Abstract

The average three-dimensional size of speckles produced by the illumination of optically rough surfaces with coherent laser light is determined from the autocorrelation function of intensity in space. Analytical and numerical results are given for rectangular and circular diffuser apertures. Important special cases are discussed and compared with available data from the literature. Several of the derived properties of the speckle patterns are stated for the first time and could lead to a greater accuracy of different speckle-interferometric measurement techniques.

© 1990 Optical Society of America

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References

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  1. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984).
  2. M. Françon, Laser Speckle and Applications in Optics (Academic, New York, 1978).
  3. E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
    [CrossRef]
  4. H. J. Tiziani, “Vibration analysis and deformation measurement,” in Speckle Metrology, R. K. Erf, ed. (Academic, New York, 1978).
    [CrossRef]
  5. D. W. Li, F. P. Chiang, “Decorrelation functions in laser speckle photography,” J. Opt. Soc. Am. A 3, 1023–1031 (1986).
    [CrossRef]
  6. B. Eliasson, F. M. Mottier, “Determination of the granular radiance distribution of a diffuser and its use for vibration analysis,” J. Opt. Soc. Am. 61, 559–565 (1971).
    [CrossRef]
  7. G. P. Weigelt, B. Stoffregen, “The longitudinal correlation of three-dimensional speckle intensity distribution,” Optik 48, 399–407 (1977).
  8. A. W. Lohmann, G. P. Weigelt, “The measurement of depth motion by speckle photography,” Opt. Commun. 17, 47–51 (1976).
    [CrossRef]
  9. J. Ohtsubo, “Statistics of speckle intensity produced by the longitudinal motion of a diffuse object,” Optik 57, 183–189 (1980).
  10. I. V. Markhvida, L. V. Tanin, “Correlation of speckle patterns produced by the longitudinal motion of a diffuse object along the optical axis,” Optik 72, 168–170 (1986).
  11. Y. Dzialowski, M. May, R. Shaw, “Measurement of axial displacements undergone by a diffusing object in speckle photography,” Opt. Commun. 21, 282–288 (1977).
    [CrossRef]
  12. C. S. Narayanamurthy, C. Joenathan, “Speckle pattern fringes produced by longitudinal motion of the diffuse object— sensitivity dependence and multiple exposures,” Opt. Commun. 65, 179–184 (1988).
    [CrossRef]
  13. I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
    [CrossRef]
  14. H. M. Pedersen, “Intensity correlation metrology: a comparative study,” Opt. Acta 29, 105–118 (1982).
    [CrossRef]
  15. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, New York, 1972).
  16. M. Born, E. Wolf, Principles of Optics, 5th ed. (Oxford U. Press, Oxford, 1975).
  17. B. R. A. Nijboer, “The diffraction theory of optical aberrations,” Physica 13, 605–620 (1947).
    [CrossRef]
  18. S. Cornbleet, Microwave Optics (Academic, London, 1976).

1988

C. S. Narayanamurthy, C. Joenathan, “Speckle pattern fringes produced by longitudinal motion of the diffuse object— sensitivity dependence and multiple exposures,” Opt. Commun. 65, 179–184 (1988).
[CrossRef]

1986

I. V. Markhvida, L. V. Tanin, “Correlation of speckle patterns produced by the longitudinal motion of a diffuse object along the optical axis,” Optik 72, 168–170 (1986).

D. W. Li, F. P. Chiang, “Decorrelation functions in laser speckle photography,” J. Opt. Soc. Am. A 3, 1023–1031 (1986).
[CrossRef]

1982

H. M. Pedersen, “Intensity correlation metrology: a comparative study,” Opt. Acta 29, 105–118 (1982).
[CrossRef]

1980

J. Ohtsubo, “Statistics of speckle intensity produced by the longitudinal motion of a diffuse object,” Optik 57, 183–189 (1980).

1977

Y. Dzialowski, M. May, R. Shaw, “Measurement of axial displacements undergone by a diffusing object in speckle photography,” Opt. Commun. 21, 282–288 (1977).
[CrossRef]

G. P. Weigelt, B. Stoffregen, “The longitudinal correlation of three-dimensional speckle intensity distribution,” Optik 48, 399–407 (1977).

1976

A. W. Lohmann, G. P. Weigelt, “The measurement of depth motion by speckle photography,” Opt. Commun. 17, 47–51 (1976).
[CrossRef]

1972

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

1971

1962

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

1947

B. R. A. Nijboer, “The diffraction theory of optical aberrations,” Physica 13, 605–620 (1947).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, New York, 1972).

