Abstract

The truncated fractional Fourier transform (FRT) is applied to a partially coherent Gaussian Schell-model (GSM) beam. The analytical propagation formula for a partially coherent GSM beam propagating through a truncated FRT optical system is derived by using a tensor method. Furthermore, we report the experimental observation of the truncated FRT for a partially coherent GSM beam. The experimental results are consistent with the theoretical results. Our results show that initial source coherence, fractional order, and aperture width (i.e., truncation parameter) have strong influences on the intensity and coherence properties of the partially coherent beam in the FRT plane. When the aperture width is large, both the intensity and the spectral degree of coherence in the FRT plane are of Gaussian distribution. As the aperture width decreases, the diffraction pattern gradually appears in the FRT plane, and the spectral degree of coherence becomes of non-Gaussian distribution. As the coherence of the initial GSM beam decreases, the diffraction pattern for the case of small aperture widths gradually disappears.

© 2008 Optical Society of America

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    [CrossRef]
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2007 (6)

2006 (5)

2005 (3)

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86, 021112 (2005).
[CrossRef]

Y. Cai and S. Zhu, “Coincidence fractional Fourier transform with partially coherent light radiation,” J. Opt. Soc. Am. A 22, 1798-1804 (2005).
[CrossRef]

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[CrossRef]

2004 (4)

2003 (5)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik (Stuttgart) 14, 504-508 (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]

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

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

2002 (3)

2001 (1)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

1999 (2)

J. Pu, H. Zhang, and S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 162, 57-63 (1999).
[CrossRef]

D. Ding and X. Liu, “Approximate description for Bessel, Bessel-Gauss, and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16, 1286-1293 (1999).
[CrossRef]

1998 (1)

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).
[CrossRef]

1997 (1)

1995 (1)

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

1993 (4)

1988 (1)

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

1986 (1)

1985 (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]

1984 (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

1981 (1)

Y. Li and E. Wolf, “Focal shift in diffracted coverging spherical waves,” Opt. Commun. 39, 211-215 (1981).
[CrossRef]

1973 (1)

J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics, Vol. XI, E.Wolf, ed. (North-Holland, 1973), pp. 247-304.
[CrossRef]

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics, Vol. XI, E.Wolf, ed. (North-Holland, 1973), pp. 247-304.
[CrossRef]

Bastiaans, M. J.

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Cai, Y.

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24, 1937-1944 (2007).
[CrossRef]

Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15, 15480-15492 (2007).
[CrossRef] [PubMed]

Y. Cai and F. Wang, “Lensless optical implementation of the coincidence fractional Fourier transform,” Opt. Lett. 31, 2278-2280 (2007).
[CrossRef]

F. Wang, Y. Cai, and S. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14, 6999-7004 (2006).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, and S. Zhu, “Coincidence subwavelength fractional Fourier transform,” J. Opt. Soc. Am. A 23, 835-841 (2006).
[CrossRef]

Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31, 685-687 (2006).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86, 021112 (2005).
[CrossRef]

Y. Cai and S. Zhu, “Coincidence fractional Fourier transform with partially coherent light radiation,” J. Opt. Soc. Am. A 22, 1798-1804 (2005).
[CrossRef]

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[CrossRef]

Y. Cai and S. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716-2718 (2004).
[CrossRef] [PubMed]

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

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

Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian Schell-model beams,” Opt. Lett. 27, 1672-1674 (2002).
[CrossRef]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian Schell-model beams,” Opt. Lett. 27, 216-218 (2002).
[CrossRef]

Chen, J.

Chu, X.

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87, 547-552 (2007).
[CrossRef]

Davidson, F. M.

Ding, D.

Du, X.

He, S.

Hu, L.

Jing, F.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik (Stuttgart) 14, 504-508 (2003).
[CrossRef]

Kato, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Kitagawa, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Korotkova, O.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Li, Y.

Y. Li, “Focal shift and focal switch in dual focus systems,” J. Opt. Soc. Am. A 14, 1297-1304 (1997).
[CrossRef]

Y. Li and E. Wolf, “Focal shift in diffracted coverging spherical waves,” Opt. Commun. 39, 211-215 (1981).
[CrossRef]

Lin, Q.

Liu, H.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik (Stuttgart) 14, 504-508 (2003).
[CrossRef]

Liu, X.

Lohmann, A. W.

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).
[CrossRef]

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

Mandel, L.

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

Mao, H.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik (Stuttgart) 14, 504-508 (2003).
[CrossRef]

Medlovic, D.

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).
[CrossRef]

Mei, Z.

Mendlovic, D.

