to maintain uniformity
across the board. The real number NINC (≥ 1) is the refractive index of themedium immediately before the Grating, through which the light passes before
impinging on the Grating’s surface. The real number PERIOD (≥ 0) is the Grating
period normalized by the vacuum wavelength λ0 of the light, and the real
numbers DLTSX and DLTSY, both belonging to the interval (0 : 1), are thesample-spacings within the SxSy-plane.

Figure 50 (next page). Fortran code fragment used to create the data-file NAME.DAT
containing the reflection/transmission coefficients (CXX, CXY, CYX, CYY) of the Grating.

Guide to DIFFRACT

120

CHARACTER GTYPEINTEGER IMIN,IMAX,JMIN,JMAX,KMIN,KMAX
REAL NSUB,NINC,PERIOD,DLTSX,DLTSYCOMPLEX CXX,CXY,CYX,CYY.
.
.

OPEN(UNIT=10,FILE='NAME.DAT',FORM='FORMATTED',STATUS='NEW')
WRITE(10,'(A1)')GTYPEWRITE(10,*)NSUB,NINC,PERIOD,DLTSX,DLTSYIF(GTYPE.EQ.'R')THEN
P=NINC*PERIODRHO=1.0
ELSE IF(GTYPE.EQ.'T')THEN
P=NSUB*PERIODRHO=NINC/NSUB
ENDIF
IMIN=0IF(RHO.LE.1)THEN
IMAX=NINT(1.0/DLTSX)IF(IMAX*DLTSX.GT.1)IMAX=IMAX-1
ELSE
IMAX=NINT(NSUB/(NINC*DLTSX))
IF(IMAX*DLTSX*NINC.GT.NSUB)IMAX=IMAX-1
ENDIF
WRITE(10,*)IMIN,IMAX
DO I=IMIN,IMAXSX=I*DLTSXSAUX=SQRT(1-SX*SX)JMAX=NINT(SAUX/DLTSY)IF(JMAX*DLTSY.GT.SAUX)JMAX=JMAX-1
JMIN=-JMAX
WRITE(10,*)JMIN,JMAX
SAUX=SQRT(1-RHO*RHO*SX*SX)DO J=JMIN,JMAXSY=J*DLTSYKMIN=NINT(-P*(SAUX+RHO*SY))IF((RHO*SX)**2+(RHO*SY+KMIN/P)**2.GT.1)KMIN=KMIN+1
KMAX=NINT(P*(SAUX-RHO*SY))IF((RHO*SX)**2+(RHO*SY+KMAX/P)**2.GT.1)KMAX=KMAX-1
WRITE(10,*)KMIN,KMAX
IF(KMIN.GT.KMAX)CYCLE.
.
.

DO K=KMIN,KMAX
WRITE(10,*)CXX(I,J,K),CXY(I,J,K),CYX(I,J,K),CYY(I,J,K)
END DOEND DOEND DOCLOSE(UNIT=10,STATUS='KEEP')

Guide to DIFFRACT

121

Note in the outer loop of the above code fragment that the parameter I ranges
from 0 to Imax, which means that only the C coefficients corresponding to Sx ≥ 0
are saved in the file. Since the grooves are assumed to be parallel to the X-axis(in the Grating’s own system of coordinates), there is symmetry with respect to
the Y-axis, which we have used to advantage in order to reduce the number ofcoefficients that need to be saved in the file.
Ideally, there should have been no need for recording in the file the values ofImin, Imax, Jmin, Jmax, Kmin, Kmax, which define the range of the various loops. In
practice, however, truncation errors and differences in numerical accuracy ofdifferent computers render ambiguous the status of those points that lie near theboundary of the unit circle in the Sx Sy-plane. Inclusion of the range of the loops
at appropriate locations throughout the file ensures that DIFFRACT assigns
correct indices to the various C coefficients when importing them from the file.
The value of Imax in the above code fragment corresponds to the largest possible
Sx that is still within the unit circle. In practice smaller values of Imax will also be
accepted by DIFFRACT, but the C coefficients of those incident plane-waves thatare thus left out of the file will be assumed to be zero. This gives the user theflexibility to record the C coefficients for a subset of the Sx Sy-plane, if he/she
happens to know in advance the range of plane-waves that are incident on theGrating. The same argument applies to the values of Imin, Jmin, Jmax, Kmin, Kmax aswell. The values used in the code fragment represent the minimum/maximum
allowed values; the user may choose to record a subset of all the C coefficientsby selecting appropriate bounds that may be above/below these limits.
The code fragment in Fig. 50 creates a “formatted” or ASCII file in a Fortranenvironment. ASCII files suitable for importation to DIFFRACT may also becreated using a number of commercially available spreadsheets, compilers, etc.
When creating such files it is important to note that the complex number C isrecorded as (Real,Imaginary), complete with its parentheses and a comma
between the real and imaginary parts.
Binary files are faster to write to and read from the hard disk, but, to assureformat compatibility, they must be created in a Fortran environment. To modify
the code fragment of Fig. 50 such that it would create a Binary file, replace‘FORMATTED’ with ‘UNFORMATTED’ in the OPEN statement, then eliminate allformat specifiers from the WRITE statements; specifically, you must replace
WRITE(10,’(A1)’) and WRITE(10,*) with WRITE(10).
The ASCII or Binary data-file thus created must be placed in DIFFRACT’s
working directory prior to running the program. DIFFRACT reads this file,122

Guide to DIFFRACT

performs a linear interpolation on the contents of the mesh in the Sx Sy-space ofincidence, and obtains continuous functions for Cxx, Cxy, Cyx, Cyy. It proceeds bysuperposing the diffracted orders and generating an emergent distribution.
Note: The option of importing reflection/transmission coefficientsfrom a data-file is not available in the Student Edition.

Guide to DIFFRACT

123

13. The Multilayer
The selection of this menu item prompts the user to define a Multilayer structurewith an arbitrary orientation in space. DIFFRACT then proceeds to decompose
the incident beam into an angular spectrum of plane-waves, compute the Fresnel
matrix of reflection (or transmission) for each such wave, and superimpose theresulting plane-waves. The reflected (or transmitted) beam thus obtained may
then be propagated through the system.
As shown in Fig. 51(a), the Multilayer has a transparent, semi-infinite substrate
and an arbitrary number of layers coated over this substrate. Individual layers
may be dielectrics, metals, birefringent materials (e.g., crystal slabs, plastic
sheets, etc.), or magneto-optical active media. The layers are numbered inincreasing order, starting at the substrate, which is considered to be layer 0. The
incident beam enters the structure from the top surface of the last layer (i.e. thelayer identified with the largest number), reaching the substrate only after
passing through all other layers. Like the substrate, the medium of incidence istransparent and could possess an arbitrary index of refraction (ninc ≥ NVIRON).
The maximum allowed number of layers is presently set to 10 in theStudent Edition and to 99 in the General Edition of DIFFRACT. One may
increase the maximum allowed number of layers in the General Edition
by modifying the corresponding parameter in the SETTINGS.LOG file.
13.1. Orientation of the Multilayer. The Multilayer is attached to thecoordinate system X'Y'Z', with its surface-normal along Z'.§ Orientation of X'Y'Z'
within the incident XYZ system is as follows: Z' has coordinates θ0 [0° : 90°] andφ0 [0° : 360°), while orientation of X' is given by ψ0 [0° : 360°] (see Fig. 22).
When ψ0 = 0, X'Y'Z' is obtained by a solid rotation of XYZ which brings Z onto
Z' by increasing θ from 0 to θ0, while keeping φ constant at φ0. When ψ0 ≠ 0, theabove operation is preceded by a rotation of the XY-plane around the Z-axis
through the angle ψ0. The freedom to choose the values of θ0, φ0, ψ0 means thatthe user can rotate and/or tilt the Multilayer in 3-dimensional space without any
constraints. (See Appendix C for a detailed description of DIFFRACT’scoordinate systems.)§

The surface normal is assumed to point into the Multilayer for both reflection-type andtransmission-type stacks. Since the incident beam generally propagates along the positive Z
axis, the surface normal is also assumed to have a positive projection on the Z-axis.

Guide to DIFFRACT

124

Environment

Index-matchedspherical cap

Transparent
medium ofincidence
Multilayer
Center of spherical surface

Spectral Compression

Beam Compression

Figure 51. (a) Multilayer stack with layers parallel to the XY-plane is sandwiched between atransparent, isotropic substrate of index nsub and a transparent, isotropic incidence medium ofindex ninc. The index NVIRON of the incident beam’s environment must satisfy the relation
1.0 ≤ NVIRON ≤ ninc. When Multilayer is used in transmission, the transmitted beam may
remain within the substrate, or it may emerge into a new environment. The index of this newenvironment (which may or may not be the same as the incident beam’s environment) mustsatisfy the relation 1.0 ≤ NVIRON ≤ nsub. (b) Spectral compression means that upon enteringand leaving the medium of incidence (or the substrate), oblique rays bend according to Snell'slaw. (c) Beam compression means that an index-matched spherical cap, placed at the interfaceof the beam’s environment with the medium of incidence (or at the junction between thesubstrate and the emergence environment), eliminates the refraction of oblique rays uponentering and/or exiting the Multilayer.

Guide to DIFFRACT125

The emergent beam's coordinates are X"Y"Z". This system is identified with
angles θ1 [0 : 180°], φ1 [0 : 360°) and ψ1 [0° : 360°] relative to the XYZ
coordinates of incidence (again see Fig. 22 for a definition of the Euler angles
θ, φ, ψ). By allowing the user to choose the emergent coordinate system,
DIFFRACT can keep track of those portions of the reflected or transmitted beamthat are of particular interest to the user.
13.2. Entering and Exiting the Multilayer. In addition to various layer
parameters, you must specify the refractive index of an incidence medium, ninc,through which light impinges upon the Multilayer (ninc ≥ NVIRON). Also, when
the Multilayer is used in transmission, the refractive index nsub of its substrate
plays an important role. When ninc happens to be exactly equal to NVIRON, therefractive index of the incident beam’s environment (and, in the transmissionmode, when nsub = 1.0) no further questions will be asked, and the program
proceeds to compute the emergent distribution.
If you choose a medium of incidence other than the environment (i.e.,ninc ≥ NVIRON), you will be forcing a change in the nature of the incident
distribution. DIFFRACT allows you to treat this problem in two different ways:• Leave the incident distribution within the XY-plane intact. Here, as shown in
Fig. 51(b), the angular spectrum of the beam will be compressed toward theorigin (in Fourier domain) in order to preserve its spatial dimensions.
• Leave the incident angular spectrum intact. In this case, as Fig. 51(c)indicates, the beam itself will be compressed toward the origin in the XYplane in order to preserve the directions of its various rays.
The Multilayer may be used either in reflection mode or in transmission mode.
In the reflection mode, the beam may stay within the incidence medium(n = ninc) or it may emerge into its original environment (n = NVIRON) by exiting
the incidence medium the same way as it had entered this medium (i.e., bydecompressing the beam or its angular spectrum). The propagation distance afterreflection is measured from the interface of the Multilayer with the medium of
incidence.
In the transmission mode the beam will enter another transparent medium, thesubstrate, whose refractive index nsub ( ≥ 1) is specified by the user. Whennsub = 1, the beam emerges directly into free-space and no further questions will
be asked. When nsub > 1, the beam may stay within the substrate, or it may exit
into yet another transparent, isotropic medium, the emergent environment,

126

Guide to DIFFRACT

which subsequently becomes the new environment for the beam. In the lattercase the user will be prompted to specify the shape of the surface through which
the light must pass in order to exit the substrate (i.e., beam or angular-spectrum
decompression, corresponding to spherical or flat exit surfaces, respectively).
The propagation distance after transmission is measured from the interface
between the Multilayer and the substrate.
When the beam reflected from the Multilayer exits the medium of incidence toreturn to its pre-incidence environment, the compression performed on theincident beam (whether done on the beam itself or on its angular spectrum) will
be automatically reversed; that is, the reflected beam will properly trace theincidence path in the reverse direction. The beam transmitted through theMultilayer will also have the option of remaining within the substrate or exiting
the substrate into a new, user-defined environment. If the index of the newenvironment happens to differ from nsub (that is, NVIRON < nsub) you may choose
to decompress the angular spectrum of the transmitted beam (in accordance with
Snell's law) in order to simulate the case of a beam exiting through a flat
substrate. Alternatively, you may leave the angular spectrum intact anddecompress the transmitted beam instead, as would be required, for instance,
when the beam exits through a hemispherical substrate.
The user may specify the top surface of the medium of incidence and the bottom
surface of the substrate as “bare” or “antireflection-coated.” When a surface isbare the various rays enter and exit through that surface with proper attenuationin accordance with Fresnel's transmission coefficients. When the surface isantireflection-coated the transmission coefficient will be assumed to be unity
regardless of the angle of incidence.
Note: DIFFRACT assumes the optical system as a whole is designed to correctaberrations that result when the beam enters and exits the Multilayer through themedium of incidence and/or through the substrate; all such aberrations,
therefore, are ignored by the program. The propagation distance you will be
asked to specify when the beam is reflected from the Multilayer and returned toits original environment (NVIRON < ninc), or when the beam is transmitted
through the Multilayer and has emerged from the substrate into a different
environment (NVIRON < nsub), is an effective distance which would have applied
had the entire propagation path been in the beam’s pre-incidence environment(reflection), or post-incidence environment (transmission).

Guide to DIFFRACT

127

13.3. Saving Multilayer Stack Parameters in Hard Disk Files: A complete
description of a given stack may be saved to the hard disk in a special file called
XXXX.STK. (The user will be prompted for the filename XXXX.) These files canbe readily imported to the program during subsequent interactive runs, or they
may be inserted into COMMAND.DAT for use during non-interactive sessions.
Several examples of .STK files may be found in the <STACKS> subdirectory of
DIFFRACT’s home directory on your hard disk.
13.4 Building XXXX.STK Files with the Help of a Template: One can create
XXXX.STK files for simulating arbitrarily complex multilayer stacks inconjunction with the Multilayer feature of DIFFRACT. In the interactive mode,
you will be given the option of creating a XXXX.STK file for the multilayer stack
under consideration. Once this text file is created (within your working
subdirectory) you may modify it by adding or removing various layers, or bychanging the parameter values of the specified layers. An alternative approach tocreating XXXX.STK files involves the use of MLYR.STK template located in the<TEMPLATES> subdirectory of DIFFRACT’s home directory. Start by copying
the MLYR.STK file to a working subdirectory, then open this text file and follow
the instructions given within the file itself.
Y
Beam Expander
Incident
Gaussian
Beam

Z

Figure 52. Anamorphic beam expander consisting of a pair of SF11 glass prisms for
operation at λ = 1.55µm. The entrance and exit facets of both prisms are coated with a thindielectric layer of refractive index n = 1.44 to reduce reflection/transmission losses. The
incident beam is a circularly symmetric Gaussian (1/e amplitude radius = 0.4mm), which islinearly polarized along the x-axis.

Example. The Multilayer feature of DIFFRACT has been used to simulate theanamorphic prism pair shown in Fig. 52. The various parameters of this beam
expander are listed in the accompanying command.dat file. Results of thesimulation are shown in Fig. 53, and also in the enclosed printout of the

128

Guide to DIFFRACT

session.dat file. In the present example, the entrance and exit facets of bothprisms are coated with a thin dielectric layer to minimize the reflection/transmission losses incurred upon entering/exiting the prisms. The bulk of eachprism is represented by either the substrate or the medium of incidence of amultilayer stack. Each such stack is tilted to represent the proper orientation of
the simulated facet, and the direction of the emergent beam in each case has
been chosen to follow the transmitted beam in that instance.

Figure 53. Plots of intensity (top) and phase (bottom) at a distance of 15mm past the exit
facet of the beam-expander depicted in Fig. 52. (Left) x-component of polarization, (right) ycomponent of polarization. Note that the y-component is nearly 10 orders-of-magnitude
weaker than the x-component. The phase profile of the x-component is quite uniform,
indicating the absence of significant misalignments and aberrations. The y-component issymmetrically split between the left- and right-hand sides.

Guide to DIFFRACT

Printout of the command.dat file corresponding to the system of Fig. 52@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 05/31/2005
Time: 15:02 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$MULTIPLY#??,ITEM?....BY: ??
#$MAKE#??,ITEM?~#??,ITEM?:#$JUMP#??_IF#??,ITEM?.GT.: ??
#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):1.550000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: mm) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.000000
0.000000
1.000000#
1/e radii Rmin,Rmax:
0.400000
0.400000#
RA,Theta:
1.000000
0.000000#
Aberrations: None#
Polarization RHO,ETA:
0.000000
0.000000#
NMAX,NMAY:
1024
1024
#
LMAX,LMAY:
8.200000
8.200000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):1.550000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-0.500000
0.500000#
Ymin,Ymax:-0.500000
0.500000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):1.550000
NVIRON: 1.000000 #_________________________________________________________________________ ##Multilayer structure: MLYR
(Length_Units: um) #Nsub:
1.744828#
Number of layers:
1#
Layer#001
Type (A-E): A#n,k:1.440000
0.000000#
Thickness:
0.360000#
Ninc:
1.000000#
Compress incident: NONE#
Entrance surface: BARE#
Reflect or Transmit: T#
Emergent environment's n:
1.744828
#

129

130

Guide to DIFFRACT

Decompress emergent: NONE#
Exit surface: BARE#
Stack's Theta,Phi,Psi:
70.73300
270.0000
0.000000#
Emergent Theta,Phi,Psi:
37.98000
270.0000
0.000000#
Propagation distance:
3000.000#
Propagation regime: DFRNL#
Smax:
0.100000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):1.550000
NVIRON: 1.744828 #_________________________________________________________________________ ##Multilayer structure: MLYR
(Length_Units: um) #Nsub:
1.000000#
Number of layers:
1#
Layer#001
Type (A-E): A#n,k:1.440000
0.000000#
Thickness:
0.270000#
Ninc:
1.744828#
Compress incident: NONE#
Entrance surface: BARE#
Reflect or Transmit: T#
Emergent environment's n:
1.000000#
Decompress emergent: NONE#
Exit surface: BARE#
Stack's Theta,Phi,Psi:
3.320000
270.0000
0.000000#
Emergent Theta,Phi,Psi:
2.480000
90.00000
0.000000#
Propagation distance:
5000.000#
Propagation regime: DFRNL#
Smax:
0.100000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):1.550000
NVIRON: 1.000000 #_________________________________________________________________________ ##Multilayer structure: MLYR
(Length_Units: um) #Nsub:
1.744828#
Number of layers:
1#
Layer#001
Type (A-E): A#n,k:1.440000
0.000000#
Thickness:
0.360000#
Ninc:
1.000000#
Compress incident: NONE#
Entrance surface: BARE#
Reflect or Transmit: T#
Emergent environment's n:
1.744828#
Decompress emergent: NONE#
Exit surface: BARE#
Stack's Theta,Phi,Psi:
70.73300
90.00000
0.000000#
Emergent Theta,Phi,Psi:
37.98000
90.00000
0.000000#
Propagation distance:
10000.00#
Propagation regime: DFRNL#
Smax:
0.100000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):1.550000
NVIRON: 1.744828 #_________________________________________________________________________ ##Multilayer structure: MLYR
(Length_Units: um) #Nsub:
1.000000#
Number of layers:
1#
Layer#001
Type (A-E): A#n,k:1.440000
0.000000#

Guide to DIFFRACT
Thickness:
0.270000#
Ninc:
1.744828#
Compress incident: NONE#
Entrance surface: BARE#
Reflect or Transmit: T#
Emergent environment's n:
1.000000#
Decompress emergent: NONE#
Exit surface: BARE#
Stack's Theta,Phi,Psi:
3.320000
90.00000
0.000000#
Emergent Theta,Phi,Psi:
2.480000
270.0000
0.000000#
Propagation distance:
15000.25#
Propagation regime: DFRNL#
Smax:
0.100000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):1.550000
NVIRON: 1.000000 #_________________________________________________________________________ ##Measure peak and FWHM: MEAS
(Length_Units: um) #Include Z-component: N#
ID numbers of PEAK I,X,Y:
1
2
3#
Compute/Return: C#
Theta:
0.0000000#
ID number for FWHM:
4#
Compute/Return: C#
Theta:
90.000000#
ID number for FWHM:
5#
Compute/Return: R#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):1.550000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-4.000000
4.000000#
Ymin,Ymax:-4.000000
4.000000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):1.550000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Phase#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-4.000000
4.000000#
Ymin,Ymax:-4.000000
4.000000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):1.550000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

131

Guide to DIFFRACT

132

Printout of the session.dat file corresponding to the system of Fig. 52$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 8.4
File SESSION.DAT $$$$ Date: 05/31/2005
Time: 16:37 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$_______________________________________________________DIMENSIONLESS LENGTHS ARE IN UNITS OF VACUUM WAVELENGTH
ALL ANGLES ARE IN DEGREES_______________________________________________________
Array dimensions: NDIM=1024, NDSP= 149**************************************
Designated Working Subdirectory: <Expander>*******************************************_________________________________________________________________
Vacuum lambda (um)= 1.0000
NVIRON= 1.0000 Length_Units: um
LMAX=
10000.0
LMAY=
10000.0
NMAX=1024
NMAY=1024_________________________________________________________________
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Length Units: mm________________
Simple Gaussian: BCX=
0.00000
BCY=
0.00000
RA=
1.00000
THETA= 0.00
RMIN= 0.400000
RMAX= 0.400000
Aberrations: None
Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________________
Vacuum lambda (um)= 1.5500
NVIRON= 1.0000 Length_Units: mm
LMAX=
8.20000
LMAY=
8.20000
NMAX=1024
NMAY=1024
Beam centroid:(X= 0.45091E-09, Y=-0.34824E-08) DIAM=0.799994E+00
XPOWER= 0.99997
YPOWER= 0.0000
Curvature= 0.00000E+00_________________________________________________________________
PLOTTING ROUTINE****************
Length Units: mm________________
Intensity distribution___________________________________________
Xmin= -0.500000E+00
Xmax= 0.500000E+00
Ymin= -0.500000E+00
Ymax= 0.500000E+00___________________________________________
IXmin=

0.1849E-07

IXmax=

0.9560E-05

Latest screen (intensity pattern) dumped in IS00.BMP
Intensity pattern for X-polrz stored in IX00.DAT
MULTILAYER**********
Length Units: um________________
Substrate:(n,k)= ( 1.744828,
Thickness= Infinite

0.000000)

Layer#001:(n,k)= ( 1.440000,
Thickness= 0.360000

0.000000)

Guide to DIFFRACT

133

Medium of incidence: (n,k)= (1.000000, 0.000000)

Transmission mode
Index of emergent environment=

1.7448Multilayer coordinates: Theta= 70.73
Emergent coordinates:
Theta= 37.98

Phi=270.00
Phi=270.00

Psi=
Psi=

0.00
0.00

Propagation distance=
3000.00
Propagation regime: Direct Fresnel_______________________________________________________NGUIDE=
99 LGUIDE= 0.159999E+05 SMAX= 0.000484________________________________________________________________________________________________________________________
Vacuum lambda (um)= 1.5500
NVIRON= 1.7448 Length_Units: um
LMAX=
8200.00
LMAY=
8200.00
NMAX=1024
NMAY=1024
Beam centroid:(X=-0.78176E-05, Y= 0.31726E+00) DIAM=0.154355E+04
XPOWER= 0.82108
YPOWER= 0.84429E-07
Curvature= 0.00000E+00_________________________________________________________________MULTILAYER**********
Length Units: um________________
Substrate:(n,k)= ( 1.000000,
Thickness= Infinite

0.000000)

Layer#001:(n,k)= ( 1.440000,
Thickness= 0.270000

0.000000)

Medium of incidence: (n,k)= (1.744828, 0.000000)

Transmission mode
Index of emergent environment=

1.0000Multilayer coordinates: Theta=
Emergent coordinates:
Theta=

3.32
2.48

Phi=270.00
Phi= 90.00

Psi=
Psi=

0.00
0.00

Propagation distance=
5000.00
Propagation regime: Direct Fresnel_______________________________________________________NGUIDE=
99 LGUIDE= 0.308710E+05 SMAX= 0.000144________________________________________________________________________________________________________________________
Vacuum lambda (um)= 1.5500
NVIRON= 1.0000 Length_Units: um
LMAX=
8200.00
LMAY=
8200.00
NMAX=1024
NMAY=1024
Beam centroid:(X=-0.30532E-04, Y= 0.32798E+00) DIAM=0.153050E+04
XPOWER= 0.80939
YPOWER= 0.72231E-07
Curvature= 0.00000E+00_________________________________________________________________MULTILAYER**********
Length Units: um________________
Substrate:(n,k)= ( 1.744828,
Thickness= Infinite

0.000000)

Layer#001:(n,k)= ( 1.440000,
Thickness= 0.360000

0.000000)

Guide to DIFFRACT

134
Medium of incidence: (n,k)= (1.000000, 0.000000)

Transmission mode
Index of emergent environment=

1.7448Multilayer coordinates: Theta= 70.73
Emergent coordinates:
Theta= 37.98

Phi= 90.00
Phi= 90.00

Psi=
Psi=

0.00
0.00

Propagation distance=
10000.0
Propagation regime: Direct Fresnel_______________________________________________________NGUIDE=
99 LGUIDE= 0.306099E+05 SMAX= 0.000253________________________________________________________________________________________________________________________
Vacuum lambda (um)= 1.5500
NVIRON= 1.7448 Length_Units: um
LMAX=
8200.00
LMAY=
8200.00
NMAX=1024
NMAY=1024
Beam centroid:(X=-0.61501E-04, Y= 0.15226E+00) DIAM=0.367242E+04
XPOWER= 0.67169
YPOWER= 0.0000
Curvature= 0.00000E+00_________________________________________________________________MULTILAYER**********
Length Units: um________________
Substrate:(n,k)= ( 1.000000,
Thickness= Infinite

0.000000)

Layer#001:(n,k)= ( 1.440000,
Thickness= 0.270000

0.000000)

Medium of incidence: (n,k)= (1.744828, 0.000000)

Transmission mode
Index of emergent environment=

1.0000Multilayer coordinates: Theta=
Emergent coordinates:
Theta=

3.32
2.48

Phi= 90.00
Phi=270.00

Psi=
Psi=

0.00
0.00

Propagation distance=
15000.3
Propagation regime: Direct Fresnel_______________________________________________________NGUIDE=
99 LGUIDE= 0.734485E+05 SMAX= 0.000060________________________________________________________________________________________________________________________
Vacuum lambda (um)= 1.5500
NVIRON= 1.0000 Length_Units: um
LMAX=
8200.00
LMAY=
8200.00
NMAX=1024
NMAY=1024
Beam centroid:(X=-0.17586E-03, Y= 0.84630E-01) DIAM=0.355751E+04
XPOWER= 0.64305
YPOWER= 0.31129E-09
Curvature= 0.00000E+00_________________________________________________________________
PEAK INTENSITY AND FWHM***********************
Length Units: um________________
Z-component of polarization excluded from PEAK & FWHM measurement
Peak Intensity=0.100E-05
XPEAK=
0.00000
YPEAK=
0.00000
Peak Intensity, XPEAK,YPEAK saved in SIGNAL array as S( 1), S( 2), S( 3)Theta=

0.00

FWHM=

472.809

Guide to DIFFRACT135

FWHM saved in SIGNAL array as S( 4)Theta= 90.00
FWHM=
2848.18
FWHM saved in SIGNAL array as S( 5)PLOTTING ROUTINE****************
Length Units: mm________________
Intensity distribution___________________________________________
Xmin= -0.400000E+01
Xmax= 0.400000E+01
Ymin= -0.400000E+01
Ymax= 0.400000E+01___________________________________________
IXmin=
IYmin=

0.1434E-17
0.3379E-27

IXmax=
IYmax=

0.1005E-05
0.3553E-15

Latest screen (intensity pattern) dumped in IS01.BMP
Intensity pattern for X-polrz stored in IX01.DAT
Intensity pattern for Y-polrz stored in IY01.DAT
PLOTTING ROUTINE****************
Length Units: mm________________
Phase distribution___________________________________________
Xmin= -0.400000E+01
Xmax= 0.400000E+01
Ymin= -0.400000E+01
Ymax= 0.400000E+01___________________________________________
PXmin= -179.985977
PYmin= -123.229103

PXmax=
PYmax=

179.979630
79.666222(degrees)(degrees)

Latest screen (phase pattern) dumped in PS01.BMP
Phase pattern for X-polrz stored in PX01.DAT
Phase pattern for Y-polrz stored in PY01.DATEnd of session: DIFFRACT terminated normally********************************************

Guide to DIFFRACT

136

14. Optical Disk Surface
This feature of DIFFRACT is useful when studying diffraction of focused beams
from patterned surfaces, in general, and from optical disk surfaces, in particular.
The incident distribution on an optical disk is generally a focused spot, centered
at a point such as (SCX, SCY) and confined to a small area in the vicinity of thatpoint. In order to save computation time, the user of DIFFRACT is asked to
specify a radius of aperture RA around the spot center (SCX, SCY), thereby
confining the calculations to the region within this aperture. Four types of
optical disks are available in DIFFRACT; these include the read-only, writable,
magneto-optical, and dye-polymer types of media. A brief description of these
media-types follows.
Note: Writable, Magneto-optical, and Dye-polymer typesof media are not available in the Student Edition.
For the read-only disk, the user specifies the amplitude reflection coefficient of
the surface, and an arbitrary number of surface relief features (bumps and/or
pits), which behave as phase-objects. The reflectivity of bumps and pits will be
the same as that of the rest of the disk surface.
Writable media are pregrooved and, in addition, they could have a number ofsurface features (e.g., bumps, pits, recorded marks). Grooves and marks in thiscase could both be amplitude- and/or phase-objects: the height of thegrooves/bumps/pits determines the phase modulation that they impart to theincident beam and, in addition, the user will be prompted for the amplitudereflection coefficients within each region. The grooves of Writable media may
be wobbled, either with a sinusoidal pattern, or with a number of equally-spaced
wiggles (blips) along the length of each groove.
The magneto-optical (MO) disk can be pregrooved, it may contain surface relief
features (i.e., bumps/pits that represent preformat marks), and it may also
contain a number of reverse-magnetized domains. Wobbled grooves of bothtypes (Sine and Blip) are also allowed for MO disks. The regular and magnetooptical reflection coefficients of the disk surface are denoted by rp and rs. For alinearly-polarized incident beam with unit amplitude, rp and rs are the reflectedcomplex amplitudes parallel and perpendicular to the incident polarization,
respectively. The surface of the MO disk is thus characterized by the reflection
coefficients’ magnitudes |rp|, |rs|, and by their relative phase φsp = Phase (rs/rp).
Dye-polymer media consist of a multilayer stack, within which grooves as wellas marks (e.g., bumps and pits) may be embedded. Multiple reflections withinGuide to DIFFRACT137

the stack and the resulting interference among the various reflected and/ortransmitted beams give rise to differing phase/amplitude values for the lightreflected from lands, grooves, and marks. As is the case with Writable andMagneto-optical media, the grooves of Dye-Polymer optical disks can be
wobbled, either sinusoidally or with isolated wiggles (blips), along their length.
In all the above cases, the medium of incidence may be specified by its
refractive index, ninc. An index of unity signifies a disk in the air-incidence
configuration. When ninc > 1, the disk is in the substrate-incidence configuration.
Note: All features on the Optical_Disk_Surface, whether they are grooves,preformat bumps/pits, recorded marks (erasable or non-erasable), or reversemagnetized domains, are treated as phase/amplitude-objects. This means that, ateach point within the clear aperture of the Optical_Disk_Surface, the incident beam
is simply multiplied by a complex reflection coefficient, and the resulting
distribution is substituted for the reflected beam. No attempts have been made to
determine the true distribution of the reflected beam by solving the Maxwell
equations, as this would have entailed massive computations far beyond thecapabilities of DIFFRACT. In reality, despite the fact that many relief featuresand magnetic domains on actual disk surfaces are small compared to thewavelength λ of the light, experiments show that approximating them asphase/amplitude objects is quite adequate for first-order calculations.
14.1. Pregrooves: A reflection grating in the XY-plane simulates the grooved
surface of optical disks. As shown in Fig. 54, the grating has a periodic patternof grooves with a trapezoidal cross-section. Orientation of the grating isdetermined by the angle Θ between X and the direction of grooves. The opticalaxis of the system crosses the XY-plane at (x, y) = (0, 0), and the center of thegrating is at (GCX, GCY). The trapezoidal cross-section of the groove is definedby parameters A, B, C, D, ζ as follows: Let the grating be centered on the opticalaxis (i.e., GCX = GCY = 0), and let the grooves run parallel to X (i.e., Θ = 0°).
Then, in the YZ-plane, the corners of the trapezoid are at ( y, z) = (0, 0), (A, ζ),(B, ζ), (C, 0), and (D, 0). Thus ζ is the groove depth, B − A is the groove width atits bottom, C is the groove width at its top, and D is the period of the grating.
For writable media, the user will be prompted for two additional parameters: theamplitude reflection coefficients RL and RG on the land and on the groove.
(DIFFRACT automatically adjusts the reflection coefficients on the side-walls of
the grooves, making a smooth, linear transition from RL to RG.)

138

Guide to DIFFRACT

Figure 54. Pregrooved Optical Disk in
the XY-plane. Grooves have trapezoidal
cross-section with depth ζ, width B − A
at the bottom, and width C at the top.
Separation between adjacent groove
centers is D. The center of the disk
surface is chosen, somewhat arbitrarily,
as a point on the straight-edge boundary
between a land and a groove; in the XYplane, the coordinates of this center are(GCX, GCY). Grooves may be arbitrarilyoriented in the XY-plane by choosing
the angle Θ. In the case of Writable
media, lands have reflection coefficient
RL, while groove bottoms’ reflection
coefficient is RG.

14.2. Pregroove wobble. Wobbles, which may be either sinusoidal (Sine) or in
the form of equally-spaced wiggles (Blip), affect the opposite edges of thegrooves differently. When wobble is specified as Sine, the central groove-edge,
i.e., that which goes through the grating center (GCX, GCY ), has an undulation of
the form A0 sin(2πx/P0), where A0 is the amplitude and P0 the period of theundulation; see Fig. 55. The phase of this edge’s wobble as measured from
(GCX, GCY ) is 0°; however, neighboring grooves may have a relative phase of
Q0. (Note that Q0 is measured with respect to the period P0 of the central grooveedge. Thus, for example, when Q0 = 180°, neighboring grooves are shifted by½P0 relative to each other.) Undulations of the groove-edge adjacent to thecentral one have the form A1 sin(2πx/P1 + Q1), with A1 being the amplitude, P1
the period, and Q1 the phase relative to that of the central groove-edge. Note thatwhile the amplitude and period of each edge’s wobble are specified in
Length_Units (cm, mm, um, nm, or normalized by the vacuum wavelength λo),the phase parameters Q0 and Q1 are angles in the range from 0° to 360°. Once
the wobble parameters are specified for both edges of a single groove,
DIFFRACT repeats the same pattern over the entire Optical_Disk_Surface.
It must be emphasized that the phase shift between the opposite edges of eachand every groove is determined solely by Q1, independently of the value chosen
for Q0. On the other hand, different grooves may be aligned with their neighbors(i.e., Q0 = 0°), or they may be phase-shifted in a uniform way (i.e., Q0 ≠ 0°). As
one moves away from the central groove-edge in a direction perpendicular to thegroove axis in steps of the groove-period D, one encounters the same wobble inGuide to DIFFRACT139

the corresponding edge of the neighboring grooves, i.e., amplitude = A0, period= P0, but with a phase that jumps in steps of Q0 between adjacent grooves.

A1
P1
Q0
Q1

D-C

D

(GCX, GCY )

P0

θ
Q0

A0
C
Figure 55. Wobbled Sine-type pregrooves on the surface of Writable, Magneto-optical, andDye-polymer optical disks. Both edges of each groove are wobbled sinusoidally, each edge
with a different amplitude, period, and phase. The central edge, which passes through thegrating center at (GCX, GCY ), has amplitude A0 and period P0, while the adjacent edge has
amplitude A1, period P1, and phase Q1. Neighboring grooves are phase shifted by Q0.

Although not emphasized in Fig. 55, each groove-edge may have a non-zero
width. In the case of the central edge, its entire width A is assumed to have thesinusoidal undulation defined by the parameters A0 and P0. Similarly, the width
C – B of an adjacent groove-edge has the undulation defined by A1, P1 and Q1.
Examples of pregrooved Optical_Disk_Surfaces with Sine-type wobbles exhibiting
radial-position modulation and groove-width modulation are shown in Fig. 56,
accompanied by command blocks containing the corresponding simulation
parameters.

Guide to DIFFRACT

140

(a)

(b)

Figure 56. Sinusoidal wobbled grooves on an Optical_Disk_Surface. (a) Grooved DyePolymer disk having parameter values A = 0.4, B = 1.4, C = 1.8, D = 2.8, ζ = 0.125,
A0 = A1 = 0.1, P0 = P1 = 6, and Q0 = Q1 = 0. This type of wobble is generally known as thegroove radial-position modulation. (b) Grooved Writable disk with the same set of parameters
as in (a) except for the phase Q1 of the adjacent groove-edge being 180° relative to the centraledge. This is commonly referred to as groove-width modulation.
Command block corresponding to the simulation depicted in Fig. 56(a)$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units:) #Type of medium: DP#
SCX,SCY,RA:
0.000000
0.000000
25.000000#(Incidence medium) Ninc:
1.560000#
Depth of focus:
2.000000#
Layer #01: n,k,Thickness:
2.200000
0.000000
0.4000000#
Layer #02: n,k,Thickness:
1.600000
0.100000
0.6000000#
Layer #03: n,k,Thickness:
2.200000
0.000000
0.2500000#
Layer #04: n,k,Thickness:
1.000000
0.000000
2.0000000#
Pregrooved medium (Y/N): Y#
Modified layers L1,L2: 2
3#
A,B,C:
0.400000
1.400000
1.800000#
D,Zeta:2.800000
0.125000#
Theta,GCX,GCY:45.00000
0.636400-0.636400#
Wobble (Sine,Blip,None): Sine#
Central wobble A0,P0,Q0:
0.100000
6.000000
0.000000#
Adjacent wobble A1,P1,Q1:
0.100000
6.000000
0.000000#
Number of marks: 3#
Modified layers L1,L2: 1
2#
Mark #01:
MCX,MCY:
0.000000
0.000000#
Orientation angle Theta:
45.00000#Round or Flat: R#
Length,Width,Height:
3.0000000.9500000-0.0500000#
Floor-to-Rim ratio:
0.000000#Mark #02:
MCX,MCY:
3.000000
3.000000#
Orientation angle Theta:
45.00000#Round or Flat: F#
Length,Width,Height:
2.0000000.9500000-0.0500000#
Floor-to-Rim ratio:
0.800000#Mark #03:
MCX,MCY:-3.000000-3.000000#

Guide to DIFFRACT
Orientation angle Theta:
45.00000Round or Flat: F
Length,Width,Height:
3.000000
Roof-to-Rim ratio:
0.8000000
Inspect surface pattern: Y
Xmin,Xmax:-5.0000000
Ymin,Ymax:-5.0000000
Color or Gray-scale: G
Save data file (Y/N): Y
File identifier: 1
Propagation distance:
5000.0000
Propagation regime: FRNHF
Smax:
0.90000000$

1410.9500000-0.050000

5.0000000
5.0000000

#
#
#
#
#
#
#
#
#
#
#
#
#
#

Command block corresponding to the simulation depicted in Fig. 56(b)$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units:) #Type of medium: WR#
SCX,SCY,RA:
0.0000000
0.0000000
25.000000#(Incidence medium) Ninc:
1.5600000#
Pregrooves: Y#
A,B,C:
0.4000000
1.4000000
1.8000000#
D,Zeta:2.8000000-0.1250000#
Reflection coeffs. RL,RG:
1.0000000
1.0000000#
Theta,GCX,GCY:45.000000
0.6364000-0.6364000#
Wobble (Sine,Blip,None): Sine#
Central wobble A0,P0,Q0:
0.1000000
6.0000000
0.0000000#
Adjacent wobble A1,P1,Q1:
0.1000000
6.0000000
180.00000#
Marks (bumps/pits): Y#
Number of marks: 3#
Mark # 1:
MCX,MCY:
0.0000000
0.0000000#
THETA:
45.000000#
Round or Flat: R#
Length, Width, Height:
3.0000000
0.9000000
0.0500000#
Reflection coefficient:
1.0000000#
Mark # 2:
MCX,MCY:
3.0000000
3.0000000#
THETA:
45.000000#
Round or Flat: F#
Length, Width, Height:
2.0000000
0.9000000
0.0500000#
Ratio:
0.8000000#
Reflection coefficient:
1.0000000#
Mark # 3:
MCX,MCY:-3.0000000-3.0000000#
THETA:
45.000000#
Round or Flat: F#
Length, Width, Height:
3.0000000
0.9000000
0.0500000#
Ratio:
0.8000000#
Reflection coefficient:
1.0000000#
Inspect surface pattern: Y#
Xmin,Xmax:-5.0000000
5.0000000#
Ymin,Ymax:-5.0000000
5.0000000#
Color or Gray-scale: G#
Save data file (Y/N): Y#
File identifier: 2#
Propagation distance:
5000.0000#
Propagation regime: FRNHF#
Smax:
0.90000000#$#

Guide to DIFFRACT142

In the case of Blip-type wobble, the central groove-edge has an array of wiggles(i.e., blips) of amplitude A0 and width W0, while the opposite groove-edge has
blips of amplitude A1 and width W1. Blips on both edges have period P0, andtheir peaks occur at the same location along the groove, as shown in Fig. 57. The
blips on one groove, however, may be collectively shifted relative to those onnearest-neighbor grooves by the phase step Q0 (0 : 360°). Figure 58 shows
examples of Blip wobble in pregrooved optical disks.

Qo

Ao
Po
Wo
W1
A1
Qo

Figure 57. Wobbled grooves with Blip type modulation on the surface of Writable, Magnetooptical, or Dye-polymer optical disks. The parameters A0, W0, A1, W1, and P0 have Length
Units, whereas Q0, a phase-shift, is specified in degrees. When Q0 = 0° the blips will be lined
up in the direction perpendicular to the grooves, but when Q0 = 180° the blips on neighboring
grooves will be displaced by ½P0.

In specifying the various parameters of a wobbled pregrooved optical disk, caremust be taken to ensure that adjacent groove-edges (as well as adjacent landedges) do not merge into each other.
Note: Pregrooves are not available in the Student Edition.

Guide to DIFFRACT

(a)

143

(b)

Figure 58. Wobbled grooves with blip-type modulation. (a) A = 0.2µm, B = 1.0µm,
C = 1.2µm, D = 3.0µm, ζ = 52 nm, A0 = A1 = 1.0µm, W0 = 2.0µm, W1 = 3.0µm, P0 = 6.0µm,
Q0 = 90°. Grooves are shown in white and lands in black. (b) A = 0.2µm, B = 1.0µm,
C = 1.2µm, D = 2.0µm, ζ = 52 nm, A0 = A1 = 0.7µm, W0 = 2.0µm, W1 = 3.0µm, P0 = 6.0µm,
Q0 = 180°. Also shown here are three marks (black) inside the central groove (gray)surrounded by lands (white).

Command block corresponding to the simulation depicted in Fig. 58(a)$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units: um) #Type of medium: WR#
SCX,SCY,RA:
0.000000
0.000000
10.00000#(Incidence medium) Ninc:
1.560000#
Pregrooves (Y/N): Y#
A,B,C:
0.200000
1.000000
1.200000#
D,Zeta:3.000000
0.052000#
Reflection coeffs. RL,RG:
1.0000000.700000#
Theta,GCX,GCY:45.00000
0.000000-2.000000#
Wobble (Sine,Blip,None): Blip#Central blip's A0,W0:
1.000000
2.000000#
Adjacent blip's A1,W1:
1.000000
3.000000#
Blip's period,phase step:
6.000000
90.00000#
Marks (bumps/pits): N#
Inspect surface pattern: Y#
Xmin,Xmax:-6.500000
6.500000#
Ymin,Ymax:-6.500000
6.500000#
Color or Gray-scale: G#
Save data file (Y/N): Y#
File identifier: 0#
Propagation distance:
0.000000#
Propagation regime: DFRNL#
Smax:
0.000000#

144

Guide to DIFFRACT

14.3. Bumps and Pits: The reflecting disk surface in the XY-plane contains
phase-objects in the form of bumps and/or pits. The amplitude reflection
coefficient for these objects is the same as that for the rest of the disk surface,
except in the cases of writable and dye-polymer media, where the reflectivity ofbumps and pits may differ from the reflectivity of the rest of the disk surface. A
bump/pit may have either a flat or a round roof /floor. The rim of a mark (bumpor pit) in the XY-plane is a rectangle capped by two semi-circles. Marks arecharacterized by three parameters (L, W, H) as follows:•••

Length L as measured from tip to tip in the XY-plane;
Width W as measured at the waist in the XY-plane;
Height H as measured at the center from the XY-plane (H > 0 for bump; H < 0 for pit).

As shown in Fig. 59(a), flat marks have a roof /floor with the same shape as therim, but their length and width are reduced by a constant factor. This factor,
which is in the interval [0 : 1], is referred to as the roof-to-rim (or floor-to-rim)
diameter ratio.
Marks with a round roof /floor may be visualized in the following way. Consider
a right circular cylinder of radius R, capped by two hemi-spheres of the sameradius, as shown in Fig. 59(b). The cylinder axis, while parallel to the XY-plane,
is a distance D below/above it. When D is less than R, a certain part of thesphero-cylinder appears above /below the plane. It is this protrusion or dent that
constitutes a mark (or bump/pit) with round roof /floor.

Figure 59. Diagrams showing the various dimensions of bumps and pits.
(a) Pit with flat floor and bump with flat roof. (b) Bump with round roof.

Guide to DIFFRACT145

For each mark the user is prompted for center coordinates (MCX, MCY), length,
width, height, and orientation angle relative to the X-axis. If flat marks arerequested, there will be an additional prompt for the roof-to-rim (or floor-to-rim)
diameter ratio. For Writable media, DIFFRACT will also inquire the reflection
coefficient at the center point of the mark. (The smooth transition of thisparameter from the center point to the rim, where mark’s reflectivity blends into
the reflectivity of its host, i.e., land or groove, automatically follows thegeometrical shape of the mark.)
Example. Figure 60(a) shows a pattern of grooves and preformat marks on a
Writable Optical_Disk_Surface. The groove period is 2λo, and the groove-depthof 0.08λo yields an effective phase shift of ~ 90° in reflection (ninc = 1.58). The
marks have different lengths, but their width and height are fixed at W = λo,
H = 0.16λo, leading to an effective phase-shift upon reflection ~ 180°. A
Gaussian laser beam of wavelength λo is brought to focus on the disk through an
objective lens having NA = 0.5, f = 6000λo; the intensity profile of the focusedlaser spot is shown in Fig. 60(b). The disk is subsequently moved under thefocused spot in such a way as to scan the central track. The reflected opticalpower that passes through the objective lens in the return path is collected by a0.1NA lens and focused onto a photodetector. Figure 61 shows the detector’s
output signal as a function of the position of the focused spot along the track.
The simulation parameters are listed in the accompanying COMMAND.DAT file.
a

−7.3

b

x/λo

7.3

−7.3

x/λo

7.3

Figure 60. (a) Pattern of grooves and preformat marks on a Writable Optical Disk Surface.
(b) Logarithmic plot of intensity distribution (scale = 4) at the focal plane of the objective
lens, which coincides with the disk surface.

Guide to DIFFRACT

146

Figure 61. Computed photodetector signal
as function of the position of the focusedlaser spot. The scan starts at (x, y) = (0, 0),
the center of the frame in Fig. 60(a), andproceeds as the disk surface moves
continuously under the focused spot,
enabling the laser beam to read the upper
half of the central track; see Fig. 60(a).
Here the scanned marks are depicted
schematically at the bottom of the frame.

Command.dat file used in the simulations depicted in Figs. 60 and 61.
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/24/2005
Time: 12:03 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$#$TAGS#20,ITEM1_#20,ITEM2:#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.0000000#
1/e radii Rmin,Rmax:
3000.0000
3000.0000#
RA,THETA:
3000.0000
0.0000000#

Guide to DIFFRACT
Aberrations: None#
Polarization RHO,ETA:
0.0000000
45.000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
60000.000
60000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: PFOC#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.50000000
6000.0000#
Aberrations: None#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance:
6000.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Log(intensity)#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-7.3000000
7.3000000#
Ymin,Ymax:-7.3000000
7.3000000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Store distribution: SAVE#
File name: SPOT.DST#$________________________________________________________________________ #10
Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ #$#Retrieve a distribution: READ#
File name: SPOT.DST#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units:) #Type of medium: WR#
SCX,SCY,RA:
0.0000000
0.0000000
12.000000#(Incidence medium) Ninc:
1.5800000#
Pregrooves: Y#
A,B,C:
0.20000000
1.2000000
1.4000000#
D,ZETA:
2.0000000-0.08000000#
Reflection coeffs. RL,RG:
0.90000000
0.90000000#
THETA,GCX,GCY:45.000000
0.49500000-0.49500000#
Wobble (Sine,Blip,None): None#
Marks (bumps/pits): Y#
Number of marks: 10#
Mark # 1:
MCX,MCY:
0.0000000
0.0000000#20147Guide to DIFFRACT

148
THETA:
45.000000
Round or Flat: R
Length, Width, Height:
2.0000000
Reflection coefficient:
0.90000000
Roof-to-Rim ratio:
0.000000Mark # 2:
MCX,MCY:
2.5000000
THETA:
45.000000
Round or Flat: R
Length, Width, Height:
1.5000000
Reflection coefficient:
0.90000000
Roof-to-Rim ratio:
0.000000Mark # 3:
MCX,MCY:
4.5000000
THETA:
45.000000
Round or Flat: R
Length, Width, Height:
3.0000000
Reflection coefficient:
0.90000000
Roof-to-Rim ratio:
0.000000Mark # 4:
MCX,MCY:-3.0000000
THETA:
45.000000
Round or Flat: R
Length, Width, Height:
1.0000000
Reflection coefficient:
0.90000000
Roof-to-Rim ratio:
0.000000Mark # 5:
MCX,MCY:-1.4142000
THETA:
45.000000
Round or Flat: R
Length, Width, Height:
4.0000000
Reflection coefficient:
0.90000000
Roof-to-Rim ratio:
0.000000Mark # 6:
MCX,MCY:
3.5355000
THETA:
45.000000
Round or Flat: R
Length, Width, Height:
2.0000000
Reflection coefficient:
0.90000000
Roof-to-Rim ratio:
0.000000Mark # 7:
MCX,MCY:-6.3640000
THETA:
45.000000
Round or Flat: R
Length, Width, Height:
1.0000000
Reflection coefficient:
0.90000000
Roof-to-Rim ratio:
0.000000Mark # 8:
MCX,MCY:
1.4142000
THETA:
45.000000
Round or Flat: R
Length, Width, Height:
2.0000000
Reflection coefficient:
0.90000000
Roof-to-Rim ratio:
0.000000Mark # 9:
MCX,MCY:
5.6569000
THETA:
45.000000
Round or Flat: R
Length, Width, Height:
2.0000000
Reflection coefficient:
0.90000000
Roof-to-Rim ratio:
0.000000Mark #10:
MCX,MCY:-2.1213000
THETA:
45.000000
Round or Flat: R
Length, Width, Height:
3.0000000
Reflection coefficient:
0.90000000
Roof-to-Rim ratio:
0.000000
Inspect surface pattern: Y
Xmin,Xmax:-7.3000000
Ymin,Ymax:-7.3000000
Color or Gray-scale: C

1.0000000

0.16000000

2.5000000

1.0000000

0.16000000

4.5000000

1.0000000

0.16000000-3.0000000

1.0000000

0.16000000

1.4142000

1.0000000

0.16000000

6.3640000

1.0000000

0.16000000-3.5355000

1.0000000

0.16000000-1.4142000

1.0000000

0.16000000

2.8284000

1.0000000

0.16000000-4.9497000

1.0000000

7.3000000
7.3000000

0.16000000

#
#
#
#
#
#21
#
#
#
#
#
#22
#
#
#
#
#
#23
#
#
#
#
#
#24
#
#
#
#
#
#25
#
#
#
#
#
#26
#
#
#
#
#
#27
#
#
#
#
#
#28
#
#
#
#
#
#29
#
#
#
#
#
#
#
#
#

Guide to DIFFRACT
Save data file (Y/N): Y#
File identifier: 0#30
Propagation distance:
6000.0000#
Propagation regime: FRNHF#
Smax:
0.10000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: COLL#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.50000000
6000.0000#
Aberrations: None#
Scalar/Quasi-vector: QV#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Propagate in environment: PROP
(Length_Units:) #Propagation distance:
25000.000#
Multiply curvature (Y/N): N#
Reposition beam (Y/N): N#
Propagation regime: DFRNL#
Smax:
0.10000000#
Scalar/Quasi-vector: SC#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: PFOC#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.10000000
40000.000#
Aberrations: None#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance:
40000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Detector: DTCT
(Length_Units:) #Quad/Split/Phi/Diff: S#
DCX,DCY,THETA:
0.0000000
0.0000000
45.000000#
LX,LY,GW:
100.00000
100.00000
0.0000000#
ID numbers for S1,S2:
12#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Process signals: PSIG#
Action: +#S(i)=S(j)+S(k)i,j,k:
3
1
2#
Action: P#
Print S(i)i:3#Action: E#$________________________________________________________________________ #

149

150

Guide to DIFFRACT

Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##$INCREMENT#20,ITEM1...BY: -0.10#$INCREMENT#20,ITEM2...BY: -0.10#$INCREMENT#21,ITEM1...BY: -0.10#$INCREMENT#21,ITEM2...BY: -0.10#$INCREMENT#22,ITEM1...BY: -0.10#$INCREMENT#22,ITEM2...BY: -0.10#$INCREMENT#23,ITEM1...BY: -0.10#$INCREMENT#23,ITEM2...BY: -0.10#$INCREMENT#24,ITEM1...BY: -0.10#$INCREMENT#24,ITEM2...BY: -0.10#$INCREMENT#25,ITEM1...BY: -0.10#$INCREMENT#25,ITEM2...BY: -0.10#$INCREMENT#26,ITEM1...BY: -0.10#$INCREMENT#26,ITEM2...BY: -0.10#$INCREMENT#27,ITEM1...BY: -0.10#$INCREMENT#27,ITEM2...BY: -0.10#$INCREMENT#28,ITEM1...BY: -0.10#$INCREMENT#28,ITEM2...BY: -0.10#$INCREMENT#29,ITEM1...BY: -0.10#$INCREMENT#29,ITEM2...BY: -0.10#$#$INCREMENT#30,ITEM1...BY: 1#40$MULTIPLY#40,ITEM1....BY: 0#$#$JUMP#10_IF#20,ITEM1.GT.: -10.0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: IX00.DAT#
Action: T#
Saving TEMP in TIFF-file: IX00.TIF#
Action: L#
Data-file loaded to TEMP: SP00.DAT#
Action: T#
Saving TEMP in TIFF-file: SP00.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Guide to DIFFRACT

151

14.4. Magnetic Domains: There can be a number of reverse-magnetizeddomains (marks) on the magneto-optical disk surface. All these marks have theshape of rectangles capped by semi-circles, as shown in Fig. 62. For each mark,the user is prompted to enter center coordinates (MCX, MCY), length L, width W,
and orientation angle Θ [-90°: 90°] relative to the X-axis. There is also a choice
between regular and chevron-type marks. (In magneto-optical data storage,
regular domains are obtained during recording by laser power modulation, while
chevron-type domains are the result of recording by magnetic field modulation.)
Note: Magnetic domains are not available in the Student Edition.

Figure 62. Regular and chevron-type marks on magneto-optical disk surface. Within the area
covered by each mark, the reflection coefficient rs of the disk is multiplied by a minus sign.
The reflection coefficient rp is the same both inside and outside the marks.

Example. Figure 63(a) shows a pattern of grooves, performat pits, and magnetic
marks (chevron type) on a magneto-optical disk surface. The disk reflection
coefficients are |rp| = 0.75, |rs| = 0.03 and φp – φs = 15°. A laser beam ofwavelength λ is focused on the disk through a 0.6NA objective lens; the incident
beam’s polarization state is linear at 45° to the groove direction. The reflected
light is collected at the exit pupil of the same objective, and the reflected
patterns of polarization rotation angle ρ and ellipticity η are shown in
Figs. 63(b, c). The various parameters of this simulation are listed in theaccompanying COMMAND.DAT file.

Guide to DIFFRACT

152

a

−6

b

x/λ

+6 −3300c

x/λ3300 −3300

x/λ

3300

Figure 63. (a) Surface pattern of an optical disk containing grooves, preformat marks, andmagnetic domains of chevron type. The track-pitch and groove-depth are 2λ and 0.08λ, while
the preformat pits’ width and depth are 0.85λ and 0.15λ, respectively. (b, c) Polarization
rotation angle ρ and ellipticity η at the exit pupil of a 0.6NA objective used to focus a laser
beam on the disk. A Gaussian beam of wavelength λ and linear polarization (at 45° to the Xaxis) is brought to focus at the disk center. Upon reflection from the disk, the beam isrecollimated by the same objective lens. The range of the polarization rotation angle in (b) is(28.04°, 66.51°), while the range of ellipticity in (c) is (-23.03°, +23.82°).
Printout of the COMMAND.DAT file used in the simulations pertaining to Fig. 63@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/20/2005
Time: 15:03 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$MULTIPLY#??,ITEM?....BY: ??
#$MAKE#??,ITEM?~#??,ITEM?:#$JUMP#??_IF#??,ITEM?.GT.: ??
#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.0000000#
1/e radii Rmin,Rmax:
3500.0000
4500.0000#
RA,THETA:
3000.0000
135.00000#
Aberrations: None#
Polarization RHO,ETA:
45.000000
0.0000000#
NMAX,NMAY:
1024
1024
#
LMAX,LMAY:
60000.000
60000.000#$________________________________________________________________________ #Guide to DIFFRACT
Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: PFOC#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.6000000
5000.0000#
Aberrations: None#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance:
5000.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units:) #Type of medium: MO#
SCX,SCY,RA:
0.0000000
0.0000000
10.000000#(Incidence medium) Ninc:
1.5800000#|Rp|,|Rs|,PHIsp:
0.7500000
0.0300000
15.000000#
Pregrooves: Y#
A,B,C:
0.2000000
1.2000000
1.4000000#
D,ZETA:
2.0000000
0.0800000#
THETA,GCX,GCY:45.000000
0.4950000-0.4950000#
Wobble (Sine,Blip,None): None#
Preformat marks: Y#
Number of marks: 4#
Mark # 1:
MCX,MCY:
0.0000000
0.0000000#
THETA:
45.000000#
Round or Flat: F#
Length, Width, Height:
3.0000000
0.8500000-0.1500000#
Ratio:
0.6000000#
Mark # 2:
MCX,MCY:
2.3000000
2.3000000#
THETA:
45.000000#
Round or Flat: F#
Length, Width, Height:
2.0000000
0.8500000-0.1500000#
Ratio:
0.6000000#
Mark # 3:
MCX,MCY:-3.0000000-3.0000000#
THETA:
45.000000#
Round or Flat: F#
Length, Width, Height:
1.4000000
0.8500000-0.1500000#
Ratio:
0.6000000#
Mark # 4:
MCX,MCY:-4.0000000-4.000000#
THETA:
45.000000#
Round or Flat: F#
Length, Width, Height:
1.0000000
0.8500000-0.1500000#
Ratio:
0.6000000#
Magnetic domains: Y#
Number of marks: 7#
Type (Regular/Chevron): C#
Mark # 1:
MCX,MCY:-1.4142000
1.4142000#
Length, Width, Angle:
1.0000000
0.8500000
45.000000#
Mark # 2:
MCX,MCY:
0.7071000
3.5355000#
Length, Width, Angle:
2.0000000
0.8500000
45.000000#
Mark # 3:
MCX,MCY:-4.2426000-1.4142000#
Length, Width, Angle:
2.5000000
0.8500000
45.000000#
Mark # 4:
MCX,MCY:
4.0000000
4.0000000#
Length, Width, Angle:
1.6000000
0.8500000
45.000000#
Mark # 5:
MCX,MCY:
1.4142000-1.4142000#
Length, Width, Angle:
2.0000000
0.8500000
45.000000#
Mark # 6:
MCX,MCY:
4.2426000
1.4142000
#

153

154

Guide to DIFFRACT

Length, Width, Angle:
1.5000000
0.8500000
45.000000#
Mark # 7:
MCX,MCY:-1.0607000-3.8891000#
Length, Width, Angle:
1.4000000
0.8500000
45.000000#
Inspect surface pattern: Y#
Xmin,Xmax:-6.0000000
6.0000000#
Ymin,Ymax:-6.0000000
6.0000000#
Color or Gray-scale: G#
Save data file (Y/N): Y#
File identifier: 0#
Propagation distance:
5000.0000#
Propagation regime: FRNHF#
Smax:
0.1000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: COLL#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.6000000
5000.0000#
Aberrations: None#
Scalar/Quasi-vector: QV#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Polarization#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-3300.0000
3300.0000#
Ymin,Ymax:-3300.0000
3300.0000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: SP00.DAT#
Action: T#
Saving TEMP in TIFF-file: SP00.TIF#
Action: L#
Data-file loaded to TEMP: RT00.DAT#
Action: T#
Saving TEMP in TIFF-file: RT00.TIF#
Action: L#
Data-file loaded to TEMP: EL00.DAT#
Action: T#
Saving TEMP in TIFF-file: EL00.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Guide to DIFFRACT

155

14.5. Multilayer Dye-Polymer Media: The dye-polymer optical disk may be
used either in reflection mode or in transmission mode. The incident beam,
initially in an environment of index NVIRON, is transplanted into an incidence
medium of refractive index ninc, where ninc ≥ NVIRON. The incidence medium
thus plays the role of a cover-layer (or transparent substrate) through which thefocused light must pass before reaching the multilayer stack of the dye-polymer
disk; see Fig. 64. The thickness of the cover-layer is ignored, since any
aberrations induced upon entering from the environment into this layer are
assumed to have been previously corrected by the focusing element. The only
effect of the incidence medium, when ninc ≠ NVIRON, is the loss of a fraction of
the incident optical power caused by reflection at its front facet.
Note: R = |(NVIRON – ninc)/(NVIRON + ninc)|2 is the front facet’s reflectance.
The user must specify the number NLYR of layers that comprise the dye-polymer
stack. These layers are numbered in increasing order, starting at the interface
with the incidence medium. For each layer the user must specify a complex
refractive index (n, k) and the layer's thickness. (Ideally the total stack thickness
should not exceed the depth-of-focus of the incident beam.) The medium beyond
the last layer of the stack, here referred to as the substrate, should also be
specified in terms of its (nsub, ksub), although for transmission type media, ksub isgenerally set to zero. The layer-numbering scheme thus assigns the number zeroto the incidence medium and the number NLYR + 1 to the substrate.
The user is also prompted for the index of the emergence environment. Uponreflection, the beam can either stay in the incidence medium (nemerge = ninc) or
return to the current environment (nemerge = NVIRON). In the transmission mode,
however, the emergence medium may be the substrate (nemerge = nsub), or anothermedium with a refractive index anywhere between unity and nsub (that is,
1 ≤ nemerge ≤ nsub). In the latter case, the thickness of the substrate is ignored andonly transmission losses in passing from the substrate to the emergenceenvironment are taken into account; R = |(nsub – nemerge)/(nsub + nemerge)|2 is the
facet’s reflectance. The substrate is thus treated like a relatively thick coverlayer, with the corresponding aberrations assumed to have been corrected by acollimator lens that will follow.
In the reflection mode, the subsequent propagation distance is measured fromthe interface between the incidence medium and Layer 1. In the transmissionmode, the propagation distance is measured from the interface between the lastlayer (NLYR) and the substrate.

