Abstract

Photon-correlation experiments have verified the theoretical prediction of Baltes et al. [ Opt. Acta 28, 11– 28 ( 1981)] that a phase grating hidden within a diffuse medium may be detected by correlation measurements. By extension of this theory to the space–time domain we have additionally verified that a simple and more reliable method of detecting the grating, valid for arbitrarily fine diffusers, is possible by temporal autocorrelation measurements of the scattered field at a single point. This method is shown to yield detailed information about the deterministic and stochastic features of the source and the source plane motion.

© 1984 Optical Society of America

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References

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  1. H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steinle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
    [CrossRef]
  2. H. P. Baltes, H. A. Ferwerda, “Inverse problems and coherence,” IEEE Trans. AP-29, 405–406 (1981).
    [CrossRef]
  3. H. P. Baltes, A. S. Glass, K. M. Jauch, “Multiplexing of coherence by beam-splitters,” Opt. Acta 28, 873–876 (1981).
    [CrossRef]
  4. H. P. Baltes, K. M. Jauch, “Multiplex version of the van Cittert–Zernike theorem,” J. Opt. Soc. Am. 71, 1434–1439 (1981).
    [CrossRef]
  5. A. S. Glass, H. P. Baltes, “The significance of far-zone coherence for sources or scatterers with hidden periodicity,” Opt. Acta 29, 169–185 (1982).
    [CrossRef]
  6. A. S. Glass, H. P. Baltes, K. M. Jauch, “The detection of hidden diffractors by coherence measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 681–686 (1982).
  7. K. M. Jauch, H. P. Baltes, “Coherence of radiation scattered by gratings covered by a diffuser. Experimental evidence,” Opt. Acta 28, 1013–1015 (1981).
    [CrossRef]
  8. A. S. Glass, “The significance of image reversal in the detection of hidden diffractors by interferometry,” Opt. Acta 29, 575–583 (1982).
    [CrossRef]
  9. K. M. Jauch, H. P. Baltes, “Reversing-wave-front interferometry of radiation from a diffusely illuminated phase grating,” Opt. Lett. 7, 127–129 (1982).
    [CrossRef] [PubMed]
  10. K. M. Jauch, H. P. Baltes, A. S. Glass, “Measurements of coherence of radiation from diffusely illuminated beamsplitters,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 687–690 (1982).
  11. J. C. Dainty, D. Newman, “Detection of gratings hidden by diffusers using photon-correlation techniques,” Opt. Lett. 8, 608–610 (1983).
    [CrossRef] [PubMed]
  12. J. W. Goodman, “The role of coherence concepts in the study of speckle,” Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86–94 (1979).
  13. J. C. Leader, “Similarities and distinctions between coherence theory relations and laser scattering phenomena,” Opt. Eng. 19, 593–601 (1980).
    [CrossRef]
  14. H. M. Pederson, “Intensity correlation metrology. a comparative study,” Opt. Acta 29, 105–118 (1982).
    [CrossRef]
  15. J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Heidelberg, 1975), p. 63.
  16. P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
    [CrossRef]
  17. B. M. Levine, J. C. Dainty, “Non-Gaussian image plane speckle: measurements for diffusers of known statistics,” Opt. Commun. 45, 252–257 (1983).
    [CrossRef]
  18. T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of a diffuse object,” Appl. Phys. 25, 179–194 (1981).
    [CrossRef]

1983 (2)

B. M. Levine, J. C. Dainty, “Non-Gaussian image plane speckle: measurements for diffusers of known statistics,” Opt. Commun. 45, 252–257 (1983).
[CrossRef]

J. C. Dainty, D. Newman, “Detection of gratings hidden by diffusers using photon-correlation techniques,” Opt. Lett. 8, 608–610 (1983).
[CrossRef] [PubMed]

1982 (6)

K. M. Jauch, H. P. Baltes, A. S. Glass, “Measurements of coherence of radiation from diffusely illuminated beamsplitters,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 687–690 (1982).

