Simulating Speckle with Mathematica

103 Pages·2022·20.864 MB·English

by  Joseph W. Goodman

Simulating
Speckle with
Mathematica
®
Joseph W. Goodman
Simulating
Speckle with
Mathematica
®
Joseph W. Goodman
SPIE PRESS
Bellingham, Washington USA
Library of Congress Cataloging-in-Publication Data

Names: Goodman, Joseph W., author.
Title: Simulating speckle with Mathematica / Joseph W. Goodman, Stanford
University.
Description: Bellingham, Washington : SPIE, [2022] | Includes
bibliographical references.
Identifiers: LCCN 2022030726 | ISBN 9781510656543 (paperback) | ISBN
9781510656550 (pdf)
Subjects: LCSH: Speckle--Computer simulation. | Speckle--Mathematical
models. | Mathematica (Computer file)
Classification: LCC QC427.8.S64 G659 2022 | DDC
535/.470113--dc23/eng20220919
LC record available at https://lccn.loc.gov/2022030726

Published by
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Copyright © 2022 Society of Photo-Optical Instrumentation Engineers (SPIE)

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The cover image was simulated from the painting “The Audience” by Gene Walch.

Printed in the United States of America.
First printing 2022.
For updates to this book, visit http://spie.org and type “PM355” in the search field.
Simulating Speckle with

