Speckle Phenomena in Optics: Theory and Applications ,TlYQ/j%h
B6-1q&
E /
Joseph W. Goodman i TLX=.M
=OamN7V=
Contents FI8 vABq
1 Origins and Manifestations of Speckle 1
}'WEqNu E
1.1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 {JlSfJw !
1.2 Intuitive Explanation of the Cause of Speckle . . . . . . . . . . . . . . . . . . . . . . . . . 2 N!%[.3o\K
1.3 Some Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Csf!I@}Z
2 Random Phasor Sums 7 {v,NNKQ4x
2.1 First and Second Moments of the Real and Imaginary Parts of the Resultant Phasor . . . . . 8 0Z1 ';A3
2.2 Random Walk with a Large Number of Independent Steps . . . . . . . . . . . . . . . . . . 9 *n x$r[Mqj
2.3 Random Phasor Sum Plus a Known Phasor . . . . . . . . . . . . . . . . . . . . . . . . . . 12 %Xe 74C"
2.4 Sums of Random Phasor Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 W4p4[&c|
2.5 Random Phasor Sums with a Finite Number of Equal-Length Components . . . . . . . . . 16 V:
n\skM
2.6 Random Phasor Sums with a Nonuniform Distribution of Phases . . . . . . . . . . . . . . . 17 l}_6 _g>6
3 First-Order Statistical Properties of Optical Speckle 23 {wh, "Ok_
3.1 Definition of Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 BF/l#)$yK
3.2 First-Order Statistics of the Intensity and Phase . . . . . . . . . . . . . . . . . . . . . . . . 24 C+%6 N@
3.2.1 Large Number of Random Phasors . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 NiB ly
3.2.2 Constant Phasor plus a Random Phasor Sum . . . . . . . . . . . . . . . . . . . . . 27 ?p& ( Af)
3.2.3 Finite Number of Equal-Length Phasors . . . . . . . . . . . . . . . . . . . . . . . . 31 &a!MT^anA~
3.3 Sums of Speckle Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 JXQh$hs
3.3.1 Sums on an Amplitude Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 0BP Ubp(
3.3.2 Sum of Two Independent Speckle Intensities . . . . . . . . . . . . . . . . . . . . . 34 ~v6]6+
3.3.3 Sum of N Independent Speckle Intensities . . . . . . . . . . . . . . . . . . . . . . 37 2#Du5d
3.3.4 Sums of Correlated Speckle Intensities . . . . . . . . . . . . . . . . . . . . . . . . 40 dZ|x `bIgs
3.4 Partially Polarized Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 iI Dun Ih
3.5 Partially Developed Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 L>&9+<-B
3.6 Speckled Speckle, or Compound Speckle Statistics . . . . . . . . . . . . . . . . . . . . . . 47 Mhu|S) hn
3.6.1 Speckle Driven by a Negative Exponential Intensity Distribution . . . . . . . . . . . 48 LV@tt&|N
3.6.2 Speckle Driven by a Gamma Intensity Distribution . . . . . . . . . . . . . . . . . . 50 {F ',e~}s
3.6.3 Sums of Independent Speckle Patterns Driven by a Gamma Intensity Distribution . . 51 !W/"Z!k
4 Higher-Order Statistical Properties of Optical Speckle 55 KnC:hus
4.1 Multivariate Gaussian Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 _)ZxD--Qg
4.2 Application to Speckle Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Ks Qn %mxS
4.3 Multidimensional Statistics of Speckle Amplitude, Phase and Intensity . . . . . . . . . . . . 58 I~ Q
G
4.3.1 Joint Density Function of the Amplitudes . . . . . . . . . . . . . . . . . . . . . . . 59 2= zw !
4.3.2 Joint Density Function of the Phases . . . . . . . . . . . . . . . . . . . . . . . . . . 60 %[x
PyqX
4.3.3 Joint Density Function of the Intensities . . . . . . . . . . . . . . . . . . . . . . . . 64 & ^;3S*p
4.4 Autocorrelation Function and Power Spectrum of Speckle . . . . . . . . . . . . . . . . . . . 66 W!V-m
4.4.1 Free-Space Propagation Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 c Rv#aV
4.4.2 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 ?izl#?