K. M. Jauch, H. P. Baltes, “Reversing-wave-front interferometry of radiation from a diffusely illuminated phase grating,” Opt. Lett. 7, 127–129 (1982).
[CrossRef] [PubMed]

A. S. Glass, H. P. Baltes, “The significance of far-zone coherence for sources or scatterers with hidden periodicity,” Opt. Acta 29, 169–185 (1982).
[CrossRef]

A. S. Glass, H. P. Baltes, K. M. Jauch, “The detection of hidden diffractors by coherence measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 681–686 (1982).

A. S. Glass, “The significance of image reversal in the detection of hidden diffractors by interferometry,” Opt. Acta 29, 575–583 (1982).
[CrossRef]

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

1981 (6)

K. M. Jauch, H. P. Baltes, “Coherence of radiation scattered by gratings covered by a diffuser. Experimental evidence,” Opt. Acta 28, 1013–1015 (1981).
[CrossRef]

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steinle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

H. P. Baltes, H. A. Ferwerda, “Inverse problems and coherence,” IEEE Trans. AP-29, 405–406 (1981).
[CrossRef]

H. P. Baltes, A. S. Glass, K. M. Jauch, “Multiplexing of coherence by beam-splitters,” Opt. Acta 28, 873–876 (1981).
[CrossRef]

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

H. P. Baltes, K. M. Jauch, “Multiplex version of the van Cittert–Zernike theorem,” J. Opt. Soc. Am. 71, 1434–1439 (1981).
[CrossRef]

1980 (1)

J. C. Leader, “Similarities and distinctions between coherence theory relations and laser scattering phenomena,” Opt. Eng. 19, 593–601 (1980).
[CrossRef]

1979 (1)

J. W. Goodman, “The role of coherence concepts in the study of speckle,” Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86–94 (1979).

1978 (1)

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

Asakura, T.

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

Baltes, H. P.

K. M. Jauch, H. P. Baltes, “Reversing-wave-front interferometry of radiation from a diffusely illuminated phase grating,” Opt. Lett. 7, 127–129 (1982).
[CrossRef] [PubMed]

K. M. Jauch, H. P. Baltes, A. S. Glass, “Measurements of coherence of radiation from diffusely illuminated beamsplitters,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 687–690 (1982).

A. S. Glass, H. P. Baltes, “The significance of far-zone coherence for sources or scatterers with hidden periodicity,” Opt. Acta 29, 169–185 (1982).
[CrossRef]

A. S. Glass, H. P. Baltes, K. M. Jauch, “The detection of hidden diffractors by coherence measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 681–686 (1982).

K. M. Jauch, H. P. Baltes, “Coherence of radiation scattered by gratings covered by a diffuser. Experimental evidence,” Opt. Acta 28, 1013–1015 (1981).
[CrossRef]

H. P. Baltes, K. M. Jauch, “Multiplex version of the van Cittert–Zernike theorem,” J. Opt. Soc. Am. 71, 1434–1439 (1981).
[CrossRef]

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steinle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

H. P. Baltes, H. A. Ferwerda, “Inverse problems and coherence,” IEEE Trans. AP-29, 405–406 (1981).
[CrossRef]

H. P. Baltes, A. S. Glass, K. M. Jauch, “Multiplexing of coherence by beam-splitters,” Opt. Acta 28, 873–876 (1981).
[CrossRef]

Dainty, J. C.

J. C. Dainty, D. Newman, “Detection of gratings hidden by diffusers using photon-correlation techniques,” Opt. Lett. 8, 608–610 (1983).
[CrossRef] [PubMed]

B. M. Levine, J. C. Dainty, “Non-Gaussian image plane speckle: measurements for diffusers of known statistics,” Opt. Commun. 45, 252–257 (1983).
[CrossRef]

Ferwerda, H. A.

H. P. Baltes, H. A. Ferwerda, “Inverse problems and coherence,” IEEE Trans. AP-29, 405–406 (1981).
[CrossRef]

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steinle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

Glass, A. S.

A. S. Glass, H. P. Baltes, K. M. Jauch, “The detection of hidden diffractors by coherence measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 681–686 (1982).

