Abstract

The fractional Fourier transform (FRT) is applied to a partially coherent off-axis Gaussian Schell-model (GSM) beam, and an analytical formula is derived for the FRT of a partially coherent off-axis GSM beam. The corresponding tensor ABCD law for performing the FRT of a partially coherent off-axis GSM beam is also obtained. As an application example, the FRT of a partially coherent linear laser array that is expanded as a sum of off-axis GSM beams is studied. The derived formulas are used to provide numerical examples. The formulas provide a convenient way to analyze and calculate the FRT of a partially coherent off-axis GSM beam.

© 2006 Optical Society of America

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  2. P. D. Santis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
    [CrossRef]
  3. J. Deschamps, D. Courjon, and J. Bulabois, "Gaussian Schell-model sources: an example and some perspectives," J. Opt. Soc. Am. A 73, 256-261 (1983).
    [CrossRef]
  4. R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants," Phys. Rev. A 31, 2419-2434 (1985).
    [CrossRef] [PubMed]
  5. J. Turunen and A. T. Friberg, "Matrix representation of Gaussian Schell-model beams in optical systems," Opt. Laser Technol. 18, 259-267 (1985).
    [CrossRef]
  6. A. T. Friberg and J. Turunen, "Imaging of Gaussian Schell-model sources," J. Opt. Soc. Am. A 5, 713-720 (1988).
    [CrossRef]
  7. D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, "Twisted Gaussian Schell-model beams: a superposition model," J. Mod. Opt. 41, 139-399 (1994).
    [CrossRef]
  8. R. Simon and N. Mukunda, "Gaussian Schell-model beams and general shape invariance," J. Opt. Soc. Am. A 16, 2465-2475 (1999).
    [CrossRef]
  9. Q. Lin and Y. Cai, "Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams," Opt. Lett. 27, 216-218 (2002).
    [CrossRef]
  10. Y. Cai and Q. Lin, "Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media," J. Opt. Soc. Am. A 19, 2036-2042 (2002).
    [CrossRef]
  11. D. Mendlovic and H. M. Ozaktas, "Fractional Fourier transforms and their optical implementation: I," J. Opt. Soc. Am. A 10, 1875-1881 (1993).
    [CrossRef]
  12. H. M. Ozaktas and D. Mendlovic, "Fractional Fourier transforms and their optical implementation: II," J. Opt. Soc. Am. A 10, 2522-2531 (1993).
    [CrossRef]
  13. A. W. Lohmann, "Image rotation, Wigner rotation, and the fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181-2186 (1993).
    [CrossRef]
  14. R. G. Dorsch and A. W. Lohmann, "Fractional Fourier transform used for a lens-design problem," Appl. Opt. 34, 4111-4112 (1995).
    [CrossRef] [PubMed]
  15. M. A. Kutay and H. M. Ozaktas, "Optimal image restoration with the fractional Fourier transform," J. Opt. Soc. Am. A 15, 825-833 (1998).
    [CrossRef]
  16. L. M. Bernardo and O. D. D. Soares, "Fractional Fourier transforms and imaging," J. Opt. Soc. Am. A 11, 2622-2626 (1994).
    [CrossRef]
  17. Z. Zalevsky and D. Mendlovic, "Fractional Wiener filter," Appl. Opt. 35, 3930-3936 (1996).
    [CrossRef] [PubMed]
  18. S. Shinde and V. M. Gadre, "An uncertainty principle for real signals in the fractional Fourier transform domain," IEEE Trans. Signal Process. 49, 2545-2548 (2001).
    [CrossRef]
  19. Y. Zhang, B. Dong, B. Gu, and G. Yang, "Beam shaping in the fractional Fourier transform domain," J. Opt. Soc. Am. A 15, 1114-1120 (1998).
    [CrossRef]
  20. Y. Cai and Q. Lin, "Properties of a flattened Gaussian beam in the fractional Fourier transform plane," J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
    [CrossRef]
  21. Y. Cai and Q. Lin, "Fractional Fourier transform for elliptical Gaussian beams," Opt. Commun. 217, 7-13 (2003).
    [CrossRef]
  22. Q. Lin and Y. Cai, "Fractional Fourier transform for partially coherent Gaussian-Schell model beams," Opt. Lett. 27, 1672-1674 (2002).
    [CrossRef]
  23. Y. Cai and Q. Lin, "Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane," J. Opt. Soc. Am. A 20, 1528-1536 (2003).
    [CrossRef]
  24. A. A. Malyutin, "Simple scheme for the astigmatic transformation of laser modes," Quantum Electron. 33, 1015-1018 (2003).
    [CrossRef]
  25. Q. Tan, Y. Yan, G. Jin, and D. Xu, "True beam smoothing in the fractional Fourier transform domain," J. Mod. Opt. 50, 2147-2153 (2003).
  26. A. R. Al-Rashed and B. E. A. Saleh, "Decentered Gaussian beams," Appl. Opt. 34, 6819-6825 (1995).
    [CrossRef] [PubMed]
  27. C. Palma, "Decentered Gaussian beams, ray bundles, and Bessel-Gaussian beams," Appl. Opt. 36, 1116-1120 (1997).
    [CrossRef] [PubMed]
  28. P. J. Cronin, P. Török, P. Varga, and C. Cogswell, "High-aperture diffraction of a scalar, off-axis Gaussian beam," J. Opt. Soc. Am. A 17, 1556-1564 (2000).
    [CrossRef]
  29. Y. Cai and Q. Lin, "Decentered elliptical Gaussian beam," Appl. Opt. 41, 4336-4340 (2002).
    [CrossRef] [PubMed]
  30. Y. Cai and Q. Lin, "Decentered elliptical Hermite-Gaussian beam," J. Opt. Soc. Am. A 20, 1111-1119 (2003).
    [CrossRef]
  31. B. Lin and B. Lu, "Characterization of off-axis superposition of partially coherent beams," J. Opt. A, Pure Appl. Opt. 5, 303-307 (2003).
    [CrossRef]

