Abstract

We derive expressions for the intensity–intensity correlations of a field produced on scattering of a random field governed by Gaussian statistics from a particle with a deterministic or a random refractive index distribution. Our results generalize the ones in [Opt. Lett. 35, 4000, 2010] to the case of an arbitrarily correlated incident field. We consider as specific examples the cases of a single plane wave and two partially correlated plane waves incident upon the scatterer.

© 2011 Optical Society of America

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  1. O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
    [CrossRef]
  2. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29(1956).
    [CrossRef]
  3. R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178, 1046–1048 (1956).
    [CrossRef]
  4. R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448(1956).
    [CrossRef]
  5. R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent beams of light,” Nature 179, 1128–1129 (1957).
    [CrossRef]
  6. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons, in coherent beams of light, detected by a coincidence counting technique,” Nature 180, 324–326 (1957).
    [CrossRef]
  7. R. Berkovits and S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123(1990).
    [CrossRef] [PubMed]
  8. H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974).
    [CrossRef]
  9. M. Nieto-Vesperinas and J. A. Sánchez-Gil, “Intensity angular correlations of light multiply scattered from random rough surfaces,” J. Opt. Soc. Am. A 10, 150–157 (1993).
    [CrossRef]
  10. Y. Xin, Y. He, Y. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett. 35, 4000–4002 (2010).
    [CrossRef] [PubMed]
  11. V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).
  12. O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330–341(2004).
    [CrossRef]
  13. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
    [CrossRef]
  14. R. K. Wang, “In vivo full range complex Fourier domain optical coherence tomography,” Appl. Phys. Lett. 90, 054103(2007).
    [CrossRef]
  15. R. A. Leitgeb, R. Michaely, T. Lasser, and S. C. Sekhar, “Complex ambiguity-free Fourier domain optical coherence tomography through transverse scanning,” Opt. Lett. 32, 3453–3455(2007).
    [CrossRef] [PubMed]
  16. M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463(2002).
    [CrossRef] [PubMed]
  17. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  18. J. W. Goodman, Statistical Optics, 1st ed. (Wiley, 1985).
  19. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  20. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  21. T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23, 1631–1638 (2006).
    [CrossRef]

2010

2008

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
[CrossRef]

2007

R. K. Wang, “In vivo full range complex Fourier domain optical coherence tomography,” Appl. Phys. Lett. 90, 054103(2007).
[CrossRef]

R. A. Leitgeb, R. Michaely, T. Lasser, and S. C. Sekhar, “Complex ambiguity-free Fourier domain optical coherence tomography through transverse scanning,” Opt. Lett. 32, 3453–3455(2007).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

2006

2004

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330–341(2004).
[CrossRef]

2002

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463(2002).
[CrossRef] [PubMed]

1993

1990

R. Berkovits and S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123(1990).
[CrossRef] [PubMed]

1983

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).

1974

H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974).
[CrossRef]

1957

R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent beams of light,” Nature 179, 1128–1129 (1957).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “Correlation between photons, in coherent beams of light, detected by a coincidence counting technique,” Nature 180, 324–326 (1957).
[CrossRef]

1956

R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29(1956).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178, 1046–1048 (1956).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448(1956).
[CrossRef]

Andrews, L. C.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330–341(2004).
[CrossRef]

Asakura, T.

H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974).
[CrossRef]

Bajraszewski, T.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463(2002).
[CrossRef] [PubMed]

Banakh, V. A.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).

Berkovits, R.

R. Berkovits and S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123(1990).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Buldakov, V. M.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).

Chen, Y.

Feng, S.

R. Berkovits and S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123(1990).
[CrossRef] [PubMed]

Fercher, A. F.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463(2002).
[CrossRef] [PubMed]

Fischer, D. G.

Fujii, H.

H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics, 1st ed. (Wiley, 1985).

Hanbury Brown, R.

R. Hanbury Brown and R. Q. Twiss, “Correlation between photons, in coherent beams of light, detected by a coincidence counting technique,” Nature 180, 324–326 (1957).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent beams of light,” Nature 179, 1128–1129 (1957).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178, 1046–1048 (1956).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29(1956).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448(1956).
[CrossRef]

He, Y.