Mathematica®

Joseph W . Goodman
Stanford University

Table of Contents

1. Introduction

1.1. Mathematica Background

1.2. Speckle Background

1.3. Methods for Simulating Speckle

2. First-Order Statistics of Speckle Amplitude

2.1. Speckle as a Sum of Independent Random Complex Phasors

2.2.Amplitude Statistics of the Sum of Many Random Phasors with Unit Lengths and

Random Phases

2.3.Amplitude Statistics of the Sum of a Large Number of Unit-Length Random

Phasors and One Large Phasor

2.4.Amplitude Statistics of the Sum of a Small Number of Unit-Length Random

Phasors

3. First-Order Statistics of Speckle Intensity

3.1. Intensity Statistics of the Sum of Many Random Phasors with Unit Lengths and

Random Phases

3.2. Intensity Statistics of the Sum of a Large Number of Unit-Length Random Phasors

and One Large Phasor

3.3. Intensity Statistics of the Sum of a Small Number of Unit-Length Random Phasors

3.4. Intensity Statistics for Sums of Independent Speckle Patterns

3.5. Intensity Statistics of Partially Developed Speckle

4. Simulation of Speckle in Optical Imaging

4.1. Generation of a Discrete DiffuserArray with Correlated Phases

4.2. Speckle in an Imaging Geometry

4.3. TheAutocorrelation Function of Speckle Intensity
vi Table of Contents

4.4. The Power Spectrum of Speckle Intensity

4.5. Effect of Speckle on Resolution

4.6. Speckle in Color Images

5. Simulation of Speckle in Free-Space Propagation

5.1. The Diffuser

5.2. The Fresnel Transfer FunctionApproach

5.3. The Fresnel Transform Approach

6. Speckle at Low Light Levels

6.1. Photocount Image of a Uniform Intensity

6.2. The Negative-Binomial Distribution

6.3. Photocount Image for a Speckle Pattern with Uniform Statistics

6.4. Photocount Image for a Diffuse Structured Object

7. Speckle Phase Vortices

7.1. Generating the Speckle Pattern

7.2. Finding the Zeros of Intensity

7.3. Phase Behavior in the Vicinity of the Zeros of Intensity

8. Polarization Speckle

8.1. The Polarization Ellipse and the Degree of Polarization

8.2. Generating the Two Polarization Components

8.3. Generating a Filtered Speckle Pattern

8.4. Generating the Polarization Ellipses

8.5. Visualizing Polarization Speckle

9. Speckle Simulation for Metrology

9.1. Measurement of In-Plane Displacement

9.2. Electronic Speckle Pattern Interferometry

9.3. Phase-Shifting Speckle Interferometry

Appendix A – Some Subtleties in Speckle Simulation With the 4f System

A.1. Effects on the Speckle Contrast

A.2. Simulation With a Smoothed Phase

A.3. Simulation With an Unsmoothed Phase

Appendix B – Some Subtleties in Dealing With Mathematica Images

B.1. Dimensions of DataArrays and Images

B.2. Images When the Data Range Exceeds (0, 1)
Table of Contents vii

B.3. Effect of Using ImageAdjust[] on an Image

B.4.Arrays With Bipolar Values

B.5.A Problem Encountered When Starting With an Image
Acknowledgements
References

1. Introduction

Thespeckle phenome nonisubi quitousi nman yfields ofsciencea ndt echnology. Specklep heno menacanbeseen inma nydifferen t im aging
modalities,includingacousticalimaging(e.g.,medicalultrasound)andmicrowaveimaging(e.g.,synthetic-apertureradarimaging).Thisbook

focusesonsimulatingopticalspecklewithMathematica®,butthesamemethodsusedcaninmanycasesbeappliedtootherimagingmodalities.

The r eade r may wonder w hy Mathema tica has be en chosen as the soft ware packa ge f or this b ook.There a re sev eral reasons f or this c hoice.
First,andmos timp ortant, Mat hematicaall ows the intersper sin gof bothc ontinuous anddis cretec alculationsun der one umbrel la.Secon d,using
Mathematic a, text, c ode, and illus trations ca n be included i n the same docum ent. Th ird, using Mat hem ati ca we c an crea te d yna mic fi gur es,
param etersof whichtheus ercan cha ngea twill.H owe ver, such man ipu lationc an notbe perform edinth epr intedv ersion ofthebook, so we
haveavoid edm anipulablefigures i nw hatfo llows and replac ed themby arr ays ofstaticfig ure s. L astly, this autho rl ovesM ath ematica for its
flexibil ity a nd c ompre hensivene ss. It seems the re a re a lm ost an infini te set of c apabi lities of t he pr ogram, many of w hich lie hidde n for t he
novi ceus erb utw hichgrad uallyarereve al edas the use ofthepro graminc reases. Thisbook ha sbeenwri ttene ntir elyinMa them atica .Itcanbe
read with the fu llprogram M ath ematic ao rw iththefree progr amWolframPlayeravailablefordownloadfromtheWolfram site.T h eMa thema t-
icafile s fo rallchapte rs can befo undatth efollowing U RL:http ://spie. org/S amples/Press boo k_Suppl emental/P M3 55_su p.zip This book is
meantas aco mp aniontoth ebo ok Speck le Phe nomenainOptics:TheoryandApplications,2ndEdition,publishedbySPIEPress(Ref.[1]).An
extensivelistofreferencescanbefoundinthatbook.

1.1 Mathematica Background

Forsomeback grou ndt ha twillbehe lpfultotheread erne wto Math em ati ca, seeR ef.[2].T hereare man ybooksth atdescri be thecapa bilitiesof
Math ematica.Ref .[3] ise spe ciallycom preh ensive .O urg oalhereist on otonlypresen tm ethod sforsim ula tingspec klein variou ssituatio ns
and applicati ons, but also to introduc e t he reader to the capabilities of Mathematica. In what follows we present some salient features of
Mathematicathatwillhelpthenovicegetstarted.

1. B uilt-in funct ionso fM athematica alwaysbeginwithupper-caseletters.Ifthecommandconsistsofaconcatenationoftwoseparatewords,
bo t hw ordsm us tbegi ninupper- casel etters.
2. In order to avoid confusion with built-in commands, user-defined functions should usually begin with a lower-case letter. However, it is

permissibletobegincommandswithanupper-caseletterinafontthatisdifferentthanSource Code Pro,theusualfontforMathematica

input.Asanexample,isnotthesameasN.

