Abstract

The mean, the variance, and the correlation function of the intensity pattern resulting from an incoherent source that has propagated through a complex ABCD optical system are derived. The number of speckle correlation cells (or modes, N) within an effective measurement area is presented and discussed. Optical field decorrelation effects with respect to secondary fringe formation in phase-shifting speckle interferometry are discussed. In particular, the correlation coefficient resulting from load-induced object tilt, in-plane translations, and displacements parallel to the optic axis as a function of the number of modes propagating through the optical system is derived and discussed. In contrast to previous research, in which the calculation of speckle interferometric decorrelation effects was restricted to direct imaging systems and the special case where N → ∞, the correlation coefficients that we derive are valid for arbitrary complex ABCD systems (i.e., nonimaging configurations) and for all finite values of N ≥ 1.

© 1993 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  2. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]
  3. H. T. Yura, S. G. Hanson, “Second-order statistics for wave propagation through complex optical systems,” J. Opt. Soc. Am. A 6, 564–575 (1989).
    [CrossRef]
  4. K. Creath, “Phaseshifting speckle interferometry,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 337–346 (1985).
    [CrossRef]
  5. M. Owner-Petersen, “Decorrelation and fringe visibility: on the limiting behavior of various electronic speckle-pattern correlation interferometers,” J. Opt. Soc. Am. A 8, 1082–1089 (1991).
    [CrossRef]
  6. J. W. Goodman, Laser Speckle and Related Phenomenon, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975).
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).
  8. H. J. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271–276 (1972).
    [CrossRef]

1991

1989

1987

1972

H. J. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271–276 (1972).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Creath, K.

K. Creath, “Phaseshifting speckle interferometry,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 337–346 (1985).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Laser Speckle and Related Phenomenon, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975).

Hanson, S. G.

Owner-Petersen, M.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Tiziani, H. J.

H. J. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271–276 (1972).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Yura, H. T.

J. Opt. Soc. Am. A

Opt. Commun.

H. J. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5, 271–276 (1972).
[CrossRef]

Other

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

J. W. Goodman, Laser Speckle and Related Phenomenon, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

K. Creath, “Phaseshifting speckle interferometry,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 337–346 (1985).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Number of intensity correlation cells for imaging and Fourier transform geometries contained in the measurement aperture as a function of the aperture radius (normalized to the lateral coherence length in the observation plane) for various values of σa. For the imaging and Fourier transform geometries, the quantities ρ0 and ω are given by Eqs. (18) and (19) and Eqs. (23) and (24), respectively.

Fig. 2
Fig. 2

Schematic representation of a clean imaging optical system.

Fig. 3
Fig. 3

Magnitude of the correlation coefficient for imaging systems as a function of normalized deformations for various values of N. For in-plane tilt and in-plane displacement, x = θ/θt and Δr/rd, respectively, where θt and rd are the respective decorrelation parameters in Table 1 for N → ∞ (e.g., for the clean imaging system, θ t = σ / 2 f 1 β and r d = 2 2 f 1 / k σ).

Fig. 4
Fig. 4

Magnitude of the correlation coefficient for a clean imaging system as a function of Δz/z0 for various values of N, where Δz is the deformation along the z axis and z0 = 4f12/2 is the depth of field.

Tables (1)

Tables Icon

Table 1 Magnitude of the Correlation Coefficient As a Function of the Number of Modes N for Three Types of Load-Induced Deformationa

Equations (67)