Korotkova, O.

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
[CrossRef]

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330–341(2004).
[CrossRef]

Kowalczyk, A.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463(2002).
[CrossRef] [PubMed]

Lasser, T.

Leitgeb, R.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463(2002).
[CrossRef] [PubMed]

Leitgeb, R. A.

Li, J.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Michaely, R.

Mironov, V. L.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).

Nieto-Vesperinas, M.

Phillips, R. L.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330–341(2004).
[CrossRef]

Sánchez-Gil, J. A.

Sekhar, S. C.

Twiss, R. Q.

R. Hanbury Brown and R. Q. Twiss, “Correlation between photons, in coherent beams of light, detected by a coincidence counting technique,” Nature 180, 324–326 (1957).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent beams of light,” Nature 179, 1128–1129 (1957).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178, 1046–1048 (1956).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448(1956).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29(1956).
[CrossRef]

Visser, T. D.

Wang, R. K.

R. K. Wang, “In vivo full range complex Fourier domain optical coherence tomography,” Appl. Phys. Lett. 90, 054103(2007).
[CrossRef]

Wojtkowski, M.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463(2002).
[CrossRef] [PubMed]

Wolf, E.

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23, 1631–1638 (2006).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Xin, Y.

Appl. Phys. Lett.

R. K. Wang, “In vivo full range complex Fourier domain optical coherence tomography,” Appl. Phys. Lett. 90, 054103(2007).
[CrossRef]

J. Biomed. Opt.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463(2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Nature

R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29(1956).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178, 1046–1048 (1956).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448(1956).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “The question of correlation between photons in coherent beams of light,” Nature 179, 1128–1129 (1957).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, “Correlation between photons, in coherent beams of light, detected by a coincidence counting technique,” Nature 180, 324–326 (1957).
[CrossRef]

Opt. Commun.

H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974).
[CrossRef]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
[CrossRef]

Opt. Eng.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330–341(2004).
[CrossRef]

Opt. Lett.

Opt. Spektrosk.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).

Phys. Rev. E

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

Phys. Rev. Lett.

R. Berkovits and S. Feng, “Theory of speckle-pattern tomography in multiple-scattering media,” Phys. Rev. Lett. 65, 3120–3123(1990).
[CrossRef] [PubMed]

Other

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

J. W. Goodman, Statistical Optics, 1st ed. (Wiley, 1985).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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Figures (3)

Fig. 1
Fig. 1

Illustration of the notation used. In the symmetric case, the incident plane waves and the z-axis lie within a single plane. The polar and azimuthal angles associated with the incident plane waves are then described by θ ( i ) and ϕ ( i ) . A point of measurement is denoted by the vector r with spherical coordinates ϕ, θ, and r = | r | . We denote these values for r 1 as ϕ 1 , θ 1 , and r 1 —likewise for r 2 . In the symmetric cases considered, we say that ϕ 1 = ϕ 2 = ϕ , θ 1 = θ 2 = θ , and r 1 = r 2 = r .

Fig. 2
Fig. 2

The normalized correlation of intensity fluctuations, [ Δ I ( r 1 , ω ) Δ I ( r 2 , ω ) / I ( r 1 , ω ) I ( r 2 , ω ) ] 1 , in the case of two partially plane waves incident upon a deterministic scatterer. Parameters: λ = 0.633 × 10 6 m ; r = 1 m , σ = 1 / k , d = 0 , a u 1 u 2 = 0.5 . In both cases, ϕ ( i ) = ϕ (see Fig. 1). (a)  θ ( i ) 0 rad . (b)  θ ( i ) = 0.15 rad . (c)  θ ( i ) π / 2 rad .

Fig. 3
Fig. 3

The normalized correlation of intensity fluctuations, [ Δ I ( r 1 , ω ) Δ I ( r 2 , ω ) / I ( r 1 , ω ) I ( r 2 , ω ) ] 1 , in the case of two plane waves incident upon a random scatterer. Parameters: λ = 0.633 × 10 6 m ; r = 1 m , a = 5 λ , γ s = λ , a u 1 u 2 = 0.5 , k γ = 1 (somewhat correlated incident waves). In all cases, ϕ ( i ) = 0 rad (see Fig. 1). (a)  θ ( i ) = 1 mrad . (b)  θ ( i ) = 50 mrad . (c)  θ ( i ) = 100 mrad .

