Abstract

Propagation properties of partially coherent electromagnetic hyperbolic-sine-Gaussian (PCESHG) vortex beams through non-Kolmogorov atmospheric turbulence, including the spectral degree of polarization and evolution behavior of coherent vortices and average intensity are investigated in detail by using the extended Huygens–Fresnel principle and the spatial power spectrum of the refractive index of non-Kolmogorov turbulence. It is shown that the motion, creation and annihilation of the coherent vortices of PCESHG vortex beams in non-Kolmogorov turbulence may appear with the increasing propagation distance, and the distance for the conservation of the topological charge depends on the turbulence parameters and beam parameters. In additions, the evolution behavior of coherent vortices, average intensity and spectral degree of polarization vary significantly for different values of the generalized exponent parameter and the generalized refractive-index structure parameter of non-Kolmogorov turbulence, and the beam parameters as well as the propagation distance.

© 2015 Optical Society of America

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2014 (2)

2013 (1)

2012 (3)

X. M. He and B. Lü, “Dynamic evolution of composite coherence vortices by superimpositions of partially coherent hyperbolic-sine-Gaussian vortex beams in non-Kolmogorov atmospheric turbulence,” Acta. Phys. Sin. 61, 054201 (2012).

Q. G. Sun, K. Y. Zhou, G. Y. Fang, G. Q. Zhang, Z. J. Liu, and S. T. Liu, “Hollow sinh-Gaussian beams and their paraxial properties,” Opt. Express 20(9), 9682–9691 (2012).
[Crossref] [PubMed]

X. Y. Liu and D. M. Zhao, “The statistical properties of anisotropic electromagnetic beams passing through the biological tissues,” Opt. Commun. 285(21-22), 4152–4156 (2012).
[Crossref]

2011 (2)

R. P. Chen, H. P. Zheng, and X. X. Chu, “Propagation properties of a sinh-Gaussian beam in a Kerr medium,” Appl. Phys. B 102(3), 695–698 (2011).
[Crossref]

Y. P. Huang, G. P. Zhao, Z. C. Duan, D. He, Z. H. Gao, and F. H. Wang, “Spreading and M2-factor of elegant Hermit-Gaussian beams through non-Kolmogorov turbulence,” J. Mod. Opt. 58(11), 912–917 (2011).
[Crossref]

2010 (2)

2009 (2)

X. L. Ji and X. W. Chen, “Changes in the polarization, the coherence and the spectrum of partially coherent electromagnetic Hermite–Gaussian beams in turbulence,” Opt. Laser Technol. 41(2), 165–171 (2009).
[Crossref]

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11(4), 045710 (2009).
[Crossref]

2008 (3)

Z. Y. Chen and J. X. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372(15), 2734–2740 (2008).
[Crossref]

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25(1), 225–230 (2008).
[Crossref] [PubMed]

X. Xiao, X. L. Ji, and B. D. Lü, “The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field,” Opt. Laser Technol. 40(1), 129–136 (2008).
[Crossref]

2007 (2)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

W. Lu, L. R. Liu, J. F. Sun, Q. G. Yang, and Y. J. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007).
[Crossref]

2005 (3)

H. T. Eyyuboğlu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A 22(12), 2709–2718 (2005).
[Crossref] [PubMed]

H. Roychowdhury and O. Korotkova, “Realizability condition for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[Crossref]

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250(4-6), 218–230 (2005).
[Crossref]

2004 (3)

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander., “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B 21(11), 1895–1900 (2004).
[Crossref]

S. Konar and S. Jana, “Linear and nonlinear propagation of sinh-Gaussian pulses in dispersive media possessing Kerr nonlinearity,” Opt. Commun. 236(1-3), 7–20 (2004).
[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(4-6), 225–230 (2004).
[Crossref]

2003 (2)

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

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003).
[Crossref]

1998 (2)

1994 (1)

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
[Crossref] [PubMed]

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Baykal, Y.

Cai, Y. J.

Casperson, L. W.

Chen, R. P.

R. P. Chen, H. P. Zheng, and X. X. Chu, “Propagation properties of a sinh-Gaussian beam in a Kerr medium,” Appl. Phys. B 102(3), 695–698 (2011).
[Crossref]

Chen, X. W.

