Abstract

Propagation of stochastic electromagnetic beams through paraxial ABCD optical systems operating through turbulent atmosphere is investigated with the help of the ABCD matrices and the generalized Huygens-Fresnel integral. In particular, the analytic formula is derived for the cross-spectral density matrix of an electromagnetic Gaussian Schell-model (EGSM) beam. We applied our analysis for the ABCD system with a single lens located on the propagation path, representing, in a particular case, the unfolded double-pass propagation scenario of active laser radar. Through a number of numerical examples we investigated the effect of local turbulence strength and lens’ parameters on spectral, coherence and polarization properties of the EGSM beam.

© 2008 Optical Society of America

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  48. Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, "Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere," Opt. Commun. 278, 157-167 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
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2008

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, "Realizability condition for electromagnetic Schell-model sources," J. Opt. Soc. Am. A 25, 1016-1021 (2008).
[CrossRef]

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

2007

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

H. Wang, X. Wang, A. Zeng, and K. Yang, "Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation," Opt. Lett. 32, 2215-2217 (2007).
[CrossRef] [PubMed]

W. Gao and O. Korotkova, "Changes in the state of polarization of a random electromagnetic beam propagating through tissue," Opt. Comm. 270, 474-478 (2007).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, "Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere," Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, "Scintillation index of elliptical Gaussian beam in turbulent atmosphere," Opt. Lett. 32, 2405-2407 (2007).
[CrossRef] [PubMed]

R. J. Noriega-Manez and J. C. Gutierrez-Vega, "Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere," Opt. Express 15, 16328-16341 (2007).
[CrossRef] [PubMed]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Complex degree of coherence for partially coherent general beams in atmospheric turbulence," J. Opt. Soc. Am. A 24, 2891-2901 (2007).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).

X. Chu, "Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere," Opt. Express 15, 17613-17618 (2007).
[CrossRef] [PubMed]

X. Du, D. Zhao, and O. Korotkova, "Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere," Opt. Express 15, 16909-16915 (2007).
[CrossRef] [PubMed]

2006

2005

Y. Cai and S. Zhu, "Ghost imaging with incoherent and partially coherent light radiation," Phys. Rev. E 71, 056607 (2005).

T. Shirai, O. Korotkova, and E. Wolf, "A method of generating electromagnetic Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 7, 232-237 (2005).
[CrossRef]

H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 249, 379-385 (2005).
[CrossRef]

D. Ge, Y. Cai and Q. Lin, "Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems," Chin. Phys. 14, 128-132 (2005).
[CrossRef]

O Korotkova, E Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation", Opt. Commun. 246, 35-43 (2005).
[CrossRef]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

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

2004

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]

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

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

H. T. Eyyuboglu and Y. Baykal, "Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere," Opt. Express 12, 4659-4674 (2004).
[CrossRef] [PubMed]

2003

T. Shirai, A. Dogariu, and E. Wolf, "Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 20, 1094-1102 (2003).
[CrossRef]

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).

Y. Cai, D. Ge, and Q. Lin, "Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams," J. Opt. A: Pure App. Opt. 5, 453-459 (2003).
[CrossRef]

2002

2001

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 3, 1-9 (2001).
[CrossRef]

1998

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

1993

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silve-halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

1992

E. Tervonen, A. T. Friberg, and J. Turunen, "Gaussian Schell-model beams generated with synthetic acousto-optic holograms," J. Opt. Soc. Am. A. 9, 796-803 (1992).
[CrossRef]

1990

Q. Lin, S. Wang, J. Alda, E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik 85, 67 (1990).

1989

H. T. Yura and S. G. Hanson, "Second-order statistics for wave propagation through complex optical systems," J. Opt. Soc. Am A 6, 564 (1989).
[CrossRef]

1988

Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

1987

H. T. Yura and S. G. Hanson, "Optical beam wave propagation through complex optical systems," J. Opt. Soc. Am A 4, 1931 (1987).
[CrossRef]

1985

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).

