Abstract

The propagation formulae for the propagation factor (known as M2-factor) and beam wander of electromagnetic Gaussian Schell-model (EGSM) array beams in non-Kolmogorov turbulence are derived by using the extended Huygens-Fresnel principle and the second-order moments of the Wigner distribution function. The results indicate that the M2-factor and beam wander depend on the beam parameters and turbulence parameters, and the relative M2-factor has a maximum when the generalized exponent parameter α is equal to 3.1. Otherwise, the changes of the separation distances (x0, y0) have great influence on the relative M2-factor. The relative beam wander increases rapidly when 3<α<3.2; however, it increases slowly when 3.2<α<4. It is also shown that the beam spreading of EGSM array beams is more affected by turbulence than the root mean square beam wander.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. Y. Ai and Y. Dan, “Range of turbulence-negligible propagation of Gaussian Schell-model array beams,” Opt. Commun. 284(13), 3216–3220 (2011).
    [Crossref]
  2. X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42(4), 604–609 (2010).
    [Crossref]
  3. X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
    [Crossref]
  4. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
    [Crossref]
  5. X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
    [Crossref]
  6. X. Ji and Z. Pu, “Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence,” Appl. Phys. B 93(4), 915–923 (2008).
    [Crossref]
  7. X. Ji, E. Zhang, and B. Lü, “Superimposed partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. B 25(5), 825–833 (2008).
    [Crossref]
  8. 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]
  9. Y. Dan and B. Zhang, “Beam Propagation Factor of Partially Coherent Flat-topped Beams in a Turbulent Atmosphere,” Opt. Express 16(20), 15563–15575 (2008).
    [Crossref]
  10. 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]
  11. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010).
    [Crossref]
  12. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 3(1), 1–9 (2001).
    [Crossref]
  13. O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
    [Crossref]
  14. E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, Cambridge, 2007).
  15. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
    [Crossref]
  16. G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000).
    [Crossref]
  17. B. Zhang, Y. Xu, Y. Dan, X. Deng, and Z. Zhao, “Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence,” Optik 149, 398–408 (2017).
    [Crossref]
  18. B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett. 33(20), 2410–2412 (2008).
    [Crossref]
  19. H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007).
    [Crossref]
  20. B. Zhang, X. Chu, and Q. Li, “Generalized beam-propagation factor of partially coherent beams propagating through hard-edged apertures,” J. Opt. Soc. Am. A 19(7), 1370–1375 (2002).
    [Crossref]
  21. X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A: Pure Appl. Opt. 11(10), 105705 (2009).
    [Crossref]
  22. G. Zhou, “Generalzied M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A: Pure Appl. Opt. 12(1), 015701 (2010).
    [Crossref]
  23. Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009).
    [Crossref]
  24. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyubolu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
    [Crossref]
  25. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed.; SPIE: Bellingham, 2005.
  26. J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46(8), 086002 (2007).
    [Crossref]
  27. Y. Huang, A. Zeng, Z. Gao, and B. Zhang, “Beam Wander of Partially Coherent Array Beams through Non-Kolmogorov Turbulence,” Opt. Lett. 40(8), 1619–1622 (2015).
    [Crossref]
  28. Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
    [Crossref]
  29. D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward Global Quantum Communication: Beam Wandering Preserves Nonclassicality,” Phys. Rev. Lett. 108(22), 220501 (2012).
    [Crossref]
  30. W. Wen and X. Chu, “Beam wande of partially coherent Airy beams,” J. Mod. Opt. 61(5), 379–384 (2014).
    [Crossref]
  31. G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: The effect of partial coherence,” Phys. Rev. E 76(5), 056606 (2007).
    [Crossref]
  32. H. T. Eyyuboğlu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93(2-3), 595–604 (2008).
    [Crossref]
  33. C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95(4), 763–771 (2009).
    [Crossref]
  34. C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J(0)- and I(0)-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98(1), 195–202 (2010).
    [Crossref]
  35. S. Yu, Z. Chen, T. Wang, G. Wu, H. Guo, and W. Gu, “Beam wander of electromagnetic Gaussian-Schell model beams propagating in atmospheric turbulence,” Appl. Opt. 51(31), 7581–7585 (2012).
    [Crossref]
  36. G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random EGSM vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
    [Crossref]

2017 (2)

