Abstract

Nonparaxial propagation theory of coherent beams in a uniaxial crystal is extended to the partially coherent case. An analytical formula for the3×3cross-spectral density matrix of a nonparaxial Gaussian Schell-model (GSM) beam propagating in a uniaxial crystal orthogonal to the optical axis is derived. Statistical properties, such as the spectral intensity and the degree of polarization, of a nonparaxial GSM beam in a uniaxial crystal are studied numerically. It is found that the statistical properties of a nonparaxial GSM beam are closely determined by its initial beam parameters and the parameters of the crystal. Uniaxial crystal can be used to modulate the spectral density and degree of polarization of a nonparaxial partially coherent beam. Our results may be useful in some applications, such as optical trapping and nonlinear optics, where a light beam with special beam profile and polarization is required.

© 2011 OSA

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2011 (4)

G. Wu, Y. Cai, and J. Chen, “Shaping the beam profile of a partially coherent beam by a phase aperture,” Opt. Commun. 284(18), 4129–4135 (2011).
[CrossRef]

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102(1), 205–213 (2011).
[CrossRef]

G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. 36(10), 1939–1941 (2011).
[CrossRef] [PubMed]

C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011).
[CrossRef] [PubMed]

2010 (2)

2009 (5)

2008 (4)

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

D. Liu and Z. Zhou, “Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 10(9), 095005 (2008).
[CrossRef]

G. Zhou, “The analytical vectorial structure of a nonparaxial Gaussian beam close to the source,” Opt. Express 16(6), 3504–3514 (2008).
[CrossRef] [PubMed]

F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25(8), 2001–2010 (2008).
[CrossRef] [PubMed]

2007 (4)

2006 (2)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (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(6), 685–687 (2006).
[CrossRef] [PubMed]

2005 (2)

K. Duan and B. Lü, “Wigner-distribution-function matrix and its application to partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. B 22(8), 1585–1593 (2005).
[CrossRef]

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[CrossRef] [PubMed]

2004 (7)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “LIDAR model for a rough-surface target: method of partial coherence,” Proc. SPIE 5237, 49–60 (2004).
[CrossRef]

M. von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head mounted retinal projection displays,” Opt. Eng. 43(7), 1552–1560 (2004).
[CrossRef]

Y. Baykal, “Average transmittance in turbulence for partially coherent sources,” Opt. Commun. 231(1-6), 129–136 (2004).
[CrossRef]

B. Lü and S. Luo, “Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36(1), 51–56 (2004).
[CrossRef]

K. Duan and B. Lü, “Partially coherent nonparaxial beams,” Opt. Lett. 29(8), 800–802 (2004).
[CrossRef] [PubMed]

K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A 21(10), 1924–1932 (2004).
[CrossRef] [PubMed]

Y. Zhang and B. Lü, “Propagation of the Wigner distribution function for partially coherent nonparaxial beams,” Opt. Lett. 29(23), 2710–2712 (2004).
[CrossRef] [PubMed]

2003 (3)

2002 (5)

2001 (2)

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 (2001).
[CrossRef] [PubMed]

J. J. Stamnes and V. Dhayalan, “Transmission of a twodimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18(7), 1662–1669 (2001).
[CrossRef]

1998 (2)

R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15(9), 2373–2382 (1998).
[CrossRef]

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
[CrossRef]

1994 (1)

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

1992 (2)

1990 (1)

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1990).
[CrossRef]

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(11), 1057–1060 (1984).
[CrossRef]

1982 (1)

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[CrossRef]

1979 (1)

1978 (1)

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[CrossRef]

1976 (1)

1975 (2)

D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am. 65(8), 887–891 (1975).
[CrossRef]

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optic,” Phys. Rev. A 11(4), 1365–1370 (1975).
[CrossRef]

Agrawal, G. P.

Ambrosini, D.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

Andrews, L. C.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “LIDAR model for a rough-surface target: method of partial coherence,” Proc. SPIE 5237, 49–60 (2004).
[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(11), 1057–1060 (1984).
[CrossRef]

Bagini, V.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

Baykal, Y.