1984

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]

1982

A. T. Friberg and R. J. Sudol, "Propagation parameters of Gaussian Schell-model beams," Opt. Commun 41, 383-387 (1982).
[CrossRef]

1979

1978

J. C. Leader, "Atmospheric propagation of partially coherent radiation", J. Opt. Soc. Am. 68, 175-185 (1978).
[CrossRef]

E. Wolf and E. Collett, "Partially coherent sources which produce same far-field intensity distribution as a laser," Opt. Commun. 25, 293-296 (1978).
[CrossRef]

F. Gori, "Collet-Wolf sources and multimode lasers," Opt. Commun. 34, 301-305 (1978).
[CrossRef]

1975

D. Kermisch, "Partially coherent image processing by laser scanning," J. Opt. Soc. Am 65, 887-891 (1975).
[CrossRef]

1970

T. L. Ho, "Coherence degradation of Gaussian beams in a turbulent atmosphere," J. Opt. Soc. Am 60, 667-673(1970).
[CrossRef]

Alda, J.

Q. Lin, S. Wang, J. Alda, E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik 85, 67 (1990).

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]

Banakh, V. A.

Baykal, Y.

Belendez, A.

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silve-halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

Bernabeu, E.

Q. Lin, S. Wang, J. Alda, E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik 85, 67 (1990).

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, "Realizability condition for electromagnetic Schell-model sources," J. Opt. Soc. Am. A 25, 1016-1021 (2008).
[CrossRef]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 3, 1-9 (2001).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

Cai, Y.

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

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Complex degree of coherence for partially coherent general beams in atmospheric turbulence," J. Opt. Soc. Am. A 24, 2891-2901 (2007).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, "Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere," Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, "Scintillation index of elliptical Gaussian beam in turbulent atmosphere," Opt. Lett. 32, 2405-2407 (2007).
[CrossRef] [PubMed]

Y. Cai and S. He, "Propagation of various dark hollow beams in a turbulent atmosphere," Opt. Express 14, 1353-1367 (2006).
[CrossRef] [PubMed]

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
[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 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. Phy. Lett. 86, 021112 (2005).
[CrossRef]

Y. Cai and S. Zhu, "Ghost imaging with incoherent and partially coherent light radiation," Phys. Rev. E 71, 056607 (2005).

D. Ge, Y. Cai and Q. Lin, "Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems," Chin. Phys. 14, 128-132 (2005).
[CrossRef]

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

Y. Cai, D. Ge, and Q. Lin, "Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams," J. Opt. A: Pure App. Opt. 5, 453-459 (2003).
[CrossRef]

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

Carretero, L.

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silve-halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

Chen, Y.

Chu, X.

Collett, E.

E. Wolf and E. Collett, "Partially coherent sources which produce same far-field intensity distribution as a laser," Opt. Commun. 25, 293-296 (1978).
[CrossRef]

Davidson, F. M.

Dogariu, A.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

T. Shirai, A. Dogariu, and E. Wolf, "Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 20, 1094-1102 (2003).
[CrossRef]

Du, X.

Eyyuboglu, H. T.

Fimia, A.

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silve-halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

Friberg, A. T.

E. Tervonen, A. T. Friberg, and J. Turunen, "Gaussian Schell-model beams generated with synthetic acousto-optic holograms," J. Opt. Soc. Am. A. 9, 796-803 (1992).
[CrossRef]

Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

A. T. Friberg and R. J. Sudol, "Propagation parameters of Gaussian Schell-model beams," Opt. Commun 41, 383-387 (1982).
[CrossRef]

Gao, W.

W. Gao and O. Korotkova, "Changes in the state of polarization of a random electromagnetic beam propagating through tissue," Opt. Comm. 270, 474-478 (2007).
[CrossRef]

Ge, D.

