Abstract

An analysis of the second-order conditional statistics of speckle patterns is developed, under the assumption of Gaussian field components. After deriving the conditional distributions and moments of intensity and phase for a joint measurement performed in the presence of partial correlation, the results of joint and conditional phase-variances are applied to determine the accuracy limits of speckle-pattern interferometric measurements.

© 1979 Optical Society of America

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References

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  1. J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
    [Crossref]
  2. J. W. Goodman, “Statistical properties of laser speckle pattern,” in Laser Speckle and Related Phenomena, Topics in Applied Phys., Vol. 9, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 9–75.
    [Crossref]
  3. D. Middleton: Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  4. J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
    [Crossref]
  5. J. S. Bendat and A. G. Piersol, Random Data (Wiley Interscience, New York, 1971).
  6. R. J. Mongeon, “Laser Vibration Probes,” CLEA Digest 1977, in IEEE J. Quant. Electron,  QE-13, 798ff. 16D (1977).
  7. R. O. Claus and C. H. Palmer, “Direct measurement of ultrasonic Stoneley waves,” Appl. Phys. Lett. 31, 547–548 (1977).
    [Crossref]
  8. S. Donati, “Laser interferometry by induced modulation of cavity field,” J. Appl. Phys. 49(2), 495–497 (1978).
    [Crossref]
  9. T. S. Trowbridge, “Retroreflection from rough surfaces,” J. Opt. Soc. Am. 68, 1225–1242 (1978).
    [Crossref]

1978 (2)

S. Donati, “Laser interferometry by induced modulation of cavity field,” J. Appl. Phys. 49(2), 495–497 (1978).
[Crossref]

T. S. Trowbridge, “Retroreflection from rough surfaces,” J. Opt. Soc. Am. 68, 1225–1242 (1978).
[Crossref]

1977 (2)

R. J. Mongeon, “Laser Vibration Probes,” CLEA Digest 1977, in IEEE J. Quant. Electron,  QE-13, 798ff. 16D (1977).

R. O. Claus and C. H. Palmer, “Direct measurement of ultrasonic Stoneley waves,” Appl. Phys. Lett. 31, 547–548 (1977).
[Crossref]

1976 (1)

1975 (1)

J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
[Crossref]

Bendat, J. S.

J. S. Bendat and A. G. Piersol, Random Data (Wiley Interscience, New York, 1971).

Claus, R. O.

R. O. Claus and C. H. Palmer, “Direct measurement of ultrasonic Stoneley waves,” Appl. Phys. Lett. 31, 547–548 (1977).
[Crossref]

Donati, S.

S. Donati, “Laser interferometry by induced modulation of cavity field,” J. Appl. Phys. 49(2), 495–497 (1978).
[Crossref]

Goodman, J. W.

J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
[Crossref]

J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
[Crossref]

J. W. Goodman, “Statistical properties of laser speckle pattern,” in Laser Speckle and Related Phenomena, Topics in Applied Phys., Vol. 9, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 9–75.
[Crossref]

Middleton, D.

D. Middleton: Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Mongeon, R. J.

R. J. Mongeon, “Laser Vibration Probes,” CLEA Digest 1977, in IEEE J. Quant. Electron,  QE-13, 798ff. 16D (1977).

Palmer, C. H.

R. O. Claus and C. H. Palmer, “Direct measurement of ultrasonic Stoneley waves,” Appl. Phys. Lett. 31, 547–548 (1977).
[Crossref]

Piersol, A. G.

J. S. Bendat and A. G. Piersol, Random Data (Wiley Interscience, New York, 1971).

Trowbridge, T. S.

Appl. Phys. Lett. (1)

R. O. Claus and C. H. Palmer, “Direct measurement of ultrasonic Stoneley waves,” Appl. Phys. Lett. 31, 547–548 (1977).
[Crossref]

IEEE J. Quant. Electron (1)

R. J. Mongeon, “Laser Vibration Probes,” CLEA Digest 1977, in IEEE J. Quant. Electron,  QE-13, 798ff. 16D (1977).

J. Appl. Phys. (1)

S. Donati, “Laser interferometry by induced modulation of cavity field,” J. Appl. Phys. 49(2), 495–497 (1978).
[Crossref]

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975).
[Crossref]

Other (3)

J. S. Bendat and A. G. Piersol, Random Data (Wiley Interscience, New York, 1971).

J. W. Goodman, “Statistical properties of laser speckle pattern,” in Laser Speckle and Related Phenomena, Topics in Applied Phys., Vol. 9, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 9–75.
[Crossref]

D. Middleton: Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

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Figures (5)

FIG. 1
FIG. 1

(a) Mean value 〈I2〉 and (b) standard deviation σ I 2 of speckle pattern intensity I2 conditioned on I1 as a function of the coherence factor μ.

