Speckle Phenomena in Optics: Theory and Applications <GlV!y
,S K6*tpI
Joseph W. Goodman S5p\J!k\B
e 2&i
Contents /R?uxhV
1 Origins and Manifestations of Speckle 1 |W[rywxx
1.1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Q |r 1.
1.2 Intuitive Explanation of the Cause of Speckle . . . . . . . . . . . . . . . . . . . . . . . . . 2 `{[C4]Ew/
1.3 Some Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 clV/i&]Qa
2 Random Phasor Sums 7 :jCaDhK
2.1 First and Second Moments of the Real and Imaginary Parts of the Resultant Phasor . . . . . 8 ;0{*V5A
2.2 Random Walk with a Large Number of Independent Steps . . . . . . . . . . . . . . . . . . 9 oMf h|B
2.3 Random Phasor Sum Plus a Known Phasor . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2(x KE _|
2.4 Sums of Random Phasor Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 `2+52q<FO
2.5 Random Phasor Sums with a Finite Number of Equal-Length Components . . . . . . . . . 16 p{GDW_
2.6 Random Phasor Sums with a Nonuniform Distribution of Phases . . . . . . . . . . . . . . . 17 U}TQXYAg
3 First-Order Statistical Properties of Optical Speckle 23 NV ~i4R*#
3.1 Definition of Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 [^ P2Kn
3.2 First-Order Statistics of the Intensity and Phase . . . . . . . . . . . . . . . . . . . . . . . . 24 D~);:}}>
3.2.1 Large Number of Random Phasors . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 "6h.6_bTw
3.2.2 Constant Phasor plus a Random Phasor Sum . . . . . . . . . . . . . . . . . . . . . 27 u>@G:kt8
3.2.3 Finite Number of Equal-Length Phasors . . . . . . . . . . . . . . . . . . . . . . . . 31 At:C4>HE@
3.3 Sums of Speckle Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 jl7-"V>j?;
3.3.1 Sums on an Amplitude Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Qk?Jy<Ra
3.3.2 Sum of Two Independent Speckle Intensities . . . . . . . . . . . . . . . . . . . . . 34 <FLc0s
3.3.3 Sum of N Independent Speckle Intensities . . . . . . . . . . . . . . . . . . . . . . 37 nQ MN2j M
3.3.4 Sums of Correlated Speckle Intensities . . . . . . . . . . . . . . . . . . . . . . . . 40 _ 3JTHf<+
3.4 Partially Polarized Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 EPfV S
3.5 Partially Developed Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 C*zdHzMj
3.6 Speckled Speckle, or Compound Speckle Statistics . . . . . . . . . . . . . . . . . . . . . . 47 DSa92:M}
3.6.1 Speckle Driven by a Negative Exponential Intensity Distribution . . . . . . . . . . . 48 cVi CWc2
3.6.2 Speckle Driven by a Gamma Intensity Distribution . . . . . . . . . . . . . . . . . . 50 WVFy Zp B
3.6.3 Sums of Independent Speckle Patterns Driven by a Gamma Intensity Distribution . . 51 A} O9e
4 Higher-Order Statistical Properties of Optical Speckle 55 U{m:{'np(H
4.1 Multivariate Gaussian Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6FAP *V;
4.2 Application to Speckle Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 1EB`6_>y
4.3 Multidimensional Statistics of Speckle Amplitude, Phase and Intensity . . . . . . . . . . . . 58 $x0F(|wxt
4.3.1 Joint Density Function of the Amplitudes . . . . . . . . . . . . . . . . . . . . . . . 59 L@uKE jR
4.3.2 Joint Density Function of the Phases . . . . . . . . . . . . . . . . . . . . . . . . . . 60 s)=7tHoqB)
4.3.3 Joint Density Function of the Intensities . . . . . . . . . . . . . . . . . . . . . . . . 64 P40 eK0 e6
4.4 Autocorrelation Function and Power Spectrum of Speckle . . . . . . . . . . . . . . . . . . . 66 <ZEll[0L
4.4.1 Free-Space Propagation Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 1NJ|%+I
4.4.2 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 %&NK|M+n
4.4.3 Speckle Size in Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 &-tf/qJ
4.5 Dependence of Speckle on Scatterer Microstructure . . . . . . . . . . . . . . . . . . . . . . 77 gE#,QOy
4.5.1 Surface vs. Volume Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 J(GLPC O$K
4.5.2 Effect of a Finite Correlation Area of the ScatteredWave . . . . . . . . . . . . . . . 78 tc Z~T
4.5.3 A Regime where Speckle Size Is Independent of Scattering Spot Size . . . . . . . . 81 n_Dhq (.
4.5.4 Relation between the Correlation Areas of the ScatteredWave and the Surface Height B(U `Zd
Fluctuations— Surface Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 83 s:{% 1 /
4.5.5 Dependence of Speckle Contrast on Surface Roughness— Surface Scattering . . . . 88 -tJ*F! w6U
4.5.6 Properties of Speckle Resulting from Volume Scattering . . . . . . . . . . . . . . . 92 GW#Wy=(_