Archbold, E.

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Oxford U. Press, Oxford, 1975).

Chiang, F. P.

Cornbleet, S.

S. Cornbleet, Microwave Optics (Academic, London, 1976).

Dzialowski, Y.

Y. Dzialowski, M. May, R. Shaw, “Measurement of axial displacements undergone by a diffusing object in speckle photography,” Opt. Commun. 21, 282–288 (1977).
[CrossRef]

Eliasson, B.

Ennos, A. E.

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

Françon, M.

M. Françon, Laser Speckle and Applications in Optics (Academic, New York, 1978).

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984).

Joenathan, C.

C. S. Narayanamurthy, C. Joenathan, “Speckle pattern fringes produced by longitudinal motion of the diffuse object— sensitivity dependence and multiple exposures,” Opt. Commun. 65, 179–184 (1988).
[CrossRef]

Li, D. W.

Lohmann, A. W.

A. W. Lohmann, G. P. Weigelt, “The measurement of depth motion by speckle photography,” Opt. Commun. 17, 47–51 (1976).
[CrossRef]

Markhvida, I. V.

I. V. Markhvida, L. V. Tanin, “Correlation of speckle patterns produced by the longitudinal motion of a diffuse object along the optical axis,” Optik 72, 168–170 (1986).

May, M.

Y. Dzialowski, M. May, R. Shaw, “Measurement of axial displacements undergone by a diffusing object in speckle photography,” Opt. Commun. 21, 282–288 (1977).
[CrossRef]

Mottier, F. M.

Narayanamurthy, C. S.

C. S. Narayanamurthy, C. Joenathan, “Speckle pattern fringes produced by longitudinal motion of the diffuse object— sensitivity dependence and multiple exposures,” Opt. Commun. 65, 179–184 (1988).
[CrossRef]

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of optical aberrations,” Physica 13, 605–620 (1947).
[CrossRef]

Ohtsubo, J.

J. Ohtsubo, “Statistics of speckle intensity produced by the longitudinal motion of a diffuse object,” Optik 57, 183–189 (1980).

Pedersen, H. M.

H. M. Pedersen, “Intensity correlation metrology: a comparative study,” Opt. Acta 29, 105–118 (1982).
[CrossRef]

Reed, I. S.

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

Shaw, R.

Y. Dzialowski, M. May, R. Shaw, “Measurement of axial displacements undergone by a diffusing object in speckle photography,” Opt. Commun. 21, 282–288 (1977).
[CrossRef]

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, New York, 1972).

Stoffregen, B.

G. P. Weigelt, B. Stoffregen, “The longitudinal correlation of three-dimensional speckle intensity distribution,” Optik 48, 399–407 (1977).

Tanin, L. V.

I. V. Markhvida, L. V. Tanin, “Correlation of speckle patterns produced by the longitudinal motion of a diffuse object along the optical axis,” Optik 72, 168–170 (1986).

Tiziani, H. J.

H. J. Tiziani, “Vibration analysis and deformation measurement,” in Speckle Metrology, R. K. Erf, ed. (Academic, New York, 1978).
[CrossRef]

Weigelt, G. P.

G. P. Weigelt, B. Stoffregen, “The longitudinal correlation of three-dimensional speckle intensity distribution,” Optik 48, 399–407 (1977).

A. W. Lohmann, G. P. Weigelt, “The measurement of depth motion by speckle photography,” Opt. Commun. 17, 47–51 (1976).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Oxford U. Press, Oxford, 1975).

IRE Trans. Inf. Theory

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory IT-8, 194–195 (1962).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

H. M. Pedersen, “Intensity correlation metrology: a comparative study,” Opt. Acta 29, 105–118 (1982).
[CrossRef]

Opt. Commun.

A. W. Lohmann, G. P. Weigelt, “The measurement of depth motion by speckle photography,” Opt. Commun. 17, 47–51 (1976).
[CrossRef]

Y. Dzialowski, M. May, R. Shaw, “Measurement of axial displacements undergone by a diffusing object in speckle photography,” Opt. Commun. 21, 282–288 (1977).
[CrossRef]

C. S. Narayanamurthy, C. Joenathan, “Speckle pattern fringes produced by longitudinal motion of the diffuse object— sensitivity dependence and multiple exposures,” Opt. Commun. 65, 179–184 (1988).
[CrossRef]

Optik

J. Ohtsubo, “Statistics of speckle intensity produced by the longitudinal motion of a diffuse object,” Optik 57, 183–189 (1980).