Guide to DIFFRACT

156

Environment (NVIRON)

Incident Laser Beam

Incidence Medium(Layer 0)
Layer 1
Layer 2
Layer 3Layer 4
Substrate(Layer 5)

Figure 64. Land/Groove structure in a multilayered dye-polymer optical disk.
DIFFRACT’s dye-polymer media can support up to four different sets of grooves

and, independently, up to four different sets of marks (bumps/pits). The
corresponding number of sets should be set to zero if no grooves or marks areembedded in the medium. Each set of grooves covers the surface area of the disk
within a circle of radius RA centered at (SCX, SCY), affecting the thickness of
two specific layers everywhere in the XY-plane inside the above circle. Different
groove-sets are completely independent of each other; their various parameters
can be specified as though the other groove-sets did not exist. In the event thattwo or more groove-sets happen to modify the thickness of a given layer, theireffects on that particular layer will be cumulative. Similarly, each set of marks
affects the thickness of two specific layers, but different mark-sets are totally
independent of each other (being independent of the grooves as well). Each set
of marks may contain an arbitrary number of bumps/pits. Again, when multiple
marks happen to modify a given layer's thickness at a specific location, theireffects will be cumulative.

Guide to DIFFRACT

157

In each groove-set exactly two layers are modified by the grooves. Within thegroove region one layer becomes thinner while the other layer becomes thicker
by the same amount (i.e., by the groove depth); all other layers retain their
thickness values. The same idea applies in the case of marks belonging to agiven set. Again two layers are modified in the presence of the mark-set, but thelayers that contain the marks are not necessarily the same as those that containthe grooves. In each case, the two specified layers are numbered L1 and L2; both
numbers must be greater than or equal to zero and less than or equal to thenumber associated with the substrate (0 ≤ L1, L2 ≤ NLYR + 1). The user may enter
L1, L2 in any order, but the program arranges them such that L1 is the smaller of
the two numbers. When L1 = L2, DIFFRACT automatically replaces L2 withNLYR + 1, the number associated with the substrate (whose infinite thickness is
not affected by the presence of grooves/marks). The program thus provides thepossibility of simulating grooves and/or marks that modify only a single layer's
thickness.
The groove-depth ζ (which may be positive or negative) is added to L1 andsubtracted from L2. The height H of a mark (positive for bump, negative for pit)
is added to L2 and subtracted from L1 (remember that L1 < L2). The user musttake care that no layer will acquire a negative thickness after being modified bygrooves and marks, although zero thickness is acceptable for a layer either
before or after modification. The following warning will be issued for each andevery layer whose thickness becomes negative upon the insertion of the relevantgrooves/marks:
Warning: Layer nn is too thin to support the specified grooves/marks.
In the dye-polymer disk having an embedded land/groove structure depicted inFig. 64, the laser beam is focused (through the incidence medium) onto a fourlayer stack. The two layers that are modified by the grooves are layers 2 and 3:
Layer 2 has become thinner and layer 3 has become thicker in the groove region;
all other layers remain intact. The same idea applies to the case of marks (bumps
and pits). Again two layers are modified in the presence of the marks, but thelayers that contain the marks are not necessarily the same layers as those whichcontain the grooves. The geometry of grooves displayed in Figs. 54, 55 and 57
applies here as well, as does the geometry of bumps/pits shown in Fig. 59.
As mentioned earlier, the dye-polymer media can support as many as four
different sets of grooves and four different sets of marks. The overlapping
profiles of all these marks/grooves may thus become confusing if displayed
simultaneously. Therefore, the user is given the option of selecting certain sets

Guide to DIFFRACT

158

of grooves/marks for inclusion in the surface pattern display. The profiles ofthe selected grooves and marks are added together (algebraically), thendisplayed as a height/depth variation over the XY-plane of the disk. Selection ofa given set of grooves/marks for inclusion in the display does not in any way
affect the computation of the overall reflectance/transmittance of the dyepolymer disk.
Note: Dye-Polymer media are not available in the Student Edition.
Example 1. Figure 65(a) shows a small section of the surface of a dye-polymer
optical disk. The grooves have period = 2λ and depth = 0.1λ. The marks havedifferent lengths and heights (or depths), but the same width W = λ. The
multilayer stack of the disk consists of four layers deposited on a polycarbonate
substrate (nsub = 1.58), and the laser beam is directly incident on the top surface(layer 1). Embedded grooves affect the thickness of layers 2 and 3, whileembedded marks affect the thickness of layers 1 and 3. A 0.6NA objective lens
is used to focus a truncated Gaussian laser beam on the disk surface (a disk
defocus of ∆z = 1.85λ is assumed). The intensity and phase distributions of thelight reflected from the disk (and collected by the same objective lens) aredisplayed in Figs. 65(b, c). The various simulation parameters are listed in theaccompanying COMMAND.DAT file.
a

−6

b

x/λ

+6 −3100cx/λ3100 −3100

x/λ

3100

Figure 65. (a) Pattern of grooves and marks (i.e., bumps and pits) on the surface a dyepolymer optical disk. A laser beam of wavelength λ = 650nm is focused onto the disk surface
through a 0.6NA objective lens; the focused spot center is at (x, y) = (0, 0). (b) Intensity
distribution of the light reflected from the disk surface as it emerges from the exit pupil of theobjective lens. (c) Phase distribution of the reflected beam at the exit pupil of the objective. In(b, c) the disk is displaced from the focal plane of the lens by ∆z = 1.85λ.

Guide to DIFFRACT

Command.dat file used in the simulation of a dye-polymer optical disk@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 04/19/2005
Time: 12:43 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$MULTIPLY#??,ITEM?....BY: ??
#$MAKE#??,ITEM?~#??,ITEM?:#$JUMP#??_IF#??,ITEM?.GT.: ??
#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
1/e radii Rmin,Rmax:
3000.0000
3000.0000#
RA,THETA:
3000.0000
0.0000000#
Aberrations: None#
Polarization RHO,ETA:
0.0000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
60000.000
60000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: PFOC#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.60000000
5000.0000#
Aberrations: None#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance:
5001.8500#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units:) #Type of medium: DP#
SCX,SCY,RA:
0.0000000
0.0000000
12.0000000#(Incidence medium) Ninc:
1.0000000#
Reflect or Transmit: R##
Number of layers:
4#
Layer #01: n,k,Thickness:
1.8000000
0.0000000
0.20000000#
Layer #02: n,k,Thickness:
2.3000000
0.1000000
0.15000000#
Layer #03: n,k,Thickness:
1.8000000
0.0000000
0.25000000#
Layer #04: n,k,Thickness:
2.0000000
7.0000000
0.10000000#
Medium beyond last (n,k):
1.5800000
0.0000000#

159

Guide to DIFFRACT

160
Emergent environment’s n:

1.000000##
Pregroove sets (0-4): 1##
G1:Modified layers L1,L2: 2
3#****************** A,B,C:
0.2000000
1.2000000
1.40000000#
D,ZETA:
2.0000000
0.1000000#
THETA,GCX,GCY:45.000000
0.4950000-0.49500000#
Wobble (Sine,Blip,None): None#
Include in display (Y/N): Y##
Mark sets (0-4): 1##
Set 1....Number of marks: 7#
Modified layers L1,L2: 1
3#
MARK #01:
MCX,MCY:
0.0000000
0.0000000#
Orientation angle THETA:
45.000000#
Round or Flat: R#
Length,Width,Height:
3.0000000
1.0000000-0.15000000#
Roof-to-Rim ratio:
0.0000000#
MARK #02:
MCX,MCY:
4.0000000
4.0000000#
Orientation angle THETA:
45.000000#
Round or Flat: R#
Length,Width,Height:
2.0000000
1.0000000
0.15000000#
Roof-to-Rim ratio:
0.0000000#
MARK #03:
MCX,MCY:-4.0000000-4.0000000#
Orientation angle THETA:
45.000000#
Round or Flat: R#
Length,Width,Height:
3.0000000
1.0000000
0.25000000#
Roof-to-Rim ratio:
0.0000000#
MARK #04:
MCX,MCY:-1.4142000
1.4142000#
Orientation angle THETA:
45.000000#
Round or Flat: R#
Length,Width,Height:
1.0000000
1.0000000
0.25000000#
Roof-to-Rim ratio:
0.0000000#
MARK #05:
MCX,MCY:
1.4142000-1.4142000#
Orientation angle THETA:
45.000000#
Round or Flat: R#
Length,Width,Height:
4.0000000
1.0000000
0.20000000#
Roof-to-Rim ratio:
0.0000000#
MARK #06:
MCX,MCY:
4.9497000
2.1213000#
Orientation angle THETA:
45.000000#
Round or Flat: R#
Length,Width,Height:
2.0000000
1.0000000-0.25000000#
Roof-to-Rim ratio:
0.0000000#
MARK #07:
MCX,MCY:
2.1213000
4.9497000#
Orientation angle THETA:
45.000000#
Round or Flat: R#
Length,Width,Height:
2.5000000
1.0000000
0.20000000#
Roof-to-Rim ratio:
0.0000000#
Include in display (Y/N): Y##
Inspect surface pattern: Y#
Xmin,Xmax:-6.0000000
6.0000000#
Ymin,Ymax:-6.0000000
6.0000000#
Color or Gray-scale: G#
Save data file (Y/N): Y#
File identifier: 0#
Propagation distance:
5001.8500#
Propagation regime: FRNHF#
Smax:
0.10000000#$________________________________________________________________________ #Remarks:#

Guide to DIFFRACT
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: COLL#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.6000000
5000.0000#
Aberrations: None#
Scalar/Quasi-vector: QV#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-3100.0000
3100.0000#
Ymin,Ymax:-3100.0000
3100.0000#
Color or Gray-scale: C#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Phase#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-3100.0000
3100.0000#
Ymin,Ymax:-3100.0000
3100.0000#
Color or Gray-scale: C#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: SP00.DAT#
Action: T#
Saving TEMP in TIFF-file: SP00.TIF#
Action: L#
Data-file loaded to TEMP: IX01.DAT#
Action: T#
Saving TEMP in TIFF-file: IX01.TIF#
Action: L#
Data-file loaded to TEMP: PX01.DAT#
Action: T#
Saving TEMP in TIFF-file: PX01.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#

161

Guide to DIFFRACT

162

Example 2. Figure 66 shows a four-layer dye-polymer disk used in thereflection mode. The objective lens focuses the beam through a plastic coverlayer onto the multilayer stack in such a way that layers 1 − 4 are all within thedepth of focus. The top facet of the cover layer is responsible for partial loss oflight in the incidence as well as reflection paths, but it does not affect the qualityof the beam in any other way. (In other words, the cover layer is thick enoughfor its front facet to be out of focus, and the lens is corrected for the cover-layerinduced spherical aberrations.) Wherever the pits are recorded, layer 1, which
has zero thickness in the initial setting, expands into the pits. The stack istherefore a 3-layer storage medium in which the 80 nm-thick storage layer (layer2) has a complex index of refraction n + ik = 1.8 + i 0.2, whereas the refractive
index of the pits is n = 1.5. The bottom two layers (layers 3 and 4) are notaffected by the recording process, but they contribute to the readout signal byplaying their part in multiple reflections and optical interference among thevarious layers.

Focusing Lens
Laser Beam
Incidence mediumn = 1.58

Thickness = 0(n = 1.5, k = 0.0)
Thickness = 0.08µm(n = 1.8, k = 0.2)

Layer 1
Pit 1

Pit 2

Layer 2n = 1.5, k = 0.0, thickness = 0.05µm

Layer 3n = 2.0, k = 6.5, thickness = 0.065µm

Layer 4

Figure 66. Reflective dye-polymer medium in a configuration where the light beam is
focused through a plastic cover layer onto a four-layer stack. There are no grooves in this disk
structure, and the two (flat-bottom) marks belong to the same mark-set.

In the stack structure depicted in Fig. 66, the sole purpose of layer 1 is toprovide a filling material (refractive index n = 1.5) for the recorded pits; where
no pits are written, the phase/amplitude reflectivity of the stack is determined bylayers 2, 3, and 4, with no contribution whatsoever from layer 1. Although thepit depths in the present example are chosen to be exactly equal to the thicknessof layer 2, the user may choose any pit depth less than this value as well.

Guide to DIFFRACT163

The simulated system parameters are listed in the accompanying command.dat
file, and the results of the simulation appear in Fig. 67. The pits embedded inlayer 2 are flat-bottom pits, having floor-to-rim ratios of 0.5 and 0.9,
respectively. The distributions of intensity and phase at the exit pupil of theobjective lens (after reflection from the dye-polymer medium and a second
passage through the lens) are shown in Figs. 67(b) and 67(c).
(a)

(b)

(c)

Figure 67. (a) Flat pits embedded in layer 2 of the system of Fig. 66; the geometry of the pits
and the optical constants of the various layers are given in the accompanying command.dat
file. (b, c) Plots of intensity and phase distribution at the exit pupil of the objective lens,
obtained after the beam is reflected from the dye-polymer disk.

In the system of Fig. 66, grooves (with or without wobble) may also be
introduced, either as stand-alone features of the medium, or in combination withpits of the kind described in Example 2. In general, the grooves in a dyepolymer medium may affect the thickness of the same pair of layers as modified
by the pits, or they may modify any other pair of layers. Needless to say, it isalso possible to introduce an extra layer of zero thickness (as was done for thepits in Fig. 66) for the sake of creating grooves inside a host layer, with thecomplex refractive index of the grooves differing from that of the host layer.
Command.dat file used in the simulation of the system depicted in Fig. 66@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 04/19/2005
Time: 01:16 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#164

Guide to DIFFRACT$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: mm) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.000000
0.000000
1.0000000#
1/e radii Rmin,Rmax:
4.000000
4.000000#
RA,Theta:
3.000000
0.000000#
Aberrations: None#
Polarization RHO,ETA:
0.000000
45.00000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
30.00000
30.00000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: mm) #Type: PFOC#
LCX,LCY:
0.000000
0.000000#
NA,FL:
0.6000000
5.000000#
Aberrations: None#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance:
5.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units: um) #Type of medium: DP#
SCX,SCY,RA:
0.000000
0.000000
25.00000#(Incidence medium) Ninc:
1.580000#
Reflect or Transmit: R##
Number of layers:
4#
Layer #01: n,k,Thickness:
1.500000
0.000000
0.000000#
Layer #02: n,k,Thickness:
1.800000
0.200000
0.080000#
Layer #03: n,k,Thickness:
1.500000
0.000000
0.050000#
Layer #04: n,k,Thickness:
2.000000
6.500000
0.065000#
Medium beyond last (n,k):
1.000000
0.000000#
Emergent environment’s n:
1.000000#
Pregroove sets (0-4): 0#
Mark sets (0-4): 1##
SET 1... Number of marks: 2#
Modified layers L1,L2: 1
2#
Mark #01:
MCX,MCY:-2.000000-2.000000#
Orientation angle Theta:
45.00000#Round or Flat: F#
Length,Width,Height:
5.000000
3.000000-0.0800000#
Floor-to-Rim ratio:
0.500000#

Guide to DIFFRACTMark #02:
MCX,MCY:
Orientation angle Theta:
Round or Flat: F
Length,Width,Height:
Floor-to-Rim ratio:
Include in display (Y/N): Y

165
2.500000
45.00000

2.500000###
3.000000
2.000000-0.0800000#
0.900000###
Inspect surface pattern: Y#
Xmin,Xmax:-5.000000
5.000000#
Ymin,Ymax:-5.000000
5.000000#
Color or Gray-scale: G#
Save data file (Y/N): Y#
File identifier: 0#
Propagation distance:
5000.000#
Propagation regime: FRNHF#
Smax:
0.1000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: mm) #Type: COLL#
LCX,LCY:
0.000000
0.000000#
NA,FL:
0.600000
5.000000#
Aberrations: None#
Scalar/Quasi-vector: QV#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-3.100000
3.100000#
Ymin,Ymax:-3.100000
3.100000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Phase#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-3.100000
3.100000#
Ymin,Ymax:-3.100000
3.100000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Guide to DIFFRACT

166

Example 3. A four-layer dye-polymer disk used in transmission mode is shownin Fig. 68. The laser beam (λo = 780nm) is focused through a 0.45NA objectivethat is corrected for the cover layer of index ninc = 1.56. The transmitted beam iscollimated by another 0.45NA objective, also corrected for the lower protective
layer (nsub = 1.56). The total thickness of the four-layer stack is 0.61 µm, well
below the depth-of-focus of the incident laser spot. The various simulation
parameters are listed in the accompanying COMMAND.DAT file.
Environment (NVIRON = 1)

Incident Laser Beam

Incidence Medium(Layer 0)
Layer 1
Layer 2
Layer 3Layer 4
Substrate(Layer 5)

Figure 68. Four layer dye-polymer disk used in the transmission mode. Two sets of grooves
are embedded in the various layers of the disk. The first groove-set affects the thickness oflayers 1 and 2, while the second groove-set modifies the thickness of layers 2 and 5 (thesubstrate). In addition, two sets of marks (not shown) are embedded in the stack. The incident
beam is focused through a 0.45NA objective into a cover layer of index ninc = 1.56 (layer 0).
Upon transmission through the stack, the beam passes through the lower protective layer ofindex nsub= 1.56, before emerging into the free-space environment (emergent environment’s
index = 1.0). The transmitted beam is subsequently collimated by a second objective lens.

Two sets of grooves and two sets of marks are embedded in the stack of Fig. 68.
The first groove-set affects the thickness of layers 1 and 2, while the second
groove-set affects the thickness of layers 2 and 5 (the substrate). The two mark-Guide to DIFFRACT

167

sets are not shown in the cross-sectional profile of Fig. 68, but their lateralprofiles in the XY-plane of the disk appear in Fig. 69(a). The first mark-set,consisting of two “flat-top” marks, affects the thickness of layers 2 and 3. The
second mark-set, consisting of three “round-bottom” marks, affects the thicknessof layers 0 (i.e., the top cover-sheet) and 1.
The transmitted beam, modulated in phase and amplitude by the various grooves
and marks, emerges into the lower cover-layer (i.e., the substrate), eventuallyreaching the free-space region below the substrate. Figures 69(b, c) show plotsof intensity and phase immediately after the beam emerges from the bottom
layer (layer 4) into the substrate. The emergent beam is subsequently collimatedby a second objective lens; Fig. 70 shows the intensity and phase profiles of thetransmitted beam at the exit pupil of this collimating lens.
(b)

(a)

(c)

Figure 69. (a) Superimposed pattern of wobbled grooves and marks in a 10 × 10 µm2 region
from the surface of the dye-polymer disk of Fig. 68. (b) Logarithmic plot of intensity and (c)the profile of phase distribution immediately after the passage of the focused beam throughthe four-layer stack of the disk.
(a)

(b)

Figure 70. Distributions of (a) intensity and (b) phase at the exit pupil of the collimator lensfollowing transmission of the focused beam through the dye-polymer disk of Fig. 68.

168

Guide to DIFFRACT

Command.dat file used in the simulation of dye-polymer disk depicted in Fig. 68$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 04/19/2005
Time: 11:35 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?: ??
#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 780.000000000
NVIRON: 1.0000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: mm) #TYPE: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW): 0.0000000 0.0000000
1.0000000#
1/e radii Rmin,Rmax: 5.0000000 5.0000000#
RA,THETA: 3.0000000 0.0000000#
Aberrations: None#
Polarization RHO,ETA: 0.0000000 45.000000#
NMAX,NMAY: 1024
1024#
LMAX,LMAY: 70.000000 70.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 780.000000000
NVIRON: 1.0000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: mm) #Type: PFOC#
LCX,LCY: 0.00000000 0.0000000#
NA,FL: 0.45000000 6.6670000#
Aberrations: None#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance: 6.66700000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 780.000000000
NVIRON: 1.0000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units: um) #Type of medium: DP#
SCX,SCY,RA: 0.00000000 0.00000000 30.000000#(Incidence medium) Ninc: 1.5600000#
Reflect or Transmit: T##
Number of layers: 4#
LAYER #01: n,k,THICKNESS: 2.8500000 0.02000000 0.18000000#
LAYER #02: n,k,THICKNESS: 1.4500000 0.01000000 0.27000000#
LAYER #03: n,k,THICKNESS: 1.7600000 0.00000000 0.15000000#
LAYER #04: n,k,THICKNESS: 2.1300000 3.15000000 0.01000000#

Guide to DIFFRACT
Medium beyond last (n,k):
Emergent environment’s n:

169
1.5600000
1.0000000

0.00000000

Pregroove sets (0-4): 2
G1:Modified layers L1,L2: 1
2****************** A,B,C: 0.2000000 0.80000000 1.1000000
D,ZETA: 1.6000000 -0.10000000
THETA,GCX,GCY: 45.000000 0.35000000 -0.3500000
Wobble (Sine,Blip,None): Sine
Central wobble A0,P0,Q0: 0.1000000 5.00000000 0.0000000
Adjacent wobble A1,P1,Q1: 0.1000000 5.00000000 0.0000000
Include in display (Y/N): Y
G2:Modified layers L1,L2: 2
5****************** A,B,C: 0.1000000 0.90000000 1.0000000
D,ZETA: 1.6000000 -0.15000000
THETA,GCX,GCY: 45.000000 0.35000000 -0.3500000
Wobble (Sine,Blip,None): Sine
Central wobble A0,P0,Q0: 0.1000000 5.00000000 0.0000000
Adjacent wobble A1,P1,Q1: 0.1000000 5.00000000 0.0000000
Include in display (Y/N): Y
Mark sets (0-4): 2
SET 1....Number of marks: 2
Modified layers L1,L2: 2
3
Mark # 01:
MCX,MCY: -2.0000000 -2.0000000
Orientation angle Theta: 45.000000
Round or Flat: F
Length, Width, Height: 2.0000000 0.50000000
Ratio: 0.5000000
Mark # 02:
MCX,MCY: 3.0000000 3.0000000
Orientation angle Theta: 45.000000
Round or Flat: F
Length, Width, Height: 1.0000000 0.50000000
Ratio: 0.5000000
Include in display (Y/N): Y
SET 2....Number of marks: 3
Modified layers L1,L2: 0
1
Mark # 01:
MCX,MCY: -1.1500000 1.1500000
Orientation angle Theta: 45.000000
Round or Flat: R
Length, Width, Height: 2.0000000 0.5000000
Ratio: 0.0000000
Mark # 02:
MCX,MCY: -4.1500000 -1.8500000
Orientation angle Theta: 45.000000
Round or Flat: R
Length, Width, Height: 1.5000000 0.5000000
Ratio: 0.0000000
Mark # 03:
MCX,MCY: 0.8500000 3.1500000
Orientation angle Theta: 45.000000
Round or Flat: R
Length, Width, Height: 1.2500000 0.5000000
Ratio: 0.00000000
Include in display (Y/N): Y
Inspect surface pattern: Y
Xmin,Xmax: -5.0000000
Ymin,Ymax: -5.0000000
Color or Gray-scale: G
Save data file (Y/N): Y

5.0000000
5.0000000

0.1000000

0.1000000-0.1000000-0.1000000-0.1000000

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170

Guide to DIFFRACT

File identifier: 0#
Propagation distance: 0.00000000#
Propagation regime: DFRNL#
Smax: 0.10000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 780.000000000
NVIRON: 1.0000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Log(intensity)#
Logarithmic SCALE: 4.0000000#
Xmin,Xmax: -4.0000000 4.0000000#
Ymin,Ymax: -4.0000000 4.0000000#
Color or Gray-scale: G#
Z-component: N#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 780.000000000
NVIRON: 1.0000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Phase#
Logarithmic SCALE: 4.0000000#
Xmin,Xmax: -4.0000000 4.0000000#
Ymin,Ymax: -4.0000000 4.0000000#
Color or Gray-scale: G#
Z-component: N#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 780.000000000
NVIRON: 1.0000000 #_________________________________________________________________________ ##Propagate in environment: PROP
(Length_Units: mm) #Propagation distance: 6.6670000#
Multiply curvature (Y/N): N#
Reposition beam (Y/N): N#
Propagation regime: FRNHF#
Smax: 0.10000000#
Scalar/Quasi-vector: SC#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 780.000000000
NVIRON: 1.0000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: mm) #Type: COLL#
LCX,LCY: 0.00000000 0.00000000#
NA,FL: 0.45000000 6.6670000#
Aberrations: None#
Scalar/Quasi-vector: QV#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 780.000000000
NVIRON: 1.0000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Intensity#
Logarithmic SCALE: 4.0000000#

Guide to DIFFRACT

171

Xmin,Xmax: -3.1000000 3.1000000#
Ymin,Ymax: -3.1000000 3.1000000#
Color or Gray-scale: G#
Z-component: N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 780.000000000
NVIRON: 1.0000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Phase#
Logarithmic SCALE: 4.0000000#
Xmin,Xmax: -3.1000000 3.1000000#
Ymin,Ymax: -3.1000000 3.1000000#
Color or Gray-scale: G#
Z-component: N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 780.000000000
NVIRON: 1.0000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

14.6. Inspecting the Surface Pattern of an Optical Disk: Once the user has
identified a set of desirable features for an Optical_Disk_Surface, he will begiven the option to inspect the resulting surface pattern. Examples of such
surface patterns are given in Figs. 56, 58, 60(a), 63(a), 65(a), 67(a) and 69(a).
The user must specify the range [Xmin : Xmax, Ymin : Ymax] within which thepattern is to be observed; both color and gray-scale displays are available.
Having viewed the surface pattern, the user will be asked whether or not heintends to save the pattern as a hard disk file. The files created for this purposeby DIFFRACT are named SPnn.DAT, where SP stands for surface pattern, and theuser is prompted for the 1-2 digit ID number nn in the range [0 : 99]. In theinteractive mode DIFFRACT rejects filenames that already exist in the home
directory; in the non-interactive mode existing files will be overwritten.
During interactive sessions the user will be given multiple opportunities toinspect and save the disk’s surface pattern. Only the last of these inspections,however, will be recorded in COMMAND.DAT. Since inspection is allowed only
once during non-interactive sessions, the viewing range and other relevant
parameters should be chosen carefully when the COMMAND.DAT file is being
prepared for a non-interactive run.

Guide to DIFFRACT

172

15. Conversion to Rays
The geometric-optical rays in DIFFRACT are determined by computing the
Poynting vector R for a given E-field distribution. Since the wavefront isassumed to be both spatially and temporally coherent, it is possible to associate
the Poynting vector at a given point on the wavefront with the unique directionof the geometric-optical ray at that point. Moreover, the magnitude of the
Poynting vector is an indicator of the strength of the ray at any given location.
Conversion_to_Rays does not affect the complex-amplitude distribution of thebeam which is being maintained in arrays AX, AY; it generates the Poynting
vector distribution over the entire cross-section of the beam, and saves the
distribution to a different location. The X- and Y-components of R are combinedin the complex number RX + i RY, and the resultant is returned in the array BX.
The Z-component and the magnitude Ro of R are combined to form RZ + i Ro,
and are returned in the array BY. (As by-products, the Z-component of the Efield, EZ and the total intensity pattern IX + IY + IZ are also generated and returnedin arrays WRKX, WRKY.) Since Conversion_ to_Rays requires the use of arrays
BX, BY, you must select the option “Enable Ray Tracing” in the beginning of
the program; otherwise the requisite memory for these arrays will not beallocated, and selecting the RAYS option will result in the following message
being issued and a return to the main menu.
Ray_tracing features have been disabled at the start of the program. To
reinstate these features quit the program, then restart it and choose the“Enable Ray Tracing” option.
When this happens you must save your distribution to a disk file, quit the
program, and then restart it, making sure that the “Enable Ray Tracing” boxwithin the “Initial Settings” screen is checked. At this point you may retrieve thecomplex-amplitude distribution that you saved earlier, and proceed to invoke the
RAYS option.
While computing the Poynting vector, in those instances where the beam’s
curvature phase-factor is non-zero, the user will be given two options:• The user may choose to incorporate the curvature into the distribution bydirect multiplication with AX, AY.

Guide to DIFFRACT173• The user may opt for collimating the beam with a fictitious lens, thenreversing the action of this lens once the rays have been obtained.
The first option is allowed only when the curvature is mild; even then, its use
requires authorization by the user. The second option is generally applicable, but
its validity depends on the degree of collimation that can be achieved with a lensthat merely eliminates the curvature phase factor.
Note: Once the RAYS option has been invoked to generate a set of rays
corresponding to a given wavefront, these rays may be visualized by callingupon the PLOT option. The most important application of RAYS, however, is in
ray_tracing through a system of optical surfaces. The RAYS option must be used
to convert a complex-amplitude distribution into a set of geometric-optical rays
before ray_tracing can be called upon.
If, after Conversion_to_Rays, the user decides to save the distribution in a disk
file, both the complex-amplitude distribution and the ray distribution will besaved. Then, in subsequent runs of the program, the Retrieve option will cause
both distributions to be loaded into memory, unless ray_tracing features havebeen disabled in the beginning of the run, in which case the ray distribution willbe ignored.

Example. Figure 71 shows (in a cross-sectional plane) the x-, y-, and zcomponents of the rays diverging from a coherent, monochromatic laser diode.
The beam’s FWHM divergence angles are ∆θx = 12°, ∆θy = 25°, and the laser’s
assumed astigmatism is 10λ, where λ is the radiation wavelength. The crosssection of the beam under examination is a distance z = 5000λ away from the
midpoint of its two waists (i.e., waists along the X- and Y-axes). In each frame,
the gray-scale encodes the range of values of Rx, Ry, Rz from minimum (black) tomaximum (white). The actual range of values of Rx, Ry, Rz are given in the
figure’s caption. The divergent nature of the beam is clearly seen in the brightand dark lobes of Rx and Ry. The various simulation parameters are listed in theaccompanying COMMAND.DAT file.

Guide to DIFFRACT

174

2500 a

c

b

y/λ

−2500

−2500

x/λ

2500 −2500

x/λ2500 −2500

x/λ

2500

Figure 71. From left to right, the x-, y-, and z-components of the ray distribution in the crosssectional plane of a beam that emanates from a semiconductor laser diode. The FWHM beam
divergence angles are ∆θx = 12°, ∆θy = 25°, and the astigmatism of the laser diode is ∆ = 10λ,where λ is the laser’s wavelength. The cross-section under examination is a distancez = 5000λ away from the midpoint of the two waists of the beam along the X- and Y-axes. In(a) the range of Rx is (−2.04 × 10 −8, +2.04 × 10 −8); in (b) the range of Ry is(−4.24 × 10 −8, +4.24 × 10 −8), while in (c) the range of Rz is (0, 3.79 × 10 −7). In each frame,
black represents the minimum and white the maximum; the gray pixels in between cover theintervening range of values of the function.

Command.dat file corresponding to the simulation depicted in Fig. 71@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.2
File COMMAND.DAT @@#$@@ Date: 03/02/2004
Time: 20:53 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Guide to DIFFRACT$________________________________________________________________________ #Remarks:##
Vacuum wavelength (um): 0.650000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: LD#
BCX,BCY:
0.0000000
0.0000000#
PHI,PSI,DELTA:
25.000000
12.000000
10.000000#
Orientation angle THETA:
0.0000000#
Distance from cavity Z0:
5000.0000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
50000.000
50000.000#$________________________________________________________________________ #Remarks:##
Vacuum wavelength (um): 0.650000
NVIRON: 1.000000 #_________________________________________________________________________ ##Ray conversion: RAYS#
Multiply curvature (Y/N): N#$________________________________________________________________________ #Remarks:##
Vacuum wavelength (um): 0.650000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Ray distribution#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-2500.0000
2500.0000#
Ymin,Ymax:-2500.0000
2500.0000#
Color or Gray-scale: G#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:##
Vacuum wavelength (um): 0.650000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: rx01.DAT#
Action: T#
Saving TEMP in TIFF-file: rx01.TIF#
Action: L#
Data-file loaded to TEMP: ry01.DAT#
Action: T#
Saving TEMP in TIFF-file: ry01.TIF#
Action: L#
Data-file loaded to TEMP: rz01.DAT#
Action: T#
Saving TEMP in TIFF-file: rz01.TIF#
Action: Q#$________________________________________________________________________ #Remarks:##
Vacuum wavelength (um): 0.650000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

175

176

Guide to DIFFRACT

16. Geometric-optical System: Definition, Inspection, and Ray_Tracing
This option allows the user to define an optical system consisting of lenses,
mirrors, prisms, gradient-index media, and apertures. It then allows aninteractive inspection of the system through the use of selective ray_tracing.
Eventually, it will trace the various rays of the incident beam through thesystem, and will construct the emergent wavefront. The ray_tracing options of
DIFFRACT are powerful tools that allow modeling of complex systems withmany optical surfaces, placed in arbitrary positions and orientations. The user
specifies the various surfaces through which light travels in the order in which
the rays encounter them. Each surface is either a Glass, a Mirror, or an Aperture.
The last surface to be specified (known as the Destination_Plane) is a flat surface
at which ray_tracing terminates. Figure 72 shows an example of a system thatcan be modeled by DIFFRACT. It consists of four glass surfaces, two mirrors(one of the metallic type, the other of the total_internal_reflection type), one
annular aperture, and a Destination_Plane. Despite being a very specific example,
Fig. 72 nevertheless shows many features of the systems that can be analyzed
using the ray_tracing options of the program.
16.1. Surface Coordinates: The location and orientation of a given surface aredetermined by the coordinates of its vertex V = (Vx, Vy, Vz), and by the directionof its surface-normal Vn at that vertex. These are specified in the Cartesian
coordinate system of the incident beam. (The incident beam is in the XY-plane at
Z = 0, and its propagation direction is generally along the positive Z-axis.) The
various surfaces of the optical system must be arranged in the order in which
they are encountered by the beam, except for the Destination plane, which is avirtual surface and can, therefore, be on either side of the last optical surface. At
the vertex of a given surface, the surface-normal Vn must have a positive
projection on rays that arrive at that surface. This is the general rule that decides
between the two possible directions of any surface-normal (see Fig. 72).
In the case of a cylindrical surface, in addition to the vertex V and the surface
normal Vn at that vertex, the user must also specify the orientation of thecylinder axis Va, which is a vector orthogonal to Vn. This is typically done byspecifying the azimuthal angle φ of the projection of Va onto the XY-plane. If,
however, Vn happens to be parallel (or nearly parallel) to the XY-plane, theprogram recognizes the insufficiency of Va’s azimuth in this instance and asks
instead for the polar angle θ of the cylinder-axis. (θ is the angle between Va andthe Z-axis, whereas φ is the angle between the X-axis and the projection of Va on
the XY-plane, measured counterclockwise from X in the right-handed XYZGuide to DIFFRACT

177

coordinates of incidence.) Either way, specification of the cylinder-axis requires
only one additional parameter, either θ or φ.
The maximum allowed number of surfaces is presently set to 10 in theStudent Edition and to 99 in the General Edition of DIFFRACT. (One may
increase the maximum allowed number of surfaces in the General Edition
by modifying the corresponding parameter in the SETTINGS.LOG file
residing in DIFFRACT’s home directory.)

Figure 72. Optical system consisting of several Glass, Mirror, and Aperture surfaces. The
location of each surface in the Cartesian coordinate system of the incident beam is specified
by its vertex coordinates V = (Vx, Vy, Vz). The surface normal at the vertex, Vn, identifies theorientation of the surface. Glass and Mirror surfaces may be flat, spherical, conical,
aspherical, cylindrical, or axiconal in shape. Apertures are confined to flat surfaces and have
either elliptical or rectangular openings. The Destination plane is a virtual plane on which
ray_tracing terminates. The location and orientation of this plane also define the coordinate
system X ' Y ' Z ' of the emergent beam.

Guide to DIFFRACT

178

16.2. Arrangement and Succession of Surfaces. The surfaces of a Geometric
Optical System are numbered sequentially from 1 up to a maximum allowed
integer. Each system must have a first surface and a last surface, the latter being
always the Destination_Plane. All remaining surfaces must be numbered between
1 and the number associated with the last surface.
For each Glass, Mirror, and Aperture the user specifies two other surfaces (within
the same system) as alternative choices for the next surface; these are referred toas Next_if_Hit and Next_if_Miss surfaces. If a ray hits a given surface, it will seek
the (user-specified) Next_if_Hit surface as a subsequent target. However, if a ray
misses a surface, its next target becomes the associated Next_if_Miss surface. Each
incident ray arriving at a Geometric-optical System thus starts by targeting
surface 1 and follows a path through the system – as dictated by the specified
sequence of next_surfaces – until it reaches the Destination_Plane. The
Destination_Plane is the only surface that has no associated next_surfaces. Figure
73 shows a simple arrangement of six Glass surfaces and the corresponding
associations with the next_surfaces.
A next_surface specified by the numeral 0 acts as an absorber. Any Glass,Mirror, or Aperture surface becomes opaque (i.e., an absorber) if its Next_if_Hit
surface is set to 0. Similarly, rays that miss a given surface can be blocked (i.e.,
absorbed) by setting the corresponding Next_if_Miss surface to 0. The above
sequencing scheme provides a powerful tool for simulating complex opticalsystems. The user, however, must take care to avoid logical inconsistencies thatwould lead to infinite loops or inaccessible surfaces. In many instances,
DIFFRACT automatically identifies such logically inconsistent arrangements andissues warnings to clarify the nature of the problem. Ultimately, however, theuser is responsible for ensuring that the specified succession sequence of thevarious surfaces is physically attainable.
Surface

Next_if_Hit

Next_if_Miss

1
2
3
4
5
6
7

2
5
4
5
6
7
–

3
0
0
0
0
0
–

Fig. 73. Arrangement of surfaces in a simple GEOP
system. The Table lists Next_if_Hit and Next_if_Miss
surfaces. The numeral 0 stands for an absorber.

X

4

7

3

6
Z

1
Incident
Beam

2

5

Destination
PlaneGuide to DIFFRACT

179

16.3. Surface Types and Shapes: Four types of surfaces are allowed in
DIFFRACT. These are Glass, Mirror, Aperture, and the Destination_Plane. Asidefrom the Destination plane, which is a flat, virtual plane for receiving the rays at
the end of tracing, the simplest surfaces are the Apertures. These have a flat
surface and either an Elliptical or a Rectangular opening with half-length LA andhalf-width WA. To obtain an Aperture with a circular opening, the user mustchoose an Elliptical Aperture, then set LA = WA = desired circle radius.
Glass and Mirror surfaces may be flat, spherical, conical, aspherical, cylindrical,
or axiconal. Spherical and conical surfaces are special cases of the generalaspheric surface. An aspheric surface is characterized by a radius of curvature Rc,
a conic constant k, twenty one aspheric coefficients A1, A2, A3, …, A20, A21, andradii of aperture Rmin, Rmax. The following equation describes asphericalsurfaces:h=r2| Rc | +√ Rc2 − (1 + k) r221+ Σ An r n.
n=1

Here h is the sag of the surface as measured from a plane tangent to the surface
at its vertex (see Fig. 74). Within the tangent plane, the radial distance from thevertex is denoted by r. The radius of curvature Rc carries a sign that depends onwhether the sag is away from the incoming beam (positive sign) or towards it(negative sign). The conic constant k can assume either a positive, a zero, or anegative value, depending on the shape of the surface. The remaining
parameters, A1, A2,…, A21, are the aspheric departure coefficients. In the absenceof the aspheric terms, the surface is a perfect sphere when k = 0, and a perfectparaboloid when k = −1. Note that the sign of Rc does not enter the above
equation, and that the sag, h, is always positive. In particular, the asphericcoefficients A1, A2,…, A21 are positive (negative) when they increase (decrease)the sag of the surface. The surface parameters, therefore, specify the magnitude
of sag, irrespective of whether the surface is convex or concave. It is only thesign of Rc that distinguishes convexity from concavity, namely, the side fromwhich the beam approaches a given surface.
Unlike surfaces of spherical symmetry, a cylindrical surface’s sag is onedimensional. In the plane perpendicular to the surface normal Vn at the vertex V,
consider a reference line drawn through V parallel to the cylinder-axis Va.

Guide to DIFFRACT

180

(a)
Incident beam
Rmax

Rmax
V

Vn

Rc > 0(b)
Incident beam
Rmax

Rmax
Vn

V

Rc < 0
(c)

WA

WA
LA

LA

Figure 74. In a Geometric-optical System, Glass and Mirror surfaces may have flat, spherical,
conical, aspherical, cylindrical, or axiconical shapes. The clear aperture of a given surface,when projected onto the tangent plane at its vertex, is an annulus of radii Rmin, Rmax, unless it
is a cylindrical surface, in which case the projection of the clear aperture is a 2 LA × 2 Rmax
rectangle from which a central 2 LA × 2 Rmin rectangle has been cut out. (a) If a curved surface
sags away from the incident beam its radius of curvature Rc will be positive (convex surface).
(b) A negative Rc is associated with a surface that sags toward the incident beam (concave
surface). (c) Elliptical and Rectangular Apertures of half-length LA and half-width WA.

Guide to DIFFRACT

181

Within this tangential plane, denote by r the distance of an arbitrary point P
from the reference line. The sag h of the cylindrical surface from the tangential
plane at P (which is a point in the tangential plane, at a distance r from thereference line) is given by the following (conic) function:h=r2| Rc | +√ Rc − (1 + k) r
2

2·

The radius of curvature Rc of the cylindrical surface is negative or positive,
depending on whether the cylindrical surface sags toward or away from theincoming beam. The surface is a circular cylinder when the conic constant k = 0,
and a parabolic cylinder when k = −1.
An axicon is a circularly symmetric cone (made of polished glass or metal, forexample). Like the other optical surfaces, the position and orientation of anaxicon are specified by its vertex V = (Vx, Vy, Vz) and by its surface normal at
the vertex, Vn, which is along the direction of the cone’s axis of symmetry. The
cone’s half-angle Θ carries its own sign: a positive Θ signifies a cone that sags
away from the incoming beam, whereas a cone with a negative Θ sags toward
the incoming beam.
The clear aperture of any Glass or Mirror surface (excepting cylindricalsurfaces), when projected onto the corresponding tangent plane at the vertex, isan annulus of radii Rmin, Rmax. (The clear aperture is a circular disk of radius
Rmax when Rmin is set to zero.) In the case of a cylindrical surface, the clear
aperture’s projection is a rectangle of length 2 LA (along cylinder-axis Va) andwidth 2Rmax (in the direction perpendicular to both Vn and Va), from which a
centered rectangle of dimensions 2 LA × 2Rmin has been removed.
Glass surfaces may be bare, antireflection coated (AR), or coated with adiffractive optical element (DOE). A Mirror may be a perfect reflector (PR), it

may be specified by a complex index of refraction, (n, k), or it may be coatedwith a diffractive optical element (DOE). Diffractive Optical Elements will be
discussed at length in Chapter 17.
For Glass surfaces, the user is prompted for the medium beyond the surface; thisis the medium into which the rays will enter upon crossing the surface. The
medium beyond a Glass surface may be free-space, homogeneous glass, or
gradient-index (GRIN) glass. If free-space is chosen as the medium, its refractive

Guide to DIFFRACT

182

index will be set to unity and no further questions will be asked. For
homogeneous glass the refractive index n of the medium needs to be specified.
GRIN media require a number of parameters for their specification.
In general, the gradient-index medium is a cylinder whose end-facets might bepolished flat, spherical, conical, aspherical, cylindrical, or axiconal. The cylinder
axis is defined by the coordinates (Ox, Oy, Oz) of its point of origin, which lieson the cylinder axis, and by the polar and azimuthal angles (θ, φ) of this axis
relative to the XYZ coordinates of incidence. The GRIN cylinder, having a radiusRo, absorbs any ray that reaches the boundary of the medium at this radius. The
index profile of GRIN media of Type 1 is given by:
126

n (r, z) = no {1 + Σ Qm (r/Ro)

2m

m=1

}{1 +mΣ= 1Wm (z/Zo) m }(Type 1)

Here no is the base index of refraction of the material, r is the radial distancefrom the cylinder axis, and z is the axial distance along the cylinder axis,
measured from the point of origin. Normalization of r by the cylinder radius Ro
renders the polynomial coefficients Qn dimensionless. Similarly, normalizationof z by the parameter Zo renders the coefficients Wn dimensionless. The scalefactor Zo must have the dimensions of length, but is otherwise arbitrary.
DIFFRACT allows the specification of a second type of GRIN medium whose

index profile is very similar to that of the Type 1 medium described above, withthe exception of n (r, z) on the left-hand side and no on the right-hand side of theequation being squared. The refractive index profile of GRIN media of Type 2 isthus given by:
126

n (r, z) = no {1 + Σ Qm (r/Ro)
2

2

m=1

2m

}{1 +mΣ= 1Wm (z/Zo) m }(Type 2)

All parameters of the above equation have exactly the same interpretation as
those given previously for Type 1 media.
16.4. Identifying an Optical System to the Program: When you select theGeometric-optical_System option from the main menu of DIFFRACT, theprogram will check to see if a system is already resident within its memory. Ifsuch a system does not exist you will be asked to specify one by defining theproperties of its various surfaces. This can be done in two different ways: You
may import the system parameters from an existing data file; alternatively, you
may enter them directly from the keyboard. In the first case, the file XXXX.SYS

Guide to DIFFRACT

183

must reside in your working directory and all you need to specify is the filename(XXXX). In the second case, you will be given the option to create a data file forlater use, then asked to specify the characteristics of your system.
Like all length parameters in DIFFRACT, the lengths specified within theGeometric-optical_System may have the Length_Units of cm, mm, um, nm, orthey may be dimensionless (that is, normalized by the vacuum wavelength λ0).
In general, the incidence medium has index of refraction NVIRON ≥ 1. The beam
emerging from the Geometric-optical_System, however, does not necessarilyfind itself in the same environment as the incidence medium: The refractive
index of the emergent environment is determined by the medium (i.e., free-spaceor homogeneous glass) within which the user has placed the Destination_Plane of
the system.
As you specify the various surfaces of your optical system, you may receive
messages to the effect that the just-entered parameter was unacceptable. This
happens, for example, when the aperture is larger than any possible cross-sectionof the specified surface, or when the refractive index of a glass is less than unity,
and so on. In such instances you will be prompted to re-enter the value of theparameter(s) in question. Once the system is fully specified, DIFFRACT performs
an analysis of the logical sequence of the various surfaces relative to each other,and proceeds to issue warning messages if, for example, a surface appears to be
unreachable by the rays, or if a surface normal appears to be pointing in thewrong direction, etc. The user should use these warning messages as a guidewhen checking the arrangement and relative orientation of various surfaces forlogical inconsistencies.
Once the rays enter a Glass surface, they will remain inside the glass medium
until they exit through another Glass surface (either into free-space or into a
different glass medium). Any mirrors or apertures that you specify while therays are inside the glass, will reflect and/or truncate the beam, but will not bring
the rays out of the glass.
You may place the Destination_Plane in free-space or inside a homogeneous
glass medium. In the latter case, the environment into which the beam will
emerge (upon leaving the Geometric-optical_System) will have the refractive
index of the last specified glass medium. Whereas the position and orientation of
the Destination_Plane define the emergent coordinate system, the refractive index
of the medium into which the Destination_Plane is placed determines the refractive
index NVIRON of the emergent environment. The one medium within which you

184

Guide to DIFFRACT

cannot place the Destination plane is the GRIN medium; an attempt to do so will
invoke the following message:
Warning: You have not yet identified an exit surface to allow the beam toleave the last GRIN medium it entered. (The Destination plane may be in
free-space or in a homogeneous glass, but not in a GRIN medium.) Please
specify an exit surface before selecting the Destination plane.
Once all the traced rays reach the Destination_Plane, the XYZ coordinate system
for the emergent rays will be automatically redefined. The Destination plane
becomes the new XY-plane, its vertex becoming the new origin, and its surfacenormal becoming the new optical-axis for the emergent rays. If the
Destination_Plane is placed inside a homogeneous glass of refractive index n > 1,
the environment of the emergent beam will automatically assume the samerefractive index, that is, NVIRON = n. Upon leaving the Geometricoptical_System, therefore, DIFFRACT expects all subsequent optical elements tobe immersed in the new (emergent) environment.
Note: Several examples, adopted from systems that have previously beenstudied with the ray_tracing option of DIFFRACT, are provided with your copy ofthe software. These files have been transferred by the Install utility to the<SYSTEMS> subdirectory of your working directory. Each file within thissubdirectory is a .SYS file that can be readily imported to DIFFRACT during
interactive sessions. Alternatively, these files may be incorporated into an
appropriate COMMAND.DAT file for use in non-interactive sessions. You may
use these examples as guides to setting up and specifying an optical system forthe ray_tracing features of DIFFRACT.

16.5. Geometric-optical System Inspection by Selective Ray_Tracing: Oncea geometric-optical system has been defined, you will be given the opportunity
to inspect it. During inspection you can select various rays in the incidence
space, and monitor the corresponding emergent rays. The information thus
provided will help you determine some important geometric-optical propertiesof the system.

Guide to DIFFRACT

185

Note: The System_Inspection option of DIFFRACT is available only in theinteractive mode. The requests issued by the user and the program’s responseto them will be recorded in the SESSION.DAT file, but the COMMAND.DAT file
will NOT keep track of these requests. Any attempts at inspecting ageometric-optical system during a non-interactive session either will beignored or will cause a failure of the program.
In general, one can fully inspect the system using two rays at a time; in
DIFFRACT these are called the reference ray and the test ray. You will beenabled to choose these rays in the space of incident rays, then monitor thecorresponding emergent rays for their exit locations and directions, as well as fortheir crossing point. By judiciously selecting the pair of incident rays, it ispossible to find important properties of the optical system such as the locationsof its focal and principal planes, its aperture stop, its entrance and exit pupils,
and so on. The example given in section 16.9 should help clarify these points.
Figure 75 shows the general case of a geometric-optical system, with thereference and test rays depicted within the incident and emergent coordinate
systems. The user identifies these rays in the space of incidence, and theprogram computes the corresponding rays in the emergent space. The
specification of a ray entails three sets of parameters: a point P on the ray, adirection vector R along the ray, and the state of polarization of the ray. P willbe identified by its Cartesian coordinates (Px, Py, Pz). R will be identified either
by its Cartesian components (Rx, Ry, Rz), or by its polar and azimuthal angles(θ, φ), measured respectively from the Z-axis and (counterclockwise) from theX-axis. The range of acceptable values of θ is (0, 90°), while that of φ is(0, 360°). In general, the ray is elliptically polarized and its ellipse ofpolarization, whose plane is perpendicular to the ray direction, may be specified
by two angles: ρ and η. The ellipticity η is the arctangent of the ratio of theminor axis of the ellipse to its major axis. Thus for linearly polarized light η = 0,
while for (left) right circular polarization η = (-) 45°; all other polarization states
have η in the interval [−45°, +45°]. To specify the other parameter, ρ, one mustfirst identify the plane of the ellipse. To this end, imagine rotating the ray
towards the Z-axis by reducing its polar angle θ to zero while keeping its
azimuthal angle φ constant. The polarization ellipse then lies in the XY-plane,and ρ will be the angle between the major axis of the ellipse and the X-axis. The
acceptable range of values for ρ is [0, 360°].

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Note: Incident rays are always specified in the coordinate system ofincidence, whereas emergent rays are referred to the coordinate systemthat is defined by the Destination plane.
Sometimes an emergent ray may be too weak for its state of polarization to be
determined accurately, in which case both ρ and η will be set to zero, and thefollowing message will be issued:
Y

Test ray
P2
X
Geometric-opticalsystem0

Z

P1
Referenceray

Coordinatesof incidenceY

Destination plane

Emergent
coordinate system
Emergent
test rayX
Z Emergent referenceray

Figure 75. The general case of a geometric-optical system, showing both the incidence andemergence coordinate systems. On the incidence side, the incoming beam is defined within
the XY-plane at Z = 0; on the emergence side, the Destination plane coincides with the XYplane at Z = 0. The two rays denoted as reference ray and test ray are chosen arbitrarily; theserays are used during system inspection and, when selected properly, they provide information
about the system’s focal points, principal planes, aperture stop, entrance and exit pupils, andso on. Within its own coordinate system, each ray is identified by a point P through which it
passes, and by a pair of angles (θ, φ) which specify its direction. As shown in the coordinate
system of incidence, θ is the polar angle of the ray with the Z-axis, while φ is its azimuthal
angle with the X-axis.

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Warning: Complex amplitude of the emergent ray is not sufficiently
strong to allow reliable computation of its state of polarization.
When you choose to inspect a geometric-optical system, you will be prompted todefine a reference ray. Usually (but not always) this will be the central ray that
goes through the system and establishes the system’s optical axis. Once a
reference ray has been defined, you will be asked to proceed with one of thefollowing actions:Define a test ray (TR)Define a new reference ray (NR)
Replace the reference ray with the latest test ray (RR)
Modify the Destination plane of the system being inspected (MD)
Quit inspecting the system (QT)
Any time that you define a ray, whether test or reference, you will receive thefollowing information:
The incident ray goes through the point (Px, Py, Pz) = (⋅⋅⋅,⋅⋅⋅,⋅⋅⋅). The
ray's propagation direction is given by (θ, φ) = (⋅⋅⋅, ⋅⋅⋅), or
equivalently, by the unit vector R = (⋅⋅⋅, ⋅⋅⋅, ⋅⋅⋅). Its state ofpolarization is given by (ρ, η) = (⋅⋅⋅, ⋅⋅⋅).
The ray emerges from the system at a point within the Destination plane,(Px, Py, Pz) = (⋅⋅⋅, ⋅⋅⋅, 0.00), propagating in the direction of (Rx, Ry, Rz) =(⋅⋅⋅, ⋅⋅⋅, ⋅⋅⋅). The emergent ray makes an angle θ = ⋅⋅⋅ with the Z-axis, its
azimuthal angle with X being φ = ⋅⋅⋅. The polarization state of the ray isgiven by the angles (ρ, η) = (⋅⋅⋅, ⋅⋅⋅).
However, if the ray happens to be blocked by one of the surfaces of the opticalsystem, instead of the above message you will receive the following:
The ray is blocked by surface # ⋅⋅⋅When, in addition to a reference ray, a test ray is also defined, the program willcompute and display the crossing point (or the point of closest encounter)

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between the two rays. This is done not only for the rays in the incidence space,but also for their corresponding rays in the emergent space. The two possible
messages issued in this instance are shown below.
In the space of incident (emergent) rays, test and reference rays are parallel.
In the space of incident (emergent) rays the test ray comes to within a distanceof ⋅⋅⋅ from the reference ray. The point of closest encounter on the test ray is(Px, Py, Pz) = (⋅⋅⋅,⋅⋅⋅,⋅⋅⋅); that on the reference ray is (⋅⋅⋅,⋅⋅⋅,⋅⋅⋅).
When a new test ray or a new reference ray is defined, the corresponding old
ray will be discarded. Sometimes it is helpful to keep a test ray in order to findits crossing points with a number of other rays which are yet to be defined. Insuch cases the user should exercise the option of replacing the reference raywith the latest test ray, then proceed to define a new test ray.
There are times when the user might like to change the Destination_Plane, andmonitor the emergent rays to see whether or not they behave in some desired
fashion; The option of modifying the Destination_Plane is provided for such
instances. Upon invoking this option, the user will be prompted for the vertex
coordinates and the surface-normal direction of the new Destination_Plane.
Subsequently, the following message will appear on the screen:
The Destination plane of the system has changed. The new vertex is at(Vx, Vy, Vz) = (⋅⋅⋅, ⋅⋅⋅, ⋅⋅⋅), and the new direction of surface normal is(θ, φ) = (⋅⋅⋅, ⋅⋅⋅). Note that this change does not affect the COMMAND.DAT
file, nor does it modify the data-file (if any) for the system. If the new
Destination plane turns out to be more suitable than the previous one, then,for future usage, you must change the COMMAND.DAT file and the system's
data-file accordingly.
Once the Destination_Plane has been modified, the old reference and test rays will
no longer be valid. The user will then be prompted for new rays.
The options to define new reference and test rays, as well as the option tomodify the Destination_Plane can be exercised as many times as desired, until all
the relevant information about a system have been obtained. The user must thenquit inspecting the system and pursue the other available options. One such
option is to discard the existing geometric-optical system altogether, and toGuide to DIFFRACT

189

define a new system. The other possibility is to return to inspecting the system inorder to find additional information about it. The third possibility is to trace
through the system all the rays that are present within the incident beam, and toconstruct the emergent wavefront at the Destination_Plane. It is with this last
option that the following sections are concerned. Of course, ray_tracing throughthe system requires that the incident wavefront be already converted to rays. If
this task has not been accomplished previously, DIFFRACT will return to themain menu and ask the user to invoke the RAYS option in preparation for
ray_tracing.
16.6. Incident and Emergent Rays during Ray_Tracing: Once the opticalsystem has been fully specified, selecting the ray_tracing option will cause theprogram to trace all the rays from the input plane to the Destination plane, and to
display some information about the rays as they cross the system. The displayed
information includes the magnitude of the brightest ray in the incident beam, themagnitude of the weakest incident ray that qualifies for tracing,† total number ofqualified rays within the incident beam, the number of emergent rays, and thegeneral neighborhood to which the emergent rays are confined. If no rays at all
make it to the Destination plane, you will receive the following message:
ABORT: No qualified rays cross the system.
On the other hand, if there are more rays arriving at the Destination plane thanthe program has the capacity to handle,‡ you will see the following:
Warning: The number of qualified rays that can cross the system is greaterthan the upper bound presently set by DIFFRACT. Certain parts of the beam
within the Destination plane will therefore be missing. You may consider
repeating the simulation later on with fewer rays, in order to avoid thisautomatic truncation by the program.

