Abstract

Laguerre-Gaussian Schell-model (LGSM) beam was proposed in theory [Opt. Lett. 38, 91 (2013 Opt. Lett. 38, 1814 (2013)] just recently. In this paper, we study the propagation of a LGSM beam in turbulent atmosphere. Analytical expressions for the cross-spectral density and the second-order moments of the Wigner distribution function of a LGSM beam in turbulent atmosphere are derived. The statistical properties, such as the degree of coherence and the propagation factor, of a LGSM beam in turbulent atmosphere are studied in detail. It is found that a LGSM beam with larger mode order n is less affected by turbulence than a LGSM beam with smaller mode order n or a GSM beam under certain condition, which will be useful in free-space optical communications.

© 2014 Optical Society of America

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2014 (1)

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

2013 (10)

F. Wang, X. Liu, L. Liu, Y. Yuan, Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[CrossRef]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[CrossRef]

Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[CrossRef] [PubMed]

Y. Gu, G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[CrossRef] [PubMed]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

Z. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[CrossRef] [PubMed]

Z. Mei, E. Schchepakina, O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
[CrossRef] [PubMed]

X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[CrossRef] [PubMed]

2012 (7)

2011 (3)

2010 (4)

2009 (3)

2008 (4)

2007 (4)

2006 (4)

2005 (1)

Y. Cai, 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 (1)

2003 (1)

2002 (1)

1999 (1)

1993 (1)

A. Belendez, L. Carretero, A. Fimia, “The use of partially coherent light to reduce the efficiency of silve-halide noise gratings,” Opt. Commun. 98(4-6), 236–240 (1993).
[CrossRef]

1984 (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, 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]

1979 (1)

1975 (1)

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, 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]

Baykal, Y.

Belendez, A.

A. Belendez, L. Carretero, A. Fimia, “The use of partially coherent light to reduce the efficiency of silve-halide noise gratings,” Opt. Commun. 98(4-6), 236–240 (1993).
[CrossRef]

Borghi, R.

Cai, Y.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

F. Wang, X. Liu, L. Liu, Y. Yuan, Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[CrossRef]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[CrossRef] [PubMed]

Y. Dong, F. Wang, C. Zhao, Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[CrossRef]

F. Wang, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
[CrossRef] [PubMed]

C. Zhao, 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]

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

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

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[CrossRef] [PubMed]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[CrossRef] [PubMed]

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

Y. Cai, Y. Chen, H. T. Eyyuboğlu, Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007).
[CrossRef] [PubMed]

F. Wang, Y. Cai, S. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14(16), 6999–7004 (2006).
[CrossRef] [PubMed]

Y. Cai, 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, S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
[CrossRef] [PubMed]

Y. Cai, 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]

Cang, J.

J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[CrossRef]

Carretero, L.

A. Belendez, L. Carretero, A. Fimia, “The use of partially coherent light to reduce the efficiency of silve-halide noise gratings,” Opt. Commun. 98(4-6), 236–240 (1993).
[CrossRef]

Chen, Y.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007).
[CrossRef] [PubMed]

Cheng, W.

Cincotti, G.

Dan, Y.

Davidson, F. M.

Dogariu, A.

Dong, Y.

Y. Dong, F. Wang, C. Zhao, Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[CrossRef]

Du, S.

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Eyyuboglu, H. T.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

F. Wang, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
[CrossRef] [PubMed]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[CrossRef] [PubMed]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[CrossRef] [PubMed]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007).
[CrossRef] [PubMed]

Y. Baykal, H. T. Eyyuboğlu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. 45(16), 3793–3797 (2006).
[CrossRef] [PubMed]

H. T. Eyyuboğlu, Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(20), 4659–4674 (2004).
[CrossRef] [PubMed]

Fimia, A.

A. Belendez, L. Carretero, A. Fimia, “The use of partially coherent light to reduce the efficiency of silve-halide noise gratings,” Opt. Commun. 98(4-6), 236–240 (1993).
[CrossRef]

Fischer, D. G.

T. van Dijk, D. G. Fischer, T. D. Visser, E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[CrossRef] [PubMed]

Gbur, G.

Gori, F.

Gu, Y.

Gutiérrez-Vega, J. C.

Haus, J. W.

He, S.