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278(1), 157–167 (2007).
[CrossRef]

Y. Baykal, “Average transmittance in turbulence for partially coherent sources,” Opt. Commun. 231(1-6), 129–136 (2004).
[CrossRef]

Cai, Y.

G. Wu, Y. Cai, and J. Chen, “Shaping the beam profile of a partially coherent beam by a phase aperture,” Opt. Commun. 284(18), 4129–4135 (2011).
[CrossRef]

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102(1), 205–213 (2011).
[CrossRef]

G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. 36(10), 1939–1941 (2011).
[CrossRef] [PubMed]

C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011).
[CrossRef] [PubMed]

F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a partially coherent field,” J. Opt. Soc. Am. A 27(5), 1120–1126 (2010).
[CrossRef] [PubMed]

F. Wang and Y. Cai, “Second-order statistics of a twisted gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010).
[CrossRef] [PubMed]

C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009).
[CrossRef] [PubMed]

F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25(8), 2001–2010 (2008).
[CrossRef] [PubMed]

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(7), 1937–1944 (2007).
[CrossRef] [PubMed]

Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278(1), 157–167 (2007).
[CrossRef]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (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(6), 685–687 (2006).
[CrossRef] [PubMed]

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[CrossRef] [PubMed]

Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27(19), 1672–1674 (2002).
[CrossRef] [PubMed]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19(10), 2036–2042 (2002).
[CrossRef] [PubMed]

L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B , doi:.
[CrossRef]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B , doi:.
[CrossRef]

Chen, J.

G. Wu, Y. Cai, and J. Chen, “Shaping the beam profile of a partially coherent beam by a phase aperture,” Opt. Commun. 284(18), 4129–4135 (2011).
[CrossRef]

Ciattoni, A.

Collett, E.

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[CrossRef]

Deng, D.

D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. B 26(11), 2044–2049 (2009).
[CrossRef]

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

Dhayalan, V.

Dong, Y.

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B , doi:.
[CrossRef]

Duan, K.

Eyyuboglu, H. T.

C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278(1), 157–167 (2007).
[CrossRef]

Fan, Z.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

Friberg, A. T.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[CrossRef] [PubMed]

E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992).
[CrossRef]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[CrossRef]

Gbur, G.

Ge, D.

Gori, F.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1990).
[CrossRef]

Guo, Q.

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[CrossRef]

Hu, L.

Kaivola, M.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[CrossRef] [PubMed]

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(11), 1057–1060 (1984).
[CrossRef]

Kermisch, D.

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(11), 1057–1060 (1984).
[CrossRef]

Korotkova, O.

Laabs, H.

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optic,” Phys. Rev. A 11(4), 1365–1370 (1975).
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Li, X.

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102(1), 205–213 (2011).
[CrossRef]

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Liu, D.

D. Liu and Z. Zhou, “Propagation of partially coherent flat-topped beams in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26(4), 924–930 (2009).
[CrossRef] [PubMed]

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxially crystals orthogonal to the optical axis,” Eur. Phys. J. D 54(1), 95–101 (2009).
[CrossRef]

D. Liu and Z. Zhou, “Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 10(9), 095005 (2008).
[CrossRef]

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M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optic,” Phys. Rev. A 11(4), 1365–1370 (1975).
[CrossRef]

Lu, X.

Lü, B.

Lukowicz, P.

M. von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head mounted retinal projection displays,” Opt. Eng. 43(7), 1552–1560 (2004).
[CrossRef]

Luo, S.

B. Lü and S. Luo, “Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36(1), 51–56 (2004).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optic,” Phys. Rev. A 11(4), 1365–1370 (1975).
[CrossRef]

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(11), 1057–1060 (1984).
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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(11), 1057–1060 (1984).
[CrossRef]

Mukunda, N.

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(11), 1057–1060 (1984).
[CrossRef]

Palma, C.