4.6 Statistics of Integrated and Blurred Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Fh ;(1X75I
4.6.1 Mean and Variance of Integrated Speckle . . . . . . . . . . . . . . . . . . . . . . . 95 ;E_{Zji_e
4.6.2 Approximate Result for the Probability Density Function of N\$6R-L
Integrated Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 R8)"M(u=l
4.6.3 “Exact” Result for the Probability Density Function of Integrated Intensity . . . . . 101 ^~$
o-IX
4.6.4 Integration of Partially Polarized Speckle Patterns . . . . . . . . . . . . . . . . . . . 106 ce\- oT
4.7 Statistics of Derivatives of Speckle Intensity and Phase . . . . . . . . . . . . . . . . . . . . 108 ;DpK* A
4.7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 ml2HA4X&$Y
4.7.2 Parameters for Various Scattering Spot Shapes . . . . . . . . . . . . . . . . . . . . 110 ^W*/!q7H
4.7.3 Derivatives of Speckle Phase: Ray Directions in a Speckle Pattern . . . . . . . . . . 111 7085&\9
4.7.4 Derivatives of Speckle Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 h
!1c(UR
4.7.5 Level Crossings of Speckle Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 116 * bK@ A2`
4.8 Zeros of Speckle Patterns: Optical Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . 118 wH.'EC
4.8.1 Conditions Required for a Zero of Intensity to Occur . . . . . . . . . . . . . . . . . 119 QO%K`}Q}
4.8.2 Properties of Speckle Phase in the Vicinity of a Zero of Intensity . . . . . . . . . . . 119 ,v+SD\7|
4.8.3 The Density of Vortices in Fully Developed Speckle . . . . . . . . . . . . . . . . . 119 fy eS )
4.8.4 The Density of Vortices for Fully Developed Speckle Plus a Coherent Background . 123 J \BdC];
5 OpticalMethods for Suppressing Speckle 125 ?CC6/bE-{
5.1 Polarization Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 U ;/ )V
5.2 Temporal Averaging with a Moving Diffuser . . . . . . . . . . . . . . . . . . . . . . . . . 127 Z:{Z&HQC
5.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 W *2SlS7
5.2.2 Smooth Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Y|L57 F
5.2.3 Rough Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 YDwn s
5.3 Wavelength and Angle Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 ~ cz t=
5.3.1 Free-Space Propagation, Reflection Geometry . . . . . . . . . . . . . . . . . . . . 136 P!/8
5.3.2 Free-Space Propagation, Transmission Geometry . . . . . . . . . . . . . . . . . . . 144 qA42f83
5.3.3 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 SF7Kb `>Y
5.4 Temporal and Spatial Coherence Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 150 * gnL0\*
5.4.1 Coherence Concepts in Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 B5hGzplS
5.4.2 Moving Diffusers and Coherence Reduction . . . . . . . . . . . . . . . . . . . . . . 152 !ibp/:x
5.4.3 Speckle Suppression by Reduction of Temporal Coherence . . . . . . . . . . . . . . 154 xnTky1zq
5.4.4 Speckle Suppression by Reduction of Spatial Coherence . . . . . . . . . . . . . . . 157 Yp8$0KK
5.5 Use of Temporal Coherence to Destroy Spatial Coherence . . . . . . . . . . . . . . . . . . 163 nHX@
5.6 Compounding Speckle Suppression Techniques . . . . . . . . . . . . . . . . . . . . . . . . 163 +q;^8d>
6 Speckle in Certain Imaging Modalities 165 BeRn9[
6.1 Speckle in the Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 )Z#7%, o
6.2 Speckle in Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 0; 9 LIL5
6.2.1 Principles of Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 AMr 9rB d
6.2.2 Speckle Suppression in Holographic Images . . . . . . . . . . . . . . . . . . . . . . 170 GU xhCoxb
6.3 Speckle in Optical Coherence Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 171 K(?7E6\vO
6.3.1 Overview of the OCT Imaging Technique . . . . . . . . . . . . . . . . . . . . . . . 172 NNT9\JRv_
6.3.2 Analysis of OCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 z{ 8!3>:E
6.3.3 Speckle and Speckle Suppression in OCT . . . . . . . . . . . . . . . . . . . . . . . 176 Kt-@a%O0
6.4 Speckle in Optical Projection Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 '"&?u8u)
6.4.1 Anatomies of Projection Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 AG ?oA328
6.4.2 Speckle Suppression in Projection Displays . . . . . . . . . . . . . . . . . . . . . . 182 VC@o]t5
6.4.3 Polarization Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 @AwH?7(b
6.4.4 A Moving Screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Ez )Go6Q
6.4.5 Wavelength Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 g!8-yri
6.4.6 Angle Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 :JlJB