X. L. Ji and X. W. Chen, “Changes in the polarization, the coherence and the spectrum of partially coherent electromagnetic Hermite–Gaussian beams in turbulence,” Opt. Laser Technol. 41(2), 165–171 (2009).
[Crossref]

Chen, Z. Y.

Z. Y. Chen and J. X. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372(15), 2734–2740 (2008).
[Crossref]

Chu, X. X.

R. P. Chen, H. P. Zheng, and X. X. Chu, “Propagation properties of a sinh-Gaussian beam in a Kerr medium,” Appl. Phys. B 102(3), 695–698 (2011).
[Crossref]

Davies, G.

Duan, Z. C.

Y. P. Huang, B. Zhang, Z. H. Gao, G. P. Zhao, and Z. C. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
[Crossref] [PubMed]

Y. P. Huang, G. P. Zhao, Z. C. Duan, D. He, Z. H. Gao, and F. H. Wang, “Spreading and M2-factor of elegant Hermit-Gaussian beams through non-Kolmogorov turbulence,” J. Mod. Opt. 58(11), 912–917 (2011).
[Crossref]

Eyyuboglu, H. T.

Fang, G. Y.

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Flossmann, F.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250(4-6), 218–230 (2005).
[Crossref]

Freund, I.

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
[Crossref] [PubMed]

Gao, Z. H.

Y. P. Huang, B. Zhang, Z. H. Gao, G. P. Zhao, and Z. C. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
[Crossref] [PubMed]

Y. P. Huang, G. P. Zhao, Z. C. Duan, D. He, Z. H. Gao, and F. H. Wang, “Spreading and M2-factor of elegant Hermit-Gaussian beams through non-Kolmogorov turbulence,” J. Mod. Opt. 58(11), 912–917 (2011).
[Crossref]

Gbur, G.

He, D.

Y. P. Huang, G. P. Zhao, Z. C. Duan, D. He, Z. H. Gao, and F. H. Wang, “Spreading and M2-factor of elegant Hermit-Gaussian beams through non-Kolmogorov turbulence,” J. Mod. Opt. 58(11), 912–917 (2011).
[Crossref]

He, X. M.

X. M. He and B. Lü, “Dynamic evolution of composite coherence vortices by superimpositions of partially coherent hyperbolic-sine-Gaussian vortex beams in non-Kolmogorov atmospheric turbulence,” Acta. Phys. Sin. 61, 054201 (2012).

Huang, Y. P.

Y. P. Huang, B. Zhang, Z. H. Gao, G. P. Zhao, and Z. C. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
[Crossref] [PubMed]

Y. P. Huang, G. P. Zhao, Z. C. Duan, D. He, Z. H. Gao, and F. H. Wang, “Spreading and M2-factor of elegant Hermit-Gaussian beams through non-Kolmogorov turbulence,” J. Mod. Opt. 58(11), 912–917 (2011).
[Crossref]

Jana, S.

S. Konar and S. Jana, “Linear and nonlinear propagation of sinh-Gaussian pulses in dispersive media possessing Kerr nonlinearity,” Opt. Commun. 236(1-3), 7–20 (2004).
[Crossref]

Ji, X. L.

X. L. Ji and X. W. Chen, “Changes in the polarization, the coherence and the spectrum of partially coherent electromagnetic Hermite–Gaussian beams in turbulence,” Opt. Laser Technol. 41(2), 165–171 (2009).
[Crossref]

X. Xiao, X. L. Ji, and B. D. Lü, “The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field,” Opt. Laser Technol. 40(1), 129–136 (2008).
[Crossref]

Jin, P.

Konar, S.

S. Konar and S. Jana, “Linear and nonlinear propagation of sinh-Gaussian pulses in dispersive media possessing Kerr nonlinearity,” Opt. Commun. 236(1-3), 7–20 (2004).
[Crossref]

Korotkova, O.

S. J. Zhu, Y. J. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010).
[Crossref] [PubMed]

E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010).
[Crossref] [PubMed]

H. Roychowdhury and O. Korotkova, “Realizability condition for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 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(4-6), 225–230 (2004).
[Crossref]

Li, J.

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11(4), 045710 (2009).
[Crossref]

Lin, J.

Liu, L. R.

W. Lu, L. R. Liu, J. F. Sun, Q. G. Yang, and Y. J. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007).
[Crossref]

Liu, S. T.

Liu, X. Y.

