Abstract

Within the framework of complex ABCD-matrix theory, exact theoretical expressions are derived for the space–time-lagged cross-covariance functions of the fields valid for arbitrary (complex) ABCD-optical systems, i.e., systems that include Gaussian-shaped apertures and partially developed speckle. Specifically, we show and discuss the results for the three generic systems, i.e., free-space propagation, Fourier transform configuration, and imaging. To cope with various surface structures of varying rms surface heights, we apply two models in addition to employing the usual model for surfaces giving rise to fully developed speckle. The theoretical results found for free-space propagation are supported by interferometrically obtained data.

© 2006 Optical Society of America

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References

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  1. T. Asakura and N. Takai, "Dynamic laser speckles and their application to velocity measurements of the diffuse object," Appl. Phys. 25, 179-194 (1981).
    [CrossRef]
  2. H. T. Yura and S. G. Hanson, "Laser-time-of flight velocimetry: analytical solution to the optical system based on ABCD matrices," J. Opt. Soc. Am. A 10, 1918-1924 (1993).
    [CrossRef]
  3. B. Rose, H. Imam, S. G. Hanson, H. T. Yura, and R. S. Hansen, "Laser-speckle angular-displacement sensor: theoretical and experimental study," Appl. Opt. 37, 2119-2129 (1998).
    [CrossRef]
  4. H. T. Yura, B. Rose, and S. G. Hanson, "Dynamic laser speckle in complex ABCD optical systems," J. Opt. Soc. Am. A 15, 1160-1166 (1998).
    [CrossRef]
  5. T. Yoshimura, "Statistical properties of dynamic speckle," J. Opt. Soc. Am. A 3, 1032-1054 (1986).
    [CrossRef]
  6. J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1984), Chap. 2.
  7. W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson,, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120-127 (2006).
    [CrossRef] [PubMed]
  8. O. V. Angelsky, S. G. Hanson, and P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Monograph PM71 (SPIE Press, 1999).
  9. A. E. Siegman, Lasers (University Science, 1986), Chap. 20.
  10. H. T. Yura and S. G. Hanson, "Optical beam wave propagation through complex optical systems," J. Opt. Soc. Am. A 4, 1931-1948 (1987).
    [CrossRef]
  11. M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. 72, 156-160 (1982).
    [CrossRef]

2006 (1)

1999 (1)

O. V. Angelsky, S. G. Hanson, and P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Monograph PM71 (SPIE Press, 1999).

1998 (2)

1993 (1)

1987 (1)

1986 (2)

1984 (1)

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1984), Chap. 2.

1982 (1)

1981 (1)

T. Asakura and N. Takai, "Dynamic laser speckles and their application to velocity measurements of the diffuse object," Appl. Phys. 25, 179-194 (1981).
[CrossRef]

Angelsky, O. V.

O. V. Angelsky, S. G. Hanson, and P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Monograph PM71 (SPIE Press, 1999).

Asakura, T.

T. Asakura and N. Takai, "Dynamic laser speckles and their application to velocity measurements of the diffuse object," Appl. Phys. 25, 179-194 (1981).
[CrossRef]

Goodman, J. W.

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1984), Chap. 2.

Hansen, R. S.

Hanson, S. G.

Imam, H.

Ina, H.

Ishijima, R.

Kobayashi, S.

Maksimyak, P. P.

O. V. Angelsky, S. G. Hanson, and P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Monograph PM71 (SPIE Press, 1999).

Miyamoto, Y.

Rose, B.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986), Chap. 20.

Takai, N.

T. Asakura and N. Takai, "Dynamic laser speckles and their application to velocity measurements of the diffuse object," Appl. Phys. 25, 179-194 (1981).
[CrossRef]

Takeda, M.

Wada, A.

Wang, W.

Yokozeki, T.

Yoshimura, T.

Yura, H. T.

Appl. Opt. (1)

Appl. Phys. (1)

T. Asakura and N. Takai, "Dynamic laser speckles and their application to velocity measurements of the diffuse object," Appl. Phys. 25, 179-194 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Express (1)

Other (3)

O. V. Angelsky, S. G. Hanson, and P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Monograph PM71 (SPIE Press, 1999).

A. E. Siegman, Lasers (University Science, 1986), Chap. 20.

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1984), Chap. 2.

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Figures (5)

Fig. 1
Fig. 1

Real, imaginary, and absolute values of Γ 1 and Γ 2 for L = 1 m , k = 10 7 m 1 , Δ x = 150 μ m , and a spot radius r s = 0.45 mm ( a = 1 ) . The phase variance of the object was σ φ 2 = 1 .