4.4.3 Speckle Size in Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 )Wy:I_F351
4.5 Dependence of Speckle on Scatterer Microstructure . . . . . . . . . . . . . . . . . . . . . . 77 #z =$ *\u
4.5.1 Surface vs. Volume Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ?T)M z
q}
4.5.2 Effect of a Finite Correlation Area of the ScatteredWave . . . . . . . . . . . . . . . 78 (W
|;gQ
4.5.3 A Regime where Speckle Size Is Independent of Scattering Spot Size . . . . . . . . 81 {ObU J3
4.5.4 Relation between the Correlation Areas of the ScatteredWave and the Surface Height \Vx_$E
Fluctuations— Surface Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 83 H;MyT Vl
4.5.5 Dependence of Speckle Contrast on Surface Roughness— Surface Scattering . . . . 88 W>(w&k]%B
4.5.6 Properties of Speckle Resulting from Volume Scattering . . . . . . . . . . . . . . . 92 :6J&%n
4.6 Statistics of Integrated and Blurred Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . 94 8'M:uI
4.6.1 Mean and Variance of Integrated Speckle . . . . . . . . . . . . . . . . . . . . . . . 95 R$bDj >8
4.6.2 Approximate Result for the Probability Density Function of Xi0 fX$-,
Integrated Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 sAS[wcOQ
4.6.3 “Exact” Result for the Probability Density Function of Integrated Intensity . . . . . 101 l4ru0V8s7
4.6.4 Integration of Partially Polarized Speckle Patterns . . . . . . . . . . . . . . . . . . . 106 *i VE O
4.7 Statistics of Derivatives of Speckle Intensity and Phase . . . . . . . . . . . . . . . . . . . . 108 h_5CWQSi
4.7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Mc{-2
4.7.2 Parameters for Various Scattering Spot Shapes . . . . . . . . . . . . . . . . . . . . 110 n d }Z[)
4.7.3 Derivatives of Speckle Phase: Ray Directions in a Speckle Pattern . . . . . . . . . . 111 Ey)ey-'\
4.7.4 Derivatives of Speckle Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 ~\+Bb8+hpJ
4.7.5 Level Crossings of Speckle Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3F32 /_`
4.8 Zeros of Speckle Patterns: Optical Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . 118 :,V&P _
4.8.1 Conditions Required for a Zero of Intensity to Occur . . . . . . . . . . . . . . . . . 119 6w~Cyu4Ov
4.8.2 Properties of Speckle Phase in the Vicinity of a Zero of Intensity . . . . . . . . . . . 119 uC%mGZ a
4.8.3 The Density of Vortices in Fully Developed Speckle . . . . . . . . . . . . . . . . . 119 KNjU!Z/4
4.8.4 The Density of Vortices for Fully Developed Speckle Plus a Coherent Background . 123 W5>emx'>
5 OpticalMethods for Suppressing Speckle 125 > D%
5.1 Polarization Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 <c$rfjM+JU
5.2 Temporal Averaging with a Moving Diffuser . . . . . . . . . . . . . . . . . . . . . . . . . 127 Xi;<O &+
5.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Vwb_$Yi+]
5.2.2 Smooth Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 C{~O!^ 2G
5.2.3 Rough Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 mrBK{@n
5.3 Wavelength and Angle Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 ;;+h4O )
5.3.1 Free-Space Propagation, Reflection Geometry . . . . . . . . . . . . . . . . . . . . 136 NAOCQDk{
5.3.2 Free-Space Propagation, Transmission Geometry . . . . . . . . . . . . . . . . . . . 144 [qU`}S2
5.3.3 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 x@3cZd0j#
5.4 Temporal and Spatial Coherence Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 150 OZEbs 7
5.4.1 Coherence Concepts in Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 $HCgawQ
5.4.2 Moving Diffusers and Coherence Reduction . . . . . . . . . . . . . . . . . . . . . . 152 1A[(R T]
5.4.3 Speckle Suppression by Reduction of Temporal Coherence . . . . . . . . . . . . . . 154 n`V? n
5.4.4 Speckle Suppression by Reduction of Spatial Coherence . . . . . . . . . . . . . . . 157 K>TEt5
5.5 Use of Temporal Coherence to Destroy Spatial Coherence . . . . . . . . . . . . . . . . . . 163 ueEf>0
5.6 Compounding Speckle Suppression Techniques . . . . . . . . . . . . . . . . . . . . . . . . 163 =1y~Qlu
6 Speckle in Certain Imaging Modalities 165 qWJHb Dd
6.1 Speckle in the Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 MT6" b
6.2 Speckle in Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 dZX;k0
6.2.1 Principles of Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Oh%p1 $H
6.2.2 Speckle Suppression in Holographic Images . . . . . . . . . . . . . . . . . . . . . . 170 Bj G fUQ
6.3 Speckle in Optical Coherence Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5fRr d;
6.3.1 Overview of the OCT Imaging Technique . . . . . . . . . . . . . . . . . . . . . . . 172 *4%%^*g.I
6.3.2 Analysis of OCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 jig3M N
6.3.3 Speckle and Speckle Suppression in OCT . . . . . . . . . . . . . . . . . . . . . . . 176 /vC!__K9:
6.4 Speckle in Optical Projection Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 +S WtHj7e
6.4.1 Anatomies of Projection Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 !E| m'_x*
