Abstract

Based on the extended Huygens-Fresnel principle and the definition of second-order moments of the Wigner distribution function (WDF), the analytical expressions for the propagation factors (M2-factors) and Strehl ratio SR of the Gaussian Schell-model (GSM) vortex beams and GSM non-vortex beams propagation through non-Kolmogorov atmospheric turbulence are derived, and used to study the influence of non-Kolmogorov atmospheric turbulence on beam quality of the GSM vortex beams. It is shown that the smaller the generalized structure constant and the outer scale of turbulence are, and the bigger the inner scale of turbulence is, the smaller the normalized propagation factor is, the bigger the Strehl ratio is, and the better the beam quality of GSM vortex beams in atmospheric turbulence is. The variation of beam quality with the generalized exponent α is nonmonotonic, when α = 3.11, the beam quality of the GSM vortex beams is the poorest through non-Kolmogorov atmospheric turbulence. GSM vortex beams is less affected by turbulence than GSM non-vortex beams under certain condition, and will be useful in long-distance free-space optical communications.

© 2016 Optical Society of America

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  8. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, 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).
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    [Crossref]
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  17. M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
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  18. F. D. Kashani, M. Alavinejad, and B. Ghafary, “Propagation properties of a non-circular partially coherent flat-topped beam in a turbulent atmosphere,” Opt. Laser Technol. 41(5), 659–664 (2009).
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  21. Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Propagation properties of partially coherent Laguerre–Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 43(4), 741–747 (2011).
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  22. K. Zhu, S. Li, Y. Tang, Y. Yu, and H. Tang, “Study on the propagation parameters of Bessel-Gaussian beams carrying optical vortices through atmospheric turbulence,” J. Opt. Soc. Am. A 29(3), 251–257 (2012).
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  25. 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).
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  26. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non-Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
    [Crossref]
  27. H. F. Xu, Z. Zhang, J. Qu, and W. Huang, “Propagation factors of cosine-Gaussian-correlated Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 22(19), 22479–22489 (2014).
    [Crossref] [PubMed]
  28. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42(4), 219–276 (2001).
    [Crossref]
  29. G. A. J. Swartzlander and G. Gbur, “Singular optical phenomena in nature,” Proc. SPIE 7057, 705703 (2008).
    [Crossref]
  30. J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
    [Crossref]
  31. J. Li, C. Ding, and B. Lü, “Generalized Stokes parameters of random electromagnetic vortex beams propagating through atmospheric turbulence,” Appl. Phys. B 103(1), 245–255 (2011).
    [Crossref]
  32. J. Li, J. Zeng, and M. Duan, “Classification of coherent vortices creation and distance of topological charge conservation in non-Kolmogorov atmospheric turbulence,” Opt. Express 23(9), 11556–11565 (2015).
    [Crossref] [PubMed]
  33. Y. Yang, Y. Dong, C. Zhao, and Y. Cai, “Generation and propagation of an anomalous vortex beam,” Opt. Lett. 38(24), 5418–5421 (2013).
    [Crossref] [PubMed]
  34. Y. Yang, Y. Dong, C. Zhao, Y. D. Liu, and Y. Cai, “Autocorrelation properties of fully coherent beam with and without orbital angular momentum,” Opt. Express 22(3), 2925–2932 (2014).
    [Crossref] [PubMed]
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    [Crossref]

2016 (1)

Y. Xu, Y. Dan, and B. Zhang, “Spreading and M2-factor based on second-order moments for partially-coherent anomalous hollow beam in turbulent atmosphere,” Optik (Stuttg.) 127(11), 4590–4595 (2016).
[Crossref]

2015 (1)

2014 (3)

2013 (4)

M. Alavinejad, N. Hadilou, and G. Taherabadi, “The influence of phase aperture on Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Commun. 311(1), 275–281 (2013).
[Crossref]

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

O. Korotkova and E. Shchepakina, “Tuning the spectral composition of random beams propagating in free space and in a turbulent atmosphere,” J. Opt. 15(7), 075714 (2013).
[Crossref]