3. Math ema ticaredu ces aninputtha tisarat ionaln um bertot h eequiva lentsim plest ra tionalnu mber p ossi ble. Th eratio 32 /6yields1 6 /3,
butdoesnotproduceanapproximatedecimalresult.To obtainadecimalresult,placeadecimalpointattheendofeitherthenumeratororthe

denominator.Thus,32./6yieldstheresult5.33333.

4. The arguments of functions must be enclosed in square brackets [ ]. Curly brackets { } are reserved for containing lists, which can

represent vectors and matrices, and are also used for variable ranges in, for example, plot commands. Parentheses ( ) are reserved for
groupingmathematics.

5. When a fun ction is first defined, e ac hinde pen dentvari ab leont he lefto fthedefinition mustbefollowedbyanunderbar_.Underbarsare
notusedontherightofthedefinition,orwhenthefunctioniscalledbylatercode.

6. Thesymbol:=isadelayedequality,forwhichtheoperationofequalityismadeonlywhenthesymbolontheleftiscalledinlaterprogram-
ming.

7. The symbol %, when used as the argument of a function, represents the last previous output, which is now used as the input to this new

function.%%representsthesecond-to-lastoutput,etc.

8. Oftencertainsymbolsmaybeusedmorethanonceinaprogram,withtheirmeaninghavingchangedbetweenuses.Ifhisthesymbol,then

thecommandClear[h]erasesthedefinition ofthesymbolhandallowsanewdefinition tobemade.To clearmorethanonesymbol,for

example, h and g, use Clear[h,g]. To clear all symbols and functions that have been previously defined, use the command
Clear["Global`*"].

9. The sy mbol ; at the e nd o f an ex press ion to be evaluated indicates that the evaluation should take place, but the result of the evaluation
shouldbesuppresseduntilitisneededlaterintheprogram.

10. The symbol I (or)isu sed inMathemati cafor the imagin ar yco nstant - 1.Th esy mbolD[] in dicatesa deriv ati veo perat ion .
11 . T o executea com mand in Math ema tica ,pl acethe cur sorinthesamecell(cellsareindicatedbyverticallinesontherightofthenotebook)
asthecommand,andpresstheshiftkeyandthereturnkeysimultaneously.

12. Thesymbols&&meanalogicaland;thesymbols||meanalogicalor.
2 Chapter 1

13. Multiplicationcanbeindicatedbyeither*orbyaspacebetweenthetwoquantitiestobemultiplied.

14. When dealingwithacomplexnumberc,thecommandAbs[c]yieldsthemagnitudeofc,andArg[c]givesthephaseangleofcin
radians.

15. Ma thematic ain put co m mand sar ealw ays writt en inSou rc e Co de Pro sem i-boldf ont. Intheev en tth ata Ma the maticacom mand
extends beyo ndtheen dof aline, you mayfindthat a hyp henhasb eeninse rtedinthe co mm and.Th ishy phe nis no tpartoft hec omm and.W hen
viewing this mat erialw ith either theMath ematic aortheWolfram Playerprograms,ifyouchangethesizeofthewindow, thelinebreakswill
change,withtheresultthatthesehyphensshoulddisappear.

At the start we d efine a cons iderable numbe r of init ializ ation c omma nds that dete rmine cer tai n parame ters, su ch as th e s ize of images, and
certainop eratorst ha twewill d escr ibe when we first use them.T he se i nitial izatio nce llsarei na hidden cel latthe start ofe achM athem at ica
notebo okcha pter.InC hapte r1yo uca nopenthisc ellb ycli ckingo nthegray linet hat ap pears jus tabov eth echa ptertit le,an dthe ng oing to the
menu above, clic ki ng Cell, the n C ell Proper tie s, and then O pen . Howev er, the re is no need to ope n that c ell unle ss y ou wish to s ee it.The
initi aliza tio nce llisrepe at ed att hebegin nin go fea chchaptertha tfoll ow ss othatyo ucana cce ss anych apterdirectlyandevaluatetheinitializa-
tioncellinthatchapter.Itisnotpossibletoseetheinitializationcellsintheprintedversionofthebook.