B. Zhang, Y. Xu, Y. Dan, X. Deng, and Z. Zhao, “Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence,” Optik 149, 398–408 (2017).
[Crossref]

Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
[Crossref]

2015 (2)

Y. Huang, A. Zeng, Z. Gao, and B. Zhang, “Beam Wander of Partially Coherent Array Beams through Non-Kolmogorov Turbulence,” Opt. Lett. 40(8), 1619–1622 (2015).
[Crossref]

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random EGSM vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

2014 (1)

W. Wen and X. Chu, “Beam wande of partially coherent Airy beams,” J. Mod. Opt. 61(5), 379–384 (2014).
[Crossref]

2012 (2)

D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward Global Quantum Communication: Beam Wandering Preserves Nonclassicality,” Phys. Rev. Lett. 108(22), 220501 (2012).
[Crossref]

S. Yu, Z. Chen, T. Wang, G. Wu, H. Guo, and W. Gu, “Beam wander of electromagnetic Gaussian-Schell model beams propagating in atmospheric turbulence,” Appl. Opt. 51(31), 7581–7585 (2012).
[Crossref]

2011 (1)

Y. Ai and Y. Dan, “Range of turbulence-negligible propagation of Gaussian Schell-model array beams,” Opt. Commun. 284(13), 3216–3220 (2011).
[Crossref]

2010 (6)

X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42(4), 604–609 (2010).
[Crossref]

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

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]

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

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J(0)- and I(0)-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98(1), 195–202 (2010).
[Crossref]

G. Zhou, “Generalzied M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A: Pure Appl. Opt. 12(1), 015701 (2010).
[Crossref]

2009 (5)

Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009).
[Crossref]

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

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A: Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95(4), 763–771 (2009).
[Crossref]

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

2008 (6)

2007 (4)

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46(8), 086002 (2007).
[Crossref]

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: The effect of partial coherence,” Phys. Rev. E 76(5), 056606 (2007).
[Crossref]

H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007).
[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]

2004 (1)

2002 (1)

2001 (1)

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

2000 (1)

1998 (1)

Agrawal, G. P.

Ai, Y.

Y. Ai and Y. Dan, “Range of turbulence-negligible propagation of Gaussian Schell-model array beams,” Opt. Commun. 284(13), 3216–3220 (2011).
[Crossref]

Andrews, L. C.

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46(8), 086002 (2007).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed.; SPIE: Bellingham, 2005.

Baykal, Y.

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J(0)- and I(0)-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98(1), 195–202 (2010).
[Crossref]

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95(4), 763–771 (2009).
[Crossref]

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

Berman, G. P.

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: The effect of partial coherence,” Phys. Rev. E 76(5), 056606 (2007).
[Crossref]

Borghi, R.

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

Cai, Y.

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J(0)- and I(0)-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98(1), 195–202 (2010).
[Crossref]

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

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]

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

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95(4), 763–771 (2009).
[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]

Chen, Z.

Chu, X.

Chumak, A. A.

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: The effect of partial coherence,” Phys. Rev. E 76(5), 056606 (2007).
[Crossref]

Cil, C. Z.

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J(0)- and I(0)-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98(1), 195–202 (2010).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95(4), 763–771 (2009).
[Crossref]

H. T. Eyyuboğlu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93(2-3), 595–604 (2008).
[Crossref]

Dai, W.

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random EGSM vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

Dan, Y.

B. Zhang, Y. Xu, Y. Dan, X. Deng, and Z. Zhao, “Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence,” Optik 149, 398–408 (2017).
[Crossref]

Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
[Crossref]

Y. Ai and Y. Dan, “Range of turbulence-negligible propagation of Gaussian Schell-model array beams,” Opt. Commun. 284(13), 3216–3220 (2011).
[Crossref]

Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009).
[Crossref]

Y. Dan and B. Zhang, “Beam Propagation Factor of Partially Coherent Flat-topped Beams in a Turbulent Atmosphere,” Opt. Express 16(20), 15563–15575 (2008).
[Crossref]

Deng, X.

B. Zhang, Y. Xu, Y. Dan, X. Deng, and Z. Zhao, “Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence,” Optik 149, 398–408 (2017).
[Crossref]

Du, X.

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
[Crossref]

Eyyuboglu, H. T.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J(0)- and I(0)-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98(1), 195–202 (2010).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95(4), 763–771 (2009).
[Crossref]

H. T. Eyyuboğlu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93(2-3), 595–604 (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]

Eyyubolu, H. T.