X. Y. Liu and D. M. Zhao, “The statistical properties of anisotropic electromagnetic beams passing through the biological tissues,” Opt. Commun. 285(21-22), 4152–4156 (2012).
[Crossref]

Liu, Z. J.

Lu, W.

W. Lu, L. R. Liu, J. F. Sun, Q. G. Yang, and Y. J. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007).
[Crossref]

Lü, B.

X. M. He and B. Lü, “Dynamic evolution of composite coherence vortices by superimpositions of partially coherent hyperbolic-sine-Gaussian vortex beams in non-Kolmogorov atmospheric turbulence,” Acta. Phys. Sin. 61, 054201 (2012).

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11(4), 045710 (2009).
[Crossref]

Lü, B. D.

X. Xiao, X. L. Ji, and B. D. Lü, “The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field,” Opt. Laser Technol. 40(1), 129–136 (2008).
[Crossref]

Ma, Y.

Maier, M.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250(4-6), 218–230 (2005).
[Crossref]

Maleev, I. D.

Marathay, A. S.

Palacios, D. M.

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Pu, J. X.

Z. Y. Chen and J. X. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372(15), 2734–2740 (2008).
[Crossref]

Roychowdhury, H.

H. Roychowdhury and O. Korotkova, “Realizability condition for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[Crossref]

Salem, M.

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(4-6), 225–230 (2004).
[Crossref]

Schwarz, U. T.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250(4-6), 218–230 (2005).
[Crossref]

Shchepakina, E.

Shvartsman, N.

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
[Crossref] [PubMed]

Sun, J. F.

W. Lu, L. R. Liu, J. F. Sun, Q. G. Yang, and Y. J. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007).
[Crossref]

Sun, Q. G.

Swartzlander, G. A.

Tan, J. B.

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Tovar, A. A.

Tyson, R. K.

Visser, T. D.

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003).
[Crossref]

Wang, F. H.

Y. P. Huang, G. P. Zhao, Z. C. Duan, D. He, Z. H. Gao, and F. H. Wang, “Spreading and M2-factor of elegant Hermit-Gaussian beams through non-Kolmogorov turbulence,” J. Mod. Opt. 58(11), 912–917 (2011).
[Crossref]

Wolf, E.

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(4-6), 225–230 (2004).
[Crossref]

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

Xiao, X.

X. Xiao, X. L. Ji, and B. D. Lü, “The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field,” Opt. Laser Technol. 40(1), 129–136 (2008).
[Crossref]

Xu, J.

J. Xu and D. Zhao, “Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence,” Opt. Laser Technol. 57, 189–193 (2014).
[Crossref]

Yang, Q. G.

W. Lu, L. R. Liu, J. F. Sun, Q. G. Yang, and Y. J. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007).
[Crossref]

Zhang, B.

Zhang, G. Q.

Zhao, D.

J. Xu and D. Zhao, “Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence,” Opt. Laser Technol. 57, 189–193 (2014).
[Crossref]

Zhao, D. M.

X. Y. Liu and D. M. Zhao, “The statistical properties of anisotropic electromagnetic beams passing through the biological tissues,” Opt. Commun. 285(21-22), 4152–4156 (2012).
[Crossref]

Zhao, G. P.

Y. P. Huang, B. Zhang, Z. H. Gao, G. P. Zhao, and Z. C. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
[Crossref] [PubMed]

Y. P. Huang, G. P. Zhao, Z. C. Duan, D. He, Z. H. Gao, and F. H. Wang, “Spreading and M2-factor of elegant Hermit-Gaussian beams through non-Kolmogorov turbulence,” J. Mod. Opt. 58(11), 912–917 (2011).
[Crossref]

Zheng, H. P.

R. P. Chen, H. P. Zheng, and X. X. Chu, “Propagation properties of a sinh-Gaussian beam in a Kerr medium,” Appl. Phys. B 102(3), 695–698 (2011).
[Crossref]

Zhou, K. Y.

Zhu, S. J.

Zhu, Y. J.

W. Lu, L. R. Liu, J. F. Sun, Q. G. Yang, and Y. J. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007).
[Crossref]

Acta. Phys. Sin. (1)

X. M. He and B. Lü, “Dynamic evolution of composite coherence vortices by superimpositions of partially coherent hyperbolic-sine-Gaussian vortex beams in non-Kolmogorov atmospheric turbulence,” Acta. Phys. Sin. 61, 054201 (2012).

