Speckle Phenomena in Optics: Theory and Applications $O kmurnn
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Joseph W. Goodman /jJD
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Contents FhY#3-jH
1 Origins and Manifestations of Speckle 1 oH6zlmqG"
1.1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7mYcO3{5{
1.2 Intuitive Explanation of the Cause of Speckle . . . . . . . . . . . . . . . . . . . . . . . . . 2 KJQ8Yhq
1.3 Some Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 N,qo/At}R[
2 Random Phasor Sums 7 9#U]?^DJ @
2.1 First and Second Moments of the Real and Imaginary Parts of the Resultant Phasor . . . . . 8 nu+K
N,3R"
2.2 Random Walk with a Large Number of Independent Steps . . . . . . . . . . . . . . . . . . 9 `.pEI q^
2.3 Random Phasor Sum Plus a Known Phasor . . . . . . . . . . . . . . . . . . . . . . . . . . 12 aXj
UDu7
2.4 Sums of Random Phasor Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 wJ2cAX;"
2.5 Random Phasor Sums with a Finite Number of Equal-Length Components . . . . . . . . . 16 &v .S_Ym
2.6 Random Phasor Sums with a Nonuniform Distribution of Phases . . . . . . . . . . . . . . . 17 K'
`qR
3 First-Order Statistical Properties of Optical Speckle 23 bdstxjJ`
3.1 Definition of Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 jw{N#QDh
3.2 First-Order Statistics of the Intensity and Phase . . . . . . . . . . . . . . . . . . . . . . . . 24 QD!NV*
3.2.1 Large Number of Random Phasors . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 M}[Q2v\
3.2.2 Constant Phasor plus a Random Phasor Sum . . . . . . . . . . . . . . . . . . . . . 27 }nPt[77U_7
3.2.3 Finite Number of Equal-Length Phasors . . . . . . . . . . . . . . . . . . . . . . . . 31 d)-ZL *o
3.3 Sums of Speckle Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 k@lJ8(i^qU
3.3.1 Sums on an Amplitude Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 D% o(HS\E
3.3.2 Sum of Two Independent Speckle Intensities . . . . . . . . . . . . . . . . . . . . . 34 G3TS?u8Q
3.3.3 Sum of N Independent Speckle Intensities . . . . . . . . . . . . . . . . . . . . . . 37 u]NsCHKlT
3.3.4 Sums of Correlated Speckle Intensities . . . . . . . . . . . . . . . . . . . . . . . . 40 I"czo9Yspd
3.4 Partially Polarized Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 0=HB!{ @
3.5 Partially Developed Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 kl:/ PM ^
3.6 Speckled Speckle, or Compound Speckle Statistics . . . . . . . . . . . . . . . . . . . . . . 47 G 0pq'7B
3.6.1 Speckle Driven by a Negative Exponential Intensity Distribution . . . . . . . . . . . 48 ]dGH
i \
3.6.2 Speckle Driven by a Gamma Intensity Distribution . . . . . . . . . . . . . . . . . . 50 [ ?7QmZK
3.6.3 Sums of Independent Speckle Patterns Driven by a Gamma Intensity Distribution . . 51 |MGT8C&^!
4 Higher-Order Statistical Properties of Optical Speckle 55 uQdH () :
4.1 Multivariate Gaussian Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 QEqYqAGzu|
4.2 Application to Speckle Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ?P[:,0_
4.3 Multidimensional Statistics of Speckle Amplitude, Phase and Intensity . . . . . . . . . . . . 58 3LfF {ED@
4.3.1 Joint Density Function of the Amplitudes . . . . . . . . . . . . . . . . . . . . . . . 59 r~T!$Tb
4.3.2 Joint Density Function of the Phases . . . . . . . . . . . . . . . . . . . . . . . . . . 60 1(Vv-bq$
4.3.3 Joint Density Function of the Intensities . . . . . . . . . . . . . . . . . . . . . . . . 64 xwsl$Rj
4.4 Autocorrelation Function and Power Spectrum of Speckle . . . . . . . . . . . . . . . . . . . 66 gBV4IQ
4.4.1 Free-Space Propagation Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 @Ik5 BT
4.4.2 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 wT! ?.Y)aj
4.4.3 Speckle Size in Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3HtM<su*h
4.5 Dependence of Speckle on Scatterer Microstructure . . . . . . . . . . . . . . . . . . . . . . 77 &_ Cc
4.5.1 Surface vs. Volume Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ."dT6u E
4.5.2 Effect of a Finite Correlation Area of the ScatteredWave . . . . . . . . . . . . . . . 78 gWU#NRRc
4.5.3 A Regime where Speckle Size Is Independent of Scattering Spot Size . . . . . . . . 81 p]S'pzh
4.5.4 Relation between the Correlation Areas of the ScatteredWave and the Surface Height F>{bVPh
VA
Fluctuations— Surface Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Qc*p+N+$