Fig. 2
Fig. 2

Mach–Zehnder interferometer used for measuring the space–time-lagged field correlations. B S 1 and B S 2 , beam splitters; M O , microscope objective; L, Lens; M, mirror.

Fig. 3
Fig. 3

Measurements of Γ 1 for free-space propagation. (a) Abs [ Γ 1 ] , (b) Re [ Γ 1 ] , and (c) Im [ Γ 1 ] .

Fig. 4
Fig. 4

Measurements of Γ 2 for free-space propagation. (a) Abs [ Γ 2 ] , (b) Re [ Γ 2 ] , and (c) Im [ Γ 2 ] .

Fig. 5
Fig. 5

Theoretical covariances derived on the basis of the parameters used for the experimental findings of Figs. 3, 4.

Equations (44)

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U ( p , t ) = d r U 0 ( r , t ) G ( r , p ) ,
G ( r , p ) = i k 2 π B exp [ i k 2 B ( A r 2 2 r p + D p 2 ) ] .
Γ 1 ( p 1 p 2 , τ ) U ( p 1 , t ) U * ( p 2 , t + τ ) ,
Γ 2 ( p 1 p 2 , τ ) U ( p 1 , t ) U ( p 2 , t + τ ) ,
U 0 ( r 1 ) U 0 * ( r 2 ) = exp { i [ φ ( r 1 Δ r 0 ) φ ( r 2 ) ] } ,
U 0 ( r 1 ) U 0 ( r 2 ) = exp { i [ φ ( r 1 Δ r 0 ) + φ ( r 2 ) ] } .
exp { i [ φ ( r 1 ) φ ( r 2 ) ] } = exp { σ φ 2 [ 1 b φ ( Δ r Δ r 0 ) ] } ,
exp { i [ φ ( r 1 ) + φ ( r 2 ) ] } = exp { σ φ 2 [ 1 + b φ ( Δ r Δ r 0 ) ] } ,
Δ r r 1 r 2 .
σ φ 2 4 k 2 h 2 , b φ ( Δ r ) h ( 0 ) h ( Δ r ) h 2 .
b φ ( Δ r ) = exp ( Δ r 2 ρ 0 2 ) .
exp { σ φ 2 [ 1 b φ ( Δ r Δ r 0 ) ] } [ 1 exp ( σ φ 2 ) ] exp [ ( Δ r Δ r 0 ) 2 ρ 0 2 σ φ 2 ] ,
exp { σ φ 2 [ 1 + b φ ( Δ r Δ r 0 ) ] } exp ( σ φ 2 ) [ 1 exp [ σ φ 2 ] ] exp [ ( Δ r Δ r 0 ) 2 ρ 0 2 ] .
exp { σ φ 2 [ 1 b φ ( Δ r Δ r 0 ) ] } ρ 0 2 σ φ 2 [ 1 exp ( σ φ 2 ) ] δ ( Δ r Δ r 0 ) ,
exp { σ φ 2 [ 1 + b φ ( Δ r Δ r 0 ) ] } ρ 0 2 exp [ σ φ 2 ] [ 1 exp ( σ φ 2 ) ] δ ( Δ r Δ r 0 ) .
exp { σ φ 2 [ 1 b φ ( Δ r Δ r 0 ) ] } 4 k 2 ρ eff 2 exp [ ( Δ r Δ r 0 ) 2 ρ eff 2 ] ,
ρ eff = ρ 0 σ φ .
exp [ σ φ 2 ( 1 b φ ( Δ r Δ r 0 ) ) ] 4 π k 2 δ ( Δ r Δ r 0 ) .
COV { u re u re u re u im u re u re u re u im u im u re u im u im u im u re u im u im u re u re u re u im u re u re u re u im u im u re u im u im u im u re u im u im } ,
u re u re = Re ( Γ 1 + Γ 2 2 ) , u im u im = Re ( Γ 1 Γ 2 2 ) ,
u im u re = Im ( Γ 1 + Γ 2 2 ) , u re u im = Im ( Γ 1 + Γ 2 2 ) .
Γ 1 U ( p ) U ( p + Δ p ) * = d r 1 d r 2 G ( r 1 , p ) G * ( r 2 , p + Δ p ) U 0 ( r 1 ) U 0 * ( r 2 ) ,