D. Ge, Y. Cai and Q. Lin, "Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems," Chin. Phys. 14, 128-132 (2005).
[CrossRef]

Y. Cai, D. Ge, and Q. Lin, "Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams," J. Opt. A: Pure App. Opt. 5, 453-459 (2003).
[CrossRef]

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, "Realizability condition for electromagnetic Schell-model sources," J. Opt. Soc. Am. A 25, 1016-1021 (2008).
[CrossRef]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 3, 1-9 (2001).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

F. Gori, "Collet-Wolf sources and multimode lasers," Opt. Commun. 34, 301-305 (1978).
[CrossRef]

Guattari, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

Gureyev, T. E.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

Gutierrez-Vega, J. C.

Hanson, S. G.

H. T. Yura and S. G. Hanson, "Second-order statistics for wave propagation through complex optical systems," J. Opt. Soc. Am A 6, 564 (1989).
[CrossRef]

H. T. Yura and S. G. Hanson, "Optical beam wave propagation through complex optical systems," J. Opt. Soc. Am A 4, 1931 (1987).
[CrossRef]

He, Q. S.

Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

He, S.

Ho, T. L.

T. L. Ho, "Coherence degradation of Gaussian beams in a turbulent atmosphere," J. Opt. Soc. Am 60, 667-673(1970).
[CrossRef]

Hu, L.

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]

Kermisch, D.

D. Kermisch, "Partially coherent image processing by laser scanning," J. Opt. Soc. Am 65, 887-891 (1975).
[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

O Korotkova, E Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation", Opt. Commun. 246, 35-43 (2005).
[CrossRef]

Korotkova, O.

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

W. Gao and O. Korotkova, "Changes in the state of polarization of a random electromagnetic beam propagating through tissue," Opt. Comm. 270, 474-478 (2007).
[CrossRef]

X. Du, D. Zhao, and O. Korotkova, "Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere," Opt. Express 15, 16909-16915 (2007).
[CrossRef] [PubMed]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 249, 379-385 (2005).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, "A method of generating electromagnetic Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 7, 232-237 (2005).
[CrossRef]

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]

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

Leader, J. C.

Lin, Q.

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, "Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere," Opt. Commun. 278, 157-167 (2007).
[CrossRef]

D. Ge, Y. Cai and Q. Lin, "Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems," Chin. Phys. 14, 128-132 (2005).
[CrossRef]

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

Y. Cai, D. Ge, and Q. Lin, "Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams," J. Opt. A: Pure App. Opt. 5, 453-459 (2003).
[CrossRef]

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

Q. Lin, S. Wang, J. Alda, E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik 85, 67 (1990).

Mayo, S. C.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

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]

Mironov, V. L.

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]

Mondello, A.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 3, 1-9 (2001).
[CrossRef]

Mukunda, N.

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).

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]

Noriega-Manez, R. J.

Paganin, D. M.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

Peschel, U.

Piquero, G.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 3, 1-9 (2001).
[CrossRef]

Plonus, M. A.

Ponomarenko, S. A.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

Ramírez-Sánchez, V.

Ricklin, J. C.

Romanini, P.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
[CrossRef]

Roychowdhury, H.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 249, 379-385 (2005).
[CrossRef]

Salem, M.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

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]

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, "Realizability condition for electromagnetic Schell-model sources," J. Opt. Soc. Am. A 25, 1016-1021 (2008).
[CrossRef]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 3, 1-9 (2001).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

Shirai, T.

T. Shirai, O. Korotkova, and E. Wolf, "A method of generating electromagnetic Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 7, 232-237 (2005).
[CrossRef]

T. Shirai, A. Dogariu, and E. Wolf, "Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 20, 1094-1102 (2003).
[CrossRef]

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 3, 1-9 (2001).
[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).

Stevenson, A. W.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[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).

Sudol, R. J.

A. T. Friberg and R. J. Sudol, "Propagation parameters of Gaussian Schell-model beams," Opt. Commun 41, 383-387 (1982).
[CrossRef]

Tervonen, E.

E. Tervonen, A. T. Friberg, and J. Turunen, "Gaussian Schell-model beams generated with synthetic acousto-optic holograms," J. Opt. Soc. Am. A. 9, 796-803 (1992).
[CrossRef]

Turunen, J.

