Speckle Phenomena in Optics: Theory and Applications >rmqBDKaQ
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Joseph W. Goodman x}I+Iggi
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Contents \aUC(K~o\;
1 Origins and Manifestations of Speckle 1 z3m85F%dR
1.1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 (Y? gn)*t
1.2 Intuitive Explanation of the Cause of Speckle . . . . . . . . . . . . . . . . . . . . . . . . . 2 6@F9G4<Z
1.3 Some Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 ;)z:fToh
2 Random Phasor Sums 7 Em
!/a$
2.1 First and Second Moments of the Real and Imaginary Parts of the Resultant Phasor . . . . . 8 Y>dzR)~3[
2.2 Random Walk with a Large Number of Independent Steps . . . . . . . . . . . . . . . . . . 9 nuMD!qu!nZ
2.3 Random Phasor Sum Plus a Known Phasor . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Vl=l?A8
2.4 Sums of Random Phasor Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 m6\E$;`
2.5 Random Phasor Sums with a Finite Number of Equal-Length Components . . . . . . . . . 16 ND#Yenye
2.6 Random Phasor Sums with a Nonuniform Distribution of Phases . . . . . . . . . . . . . . . 17 jTtu0Q|
3 First-Order Statistical Properties of Optical Speckle 23 ;LPfXpR
3.1 Definition of Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 b)5uf'?-
3.2 First-Order Statistics of the Intensity and Phase . . . . . . . . . . . . . . . . . . . . . . . . 24 4ber!rJM
3.2.1 Large Number of Random Phasors . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 aU "8{
3.2.2 Constant Phasor plus a Random Phasor Sum . . . . . . . . . . . . . . . . . . . . . 27 IT7wT+
3.2.3 Finite Number of Equal-Length Phasors . . . . . . . . . . . . . . . . . . . . . . . . 31 yT"Eq"7/Y#
3.3 Sums of Speckle Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ;oKZ!ND
3.3.1 Sums on an Amplitude Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 /}fHt^2H
3.3.2 Sum of Two Independent Speckle Intensities . . . . . . . . . . . . . . . . . . . . . 34 (!7sE9rP
3.3.3 Sum of N Independent Speckle Intensities . . . . . . . . . . . . . . . . . . . . . . 37 2M#Q.F
3.3.4 Sums of Correlated Speckle Intensities . . . . . . . . . . . . . . . . . . . . . . . . 40 GxI!{oi2
3.4 Partially Polarized Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 y@: h4u"3
3.5 Partially Developed Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 #64-~NVL_
3.6 Speckled Speckle, or Compound Speckle Statistics . . . . . . . . . . . . . . . . . . . . . . 47 lH x^D;m6
3.6.1 Speckle Driven by a Negative Exponential Intensity Distribution . . . . . . . . . . . 48 $m{:C;UH
3.6.2 Speckle Driven by a Gamma Intensity Distribution . . . . . . . . . . . . . . . . . . 50 uLL]A>vR
3.6.3 Sums of Independent Speckle Patterns Driven by a Gamma Intensity Distribution . . 51 n&;85IF1
4 Higher-Order Statistical Properties of Optical Speckle 55 0$)>D==
4.1 Multivariate Gaussian Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Ky!Y"
4.2 Application to Speckle Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 i$:*Pb3mV
4.3 Multidimensional Statistics of Speckle Amplitude, Phase and Intensity . . . . . . . . . . . . 58 p{Yv3dNl
4.3.1 Joint Density Function of the Amplitudes . . . . . . . . . . . . . . . . . . . . . . . 59 ^Y>F|;M#
4.3.2 Joint Density Function of the Phases . . . . . . . . . . . . . . . . . . . . . . . . . . 60 L~rBAIdD
4.3.3 Joint Density Function of the Intensities . . . . . . . . . . . . . . . . . . . . . . . . 64 p;59?
4.4 Autocorrelation Function and Power Spectrum of Speckle . . . . . . . . . . . . . . . . . . . 66 ^w@%cVh
4.4.1 Free-Space Propagation Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ]}-7_n#cC
4.4.2 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 ^T;*M_
4.4.3 Speckle Size in Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 iohop(LZ
4.5 Dependence of Speckle on Scatterer Microstructure . . . . . . . . . . . . . . . . . . . . . . 77 kHghPn?8]
4.5.1 Surface vs. Volume Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 0w\zLU
4.5.2 Effect of a Finite Correlation Area of the ScatteredWave . . . . . . . . . . . . . . . 78 ~ Ei $nV
4.5.3 A Regime where Speckle Size Is Independent of Scattering Spot Size . . . . . . . . 81 g1/[eoZzk
4.5.4 Relation between the Correlation Areas of the ScatteredWave and the Surface Height `iAF3:
Fluctuations— Surface Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 83 h-#6av:
4.5.5 Dependence of Speckle Contrast on Surface Roughness— Surface Scattering . . . . 88 t7dt*D_YqK
4.5.6 Properties of Speckle Resulting from Volume Scattering . . . . . . . . . . . . . . . 92 Ustv{:7v
4.6 Statistics of Integrated and Blurred Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . 94 ,.83m%i
4.6.1 Mean and Variance of Integrated Speckle . . . . . . . . . . . . . . . . . . . . . . . 95 X<`
4.6.2 Approximate Result for the Probability Density Function of &Fzb6/
Integrated Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 @uqd.Q
4.6.3 “Exact” Result for the Probability Density Function of Integrated Intensity . . . . . 101 I {S;L
4.6.4 Integration of Partially Polarized Speckle Patterns . . . . . . . . . . . . . . . . . . . 106
nzuX&bSw
4.7 Statistics of Derivatives of Speckle Intensity and Phase . . . . . . . . . . . . . . . . . . . . 108 1MP~dRZ$
4.7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 iZ3IdiZ
4.7.2 Parameters for Various Scattering Spot Shapes . . . . . . . . . . . . . . . . . . . . 110 DnMwUykF>0
4.7.3 Derivatives of Speckle Phase: Ray Directions in a Speckle Pattern . . . . . . . . . . 111 W#4 7h7M
4.7.4 Derivatives of Speckle Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 SIF/-{i(X
4.7.5 Level Crossings of Speckle Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 116 J{p1|+h%
4.8 Zeros of Speckle Patterns: Optical Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . 118 +qtJaYf/0
4.8.1 Conditions Required for a Zero of Intensity to Occur . . . . . . . . . . . . . . . . . 119 L3u&/Tn2
4.8.2 Properties of Speckle Phase in the Vicinity of a Zero of Intensity . . . . . . . . . . . 119 ^pAAzr"hv
4.8.3 The Density of Vortices in Fully Developed Speckle . . . . . . . . . . . . . . . . . 119 UUYSFa%
4.8.4 The Density of Vortices for Fully Developed Speckle Plus a Coherent Background . 123 axv>6k
5 OpticalMethods for Suppressing Speckle 125 n/;WxnnQ
5.1 Polarization Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 t9k zw*U9
5.2 Temporal Averaging with a Moving Diffuser . . . . . . . . . . . . . . . . . . . . . . . . . 127 W7R<