Mima, K.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Miyanaga, N.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Mukunda, N.

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95-109 (1993).
[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]

Nakatsuka, M.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Nemoto, S.

J. Pu, H. Zhang, and S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 162, 57-63 (1999).
[CrossRef]

Ni, Y.

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87, 547-552 (2007).
[CrossRef]

Ozaktas, H. M.

Peschel, U.

Pu, J.

J. Pu, H. Zhang, and S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 162, 57-63 (1999).
[CrossRef]

Ricklin, J. C.

Salem, M.

Shen, M.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

Simon, R.

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95-109 (1993).
[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]

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]

Wang, D.

Wang, F.

Wei, X.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik (Stuttgart) 14, 504-508 (2003).
[CrossRef]

Wen, J. J.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Wolf, E.

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29, 1173-1175 (2004).
[CrossRef] [PubMed]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

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

Y. Li and E. Wolf, “Focal shift in diffracted coverging spherical waves,” Opt. Commun. 39, 211-215 (1981).
[CrossRef]

Yamanaka, C.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Yan, J.

Yu, Y.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).
[CrossRef]

Zhang, H.

J. Pu, H. Zhang, and S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 162, 57-63 (1999).
[CrossRef]

Zhao, D.

X. Du and D. Zhao, “Fractional Fourier transform of truncated elliptical Gaussian beam,” Appl. Opt. 45, 9049-9052 (2006).
[CrossRef] [PubMed]

Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21, 2375-2381 (2004).
[CrossRef]

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik (Stuttgart) 14, 504-508 (2003).
[CrossRef]

Zheng, C.

C. Zheng, “Fractional Fourier transform for a hollow Gaussian beam,” Phys. Lett. A 355, 156-161 (2006).
[CrossRef]

Zhou, G.

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87, 547-552 (2007).
[CrossRef]

Zhu, Q.

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of Hermite-Gaussian beams in apertured fractional Fourier transforming systems,” Optik (Stuttgart) 14, 504-508 (2003).
[CrossRef]

Zhu, S.

Appl. Opt. (1)

Appl. Phys. B (1)

X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B 87, 547-552 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phys. Lett. 86, 021112 (2005).
[CrossRef]

Chin. Opt. Lett. (1)

J. Acoust. Soc. Am. (1)

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

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

D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, and X. Wei, “Propagation of flattened Gaussian beams in apertured fractional Fourier transforming systems,” J. Opt. A, Pure Appl. Opt. 6, 148-154 (2004).
[CrossRef]

Y. Cai and Q. Lin, “Properties of 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 (14)

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Figures (9)

Fig. 1
Fig. 1

Optical system for performing the truncated FRT for a light beam.

Fig. 2
Fig. 2

Normalized intensity distribution (cross line ν = 0 ) of the partially coherent GSM beam in the FRT plane for different values of the fractional order p.

Fig. 3
Fig. 3

Experimental setup for observing the truncated FRT for a partially coherent GSM beam.

Fig. 4
Fig. 4

Experimental results of (a) the intensity distribution of the partially coherent beam, and (b) the corresponding normalized intensity distribution at cross line y = 0 in the input plane (dotted curve). The solid curve is the result of a Gaussian fit of the experimental data.

Fig. 5
Fig. 5

Experimental result of the modulus of the square of the spectral degree of coherence g 2 ( x 1 x 2 ) (along x 1 x 2 ) for the partially coherent GSM beam in the input plane. The solid curve is the result of a Gaussian fit of the experimental data.

Fig. 6
Fig. 6

Experimental results (dotted curves) of the normalized intensity distribution (cross line ν = 0 ) of the partially coherent GSM beam in the FRT plane for different values of the aperture radius.

Fig. 7
Fig. 7

Experimental results (dotted curves) of the normalized intensity distribution (cross line ν = 0 ) of the partially coherent GSM beam in the FRT plane for different values of the initial coherence width σ g 0 .

Fig. 8
Fig. 8

Experimental results (dotted curves) of the normalized intensity distribution (cross line ν = 0 ) of the partially coherent GSM beam in the FRT plane for different values of fractional order p.

Fig. 9
Fig. 9

Experimental results (dotted curves) of the modulus of the square of the spectral degree of coherence of the partially coherent GSM beam in the FRT plane for different values of aperture radius a and initial coherence width σ g 0 .