Feng, H.

Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
[Crossref]

Gao, Z.

Gori, F.

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

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
[Crossref]

Gorshkov, V. N.

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: The effect of partial coherence,” Phys. Rev. E 76(5), 056606 (2007).
[Crossref]

Gu, W.

Guo, H.

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random EGSM vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

S. Yu, Z. Chen, T. Wang, G. Wu, H. Guo, and W. Gu, “Beam wander of electromagnetic Gaussian-Schell model beams propagating in atmospheric turbulence,” Appl. Opt. 51(31), 7581–7585 (2012).
[Crossref]

Huang, Y.

Ji, X.

X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42(4), 604–609 (2010).
[Crossref]

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A: Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

X. Ji and Z. Pu, “Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence,” Appl. Phys. B 93(4), 915–923 (2008).
[Crossref]

X. Ji, E. Zhang, and B. Lü, “Superimposed partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. B 25(5), 825–833 (2008).
[Crossref]

Jia, X.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A: Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

Kandpal, H. C.

Kanseri, B.

Korotkova, O.

Li, Q.

Li, X.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42(4), 604–609 (2010).
[Crossref]

Lin, Q.

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]

Lü, B.

Mondello, A.

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

Phillips, R. L.

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46(8), 086002 (2007).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed.; SPIE: Bellingham, 2005.

Piquero, G.

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

Pu, Z.

X. Ji and Z. Pu, “Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence,” Appl. Phys. B 93(4), 915–923 (2008).
[Crossref]

Qu, J.

Recolons, J.

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46(8), 086002 (2007).
[Crossref]

Salem, M.

Santarsiero, M.

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

Semenov, A. A.

D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward Global Quantum Communication: Beam Wandering Preserves Nonclassicality,” Phys. Rev. Lett. 108(22), 220501 (2012).
[Crossref]

Simon, R.

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

Tang, H.

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random EGSM vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

Tian, H.

Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
[Crossref]

Vasylyev, D. Y.

D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward Global Quantum Communication: Beam Wandering Preserves Nonclassicality,” Phys. Rev. Lett. 108(22), 220501 (2012).
[Crossref]

Vogel, W.

D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward Global Quantum Communication: Beam Wandering Preserves Nonclassicality,” Phys. Rev. Lett. 108(22), 220501 (2012).
[Crossref]

Wang, F.

Wang, H.

Wang, S.

Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
[Crossref]

Wang, T.

Wang, X.

Wen, W.

W. Wen and X. Chu, “Beam wande of partially coherent Airy beams,” J. Mod. Opt. 61(5), 379–384 (2014).
[Crossref]

Wolf, E.

Wu, G.

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random EGSM vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

S. Yu, Z. Chen, T. Wang, G. Wu, H. Guo, and W. Gu, “Beam wander of electromagnetic Gaussian-Schell model beams propagating in atmospheric turbulence,” Appl. Opt. 51(31), 7581–7585 (2012).
[Crossref]

Xu, Y.

B. Zhang, Y. Xu, Y. Dan, X. Deng, and Z. Zhao, “Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence,” Optik 149, 398–408 (2017).
[Crossref]

Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
[Crossref]

Yang, F.

X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42(4), 604–609 (2010).
[Crossref]

Yang, K.

Yu, S.

Yuan, Y.

Zeng, A.

Zhang, B.

Zhang, E.

Zhang, T.

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A: Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

Zhao, D.

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
[Crossref]

Zhao, Z.

B. Zhang, Y. Xu, Y. Dan, X. Deng, and Z. Zhao, “Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence,” Optik 149, 398–408 (2017).
[Crossref]

Zhou, G.

G. Zhou, “Generalzied M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A: Pure Appl. Opt. 12(1), 015701 (2010).
[Crossref]

Zhu, S.

Zhu, Y.