Pattanayak, D. N.

Peschel, U.

Phillips, R. L.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “LIDAR model for a rough-surface target: method of partial coherence,” Proc. SPIE 5237, 49–60 (2004).
[CrossRef]

Ponomarenko, S. A.

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 (2001).
[CrossRef] [PubMed]

Santarsiero, M.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

Seshadri, S. R. S.

S. R. S. Seshadri, “Virtual source for the Bessel-Gaussian beam,” Opt. Lett. 27, 988–1000 (2002).

Setälä, T.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[CrossRef] [PubMed]

Shao, J.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

Sherman, G.

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[CrossRef] [PubMed]

Simon, R.

Stamnes, J.

Stamnes, J. J.

Sudol, R. J.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[CrossRef]

Tang, B.

Tervonen, E.

Troster, G.

M. von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head mounted retinal projection displays,” Opt. Eng. 43(7), 1552–1560 (2004).
[CrossRef]

Turunen, J.

von Waldkirch, M.

M. von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head mounted retinal projection displays,” Opt. Eng. 43(7), 1552–1560 (2004).
[CrossRef]

Wang, F.

Wolf, E.

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[CrossRef]

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[CrossRef]

Wu, G.

G. Wu, Y. Cai, and J. Chen, “Shaping the beam profile of a partially coherent beam by a phase aperture,” Opt. Commun. 284(18), 4129–4135 (2011).
[CrossRef]

G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. 36(10), 1939–1941 (2011).
[CrossRef] [PubMed]

Wünsche, A.

Xu, S.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[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(11), 1057–1060 (1984).
[CrossRef]

Yu, H.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

Zhang, L.

L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B , doi:.
[CrossRef]

Zhang, Y.

Zhao, C.

Zhou, G.

Zhou, Z.

D. Liu and Z. Zhou, “Propagation of partially coherent flat-topped beams in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26(4), 924–930 (2009).
[CrossRef] [PubMed]

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxially crystals orthogonal to the optical axis,” Eur. Phys. J. D 54(1), 95–101 (2009).
[CrossRef]

D. Liu and Z. Zhou, “Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 10(9), 095005 (2008).
[CrossRef]

Zhu, S. Y.

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[CrossRef] [PubMed]

Appl. Phys. B (3)

L. Zhang and Y. Cai, “Propagation of a twisted anisotropic Gaussian-Schell model beam beyond the paraxial approximation,” Appl. Phys. B , doi:.
[CrossRef]

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102(1), 205–213 (2011).
[CrossRef]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B , doi:.
[CrossRef]

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[CrossRef]

Eur. Phys. J. D (1)

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxially crystals orthogonal to the optical axis,” Eur. Phys. J. D 54(1), 95–101 (2009).
[CrossRef]

J. Mod. Opt. (1)

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

D. Liu and Z. Zhou, “Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 10(9), 095005 (2008).
[CrossRef]

J. Opt. Soc. Am. (3)

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

A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20(11), 2163–2171 (2003).
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K. Duan and B. Lü, “Partially coherent vectorial nonparaxial beams,” J. Opt. Soc. Am. A 21(10), 1924–1932 (2004).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19(10), 2036–2042 (2002).
[CrossRef] [PubMed]

G. Gbur and O. Korotkova, “Angular spectrum representation for the propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence,” J. Opt. Soc. Am. A 24(3), 745–752 (2007).
[CrossRef] [PubMed]

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(7), 1937–1944 (2007).
[CrossRef] [PubMed]

F. Wang, Y. Cai, and Q. Lin, “Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam,” J. Opt. Soc. Am. A 25(8), 2001–2010 (2008).
[CrossRef] [PubMed]

B. Tang, “Hermite-cosine-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26(12), 2480–2487 (2009).
[CrossRef] [PubMed]

F. Wang, Y. Cai, and O. Korotkova, “Degree of paraxiality of a partially coherent field,” J. Opt. Soc. Am. A 27(5), 1120–1126 (2010).
[CrossRef] [PubMed]