Equations (31)

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Γ l ( ρ ̃ ) = Γ l ( ρ 1 , ρ 2 ) = 1 λ 2 [ det ( B ¯ 1 ) ] 1 2 Γ i ( r 1 , r 2 ) × exp { i k 2 [ r ̃ T B ¯ 1 1 A ¯ 1 r ̃ 2 r ̃ T B ¯ 1 1 ρ ̃ + ρ ̃ T D ¯ 1 B ¯ 1 1 ρ ̃ ] } d r 1 d r 2 ,
A ¯ 1 = I ¯ , B ¯ 1 = f tan ( ϕ 2 ) [ I 0 0 I ] , C ¯ 1 = 0 I ¯ , D ¯ 1 = I ¯ ,
Γ f ( u ̃ ) = Γ f ( u 1 , u 2 ) = 1 λ 2 [ det ( B ¯ 2 ) ] 1 2 H ( ρ 1 ) H * ( ρ 2 ) Γ l ( ρ 1 , ρ 2 ) exp { i k 2 [ ρ ̃ T B ¯ 2 1 A ¯ 2 ρ ̃ 2 ρ ̃ T B ¯ 2 1 u ̃ + u ̃ T D ¯ 2 B ¯ 2 1 u ̃ ] } d ρ 1 d ρ 2 ,
A ¯ 2 = cos ϕ I ¯ , B ¯ 2 = f tan ( ϕ 2 ) [ I 0 0 I ] ,
C ¯ 2 = sin ϕ f [ I 0 0 I ] , D ¯ 2 = I ¯ .
Γ i ( r ̃ ) = exp ( i k 2 r ̃ T M i 1 r ̃ ) ,
M i 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 l ( ρ ̃ ) = Γ l ( ρ 1 , ρ 2 ) = [ det ( A ¯ 1 + B ¯ 1 M i 1 ) ] 1 2 exp [ i k 2 ρ ̃ T M l 1 ρ ̃ ] ,
M l 1 = ( C ¯ 1 + D ¯ 1 M i 1 ) ( A ¯ 1 + B ¯ 1 M i 1 ) 1 = ( B ¯ 1 + M i ) 1 .
H ( ρ ) = { 1 , ρ a 0 , ρ > a } .
H ( ρ ) = m = 1 M A m exp ( B m a 2 ρ 2 ) ,
H ( ρ 1 ) H * ( ρ 2 ) = m = 1 M n = 1 M A m A n * exp [ i k 2 ρ ̃ T B mn ρ ̃ ] ,
B mn = 2 i k a 2 [ B m I 0 0 B n * I ] ,
Γ f ( u ̃ ) = m = 1 M n = 1 M A m A n * [ det ( A ¯ 2 + B ¯ 2 ( B ¯ 1 + M i ) 1 + B ¯ 2 B mn ) ] 1 2 × [ det ( I ¯ + B ¯ 1 M i 1 ) ] 1 2 exp [ i k 2 u ̃ T M fmn 1 u ̃ ] ,
M fmn 1 = [ C ¯ 2 + ( B ¯ 1 + M i ) 1 + B mn ] [ A ¯ 2 + B ¯ 2 ( B ¯ 1 + M i ) 1 + B ¯ 2 B mn ] 1 .
H ( ρ ) = H ( ρ x , ρ y ) = { 1 , ρ x a , ρ y b 0 , ρ x > a , ρ y > b } ,
H ( ρ x , ρ y ) = m = 1 M A m exp ( B m a 2 ρ x 2 ) n = 1 M A n exp ( B n b 2 ρ y 2 ) ,
H ( ρ 1 ) H * ( ρ 2 ) = m = 1 M n = 1 M p = 1 M l = 1 M A m A n A p * A l * exp [ i k 2 ρ ̃ T B mnpl ρ ̃ ] ,
B mnpl = 2 i k [ B m a 2 0 0 0 0 B n b 2 0 0 0 0 B p * a 2 0 0 0 0 B l * b 2 ] .
Γ f ( u ̃ ) = m = 1 M n = 1 M p = 1 M l = 1 M A m A n A p * A l * [ det ( A ¯ 2 + B ¯ 2 ( B ¯ 1 + M i ) 1 + B ¯ 2 B mnpl ) ] 1 2 [ det ( I ¯ + B ¯ 1 M i 1 ) ] 1 2 × exp [ i k 2 u ̃ T M fmnpl 1 u ̃ ] ,