†

Incident rays must qualify for tracing before they can be admitted into the system.
DIFFRACT compares the magnitude of each ray with that of the brightest ray in the incident
beam, and disqualifies a ray if the ratio of the magnitudes is less than a certain cutoff value.
The cutoff is presently set to 10−4.
‡

Presently, the number of rays that are allowed to cross the system is limited to 11250.

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Guide to DIFFRACT

16.7. Searching for Curvature/Tilt Phase-factor: With the emergent rays
(within the Destination plane) at hand, the program will perform a search todetermine the center of curvature (Xc, Yc) and the radius of curvature Zc of theemergent beam. This is done by trying to fit the optical-path-length (OPL)
distribution of the emergent rays to the phase-front of a converging or diverging
spherical wave. The user provides best-estimate values for these parameters,
which will serve as the starting point for the search. These starting values neednot be very accurate, but they should not be too far off either. The sign of Zc, inparticular, is very important: for a converging beam the starting value of Zc
should be positive, whereas for a diverging beam it must be negative. Once theprogram obtains its best-estimate values, the results will be displayed, and theuser will be given the option to accept or reject them. If rejected, the process will
begin anew, at another user-defined starting point. At any point during thisprocess the user may abort the search by supplying his own (Xc, Yc, Zc) values.
Once the search results are accepted (or the user has supplied his own estimateof the center and radius of curvature), the program will enter its final stage,
where the emergent wavefront will be reconstructed from the set of rays thathave crossed the system. At this point, the following message (containing actual
numerical values for the parameters) will appear:
The emergent beam is converging towards (diverging away from) the point(Xc, Yc, Zc), with an r.m.s. wavefront error of ⋅⋅⋅ . If the beam's crosssectional area is sufficiently small, it may be considered to be a collimatedbeam, traveling along the direction (θ, φ) = (⋅⋅⋅, ⋅⋅⋅).
16.8. Phase-factor Stripping and Wavefront Reconstruction: The curvatureof the emergent beam, if significant, must be stripped from the wavefront andcarried on the side. On the other hand, if the curvature is small enough for thecorresponding phase-factor to be adequately sampled by the mesh, it might beleft alone. Sometimes, the radius of curvature is so large (compared to the beam
diameter) that it is more appropriate to consider the corresponding phase a tilt
rather than a curvature. This happens, for instance, when the optical system
under consideration consists not of lenses and curved mirrors, but of flat mirrors
and prisms. In this case the system has no focusing power, but it can bend andredirect the beam, endowing it with a tilt phase-factor. In such cases, if the tilt
angle is small, the corresponding phase-factor should be left on the beam. If thetilt is large, however, the simulation must be repeated after adjusting theGuide to DIFFRACT

191

orientation of the Destination plane in order to bring the direction of propagationof the emergent beam into alignment with the Z-axis.
After the user has given instructions on whether or not to strip the curvature/tilt
phase-factor from the emergent wavefront, DIFFRACT proceeds to incorporate
the optical-path-length (OPL) of the rays into the mesh as a phase-factor. (Thecurvature will be removed if so requested.) Subsequently, the emergent
wavefront will be fully reconstructed from the set of polarized rays within the
Destination plane. (The user will also be prompted to modify the meshparameters NMAX, NMAY, LMAX, LMAY if the default values are deemed
unsatisfactory.) Ray_tracing will end with the following message:
Construction of the emergent wavefront within the Destination
plane is now complete.
(Total energy flux = ··· )Needless to say, the reconstructed wavefront can be treated like any otherwavefront in DIFFRACT. It can be viewed using the PLOT option or analyzed
using the WAVE and MEASURE options, it can be propagated within theenvironment, it can be sent through various optical elements, and so on.
16.9. Examining a Geometric-optical System: This section’s first example
concerns the system of Fig. 76(a), analyzed with the various tools provided by
the Geometric-optical_System option of DIFFRACT. The file TEST.SYS
containing the specifications of this system is shown in Fig. 76(b). The
wavelength of the incident beam, specified in the file as λo = 632.8 nm, is the
wavelength of a red HeNe laser; all input/output lengths are in millimeters.
Destination plane has been chosen such that a ray entering along the optical axis
will exit at the vertex of the Destination plane and perpendicular to it.§We define a reference ray (in incidence space) at the point P = (Px, Py, Pz) =(0, 0, 0) in the direction (θ, φ) = (0, 0); this, of course, is the optical axis of thesystem. We then define a test ray parallel to the optical axis and at a fairly shortdistance from it, so that it would qualify as a paraxial ray.

§This was achieved using the Inspect option (within the Geometric-optical_System option) of
the program. By observing the direction of the emergent ray, and by modifying the parametersof the Destination plane accordingly, we arrived at the location and orientation that made theemergent ray orthogonal to the Destination plane at its vertex.

Guide to DIFFRACT

192

Bi-convex spherical lens(n = 1.5)

Y
X

Flat mirror(n, k) = (2, 7)

Z

Aperture(RA = 4 mm)
Bi-convex spherical lens(n = 1.8)

Aperture(RA = 2.5 mm)

Prism (n = 1.5)
V

Destination plane
Figure 76(a). Diagram of an optical system consisting of two ordinary spherical lenses, twocircular apertures, a flat aluminized mirror, and a glass prism. In a system such as this, theincident beam is specified by its distribution in the XY-plane at Z = 0, whereas the
distribution of the emergent beam is to be found in the Destination plane. The vertex(Vx, Vy, Vz) of the Destination plane, and its surface-normal direction (θ, φ) are both definedwith respect to the XYZ coordinates of incidence. To obtain the emergent coordinate system,
one must translate the XYZ system of incidence until its point of origin coincides with thevertex of the Destination plane. One must then perform a solid rotation of the XYZ system insuch a way as to move θ towards zero while keeping φ constant. The Z-axis thus becomesaligned with the Destination plane’s surface-normal, and the X- and Y-axes, in their new
orientation, become the X- and Y-axes of the emergent system.

Guide to DIFFRACT@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File TEST.SYS @@#$@@ Date: 06/22/2005
Time: 23:07 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$ Notes: This TEST system consists of two ordinary spherical lenses, two #$circular apertures, a flat aluminized mirror, and a glass prism. #$The Destination plane is chosen to be centered on and orthogonal #$to the optical axis of the system on the emergent beam side.
#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#
Vacuum wavelength: 632.800E-6(Length_Units: mm) ##Name of the system: TEST##**** Surface #01:
Type: Glass#
Shape: S#
Vertex (Vx,Vy,Vz):
0.0000000
0.0000000
20.000000#
Normal (Theta,Phi):
0.0000000
0.0000000#
Radius of curvature (RC):10.000000#
Aperture's Rmin,Rmax:
0.0000000
5.0000000#
Coating (AR/DOE/BARE): BARE#
Medium beyond surface: HOMO#
Refractive index (n):1.5000000#
Next surface (Hit,Miss):20##**** Surface #02:
Type: Glass#
Shape: S#
Vertex (Vx,Vy,Vz):
0.0000000
0.0000000
23.000000#
Normal (Theta,Phi):
0.0000000
0.0000000#
Radius of curvature (RC):-30.000000#
Aperture's Rmin,Rmax:
0.0000000
5.0000000#
Coating (AR/DOE/BARE): BARE#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
30##**** Surface #03:
Type: Aperture#
Vertex (Vx,Vy,Vz):
0.0000000
0.0000000
24.000000#
Normal (Theta,Phi):
0.0000000
0.0000000#
Aperture shape: Ellipse#
Major-axis Phi,LA,WA:
0.0000000
4.0000000
4.0000000#
Next surface (Hit,Miss):
40##**** Surface #04:
Type: Mirror#
Shape: F#
Vertex (Vx,Vy,Vz):
0.0000000
0.0000000
31.000000#
Normal (Theta,Phi):
45.000000
90.000000#
Aperture's Rmin,Rmax:
0.0000000
8.0000000#
Coating (PR/DOE/BARE): BARE#
Complex index (n,k):
2.0000000
7.0000000#
Next surface (Hit,Miss):
50##**** Surface #05:
Type: Glass#
Shape: S#
Vertex (Vx,Vy,Vz):.00000000-14.000000
31.000000#
Normal (Theta,Phi):
90.000000
270.00000#
Radius of curvature (RC):15.000000#
Aperture's Rmin,Rmax:
0.0000000
2.5000000#
Coating (AR/DOE/BARE): BARE#
Medium beyond surface: HOMO#
Refractive index (n):1.8000000#
Next surface (Hit,Miss):
60##**** Surface #06:
Type: Glass#
Shape: S#

193

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194
Vertex (Vx,Vy,Vz):
0.0000000
Normal (Theta,Phi):
90.000000
Radius of curvature (RC):-10.000000
Aperture's Rmin,Rmax:
0.0000000
Coating (AR/DOE/BARE): BARE
Medium beyond surface: FREE
Next surface (Hit,Miss):7-16.000000
270.00000

31.000000###
2.5000000
#
#
#
0
#
#
**** Surface #07:
Type: Aperture#
Vertex (Vx,Vy,Vz):
0.0000000-17.000000
31.000000#
Normal (Theta,Phi):
90.000000
270.00000#
Aperture shape: Ellipse#
Major-axis Phi,LA,WA:
0.0000000
2.5000000
2.5000000#
Next surface (Hit,Miss):
80##**** Surface #08:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.0000000-20.000000
31.000000#
Normal (Theta,Phi):
90.000000
270.00000#
Aperture's Rmin,Rmax:
0.0000000
10.000000#
Coating (AR/DOE/BARE): BARE#
Medium beyond surface: HOMO#
Refractive index (n):1.5000000#
Next surface (Hit,Miss):
90##**** Surface #09:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.0000000-25.000000
31.000000#
Normal (Theta,Phi):
70.000000
270.00000#
Aperture's Rmin,Rmax:
0.0000000
10.000000#
Coating (AR/DOE/BARE): BARE#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
100##**** Surface #10:
Type: Destination#
Vertex (Vx,Vy,Vz):
0.0000000-30.000000
30.040230#
Normal (Theta,Phi):
100.87000
270.00000##@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Figure 76(b). The file TEST.SYS containing various parameters of the optical system ofFig. 76(a). The position and orientation of the Destination plane are chosen such that a rayentering the system along the Z-axis would exit at the vertex of the Destination plane andperpendicular to it.

This incident test ray originates at P = (0.1, 0, 0) along the direction(θ, φ) = (0, 0). The two emergent rays cross each other at a distance of 50 mm
from the Destination plane, which is, therefore, the location of the back focal
point of the system. Since the prism induces a certain amount of astigmatism inthe beam, if we change the origin of the test ray to P = (0, 0.1, 0), we find thecorresponding focal point at 41 mm from the Destination plane. The astigmatic
separation between the two (back) foci is thus around 9 mm.
The latter test ray, when emerging from the Destination plane, has a height of
0.041 mm. Since the height of this ray at the second principal plane must equal
its height upon entering the system (namely, 0.1 mm), we conclude that theGuide to DIFFRACT

195

second principal plane is -59 mm away from the Destination plane.** This yields
a back focal length of 100 mm for rays that enter the system parallel to theoptical axis within the YZ-plane. (Similarly, the back focal length for rays
entering within the XZ-plane is found to be ≅ 109 mm.)To determine the front focal point of the system, we assume for the incident test
ray a fixed direction, (θ, φ) = (0.1°, 0°), and find by trial and error the locationof an on-axis point P = (0, 0, Pz) through which this incident ray must pass inorder for the emergent ray to become parallel to the optical axis.
This procedure yields Pz = -223 mm for the location of the front focal point.†† To
find the first principal plane, we note that the emergent ray has a height of
0.183 mm at the Destination plane. This must also be the height of the incident
ray at the first principal plane and, therefore, the first principal plane is at
Z = -118 mm. The front focal length is thus around 105 mm.
The aperture stop may be found by placing an incident test ray at P = (0, 0, 0),fixing its azimuthal angle φ at 0°, and increasing its polar angle θ from 0°, until
one of the surfaces of the system blocks the ray. The surface that blocks the rayin this instance turns out to be surface #3, i.e., the circular aperture immediately
behind the first lens. The largest value of θ at which the ray passes through thesystem is 11.34°. Surface #3 is, therefore, the aperture stop of the system.
As for the entrance and exit pupils, recall that these are the images of theaperture stop produced by those parts of the system located, respectively, before
and after the stop. For the next set of calculations, therefore, we assign the roleof reference ray to the marginal test ray found above. Thus the ray originating at
P = (0, 0, 0) and propagating along (θ, φ) = (11.34°, 0°), which just clears theaperture stop, becomes the new reference ray. We then define a new test ray,originating at a slightly different point P = (0.1, 0, 0) and having the sameazimuthal angle φ = 0°. Again, we increase the polar angle θ of this new test ray
from 0°, until it is blocked by the aperture stop; the marginal angle this time**

The concept of principal planes may not be applicable to the system under consideration
here. Our intention, however, is merely to describe the rules and procedures for finding suchplanes, whether or not they are technically admissible.
††

Had we fixed the direction of the incident test ray at (θ, φ) = (0.1°, 90°), or at any othervalue of φ for that matter, we would have found exactly the same answer; the reason beingthat the emergent rays are now parallel to the (emergent) optical axis, and the prism cannot
affect their parallelism in any way; in other words, there is no astigmatism here.

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Guide to DIFFRACT

turns out to be θ = 11.11°. Since these two rays must have met at the rim of theaperture stop, their crossing point in the incident (emergent) space must be on
the rim of the entrance (exit) pupil. The entrance pupil is thus found to be at
Z = 24 mm (in the incident space), having a radius of 4.8 mm, while the exit
pupil is at Z = -3.6 mm (in the emergent space), having a radius of 1.54 mm.‡‡
Having thus analyzed the system of Fig. 76 and obtained its importantcharacteristics, we now simulate the propagation of a collimated beam of lightfrom a red HeNe laser through this system, and determine the various
distributions of the emergent beam. The incident beam is assumed to becollimated and uniform, having a 5 mm diameter circular aperture. In theincidence space the beam is directed along the optical axis of the system, and iscentered at the origin of the XY-plane. Frames (a) and (b) of Fig. 77 show the
emergent intensity and phase profiles at the Destination plane. The ellipticalcross-section of the beam is due to the bending action of the prism in onedimension only. Since the emergent curvature phase-factor has already been
removed, the phase profile shown in the figure is caused by residual aberrations,
namely, spherical aberration and astigmatism, balanced by an optimum amountof defocus.
When the emergent beam is allowed to propagate a distance of 31.16 mm§§ tothe plane of least confusion, the intensity profile shown in Fig. 77(c) is obtained.
[Frame (d) of the figure is just a logarithmic plot of the same distribution as inframe (c).] This diamond-like pattern is typical of an astigmatic distribution
within the plane of least confusion.

‡‡

In fact, due to the presence of prism in the system, the exit pupil is not exactly a circle, butrather resembles an ellipse. The aforementioned radius of 1.54 mm, therefore, is only the halfdiameter of the ellipse in the direction of the X-axis.
§§

This is the radius of curvature corresponding to the curvature phase-factor removed fromthe emergent beam during wavefront reconstruction. Since DIFFRACT computes thecurvature by minimizing the residual r.m.s. wavefront error, the resulting radius of curvature
should be the distance to the plane of least confusion.

Guide to DIFFRACT

197

2000 a

b

y/λ0

−2000
−2000

x/λ0

2000

−2000

x/λ0

2000

x/λ0

300

d

300 c

y/λ0

−300

−300

x/λ0

300

−300

Figure 77. Emergent light intensity/phase distributions from the system of Fig. 76, when acollimated, uniform beam having a 5 mm diameter circular aperture is incident along theoptical axis at Z = 0.
(a) Pattern of intensity at the Destination plane.
(b) Phase distribution at the Destination plane.
(c) Intensity distribution in the plane of least confusion, located 35.16 mm after the
Destination plane.
(d) Distribution of log (intensity) in the plane of least confusion.

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198

Example. Figure 78 shows a collimator lens with two aspheric surfaces, thesecond of which has both a convex and a concave part. These two parts of thesecond surface must be treated as separate surfaces in DIFFRACT, as seen in theprintout of the COLL.SYS file in Fig. 79. The COMMAND.DAT file used for
simulating the system of Fig. 78 (from the laser diode to the Destination planelocated 1.0 mm to the right of the last vertex) is shown in Fig. 80.

R2
R1

Figure 78. This collimator for a diode laser beam consists of two aspherical surfaces. The
parameters of the second surface are such that they produce both a convex and a concave
segment. The second surface must therefore be treated as consisting of two separate surfaces
with identical parameters except for the inner and outer radii Rmin, Rmax. For the central,
convex part Rmin = 0, Rmax = R1, while for the concave part Rmin = R1, Rmax = R2. An incoming
ray, after going through the front surface, will attempt to go through the convex part of therear surface. If it succeeds, it will proceed to the Destination plane; if not, it will attempt to go
through the concave part before heading for the Destination plane.

The laser diode beam is converted to rays just before entering the first surface ofthe lens. These rays are traced through the collimator and collected at the
Destination plane, where they are used to generate the emergent wavefront. For
the X-, Y-, and Z-components of the emergent E-field, computed plots ofintensity and phase are shown in Fig. 81. In this simulation the curvature of theemergent beam, being fairly small, did not have to be stripped away. The
elongated profile of the laser diode radiation is evident in the intensity plot for
the X-polarized component of the emergent beam (top row, leftmost panel). The
imperfect correction for spherical aberration, compounded perhaps with someresidual defocus, results in the ~ 60° phase variation across the aperture seen in
the Ex phase plot (bottom row, leftmost panel).

Guide to DIFFRACT

Figure 79. Printout of COLL.SYS corresponding to the collimator depicted in Fig. 78.
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COLL.SYS @@#$@@ Date: 06/22/2005
Time: 23:23 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$ Notes: Collimator lens with two aspheric surfaces, the second surface#$
having both convex and concave segments, thus treated separately #$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#
Vacuum wavelength: 660.000E-6(Length_Units: mm) ##Name of the system: COLL##**** Surface #01:
Type: Glass#
Shape: A#
Vertex (Vx,Vy,Vz): 0.00000000 0.00000000 0.00000000#
Normal (Theta,Phi): 0.00000000 0.00000000#
RC,K: 15.6720000 14.9336000#
A1,A2,A3: 0.00000000 0.00000000 0.00000000#
A4,A5,A6: 0.00570806 0.00000000 -0.00011180#
A7,A8,A9: 0.00000000 0.00000000 0.00000000#
A10,A11,A12: 0.00000000 0.00000000 0.00000000#
A13,A14,A15: 0.00000000 0.00000000 0.00000000#
A16,A17,A18: 0.00000000 0.00000000 0.00000000#
A19,A20,A21: 0.00000000 0.00000000 0.00000000#
Aperture's Rmin,Rmax: 0.00000000 2.50000000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: HOMO#
Refractive index (n): 1.5270100#
Next surface (Hit,Miss): 20##**** Surface #02:
Type: Glass#
Shape: A#
Vertex (Vx,Vy,Vz): 0.00000000 0.00000000 5.00000000#
Normal (Theta,Phi): 0.00000000 0.00000000#
RC,K: -23.5814000 14.3342000#
A1,A2,A3: 0.00000000 0.00000000 0.00000000#
A4,A5,A6: -0.00484200 0.00000000 -0.00027133#
A7,A8,A9: 0.00000000 -3.98128E-5 0.00000000#
A10,A11,A12: 3.19340E-6 0.00000000 -1.03122E-6#
A13,A14,A15: 0.00000000 0.00000000 0.00000000#
A16,A17,A18: 0.00000000 0.00000000 0.00000000#
A19,A20,A21: 0.00000000 0.00000000 0.00000000#
Aperture's Rmin,Rmax: 0.00000000 1.37000000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: FREE#
Next surface (Hit,Miss): 43##**** Surface #03:
Type: Glass#
Shape: A#
Vertex (Vx,Vy,Vz): 0.00000000 0.00000000 5.00000000#
Normal (Theta,Phi): 0.00000000 0.00000000#
RC,K: -23.5814000 14.3342000#
A1,A2,A3: 0.00000000 0.00000000 0.00000000#
A4,A5,A6: -0.00484200 0.00000000 -0.00027133#
A7,A8,A9: 0.00000000 -3.98128E-5 0.00000000#
A10,A11,A12: 3.19340E-6 0.00000000 -1.03122E-6#
A13,A14,A15: 0.00000000 0.00000000 0.00000000#
A16,A17,A18: 0.00000000 0.00000000 0.00000000#
A19,A20,A21: 0.00000000 0.00000000 0.00000000#
Aperture's Rmin,Rmax: 1.37000000 2.30000000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: FREE#
Next surface (Hit,Miss): 4
0
#

199

Guide to DIFFRACT

200

#
#
0.00000000 6.00000000#
0.00000000##!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#**** Surface #04:
Type: Destination
Vertex (Vx,Vy,Vz): 0.00000000
Normal (Theta,Phi): 0.00000000

Figure 80. Command.dat file corresponding to the system depicted in Fig. 78.
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/22/2005
Time: 17:06 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #TYPE: UB/SG/GG/LG/HG/LD: LD#
BCX,BCY,Power(mW): 0.0000000
0.0000000
1.0000000#
PHI,PSI,DELTA: 9.2000000
19.000000
1.5000000#
Orientation Angle THETA: 0.0000000#
Distance from cavity Z0: 26212.000#
NMAX,NMAY: 512
512#
LMAX,LMAY: 40000.000
40000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Ray conversion: RAYS#
Multiply curvature (Y/N): N#$________________________________________________________________________ #Guide to DIFFRACT
Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Rays#
Logarithmic SCALE: 4.0000000#
Xmin,Xmax: -8500.000
8500.000#
Ymin,Ymax: -8500.000
8500.000#
Color or Gray-scale: G#
Z-component: N#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Define a new system: NSYS#
Name of the system: COLL#**** Surface #01:
Type: Glass#
Shape: A#
Vertex (Vx,Vy,Vz): 0.00000000 0.00000000 0.00000000#
Normal (Theta,Phi): 0.00000000 0.00000000#
RC,K: 15.6720000 14.9336000#
A1,A2,A3: 0.00000000 0.00000000 0.00000000#
A4,A5,A6: 0.00570806 0.00000000 -0.00011180#
A7,A8,A9: 0.00000000 0.00000000 0.00000000#
A10,A11,A12: 0.00000000 0.00000000 0.00000000#
A13,A14,A15: 0.00000000 0.00000000 0.00000000#
A16,A17,A18: 0.00000000 0.00000000 0.00000000#
A19,A20,A21: 0.00000000 0.00000000 0.00000000#
Aperture's Rmin,Rmax: 0.00000000 2.50000000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: HOMO#
Refractive index (n): 1.5270100#
Next surface (Hit,Miss): 20#**** Surface #02:
Type: Glass#
Shape: A#
Vertex (Vx,Vy,Vz): 0.00000000 0.00000000 5.00000000#
Normal (Theta,Phi): 0.00000000 0.00000000#
RC,K: -23.5814000 14.3342000#
A1,A2,A3: 0.00000000 0.00000000 0.00000000#
A4,A5,A6: -0.00484200 0.00000000 -0.00027133#
A7,A8,A9: 0.00000000 -3.98128E-5 0.00000000#
A10,A11,A12: 3.19340E-6 0.00000000 -1.03122E-6#
A13,A14,A15: 0.00000000 0.00000000 0.00000000#
A16,A17,A18: 0.00000000 0.00000000 0.00000000#
A19,A20,A21: 0.00000000 0.00000000 0.00000000#
Aperture's Rmin,Rmax: 0.00000000 1.37000000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: FREE#
Next surface (Hit,Miss): 43#**** Surface #03:
Type: Glass#
Shape: A#
Vertex (Vx,Vy,Vz): 0.00000000 0.00000000 5.00000000#
Normal (Theta,Phi): 0.00000000 0.00000000#
RC,K: -23.5814000 14.3342000#
A1,A2,A3: 0.00000000 0.00000000 0.00000000#
A4,A5,A6: -0.00484200 0.00000000 -0.00027133#
A7,A8,A9: 0.00000000 -3.98128E-5 0.00000000#
A10,A11,A12: 3.19340E-6 0.00000000 -1.03122E-6#
A13,A14,A15: 0.00000000 0.00000000 0.00000000#
A16,A17,A18: 0.00000000 0.00000000 0.00000000#
A19,A20,A21: 0.00000000 0.00000000 0.00000000#
Aperture's Rmin,Rmax: 1.37000000 2.30000000
#

201

202

Guide to DIFFRACT
Coating (AR/DOE/BARE): AR#
Medium beyond surface: FREE#
Next surface (Hit,Miss): 40#**** Surface #04:
Type: Destination#
Vertex (Vx,Vy,Vz): 0.00000000 0.00000000 6.00000000#
Normal (Theta,Phi): 0.00000000 0.00000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units:) #Trace all incident rays: TRAY#
Initial guess (Xc,Yc,Zc): 0.00000000 0.00000000 0.00000000#
Accept or Search: S#
Strip phase-factor (Y/N): N#
NMAX,NMAY: 512
512#
LMAX,LMAY: 35000.000 35000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE: 4.0000000#
Xmin,Xmax: -3400.0000 3400.0000#
Ymin,Ymax: -3400.0000 3400.0000#
Color or Gray-scale: G#
Z-component: N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Phase#
Logarithmic SCALE: 4.0000000#
Xmin,Xmax: -3400.0000 3400.0000#
Ymin,Ymax: -3400.0000 3400.0000#
Color or Gray-scale: G#
Z-component: N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#

Guide to DIFFRACT

203

Figure 81. Plots of intensity (top) and phase (bottom) at the Destination plane in the system
of Fig. 78. From left to right, the three panels represent the E-field components along the X-,
Y-, and Z-axes.

16.10. Building XXXX.SYS Files from a Template: One can create XXXX.SYS
files for simulating arbitrarily complex geometric-optical systems in conjunction
with the ray-tracing feature of DIFFRACT. In the interactive mode, you will begiven the option of creating a XXXX.SYS file for the geometric-optical systems
under consideration. Once this text file is created (within your working
subdirectory) you may modify it by adding or removing various elements, or bychanging the parameter values of the specified elements. An alternative
approach to creating XXXX.SYS files involves the use of GEOP.SYS template
located in the <TEMPLATES> subdirectory of DIFFRACT’s home directory. Startby copying the GEOP.SYS file to a working subdirectory, then open this text fileand follow the instructions given within the file itself.

204

Guide to DIFFRACT

17. Diffractive Optical Elements
Within a Geometric_Optical_System (GEOP), Glass surfaces may be Bare or
Antireflection coated (AR), or they may act as a Diffractive Optical Element (DOE).
Mirrors may be specified either as perfect reflectors (PR), or in terms of theircomplex refractive index (n, k); A Mirror surface may also act as a DOE. In
general, any surface within the GEOP option, whether Flat, Spherical, Conical,
Aspherical, Cylindrical, or Axiconal, may be specified as coated with a DOE.
The incident rays enter a Bare Glass surface in accordance with Snell's law ofrefraction, and they are attenuated by the Fresnel transmission coefficients.
These coefficients will be set to unity when the surface is Antireflection coated.
Similarly, for a Bare Mirror surface whose complex refractive index is specified
as (n, k), the program assigns the corresponding Fresnel reflection coefficients toeach incident ray. These coefficients are set to unity when the Mirror is a perfectreflector. DIFFRACT™ fully accounts for the polarization state of every ray that
undergoes refraction or reflection, by determining the p- and s-components of
the ray and multiplying them with the proper Fresnel transmission or reflection
coefficient.
When a Glass or Mirror surface is specified as coated with a DOE, the user is
prompted for the DOE's construction wavelength λc and the desired diffraction
order m. If the incident beam's wavelength λo coincides with λc, the +1st
diffracted order will have 100% efficiency and all other orders will vanish.
When λo ≠ λc, the various diffracted orders will coexist, albeit with different
efficiencies as given by the scalar theory of diffraction gratings (Appendix K).
For Flat, Spherical, Conical, Aspherical, Axiconal surfaces, the phase polynomial
specifying the DOE (at the construction wavelength λc) is defined as follows:
21

F(r) = 2π Σ bn r n.

(1)

n=1

Here bn are the (user-defined) polynomial coefficients, and r is the radial
distance from the Glass/Mirror’s vertex (measured in the tangential plane of thesurface through the vertex). For Cylindrical surfaces the phase polynomial is also
given by Eq. (1), but r is the vertical distance above/below the plane containing
the surface normal Vn and the cylinder axis. The phase function of a DOE coatedon a Cylindrical surface thus varies in the direction of the sag but remains
constant along the cylinder axis. For Flat surfaces the user is given the option of
specifying the phase profile as the circularly symmetric function F(r) of Eq. (1),or as the N th order xy polynomial defined below:

Guide to DIFFRACT

205
N

F(x, y) = 2π ΣN−m

Σ bmn x m y n.

(2)

m=0 n=0

As shown in Fig. 82, x and y in Eq. (2) are the Cartesian coordinates in the X'Y'plane of the Flat Glass/Mirror surface. The origin of X'Y' is the vertex V of thecorresponding surface, but the user must specify the orientation of X' within the
XYZ coordinates of incidence. Typically, the projection of the DOE’s X'-axis in
the XY-plane of incidence has azimuth φ, which angle may be used to identify
the DOE's coordinates. If, however, the Glass/Mirror’s surface-normal happens
to be orthogonal (or nearly so) to the Z-axis of incidence, then the polar angle θ(i.e., the angle between X' and Z) should be used to specify the DOE's orientation.
In the DOE polynomial F(x, y), the sum over m and n covers all non-negative
integers that land n + m between 1 and 6.
Y
X

φ

Y′
X′

θ

Z

Vn

Fig. 82. A Flat Glass/Mirror surface having surface normal Vn is coated with a Diffractive
Optical Element (DOE). When the DOE’s phase profile is given by the circularly symmetricpolynomial F (r ), the orientation of X' Y' within the XYZ coordinates of incidence becomes
irrelevant. On the other hand, if the phase profile is defined by the F(x, y) polynomial of
Eq. (2), then the orientation of X' must be specified by its polar and azimuthal angles (θ, φ ).
In general, one need not specify both θ and φ, as the knowledge of Vn renders one of the twoorientation angles redundant.

When the user specifies a Flat Glass or Mirror surface as coated with a DOE,
the program will inquire the orientation angle of the X'-axis (either θ or φ ,depending on the direction of the surface-normal Vn). If, in response to thisquestion, the user enters –1, the program will assume a circularly symmetricphase profile, and proceeds to prompt the user for the radial polynomial
coefficients bn; see Eq.(1). If, on the other hand, the user enters a legitimate
value for the orientation angle, the program will prompt for the xy-polynomial
coefficients bmn; see Eq.(2).

206

Guide to DIFFRACT

17.1. Transmissive DOE on flat substrate. Figure 83 shows the diagram of asystem consisting of a flat DOE on a 0.5mm-thick glass substrate, illuminated by
a Gaussian beam. The parameters of the DOE are listed in the caption, anddetails of the system appear in the accompanying COMMAND.DAT file. Figure
84 shows the transmitted intensity and phase profiles at the Destination_Plane.

Gaussian
Beam

9.5mm
0.5mm

Destination
Plane

Figure 83. A Gaussian beam (λo = 0.65µm, e−1 radius Ro = 2.0 mm, diameter D = 3.0 mm) is
normally incident on a DOE that coats the front facet of a 0.5mm-thick glass substrate(nsub = 1.50). The incident beam, whose intensity profile is shown on the left-hand side, islinearly polarized along the x-axis. Since the DOE’s construction wavelength λc is the same asλo, the only emergent diffracted beam is the +1st order. The phase profile of the DOE (whose
x-axis is at 30° to the coordinate system’s x-axis) is F(x, y) = 0.93 x −2.4 x2 + 1.75 xy + 2.9 y2;both x and y are in millimeters. The rear facet of the substrate is bare (i.e., uncoated), and the
Destination Plane is a distance of ∆ z = 9.5 mm behind this rear facet.

Guide to DIFFRACT

207

Figure 84. Plots of intensity (top) and phase (bottom) at the Destination Plane of the system
of Fig. 83. Only the x- and z-components of polarization are shown (left and right,
respectively), as the emergent y-component is too weak to make a significant contribution to
the system’s output. Note that the intensity of the z-component is five orders-of-magnitude
below that of the x-component.
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/12/2005
Time: 13:27 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$MULTIPLY#??,ITEM?....BY: ??
#$MAKE#??,ITEM?~#??,ITEM?:#$JUMP#??_IF#??,ITEM?.GT.: ??
#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

208

Guide to DIFFRACT$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: mm) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.000000
0.000000
1.000000#
1/e radii Rmin,Rmax:
2.000000
2.000000#
RA,Theta:
1.500000
0.000000#
Aberrations: None#
Polarization RHO,ETA:
0.000000
0.000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
10.00000
10.00000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Ray conversion: RAYS#
Multiply curvature (Y N): N#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Ray distribution#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-1.600000
1.600000#
Ymin,Ymax:-1.600000
1.600000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Define a new system: NSYS#
Name of the system: DOE1#**** Surface #01:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
0.000000#
Normal (Theta,Phi):
0.000000
0.000000#
Aperture's Rmin,Rmax:
0.000000
1.500000#
Coating (AR/DOE/BARE): DOE#
Radial(-1); X-axis Phi:
30.00000#
ConstructionLambda,Order:
0.650000E-03
1
#
B1_0,
B0_1,
B2_0:
0.930000
0.000000-2.400000
#
B1_1,
B0_2,
B3_0:
1.750000
2.900000
0.000000
#
B2_1,
B1_2,
B0_3:
0.000000
0.000000
0.000000
#
B4_0,
B3_1,
B2_2:
0.000000
0.000000
0.000000
#
B1_3,
B0_4,
B5_0:
0.000000
0.000000
0.000000
#
B4_1,
B3_2,
B2_3:
0.000000
0.000000
0.000000
#
B1_4,
B0_5,
B6_0:
0.000000
0.000000
0.000000
#
B5_1,
B4_2,
B3_3:
0.000000
0.000000
0.000000
#
B2_4,
B1_5,
B0_6:
0.000000
0.000000
0.000000#
Medium beyond surface: HOMO#
Refractive index (n):1.500000#
Next surface (Hit,Miss):
20#Guide to DIFFRACT**** Surface #02:

Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
0.500000#
Normal (Theta,Phi):
0.000000
0.000000#
Aperture's Rmin,Rmax:
0.000000
2.500000#
Coating (AR/DOE/BARE): BARE#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
30#**** Surface #03:
Type: Destination#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
10.00000#
Normal (Theta,Phi):
0.000000
0.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Trace all incident rays: TRAY#
Initial guess (Xc,Yc,Zc):
0.000000
0.000000
0.0000000#
Accept or Search: Search#
Strip phase-factor (Y/N): N#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
5.000000
5.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-1.600000
1.600000#
Ymin,Ymax:-1.600000
1.600000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Phase#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-1.600000
1.600000#
Ymin,Ymax:-1.600000
1.600000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Command.dat file corresponding to the system of Fig. 83

209

210

Guide to DIFFRACT

17.2. Reflective DOE on flat substrate. Figure 85 shows the diagram of asystem consisting of a flat DOE on the rear facet of a glass prism, illuminated by
a Gaussian beam. The parameters of the DOE are listed in the caption, anddetails of the system appear in the accompanying COMMAND.DAT file. Figure
86 shows the reflected intensity and phase profiles at the Destination_Plane.
2.0mm
Gaussian
Beam
DOE

10mm
Destination
Plane
Figure 85. Gaussian beam (λo = 0.65µm, e−1 radius Ro = 2.0mm, diameter D = 3.0mm) entersa glass prism of refractive index n = 1.65 whose rear facet is coated with a DOE. The incident
beam, linearly polarized along the x-axis, has the intensity profile shown on the left-hand side.
Since the DOE’s construction wavelength λc is the same as λo, the only emergent diffracted
beam is the +1st order. The DOE’s aperture diameter is 5.0mm, and its phase profile within its
own plane is F (x, y) = 3.0 ( x2 + y2 ); both x and y are in millimeters. The entrance and exit
facets of the prism are antireflection-coated, and the Destination Plane is a distance∆ y = 10mm below the prism’s exit facet.

Figure 86, top row; see caption on the next page …

Guide to DIFFRACT

211

Figure 86. Plots of intensity (preceding page), phase (middle) and phase_minus_curvature(bottom) at the Destination Plane of the system of Fig. 85. From left to right: x-, y-, and zcomponents of polarization. Note that the y- and z-components are several orders-ofmagnitude weaker than the x-component.
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/10/2005
Time: 17:21 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

212

Guide to DIFFRACT$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: mm) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.000000
0.000000
1.000000#
1/e radii Rmin,Rmax:
2.000000
2.000000#
RA,Theta:
1.500000
0.000000#
Aberrations: None#
Polarization RHO,ETA:
0.000000
0.000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
10.00000
10.00000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Ray conversion: RAYS#
Multiply curvature (Y/N): N#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Define a new system: NSYS#
Name of the system: DOE2#**** Surface #01:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
0.000000#
Normal (Theta,Phi):
0.000000
0.000000#
Aperture's Rmin,Rmax:
0.000000
2.000000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: HOMO#
Refractive index (n):1.650000#
Next surface (Hit,Miss):20#**** Surface #02:
Type: Mirror#
Shape: F#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
2.000000#
Normal (Theta,Phi):
45.00000
90.00000#
Aperture's Rmin,Rmax:
0.000000
2.500000#
Coating (PR/DOE/BARE): DOE#
Radial(-1); X-axis Phi:
-1.000000#
ConstructionLambda,Order:
0.650000E-03
1#
B01,B02,B03:
0.000000
3.000000
0.000000#
B04,B05,B06:
0.000000
0.000000
0.000000#
B07,B08,B09:
0.000000
0.000000
0.000000#
B10,B11,B12:
0.000000
0.000000
0.000000#
B13,B14,B15:
0.000000
0.000000
0.000000#
B16,B17,B18:
0.000000
0.000000
0.000000#
B19,B20,B21:
0.000000
0.000000
0.000000#
Next surface (Hit,Miss):
30#**** Surface #03:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.000000-2.000000
2.000000#
Normal (Theta,Phi):
90.00000
270.0000#
Aperture's Rmin,Rmax:
0.000000
2.000000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
40#**** Surface #04:
Type: Destination#

Guide to DIFFRACT
Vertex (Vx,Vy,Vz):
0.000000-12.00000
2.000000#
Normal (Theta,Phi):
90.00000
270.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Trace all incident rays: TRAY#
Initial guess (Xc,Yc,Zc):
0.000000
0.000000-1.000000#
Accept or Search: Search#
Strip phase-factor (Y/N): N#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
5.000000
5.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-1.750000
1.750000#
Ymin,Ymax:-1.750000
1.750000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Phase#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-1.750000
1.750000#
Ymin,Ymax:-1.750000
1.750000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Command.dat file corresponding to the system of Fig. 85

213

214

Guide to DIFFRACT

17.3. Reflective DOE on a parabolic mirror. Figure 87 shows the diagram of a
DOE-coated parabolic mirror illuminated with a Gaussian beam. Two cases are
examined; the parameters of the DOE for both cases are listed in the caption, anddetails of the system appear in the accompanying COMMAND.DAT file. Figures
88 and 89 show the reflected intensity and phase profiles at the Destination
Plane for Cases I and II, respectively.

Destination
Plane
Parabolic
Mirror

DOE
Gaussian
Beam

10mm

Figure 87. A Gaussian beam (λo = 0.65µm, e−1 radius Ro = 2.0mm, diameter D = 3.0mm) is
reflected from a DOE-coated parabolic mirror (Rc = 40mm, k = −1, aperture diameter =3.0mm). The incident beam, linearly polarized along the x-axis, has the intensity profile
shown on the left-hand side. Two different phase profiles are considered for the DOE: In CaseI the profile is F1 (r ) = r 2 – 0.6r 3 – 1.25r 4 + 0.7r 5, whereas in Case II it is F2 (r ) = r 2 –1.25r 4 + 0.35r 6 + 0.1r 8. In both cases r is in millimeters. Since the DOE’s construction
wavelength λc is 0.55µm, various diffracted orders exist, although the most intense beam,shown in Figs. 88 and 89, is the +1st order. The Destination Plane is a distance ∆ z = 10.0 mm
from the vertex of the paraboloid; this also happens to be 10.0 mm before the nominal focal
plane of the parabolic mirror.

Guide to DIFFRACT

215

Figure 88. Plots of intensity (top) and phase (bottom) at the Destination Plane of the system
of Fig. 87 for Case I, F1 (r ) = r 2 – 0.6r 3 – 1.25r 4 + 0.7r 5. From left to right: x-, y-, and zcomponents of polarization. Note that the y-component is nearly six orders-of-magnitude
weaker than the x-component, but the z-component is only ~500 times weaker.

216

Guide to DIFFRACT

Figure 89. Plots of intensity (top) and phase (bottom) at the Destination Plane of the system
of Fig. 87 for Case II, F2 (r ) = r 2 – 1.25r 4 + 0.35r 6 + 0.1r 8. From left to right: x-, y-, and zcomponents of polarization. Note that the y-component is nearly six orders-of-magnitude
weaker than the x-component, but the z-component is only ~600 times weaker.

Guide to DIFFRACT@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/14/2005
Time: 20:57 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: mm) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.000000
0.000000
1.000000#
1/e radii Rmin,Rmax:
2.000000
2.000000#
RA,Theta:
1.500000
0.000000#
Aberrations: None#
Polarization RHO,ETA:
0.000000
0.000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
10.00000
10.00000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Ray conversion: RAYS#
Multiply curvature (Y/N): N#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Define a new system: NSYS#
Name of the system: DOE3#**** Surface #01:
Type: Mirror#
Shape: C#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
10.00000#
Normal (Theta,Phi):
0.000000
0.000000#
RC,K:-40.00000-1.000000#
Aperture's Rmin,Rmax:
0.000000
1.500000#
Coating (PR/DOE/BARE): DOE#
ConstructionLambda,Order:
0.550E-03
1#
B01,B02,B03:
0.000000
1.000000-0.600000#
B04,B05,B06:-1.250000
0.700000
0.000000#
B07,B08,B09:
0.000000
0.000000
0.000000
#

217

218

Guide to DIFFRACT
B10,B11,B12:
0.000000
0.000000
0.000000#
B13,B14,B15:
0.000000
0.000000
0.000000#
B16,B17,B18:
0.000000
0.000000
0.000000#
B19,B20,B21:
0.000000
0.000000
0.000000#
Next surface (Hit,Miss):20#**** Surface #02:
Type: Destination#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
0.000000#
Normal (Theta,Phi):
180.0000
0.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Trace all incident rays: TRAY#
Initial guess (Xc,Yc,Zc):
0.000000
0.000000
0.000000#
Accept or Search: Search#
Strip phase-factor (Y/N): Y#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
3.000000
3.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax: -0.8500000
0.8500000#
Ymin,Ymax: -0.8500000
0.8500000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Phase#
Logarithmic SCALE:
4.000000#
Xmin,Xmax: -0.8500000
0.8500000#
Ymin,Ymax: -0.8500000
0.8500000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Command.dat file corresponding to the system of Fig. 87, Case I

Guide to DIFFRACT

219

17.4. Transmissive DOE on an aspheric glass lens. Figure 90 shows a DOEcoated aspheric lens illuminated with a Gaussian beam. The parameters of the
DOE are listed in the caption, and details of the system appear in theaccompanying COMMAND.DAT file. Figure 91 shows the intensity distribution at
the focal plane, while Fig. 92 shows the transmitted intensity and phase profiles
at the exit pupil.

GaussianBeam

Destination Plane

DOE

Aspheric
Lens

Cover
Slip

Figure 90. Gaussian beam (λo = 0.78µm, e−1 radius Ro = 2.0mm, diameter D = 3.0mm) is
focused through a DOE-coated bi-aspheric lens (aperture diameter = 3.0mm). The incident
beam, linearly polarized along the x-axis, has the intensity profile shown on the left-hand side.
The DOE’s phase profile is F (r ) = 4.2r 2 – 2.5r 4 + 0.25r 6 (r in mm). The DOE’sconstruction wavelength λc is the same as λo; thus the emergent diffracted beam is the +1storder. The Destination Plane is at the exit pupil of the aspheric singlet, and the beam issubsequently propagated to the focal plane.

Figure 91. Intensity distribution at the focal plane of the lens in the system of Fig. 90; (left) xcomponent, (right) y-component of polarization. The y-component is more than three ordersof-magnitude weaker than the x-component.

220

Guide to DIFFRACT

Figure 92. Plots of intensity (top) and phase (bottom) at the exit pupil of the aspheric lens inthe system of Fig. 90. From left to right: x-, y-, and z-components of polarization. Note thatthe y-component is nearly three orders-of-magnitude weaker than the x-component, but the zcomponent is not so weak.

Guide to DIFFRACT@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/11/2005
Time: 18:56 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: mm) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.000000
0.000000
1.000000#
1/e radii Rmin,Rmax:
2.000000
2.000000#
RA,Theta:
1.500000
0.000000#
Aberrations: None#
Polarization RHO,ETA:
0.000000
0.000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
15.00000
15.00000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Ray conversion: RAYS#
Multiply curvature (Y/N): N#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Define a new system: NSYS#
Name of the system: GEL1#**** Surface #01:
Type: Glass#
Shape: A#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
0.000000#
Normal (Theta,Phi):
0.000000
0.000000#
RC,K:
1.930000-0.655844#
A1,A2,A3:
0.000000
0.000000
0.000000#
A4,A5,A6:
2.832980E-03
0.000000-4.388600E-05#
A7,A8,A9:0.000000
1.523680E-04
0.000000#
A10,A11,A12:-1.17709E-04
0.000000
0.000000#
A13,A14,A15:
0.000000
0.000000
0.000000#
A16,A17,A18:
0.000000
0.000000
0.000000
#

221

222

Guide to DIFFRACT
A19,A20,A21:
0.000000
0.000000
0.000000#
Aperture's Rmin,Rmax:
0.000000
1.500000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: HOMO#
Refractive index (n):1.597075#
Next surface (Hit,Miss):20#**** Surface #02:
Type: Glass#
Shape: A#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
1.825600#
Normal (Theta,Phi):
0.000000
0.000000#
RC,K:-6.743680-31.75381#
A1,A2,A3:
0.000000
0.000000
0.000000#
A4,A5,A6:-7.358060E-03
0.000000
2.507670E-03#
A7,A8,A9:0.000000
1.105950E-03
0.000000#
A10,A11,A12:-3.871010E-04
0.000000
0.000000#
A13,A14,A15:
0.000000
0.000000
0.000000#
A16,A17,A18:
0.000000
0.000000
0.000000#
A19,A20,A21:
0.000000
0.000000
0.000000#
Aperture's Rmin,Rmax:
0.000000
1.500000#
Coating (AR/DOE/BARE): DOE#
ConstructionLambda,Order:
0.780E-03
1#
B01,B02,B03:
0.000000
4.200000
0.000000#
B04,B05,B06:-2.500000
0.000000
0.250000#
B07,B08,B09:
0.000000
0.000000
0.000000#
B10,B11,B12:
0.000000
0.000000
0.000000#
B13,B14,B15:
0.000000
0.000000
0.000000#
B16,B17,B18:
0.000000
0.000000
0.000000#
B19,B20,B21:
0.000000
0.000000
0.000000#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
30#**** Surface #03:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
2.826100#
Normal (Theta,Phi):
0.000000
0.000000#
Aperture's Rmin,Rmax:
0.000000
1.500000#
Coating (AR/DOE/BARE): BARE#
Medium beyond surface: HOMO#
Refractive index (n):1.573456#
Next surface (Hit,Miss):
40#**** Surface #04:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
4.026100#
Normal (Theta,Phi):
0.000000
0.000000#
Aperture's Rmin,Rmax:
0.000000
1.500000#
Coating (AR/DOE/BARE): BARE#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
50#**** Surface #05:
Type: Destination#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
1.298300#
Normal (Theta,Phi):
0.000000
0.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Trace all incident rays: TRAY#
Initial guess (Xc,Yc,Zc):
0.000000
0.000000
0.000000#
Accept or Search: Search#
Strip phase-factor (Y/N): Y#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
18.00000
18.00000#$________________________________________________________________________ #Guide to DIFFRACT
Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-2.000000
2.000000#
Ymin,Ymax:-2.000000
2.000000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Phase#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-2.000000
2.000000#
Ymin,Ymax:-2.000000
2.000000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Propagate in environment: PROP
(Length_Units: mm) #Propagation distance:
2.737000#
Multiply curvature (Y/N): N#
Reposition beam (Y/N): N#
Propagation regime: FRNHF#
Smax:
0.000000#
Scalar/Quasi-vector: SC#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-4.000000
4.000000#
Ymin,Ymax:-4.000000
4.000000#
Color or Gray-scale: C#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Command.dat file corresponding to the system of Fig. 90

223

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Guide to DIFFRACT

17.5. Transmissive DOE on a cylindrical GRIN lens. Figure 93 shows a DOEcoated cylindrical GRIN lens used as a fast-axis collimator in front of asemiconductor laser diode. The GRIN lens is followed by a plano-cylindrical
singlet for collimating the slow axis. The parameters of the DOE are listed in thecaption, and details of the system appear in the accompanying COMMAND.DAT
file. The intensity and phase plots of the emergent beam at the Destination_Plane
appear in Fig. 94.
Plano-Cylindrical Lens

Laser
Diode

DOE
GRIN
Cylindrical Lens

Destination Plane

Figure 93. A laser-diode beam (λo = 0.98µm, FWHM divergence angles = 35° and 7°,
astigmatism = 0) is collimated by a DOE-coated GRIN cylindrical lens (aperture diameter =10 × 1.2 mm2), followed by a plano-cylindrical singlet. The laser beam is linearly polarized
in the horizontal direction. The DOE’s phase profile is F (r ) = −2.0 × 10 −5r 2 (r in µm), andits construction wavelength λc is the same as λo; thus the emergent diffracted beam is the +1storder. The Destination plane is 0.3mm beyond the last glass surface.

Figure 94, top row; see caption on the next page …

Guide to DIFFRACT

225

Figure 94. Plots of the Intensity (top) and Phase (bottom) at the Destination plane of thesystem of Fig. 93. From left to right: x-, y-, and z-components of polarization. The y- and zcomponents are five to six orders-of-magnitude weaker than the x-component.
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/15/2005
Time: 13:38 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$MULTIPLY#??,ITEM?....BY: ??
#$MAKE#??,ITEM?~#??,ITEM?:#$JUMP#??_IF#??,ITEM?.GT.: ??
#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):0.9800000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: um) #Type: UB/SG/GG/LG/HG/LD: LD#
BCX,BCY,Power(mW):
0.000000
0.000000
1.000000#
PHI,PSI,DELTA:
7.000000
35.00000
0.000000#
Orientation angle THETA:
0.000000#
Distance from waist Z0:
245.0000#
NMAX,NMAY:
1024
1024
#
LMAX,LMAY:
1000.000
1000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):0.9800000
NVIRON: 1.000000 #_________________________________________________________________________ ##Ray conversion: RAYS#
Multiply curvature (Y/N): N#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):0.9800000
NVIRON: 1.000000 #_________________________________________________________________________ ##226Guide to DIFFRACT
Plot distribution: PLOT
(Length_Units: um) #Type: Ray distribution#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-200.0000
200.0000#
Ymin,Ymax:-200.0000
200.0000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): N#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):0.9800000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: um) #Define a new system: NSYS#
Name of the system: DOCY#**** Surface #01:
Type: Glass#
Shape: L#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
103.0000#
Normal (Theta,Phi):
0.000000
0.000000#
RC,K:
1500.000
0.000000#
Cylinder axis Phi,LA:
0.000000
5000.000#
Aperture's Rmin,Rmax:
0.000000
600.0000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: GRIN#
Type of GRIN medium: 1#
Origin (Ox,Oy,Oz):
0.000000
0.000000
1603.000#
Axis (Theta,Phi):
90.00000
0.000000#
N0,R0,Z0:
1.590100
1501.000
1.000000#
R coefficients: Q1,Q2,Q3:-4.455000E-02
0.000000
0.000000#
Q4,Q5,Q6:
0.000000
0.000000
0.000000#
Z coefficients: W1,W2,W3:
0.000000
0.000000
0.000000#
W4,W5,W6:
0.000000
0.000000
0.000000#
W7,W8,W9:0.000000
0.000000
0.000000#
W10,W11,W12:0.000000
0.000000
0.000000#
Next surface (Hit,Miss):20#**** Surface #02:
Type: Glass#
Shape: L#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
3103.000#
Normal (Theta,Phi):
0.000000
0.000000#
RC,K:-1500.000
0.000000#
Cylinder axis Phi,LA:
0.000000
5000.000#
Aperture's Rmin,Rmax:
0.000000
600.0000#
Coating (AR/DOE/BARE): DOE#
ConstructionLambda,Order:
0.980000
1#
B01,B02,B03:
0.000000-2.000000E-5
0.000000#
B04,B05,B06:
0.000000
0.000000
0.000000#
B07,B08,B09:
0.000000
0.000000
0.000000#
B10,B11,B12:
0.000000
0.000000
0.000000#
B13,B14,B15:
0.000000
0.000000
0.000000#
B16,B17,B18:
0.000000
0.000000
0.000000#
B19,B20,B21:
0.000000
0.000000
0.000000#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
30#**** Surface #03:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
3500.000#
Normal (Theta,Phi):
0.000000
0.000000#
Aperture's Rmin,Rmax:
0.000000
750.0000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: HOMO#
Refractive index (n):1.650000#

Guide to DIFFRACT
Next surface (Hit,Miss):
40#**** Surface #04:
Type: Glass#
Shape: L#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
6700.000#
Normal (Theta,Phi):
0.000000
0.000000#
RC,K:-3000.000
0.000000#
Cylinder axis Phi,LA:
90.00000
2500.000#
Aperture's Rmin,Rmax:
0.000000
750.0000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
50#**** Surface #05:
Type: Destination#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
7000.000#
Normal (Theta,Phi):
0.000000
0.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):0.9800000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: um) #Trace all incident rays: TRAY#
Initial guess (Xc,Yc,Zc):
0.000000
0.000000
0.000000#
Accept or Search: Search#
Strip phase-factor (Y/N): N#
NMAX,NMAY:
1024
1024
#
LMAX,LMAY:
3000.000
3000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):0.9800000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-850.000
850.000#
Ymin,Ymax:-850.000
850.000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):0.9800000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Phase#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-850.000
850.000#
Ymin,Ymax:-850.000
850.000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):0.9800000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Command.dat file corresponding to the system of Fig. 93

227

228

Guide to DIFFRACT

17.6. Reflective DOE on a conical mirror surface. Figure 95 shows a DOEcoated conical mirror in a system that also contains a spherical mirror and abeam-splitter. The system parameters are given in the caption, with detailappearing in the accompanying COMMAND.DAT file. The intensity and phaseplots of the emergent beam at the Destination_Plane appear in Fig. 96.
Beam-splitter
Spherical
MirrorConicalMirrorDestination
Plane
DOE

Figure 95. A Gaussian beam (λo = 0.78µm, e−1 radius = 2.0mm, diameter = 3.0mm) istransmitted through a beam-splitter to a spherical mirror having Rc = 300 mm, aperture
diameter = 4.0mm. The reflected beam is redirected at the beam-splitter to a conical mirror(Rc = 250mm, k = −0.5, aperture diameter = 5.0mm). The conical mirror is coated with a DOE
having a phase profile F (r ) = 2.0 r 2 (r in mm) and construction wavelength λc = 0.65µm; thetraced emergent beam is the +1st diffracted order. The incident beam is linearly polarizedalong the x-axis, the spherical mirror is 50mm away from the beam-splitter, the DOE-coatedmirror is 10mm below the splitter, and the Destination plane is 50mm in front of the conicalmirror’s vertex.

Guide to DIFFRACT

229

Figure 96. Plots of the Intensity (top) and Phase (bottom) at the Destination plane of thesystem of Fig. 95. From left to right: x-, y-, and z-components of polarization. The y- and zcomponents are nearly four orders-of-magnitude weaker than the x-component. The emergent
beam’s curvature, which corresponds to a virtual focus at a distance of 124.1mm to the left of
the Destination plane, has been subtracted from the phase profile.

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/13/2005
Time: 09:45 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$MULTIPLY#??,ITEM?....BY: ??
#$MAKE#??,ITEM?~#??,ITEM?:#$JUMP#??_IF#??,ITEM?.GT.: ??
#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

230

Guide to DIFFRACT$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: mm) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.000000
0.000000
1.000000#
1/e radii Rmin,Rmax:
2.000000
2.000000#
RA,Theta:
1.500000
0.000000#
Aberrations: None#
Polarization RHO,ETA:
0.000000
0.000000#
NMAX,NMAY:
1024 1024#
LMAX,LMAY:
15.00000
15.00000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Ray conversion: RAYS#
Multiply curvature (Y/N): N#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Define a new system: NSYS#
Name of the system: DOE6#**** Surface #01:
Type: Mirror#
Shape: S#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
100.0000#
Normal (Theta,Phi):
0.000000
0.000000#
Radius of curvature (RC):
300.0000#
Aperture's Rmin,Rmax:
0.000000
2.000000#
Coating (PR/DOE/BARE): BARE#
Complex index (n,k):0.300000
6.900000#
Next surface (Hit,Miss):20#**** Surface #02:
Type: Mirror#
Shape: F#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
50.00000#
Normal (Theta,Phi):
135.0000
90.00000#
Aperture's Rmin,Rmax:
0.000000
3.000000#
Coating (PR/DOE/BARE): PR#
Next surface (Hit,Miss):
30#**** Surface #03:
Type: Mirror#
Shape: C#
Vertex (Vx,Vy,Vz):
0.000000-10.000000
50.00000#
Normal (Theta,Phi):
135.0000
270.000000#
RC,K:
250.0000-0.500000#
Aperture's Rmin,Rmax:
0.000000
2.500000#
Coating (PR/DOE/BARE): DOE#
ConstructionLambda,Order:
0.650000E-03
1#
B01,B02,B03:
0.000000
0.200000
0.000000#
B04,B05,B06:
0.000000
0.000000
0.000000#
B07,B08,B09:
0.000000
0.000000
0.000000#
B10,B11,B12:
0.000000
0.000000
0.000000#
B13,B14,B15:
0.000000
0.000000
0.000000#
B16,B17,B18:
0.000000
0.000000
0.000000#
B19,B20,B21:
0.000000
0.000000
0.000000#
Next surface (Hit,Miss):
40#**** Surface #04:
Type: Destination#
Vertex (Vx,Vy,Vz):
0.000000-10.000000
100.0000#

Guide to DIFFRACT
Normal (Theta,Phi):
0.000000
0.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Trace all incident rays: TRAY#
Initial guess (Xc,Yc,Zc):
0.000000
0.000000-1.000000#
Accept or Search: Search#
Strip phase-factor (Y/N): Y#
NMAX,NMAY:
1024
1024
#
LMAX,LMAY:
12.00000
12.00000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-3.600000
3.600000#
Ymin,Ymax:-3.600000
3.600000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Phase#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-3.600000
3.600000#
Ymin,Ymax:-3.600000
3.600000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Command.dat file corresponding to the system of Fig. 95

231

232

Guide to DIFFRACT

17.7. Transmissive DOE on an axicon. Figure 97 shows an axicon, coated with
DOEs on both its entrance and exit facets. The system parameters are given in
the caption, with detail appearing in the accompanying COMMAND.DAT file. The
intensity and phase profiles of the emergent beam at the Destination_Plane areshown in Fig. 98.

AxiconDOE
DOE

Destination
Plane

Figure 97. A Gaussian beam (λo = 0.65µm, e−1 radius = 2.0mm, diameter = 3.0mm) istransmitted through an axicon having a full cone angle of 179° and an aperture diameter of
4.0mm. The DOE coating the entrance facet of the axicon has a phase profile F (r ) =r 2 − 0.6 r 3 (r in mm). On the exit facet, the phase profile is given by
F (x, y) = − 0.5 x 2 − 1.5 y 2, where x and y are in mm, and where the x y-axes are rotatedthrough an angle φ = 45° relative to the x y coordinate system. The DOEs are constructed with
λc = 0.65µm, and the traced beam is the +1st diffracted order. The incident beam is linearlypolarized along the x-axis, the refractive index and thickness of the axicon are 1.78 and2.0mm, respectively, and the Destination plane is positioned 3.0mm past the axicon’s flat
surface.