E. Tervonen, A. T. Friberg, and J. Turunen, "Gaussian Schell-model beams generated with synthetic acousto-optic holograms," J. Opt. Soc. Am. A. 9, 796-803 (1992).
[CrossRef]

Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

Wang, F.

Wang, H.

Wang, S.

Q. Lin, S. Wang, J. Alda, E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik 85, 67 (1990).

Wang, S. C. H.

Wang, X.

Wilkin, S. W.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

Wolf, E

O Korotkova, E Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation", Opt. Commun. 246, 35-43 (2005).
[CrossRef]

Wolf, E.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, "A method of generating electromagnetic Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 7, 232-237 (2005).
[CrossRef]

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]

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).

T. Shirai, A. Dogariu, and E. Wolf, "Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 20, 1094-1102 (2003).
[CrossRef]

E. Wolf and E. Collett, "Partially coherent sources which produce same far-field intensity distribution as a laser," Opt. Commun. 25, 293-296 (1978).
[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]

Yang, K.

Yura, H. T.

H. T. Yura and S. G. Hanson, "Second-order statistics for wave propagation through complex optical systems," J. Opt. Soc. Am A 6, 564 (1989).
[CrossRef]

H. T. Yura and S. G. Hanson, "Optical beam wave propagation through complex optical systems," J. Opt. Soc. Am A 4, 1931 (1987).
[CrossRef]

Zeng, A.

Zhao, D.

Zhu, S.

Y. Cai and S. Zhu, "Ghost imaging with incoherent and partially coherent light radiation," Phys. Rev. E 71, 056607 (2005).

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

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

Appl. Phy. Lett.

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

Appl. Phys. B

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).

Appl. Phys. Lett.

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

Chin. Phys.

D. Ge, Y. Cai and Q. Lin, "Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems," Chin. Phys. 14, 128-132 (2005).
[CrossRef]

J. Mod. Opt.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

J. Opt. A: Pure App. Opt.

Y. Cai, D. Ge, and Q. Lin, "Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams," J. Opt. A: Pure App. Opt. 5, 453-459 (2003).
[CrossRef]

J. Opt. A: Pure Appl. Opt.

T. Shirai, O. Korotkova, and E. Wolf, "A method of generating electromagnetic Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 7, 232-237 (2005).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A: Pure Appl. Opt. 3, 1-9 (2001).
[CrossRef]

J. Opt. Soc. Am

D. Kermisch, "Partially coherent image processing by laser scanning," J. Opt. Soc. Am 65, 887-891 (1975).
[CrossRef]

T. L. Ho, "Coherence degradation of Gaussian beams in a turbulent atmosphere," J. Opt. Soc. Am 60, 667-673(1970).
[CrossRef]

J. Opt. Soc. Am A

H. T. Yura and S. G. Hanson, "Optical beam wave propagation through complex optical systems," J. Opt. Soc. Am A 4, 1931 (1987).
[CrossRef]

H. T. Yura and S. G. Hanson, "Second-order statistics for wave propagation through complex optical systems," J. Opt. Soc. Am A 6, 564 (1989).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A.

E. Tervonen, A. T. Friberg, and J. Turunen, "Gaussian Schell-model beams generated with synthetic acousto-optic holograms," J. Opt. Soc. Am. A. 9, 796-803 (1992).
[CrossRef]

Opt. Comm.

W. Gao and O. Korotkova, "Changes in the state of polarization of a random electromagnetic beam propagating through tissue," Opt. Comm. 270, 474-478 (2007).
[CrossRef]

Opt. Commun

A. T. Friberg and R. J. Sudol, "Propagation parameters of Gaussian Schell-model beams," Opt. Commun 41, 383-387 (1982).
[CrossRef]

Opt. Commun.

Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

E. Wolf and E. Collett, "Partially coherent sources which produce same far-field intensity distribution as a laser," Opt. Commun. 25, 293-296 (1978).
[CrossRef]

F. Gori, "Collet-Wolf sources and multimode lasers," Opt. Commun. 34, 301-305 (1978).
[CrossRef]

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silve-halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 249, 379-385 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

O Korotkova, E Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation", Opt. Commun. 246, 35-43 (2005).
[CrossRef]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi and A. Mondello, "Synthesis of partially polarized Gaussian Schell-model sources," Opt. Commun. 208, 9-16 (2002).
[CrossRef]

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

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, "Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere," Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Optik

Q. Lin, S. Wang, J. Alda, E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik 85, 67 (1990).

Phys. Lett. A

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).