2003 (7)

Y. Cai and Q. Lin, "Properties of a flattened Gaussian beam in the fractional Fourier transform plane," J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Y. Cai and Q. Lin, "Fractional Fourier transform for elliptical Gaussian beams," Opt. Commun. 217, 7-13 (2003).
[CrossRef]

A. A. Malyutin, "Simple scheme for the astigmatic transformation of laser modes," Quantum Electron. 33, 1015-1018 (2003).
[CrossRef]

Q. Tan, Y. Yan, G. Jin, and D. Xu, "True beam smoothing in the fractional Fourier transform domain," J. Mod. Opt. 50, 2147-2153 (2003).

B. Lin and B. Lu, "Characterization of off-axis superposition of partially coherent beams," J. Opt. A, Pure Appl. Opt. 5, 303-307 (2003).
[CrossRef]

Y. Cai and Q. Lin, "Decentered elliptical Hermite-Gaussian beam," J. Opt. Soc. Am. A 20, 1111-1119 (2003).
[CrossRef]

Y. Cai and Q. Lin, "Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane," J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

2002 (4)

2001 (1)

S. Shinde and V. M. Gadre, "An uncertainty principle for real signals in the fractional Fourier transform domain," IEEE Trans. Signal Process. 49, 2545-2548 (2001).
[CrossRef]

2000 (1)

1999 (1)

1998 (2)

1997 (1)

1996 (1)

1995 (3)

1994 (2)

L. M. Bernardo and O. D. D. Soares, "Fractional Fourier transforms and imaging," J. Opt. Soc. Am. A 11, 2622-2626 (1994).
[CrossRef]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, "Twisted Gaussian Schell-model beams: a superposition model," J. Mod. Opt. 41, 139-399 (1994).
[CrossRef]

1993 (3)

1988 (1)