4.5.5 Dependence of Speckle Contrast on Surface Roughness— Surface Scattering . . . . 88 &cyB}Gv
4.5.6 Properties of Speckle Resulting from Volume Scattering . . . . . . . . . . . . . . . 92 DDmC3
4.6 Statistics of Integrated and Blurred Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . 94 I(VqtC:K.
4.6.1 Mean and Variance of Integrated Speckle . . . . . . . . . . . . . . . . . . . . . . . 95 !%R):^R8
4.6.2 Approximate Result for the Probability Density Function of vT{(7m!Ra
Integrated Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 >(H:eRKq
4.6.3 “Exact” Result for the Probability Density Function of Integrated Intensity . . . . . 101 <$?#P#A
4.6.4 Integration of Partially Polarized Speckle Patterns . . . . . . . . . . . . . . . . . . . 106 Cu%BU}(
4.7 Statistics of Derivatives of Speckle Intensity and Phase . . . . . . . . . . . . . . . . . . . . 108 4 `l$0m@>
4.7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 _Ml?cT/J.O
4.7.2 Parameters for Various Scattering Spot Shapes . . . . . . . . . . . . . . . . . . . . 110 cG0)F%?X?
4.7.3 Derivatives of Speckle Phase: Ray Directions in a Speckle Pattern . . . . . . . . . . 111 ^n~ Kr1}nj
4.7.4 Derivatives of Speckle Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 YvG$2F |_)
4.7.5 Level Crossings of Speckle Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 116 -n"f>c_{>
4.8 Zeros of Speckle Patterns: Optical Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . 118 ~N9k8eT
4.8.1 Conditions Required for a Zero of Intensity to Occur . . . . . . . . . . . . . . . . . 119 YkF LNCg4}
4.8.2 Properties of Speckle Phase in the Vicinity of a Zero of Intensity . . . . . . . . . . . 119 R-pON4D"*
4.8.3 The Density of Vortices in Fully Developed Speckle . . . . . . . . . . . . . . . . . 119 yj_/:eX
4.8.4 The Density of Vortices for Fully Developed Speckle Plus a Coherent Background . 123 KF
*F
5 OpticalMethods for Suppressing Speckle 125 )( 3)^/Xz
5.1 Polarization Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Ma?uB8o+~
5.2 Temporal Averaging with a Moving Diffuser . . . . . . . . . . . . . . . . . . . . . . . . . 127 bl!pKOY
5.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 2UYtEJ(?`{
5.2.2 Smooth Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 hO> q|+mC
5.2.3 Rough Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 ^- ?^iWQ G
5.3 Wavelength and Angle Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 z<t2yh(DF
5.3.1 Free-Space Propagation, Reflection Geometry . . . . . . . . . . . . . . . . . . . . 136 'g<{l&u
5.3.2 Free-Space Propagation, Transmission Geometry . . . . . . . . . . . . . . . . . . . 144 <k1muSe
5.3.3 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 uu]C;wl
5.4 Temporal and Spatial Coherence Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 150 nl 2Lqu1
5.4.1 Coherence Concepts in Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 !Usmm8!K
5.4.2 Moving Diffusers and Coherence Reduction . . . . . . . . . . . . . . . . . . . . . . 152 (&*Bl\YoX
5.4.3 Speckle Suppression by Reduction of Temporal Coherence . . . . . . . . . . . . . . 154 IW n G@!
5.4.4 Speckle Suppression by Reduction of Spatial Coherence . . . . . . . . . . . . . . . 157 t pzWi
W/
5.5 Use of Temporal Coherence to Destroy Spatial Coherence . . . . . . . . . . . . . . . . . . 163 )2E vZn
5.6 Compounding Speckle Suppression Techniques . . . . . . . . . . . . . . . . . . . . . . . . 163 4
w*m]D{
6 Speckle in Certain Imaging Modalities 165 S`[(y?OF?
6.1 Speckle in the Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 B7C<;`5TiD
6.2 Speckle in Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Se[=$W
6.2.1 Principles of Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 H{E(=S
6.2.2 Speckle Suppression in Holographic Images . . . . . . . . . . . . . . . . . . . . . . 170 ^0_ >
6.3 Speckle in Optical Coherence Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 171 !4^Lv{1QZ
6.3.1 Overview of the OCT Imaging Technique . . . . . . . . . . . . . . . . . . . . . . . 172 =1>G *
,
6.3.2 Analysis of OCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 R+0fs$s u
6.3.3 Speckle and Speckle Suppression in OCT . . . . . . . . . . . . . . . . . . . . . . . 176 8}[<3K%*g
6.4 Speckle in Optical Projection Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 ,>(X}Q
6.4.1 Anatomies of Projection Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6& E[hvu
6.4.2 Speckle Suppression in Projection Displays . . . . . . . . . . . . . . . . . . . . . . 182 gmTBp}3
6.4.3 Polarization Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 sDgo G
6.4.4 A Moving Screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 hQ:wW}HWW