Y. Yang, Y. Dong, C. Zhao, and Y. Cai, “Generation and propagation of an anomalous vortex beam,” Opt. Lett. 38(24), 5418–5421 (2013).
[Crossref] [PubMed]

2012 (2)

K. Zhu, S. Li, Y. Tang, Y. Yu, and H. Tang, “Study on the propagation parameters of Bessel-Gaussian beams carrying optical vortices through atmospheric turbulence,” J. Opt. Soc. Am. A 29(3), 251–257 (2012).
[Crossref] [PubMed]

Y. Yuan, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

2011 (8)

S. Zhu and Y. Cai, “M2-factor of a truncated electromagnetic Gaussian Schell-model beam,” Appl. Phys. B 103(4), 971–984 (2011).
[Crossref]

F. Cheng and Y. Cai, “Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 284(1), 30–37 (2011).
[Crossref]

X. Ji and X. Li, “M2-factor of truncated partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 28(6), 970–975 (2011).
[Crossref] [PubMed]

X. Ji, X. Li, and G. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
[Crossref]

X. Ji and X. Li, “Propagation properties of apertured laser beams with amplitude modulations and phase fluctuations through atmospheric turbulence,” Appl. Phys. B 104(1), 207–213 (2011).
[Crossref]

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Propagation properties of partially coherent Laguerre–Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 43(4), 741–747 (2011).
[Crossref]

J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
[Crossref]

J. Li, C. Ding, and B. Lü, “Generalized Stokes parameters of random electromagnetic vortex beams propagating through atmospheric turbulence,” Appl. Phys. B 103(1), 245–255 (2011).
[Crossref]

2009 (3)

2008 (4)

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] [PubMed]

X. Xiao, X. Ji, and B. 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]

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
[Crossref]

G. A. J. Swartzlander and G. Gbur, “Singular optical phenomena in nature,” Proc. SPIE 7057, 705703 (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]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non-Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[Crossref]

2001 (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42(4), 219–276 (2001).
[Crossref]

1995 (2)

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995).
[Crossref]

1990 (1)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[Crossref]

1989 (1)

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70(5), 361–364 (1989).
[Crossref]

Alavinejad, M.

M. Alavinejad, N. Hadilou, and G. Taherabadi, “The influence of phase aperture on Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Commun. 311(1), 275–281 (2013).
[Crossref]

F. D. Kashani, M. Alavinejad, and B. Ghafary, “Propagation properties of a non-circular partially coherent flat-topped beam in a turbulent atmosphere,” Opt. Laser Technol. 41(5), 659–664 (2009).
[Crossref]

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
[Crossref]

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]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non-Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[Crossref]

Baykal, Y.

Y. Yuan, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, 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] [PubMed]

Beland, R. R.

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995).
[Crossref]

Cai, Y.

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[Crossref] [PubMed]

Y. Yang, Y. Dong, C. Zhao, Y. D. Liu, and Y. Cai, “Autocorrelation properties of fully coherent beam with and without orbital angular momentum,” Opt. Express 22(3), 2925–2932 (2014).
[Crossref] [PubMed]

Y. Yang, Y. Dong, C. Zhao, and Y. Cai, “Generation and propagation of an anomalous vortex beam,” Opt. Lett. 38(24), 5418–5421 (2013).
[Crossref] [PubMed]

S. Du, Y. Yuan, C. Liang, and 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, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

S. Zhu and Y. Cai, “M2-factor of a truncated electromagnetic Gaussian Schell-model beam,” Appl. Phys. B 103(4), 971–984 (2011).
[Crossref]

F. Cheng and Y. Cai, “Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 284(1), 30–37 (2011).
[Crossref]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, 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] [PubMed]

Chen, J.

Y. Yuan, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Chen, R.

Cheng, F.

F. Cheng and Y. Cai, “Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 284(1), 30–37 (2011).
[Crossref]

Cui, Z.

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Propagation properties of partially coherent Laguerre–Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 43(4), 741–747 (2011).
[Crossref]

Dan, Y.

Ding, C.