6.4.2 Speckle Suppression in Projection Displays . . . . . . . . . . . . . . . . . . . . . . 182 BkP4.XRI
6.4.3 Polarization Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 lOM8%{.'_x
6.4.4 A Moving Screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 R{ u dV
6.4.5 Wavelength Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 8Ltl32JSB[
6.4.6 Angle Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 _-rC]iQJ55
6.4.7 Over-Design of the Projection Optics . . . . . . . . . . . . . . . . . . . . . . . . . 186 tu( ^D23
6.4.8 Changing Diffuser Projected onto the Screen . . . . . . . . . . . . . . . . . . . . . 188 q] Kv.x]$R
6.4.9 Specially Designed Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 C DoD9Hq,
6.5 Speckle in Projection Microlithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 'Va<GHr>+
6.5.1 Coherence Properties of Excimer Lasers . . . . . . . . . . . . . . . . . . . . . . . 200 #lc6-K#
6.5.2 Temporal Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 _%Yi ^ ^
6.5.3 From Exposure Fluctuations to Line Position Fluctuations . . . . . . . . . . . . . . 202 /pV N1Yt
7 Speckle in Certain Non-imagingModalities 205 3Yo)K
7.1 Speckle in Multimode Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 ;G[0%z+*
7.1.1 Modal Noise in Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 \XwC |[%P
7.1.2 Statistics of Constrained Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 E M`'=<)V
7.1.3 Frequency Dependence of Modal Noise . . . . . . . . . . . . . . . . . . . . . . . . 211 >TiE Y MW
7.2 Effects of Speckle on Optical Radar Performance . . . . . . . . . . . . . . . . . . . . . . . 216 Mdh]qKw
7.2.1 Spatial Correlation of the Speckle Returned from Distant Targets . . . . . . . . . . . 217 : dNJ2&kJ
7.2.2 Speckle at Low Light Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 :@a0h
7.2.3 Detection Statistics—Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . 222 R+_!FnOJ
7.2.4 Detection Statistics— Heterodyne Detection . . . . . . . . . . . . . . . . . . . . . 227 bZr,jLEf
7.2.5 Comparison of Direct Detection and Heterodyne Detection . . . . . . . . . . . . . . 234 n_:EWm$\
7.2.6 Reduction of the Effects of Speckle in Optical Radar Detection . . . . . . . . . . . . 235 v+}${h9
7.3 Speckle and Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 e=aU9v
L
8 Speckle in Imaging Through the Atmosphere 239 wS+!>Q_]w
8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 / DP0K
@%
8.1.1 Refractive Index Fluctuations in the Atmosphere . . . . . . . . . . . . . . . . . . . 239 g* `xEb= '
8.2 Short-Exposure and Long-Exposure Point-Spread Functions . . . . . . . . . . . . . . . . . 240 5W$Jxuy qj
8.3 Long-Exposure and Short-Exposure Average Optical Transfer Functions . . . . . . . . . . . 242 i+pQ 7wx
8.4 Statistical Properties of the Short-Exposure OTF and MTF . . . . . . . . . . . . . . . . . . 243 BO7XN;
8.5 Astronomical Speckle Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 }!?RB v'W
8.5.1 Object Information that Is Retrievable . . . . . . . . . . . . . . . . . . . . . . . . . 248 tCm]1ZgRW
8.5.2 Results of a More Complete Analysis of the Form of the Speckle Transfer Function . 250 RWX!d54&
8.6 The Cross-Spectrum or Knox–Thompson Technique . . . . . . . . . . . . . . . . . . . . . 252 <1B+@
8.6.1 The Cross-Spectrum Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . 253 ~mwI r
8.6.2 Recovering Full Object Information from the Cross-Spectrum . . . . . . . . . . . . 254 8!HB$vdw7
8.7 The Bispectrum Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 7 \[fjCg\w
8.7.1 The Bispectrum Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 bwcr/J( Nb
8.7.2 Recovering Full Object Information from the Bispectrum . . . . . . . . . . . . . . . 258 t\ a|Gp W
8.8 Speckle Correlography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 MT:VQ>f C
A Linear Transformations 263 OZd
(~E
B Contrast of Partially Developed Speckle 267 dsj}GgG?Z
C Statistics of Derivatives of Speckle 271 W/b)OlG"2
C.1 The Correlation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Jgg< u#
C.2 Joint Density Function of the Derivatives of Phase . . . . . . . . . . . . . . . . . . . . . . . 274 t%J1 (H
C.3 Joint Density Function of the Derivatives of Intensity . . . . . . . . . . . . . . . . . . . . . 274 Lis>Qr
D Wavelength and Angle Dependence 277 bo(w$&
VW
D.1 Free-Space Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 T I t\
D.2 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 0 q 1x+
E Speckle Contrast when a Dynamic Diffuser is Projected onto a Random Screen 285 T ,jb%uPcE
E.1 Random Phase Diffusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 0FY-e~xr
E.2 Diffuser that Just Fills the Projection Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 287 /AyxkXq
E.3 Diffuser Overfills the Projection Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 +GL$[ 5G
F Statistics of Constrained Speckle 289 I8`$a
Bibliography 291 %4Qs|CM)m
Index 299