A. S. Glass, “The significance of image reversal in the detection of hidden diffractors by interferometry,” Opt. Acta 29, 575–583 (1982).
[CrossRef]

A. S. Glass, H. P. Baltes, “The significance of far-zone coherence for sources or scatterers with hidden periodicity,” Opt. Acta 29, 169–185 (1982).
[CrossRef]

K. M. Jauch, H. P. Baltes, A. S. Glass, “Measurements of coherence of radiation from diffusely illuminated beamsplitters,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 687–690 (1982).

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steinle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

H. P. Baltes, A. S. Glass, K. M. Jauch, “Multiplexing of coherence by beam-splitters,” Opt. Acta 28, 873–876 (1981).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “The role of coherence concepts in the study of speckle,” Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86–94 (1979).

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Heidelberg, 1975), p. 63.

Gray, P. F.

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

Jauch, K. M.

K. M. Jauch, H. P. Baltes, A. S. Glass, “Measurements of coherence of radiation from diffusely illuminated beamsplitters,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 687–690 (1982).

K. M. Jauch, H. P. Baltes, “Reversing-wave-front interferometry of radiation from a diffusely illuminated phase grating,” Opt. Lett. 7, 127–129 (1982).
[CrossRef] [PubMed]

A. S. Glass, H. P. Baltes, K. M. Jauch, “The detection of hidden diffractors by coherence measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 681–686 (1982).

K. M. Jauch, H. P. Baltes, “Coherence of radiation scattered by gratings covered by a diffuser. Experimental evidence,” Opt. Acta 28, 1013–1015 (1981).
[CrossRef]

H. P. Baltes, K. M. Jauch, “Multiplex version of the van Cittert–Zernike theorem,” J. Opt. Soc. Am. 71, 1434–1439 (1981).
[CrossRef]

H. P. Baltes, A. S. Glass, K. M. Jauch, “Multiplexing of coherence by beam-splitters,” Opt. Acta 28, 873–876 (1981).
[CrossRef]

Leader, J. C.

J. C. Leader, “Similarities and distinctions between coherence theory relations and laser scattering phenomena,” Opt. Eng. 19, 593–601 (1980).
[CrossRef]

Levine, B. M.

B. M. Levine, J. C. Dainty, “Non-Gaussian image plane speckle: measurements for diffusers of known statistics,” Opt. Commun. 45, 252–257 (1983).
[CrossRef]

Newman, D.

Pederson, H. M.

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

Steinle, B.

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steinle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

Takai, N.

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

Appl. Phys. (1)

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

IEEE Trans. (1)

H. P. Baltes, H. A. Ferwerda, “Inverse problems and coherence,” IEEE Trans. AP-29, 405–406 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (7)

H. P. Baltes, A. S. Glass, K. M. Jauch, “Multiplexing of coherence by beam-splitters,” Opt. Acta 28, 873–876 (1981).
[CrossRef]

A. S. Glass, H. P. Baltes, “The significance of far-zone coherence for sources or scatterers with hidden periodicity,” Opt. Acta 29, 169–185 (1982).
[CrossRef]

K. M. Jauch, H. P. Baltes, “Coherence of radiation scattered by gratings covered by a diffuser. Experimental evidence,” Opt. Acta 28, 1013–1015 (1981).
[CrossRef]

A. S. Glass, “The significance of image reversal in the detection of hidden diffractors by interferometry,” Opt. Acta 29, 575–583 (1982).
[CrossRef]

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steinle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

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

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

Opt. Commun. (1)

B. M. Levine, J. C. Dainty, “Non-Gaussian image plane speckle: measurements for diffusers of known statistics,” Opt. Commun. 45, 252–257 (1983).
[CrossRef]

Opt. Eng. (1)

J. C. Leader, “Similarities and distinctions between coherence theory relations and laser scattering phenomena,” Opt. Eng. 19, 593–601 (1980).
[CrossRef]

Opt. Lett. (2)

Proc. Soc. Photo-Opt. Instrum. Eng. (3)

K. M. Jauch, H. P. Baltes, A. S. Glass, “Measurements of coherence of radiation from diffusely illuminated beamsplitters,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 687–690 (1982).