Guide to DIFFRACT

233

Figure 98. Plots of the Intensity (top) and Phase (bottom) at the Destination plane of thesystem of Fig. 97. (Left) x-component of polarization; (right) z-component of polarization.
The z-component is nearly five orders-of-magnitude weaker than the x-component.
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/15/2005
Time: 16:22 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$MULTIPLY#??,ITEM?....BY: ??
#$MAKE#??,ITEM?~#??,ITEM?:#

234

Guide to DIFFRACT$JUMP#??_IF#??,ITEM?.GT.: ??
#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: mm) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.000000
0.000000
1.000000#
1/e radii Rmin,Rmax:
2.000000
2.000000#
RA,Theta:
1.500000
0.000000#
Aberrations: None#
Polarization RHO,ETA:
0.000000
0.000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
10.00000
10.00000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Ray conversion: RAYS#
Multiply curvature (Y/N): N#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Define a new system: NSYS#
Name of the system: DOE7#**** Surface #01:
Type: Glass#
Shape: X#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
10.00000#
Normal (Theta,Phi):
0.000000
0.000000#
Axicon's half-angle:-89.50000#
Aperture's Rmin,Rmax:
0.000000
2.000000#
Coating (AR/DOE/BARE): DOE#
ConstructionLambda,Order:
0.650000E-03
1#
B01,B02,B03:
0.000000
1.000000-0.600000#
B04,B05,B06:
0.000000
0.000000
0.000000#
B07,B08,B09:
0.000000
0.000000
0.000000#
B10,B11,B12:
0.000000
0.000000
0.000000#
B13,B14,B15:
0.000000
0.000000
0.000000#
B16,B17,B18:
0.000000
0.000000
0.000000#
B19,B20,B21:
0.000000
0.000000
0.000000#
Medium beyond surface: HOMO#
Refractive index (n):1.780000#
Next surface (Hit,Miss):20#**** Surface #02:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
12.00000#
Normal (Theta,Phi):
0.000000
0.000000#
Aperture's Rmin,Rmax:
0.000000
2.000000#
Coating (AR/DOE/BARE): DOE#
Radial(-1); X-axis Phi:
45.00000#
ConstructionLambda,Order:
0.650000E-03
1
#
B1_0,
B0_1,
B2_0:
0.000000
0.000000-0.500000
#
B1_1,
B0_2,
B3_0:
0.000000-1.500000
0.000000
#
B2_1,
B1_2,
B0_3:
0.000000
0.000000
0.000000
#
B4_0,
B3_1,
B2_2:
0.000000
0.000000
0.000000#

Guide to DIFFRACT
B1_3,
B0_4,
B5_0:
0.000000
0.000000
0.000000
#
B4_1,
B3_2,
B2_3:
0.000000
0.000000
0.000000
#
B1_4,
B0_5,
B6_0:
0.000000
0.000000
0.000000
#
B5_1,
B4_2,
B3_3:
0.000000
0.000000
0.000000
#
B2_4,
B1_5,
B0_6:
0.000000
0.000000
0.000000#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
30#**** Surface #03:
Type: Destination#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
15.00000#
Normal (Theta,Phi):
0.000000
0.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Trace all incident rays: TRAY#
Initial guess (Xc,Yc,Zc):
0.000000
0.000000
0.000000#
Accept or Search: Search#
Strip phase-factor (Y/N): N#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
5.000000
5.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-1.750000
1.750000#
Ymin,Ymax:-1.750000
1.750000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Phase#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-1.750000
1.750000#
Ymin,Ymax:-1.750000
1.750000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Command.dat file corresponding to the system of Fig. 97

235

236

Guide to DIFFRACT

17.8. Transmissive DOE sandwiched between two flat substrates. Figure 99
shows an aspheric lens illuminated with a (truncated) Gaussian beam. The
emerging convergent beam passes through a glass plate on its way to a flat DOE
sandwiched between two glass plates. The parameters of the DOE are listed in
the caption, and details of the system appear in the accompanying
COMMAND.DAT file. Figure 100 shows the intensity and phase distribution at
the Destination_Plane, while Fig. 101 shows plots of log_intensity, intensity, andphase in the plane of best focus.
X

X
Gaussian
Beam

Glass
Plate

X
DOE

Z

Cover Substrate
Slip
Aspheric
Lens

Destination
Plane

Figure 99. Gaussian beam (λo = 0.66µm, e−1 radius Ro = 2.5mm, diameter D = 4.0mm) is
focused by a 4.0mm diameter aspheric lens. The incident beam, linearly polarized along the xaxis, has the intensity profile shown on the left-hand side. The DOE’s phase profile is
F (x, y) = 639.77x + 17.47x 2 – 19.76 y 2 – 30.18 x 3 – 0.0042 x2y – 33.69 xy 2 + 0.0021 y 3 –3.25 x 4 (x, y in mm). The DOE’s construction wavelength λc is the same as λo; thus the
emergent diffracted beam is the +1st order. The Destination plane is at z = 10.0 mm (measured
from the first vertex of the lens), and is tilted by θ = 6.03°, as shown. The beam issubsequently propagated a distance of 10.468 mm along the normal to the Destination plane,
which brings the beam to its plane of best focus.

Guide to DIFFRACT

237

Figure 100. Distributions of intensity (top) and phase (bottom) at the Destination plane in thesystem of Fig. 99; from left to right, x-, y-, and z-components of polarization. Note that the
emergent beam is centered at x = −3.6 mm. The y-component is nearly four orders-ofmagnitude weaker than the x-component, but the power content of the z-component is nonnegligible. The phase profile is quite uniform, corresponding to a small residual aberration(r.m.s. wavefront error ≈ 0.003λo).

238

Guide to DIFFRACT

Figure 101. Plots of log_intensity (top), intensity (middle), and phase (bottom) at the plane ofbest focus in the system of Fig. 99. From left to right: x-, y-, and z-components ofpolarization. Note that the y-component is nearly four orders-of-magnitude weaker than the xcomponent, but the z-component is not so weak. The observed linear phase profile is due tothe 6.03° tilt of the focal plane relative to the incident beam coordinates (see the focal plane
coordinates in Fig. 99).

Guide to DIFFRACT@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/10/2005
Time: 09:55 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: mm) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.000000
0.000000
1.000000#
1/e radii Rmin,Rmax:
2.500000
2.500000#
RA,Theta:
2.000000
0.000000#
Aberrations: None#
Polarization RHO,ETA:
0.000000
0.000000#
NMAX,NMAY:
1024 1024#
LMAX,LMAY:
15.00000
15.00000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Ray conversion: RAYS#
Multiply curvature (Y/N): N#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Define a new system: NSYS#
Name of the system: NSG1#**** Surface #01:
Type: Glass#
Shape: A#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
0.000000#
Normal (Theta,Phi):
0.000000
0.000000#
RC,K:
11.40000-0.733000#
A1,A2,A3:
0.000000
0.000000
0.000000#
A4,A5,A6:
2.820000E-07
0.000000-3.750000E-08#
A7,A8,A9:0.000000-1.500000E-09
0.000000#
A10,A11,A12:
0.000000
0.000000
0.000000#
A13,A14,A15:
0.000000
0.000000
0.000000#
A16,A17,A18:
0.000000
0.000000
0.000000#
A19,A20,A21:
0.000000
0.000000
0.000000
#

239

240

Guide to DIFFRACT
Aperture's Rmin,Rmax:
0.000000
2.000000
Coating (AR/DOE/BARE): AR
Medium beyond surface: HOMO
Refractive index (n):1.540440
Next surface (Hit,Miss):20**** Surface #02:
Type: Glass
Shape: S
Vertex (Vx,Vy,Vz):
0.000000
0.000000
Normal (Theta,Phi):
0.000000
0.000000
Radius of curvature (RC):-98.00000
Aperture's Rmin,Rmax:
0.000000
2.000000
Coating (AR/DOE/BARE): AR
Medium beyond surface: FREE
Next surface (Hit,Miss):
30**** Surface #03:
Type: Glass
Shape: F
Vertex (Vx,Vy,Vz):
0.000000
0.000000
Normal (Theta,Phi):
0.000000
0.000000
Aperture's Rmin,Rmax:
0.000000
2.000000
Coating (AR/DOE/BARE): AR
Medium beyond surface: HOMO
Refractive index (n):1.520168
Next surface (Hit,Miss):
40**** Surface #04:
Type: Glass
Shape: F
Vertex (Vx,Vy,Vz):
0.000000
0.000000
Normal (Theta,Phi):
0.000000
0.000000
Aperture's Rmin,Rmax:
0.000000
2.000000
Coating (AR/DOE/BARE): AR
Medium beyond surface: FREE
Next surface (Hit,Miss):
50**** Surface #05:
Type: Glass
Shape: F
Vertex (Vx,Vy,Vz):
0.000000
0.000000
Normal (Theta,Phi):
0.000000
0.000000
Aperture's Rmin,Rmax:
0.000000
1.000000
Coating (AR/DOE/BARE): AR
Medium beyond surface: HOMO
Refractive index (n):1.520168
Next surface (Hit,Miss):
60**** Surface #06:
Type: Glass
Shape: F
Vertex (Vx,Vy,Vz):
0.000000
0.000000
Normal (Theta,Phi):
0.000000
0.000000
Aperture's Rmin,Rmax:
0.000000
1.000000
Coating (AR/DOE/BARE): DOE
Radial(-1); X-axis Phi:
0.000000
ConstructionLambda,Order:
660.0000E-06
1
B1_0,
B0_1,
B2_0:
639.7700
0.000000
B1_1,
B0_2,
B3_0:
-0.000003-19.76000
B2_1,
B1_2,
B0_3:
-0.004200-33.69000
B4_0,
B3_1,
B2_2:
-3.250000
0.000000
B1_3,
B0_4,
B5_0:
0.000000
0.000000
B4_1,
B3_2,
B2_3:
0.000000
0.000000
B1_4,
B0_5,
B6_0:
0.000000
0.000000
B5_1,
B4_2,
B3_3:
0.000000
0.000000
B2_4,
B1_5,
B0_6:
0.000000
0.000000
Medium beyond surface: HOMO
Refractive index (n):1.520168
Next surface (Hit,Miss):
70**** Surface #07:
Type: Glass
Shape: F
Vertex (Vx,Vy,Vz):
0.000000
0.000000

1.700000

1.800000

2.410000

16.79000

17.29000

17.42000-30.18000
0.002100
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

19.29000

#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
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#
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#
#
#
#
#
#
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#
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#
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#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#

Guide to DIFFRACT
Normal (Theta,Phi):
0.000000
0.000000#
Aperture's Rmin,Rmax:
0.000000
1.000000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
80#**** Surface #08:
Type: Destination#
Vertex (Vx,Vy,Vz):
0.000000
0.000000
10.00000#
Normal (Theta,Phi):
6.030000
0.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Trace all incident rays: TRAY#
Initial guess (Xc,Yc,Zc):
0.000000
0.000000
0.000000#
Accept or Search: Search#
Strip phase-factor (Y/N): Y#
NMAX,NMAY:
1024 1024#
LMAX,LMAY:
30.00000
30.00000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-5.000000-2.000000#
Ymin,Ymax:-1.500000
1.500000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Phase#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-5.000000-2.000000#
Ymin,Ymax:-1.500000
1.500000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Propagate in environment: PROP
(Length_Units: mm) #Propagation distance:
10.46800#
Multiply curvature (Y/N): N#
Reposition beam (Y/N): N#
Propagation regime: FRNHF#
Smax:
0.000000#
Scalar/Quasi-vector: SC#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #

241

242

Guide to DIFFRACT_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Log(Intensity)#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-15.00000
15.00000#
Ymin,Ymax:-15.00000
15.00000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-5.000000
5.000000#
Ymin,Ymax:-5.000000
5.000000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 2#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Phase#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-5.000000
5.000000#
Ymin,Ymax:-5.000000
5.000000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 2#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):660.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Command.dat file corresponding to the system of Fig. 99

Guide to DIFFRACT243

Appendix A
Frequently asked questions about DIFFRACT
Question: How can one obtain printouts of the various plots of the beam crosssection that DIFFRACT displays on the monitor?
Answer: A quick printout of the entire Graphics Window may be obtained byfirst selecting that window, then choosing Print from the File menu displayed at
the top of the screen. However, if you want to incorporate the graphics into adocument you must save the plot files to the hard disk. Saving will create, in
addition to the graphic data-files, a bitmap file containing the latest screen
dump. You may load these .BMP files to any Windows-based utility package(such as Word® and Paint®) in order to manipulate and/or print them. As forthe graphic .DAT files, using the File_Management/Graphics option of
DIFFRACT, you may convert these files into TIFF format, and save them again.
The TIFF files may subsequently be ported to a word processor and incorporated
within your documents. The .DAT graphic files are simple 2-D arrays of ASCII
numbers; these files may therefore be ported to many commercially available
graphics programs (e.g., Matlab®, Mathematica®, Origin®, etc.), and renderedas 3-D plots, contour plots, color-coded plots, etc.
Question: How do I abort the program?
Answer: If DIFFRACT’s main menu is on the screen and you are being prompted
to select an option, simply choose the QUIT option. This will end the program
normally, and all your data files (such as command.dat, session.dat, signals.dat,etc.) will be preserved. If, on the other hand, the program has stopped
responding to you, or if it is in the middle of a lengthy calculation which you
want to terminate, you can do one of two things:• Close the main window by clicking the mouse on the upper right corner of thescreen.
• Type Ctrl + C at the keyboard, that is, hold down the Ctrl key and type C.
Either way the program will abort and your data files will be fully or partiallylost.
Question: Is there a good way to determine whether the mesh is adequate for the
problem at hand?

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Answer: The mesh parameters NMAX and NMAY are chosen by the user, typicallyin the beginning of a session, and do not change their values throughout thesession, unless forced by the user. These parameters represent the number ofpixels along the X- and Y-axes, and their values are limited by the available
memory of the computer, as well as by the desired speed of execution of the
program. (In the Student Edition the maximum allowed value of NMAX and NMAY
is 256, and the user cannot increase this upper bound. In the General Edition
these parameters are initially limited to 1024, but the user may change this upperbound by modifying the corresponding parameter in the SETTINGS.LOG file.)
The other two parameters of the mesh, LMAX and LMAY, are also chosen by theuser at the outset, but their values keep changing as the beam propagates throughthe optical system under consideration. LMAX and LMAY are the actual lengths ofthe mesh along the X- and Y-axes (in user-selected Length_Units or as
dimensionless lengths, i.e., normalized by the vacuum wavelength λ0), and they
expand/shrink with the beam in order to accommodate the beam at various
cross-sections of the system. DIFFRACT adjusts the values of LMAX and LMAY
automatically throughout the simulation, and displays the adjusted values atappropriate points.
Two important mesh parameters for the user to keep track of are∆x = LMAX/NMAX and ∆y = LMAY/NMAY; these are the intervals between adjacentsamples along the X and Y directions of the mesh. The accuracy of a given
simulation depends on whether ∆x and ∆y are small compared to the typicallength-scale over which the wavefront (i.e., the complex-amplitude distribution
within the XY-plane) undergoes significant variations. Since ∆x and ∆y change
automatically along the propagation path of the beam, the user must monitor
their values at every cross-section, in order to ensure that the discretization of
the wavefront by the mesh is accurate.
It is possible, of course, to use the Rescale option from the main menu atdifferent cross-sections, and to force the mesh to not only accommodate thebeam, but also to sample it properly. This action is not recommended, however,since Rescale simply interpolates between the existing pixels, without adding
any physical information. The proper course of action is to choose NMAX, NMAY,
LMAX, LMAY only once, in the beginning of a session, but with the entire optical
path in mind. If at some point during the simulation, the mesh turns out to be toocoarse or too small for the beam, the user is advised to repeat the entire
simulation with a different set of parameters for the starting mesh.

Guide to DIFFRACT

245

With regard to the choice of parameters for an initial mesh, it is helpful torecognize the inverse relationships inherent to the Fourier transform. For
instance, if a beam is focused by a lens, its distribution at the focal plane will be
the Fourier transform of the distribution at the entrance pupil of the lens. Thus, if
the mesh at the entrance pupil has a length of LMAX, its length at the focal plane
will be inversely proportional to LMAX. Consequently, increasing LMAX at theentrance pupil will reduce the value of LMAX at the focal plane. Similar
considerations apply to far field propagation of a beam, since far field
distribution is the Fourier transform of the initial distribution.
Question: I have created a COMMAND.DAT file and started running DIFFRACT in
the non-interactive mode using this COMMAND.DAT. At first I chose the STEP
mode to go through the program step by step and check its operation. Now that I
am satisfied with the setup, can I switch to the ZOOM mode and let the program
run its course without my sitting at the keyboard?
Answer: No, unfortunately there is no way to go back and forth between the
STEP and ZOOM modes during non-interactive sessions. Once you are satisfied
with your COMMAND.DAT, you must either continue in the STEP mode until thecommand sequence is exhausted, or abort the program (by closing the main
window or by typing Ctrl + C at the keyboard). Only then can you restart the
program and choose to ZOOM through the command.dat.
Question: I am running DIFFRACT in the non-interactive mode when I realizethat I need to modify one or more command blocks within the COMMAND.DAT
file. Is it possible to switch to the interactive mode and enter the desiredcommands from the keyboard, or is there any way at all to pause the noninteractive session in order to modify its COMMAND.DAT file?
Answer: No, unfortunately there is no way to modify the COMMAND.DAT file inthe midst of a non-interactive session. You must finish the session (or abort it byclosing the main window or by typing Ctrl + C), modify your COMMAND.DAT
file, then repeat the simulation from the start.
Question: DIFFRACT is running very slowly on my machine. How can I improve
its speed?
Answer: The speed of execution of the program during any given session isrelated to the chosen mesh size, namely, the value of NMAX × NMAY. You should

246

Guide to DIFFRACT

try to use the smallest possible mesh without going below the required accuracy
for the discretization of your wavefronts.
Occasionally, the chosen mesh is larger than the available memory of themachine. In such situations, the program swaps segments of the mesh between
the RAM and the hard disk and, in doing so, slows down the computations
drastically. Since a concurrently running program might be occupying a sizable
chunk of memory, it might be helpful to terminate all other programs whilerunning DIFFRACT. You should also ensure that the value chosen for NDIM in the
beginning of the program is not excessively large; NDIM must be equal to thelargest value of NMAX and NMAY that is used during the session, but it need not be
any larger.
Also, if you do not intend to use the ray_tracing features of DIFFRACT, you canreduce memory requirements by unchecking the “Enable_Ray_Tracing” box at theoutset. If all fails, of course, the only remaining solution is to increase thememory of your machine by adding a few megabytes of RAM.
Question: I want to simulate a fairly large and complex optical system. How do
I minimize the amount of time spent in the interactive mode for setting up anddebugging the command.dat file?
Answer: Upon starting the program in the interactive mode, choose a very small
mesh, say, one having NMAX = NMAY = 128. As you build your optical systemsimulator by entering parameters and options from the keyboard, DIFFRACT runs
the beam through this system and gives answers that are not necessarily correct,
since the chosen mesh is too coarse. Ignore these results and continue building
your simulation until all components of your system are in place. Then quit the
program and edit the COMMAND.DAT file using a text editor. Within thisCOMMAND.DAT, you can now reset NMAX and NMAY to their realistic values,
modify any other parameter values that you want, add any necessary looping
commands ($Increment, $Multiply, $Make, $Jump, $Tags), and save themodified version of the file as the new COMMAND.DAT. DIFFRACT may now be
run in the non-interactive mode from this new COMMAND.DAT file, and you
may choose either the STEP option (if you want to examine intermediate results),or the ZOOM option (if you want to leave the computer and let it run the programin the background).
Question: What is the fastest way to build a COMMAND.DAT file?

Guide to DIFFRACT

247

Answer: If you are already familiar with the basic structure of DIFFRACT, use
the <TEMPLATES > sub-directory (located in DIFFRACT’s home directory) tocreate your own COMMAND.DAT files. To avoid accidental deletion or
modification of the files contained in <TEMPLATES >, copy the text files
TEMPLATE.DAT and COMMAND.DAT to another (working) sub-directory, thenopen both files inside your working subdirectory and place them side by side on
the monitor. Copy desired command blocks from TEMPLATE.DAT and pastethem (in a logical order) onto the empty COMMAND.DAT file. Proceed to modify
the various command parameters until the elements of your system are properly
represented. When all the elements of the system are in place, you may want tocopy one or more $commands to appropriate locations within the completed
COMMAND.DAT file. These $commands enable the creation of loops (or otherprogramming constructs) for the purpose of repeating a given simulation withdifferent sets of parameters.
Question: How can one create Geometric-optical_System files (XXXX.SYS)?
Answer: When you run DIFFRACT in the interactive mode, you will be given theoption of creating a XXXX.SYS file for the geometric-optical system underconsideration. Once this text file is created (within your working subdirectory)
you may modify it by adding or removing various elements, or by changing theparameter values of the specified elements. An alternative approach to creating
XXXX.SYS files involves the use of GEOP.SYS template located in the<TEMPLATES> subdirectory of DIFFRACT’s home directory. Start by copying
the GEOP.SYS file to a working subdirectory, then open this text file and follow
the instructions given within the file itself.
Question: How can one create XXXX.STK files for simulating multilayer stacks
in conjunction with the Multilayer feature of the program?
Answer: When you run DIFFRACT in the interactive mode, you will be given theoption of creating a XXXX.STK file for the multilayer stack under consideration.
Once this text file is created (within your working subdirectory) you may
modify it by adding or removing various layers, or by changing the parametervalues of the specified layers. An alternative approach to creating XXXX.STK
files involves the use of MLYR.STK template located in the <TEMPLATES>
subdirectory of DIFFRACT’s home directory. Start by copying the MLYR.STK file
to a working subdirectory, then open this text file and follow the instructions
given within the file itself.

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Guide to DIFFRACT

Question: Can I change the default values of some of the parameters that are
seemingly set internally by DIFFRACT?
Answer: Yes, unless you are using the Student Edition of the program. The
Install routine for the General Edition copies two files to the working directoryof DIFFRACT; these are called settings.log and settings.exe. You may examine
the contents of settings.log with a text editor, to find out which parameters are
set to which default values. For example, the upper bound on the value of NDIM
is set here, as is the size of the display arrays (NDSP0), or the maximum numberof surfaces allowed in geometric-optical systems, or the maximum allowed
number of layers for multilayer stacks. Whenever you run DIFFRACT, it reads (inthe beginning) the file settings.log, and sets the default values accordingly. You
may change these values directly by modifying the SETTINGS.LOG file.
Alternatively, you may invoke the routine SETTINGS.EXE (by double-clicking on
its icon or by typing “settings” at the DOS prompt within DIFFRACT’s home
directory), then answer the questions posed. In this way the contents of
SETTINGS.LOG file will be changed according to your instructions.
Question: How do I prevent DIFFRACT’s logo from appearing in the beginningof each session?
Answer: In the home directory of DIFFRACT there is a bitmap file called
LOGO.BMP. Removing or renaming this file will cause the program to skip thelogo screen.
Question: How do I navigate among DIFFRACT’s three windows, that is, how do
I switch from, say, the Dialogue Window to the Graphics Window or to thewindow that contains the Guiding Information?
Answer: In their default positions, the three windows are either fully or partiallyvisible at all times. You may point and click the mouse on the desired window toswitch to that window. Alternatively, you might use the Ctrl + TAB key
combination to move sequentially among the windows. The latter option also
works if you have moved the windows or changed their sizes, thus causing thedesired window to be hidden from view.
You switch to the Guiding Information window to read the instructions or to get
detailed information about a command or option with which you are notfamiliar. You switch to the Graphics Window to view the displayed graphics, orGuide to DIFFRACT

249

to enlarge the window in order to see more of the screen in more detail.
Whatever the reasons may be for switching to another window, you must always
return to the Dialogue Window in order to continue your interactions with theprogram.
You can enlarge, shrink, resize, or move a window by dragging its various
boundaries, or by clicking the buttons at the upper right corner of the window.
These changes remain in effect for a short while, until the main menu of
DIFFRACT returns to the Dialogue Window, at which point the three windows
will be reset to their default positions.
Question: When using the PLOT option, what is the fastest way to specify theboundaries [Xmin : Xmax, Ymin : Ymax] of a plot?
Answer: Start by typing a value for Xmin; the program assumes that you want toplot a square region centered at the origin, (X, Y) = (0, 0), and enters thecorresponding values for Xmax, Ymin, and Ymax automatically. If these values
are acceptable, you do not need to do anything else, otherwise, type in thedesired values for the remaining parameters. (If the first parameter that you enterfor the plot boundary is not Xmin, you will have to type all four parameters, evenwhen you intend to plot a square region centered at the origin.)
Question: How do I run two or more COMMAND.DAT files in the ZOOM mode,
without having to stay around to initialize DIFFRACT for each run?
Answer: In the beginning of each session, the user is asked to specify a working
subdirectory. You can specify as many as 20 different subdirectories at thispoint, simply by entering their names sequentially (separated by blanks orcommas). The program then begins in the first specified directory and, uponcompleting the run, moves to the second directory, where it starts a new run. The
process thus continues until all the specified subdirectories have been exhausted.
In this way, all the various COMMAND.DAT files, initially placed in separate
subdirectories within DIFFRACT’s home directory, are executed independently,
each within its own directory. The data-files created during each one of these
multiple sessions are recorded in the corresponding subdirectory. All data-files
needed by the program for a given run (including COMMAND.DAT) must residewithin the same subdirectory.

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Guide to DIFFRACT

Question: I used the TIFF option under FMAN, but it outputs only black-andwhite images. How can I specify a color option for the TIFF output?
Answer: TIFF always yields black-and-white images. To get color pictures,
while in FMAN, use the Display option (D) with Color sub-option (C).
DIFFRACT will display the color image on the monitor. You may save this as a
Bitmap file using the B option. Later, you may have to double-click on thexxxx.bmp file thus created in order to open it (with MS_Paint) and change or
remove some of the text, labels, white space, etc.
Question: How can I compute the total intensity |Ex|2 + |Ey|2 in DIFFRACT?
Answer: First, PLOT the desired intensity distribution for the x- and ycomponents separately and save them to file by specifying a file number nn. At
this point you will have the intensity plots IXnn.DAT and IYnn.DAT in yourworking directory. After this, call FMAN and use the Load option (L) to load
one of the files, say, IXnn. While IXnn is in FMAN's buffer, use the
Load_and_Add option (A) to add IYnn to the buffer (set weight-factor = 1). You
now have the sum of the two intensity profiles in FMAN's buffer, which can besaved to file in Tiff format (T option), Bitmap format (B option), or ASCII
format (F option). To save as bitmap, however, note that you will have todisplay the content of the buffer (D option) before calling upon the B option.

Guide to DIFFRACT251

Appendix B
Relationships among various Menu Options
The general rule concerning the menu options of DIFFRACT is that, once a
distribution has been placed in the mesh, any option can be called upon. The
options basically take the distribution from the mesh, operate on it, and place itback in the mesh. Each option may be called an arbitrary number of times,
independently of all the other options called (exceptions are listed below). Theremay be physical constraints, such as two optical elements bumping into eachother if placed too closely, but there are no logical constraints as far as
DIFFRACT is concerned. Certain features of the various options, of which theuser should be aware, are described in the following paragraphs.
HELP: This option may be called at any time during a session. It displays a listof topics for which guidance information is available. Type the four-letter nameof the topic and view the available information on the screen.
FMAN: This option may be called at any time during a session. It does notchange the mesh or its contents, but allows access to and manipulation of thegraphic files.
GEOP: This option may be called at any time during a session. The first time
GEOP is invoked, the user must define a geometric-optical system. This systemstays resident within DIFFRACT’s memory throughout the entire session and,
subsequently, may be inspected, modified or used for ray_tracing. Once in the
GEOP environment, the user may define a new geometric-optical system (NSYS
option), inspect the resident system (INSP option), or ask the program to trace
through the resident system the entire set of rays corresponding to the incident
beam (TRAY option). The first two options, NSYS and INSP, do not modify themesh; in fact, they do not even require the existence of a distribution within themesh. The TRAY option, however, needs not only the presence of an incident
beam, but the beam must also have been converted to geometric-optical raysprior to the invocation of GEOP.
QUIT: This option may be called at any time during a session. Once you quit
the program, the mesh and its contents will be lost, unless you SAVE them aheadof time.

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BEAM and READ: To initiate the mesh, you must call one of these two options.
You can also call these options at other times, when the mesh is alreadyoccupied. In the latter case, however, you will be warned that if you proceed thecurrent contents of the mesh will be lost.
PLOT, SAVE, MEAS, WAVE, HELP, FMAN, PSIG, RAYS, DTCT: These
options do not change the mesh or its contents. Unless the mesh is empty, they
can be called at any time. (The only exception to this rule occurs in the case of
the PLOT option, if you want to plot the distribution of geometric-optical rays.
In this instance you must first call RAYS to generate the ray_distribution, before
calling the PLOT.)
READ, XOBJ, INTF: These options require that a certain data-file exist within
the working directory, so they can import it to the mesh. The files needed by
READ and INTF must be created by the SAVE option within DIFFRACT. The
files needed by XOBJ (either in IMPORT mode, or for producing doubleexposure plates in CREATE mode) may be produced using the CREATE suboption of the XOBJ option. Alternatively, such files may be created by someother utility software, then placed in DIFFRACT’s working directory for
subsequent importation as external objects.
MLYR and GEOP: During interactive sessions, the parameter sets for these
options may be entered from the keyboard or, alternatively, from special files
residing in the home directory. The input files for MLYR have names like
XXXX.STK, while those for GEOP have names such as XXXX.SYS.
INSP option within GEOP: To inspect a geometric-optical system, the user
selects GEOP from the main menu, then proceeds to define the system, unlessthe system has already been defined (and is therefore resident in DIFFRACT’smemory). In any event, once the geometric-optical system has been identified,the user may call upon the INSP option to inspect the system. The inspection
process is highly interactive, and requires that the user ask questions based on
the outcome of his previous queries. Consequently, the INSP option within
GEOP is allowed only in the interactive mode of DIFFRACT. The keyboard
entries during inspection of a geometric-optical system do not appear in the
COMMAND.DAT file; any attempts at invoking the INSP option during a noninteractive session will either be ignored or, more likely, will cause the program
to crash.

Guide to DIFFRACT

253

Appendix C
Coordinate System(s) of DIFFRACT
The beam in DIFFRACT™ is generally described with reference to a Cartesian
coordinate system xyz. Usually, the propagation direction is either the z-axis or
has a slight deviation from z. This xyz system is sometimes referred to as thecoordinates of incidence. The Change_Coordinate_System (CSYS) option
allows the user to shift the origin of the xyz system within the xy-plane, rotate
the xy-plane around the z-axis, or tilt xyz through the Euler angles (θ, φ, ψ). To
describe the Euler angles, we denote by x´y´z´ the coordinate system that resultsfrom the tilt operation. As shown in Fig. C1, the origins of the initial and final
coordinate systems are coincident.
(a)

x'

θ

y

ψ

z'

φ

x'

x

(b)

x

y'

z , z'zyy'Fig. C1. An arbitrarily tilted coordinate system x´y´z´ within a reference system xyz. (a) The
first two Euler angles (θ, φ) are the polar coordinates of the z´-axis within the xyz system. (b)When z´ is brought into alignment with z by keeping φ constant while shrinking θ to zero, thex´y´-plane becomes a rotated version (around the common z-axis) of the xy-plane; thecorresponding rotation angle (counterclockwise around z) is the third Euler angle ψ.

Any arbitrary orientation of x´y´z´ relative to xyz can be achieved with only tworotations in 3D-space (note: this is 2 rotations, not 3). To understand theconcept, one might think about the problem in reverse, assuming that at firstx´y´z´ has some arbitrary orientation with respect to xyz; see Fig. C1(a). Thus z´makes an angle θ with z, while its azimuthal angle in the xy-plane is φ (measured
counterclockwise from the x-axis). To bring the two coordinate systems intoalignment (while keeping their origins coincident) one must first rotate z´ toward
the z-axis. But there are many ways to do so; the way the Euler angles aredefined in DIFFRACT™ requires the azimuthal angle φ to remain constant whileθ is being reduced to zero. In this way z´ is moved to overlap with z in a uniqueway, by ensuring that the shadow of z´ on the xy-plane of the xyz system

254

Guide to DIFFRACT

maintains the same angle φ with the x–axis at all times. (φ is undefined when θ =0, but then the two z-axes are already aligned.)
After the z-axes overlap, the two x-axes may still have an angle ψ between them;in DIFFRACT™ this is Euler’s third angle. If one rotates, through the angle ψ, thex´y´-plane around the common z-axis, the two coordinates will completely
overlap. Therefore, once the z-axes are aligned, ψ will be the deviation angle ofx´ measured in the xy-plane from the x-axis. As is standard practice for righthanded coordinate systems, ψ is measured counterclockwise from the x-axis,
with the observer standing along the (common) z-axis.
Note: There exists a slight difference between DIFFRACT’s definition and what
textbooks typically define as Euler angles. In our notation, the first rotation
depends on ψ only, while the second rotation is determined by (θ, φ), everything
being referred to the incidence coordinates. In textbook definitions, our φ isreplaced with φ − ψ.
Tilted Optical Elements. In several places in DIFFRACT™, the coordinates aretilted/rotated in accordance with the aforementioned Euler angle rules. Asidefrom the Change_Coordinate_System's TILT option, these instances include:
Flat Mirror (MIRR), Diffraction Grating (GRAT), and Multilayer Stack(MLYR). The optical elements MIRR, GRAT, MULT as well as the emergent
coordinate system can be arbitrarily oriented within the xyz coordinates ofincidence. The incidence coordinate system and its point of origin are used as
reference. The Euler angles (θo, φo, ψo) specifying the coordinates of the element(MIRR, GRAT, MULT) as well as those of the emergent coordinates (θ1, φ1, ψ1)
are specified relative to the incidence coordinates.
Flat Mirror. The case of Flat Mirror (MIRR) is fairly straightforward. The
mirror’s orientation is specified in terms of its surface normal N in theincidence coordinates. N comes out of the shiny side of the mirror (as opposed togoing into the shiny side); see Fig. C2. Since the mirror is flat and isotropic, weneed only two angles to specify its surface normal: θo in the interval (90 : 180°],and φo belonging to (0 : 360°). Note that θo must be greater than 90°, otherwise
the incident beam will not reflect from the mirror. By choosing (θo, φo) the user
can set the orientation of the mirror in the xyz coordinates of incidence.
For the most part, the incident beam in DIFFRACT™ travels along the positive zaxis, but it is possible to introduce a linear phase profile on the beam (e.g.,
Seidel distortion, with coefficient C11 and orientation angle φ11), which

Guide to DIFFRACT

255

effectively sends the beam in a different direction. The user may thus specify,
with total freedom, the incident beam’s propagation direction (typicallydisplayed as an arrow at the center of the coordinate system). This arrow, whose
default direction is along z, may be given any arbitrary orientation with polar
and azimuthal angles (θ, φ).
xNFig. C2. Flat Mirror (MIRR)
in the xyz coordinates ofincidence. The mirror’ssurface normal N, whichcomes out of the reflective
facet, has polar andazimuthal angles (θo, φo).

φo

θo

z

y

Finally, the user must specify an Emergent_Coordinate_System, to which
DIFFRACT™ will switch once the beam has been reflected from the Mirror. The
emergent system is denoted by x"y"z", its origin being coincident with that of thexyz coordinates of incidence, while its orientation is given by the Euler angles(θ1, φ1, ψ1) referred to the incidence system. Ideally the emergent z-axis shouldbe aligned with the direction of the reflected beam, but the user may choose togo slightly away from this direction.
Multilayer Stack and Diffraction Grating. Almost everything described below
applies to both Multilayer (MLYR) and Grating (GRAT) features of DIFFRACT™.
For simplicity, the description will refer only to MLYR, but the effects of
NVIRON, Incidence_Medium (index = Ninc), and Substrate (index = Nsub), arethe same for MLYR and GRAT. The laws of reflection and transmission for MLYR
are fairly simple, so the following description completely defines the behavior of
MLYR. For GRAT, however, one will have to take into consideration Bragg's lawof diffraction from gratings (in reflection or in transmission).
We depict the incident beam with an arrow in the xyz coordinates of incidence,
having polar angle θ and azimuthal angle φ. Here θ [0: 90°] is measured fromthe positive z-axis, while the projection of the arrow in the xy-plane makes anangle φ [0: 360°] with x (measured counterclockwise from the positive x-axis).
This incident beam is in a homogenous medium, referred to in DIFFRACT™ as

256

Guide to DIFFRACT

the Environment (index of refraction = NVIRON). Orientation of the opticalelement (i.e., multilayer or grating) is specified by the Euler angles (θo, φo, ψo)of the x´y´z´ system attached to the element, as shown in Fig. C3. Thus the
optical element may be arbitrarily oriented within the xyz coordinates ofincidence. Because MLYR can contain birefringent and other anisotropic layers,and because GRAT is inherently anisotropic (by virture of the directionality of its
grooves), it is not sufficient to specify the optical element’s orientation simply
by specifying a surface-normal (as was the case for the Flat Mirror in thepreceding section). One must, therefore, specify the full coordinates x´y´z´ of
MLYR (or GRAT) with all three Euler angles. The allowed range of θo in thereflection mode is [0: 90°).
Fig. C3. Multilayer placed in thexyz coordinates of incidence. The
stack’s orientation within the xyz
system is specified by its own
coordinates x´y´z´ through the
Euler angles (θo, φo, ψo). z´ is thestacks’s surface-normal, entering
the stack from the side of theincidence medium and exitingfrom the substrate side. Unlike
the case of a Flat Mirror, where
the third Euler angle is irrelevant,here ψo plays a crucial role, assome of the layers may beanisotropic.

x
x'
z'

φo

Incident
Beam

θo

z

Multilayeryy'With reference to Fig. C4, the incident ray, before entering the MLYR proper,
must enter the Medium of Incidence whose refractive index is Ninc
(DIFFRACT™ requires Ninc ≥ NVIRON). The incidence medium may come into
contact with the environment (index = NVIRON) at a flat interface. This interface
is parallel to the MLYR surface, but its distance from MLYR’s front facet is not
specified in DIFFRACT. The assumption, therefore, is that the incident ray entersthrough a flat interface parallel to the MLYR surface. Before this (flat) interface,
the index of the medium in which the incident beam resides is NVIRON; after
entering through the interface the index becomes Ninc. Snell's law of refraction,
NVIRON × sin (incidence_angle) = Ninc × sin(transmitted_angle),

is used to determine the direction of the incident ray inside the incidence
medium.

Guide to DIFFRACT

257

Note: The reason the incidence_medium’s thickness is not specified in DIFFRACT™ is thatvarious thicknesses produce different amounts of aberrations on the beam. DIFFRACT™
implicitly assumes that all such aberrations are automatically corrected by the opticalelements that either precede or follow MLYR; the program, therefore, does not need to know
the thickness of the incidence medium.

Reflected BeamMultilayer
Transmitted Beam

Incidence Environment(NVIRON)

Emergence Environment(NVIRON)

Incidence
MediumSubstrate(Nsub)

Fig. C4. Multilayer stack sandwiched between a substrate of refractive index Nsub and an
incidence medium of refractive index Ninc. The stack is illuminated with a beam residing inan environment of refractive index NVIRON. Since the interface between the incidence
medium and the incidence environment is flat, the depicted situation represents the case ofspectral compression (SPEC). The reflected beam may stay inside the incidence medium (inwhich case decompression is not required), or return to the incidence environment (in whichcase spectral decompression will be automatically applied). In the transmission mode, thebeam may stay in the substrate (no decompression required), or emerge into a newenvironment. Since the depicted interface between the substrate and the emergenceenvironment is flat, spectral decompression (SPEC) will have to be specified in this case.
Note that DIFFRACT™ does not require the refractive indices of the incidence andemergence environments to be the same, even though, in the present diagram, both have been
denoted by NVIRON. If the reflected beam stays within the medium of incidence, then, uponcompletion of MLYR calculations, the new environment of the beam will have
NVIRON = Ninc. Similarly, in the transmission mode, if the emergent beam stays within thesubstrate, the new environment of the beam will have NVIRON = Nsub.

In summary, DIFFRACT™ assumes that the incident rays are bent according to
Snell's law when they enter the incidence medium (index = Ninc), but it ignores
any phase-shifts suffered by these rays as they travel through the incidence
medium to reach the MLYR surface. When the incident ray enters the incidence
medium through a flat interface, as is the case in Fig. C4, Spectral
Compression is said to have been applied. (In command.dat language,

258

Guide to DIFFRACT

Compression: SPEC.) In this context, spectrum refers to spatial-frequencies ofthe beam, not its temporal-frequencies.
Multilayer

Reflected BeamTransmitted Beam

Incidence Environment(NVIRON)

Emergence Environment(NVIRON)

Incidence
MediumSubstrate(Nsub)

Fig. C5. Similar to Fig. C4, except for the shape of the interface between the incidence
environment and the incidence medium, and also that between the substrate and theemergence environment, which are both hemispherical in the present illustration. Upon
entering the incidence medium, beam compression (BEAM) must be applied to the incident
beam. No decompression will be required if the beam stays within the incidence medium (inReflection mode), or within the substrate (in Transmission mode). If the reflected beam
returns to the incidence environment, beam decompression will be automatically applied. In
Transmission mode, if the beam exits the substrate and enters an emergence environment
where NVIRON ≠ Nsub, then beam decompression (BEAM) will have to be specified. Notethat DIFFRACT™ does not require the refractive indices of the incidence and emergenceenvironments to be the same, even though, in the present diagram, both have been denoted by
NVIRON.
DIFFRACT™ allows an alternative method of compression of the incident rays,
known as Beam Compression. (In command.dat language, Compression:
BEAM.) In this case the medium of incidence is a dielectric hemisphere (refractive
index = Ninc) whose flat facet is in contact with MLYR’s top surface, as shown in
Fig. C5. The ray in the environment (index = NVIRON) is assumed to be pointing

toward the center of the sphere (from which the hemisphere is cut). Therefore,
the incident ray does not bend as it enters this hemisphere, but upon entering thehemisphere the index of the medium in which the ray travels changes fromNVIRON to Ninc.
One may assume that in both cases discussed above (i.e., flat or hemisphericalmedium of incidence, corresponding to SPEC or BEAM compression of theGuide to DIFFRACT

259

incident beam) the incident ray makes contact with MLYR’s top surface at theorigin of the xyz coordinates of incidence.
Suppose MLYR is specified as being in the reflection mode ( R ). Then the ray isreflected from the MLYR surface in accordance with the laws of reflection,
namely, angles of incidence and reflection are the same, and the reflected andincident rays form a plane which also contains the surface normal of MLYR.
(Note: the law of reflection is different for MLYR and GRAT; the above rule isonly valid for the zero-order of diffraction from a grating).
The ray reflected from MLYR is initially in the incidence medium (index = Ninc).
The index of the emergent medium in the Reflection mode ( R ) can only be Nincor NVIRON. Depending on which emergent medium the user specifies, thereflected ray must either stay inside the incidence medium (Ninc) or return to theincidence environment (NVIRON). If SPEC was chosen as the method ofcompression of the incident beam (i.e., flat interface between NVIRON and Ninc)then the reflected ray must exit through the same flat interface to return to the
NVIRON. On the other hand, if BEAM compression was specified for entering theincidence medium, then the same hemispherical interface must be assumed to bepresent when the reflected ray leaves the incidence medium to enter the
emergence environment, which in this case (i.e., in Reflection mode) is the sameas the incidence environment (NVIRON). If the emergent environment is
specified as Ninc, Decompression is set automatically to NONE, with thereflected beam remaining within the incidence medium. In the end, the emergent
beam is either going to be in the medium of incidence (no decompression) or
returning to the environment of the incident beam (NVIRON), either with BEAM
or with SPEC decompression.
Finally, the user must specify the emergence coordinates with Euler angles (θ1,φ1, ψ1). The direction of the emergent ray (after reflection from MLYR) is now
going to be specified in the emergent coordinate system. For the most part, theuser wants to have the z-axis of the emergent system aligned with the directionof the emergent beam, but there may be cases when the user may want the z-axisof the emergent coordinates to deviate somewhat from the direction of theemergent beam. The MLYR option of DIFFRACT™ is flexible enough to allow
both possibilities.
If MLYR is operated in the transmission mode ( T ), then, instead of the reflected
beam, the program follows the transmitted beam. In this case the MLYR substrate(index = Nsub) plays the role of the incidence medium (index = Ninc) in the case

260

Guide to DIFFRACTof the reflection mode. The beam emerging from MLYR at first enters MLYR’s
substrate.
If the user decides to keep the transmitted beam in the substrate, then theemergent environment's refractive index will be specified as Nsub (i.e., emergent
NVIRON = Nsub), and no decompression will be needed for the transmitted beam(in command.dat language, Decompression: None). However, the user has theoption of specifying an alternative emergent environment, with a refractive
index smaller than Nsub. (If Nsub is already equal to 1.0, then emergent NVIRON
must be 1.0 also, as no media with refractive index below unity are allowed.)
Assuming that the emergent NVIRON (in the transmission mode) is less thanNsub, the user must specify a decompression method (SPEC or BEAM). Again,
as in the previous case, SPEC means that the interface between substrate and theemergence environment is flat, while BEAM decompression signifies a hemispherical interface between the two media, with the flat facet of the hemispherein contact with the bottom layer of MLYR, and also with the center of thehemisphere at the origin of the emergence coordinates. The transmitted ray thus
emerges from MLYR and enters either the environment of the substrate(emergent NVIRON = Nsub) or a new emergent environment (NVIRON < Nsub). In
either case, the program determines the direction of the emergent beam within
the emergence environment, then determines the orientation of this beam in theemergence coordinate system, which is specified by the Euler angles (θ1, φ1, ψ1).
Note that, in the transmission mode, the range of θ1 can only be [0: 90°).

Guide to DIFFRACT

261

Appendix D
Command blocks as they appear in COMMAND.DAT
Every time you run DIFFRACT in the interactive mode, you are given the option
to create a file called COMMAND.DAT. This file is essentially a listing of thesequence of commands and data that you enter from the keyboard. For each
option chosen from the main menu, a block of commands and data (hereinafterreferred to as command block) is entered into the COMMAND.DAT file. Some of
the menu options may have sub-options of their own, leading to different typesof command blocks depending on the particular sub-option(s) chosen.
This appendix contains a collection of all possible command blocks that can becreated in response to the user’s choices. The blocks corresponding to the 29
options offered by DIFFRACT’s main menu are listed in the same order in which
they appear in the menu. (The HELP option does not have a command block.)
Some of these options generate only one type of command block, while others
give rise to two or more different types. All possible types of command blocks
associated with a given menu option are grouped together and are listed underthat option.
In rare occasions the command blocks contain items that may not have beenentered from the keyboard during an interactive session. This is because,
depending on the choices made by the user, an item that is required under one
set of circumstances may not be needed under another. By consistently placingsuch items in COMMAND.DAT, we ensure that they will be there when needed(for instance, when the user edits the command block and alters some of its
parameters). Of course, during non-interactive runs all unnecessary parameters
encountered within COMMAND.DAT will be ignored.
The command blocks listed in this appendix are also available electronically inthe text file TEMPLATE.DAT located within the <TEMPLATES> subdirectory of
DIFFRACT’s home directory. By copying a logically-ordered sequence of thesecommand blocks from the template to an empty COMMAND.DAT file (alsoavailable in the <TEMPLATES> subdirectory), one can build simulations of any
degree of complexity, and proceed to run them in the non-interactive STEP or
ZOOM modes of the program.
1) BEAM offers six possibilities for the initial distribution: Uniform Beam (UB),
Simple Gaussian (SG), General Gaussian (GG), Laguerre-Gaussian (LG),Hermite-Gaussian (HG), and Laser Diode (LD). The corresponding blocks of262

Guide to DIFFRACTCOMMAND.DAT are listed below. Note that aberration coefficients (in the case of
UB, SG, GG, LG, and HG) appear only when the answer to Aberrations is
Seidel or Zernike. If the answer is None, no aberration coefficients will appear

in the command block. In the case of the Laser Diode beam, the only allowed
aberration is astigmatism (∆), which must be specified along with the FWHM
angular beam diameters φ and ψ.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: UB#
BCX,BCY,POWER(mW):
0.0000000
0.0000000
2.400000#
Radius of aperture:
3000.0000#
Aberrations: Zernike#
C11,Phi11:0.0000000
0.0000000#
C20:
0.2000000#
C22,Phi22:0.0000000
0.0000000#
C31,Phi31:
0.0000000
0.0000000#
C33,Phi33:
0.0000000
0.0000000#
C40:
0.0000000#
C42,Phi42:0.2500000
0.0000000#
C44,Phi44:0.5000000
10.000000#
C51,Phi51:
0.0000000
0.0000000#
C53,Phi53:
0.0000000
0.0000000#
C55,Phi55:
0.0000000
0.0000000#
C60:
0.0000000#
C62,Phi62:0.0000000
0.0000000#
C64,Phi64:
0.0000000
0.0000000#
C66,Phi66:
0.0000000
0.0000000#
C71,Phi71:
0.0000000
0.0000000#
C73,Phi73:
0.0000000
0.0000000#
C75,Phi75:
0.0000000
0.0000000#
C77,Phi77:
1.0000000
20.000000#
C80:
0.0000000#
C82,Phi82:
0.0000000
0.0000000#
C84,Phi84:
0.0000000
0.0000000#
C86,Phi86:
0.0000000
0.0000000#
C88,Phi88:
0.1000000
40.000000#
C91,Phi91:
0.0000000
0.0000000#
C93,Phi93:
0.0000000
0.0000000#
C95,Phi95:
0.0000000
0.0000000#
C97,Phi97:0.1000000
0.0000000#
C99,Phi99:
0.0000000
0.0000000#
C10,00:0.0000000#
C10,02;Phi10,02:
0.0000000
0.0000000#
C10,04;Phi10,04:
0.0000000
0.0000000#
C10,06;Phi10,06:
0.0000000
0.0000000#
C10,08;Phi10,08:
0.0000000
0.0000000#
C10,10;Phi10,10:
0.0000000
0.0000000#
C11,01;Phi11,01:
0.0000000
0.0000000#
C11,03;Phi11,03:
0.0000000
0.0000000#
C11,05;Phi11,05:
0.0000000
0.0000000#
C11,07;Phi11,07:
0.0000000
0.0000000#
C11,09;Phi11,09:
0.0000000
0.0000000#
C11,11;Phi11,11:0.0000000
0.0000000#
C12,00:0.0000000#
C12,02;Phi12,02:0.0000000
0.0000000#
C12,04;Phi12,04:0.0000000
0.0000000#
C12,06;Phi12,06:
0.0000000
0.0000000#
C12,08;Phi12,08:0.0000000
0.0000000#

Guide to DIFFRACT
C12,10;Phi12,10:
C12,12;Phi12,12:
Polarization RHO,ETA:
NMAX,NMAY:
LMAX,LMAY:

263
0.0000000
0.0000000
12.000000
512
512
30000.000

0.0000000
0.0000000
34.000000
30000.000

#
#
#
#
#

$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.7800
NVIRON: 1.550000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,POWER(mW):
120.00000
0.0000000
1.500000#
1/e radii Rmin,Rmax:
2000.0000
3500.0000#
RA,THETA:
5000.0000
45.000000#
Aberrations: Seidel#
Spherical C40:
1.0000000#
Coma C31,Phi31:
0.0000000
0.0000000#
Astigmatism C22,Phi22:
1.0000000
30.000000#
Curvature C20:0.50000000#
Distortion C11,Phi11:0.25000000
30.000000#
Polarization RHO,ETA:
0.0000000
45.000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
40000.000
40000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: GG#
BCX,BCY,POWER(mW):
0.0000000
0.0000000
1.000000#(a1,a2):0.1000000
0.5000000#(b1,b2):-0.0500000-0.2000000#(c1,c2):0.2000000-1.0000000#
Aperture radius RA:
5.0000000#
Aberrations: None#
Polarization RHO,ETA:
22.500000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
25.000000
25.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 1.5500
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: LG#
BCX,BCY,POWER(mW):
0.0000000
0.0000000
10.00000#
Order parameters (Nr,Na): 50#
R0,Theta:
1000.0000
45.000000#
Radius of aperture:
5000.0000#
Aberrations: None#
Polarization RHO,ETA:
0.0000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
30000.000
30000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #264

Guide to DIFFRACT_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: HG#
BCX,BCY,POWER(mW):
0.0000000
0.0000000
1.500000#
Order parameters (Nx,Ny): 5
3#Base 1/e radii RX0,RY0:2000.0000
3000.0000#
RA,THETA:
10000.000
45.000000#
Aberrations: None#
Polarization RHO,ETA:
0.0000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
30000.000
30000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):532.0000
NVIRON: 1.500000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: LD#
BCX,BCY,POWER(mW):
0.0000000
0.0000000
3.000000#
PHI,PSI,DELTA:
11.000000
27.000000
5.000000#
Orientation angle THETA:
0.0000000#
Distance from cavity Z0:
6000.0000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
30000.000
30000.000#

2) READ is used when a previously saved cross-section of the beam is to beimported from the hard disk file XXXX.DST. The upper-bound NDIM on the meshdimensions must be greater than or equal to NMAX, NMAY of the saved
distribution. If the saved cross-section of the beam happens to contain thecorresponding ray distribution, then the ray distribution will also be read into themesh provided that the ray-trace option of the program has been enabled.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Retrieve a distribution: READ#
File name: ABCD.DST#

3) Propagate_in_Environment allows the user to propagate the beam in one ofseveral modes. Forward and backward propagation (i.e., positive and negative
propagation distances) are allowed in all cases. If the beam happens to carry acurvature outside the mesh, the user will be given the option to multiply thecorresponding curvature phase-factor into the mesh before proceeding further. If
the user decides to keep the curvature out of the mesh, or if the propagation
distance is zero, no further questions will be asked. If, on the other hand, themultiplication option is chosen, or if the beam has no outside curvature to begin
with, DIFFRACT will want to know the propagation regime and the desired modeof treatment for polarization (i.e., Scalar or Quasi-vector).

Guide to DIFFRACT

265

When the propagation regime is specified as Extended Fresnel (XFRNL) the user
should specify a value in the interval [0.0, 1.0) for the parameter Smax, which is
the magnitude of the largest spatial frequency contained in the beam. (Setting
Smax = 0.0 would allow the program to proceed with its internal estimate of thisparameter.) When the propagation regime is specified as Direct Fresnel (DFRNL)
of Fraunhofer (FRNHF), Smax is not needed, and the program will simply ignore
the specified value.
Another question concerns the positioning of the beam center. If the beam
centroid happens to be at the origin of the XY-plane, or if the user’s answer tothe question Reposition the beam? is No, the beam will be propagated as is.
If, on the other hand, the beam centroid is at (xc, yc) ≠ (0, 0) and the user’s
answer is Yes, the beam will be shifted to the center of coordinates, propagated
the specified distance, then shifted back to its original position.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Propagate in environment: PROP
(Length_Units: mm) #Propagation distance:
10.0000#
Multiply curvature (Y/N): N#
Reposition beam (Y/N): N#
Propagation regime: DFRNL#
Smax:
0.0000000#
Scalar/Quasi-vector: SC#

4) Lens_Action enables the user to choose a lens from a list of seven different
lenses for placing in the optical path.
Positive_focusing lens brings a collimated beam to focus. The user must choose
the method of calculation, which is either the “exact”, multiple-FFT method(EXCT), or the stationary-phase approximation (APRX). In the Student Edition,
however, the EXCT method is not available. In general, the Positive_focusinglens satisfies Abbe’s sine condition. The lens may be aberrated, with the Seidel
or Zernike aberration coefficients appearing in the command block. When theanswer to the Aberrations question is None, these coefficients must be
removed from the block.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: mm) #Type: PFOC#
LCX,LCY:
0.0000000
0.0000000#

266

Guide to DIFFRACT
NA,FL:
0.5000000
Aberrations: Seidel
Spherical C40:
0.0000000
Coma C31,PHI31:
0.5000000
Astigmatism C22,PHI22:
1.2000000
Curvature C20:
0.0000000
Distortion C11,PHI11:
0.0000000
Scalar/Quasi-vector: QV
Calculation method: APRX
Propagation distance:
7.0050000$
Lens: LENS
Type: PFOC
LCX,LCY:
0.0000000
NA,FL:
0.5000000
Aberrations: Zernike
C11,Phi11:0.0000000
C20:
0.0000000
C22,Phi22:0.0000000
C31,Phi31:-0.3400000
C33,Phi33:
0.0000000
C40:
0.0000000
C42,Phi42:0.0000000
C44,Phi44:0.0000000
C51,Phi51:
0.0000000
C53,Phi53:
0.0000000
C55,Phi55:
0.0000000
C60:
0.0000000
C62,Phi62:0.3400000
C64,Phi64:
0.0000000
C66,Phi66:
0.0000000
C71,Phi71:
0.0000000
C73,Phi73:
0.0000000
C75,Phi75:-0.1000000
C77,Phi77:
0.0000000
C80:
0.0000000
C82,Phi82:
0.0000000
C84,Phi84:
0.0000000
C86,Phi86:
8.00000000E-02
C88,Phi88:
0.0000000
C91,Phi91:
0.0000000
C93,Phi93:
0.0000000
C95,Phi95:
0.0000000
C97,Phi97:0.1000000
C99,Phi99:
0.0000000
C10,00:0.0000000
C10,02;Phi10,02:
0.0000000
C10,04;Phi10,04:
0.0000000
C10,06;Phi10,06:
0.0000000
C10,08;Phi10,08:
0.0000000
C10,10;Phi10,10:
0.0000000
C11,01;Phi11,01:
0.0000000
C11,03;Phi11,03:
0.0000000
C11,05;Phi11,05:
0.0000000
C11,07;Phi11,07:
0.0000000
C11,09;Phi11,09:
0.0000000
C11,11;Phi11,11:0.0000000
C12,00:0.0000000
C12,02;Phi12,02:0.0000000
C12,04;Phi12,04:0.0000000
C12,06;Phi12,06:
0.0000000
C12,08;Phi12,08:0.0000000
C12,10;Phi12,10:0.0000000
C12,12;Phi12,12:0.0000000
Scalar/Quasi-vector: QV
Calculation method: APRX
Propagation distance:
6000.0000

7.0000000
30.000000
45.000000
0.0000000

#
#
#
#
#
#
#
#
#
#

#
(Length_Units: um) ##0.0000000#
6000.0000##
0.0000000##
0.0000000#
0.0000000#
0.0000000##
0.0000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000##
23.000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000#
45.000000#
0.0000000##
0.0000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000##
0.0000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000##
0.0000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000#
0.0000000####

Guide to DIFFRACT

267

Collimator is treated essentially as a phase-object. The sign of the focal length
FL determines whether the lens is positive or negative. The curvature of theincident beam must be very close to 1/FL, or else the sampling of the emergent
beam’s curvature may be too coarse. It is assumed that the lens satisfies Abbe’s
sine condition. The Collimator may be aberrated, with the Seidel or Zernike
aberration coefficients appearing in the command block. When the answer to the
Aberrations question is None, these coefficients must be removed from theblock.
The beam enters the lens at the entrance pupil and emerges at the exit pupil (i.e.,
the propagation distance is zero). To propagate the beam beyond the exit pupil,
one must use the PROP option.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: mm) #Type: COLL#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.5000000
5.5000000#
Aberrations: Seidel#
Spherical C40:
0.0000000#
Coma C31,Phi31:
1.0000000
45.000000#
Astigmatism C22,Phi22:0.0000000
0.0000000#
Curvature C20:
0.0000000#
Distortion C11,Phi11:0.0000000
0.0000000#
Scalar/Quasi-vector: QV#$
Lens: LENS
Type: COLL
LCX,LCY:.00000000
NA,FL:.50000000
Aberrations: None
Scalar/Quasi-vector: QV#(Length_Units: um) ##.00000000#
6000.0000###

General_purpose_Lens is treated as a phase-object. It is assumed that the lensis corrected for given object and image distances, and that it satisfies Abbe’s
sine condition. The sign of the focal length FL determines whether the lens is
positive or negative. A point source object, located on-axis and at the correct
distance from the lens, will emit a cone of light that is captured by the lens andis refocused to the conjugate image point. The numerical aperture of thecaptured incident cone is NA1, and that of the image cone is NA2. The values of
NA1 and NA2 may be positive or negative, depending on whether the point
source and its image are real or virtual. The only physical constraint on theselens parameters is that FL (NA1 + NA2) must be positive. The lens may be268

Guide to DIFFRACT

aberrated, with the Seidel or Zernike aberration coefficients appearing in thecommand block. When the answer to the Aberrations question is None, thesecoefficients must be removed from the block.
The beam going through the General_purpose lens emerges at the exit pupil, thatis, the propagation distance is zero. The user has the option of multiplying the
emergent beam’s curvature into the mesh. To propagate the beam beyond theexit pupil, one must use the PROP option.
Note: The General_purpose lens is not available in the Student Edition.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: um) #Type: GENL#
LCX,LCY:
0.0000000
0.0000000#
NA1,NA2,FL:
0.5000000
0.2000000
6000.0000#
Aberrations: Seidel#
Spherical C40:
0.0000000#
Coma C31,Phi31:
0.1000000
45.000000#
Astigmatism C22,Phi22:0.0000000
0.0000000#
Curvature C20:0.2300000#
Distortion C11,Phi11:0.0000000
0.0000000#
Scalar/Quasi-vector: QV#Multiply curvature (Y/N): N#

Cylindrical_Lens simulates the action of an ideal, positive cylindrical lens on
the incident beam. The clear aperture of the lens is a circle of radius RA, and theorientation angle THETA is in the interval (-90º, 90º). The cylindrical lens istreated in the stationary-phase approximation.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: um) #Type: CYLN#
LCX,LCY,THETA:.00000000.00000000
90.000000#
RA,FL:
3000.0000
10000.000#
Propagation distance:
10005.000#
Scalar/Quasi-vector: QV#

Astigmatic_Lens simulates the action of an astigmat on the incident beam. Thislens is ideal, in the sense that it has no other aberrations, and is treated in thestationary-phase approximation.

