Abstract

Based on the second-order moments and the non-Kolmogorov turbulence spectrum, the general analytical expression for the turbulence distance of laser beams propagating through non-Kolmogorov turbulence is derived, which depends on the non-Kolmogorov turbulence parameters including the generalized exponent parameter α, inner scale l0, and outer scale L0 and the initial second-order moments of the beams at the plane of z=0. Taking the partially coherent Hermite–Gaussian linear array (PCHGLA) beam as an illustrative example, the effects of non-Kolmogorov turbulence and array parameters on the turbulence distance are discussed in detail. The results show that the turbulence distance zMx(α) of PCHGLA beams through non-Kolmogorov turbulence first decreases to a dip and then increases with increasing α, and the value of zMx(α) increases with increasing beam number and beam order and decreasing coherence parameter, meaning less influence of non-Kolmogorov turbulence on partially coherent array beams than that of fully coherent array beams and a single partially coherent beam. However, the value of zMx(α) for PCHGLA beams first increases nonmonotonically with the increasing of the relative beam separation x0 for x01 and increases monotonically as x0 increases for x0>1. Moreover, the variation behavior of the turbulence distance with the generalized exponent parameter, inner scale, and outer scale of the turbulence and the beam number is similar, but different with the relative beam separation for coherent and incoherent combination cases.

© 2013 Optical Society of America

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  5. X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
    [CrossRef]
  6. X. Ji and X. Shao, “Influence of turbulence on the beam propagation factor of Gaussian Schell-model array beams,” Opt. Commun. 283, 869–873 (2010).
    [CrossRef]
  7. X. Ji and Z. Pu, “Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence,” Appl. Phys. B 93, 915–923 (2008).
    [CrossRef]
  8. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
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  9. X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98, 557–565 (2010).
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    [CrossRef]
  12. H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
    [CrossRef]
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  14. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16, 18437–18442 (2008).
    [CrossRef]
  15. X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93, 901–905 (2008).
    [CrossRef]
  16. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91, 265–271 (2008).
    [CrossRef]
  17. X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite–Gaussian array beams,” J. Opt. A 11, 105705 (2009).
    [CrossRef]
  18. X. Ji and Z. Pu, “Effective Rayleigh range of Gaussian array beams propagating through atmospheric turbulence,” Opt. Commun. 283, 3884–3890 (2010).
    [CrossRef]
  19. Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
    [CrossRef]
  20. Y. L. Ai and Y. Q. Dan, “Range of turbulence-negligible propagation of Gaussian Schell-model array beams,” Opt. Commun. 284, 3216–3220 (2011).
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  21. M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
    [CrossRef]
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    [CrossRef]
  23. 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]
  24. Y. P. Huang, Z. H. Gao, and B. Zhang, “The Rayleigh range of elegant Hermite–Gaussian beams in non-Kolmogorov turbulence,” Opt. Lasers Technol. 50, 125–129 (2013).
    [CrossRef]
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  26. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
    [CrossRef]
  27. Y. P. Huang, G. P. Zhao, D. He, and Z. C. Duan, “Spreading and M2-factor of elegant Hermite-Gaussian beams through non-Kolmogorov turbulence,” J. Mod. Opt. 58, 912–917 (2011).
    [CrossRef]

2013

Y. P. Huang, Z. H. Gao, and B. Zhang, “The Rayleigh range of elegant Hermite–Gaussian beams in non-Kolmogorov turbulence,” Opt. Lasers Technol. 50, 125–129 (2013).
[CrossRef]

2011

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

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

2010

Y. Q. Dan, S. G. Zeng, B. Y. Hao, and B. Zhang, “Range of turbulence-independent propagation and Rayleigh range of partially coherent beams in atmospheric turbulence,” J. Opt. Soc. Am. A 27, 426–434 (2010).

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
[CrossRef]

X. Ji and X. Shao, “Influence of turbulence on the beam propagation factor of Gaussian Schell-model array beams,” Opt. Commun. 283, 869–873 (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, 557–565 (2010).
[CrossRef]

X. Ji and Z. Pu, “Effective Rayleigh range of Gaussian array beams propagating through atmospheric turbulence,” Opt. Commun. 283, 3884–3890 (2010).
[CrossRef]

2009

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

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

2008

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

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

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93, 901–905 (2008).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91, 265–271 (2008).
[CrossRef]

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

2007

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (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]

2003

2002

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[CrossRef]

S. A. Ponomarenko, J. J. Greffet, and E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

2000

1999

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect on non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

1997

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

W. D. Bilida, J. D. Strohschein, and H. J. J. Seguin, “High-power 24 channel radial array slab RF-excited carbon dioxide laser,” Proc. SPIE 2987, 13–21 (1997).
[CrossRef]

1996

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

1992

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

1991

Ai, Y. L.

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

Amarande, S.

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]

Baker, H. J.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

Barchers, J. D.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect on non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

Baykal, Y.

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (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, 557–565 (2010).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91, 265–271 (2008).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

Belen’kii, M. S.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect on non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

Bilida, W. D.

W. D. Bilida, J. D. Strohschein, and H. J. J. Seguin, “High-power 24 channel radial array slab RF-excited carbon dioxide laser,” Proc. SPIE 2987, 13–21 (1997).
[CrossRef]

Brown, J. M.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect on non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

Cai, Y.

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91, 265–271 (2008).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

Chen, Y.

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

Dan, Y. Q.

Dogariu, A.

Du, X.

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93, 901–905 (2008).
[CrossRef]

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

Duan, Z. C.