I. V. Markhvida, L. V. Tanin, “Correlation of speckle patterns produced by the longitudinal motion of a diffuse object along the optical axis,” Optik 72, 168–170 (1986).

G. P. Weigelt, B. Stoffregen, “The longitudinal correlation of three-dimensional speckle intensity distribution,” Optik 48, 399–407 (1977).

Physica

B. R. A. Nijboer, “The diffraction theory of optical aberrations,” Physica 13, 605–620 (1947).
[CrossRef]

Other

S. Cornbleet, Microwave Optics (Academic, London, 1976).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984).

M. Françon, Laser Speckle and Applications in Optics (Academic, New York, 1978).

H. J. Tiziani, “Vibration analysis and deformation measurement,” in Speckle Metrology, R. K. Erf, ed. (Academic, New York, 1978).
[CrossRef]

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, New York, 1972).

M. Born, E. Wolf, Principles of Optics, 5th ed. (Oxford U. Press, Oxford, 1975).

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Figures (4)

Fig. 1
Fig. 1

Common free-space diffuser geometry with coordinate systems used in Section 2.

Fig. 2
Fig. 2

Correlation coefficient for a circular aperture with radius r0. The contour lines show μI(γ, δ, ∊; x, y, z) for (a) δ = 0, x = y = 0, z = 20r0 and (b) δ = 0, x = 0.2z, y = 0, z = 20r0. The contours 0.5 and 1/e2 are emphasized.

Fig. 3
Fig. 3

Correlation coefficient for a quadratic aperture of dimensions Dx = Dy = D. The contour lines show μI(γ, δ, ∊; x, y, z) with δ = 0, x = y = 0, z = 10D. The contours 0.5 and 1/e2 are emphasized.

Fig. 4
Fig. 4

Correlation coefficient for a quadratic aperture of dimensions Dx = Dy = D. The three-dimensional plot shows the cross section μI(γ, δ, ∊; x, y, z) with x = y = 0, z = 10D, ∊ = 1. The peak height of the main lobe is 0.16, and that of highest sidelobes is 0.096.

Equations (50)