D. Liu and Z. Zhou, “Propagation of partially coherent flat-topped beams in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26(4), 924–930 (2009).
[CrossRef] [PubMed]

A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9(5), 765–774 (1992).
[CrossRef]

J. J. Stamnes and V. Dhayalan, “Transmission of a twodimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18(7), 1662–1669 (2001).
[CrossRef]

R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15(9), 2373–2382 (1998).
[CrossRef]

E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (8)

Y. Baykal, “Average transmittance in turbulence for partially coherent sources,” Opt. Commun. 231(1-6), 129–136 (2004).
[CrossRef]

G. Wu, Y. Cai, and J. Chen, “Shaping the beam profile of a partially coherent beam by a phase aperture,” Opt. Commun. 284(18), 4129–4135 (2011).
[CrossRef]

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboğlu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278(1), 157–167 (2007).
[CrossRef]

H. Laabs, “Propagation of Hermite–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998).
[CrossRef]

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[CrossRef]

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1990).
[CrossRef]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[CrossRef]

Opt. Eng. (1)

M. von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head mounted retinal projection displays,” Opt. Eng. 43(7), 1552–1560 (2004).
[CrossRef]

Opt. Express (5)

Opt. Laser Technol. (1)

B. Lü and S. Luo, “Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36(1), 51–56 (2004).
[CrossRef]

Opt. Lett. (8)

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[CrossRef]

Phys. Rev. A (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optic,” Phys. Rev. A 11(4), 1365–1370 (1975).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (3)

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[CrossRef] [PubMed]

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 (2001).
[CrossRef] [PubMed]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(1), 016615 (2002).
[CrossRef] [PubMed]

Phys. Rev. Lett. (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(11), 1057–1060 (1984).
[CrossRef]

Proc. SPIE (1)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “LIDAR model for a rough-surface target: method of partial coherence,” Proc. SPIE 5237, 49–60 (2004).
[CrossRef]

Other (3)

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation in the Turbulent Atmosphere, 2nd ed. (SPIE, 2005).

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).

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Figures (6)

Fig. 1
Fig. 1

Geometry of the propagation of a laser beam in a uniaxial crystal orthogonal to the optical axis.

Fig. 1
Fig. 1

Normalized intensity distribution (contour graph) of a paraxial GSM beam in a uniaxial crystal at several propagation distances for different values of the initial coherence width σ g .

Fig. 2
Fig. 2

Intensity distributions (contour graphs) I x ( r , z ) , I y ( r , z ) and I z ( r , z ) of a nonparaxial GSM beam in a uniaxial crystal for different values of the initial coherence width σ g at z = 20 z r .

Fig. 3
Fig. 3

Intensity distributions I x ( r , z ) , I y ( r , z ) and I z ( r , z ) of a nonparaxial GSM beam in a uniaxial crystal for different values of the ratio of extraordinary index to ordinary refractive index n e / n o at z = 20 z r .

Fig. 4
Fig. 4

Distribution of the degree of polarization of a nonparaxial GSM beam in a uniaxial crystal at several propagation distances for different values of the initial coherence width σ g

Fig. 5
Fig. 5

Distribution of the degree of polarization of a nonparaxial GSM beam in a uniaxial crystal at several propagation distances for different values of the ratio of extraordinary index to ordinary refractive index n e / n o

Equations (45)