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U ( p ) = ( i k t 0 2 π B ) exp ( iKL ) d 2 r U s ( r ) × exp [ i k 2 B ( D p 2 2 r · p + A r 2 ) ] ,
I M = 1 S d 2 p W ( p ) I ( p ) ,
I M = 1 S d 2 p W ( p ) I ( p ) ,
I ( p ) = 1 | B | 2 exp [ k Im ( D / B ) p 2 ] d 2 r I s ( r ) × exp [ 2 k Im ( 1 / B ) p · r ] × exp [ k Im ( A / B ) r 2 ] ,
I M 2 = 1 S 2 d 2 p 1 d 2 p 2 W ( p 1 ) W ( p 2 ) I ( p 1 ) I ( p 2 ) .
I ( p 1 ) I ( p 2 ) = I ( p 1 ) I ( p 2 ) [ 1 + | γ ( p 1 , p 2 ) | 2 ] ,
| γ ( p 1 , p 2 ) | = | Γ ( p 1 , p 2 ) | [ I ( p 1 ) I ( p 2 ) ] 1 / 2
Γ ( p 1 , p 2 ) = U ( p 1 ) U * ( p 2 ) = | i k 2 π B | 2 exp [ i k 2 ( D B p 1 2 D * B * p 2 2 ) ] × d 2 r I s ( r ) × exp [ i k r · ( p 1 B p 2 B * ) ] exp [ k Im ( A / B ) r 2 ] .
N = I M 2 Var ( I M ) = I M 2 I M 2 I M 2 .
N = [ W ( p ) I ( p ) d 2 p ] 2 W ( p 1 ) W ( p 2 ) I ( p 1 ) I ( p 2 ) | γ ( p 1 , p 2 ) | 2 d 2 p 1 d 2 p 2 ,
Γ ( p 1 , p 2 ) = 2 I 0 k 2 ρ 0 2 exp [ ( p 1 p 2 ) 2 ρ 0 2 ] exp [ ( p 1 2 + p 2 2 ) ω 2 ] × exp { i [ Im ( B ) 2 ρ 0 2 | B | 2 k 2 Re ( D / B ) } ( p 1 2 p 2 2 ) } ,
ρ 0 2 = 8 | B | 2 k 2 r s 2 + 4 k Im ( B A * ) ,
ω 2 = ρ 0 2 { | B | 2 2 [ Im ( B D * ) Im ( B A * ) ( Im B ) 2 ] + 4 | B | 2 k r s 2 Im ( B D * ) } .
I ( p ) = 2 I 0 k 2 ρ 0 2 exp [ 2 p 2 / ω 2 ] ,
| γ | = exp [ ( p 1 p 2 ) 2 / ρ 0 2 ] .
N = 1 + 2 ω 2 ρ 0 2 .
A = L 2 L 1 2 i L 2 k σ 2 ,
B = 2 i L 1 L 2 k σ 2 ,
D = L 1 L 2 i L 1 σ 2 4 L 1 L 2 ( k σ a σ ) 2 .
ρ 0 2 = 8 L 2 2 k 2 σ 2 ( 1 + 4 L 1 2 k 2 r s 2 σ 2 ) ,
ω 2 = ( 1 σ a 2 + 1 ω 2 ) 1 ,
ω = [ ( L 2 L 1 ) 2 r s 2 + ( 2 L 2 k σ ) 2 ] 1 / 2
N = 1 + 2 ( σ a / ρ 0 ) 2 1 + ( σ a / ω ) 2
A = 2 i f k σ 2 ,
B = f 2 f 2 i k σ 2 ,
D = 4 f 2 ( k σ σ a ) 2 2 f i k ( 1 σ 2 + 1 σ a 2 ) .
ρ 0 2 = 8 f 2 k 2 r s 2 [ 1 + ( 2 f k σ 2 ) 2 ] + 8 f 2 k 2 σ 2 ;
ω = [ r s 2 + σ 2 + ( 2 f k σ ) 2 ] 1 / 2 .
N max = 1 + | B | 2 Im ( B A * ) Im ( B D * ) ( Im B ) 2 .
N max = 1 + ( k σ σ a 2 d ) 2 ,
( I a I b ) 2 = I a 2 + I b 2 2 I a I b ,
I = | U 1 + U 2 | 2 = I 1 + I 2 + 2 Re ( U 1 U 2 * ) ,
γ = U b U a * | U a | 2 1 / 2 | U b | 2 1 / 2 ,
U a ( r ) = exp ( i k β θ · r ) U b ( r ) ,
β = ( 1 + cos α ) ,
γ = | γ | exp ( i Φ t ) ,
| γ | = exp ( θ 2 / θ t 2 ) ,
θ t = ρ 0 2 | B | β ,
Φ t = 4 β Im ( B ) p x θ ρ 0 2 .
U a ( r ) = U b ( r Δ r ) .
γ = | γ | exp ( i Φ ) exp [ i ( k i k o ) · Δ r ] ,
| γ | = exp [ ( Δ r ) 2 r d 2 ] ,
r d = ρ 0 | A | ,
Φ = c 1 Δ r 2 + c 2 p · Δ r
c 1 = 4 Re ( B A * ) k r s 2 p 0 2
c 2 = k Re ( 1 / B ) 4 Im ( 1 / B ) Re ( B A * ) ρ 0 2 .
[ Ã B C D ] = [ A B C D ] [ 1 Δ z 0 1 ] .
γ = a exp [ i ( k i k o ) z Δ z ] × exp [ i k [ Re ( D / B ) Re ( D / B ) ] p 2 / 2 ] × exp [ c 3 p 2 / ω 2 ] ,
a = ρ 0 a ρ 0 b | ρ 0 2 | ,
ρ 0 2 = ρ 0 a 2 + 2 Δ z [ 4 A B * ( k r s ) 2 i | A | 2 k ] ,
ρ 0 = ( 2 f 2 k σ ) ( 2 N N 1 ) 1 / 2 ,
ω = ( 2 f 2 k σ ) N 1 / 2 ,
N = 1 + ( k σ r s 2 f 1 ) 2 .
ρ 0 = ( 2 f k σ ) ( 2 N N 1 ) 1 / 2 [ 1 + ( k σ 2 / 2 f ) 2 N ] ,
ω 2 = ( 2 f k σ ) 2 N + σ 2 ,
r d = 2 2 f 1 k σ ( N N 1 ) 1 / 2
θ t = 1 2 ( σ L 1 ) ( N N 1 ) 1 / 2 β 1 ,
r d = 2 2 L 1 k σ ( N N 1 ) 1 / 2 × [ 1 + ( 2 L 1 / k σ 2 ) 2 ] 1 / 2
θ t = 1 2 ( σ f ) ( N N 1 ) β 1 × [ 1 + ( k σ 2 / 2 f ) 2 / N 1 + ( k σ 2 / 2 f ) 2 ] 1 / 2
r d = 2 σ ( N N 1 ) 1 / 2 × [ 1 + ( k σ 2 / 2 f ) 2 / N ]
a = { 1 + ( N 1 N ) 2 ( Δ z / k σ 2 1 + ( k σ 2 / 2 f ) 2 / N ) 2 } 1 / 2
A ρ ( q ) = d 2 r ρ ( r ) exp ( i q · r ) ,
q = k r p L ,
A ρ ( q 1 ) A ρ * ( q 2 ) = constant δ ( q 1 q 2 ) .
A ρ ( q 1 ) A ρ * ( q 2 ) = d 2 r 1 d 2 r 2 ρ ( r 1 ) ρ * ( r 2 ) × exp [ i ( q 1 · r 1 q 2 · r 2 ) ] .
A ρ ( q 1 ) A ρ * ( q 2 ) = constant d 2 r | ρ ( r ) | 2 × exp [ i ( q 1 q 2 ) · r ] .
| ( q 1 q 2 ) · r | max k D r s L 1 ,

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