Guide to DIFFRACT

269$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: um) #Type: ASTG#
LCX,LCY,THETA:
10.000000-20.000000
45.000000#
NA,FL1,FL2:
0.1500000
20000.000
21000.000#
Propagation distance:
20500.000#
Scalar/Quasi-vector: SC#

Toric_Ring_Lens simulates the action of an aberration-free toric_ring lens on
the incident beam. In general, the toric_ring is treated as a phase-object,
although two methods are available for beam propagation beyond the lens.
When the lensing action is mild (i.e., when the numerical aperture of the lens issmall), the beam may be readily treated in its appropriate regime of propagationsuch as near-field (DFRNL), intermediate-field (XFRNL), or far-field (FRNHF).
Under certain circumstances, however, if the phase-factor that the lens imparts
to the beam is factorized, the constraint of small numerical aperture may berelaxed. These two methods of computing beam propagation, direct phase-object(DPHOB) and extended phase-object (XPHOB), require somewhat different
command blocks, as shown below.
Note: The Toric_ring lens is not available in the Student Edition.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: um) #Type: TRNG#
LCX,LCY:.00000000.00000000#
R0,RING,RA:.00000000
1500.0000
3000.0000#
FL:
300000.00#
Calculation method: DPHOB#
Propagation distance: 300000.00#
Propagation regime: DFRNL#
Smax: 1.51649800E-02#
Scalar/Quasi-vector: SC#$
Lens: LENS
Type: TRNG
LCX,LCY:.00000000
R0,RING,RA:
50.000000
FL:
14000.000
Calculation method: XPHOB
Propagation distance: 14010.000
Scalar/Quasi-vector: QV#(Length_Units: um) ##.00000000#
70.000000
3000.0000
#
#
#
#
#

270

Guide to DIFFRACT

Split_Lens simulates the action of a split lens on the incident beam. The two
halves of this lens, assumed to be free of all aberrations, are treated separately
and independently, in the stationary-phase approximation.
Note: The Split _Lens is not available in the Student Edition.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: um) #Type: SPLN#THETA:
45.000000#
NA,FL:
0.5000000
6000.0000#
Upper half: LCX,LCY:
10.000000
10.000000#
Lower half: LCX,LCY:-10.000000-10.000000#
Propagation distance:
6000.0000#
Scalar/Quasi-vector: QV#5) Measure allows the measurement of peak_intensity and full width at halfmaximum intensity (FWHM) of the beam. Before returning to the main menu, theuser may compute the FWHM in as many directions within the XY-plane as isdesired. Typically, the intensity is the sum of the intensities of the X- and Ycomponents of polarization (these are always present in the mesh), but the user
may request to include the Z-component, if deemed necessary. The Zcomponent’s intensity is then added to those of the X- and Y-components before
the beam’s peak_intensity and FWHM are computed.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Measure peak and FWHM: MEAS
(Length_Units: um) #Include Z-component: Y#
ID numbers of PEAK I,X,Y:
1
2
3#
Compute/Return: C#
Theta:
0.0000000#
ID number for FWHM:
4#
Compute/Return: C#
Theta:
90.000000#
ID number for FWHM:
5#
Compute/Return: R#

6) MASK allows the insertion of elliptical, annular, and rectangular masks inthe beam’s path. Each of the various regions of the mask multiplies the incident
beam amplitude with a constant positive coefficient, and shifts the phase of thebeam by an angle in the range of [-180° : 180°]. In addition to the mask center
coordinates (MCX, MCY) and various dimensions, the parameter α belonging to
the interval [0, 1] is used to specify the range over which the smooth transition

Guide to DIFFRACT

271

from one region of the mask to a neighboring region takes place. For example, if
the mask is a circle of radius R, the transition from the inside amplitude
A1exp(iφ1) to the outside amplitude A2exp(iφ2) occurs between the radii (1 – α) R
and (1 + α) R.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Amplitude/phase mask: MASK
(Length_Units: mm) #Shape: Circle#
MCX,MCY: 0.00000000 0.00000000#
Circle radius,Alpha: 6.15000000 0.20000000#
Inside amplitude,phase: 1.00000000 23.0000000#
Outside amplitude,phase: 0.50000000 45.0000000#
Aberrations: None#$
Amplitude/phase mask: MASKShape: Circle
MCX,MCY: 0.00000000
Circle radius,Alpha: 5.30000000
Inside amplitude,phase: 1.00000000
Outside amplitude,phase: 0.00000000
Aberrations: Seidel
Spherical C40: 0.00000000
Coma C31,PHI31: 0.10000000
Astigmatism C22,PHI22: 0.00000000
Curvature C20: 0.00000000
Distortion C11,PHI11: 0.25000000$

0.30000000
0.00000000
34.0000000
0.00000000
0.00000000
0.00000000
66.0000000#(Length_Units: mm) #
#
#
#
#
#
#
#
#
#
#
#

Amplitude/phase mask: MASKShape: Ellipse
MCX,MCY:
100.00000
Rmax,Rmin,Alpha:2000.0000
Theta:
45.000000
Inside amplitude,phase:
1.0000000
Outside amplitude,phase:
0.5000000#(Length_Units: um) ##100.00000#
1500.0000
0.10000000##
30.000000#
90.000000#

Amplitude/phase mask: MASKShape: Annulus
MCX,MCY:
0.0000000
Inner circle R1,Alpha1:1500.0000
Outer circle R2,Alpha2:
3000.0000
Inside amplitude,phase:
1.0000000
Annulus amplitude,phase:
0.5000000
Outside amplitude,phase:
0.7000000(Length_Units: um) ##0.0000000#
0.1000000#
0.0500000#
0.0000000#
45.000000#
90.000000#

Amplitude/phase mask: MASKShape: Rectangle
MCX,MCY:
10.000000
Length,Width,Alpha:4000.0000
Orientation angle Theta:
45.000000
Inside amplitude,phase:
1.0000000
Outside amplitude,phase:
0.8000000(Length_Units: um) ##-30.000000#
2000.0000
0.15000000##
0.0000000#
90.000000#

272

Guide to DIFFRACT

7) Polarization_Element allows the simulation of polarizers, analyzers, waveplates, and leaky polarizing beam-splitters.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Polarization element: POLE
(Length_Units: um) #ACX,ACY:
10.000000
10.000000#
RA,THETA:
3000.0000
56.000000#
ALPHA,PHIA:.70000000
23.000000#
BETA,PHIB:.90000000
165.00000#

8) Flat_Mirror simulates the action on the incident beam of a flat metallic or
dielectric mirror. The mirror may have any arbitrary orientation in threedimensional space. The emergent coordinate system must be chosen such thatthe reflected beam will continue to propagate either along or very close to the(new) optical axis.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):1.000000
NVIRON: 1.000000 #_________________________________________________________________________ ##Flat mirror: MIRR
(Length_Units: mm) #Complex index (n,k):0.6550000
6.9900000#
Normal Theta0,Phi0:
135.00000
0.0000000#
Emergent Theta,Phi,Psi:
90.000000
0.0000000
0.0000000#
Propagation distance:
0.3250000#
Propagation regime: DFRNL#
Smax:
0.0010000#

9) Multilayer_Structure places a multilayer stack in the beam path. The
maximum number of layers allowed is presently limited to 10 in the Student
Edition and to 99 in the General Edition. Each layer can be a dielectric, metal,
birefringent, magneto-optical, or an arbitrary material with a known dielectric
tensor. When the material of a given layer is anisotropic (i.e., type C, D, or E),
the user will have the option to reorient the principal axes of the layer’s
dielectric tensor. This is done by specifying the Euler angles (θ, φ, ψ)
immediately after the tensor elements. When the principal axes are not to berotated, the Euler angles will be set to zero.
In Reflection mode, DIFFRACT automatically applies the same decompression to
the reflected beam that the user has specified for the incident beam. The answer
to the query Decompress emergent: is, therefore, ignored. In transmissionmode, compression of the incident beam and decompression of the transmitted

Guide to DIFFRACT

273

beam, being independent of each other, are carried out exactly as requested by
the user. The top surface of the incident medium and the bottom surface of thesubstrate may be specified as BARE (for uncoated) or COAT (for antireflection
coated).
Orientation of the Multilayer within the incident coordinates is specified by thestack’s Euler angles (θ, φ, ψ). Similarly, the emergent coordinate system is
specified by its Euler angles within the incident coordinates.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Multilayer structure: MLYR
(Length_Units: nm) #Nsub:
1.5000000#
Number of layers: 5#
Layer#01
Type (A-E): A#n,k:2.0000000
7.0000000#
Thickness:
100.00000#
Layer#02
Type (A-E): B#
Real(E),Imag(E):2.2500000.00000000#
Thickness:
120.00000#
Layer#03:
Type (A-E): C#
Real(EXX),Imag(EXX):
2.7000000.00000000#
Real(EYY),Imag(EYY):
2.6000000.10000000#
Real(EZZ),Imag(EZZ):
2.4500000
1.00000000E-02#
Thickness:
500.00000#
Euler Theta,Phi,Psi:
45.000000
90.000000
30.000000#
Layer#04:
Type (A-E): D#
Real(EXX),Imag(EXX):-5.0000000
20.000000#
Real(EXY),Imag(EXY):.50000000
5.00000000E-03#
Real(EXZ),Imag(EXZ):.00000000.00000000#
Real(EYZ),Imag(EYZ):.00000000.00000000#
Thickness:
15.000000#
Euler Theta,Phi,Psi:.00000000.00000000.00000000#
Layer#05:
Type (A-E): E#
Real(EXX),Imag(EXX):
2.7000000.10000000#
Real(EXY),Imag(EXY):.00000000.00000000#
Real(EXZ),Imag(EXZ):.00000000.00000000#
Real(EYX),Imag(EYX):.00000000.00000000#
Real(EYY),Imag(EYY):
2.7000000.10000000#
Real(EYZ),Imag(EYZ):.00000000.00000000#
Real(EZX),Imag(EZX):.00000000.00000000#
Real(EZY),Imag(EZY):.00000000.00000000#
Real(EZZ),Imag(EZZ):
2.9000000
5.00000000E-02#
Thickness:
100.00000#
Euler Theta,Phi,Psi:
30.000000
70.000000
90.000000#
Ninc:
1.5000000#
Compress incident: BEAM#
Entrance surface: BARE#
Reflect or Transmit: R#
Emergent environment's n:
1.0000000#
Decompress emergent: BEAM#
Exit surface: BARE#
Stack's Theta,Phi,Psi:
45.000000
180.00000
234.00000#
Emergent Theta,Phi,Psi:
90.000000.00000000.00000000#
Propagation distance:
6000.0000#
Propagation regime: FRNHF#
Smax:.12500000#

274

Guide to DIFFRACT

10) Optical_Disk_Surface allows the user to place the surface of an optical disk
in the path of the beam. Four types of optical disks are allowed: Read-only,
Writable, Magneto-optical, and Dye-polymer. The user must define the various
features of the disk surface such as grooves, preformat pits, and data marks.
(These features can be inspected by requesting a display of the disk surface on
the monitor.) Subsequently, the user specifies the propagation distance and theappropriate regime for propagation. The mode of polarization propagation in thecase of an Optical_Disk_Surface is Scalar; the user will not be given the optionof Quasi-vectorial treatment.
Note: Writable, Magneto-optical, and Dye-polymer typesof media are not available in the Student Edition.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):685.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units: um) #Type of medium: RO#
SCX,SCY,RA:
0.0000000
0.0000000
20.000000#(Incidence medium) Ninc:
1.5700000#
Reflection coefficient:
0.9400000#
Number of marks: 2#
Mark # 1:
MCX,MCY:.00000000.00000000#
THETA:
45.000000#
Round or Flat: R#
Length, Width, Height:
3.0000000
1.0000000
0.1000000#
Ratio:
0.0000000#
Mark # 2:
MCX,MCY:
3.0000000
3.0000000#
THETA:
45.000000#
Round or Flat: F#
Length, Width, Height:
2.0000000
0.9000000-0.1000000#
Ratio:
0.0000000#
Inspect surface pattern: Y#
Xmin,Xmax:-5.0000000
5.0000000#
Ymin,Ymax:-5.0000000
5.0000000#
Color or Gray-scale: C#
Save data file (Y/N): N#
File identifier: 0#
Propagation distance:
6000.0000#
Propagation regime: FRNHF#
Smax: 0.100000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units: um) #Type of medium: WR#
SCX,SCY,RA:
0.0000000
0.0000000
20.000000#(Incidence medium) Ninc:
1.5700000#
Pregrooves: Y#
A,B,C:
0.2000000
0.4000000
0.6000000#
D,ZETA:
2.0000000
0.1250000#
Reflection coeffs. RL,RG:
1.0000000
1.0000000#
THETA,GCX,GCY:45.000000-0.5000000
0.5000000#
Wobble (Sine,Blip,None): Sine#Guide to DIFFRACT
Central wobble A0,P0,Q0:
0.1000000
Adjacent wobble A1,P1,Q1:
0.1000000
Marks (bumps/pits): Y
Number of marks: 2
Mark # 1:
MCX,MCY:
0.0000000
THETA:
45.000000
Round or Flat: R
Length, Width, Height:
3.0000000
Roof-to-Rim Ratio:
0.0000000
Reflection coefficient:
0.7000000
Mark # 2:
MCX,MCY:
4.0000000
THETA:
45.000000
Round or Flat: F
Length, Width, Height:
2.0000000
Roof-to-Rim Ratio:
0.8000000
Reflection coefficient:
0.5000000
Inspect surface pattern: Y
Xmin,Xmax:-6.0000000
Ymin,Ymax:-6.0000000
Color or Gray-scale: C
Save data file (Y/N): N
File identifier: 0
Propagation distance:
6000.0000
Propagation regime: FRNHF
Smax:
0.1300000

275
8.0000000
8.0000000

0.0000000
180.00000

0.0000000
1.2000000

0.2500000

4.0000000
1.2000000

6.0000000
6.0000000

0.2500000

#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#

$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units: um) #Type of medium: MO#
SCX,SCY,RA:
0.0000000
0.0000000
20.000000#(Incidence medium) Ninc:
1.5700000#|Rp|,|Rs|,PHIsp:
0.5000000
0.0070000
12.000000#
Pregrooves: Y#
A,B,C:
0.2000000
0.4000000
0.6000000#
D,ZETA:
2.0000000
0.1250000#
THETA,GCX,GCY:45.000000-0.5000000
0.5000000#
Wobble (Sine,Blip,None): Blip#Central blip's A0,W0:
0.250000
1.000000#
Adjacent blip's A1,W1:
0.350000
1.800000#
Blip's period,phase step:
8.000000
90.00000#
Preformat marks: Y#
Number of marks: 2#
Mark # 1:
MCX,MCY:
0.0000000
0.0000000#
THETA:
45.000000#
Round or Flat: R#
Length, Width, Height:
3.0000000
1.2000000
0.2500000#
Roof-to-Rim Ratio:
0.0000000#
Mark # 2:
MCX,MCY:
3.0000000
3.0000000#
THETA:
45.000000#
Round or Flat: F#
Length, Width, Height:
2.0000000
1.2000000
0.2500000#
Roof-to-Rim Ratio:
0.9000000#
Magnetic domains: Y#
Number of marks: 2#
Type (Regular/Chevron): R#
Mark # 1:
MCX,MCY:-1.4000000
1.4000000#
Length, Width, Angle:
4.0000000
1.2000000
45.000000#
Mark # 2:
MCX,MCY:
1.4000000-1.4000000#
Length, Width, Angle:
5.0000000
1.2000000
45.000000#
Inspect surface pattern: Y#
Xmin,Xmax:-6.0000000
6.0000000#
Ymin,Ymax:-6.0000000
6.0000000#
Color or Gray-scale: C#
Save data file (Y/N): N#
File identifier: 0#
Propagation distance:
6000.0000#
Propagation regime: FRNHF#
Smax:
0.8300000#

276

Guide to DIFFRACT$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):405.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units: um) #Type of medium: DP#
SCX,SCY,RA:
0.0000000
0.0000000
20.000000#(Incidence medium) Ninc:
1.5600000#
Reflect or Transmit: R##
Number of layers: 2#
Layer #01: n,k,THICKNESS:
2.3000000
0.0000000
0.0360000#
Layer #02: n,k,THICKNESS:
1.4000000
0.0000000
0.5300000#
Medium beyond last (n,k):1.6000000
0.1000000#
Emergent environment's n:
1.0000000##
Pregroove sets (0-4): 1##
G1:Modified layers L1,L2: 0
2#****************** A,B,C:
0.0000000
2.0000000
2.0000000#
D,Zeta:4.0000000
0.1250000#
Theta,GCX,GCY:45.000000
0.7070000-0.7070000#
Wobble (Sine,Blip,None): None#
Include in display (Y/N): Y##
Mark sets (0-4): 1##
SET 1....Number of marks: 2#
Modified layers L1,L2: 2
4#
Mark #01:
MCX,MCY:
2.0000000
2.0000000#
Orientation angle THETA:
45.000000#
Round or Flat: R#
Length,Width,Height:
3.0000000
1.5000000
0.1250000#
Roof-to-Rim ratio:
0.0000000#
Mark #02:
MCX,MCY:-3.0000000-2.0000000#
Orientation angle Theta:
45.000000#
Round or Flat: F#
Length,Width,Height:
4.0000000
2.0000000-0.1200000#
Floor-to-Rim ratio:
0.6500000#
Include in display (Y/N): Y##
Inspect surface pattern: Y#
Xmin,Xmax:-8.0000000
8.0000000#
Ymin,Ymax:-8.0000000
8.0000000#
Color or Gray-scale: C#
Save data file (Y/N): N#
File identifier: 0#
Propagation distance:
6000.0000#
Propagation regime: FRNHF#
Smax:
0.0600000#

11) Diffraction_Grating enables the user to place in the path of the incident
beam a phase/amplitude grating. Both Reflection and Transmission type gratings
are allowed. Two methods are available for specifying the grating parameters.
The first method requires the importation of grating’s reflection/transmission
coefficients from a data-file XXXX.DAT, which may be an ASCII or a Binary
file. The second method (which treats diffraction from gratings in the scalar
approximation) prompts the user for the trapezoidal cross-section of the grating.
In both cases the user specifies the refractive indices of the medium of incidence
and of the substrate. If the indices differ from unity, the user will be promptedGuide to DIFFRACT

277

for the shape of the entrance/exit surfaces: flat surfaces cause compression/
decompression of the spectrum, whereas hemispherical surfaces cause beamcompression/ decompression. Both types of entrance/exit surfaces may be
specified as BARE (for uncoated) or COAT (for antireflection-coated).
The grating’s surface-normal may be specified in an arbitrary direction in spaceby the Euler angles (θ0, φ0, ψ0) relative to the coordinates of incidence. The user
must also specify the orientation of the emergent coordinate system by its
corresponding Euler angles (θ1, φ1, ψ1).
Note: The method of importing reflection/transmission coefficientsfrom a data-file is not available in the Student Edition.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Diffraction grating: GRAT
(Length_Units: um) #Filename: ABCD.DAT#
Type of file: Binary#Compress incident: NONE#
Entrance surface: BARE#
Emergent environment's n:
1.5000000#
Decompress emergent: NONE#
Exit surface: BARE#
Grating Theta,Phi,Psi:
45.000000
180.000000.00000000#
Emergent Theta,Phi,Psi:
90.000000.00000000.00000000#
Propagation distance: 3750.0000#
Propagation regime: FRNHF#
Smax: 0.99000000#$
Diffraction grating: GRAT
Filename: TEST.DAT
Type of file: ASCII
Compress incident: SPEC
Entrance surface: BARE
Emergent environment's n:
1.0000000
Decompress emergent: BEAM
Exit surface: COAT
Grating Theta,Phi,Psi:.00000000
Emergent Theta,Phi,Psi:.00000000
Propagation distance: 4000.0000
Propagation regime: FRNHF
Smax: 0.70311100$
Diffraction grating: GRAT
Reflection/Transmission: R
Parameters: A,B,C:.00000000
D,Period,Zeta:
2.0000000
AL,AG:.70000000
Nsub,Ninc:
1.0000000
Compress incident: BEAM
Entrance surface: COAT
Emergent environment's n:
1.0000000
Decompress emergent: BEAM#
(Length_Units: um) #
#
#
#
#
#
#
#
.00000000
45.000000#.00000000.00000000#####(Length_Units: um) ##1.0000000
1.5000000#
4.0000000.12500000#.80000000#
1.5000000
#
#
#
#
#

278

Guide to DIFFRACT
Exit surface: COAT
Grating Theta,Phi,Psi:.00000000
Emergent Theta,Phi,Psi:
180.00000
Propagation distance: 3000.0000
Propagation regime: FRNHF
Smax: 0.16000000$
Diffraction grating: GRAT
Reflection/Transmission: T
Parameters: A,B,C:.00000000
D,Period,Zeta:
2.0000000
AL,AG:.50000000
Nsub,Ninc:
1.5000000
Compress incident: BEAM
Entrance surface: COAT
Emergent environment's n:
1.3300000
Decompress emergent: SPEC
Exit surface: BARE
Grating Theta,Phi,Psi:.00000000
Emergent Theta,Phi,Psi:.00000000
Propagation distance: 3750.0000
Propagation regime: FRNHF
Smax: 0.16000000.00000000.00000000

45.000000.00000000

#
#
#
#
#
#

#
(Length_Units: um) ##1.0000000
1.5000000#
4.0000000.25000000#.60000000#
2.0000000
#
#
#
#
#
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.00000000
45.000000#.00000000.00000000####

12) Launch Guided Mode creates either a Laguerre_Gaussian or a Hermite_
Gaussian beam having user-specified parameters, and computes the overlap
integral between the incident beam (i.e., the present content of the mesh) and thespecified mode. The program then proceeds to replace the contents of the meshwith the specified Laguerre_Gaussian (or Hermite_Gaussian) mode, setting themode’s complex amplitude equal to the overlap integral. In this way the
specified mode is launched with the proper amplitude and phase, as it would in a
waveguide that supports the desired mode.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Launch guided mode: MODE
(Length_Units: um) #Laguerre/Hermite Gauss: LG#
MCX,MCY:
0.0000000
0.0000000#
Order parameters (Nr,Na): 4
2#
1/e radius R0,Theta:
20.000000
12.000000#
Radius of aperture:
350.00000#
Multiply curvature (Y/N): N#$
Launch guided mode: MODE
Laguerre/Hermite Gauss: HG
MCX,MCY:
10.000000
Order parameters (Nx,Ny): 5
7Base 1/e radii RX0,RY0:
10.000000
RA,THETA:
150.00000
Multiply curvature (Y/N): N#(Length_Units: um) ##20.000000##
15.000000#
45.000000##

Guide to DIFFRACT

279

13) Rescale_the_Mesh allows changing of the mesh in the XY-plane. The
number of pixels NMAX0 and NMAY0 along the X- and Y-axes must be less than orequal to the declared upper bound NDIM on mesh size.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Rescale the mesh: RSCL
(Length_Units: um) #XC,YC,THETA:
10.000000-5.0000000
45.000000#
NMAX0,NMAY0: 256
256#
LMAX0,LMAY0:30.000000
30.000000#

14) Change_Coordinate_System allows the user to switch to a new Cartesian
coordinate system by Shifting, Rotating, or Tilting the current system of
coordinates.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Change coordinate system: CSYS
(Length_Units: mm) #Shift/Rotate/Tilt: S#
XC,YC:
0.0200000-0.1000000#$
Change coordinate system: CSYS
Shift/Rotate/Tilt: R
THETA:
300.00000

#
#
#
#

$
Change coordinate system: CSYS
Shift/Rotate/Tilt: T
THETA0,PHI0,PSI0:.10000000####

90.000000

20.000000

15) Weak_Biprism enables the user to place a mild biprism in the XY-plane.
The two prisms must be non-overlapping, but otherwise they are independent ofeach other.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Weak biprism: BIPR
(Length_Units: um) #Prism #1:
PCX,PCY:
1000.0000
1000.0000#
L,W,PSI:
1200.0000
700.00000
45.000000#
THETA,PHI:.10000000
90.000000#
Prism #2:
PCX,PCY:-1000.0000-1000.0000#
L,W,PSI:
1400.0000
600.00000-45.000000#
THETA,PHI:.10000000
225.00000#

280

Guide to DIFFRACT

16) Geometric-optical_System enables the user to define an optical system
consisting of an arbitrary number of Glass, Mirror, and Aperture surfaces, plus a
Destination plane, and to trace the incident beam’s rays through this system.
(The maximum allowed number of surfaces is 10 in the Student Edition and 99
in the General Edition of DIFFRACT.) There exist two types of command blockwith the GEOP heading. The first type is used to define a geometric-opticalsystem, by invoking the NSYS sub-option. The second type invokes the TRAY
sub-option to trace the entire set of rays (corresponding to an incident beam)through a previously-defined system.
Within the NSYS sub-option, any number of surfaces may be placed in thegeometric-optical system. The four types of surfaces that are presently allowed
are: Glass (G), Mirror (M), Aperture (A), and Destination plane (D). The
Destination plane must be the last surface identified. The allowed shapes for
Glass and Mirror surfaces are: Flat (F), Spherical (S), Conical (C), Aspherical(A), Cylindrical (L), and Axiconical (X). All vertices and surface-normals aredefined relative to the Cartesian coordinate system of the incident beam, where
the incident wavefront is in the XY-plane at Z = 0.
In the TRAY sub-option, the initial guess for the emergent beam’s center ofcurvature (Xc,Yc,Zc) must be specified within the emergent coordinates.
$________________________________________________________________________ #Remarks:##
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Define a new system: NSYS#
Name of the system: ABCD##**** Surface #01:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.0000000
0.0000000
10.000000#
Normal (Theta,Phi):
0.0000000
0.0000000#
Aperture's Rmin,Rmax:
0.0000000
10.000000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: HOMO#
Refractive index (n):1.5000000#
Next surface (Hit,Miss):20##**** Surface #02:
Type: Glass#
Shape: S#
Vertex (Vx,Vy,Vz):
0.0000000
0.0000000
20.000000#
Normal (Theta,Phi):
0.0000000
0.0000000#
Radius of curvature (RC):-30.000000#
Aperture's Rmin,Rmax:
0.0000000
1.0000000#
Coating (AR/DOE/BARE): BARE#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
30##**** Surface #03:
Type: Glass#Guide to DIFFRACT
Shape: C
Vertex (Vx,Vy,Vz):
0.0000000
Normal (Theta,Phi):
0.0000000
RC,K:
40.000000
Aperture's Rmin,Rmax:
0.0000000
Coating (AR/DOE/BARE): AR
Medium beyond surface: GRIN
Type of GRIN medium: 2
Origin (Ox,Oy,Oz):
0.0000000
Axis (Theta,Phi):
0.0000000
N0,R0,Z0:
1.5700000
R coefficients: Q1,Q2,Q3:
0.1000000
Q4,Q5,Q6:
0.0000000
Z coefficients: W1,W2,W3:
0.0000000
W4,W5,W6:
0.0000000
W7,W8,W9:0.0000000
W10,W11,W12:0.0000000
Next surface (Hit,Miss):
4281
0.0000000
0.0000000-12.000000
10.000000

30.000000

0.0000000
0.0000000
6.0000000
0.0200000
0.0000000
0.1000000
0.0000000
0.0000000
0.0000000

0.0000000
10.000000-0.0040000
0.0000000-0.0200000
0.0000000
0.0000000
0.0000000

0**** Surface #04:

Type: Glass
Shape: A
Vertex (Vx,Vy,Vz):
0.0000000
Normal (Theta,Phi):
0.0000000
RC,K:-24.000000
A1,A2,A3:
0.0000000
A4,A5,A6:
1.10000E-04
A7,A8,A9:0.0000000
A10,A11,A12:
1.124500E-06
A13,A14,A15:
0.0000000
A16,A17,A18:
0.0000000
A19,A20,A21:
0.0000000
Aperture's Rmin,Rmax:
2.0000000
Coating (AR/DOE/BARE): BARE
Medium beyond surface: HOMO
Refractive index (n):1.5000000
Next surface (Hit,Miss):
50Type: Glass
Shape: L
Vertex (Vx,Vy,Vz):
0.0000000
Normal (Theta,Phi):
0.0000000
RC,K:-1500.0000
Cylinder axis Phi,LA:
0.0000000
Aperture's Rmin,Rmax:
0.0000000
Coating (AR/DOE/BARE): AR
Medium beyond surface: FREE
Next surface (Hit,Miss):
60.0000000
0.0000000-0.1400000
0.0000000
0.0000000
2.123400E-05
0.0000000
0.0000000
0.0000000
0.0000000
8.0000000

40.000000
0.0000000-5.76670E-06
0.0000000
0.0000000
0.0000000
0.0000000
0.0000000**** Surface #05:

Type: Glass
Shape: X
Vertex (Vx,Vy,Vz):
0.0000000
Normal (Theta,Phi):
0.0000000
Axicon's half-angle:
34.000000
Aperture's Rmin,Rmax:
0.000000
Coating (AR/DOE/BARE): AR
Medium beyond surface: HOMO
Refractive index (n):1.5000000
Next surface (Hit,Miss):
7

0.0000000
0.0000000
0.0000000
5000.0000
600.00000

3103.0000**** Surface #06:

0.0000000
0.0000000

200.00000

10.000000

0**** Surface #07:

Type: Mirror
Shape: F
Vertex (Vx,Vy,Vz):.00000000
Normal (Theta,Phi):
45.000000
Aperture's Rmin,Rmax:
0.0000000
Coating (AR/DOE/BARE): DOE
Radial(-1); X-axis Phi:
25.00000
ConstructionLambda,Order:
0.650000E-03.00000000
270.00000
10.000000
1

50.000000

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282

Guide to DIFFRACT
B1_0,
B0_1,
B2_0:
B1_1,
B0_2,
B3_0:
B2_1,
B1_2,
B0_3:
B4_0,
B3_1,
B2_2:
B1_3,
B0_4,
B5_0:
B4_1,
B3_2,
B2_3:
B1_4,
B0_5,
B6_0:
B5_1,
B4_2,
B3_3:
B2_4,
B1_5,
B0_6:
Next surface (Hit,Miss):

0.930000
1.750000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.0000008

Type: Mirror
Shape: S
Vertex (Vx,Vy,Vz):.00000000
Normal (Theta,Phi):
90.000000
Radius of curvature (RC):-30.000000
Aperture's Rmin,Rmax:
0.0000000
Coating (PR/DOE/BARE): PR
Next surface (Hit,Miss):
90.000000
2.900000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000-2.400000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.0000000**** Surface #08:

30.000000
90.000000

50.000000

20.000000
0**** Surface #09:

Type: Mirror
Shape: C
Vertex (Vx,Vy,Vz):.00000000
Normal (Theta,Phi):
90.000000
RC,K:-20.000000
Aperture's Rmin,Rmax:
0.0000000
Coating (AR/DOE/BARE): DOE
ConstructionLambda,Order:
0.65000E-03
B01,B02,B03:
0.000000
B04,B05,B06:-2.500000
B07,B08,B09:
0.000000
B10,B11,B12:
0.000000
B13,B14,B15:
0.000000
B16,B17,B18:
0.000000
B19,B20,B21:
0.000000
Next surface (Hit,Miss):
100
Type: Mirror
Shape: A
Vertex (Vx,Vy,Vz):
0.0000000
Normal (Theta,Phi):
90.000000
RC,K:-15.000000
A1,A2,A3:
0.0000000
A4,A5,A6:
1.123000E-04
A7,A8,A9:0.0000000
A10,A11,A12:
0.216500E-05
A13,A14,A15:
0.0000000
A16,A17,A18:
0.0000000
A19,A20,A21:
0.0000000
Aperture's Rmin,Rmax:
0.0000000
Coating (PR/DOE/BARE): BARE
Complex index (n,k):
2.0000000
Next surface (Hit,Miss):
11
0

20.000000
270.00000-1.0000000
12.000000
14.200000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

50.000000

0.000000
0.250000
0.000000
0.000000
0.000000
0.000000
0.000000**** Surface #10:

Type: Mirror
Shape: L
Vertex (Vx,Vy,Vz):
0.0000000
Normal (Theta,Phi):
0.0000000
RC,K:
123.00000
Cylinder axis Phi,LA:
67.000000
Aperture's Rmin,Rmax:
0.0000000
Coating (PR/DOE/BARE): BARE
Complex index (n,k):2.2300000
Next surface (Hit,Miss):
12

25.000000
90.000000
0.1540000
0.0000000
0.0000000
7.513400E-04
0.0000000
0.0000000
0.0000000
0.0000000
15.000000

50.000000
0.0000000-1.76670E-06
0.0000000
0.0000000
0.0000000
0.0000000
0.0000000

7.0000000**** Surface #11:**** Surface #12:

Type: Mirror
Shape: X

0.0000000
0.0000000
0.1000000
340.00000
100.00000
5.6500000
01200.0000

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Guide to DIFFRACT
Vertex (Vx,Vy,Vz):
0.0000000
Normal (Theta,Phi):
180.00000
Axicon's half-angle:
45.000000
Aperture's Rmin,Rmax:
0.0000000
Coating (PR/DOE/BARE): BARE
Complex index (n,k):0.8700000
Next surface (Hit,Miss):13**** Surface #13:
Type: Aperture
Vertex (Vx,Vy,Vz):
0.0000000
Normal (Theta,Phi):
90.000000
Aperture shape: Ellipse
Major-axis Theta,LA,WA:
0.0000000
Next surface (Hit,Miss):14**** Surface #14:
Type: Aperture
Vertex (Vx,Vy,Vz):
0.0000000
Normal (Theta,Phi):
0.0000000
Aperture shape: Rectangle
Major-axis Phi,LA,WA:
45.000000
Next surface (Hit,Miss):15**** Surface #15:
Type: Destination
Vertex (Vx,Vy,Vz):
0.0000000
Normal (Theta,Phi):
90.000000

283
0.0000000
0.0000000

0.0000000

1000.0000
4.3500000
021.000000
270.00000

50.000000

5.0000000

3.5000000

0.0000000
0.0000000

23.000000

20.000000

12.000000

0.0000000
270.00000

50.000000

0

0

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$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):1.530000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Trace all incident rays: TRAY#
Initial guess (Xc,Yc,Zc):
0.0000000
0.0000000
0.0000000#
Accept or Search: Search#
Strip phase-factor (Y/N): Y#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
30.000000
30.000000#

17) RAYS generates geometric-optical rays for the current cross-section of thebeam. The request for multiplication of the curvature phase-factor into the meshis acknowledged only when the beam happens to carry a curvature outside themesh; it will be ignored otherwise.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.632800
NVIRON: 1.560000 #_________________________________________________________________________ ##Ray conversion: RAYS#
Multiply curvature (Y/N): N#

18) Extract_Phase/Amplitude allows the user to either remove the phase of the
distribution at the current cross-section of the beam (Phi = 0), or to eliminate
the non-uniformities of the amplitude while maintaining the phase(Amplitude = flat), or to maintain the amplitude while reversing the sign ofthe phase (Conjugate) .

284

Guide to DIFFRACT$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Extract phase/amplitude: XTRC#
Phi=0/Amp=flat/Conjugate: P#$
Extract phase/amplitude: XTRC
Phi=0/Amp=flat/Conjugate: A

#
#
#

$
Extract phase/amplitude: XTRC
Phi=0/Amp=flat/Conjugate: C###19) Photodetector allows the measurement of total integrated intensity on theactive regions of a number of different detectors. The four detectors presently
available in DIFFRACT are Quad, Split, Phi, and Differential. The ID of eachsignal signifies its assigned location within the SIGNAL array. The ID’s must bein the range [0 : 99].
Note: Phi and Differential detectors are not available in the Student Edition$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Detector: DTCT
(Length_Units: um) #Quad/Split/Phi/Diff: Q#
DCX,DCY,THETA:.00000000.00000000
45.000000#
LX,LY,GW:
1000.0000
1000.0000
100.00000#
ID numbers for S1,S2:
1
2#
ID numbers for S3,S4:
34#$
Detector: DTCT
Quad/Split/Phi/Diff: S
DCX,DCY,THETA:.00000000
LX,LY,GW:
2000.0000
ID numbers for S1,S2:
5
6#(Length_Units: um) ##.00000000
20.000000#
2500.0000
10.000000##

Detector: DTCT
Quad/Split/Phi/Diff: P
DCX,DCY:.00000000
R,THETA,GW:
10000.000
ID numbers for S1,S2:
7
8
ID numbers for S3,S4:
9
10#(Length_Units: um) ##.00000000#
45.000000
20.000000###

Detector: DTCT
Quad/Split/Phi/Diff: D
THETA:
45.000000
ID numbers for S1,S2:
11
12

#
#
#
#
#

$

$

Guide to DIFFRACT

28520) External_Object provides the following two options: (i) IMPORT aphase/amplitude mask from a file named XXXX.DAT, and place this mask in the
beam path. The file that contains the external object (i.e., the mask) must resideon the hard disk within the home directory of DIFFRACT. Both ASCII and Binary
files are allowed. (ii) CREATE a phase/amplitude mask by exposing aphotographic plate to the intensity profile of the beam currently residing within
the mesh. Double-exposure is also allowed. Both positive and negativetransparencies may be created within this environment. The mask is eventually
saved as an ASCII or Binary file to the hard disk.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##External object: XOBJ
(Length_Units: mm) #Import or Create: I#
Filename: ABCD.DAT#
OCX,OCY,RA:.00000000.00000000
1.5300000#
Periodic repeat (Y/N): N#$
External object: XOBJImport or Create: C
LX,LY:
3.2000000
NX,NY:
256
256
PCX,PCY,RA:
0.0000000
Include Z-component: N
Exposure (I0,T0,Phi0):1.00000000E-06
Expose FILM/filename: FILM
Positive or Negative: N
Save to filename: ABCD.DATASCII or Binary file: Binary#(Length_Units: mm) ##3.2000000##
0.0000000
1.5300000##
1.0000000-36.000000
#
#
#
#
#

21) Interfere enables the user to combine (interferometrically) the current crosssection of the beam with another beam. The latter must have been previously
computed and saved on a hard disk file XXXX.DST. Prior to superposition, thebeam cross-section imported from the file will be multiplied by the complex
scale-factor A exp(iφ), where A is a positive real constant, and φ is a phase angle
in the range (0°, 360°).
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Two-beam interference: INTF#
Input-file name: AB01.DST#
Scale-factor (A,PHI):
1.0000000
12.000000#

286

Guide to DIFFRACT

22) SAVE allows the user to save the current cross-section of the beam in thehard disk file XXXX.DST. The 4-letter filename is chosen by the user.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Store distribution: SAVE#
File name: AA01.DST#

23) Process_Signals provides the means of processing the various data-items
stored in the SIGNAL array. The signal processing actions (i.e., Add, Subtract,
Multiply, Divide, Assign, Amplify, and Print) may be used any number of times
and in any desired order. The i, j, k indices belong to the interval [0 : 99]. The
Assignment action (=) assigns either a constant value C or the current value ofthe mesh parameters NMAX, NMAY, LMAX, LMAY to a specific location of the
SIGNAL array. To assign a particular mesh parameter, simply type the name ofthe parameter (e.g., NMAX) instead of a constant value. Signal processing
terminates with the End command (Action: E).
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Process signals: PSIG#
Action: +#S(i)=S(j)+S(k)i,j,k:
20
1
2#
Action: #S(i)=S(j)-S(k)i,j,k:
21
3
4#
Action: *#S(i)=S(j)*S(k)i,j,k:
22
5
6#
Action: /#S(i)=S(j)/S(k)i,j,k:
23
7
8#
Action: =#S(i)=Ci,C:
24
1.4500000#
Action: =#S(i)=C
i,C:
25
NMAX
#
Action: A#
S(i)=G*S(j)i,j,G:
26
9
10.000000#
Action: P#
Print S(i)i:20#Action: P#
Print S(i)i:21#Action: P#
Print S(i)i:22#Action: P#
Print S(i)i:23#
Action: P#
Print S(i)i:24#
Action: P#
Print S(i)i:25#Action: P#
Print S(i)i:26#Action: E#

Guide to DIFFRACT

28724) Wavefront Analysis computes the effective wavefront error for X-, Y-, andZ-components of polarization, then proceeds to save the results in user-specified
locations within the SIGNAL array. (Note: The wavefront error for the Zcomponent is computed only when this component happens to reside in themesh prior to invoking the WAVE option.)$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Wavefront analysis: WAVE#
SIGNAL ID for WFE X,Y,Z:
123#25) Spatial Filter provides two methods of filtering the beam’s cross-section inthe (Fourier) domain of spatial frequencies. When the Fast Fourier Transform(FFT) method is chosen, DIFFRACT uses the existing mesh parameters, ignoring
the New Mesh Parameters (NMAX, NMAY, LMAX, LMAY) specified in thecommand block. The New Mesh Parameters will be used only when the chosencomputation method is Discrete Fourier Transform (DFT).
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):0.532000
NVIRON: 1.650000 #_________________________________________________________________________ ##Spatial Filter: FLTR
(Length_Units: mm) #Computation Method: FFT#
CSX,CSY,S0:0.0000000
0.0000000
0.70000000#
Inside (A0,Phi0):1.0000000
0.0000000#
Outside (A1,Phi1):0.0000000
0.0000000#
New Mesh NMAX,NMAY:
256
256
#
New Mesh LMAX,LMAY:
2.1200000
2.1200000#

26) FDTD Interface operates in either of two modes: Export and Import. In
Export mode, the FDTD mesh parameters NX, NY (both ≤ NDIM), as well as LX,
LY, ∆z must be specified. The exported mesh will be Staggered or not, in
accordance with the user’s request. The user must also specify the 4-letter nameof the exported file, and whether it should be formatted as ASCII or Binary. In
the Import mode, the user specifies the name of the data file that contains the
cross-section of the beam computed by an external FDTD routine. (DIFFRACT
automatically determines the ASCII / Binary format of this file.)$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##FDTD Interface: FDTD
(Length_Units: um) #Export/Import: Export#288

Guide to DIFFRACT
NX,NY:
256
256
LX,LY:
20.000000
DeltaZ:
0.1000000
Staggered mesh (Y/N): Y
Filename: ABCD.DATASCII or Binary: Binary$
FDTD Interface: FDTD
Export/Import: Import
Filename: ABCD.DAT

20.000000

#
#
#
#
#
#
#
(Length_Units: um) ###27) File_Management/Graphics enables the user to temporarily switch to the
MS-DOS environment, to load, add, compute logarithm, display, and save certain
graphic files, and to convert loaded files into TIFF format. Any number ofactions may be taken while in the file_management/graphics environment. The
last action, of course, must be Quit, which causes a return to the main menu.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um):
0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: S#
Action: L#
Data-file loaded to TEMP: IX00.DAT#
Action: A#
Data-file added to TEMP: IX01.DAT#
Weight Factor:
2.450000#
Action: B#
Dump screen to BMP-file: JUNK.BMP#
Action: C#
Logarithmic SCALE: 3.5000000#
Action: D#
Color or Gray-scale: C#
Action: F#
Saving TEMP in data-file: ABCD.DAT#
Action: T#
Saving TEMP in TIFF-file: IX00.TIF#
Action: Q#

28) PLOT provides several possibilities for displaying the data on the monitor
and, subsequently, saving the graphic files on hard disk. At any given crosssection of the system, you may choose to display Intensity, Log(intensity),
Phase, Polarization, or the distribution of Rays within the rectangular area
[Xmin:Xmax, Ymin:Ymax]. Note that the command blocks for these various
options are identical, even though the questions asked by the program during aninteractive session may not be the same. For example, when you request apolarization plot in the interactive mode, you will not be asked about the Zcomponent. However, the question about the Z-component automaticallyappears in the COMMAND.DAT. Of course, during non-interactive sessions thisquestion (and your answer to it) will be ignored, but if you decide to edit theGuide to DIFFRACT

289

command block and, for example, change the polarization plot into a phase plot,
the relevant questions will already be there for you to answer.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):1550.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-3200.0000
3200.0000#
Ymin,Ymax:-3200.0000
3200.0000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 13#$
Plot distribution: PLOT
Type: Log(Intensity)
Logarithmic SCALE:
4.0000000
Xmin,Xmax:-5.0000000
Ymin,Ymax:-3.5000000
Color or Gray-scale: C
Z-component (Y/N): Y
Save data files (Y/N): N
File identifier: 0#(Length_Units: cm) ###2.0000000#
3.5000000
#
#
#
#
#

Plot distribution: PLOT
Type: Phase
Logarithmic SCALE:
4.0000000
Xmin,Xmax:-3.5000000
Ymin,Ymax:-3.5000000
Color or Gray-scale: C
Z-component (Y/N): Y
Save data files (Y/N): Y
File identifier: 23#(Length_Units: mm) ###3.5000000#
3.5000000
#
#
#
#
#

Plot distribution: PLOT
Type: Polarization
Logarithmic SCALE:
4.0000000
Xmin,Xmax:-3.5000000
Ymin,Ymax:-3.5000000
Color or Gray-scale: G
Z-component (Y/N): N
Save data files (Y/N): Y
File identifier: 1#(Length_Units: mm) ###3.5000000#
3.5000000
#
#
#
#
#

$

$

29) QUIT appears only once at the end of the COMMAND.DAT file. Anycommand blocks appearing after this command will be ignored.
$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#290

Guide to DIFFRACT

Guide to DIFFRACT

291

Appendix E
Selected Test Runs
This Appendix contains several test runs of DIFFRACT. The selection is
not intended to be exhaustive; rather, it shows a few simple systems
that can be simulated using available options in the latest edition of
DIFFRACT. A more extensive set of test runs covering many features ofthe program is provided on your distribution CD-ROM within the< TESTRUNS > directory. A list of all test runs available on the CD-ROM
appears in Appendix F. For additional supplementary resource files
see Appendix G and the < OPN > directory on your distribution CDROM.

1. Focusing a beam of polarized light through a slab of aragonite crystal....................292
2. Ray tracing through a multi-element objective lens..................................................298
3. Focusing a beam of polarized light at high numerical aperture ................................310
4. The Goos–Hänchen effect .........................................................................................318
5. Gradient-index (GRIN) lens ......................................................................................323
6. Coherent imaging of a phase object .......................................................................... 331
7. Internal conical refraction ......................................................................................... 337
8. Geometric-optical system.......................................................................................... 350
9. Interferograms showing various Seidel aberrations ..................................................359
10. Astigmatic focus-error and push-pull track-error detection scheme .........................371
11. GRIN lens, anamorphic prism pair, and geometric-optical ray tracing ....................381
12. Focusing through high-NA lens using EXCT and APRX methods ..........................391

292

Guide to DIFFRACT

Test Run 1
Focusing a beam of polarized light through a slab of aragonite crystal
Featuring: Polarization-sensitive element, Positive_focusing lens, Multilayer structure

Figure A1.1. Schematic diagram of the simulated system. Double passage of the focused
beam through the birefringent crystal slab causes variation of the polarization rotation angle
over the beam’s cross-section. Crossed polarizers convert polarization rotation into intensity
variation.

Figure A1.2. Plots of intensity (left) and log_intensity (right ) at the observation plane.

Guide to DIFFRACT

293

COMMAND.DAT file corresponding to Test Run 1@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 03/02/2006
Time: 08:54 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
1/e radii Rmin,Rmax:
20000.000
20000.000#
RA,THETA:
7500.0000
0.0000000#
Aberrations: None#
Polarization RHO,ETA:
45.000000
0.0000000#
NMAX,NMAY:
1024
1024
#
LMAX,LMAY:
40000.000
40000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: PFOC#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.3750000
20000.000#
Aberrations: None#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance: 19750.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Multilayer structure: MLYR
(Length_Units:
) #
Nsub:
1.0000000#
Number of layers: 2#
Layer#01
Type (A-E): A#n,k:2.0000000
7.0000000#
Thickness:
0.1000000#
Layer#02
Type (A-E): C#
Real(EXX),Imag(EXX):
2.8426000
0.0000000#
Real(EYY),Imag(EYY):
2.8291000
0.0000000#
Real(EZZ),Imag(EZZ):
2.3440000
0.0000000#
Thickness:
430.00000#

294

Guide to DIFFRACT
Euler Theta,Phi,Psi:
0.0000000
0.0000000
347.0000#
Ninc:
1.0000000#
Compress incident: NONE#
Entrance surface: BARE#
Reflect or Transmit: R#
Emergent environment's n:
1.0000000#
Decompress emergent: NONE#
Exit surface: BARE#
Stack's Theta,Phi,Psi:
0.0000000
0.0000000
0.0000000#
Emergent Theta,Phi,Psi:
180.00000
0.0000000
0.0000000#
Propagation distance:
19750.000#
Propagation regime: FRNHF#
Smax:
0.1000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: COLL#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.3750000
20000.000#
Aberrations: NONE#
Scalar/Quasi-vector: QV#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Polarization element: POLE
(Length_Units:) #ACX,ACY:
0.0000000
0.0000000#
RA,THETA:
7500.0000
45.000000#
ALPHA,PHIA:
1.0000000
0.0000000#
BETA,PHIB:
0.0000000
0.0000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-8000.0000
8000.0000#
Ymin,Ymax:-8000.0000
8000.0000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Log(intensity)#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-8000.0000
8000.0000#
Ymin,Ymax:-8000.0000
8000.0000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Guide to DIFFRACT

295

File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: IX00.DAT#
Action: T#
Saving TEMP in TIFF-file: IX00.TIF#
Action: L#
Data-file loaded to TEMP: IX01.DAT#
Action: T#
Saving TEMP in TIFF-file: IX01.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@SESSION.DAT file corresponding to Test Run 1$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 8.4
File SESSION.DAT $$$$ Date: 04/26/2006
Time: 09:53 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ALL LENGTHS ARE IN UNITS OF WAVELENGTH; ALL ANGLES ARE IN DEGREES_________________________________________________________________
Array dimensions: NDIM=1024, NDSP= 149**************************************
Designated Working Subdirectory: <Test01>*****************************************
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Simple Gaussian: BCX=
0.0
BCY=
0.0
RA=
7500.0
THETA= 0.00
RMIN= 20000.0 RMAX= 20000.0
Aberrations: None
Polarization state: Rho= 45.0000
Eta= 0.0000_________________________________________________________
NMAX=1024
NMAY=1024
LMAX= 40000.00
LMAY= 40000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
14642.8
XPOWER=0.500E+00
YPOWER=0.500E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________LENS ACTION***********
TYPE: Positive_Focusing
LCX=
0.000
LCY=
0.000
NA=0.375000
FL=
20000.00
Aberrations: None
Quasi-vector treatment
Propagation distance=
19750.00__________________________________________________APRX method: NGUIDE=
758 LGUIDE=
75000.0___________________________________________________________________________________________________________
NMAX=1024
NMAY=1024LMAX=
512.00LMAY=
512.00Beam centroid:(X=0.00,Y=
0.00)
DIAM=
193.8
XPOWER=0.500E+00
YPOWER=0.500E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________MULTILAYER**********_____________________________________________________________

296

Guide to DIFFRACT
Vacuum wavelength of the incident light (LAMBDA) in the sameunits as used below to specify layer thicknesses: .100000E+01_____________________________________________________________
Substrate:(n,k)= ( 1.000000, 0.000000)
Thickness= Infinite
Layer# 1:(n,k)= (Thickness=

2.000000, 7.000000)
0.100000

Layer# 2:

EXX= ( 2.842600, 0.000000)
EYY= ( 2.829100, 0.000000)
EZZ= ( 2.344000, 0.000000)
Thickness=
430.000000____________________________________________
Euler angles of rotated coordinates:(Theta=0.000 Phi=
0.000 Psi= 347.000)____________________________________________

Medium of incidence: (n,k)= (1.000000, 0.000000)

Reflection mode
Index of emergent environment= 1.0000______________________________________________________________DIFFRACT now returns to its standard mode, where all lengths
are normalized by the wavelength (LAMBDA) of the light within
its present environment.
(Refractive index= 1.0000)______________________________________________________________
Multilayer coordinates: Theta= 0.00
Emergent coordinates:
Theta=180.00

Phi=
Phi=

0.00
0.00

Psi=
Psi=

0.00
0.00

Propagation distance=
19750.00
Propagation regime: Fraunhofer_____________________________________________________NGUIDE=
100 LGUIDE=
3876.91 SMAX= 0.001290______________________________________________________________________________________________________________
NMAX=1024
NMAY=1024
LMAX= 39500.00
LMAY= 39500.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
14654.9
XPOWER=0.283E+00
YPOWER=0.368E+00
Curvature= 0.51E-04(Environment's index of refraction= 1.0000)_________________________________________________________LENS ACTION***********
TYPE: Collimator
LCX=
0.000
LCY=
0.000
NA=0.375000
FL=
20000.00
Aberrations: None
Quasi-vector treatment_________________________________________________________
NMAX=1024
NMAY=1024
LMAX= 39500.00
LMAY= 39500.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
13699.1
XPOWER=0.250E+00
YPOWER=0.326E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________POLARIZATION ELEMENT********************
ACX=
0.00
ACY=
0.00
RA=
7500.00
THETA= 45.000(ALPHA,PHIA)=(1.000000, 0.000)(BETA,PHIB)=(0.000000, 0.000)_________________________________________________________
NMAX=1024
NMAY=1024
LMAX= 39500.00
LMAY= 39500.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
12560.8
XPOWER=0.858E-01
YPOWER=0.858E-01
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________

Guide to DIFFRACT

297
PLOTTING ROUTINE****************

Intensity distribution___________________________________________
Xmin=-8000.00
Xmax=
8000.00
Ymin=-8000.00
Ymax=
8000.00___________________________________________
IXmin= 0.0000E+00
IXmax= 0.2178E-08
IYmin= 0.0000E+00
IYmax= 0.2178E-08
Latest screen (intensity pattern) dumped in IS00.BMP
Intensity pattern for X-polrz stored in IX00.DAT
Intensity pattern for Y-polrz stored in IY00.DAT
PLOTTING ROUTINE****************
Base 10 logarithm of normalized intensity distribution(Range: from 0 at peak_intensity to -4.0000 at cutoff)___________________________________________
Xmin=-8000.00
Xmax=
8000.00
Ymin=-8000.00
Ymax=
8000.00___________________________________________
IXmin= 0.0000E+00
IXmax= 0.2178E-08
IYmin= 0.0000E+00
IYmax= 0.2178E-08
Latest screen (intensity pattern) dumped in IS01.BMP
Intensity pattern for X-polrz stored in IX01.DAT
Intensity pattern for Y-polrz stored in IY01.DAT
FILE MANAGEMENT/GRAPHICS************************
Action: Load data-file IX00.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.000000E+00
Maximum Value: 0.217784E-08****************************
Action: Save TIFF version of TEMP array in IX00.TIF
IX00.TIF is successfully created and written to disk____________________________________________________
Action: Load data-file IX01.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: -0.400000E+01
Maximum Value: 0.000000E+00****************************
Action: Save TIFF version of TEMP array in IX01.TIF
IX01.TIF is successfully created and written to disk____________________________________________________End of session: DIFFRACT terminated normally********************************************

298

Guide to DIFFRACT

Test Run 2
Ray tracing through a multi-element objective lensFeaturing: Gaussian beam, Conversion to rays, and Geometric-optical system.

Figure A2.1. Schematic diagram of the simulated system, showing a collimated beam
entering a compound objective lens. The beam is brought to diffraction-limited focus at the
focal plane of the lens.

4000

y/λ

−4000

−4000

x/λ

4000

Figure A2.2. Incident beam’s intensity distribution.

Guide to DIFFRACT

299(a) x-component
4000

y/λ

-4000
(b) y-component
4000

y/λ

-4000
(c) z-component
4000

y/λ

-4000
-4000

x/λ

4000 -4000

x/λ

4000

Figure A2.3. Distribution of intensity (left) and phase (right) at the exit pupilof the lens for the three components of polarization.

300

Guide to DIFFRACT

3

a

y/λ

-3
3

b

y/λ

-3
3

c

y/λ

-3
-3
x/λ
3
Figure A2.4. Intensity distribution at the focal plane of the lens for the three componentsof polarization. (a) x-component, (b) y-component, (c) z-component.