Appl. Phys. B (1)

R. P. Chen, H. P. Zheng, and X. X. Chu, “Propagation properties of a sinh-Gaussian beam in a Kerr medium,” Appl. Phys. B 102(3), 695–698 (2011).
[Crossref]

J. Mod. Opt. (1)

Y. P. Huang, G. P. Zhao, Z. C. Duan, D. He, Z. H. Gao, and F. H. Wang, “Spreading and M2-factor of elegant Hermit-Gaussian beams through non-Kolmogorov turbulence,” J. Mod. Opt. 58(11), 912–917 (2011).
[Crossref]

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

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11(4), 045710 (2009).
[Crossref]

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (1)

Opt. Commun. (7)

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250(4-6), 218–230 (2005).
[Crossref]

H. Roychowdhury and O. Korotkova, “Realizability condition for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[Crossref]

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003).
[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(4-6), 225–230 (2004).
[Crossref]

W. Lu, L. R. Liu, J. F. Sun, Q. G. Yang, and Y. J. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Evolution of normalized intensity profiles of a PCESHG vortex beam with m = + 1 through non-Kolmogorov turbulence, (a) α = 3.9, C ˜ n 2 = 10−14m3-α ; (b) α = 3.11, C ˜ n 2 = 10−14m3-α; (c) α = 3.11, C ˜ n 2 = 10−16m3-α.
Fig. 2
Fig. 2 Changes in the spectral degree of polarization of a PCESHG vortex beam with m = + 1 through non-Kolmogorov turbulence for different parameters C ˜ n 2 , σxx andσyy.
Fig. 3
Fig. 3 Changes in the degree of polarization of a PCESHG vortex beam with m = + 1 through non-Kolmogorov turbulence versus α for different values of z, σxx and σyy, C ˜ n 2 and Ω0.
Fig. 4
Fig. 4 Curves of Reμ = 0 and Imμ = 0 of a PCESHG vortex beam with m = + 1 through non-Kolmogorov turbulence at the propagation distance (a) z = 25m and (b) z = 300m.
Fig. 5
Fig. 5 Contour lines of phase of a PCESHG vortex beam with m = + 1 through non-Kolmogorov turbulence at the plane (a) z = 25m and (b) z = 300m.
Fig. 6
Fig. 6 Position and number of coherent vortices of a PCESHG vortex beam with m = + 1 through non-Kolmogorov turbulence for different values ofα and C ˜ n 2 .
Fig. 7
Fig. 7 Curves of Reμ = 0 and Imμ = 0 of a PCESHG vortex beam with m = + 1 through non-Kolmogorov turbulence for the different values of α, C ˜ n 2 , w0 and Ω0 at the different propagation distance z.

Equations (51)