6.4.7 Over-Design of the Projection Optics . . . . . . . . . . . . . . . . . . . . . . . . . 186 LakP'P6`E
6.4.8 Changing Diffuser Projected onto the Screen . . . . . . . . . . . . . . . . . . . . . 188 p=zjJ~DVd
6.4.9 Specially Designed Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 ??F{Gli"C`
6.5 Speckle in Projection Microlithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 ^6=nL <L
6.5.1 Coherence Properties of Excimer Lasers . . . . . . . . . . . . . . . . . . . . . . . 200 1~+w7Ar =(
6.5.2 Temporal Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 h E;
6.5.3 From Exposure Fluctuations to Line Position Fluctuations . . . . . . . . . . . . . . 202 Bn.8wMB
7 Speckle in Certain Non-imagingModalities 205 8WvT0q>]
7.1 Speckle in Multimode Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 ~Oq(JM
$M
7.1.1 Modal Noise in Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 K
\O ,AE
7.1.2 Statistics of Constrained Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 0wx`y$~R
7.1.3 Frequency Dependence of Modal Noise . . . . . . . . . . . . . . . . . . . . . . . . 211 S&a 44i
7.2 Effects of Speckle on Optical Radar Performance . . . . . . . . . . . . . . . . . . . . . . . 216 8WAg{lVs
7.2.1 Spatial Correlation of the Speckle Returned from Distant Targets . . . . . . . . . . . 217 'F@'4[uda
7.2.2 Speckle at Low Light Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 76
y}1aa
7.2.3 Detection Statistics—Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . 222 6wGf47
7.2.4 Detection Statistics— Heterodyne Detection . . . . . . . . . . . . . . . . . . . . . 227 &0
@2JS/!
7.2.5 Comparison of Direct Detection and Heterodyne Detection . . . . . . . . . . . . . . 234 =0Nd\
7.2.6 Reduction of the Effects of Speckle in Optical Radar Detection . . . . . . . . . . . . 235 bNXT*HOZb3
7.3 Speckle and Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 /as1
8 Speckle in Imaging Through the Atmosphere 239 qZ4DO*%b3
8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 TY? F s-
8.1.1 Refractive Index Fluctuations in the Atmosphere . . . . . . . . . . . . . . . . . . . 239 P%1s6fjU
8.2 Short-Exposure and Long-Exposure Point-Spread Functions . . . . . . . . . . . . . . . . . 240 ZY83, :<
8.3 Long-Exposure and Short-Exposure Average Optical Transfer Functions . . . . . . . . . . . 242 BM3nZ<%3
8.4 Statistical Properties of the Short-Exposure OTF and MTF . . . . . . . . . . . . . . . . . . 243 /t816,i
8.5 Astronomical Speckle Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 [j5L}e!T
8.5.1 Object Information that Is Retrievable . . . . . . . . . . . . . . . . . . . . . . . . . 248 GR"Jk[W9
8.5.2 Results of a More Complete Analysis of the Form of the Speckle Transfer Function . 250 x{=t y*E
8.6 The Cross-Spectrum or Knox–Thompson Technique . . . . . . . . . . . . . . . . . . . . . 252 ;&iQNXL
8.6.1 The Cross-Spectrum Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . 253 1e } wDMU(
8.6.2 Recovering Full Object Information from the Cross-Spectrum . . . . . . . . . . . . 254 3N;X|pa
8.7 The Bispectrum Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 6%)dsTAB
8.7.1 The Bispectrum Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 5@i/4%S
8.7.2 Recovering Full Object Information from the Bispectrum . . . . . . . . . . . . . . . 258 ]g: VvTJ;?
8.8 Speckle Correlography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 mX
%;
A Linear Transformations 263 [ z?<'Tj
B Contrast of Partially Developed Speckle 267 #SO9e.yhI
C Statistics of Derivatives of Speckle 271 NbSwn }e_
C.1 The Correlation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 I &4|T<j
C.2 Joint Density Function of the Derivatives of Phase . . . . . . . . . . . . . . . . . . . . . . . 274 Nl1& na)K}
C.3 Joint Density Function of the Derivatives of Intensity . . . . . . . . . . . . . . . . . . . . . 274 !.9NJ2'8
D Wavelength and Angle Dependence 277 gz eG5p
D.1 Free-Space Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Oq[t gmf
D.2 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 d aaurT
E Speckle Contrast when a Dynamic Diffuser is Projected onto a Random Screen 285 O4 [[9
E.1 Random Phase Diffusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 y6$5meh.T
E.2 Diffuser that Just Fills the Projection Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 287 `/"TYR%
E.3 Diffuser Overfills the Projection Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 l r K5q
F Statistics of Constrained Speckle 289 n:HF&j4C,
Bibliography 291 kYx|`-PA<r
Index 299