Phys. Rev. A

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).

Phys. Rev. E

Y. Cai and S. Zhu, "Ghost imaging with incoherent and partially coherent light radiation," Phys. Rev. E 71, 056607 (2005).

Phys. Rev. Lett.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, "Generalized eikonal of partially coherent beams and its use in quantitative imaging," Phys. Rev. Lett. 93, 068103 (2004).
[CrossRef] [PubMed]

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]

Pure Appl. Opt.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi and G. Guattari, "Beam coherence-polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

Waves Random Complex Media

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

Other

E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).

C. Brosseau, Fundamentals of polarized light-a statistical approach (Wiley, New York, 1998).

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

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1964).

V. A. Banakh and L. V. Mironov, LIDAR in a turbulent atmosphere (Artech House, Dedham 1987).

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

A. Gerrard and J. M. Burch, Introduction to matrix methods in optics (John Wiley and Sons, 1975).

A. Ishimaru, Wave propagation and scattering in random media, (Academic Press, New York, 1978) Vol. 2.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Washington, 2001).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

(a). Focusing geometry, (b). Schematic of laser radar configuration

Fig. 2.
Fig. 2.

Normalized intensity distribution and corresponding cross line (y=0) of an EGSM beam at the geometrical focal plane for three different values of the structure constant of turbulent atmosphere.

Fig. 3.
Fig. 3.

Normalized intensity distribution (cross line, y=0) of an EGSM beam at the geometrical focal plane for different values of the structure constant of turbulent atmosphere and the source correlation coefficients

Fig. 4.
Fig. 4.

Degree of polarization and corresponding cross line (y=0) of an EGSM beam at the geometrical focal plane for three different values of the structure constant of turbulent atmosphere.

Fig. 5.
Fig. 5.

Degree of polarization (cross line, y=0) of an EGSM beam at the geometrical focal plane for different values of the structure constant of turbulent atmosphere and the source correlation coefficients.

Fig. 6.
Fig. 6.

Spectral degree of coherence and corresponding cross line (y1-y2=0) of an EGSM beam at the geometrical focal plane for three different values of the structure constant of turbulent atmosphere.

Fig. 7.
Fig. 7.

Spectral degree of coherence (cross line, y1-y2=0) of an EGSM beam at the geometrical focal lane for different values of the structure constant of turbulent atmosphere and the source correlation coefficients.

Equations (42)