1985 (2)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants," Phys. Rev. A 31, 2419-2434 (1985).
[CrossRef] [PubMed]

J. Turunen and A. T. Friberg, "Matrix representation of Gaussian Schell-model beams in optical systems," Opt. Laser Technol. 18, 259-267 (1985).
[CrossRef]

1983 (1)

J. Deschamps, D. Courjon, and J. Bulabois, "Gaussian Schell-model sources: an example and some perspectives," J. Opt. Soc. Am. A 73, 256-261 (1983).
[CrossRef]

1979 (1)

P. D. Santis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Al-Rashed, A. R.

Ambrosini, D.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, "Twisted Gaussian Schell-model beams: a superposition model," J. Mod. Opt. 41, 139-399 (1994).
[CrossRef]

Bagini, V.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, "Twisted Gaussian Schell-model beams: a superposition model," J. Mod. Opt. 41, 139-399 (1994).
[CrossRef]

Bernardo, L. M.

Bulabois, J.

J. Deschamps, D. Courjon, and J. Bulabois, "Gaussian Schell-model sources: an example and some perspectives," J. Opt. Soc. Am. A 73, 256-261 (1983).
[CrossRef]

Cai, Y.

Cogswell, C.

Courjon, D.

J. Deschamps, D. Courjon, and J. Bulabois, "Gaussian Schell-model sources: an example and some perspectives," J. Opt. Soc. Am. A 73, 256-261 (1983).
[CrossRef]

Cronin, P. J.

Deschamps, J.

J. Deschamps, D. Courjon, and J. Bulabois, "Gaussian Schell-model sources: an example and some perspectives," J. Opt. Soc. Am. A 73, 256-261 (1983).
[CrossRef]

Dong, B.

Dorsch, R. G.

Friberg, A. T.

A. T. Friberg and J. Turunen, "Imaging of Gaussian Schell-model sources," J. Opt. Soc. Am. A 5, 713-720 (1988).
[CrossRef]

J. Turunen and A. T. Friberg, "Matrix representation of Gaussian Schell-model beams in optical systems," Opt. Laser Technol. 18, 259-267 (1985).
[CrossRef]

Gadre, V. M.

S. Shinde and V. M. Gadre, "An uncertainty principle for real signals in the fractional Fourier transform domain," IEEE Trans. Signal Process. 49, 2545-2548 (2001).
[CrossRef]

Gori, F.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, "Twisted Gaussian Schell-model beams: a superposition model," J. Mod. Opt. 41, 139-399 (1994).
[CrossRef]

P. D. Santis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Gu, B.

Guattari, G.

P. D. Santis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Jin, G.

Q. Tan, Y. Yan, G. Jin, and D. Xu, "True beam smoothing in the fractional Fourier transform domain," J. Mod. Opt. 50, 2147-2153 (2003).

Kutay, M. A.

Lin, B.

B. Lin and B. Lu, "Characterization of off-axis superposition of partially coherent beams," J. Opt. A, Pure Appl. Opt. 5, 303-307 (2003).
[CrossRef]

Lin, Q.

Lohmann, A. W.

Lu, B.

B. Lin and B. Lu, "Characterization of off-axis superposition of partially coherent beams," J. Opt. A, Pure Appl. Opt. 5, 303-307 (2003).
[CrossRef]

Malyutin, A. A.

A. A. Malyutin, "Simple scheme for the astigmatic transformation of laser modes," Quantum Electron. 33, 1015-1018 (2003).
[CrossRef]

Mandel, L.

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

Mendlovic, D.

Mukunda, N.

R. Simon and N. Mukunda, "Gaussian Schell-model beams and general shape invariance," J. Opt. Soc. Am. A 16, 2465-2475 (1999).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants," Phys. Rev. A 31, 2419-2434 (1985).
[CrossRef] [PubMed]

Ozaktas, H. M.

Palma, C.