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ε = ( n e 2 0 0 0 n o 2 0 0 0 n o 2 ) ,
2 E ( E ) + k 0 2 ε E = 0 ,
E ( r , z ) = d 2 k exp ( i k r ) exp ( i k e z z ) ( E ˜ x ( k ) k x k y k 0 2 n o 2 k x 2 E ˜ x ( k ) k e z k x k 0 2 n o 2 k x 2 E ˜ x ( k ) )               + d 2 k exp ( i k r ) exp ( i k o z z ) ( 0 k x k y k 0 2 n o 2 k x 2 E ˜ x ( k ) + E ˜ y ( k ) k y k o z [ k x k y k 0 2 n o 2 k x 2 E ˜ x ( k ) + E ˜ y ( k ) ] ) ,    
E ˜ s ( k ) = 1 2 π 2 d 2 r exp ( i k r ) E s ( r , 0 ) ,           ( s = x , y )
k o z ( k ) = ( k 0 2 n o 2 k 2 ) 1 / 2 , k e z ( k ) = [ k 0 2 n e 2 ( n e 2 / n o 2 ) k x 2 k y 2 ] 1 / 2 .      
E p ( r , z ) = d 2 k exp ( i k r ) exp ( i k 0 n e z ) exp ( i n e 2 k x 2 + n o 2 k y 2 2 k 0 n e n o 2 z ) ( E ˜ x ( k ) 0 0 )                 + d 2 k exp ( i k r ) exp ( i k 0 n o z ) exp ( i k x 2 + k y 2 2 k 0 n o z ) ( 0 E ˜ y ( k ) 0 ) .
E p x ( r , z ) = k 0 n o 2 π i z d 2 r 0 Γ e ( r 0 , r , 0 ) E x ( r 0 , 0 ) ,   E p y ( r , z ) = k 0 n o 2 π i z d 2 r 0 Γ o ( r 0 , 0 ) E y ( r 0 , r , 0 ) , E p z ( r , z )     = 0 ,            
Γ e ( r 0 , r , 0 ) = exp ( i k 0 n e z ) exp { k 0 2 i z n e [ n o 2 ( x x 0 ) 2 + n e 2 ( y y 0 ) 2 ] } ,
Γ o ( r 0 , r , 0 ) = exp ( i k 0 n o z ) exp { k 0 n o 2 i z [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } .
E ( r , z ) = exp ( i k 0 n e z ) d 2 k exp ( i k r ) × exp ( i n e 2 k x 2 + n o 2 k y 2 2 k 0 n e n o 2 z ) ( E ˜ x ( k ) k x k y k 0 2 n o 2 E ˜ x ( k ) n e k x k 0 n o 2 E ˜ x ( k ) )                 + exp ( i k 0 n o z ) d 2 k exp ( i k r ) exp ( i k x 2 + k y 2 2 k 0 n o z ) ( 0 k x k y k 0 2 n o 2 E ˜ x ( k ) + E ˜ y ( k ) k y k 0 n o E ˜ y ( k ) ) .
E x ( r , z ) = k 0 n o 2 π i z d 2 r 0 Γ e ( r 0 , r , 0 ) E x ( r 0 , 0 ) ,
E y ( r , z ) = i k 0 n o 2 π z 3 d 2 r 0 ( x x 0 ) ( y y 0 ) [ Γ e ( r 0 , r , 0 ) Γ o ( r 0 , r , 0 ) ] E x ( r 0 , 0 )                 + k 0 n o 2 π i z d 2 r 0 Γ o ( r 0 , r , 0 ) E y ( r 0 , 0 ) ,
E z ( r , z )     = i k 0 n o 2 π z 2 d 2 r 0 ( x x 0 ) Γ e ( r 0 , r , 0 ) E x ( r 0 , 0 )                 + i k 0 n o 2 π z 2 d 2 r 0 ( y y 0 ) Γ o ( r 0 , r , 0 ) E y ( r 0 , 0 ) .