J. Li, C. Ding, and B. Lü, “Generalized Stokes parameters of random electromagnetic vortex beams propagating through atmospheric turbulence,” Appl. Phys. B 103(1), 245–255 (2011).
[Crossref]

Dong, Y.

Du, S.

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

Duan, M.

Eyyuboglu, H. T.

Y. Yuan, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, 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] [PubMed]

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]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non-Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[Crossref]

Gbur, G.

G. A. J. Swartzlander and G. Gbur, “Singular optical phenomena in nature,” Proc. SPIE 7057, 705703 (2008).
[Crossref]

Ghafary, B.

F. D. Kashani, M. Alavinejad, and B. Ghafary, “Propagation properties of a non-circular partially coherent flat-topped beam in a turbulent atmosphere,” Opt. Laser Technol. 41(5), 659–664 (2009).
[Crossref]

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
[Crossref]

Hadilou, N.

M. Alavinejad, N. Hadilou, and G. Taherabadi, “The influence of phase aperture on Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Commun. 311(1), 275–281 (2013).
[Crossref]

Huang, W.

Ji, G.

X. Ji, X. Li, and G. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
[Crossref]

Ji, X.

X. Ji, X. Li, and G. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
[Crossref]

X. Ji and X. Li, “Propagation properties of apertured laser beams with amplitude modulations and phase fluctuations through atmospheric turbulence,” Appl. Phys. B 104(1), 207–213 (2011).
[Crossref]

X. Ji and X. Li, “M2-factor of truncated partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 28(6), 970–975 (2011).
[Crossref] [PubMed]

X. Xiao, X. Ji, and B. 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]

Kashani, F. D.

F. D. Kashani, M. Alavinejad, and B. Ghafary, “Propagation properties of a non-circular partially coherent flat-topped beam in a turbulent atmosphere,” Opt. Laser Technol. 41(5), 659–664 (2009).
[Crossref]

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
[Crossref]

Korotkova, O.

O. Korotkova and E. Shchepakina, “Tuning the spectral composition of random beams propagating in free space and in a turbulent atmosphere,” J. Opt. 15(7), 075714 (2013).
[Crossref]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, 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] [PubMed]

Li, J.

J. Li, J. Zeng, and M. Duan, “Classification of coherent vortices creation and distance of topological charge conservation in non-Kolmogorov atmospheric turbulence,” Opt. Express 23(9), 11556–11565 (2015).
[Crossref] [PubMed]

J. Li, C. Ding, and B. Lü, “Generalized Stokes parameters of random electromagnetic vortex beams propagating through atmospheric turbulence,” Appl. Phys. B 103(1), 245–255 (2011).
[Crossref]

J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
[Crossref]

Li, S.

Li, X.

X. Ji and X. Li, “M2-factor of truncated partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 28(6), 970–975 (2011).
[Crossref] [PubMed]

X. Ji, X. Li, and G. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
[Crossref]

X. Ji and X. Li, “Propagation properties of apertured laser beams with amplitude modulations and phase fluctuations through atmospheric turbulence,” Appl. Phys. B 104(1), 207–213 (2011).
[Crossref]

Liang, C.

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

Liu, L.

Liu, Y. D.

Lü, B.

J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
[Crossref]

J. Li, C. Ding, and B. Lü, “Generalized Stokes parameters of random electromagnetic vortex beams propagating through atmospheric turbulence,” Appl. Phys. B 103(1), 245–255 (2011).
[Crossref]

X. Xiao, X. Ji, and B. 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]

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non-Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[Crossref]

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]

Qu, J.

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Shchepakina, E.

O. Korotkova and E. Shchepakina, “Tuning the spectral composition of random beams propagating in free space and in a turbulent atmosphere,” J. Opt. 15(7), 075714 (2013).
[Crossref]

Shi, J.

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Propagation properties of partially coherent Laguerre–Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 43(4), 741–747 (2011).
[Crossref]

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[Crossref]

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42(4), 219–276 (2001).
[Crossref]

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Swartzlander, G. A. J.