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W ( 0 ) ( ρ 1 , ρ 2 ,0 )=[ W xx ( 0 ) ( ρ 1 , ρ 2 ,0 ) W xy ( 0 ) ( ρ 1 , ρ 2 ,0 ) W yx ( 0 ) ( ρ 1 , ρ 2 ,0 ) W yy ( 0 ) ( ρ 1 , ρ 2 ,0 ) ],
U(ρ,z=0)=u(ρ) [ ρ x +isgn(m) ρ y ] | m | ,
W ij ( 0 ) ( ρ 1 , ρ 2 ,0)= A i A j B ij [( ρ 1x ρ 2x + ρ 1y ρ 2y )+isgn( m )( ρ 1x ρ 2y ρ 2x ρ 1y )] m ×exp( ρ 1 2 + ρ 2 2 w 0 2 )sinh[ Ω 0 ( ρ 1x + ρ 1y )]sinh[ Ω 0 ( ρ 2x + ρ 2y )]exp( | ρ 1 ρ 2 | 2 2 σ 0 2 ).
A x 2 σ xx 2 σ xx 2 +4 w 0 2 2 A x A y | B xy | σ xy 2 σ xy 2 +4 w 0 2 + A y 2 σ yy 2 σ yy 2 +4 w 0 2 0,
σ xx 2 σ xx 2 +4 w 0 2 2 σ xy 2 σ xy 2 +4 w 0 2 + σ yy 2 σ yy 2 +4 w 0 2 0,
1 4 w 0 2 + 1 σ xx 2 << 2π λ 2 , 1 4 w 0 2 + 1 σ yy 2 << 2π λ 2 ,
max{ σ xx , σ yy } σ xy min{ σ xx B xy , σ yy B xy },
W ij ( ρ 1 , ρ 2 ,z)= ( k 2πz ) 2 d 2 ρ 1 d 2 ρ 2 W ij ( 0 ) ( ρ 1 , ρ 2 ,0) ×exp{ ik 2z [ ( ρ 1 ρ 1 ) 2 ( ρ 2 ρ 2 ) 2 ] } ×exp[ ψ ( ρ 1 , ρ 1 )+ψ( ρ 2 , ρ 2 )],
exp[ ψ ( ρ 1 , ρ 1 )+ψ( ρ 2 , ρ 2 )] =exp{ π 2 k 2 z 3 [ ( ρ 1 ρ 2 ) 2 +( ρ 1 ρ 2 )( ρ 1 ρ 2 )+ ( ρ 1 ρ 2 ) 2 ] 0 κ 3 Φ n ( κ,α )dκ },
Φ n (κ,α)= C ˜ n 2 A(α) exp[( κ 2 / κ m 2 )] ( κ 2 + κ 2 2 ) α/2 ,0κ<,3<α<4,
T( α )= π 2 k 2 3 0 κ 3 Φ n ( κ,α )dκ = π 2 k 2 C ˜ n 2 A(α) 6(α2) { [ c(α) l 0 ] 2α [ 8 π 2 L 0 2 + (α2) c 2 (α) l 0 2 ] ×exp( 4 π 2 c 2 (α) l 0 2 L 0 2 )Γ[2 α 2 , 4 π 2 l 0 2 c 2 (α) L 0 2 ]2 ( 2π L 0 ) 4α }.
W ij± ( ρ 1 , ρ 2 ,z)= A i A j B ij ( k 4z ) 2 1 a ij C ij exp[ ik 2z ( ρ 1 2 ρ 2 2 )] ×exp[ T( α )z ( ρ 1 ρ 2 ) 2 ]( V 1ij + V 2ij V 3ij V 4ij ),
V 1ij =exp( E x 2 + E y 2 4 a ij + F x 2 + F y 2 C ij )( F x 2 + F y 2 C ij 2 + 1 C ij 1 4Q D x 2 + D y 2 4 Q 2 ±i D x F y F x D y C ij Q ),
V 2ij =exp( E x 2 + E y 2 4 a ij + F x 2 + F y 2 C ij )( F x 2 + F y 2 C ij 2 + 1 C ij 1 4Q D x 2 + D y 2 4 Q 2 ±i D x F y F x D y C ij Q ),
V 3ij =exp( R x 2 + R y 2 4 a ij + G x 2 + G y 2 C ij )( G x 2 + G y 2 C ij 2 + 1 C ij 1 4Q H x 2 + H y 2 4 Q 2 ±i H x G y G x H y C ij Q ),
V 4ij =exp( R x 2 + R y 2 4 a ij + G x 2 + G y 2 C ij )( G x 2 + G y 2 C ij 2 + 1 C ij 1 4Q H x 2 + H y 2 4 Q 2 ±i H x G y G x H y C ij Q ),
a ij = 1 2 w 0 2 +T( α )z+ 1 2 σ ij 2 ,
b= ik z ,
C ij = 2 w 0 2 b 2 4 a ij ,
E x = b 2 ( ρ 1x + ρ 2x )T( α )z( ρ 1x ρ 2x ),
E y = b 2 ( ρ 1y + ρ 2y )T( α )z( ρ 1y ρ 2y ),