The astute rea der may notic e tha t som etimes we bun dle severa l M ath emati ca co mm ands in on e in put cell, wh ile in oth er c ases we list the
comm andsin separ atece lls .Th e sam ecomm an ds areexecu tedinthesam eor der inbot h cases, but some tim esbre aking on ecellinto se veral
helpsreducewhitespaceinthePDFversionofthechapters.MathematicawillnotbreakasinglecellanditsoutputwhenitpaginatesthePDF
version.

Fourse tsofuser-d efined M a them aticaco m mandsareus edma ny timesin th isbo ok.T hefirst arec omman dsforforwardandinversediscrete
Fouriertransforms(DFTs)inanynumberofdimensions,whichareappliedtolists,i.e.,discretedataarrays:
dft[arg_]:=Fourierarg, FourierParameters{1,-1}
idft[arg_]:=InverseFourierarg, FourierParameters {1,-1}.

There are cor responding use r-defin ed commands for Fourier transforms and inverse Fourier transforms of two-dimensional (2D) continuous
functions(bycontinuous,wemeannotdiscrete):
ft2[arg_]:=
FullSimplifyFourierTransformarg,{x,y}, {u,v},FourierParameters 0,-2*Pi
ift2[arg_] := FullSimplify
InverseFourierTransformarg,{u,v}, {x,y},FourierParameters 0,-2*Pi.

Anadditionalpairofcommandsconvertsa2Drealnumericalarraytoimageformat:Image[array],andacommandtoconverta2Dimage

(inimageformat)backtonumericaldata:ImageData[image].Hopefully, thereaderwillrememberandunderstandthesecommandswhen

theyareencounteredinthechaptersthatfollow.SeeAppendixBforadiscussionofsomesubtletiesoftheselattertwocommands.

Lastlyaresomedefinitionsoffunctionsthatareusedanumberoftimesinthechaptersthatfollow.Theseinclude:
rect[x] = UnitBox[x]

whichisunityforx< 1 and0otherwise,
2
circ[r]= UnitBox[r - 1/2]

whichisacircularlysymmetricfunctionthatisunityforradiuslessthan 1 andzerootherwise,and
2
gaus[x]= Exp[-π*x^2].

ForthereaderwhoisusingeithertheMathematicaorWolframPlayertoreadthenotebooks,theauthorrecommendsthatyousetthemagnifica-

tion(intheupperright-handcornerofthenotebook)to150%foreasierreading.IntheMathematicanotebooks,theTable ofContentsprovides

linkstoeachofthechapters.

Theauthor wis hestoe mphasizeth ath eisno tan “exper t” M athemat icaus er,b uthasm aster edthe prog ram wellenoughto per form almost any
calcula tion hew ishes .Hope fully, t here ad erw ill arr iveat as imilarlevelafterreadingthisbook.ButwithMathematica,onecanalwaysdive
deeperandlearnnewtricks.That’spartofthefunofusingtheprogram.

1.2 Speckle Background

Speckle ari ses fu ndamen tally wheneve r a sum of randomly phased complex qua ntities (i .e., p ha so rs, which a re qua ntiti es h aving bot h a
magnitude a nd a p hase) are summ ed tog ether. In co her ent imag ing ap plicati ons, spec kl e can be a si gnifi cant no ise. A s we shall se e, the
fluctuat ion s ofthe ima gein tensitycanbecomparableinstrengthtothemeanintensityateachpointintheimage.Foramoredetailedback-
groundonspeckle,seeRef.[1].

Itisimport antt odistingui sh betw een theampli tudea nd the intensity ofas um ofphasors be cau seth ese twoqua ntitiesha vediff eren tstatistic al
properties.Theamplitudeofthesumgenerallyreferstotheabsolutevalueormagnitudeofthesumofcomplexphasors,whiletheintensityof