Guide to DIFFRACT301

COMMAND.DAT file corresponding to Test Run 2@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 06/12/2006
Time: 19:34 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.67000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
1/e radii Rmin,Rmax:
5000.0000
5000.0000#
RA,THETA:
3358.0000
0.0000000#
Aberrations: None#
Polarization RHO,ETA:
0.0000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
40000.000
40000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.67000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-4000.0000
4000.0000#
Ymin,Ymax:-4000.0000
4000.0000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.67000
NVIRON: 1.000000 #_________________________________________________________________________ ##Ray conversion: RAYS#
Multiply curvature (Y/N): N#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.67000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Define a new system: NSYS#
Name of the system: AMTL#302

Guide to DIFFRACT**** Surface #01:

Type: Glass
Shape: S
Vertex (Vx,Vy,Vz):.00000000
Normal (Theta,Phi):.00000000
Radius of curvature (RC):9.2520000
Aperture's Rmin,Rmax:
0.0000000
Coating (AR/DOE/BARE): BARE
Medium beyond surface: HOMO
Refractive index (n):1.7971240
Next surface (Hit,Miss):2**** Surface #02:
Type: Glass
Shape: S
Vertex (Vx,Vy,Vz):.00000000
Normal (Theta,Phi):.00000000
Radius of curvature (RC):-15.509000
Aperture's Rmin,Rmax:
0.0000000
Coating (AR/DOE/BARE): BARE
Medium beyond surface: FREE
Next surface (Hit,Miss):3**** Surface #03:
Type: Glass
Shape: S
Vertex (Vx,Vy,Vz):.00000000
Normal (Theta,Phi):.00000000
Radius of curvature (RC):-4.3400000
Aperture's Rmin,Rmax:
0.0000000
Coating (AR/DOE/BARE): BARE
Medium beyond surface: HOMO
Refractive index (n):1.5139050
Next surface (Hit,Miss):4**** Surface #04:
Type: Glass
Shape: S
Vertex (Vx,Vy,Vz):.00000000
Normal (Theta,Phi):.00000000
Radius of curvature (RC):-4.7610000
Aperture's Rmin,Rmax:
0.0000000
Coating (AR/DOE/BARE): BARE
Medium beyond surface: FREE
Next surface (Hit,Miss):5**** Surface #05:
Type: Glass
Shape: S
Vertex (Vx,Vy,Vz):.00000000
Normal (Theta,Phi):.00000000
Radius of curvature (RC):3.6200000
Aperture's Rmin,Rmax:
0.0000000
Coating (AR/DOE/BARE): BARE
Medium beyond surface: HOMO
Refractive index (n):1.7971240
Next surface (Hit,Miss):6**** Surface #06:
Type: Glass
Shape: S
Vertex (Vx,Vy,Vz):.00000000
Normal (Theta,Phi):.00000000
Radius of curvature (RC):8.5000000
Aperture's Rmin,Rmax:
0.0000000
Coating (AR/DOE/BARE): BARE
Medium beyond surface: FREE
Next surface (Hit,Miss):7**** Surface #07:
Type: Glass
Shape: F
Vertex (Vx,Vy,Vz):.00000000
Normal (Theta,Phi):.00000000
Aperture's Rmin,Rmax:
0.0000000
Coating (AR/DOE/BARE): BARE
Medium beyond surface: HOMO
Refractive index (n):1.5789620
Next surface (Hit,Miss):8**** Surface #08:
Type: Glass.00000000.00000000.00000000

2.2500000

0.00000000.00000000

1.7500000

2.2500000
0.00000000.00000000

2.1090000

2.2500000

0.00000000.00000000

5.1090000

2.2500000
0.00000000.00000000

5.2090000

2.2500000

0.00000000.00000000

6.9590000

2.2500000
0.00000000.00000000
2.2500000

08.2843240

#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#

Guide to DIFFRACT
Shape: F#
Vertex (Vx,Vy,Vz):.00000000.00000000
9.4843240#
Normal (Theta,Phi):.00000000.00000000#
Aperture's Rmin,Rmax:
0.0000000
2.2500000#
Coating (AR/DOE/BARE): BARE#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
90#**** Surface #09:
Type: Destination#
Vertex (Vx,Vy,Vz):.00000000.00000000
5.4036000#
Normal (Theta,Phi):.00000000.00000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.67000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units:) #Trace all incident rays: TRAY#
Initial guess (Xc,Yc,Zc):.00000000.00000000.00000000#
Accept or Search: Search#
Strip phase-factor (Y/N): Y#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
40000.000
40000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.67000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-4000.0000
4000.0000#
Ymin,Ymax:-4000.0000
4000.0000#
Color or Gray-scale: G#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 2#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.67000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Phase#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-4000.0000
4000.0000#
Ymin,Ymax:-4000.0000
4000.0000#
Color or Gray-scale: G#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 2#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.67000
NVIRON: 1.000000 #_________________________________________________________________________ ##Free space propagation: PROP
(Length_Units:) #Propagation distance: 6090.6500#
Multiply curvature (Y/N): N#
Reposition beam (Y/N): N#
Propagation regime: FRNHF#
Smax:
0.1000000#
Scalar/Quasi-vector: SC#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.67000
NVIRON: 1.000000 #_________________________________________________________________________ #
#

303

304

Guide to DIFFRACT
Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-3.0000000
3.0000000#
Ymin,Ymax:-3.0000000
3.0000000#
Color or Gray-scale: G#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 3#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.67000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: IX01.DAT#
Action: T#
Saving TEMP in TIFF-file: IX01.TIF#
Action: L#
Data-file loaded to TEMP: IX02.DAT#
Action: T#
Saving TEMP in TIFF-file: IX02.TIF#
Action: L#
Data-file loaded to TEMP: IY02.DAT#
Action: T#
Saving TEMP in TIFF-file: IY02.TIF#
Action: L#
Data-file loaded to TEMP: IZ02.DAT#
Action: T#
Saving TEMP in TIFF-file: IZ02.TIF#
Action: L#
Data-file loaded to TEMP: PX02.DAT#
Action: T#
Saving TEMP in TIFF-file: PX02.TIF#
Action: L#
Data-file loaded to TEMP: PY02.DAT#
Action: T#
Saving TEMP in TIFF-file: PY02.TIF#
Action: L#
Data-file loaded to TEMP: PZ02.DAT#
Action: T#
Saving TEMP in TIFF-file: PZ02.TIF#
Action: L#
Data-file loaded to TEMP: IX03.DAT#
Action: T#
Saving TEMP in TIFF-file: IX03.TIF#
Action: L#
Data-file loaded to TEMP: IY03.DAT#
Action: T#
Saving TEMP in TIFF-file: IY03.TIF#
Action: L#
Data-file loaded to TEMP: IZ03.DAT#
Action: T#
Saving TEMP in TIFF-file: IZ03.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.67000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Guide to DIFFRACT305

SESSION.DAT file corresponding to Test Run 2$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 8.4
File SESSION.DAT $$$$ Date: 04/26/2006
Time: 11:09 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ALL LENGTHS ARE IN UNITS OF WAVELENGTH; ALL ANGLES ARE IN DEGREES_________________________________________________________________
Array dimensions: NDIM= 512, NDSP= 149**************************************
Designated Working Subdirectory: <Test02>*****************************************
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Simple Gaussian: BCX=
0.0
BCY=
0.0
RA=
3358.0
THETA= 0.00
RMIN=
5000.0 RMAX=
5000.0Aberrations: None
Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 40000.00
LMAY= 40000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
6189.4
XPOWER=0.100E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=-4000.00
Xmax=
4000.00
Ymin=-4000.00
Ymax=
4000.00___________________________________________
IXmin=

0.0000E+00

IXmax=

0.4294E-07

Latest screen (intensity pattern) dumped in IS01.BMP
Intensity pattern for X-polrz stored in IX01.DATCONVERSION TO RAYS******************
Ray components RX,RY,RZ are computed and stored in BX and BY
GEOMETRIC-OPTICAL SYSTEM DEFINITION***********************************Name of the system: AMTL
Vacuum wavelength of the incident light (LAMBDA) in the sameunits as used below to specify system parameters: .670000E-03
Surface #01***********
Type: Glass
Shape: Spherical
Vertex coordinates: (0.000000E+00,0.000000E+00,0.000000E+00)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Radius of curvature, including sign (RC):9.2520Radius of aperture (RA):2.2500Antireflection: No
Homogeneous medium beyond the surface: n= 1.797124

306

Guide to DIFFRACT
Surface #02***********
Type: Glass
Shape: Spherical
Vertex coordinates: (0.000000E+00,0.000000E+00,0.175000E+01)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Radius of curvature, including sign (RC): -15.509
Radius of aperture (RA):2.2500Antireflection: No
Free space medium beyond the surface: n = 1.00
Surface #03***********
Type: Glass
Shape: Spherical
Vertex coordinates: (0.000000E+00,0.000000E+00,0.210900E+01)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Radius of curvature, including sign (RC): -4.3400
Radius of aperture (RA):2.2500Antireflection: No
Homogeneous medium beyond the surface: n= 1.513905
Surface #04***********
Type: Glass
Shape: Spherical
Vertex coordinates: (0.000000E+00,0.000000E+00,0.510900E+01)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Radius of curvature, including sign (RC): -4.7610
Radius of aperture (RA):2.2500Antireflection: No
Free space medium beyond the surface: n = 1.00
Surface #05***********
Type: Glass
Shape: Spherical
Vertex coordinates: (0.000000E+00,0.000000E+00,0.520900E+01)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Radius of curvature, including sign (RC):3.6200Radius of aperture (RA):2.2500Antireflection: No
Homogeneous medium beyond the surface: n= 1.797124
Surface #06***********
Type: Glass
Shape: Spherical
Vertex coordinates: (0.000000E+00,0.000000E+00,0.695900E+01)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Radius of curvature, including sign (RC):8.5000Radius of aperture (RA):2.2500Antireflection: No
Free space medium beyond the surface: n = 1.00
Surface #07***********
Type: Glass
Shape: Flat
Vertex coordinates: (0.000000E+00,0.000000E+00,0.828432E+01)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Radius of aperture (RA):2.2500Antireflection: No
Homogeneous medium beyond the surface: n= 1.578962

Guide to DIFFRACT
Surface #08***********
Type: Glass
Shape: Flat
Vertex coordinates: (0.000000E+00,0.000000E+00,0.948432E+01)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Radius of aperture (RA):2.2500Antireflection: No
Free space medium beyond the surface: n = 1.00
Surface #09***********
Type: Destination
Shape: Flat
Vertex coordinates: (0.000000E+00,0.000000E+00,0.540360E+01)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)______________________________________________________________DIFFRACT now returns to its standard mode, where all lengths
are normalized by the wavelength (LAMBDA) of the light within
its present environment.
(Refractive index= 1.0000)______________________________________________________________GEOMETRIC-OPTICAL RAY TRACING*****************************_______________________________________________________________
Note: Presently all lengths are in the same units as the vacuum
wavelength.
(LAMBDA=.670000E-03)_______________________________________________________________
Incident Beam:**************
Magnitude of the brightest ray in the beam:0.429E-07
Magnitude of the weakest ray to qualify for tracing:0.174E-07
Total number of qualified rays within the incident beam:
5785(Upper bound on the number of rays presently allowed to cross: 45000)
Emergent Beam:**************
Number of qualified rays that traverse the system: 5249
Region within destination plane to which rays are confined:
Circle: Xc,Yc,Radius =0.00000.00002.5197
Rectangle: Xmin,Xmax =-2.4655
2.4655
Ymin,Ymax =-2.4655
2.4655______________________________________________________________DIFFRACT now returns to its standard mode, where all lengths
are normalized by the wavelength (LAMBDA) of the light within
its present environment.
(Refractive index= 1.0000)______________________________________________________________The emergent beam is converging towards the point (Xc,Yc,Zc)=( 0.307975E-04,-0.714219E-04, 6090.63), with an rms wavefront error of 0.25552E-01. If the beam's cross-sectional area
is sufficiently small, it may be considered to be a collimatedbeam, traveling along the direction (Theta= 0.00, Phi=293.33).
______________________________________________________________
Emergent wavefront stripped from curvature/tilt phase-factor**************************************************************
Construction of the emergent wave-front within the destination
plane is now complete.
(Total energy flux = 0.5537)**************************************************************_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 40000.00
LMAY= 40000.00
Beam centroid:(X=-3.67,Y=-0.49)
DIAM=
6561.5
XPOWER=0.553E+00
YPOWER=0.569E-06
Curvature=-0.16E-03(Environment's index of refraction= 1.0000)_________________________________________________________

307

308

Guide to DIFFRACT
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=-4000.00
Xmax=
4000.00
Ymin=-4000.00
Ymax=
4000.00___________________________________________
IXmin=
IYmin=
IZmin=

0.0000E+00
0.0000E+00
0.0000E+00

IXmax=
IYmax=
IZmax=

0.2863E-07
0.9740E-13
0.2485E-08

Latest screen (intensity pattern) dumped in IS02.BMP
Intensity pattern for X-polrz stored in IX02.DAT
Intensity pattern for Y-polrz stored in IY02.DAT
Intensity pattern for Z-polrz stored in IZ02.DAT
Peak intensity=0.301E-07 XPEAK=(Total energy flux=0.554E+00-156.25 YPEAK=ZPOWER=0.401E-01)-312.50

PLOTTING ROUTINE****************
Phase distribution********************************************************************
Warning: The beam carries a curvature phase-factor outside the mesh.
This curvature, whose center is the point (Xc=
0.00, Yc=0.00,
Zc=0.609063E+04), will NOT be included in the following display(s).
Therefore, when examining the phase plot(s), it must be rememberedthat a parabolic term needs to be added to the plot(s) in order tocomplete the picture.
********************************************************************___________________________________________
Xmin=-4000.00
Xmax=
4000.00
Ymin=-4000.00
Ymax=
4000.00___________________________________________
PXmin=
0.000000
PYmin= -103.339851
PZmin= -102.230255

PXmax=
PYmax=
PZmax=

104.647064
104.544838
104.635803(degrees)(degrees)(degrees)

Latest screen (phase pattern) dumped in PS02.BMP
Phase pattern for X-polrz stored in PX02.DAT
Phase pattern for Y-polrz stored in PY02.DAT
Phase pattern for Z-polrz stored in PZ02.DAT
Peak intensity=0.301E-07 XPEAK=(Total energy flux=0.554E+00-156.25 YPEAK=ZPOWER=0.401E-01)-312.50

PROPAGATION IN ENVIRONMENT**************************
Propagation distance=

6090.6499(Forward)********************************************************
Note: Mesh is too coarse for the curvature phase-factor.
Multiplication is not recommended.
********************************************************
Curvature phase-factor is NOT multiplied in AX,AY*********************************************************************WARNING: The beam appears to be converging towards (or diverging awayfrom) the point (Xc,Yc,Zc)=(0.307975E-04,-.714219E-04,0.609063E+04).

Guide to DIFFRACT309

The computation method used here will yield accurate results only if
the beam is brought to the vicinity of the above point. If the chosen
propagation distance is much different from Zc the results are likely
to be inaccurate.
*********************************************************************_________________________________________________________
NMAX= 512
NMAY= 512
LMAX=
77.96
LMAY=
77.96
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5.3
XPOWER=0.551E+00
YPOWER=0.560E-06
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=
-3.00
Xmax=
3.00
Ymin=
-3.00
Ymax=
3.00
___________________________________________
IXmin=
IYmin=
IZmin=

0.3950E-04
0.4055E-16
0.4837E-09

IXmax=
IYmax=
IZmax=

0.4783E+00
0.6186E-07
0.1877E-01

Latest screen (intensity pattern) dumped in IS03.BMP
Intensity pattern for X-polrz stored in IX03.DAT
Intensity pattern for Y-polrz stored in IY03.DAT
Intensity pattern for Z-polrz stored in IZ03.DAT
Peak intensity=0.478E+00 XPEAK=(Total energy flux=0.553E+00

0.00 YPEAK=ZPOWER=0.399E-01)

FILE MANAGEMENT/GRAPHICS************************
Action: Load data-file IX01.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.000000E+00
Maximum Value: 0.429367E-07****************************
Action: Save TIFF version of TEMP array in IX01.TIF
IX01.TIF is successfully created and written to disk____________________________________________________
Action: Load data-file IX02.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.000000E+00
Maximum Value: 0.286346E-07****************************End of session: DIFFRACT terminated normally********************************************(Truncated)

0.00

310

Guide to DIFFRACT

Test Run 3
Focusing a polarized beam of light at high numerical aperture
Featuring: Gaussian beam (with linear and circular polarization), Positive_focusing lens.
X

Figure A3.1. A truncated Gaussian beamis brought to diffraction-limited focus bya high-NA lens. In the two cases
considered, the incident beam is either
linearly or circularly polarized.
(a) x-polarization

−4

x/λ

NA = 0.85

Z

Truncated
Gaussian Beam(b) y-polarization

4 −4

x/λ

Focal Plane(c) z-polarization

4 −4

x/λ

4

Figure A3.2. Plots of log (intensity) distribution for the three components of polarization at
the focal plane of a high-NA lens. The lens parameters are NA = 0.85, f = 5000λ. Τheincident Gaussian beam is truncated at its full-width at half-maximum (FWHM) radius.
Top row: The incident beam is linearly polarized in the X-direction.
Bottom row: The incident beam is circularly polarized.

Guide to DIFFRACT

311

COMMAND.DAT file corresponding to Test Run 3(linearly-polarized incident beam)@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 12/07/2006
Time: 03:51 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
1/e radii Rmin,Rmax:
7220.0000
7220.0000#
RA,THETA:
4250.0000
0.0000000#
Aberrations: None#
Polarization RHO,ETA:
0.0000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
50000.000
50000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: PFOC#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.8500000
5000.0000#
Aberrations: None#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance:
5000.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Log(intensity)#
Logarithmic Scale:
4.0000000#
Xmin,Xmax:-4.0000000
4.0000000#
Ymin,Ymax:-4.0000000
4.0000000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #312

Guide to DIFFRACT
Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: IX00.DAT#
Action: T#
Saving TEMP in TIFF-file: IX00.TIF#
Action: L#
Data-file loaded to TEMP: IY00.DAT#
Action: T#
Saving TEMP in TIFF-file: IY00.TIF#
Action: L#
Data-file loaded to TEMP: IZ00.DAT#
Action: T#
Saving TEMP in TIFF-file: IZ00.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@SESSION.DAT file corresponding to Test Run 3(linearly-polarized incident beam)$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 8.4
File SESSION.DAT $$$$ Date: 04/26/2006
Time: 22:36 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ALL LENGTHS ARE IN UNITS OF WAVELENGTH; ALL ANGLES ARE IN DEGREES_________________________________________________________________
Array dimensions: NDIM= 512, NDSP= 149**************************************
Designated Working Subdirectory: <Test03>*****************************************
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Simple Gaussian: BCX=
0.0
BCY=
0.0
RA=
4250.0
THETA= 0.00
RMIN=
7220.0 RMAX=
7220.0Aberrations: None
Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 50000.00
LMAY= 50000.00

Guide to DIFFRACT

313

Beam centroid:(X=0.00,Y=
0.00)
DIAM=
8002.2
XPOWER=0.100E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________LENS ACTION***********
TYPE: Positive_Focusing
LCX=
0.000
LCY=
0.000
NA=0.850000
FL=
5000.00
Aberrations: None
Quasi-vector treatment
Propagation distance=
5000.00__________________________________________________APRX method: NGUIDE=
99 LGUIDE=
42500.0___________________________________________________________________________________________________________
NMAX= 512
NMAY= 512
LMAX=
51.20
LMAY=
51.20
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5.9
XPOWER=0.103E+01
YPOWER=0.985E-02
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Base 10 logarithm of normalized intensity distribution(Range: from 0 at peak_intensity to -4.0000 at cutoff)___________________________________________
Xmin=
-4.00
Xmax=
4.00
Ymin=
-4.00
Ymax=
4.00
___________________________________________
IXmin=
IYmin=
IZmin=

0.1520E-05
0.0000E+00
0.3461E-17

IXmax=
IYmax=
IZmax=

0.2299E+01
0.6159E-02
0.2585E+00

Intensity pattern for X-polrz stored in IX00.DAT
Intensity pattern for Y-polrz stored in IY00.DAT
Intensity pattern for Z-polrz stored in IZ00.DAT
Peak intensity=0.230E+01 XPEAK=(Total energy flux=0.100E+01

0.00 YPEAK=ZPOWER=0.225E+00)

FILE MANAGEMENT/GRAPHICS************************
Action: Load data-file IX00.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: -0.400000E+01
Maximum Value: 0.000000E+00****************************
Action: Save TIFF version of TEMP array in IX00.TIF
IX00.TIF is successfully created and written to disk____________________________________________________
Action: Load data-file IY00.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149

0.00

314

Guide to DIFFRACT
Minimum Value: -0.400000E+01
Maximum Value: 0.000000E+00****************************
Action: Save TIFF version of TEMP array in IY00.TIF
IY00.TIF is successfully created and written to disk____________________________________________________
Action: Load data-file IZ00.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: -0.400000E+01
Maximum Value: 0.000000E+00****************************
Action: Save TIFF version of TEMP array in IZ00.TIF
IZ00.TIF is successfully created and written to disk____________________________________________________End of session: DIFFRACT terminated normally********************************************COMMAND.DAT file corresponding to Test Run 3(circularly-polarized incident beam)@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 03/03/2006
Time: 08:33 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#

Guide to DIFFRACT

315

Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
1/e radii Rmin,Rmax:
7220.0000
7220.0000#
RA,THETA:
4250.0000
0.0000000#
Aberrations: None#
Polarization RHO,ETA:
0.0000000
45.000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
50000.000
50000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: PFOC#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.8500000
5000.0000#
Aberrations: None#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance:
5000.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Log(intensity)#
Logarithmic Scale:
4.0000000#
Xmin,Xmax:-4.0000000
4.0000000#
Ymin,Ymax:-4.0000000
4.0000000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: IX01.DAT#
Action: T#
Saving TEMP in TIFF-file: IX01.TIF#
Action: L#
Data-file loaded to TEMP: IY01.DAT#
Action: T#
Saving TEMP in TIFF-file: IY01.TIF#
Action: L#
Data-file loaded to TEMP: IZ01.DAT#
Action: T#
Saving TEMP in TIFF-file: IZ01.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

316

Guide to DIFFRACTSESSION.DAT file corresponding to Test Run 3(circularly-polarized incident beam)$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 8.4
File SESSION.DAT $$$$ Date: 04/26/2002
Time: 22:39 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ALL LENGTHS ARE IN UNITS OF WAVELENGTH; ALL ANGLES ARE IN DEGREES_________________________________________________________________
Array dimensions: NDIM= 512, NDSP= 149**************************************
Designated Working Subdirectory: <Test03>*****************************************
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Simple Gaussian: BCX=
0.0
BCY=
0.0
RA=
4250.0
THETA= 0.00
RMIN=
7220.0 RMAX=
7220.0Aberrations: None
Polarization state: Rho= 0.0000
Eta= 45.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 50000.00
LMAY= 50000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
8002.2
XPOWER=0.500E+00
YPOWER=0.500E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________LENS ACTION***********
TYPE: Positive_Focusing
LCX=
0.000
LCY=
0.000
NA=0.850000
FL=
5000.00
Aberrations: None
Quasi-vector treatment
Propagation distance=
5000.00__________________________________________________APRX method: NGUIDE=
99 LGUIDE=
42500.0___________________________________________________________________________________________________________
NMAX= 512
NMAY= 512
LMAX=
51.20
LMAY=
51.20
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5.9
XPOWER=0.519E+00
YPOWER=0.519E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Base 10 logarithm of normalized intensity distribution(Range: from 0 at peak_intensity to -4.0000 at cutoff)___________________________________________
Xmin=
-4.00
Xmax=
4.00
Ymin=
-4.00
Ymax=
4.00
___________________________________________
IXmin=
IYmin=
IZmin=

0.9484E-06
0.9484E-06
0.1327E-13

IXmax=
IYmax=
IZmax=

0.1150E+01
0.1150E+01
0.1301E+00

Intensity pattern for X-polrz stored in IX01.DAT
Intensity pattern for Y-polrz stored in IY01.DAT
Intensity pattern for Z-polrz stored in IZ01.DATGuide to DIFFRACT

317

Peak intensity=0.230E+01 XPEAK=(Total energy flux=0.100E+01

0.00 YPEAK=ZPOWER=0.225E+00)

FILE MANAGEMENT/GRAPHICS************************
Action: Load data-file IX01.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: -0.400000E+01
Maximum Value: 0.000000E+00****************************
Action: Save TIFF version of TEMP array in IX01.TIF
IX01.TIF is successfully created and written to disk____________________________________________________
Action: Load data-file IY01.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: -0.400000E+01
Maximum Value: 0.000000E+00****************************
Action: Save TIFF version of TEMP array in IY01.TIF
IY01.TIF is successfully created and written to disk____________________________________________________
Action: Load data-file IZ01.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: -0.400000E+01
Maximum Value: 0.000000E+00****************************
Action: Save TIFF version of TEMP array in IZ01.TIF
IZ01.TIF is successfully created and written to disk____________________________________________________End of session: DIFFRACT terminated normally********************************************

0.00

318

Guide to DIFFRACT

Test Run 4
The Goos-Hänchen EffectFeaturing: Positive_focusing Lens and Flat Mirror (simulating the TIR prism).

Figure A4.1. Schematic diagram showing the focusing of a collimated beam of light througha 0.05 NA positive_focusing lens. Total internal reflection (TIR) within a glass prism placed
before the focal plane causes a small shift of the focused spot in the +Z direction; the shift isgenerally referred to as the Goos-Hänchen effect.
30

y/λ

-30
-30

x/λ

+30 -30x/λ+30 -30x/λ+30Figure A4.2. Logarithmic plots of intensity distribution at the observation plane. From left toright: focused spot’s components of polarization along the X-, Y-, and Z-axes. The majorityof the optical power (more than 99%) is contained in the middle frame. The diameter of the
Airy disk is approximately 25λ. The Goos-Hänchen effect is manifest in the 1.7λ downward
shift of the distributions.

Guide to DIFFRACT

319

COMMAND.DAT file corresponding to Test Run 4@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 03/03/2006
Time: 08:37 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.5320
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
1/e radii Rmin,Rmax:
4000.0000
4000.0000#
RA,THETA:
3000.0000
0.0000000#
Aberrations: None#
Polarization RHO,ETA:
90.000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
35000.000
35000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.5320
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: PFOC#
LCX,LCY:
0.00000000
0.0000000#
NA,FL:
0.05000000
60000.000#
Aberrations: None#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance: 57500.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.5320
NVIRON: 1.000000 #_________________________________________________________________________ ##Flat mirror: MIRR
(Length_Units:) #Complex index (n,k):0.6600000
0.0000000#
Normal THETA0,PHI0:
135.00000
90.000000#
Emergent THETA,PHI,PSI:
90.000000
90.000000
0.0000000#
Propagation distance:
2500.0000#
Propagation regime: DFRNL#
Smax:
0.1000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.5320
NVIRON: 1.000000 #_________________________________________________________________________ #320Guide to DIFFRACT#
Plot distribution: PLOT
(Length_Units:) #Type: Log(intensity)#
Logarithmic Scale:
4.0000000#
Xmin,Xmax:-30.000000
30.000000#
Ymin,Ymax:-30.000000
30.000000#
Color or Gray-scale: G#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.5320
NVIRON: 1.000000 #_________________________________________________________________________ ##Measure peak and FWHM: MEAS#
Include Z-component: Y#
ID numbers of PEAK I,X,Y:
1
2
3#
Compute/Return: C#
THETA:
0.0000000#
ID number for FWHM:
4#
Compute/Return: C#
THETA:
90.000000#
ID number for FWHM:
5#
Compute/Return: R#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.5320
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: IX00.DAT#
Action: T#
Saving TEMP in TIFF-file: IX00.TIF#
Action: L#
Data-file loaded to TEMP: IY00.DAT#
Action: T#
Saving TEMP in TIFF-file: IY00.TIF#
Action: L#
Data-file loaded to TEMP: IZ00.DAT#
Action: T#
Saving TEMP in TIFF-file: IZ00.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.5320
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Guide to DIFFRACT321

SESSION.DAT file corresponding to Test Run 4$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 7.9
File SESSION.DAT $$$$ Date: 04/26/2002
Time: 23:08 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ALL LENGTHS ARE IN UNITS OF WAVELENGTH; ALL ANGLES ARE IN DEGREES_________________________________________________________________
Array dimensions: NDIM= 512, NDSP= 99**************************************
Designated Working Subdirectory: <Test04>*****************************************
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Simple Gaussian: BCX=
0.0
BCY=
0.0
RA=
3000.0
THETA= 0.00
RMIN=
4000.0 RMAX=
4000.0Aberrations: None
Polarization state: Rho= 90.0000
Eta= 0.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 35000.00
LMAY= 35000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5417.7
XPOWER=0.000E+00
YPOWER=0.100E+01
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________LENS ACTION***********
TYPE: Positive_Focusing
LCX=
0.000
LCY=
0.000
NA=0.050000
FL=
60000.00
Aberrations: None
Quasi-vector treatment
Propagation distance=
57500.00__________________________________________________APRX method: NGUIDE=
125 LGUIDE=
30000.0___________________________________________________________________________________________________________
NMAX= 512
NMAY= 512LMAX=
877.71LMAY=
877.71
Beam centroid:(X=-0.01,Y=-0.01)DIAM=
239.3
XPOWER=0.480E-07
YPOWER=0.999E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________FLAT MIRROR***********
Mirror's refractive index: (n,k)=( 0.6600, 0.0000)
Direction of mirror normal: Theta=135.0000
Phi= 90.0000
Emergent coordinates: Theta= 90.0000 Phi= 90.0000 Psi= 0.0000
Propagation distance=
2500.00
Propagation regime: Direct Fresnel_____________________________________________________NGUIDE=
50 LGUIDE=
4786.42 SMAX= 0.002089______________________________________________________________________________________________________________
NMAX= 512
NMAY= 512LMAX=
877.71LMAY=
877.71
Beam centroid:(X=-0.01,Y=-1.48)DIAM=
83.0
XPOWER=0.181E-02
YPOWER=0.997E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________

322

Guide to DIFFRACT

PLOTTING ROUTINE****************
Base 10 logarithm of normalized intensity distribution(Range: from 0 at peak_intensity to -4.0000 at cutoff)___________________________________________
Xmin=-30.00
Xmax=
30.00
Ymin=
-30.00Ymax=
30.00___________________________________________
IXmin= 0.2288E-19
IXmax= 0.7366E-05
IYmin= 0.8990E-07
IYmax= 0.7560E-02
IZmin= 0.4882E-10
IZmax= 0.2192E-05
Intensity pattern for X-polrz stored in IX00.DAT
Intensity pattern for Y-polrz stored in IY00.DAT
Intensity pattern for Z-polrz stored in IZ00.DAT
Peak intensity=0.760E-02 XPEAK=(Total energy flux=0.100E+01

0.00 YPEAK=ZPOWER=0.509E-03)-1.71

PEAK INTENSITY AND FWHM***********************
Z-component of polarization included in PEAK & FWHM measurement
Peak Intensity=0.760E-02
XPEAK= 0.00
YPEAK= -1.71
Peak Intensity, XPEAK,YPEAK saved in SIGNAL array as S(1), S(2), S(3)Theta= 0.00
FWHM=
10.89
FWHM saved in SIGNAL array as S( 4)Theta= 90.00
FWHM=
10.89
FWHM saved in SIGNAL array as S( 5)FILE MANAGEMENT/GRAPHICS************************
Action: Load data-file IX00.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 99
Number of Columns: 99
Minimum Value: -0.400000E+01
Maximum Value: 0.000000E+00****************************
Action: Save TIFF version of TEMP array in IX00.TIF
IX00.TIF is successfully created and written to disk____________________________________________________End of session: DIFFRACT terminated normally********************************************(TRUNCATED)Guide to DIFFRACT

323

Test Run 5
Gradient-index Lens
Featuring: Laser diode, Geometric-optical System, and Propagation in the Environment.

Figure A5.1. Schematic diagram of the simulated system. A laser diode beam (λ = 680 nm)
having a circular cross-section with a divergence angle of 18° (FWHM) illuminates a
gradient-index lens. When the laser diode is on-axis, its distance from the front facet of thelens is 920λ. In this instance the focused spot appears a distance of 41.4λ beyond the flat facetof the lens.

324

Guide to DIFFRACT

+5

+3

y/λ

y/λ

-5

-3
-3

x/λ

+3

-5

x/λ

+5

Figure A5.2. Distributions of intensity (left) and log_intensity (right) at the focal plane of the
GRIN lens for an on-axis object point. The source point is 920λ in front of the lens, while its
image is a distance of 41.4λ beyond the flat facet of the lens.

-14

-12

y/λ

y/λ

-20

-22
-20

x/λ

-14

-22

x/λ

-12

Figure A5.3. Distributions of intensity (left) and log_intensity (right) at the focal plane of the
GRIN lens for an off-axis object point. The source point is at (x, y) = (60λ, 60λ) in the object
plane, while its image is located a distance of 40λ beyond the flat facet of the lens at(x, y) = (-17.5λ, -17.5λ).

Guide to DIFFRACT325

COMMAND.DAT file corresponding to Test Run 5@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 03/03/2006
Time: 08:43 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.68000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: LD#
BCX,BCY,Power(mW):
60.000000
60.000000
1.000000#
PHI,PSI,DELTA:
18.000000
18.000000
0.000000#
Orientation angle THETA:
0.0000000#
Distance from cavity Z0:
920.00000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
3000.0000
3000.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.68000
NVIRON: 1.000000 #_________________________________________________________________________ ##Ray conversion: RAYS#
Multiply curvature (Y/N): N#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.68000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: um) #Define a new system: NSYS#
Name of the system: GR01#**** Surface #01:
Type: Glass#
Shape: S#
Vertex (Vx,Vy,Vz):.0000000.0000000.0000000#
Normal (Theta,Phi):.0000000.0000000#
Radius of curvature (RC):
235.0000#
Aperture's Rmin,Rmax:
0.0000000
105.0000#
Coating (AR/DOE/BARE): BARE#
Medium beyond surface: GRIN#
Type of GRIN medium: 1#
Origin (Ox,Oy,Oz):.0000000.0000000.0000000#
Axis (Theta,Phi):.0000000.0000000#
N0,R0,Z0:
1.666000
105.0000.0000000#
R coefficients: Q1,Q2,Q3:-0.054300.0000000.0000000#
Q4,Q5,Q6:.0000000.0000000.0000000#
Z coefficients: W1,W2,W3:.0000000.0000000.0000000#

326

Guide to DIFFRACT
W4,W5,W6:.0000000.0000000.0000000#
W7,W8,W9:.0000000.0000000.0000000#
W10,W11,W12:.0000000.0000000.0000000#
Next surface (Hit,Miss):20#**** Surface #02:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):.0000000.0000000
380.0000#
Normal (Theta,Phi):.0000000.0000000#
Aperture's Rmin,Rmax:.0000000
105.0000#
Coating (AR/DOE/BARE): BARE#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
30#**** Surface #03:
Type: Destination#
Vertex (Vx,Vy,Vz):.0000000.0000000
250.0000#
Normal (Theta,Phi):.0000000.0000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.68000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units:) #Trace all incident rays: TRAY#
Initial guess (Xc,Yc,Zc):
0.000000
0.000000
0.000000#
Accept or Search: Search#
Strip phase-factor (Y/N): Y#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
2000.0000
2000.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.68000
NVIRON: 1.000000 #_________________________________________________________________________ ##Propagate in environment: PROP
(Length_Units:) #Propagation distance: 231.00000#
Multiply curvature (Y/N): N#
Reposition beam (Y/N): N#
Propagation regime: DFRNL#
Smax:
0.0000000#
Scalar/Quasi-vector: SC#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.68000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic Scale:
4.0000000#
Xmin,Xmax:-20.0000000-14.0000000#
Ymin,Ymax:-20.0000000-14.0000000#
Color or Gray-scale: C#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.68000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Log(intensity)#
Logarithmic Scale:
4.0000000#
Xmin,Xmax:-22.0000000-12.0000000#
Ymin,Ymax:-22.0000000-12.0000000#
Color or Gray-scale: C#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#

Guide to DIFFRACT

327$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.68000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: IX00.DAT#
Action: T#
Saving TEMP in TIFF-file: IX00.TIF#
Action: L#
Data-file loaded to TEMP: IX01.DAT#
Action: T#
Saving TEMP in TIFF-file: IX01.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.68000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@SESSION.DAT file corresponding to Test Run 5$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 7.9
File SESSION.DAT $$$$ Date: 04/26/2002
Time: 23:54 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ALL LENGTHS ARE IN UNITS OF WAVELENGTH; ALL ANGLES ARE IN DEGREES_________________________________________________________________
Array dimensions: NDIM= 512, NDSP= 149**************************************
Designated Working Subdirectory: <Test05>*****************************************
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Laser Diode: BCX=
60.0
BCY=
60.0
THETA= 0.00
PHI= 18.00
PSI= 18.00
DELTA=
0.0000
Propagation distance Z0=
920.00_________________________________________________________
NMAX= 512
NMAY= 512
LMAX=
3000.00
LMAY=
3000.00
Beam centroid:(X=60.00,Y=
60.00)DIAM=
495.0
XPOWER=0.100E+01
YPOWER=0.000E+00
Curvature= 0.11E-02(Environment's index of refraction= 1.0000)_________________________________________________________
CONVERSION TO RAYS******************
Ray components RX,RY,RZ are computed and stored in BX and BY

328

Guide to DIFFRACTGEOMETRIC-OPTICAL SYSTEM DEFINITION***********************************Name of the system: GR01
Vacuum wavelength of the incident light (LAMBDA) in the sameunits as used below to specify system parameters: .680000E+00
Surface #01***********
Type: Glass
Shape: Spherical
Vertex coordinates: (0.000000E+00,0.000000E+00,0.000000E+00)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Radius of curvature, including sign (RC):
235.00Radius of aperture (RA):105.00Antireflection: No
Gradient-index (GRIN) medium beyond the surface:
Point of origin (Ox,Oy,Oz): ( 0.0000E+00, 0.0000E+00, 0.0000E+00)
Axis of symmetry (Theta,Phi): ( 0.000, 0.000)
Base index of refraction (N0)= 1.666000
Radius of cylinder (R0)= 0.10500E+03
Axial length-scale (Z0)= 0.00000E+00
Radial coefficients (Q1,Q2,Q3): -0.5430E-01, 0.000E+00, 0.000E+00(Q4,Q5,Q6): 0.0000E+00, 0.000E+00, 0.000E+00
Axial coefficients (W1,W2,W3): 0.0000E+00, 0.000E+00, 0.000E+00(W4,W5,W6): 0.0000E+00, 0.000E+00, 0.000E+00(W7,W8,W9): 0.0000E+00, 0.000E+00, 0.000E+00(W10,W11,W12): 0.0000E+00, 0.000E+00, 0.000E+00
Surface #02***********
Type: Glass
Shape: Flat
Vertex coordinates: (0.000000E+00,0.000000E+00,0.380000E+03)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Radius of aperture (RA):105.00Antireflection: No
Free space medium beyond the surface: n = 1.00
Surface #03***********
Type: Destination
Shape: Flat
Vertex coordinates: (0.000000E+00,0.000000E+00,0.250000E+03)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)______________________________________________________________DIFFRACT now returns to its standard mode, where all lengths
are normalized by the wavelength (LAMBDA) of the light within
its present environment.
(Refractive index= 1.0000)______________________________________________________________GEOMETRIC-OPTICAL RAY TRACING*****************************_______________________________________________________________
Note: Presently all lengths are in the same units as the vacuum
wavelength.
(LAMBDA=.680000E+00)_______________________________________________________________
Incident Beam:**************
Magnitude of the brightest ray in the beam:0.104E-04
Magnitude of the weakest ray to qualify for tracing:0.104E-08
Total number of qualified rays within the incident beam: 26241(Upper bound on the number of rays presently allowed to cross: 45000)

Guide to DIFFRACT

329

Emergent Beam:**************
Number of qualified rays that traverse the system: 2010
Region within destination plane to which rays are confined:
Circle: Xc,Yc,Radius =-6.6243-6.6243
105.11
Rectangle: Xmin,Xmax =-105.2995.565
Ymin,Ymax =-105.2995.565______________________________________________________________DIFFRACT now returns to its standard mode, where all lengths
are normalized by the wavelength (LAMBDA) of the light within
its present environment.
(Refractive index= 1.0000)______________________________________________________________The emergent beam is converging towards the point (Xc,Yc,Zc)=( -17.5428, -17.5431, 231.173), with an rms wavefront error of 0.59040E-01. If the beam's cross-sectional area
is sufficiently small, it may be considered to be a collimatedbeam, traveling along the direction (Theta= 6.13, Phi=225.00).
______________________________________________________________
Emergent wavefront stripped from curvature/tilt phase-factor**************************************************************
Construction of the emergent wave-front within the destination
plane is now complete.
(Total energy flux = 0.3987)**************************************************************_________________________________________________________
NMAX= 512
NMAY= 512
LMAX=
2000.00
LMAY=
2000.00
Beam centroid:(X=6.04,Y=
9.21)
DIAM=
264.2
XPOWER=0.402E+00
YPOWER=0.267E-03
Curvature=-0.43E-02(Environment's index of refraction= 1.0000)_________________________________________________________PROPAGATION IN ENVIRONMENT**************************
Propagation distance=

231.00000(Forward)********************************************************
Note: Mesh is too coarse for the curvature phase-factor.
Multiplication is not recommended.
********************************************************
Curvature phase-factor is NOT multiplied in AX,AY*********************************************************************WARNING: The beam appears to be converging towards (or diverging awayfrom) the point (Xc,Yc,Zc)=(-.175428E+02,-.175431E+02,0.231173E+03).
The computation method used here will yield accurate results only if
the beam is brought to the vicinity of the above point. If the chosen
propagation distance is much different from Zc the results are likely
to be inaccurate.
*********************************************************************_________________________________________________________
NMAX= 512
NMAY= 512
LMAX=
59.18
LMAY=
59.18
Beam centroid:(X=-17.41,Y=-17.38)DIAM=
4.3
XPOWER=0.395E+00
YPOWER=0.257E-03
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=-20.00
Xmax=-14.00Ymin=-20.00Ymax=-14.00___________________________________________

330

Guide to DIFFRACT
IXmin=
IYmin=

0.1229E-05
0.3563E-08

IXmax=
IYmax=

0.3235E+00
0.8054E-04

Latest screen (intensity pattern) dumped in IS00.BMP
Intensity pattern for X-polrz stored in IX00.DAT
Intensity pattern for Y-polrz stored in IY00.DAT
PLOTTING ROUTINE****************
Base 10 logarithm of normalized intensity distribution(Range: from 0 at peak_intensity to -4.0000 at cutoff)___________________________________________
Xmin=-22.00
Xmax=-12.00Ymin=-22.00Ymax=-12.00___________________________________________
IXmin=
IYmin=

0.2328E-06
0.8714E-10

IXmax=
IYmax=

0.3206E+00
0.8044E-04

Latest screen (intensity pattern) dumped in IS01.BMP
Intensity pattern for X-polrz stored in IX01.DAT
Intensity pattern for Y-polrz stored in IY01.DAT
FILE MANAGEMENT/GRAPHICS************************
Action: Load data-file IX00.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.122931E-05
Maximum Value: 0.323541E+00****************************
Action: Save TIFF version of TEMP array in IX00.TIF
IX00.TIF is successfully created and written to disk____________________________________________________
Action: Load data-file IX01.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: -0.400000E+01
Maximum Value: 0.000000E+00****************************
Action: Save TIFF version of TEMP array in IX01.TIF
IX01.TIF is successfully created and written to disk____________________________________________________End of session: DIFFRACT terminated normally********************************************

Guide to DIFFRACT

331

Test Run 6
Coherent imaging of a phase object
Featuring: Optical disk, Positive_focusing lens, and Collimator.

Condenser( f = 10000λ)Gaussian laser beam
(1/e radius = 275λ)Imaging lens(NA = 0.5, M = 2×)
Image plane

Phase object(transparent)

Figure A6.1. Schematic diagram of the simulated system. The imaging lens is modeled as the
combination of a collimating lens followed by a positive_focusing lens.

a

−7

b

x/λ

+7

−16

x/λ

+16

Figure A6.2. (a) Phase objects of different dimensions within the object plane. The gray-scale
depicts the height of the marks only; the two white marks have a height of 0.2λ, the gray markis 0.1λ high, and the black mark is 0.2λ deep. (b) Intensity distribution in the image plane.

332

Guide to DIFFRACTCOMMAND.DAT file corresponding to Test Run 6@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 03/03/2006
Time: 08:55 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 680.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
1/e radii Rmin,Rmax:
275.00000
275.00000#
RA,THETA:
5000.0000
0.0000000#
Aberrations: None#
Polarization RHO,ETA:
0.0000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
35000.000
35000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 680.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: PFOC#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.5000000
10000.000#
Aberrations (Y/N): N#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance: 10000.00#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 680.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units:) #Type of medium: WR#
SCX,SCY,RA:.00000000.00000000
20.000000#(Incidence medium) Ninc:
1.0000000#
Pregrooves: N#
Reflection coefficient:
1.0000000#
Marks (bumps/pits): Y#
Number of marks: 4#
Mark # 1:
MCX,MCY:.00000000.00000000#
THETA:
45.000000#
Round or Flat: R#

Guide to DIFFRACT

333

Length, Width, Height:
3.0000000
3.0000000
0.2000000#
Roof-to-Rim Ratio:
0.0000000#
Reflection coefficient:
1.0000000#
Mark # 2:
MCX,MCY:
3.0000000
3.0000000#
THETA:
45.000000#
Round or Flat: R#
Length, Width, Height:
3.0000000
2.0000000
0.1000000#
Roof-to-Rim Ratio:
0.0000000#
Reflection coefficient:
1.0000000#
Mark # 3:
MCX,MCY:-3.3000000-3.3000000#
THETA:
45.000000#
Round or Flat: R#
Length, Width, Height:
4.0000000
2.0000000-0.2000000#
Roof-to-Rim Ratio:
0.0000000#
Reflection coefficient:
1.0000000#
Mark # 4:
MCX,MCY:-2.7500000
2.7500000#
THETA:
45.000000#
Round or Flat: R#
Length, Width, Height:
6.0000000
2.0000000
0.2000000#
Roof-to-Rim Ratio:
0.0000000#
Reflection coefficient:
1.0000000#
Inspect surface pattern: Y#
Xmin,Xmax:-7.0000000
7.0000000#
Ymin,Ymax:-7.0000000
7.0000000#
Color or Gray-scale: G#
Save data file (Y/N): Y#
File identifier: 0#
Propagation distance:
10000.000#
Propagation regime: FRNHF#
Smax:
0.1000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 680.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: COLL#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.5000000
10000.000#
Aberrations (Y/N): N#
Scalar/Quasi-vector: QV#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 680.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: PFOC#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.2500000
20000.000#
Aberrations (Y/N): N#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance: 20000.00#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 680.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-16.000000
16.000000#
Ymin,Ymax:-16.000000
16.000000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#

334

Guide to DIFFRACT
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 680.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: SP00.DAT#
Action: T#
Saving TEMP in TIFF-file: SP00.TIF#
Action: L#
Data-file loaded to TEMP: IX00.DAT#
Action: T#
Saving TEMP in TIFF-file: IX00.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 680.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@SESSION.DAT file corresponding to Test Run 6$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 7.9
File SESSION.DAT $$$$ Date: 04/27/2002
Time: 07:58 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ALL LENGTHS ARE IN UNITS OF WAVELENGTH; ALL ANGLES ARE IN DEGREES_________________________________________________________________
Array dimensions: NDIM= 512, NDSP= 149**************************************
Designated Working Subdirectory: <Test06>*****************************************
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Simple Gaussian: BCX=
0.0
BCY=
0.0
RA=
5000.0
THETA= 0.00
RMIN=
275.0 RMAX=
275.0
Aberrations: None
Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 35000.00
LMAY= 35000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
550.0
XPOWER=0.100E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________LENS ACTION***********
TYPE: Positive_Focusing
LCX=
0.000
LCY=
0.000
NA=0.500000
FL=
10000.00
Aberrations: None
Quasi-vector treatment
Propagation distance=
10000.00__________________________________________________APRX method: NGUIDE=
100 LGUIDE=
50000.0___________________________________________________________________________________________________________
NMAX= 512
NMAY= 512LMAX=
146.29LMAY=
146.29Beam centroid:(X=0.00,Y=
0.00)
DIAM=
23.1

Guide to DIFFRACT

335

XPOWER=0.100E+01
YPOWER=0.895E-08
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________OPTICAL DISK SURFACE********************
Disk type: WRITABLE
Aperture: SCX=
0.00
SCY=
0.00
RA=
20.00
Air-incident configuration (Ninc= 1.000000)
Amplitude reflection coefficient of the disk surface = 1.0000
BUMPS AND PITS______________Mark # 1:

MCX= 0.0000
MCY= 0.0000
Theta=
45.00
Length=3.000
Width=
3.000
Height=0.200
Amplitude reflection coefficient at mark center=1.0000(Mark roof is round)Mark # 2:

MCX= 3.0000
MCY= 3.0000
Theta=
45.00
Length=3.000
Width=
2.000
Height=0.100
Amplitude reflection coefficient at mark center=1.0000(Mark roof is round)Mark # 3:

MCX= -3.3000
MCY= -3.3000
Theta=
45.00
Length=4.000
Width=
2.000
Height= -0.200
Amplitude reflection coefficient at mark center=1.0000(Mark floor is round)Mark # 4:

MCX= -2.7500
MCY= 2.7500
Theta=
45.00
Length=6.000
Width=
2.000
Height=0.200
Amplitude reflection coefficient at mark center=1.0000(Mark roof is round)

Latest screen (disk_surface_pattern) dumped in DS00.BMP_________________________________
Surface pattern saved in SP00.DAT
Xmin=
-7.00
Xmax=
7.00
Ymin=
-7.00
Ymax=
7.00
_________________________________
Propagation distance=
10000.00
Propagation regime: Fraunhofer_____________________________________________________NGUIDE=
999 LGUIDE=
200.00 SMAX= 0.500000______________________________________________________________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 35000.00
LMAY= 35000.00
Beam centroid:(X=0.45,Y=-16.31)
DIAM=
4468.5
XPOWER=0.993E+00
YPOWER=0.834E-08
Curvature= 0.10E-03(Environment's index of refraction= 1.0000)_________________________________________________________LENS ACTION***********
TYPE: Collimator
LCX=
0.000
LCY=
0.000
NA=0.500000
FL=
10000.00
Aberrations: None
Quasi-vector treatment_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 35000.00
LMAY= 35000.00
Beam centroid:(X=3.16,Y=-13.27)
DIAM=
2851.5
XPOWER=0.979E+00
YPOWER=0.509E-04
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________LENS ACTION

336

Guide to DIFFRACT***********
TYPE: Positive_Focusing
LCX=
0.000
LCY=
0.000
NA=0.250000
FL=
20000.00
Aberrations: None
Quasi-vector treatment
Propagation distance=
20000.00__________________________________________________APRX method: NGUIDE=
100 LGUIDE=
50000.0___________________________________________________________________________________________________________
NMAX= 512
NMAY= 512LMAX=
292.57LMAY=
292.57
Beam centroid:(X=-0.08,Y=
0.02)
DIAM=
46.4
XPOWER=0.979E+00
YPOWER=0.298E-04
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=-16.00
Xmax=
16.00
Ymin=
-16.00Ymax=
16.00___________________________________________
IXmin=
IYmin=

0.1762E-03
0.1147E-10

IXmax=
IYmax=

0.2496E-02
0.5522E-06

Latest screen (intensity pattern) dumped in IS00.BMP
Intensity pattern for X-polrz stored in IX00.DAT
Intensity pattern for Y-polrz stored in IY00.DAT
FILE MANAGEMENT/GRAPHICS************************
Action: Load data-file SP00.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: -0.199009E+00
Maximum Value: 0.200000E+00****************************
Action: Save TIFF version of TEMP array in SP00.TIF
SP00.TIF is successfully created and written to disk____________________________________________________
Action: Load data-file IX00.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.176215E-03
Maximum Value: 0.249593E-02****************************
Action: Save TIFF version of TEMP array in IX00.TIF
IX00.TIF is successfully created and written to disk____________________________________________________End of session: DIFFRACT terminated normally********************************************

Guide to DIFFRACT

337

Test Run 7
Internal conical refraction
Featuring: Biaxially birefringent crystal within the Multilayer option.

Figure A7.1. A collimated beam enters a slab of biaxially birefringent crystal. The crystal iscut with one of its optical axes perpendicular to the surface. Upon entering the slab, the beam
spreads into a cone, and emerges in the form of two concentric cylinders.

1400 a

b

c

y/λ

-400
-900

x/λ

900 -900x/λ

900 -900x/λ

900Figure A7.2. Distributions at the exit facet of the crystal slab for a circularly polarized
incident beam. (a) Total intensity (Ix + Iy), (b) polarization rotation angle ρ, (c) polarization
ellipticity η.

338

Guide to DIFFRACT

1400 a

b

y/λ

-400

−900

x/λ

900

−900

x/λ

900

Figure A7.3. Distributions of intensity for (a) x-component and (b) y-component ofpolarization at the exit facet of the crystal when the incident beam is linearly polarized.

1400

y/λ

-400
-900

x/λ

900

Figure A7.4. Distribution of total intensity (Ix + Iy) at the exit
facet of the crystal slab for an unpolarized incident beam.

Guide to DIFFRACT

339

COMMAND.DAT file corresponding to Test Run 7@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 03/03/2006
Time: 09:17 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 650.000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
1/e radii Rmin,Rmax:
75.000000
75.000000#
RA,THETA:
150.00000
0.0000000#
Aberrations: None#
Polarization RHO,ETA:
0.0000000
45.000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
3000.0000
3000.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 650.000
NVIRON: 1.000000 #_________________________________________________________________________ ##Multilayer structure: MLYR
(Length_Units:
) #
Nsub:
1.0000000#
Number of layers: 1#
Layer#01
Type (A-E): C#
Real(EXX),Imag(EXX):
2.3500000
0.0000000#
Real(EYY),Imag(EYY):
2.2500000
0.0000000#
Real(EZZ),Imag(EZZ):
2.4500000
0.0000000#
Thickness:
25000.000#
Euler Theta,Phi,Psi:
46.220000
270.00000
0.000000#
Ninc:
1.0000000#
Compress incident: NONE#
Entrance surface: BARE#
Reflect or Transmit: T#
Emergent environment's n:
1.0000000#
Decompress emergent: NONE#
Exit surface: BARE#
Stack's Theta,Phi,Psi:
0.0000000
0.0000000
0.0000000#
Emergent Theta,Phi,Psi:
0.0000000
0.0000000
0.0000000#
Propagation distance:
0.0000000#
Propagation regime: DFRNL#
Smax:
0.1000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 650.000
NVIRON: 1.000000 #340

Guide to DIFFRACT_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-900.00000
900.00000#
Ymin,Ymax:-400.00000
1400.0000#
Color or Gray-scale: C#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 650.000
NVIRON: 1.000000 #_________________________________________________________________________ ##Amplitude/phase mask: MASK
(Length_Units:) #Shape: Annulus#
MCX,MCY:
0.0000000
543.00000#
Inner circle R1,Alpha1:
314.00000
0.0000000#
Outer circle R2,Alpha2:657.00000
0.0000000#
Inside amplitude,phase:
0.0000000
0.0000000#
Annulus amplitude,phase:
1.0000000
0.0000000#
Outside amplitude,phase:
0.0000000
0.0000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 650.000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Polarization#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-900.00000
900.00000#
Ymin,Ymax:-400.00000
1400.0000#
Color or Gray-scale: C#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 650.000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: IX01.DAT#
Action: A#
Data-file added to TEMP: IY01.DAT#
Weight Factor: 1.00000#
Action: T#
Saving TEMP in TIFF-file: IT01.TIF#
Action: L#
Data-file loaded to TEMP: RT01.DAT#
Action: T#
Saving TEMP in TIFF-file: RT01.TIF#
Action: L#
Data-file loaded to TEMP: EL01.DAT#
Action: T#
Saving TEMP in TIFF-file: EL01.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 650.000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#

Guide to DIFFRACT

341

BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
1/e radii Rmin,Rmax:
75.000000
75.000000#
RA,THETA:
150.00000
0.0000000#
Aberrations: None#
Polarization RHO,ETA:
0.0000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
3000.0000
3000.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 650.000
NVIRON: 1.000000 #_________________________________________________________________________ ##Multilayer structure: MLYR
(Length_Units:
) #
Nsub:
1.0000000#
Number of layers: 1#
Layer#01
Type (A-E): C#
Real(EXX),Imag(EXX):
2.3500000
0.0000000#
Real(EYY),Imag(EYY):
2.2500000
0.0000000#
Real(EZZ),Imag(EZZ):
2.4500000
0.0000000#
Thickness:
25000.000#
Euler Theta,Phi,Psi:
46.220000
270.00000
0.000000#
Ninc:
1.0000000#
Compress incident: NONE#
Entrance surface: BARE#
Reflect or Transmit: T#
Emergent environment's n:
1.0000000#
Decompress emergent: NONE#
Exit surface: BARE#
Stack's Theta,Phi,Psi:
0.0000000
0.0000000
0.0000000#
Emergent Theta,Phi,Psi:
0.0000000
0.0000000
0.0000000#
Propagation distance:
0.0000000#
Propagation regime: DFRNL#
Smax:
0.1000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 650.000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-900.00000
900.00000#
Ymin,Ymax:-400.00000
1400.0000#
Color or Gray-scale: C#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 2#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 650.000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: IX02.DAT#
Action: T#
Saving TEMP in TIFF-file: IX02.TIF#
Action: L#
Data-file loaded to TEMP: IY02.DAT#
Action: T#
Saving TEMP in TIFF-file: IY02.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 650.000
NVIRON: 1.000000 #_________________________________________________________________________ ##342

Guide to DIFFRACT
Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
1/e radii Rmin,Rmax:
75.000000
75.000000#
RA,THETA:
150.00000
0.0000000#
Aberrations: None#
Polarization RHO,ETA:
90.000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
3000.0000
3000.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 650.000
NVIRON: 1.000000 #_________________________________________________________________________ ##Multilayer structure: MLYR
(Length_Units:
) #
Nsub:
1.0000000#
Number of layers: 1#
Layer#01
Type (A-E): C#
Real(EXX),Imag(EXX):
2.3500000
0.0000000#
Real(EYY),Imag(EYY):
2.2500000
0.0000000#
Real(EZZ),Imag(EZZ):
2.4500000
0.0000000#
Thickness:
25000.000#
Euler Theta,Phi,Psi:
46.220000
270.00000
0.000000#
Ninc:
1.0000000#
Compress incident: NONE#
Entrance surface: BARE#
Reflect or Transmit: T#
Emergent environment's n:
1.0000000#
Decompress emergent: NONE#
Exit surface: BARE#
Stack's Theta,Phi,Psi:
0.0000000
0.0000000
0.0000000#
Emergent Theta,Phi,Psi:
0.0000000
0.0000000
0.0000000#
Propagation distance:
0.0000000#
Propagation regime: DFRNL#
Smax:
0.1000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 650.000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-900.00000
900.00000#
Ymin,Ymax:-400.00000
1400.0000#
Color or Gray-scale: C#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 3#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 650.000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: IX02.DAT#
Action: A#
Data-file added to TEMP: IY02.DAT#
Weight Factor: 1.00000#
Action: A#
Data-file added to TEMP: IX03.DAT#
Weight Factor: 1.00000#
Action: A#
Data-file added to TEMP: IY03.DAT#
Weight Factor: 1.00000#
Action: T#

Guide to DIFFRACT

343

Saving TEMP in TIFF-file: IT02.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm): 650.000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Printout of the file CONE.STK containing the multilayer stack parameters@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File CONE.STK @@#$@@ Date: 03/03/2006
Time: 09:33 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$ Notes: Slab of biaxially birefringent crystal, cut with one of its#$
optical axes perpendicular to the slab's surface.
#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#
Vacuum wavelength:1.0000000(Length_Units:
) #
Nsub:
1.0000000#
Number of layers: 1#
Layer#01
Type (A-E): C#
Real(EXX),Imag(EXX):
2.3500000
0.0000000#
Real(EYY),Imag(EYY):
2.2500000
0.0000000#
Real(EZZ),Imag(EZZ):
2.4500000
0.0000000#
Thickness:
25000.000#
Euler Theta,Phi,Psi:
46.220000
270.00000
0.000000#
Ninc:
1.0000000##@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

344

Guide to DIFFRACTSESSION.DAT file corresponding to Test Run 7$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 7.9
File SESSION.DAT $$$$ Date: 04/27/2002
Time: 09:14 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ALL LENGTHS ARE IN UNITS OF WAVELENGTH; ALL ANGLES ARE IN DEGREES_________________________________________________________________
Array dimensions: NDIM= 512, NDSP= 149**************************************
Designated Working Subdirectory: <Test07>*****************************************
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Simple Gaussian: BCX=
0.0
BCY=
0.0
RA=
150.0
THETA= 0.00
RMIN=
75.0 RMAX=
75.0
Aberrations: None
Polarization state: Rho= 0.0000
Eta= 45.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX=
3000.00
LMAY=
3000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
149.8
XPOWER=0.500E+00
YPOWER=0.500E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________MULTILAYER**********_____________________________________________________________
Vacuum wavelength of the incident light (LAMBDA) in the sameunits as used below to specify layer thicknesses: .100000E+01_____________________________________________________________
Substrate:(n,k)= ( 1.000000, 0.000000)
Thickness= Infinite

Layer# 1:

EXX= ( 2.350000, 0.000000)
EYY= ( 2.250000, 0.000000)
EZZ= ( 2.450000, 0.000000)
Thickness=
25000.000000____________________________________________
Euler angles of rotated coordinates:(Theta= 46.220 Phi= 270.000 Psi=
0.000)____________________________________________

Medium of incidence: (n,k)= ( 1.000000, 0.000000)
Transmission mode
Index of emergent environment= 1.0000______________________________________________________________DIFFRACT now returns to its standard mode, where all lengths
are normalized by the wavelength (LAMBDA) of the light within
its present environment.
(Refractive index= 1.0000)______________________________________________________________
Multilayer coordinates: Theta= 0.00
Phi= 0.00
Psi= 0.00
Emergent coordinates:
Theta= 0.00
Phi= 0.00
Psi= 0.00
Propagation distance=
0.00_________________________________________________________
NMAX= 512
NMAY= 512
LMAX=
3000.00
LMAY=
3000.00
Beam centroid:(X=-2.77,Y=524.71)
DIAM=
1609.1XPOWER=0.432E+00
YPOWER=0.430E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________

Guide to DIFFRACT345
PLOTTING ROUTINE****************

Intensity distribution___________________________________________
Xmin=-900.00
Xmax=
900.00Ymin=-400.00
Ymax=
1400.00___________________________________________
IXmin=
IYmin=

0.2339E-11
0.1145E-12

IXmax=
IYmax=

0.1751E-05
0.1780E-05

Intensity pattern for X-polrz stored in IX01.DAT
Intensity pattern for Y-polrz stored in IY01.DAT
PHASE/AMPLITUDE MASK********************
Annular: MCX=
R1=
R2=

0.00
314.00657.00MCY=
543.00
ALPHA1=0.0000
ALPHA2=0.0000

Inside Amplitude = 0.000000
Phase=
0.00
Annulus Amplitude= 1.000000
Phase=
0.00
Outside Amplitude= 0.000000
Phase=
0.00_________________________________________________________
NMAX= 512
NMAY= 512
LMAX=
3000.00
LMAY=
3000.00
Beam centroid:(X=-2.36,Y=
539.25)
DIAM=
1531.6
XPOWER=0.404E+00
YPOWER=0.413E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Distributions of polarization rotation angle and ellipticity___________________________________________
Xmin=-900.00
Xmax=
900.00Ymin=-400.00
Ymax=
1400.00___________________________________________
RTmin=
ELmin=-89.995323-8.218790

RTmax=
ELmax=

89.971657
15.489133(degrees)(degrees)

Polarization rotation angle is stored in RT01.DAT
Polarization ellipticity is stored in EL01.DAT
FILE MANAGEMENT/GRAPHICS************************
Action: Load data-file IX01.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.233945E-11
Maximum Value: 0.175114E-05****************************
Action: Load-and-add data-file IY01.DAT into the TEMP array(Weight Factor= 0.100000E+01)****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.264157E-09

346

Guide to DIFFRACT
Maximum Value: 0.179396E-05****************************
Action: Save TIFF version of TEMP array in IT01.TIF
IT01.TIF is successfully created and written to disk____________________________________________________
Action: Load data-file RT01.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: -0.899953E+02
Maximum Value: 0.899717E+02****************************
Action: Save TIFF version of TEMP array in RT01.TIF
RT01.TIF is successfully created and written to disk____________________________________________________
Action: Load data-file EL01.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: -0.821879E+01
Maximum Value: 0.154891E+02****************************
Action: Save TIFF version of TEMP array in EL01.TIF
EL01.TIF is successfully created and written to disk____________________________________________________
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Simple Gaussian: BCX=
0.0
BCY=
0.0
RA=
150.0
THETA= 0.00
RMIN=
75.0 RMAX=
75.0
Aberrations: None
Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX=
3000.00
LMAY=
3000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
149.8
XPOWER=0.100E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________MULTILAYER**********_____________________________________________________________
Vacuum wavelength of the incident light (LAMBDA) in the sameunits as used below to specify layer thicknesses: .100000E+01_____________________________________________________________
Substrate:
Layer# 1:(n,k)= ( 1.000000, 0.000000)
Thickness= Infinite

EXX= ( 2.350000, 0.000000)
EYY= ( 2.250000, 0.000000)
EZZ= ( 2.450000, 0.000000)
Thickness=
25000.000000____________________________________________
Euler angles of rotated coordinates:(Theta= 46.220 Phi= 270.000 Psi=
0.000)____________________________________________

Guide to DIFFRACT

347

Medium of incidence: (n,k)= ( 1.000000, 0.000000)
Transmission mode
Index of emergent environment= 1.0000______________________________________________________________DIFFRACT now returns to its standard mode, where all lengths
are normalized by the wavelength (LAMBDA) of the light within
its present environment.
(Refractive index= 1.0000)______________________________________________________________
Multilayer coordinates: Theta= 0.00
Phi= 0.00
Psi= 0.00
Emergent coordinates:
Theta= 0.00
Phi= 0.00
Psi= 0.00
Propagation distance=
0.00_________________________________________________________
NMAX= 512
NMAY= 512
LMAX=
3000.00
LMAY=
3000.00
Beam centroid:(X=-0.07,Y=
257.05)
DIAM=
1433.3
XPOWER=0.653E+00
YPOWER=0.211E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=-900.00
Xmax=
900.00Ymin=-400.00
Ymax=
1400.00___________________________________________
IXmin=
IYmin=

0.1628E-10
0.5183E-23

IXmax=
IYmax=

0.3501E-05
0.8953E-06

Intensity pattern for X-polrz stored in IX02.DAT
Intensity pattern for Y-polrz stored in IY02.DAT
FILE MANAGEMENT/GRAPHICS************************
Action: Load data-file IX02.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.162812E-10
Maximum Value: 0.350061E-05****************************
Action: Save TIFF version of TEMP array in IX02.TIF
IX02.TIF is successfully created and written to disk____________________________________________________
Action: Load data-file IY02.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.518339E-23
Maximum Value: 0.895285E-06****************************
Action: Save TIFF version of TEMP array in IY02.TIF
IY02.TIF is successfully created and written to disk____________________________________________________

348

Guide to DIFFRACTINCIDENT AMPLITUDE DISTRIBUTION*******************************
Simple Gaussian: BCX=
0.0
BCY=
0.0
RA=
150.0
THETA= 0.00
RMIN=
75.0 RMAX=
75.0
Aberrations: None
Polarization state: Rho= 90.0000
Eta= 0.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX=
3000.00
LMAY=
3000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
149.8
XPOWER=0.000E+00
YPOWER=0.100E+01
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________MULTILAYER**********_____________________________________________________________
Vacuum wavelength of the incident light (LAMBDA) in the sameunits as used below to specify layer thicknesses: .100000E+01_____________________________________________________________
Substrate:(n,k)= ( 1.000000, 0.000000)
Thickness= Infinite

Layer# 1:

EXX= ( 2.350000, 0.000000)
EYY= ( 2.250000, 0.000000)
EZZ= ( 2.450000, 0.000000)
Thickness=
25000.000000____________________________________________
Euler angles of rotated coordinates:(Theta= 46.220 Phi= 270.000 Psi=
0.000)____________________________________________

Medium of incidence: (n,k)= ( 1.000000, 0.000000)
Transmission mode
Index of emergent environment= 1.0000______________________________________________________________DIFFRACT now returns to its standard mode, where all lengths
are normalized by the wavelength (LAMBDA) of the light within
its present environment.
(Refractive index= 1.0000)______________________________________________________________
Multilayer coordinates: Theta= 0.00
Phi= 0.00
Psi= 0.00
Emergent coordinates:
Theta= 0.00
Phi= 0.00
Psi= 0.00
Propagation distance=
0.00_________________________________________________________
NMAX= 512
NMAY= 512
LMAX=
3000.00
LMAY=
3000.00
Beam centroid:(X=-0.04,Y=
793.70)DIAM=
1402.3
XPOWER=0.211E+00
YPOWER=0.650E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=-900.00
Xmax=
900.00Ymin=-400.00
Ymax=
1400.00___________________________________________
IXmin=
IYmin=

0.6344E-23
0.5565E-13

IXmax=
IYmax=

0.8953E-06
0.3539E-05

Intensity pattern for X-polrz stored in IX03.DAT
Intensity pattern for Y-polrz stored in IY03.DAT

Guide to DIFFRACT

349
FILE MANAGEMENT/GRAPHICS************************

Action: Load data-file IX02.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.162812E-10
Maximum Value: 0.350061E-05****************************
Action: Load-and-add data-file IY02.DAT into the TEMP array(Weight Factor= 0.100000E+01)****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.371956E-09
Maximum Value: 0.350103E-05****************************
Action: Load-and-add data-file IX03.DAT into the TEMP array(Weight Factor= 0.100000E+01)****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.371956E-09
Maximum Value: 0.350632E-05****************************
Action: Load-and-add data-file IY03.DAT into the TEMP array(Weight Factor= 0.100000E+01)****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.129510E-08
Maximum Value: 0.356877E-05****************************
Action: Save TIFF version of TEMP array in IT02.TIF
IT02.TIF is successfully created and written to disk____________________________________________________End of session: DIFFRACT terminated normally********************************************

350

Guide to DIFFRACT

Test Run 8
Geometric-optical System
Featuring: Ray_tracing for laser diode beam, Flat mirror, Conical mirror, TIR prism.