Kato, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, 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, 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.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[CrossRef] [PubMed]

Z. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[CrossRef] [PubMed]

Z. Mei, E. Schchepakina, O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
[CrossRef] [PubMed]

Z. Tong, O. Korotkova, “Non-uniformly correlated beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[CrossRef] [PubMed]

S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[CrossRef] [PubMed]

Z. Tong, O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012).
[CrossRef] [PubMed]

O. Korotkova, S. Sahin, E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[CrossRef]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[CrossRef] [PubMed]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[CrossRef] [PubMed]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).
[CrossRef]

Lajunen, H.

Liang, C.

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Lin, Q.

Liu, L.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

F. Wang, X. Liu, L. Liu, Y. Yuan, Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[CrossRef]

X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[CrossRef] [PubMed]

Liu, X.

X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[CrossRef] [PubMed]

J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[CrossRef]

F. Wang, X. Liu, L. Liu, Y. Yuan, Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

Liu, Z.

Ma, Y.

Mei, Z.

Mima, K.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, 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]

Miyanaga, N.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, 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]

Nakatsuka, M.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, 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]

Noriega-Manez, R. J.

Peschel, U.

Plonus, M. A.

Qu, J.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[CrossRef] [PubMed]

Ricklin, J. C.

Saastamoinen, T.

Sahin, S.

Sanchez, V. R.

F. Gori, V. R. Sanchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

Santarsiero, M.

Schchepakina, E.

Shchepakina, E.

Shen, Y.

Shirai, T.

F. Gori, V. R. Sanchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

T. Shirai, A. Dogariu, E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20(6), 1094–1102 (2003).
[CrossRef] [PubMed]

Tong, Z.

Vahimaa, P.

van Dijk, T.

T. van Dijk, D. G. Fischer, T. D. Visser, E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[CrossRef] [PubMed]

Visser, T. D.

T. van Dijk, D. G. Fischer, T. D. Visser, E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[CrossRef] [PubMed]

Wang, F.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

F. Wang, X. Liu, L. Liu, Y. Yuan, Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[CrossRef] [PubMed]

F. Wang, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
[CrossRef] [PubMed]

Y. Dong, F. Wang, C. Zhao, Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[CrossRef]

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

F. Wang, Y. Cai, S. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14(16), 6999–7004 (2006).
[CrossRef] [PubMed]

Wang, S. C. H.

Wang, X.

Wolf, E.

T. van Dijk, D. G. Fischer, T. D. Visser, E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[CrossRef] [PubMed]

T. Shirai, A. Dogariu, E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20(6), 1094–1102 (2003).
[CrossRef] [PubMed]

Wu, G.

Xiu, P.

J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[CrossRef]

Yamanaka, C.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, 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]

Yuan, Y.

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

F. Wang, X. Liu, L. Liu, Y. Yuan, Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[CrossRef]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[CrossRef] [PubMed]

Zhan, Q.

Zhang, B.

Zhao, C.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Y. Dong, F. Wang, C. Zhao, Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[CrossRef]

C. Zhao, 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]

Zhao, H.

Zhou, P.

Zhu, S. Y.

Y. Cai, 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. Opt. (1)

Appl. Phys. Lett. (2)

Y. Cai, 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]

F. Wang, X. Liu, L. Liu, Y. Yuan, Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[CrossRef]

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

F. Gori, V. R. Sanchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Opt. Commun. (3)

A. Belendez, L. Carretero, A. Fimia, “The use of partially coherent light to reduce the efficiency of silve-halide noise gratings,” Opt. Commun. 98(4-6), 236–240 (1993).
[CrossRef]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

Opt. Express (12)

H. T. Eyyuboğlu, Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(20), 4659–4674 (2004).
[CrossRef] [PubMed]

Y. Cai, S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
[CrossRef] [PubMed]

F. Wang, Y. Cai, S. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14(16), 6999–7004 (2006).
[CrossRef] [PubMed]

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

R. J. Noriega-Manez, J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
[CrossRef] [PubMed]

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

Z. Mei, E. Schchepakina, O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[CrossRef] [PubMed]

Y. Dan, B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008).
[CrossRef] [PubMed]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[CrossRef] [PubMed]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[CrossRef] [PubMed]

W. Cheng, J. W. Haus, Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009).
[CrossRef] [PubMed]

Opt. Laser Technol. (2)

J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013).
[CrossRef]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Opt. Lett. (16)

G. Wu, 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, 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]

H. Lajunen, T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[CrossRef] [PubMed]

F. Wang, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
[CrossRef] [PubMed]

Y. Gu, G. Gbur, “Reduction of turbulence-induced scintillation by nonuniformly polarized beam arrays,” Opt. Lett. 37(9), 1553–1555 (2012).
[CrossRef] [PubMed]