C. Palma, "Decentered Gaussian beams, ray bundles, and Bessel-Gaussian beams," Appl. Opt. 36, 1116-1120 (1997).
[CrossRef] [PubMed]

P. D. Santis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Saleh, B. E. A.

Santarsiero, M.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, "Twisted Gaussian Schell-model beams: a superposition model," J. Mod. Opt. 41, 139-399 (1994).
[CrossRef]

Santis, P. D.

P. D. Santis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Shinde, S.

S. Shinde and V. M. Gadre, "An uncertainty principle for real signals in the fractional Fourier transform domain," IEEE Trans. Signal Process. 49, 2545-2548 (2001).
[CrossRef]

Simon, R.

R. Simon and N. Mukunda, "Gaussian Schell-model beams and general shape invariance," J. Opt. Soc. Am. A 16, 2465-2475 (1999).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants," Phys. Rev. A 31, 2419-2434 (1985).
[CrossRef] [PubMed]

Soares, O. D. D.

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants," Phys. Rev. A 31, 2419-2434 (1985).
[CrossRef] [PubMed]

Tan, Q.

Q. Tan, Y. Yan, G. Jin, and D. Xu, "True beam smoothing in the fractional Fourier transform domain," J. Mod. Opt. 50, 2147-2153 (2003).

Török, P.

Turunen, J.

A. T. Friberg and J. Turunen, "Imaging of Gaussian Schell-model sources," J. Opt. Soc. Am. A 5, 713-720 (1988).
[CrossRef]

J. Turunen and A. T. Friberg, "Matrix representation of Gaussian Schell-model beams in optical systems," Opt. Laser Technol. 18, 259-267 (1985).
[CrossRef]

Varga, P.

Wolf, E.

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

Xu, D.

Q. Tan, Y. Yan, G. Jin, and D. Xu, "True beam smoothing in the fractional Fourier transform domain," J. Mod. Opt. 50, 2147-2153 (2003).

Yan, Y.

Q. Tan, Y. Yan, G. Jin, and D. Xu, "True beam smoothing in the fractional Fourier transform domain," J. Mod. Opt. 50, 2147-2153 (2003).

Yang, G.

Zalevsky, Z.

Zhang, Y.

Appl. Opt. (5)

IEEE Trans. Signal Process. (1)

S. Shinde and V. M. Gadre, "An uncertainty principle for real signals in the fractional Fourier transform domain," IEEE Trans. Signal Process. 49, 2545-2548 (2001).
[CrossRef]

J. Mod. Opt. (2)

Q. Tan, Y. Yan, G. Jin, and D. Xu, "True beam smoothing in the fractional Fourier transform domain," J. Mod. Opt. 50, 2147-2153 (2003).

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, "Twisted Gaussian Schell-model beams: a superposition model," J. Mod. Opt. 41, 139-399 (1994).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (2)

B. Lin and B. Lu, "Characterization of off-axis superposition of partially coherent beams," J. Opt. A, Pure Appl. Opt. 5, 303-307 (2003).
[CrossRef]

Y. Cai and Q. Lin, "Properties of a flattened Gaussian beam in the fractional Fourier transform plane," J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

J. Opt. Soc. Am. A (13)

J. Deschamps, D. Courjon, and J. Bulabois, "Gaussian Schell-model sources: an example and some perspectives," J. Opt. Soc. Am. A 73, 256-261 (1983).
[CrossRef]

L. M. Bernardo and O. D. D. Soares, "Fractional Fourier transforms and imaging," J. Opt. Soc. Am. A 11, 2622-2626 (1994).
[CrossRef]

R. Simon and N. Mukunda, "Gaussian Schell-model beams and general shape invariance," J. Opt. Soc. Am. A 16, 2465-2475 (1999).
[CrossRef]

M. A. Kutay and H. M. Ozaktas, "Optimal image restoration with the fractional Fourier transform," J. Opt. Soc. Am. A 15, 825-833 (1998).
[CrossRef]