Figure A8.1. Schematic diagram of the simulated system.
450

y (µm)−450−450

x (µm)

450

Figure A8.2. Incident beam’s intensity distribution (x-component of polarization)

Guide to DIFFRACT

7.0

351

a

b

y (mm)-7.0-7.0

x (mm)

7.0-7.0

x (mm)

7.0

Figure A8.3. Distributions of intensity for (a) x-component and (b) y-componentof polarization at the destination plane.

7.0

a

b

y (mm)-7.0-7.0

x (mm)

7.0-7.0

x (mm)

7.0

Figure A8.4. Distributions of phase for (a) x-component and (b) y-componentof polarization at the destination plane.

352

Guide to DIFFRACTCOMMAND.DAT file corresponding to Test Run 8@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 03/03/2006
Time: 14:11 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: um) #Type: UB/SG/GG/LG/HG/LD: LD#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
PHI,PSI,DELTA:
30.000000
20.000000
2.340000#
Orientation angle THETA:
0.0000000#
Distance from cavity Z0:
780.00000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
4500.0000
4500.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Ray conversion: RAYS#
Multiply curvature (Y/N): N#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-450.00000
450.00000#
Ymin,Ymax:-450.00000
450.00000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Define a new system: NSYS#
Name of the system: TR08#

Guide to DIFFRACT**** Surface #01:
Type: Aperture#
Vertex (Vx,Vy,Vz):
0.0000000
0.0000000
0.5000000#
Normal (Theta,Phi):
45.000000
270.00000#
Aperture shape: Ellipse#
Major-axis Phi,LA,WA:
0.0000000
2.5000000
2.5000000#
Next surface (Hit,Miss):20#**** Surface #02:
Type: Mirror#
Shape: C#
Vertex (Vx,Vy,Vz):
0.0000000
0.0000000
10.000000#
Normal (Theta,Phi):
0.0000000
0.0000000#
RC,K:-21.560000-1.0000000#
Aperture's Rmin,Rmax:
0.0000000
8.0000000#
Coating (PR/DOE/BARE): BARE#
Complex index (n,k):
2.0000000
7.0000000#
Next surface (Hit,Miss):
30#**** Surface #03:
Type: Mirror#
Shape: F#
Vertex (Vx,Vy,Vz):
0.0000000
0.0000000
0.5000000#
Normal (Theta,Phi):
135.00000
90.000000#
Aperture's Rmin,Rmax:
2.5000000
10.000000#
Coating (PR/DOE/BARE): BARE#
Complex index (n,k):
2.0000000
7.0000000#
Next surface (Hit,Miss):
40#**** Surface #04:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.0000000-20.000000
0.5000000#
Normal (Theta,Phi):
90.000000
270.00000#
Aperture's Rmin,Rmax:
0.0000000
8.0000000#
Coating (AR/DOE/BARE): BARE#
Medium beyond surface: HOMO#
Refractive index (n):1.6000000#
Next surface (Hit,Miss):
50#**** Surface #05:
Type: Mirror#
Shape: F#
Vertex (Vx,Vy,Vz):
0.0000000-30.000000
0.5000000#
Normal (Theta,Phi):
135.00000
270.00000#
Aperture's Rmin,Rmax:
0.0000000
10.000000#
Coating (PR/DOE/BARE): BARE#
Complex index (n,k):
1.0000000
0.0000000#
Next surface (Hit,Miss):
60#**** Surface #06:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.0000000-30.000000
10.500000#
Normal (Theta,Phi):
0.0000000
0.0000000#
Aperture's Rmin,Rmax:
0.0000000
8.0000000#
Coating (AR/DOE/BARE): BARE#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
70#**** Surface #07:
Type: Destination#
Vertex (Vx,Vy,Vz):
0.0000000-30.000000
20.000000#
Normal (Theta,Phi):
0.0000000
0.0000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Trace all incident rays: TRAY#
Initial guess (Xc,Yc,Zc):
0.0000000
0.0000000
0.0000000#
Accept or Search: Search#
Strip phase-factor (Y/N): N#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
30.000000
30.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.780000
NVIRON: 1.000000 #

353

354

Guide to DIFFRACT_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-7.0000000
7.0000000#
Ymin,Ymax:-7.0000000
7.0000000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 2#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Phase#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-7.0000000
7.0000000#
Ymin,Ymax:-7.0000000
7.0000000#
Color or Gray-scale: C#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 2#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: IX01.DAT#
Action: T#
Saving TEMP in TIFF-file: IX01.TIF#
Action: L#
Data-file loaded to TEMP: IX02.DAT#
Action: T#
Saving TEMP in TIFF-file: IX02.TIF#
Action: L#
Data-file loaded to TEMP: IY02.DAT#
Action: T#
Saving TEMP in TIFF-file: IY02.TIF#
Action: L#
Data-file loaded to TEMP: PX02.DAT#
Action: T#
Saving TEMP in TIFF-file: PX02.TIF#
Action: L#
Data-file loaded to TEMP: PY02.DAT#
Action: T#
Saving TEMP in TIFF-file: PY02.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (um): 0.780000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Guide to DIFFRACT

355

SESSION.DAT file corresponding to Test Run 8$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 8.2
File SESSION.DAT $$$$ Date: 03/03/2004
Time: 14:17 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$_______________________________________________________DIMENSIONLESS LENGTHS ARE IN UNITS OF VACUUM WAVELENGTH
ALL ANGLES ARE IN DEGREES_______________________________________________________
Array dimensions: NDIM= 512, NDSP= 149**************************************
Designated Working Subdirectory: <Test08>*****************************************_________________________________________________________________
Vacuum lambda (um)= 1.0000
NVIRON= 1.0000 Length_Units: um
LMAX=
10000.0
LMAY=
10000.0
NMAX= 512
NMAY= 512_________________________________________________________________
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Length Units: um________________
Laser Diode: BCX=
0.00000
BCY=
0.00000
THETA= 0.00
PHI= 30.00
PSI= 20.00
DELTA=
2.34000
Propagation distance Z0=
780.000_________________________________________________________________
Vacuum lambda (um)= 0.78000
NVIRON= 1.0000 Length_Units: um
LMAX=
4500.00
LMAY=
4500.00
NMAX= 512
NMAY= 512
Beam centroid:(X=-0.40682E-07, Y=-0.20524E-05) DIAM=0.600643E+03
XPOWER= 0.99999
YPOWER= 0.0000
Curvature= 0.12821E-02_________________________________________________________________
CONVERSION TO RAYS******************
Ray components RX,RY,RZ are computed and stored in BX and BY
PLOTTING ROUTINE****************
Length Units: um________________
Intensity distribution___________________________________________
Xmin= -0.450000E+03
Xmax= 0.450000E+03
Ymin= -0.450000E+03
Ymax= 0.450000E+03___________________________________________
IXmin=
IZmin=

0.1146E-09
0.0000E+00

IXmax=
IZmax=

0.4670E-05
0.7694E-07

Latest screen (intensity pattern) dumped in IS01.BMP
Intensity pattern for X-polrz stored in IX01.DAT
Intensity pattern for Z-polrz stored in IZ01.DATPeak intensity=0.467E-05 XPEAK=
0.00000
YPEAK=
0.00000(Total energy flux=0.988E+00
ZPOWER=0.224E-01)
GEOMETRIC-OPTICAL SYSTEM DEFINITION***********************************
Length Units: mm________________

356

Guide to DIFFRACTName of the system: TR08
Surface #01***********
Type: Aperture
Shape: Flat
Vertex coordinates: (0.000000E+00,0.000000E+00,0.500000E+00)
Direction of surface normal (Theta,Phi): ( 45.000,270.000)
Aperture shape: Ellipse
Major Axis Orientation (Theta,Phi): ( 90.000, 0.000)
Half-length, Half-width:2.5000, 2.5000
Next surface (Hit,Miss):
2,0
Surface #02***********
Type: Mirror
Shape: Conical
Vertex coordinates: (0.000000E+00,0.000000E+00,0.100000E+02)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Radius of curvature, including sign (RC): -21.560
Conic constant (K): -1.0000
Aperture radii Rmin,Rmax:
0.0000, 8.0000
Complex index (n,k): ( 2.000000, 7.000000)
Next surface (Hit,Miss):
3,0
Surface #03***********
Type: Mirror
Shape: Flat
Vertex coordinates: (0.000000E+00,0.000000E+00,0.500000E+00)
Direction of surface normal (Theta,Phi): (135.000, 90.000)
Aperture radii Rmin,Rmax:
2.5000, 10.000
Complex index (n,k): ( 2.000000, 7.000000)
Next surface (Hit,Miss):
4,0
Surface #04***********
Type: Glass
Shape: Flat
Vertex coordinates: (0.000000E+00,-.200000E+02,0.500000E+00)
Direction of surface normal (Theta,Phi): ( 90.000,270.000)
Aperture radii Rmin,Rmax:
0.0000, 8.0000
Antireflection: No
Homogeneous medium beyond the surface: n= 1.600000
Next surface (Hit,Miss):
5,0
Surface #05***********
Type: Mirror
Shape: Flat
Vertex coordinates: (0.000000E+00,-.300000E+02,0.500000E+00)
Direction of surface normal (Theta,Phi): (135.000,270.000)
Aperture radii Rmin,Rmax:
0.0000, 10.000
Complex index (n,k): ( 1.000000, 0.000000)
Next surface (Hit,Miss):
6,0
Surface #06***********
Type: Glass
Shape: Flat
Vertex coordinates: (0.000000E+00,-.300000E+02,0.105000E+02)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Aperture radii Rmin,Rmax:
0.0000, 8.0000
Antireflection: No
Free space medium beyond the surface: n = 1.00
Next surface (Hit,Miss):
7,0Guide to DIFFRACT

357

Surface #07***********
Type: Destination
Shape: Flat
Vertex coordinates: (0.000000E+00,-.300000E+02,0.200000E+02)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Surface Arrangement Analysis for Geometric-Optical System TR08**************************************************************
System TR08 is logically consistent***********************************GEOMETRIC-OPTICAL RAY TRACING*****************************
Length Units: mm________________
Incident Beam:**************
Magnitude of the brightest ray in the beam:0.467E-05
Magnitude of the weakest ray to qualify for tracing:0.468E-09
Total number of qualified rays within the incident beam: 15857(Upper bound on the number of rays presently allowed to cross: 20000)
Emergent Beam:**************
Number of qualified rays that traverse the system:10591
Region within destination plane to which rays are confined:
Circle: Xc,Yc,Radius =0.20938E-06
0.337817.4770
Rectangle: Xmin,Xmax =-6.4030
6.4030Ymin,Ymax =-5.7683
6.2501______________________________________________________________The emergent beam is diverging away from the point (Xc,Yc,Zc)=(-0.204913E-02, 0.184503E+00,-0.608159E-02), with an rms wavefront error of 0.16506E+04. If the beam's cross-sectional area
is sufficiently small, it may be considered to be a collimatedbeam, traveling along the direction (Theta=88.11, Phi=270.64).
______________________________________________________________
Emergent wavefront is not stripped**************************************************************
Construction of the emergent wave-front within the destination
plane is now complete.
(Total energy flux = 0.3986)**************************************************************_________________________________________________________________
Vacuum lambda (um)= 0.78000
NVIRON= 1.0000 Length_Units: mm
LMAX=
30.0000
LMAY=
30.0000
NMAX= 512
NMAY= 512
Beam centroid:(X=-0.42094E-02, Y= 0.26294E-01) DIAM=0.959999E+01
XPOWER= 0.40154
YPOWER= 0.13012E-03
Curvature= 0.00000E+00_________________________________________________________________
PLOTTING ROUTINE****************
Length Units: mm________________
Intensity distribution___________________________________________
Xmin= -0.700000E+01
Xmax= 0.700000E+01
Ymin= -0.700000E+01
Ymax= 0.700000E+01___________________________________________
IXmin=
IYmin=

0.0000E+00
0.0000E+00

IXmax=
IYmax=

0.1583E-07
0.2728E-11

358

Guide to DIFFRACT
IZmin=

0.0000E+00

IZmax=

0.4773E-17

Latest screen (intensity pattern) dumped in IS02.BMP
Intensity pattern for X-polrz stored in IX02.DAT
Intensity pattern for Y-polrz stored in IY02.DAT
Intensity pattern for Z-polrz stored in IZ02.DAT
Peak intensity=0.167E-07 XPEAK= 0.468750
YPEAK=
1.93359(Total energy flux=0.399E+00
ZPOWER=0.173E-09)
PLOTTING ROUTINE****************
Length Units: mm________________
Phase distribution___________________________________________
Xmin= -0.700000E+01
Xmax= 0.700000E+01
Ymin= -0.700000E+01
Ymax= 0.700000E+01___________________________________________
PXmin= -139.926819
PYmin= -179.880493
PZmin= -179.925278

PXmax=
PYmax=
PZmax=

81.536011
179.998825
179.934845(degrees)(degrees)(degrees)

Latest screen (phase pattern) dumped in PS02.BMP
Phase pattern for X-polrz stored in PX02.DAT
Phase pattern for Y-polrz stored in PY02.DAT
Phase pattern for Z-polrz stored in PZ02.DAT
Peak intensity=0.167E-07 XPEAK= 0.468750
YPEAK=
1.93359(Total energy flux=0.399E+00
ZPOWER=0.173E-09)
FILE MANAGEMENT/GRAPHICS************************
Action: Load data-file IX01.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.114622E-09
Maximum Value: 0.467040E-05****************************
Action: Save TIFF version of TEMP array in IX01.TIF
IX01.TIF is successfully created and written to disk____________________________________________________
Action: Load data-file IX02.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.000000E+00
Maximum Value: 0.158348E-07****************************End of session: DIFFRACT terminated normally********************************************)(Truncated)Guide to DIFFRACT

359

Test Run 9
Interferograms showing various Seidel aberrations
Featuring: Seidel aberrations and interferometric beam combination

Figure A9.1. Diagram of a Michelson interferometer, showing a uniform, linearly polarized
beam of light divided by the beam-splitter between the two arms of the device. One beam isreflected from a flat mirror and used as reference. The other beam is reflected from a distorted
mirror and acquires wavefront aberrations. The two beams are combined again at the beamsplitter, and their interference pattern is monitored at the observation plane.

360

Guide to DIFFRACT

a

b

c

d

e

f

−3200

x/λ

3200

−3200

x/λ

3200

Figure A9.2. Intensity distributions at the observation plane when the aberrated mirror has (a)spherical, (b) curvature, (c) coma, (d) astigmatism plus curvature, (e) distortion, (f) a mixtureof spherical, coma, curvature, and astigmatic aberrations.

Guide to DIFFRACT

361

COMMAND.DAT file corresponding to Test Run 9@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.2
File COMMAND.DAT @@#$@@ Date: 03/03/2004
Time: 10:10 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: UB#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
Radius of aperture:
3000.0000#
Aberrations: None#
Polarization RHO,ETA:
0.000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
10000.000
10000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Store distribution: SAVE#
File name: AAAA.DST#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: UB#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
Radius of aperture:
3000.0000#
Aberrations (Y/N): Seidel#
Spherical C40:
2.0000000#
Coma C31,PHI31:.00000000.00000000#
Astigmatism C22,PHI22:.00000000.00000000#
Curvature C20:.00000000#
Distortion C11,PHI11:.00000000.00000000#
Polarization RHO,ETA:.00000000.00000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
10000.000
10000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##362

Guide to DIFFRACT

Two-beam interference: INTF#
Input-file name: AAAA.DST#
Scale-factor (A,PHI):
1.0000000.00000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-3200.0000
3200.0000#
Ymin,Ymax:-3200.0000
3200.0000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: UB#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
Radius of aperture:
3000.0000#
Aberrations (Y/N): Seidel#
Spherical C40:
.00000000#
Coma C31,PHI31:.00000000.00000000#
Astigmatism C22,PHI22:.00000000.00000000#
Curvature C20:
2.0000000#
Distortion C11,PHI11:.00000000.00000000#
Polarization RHO,ETA:.00000000.00000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
10000.000
10000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Two-beam interference: INTF#
Input-file name: AAAA.DST#
Scale-factor (A,PHI):
1.0000000.00000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-3200.0000
3200.0000#
Ymin,Ymax:-3200.0000
3200.0000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 2#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: UB#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#

Guide to DIFFRACT
Radius of aperture:
3000.0000#
Aberrations (Y/N): Seidel#
Spherical C40:
0.0000000#
Coma C31,PHI31:
2.0000000
120.00000#
Astigmatism C22,PHI22:
0.0000000
0.0000000#
Curvature C20:
0.0000000#
Distortion C11,PHI11:
0.0000000
0.0000000#
Polarization RHO,ETA:
0.0000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
10000.000
10000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Two-beam interference: INTF#
Input-file name: AAAA.DST#
Scale-factor (A,PHI):
1.0000000.00000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-3200.0000
3200.0000#
Ymin,Ymax:-3200.0000
3200.0000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 3#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: UB#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
Radius of aperture:
3000.0000#
Aberrations (Y/N): Seidel#
Spherical C40:
0.0000000#
Coma C31,PHI31:
0.0000000
0.0000000#
Astigmatism C22,PHI22:
6.0000000-45.000000#
Curvature C20:
-3.0000000#
Distortion C11,PHI11:
0.0000000
0.0000000#
Polarization RHO,ETA:
0.0000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
10000.000
10000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Two-beam interference: INTF#
Input-file name: AAAA.DST#
Scale-factor (A,PHI):
1.0000000.00000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-3200.0000
3200.0000
#

363

364

Guide to DIFFRACT
Ymin,Ymax:-3200.0000
3200.0000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 4#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: UB#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
Radius of aperture:
3000.0000#
Aberrations (Y/N): Seidel#
Spherical C40:
0.0000000#
Coma C31,PHI31:
0.0000000
0.0000000#
Astigmatism C22,PHI22:
0.0000000
0.0000000#
Curvature C20:
0.0000000#
Distortion C11,PHI11:
2.0000000
45.000000#
Polarization RHO,ETA:
0.0000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
10000.000
10000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Two-beam interference: INTF#
Input-file name: AAAA.DST#
Scale-factor (A,PHI):
1.0000000.00000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-3200.0000
3200.0000#
Ymin,Ymax:-3200.0000
3200.0000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 5#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: UB#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
Radius of aperture:
3000.0000#
Aberrations (Y/N): Seidel#
Spherical C40:
2.0000000#
Coma C31,PHI31:
2.0000000
120.00000#
Astigmatism C22,PHI22:
6.0000000-45.000000#
Curvature C20:
-3.0000000#
Distortion C11,PHI11:
0.0000000
0.0000000#
Polarization RHO,ETA:
0.0000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
10000.000
10000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ #Guide to DIFFRACT#
Two-beam interference: INTF#
Input-file name: AAAA.DST#
Scale-factor (A,PHI):
1.0000000.00000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-3200.0000
3200.0000#
Ymin,Ymax:-3200.0000
3200.0000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 6#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: IX01.DAT#
Action: T#
Saving TEMP in TIFF-file: IX01.TIF#
Action: L#
Data-file loaded to TEMP: IX02.DAT#
Action: T#
Saving TEMP in TIFF-file: IX02.TIF#
Action: L#
Data-file loaded to TEMP: IX03.DAT#
Action: T#
Saving TEMP in TIFF-file: IX03.TIF#
Action: L#
Data-file loaded to TEMP: IX04.DAT#
Action: T#
Saving TEMP in TIFF-file: IX04.TIF#
Action: L#
Data-file loaded to TEMP: IX05.DAT#
Action: T#
Saving TEMP in TIFF-file: IX05.TIF#
Action: L#
Data-file loaded to TEMP: IX06.DAT#
Action: T#
Saving TEMP in TIFF-file: IX06.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):633.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

365

366

Guide to DIFFRACTSESSION.DAT file corresponding to Test Run 9$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 7.9
File SESSION.DAT $$$$ Date: 04/27/2002
Time: 12:13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ALL LENGTHS ARE IN UNITS OF WAVELENGTH; ALL ANGLES ARE IN DEGREES_________________________________________________________________
Array dimensions: NDIM= 512, NDSP= 149**************************************
Designated Working Subdirectory: <Test09>*****************************************
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Uniform Beam: BCX=
0.0
BCY=
0.0
RA=
3000.0Aberrations: None
Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 10000.00
LMAY= 10000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5999.5
XPOWER=0.100E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________STORE DISTRIBUTION******************
Distribution stored in AAAA.DST
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Uniform Beam: BCX=

0.0

BCY=

Aberrations: Seidel
Spherical: C40= 2.000000
Coma: C31= 0.000000
Phi31=
0.00
Astigmatism: C22= 0.000000
Phi22=
Curvature: C20= 0.000000
Distortion: C11= 0.000000
Phi11=

0.0

RA=

3000.00.00
0.00

Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 10000.00
LMAY= 10000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5999.5
XPOWER=0.100E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
INTERFEROMETRIC BEAM COMBINATION********************************
Input distribution read from AAAA.DST
Scale-factor Alpha=A exp(i PHI), where A=
1.0000, PHI= 0.00(deg)_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 10000.00
LMAY= 10000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5536.6
XPOWER=0.235E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________

Guide to DIFFRACT367
PLOTTING ROUTINE****************

Intensity distribution___________________________________________
Xmin=-3200.00
Xmax=
3200.00
Ymin=-3200.00
Ymax=
3200.00___________________________________________
IXmin= 0.0000E+00
IXmax= 0.1414E-06
Intensity pattern for X-polrz stored in IX01.DAT
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Uniform Beam: BCX=

0.0

BCY=

Aberrations: Seidel
Spherical: C40= 0.000000
Coma: C31= 0.000000
Phi31=
0.00
Astigmatism: C22= 0.000000
Phi22=
Curvature: C20= 2.000000
Distortion: C11= 0.000000
Phi11=

0.0

RA=

3000.00.00
0.00

Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 10000.00
LMAY= 10000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5999.5
XPOWER=0.100E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
INTERFEROMETRIC BEAM COMBINATION********************************
Input distribution read from AAAA.DST
Scale-factor Alpha=A exp(i PHI), where A=
1.0000, PHI= 0.00(deg)_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 10000.00
LMAY= 10000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
6001.7
XPOWER=0.200E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=-3200.00
Xmax=
3200.00
Ymin=-3200.00
Ymax=
3200.00___________________________________________
IXmin= 0.0000E+00
IXmax= 0.1414E-06
Intensity pattern for X-polrz stored in IX02.DAT
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Uniform Beam: BCX=

0.0

BCY=

Aberrations: Seidel
Spherical: C40= 0.000000
Coma: C31= 2.000000
Phi31= 120.00
Astigmatism: C22= 0.000000
Phi22=
Curvature: C20= 0.000000
Distortion: C11= 0.000000
Phi11=

0.0

0.00
0.00

RA=

3000.0368

Guide to DIFFRACT
Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 10000.00
LMAY= 10000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5999.5
XPOWER=0.100E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
INTERFEROMETRIC BEAM COMBINATION********************************
Input distribution read from AAAA.DST
Scale-factor Alpha=A exp(i PHI), where A=
1.0000, PHI= 0.00(deg)_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 10000.00
LMAY= 10000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5545.8
XPOWER=0.237E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=-3200.00
Xmax=
3200.00
Ymin=-3200.00
Ymax=
3200.00___________________________________________
IXmin= 0.0000E+00
IXmax= 0.1414E-06
Intensity pattern for X-polrz stored in IX03.DAT
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Uniform Beam: BCX=

0.0

BCY=

0.0

RA=

3000.0Aberrations: Seidel
Spherical: C40= 0.000000
Coma: C31= 0.000000
Phi31=
0.00
Astigmatism: C22= 6.000000
Phi22= -45.00
Curvature: C20= -3.000000
Distortion: C11= 0.000000
Phi11=
0.00
Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 10000.00
LMAY= 10000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5999.5
XPOWER=0.100E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
INTERFEROMETRIC BEAM COMBINATION********************************
Input distribution read from AAAA.DST
Scale-factor Alpha=A exp(i PHI), where A=
1.0000, PHI= 0.00(deg)_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 10000.00
LMAY= 10000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5827.6
XPOWER=0.209E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________

Guide to DIFFRACT369
PLOTTING ROUTINE****************

Intensity distribution___________________________________________
Xmin=-3200.00
Xmax=
3200.00
Ymin=-3200.00
Ymax=
3200.00___________________________________________
IXmin= 0.0000E+00
IXmax= 0.1414E-06
Intensity pattern for X-polrz stored in IX04.DAT
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Uniform Beam: BCX=

0.0

BCY=

0.0

RA=

3000.0Aberrations: Seidel
Spherical: C40= 0.000000
Coma: C31= 0.000000
Phi31=
0.00
Astigmatism: C22= 0.000000
Phi22=
0.00
Curvature: C20= 0.000000
Distortion: C11= 2.000000
Phi11= 45.00
Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 10000.00
LMAY= 10000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5999.5
XPOWER=0.100E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
INTERFEROMETRIC BEAM COMBINATION********************************
Input distribution read from AAAA.DST
Scale-factor Alpha=A exp(i PHI), where A=
1.0000, PHI= 0.00(deg)_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 10000.00
LMAY= 10000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5953.8
XPOWER=0.195E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=-3200.00
Xmax=
3200.00
Ymin=-3200.00
Ymax=
3200.00___________________________________________
IXmin= 0.0000E+00
IXmax= 0.1414E-06
Intensity pattern for X-polrz stored in IX05.DAT
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Uniform Beam: BCX=

0.0

BCY=

0.0

Aberrations: Seidel
Spherical: C40= 2.000000
Coma: C31= 2.000000
Phi31= 120.00
Astigmatism: C22= 6.000000
Phi22= -45.00
Curvature: C20= -3.000000
Distortion: C11= 0.000000
Phi11=
0.00

RA=

3000.0370

Guide to DIFFRACT
Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 10000.00
LMAY= 10000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5999.5
XPOWER=0.100E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
INTERFEROMETRIC BEAM COMBINATION********************************
Input distribution read from AAAA.DST
Scale-factor Alpha=A exp(i PHI), where A=
1.0000, PHI= 0.00(deg)_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 10000.00
LMAY= 10000.00
Beam centroid:(X=-10.23,Y=
1.79)
DIAM=
5801.2
XPOWER=0.205E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=-3200.00
Xmax=
3200.00
Ymin=-3200.00
Ymax=
3200.00___________________________________________
IXmin= 0.0000E+00
IXmax= 0.1414E-06
Intensity pattern for X-polrz stored in IX06.DAT
FILE MANAGEMENT/GRAPHICS************************
Action: Load data-file IX01.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.000000E+00
Maximum Value: 0.141430E-06****************************
Action: Save TIFF version of TEMP array in IX01.TIF
IX01.TIF is successfully created and written to disk____________________________________________________End of session: DIFFRACT terminated normally********************************************(Truncated)Guide to DIFFRACT

371

Test Run 10
Astigmatic focus-error and push-pull track-error detection scheme
Featuring: Simple Gaussian beam, Positive_focusing lens, Optical disk, Astigmatic lens,Quad detector.

Figure A10.1. Schematic diagram of the simulated system. A Gaussian beam is brought by anobjective lens to diffraction-limited focus on the grooved surface of an optical disk. The beamreflected from the disk surface is directed by a beam-splitter towards an astigmatic lens,
which concentrates the beam on a quad detector. The various combinations of the signals
from this detector produce focus- and track-error signals.

372

Guide to DIFFRACT

3500 a

350

b

y/λ

y/λ

−3500

−350
−3500

x/λ

3500

−350

x/λ

350

Figure A10.2. Distributions of intensity (a) at the entrance pupil of the astigmatic lens and (b)at the quad detector. The beam is in focus and on track.

3500 a

350

b

y/λ

y/λ

-3500

-350
-3500

x/λ

3500

-350

x/λ

350

FigureA10.3. Distribution of intensity (a) at the entrance pupil of the astigmatic lens and (b)at the quad detector. The beam is in focus and off track.

Guide to DIFFRACT

373

3500 a

350

b

y/λ

y/λ

−3500

−350
−3500

x/λ

3500

−350

x/λ

350

Figure A10.4. Distributions of intensity (a) at the entrance pupil of the astigmatic lens and (b)at the quad detector. The beam is +10λ out of focus and on track.

3500 a

450

b

y/λ

y/λ

−3500

−450
−3500

x/λ

3500

−450

x/λ

450

Figure A10.5. Distributions of intensity (a) at the entrance pupil of the astigmatic lens,(b) at the quad detector. The beam is -5λ out of focus and on track.

374

Guide to DIFFRACT

0.8

(a)
0.6

0.4

0.2

FES
0.0

-0.2

-0.4

-0.6
-10

-5

0

5

10

1.5

2.0

Defocus Distance (λ)
0.8

(b)

0.6
0.4
0.2
0.0

TES
-0.2
-0.4
-0.6
-0.8
0.0

0.5

1.0

Off-track Error (λ)
Figure A10.6. (a) Calculated focus-error signal versus disk defocus.
(b) Calculated track-error signal versus off-track distance.

Guide to DIFFRACT

375

COMMAND.DAT file corresponding to Test Run 10@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 03/03/2006
Time: 10:23 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$MULTIPLY#??,ITEM?....BY: ??
#$MAKE#??,ITEM?~#??,ITEM?:#$JUMP#??_IF#??,ITEM?.GT.: ??
#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units:) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
1/e radii Rmin,Rmax:
4397.0000
4397.0000#
RA,THETA:
3146.0000
0.0000000#
Aberrations: None#
Polarization RHO,ETA:
0.0000000
0.0000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
45000.000
45000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: PFOC#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.5500000
5720.6000#
Aberrations (Y/N): None#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance:
5720.6000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Optical disk surface: DISK
(Length_Units:) #Type of medium: WR#
SCX,SCY,RA:
0.0000000
0.0000000
30.000000#(Incidence medium) Ninc:
1.0000000#
Pregrooves: Y#
A,B,C:
0.0000000
0.8000000
0.8000000#
D,ZETA:
2.0000000
0.1250000#
Reflection coeffs. RL,RG:
1.0000000
1.0000000#
THETA,GCX,GCY:45.000000-0.4242600
0.4242600#
Wobble (Sine,Blip,None): None#
Marks (bumps/pits): N#
Inspect surface pattern: N#
Xmin,Xmax:-6.0000000
6.0000000#
Ymin,Ymax:-6.0000000
6.0000000#
Color or Gray-scale: G#
Save data file (Y/N): N#

376

Guide to DIFFRACT
File identifier: 0#
Propagation distance:
5720.6000#
Propagation regime: FRNHF#
Smax:
0.1000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: COLL#
LCX,LCY:
0.0000000
0.0000000#
NA,FL:
0.5500000
5720.6000#
Aberrations (Y/N): None#
Scalar/Quasi-vector: QV#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Propagate in environment: PROP
(Length_Units:) #Propagation distance:
400000.00#
Multiply curvature (Y/N): N#
Reposition beam (Y/N): N#
Propagation regime: DFRNL#
Smax:
0.1000000#
Scalar/Quasi-vector: SC#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-3500.0000
3500.0000#
Ymin,Ymax:-3500.0000
3500.0000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units:) #Type: ASTG#
LCX,LCY,THETA:
0.0000000
0.0000000
0.0000000#
NA,FL1,FL2:
0.1000000
62765.000
74706.000#
Propagation distance:
68387.000#
Scalar/Quasi-vector: QV#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units:) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-350.00000
350.00000#
Ymin,Ymax:-350.00000
350.00000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 2#$________________________________________________________________________ #Guide to DIFFRACT
Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Detector: DTCT
(Length_Units:) #Quad/Split/Phi/Diff: Q#
DCX,DCY,THETA:
0.0000000
0.0000000
45.000000#
LX,LY,GW:
1000.0000
1000.0000
0.0000000#
ID numbers for S1,S2:
1
2#
ID numbers for S3,S4:
34#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Process signals: PSIG
(Length_Units:) #Action: +#S(i)=S(j)+S(k)i,j,k:
5
1
3#
Action: +#S(i)=S(j)+S(k)i,j,k:
6
2
4#
Action: +#S(i)=S(j)+S(k)i,j,k:
10
1
4#
Action: +#S(i)=S(j)+S(k)i,j,k:
11
2
3#
Action: +#S(i)=S(j)+S(k)i,j,k:
7
5
6#
Action: #S(i)=S(j)-S(k)i,j,k:
8
5
6#
Action: #S(i)=S(j)-S(k)i,j,k:
12
10
11#
Action: /#S(i)=S(j)/S(k)i,j,k:
98
7#
Action: /#S(i)=S(j)/S(k)i,j,k:
13
12
7#
Action: P#
Print S(i)i:7#Action: P#
Print S(i)i:8#Action: P#
Print S(i)i:12#Action: P#
Print S(i)i:9#Action: P#
Print S(i)i:13#Action: E#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: IX01.DAT#
Action: T#
Saving TEMP in TIFF-file: IX01.TIF#
Action: L#
Data-file loaded to TEMP: IX02.DAT#
Action: T#
Saving TEMP in TIFF-file: IX02.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):780.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

377

378

Guide to DIFFRACTSESSION.DAT file corresponding to Test Run 10$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 7.9
File SESSION.DAT $$$$ Date: 04/27/2002
Time: 13:11 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ALL LENGTHS ARE IN UNITS OF WAVELENGTH; ALL ANGLES ARE IN DEGREES_________________________________________________________________
Array dimensions: NDIM= 512, NDSP= 149**************************************
Designated Working Subdirectory: <Test10>*****************************************
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Simple Gaussian: BCX=
0.0
BCY=
0.0
RA=
3146.0
THETA= 0.00
RMIN=
4397.0 RMAX=
4397.0Aberrations: None
Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 45000.00
LMAY= 45000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5735.4
XPOWER=0.100E+01
YPOWER=0.000E+00
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________LENS ACTION***********
TYPE: Positive_Focusing
LCX=
0.000
LCY=
0.000
NA=0.550000
FL=
5720.60
Aberrations: None
Quasi-vector treatment
Propagation distance=
5720.60__________________________________________________APRX method: NGUIDE=
99 LGUIDE=
31463.3___________________________________________________________________________________________________________
NMAX= 512
NMAY= 512
LMAX=
65.09
LMAY=
65.09
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
7.1
XPOWER=0.100E+01
YPOWER=0.916E-03
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________OPTICAL DISK SURFACE********************
Disk type: WRITABLE
Aperture: SCX=
0.00
SCY=
0.00
RA=
Air-incident configuration (Ninc= 1.000000)

30.00

PREGROOVES__________
A=
ZETA=

0.00000.1250B=
THETA=

0.8000
45.00

C=
GCX=

0.8000-0.424D=
GCY=

2.00000.424

Amplitude reflection coefficient on land center (RL): 1.0000
Amplitude reflection coefficient on groove center (RG): 1.0000
Propagation distance=
5720.60
Propagation regime: Fraunhofer_____________________________________________________NGUIDE=
1499 LGUIDE=
300.00 SMAX= 0.500000

Guide to DIFFRACT

379______________________________________________________________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 45000.00
LMAY= 45000.00
Beam centroid:(X=-0.01,Y=-0.05)
DIAM=
9798.4
XPOWER=0.910E+00
YPOWER=0.789E-03
Curvature= 0.17E-03(Environment's index of refraction= 1.0000)_________________________________________________________LENS ACTION***********
TYPE: Collimator
LCX=
0.000
LCY=
0.000
NA=0.550000
FL=
5720.60
Aberrations: None
Quasi-vector treatment_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 45000.00
LMAY= 45000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5774.0
XPOWER=0.782E+00
YPOWER=0.327E-03
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________PROPAGATION IN ENVIRONMENT**************************
Propagation distance=
400000.00(Forward)
Beam is NOT repositioned
Propagation regime: Direct Fresnel_____________________________________________________NGUIDE=91 LGUIDE=
28869.80 SMAX= 0.000159_____________________________________________________
Scalar treatment_________________________________________________________
NMAX= 512
NMAY= 512
LMAX= 45000.00
LMAY= 45000.00
Beam centroid:(X=0.00,Y=
0.00)
DIAM=
5801.2
XPOWER=0.782E+00
YPOWER=0.327E-03
Curvature= 0.00E+00(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=-3500.00
Xmax=
3500.00
Ymin=-3500.00
Ymax=
3500.00___________________________________________
IXmin=
IYmin=

0.3453E-11
0.9948E-17

IXmax=
IYmax=

0.5578E-07
0.6894E-10

Intensity pattern for X-polrz stored in IX01.DAT
Intensity pattern for Y-polrz stored in IY01.DATLENS ACTION***********
TYPE: Astigmatic
LCX=
0.000
LCY=
0.000
THETA=
0.00
NA=0.100000
FL1=62765.00
FL2=74706.00
Propagation distance=
68387.00
Quasi-vector treatment__________________________________________NGUIDX=
1165
LGUIDX=71387.7500
NGUIDY=1234
LGUIDY=65434.1914___________________________________________________________________________________________________
NMAX= 512
NMAY= 512LMAX=
849.99LMAY=
714.13
Beam centroid:(X=-0.03,Y=-0.01)DIAM=
507.4
XPOWER=0.782E+00
YPOWER=0.324E-03
Curvature= 0.00E+00

380

Guide to DIFFRACT(Environment's index of refraction= 1.0000)_________________________________________________________
PLOTTING ROUTINE****************
Intensity distribution___________________________________________
Xmin=-350.00
Xmax=
350.00Ymin=-350.00
Ymax=
350.00___________________________________________
IXmin=
IYmin=

0.8512E-11
0.2513E-13

IXmax=
IYmax=

0.7692E-05
0.7764E-08

Intensity pattern for X-polrz stored in IX02.DAT
Intensity pattern for Y-polrz stored in IY02.DAT
PHOTODETECTOR*************
Detector type: Quad
DCX=
LX=

0.00
1000.00

DCY=
LY=

0.00
1000.00

THETA= 45.00
GW=
0.000

Signals S1,S2,S3,S4 saved in SIGNAL array as S(1), S(2), S(3), S(4)S1=0.197E+00
SUM=0.782E+00

S2=0.194E+00
P12M34= 0.00001

S3=0.197E+00
P13M24= 0.00823

S4=0.194E+00
P14M23= -0.00013

SIGNAL PROCESSING*****************
Addition:
Addition:
Addition:
Addition:
Addition:
Subtraction:
Subtraction:
Division:
Division:
Printing:
Printing:
Printing:
Printing:
Printing:

S( 5) = S( 1) + S( 3)S( 6) = S( 2) + S( 4)
S(10) = S( 1) + S( 4)S(11) = S( 2) + S( 3)S( 7) = S( 5) + S( 6)S( 8) = S( 5) - S( 6)S(12) = S(10) - S(11)S( 9) = S( 8) / S( 7)
S(13) = S(12) / S( 7)
S( 7) =
0.782280E+00
S( 8) =
0.644198E-02
S(12) = -0.996292E-04
S( 9) =
0.823488E-02
S(13) = -0.127357E-03
FILE MANAGEMENT/GRAPHICS************************

Action: Load data-file IX01.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.345322E-11
Maximum Value: 0.557788E-07****************************
Action: Save TIFF version of TEMP array in IX01.TIF
IX01.TIF is successfully created and written to disk____________________________________________________End of session: DIFFRACT terminated normally********************************************(Truncated)Guide to DIFFRACT

381

Test Run 11
GRIN Lens, Anamorphic Prism Pair, and Geometric-optical Ray Tracing
Featuring: Simple Gaussian beam at the output of a single-mode optical fiber, plano-convex
GRIN lens (collimator), anamorphic prism pair (beam-expander), and geometric-optical raytracing.

Y
Fiber

Collimator

SF11 Expander

Z

Figure A11.1. The output of a single-mode optical fiber at λ = 1550nm is collimated by aplano-convex gradient-index (GRIN) lens, having a refractive index gradient along the lengthof the lens. The collimated beam is expanded along the Y-axis using a pair of SF11 glass
prisms (anamorphic magnification factor ~ 6.1). The output of the fiber is a circularly
symmetric Gaussian beam, linearly polarized along the x-axis.

Figure A11.2 Plots of the ray distribution at the entrance facet of the GRIN lens, located3.551mm from the exit facet of the fiber. Left to right: x, y, and z-components of the rays.

382

Guide to DIFFRACT

Figure A11.3. Plots of intensity (top) and phase (bottom) at a distance of 25mm past the exit
face of the second prism of the anamorphic prism pair. (Left) x-component of polarization;(right) y-component of polarization. Note that the x-component is about six orders-ofmagnitude brighter than the y-component. The phase profile of the x-component is quite
uniform (i.e., the collimated/expanded beam is free from aberrations). The beam’s ycomponent of polarization shows a typical four-fold symmetry.

Guide to DIFFRACT

383

COMMAND.DAT file corresponding to Test Run 11@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 01/31/2006
Time: 10:31 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):1550.0000
NVIRON: 1.00000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: um) #Type: UB/SG/GG/LG/HG/LD: SG#
BCX,BCY,Power(mW):
0.0000000
0.0000000
1.000000#
1/e radii RX0,RY0:5.0000000
5.0000000#
RA,THETA:
15.000000
0.0000000#
Aberrations: None#
Polarization RHO,ETA:
0.0000000
0.0000000#
NMAX,NMAY:
1024
1024
#
LMAX,LMAY:
450.00000
450.00000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):1550.0000
NVIRON: 1.00000 #_________________________________________________________________________ ##Propagate in environment: PROP
(Length_Units: mm) #Propagation distance: 3.55100#
Multiply curvature (Y/N): N#
Reposition beam (Y/N): N#
Propagation regime: FRNHF#
Smax:
0.10000000#
Scalar/Quasi-vector: SC#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):1550.0000
NVIRON: 1.00000 #_________________________________________________________________________ ##Ray conversion: RAYS#
Multiply curvature (Y/N): N#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):1550.0000
NVIRON: 1.00000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Ray distribution#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-500.00000
500.00000#
Ymin,Ymax:-500.00000
500.00000#

384

Guide to DIFFRACT
Color or Gray-scale: G#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):1550.0000
NVIRON: 1.00000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: um) #Define a new system: NSYS#
Name of the system: XPND##**** Surface #01:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.0000000
0.0000000
0.0000000#
Normal (Theta,Phi):
0.0000000
0.0000000#
Aperture's Rmin,Rmax:
0.0000000
850.00000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: GRIN#
Type of GRIN medium: 1#
Origin (Ox,Oy,Oz):
0.0000000
0.0000000-50.000000#
Axis (Theta,Phi):
0.0000000
0.0000000#
N0,R0,Z0:
1.6123200
850.00000
5800.0000#
R coefficients: Q1,Q2,Q3:
0.0000000
0.0000000
0.0000000#
Q4,Q5,Q6:
0.0000000
0.0000000
0.0000000#
Z coefficients: W1,W2,W3:
0.0195047
0.1516540-2.1585786#
W4,W5,W6:
20.635024-76.617864
151.00743#
W7,W8,W9:-176.56517
125.41745-52.243669#
W10,W11,W12:
11.434087-0.9852130
0.0000000#
Next surface (Hit,Miss):20##**** Surface #02:
Type: Glass#
Shape: S#
Vertex (Vx,Vy,Vz):
0.0000000
0.0000000
2900.0000#
Normal (Theta,Phi):
0.0000000
0.0000000#
Radius of curvature (RC):-3715.0000#
Aperture's Rmin,Rmax:
0.0000000
850.00000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
30##**** Surface #03:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.0000000
0.0000000
10900.000#
Normal (Theta,Phi):
70.733000
270.00000#
Aperture's Rmin,Rmax:
0.0000000
2500.0000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: HOMO#
Refractive index (n):1.7448280#
Next surface (Hit,Miss):
40##**** Surface #04:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.0000000-1846.1500
13264.690#
Normal (Theta,Phi):
41.300000
270.00000#
Aperture's Rmin,Rmax:
0.0000000
2500.0000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
50##**** Surface #05:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.0000000-4749.6650
17335.268#
Normal (Theta,Phi):
35.233000
90.000000#
Aperture's Rmin,Rmax:
0.0000000
7500.0000#
Coating (AR/DOE/BARE): AR#

Guide to DIFFRACT
Medium beyond surface: HOMO
Refractive index (n):1.7448280
Next surface (Hit,Miss):
6

#
#
0
#
#
**** Surface #06:
Type: Glass#
Shape: F#
Vertex (Vx,Vy,Vz):
0.0000000-4317.0000
27325.900#
Normal (Theta,Phi):
5.8000000
90.000000#
Aperture's Rmin,Rmax:
0.0000000
7500.0000#
Coating (AR/DOE/BARE): AR#
Medium beyond surface: FREE#
Next surface (Hit,Miss):
70##**** Surface #07:
Type: Destination#
Vertex (Vx,Vy,Vz):
0.0000000-4317.0000
40000.000#
Normal (Theta,Phi):
0.0000000
0.0000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):1550.0000
NVIRON: 1.00000 #_________________________________________________________________________ ##Geometric-optical system: GEOP
(Length_Units: mm) #Trace all incident rays: TRAY#
Initial guess (Xc,Yc,Zc):
0.0000000
0.0000000
0.0000000#
Accept or Search: Search#
Strip phase-factor (Y/N): N#
NMAX,NMAY:
1024
1024
#
LMAX,LMAY:
30.000000
30.000000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):1550.0000
NVIRON: 1.00000 #_________________________________________________________________________ ##Propagate in environment: PROP
(Length_Units: mm) #Propagation distance:
25.000000#
Multiply curvature (Y/N): N#
Reposition beam (Y/N): N#
Propagation regime: DFRNL#
Smax:
0.1000000#
Scalar/Quasi-vector: SC#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):1550.0000
NVIRON: 1.00000 #_________________________________________________________________________ ##Measure peak and FWHM: MEAS
(Length_Units: um) #Include Z-component: N#
ID numbers of PEAK I,X,Y:
1
2
3#
Compute/Return: C#
Theta:
0.0000000#
ID number for FWHM:
4#
Compute/Return: C#
Theta:
90.000000#
ID number for FWHM:
5#
Compute/Return: R#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):1550.0000
NVIRON: 1.00000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Intensity#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-5.0000000
5.0000000#
Ymin,Ymax:-5.0000000
5.0000000#
Color or Gray-scale: G#
Z-component (Y/N): N
#

385

386

Guide to DIFFRACT
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):1550.0000
NVIRON: 1.00000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: mm) #Type: Phase#
Logarithmic SCALE:
4.0000000#
Xmin,Xmax:-5.0000000
5.0000000#
Ymin,Ymax:-5.0000000
5.0000000#
Color or Gray-scale: G#
Z-component (Y/N): N#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):1550.0000
NVIRON: 1.00000 #_________________________________________________________________________ ##End of session: QUIT##@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@SESSION.DAT file corresponding to Test Run 11$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 8.2
File SESSION.DAT $$$$ Date: 01/31/2004
Time: 11:14 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$_______________________________________________________DIMENSIONLESS LENGTHS ARE IN UNITS OF VACUUM WAVELENGTH
ALL ANGLES ARE IN DEGREES_______________________________________________________
Array dimensions: NDIM=1024, NDSP= 149**************************************
Designated Working Subdirectory: <TR011>****************************************_________________________________________________________________
Vacuum lambda (um)= 1.0000
NVIRON= 1.0000 Length_Units: um
LMAX=
10000.0
LMAY=
10000.0
NMAX=1024
NMAY=1024_________________________________________________________________
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Length Units: um________________
Simple Gaussian: BCX=
0.00000
BCY=
0.00000
RA=
15.0000
THETA= 0.00
RMIN= 5.00000
RMAX= 5.00000
Aberrations: None
Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________________
Vacuum lambda (nm)= 1550.0
NVIRON= 1.0000 Length_Units: um
LMAX=
450.000
LMAY=
450.000
NMAX=1024
NMAY=1024
Beam centroid:(X=-0.29475E-08, Y=-0.24003E-07) DIAM=0.999999E+01
XPOWER= 1.0000
YPOWER= 0.0000
Curvature= 0.00000E+00_________________________________________________________________

Guide to DIFFRACT

387
PROPAGATION IN ENVIRONMENT**************************

Length Units: mm________________
Propagation distance=
3.551000(Forward)
Beam is NOT repositioned
Propagation regime: Fraunhofer_______________________________________________________NGUIDE=
99 LGUIDE= 0.500000E-01 SMAX= 0.155000_______________________________________________________
Scalar treatment_________________________________________________________________
Vacuum lambda (nm)= 1550.0
NVIRON= 1.0000 Length_Units: mm
LMAX=
12.5248
LMAY=
12.5248
NMAX=1024
NMAY=1024
Beam centroid:(X=-0.34498E-07, Y=-0.26518E-07) DIAM=0.704344E+00
XPOWER= 1.0000
YPOWER= 0.0000
Curvature= 0.28161E+00_________________________________________________________________
CONVERSION TO RAYS******************
Ray components RX,RY,RZ are computed and stored in BX and BY
PLOTTING ROUTINE****************
Length Units: um________________
Ray distribution___________________________________________
Xmin= -0.500000E+03
Xmax= 0.500000E+03
Ymin= -0.500000E+03
Ymax= 0.500000E+03___________________________________________
RXmin= -0.3715E-06
RYmin= -0.3706E-06
RZmin= 0.4584E-08

RXmax=
RYmax=
RZmax=

0.3715E-06
0.3706E-06
0.1245E-04

Latest screen (ray distribution) dumped in RS00.BMP
X-component of ray distribution stored in RX00.DAT
Y-component of ray distribution stored in RY00.DAT
Z-component of ray distribution stored in RZ00.DAT
Peak intensity=0.125E-04 XPEAK=
0.00000
YPEAK=
0.00000(Total energy flux=0.100E+01
ZPOWER=0.246E-02)
GEOMETRIC-OPTICAL SYSTEM DEFINITION***********************************
Length Units: um________________Name of the system: XPND
Surface #01***********
Type: Glass
Shape: Flat
Vertex coordinates: (0.000000E+00,0.000000E+00,0.000000E+00)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Aperture radii Rmin,Rmax:
0.0000, 850.00
Antireflection: Yes
Gradient-index (GRIN) medium beyond the surface (Type 1):
Point of origin (Ox,Oy,Oz): ( 0.0000E+00, 0.0000E+00,-0.5000E+02)

388

Guide to DIFFRACT
Axis of symmetry (Theta,Phi): ( 0.000, 0.000)
Base index of refraction (N0)= 1.612320
Radius of cylinder (R0)= 0.85000E+03
Axial length-scale (Z0)= 0.58000E+04
Radial coefficients (Q1,Q2,Q3): 0.0000E+00, 0.0000E+00, 0.0000E+00(Q4,Q5,Q6): 0.0000E+00, 0.0000E+00, 0.0000E+00
Axial coefficients (W1,W2,W3): 0.19505E-01, 0.15165E+00,-0.21586E+01(W4,W5,W6): 0.20635E+02,-0.76618E+02, 0.15101E+03(W7,W8,W9):-0.17657E+03, 0.12542E+03,-0.52244E+02(W10,W11,W12): 0.11434E+02,-0.98521E+00, 0.00000E+00
Next surface (Hit,Miss):
2,0
Surface #02***********
Type: Glass
Shape: Spherical
Vertex coordinates: (0.000000E+00,0.000000E+00,0.290000E+04)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Radius of curvature, including sign (RC): -3715.0
Aperture radii Rmin,Rmax:
0.0000, 850.00
Antireflection: Yes
Free space medium beyond the surface: n = 1.00
Next surface (Hit,Miss):
3,0
Surface #03***********
Type: Glass
Shape: Flat
Vertex coordinates: (0.000000E+00,0.000000E+00,0.109000E+05)
Direction of surface normal (Theta,Phi): ( 70.733,270.000)
Aperture radii Rmin,Rmax:
0.0000, 2500.0
Antireflection: Yes
Homogeneous medium beyond the surface: n= 1.744828
Next surface (Hit,Miss):
4,0
Surface #04***********
Type: Glass
Shape: Flat
Vertex coordinates: (0.000000E+00,-.184615E+04,0.132647E+05)
Direction of surface normal (Theta,Phi): ( 41.300,270.000)
Aperture radii Rmin,Rmax:
0.0000, 2500.0
Antireflection: Yes
Free space medium beyond the surface: n = 1.00
Next surface (Hit,Miss):
5,0
Surface #05***********
Type: Glass
Shape: Flat
Vertex coordinates: (0.000000E+00,-.474967E+04,0.173353E+05)
Direction of surface normal (Theta,Phi): ( 35.233, 90.000)
Aperture radii Rmin,Rmax:
0.0000, 7500.0
Antireflection: Yes
Homogeneous medium beyond the surface: n= 1.744828
Next surface (Hit,Miss):
6,0
Surface #06***********
Type: Glass
Shape: Flat
Vertex coordinates: (0.000000E+00,-.431700E+04,0.273259E+05)
Direction of surface normal (Theta,Phi): ( 5.800, 90.000)
Aperture radii Rmin,Rmax:
0.0000, 7500.0
Antireflection: Yes
Free space medium beyond the surface: n = 1.00
Next surface (Hit,Miss):
7,0Guide to DIFFRACT

389

Surface #07***********
Type: Destination
Shape: Flat
Vertex coordinates: (0.000000E+00,-.431700E+04,0.400000E+05)
Direction of surface normal (Theta,Phi): ( 0.000, 0.000)
Surface Arrangement Analysis for Geometric-Optical System XPND**************************************************************
System XPND is logically consistent***********************************GEOMETRIC-OPTICAL RAY TRACING*****************************
Length Units: mm________________
Incident Beam:**************
Magnitude of the brightest ray in the beam:0.125E-04
Magnitude of the weakest ray to qualify for tracing:0.125E-08
Total number of qualified rays within the incident beam: 12337(Upper bound on the number of rays presently allowed to cross: 31250)
Emergent Beam:**************
Number of qualified rays that traverse the system: 6557
Region within destination plane to which rays are confined:
Circle: Xc,Yc,Radius =0.15650E-05 -0.27963
5.0189Rectangle: Xmin,Xmax =-0.80557
0.80558
Ymin,Ymax =-5.2985
4.2837______________________________________________________________The emergent beam is converging towards the point (Xc,Yc,Zc)=(-0.843934E-02, 0.133974E+00, 0.107031E+06), with an rms wavefront error of 0.34466E-01. If the beam's cross-sectional area
is sufficiently small, it may be considered to be a collimatedbeam, traveling along the direction (Theta= 0.00, Phi= 93.60).
______________________________________________________________
Emergent wavefront is not stripped**************************************************************
Construction of the emergent wave-front within the destination
plane is now complete.
(Total energy flux = 1.043)**************************************************************_________________________________________________________________
Vacuum lambda (nm)= 1550.0
NVIRON= 1.0000 Length_Units: mm
LMAX=
30.0000
LMAY=
30.0000
NMAX=1024
NMAY=1024
Beam centroid:(X=-0.24801E-03, Y=-0.18385E-01) DIAM=0.469090E+01
XPOWER= 1.0423
YPOWER= 0.13914E-05
Curvature= 0.00000E+00_________________________________________________________________PROPAGATION IN ENVIRONMENT**************************
Length Units: mm________________
Propagation distance=
25.00000(Forward)
Beam is NOT repositioned
Propagation regime: Direct Fresnel_______________________________________________________NGUIDE=46909 LGUIDE= 0.234545E+02 SMAX= 0.155000_______________________________________________________
Scalar treatment

390

Guide to DIFFRACT_________________________________________________________________
Vacuum lambda (nm)= 1550.0
NVIRON= 1.0000 Length_Units: mm
LMAX=
30.0000
LMAY=
30.0000
NMAX=1024
NMAY=1024
Beam centroid:(X=-0.24711E-03, Y=-0.18363E-01) DIAM=0.468929E+01
XPOWER= 1.0422
YPOWER= 0.13913E-05
Curvature= 0.00000E+00_________________________________________________________________
PEAK INTENSITY AND FWHM***********************
Length Units: um________________
Z-component of polarization excluded from PEAK & FWHM measurement
Peak Intensity=0.934E-06
XPEAK=
0.00000
YPEAK=
0.00000
Peak Intensity, XPEAK,YPEAK saved in SIGNAL array as S(1), S(2), S(3)Theta= 0.00
FWHM=
639.607
FWHM saved in SIGNAL array as S( 4)Theta= 90.00
FWHM=
3901.08
FWHM saved in SIGNAL array as S( 5)PLOTTING ROUTINE****************
Length Units: mm________________
Intensity distribution___________________________________________
Xmin= -0.500000E+01
Xmax= 0.500000E+01
Ymin= -0.500000E+01
Ymax= 0.500000E+01___________________________________________
IXmin=
IYmin=

0.1029E-17
0.1266E-23

IXmax=
IYmax=

0.9338E-06
0.8074E-12

Latest screen (intensity pattern) dumped in IS01.BMP
Intensity pattern for X-polrz stored in IX01.DAT
Intensity pattern for Y-polrz stored in IY01.DAT
PLOTTING ROUTINE****************
Length Units: mm________________
Phase distribution___________________________________________
Xmin= -0.500000E+01
Xmax= 0.500000E+01
Ymin= -0.500000E+01
Ymax= 0.500000E+01___________________________________________
PXmin= -179.977844
PYmin= -179.960892

PXmax=
PYmax=

179.999374
179.997925(degrees)(degrees)

Latest screen (phase pattern) dumped in PS01.BMP
Phase pattern for X-polrz stored in PX01.DAT
Phase pattern for Y-polrz stored in PY01.DATEnd of session: DIFFRACT terminated normally********************************************

Guide to DIFFRACT

391

Test Run 12
Focusing through High-NA Lens using EXCT and APRX Methods
Featuring: Uniform Beam, Positive_Focusing Lens (APRX and EXCT), Measure FWHM.