6.4.5 Wavelength Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 g Yh o$E
6.4.6 Angle Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 s! gVY!0
6.4.7 Over-Design of the Projection Optics . . . . . . . . . . . . . . . . . . . . . . . . . 186 !2B~.!&
6.4.8 Changing Diffuser Projected onto the Screen . . . . . . . . . . . . . . . . . . . . . 188 xK[ [b
6.4.9 Specially Designed Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 -2laM9Ed
6.5 Speckle in Projection Microlithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7{"F%`7L
6.5.1 Coherence Properties of Excimer Lasers . . . . . . . . . . . . . . . . . . . . . . . 200 yp]vDm
6.5.2 Temporal Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 b't6ekkN
6.5.3 From Exposure Fluctuations to Line Position Fluctuations . . . . . . . . . . . . . . 202 D)L~vA/8b
7 Speckle in Certain Non-imagingModalities 205 Z< C39s
7.1 Speckle in Multimode Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 :O,,fJ<x.O
7.1.1 Modal Noise in Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 #WDpiV7B
7.1.2 Statistics of Constrained Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 - =:tlH
n
7.1.3 Frequency Dependence of Modal Noise . . . . . . . . . . . . . . . . . . . . . . . . 211 n ~>b}DY
7.2 Effects of Speckle on Optical Radar Performance . . . . . . . . . . . . . . . . . . . . . . . 216 CO
ZfR~}
7.2.1 Spatial Correlation of the Speckle Returned from Distant Targets . . . . . . . . . . . 217 t,w/L*r+w
7.2.2 Speckle at Low Light Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 mOjjw_3gq
7.2.3 Detection Statistics—Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . 222 'q/C: Yo
7.2.4 Detection Statistics— Heterodyne Detection . . . . . . . . . . . . . . . . . . . . . 227 &% (1?\~u
7.2.5 Comparison of Direct Detection and Heterodyne Detection . . . . . . . . . . . . . . 234 *
kL>9
7.2.6 Reduction of the Effects of Speckle in Optical Radar Detection . . . . . . . . . . . . 235 9-A@2&J1
7.3 Speckle and Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 q oph#\
8 Speckle in Imaging Through the Atmosphere 239 YuoErP=P
8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 e7Gb7c~
8.1.1 Refractive Index Fluctuations in the Atmosphere . . . . . . . . . . . . . . . . . . . 239 3M^`6W[;
8.2 Short-Exposure and Long-Exposure Point-Spread Functions . . . . . . . . . . . . . . . . . 240 6>)KiigZ\
8.3 Long-Exposure and Short-Exposure Average Optical Transfer Functions . . . . . . . . . . . 242 {9@E[bWp#
8.4 Statistical Properties of the Short-Exposure OTF and MTF . . . . . . . . . . . . . . . . . . 243 H- WNu+
8.5 Astronomical Speckle Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 G'HLnx}Yi
8.5.1 Object Information that Is Retrievable . . . . . . . . . . . . . . . . . . . . . . . . . 248 02^ \np
8.5.2 Results of a More Complete Analysis of the Form of the Speckle Transfer Function . 250 rP 6k}
8.6 The Cross-Spectrum or Knox–Thompson Technique . . . . . . . . . . . . . . . . . . . . . 252 f~jd N~
8.6.1 The Cross-Spectrum Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . 253 v+C D{Tc
8.6.2 Recovering Full Object Information from the Cross-Spectrum . . . . . . . . . . . . 254 ,pZz`B#
8.7 The Bispectrum Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 >9g^-~X;v
8.7.1 The Bispectrum Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 4Im}!q5;:<
8.7.2 Recovering Full Object Information from the Bispectrum . . . . . . . . . . . . . . . 258 E}36
8.8 Speckle Correlography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 u91
A Linear Transformations 263 &$qIJvMiK
B Contrast of Partially Developed Speckle 267 u0u z~ s
C Statistics of Derivatives of Speckle 271 ,:81DA
C.1 The Correlation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 La^Zr,T!
C.2 Joint Density Function of the Derivatives of Phase . . . . . . . . . . . . . . . . . . . . . . . 274 2<2a3'pG
C.3 Joint Density Function of the Derivatives of Intensity . . . . . . . . . . . . . . . . . . . . . 274 4g.S!-H@R
D Wavelength and Angle Dependence 277 5 (\[Gke
D.1 Free-Space Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 HqOnZ>D
D.2 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 -x/g+T-
E Speckle Contrast when a Dynamic Diffuser is Projected onto a Random Screen 285 cwUor}<|
E.1 Random Phase Diffusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 b]8\% =d
E.2 Diffuser that Just Fills the Projection Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 287 ws]d ,]
E.3 Diffuser Overfills the Projection Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 ybB<AkYc
F Statistics of Constrained Speckle 289 !<-+}X+o8$
Bibliography 291 Y k"yup@3
Index 299