G. A. J. Swartzlander and G. Gbur, “Singular optical phenomena in nature,” Proc. SPIE 7057, 705703 (2008).
[Crossref]

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M. Alavinejad, N. Hadilou, and G. Taherabadi, “The influence of phase aperture on Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Commun. 311(1), 275–281 (2013).
[Crossref]

Tang, H.

Tang, Y.

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]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non-Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
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[Crossref]

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Y. Xu, Y. Dan, and B. Zhang, “Spreading and M2-factor based on second-order moments for partially-coherent anomalous hollow beam in turbulent atmosphere,” Optik (Stuttg.) 127(11), 4590–4595 (2016).
[Crossref]

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S. Du, Y. Yuan, C. Liang, and 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, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, 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).
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Appl. Phys. B (3)

S. Zhu and Y. Cai, “M2-factor of a truncated electromagnetic Gaussian Schell-model beam,” Appl. Phys. B 103(4), 971–984 (2011).
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J. Li, C. Ding, and B. Lü, “Generalized Stokes parameters of random electromagnetic vortex beams propagating through atmospheric turbulence,” Appl. Phys. B 103(1), 245–255 (2011).
[Crossref]

J. Opt. (1)

O. Korotkova and E. Shchepakina, “Tuning the spectral composition of random beams propagating in free space and in a turbulent atmosphere,” J. Opt. 15(7), 075714 (2013).
[Crossref]

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

New J. Phys. (1)

X. Ji, X. Li, and G. Ji, “Propagation of second-order moments of general truncated beams in atmospheric turbulence,” New J. Phys. 13(10), 103006 (2011).
[Crossref]

Opt. Commun. (4)

F. Cheng and Y. Cai, “Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere,” Opt. Commun. 284(1), 30–37 (2011).
[Crossref]

M. Alavinejad, N. Hadilou, and G. Taherabadi, “The influence of phase aperture on Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Commun. 311(1), 275–281 (2013).
[Crossref]

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70(5), 361–364 (1989).
[Crossref]

J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
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Opt. Express (6)

Opt. Laser Technol. (4)

X. Xiao, X. Ji, and B. 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]

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

F. D. Kashani, M. Alavinejad, and B. Ghafary, “Propagation properties of a non-circular partially coherent flat-topped beam in a turbulent atmosphere,” Opt. Laser Technol. 41(5), 659–664 (2009).
[Crossref]

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Propagation properties of partially coherent Laguerre–Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 43(4), 741–747 (2011).
[Crossref]

Opt. Lasers Eng. (2)

M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. 46(1), 1–5 (2008).
[Crossref]

Y. Yuan, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Opt. Lett. (2)

Optik (Stuttg.) (1)

Y. Xu, Y. Dan, and B. Zhang, “Spreading and M2-factor based on second-order moments for partially-coherent anomalous hollow beam in turbulent atmosphere,” Optik (Stuttg.) 127(11), 4590–4595 (2016).
[Crossref]

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A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[Crossref]

G. A. J. Swartzlander and G. Gbur, “Singular optical phenomena in nature,” Proc. SPIE 7057, 705703 (2008).
[Crossref]

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995).
[Crossref]

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]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non-Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
[Crossref]

Prog. Opt. (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42(4), 219–276 (2001).
[Crossref]

Other (4)

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Figures (3)

Fig. 1
Fig. 1 Normalized propagation factor of GSM vortex beams and GSM non-vortex beams versus (a) the generalized exponent α and (b) the propagation distance z.

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Fig. 2
Fig. 2 Normalized propagation factor of GSM vortex beams versus the propagation distance z for different (a) the generalized structure constant, (b) the inner scale of turbulence l0 and (c) the outer scale of turbulence L0.

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Fig. 3
Fig. 3 Strehl ratio of GSM vortex beams and GSM non-vortex beams propagation through non-Kolmogorov atmospheric turbulence versus (a) the generalized exponent α, (b) the generalized structure constant C ˜ n 2 , (c) the inner scale of turbulence l0 and (d) the outer scale of turbulence L0.