Appl. Opt. (1)

Appl. Phys. B (5)

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

X. Ji and Z. Pu, “Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence,” Appl. Phys. B 93(4), 915–923 (2008).
[Crossref]

H. T. Eyyuboğlu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93(2-3), 595–604 (2008).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95(4), 763–771 (2009).
[Crossref]

C. Z. Cil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J(0)- and I(0)-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98(1), 195–202 (2010).
[Crossref]

J. Mod. Opt. (2)

Y. Xu, H. Tian, Y. Dan, H. Feng, and S. Wang, “Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(8), 844–854 (2017).
[Crossref]

W. Wen and X. Chu, “Beam wande of partially coherent Airy beams,” J. Mod. Opt. 61(5), 379–384 (2014).
[Crossref]

J. Opt. A: Pure Appl. Opt. (3)

X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A: Pure Appl. Opt. 11(10), 105705 (2009).
[Crossref]

G. Zhou, “Generalzied M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A: Pure Appl. Opt. 12(1), 015701 (2010).
[Crossref]

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

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

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

Opt. Commun. (4)

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random EGSM vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[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]

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

Y. Ai and Y. Dan, “Range of turbulence-negligible propagation of Gaussian Schell-model array beams,” Opt. Commun. 284(13), 3216–3220 (2011).
[Crossref]

Opt. Eng. (1)

J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. 46(8), 086002 (2007).
[Crossref]

Opt. Express (5)

Opt. Laser Technol. (1)

X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42(4), 604–609 (2010).
[Crossref]

Opt. Lett. (6)

Optik (1)

B. Zhang, Y. Xu, Y. Dan, X. Deng, and Z. Zhao, “Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence,” Optik 149, 398–408 (2017).
[Crossref]

Phys. Rev. E (1)

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: The effect of partial coherence,” Phys. Rev. E 76(5), 056606 (2007).
[Crossref]

Phys. Rev. Lett. (1)

D. Y. Vasylyev, A. A. Semenov, and W. Vogel, “Toward Global Quantum Communication: Beam Wandering Preserves Nonclassicality,” Phys. Rev. Lett. 108(22), 220501 (2012).
[Crossref]

Other (2)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed.; SPIE: Bellingham, 2005.

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

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Figures (5)

Fig. 1.
Fig. 1. Schematic drawing of EGSM array beams.
Fig. 2.
Fig. 2. Relative M2-factor vs. propagation distance z, λ=632.8 nm, δ0xx=5 mm, δ0yy=3 mm, (a) N = 3, l0=20 mm, L0=50 m, Cn2=10−14m3-α, x0=y0=10 mm, α=11/3, P0=0.5. (b) σ0x=10 mm, σ0y=5 mm, l0=20 mm, L0=50 m, Cn2=10−14m3-α, x0=y0=10 mm, α=11/3, P0=0.5. (c) σ0x=10 mm, σ0y=5 mm, N = 3, L0=50 m, Cn2=10−14m3-α, x0=y0=10 mm, α=11/3, P0=0.5. (d) σ0x=10 mm, σ0y=5 mm, N = 3, l0=20 mm, Cn2=10−14m3-α, x0=y0=10 mm, α=11/3, P0=0.5. (e) σ0x=10 mm, σ0y=5 mm, N = 3, l0=20 mm, L0=50 m, x0=y0=10 mm, α=11/3, P0=0.5. (f) σ0x=10 mm, σ0y=5 mm, N = 3, l0=20 mm, L0=50 m, Cn2=10−14m3-α, α=11/3, P0=0.5. (g) σ0x=10 mm, σ0y=5 mm, N = 3, l0=20 mm, L0=50 m, Cn2=10−14m3-α, x0=y0=10 mm, P0=0.5. (h) σ0x=10 mm, σ0y=5 mm, N = 3, l0=20 mm, L0=50 m, Cn2=10−14m3-α, x0=y0=10 mm, α=11/3.
Fig. 3.
Fig. 3. Relative M2-factor vs. the generalized exponent parameter α, λ=632.8 nm, N = 3, δ0xx=5 mm, δ0yy=3 mm, σ0x=10 mm, σ0y=5 mm, z = 10 km, (a) P0=0.5, x0=y0=10 mm, L0=50 m, Cn2=10−14m3-α. (b) P0=0.5, x0=y0=10 mm, l0=20 mm, Cn2=10−14m3-α. (c) P0=0.5, Cn2=10−14m3-α, l0=20 mm, L0=50 m. (d) Cn2=10−14m3-α, x0=y0=10 mm, l0=20 mm, L0=50 m.
Fig. 4.
Fig. 4. The rms beam wander Bw and the relative beam wander Bwr vs. the generalized exponent parameter α, λ=632.8 nm, N = 3, δ0xx=5 mm, δ0yy=3 mm, σ0x=10 mm, σ0y=5 mm, l0=20 mm, L0=50 m, L = 10 km (a) P0=0.5, x0=y0=10 mm. (b) x0=y0=10 mm, Cn2=10−14m3-α. (c) and (d) P0=0.5, Cn2=10−14m3-α.
Fig. 5.
Fig. 5. The beam wander and the relative rms beam wander Bwr vs. propagation distance L λ=632.8 nm, N = 3, δ0xx=5 mm, δ0yy=3 mm, l0=20 mm, L0=50 m. (a) and (c) Cn2=10−14m3-α, σ0x=10 mm, σ0y=5 mm, P0=0.5, x0=y0=10 mm. (b) α=11/3, σ0x=10 mm, σ0y=5 mm, P0=0.5, x0=y0=10 mm, (d) α=11/3, Cn2=10−14m3-α, P0=0.5, x0=y0=10 mm. (e) α=11/3, Cn2=10−14m3-α, σ0x=10 mm, σ0y=5 mm, x0=y0=10 mm. (f) α=11/3, Cn2=10−14m3-α, σ0x=10 mm, σ0y=5 mm, P0=0.5.