FIG. 2
FIG. 2

Mean value of intensity I2 conditioned on intensity I1 and phase difference ϑ = ϑ1ϑ2 + φ, as a function of the coherence factor. On the abscissa, μ is multiplied by the sign of cosϑ to allow for any sign combination of μ and cosϑ.

FIG. 3
FIG. 3

Standard deviation of intensity I2 conditioned on I1 and ϑ versus the coherence factor.

FIG. 4
FIG. 4

Variance of the speckle phase ϑ2 conditioned on ϑ1. Thick line is for the free distribution (not conditioned on intensities); thin lines are for the variance conditioned also on intensities I1 and I2 entering as parameters.

FIG. 5
FIG. 5

Free and conditional phase standard deviation for coherence factors μ = 1 − ζ2 near unity, plotted against ζ.

Equations (22)

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p ( I 1 , I 2 , ϑ 1 , ϑ 2 ) = 1 16 π 2 σ 4 ( 1 μ 2 ) × exp ( I 1 + I 2 2 I 1 I 2 μ cos ( ϑ 1 ϑ 2 + φ ) 2 σ 2 ( 1 μ 2 ) ) ,
p ( I 1 , I 2 ) = 1 4 σ 4 ( 1 μ 2 ) × exp ( I 1 + I 2 2 σ 2 ( 1 μ 2 ) ) I 0 ( μ I 1 I 2 σ 2 ( 1 μ 2 ) ) ,
p ( ϑ 1 , ϑ 2 ) = 1 μ 2 4 π 2 1 β 2 + β arcsin β + ( π / 2 ) β ( 1 β 2 ) 3 / 2 ,
p ( I 2 | I 1 ) = 1 2 σ 2 ( 1 μ 2 ) × exp ( I 2 + μ 2 I 1 2 σ 2 ( 1 μ 2 ) ) I 0 ( μ I 1 I 2 σ 2 ( 1 μ 2 ) )
p ( ϑ 2 | ϑ 1 ) = p ( ϑ 2 , ϑ 1 ) 2 π .
I 2 = 2 σ 2 , σ I 2 2 = ( 2 σ 2 ) 2 , ϑ 2 = 0 , σ ϑ 2 2 = π 2 / 3 .
I 2 | I 1 = 2 σ 2 ( 1 μ 2 ) + μ 2 I 1 ,
σ I 2 2 | I 1 = [ 2 σ 2 ( 1 μ 2 ) ] 2 + 4 σ 2 μ 2 ( 1 μ 2 ) I 1 .
ϑ 2 | ϑ 1 = ϑ 1 + φ ( μ > 0 ) = ϑ 1 + φ π ( μ < 0 ) ,
P ( ϑ ) = π ϑ p ( ϑ ) d ϑ = 1 2 + ϑ 2 π + π / 2 + arcsin β 2 π 1 β 2 μ sin ϑ ,
σ ϑ 2 2 | ϑ 1 = π 2 3 π arcsin | μ | + arcsin 2 | μ | 1 2 n = 1 μ 2 n n 2 .
p ( I 2 | I 1 , ϑ 1 , ϑ 2 ) = [ 2 σ 2 ( 1 μ 2 ) ] 1 × exp ( I 2 2 μ I 1 I 2 cos ϑ 2 σ 2 ( 1 μ 2 ) ) D ( δ ) ,
1 / D ( δ ) = 1 + π δ exp ( δ 2 ) ( 1 + erf δ ) ,
I 2 | I 1 , ϑ = σ 2 ( 1 μ 2 ) [ 3 D ( δ ) + 2 δ 2 ] ,
σ I 2 2 | I 1 , ϑ = σ 4 ( 1 μ 2 ) 2 [ 6 ( 1 2 δ 2 ) D ( δ ) D 2 ( δ ) + 8 δ 2 ] ,
p ( ϑ 2 | I 1 , I 2 , ϑ 1 ) = 1 4 π 2 exp μ I 1 I 2 cos ( ϑ 1 ϑ 2 ) σ 2 ( 1 μ 2 ) / I 0 [ μ I 1 I 2 σ 2 ( 1 μ 2 ) ] ,
σ ϑ 2 2 | I 1 , I 2 , ϑ 1 = π 2 3 + 4 I 0 ( z ) n = 1 ( 1 ) n n 2 I n ( z ) ,
σ ϑ 2 = ζ 2 ( 3 ln 2 ζ 2 ) , σ ϑ 2 | I 1 I 2 = ζ 2 / ( I 1 I 2 / 2 σ 2 ) .
μ c = exp ( r 2 / 2 w 2 ) ,
NED = C 2 2 r k w .
μ c = exp { 2 i k ( z 2 z 1 ) + i arctan [ k ( z 2 z 1 ) w 2 / 4 z 2 z 1 ] } { 1 + [ k ( z 2 z 1 ) w 2 / 4 z 2 z 1 ] 2 } 1 / 2 ,
NED = C 8 2 ( z 2 z 1 ) w 2 z 2 z 1 .