Speckle Phenomena in Optics: Theory and Applications xc;DdK=1X
BwxnDe G)
Joseph W. Goodman !
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JrDHRIkgm
Contents !v9lk9SV
1 Origins and Manifestations of Speckle 1 =Vb~s+YW
1.1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 z(me @P!D~
1.2 Intuitive Explanation of the Cause of Speckle . . . . . . . . . . . . . . . . . . . . . . . . . 2 J &YQ]l
1.3 Some Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Ma6W@S
2 Random Phasor Sums 7 @2
=z}S3O
2.1 First and Second Moments of the Real and Imaginary Parts of the Resultant Phasor . . . . . 8 Oz{%k#X-
2.2 Random Walk with a Large Number of Independent Steps . . . . . . . . . . . . . . . . . . 9 #Fs|f3-@
2.3 Random Phasor Sum Plus a Known Phasor . . . . . . . . . . . . . . . . . . . . . . . . . . 12 #Qh>z%Mn^3
2.4 Sums of Random Phasor Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 :.uk$jx
2.5 Random Phasor Sums with a Finite Number of Equal-Length Components . . . . . . . . . 16 yNa;\UF
2.6 Random Phasor Sums with a Nonuniform Distribution of Phases . . . . . . . . . . . . . . . 17 `T"rG }c
3 First-Order Statistical Properties of Optical Speckle 23 J} TfRrf
3.1 Definition of Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 YEv
Lhh
3.2 First-Order Statistics of the Intensity and Phase . . . . . . . . . . . . . . . . . . . . . . . . 24 S~)w\(r
3.2.1 Large Number of Random Phasors . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5mgHlsDzu
3.2.2 Constant Phasor plus a Random Phasor Sum . . . . . . . . . . . . . . . . . . . . . 27 [i7YVwG4
3.2.3 Finite Number of Equal-Length Phasors . . . . . . . . . . . . . . . . . . . . . . . . 31 LA4<#KP
3.3 Sums of Speckle Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ~W03{9(Vp8
3.3.1 Sums on an Amplitude Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 rk|@B{CA;
3.3.2 Sum of Two Independent Speckle Intensities . . . . . . . . . . . . . . . . . . . . . 34 _1|$P|$P.
3.3.3 Sum of N Independent Speckle Intensities . . . . . . . . . . . . . . . . . . . . . . 37 ?Elg ?)os
3.3.4 Sums of Correlated Speckle Intensities . . . . . . . . . . . . . . . . . . . . . . . . 40 rh%m;i<b
3.4 Partially Polarized Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ? \m3~6y
3.5 Partially Developed Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 pQWHG#?7
3.6 Speckled Speckle, or Compound Speckle Statistics . . . . . . . . . . . . . . . . . . . . . . 47 por/^=e{ Y
3.6.1 Speckle Driven by a Negative Exponential Intensity Distribution . . . . . . . . . . . 48 cozXb$bBY
3.6.2 Speckle Driven by a Gamma Intensity Distribution . . . . . . . . . . . . . . . . . . 50 v`DI<Lt
3.6.3 Sums of Independent Speckle Patterns Driven by a Gamma Intensity Distribution . . 51 gR Nv-^
4 Higher-Order Statistical Properties of Optical Speckle 55 ~R]35Cp-#
4.1 Multivariate Gaussian Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 =X(%Svnp
4.2 Application to Speckle Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Z4hLdHo_
4.3 Multidimensional Statistics of Speckle Amplitude, Phase and Intensity . . . . . . . . . . . . 58 UE :HMn6
4.3.1 Joint Density Function of the Amplitudes . . . . . . . . . . . . . . . . . . . . . . . 59 \9:wfLF8!
4.3.2 Joint Density Function of the Phases . . . . . . . . . . . . . . . . . . . . . . . . . . 60 OP! R[27>
4.3.3 Joint Density Function of the Intensities . . . . . . . . . . . . . . . . . . . . . . . . 64 -rSIBc:$8
4.4 Autocorrelation Function and Power Spectrum of Speckle . . . . . . . . . . . . . . . . . . . 66
bwi D$
4.4.1 Free-Space Propagation Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 1;lmu]I>)
4.4.2 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 k V&9`c+
4.4.3 Speckle Size in Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 s B
20/F
4.5 Dependence of Speckle on Scatterer Microstructure . . . . . . . . . . . . . . . . . . . . . . 77 ;inzyFbL=
4.5.1 Surface vs. Volume Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 +dW|^I{H}
4.5.2 Effect of a Finite Correlation Area of the ScatteredWave . . . . . . . . . . . . . . . 78 6bO~/mpWT~
4.5.3 A Regime where Speckle Size Is Independent of Scattering Spot Size . . . . . . . . 81 H !) =y
4.5.4 Relation between the Correlation Areas of the ScatteredWave and the Surface Height 9$1)k;ChP/
Fluctuations— Surface Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 83 \9{F5S z