Y. Zhang, B. Dong, B. Gu, and G. Yang, "Beam shaping in the fractional Fourier transform domain," J. Opt. Soc. Am. A 15, 1114-1120 (1998).
[CrossRef]

A. T. Friberg and J. Turunen, "Imaging of Gaussian Schell-model sources," J. Opt. Soc. Am. A 5, 713-720 (1988).
[CrossRef]

D. Mendlovic and H. M. Ozaktas, "Fractional Fourier transforms and their optical implementation: I," J. Opt. Soc. Am. A 10, 1875-1881 (1993).
[CrossRef]

A. W. Lohmann, "Image rotation, Wigner rotation, and the fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181-2186 (1993).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, "Fractional Fourier transforms and their optical implementation: II," J. Opt. Soc. Am. A 10, 2522-2531 (1993).
[CrossRef]

Y. Cai and Q. Lin, "Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media," J. Opt. Soc. Am. A 19, 2036-2042 (2002).
[CrossRef]

P. J. Cronin, P. Török, P. Varga, and C. Cogswell, "High-aperture diffraction of a scalar, off-axis Gaussian beam," J. Opt. Soc. Am. A 17, 1556-1564 (2000).
[CrossRef]

Y. Cai and Q. Lin, "Decentered elliptical Hermite-Gaussian beam," J. Opt. Soc. Am. A 20, 1111-1119 (2003).
[CrossRef]

Y. Cai and Q. Lin, "Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane," J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

Opt. Commun. (2)

P. D. Santis, F. Gori, G. Guattari, and C. Palma, "An example of a Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Y. Cai and Q. Lin, "Fractional Fourier transform for elliptical Gaussian beams," Opt. Commun. 217, 7-13 (2003).
[CrossRef]

Opt. Laser Technol. (1)

J. Turunen and A. T. Friberg, "Matrix representation of Gaussian Schell-model beams in optical systems," Opt. Laser Technol. 18, 259-267 (1985).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants," Phys. Rev. A 31, 2419-2434 (1985).
[CrossRef] [PubMed]

Quantum Electron. (1)

A. A. Malyutin, "Simple scheme for the astigmatic transformation of laser modes," Quantum Electron. 33, 1015-1018 (2003).
[CrossRef]

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Optical system used to perform the FRT.

Fig. 2
Fig. 2

Normalized intensity profile of a partially coherent linear laser array consisting of five individual off-axis GSM beams with the same separation x d = 2 mm along the x axis in the plane z = 0 with σ 10 = 0.5 mm .

Fig. 3
Fig. 3

Normalized intensity profile of partially coherent off-axis GSM beams in the FRT plane for y = 0 with different transverse coherence widths σ g 0 : (a) σ g 0 = 0.005 mm , (b) σ g 0 = 0.01 mm , (c) σ g 0 = 0.1 mm .

Fig. 4
Fig. 4

Normalized intensity profile of partially coherent off-axis GSM beams in the FRT plane for y = 0 with different fractional orders p: (a) p = 0.2 , (b) p = 0.8 , (c) p = 1.2 , (c) p = 1.8 .

Equations (18)