Γ 2 U ( p ) U ( p + Δ p ) = d r 1 d r 2 G ( r 1 , p ) G ( r 2 , p + Δ p ) U 0 ( r 1 ) U 0 ( r 2 ) ,
Γ 1 x exp [ i k ( ( B D * ( A * B A B * ) ) Δ p x 2 2 A B * Δ x Δ p x + A A * B * Δ x 2 ) 2 B * ( A B * A * B ) ] ,
Γ 2 x σ φ 2 exp [ σ φ 2 ] ( A * 2 A B * 2 B ) exp [ i k ( ( 2 A D 1 ) Δ p x 2 2 A Δ x Δ p x + A 2 Δ x 2 ) 4 A B ] .
M ̂ free space = [ 1 i a L i a L 1 ] ,
Γ 1 = exp [ i Δ p x ( Δ p x + Δ x ) a r s 2 ] exp [ ( Δ p x Δ x ) 2 2 a 2 r s 2 ] exp [ Δ x 2 2 r s 2 ] ,
Γ 2 = σ φ 2 exp [ σ φ 2 ] ( a a + i ) exp [ ( ( 1 + 2 i a ) Δ p x 2 + ( a + i ) Δ x 2 + 2 Δ p x Δ x ( 1 i a ) ) 2 a ( a + i ) r s 2 ] .
Γ 1 2 = exp [ ( Δ p x Δ x ) 2 a 2 r s 2 ] exp [ Δ x 2 r s 2 ] ,
Γ 2 2 = σ φ 4 exp [ 2 σ φ 2 ] a 2 a 2 + 1 exp [ Δ p x 2 2 ( 1 + a 2 ) r s 2 ] exp [ Δ x 2 r s 2 ] .
[ 2 i f k r s 2 f 2 i f 2 k s 2 1 f 2 i k s 2 2 i f k s 2 ] .
Γ 1 = exp [ ( β + i ) γ 2 + 2 i β 2 4 β ( β i ) f Δ p x 2 ] exp ( Δ x 2 2 r s 2 ) exp ( i k Δ x Δ p x 2 f ) ,
Γ 2 = σ φ 2 exp ( σ φ 2 ) ( 1 1 i β ) exp [ i k ( γ 2 + 2 β 2 ) 4 β ( β + i ) f Δ p x 2 ] exp [ i k β 4 γ 2 ( β + i ) f Δ x 2 ] exp [ k Δ x Δ p x 2 f ( β + i ) ] ,
β 2 f k s 2 , γ r s s .
Γ 1 2 = exp { k [ ( β 1 ) 2 γ 2 2 β 2 ] Δ p x 2 2 f β ( 1 + β 2 ) } exp ( Δ x 2 2 r s 2 ) ,
Γ 2 2 = σ φ 4 exp ( 2 σ φ 2 ) 1 ( 1 + β 2 ) exp [ k ( γ 2 + 2 β 2 ) Δ p x 2 2 f β ( 1 + β 2 ) ] exp [ β k Δ x 2 2 f γ 2 ( 1 + β 2 ) ] exp [ β k Δ p x Δ x f ( 1 + β 2 ) ] .
M ̂ imaging = [ f 2 f 1 2 i f 1 f 2 k s 2 2 i f 1 f 2 k r s 2 f 1 f 2 ] ,
Γ 1 exp [ k ( Δ p x M Δ x ) 2 4 κ M ] ,
Γ 2 σ φ 2 exp [ σ φ 2 ] exp [ k ( Δ p x M Δ x ) 2 4 κ M ] ,
κ 2 f 1 f 2 k s 2 , M f f 1 .
Phase factor for Γ 1 : exp [ i k ( Δ p x Δ x ) p x L ] ,
Phase factor for Γ 2 : exp [ i a k Δ p x p x ( 1 + a ) L ] .
Γ 1 x exp ( i k { [ B D * ( A * B A B * ) ] Δ p x 2 2 A B * Δ x Δ p x + A A * B * Δ x 2 } 2 B * ( A B * A * B ) ) exp [ A 2 ( Δ p x A * Δ x ) 2 4 ( A * B A B * ) 2 k 2 ρ eff 2 ] .
Γ 2 x σ φ 2 exp ( σ φ 2 ) [ A * 2 A B * 2 B + i k ρ eff 2 ( A * B + A B * ) 8 B 2 ] exp { i k [ ( 2 A D 1 ) Δ p x 2 2 A Δ x Δ p x + A 2 Δ x 2 ] 4 A B } exp [ ( Δ p x A Δ x ) 2 ( k ρ eff 4 B ) 2 ] .

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