Equations (27)

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W ( ρ 1 , ρ 2 , 0 ) = [ W x x ( ρ 1 , ρ 2 , 0 ) W x y ( ρ 1 , ρ 2 , 0 ) W y x ( ρ 1 , ρ 2 , 0 ) W y y ( ρ 1 , ρ 2 , 0 ) ] ,
W p q ( ρ 1 , ρ 2 , 0 ) = A p A q B p q × i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 exp ( ( x 1 i x 0 ) 2 + ( y 1 j y 0 ) 2 4 σ 0 p 2 ) × exp ( ( x 2 i x 0 ) 2 + ( y 2 j y 0 ) 2 4 σ 0 q 2 ) exp ( ρ d 2 2 δ 0 p q 2 ) ,
W T r ( ρ 1 , ρ 2 , 0 ) = W x x ( ρ 1 , ρ 2 , 0 ) + W y y ( ρ 1 , ρ 2 , 0 ) ,
ρ 1 = ρ + ρ d / 2 ; ρ 2 = ρ ρ d / 2 ,
W T r ( ρ , ρ d , 0 ) = A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 exp ( 2 x 2 x d 2 / 2 2 i 2 x 0 2 + 4 i x 0 x 4 σ 0 x 2 ) × exp ( 2 y 2 y d 2 / 2 2 j 2 y 0 2 + 4 j y 0 y 4 σ 0 x 2 ) × exp ( ρ d 2 2 δ 0 x x 2 ) + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 exp ( 2 x 2 x d 2 / 2 2 i 2 x 0 2 + 4 i x 0 x 4 σ 0 y 2 ) × exp ( 2 y 2 y d 2 / 2 2 j 2 y 0 2 + 4 j y 0 y 4 σ 0 y 2 ) × exp ( ρ d 2 2 δ 0 y y 2 ) .
h ( ρ , θ , 0 ) = ( k 2 π ) 2 W T r ( ρ , ρ d , 0 ) exp ( i k θ ρ d ) d 2 ρ d ,
x n 1 y n 2 θ x m 1 θ y m 2 0 = 1 P x n 1 y n 2 θ x m 1 θ y m 2 h ( ρ , θ , 0 ) d 2 ρ d 2 θ ,
δ ( s ) = 1 2 π exp ( i s x ) d x ,
δ ( n ) ( s ) = 1 2 π ( i x ) n exp ( i s x ) d x ( n = 0 ,   1 ,   2 ) ,
f ( x ) δ ( n ) ( x ) d x = ( 1 ) n f ( n ) ( 0 ) ( n = 0 ,   1 ,   2 ) ,
ρ 2 = ρ 2 0 + 2 ρ θ 0 z + θ 2 0 z 2 + 4 3 π 2 T z 3 = 1 P [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 x 4 + 2 i 2 x 0 2 σ 0 x 2 π ) + A y 2 B y y × i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 y 4 + 2 i 2 x 0 2 σ 0 y 2 π ) ] + 1 P [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 x 4 + 2 j 2 y 0 2 σ 0 x 2 π ) + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 y 4 + 2 j 2 y 0 2 σ 0 y 2 π ) ] + 2 z 2 P k 2 [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 x 2 + 1 δ 0 x x 2 ) + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 y 2 + 1 δ 0 y y 2 ) ] + 4 3 π 2 T z 3 ,
θ 2 = θ x 2 0 + θ y 2 0 + 4 π 2 T z = 2 P k 2 [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 x 2 + 1 δ 0 x x 2 ) + A y 2 B y y × i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 y 2 + 1 δ 0 y y 2 ) ] + 4 π 2 T z ,
ρ θ = θ 2 0 z + 2 π 2 T z 2 = 2 z P k 2 [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 x 2 + 1 δ 0 x x 2 ) + A y 2 B y y × i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 y 2 + 1 δ 0 y y 2 ) ] + 2 π 2 T z 2 ,