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Γ 0 ( r 1 , r 2 ) = exp [ r 1 2 + r 2 2 4 σ I 0 2 ( r 1 r 2 ) 2 2 σ g 0 2 ] ,
Γ 0 ( r ̃ ) = exp [ i k 2 r ̃ T M 1 1 r ̃ ] ,
M 1 1 = [ ( i 2 k σ I 0 2 i k σ g 0 2 ) I i k σ g 0 2 I i k σ g 0 2 I ( i 2 k σ I 0 2 i k σ g 0 2 ) I ] ,
Γ 0 ( r ̃ ) = exp [ i k 2 ( r ̃ r ̃ 0 ) T M 1 1 ( r ̃ r ̃ 0 ) ] ,
Γ p ( u 1 , u 2 ) = 1 ( λ f sin ϕ ) 2 Γ 0 ( r 1 , r 2 ) exp [ i π ( r 1 2 + u 1 2 r 2 2 u 2 2 ) λ f tan ϕ ] exp [ 2 π i ( r 1 u 1 r 2 u 2 ) λ f sin ϕ ] d r 1 , d r 2 ,
Γ p ( u ̃ ) = 1 ( λ f sin ϕ ) 2 Γ 0 ( r ̃ ) exp [ i π λ ( r ̃ T N 11 r ̃ + u ̃ T N 11 u ̃ 2 r ̃ T N 12 u ̃ ) ] d 2 r ̃ ,
N 11 = 1 f tan ϕ [ I 0 0 I ] , N 12 = 1 f sin ϕ [ I 0 0 I ] ,
Γ p ( u ̃ ) = 1 ( f sin ϕ ) 2 [ det ( M 1 1 + N 11 ) ] 1 2 exp { i π λ u ̃ T [ N 11 N 12 T ( M 1 1 + N 11 ) 1 N 12 ] u ̃ } × exp [ i k 2 r ̃ 0 T ( M 1 + N 11 ) 1 r ̃ 0 ] exp [ i k r ̃ 0 T ( N 12 1 N 11 M 1 + N 12 1 ) 1 u ̃ ] .
A ¯ = N 12 1 N 11 = cos ϕ [ I 0 0 I ] ,
B ¯ = N 12 1 = f sin ϕ [ I 0 0 I ] ,
C ¯ = N 11 N 12 1 N 11 N 12 T = sin ϕ f [ I 0 0 I ] , D ¯ = N 11 N 12 1 = cos ϕ [ I 0 0 I ] .
Γ p ( u ̃ ) = [ det ( A ¯ + B ¯ M 1 1 ) ] 1 2 exp [ i k 2 u ̃ T M p 1 u ̃ ] exp [ i k 2 r ̃ 0 T ( M 1 + A ¯ 1 B ¯ ) 1 r ̃ 0 ] × exp [ i k r ̃ 0 T ( A ¯ M 1 + B ¯ ) 1 u ̃ ] ,
M p 1 = ( C ¯ + D ¯ M 1 1 ) ( A ¯ + B ¯ M 1 1 ) 1 .
Γ 0 N ( r 1 , r 2 ) = n = N 1 2 N 1 2 exp [ ( r 1 r 0 n ) 2 + ( r 2 r 0 n ) 2 4 σ I 0 2 ( r 1 r 2 ) 2 2 σ g 0 2 ] ,
Γ 0 N ( r ̃ ) = n = N 1 2 N 1 2 exp [ i k 2 ( r ̃ r ̃ 0 n ) T M 1 1 ( r ̃ r ̃ 0 n ) ] ,
Γ 0 N ( r ̃ ) = n = 1 N 2 N 2 exp [ i k 2 ( r ̃ r ̃ 0 n ) T M 1 1 ( r ̃ r ̃ 0 n ) ] ,
Γ p N ( u ̃ ) = n = N 1 2 N 1 2 [ det ( A ¯ + B ¯ M 1 1 ) ] 1 2 exp [ i k 2 u ̃ T M p 1 u ̃ ] exp [ i k 2 r ̃ 0 n T ( M 1 + A ¯ 1 B ¯ ) 1 r ̃ 0 n ] × exp [ i k r ̃ 0 n T ( A ¯ M 1 + B ¯ ) 1 u ̃ ] ,
Γ p N ( u ̃ ) = n = 1 N 2 N 2 [ det ( A ¯ + B ¯ M 1 1 ) ] 1 2 exp [ i k 2 u ̃ T M p 1 u ̃ ] exp [ i k 2 r ̃ 0 n T ( M 1 + A ¯ 1 B ¯ ) 1 r ̃ 0 n ] × exp [ i k r ̃ 0 n T ( A ¯ M 1 + B ¯ ) 1 u ̃ ] .

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