4.5.5 Dependence of Speckle Contrast on Surface Roughness— Surface Scattering . . . . 88 Bh]!WMA w.
4.5.6 Properties of Speckle Resulting from Volume Scattering . . . . . . . . . . . . . . . 92 aFj. i8+
4.6 Statistics of Integrated and Blurred Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3
hKBc0
4.6.1 Mean and Variance of Integrated Speckle . . . . . . . . . . . . . . . . . . . . . . . 95 K@u &(}
4.6.2 Approximate Result for the Probability Density Function of u0o'K9.r
Integrated Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 pyZ9OA!PD
4.6.3 “Exact” Result for the Probability Density Function of Integrated Intensity . . . . . 101 =!b6FjsiG
4.6.4 Integration of Partially Polarized Speckle Patterns . . . . . . . . . . . . . . . . . . . 106 aP-<4uGx
4.7 Statistics of Derivatives of Speckle Intensity and Phase . . . . . . . . . . . . . . . . . . . . 108 Sq2P-y!w
4.7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 FjFMR
63
4.7.2 Parameters for Various Scattering Spot Shapes . . . . . . . . . . . . . . . . . . . . 110 ) R2XU
4.7.3 Derivatives of Speckle Phase: Ray Directions in a Speckle Pattern . . . . . . . . . . 111 3Q By\1h.
4.7.4 Derivatives of Speckle Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 ;_?MX/w|&
4.7.5 Level Crossings of Speckle Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 116 #{J,kcxS
4.8 Zeros of Speckle Patterns: Optical Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . 118 qu |i;WZE
4.8.1 Conditions Required for a Zero of Intensity to Occur . . . . . . . . . . . . . . . . . 119 DcD{*t?x
4.8.2 Properties of Speckle Phase in the Vicinity of a Zero of Intensity . . . . . . . . . . . 119 1zxq ^BI
4.8.3 The Density of Vortices in Fully Developed Speckle . . . . . . . . . . . . . . . . . 119 oG oK,
4.8.4 The Density of Vortices for Fully Developed Speckle Plus a Coherent Background . 123 g);^NAA
5 OpticalMethods for Suppressing Speckle 125 B 0ee?VC
5.1 Polarization Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 {Q Av~S>4
5.2 Temporal Averaging with a Moving Diffuser . . . . . . . . . . . . . . . . . . . . . . . . . 127 Iah[j,]r
5.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 aA'TD:&p1
5.2.2 Smooth Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 W"q@Qa`Bm
5.2.3 Rough Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 G$M9=@Ug
5.3 Wavelength and Angle Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Nw_@A8-r
5.3.1 Free-Space Propagation, Reflection Geometry . . . . . . . . . . . . . . . . . . . . 136 XMIbUbU k-
5.3.2 Free-Space Propagation, Transmission Geometry . . . . . . . . . . . . . . . . . . . 144 /=?x{(B>
5.3.3 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 c=O,;lWFqm
5.4 Temporal and Spatial Coherence Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 150 S>Yj@ L
5.4.1 Coherence Concepts in Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 XXZaK gsq
5.4.2 Moving Diffusers and Coherence Reduction . . . . . . . . . . . . . . . . . . . . . . 152 23F/\2MSG
5.4.3 Speckle Suppression by Reduction of Temporal Coherence . . . . . . . . . . . . . . 154 uQ1@b-e`5
5.4.4 Speckle Suppression by Reduction of Spatial Coherence . . . . . . . . . . . . . . . 157 &53]sFZ
5.5 Use of Temporal Coherence to Destroy Spatial Coherence . . . . . . . . . . . . . . . . . . 163 <O<LYN+(
5.6 Compounding Speckle Suppression Techniques . . . . . . . . . . . . . . . . . . . . . . . . 163 = +:{P?*}
6 Speckle in Certain Imaging Modalities 165 .Pndx%X9s
6.1 Speckle in the Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 YI RZ+H<Q
6.2 Speckle in Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 D ;}xr_
6.2.1 Principles of Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 A7_4 .VH
6.2.2 Speckle Suppression in Holographic Images . . . . . . . . . . . . . . . . . . . . . . 170 g|X ;ahTT
6.3 Speckle in Optical Coherence Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 171 21X`h3+=
6.3.1 Overview of the OCT Imaging Technique . . . . . . . . . . . . . . . . . . . . . . . 172 {Ro2ouQ!V
6.3.2 Analysis of OCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 thlY0XCq,%
6.3.3 Speckle and Speckle Suppression in OCT . . . . . . . . . . . . . . . . . . . . . . . 176 [lzN !!B!
6.4 Speckle in Optical Projection Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 d*8 $>GA
6.4.1 Anatomies of Projection Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 OR1DYHHT/1
6.4.2 Speckle Suppression in Projection Displays . . . . . . . . . . . . . . . . . . . . . . 182 ZUm?*.g\^
6.4.3 Polarization Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 B!]2Se2G
6.4.4 A Moving Screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 n.MRz WJpZ