J. W. Goodman, “The role of coherence concepts in the study of speckle,” Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86–94 (1979).

A. S. Glass, H. P. Baltes, K. M. Jauch, “The detection of hidden diffractors by coherence measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 369, 681–686 (1982).

Other (1)

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Heidelberg, 1975), p. 63.

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Figures (9)

Fig. 1
Fig. 1

Experimental arrangement.

Fig. 2
Fig. 2

Theoretical plot of the average intensity distribution in the far field of the grating/diffuser combinations: a, L/b = 1.0, b, L/b = 0.5, and c, L/b = 0.33. (L is the 1/e correlation length of the complex amplitude transmittance of the diffuser and b is the spatial period of the sinusoidal phase grating).

Fig. 3
Fig. 3

Experimental measurement of the angular width of the cross-correlation peak in the antisymmetric scan, σ = 2λ/b + Δσ versus Δσ. The 1/e width of this peak occurs at Δσ ≅ 0.26 × 10−3 rads, which is approximately equal to 1/ka, where k ≅ 10+4/mm is the He–Ne wave number and a = 0.33 mm is the laser-beam radius (L/b ≅ 1.2 for this measurement).

Fig. 4
Fig. 4

Peak values of the intensity cross correlation measured at angles (−θ0, θ0) as a function of L/b (sin θ0/b).

Fig. 5
Fig. 5

Theoretical plots of the temporal-intensity autocorrelation at the center of the diffraction field (s = 0, σ = 0). The diffuser velocity is vξ = 1.8 mm/sec, and vη = 30 mm/sec. The amplitude correlation length L = 2 μm, and a, b = 5.1 μm; b, b = 9.2 μm.

Fig. 6
Fig. 6

Experimental measurements of the intensity autocorrelation function at the center of the diffraction field (s = 0, σ = 0). The diffuser velocity is vξ = 0.5 mm/sec, vη = 0 (pure translation along ξ ^). The diffuser has an amplitude correlation length of L ≅ 0.9 μm, and a, b = 9.2 μm; b, b = 5.1 μm. Note that the period of oscillation varies as b/vξ to a high degree of accuracy and the modulation strength increases as L/b is reduced.

Fig. 7
Fig. 7

Experimental measurement of the temporal-intensity autocorrelation function at the center of the diffraction field (s = 0, σ = 0) with parameters L ≅ 0.9 μm, b = 9.2 μm, a = 0.33 mm, v ≅ 50 mm sec−1, vη ≅ 560 mm sec−1. Note the modulation period tm ≅ 1.8 × 10−2 sec, and the Gaussian envelope has a 1/e time of to = 1.2 × 10−3 sec, as predicted by theory.

Fig. 8
Fig. 8

Modulation strength as a function of intensity overlap L/b at fixed scan angle: a, K = 0; b, K = 1. The solid curve is a theoretical plot; the +’s refer to experimental measurements.

Fig. 9
Fig. 9

Modulation strength as a function of scan angle K = s/sin θ0 (where sin θ0 ≡ λ/b) for fixed overlap parameter L/b = a, 0.18; b, 0.25; c, 0.30. The solid curve refers to theory, the +’s refer to experimental measurements.

Equations (38)