W ( r 1 , r 2 , z ) = ( W x x ( r 1 , r 2 , z ) W x y ( r 1 , r 2 , z ) W x z ( r 1 , r 2 , z ) W x y * ( r 1 , r 2 , z ) W y y ( r 1 , r 2 , z ) W y z ( r 1 , r 2 , z ) W x z * ( r 1 , r 2 , z ) W y z * ( r 1 , r 2 , z ) W z z ( r 1 , r 2 , z ) ) ,
W α β ( r 1 , r 2 , z ) = E α * ( r 1 , z ) E β ( r 2 , z ) , ( α , β = x , y , z ) .
W x x ( r 1 , r 2 , z ) = k 0 2 n o 2 4 π 2 z 2 d 2 r 10 d 2 r 20 Π e e W 0 x x ( r 10 , r 20 , 0 ) ,
W y y ( r 1 , r 2 , z ) = k 0 2 n o 2 4 π 2 z 4 d 2 r 10 d 2 r 20 ( Π o o Π o e * ) W 0 x y ( r 10 , r 20 , 0 ) ( x 1 x 10 ) ( y 1 y 10 ) + k 0 2 n o 2 4 π 2 z 6 d 2 r 10 d 2 r 20 ( Π e e Π o e Π o e * + Π o o ) W 0 x x ( r 10 , r 20 , 0 ) ( x 1 x 10 ) ( y 1 y 10 ) × ( x 2 x 20 ) ( y 2 y 20 ) + k 0 2 n o 2 4 π 2 z 4 d 2 r 10 d 2 r 20 ( Π o o Π o e ) W 0 y x ( r 10 , r 20 , 0 ) × ( x 2 x 20 ) ( y 2 y 20 ) + k 0 2 n o 2 4 π 2 z 2 d 2 r 10 d 2 r 20 Π o o W 0 y y ( r 10 , r 20 , 0 ) ,
W z z ( r 1 , r 2 , z ) = k 0 2 n o 2 4 π 2 z 4 d 2 r 10 d 2 r 20 Π e e W 0 x x ( r 10 , r 20 , 0 ) ( x 1 x 10 ) ( x 2 x 20 ) + k 0 2 n o 2 4 π 2 z 4 d 2 r 10 d 2 r 20 Π o e * W 0 x y ( r 10 , r 20 , 0 ) ( x 1 x 10 ) ( y 2 y 20 ) + k 0 2 n o 2 4 π 2 z 4 d 2 r 10 d 2 r 20 Π o e W 0 y x ( r 10 , r 20 , 0 ) ( y 1 y 10 ) ( x 2 x 20 ) + k 0 2 n o 2 4 π 2 z 4 d 2 r 10 d 2 r 20 Π o o W 0 y y ( r 10 , r 20 , 0 ) ( y 1 y 10 ) ( y 2 y 20 )
W x y ( r 1 , r 2 , z ) = k 0 2 n o 2 4 π 2 z 4 d 2 r 10 d 2 r 20 ( Π o e * Π e e ) W 0 x x ( r 10 , r 20 , 0 ) × ( x 2 x 20 ) ( y 2 y 20 ) k 0 2 n o 2 4 π 2 z 2 d 2 r 10 d 2 r 20 Π o e * W 0 x y ( r 10 , r 20 , 0 ) ,
W x z ( r 1 , r 2 , z ) = k 0 2 n o 2 4 π 2 z 3 d 2 r 10 d 2 r 20 Π e e W 0 x x ( r 10 , r 20 , 0 ) ( x 2 x 20 ) k 0 2 n o 2 4 π 2 z 3 d 2 r 10 d 2 r 20 Π o e * W 0 x y ( r 10 , r 20 , 0 ) ( y 2 y 20 ) ,
W y z ( r 1 , r 2 , z ) = k 0 2 n o 2 4 π 2 z 5 d 2 r 10 d 2 r 20 ( Π e e Π o e ) W 0 x x ( r 10 , r 20 , 0 ) ( x 1 x 10 ) × ( y 1 y 10 ) ( x 2 x 20 ) + k 0 2 n o 2 4 π 2 z 5 d 2 r 10 d 2 r 20 ( Π o e * Π o o ) W 0 x y ( r 10 , r 20 , 0 ) × ( x 1 x 10 ) ( y 1 y 10 ) ( y 2 y 20 ) k 0 2 n o 2 4 π 2 z 3 d 2 r 10 d 2 r 20 Π o e W 0 y x ( r 10 , r 20 , 0 ) × ( x 2 x 20 ) k 0 2 n o 2 4 π 2 z 3 d 2 r 10 d 2 r 20 Π o o W 0 y y ( r 10 , r 20 , 0 ) ( y 2 y 20 ) ,