S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[CrossRef] [PubMed]

Z. Tong, O. Korotkova, “Non-uniformly correlated beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[CrossRef] [PubMed]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007).
[CrossRef] [PubMed]

P. Zhou, Y. Ma, X. Wang, H. Zhao, Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010).
[CrossRef] [PubMed]

Y. Gu, G. Gbur, “Scintillation of airy beam arrays in atmospheric turbulence,” Opt. Lett. 35(20), 3456–3458 (2010).
[CrossRef] [PubMed]

Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[CrossRef] [PubMed]

Y. Gu, G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[CrossRef] [PubMed]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

Z. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[CrossRef] [PubMed]

X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[CrossRef] [PubMed]

Phys. Rev. A (2)

Y. Dong, F. Wang, C. Zhao, Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[CrossRef]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

Y. Cai, 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]

Phys. Rev. Lett. (2)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, 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]

T. van Dijk, D. G. Fischer, T. D. Visser, E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010).
[CrossRef] [PubMed]

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C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. Doc. ID 202151 (2014).

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Figures (6)

Fig. 1
Fig. 1

Modulus of the degree of coherence of a LGSM beam at several propagation distances in free space.

Fig. 2
Fig. 2

Modulus of the degree of coherence of a LGSM beam at several propagation distances in turbulent atmosphere for different values of the structure constant C n 2 .

Fig. 3
Fig. 3

Modulus of the degree of coherence of a LGSM beam at several propagation distances in turbulent atmosphere for different values of the mode order n.

Fig. 4
Fig. 4

Normalized propagation factor of a LGSM beam versus the propagation distance in turbulent atmosphere for different values of the mode order n and the structure constant C n 2 .

Fig. 5
Fig. 5

Normalized propagation factor of a LGSM beam versus the propagation distance in turbulent atmosphere for different values of the mode order n and the coherence width σ g .

Fig. 6
Fig. 6

Normalized propagation factor of a LGSM beam versus the propagation distance in turbulent atmosphere for different values of the mode order n and the wavelengthλ.

Equations (38)