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U ( ξ , η , t ) = P ( ξ , η ) exp [ i ϕ ( ξ , η , t ) ] T ( ξ , η ) ,
P ( ξ , η ) = I 0 1 / 2 exp [ - ( ξ 2 + η 2 ) / 4 a 2 ] ,
T ( ξ , η ) = exp [ i α ( sin 2 π ξ / b ) ] ,
ϕ ( ξ , η , t ) = ϕ ( ξ - v ξ t , η - v η t )
exp [ i α sin ( 2 π ξ / b ) ] = n = - g n exp ( i n 2 π ξ / b ) ,
Γ ( x 1 , y 1 ; x 2 , y 2 , τ ) = U ( x 1 , y 1 , t ) U * ( x 2 , y 2 , t + τ ) ,
k a 1 ,
R k a 2 ,
Γ ( x 1 , y 1 , x 2 , y 2 , τ ) = - d ξ 1 d ξ 2 × - d η 1 d η 2 U ( ξ 1 , η 1 , t ) U * ( ξ 2 , η 2 , t + τ ) × exp [ - 2 π i λ R ( x 1 ξ 1 - x 2 ξ 2 + y 1 η 1 - y 2 η 2 ) ] ,
exp { i [ ϕ ( ξ 1 , η 1 , t ) - ϕ ( ξ 2 , η 2 , t + τ ) ] } .
exp { i [ ϕ ( ξ 1 , η 1 , t ) - ϕ ( ξ 2 , η 2 , t + τ ) ] } exp { - [ ( ξ 1 - ξ 2 - v ξ τ ) 2 + ( η 1 - η 2 - v η τ ) 2 ] / 2 L 2 } ,
L = l h / [ 2 π ( n - 1 ) σ h / λ ]
U ( ξ 1 , η 1 , t ) U * ( ξ 2 , η 2 , t + τ ) = exp { - [ ( ξ 1 - ξ 2 - v ξ τ ) 2 + ( η 1 - η 2 - v η τ ) 2 2 L 2 ] } × [ P ( ξ 1 , η 1 ) T ( ξ 1 ) ] [ P ( ξ 2 , η 2 ) T ( ξ 2 ) ] * .
ξ 0 = ( ξ 1 + ξ 2 ) / 2 , η 0 = ( η 1 + η 2 ) / 2 , ξ = ξ 1 - ξ 2 , η = η 1 - η 2 ,
[ P ( ξ 1 , η 1 ) T ( ξ 1 ) ] [ P ( ξ 2 , η 2 ) T ( ξ 2 ) ] * = n , m = - g n g m * exp [ - ( ξ 2 + η 2 ) / 8 a 2 ] exp [ i 2 π ( n + m ) ξ / b ] × exp [ - ( ξ 0 2 + η 0 2 ) / 2 a 2 ] exp [ i 2 π ( n - m ) ξ 0 / b ] .
Γ ( s , σ , 0 , 0 , τ ) = exp ( - v 2 τ 2 / 8 a 2 ) × n , m = - g n g m * exp [ - i 2 v ξ τ b ( s / sin θ 0 - n + m 2 ) ] × exp { - ½ k 2 a 2 [ σ - ( n - m ) λ b ] 2 } × exp { - ½ k 2 L 2 [ s - ½ ( n + m ) λ b ] 2 } ,
s x 1 + x 2 2 R = sin θ 1 + sin θ 2 2
σ x 1 - x 2 R = sin θ 1 - sin θ 2
C I ( x 1 , x 2 , τ ) = I ( x 1 , t ) I * ( x 2 , t + τ ) I ( x 1 , 0 ) I ( x 2 , 0 ) - 1 ,
C I ( x 1 , x 2 , τ ) = Γ ( x 1 , x 2 , τ ) 2 Γ ( x 1 , x 1 , 0 ) Γ ( x 2 , x 2 , 0 ) γ ( x 1 , x 2 , τ ) 2 ,
g 0 0.826 , g 1 0.385 , g 2 0.08 , g 3 0.012 ,
γ ( s , σ , 0 ) = n , m g n g m exp { - ½ k 2 a 2 [ σ - ( n - m ) λ b ] 2 } exp { - ½ k 2 L 2 [ s - ½ ( n + m ) λ 2 ] 2 } ( { n g n 2 exp [ - 2 π 2 L 2 b 2 ( s + σ / 2 - n λ / b ) 2 ] } { n g n 2 exp [ - 2 π 2 L 2 b 2 ( s - σ / 2 - n λ / b ) 2 ] } ) 1 / 2 ,