Equations (36)

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Δ I ( r , ω ) = I ( r , ω ) I ( r , ω ) ,
I ( r , ω ) = U * ( r , ω ) U ( r , ω ) .
Δ I ( r 1 , ω ) Δ I ( r 2 , ω ) = I ( r 1 , ω ) I ( r 2 , ω ) I ( r 1 , ω ) I ( r 2 , ω ) = U * ( r 1 , ω ) U ( r 1 , ω ) U * ( r 2 , ω ) U ( r 2 , ω ) U * ( r 1 , ω ) U ( r 1 , ω ) U * ( r 2 , ω ) U ( r 2 , ω ) .
Δ I ( r 1 , ω ) Δ I ( r 2 , ω ) = U * ( r 1 , ω ) U ( r 2 , ω ) U * ( r 2 , ω ) U ( r 1 , ω ) = | W ( r 1 , r 2 , ω ) | 2 .
I ( r 1 , ω ) I ( r 2 , ω ) = | W ( r 1 , r 2 , ω ) | 2 + S ( r 1 , ω ) S ( r 2 , ω ) ,
U ( i ) ( r , ω ) = a ( i ) ( u , ω ) e i k ( u · r + u z z ) d 2 u ,
u z = 1 u 2 when u 2 1 ; u z = i u 2 1 when u 2 > 1 ,
U ( t ) ( r , ω ) = a ( t ) ( u , ω ) e i k ( u · r ± u z z ) d 2 u .
a ( t ) ( u , u , ω ) = S ( u , u , ω ) a ( i ) ( u , ω ) d 2 u ,
W ( t ) ( r 1 u 1 , r 2 u 2 , ω ) = M ( u 1 , u , u 2 , u , ω ) A ( i ) ( u , u , ω ) × e i k ( u 2 · r 2 u 1 · r 1 ± ( u 2 z z 2 u 1 z z 1 ) ) d 2 u d 2 u d 2 u 1 d 2 u 2 ,
A ( i ) ( u , u , ω ) = a ( i ) * ( u , ω ) a ( i ) ( u , ω ) ,
M ( u 1 , u , u 2 , u , ω ) = S * ( u 1 , u , ω ) S ( u 2 , u , ω )
M ( u 1 , u , u 2 , u , ω ) m = S * ( u 1 , u , ω ) S ( u 2 , u , ω ) m
W ( t ) ( r 1 u 1 , r 2 u 2 , ω ) ± 4 π 2 u 1 z u 2 z k 2 r 1 r 2 M ( u 1 , u , u 2 , u , ω ) A ( i ) ( u , u , ω ) d 2 u d 2 u .
F ( r , ω ) = k 2 4 π [ n 2 ( r , ω ) 1 ] when r D ; F ( r , ω ) = 0 when r D ,
M ( 1 ) ( u 1 , u , u 2 , u , ω ) = F ˜ * ( K 1 , ω ) F ˜ ( K 2 , ω ) ,
C F ( r 1 , r 2 , ω ) = F * ( r 1 , ω ) F ( r 2 , ω ) m .
M ( 1 ) ( u 1 , u , u 2 , u , ω ) m = C ˜ F ( K 1 , K 2 , ω ) = F ˜ * ( K 1 , ω ) F ˜ ( K 2 , ω ) m .
Δ I ( r 1 , ω ) Δ I ( r 2 , ω ) ( t ) = ( 4 π 2 u 1 z u 2 z k 2 r 1 r 2 ) 2 M ( u 1 , u 2 , u , u , u , u , ω ) × A ( u , u , u , u , ω ) d 2 u d 2 u d 2 u d 2 u ,
M ( u 1 , u 2 , u , u , u , u , ω ) = M * ( u 1 , u , u 2 , u , ω ) M ( u 1 , u , u 2 , u , ω ) ,