M fmnpl 1 = [ C ¯ 2 + ( B ¯ 1 + M i ) 1 + B mnpl ] [ A ¯ 2 + B ¯ 2 ( B ¯ 1 + M i ) 1 + B ¯ 2 B mnpl ] 1 .
g ( u 1 u 2 ) = Γ f ( u 1 , u 2 ) Γ f ( u 1 , u 1 ) Γ f ( u 2 , u 2 ) .
g ( r 1 r 2 ) = exp [ ( r 1 r 2 ) 2 2 σ g 0 2 ] .
Γ f ( u ̃ ) = Γ f ( u 1 , u 2 ) = 1 λ 2 [ det ( B ¯ 2 ) ] 1 2 [ det ( A ¯ 1 + B ¯ 1 M i 1 ) ] 1 2 m = 1 M n = 1 M A m A n * exp [ i k 2 u ̃ T D ¯ 2 B ¯ 2 1 u ̃ ] × exp [ i k 2 ρ ̃ T ( M l 1 + B ¯ 2 1 + B mn ) ρ ̃ + i k ρ ̃ T B ¯ 2 1 u ̃ ] d ρ 1 d ρ 2 = 1 λ 2 [ det ( B ¯ 2 ) ] 1 2 [ det ( A ¯ 1 + B ¯ 1 M i 1 ) ] 1 2 m = 1 M n = 1 M A m A n * exp [ i k 2 u ̃ T D ¯ 2 B ¯ 2 1 u ̃ ] × exp [ i k 2 u ̃ T B ¯ 2 1 T ( M l 1 + B ¯ 2 1 + B mn ) 1 B ¯ 2 1 u ̃ ] × exp [ i k 2 ( M l 1 + B ¯ 2 1 + B mn ) 1 2 ρ ̃ ( M l 1 + B ¯ 2 1 + B mn ) 1 2 B ¯ 2 1 u ̃ 2 ] d ρ 1 d ρ 2 .
exp ( a x 2 ) d x = π a ,
Γ f ( u ̃ ) = m = 1 M n = 1 M A m A n * [ det ( B ¯ 2 ) ] 1 2 [ det ( A ¯ 1 + B ¯ 1 M i 1 ) ] 1 2 [ det ( M l 1 + B ¯ 2 1 A 2 + B mn ) ] 1 2 exp [ i k 2 u ̃ T D ¯ 2 B ¯ 2 1 u ̃ ] exp [ i k 2 u ̃ T B ¯ 2 1 T ( M l 1 + B ¯ 2 1 A 2 + B mn ) 1 B ¯ 2 1 u ̃ ] .
[ det ( B ¯ 2 ) ] 1 2 [ det ( A ¯ 1 + B ¯ 1 M i 1 ) ] 1 2 [ det ( M l 1 + B ¯ 2 1 A ¯ 2 + B mn ) ] 1 2 = [ det ( A ¯ 1 + B ¯ 1 M i 1 ) ] 1 2 [ det ( B ¯ 2 M l 1 + A ¯ 2 + B ¯ 2 B mn ) ] 1 2 = [ det ( I ¯ + B ¯ 1 M i 1 ) ] 1 2 [ det ( A ¯ 2 + B ¯ 2 ( B ¯ 1 + M i ) 1 + B ¯ 2 B mn ) ] 1 2 ,
B ¯ 2 1 T ( M l 1 + B ¯ 2 1 A ¯ 2 + B mn ) 1 B ¯ 2 1 + D ¯ 2 B ¯ 2 1 = B ¯ 2 1 T ( B ¯ 2 M l 1 + A ¯ 2 + B ¯ 2 B mn ) 1 + D ¯ 2 B ¯ 2 1 = [ B ¯ 2 1 T + D ¯ 2 B ¯ 2 1 ( B ¯ 2 M l 1 + A ¯ 2 + B ¯ 2 B mn ) ] ( B ¯ 2 M l 1 + A ¯ 2 + B ¯ 2 B mn ) 1 = ( C ¯ 2 + D ¯ 2 M l 1 + D ¯ 2 B mn ) ( B ¯ 2 M l 1 + A ¯ 2 + B ¯ 2 B mn ) 1 = [ C ¯ 2 + ( B ¯ 1 + M i ) 1 + B mn ] [ B ¯ 2 ( B ¯ 1 + M i ) 1 + A ¯ 2 + B ¯ 2 B mn ] 1 ,
M fmn 1 = [ C ¯ 2 + ( B ¯ 1 + M i ) 1 + B mn ] [ B ¯ 2 ( B ¯ 1 + M i ) 1 + A ¯ 2 + B ¯ 2 B mn ] 1 ,
( B ¯ 2 1 A ¯ 2 ) T = B ¯ 2 1 A ¯ 2 , ( D ¯ 2 B ¯ 2 1 ) T = D ¯ 2 B ¯ 2 1 ,
C ¯ 2 D ¯ 2 B ¯ 2 1 A ¯ 2 = ( B ¯ 2 1 ) T .

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