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Equations (34)

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U(s,z=0)= [ s x +isgn(m) s y ] | m | exp(− s x 2 + s y 2 w 0 2 ),
W (0) ( s 1 , s 2 ,0)= [ s 1x s 2x + s 1y s 2y +isgn(m) s 1x s 2y −isgn(m) s 2x s 1y ] | m | ×exp(− s 1 2 + s 2 2 w 0 2 )exp[− ( s 1 − s 2 ) 2 2 σ 0 2 ],
W( ρ, ρ d ,z )= ( k 2πz ) 2 ∫ −∞ ∞ ∫ −∞ ∞ ∫ −∞ ∞ ∫ −∞ ∞ W (0) ( s, s d ,0 ) ×exp[ ik z ( ρ−s )( ρ d − s d )−H( ρ d , s d ,z) ] d 2 s d 2 s d ,
W (0) ( s, s d ,0 )= W (0) ( s 1 , s 2 ,0 )= W (0) ( s+ s d 2 ,s− s d 2 ,0 ),
ρ= ρ 1 + ρ 2 2 , ρ d = ρ 1 − ρ 2 ,s= s 1 + s 2 2 , s d = s 1 − s 2 ,
H( ρ d , s d ,z)=4 π 2 k 2 z ∫ 0 1 dξ ∫ 0 ∞ [1− J 0 (κ| s d ξ+(1−ξ) ρ d | )] Φ n (κ,α)κdκ,
W( ρ, ρ d ,z )= ( 1 2π ) 2 ∫ −∞ ∞ ∫ −∞ ∞ ∫ −∞ ∞ ∫ −∞ ∞ W(s', ρ d + z k κ d ,0) ×exp[−iρ⋅ κ d +is'⋅ κ d −H( ρ d , s d ,z)] d 2 s' d 2 κ d ,
W(s', ρ d + z k κ d ,0)=exp[− 2s ' 2 w 0 2 −( 1 2 w 0 2 + 1 2 σ 0 2 ) ( ρ d + z k κ d ) 2 ]{ s ' 2 − 1 4 ( ρ d + z k κ d ) 2 ∓[ s ′ x ( ρ dy + z k κ dy )- s ′ y ( ρ dx + z k κ dx ) ] } ,
W( ρ, ρ d ,z )= ( 1 2π ) 2 π w 0 2 2 ∫ −∞ ∞ ∫ −∞ ∞ exp[− w 0 2 κ d 2 8 −iρ⋅ κ d −( 1 w 0 2 + 1 σ 0 2 ) s d 2 2 −H( ρ d , s d ,z)] ×[ w 0 2 2 (1− w 0 2 κ d 2 8 )∓ w 0 2 4 ( s dx κ dy − s dy κ dx )− s d 2 4 ] d 2 κ d .
h(ρ,θ,z)= ( k 2π ) 2 ∫ −∞ ∞ ∫ -∞ ∞ W(ρ, ρ d ,z)exp(−ikθ⋅ρ ) d d 2 ρ d ,
h(ρ,θ,z)= ( 1 2π ) 2 π w 0 2 2 ( k 2π ) 2 ∫ −∞ ∞ ∫ −∞ ∞ ∫ −∞ ∞ ∫ -∞ ∞ { w 0 2 2 (1− w 0 2 κ d 2 8 ) ∓ w 0 2 4 [( ρ dx + z k κ dx ) κ dy −( ρ dy + z k κ dy ) κ dx ]− ( ρ d + z k κ d ) 2 4 }exp[− w 0 2 κ d 2 8 −ikθ⋅ρ − d iρ⋅ κ d −( 1 w 0 2 + 1 σ 0 2 ) ( ρ d + z k κ d ) 2 2 −H( ρ d , ρ d + z k κ d ,z)] d 2 κ d d 2 ρ d ,