F x = 1 2 [b( ρ 1x ρ 2x ) b E x 2 a ij +2 Ω 0 ],
F y = 1 2 [b( ρ 1y ρ 2y ) b E y 2 a ij +2 Ω 0 ],
F x = 1 2 [b( ρ 1x ρ 2x ) b E x 2 a ij 2 Ω 0 ],
F y = 1 2 [b( ρ 1y ρ 2y ) b E y 2 a ij 2 Ω 0 ].
Q ij = a ij b 2 w 0 2 8 ,
D x = E x 2 b w 0 2 8 [b( ρ 1x ρ 2x )+2 Ω 0 ],
D y = E y 2 b w 0 2 8 [b( ρ 1y ρ 2y )+2 Ω 0 ],
D x = E x 2 b w 0 2 8 [b( ρ 1x ρ 2x )-2 Ω 0 ],
D y = E y 2 b w 0 2 8 [b( ρ 1y ρ 2y )-2 Ω 0 ],
R x = E x + Ω 0 , R y = E y + Ω 0 .
R x = E x Ω 0 , R y = E y Ω 0 .
G x = 1 2 [b( ρ 1x ρ 2x ) b R x 2 a ij ], G y = 1 2 [b( ρ 1y ρ 2y ) b R y 2 a ij ],
G x = 1 2 [b( ρ 1x ρ 2x ) b R x 2 a ij ], G y = 1 2 [b( ρ 1y ρ 2y ) b R y 2 a ij ],
H x = 1 2 [ b 2 w 0 2 4 ( ρ 2x - ρ 1x )+ R x ],
H y = 1 2 [ b 2 w 0 2 4 ( ρ 2y - ρ 1y )+ R y ],
H x = 1 2 [ b 2 w 0 2 4 ( ρ 2x - ρ 1x )+ R x ],
H y = 1 2 [ b 2 w 0 2 4 ( ρ 2y - ρ 1y )+ R y ].
W ij ( ρ , ρ ,z)= A i A j B ij ( k 4z ) 2 1 a ij C ij ( V 1ij + V 2ij V 3ij V 4ij ) = A i A j B ij ( k 4z ) 2 1 a ij C ij [ p 1ij q 1ij + p 2ij q 2ij +2 q 3ij t 3ij 2 p 3ij s 3ij ± p 1ij s 1ij ± p 2ij s 2ij ],
p 1ij =exp[ b 2 4 a ij ( ρ x 2 + ρ y 2 )]exp{ 1 4 C ij [ ( 2 Ω 0 + k 2 2 a ij z 2 ρ x ) 2 + ( 2 Ω 0 + k 2 2 a ij z 2 ρ y ) 2 ] },
q 1ij = 1 4 C ij 2 [ ( 2 Ω 0 + k 2 2 a ij z 2 ρ x ) 2 + ( 2 Ω 0 + k 2 2 a ij z 2 ρ y ) 2 ] + 1 C ij + k 2 16 Q ij 2 z 2 [ ( ρ x w 0 2 Ω 0 2 ) 2 + ( ρ y w 0 2 Ω 0 2 ) 2 ] 1 4 Q ij ,
s 1ij = k 4 C ij Q ij z [ ( ρ x w 0 2 Ω 0 2 ) ( 2 Ω 0 + k 2 2 a ij z 2 ρ y )( 2 Ω 0 + k 2 2 a ij z 2 ρ x ) ( ρ y w 0 2 Ω 0 2 ) ],
p 3ij =exp{ 1 4 a ij ( 1 k 2 4 C ij a ij z 2 )[ k 2 z 2 ( ρ x 2 + ρ y 2 )+2 Ω 0 2 ] }cos[ 2k Ω 0 z ( ρ x + ρ y ) ],
q 3ij =exp{ 1 4 a ij ( 1 k 2 4 C ij a ij z 2 )[ k 2 z 2 ( ρ x 2 + ρ y 2 )+2 Ω 0 2 ] }sin[ 2k Ω 0 z ( ρ x + ρ y ) ],
s 3ij = 1 C ij 1 4 Q ij 1 16 ( k 2 C ij 2 a ij 2 z 2 + 1 Q ij 2 )[ k 2 z 2 ( ρ x 2 + ρ y 2 )+2 Ω 0 2 ],
t 3ij = k Ω 0 8z ( k 2 C ij 2 a ij 2 z 2 + 1 Q ij 2 )( ρ x + ρ y ),
I( ρ ,z )=Tr W ( ρ , ρ ,z )= W xx ( ρ , ρ ,z )+ W yy ( ρ , ρ ,z ).
P( ρ ,z )= 1 4Det W ( ρ , ρ ,z ) [ Tr W ( ρ , ρ ,z ) ] 2 ,
μ( ρ 1 , ρ 2 ,z )= Tr W ( ρ 1 , ρ 2 ,z ) [Tr W ( ρ 1 , ρ 1 ,z )Tr W ( ρ 2 , ρ 2 ,z )] 1/2 ,
Re[μ( ρ 1 , ρ 2 ,z)]=0,
Im[μ( ρ 1 , ρ 2 ,z)]=0.

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