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E ( ρ 1 , l ) = i λ [ det ( B ) ] 1 / 2 E ( r 1 , 0 )
× exp [ i k 2 ( r 1 T B 1 A r 1 2 r 1 T B 1 ρ 1 + ρ 1 T D B 1 ρ 1 ) + Ψ ( r 1 , ρ 1 ) ] d r 1 ,
( B 1 A ) T = B 1 A , ( B 1 ) T = ( C D B 1 A ) , ( D B 1 ) T = D B 1 .
W ( r 1 , r 2 0 ) = E ( r 1 ,0 ) E * ( r 2 ,0 ) , W ( ρ 1 , ρ 2 , l ) = E ( ρ 1 , l ) E * ( ρ 2 ,l ) .
W ( ρ 1 , ρ 2 , l ) = 1 λ 2 [ det ( B ) ] 1 / 2 [ det ( B * ) ] 1 / 2 W ( r 1 , r 2 , 0 ) exp [ i k 2 ( r 1 T B 1 Ar 1 2 r 1 T B 1 ρ 1 + ρ 1 T D B 1 ρ 1 ) ]
× exp [ i k 2 ( r 2 T ( B * ) 1 A * r 2 2 r 2 T ( B * ) 1 ρ 2 + ρ 2 T D * ( B * ) 1 ρ 2 ) ] exp [ Ψ ( r 1 , ρ 1 ) + Ψ * ( r 2 , ρ 2 ) ] d r 1 d r 2 ,
exp [ ψ ( r 1 , ρ 1 ) + ψ * ( r 2 , ρ 2 ) ] = exp [ ( r 1 r 2 ) 2 ρ 0 2 ( r 1 r 2 ) ( ρ 1 ρ 2 ) ρ 0 2 ( ρ 1 ρ 2 ) 2 ρ 0 2 ] ,
ρ 0 = Det [ B ] 1 2 ( 1 . 46 k 2 C n 2 0 l Det [ B ( z ) ] 5 6 d z ) 3 5 ,
W ( ρ , l ) = k 2 4 π 2 [ det ( B ) ] 1 2 W ( r , 0 ) exp [ i k 2 ( r T B 1 A r 2 r T B 1 ρ + ρ T D B 1 ρ ) ]
× exp [ i k 2 r T P r i k 2 r T P ρ i k 2 ρ T P ρ ] d r ,
A = ( A 0 I 0 I A * ) , B = ( B 0 I 0 I B * ) , C = ( C 0 I 0 I C * ) , D = ( D 0 I 0 I D * ) , P = 2 i k ρ 0 2 ( I I I I ) ,
W a β ( r 1 , r 2 , 0 ) = A a A β B a β exp [ r 1 2 4 σ a 2 r 2 2 4 σ β 2 ( r 1 r 2 ) 2 2 δ a β 2 ] , ( a = x , y ; β = x , y )
W a β ( r ) = A α A β B a β exp [ i k 2 r T M 0 α β 1 r ] , ( α = x , y ; β = x , y ) ,
M 0 α β 1 = [ 1 i k ( 1 2 σ a 2 + 1 δ α β 2 ) I i k δ α β 2 I i k δ α β 2 I 1 i k ( 1 2 σ β 2 + 1 δ α β 2 ) I ] ,
W α β ( ρ , l ) = A α A β B α β [ det ( A + B M 0 α β 1 + B P ) ] 1 2 exp [ i k 2 ρ T M 1 α β 1 ρ ]
exp [ i k 2 ρ T P ρ i k 2 ρ T ( B 1 T 1 4 P T ) ( M 0 α β 1 + B 1 A + P ) 1 P ρ ] ,
M 1 α β 1 = ( C + D M 0 α β 1 + D P ) ( A + B M 0 α β 1 + B P ) 1 ,
M 0 α β 1 = ( 1 2 i k ( σ a 2 ) 1 + 1 i k ( δ α β 2 ) 1 i k ( δ α β 2 ) 1 i k ( δ α β 2 ) 1 1 2 i k ( σ β 2 ) 1 + 1 i k ( δ α β 2 ) 1 ) ,
( A B C D ) = ( I l I 0 I ) ,
W α β ( ρ , l ) = A α A β B α β [ det ( I + B M 0 α β 1 + B P ) ] 1 2 exp [ i k 2 ρ T ( P + B ) ρ ]