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W( r 1 , r 2 )=exp[ r 1 2 + r 2 2 4 σ 0 2 ( r 1 r 2 ) 2 2 σ g 2 ] L n 0 [ ( r 1 r 2 ) 2 2 σ g 2 ],
μ( r 1 , r 2 )= W( r 1 , r 2 ) W( r 1 , r 1 )W( r 2 , r 2 ) =exp[ ( r 1 r 2 ) 2 2 σ g 2 ] L n 0 [ ( r 1 r 2 ) 2 2 σ g 2 ].
W( ρ 1 , ρ 2 )= 1 λ 2 z 2 W( r 1 , r 2 ) ×exp[ ik 2z ( r 1 ρ 1 ) 2 + ik 2z ( r 2 ρ 2 ) 2 ] × exp[ Ψ( r 1 , ρ 1 )+ Ψ * ( r 2 , ρ 2 ) ] d 2 r 1 d 2 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 ],
W( ρ 1 , ρ 2 )= 1 λ 2 z 2 exp[ ik 2z ( ρ 1 2 ρ 2 2 ) ( ρ 1 ρ 2 ) 2 ρ 0 2 ] ×exp[ ( ik 2z 1 4 σ 0 2 ) r 1 2 +( ik 2z 1 4 σ 0 2 ) r 2 2 + r 1 ( ik z ρ 1 ρ 1 ρ 2 ρ 0 2 ) ] ×exp[ r 2 ( ik z ρ 2 ρ 1 ρ 2 ρ 0 2 )( 1 2 σ g 2 + 1 ρ 0 2 ) ( r 1 r 2 ) 2 ] L n 0 [ ( r 1 r 2 ) 2 2 σ g 2 ] d 2 r 1 d 2 r 2 .
ik z ρ 1 ' = ik z ρ 1 ρ 1 ρ 2 ρ 0 2 , ik z ρ 2 ' = ik z ρ 2 ρ 1 ρ 2 ρ 0 2 ,
A( ρ 1 , ρ 2 )=exp[ ik 2z ( ρ 1 2 ρ 2 2 ) ( ρ 1 ρ 2 ) 2 ρ 0 2 ],
W( ρ 1 ' , ρ 2 ' )= A( ρ 1 , ρ 2 ) λ 2 z 2 exp[ ( 1 2 σ g 2 + 1 ρ 0 2 ) ( r 1 r 2 ) 2 ] L n 0 [ ( r 1 r 2 ) 2 2 σ g 2 ] ×exp[ ( ik 2z 1 4 σ 0 2 ) r 1 2 +( ik 2z 1 4 σ 0 2 ) r 2 2 + ik z ( r 1 ρ 1 ' r 2 ρ 2 ' ) ] d 2 r 1 d 2 r 2 .
r= r 1 + r 2 2 , r d = r 1 r 2 , ρ ' = ρ 1 ' + ρ 2 ' 2 , ρ d ' = ρ 1 ' ρ 2 ' ,
W( ρ ' , ρ d ' )= A( ρ 1 , ρ 2 ) λ 2 z 2 exp[ ( 1 2 σ g 2 + 1 ρ 0 2 + 1 8 σ 0 2 ) r d 2 + ik z r d ρ ' ] L n 0 ( r d 2 2 σ g 2 ) d 2 r d × exp[ 1 2 σ 0 2 r 2 +r( ik z r d + ik z ρ d ' ) ] d 2 r .
W( ρ ' , ρ d ' )= A( ρ 1 , ρ 2 ) λ 2 z 2 2π σ 0 2 exp( k 2 σ 0 2 2 z 2 ρ d '2 ) L n 0 ( r d 2 2 σ g 2 ) ×exp[ ( 1 2 σ g 2 + 1 ρ 0 2 + 1 8 σ 0 2 + k 2 σ 0 2 2 z 2 ) r d 2 + r d ( ik z ρ ' + k 2 σ 0 2 z 2 ρ d ' ) ] d 2 r d .
a= 1 2 σ g 2 + 1 ρ 0 2 + 1 8 σ 0 2 + k 2 σ 0 2 2 z 2 , b= ik z , c= k 2 σ 0 2 z 2 ,
W( ρ ' , ρ d ' )= A( ρ 1 , ρ 2 ) λ 2 z 2 2π σ 0 2 exp( k 2 σ 0 2 2 z 2 ρ d '2 ) × exp[ a r d 2 + r d ( b ρ ' +c ρ d ' ) ] L n 0 ( r d 2 2 σ g 2 ) d 2 r d .
L n 0 (x)= p=0 n ( n p ) (1) p p! x p ,
( x 2 + y 2 ) p = m=0 p ( p m ) x 2( pm ) y 2m ,
W( ρ x ' , ρ y ' , ρ dx ' , ρ dy ' )= 2π σ 0 2 λ 2 z 2 A( ρ 1 , ρ 2 )exp[ k 2 σ 0 2 2 z 2 ( ρ dx '2 + ρ dy '2 ) ] p=0 n m=0 p ( n p ) ( p m ) (1) p 2 p p! σ g 2p × exp[ a r dx 2 + r dx ( b ρ x ' +c ρ dx ' )a r dy 2 + r dy ( b ρ y ' +c ρ dy ' ) ] r dx 2( pm ) r dy 2m d r dx d r dy .
x n exp[ ( xβ ) 2 ]dx= ( 2i ) n π H n ( iβ ),