6.4.5 Wavelength Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 '@|_OmcY
6.4.6 Angle Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 }Db[ 4
6.4.7 Over-Design of the Projection Optics . . . . . . . . . . . . . . . . . . . . . . . . . 186 Cwa ^"r3P1
6.4.8 Changing Diffuser Projected onto the Screen . . . . . . . . . . . . . . . . . . . . . 188 3f.b\4 U
6.4.9 Specially Designed Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 yF}OfK?0f
6.5 Speckle in Projection Microlithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 2j JmE&)7,
6.5.1 Coherence Properties of Excimer Lasers . . . . . . . . . . . . . . . . . . . . . . . 200 <*ZJaBwWU~
6.5.2 Temporal Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 z;f2*F
6.5.3 From Exposure Fluctuations to Line Position Fluctuations . . . . . . . . . . . . . . 202 {PP9$>4`l
7 Speckle in Certain Non-imagingModalities 205 a ] 17qMl
7.1 Speckle in Multimode Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 \*hrW(
7.1.1 Modal Noise in Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 5^<h}u9
7.1.2 Statistics of Constrained Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 SvM6iZ]
7.1.3 Frequency Dependence of Modal Noise . . . . . . . . . . . . . . . . . . . . . . . . 211 (_kp{0r#
7.2 Effects of Speckle on Optical Radar Performance . . . . . . . . . . . . . . . . . . . . . . . 216 (tah]Bx
7.2.1 Spatial Correlation of the Speckle Returned from Distant Targets . . . . . . . . . . . 217 2"P 99$"
7.2.2 Speckle at Low Light Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 -H^oXeN
7.2.3 Detection Statistics—Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . 222 Bs+(L [Z
7.2.4 Detection Statistics— Heterodyne Detection . . . . . . . . . . . . . . . . . . . . . 227 Y`7~Am/r;&
7.2.5 Comparison of Direct Detection and Heterodyne Detection . . . . . . . . . . . . . . 234
/-_=nf}w
7.2.6 Reduction of the Effects of Speckle in Optical Radar Detection . . . . . . . . . . . . 235 )~4II.`%^
7.3 Speckle and Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 /n~\\9#3
8 Speckle in Imaging Through the Atmosphere 239 :NHH
Dl
8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 9O`
m,t
8.1.1 Refractive Index Fluctuations in the Atmosphere . . . . . . . . . . . . . . . . . . . 239 {^ mNJ
8.2 Short-Exposure and Long-Exposure Point-Spread Functions . . . . . . . . . . . . . . . . . 240 $E<Esf$
8.3 Long-Exposure and Short-Exposure Average Optical Transfer Functions . . . . . . . . . . . 242 d<!3`qe
8.4 Statistical Properties of the Short-Exposure OTF and MTF . . . . . . . . . . . . . . . . . . 243
/tV/85r
8.5 Astronomical Speckle Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 ? &G`{Ey
8.5.1 Object Information that Is Retrievable . . . . . . . . . . . . . . . . . . . . . . . . . 248 6vuq1
8.5.2 Results of a More Complete Analysis of the Form of the Speckle Transfer Function . 250 S:4cr I
8.6 The Cross-Spectrum or Knox–Thompson Technique . . . . . . . . . . . . . . . . . . . . . 252 q5h*` 7f
8.6.1 The Cross-Spectrum Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . 253 Ds #/
8.6.2 Recovering Full Object Information from the Cross-Spectrum . . . . . . . . . . . . 254 4a0:2 kIKa
8.7 The Bispectrum Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 fx=Awba
8.7.1 The Bispectrum Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 'w>_+jLT
8.7.2 Recovering Full Object Information from the Bispectrum . . . . . . . . . . . . . . . 258 d2oh/j6`TA
8.8 Speckle Correlography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 O ,rwP
A Linear Transformations 263 H1<>NWm!v7
B Contrast of Partially Developed Speckle 267 0i ULCK
C Statistics of Derivatives of Speckle 271 7{e{9QbJ4
C.1 The Correlation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 {FRAv(,\
C.2 Joint Density Function of the Derivatives of Phase . . . . . . . . . . . . . . . . . . . . . . . 274 4c
oJRqf=
C.3 Joint Density Function of the Derivatives of Intensity . . . . . . . . . . . . . . . . . . . . . 274 3czeTj
D Wavelength and Angle Dependence 277 9TF f8'?d
D.1 Free-Space Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Qy<[7
D.2 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 *n]7
E Speckle Contrast when a Dynamic Diffuser is Projected onto a Random Screen 285 s2?T5oWU
E.1 Random Phase Diffusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 V/ cP4{L
E.2 Diffuser that Just Fills the Projection Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 287 'O5'i\uz
E.3 Diffuser Overfills the Projection Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Y2xL>F
F Statistics of Constrained Speckle 289 .Ha'p.
Bibliography 291 0TfS =scT
Index 299