× exp [ i k 2 ρ T ( B 1 1 2 P ) T ( M 1 1 + B 1 + P ) 1 ( B 1 1 2 P ) ρ ] ,
( A B C D ) = ( I f I 0 I ) ( I 0 I ( 1 f ) I I ) ( I l 1 I 0 I I ) = ( 0 I f I ( 1 f ) I ( 1 l 1 f ) I ) .
( A ( z ) B ( z ) C ( z ) D ( z ) ) = ( I ( l 1 z ) I 0 I ) ( I 0 I ( 1 f ) I I ) ( I f I 0 I I ) = ( ( 1 + z l 1 f ) I f I ( 1 f ) I 0 I ) .
( A ( z ) B ( z ) C ( z ) D ( z ) ) = ( I ( f + l 1 z ) I 0 I I ) .
ρ 0 = [ 0 . 1825 C n 2 k 2 ( 3 f + 8 l 1 ) ] 3 5 .
I ( ρ 1 , l ) = T r W ( ρ 1 , ρ 1 , l )
P ( ρ 1 , l ) = 1 4 Det W ( ρ 1 , ρ 1 , l ) [ Tr W ( ρ 1 , ρ 1 , l ) ] 2 .
μ ( ρ 1 , ρ 2 , l ) = Tr W ( ρ 1 , ρ 2 , l ) Tr W ( ρ 1 , ρ 1 , l ) Tr W ( ρ 2 , ρ 2 , l ) .
W ( ρ ˜ , l ) = k 2 A α A β B α β 4 π 2 [ det ( B ˜ ) ] 1 2 exp [ i k 2 ρ ˜ T ( D ˜ B ˜ 1 + P ˜ ) ρ ˜ ]
× exp [ i k 2 r ˜ T ( M 0 α β 1 + B ˜ 1 A ˜ + P ˜ ) r ˜ ] exp [ i k r ˜ T ( B ˜ 1 1 2 P ˜ ) ρ ˜ ] d r ˜
= k 2 A α A β B α β 4 π 2 [ det ( B ˜ ) ] 1 2 exp [ i k 2 ρ ˜ T ( D ˜ B ˜ 1 + P ˜ ) ρ ˜ ]
× exp [ i k 2 ρ ˜ T ( B ˜ 1 1 2 P ˜ ) T ( M 0 α β 1 + B ˜ 1 A ˜ + P ˜ ) 1 ( B ˜ 1 1 2 P ˜ ) ρ ˜ ]
× exp [ i k 2 ( M 0 α β 1 + B ˜ 1 A ˜ + P ˜ ) 1 2 r ˜ ( M 0 α β 1 + B ˜ 1 A ˜ + P ˜ ) 1 2 ( B ˜ 1 1 2 P ˜ ) ρ ˜ ] d r ˜ , ( A 1 )
exp ( a x 2 ) d x = π / a ,
W ( ρ ˜ , l ) = A α A β B αβ [ det ( B ˜ ) ] 1 2 [ det ( M 0 αβ 1 + B ˜ 1 A ˜ + P ˜ ) ] 1 2 exp [ i k 2 ρ ˜ T ( D ˜ B ˜ 1 + P ˜ ) ρ ˜ ]
exp [ i k 2 ρ ˜ T ( B ˜ 1 1 2 P ˜ ) T ( M 0 αβ 1 + B ˜ 1 A ˜ + P ˜ ) 1 ( B ˜ 1 1 2 P ˜ ) ρ ˜ ] ,
[ det ( B ˜ ) ] 1 2 [ det ( M 0 αβ 1 + B ˜ 1 A ˜ + P ˜ ) ] 1 2 = [ det ( A ˜ + B ˜ M 0 αβ 1 + B ˜ P ˜ ) ] 1 2 ,
D ˜ B ˜ 1 B ˜ 1 T ( M 0 αβ 1 + B ˜ 1 A ˜ + P ˜ ) 1 B ˜ 1 = D ˜ B ˜ 1 B ˜ 1 ( A ˜ + B ˜ M 0 αβ 1 + B ˜ P ˜ ) 1
= [ D ˜ B ˜ 1 ( A ˜ + B ˜ M 0 αβ 1 + B ˜ P ˜ ) B ˜ 1 ] ( A ˜ + B ˜ M 0 αβ 1 + B ˜ P ˜ ) 1
= [ D ˜ B ˜ 1 A ˜ + D ˜ M 0 αβ 1 + D ˜ P ˜ B ˜ 1 ] ( A ˜ + B ˜ M 0 αβ 1 + B ˜ P ˜ ) 1
= ( C ˜ + D ˜ M 0 αβ 1 + D ˜ P ˜ ) ( A ˜ + B ˜ M 0 αβ 1 + B ˜ P ˜ ) 1 ,
M 1 αβ 1 = ( C ˜ + D ˜ M 0 αβ 1 + D ˜ P ˜ ) ( A ˜ + B ˜ M 0 αβ 1 + B ˜ P ˜ ) 1 ,

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