A ( u , u , u , u , ω ) = A ( i ) * ( u , u , ω ) A ( i ) ( u , u , ω ) .
A ( t ) ( u 1 , u 2 , ω ) = M ( 1 ) ( u 1 , u , u 2 , u , ω ) m | a ( i ) ( u , ω ) | 2 .
Δ I ( r 1 , ω ) Δ I ( r 2 , ω ) ( t ) = ( 4 π 2 u 1 z u 2 z k 2 r 1 r 2 ) 2 | M ( 1 ) ( u 1 , u , u 2 , u , ω ) m | 2 .
| F ˜ * [ K 1 , ω ] F ˜ [ K 2 , ω ] m | 2 = ( k 2 r 1 r 2 4 π 2 u 1 z u 2 z ) 2 Δ I ( r 1 u 1 , ω ) Δ I ( r 2 u 2 , ω ) ( t ) .
F ˜ ( K 1 , ω ) F ˜ * ( K 2 , ω ) m F ˜ * ( K 1 , ω ) F ˜ ( K 2 , ω ) m = F ˜ * ( K 1 , ω ) F ˜ ( K 1 , ω ) F ˜ * ( K 2 , ω ) F ˜ ( K 2 , ω ) m 1 .
F ˜ * ( K 1 , ω ) F ˜ ( K 1 , ω ) F ˜ * ( K 2 , ω ) F ˜ ( K 2 , ω ) m = ( k 2 r 1 r 2 4 π 2 u 1 z u 2 z ) 2 Δ I ( r 1 , ω ) Δ I ( r 2 , ω ) ( t ) + 1 .
Δ I ( r 1 , ω ) Δ I ( r 2 , ω ) ( t ) = ( 4 π 2 u 1 z u 2 z k 2 r 1 r 2 ) 2 u u u u M ( u 1 , u 2 , u , u , u , u , ω ) × A ( u , u , u , u , ω ) ,
A ( u , u , u , u , ω ) = a u u a u u exp [ k 2 γ 2 2 ( u u ) 2 ] exp [ k 2 γ 2 2 ( u u ) 2 ] ,
M ( u 1 , u 2 , u , u , u , u , ω ) = F ˜ ( K 1 , ω ) F ˜ * ( K 2 , ω ) F ˜ * ( K 1 , ω ) F ˜ ( K 2 , ω ) .
F ( r , ω ) = B exp [ x 2 + y 2 + ( z d ) 2 2 σ 2 ] ,
M ( u 1 , u 2 , u , u , u , u , ω ) = B 4 ( 2 π ) 6 σ 12 exp [ k 2 σ 2 2 [ ( u 1 u ) 2 + ( u 2 u ) 2 + ( u 1 u ) 2 + ( u 2 u ) 2 ] ] × exp [ i k d [ ( u 1 z u z ) ( u 2 z u z ) ( u 1 z u z ) + ( u 2 z u z ) ] ] .
C F ( r 1 , r 2 , ω ) = [ L ( r 1 , ω ) ] 1 / 2 [ L ( r 2 , ω ) ] 1 / 2 L ( r 1 , r 2 , ω )
L ( r , ω ) = L ( 0 ) exp [ r 2 2 a 2 ] ,
L ( r 1 , r 2 , ω ) = exp [ ( r 1 r 2 ) 2 2 γ s ] .
C ˜ F ( u 1 , u , u 2 , u , ω ) = 64 π 3 L ( 0 ) B a 4 γ s 2 ( 4 a 2 + γ s 2 ) ( 1 a 2 + 2 γ s 2 ) ( 1 a 2 + 2 2 a 2 + γ s 2 ) × exp [ 2 a 4 k 2 ( ( u 1 u ) 2 + ( u 2 u ) 2 ) 4 a 2 + γ s 2 ] exp [ a 2 k 2 ( u u 1 + u 2 u ) 2 4 a 2 + γ s 2 ] .
M ( u 1 , u 2 , u , u , u , u , ω ) = C ˜ F * ( u 1 , u , u 2 , u , ω ) C ˜ F ( u 1 , u , u 2 , u , ω ) ,

Metrics