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A ( r ) = ( 1 / i λ ) α ( p ) z | r p | exp ( i k | r p | ) | r p | d ξ d η ,
R I ( r , r ) = I ( r ) I ( r ) = A ( r ) A * ( r ) A ( r ) A * ( r ) ,
R I ( r , r ) = I ( r ) I ( r ) + | J A ( r , r ) | 2 .
J A ( r , r ) = A ( r ) A * ( r )
J A ( r , r ) = ( z / λ ) 2 J α ( p 1 , p 2 ) × exp [ i k ( | r p 1 | | r p 2 | ) ] | r p 1 | 2 | r p 2 | 2 d ξ 1 d η 1 d ξ 2 d η 2 .
J α ( p 1 , p 2 ) = α ( p 1 ) α * ( p 2 ) .
J α ( p 1 , p 2 ) = κ u ( p 1 ) u * ( p 2 ) P ( p 1 ) P ( p 2 ) δ ( p 1 p 2 ) ,
J A ( r , r ) = κ ( z / λ ) 2 | u ( p ) | 2 | P ( p ) | 2 × exp [ i k ( | r p | ) | r p | ] | r p | 2 | r p | 2 d ξ d η .
μ A ( r , r ) = J A ( r , r ) [ J A ( r , r ) J A ( r , r ) ] 1 / 2
R 1 ( r , r ) = I ( r ) I ( r ) [ 1 + | μ A ( r , r ) | 2 ] .
μ I ( r , r ) = | | P ( p ) | 2 exp [ i k ( | r p | | r p | ) ] d ξ d η | P ( p ) | 2 d ξ d η | 2 .
| r p | 2 | r p | 2 z 2 ,
p = ( ξ , η ) , r = ( x , y , z ) , r = ( x + γ , y + δ , z + )
μ I ( r , r ) = | | P ( ξ , η ) | 2 exp [ i τ ( ξ 2 + η 2 ) exp [ i ( α ξ + β η ) ] d ξ d η | P ( ξ , η ) | 2 d ξ d η | 2 ,
τ = k / 2 z 2 = π / z 2 , α = 2 k ( z γ x ) / 2 z 2 = ( 2 π / z 2 ) ( z γ x ) , β = 2 k ( z δ y ) / 2 z 2 = ( 2 π / z 2 ) ( z δ y ) ,
μ I ( γ , δ , ; x , y , z ) = μ I ( γ γ 0 , δ δ 0 , ; 0 , 0 , z ) ,
P ( ξ , η ) = rect ( ξ / D x ) rect ( η / D y )
| P ( ξ , η ) | 2 d ξ d η = D x D y
a = α / 2 π τ = 2 / ( γ x / z ) , b = τ D x / 2 π τ = / 2 ( D x / z ) , c = β / 2 π τ = z / ( δ y / z ) , d = τ D y / 2 π τ = / 2 ( D y / z ) ,
μ I = ( 1 / 16 b 2 d 2 ) { [ C ( a + b ) C ( a b ) ] 2 + [ S ( a + b ) S ( a b ) ] 2 } { [ C ( c + d ) C ( c d ) ] 2 [ S ( c + d ) S ( c d ) ] 2 } ,
C ( z ) = 0 z cos ( π t 2 / 2 d t , S ( z ) = 0 z sin ( π t 2 / 2 ) d t
C ( z ) = 1 / 2 + ( 1 / π z ) sin ( π z 2 / 2 ) , S ( z ) = 1 / 2 ( 1 / π z ) cos ( π z 2 / 2 ) ,
μ I = sin 2 ( π a b ) ( π a b ) 2 sin 2 ( π c d ) ( π c d ) 2 = [ sin ( γ D x / z ) sinc ( δ D y / z ) ] 2 .
( x 2 + y 2 ) / z 2 = ( γ 2 + δ 2 ) / 2
C ( z ) = C ( z ) , S ( z ) = S ( z ) ,
μ I = C 2 ( b ) + S 2 ( b ) b 2 C 2 ( d ) + S 2 ( d ) d 2 ,
μ A = [ C 2 ( b ) + S 2 ( b ) b 2 ] .
s z = 7.31 λ ( z / D ) 2 .
s r = s z r / z = 7.31 λ ( z / D 2 ) x 2 + y 2 + z 2 .
μ I = C 2 ( 2 b ) + S 2 ( 2 b ) 4 b 2 C 2 ( 2 d ) + S 2 ( 2 d ) 4 d 2 ,
s e = s r / 4 = ( 7.31 / 4 ) λ ( z / D 2 ) x 2 + y 2 + z 2 .
P ( t , ψ ) = circ ( t / r 0 ) ,
μ I ( r , r ) = 1 ( π r 0 2 ) 2 | 0 2 π 0 r 0 exp ( i τ t 2 ) exp { it [ α cos ( ψ ) + β sin ( ψ ) ] } t d t d ψ | 2 .
μ I = | 2 0 1 exp [ i ( u / 2 ) s 2 ] J 0 ( υ s ) s d s | 2 ,
s = r 0 t , u = 2 r 0 2 τ , υ = r 0 α 2 + β 2 .
μ I = ( 4 / u 2 ) [ U 1 2 ( u , υ ) + U 2 2 ( u , υ ) ]
μ I = ( 4 / u 2 ) { 1 + V 0 2 ( u , υ ) + V 1 2 ( u , υ ) + 2 V 0 ( u , υ ) × cos [ 1 / 2 ( u + υ 2 / u ) 2 V 1 ( u , υ ) sin [ 1 / 2 ( u + υ 2 / u ) ] } .
μ I ( r , r ) = | 2 n = 0 i n ( 2 n + 1 ) j n ( u / 4 ) J 2 n + 1 ( υ ) / υ | 2 ,
j n ( z ) = π / 2 z J n + 1 / 2 ( z )
j n ( 0 ) = { 1 n = 0 0 n 0 ,
μ I ( γ , δ , 0 ; x , y , z ) = | 2 J 1 ( υ ) / υ | 2 ,
υ = 2 π ( r 0 / z ) γ 2 + δ 2 = : π ( D 0 / z ) ρ .
s ρ = 1.22 λ ( z / D 0 )
lim υ 0 J m ( υ ) υ = { 1 / 2 m = 1 0 m > 1
j 0 ( z ) = sin ( z ) / z ,
μ I = | sinc ( u / 4 π ) | 2 = | sinc ( D 0 2 / 8 z 2 ) | 2 .
s z = 8 λ ( z / D 0 ) 2 .
μ I = ( 1 / u 2 ) [ 1 2 cos ( u ) J 0 ( u ) + J 0 2 ( u ) ] .
s e = 2.29 λ ( z / D 0 2 ) x 2 + y 2 + z 2 .
s z = 7.31 2 λ z 2 / D x 4 + D y 4 .

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