Π e e = Γ e ( r 10 , r 1 , 0 ) Γ e * ( r 20 , r 2 , 0 ) = exp { k 0 2 i z n e [ n o 2 ( x 1 x 10 ) 2 + n e 2 ( y 1 y 10 ) 2 ] }           × exp { k 0 2 i z n e [ n o 2 ( x 2 x 20 ) 2 + n e 2 ( y 2 y 20 ) 2 ] } ,
Π o o = Γ o ( r 10 , r 1 , 0 ) Γ o * ( r 20 , r 2 , 0 ) = exp { k 0 n o 2 i z [ ( x 1 x 10 ) 2 + ( y 1 y 10 ) 2 ] }           × exp { k 0 n o 2 i z [ ( x 2 x 20 ) 2 + ( y 2 y 20 ) 2 ] } ,
Π o e = Γ o ( r 10 , r 1 , 0 ) Γ e * ( r 20 , r 2 , 0 ) = exp { k 0 n o 2 i z [ ( x 1 x 10 ) 2 + ( y 1 y 10 ) 2 ] }           × exp { k 0 2 i z n e [ n o 2 ( x 2 x 20 ) 2 + n e 2 ( y 2 y 20 ) 2 ] + i k 0 ( n o n e ) z } .
W 0 ( r 10 , r 20 , 0 ) = ( W 0 x x ( r 10 , r 20 , 0 ) 0 0 0 0 0 0 0 0 ) ,
W 0 x x ( r 10 , r 20 , 0 ) = exp ( x 10 2 + y 10 2 + x 20 2 + y 20 2 4 σ I 2 ( x 10 x 20 ) 2 + ( y 10 y 20 ) 2 2 σ g 2 ) ,
W x x ( r 1 , r 2 , z ) = a o e 2 H x o o H y e e A o o A e e B o o o o B e e e e ,
W y y ( r 1 , r 2 , z ) = a o e 2 E x o o o o E y e e e e H x o o H y e e z 4 A o o A e e B o o o o B e e e e + a o e 2 F x o e o o G y o e e e E o e * L x o e o o J y o e e e z 4 A o o A e e B o e o o B o e e e                         + a o e 2 G x o o o e F y e e o e E o e L y e e o e J x o o o e z 4 A o e B o o o e B e e o e a o e 2 E x o e o e E y o e o e H x o e H y o e z 4 A o e B o e o e ,
W z z ( r 1 , r 2 , z ) = a o e 2 E x o o o o H x o o H y e e z 2 A o o A e e B o o o o B e e e e ,
W x y ( r 1 , r 2 , z ) = a o e 2 C x o o o o C y e e e e H x o o H y e e z 2 A o o A e e B o o o o B e e e e a o e 2 C x o e o o C y o e e e E o e * L x o e o o J y o e e e z 2 A o o A e e B o e o o B o e e e ,
W x z ( r 1 , r 2 , z ) = a o e 2 C x o o o o H x o o H y e e z A o o A e e B o o o o B e e e e ,
W y z ( r 1 , r 2 , z ) = a o e 2 E x o o o o D y H x o o H y e e z 3 A o o A e e B o o o o B e e e e + a o e 2 G x o o o e D y o e E o e L y e e o e J x o o o e z 3 A o e B o o o e B e e o e ,
a μ γ = k 0 n μ n γ 2 i z n e ,   A μ γ = 1 4 σ I 2 + 1 2 σ g 2 + a μ γ ,   B μ γ τ ν = 1 4 σ I 2 + 1 2 σ g 2 a μ γ 1 4 A τ ν σ g 4 ,
C s μ γ τ ν = a μ γ s 2 B μ γ τ ν a τ ν s 1 2 A τ ν B μ γ τ ν σ g 2 + s 2 ,    D y = y 1 a e e y 1 A e e a e e 2 A e e B e e e e σ g 2 ( y 1 2 A e e σ g 2 y 2 ) ,