W( ρ 1x , ρ 1y , ρ 2x , ρ 2y )= p=0 n m=0 p ( n p )( p m ) 1 2 3p p! σ g 2p a p+1 2 π 2 σ 0 2 λ 2 z 2 exp{ k 2 σ 0 2 2 z 2 [ ( ρ 1x ' ρ 2x ' ) 2 + ( ρ 1y ' ρ 2y ' ) 2 ] } ×exp[ ik 2z ( ρ 1x 2 + ρ 1y 2 ρ 2x 2 ρ 2y 2 ) ( ρ 1x ρ 2x ) 2 + ( ρ 1y ρ 2y ) 2 ρ 0 2 ] ×exp{ 1 4a [ b( ρ 1x ' + ρ 2x ' 2 )+c( ρ 1x ' ρ 2x ' ) ] 2 } H 2(pm) [ i b( ρ 1x ' + ρ 2x ' 2 )+c( ρ 1x ' ρ 2x ' ) 2 a ] ×exp{ 1 4a [ b( ρ 1y ' + ρ 2y ' 2 )+c( ρ 1y ' ρ 2y ' ) ] 2 } H 2m [ i b( ρ 1y ' + ρ 2y ' 2 )+c( ρ 1y ' ρ 2y ' ) 2 a ],
ρ 1x ' =( 1 z ik ρ 0 2 ) ρ 1x + z ik ρ 0 2 ρ 2x , ρ 1y ' =( 1 z ik ρ 0 2 ) ρ 1y + z ik ρ 0 2 ρ 2y ,
ρ 2x ' =( 1+ z ik ρ 0 2 ) ρ 2x z ik ρ 0 2 ρ 1x , ρ 2y ' =( 1+ z ik ρ 0 2 ) ρ 2y z ik ρ 0 2 ρ 1y .
μ( ρ 1x , ρ 1y , ρ 2x , ρ 2y )= W( ρ 1x , ρ 1y , ρ 2x , ρ 2y ) W( ρ 1x , ρ 1y , ρ 1x , ρ 1y )W( ρ 2x , ρ 2y , ρ 2x , ρ 2y ) .
r= r 1 + r 2 2 , r d = r 1 r 2 , ρ= ρ 1 + ρ 2 2 , ρ d = ρ 1 ρ 2 ,
W( ρ, ρ d )= ( k 2πz ) 2 W( r, r d ) exp[ ik z ( ρr )( ρ d r d ) ] ×exp( r d 2 ρ 0 2 r d ρ d ρ 0 2 ρ d 2 ρ 0 2 ) d 2 r d 2 r d ,
W( r, r d )=W( r 1 , r 2 )=W( r+ r d 2 ,r r d 2 ).
W( ρ, ρ d )= ( 1 2π ) 2 W( r ' , ρ d + z k κ d ) d 2 r ' d 2 κ d ×exp( iρ κ d i r ' κ d 3 ρ 0 2 ρ d 2 z 2 ρ 0 2 k 2 κ d 2 3z ρ 0 2 k ρ d κ d ),
W( r ' , ρ d + z k κ d )=exp[ 1 2 σ 0 2 r ' 2 ( 1 8 σ 0 2 + 1 2 σ g 2 ) ( ρ d + z k κ d ) 2 ] L n 0 [ 1 2 σ g 2 ( ρ d + z k κ d ) 2 ].
h( ρ,θ )= ( k 2π ) 2 W( ρ, ρ d )exp( ikθ ρ d ) d 2 ρ d ,
h( ρ,θ )= k 2 16 π 4 2π σ 0 2 L n 0 [ 1 2 σ g 2 ( ρ d + z k κ d ) 2 ] ×exp( a ρ d 2 b κ d 2 c ρ d κ d ikθ ρ d +iρ κ d ) d 2 κ d d 2 ρ d .
a= 1 8 σ 0 2 + 1 2 σ g 2 + 3 ρ 0 2 , b= z 2 8 k 2 σ 0 2 + z 2 2 k 2 σ g 2 + σ 0 2 2 + z 2 k 2 ρ 0 2 , c= z 4k σ 0 2 + z k σ g 2 + 3z k ρ 0 2 .
< x n 1 y n 2 θ x m 1 θ y m 2 >= 1 P x n 1 y n 2 θ x m 1 θ y m 2 h(ρ,θ) d 2 ρ d 2 θ,
P= h(ρ,θ,z) d 2 ρ d 2 θ .
ρ 2 = z 2 k 2 [ 1 2 σ 0 2 + 2 σ g 2 ( 1+n ) ]+2 σ 0 2 + 4 z 2 k 2 ρ 0 2 ,
θ 2 = 1 k 2 [ 1 2 σ 0 2 + 2 σ g 2 ( 1+n ) ]+ 12 k 2 ρ 0 2 ,
ρθ = z k 2 [ 1 2 σ 0 2 + 2 σ g 2 ( 1+n ) ] 6z k 2 ρ 0 2 .
M 2 (z)=k ( ρ 2 θ 2 ρθ 2 ) 1/2 .
M 2 ( z )={ [ z 2 2 k 2 σ 0 2 + 2 z 2 k 2 σ g 2 ( 1+n )+2 σ 0 2 + 4 z 2 k 2 ρ 0 2 ][ 1 2 σ 0 2 + 2 σ g 2 ( 1+n )+ 12 ρ 0 2 ] [ z 2 σ 0 2 + 2z σ g 2 ( 1+n )+ 6z ρ 0 2 ] 2 }
M 2 = [ 1+ 4 σ 0 2 σ g 2 ( 1+n ) ] 1/2 .
M 2 = ( 1+ 4 σ 0 2 σ g 2 ) 1/2 .

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