exp { - ½ k 2 a 2 [ σ - ( n - m ) λ / b ] 2 } = exp { - 2 π 2 a 2 b 2 [ 2 - ( n - m ) ] 2 } δ 2 , n - m
γ ( 0 , 2 λ / b ) 2 g 0 g 2 exp ( - 2 π 2 L 2 / b 2 ) - g 1 2 + 2 g 1 g 3 exp ( - 8 π 2 L 2 / b 2 ) g 1 2 + ( g 0 2 + g 1 2 ) exp ( - 2 π 2 L 2 / b 2 ) ,
0 = k = 0 2 n ( - ) k J k ( z ) J 2 n - k ( z ) + 2 k = 1 J k ( z ) J 2 n + k ( z ) ( n 1 ) . ]
γ ( s , σ , τ ) 2 = Γ ( s , σ , τ ) 2 Γ ( s + σ / 2 , 0 , 0 ) Γ ( s - σ / 2 , 0 , 0 ) ,
γ ( s , σ , τ ) 2 = exp ( - v 2 τ 2 / 4 a 2 ) | n , m A m n ( s , σ ) exp ( - i 2 π v ξ τ b ) [ s / sin θ 0 - ½ ( n + m ) ] | 2 { n g n 2 exp [ - 2 π 2 L 2 b 2 ( s + σ / 2 - n λ / b ) 2 ] } { n g n 2 exp [ - 2 π 2 L 2 b 2 ( s - σ / 2 - n λ / b ) 2 ] } .
γ ( s , σ , τ ) 2 exp ( - v 2 τ 2 / 4 a 2 ) { n , m A n m ( s , σ ) × cos [ ω n m ( s ) τ ] 2 + n , m A n m ( s , σ ) sin [ ω n m ( s ) τ ] 2 } ,
A n m ( s , σ ) = g n g m exp { - ½ k 2 a 2 [ σ - ( n - m ) λ b ] 2 } × exp { - ½ k 2 L 2 [ s - ½ ( n + m ) λ b ] 2 } ,
ω n m ( s ) = 2 π v ξ b ( s / sin θ 0 - n + m 2 )
A n m ( s , 0 ) g n g m exp { - ½ k 2 L 2 [ s - ½ ( n + m ) λ b ] 2 } δ n , m ,
q 2 2 π 2 L 2 / b 2 , ω 0 2 π v ξ / b , K = s / sin θ 0 = sin θ / sin θ 0 ,
γ ( s , 0 , τ ) 2 = exp ( v 2 τ 2 / 4 a 2 ) n g n 2 exp [ - q 2 ( K - n ) 2 ] cos [ ω 0 τ ( K - n ) ] 2 + n g n 2 exp [ - q 2 ( K - n ) 2 ] sin [ ω 0 τ ( K - n ) ] 2 g n 2 exp [ - q 2 ( K - n ) 2 ] 2 .
γ ( s , 0 , τ ) 2 exp ( - v 2 τ 2 / 4 a 2 ) / N ( K , q ) × [ g 0 4 exp ( - 2 q 2 K 2 ) + g 1 4 { exp [ - 2 q 2 ( K + 1 ) 2 ] + exp [ - 2 q 2 ( K - 1 ) 2 ] } + 2 g 0 2 g 1 2 ( exp { - q 2 [ K 2 + ( K + 1 ) 2 ] } + 2 exp { - q 2 [ K 2 + ( K - 1 ) 2 ] } ) cos ( ω 0 τ ) ] ,
N ( K , q ) = ( g 0 2 exp ( - q 2 K 2 ) + g 1 2 { exp [ - q 2 ( K + 1 ) 2 ] + exp [ - q 2 ( K - 1 ) 2 ] } ) 2
γ ( s , 0 , τ ) 2 exp ( - v 2 τ 2 / 4 a 2 ) [ A ( q , K ) + B ( q , K ) cos ( ω 0 τ ) ] A ( q , K ) + B ( q , K ) ,
A ( q , K ) = A ( q , - K ) , B ( q , K ) = B ( q , - K ) ,
S ( q , K ) = B ( q , K ) / A ( q , K ) ,

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