D y o e = y 1 a e e y 1 A e e a e e 2 A o e B e e o e σ g 2 ( n o y 1 2 n e A o e σ g 2 y 2 ) ,   E o e = exp [ i k 0 ( n o n e ) z ] ,
E s μ γ τ ν = s 1 s 2 a τ ν s 1 s 2 A τ ν + 1 4 A τ ν B μ γ τ ν σ g 2 + a μ γ 2 2 A τ ν B μ γ τ ν 2 σ g 2 ( s 1 2 A τ ν σ g 2 s 2 ) 2                                                                         + a μ γ B μ γ τ ν ( s 1 2 A τ ν σ g 2 s 2 ) ( a τ ν s 1 A τ ν s 1 s 2 2 A τ ν σ g 2 ) ,
F s μ γ τ ν = s 1 s 2 a τ ν s 1 s 2 A τ ν + 1 4 A τ ν B μ γ τ ν σ g 2 + a μ γ 2 2 A τ ν B μ γ τ ν 2 σ g 2 ( n o s 1 2 n e A τ ν σ g 2 s 2 ) 2                                                                       + a μ γ B μ γ τ ν ( n o s 1 2 n e A τ ν σ g 2 s 2 ) ( a τ ν σ 1 A τ ν s 1 s 2 2 A τ ν σ g 2 ) ,
G s μ γ τ ν = s 1 s 2 a τ ν s 1 s 2 A τ ν + 1 4 A τ ν B μ γ τ ν σ g 2 + a μ γ 2 2 A τ ν B μ γ τ ν 2 σ g 2 ( n e s 1 2 n o A τ ν σ g 2 s 2 ) 2                                                                       + a μ γ B μ γ τ ν ( n e s 1 2 n o A τ ν σ g 2 s 2 ) ( a τ ν s 1 A τ ν s 1 s 2 2 A τ ν σ g 2 ) ,
H s μ γ = exp [ a μ γ s 1 2 + a μ γ s 2 2 + a μ γ 2 s 1 2 A μ γ + a μ γ 2 B μ γ μ γ ( s 2 s 1 2 A μ γ σ g 2 ) 2 ] ,
L s μ γ τ ν = exp [ a τ ν s 1 2 + a μ γ s 2 2 + a τ ν 2 s 1 2 A τ ν + a μ γ 2 B μ γ τ ν ( s 2 n o s 1 2 n e A τ ν σ g 2 ) 2 ] ,
J s μ γ τ ν = exp [ a τ ν s 1 2 + a μ γ s 2 2 + a τ ν 2 s 1 2 A τ ν + a μ γ 2 B μ γ τ ν ( s 2 n e s 1 2 n o A τ ν σ g 2 ) 2 ] ,   ( s = x       or     y ;     μ , γ , τ , ν = o       or     e )
( a x 2 + b x + c ) exp ( p x 2 + q x ) d x = 1 4 p 2 ( 2 a p + a q 2 2 b p q + 4 c p 2 ) π p exp ( q 2 4 p )               ( p < 0 ) .
W α β ( r 1 , r 2 , z ) = { a o e 2 H x o o H y e e A o o A e e B o o o o B e e e e α = β = x 0 otherwise .
I ( r , z ) = I x ( r , z ) + I y ( r , z ) + I z ( r , z )            = W x x ( r , r , z ) + W y y ( r , r , z ) + W z z ( r , r , z ) ,
P ( r , z ) = 3 2 W x x ( r , r , z ) 2 + W y y ( r , r , z ) 2 + W z z ( r , r , z ) 2 [ W x x ( r , r , z ) + W y y ( r , r , z ) + W z z ( r , r , z ) ] 2 1 2     .

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