X

Focal Plane
λ = 0.65µm
Z
E
100µm
NA = 0.9

Figure A12.1. A uniform plane wave (λo = 0.65µm), linearly-polarized along X, is focusedthrough a 0.9NA Positive_Focusing micro-lens. The lens has focal length f = 100µm, entrance
pupil diameter 2f NA = 180µm, exit pupil diameter 2f tan[asin(NA)] = 413µm.

(a)

−2.0

(b)

x (µm)+2.0 −2.0

x (µm)+2.0

Figure A12.2. Logarithmic plots of total intensity distribution (Ix + Iy + Iz) at the focal planeof the 0.9NA lens of Fig. 1. (a) APRX method: Peak_Intensity = 2.63, FWHM = 0.475µm
along X, 0.350µm along Y. (b) EXCT method: Peak_Intensity = 2.61, FWHM = 0.475µm
along X, 0.352µm along Y.

392

Guide to DIFFRACT

For most practical applications, the APRX method of computing the distribution at andaround the focus of a positive_focusing lens is quite satisfactory. At very high or very low
NA, or in the presence of substantial amounts of aberration, however, one may want to resort
to the EXCT method. Unfortunately, the mesh size and the sampling requirements of the
EXCT method grow rapidly with both the numerical aperture and the focal length of the lens.
Nevertheless, in the case of micro-lenses, since the focal length and the clear aperture
diameter are fairly small (in units of the wavelength λ), the mesh requirements are generally
reasonable.
Figures 2-4 show that, in the case of an aberration-free 0.9NA micro-lens, the Airy patterns
calculated by the APRX and EXCT treatments are essentially identical. Note that for APRX
the initial mesh parameters are quite reasonable (NMAX = NMAY = 512, LMAX = LMAY =1000µm), whereas the EXCT method requires very large values of these parameters (NMAX= NMAY = 2048, LMAX = LMAY = 3000µm).

Figure A12.3. Distributions of intensity (top) and phase (bottom) for the x, y, and zcomponents of polarization obtained by the APRX method.

Guide to DIFFRACT

393

Figure A12.4. Distributions of intensity (top) and phase (bottom) for the x, y, and zcomponents of polarization obtained by the EXCT method.

394

Guide to DIFFRACT

Command.dat for Test run 12, APRX Method@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.4
File COMMAND.DAT @@#$@@ Date: 05/12/2006
Time: 12:40 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: um) #Type: UB/SG/GG/LG/HG/LD: UB#
BCX,BCY,Power(mW):
0.000000
0.000000
1.000000#
Radius of aperture:
90.00000#
Aberrations: None#
Polarization RHO,ETA:
0.000000
0.000000#
NMAX,NMAY:
512
512
#
LMAX,LMAY:
1000.000
1000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: um) #Type: PFOC#
LCX,LCY:
0.000000
0.000000#
NA,FL:
0.900000
100.0000#
Aberrations: Seidel#
Spherical C40:
0.000000#
Coma C31,Phi31:
0.000000
0.000000#
Astigmatism C22,Phi22:0.000000
0.000000#
Curvature C20:
0.000000#
Distortion C11,Phi11:0.000000
0.000000#
Scalar/Quasi-vector: QV#
Calculation method: APRX#
Propagation distance:
100.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-2.000000
2.000000#
Ymin,Ymax:-2.000000
2.000000#
Color or Gray-scale: G#
Z-component (Y/N): Y#

Guide to DIFFRACT
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Log(intensity)#
Logarithmic SCALE:
3.000000#
Xmin,Xmax:-2.000000
2.000000#
Ymin,Ymax:-2.000000
2.000000#
Color or Gray-scale: G#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Phase#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-2.000000
2.000000#
Ymin,Ymax:-2.000000
2.000000#
Color or Gray-scale: G#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: IX00.DAT#
Action: A#
Data-file added to TEMP: IY00.DAT#
Weight Factor:
1.000000#
Action: A#
Data-file added to TEMP: IZ00.DAT#
Weight Factor:
1.000000#
Action: D#
Color or Gray-scale: G#
Action: B#
Dump screen to BMP-file: ITOT.BMP#
Action: T#
Saving TEMP in TIFF-file: ITOT.TIF#
Action: C#
Logarithmic SCALE:
3.000000#
Action: D#
Color or Gray-scale: G#
Action: B#
Dump screen to BMP-file: ILOG.BMP#
Action: T#
Saving TEMP in TIFF-file: ILOG.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Measure peak and FWHM: MEAS
(Length_Units: um) #Include Z-component: Y#
ID numbers of PEAK I,X,Y:
1
2
3
#

395

396

Guide to DIFFRACT
Compute/Return: C#
Theta:
0.000000#
ID number for FWHM:
4#
Compute/Return: C#
Theta:
45.00000#
ID number for FWHM:
5#
Compute/Return: C#
Theta:
90.00000#
ID number for FWHM:
6#
Compute/Return: R#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Session.dat for Test Run 12, APRX Method$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 8.4
File SESSION.DAT $$$$ Date: 05/12/2005
Time: 12:40 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$_______________________________________________________DIMENSIONLESS LENGTHS ARE IN UNITS OF VACUUM WAVELENGTH
ALL ANGLES ARE IN DEGREES_______________________________________________________
Array dimensions: NDIM= 512, NDSP= 149**************************************
Designated Working Subdirectory: <Test12>*****************************************_________________________________________________________________
Vacuum lambda (um)= 1.0000
NVIRON= 1.0000 Length_Units: um
LMAX=
10000.0
LMAY=
10000.0
NMAX= 512
NMAY= 512_________________________________________________________________
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Length Units: um________________
Uniform Beam: BCX=
0.00000
BCY=
0.00000
RA=
90.000Aberrations: None
Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________________
Vacuum lambda (nm)= 650.00
NVIRON= 1.0000 Length_Units: um
LMAX=
1000.00
LMAY=
1000.00
NMAX= 512
NMAY= 512
Beam centroid:(X=-0.42849E-07, Y= 0.16608E-06) DIAM=0.179932E+03
XPOWER= 0.99994
YPOWER= 0.0000
Curvature= 0.00000E+00_________________________________________________________________LENS ACTION

Guide to DIFFRACT

397***********

Length Units: um________________
TYPE: Positive_Focusing(LCX,LCY)=( 0.0000, 0.0000)
NA=0.900000
FL=
100.000
Aberrations: Seidel
Spherical: C40= 0.000000
Coma: C31= 0.000000
Phi31=
0.00
Astigmatism: C22= 0.000000
Phi22=
0.00
Curvature: C20= 0.000000
Distortion: C11= 0.000000
Phi11=
0.00
Quasi-vector treatment
Propagation distance=
100.000____________________________________________________APRX method: NGUIDE=100
LGUIDE= 0.90000E+03_____________________________________________________________________________________________________________________
Vacuum lambda (nm)= 650.00
NVIRON= 1.0000 Length_Units: um
LMAX=
33.2800
LMAY=
33.2800
NMAX= 512
NMAY= 512
Beam centroid:(X=-0.41015E-03, Y=-0.66020E-03) DIAM=0.466860E+01
XPOWER= 1.0547
YPOWER= 0.18476E-01
Curvature= 0.00000E+00_________________________________________________________________
PLOTTING ROUTINE****************
Length Units: um________________
Intensity distribution___________________________________________
Xmin= -0.200000E+01
Xmax= 0.200000E+01
Ymin= -0.200000E+01
Ymax= 0.200000E+01___________________________________________
IXmin=
IYmin=
IZmin=

0.1584E-04
0.0000E+00
0.5054E-17

IXmax=
IYmax=
IZmax=

0.2635E+01
0.1175E-01
0.3913E+00

Latest screen (intensity pattern) dumped in IS00.BMP
Intensity pattern for X-polrz stored in IX00.DAT
Intensity pattern for Y-polrz stored in IY00.DAT
Intensity pattern for Z-polrz stored in IZ00.DAT
Peak intensity=0.263E+01 XPEAK=
0.00000
YPEAK=
0.00000(Total energy flux=0.100E+01
ZPOWER=0.318E+00)
PLOTTING ROUTINE****************
Length Units: um________________
Base 10 logarithm of normalized intensity distribution(Range: from 0 at peak_intensity to -3.0000 at cutoff)___________________________________________
Xmin= -0.200000E+01
Xmax= 0.200000E+01
Ymin= -0.200000E+01
Ymax= 0.200000E+01___________________________________________
IXmin=
IYmin=
IZmin=

0.1584E-04
0.0000E+00
0.5054E-17

IXmax=
IYmax=
IZmax=

0.2635E+01
0.1175E-01
0.3913E+00

Latest screen (intensity pattern) dumped in IS01.BMP
Intensity pattern for X-polrz stored in IX01.DAT398

Guide to DIFFRACT
Intensity pattern for Y-polrz stored in IY01.DAT
Intensity pattern for Z-polrz stored in IZ01.DATPeak intensity=0.263E+01 XPEAK=
0.00000
YPEAK=
0.00000(Total energy flux=0.100E+01
ZPOWER=0.318E+00)
PLOTTING ROUTINE****************
Length Units: um________________
Phase distribution___________________________________________
Xmin= -0.200000E+01
Xmax= 0.200000E+01
Ymin= -0.200000E+01
Ymax= 0.200000E+01___________________________________________
PXmin= -180.000000
PYmin= -180.000000
PZmin= -174.949753

PXmax=
PYmax=
PZmax=

180.000000
180.000000
178.610291(degrees)(degrees)(degrees)

Latest screen (phase pattern) dumped in PS00.BMP
Phase pattern for X-polrz stored in PX00.DAT
Phase pattern for Y-polrz stored in PY00.DAT
Phase pattern for Z-polrz stored in PZ00.DAT
Peak intensity=0.263E+01 XPEAK=
0.00000
YPEAK=
0.00000(Total energy flux=0.100E+01
ZPOWER=0.318E+00)
FILE MANAGEMENT/GRAPHICS************************
Action: Load data-file IX00.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.158364E-04
Maximum Value: 0.263493E+01****************************
Action: Load-and-add data-file IY00.DAT into the TEMP array(Weight Factor= 0.100000E+01)****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.241656E-04
Maximum Value: 0.263493E+01****************************
Action: Load-and-add data-file IZ00.DAT into the TEMP array(Weight Factor= 0.100000E+01)****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.876708E-04
Maximum Value: 0.263493E+01****************************
Action: Save the contents of Graphics Window in ITOT.BMP
Action: Save TIFF version of TEMP array in ITOT.TIF

Guide to DIFFRACT

399

ITOT.TIF is successfully created and written to disk____________________________________________________
Action: Compute logarithm of normalized TEMP****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: -0.300000E+01
Maximum Value: 0.000000E+00****************************
Action: Save the contents of Graphics Window in ILOG.BMP
Action: Save TIFF version of TEMP array in ILOG.TIF
ILOG.TIF is successfully created and written to disk____________________________________________________
PEAK INTENSITY AND FWHM***********************
Length Units: um________________
Z-component of polarization included in PEAK & FWHM measurement
Peak Intensity=0.263E+01
XPEAK=
0.00000
YPEAK=
0.00000
Peak Intensity, XPEAK,YPEAK saved in SIGNAL array as S(1), S(2), S(3)Theta= 0.00
FWHM= 0.475419
FWHM saved in SIGNAL array as S( 4)Theta= 45.00
FWHM= 0.402885
FWHM saved in SIGNAL array as S( 5)Theta= 90.00
FWHM= 0.350460
FWHM saved in SIGNAL array as S( 6)End of session: DIFFRACT terminated normally********************************************

400

Guide to DIFFRACT

Command.dat for Test Run 12, EXCT Method@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$@@ DIFFRACT 8.3
File COMMAND.DAT @@#$@@ Date: 07/10/2004
Time: 12:40 @@#$@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@#$#$All dimensionless lengths are in units of vacuum wavelength#$
All angles are in degrees#$#$
Syntax of $ commands:#$#$INCREMENT#??,ITEM?...BY: ??
#$#$MULTIPLY#??,ITEM?....BY: ??
#$#$MAKE#??,ITEM?~#??,ITEM?:#$#$JUMP#??_IF#??,ITEM?.GT.: ??
#$#$TAGS#??,ITEM?_#??,ITEM?:#$#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Header @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Initial distribution: BEAM
(Length_Units: um) #Type: UB/SG/GG/LG/HG/LD: UB#
BCX,BCY,Power(mW):
0.000000
0.000000
1.000000#
Radius of aperture:
90.00000#
Aberrations: None#
Polarization RHO,ETA:
0.000000
0.000000#
NMAX,NMAY:
2048
2048
#
LMAX,LMAY:
3000.000
3000.000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Lens: LENS
(Length_Units: um) #Type: PFOC#
LCX,LCY:
0.000000
0.000000#
NA,FL:
0.900000
100.0000#
Aberrations: Seidel#
Spherical C40:
0.000000#
Coma C31,Phi31:
0.000000
0.000000#
Astigmatism C22,Phi22:0.000000
0.000000#
Curvature C20:
0.000000#
Distortion C11,Phi11:0.000000
0.000000#
Scalar/Quasi-vector: QV#
Calculation method: EXCT#
Propagation distance:
100.0000#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Intensity#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-2.000000
2.000000#
Ymin,Ymax:-2.000000
2.000000#
Color or Gray-scale: G#
Z-component (Y/N): Y#
Save data files (Y/N): Y#

Guide to DIFFRACT
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Log(intensity)#
Logarithmic SCALE:
3.000000#
Xmin,Xmax:-2.000000
2.000000#
Ymin,Ymax:-2.000000
2.000000#
Color or Gray-scale: G#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 1#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Plot distribution: PLOT
(Length_Units: um) #Type: Phase#
Logarithmic SCALE:
4.000000#
Xmin,Xmax:-2.000000
2.000000#
Ymin,Ymax:-2.000000
2.000000#
Color or Gray-scale: G#
Z-component (Y/N): Y#
Save data files (Y/N): Y#
File identifier: 0#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##File management/Graphics: FMAN#
Action: L#
Data-file loaded to TEMP: IX00.DAT#
Action: A#
Data-file added to TEMP: IY00.DAT#
Weight Factor:
1.000000#
Action: A#
Data-file added to TEMP: IZ00.DAT#
Weight Factor:
1.000000#
Action: D#
Color or Gray-scale: G#
Action: B#
Dump screen to BMP-file: ITOT.BMP#
Action: T#
Saving TEMP in TIFF-file: ITOT.TIF#
Action: C#
Logarithmic SCALE:
3.000000#
Action: D#
Color or Gray-scale: G#
Action: B#
Dump screen to BMP-file: ILOG.BMP#
Action: T#
Saving TEMP in TIFF-file: ILOG.TIF#
Action: Q#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##Measure peak and FWHM: MEAS
(Length_Units: um) #Include Z-component: Y#
ID numbers of PEAK I,X,Y:
1
2
3#
Compute/Return: C
#

401

402

Guide to DIFFRACT
Theta:
0.000000#
ID number for FWHM:
4#
Compute/Return: C#
Theta:
45.00000#
ID number for FWHM:
5#
Compute/Return: C#
Theta:
90.00000#
ID number for FWHM:
6#
Compute/Return: R#$________________________________________________________________________ #Remarks:#
Vacuum wavelength (nm):650.0000
NVIRON: 1.000000 #_________________________________________________________________________ ##End of session: QUIT#@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Session.dat for Test Run 12, EXCT Method$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ DIFFRACT 8.4
File SESSION.DAT $$$$ Date: 05/12/2005
Time: 16:05 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$_______________________________________________________DIMENSIONLESS LENGTHS ARE IN UNITS OF VACUUM WAVELENGTH
ALL ANGLES ARE IN DEGREES_______________________________________________________
Array dimensions: NDIM=2048, NDSP= 149**************************************
Designated Working Subdirectory: <Test12>*****************************************_________________________________________________________________
Vacuum lambda (um)= 1.0000
NVIRON= 1.0000 Length_Units: um
LMAX=
10000.0
LMAY=
10000.0
NMAX=2048
NMAY=2048_________________________________________________________________
INCIDENT AMPLITUDE DISTRIBUTION*******************************
Length Units: um________________
Uniform Beam: BCX=

0.00000

BCY=

0.00000

RA=

90.0000

Aberrations: None
Polarization state: Rho= 0.0000
Eta= 0.0000_________________________________________________________________
Vacuum lambda (nm)= 650.00
NVIRON= 1.0000 Length_Units: um
LMAX=
3000.00
LMAY=
3000.00
NMAX=2048
NMAY=2048
Beam centroid:(X= 0.52727E-07, Y= 0.70390E-07) DIAM=0.180099E+03
XPOWER= 1.0001
YPOWER= 0.0000
Curvature= 0.00000E+00_________________________________________________________________LENS ACTION***********

Guide to DIFFRACT

403

Length Units: um________________
TYPE: Positive_Focusing(LCX,LCY)=( 0.0000, 0.0000)
NA=0.900000
FL=
100.000
Aberrations: Seidel
Spherical: C40= 0.000000
Coma: C31= 0.000000
Phi31=
0.00
Astigmatism: C22= 0.000000
Phi22=
0.00
Curvature: C20= 0.000000
Distortion: C11= 0.000000
Phi11=
0.00
Quasi-vector treatment
Propagation distance=
100.000____________________________________________________
EXCT method: NGUIDE=1274
LGUIDE= 0.20647E+04_____________________________________________________________________________________________________________________
Vacuum lambda (nm)= 650.00
NVIRON= 1.0000 Length_Units: um
LMAX=
83.2498
LMAY=
83.2498
NMAX=2048
NMAY=2048
Beam centroid:(X=-0.28989E-04, Y=-0.57624E-04) DIAM=0.535369E+01
XPOWER= 1.0409
YPOWER= 0.17597E-01
Curvature= 0.00000E+00_________________________________________________________________
PLOTTING ROUTINE****************
Length Units: um________________
Intensity distribution___________________________________________
Xmin= -0.200000E+01
Xmax= 0.200000E+01
Ymin= -0.200000E+01
Ymax= 0.200000E+01___________________________________________
IXmin=
IYmin=
IZmin=

0.9653E-05
0.6413E-20
0.3087E-17

IXmax=
IYmax=
IZmax=

0.2610E+01
0.1164E-01
0.3893E+00

Latest screen (intensity pattern) dumped in IS00.BMP
Intensity pattern for X-polrz stored in IX00.DAT
Intensity pattern for Y-polrz stored in IY00.DAT
Intensity pattern for Z-polrz stored in IZ00.DAT
Peak intensity=0.261E+01 XPEAK=
0.00000
YPEAK=
0.00000(Total energy flux=0.991E+00
ZPOWER=0.310E+00)
PLOTTING ROUTINE****************
Length Units: um________________
Base 10 logarithm of normalized intensity distribution(Range: from 0 at peak_intensity to -3.0000 at cutoff)___________________________________________
Xmin= -0.200000E+01
Xmax= 0.200000E+01
Ymin= -0.200000E+01
Ymax= 0.200000E+01___________________________________________
IXmin=
IYmin=
IZmin=

0.9653E-05
0.6413E-20
0.3087E-17

IXmax=
IYmax=
IZmax=

0.2610E+01
0.1164E-01
0.3893E+00

Latest screen (intensity pattern) dumped in IS01.BMP
Intensity pattern for X-polrz stored in IX01.DAT
Intensity pattern for Y-polrz stored in IY01.DAT404

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Intensity pattern for Z-polrz stored in IZ01.DATPeak intensity=0.261E+01 XPEAK=
0.00000
YPEAK=
0.00000(Total energy flux=0.991E+00
ZPOWER=0.310E+00)
PLOTTING ROUTINE****************
Length Units: um________________
Phase distribution___________________________________________
Xmin= -0.200000E+01
Xmax= 0.200000E+01
Ymin= -0.200000E+01
Ymax= 0.200000E+01___________________________________________
PXmin= -179.999100
PYmin= -179.999985
PZmin= -177.279938

PXmax=
PYmax=
PZmax=

179.999283
179.997925
178.580887(degrees)(degrees)(degrees)

Latest screen (phase pattern) dumped in PS00.BMP
Phase pattern for X-polrz stored in PX00.DAT
Phase pattern for Y-polrz stored in PY00.DAT
Phase pattern for Z-polrz stored in PZ00.DAT
Peak intensity=0.261E+01 XPEAK=
0.00000
YPEAK=
0.00000(Total energy flux=0.991E+00
ZPOWER=0.310E+00)
FILE MANAGEMENT/GRAPHICS************************
Action: Load data-file IX00.DAT into an empty TEMP array****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.965303E-05
Maximum Value: 0.261013E+01****************************
Action: Load-and-add data-file IY00.DAT into the TEMP array(Weight Factor= 0.100000E+01)****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.144271E-04
Maximum Value: 0.261013E+01****************************
Action: Load-and-add data-file IZ00.DAT into the TEMP array(Weight Factor= 0.100000E+01)****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: 0.780812E-04
Maximum Value: 0.261013E+01****************************
Action: Save the contents of Graphics Window in ITOT.BMP
Action: Save TIFF version of TEMP array in ITOT.TIF
ITOT.TIF is successfully created and written to disk

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405____________________________________________________
Action: Compute logarithm of normalized TEMP****************************TEMP array: Size and Range****************************
Number of Rows: 149
Number of Columns: 149
Minimum Value: -0.300000E+01
Maximum Value: 0.000000E+00****************************
Action: Save the contents of Graphics Window in ILOG.BMP
Action: Save TIFF version of TEMP array in ILOG.TIF
ILOG.TIF is successfully created and written to disk____________________________________________________
PEAK INTENSITY AND FWHM***********************
Length Units: um________________
Z-component of polarization included in PEAK & FWHM measurement
Peak Intensity=0.261E+01
XPEAK=
0.00000
YPEAK=
0.00000
Peak Intensity, XPEAK,YPEAK saved in SIGNAL array as S(1), S(2), S(3)Theta= 0.00
FWHM= 0.475165
FWHM saved in SIGNAL array as S( 4)Theta= 45.00
FWHM= 0.403306
FWHM saved in SIGNAL array as S( 5)Theta= 90.00
FWHM= 0.352107
FWHM saved in SIGNAL array as S( 6)End of session: DIFFRACT terminated normally********************************************

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407

Appendix F
Test Runs available on DIFFRACT’s distribution CD-ROM
The following test runs cover available options in the main menu of
DIFFRACT. These test runs are now available in the <TESTRUNS > directory
located on DIFFRACT’s distribution CD-ROM. To minimize the required
disk space, the Install routine does NOT automatically transfer these files toyour home directory on the hard disk. However, you may do so yourself bydragging the entire < TESTRUNS > directory (or selected portions of it) tothe hard drive.
Since the test runs have been collected over a period of time with different
versions of the program, there exist minor differences between thesequence of commands in the COMMAND.DAT files supplied with the test
runs and the sequence of commands that the current version of DIFFRACT
will actually accept. Do not attempt to run DIFFRACT 8.4 with the exact
COMMAND.DAT files supplied; instead, use the various test runs as guides
to setting up and running the program.

TR1.

Diode Laser, Collimator Lens, Free-space Propagation, Polarizer, Clear Aperture,
Mask, Quad Detector, and Split Detector

TR2a.
TR2b.
TR2c.

Ray Tracing (Parabolic Mirror and Flat Mirror)
Ray Tracing (Simple Lens with Four Spherical Surfaces)
Ray Tracing (Spherical Lens, Aperture, Flat Mirror, and Glass Prism)

TR3.

Uniform Beam, High-NA Positive Lens, Leaky Polarizer, Quarter-wave Plate, andBackward Propagation.

TR4a.
TR4b.
TR4c.
TR4d.
TR4e.
TR4f.

Read-only Optical Disk
Writable Optical Disk
Magneto-optical Disk with Regular Domains
Magneto-optical Disk with Chevron Domains
Dye-polymer Optical Disk (Substrate Incidence)
Dye-polymer Optical Disk (Air Incidence)

TR5.

Interferometric Beam Combination and Positive_focusing Lens

TR6.

Reflection of a Beam from a Front-surface Flat Mirror

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TR7.

Imaging with a Coherent Beam of Light

TR8a.
TR8b.

Reflecting Focused Beam from Back-surface Mirror (birefringent substrate)
Transmitting Focused Beam through Vertically Birefringent Crystal

TR9a.
TR9b.

Reflecting Focused Beam from Magneto-optic film through Glass Substrate
Reflecting Focused Beam from MO film through Birefringent Substrate

TR10a. Astigmatic Lens
TR10b. Cylindrical Lens
TR10c. Split Lens
TR10d. Toric Ring Lens and Phi Detector
TR11.

Weak Biprism

TR12a. Near Field and Far Field of Gaussian Beam from a Diffraction Grating
TR12b. Near Field and Far Field of 2-D Grating with a Sinusoidal Pattern
TR12c. Diffraction Grating in the Path of a Converging Beam
TR12d. Diffraction Grating (reflection)
TR13a. Positive_focusing Lens (the exact method)
TR13b. Positive_focusing Lens (the approximate method)
TR13c. Uniform Beam, Positive_focusing Lens, Split Detector, and Signal Processing
TR13d. Collimating Lens and Propagation in Extended Fresnel Regime
TR14a. Focusing and Recollimation through Aberrated Lens Pair (scalar treatment)
TR14b. Focusing and Recollimation through Aberrated Lens Pair (quasi-vectorial)
TR15a. Gaussian Beam, Split Detector, Polarization Element, and Clear Aperture
TR15b. Clear Aperture, Positive_focusing Lens, and Free_space Propagation
TR15c. Clear Aperture, Positive_focusing Lens, and Free_space Propagation
TR16.

Laser Diode, Lens, Polarization Element, and Multilayer

TR17.

Focusing through a Tilted Glass Slab

TR18.

Geometric-optical Rays Corresponding to a Complex_Amplitude Distribution

TR19.

The Talbot Effect

TR20.

Focusing and Recollimation through Tilted Lens Pair

TR21.

Focused Beam Reflecting from a Multilayer Structure

TR22.

Double Refraction through a Slab of Birefringent Crystal

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409

Appendix G
OPN articles (and their resource files) available on DIFFRACT’s CD-ROM

The resource files for the following articles – originally published in Optics &Photonics News (OPN) – are now available in the < OPN > directory located on
DIFFRACT’s distribution CD-ROM. To minimize the required disk space, theInstall routine does NOT automatically transfer these files to your home
directory on the hard disk. However, you may do so yourself by dragging theentire < OPN > directory (or selected portions of it) to the hard drive. For easeof reference, a bound collection of the original articles is also included in your
DIFFRACT package.
Since the OPN articles have been written over a period of several years using
different versions of the program, there exist minor differences between thesequence of commands in the COMMAND.DAT files supplied in the < OPN >directory and the sequence of commands that the current version of DIFFRACT
will actually accept. Do not attempt to run DIFFRACT 8.4 with the exact
COMMAND.DAT files supplied; instead, use the various resource files as
guides to setting up and running the program.

1. Some Quirks of Total Internal Reflection, Optics & Photonics News, 36-39, February
1997.
2. Fabry-Perot Etalons in Polarized Light, Optics & Photonics News, 39-44, March 1997.
3. The Talbot Effect, Optics & Photonics News, 42-47, April 1997.
4. What in the World are Surface Plasmons? Optics & Photonics News, 50-55, May 1997.
5. Internal Conical Refraction, Optics & Photonics News, 43-45, June 1997.
6. The Ronchi Test, Optics & Photonics News, 42-46, July 1997.
7. External Conical Refraction, Optics & Photonics News, 50-52, August 1997.
8. The Method of Fox and Li, Optics & Photonics News, 38-41, September 1997.
9. Coherent and Incoherent Imaging, Optics & Photonics News, 40-44, October 1997.
10. Zernike’s Method of Phase Contrast, Optics & Photonics News, 40-43, November 1997.
11. Michelson’s Stellar Interferometer, Optics & Photonics News, 40-43, January 1998.
12. Abbe’s Sine Condition, Optics & Photonics News, 56-60, February 1998.
13. Holography and Holographic Interferometry, Optics & Photonics News, 41-45, March
1998.
14. Self-focusing in Nonlinear Optical Media, Optics & Photonics News, 43-47, April 1998.
15. Scanning Optical Microscopy: Parts I and II, Optics & Photonics News, 56-59 May 1998,
and 42-45, June 1998.

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16. Reciprocity in Classical Linear Optics, Optics & Photonics News, 53-58, July 1998.
17. The Ewald-Oseen Extinction Theorem, Optics & Photonics News, 50-55, August 1998.
18. Evanescent Coupling: Parts I and II, Optics & Photonics News, 59-62, September 1998,
and 35-38, October 1998.
19. Polarization Microscopy, Optics & Photonics News, November 1998.
20. First-order Temporal Coherence in Classical Optics, Optics & Photonics News 10, 44-49(January 1999).
21. Optical Vortices, Optics & Photonics News 10, (February 1999).
22. Geometric-optical Rays, Poynting’s Vector, and Field Momenta, Optics & Photonics
News 10 (March 1999).
23. The Shack-Hartmann Wavefront Sensor, Optics & Photonics News 10 (April 1999).
24. Laser-induced Heating of Multilayers, Optics & Photonics News 10 (May 1999).
25. The Van Cittert-Zernike Theorem, Optics & Photonics News 10 (June 1999).
26. Diffraction Gratings: Parts 1 and 2, Optics & Photonics News 10 (July, August 1999).
27. Nomarski's Differential Interference Contrast Microscope, Optics & Photonics News 10,
34-37 (September 1999).
28. The Van Leeuwenhoek Microscope, Optics & Photonics News 10, 39-42, (October 1999).
29. The Faraday Effect, Optics & Photonics News 10, (November 1999).
30. Partial polarization, Stokes parameters, and the Poincarè sphere, Optics & Photonics
News 11, (January 2000).
31. Projection Photolithography, Optics & Photonics News 11, (February 2000).
32. Bracewell’s Interferometric Telescope, Optics & Photonics News 11, (March 2000).
33. Ellipsometry, Optics & Photonics News 11, (April 2000).
34. Fourier Optics: Parts 1 and 2, Optics & Photonics News 11, (May, June 2000).
35. The Beam Propagation Method, Optics & Photonics News 11, (July 2000).
36. The Magneto-optical Kerr Effect, Optics & Photonics News 11, (October 2000).
37. Gaussian Beam Optics, Optics & Photonics News 12, (January 2001).
38. Spatial Optical Solitons, Optics & Photonics News 12, (April 2001).
39. Launching light into a fiber, Optics & Photonics News 12, 56-59 (August 2001).
40. Omni-directional dielectric mirrors, Optics & Photonics News 12, 46-50 (September2001).
41. The uncertainty principle in classical optics, Optics & Photonics News 13, 44-48 (January2002).
42. Doppler shift, stellar aberration, and the convection of light by moving media, Optics &Photonics News 13, 52-56 (April 2002).
43. The Optics of Semiconductor Diode Lasers, Optics & Photonics News 13, 57-61 (July
2002).
44. Interaction of Light with Subwavelength Structures, Optics & Photonics News 14, 56-61(March 2003).
45. Transmission of Light through Small Elliptical Apertures, Parts 1 and 2, Optics &Photonics News 15, 38-43 (March 2004), 44-48 (April 2004).

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411

Appendix H
Reprint of the article, "Certain Computational Aspects of Vector Diffraction
Problems," by M. Mansuripur, originally published in Journal of the OpticalSociety of America A, Vol.6, No.5, pp 786-805 (1989). Erratum appeared in J.
Opt. Soc. Am. A, Vol.10, No.2, pp 382-383 (1993).

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413

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431

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435

Appendix I
Reprint of the article, “Analysis of Multilayer Thin Film Structures containing
Magneto-optic and Anisotropic Media at Oblique Incidence using 2 × 2
Matrices," by M. Mansuripur, originally published in the Journal of Applied
Physics, Vol. 67, pp 6466-6475 (1990).

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437

438

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439

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442

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443

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445

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447

Appendix J
Reprint of the article, "Abbe’s Sine Condition," by M. Mansuripur, originallypublished in Optics & Photonics News, Vol.9, pp 56-60 (1998).

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455

Appendix K
Theory of Diffractive Optical Elements

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457

Diffractive Optical Elements
Transmissive Diffractive Optical Element. Figure A1(a) shows a geometricoptical ray (vacuum wavelength = λo) arriving through a medium of refractive
index n1 at the surface of a substrate (refractive index = n2) coated with avariable thickness layer; the angle and the azimuth of incidence are θ1, φ1, thoseof the transmitted ray are θ2, φ2. The incident wavefront at the front facet of thesubstrate may be written as A(x, y) = Ao exp[i(2πn1/λo)(xσx + yσy)], whereσx = sinθ1 cosφ1 and σy = sinθ1 sinφ1.
n2

n1

n1

n2

t (x, y)
θ2

θ2

θ1

θ1
n

Figure A1. (a) A ray of light (vacuum wavelength = λo) is incident at an oblique angle(θ1, φ1) from a medium of refractive index n1 onto a substrate of index n2. The substrate iscoated with a layer of index n and variable thickness t (x, y), where n is assumed to be large
and t(x, y) very small, so that only the optical path difference, OPD = (n − n1) t (x, y), has afinite value. (b) The variable thickness layer is converted to a DOE by reducing the coating
layer’s thickness wherever the OPD contains an integer multiple of the construction
wavelength λc. The characteristic function of the DOE is thus the fractional part f (x, y) of thecharacteristic function of the coating layer in (a), defined as F(x, y) = (n − n1) t (x, y) /λc.

The coating layer has thickness t(x, y) and refractive index n. To avoid certain
complications in the following analysis we shall assume that n is very large andt(x, y) very small, so that only the product (n − n1) t(x, y), known as the optical
path difference (OPD), has a finite value. The characteristic function of thecoating layer is thus the dimensionless function F(x, y) = (n − n1) t(x, y) /λc,where λc is some fixed “construction wavelength.” The characteristic function is
generally specified by a polynomial such as
N N− mF(x, y) = ΣΣ bmn xm yn.

(1a)m=0 n=0F(x, y) must be greater than or equal to zero across the surface since n − n1,t(x, y) and λc are all non-negative. For later reference, the gradient of F(x, y) is
written below:

458

Guide to DIFFRACTN

N− m∇F(x, y) = (∂F /∂x, ∂F /∂y) = [Σ m ( Σ bmn y ) x
n

m=1

n=0

N

N− n

n=1

m=0

, Σ n ( Σ bmn xm) y n −1]. (1b)m −1

A Diffractive Optical Element (DOE) is constructed from the above coating layer
by reducing the layer’s thickness whenever F(x, y) happens to be greater thanunity. By removing from t(x, y) all integer multiples of λc /(n − n1), one obtains a
coating such as that in Fig. A1(b), for which the integer part of F(x, y), if
any, has been eliminated in all locations. The characteristic function f (x, y) of
the DOE, with values confined to [0, 1], is simply the fractional part of F(x, y).
As shown in Fig. A2, the coating layer’s F(x, y) is truncated at contours where
the function acquires integer values, so the local period (∆x, ∆y) of the DOE at apoint such as (xo, yo) is the shortest line segment through (xo, yo) that satisfies theequation∇F(x, y) · (∆x x + ∆y y ) = (∂F/∂x) ∆x + (∂F /∂y) ∆y = 1.

(2)

In Eq. (2) x and y are unit vectors along the coordinate axes. Noting that|∇F | 2 = (∂F/∂x)2 + (∂F/∂y)2, we find (∆x, ∆y) = ∇F/ |∇F |2. This is the local
period of the grating at (xo, yo), which is directed along ∇F and has magnitude
1/ |∇F |. In the linear approximation, a single period of the grating begins at(x, y) = (xo, yo) − f (xo, yo)∇F/ |∇F |2, where f (x, y) = 0, and ends at (x, y) =(xo, yo) + [1 − f (xo, yo)] ∇F/ |∇F |2, where f (x, y) = 1.

Figure A2. Diagram of a DOE showing theslicing contours where the function F(x, y)
assumes integer values. The DOE’s
characteristic function f (x, y) is the fractionalpart of F(x, y). Thus while F(x, y) is continuous
across the X Y-plane, f (x, y) jumps by one unit
at each contour. The space between each pairof adjacent contours contains a single grooveof the DOE, where f (x, y) varies continuously
between the values of 0 and 1. At an arbitrary
location (xo, yo) in the X Y-plane, the separation
between adjacent contours is given by(∆x, ∆ y) = ∇F/ |∇F |2, which is a vector of
magnitude 1/ |∇F | oriented orthogonal to thecontours.

y∇F/ |∇F |2(xo, yo)×xGuide to DIFFRACT

459

Since n is assumed to be large, inside the coating layer of Fig. A1(a) the ray
travels along the Z-axis and acquires an extra phase Ψ (x, y) =2π(n − n1) t(x, y)/λo = 2π(λc/λo)F(x, y). As long as λo = λc, the truncation ofF(x, y), i.e., removal of its integer part, does not affect the acquired phaseshift Ψ (x, y); in other words, eliminating 2π multiples does not change thetransmitted beam’s phase profile. However, when λo ≠ λc, the X Y-plane may be
divided into segments, defined by the contours of truncation, where the phase of
the transmitted beam over each segment differs from Ψ (x, y) by some integermultiple of 2π(λc/λo); the DOE thus modulates the incident phase byψ (x, y) = 2π(λc/λo) f (x, y). In the vicinity of an arbitrary point (xo, yo),considering the local periodicity of the grating along the direction ∇F, themodulating phase function exp[iψ (x, y)] may be expanded in the following(one-dimensional) Fourier series:
exp [i2π(λc/λo) f (x, y)] = Σ Cm exp{i2πm[(∂F/∂x) (x – xo) + (∂F/∂y) (y – yo)]}, (3a)mwhere the Fourier coefficients are given by
Cm = |∇F | ∫ exp [i2π(λc/λo) f (x, y)] exp(−i2πm|∇F | s) ds.

(3b)

In Eq.(3b), the one-dimensional integral is taken in the X Y-plane along a
straight line segment drawn parallel to ∇F through (xo, yo); the range of
integration, starting at (x, y) = (xo, yo) − f (xo, yo)∇F/ |∇F |2 and ending at(x, y) = (xo, yo) + [1 − f (xo, yo)] ∇F/ |∇F |2, covers one full period of the grating;see Fig. A2. Expanding f (x, y) to first order in Taylor series yields
f (x, y) = f (xo, yo) + (∂F/∂x) (x – xo) + (∂F/∂y) (y – yo).

(4)

Substituting for f (x, y) in Eq.(3b) from Eq.(4) and carrying out the integration,
we findCm = exp[i2πmf (xo, yo)] exp{iπ[(λc/λo) – m]} sinc[(λc/λo) – m],(5)

where sinc(x) = sin(πx)/πx. The mth order diffraction efficiency is thus found tohave the constant amplitude |Cm| = sinc[(λc/λo) – m] across the X Y-plane for any
given λo. When λo happens to be the same as the construction wavelength λc, thefirst order beam will have 100% efficiency while all other orders vanish. Also, if
λc is an integer-multiple of λo, only one order will emerge, unattenuated, from

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the DOE. For all other values of λo, the various orders m = 0, ±1, ±2, etc., will
coexist. The second term in Eq.(6) corresponds to a constant phase, π[(λc/λo) –m], which is independent of (xo, yo) and may thus be ignored in practice. The
remaining phase, 2πmf (xo, yo), varies continuously across the X Y-plane withabsolutely no dependence on λo. Since f (xo, yo) is the fractional part of F(xo, yo),
the two functions may be exchanged and the phase acquired by the mth orderrays written as 2πmF(xo, yo). In practice the lack of any discontinuous jumps inthis phase profile of the mth order beam is extremely important, since it meansthat the wavefront associated with each and every diffraction order is wellbehaved. In other words, if one assembles all the mth order rays from across theth
DOE to construct the m order transmitted beam, the beam will have acontinuous wavefront.
The transmitted wavefront around (xo, yo), the foot of the incident ray, can now
be writtenA(x, y) = Σ A′m exp[i (2πn2/λo) (xσ′x m + yσ′y m)]m= Ao exp[i (2πn1/λo) (xσx + yσy)] exp[iψ (x, y)]= Σ Cm Ao exp{i (2π/λo) [(n1σx + mλo∂F/∂x) x + (n1σy + mλo∂F/∂y) y]}.(6)mThe (complex) amplitude and the direction of the mth order transmitted ray arethus given by
A′m = Cm Ao,(7a)(σ′x , σ′y) m = (n1σx + mλo∂F/∂x, n1σy + mλo∂F/∂y) /n2.

(7b)Note that the mismatch between the refractive indices n1, n, and n2 is not taken
into consideration in Eq.(7a) as far as reflection losses at the various interfaces
are concerned. Also ignored in this analysis are the effects of incident
polarization on the transmission coefficient Cm, which would have required arigorous vector diffraction treatment.
For m ≠ 0, the direction of the mth order transmitted ray, (σ′x , σ′y) m, is seen from
Eq.(7b) to depend on the illumination wavelength λo in a way that gives rise to a
substantial amount of chromatic aberration; this provides the basis for correcting
the chromatic aberrations of conventional refractive lenses by incorporating
diffractive optical elements in the so-called hybrid designs. In going frommedium 1 to medium 2 of Fig. A1, the undiffracted 0th order ray follows theGuide to DIFFRACT

461

Snell’s law since, according to Eq.(7b), (n2σ′xo, n2σ′yo) = (n1σx , n1σy). For otherdiffraction orders, one must add mλo∇F to the incident beam’s (n1σx , n1σy) inorder to obtain the transmitted beam’s (n2σ′x, n2σ′y) m.
Having exploited the localized ray picture to build the transmitted wavefront(s)across the DOE surface, we now abandon the rays and concentrate instead on thetransmitted wavefronts (one for each diffracted order). When the incident
wavelength λo differs from the construction wavelength λc, the various orders
will be present in the mix in different amounts, with the magnitude of the mthbeam, |Cm| , being a function of m and the wavelength ratio λc/λo. Although thephase profile of each diffracted order is independent of the incident wavelength
λo, this does not imply that a given diffracted order behaves identically inresponse to different incident wavelengths. Remember that the mth order phaseprofile is exp[i2πmF(x, y)], so, for simplicity’s sake, let us assume that
F(x, y) = αx + βy, where α and β are arbitrary constants. This phase profile maythen be written as exp[i(2π/λ)(mλαx + mλβy)], where λ = λo n2 is the
wavelength within the medium of refractive index n2. This represents a plane
wave having direction cosines (σx , σy) = (mλα, mλβ), whose propagation
direction evidently depends on λo, even though its phase profile is independentof the incident wavelength. The bottom line is that the rays and the wavefronts
that emerge from the above analysis paint a consistent picture, both leading to
the same conclusions concerning the diffraction efficiency and the chromaticaberrations associated with each diffracted order of the transmitted beam.
Reflective Diffractive Optical Element. The arguments of the preceding
section may be extended to cover the case of an ideal reflective DOE shown in
Fig. A3. As before, the incidence medium has refractive index n1, but the DOE’s
substrate is a perfect reflector. We assume once again that the variable-thickness
layer has a large refractive index n and a correspondingly small thickness t(x, y).
The optical path difference upon transmission through the layer and reflection at
the substrate interface is thus given by OPD = 2(n − n1) t(x, y), which yields thecharacteristic function F(x, y) = 2(n − n1) t(x, y)/λc, with λc being theconstruction wavelength. Once again, the DOE is constructed from the above
coating layer by reducing the layer’s thickness whenever F(x, y) exceeds unity.
Note that the above factor of 2 in the expression for the OPD – representing theeffect of double-path through the coating layer – does not affect any of thesubsequent results, since the starting point of our derivations is the function
F(x, y), which already incorporates this factor. The formal derivations for areflective DOE parallel those of the transmissive DOE in the preceding section,

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until we reach Eq.(6), at which point the refractive index n2 of the medium into
which the beam emerges (upon transmission through the DOE) must be replacedwith n1, reflecting the fact that the incidence and emergence media are now thesame. Therefore, for reflective DOEs, the only equation that needs to be modified
is Eq.(7b), which assumes the following form:(σ′x , σ′y) m = [σx + (mλo /n1) ∂F/∂x, σy + (mλo /n1) ∂F/∂y].

(8)

All the considerations discussed in the case of transmissive DOEs apply equally
to reflective elements as well.
n1

Figure A3. The case of a reflective DOE differs from that ofa transmissive DOE in that the transparent substrate is now
replaced with a perfect reflector. The incident rays, after
traveling through the coating layer, bouncing back at thesubstrate interface, and returning through the same thicknessof the coating layer, re-emerge into the incidence medium(refractive index = n1). The DOE is constructed from thecoating layer by removing from t (x, y) all integer multiplesof ½λc /(n − n1).

t (x, y)Perfect
Reflector

θ2
θ1n

DOE on a curved surface. Curved surfaces may also be coated with DOEs, andthe method of calculating reflected transmitted rays is essentially the same asthat described in conjunction with flat surfaces in the preceding sections. The
reason is that all such calculations are based on the properties of the surface andof the incident and emergent rays over small patches, where curved surfaces areflat locally. The only complication arises from the fact that the DOE’s
characteristic function is usually defined with respect to a coordinate system
whose axes do not follow the profile of the surface. We limit the present
discussion to the case of a curved surface of revolution, such as that in Fig. A4,
where the axis of symmetry is z, and the sag is a given function h(r) of r. The
characteristic function of such a DOE is usually defined by a radial polynomial,N

F(r) = Σ bn r n.

(9)

n=0

Consider the local surface coordinate s shown in Fig. A4. The value of s at eachpoint on the surface is the length of the curve measured from some point ofreference such as the vertex at (r, z) = (0, 0). What we need is the characteristic
function’s gradient over a short distance ∆s, namely, ∆F/∆s. But

Guide to DIFFRACT

Therefore,463∆s = √ (∆r)2 + (∆h)2 = ∆r √ 1 + (dh/dr)2 .

(10)∂F/∂s = (∂F/∂r)/√ 1 + (dh/dr)2(11)

Equation (11), in conjunction with the equations derived previously for flat
surfaces, is all that one needs in order to compute the various diffracted rays andwavefronts associated with DOEs on curved substrates.
rs

θ1
Figure A4. A surface of revolution around the z-axis
is defined by its sag h (r), which is the distance of thesurface (along z) from the plane tangent to the surface
at its vertex. The curvilinear coordinate s follows thetangent to the surface in the r z-plane. The value of s
at each point is the length of the curve measured fromsome point of reference, such as the vertex at(r, z) = (0, 0). Also shown is a pair of incident andrefracted rays at the surface.

θ2
z

h(r)

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Guide to DIFFRACT

Guide to DIFFRACT

465

References
1. M. Mansuripur, Classical Optics and its Applications, Cambridge University Press,
London, 2002. Expanded Japanese edition, New Technology Communications, Co.,Tokyo, 2006.
2. M. Mansuripur, The Physical Principles of Magneto-optical Recording, Cambridge
University Press, London, 1995.
3. M. Mansuripur, "Certain Computational Aspects of Vector Diffraction Problems",
Journal of the Optical Society of America A, Vol.6, No.5, 786-805 (1989). See also
erratum, J. Opt. Soc. Am. A, Vol.10, No.2, 382-383 (1993).
4. M. Mansuripur, "Distribution of Light at and Near the Focus of High Numerical Aperture
Objectives", Journal of the Optical Society of America A, Vol.3, No.12, 2086-2093(1986). See also erratum, J. Opt. Soc. Am. A, Vol. 10, 382-383 (1993).
5. M. Mansuripur, "Analysis of multilayer thin film structures containing magneto-optic andanisotropic media at oblique incidence using 2 × 2 matrices," J. Appl. Phys. 67, 6466-6475(1990).
6. M. Mansuripur, "Analysis of Astigmatic Focusing and Push-Pull Tracking Error Signals
in Magneto-Optical Disk Systems", Applied Optics 26, No.18, 3981-3986 (1987).
7. M. Mansuripur and C. Pons, "Diffraction Modeling of Optical Path for Magneto-Optical
Disk Systems", Proceedings of SPIE, Vol.899, 56-60 (1988).
8. M. Mansuripur, "Computer Modeling of Optical Storage Media and Systems",
Proceedings of SPIE, Vol.1316, 70-80 (1990).
9. M. Mansuripur, "Effects of High-Numerical-Aperture Focusing on the State ofPolarization in Optical and Magneto-Optic Data Storage Systems," Applied Optics,
Vol.30, No.22, 3154-3162 (1991).
10. J.S. Hartman, M.A. Lind, and M. Mansuripur, "Read Channel Optical Modeling for a
Bump Forming Dye-Polymer Optical Data Storage Medium", Proceedings of SPIE, Vol.
1078, 308-323 (1989).
11. B.E. Bernacki and M. Mansuripur, "Investigation of substrate birefringence effects on
optical-disk performance", Applied Optics 32, 6547-6555 (1993).
12. B.E. Bernacki and M. Mansuripur, "Causes of focus error feedthrough in optical disk
systems: astigmatic and obscuration methods," Applied Optics 33, 735-743 (1994).

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Guide to DIFFRACT

13. S. Sugaya and M. Mansuripur, "Effects of substrate birefringence on focusing andtracking servo signals in magneto-optical disk data storage," Applied Optics 33, 50735079 (1994).
14. S. Sugaya and M. Mansuripur, "Effect of tilted ellipsoid of birefringence on readout signal
in magneto-optical disk data storage," Applied Optics 33, 5999-6008 (1994).
15. R.E. Gerber and M. Mansuripur, "Effects of substrate birefringence and tilt on theirradiance and phase patterns of the return beam in magneto-optical disk data storage,"
Applied Optics 34, 4780-4787 (1995).
16. T.D. Goodman and M. Mansuripur, "Optimization of groove depth for cross-talkcancellation in the scheme of land–groove recording in magneto-optic disk systems,"
Applied Optics 35, 1107-1119 (1996).
17. T.D. Goodman and M. Mansuripur, "Subtle effects of the substrate in optical disk data
storage systems," Applied Optics 35, 6747-6753 (1996).
18. R.E. Gerber and M. Mansuripur, “Tilt correction in an optical disk system,” Applied
Optics 35, 7000-7007 (1996).
19. Y.C. Hsieh and M. Mansuripur, “Image contrast in polarization microscopy of magnetooptical disk data storage media through birefringent plastic substrates,” Applied Optics 36,
4839-4852 (1997).
20. C.L. Bartlett, D. Kay, and M. Mansuripur, “Computer simulations of effects of disk tilt
and lens tilt on push-pull tracking error signal in an optical disk drive,” Applied Optics 36,
8467-8473 (1997).
21. C. Peng, M. Mansuripur, W.M. Kim, and S.G. Kim, “Edge detection in phase-change
optical data storage,” Applied Physics Letters 71, 2088-2090 (1997).
22. J. H. Yoo, C. W. Lee, D. H. Shin, C. Bartlett, K. Cheong, J. K. Erwin, and M. Mansuripur,“Investigation of certain diffraction effects in an optical disk,” Applied Optics 36, 92879295 (1997).
23. E. M. Wright, M. Mansuripur, V. Liberman, and K. Bates, “Spatial pattern ofmicrochannel formation in fused silica irradiated by nanosecond ultraviolet pulses,”
Applied Optics 38, 5785-5788 (1999).
24. Wei-Hung Yeh, Lifeng Li, and M. Mansuripur, “Computation of effective groove depth inan optical disk with vector diffraction theory,” Applied Optics 39, 316-323 (2000).
25. R. Liang, J. K. Erwin, and M. Mansuripur, “Variation on Zernike’s phase-contrast
microscope,” Applied Optics 39, 2152-2158 (2000).
26. J. Tesar, R. Liang, and M. Mansuripur, “Optical modeling combining geometrical ray
tracing and physical-optics software,” Opt. Eng. 39, 1845-1849 (2000).

Guide to DIFFRACT

467

27. M. Mansuripur, R. Liang, and J. Tsujiuchi, “Simulation software for optical systems:
DIFFRACT™ and its applications,” (in Japanese), Optical Technology Contact 38, 147156 (2000).
28. R. Liang, J. Carriere, and M. Mansuripur, “Intensity, polarization, and phase informationin optical disk systems,” Applied Optics, 2002.
29. M. Mansuripur, “Vector diffraction theory of focusing in systems of high numericalaperture,” Optics & Photonics News, 72-75, March 1992.
30. M. Mansuripur, “Some Quirks of Total Internal Reflection,” Optics & Photonics News,
36-39, February 1997.
31. M. Mansuripur, “Fabry-Perot Etalons in Polarized Light,” Optics & Photonics News, 3944, March 1997.
32. M. Mansuripur, “The Talbot Effect,” Optics & Photonics News, 42-47, April 1997.
33. M. Mansuripur, “What in the World are Surface Plasmons?” Optics & Photonics News,
50-55, May 1997.
34. M. Mansuripur, “Internal Conical Refraction,” Optics & Photonics News, 43-45, June
1997.
35. M. Mansuripur, “The Ronchi Test,” Optics & Photonics News, 42-46, July 1997.
36. M. Mansuripur, “External Conical Refraction,” Optics & Photonics News, 50-52, August
1997.
37. M. Mansuripur, “The Method of Fox and Li,” Optics & Photonics News, 38-41,
September 1997.
38. M. Mansuripur, “Coherent and Incoherent Imaging,” Optics & Photonics News, 40-44,
October 1997.
39. M. Mansuripur, “Zernike’s Method of Phase Contrast,” Optics & Photonics News, 40-43,
November 1997.
40. M. Mansuripur, “Michelson’s Stellar Interferometer,” Optics & Photonics News, 40-43,
January 1998.
41. M. Mansuripur, “Abbe’s Sine Condition,” Optics & Photonics News, 56-60, February
1998.
42. M. Mansuripur, “Holography and Holographic Interferometry,” Optics & Photonics
News, 41-45, March 1998.

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Guide to DIFFRACT

43. M. Mansuripur and E. M. Wright, "Self-focusing in Nonlinear Optical Media," Optics &Photonics News, 43-47, April 1998.
44. M. Mansuripur, L. Li and Y. H. Yeh, "Scanning Optical Microscopy: Parts I and II,"
Optics & Photonics News, 56-59 May 1998, and 42-45, June 1998.
45. M. Mansuripur, "Reciprocity in Classical Linear Optics," Optics & Photonics News, 5358, July 1998.
46. M. Mansuripur, "The Ewald-Oseen Extinction Theorem," Optics & Photonics News, 5055, August 1998.
47. M. Mansuripur, "Evanescent Coupling: Parts I and II," Optics & Photonics News, 59-62,
September 1998, and 35-38, October 1998.
48. M. Mansuripur, "Polarization Microscopy," Optics & Photonics News, November 1998.
49. M. Mansuripur and E. M. Wright, “First-order Temporal Coherence in Classical Optics,"
Optics & Photonics News 10, 44-49(January 1999).
50. M. Mansuripur and E. M. Wright, “Optical Vortices," Optics & Photonics News 10,(February 1999).
51. M. Mansuripur, “Geometric-optical Rays, Poynting’s Vector, and Field Momenta," Optics& Photonics News 10 (March 1999).
52. M. Mansuripur, “The Shack-Hartmann Wavefront Sensor," Optics & Photonics News 10(April 1999).
53. M. Mansuripur, “Laser-induced Heating of Multilayers," Optics & Photonics News 10(May 1999).
54. M. Mansuripur, “The Van Cittert-Zernike Theorem," Optics & Photonics News 10 (June1999).
55. M. Mansuripur, Lifeng Li, and Y. H. Yeh, “Diffraction Gratings: Parts 1 and 2," Optics &Photonics News 10 (July, August 1999).
56. M. Mansuripur, "Nomarski's Differential Interference Contrast Microscope," Optics &Photonics News 10, 34-37 (September 1999).
57. M. Mansuripur, "The Van Leeuwenhoek Microscope," Optics & Photonics News 10, 3942, (October 1999).
58. M. Mansuripur, "The Faraday Effect," Optics & Photonics News 10, (November 1999).

Guide to DIFFRACT

469

59. M. Mansuripur, "Partial polarization, Stokes parameters, and the Poincarè sphere," Optics& Photonics News 11, (January 2000).
60. M. Mansuripur and R. Liang, "Projection Photolithography," Optics & Photonics News11, (February 2000).
61. M. Mansuripur, "Bracewell’s Interferometric Telescope," Optics & Photonics News 11,(March 2000).
62. M. Mansuripur, "Ellipsometry," Optics & Photonics News 11, (April 2000).
63. M. Mansuripur, "Fourier Optics: Parts 1 and 2," Optics & Photonics News 11, (May, June
2000).
64. M. Mansuripur, E. M. Wright, and M. Fallahi, "The Beam Propagation Method," Optics& Photonics News 11, (July 2000).
65. M. Mansuripur, "The Magneto-optical Kerr Effect," Optics & Photonics News 11,(October 2000).
66. M. Mansuripur, "Gaussian Beam Optics," Optics & Photonics News 12, (January 2001).
67. M. Mansuripur and E. M. Wright, "Spatial Optical Solitons," Optics & Photonics News
12, (April 2001).
68. M. Mansuripur, "Launching light into a fiber," Optics & Photonics News 12, 56-59(August 2001).
69. M. Mansuripur, "Omni-directional dielectric mirrors," Optics & Photonics News 12, 4650 (September 2001).
70. M. Mansuripur, "The uncertainty principle in classical optics," Optics & Photonics News13, 44-48 (January 2002).
71. M. Mansuripur, "Doppler shift, stellar aberration, and the convection of light by moving
media," Optics & Photonics News 13, 52-56 (April 2002).
72. M. Mansuripur and E. M. Wright, "The optics of semiconductor diode lasers," Optics &Photonics News 13, 57-61 (July 2002).
73. M. Mansuripur, A. R. Zakharian, and J. V. Moloney, "Interaction of light withsubwavelength structures," Optics & Photonics News 13, 56-61 (March 2003).
74. A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Simulating near-field effects inhigh-density optical disk data storage,” (invited feature article), Computing in Science &Engineering 5, No. 6, pp 15-21 (November/December 2003).

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Guide to DIFFRACTMM Research, Inc.
5748 N. Camino del CondeTucson, Arizona 85718