P = A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 2 π σ 0 x 2 + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 2 π σ 0 y 2 .
T = 0 κ 3 Φ n ( κ , α ) d κ ,
Φ n ( κ , α ) = A ( α ) C n 2 [ κ 2 + ( 2 π / L 0 ) 2 ] α / 2 exp { κ 2 [ c ( α ) / l 0 ] 2 } ,
A ( α ) = 1 4 π 2 Γ ( α 1 ) cos ( α π 2 ) ,
c ( α ) = [ 2 π 3 A ( α ) Γ ( 5 α 2 ) ] 1 / α 5 .
T ( α ) = A ( α ) C ~ n 2 2 ( α 2 ) { [ 2 κ 0 2 + ( α 2 ) κ m 2 ] κ m 2 α exp ( κ 0 2 κ m 2 ) Γ ( 2 α 2 , κ 0 2 κ m 2 ) 2 κ 0 4 α } ,
M 2 ( z ) = k [ ρ 2 θ 2 ( ρ θ ) 2 ] 1 / 2 = k { { 1 P [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 x 4 + 2 i 2 x 0 2 σ 0 x 2 π ) + A y 2 B y y × i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 y 4 + 2 i 2 x 0 2 σ 0 y 2 π ) ] + 1 P [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 x 4 + 2 j 2 y 0 2 σ 0 x 2 π ) + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 2 π σ 0 y 4 + 2 j 2 y 0 2 σ 0 y 2 π ) ] + 2 z 2 P k 2 [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 x 2 + 1 δ 0 x x 2 ) + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 y 2 + 1 δ 0 y y 2 ) ] + 4 3 π 2 T z 3 } × { 2 P k 2 [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 x 2 + 1 δ 0 x x 2 ) + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 y 2 + 1 δ 0 y y 2 ) ] + 4 π 2 T z } { 2 z P k 2 [ A x 2 B x x i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 x 2 + 1 δ 0 x x 2 ) + A y 2 B y y i = ( N 1 ) / 2 ( N 1 ) / 2 j = ( N 1 ) / 2 ( N 1 ) / 2 ( 1 4 σ 0 y 2 + 1 δ 0 y y 2 ) ] + 2 π 2 T z 2 } 2 } 1 / 2 .
M r 2 ( z ) = M 2 ( z ) M f 2 ( 0 ) .
r c 2 = 4 π 2 k 2 W FS 2 0 L 0 κ Φ n ( κ ) exp ( κ 2 W LT 2 ) [ 1 exp ( 2 L 2 κ 2 ( 1 z / L ) 2 k 2 W FS 2 ) ] d κ d z .
r c 2 = 4 π 2 C n 2 A ( α ) L 2 α 2 κ 0 α 0 L ( 1 z L ) 2 { ( κ 0 κ m ) 2 2 κ 0 4 + κ 0 α κ m 2 ( W L T 2 + κ m 2 ) α / 2 [ 2 κ 0 2 ( 1 + κ m 2 W L T 2 ) + ( α 2 ) κ m 2 ] × ( 1 + κ m 2 W L T 2 ) 2 exp [ ( κ 0 κ m ) 2 + κ 0 2 W L T 2 ] Γ [ 2 α 2 , ( κ 0 κ m ) 2 + κ 0 2 W L T 2 ] } d z .
W L T 2 = ρ 2 = ρ 2 0 + 2 ρ θ 0 z + θ 2 0 z 2 + 4 3 T π 2 z 3 .
B w = [ r c 2 ] 1 / 2 ,
B w r = [ r c 2 / W L T 2 ] 1 / 2 .
P 0 ( ρ ; 0 ) = 1 4 D e t W ( ρ , ρ ; 0 ) [ T r W ( ρ , ρ ; 0 ) ] 2 ,

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