< ρ x n 1 ρ y n 2 θ x m 1 θ y m 2 >= 1 P ∫ −∞ ∞ ∫ −∞ ∞ ρ x n 1 ρ y n 2 θ x m 1 θ y m 2 h(ρ,θ,z) d 2 ρ d 2 θ,
P= ∫ −∞ ∞ ∫ −∞ ∞ h(ρ,θ,z) d 2 ρ d 2 θ,
M 2 (z)=k (< ρ 2 >< θ 2 >−<ρθ > 2 ) 1/2 ,
< ρ 2 >=< ρ x 2 >+< ρ y 2 >,
< θ 2 >=< θ x 2 >+< θ y 2 >,
<ρθ>=< ρ x θ x >+< ρ y θ y >.
< ρ 2 >= w 0 2 +( 2 w 0 2 + 1 σ 0 2 ) 2 z 2 k 2 + 4 z 2 k 2 T,
< θ 2 >=( 2 w 0 2 + 1 σ 0 2 ) 2 k 2 + 12 k 2 T,
<ρθ>=( 2 w 0 2 + 1 σ 0 2 ) 2z k 2 + 6z k 2 T,
T( α,z )= π 2 k 2 z 3 ∫ 0 ∞ κ 3 Φ n ( κ,α )dκ,
Φ n (κ)=A(α) C ˜ n 2 exp[−( κ 2 / κ m 2 )] ( κ 2 + κ 0 2 ) α/2 , (0≤κ<∞, 3<α<4 ),
A(α)= Γ(α−1)cos( απ /2 ) / (4 π 2 ) , κ 0 = 2π / L 0 , κ m = c( α ) / l 0 , c( α )= {Γ[ (5−α) /2 ]A( α ) 2π /3 } 1/( α−5 ) ,
T(α,z)= π 2 k 2 z 6( α−2 ) A( α ) C ˜ n 2 { exp( κ 0 2 κ m 2 ) κ m ( 2−α ) ×[ ( α−2 ) κ m 2 +2 κ 0 2 ]Γ( 2− α 2 , κ 0 2 κ m 2 )−2 κ 0 4−α },
M 2 (z)=k{ [ w 0 2 +( 2 w 0 2 + 1 σ 0 2 ) 2 z 2 k 2 + 4 z 2 k 2 T] [( 2 w 0 2 + 1 σ 0 2 ) 2 k 2 + 12 k 2 T] − [( 2 w 0 2 + 1 σ 0 2 ) 2z k 2 + 6z k 2 T] 2 } 1/2 .
M ′ 2 (z)=k{ [ w 0 2 2 +( 1 w 0 2 + 1 σ 0 2 ) 2 z 2 k 2 + 4 z 2 k 2 T][( 1 w 0 2 + 1 σ 0 2 ) 2 k 2 + 12 k 2 T] − [( 1 w 0 2 + 1 σ 0 2 ) 2z k 2 + 6z k 2 T] 2 } 1/2 .
S R = I max I 0max ,
I(ρ,z)= k 2 w 0 4 32 z 2 A+4 k 2 w 0 2 [2−B w 0 2 (1−B ρ 2 )−C(1−B ρ 2 )]exp[−B ρ 2 ],
A= 1 2 w 0 2 + 1 2 σ 0 2 +T,
B= 2 k 2 8 z 2 A+ k 2 w 0 2 ,
C= 4 z 2 8 z 2 w 0 2 A+ k 2 w 0 4 .
I 0 (ρ,z)= k 2 w 0 4 32 z 2 A 0 +4 k 2 w 0 2 [2− B 0 w 0 2 (1− B 0 ρ 2 )− C 0 (1− B 0 ρ 2 )]exp[− B 0 ρ 2 ],
S R = (8 z 2 A 0 + k 2 w 2 )[2−B w 0 2 (1−B ρ 2 )−C(1−B ρ 2 )]exp[−B ρ 2 ] (8 z 2 A+ k 2 w 2 )[2− B 0 w 0 2 (1− B 0 ρ 2 )− C 0 (1− B 0 ρ 2 )]exp[− B 0 ρ 2 ] .
S R = (8 z 2 A 0 + k 2 w 0 2 )exp[−B ρ 2 ] (8 z 2 A+ k 2 w 0 2 )exp[− B 0 ρ 2 ] .

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