Y. P. Huang, G. P. Zhao, D. He, and Z. C. Duan, “Spreading and M2-factor of elegant Hermite-Gaussian beams through non-Kolmogorov turbulence,” J. Mod. Opt. 58, 912–917 (2011).
[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, 557–565 (2010).
[CrossRef]

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91, 265–271 (2008).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

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]

Fugate, R. Q.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect on non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

Gao, Z. H.

Y. P. Huang, Z. H. Gao, and B. Zhang, “The Rayleigh range of elegant Hermite–Gaussian beams in non-Kolmogorov turbulence,” Opt. Lasers Technol. 50, 125–129 (2013).
[CrossRef]

Gbur, G.

Greffet, J. J.

S. A. Ponomarenko, J. J. Greffet, and E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

Hall, D. R.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

Hao, B. Y.

He, D.

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

Hornby, A. M.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

Huang, Y. P.

Y. P. Huang, Z. H. Gao, and B. Zhang, “The Rayleigh range of elegant Hermite–Gaussian beams in non-Kolmogorov turbulence,” Opt. Lasers Technol. 50, 125–129 (2013).
[CrossRef]

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

Ji, X.

X. Ji and X. Shao, “Influence of turbulence on the beam propagation factor of Gaussian Schell-model array beams,” Opt. Commun. 283, 869–873 (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, 557–565 (2010).
[CrossRef]

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
[CrossRef]

X. Ji and Z. Pu, “Effective Rayleigh range of Gaussian array beams propagating through atmospheric turbulence,” Opt. Commun. 283, 3884–3890 (2010).
[CrossRef]

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

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

Jia, X.

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

Karis, S. J.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect on non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

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, 557–565 (2010).
[CrossRef]

Lü, B.

Ma, H.

Martínez-Herrero, R.

Mejías, P. M.

Morley, R. J.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

Osmon, C. L.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect on non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

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]

Ponomarenko, S. A.

S. A. Ponomarenko, J. J. Greffet, and E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

Pu, Z.

X. Ji and Z. Pu, “Effective Rayleigh range of Gaussian array beams propagating through atmospheric turbulence,” Opt. Commun. 283, 3884–3890 (2010).
[CrossRef]

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

Seguin, H. J. J.

W. D. Bilida, J. D. Strohschein, and H. J. J. Seguin, “High-power 24 channel radial array slab RF-excited carbon dioxide laser,” Proc. SPIE 2987, 13–21 (1997).
[CrossRef]

Serna, J.

Shao, X.

X. Ji and X. Shao, “Influence of turbulence on the beam propagation factor of Gaussian Schell-model array beams,” Opt. Commun. 283, 869–873 (2010).
[CrossRef]

Strohschein, J. D.

W. D. Bilida, J. D. Strohschein, and H. J. J. Seguin, “High-power 24 channel radial array slab RF-excited carbon dioxide laser,” Proc. SPIE 2987, 13–21 (1997).
[CrossRef]

Taghizadeh, M. R.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

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]

Weber, H.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

Wolf, E.

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[CrossRef]

S. A. Ponomarenko, J. J. Greffet, and E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

Yelden, E. F.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996).
[CrossRef]

Zeng, S. G.

Zhang, B.

Zhang, T.

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

Zhao, D.

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

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93, 901–905 (2008).
[CrossRef]

Zhao, G. P.

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

Zhu, Y.

Appl. Phys. B

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

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

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93, 901–905 (2008).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91, 265–271 (2008).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[CrossRef]

IEEE J. Quantum Electron.

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Figures (7)

Fig. 1.
Fig. 1.

Schematic diagram of the one-dimensional PCHGLA beams.

Fig. 2.
Fig. 2.

Turbulence distance zMx(α) of PCHGLA beams in non-Kolmogorov turbulence versus the generalized exponent parameter α for different values of (a) the relative beam separation x0 and the beam number N, (b) the beam order m, and (c) the coherence parameter β.

Fig. 3.
Fig. 3.

zMx(α) of PCHGLA beams in non-Kolmogorov turbulence versus the relative beam separation x0 for different values of (a) beam number N and (b) the generalized exponent parameter α.

Fig. 4.
Fig. 4.

zMx(α) of PCHGLA beams in non-Kolmogorov turbulence for the coherent combination and incoherent combination cases versus the generalized exponent parameter α.

Fig. 5.
Fig. 5.

zMx(α) of PCHGLA beams in non-Kolmogorov turbulence for the coherent combination and incoherent combination cases versus (a) inner scale and (b) outer scale.

Fig. 6.
Fig. 6.

zMx(α) of PCHGLA beams in non-Kolmogorov turbulence for the coherent combination and incoherent combination cases versus beam number N.

Fig. 7.
Fig. 7.

zMx(α) of PCHGLA beams in non-Kolmogorov turbulence for the coherent combination and incoherent combination cases versus the relative beam separation x0.

Equations (28)

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x2=x20+θx20z2+T(α)z3,
θx2=θx20+3T(α)z,
xθx=θx20z+3T(α)z2/2,
T(α)=2π230Φn(κ,α)κ3dκ
Φn(κ,α)=A(α)C˜n2exp[(κ2/κm2)](κ2+κ02)α/2,0κ<,3<α<4,
T(α)=23π20Φn(κ)κ3dκ=A(α)π2C˜n23(α2){[c(α)/l0]2α[8π2L02+(α2)c2(α)l02]exp(4π2c2(α)l02L02)Γ[2α2,4π2l02c2(α)L02]2(2πL0)4α}.
Mx2(z)=2kx2θx2xθx2.
Mx2(α,z)=k[4x20θx20+12x20T(α)z+4θx20T(α)z3+3T2(α)z4]1/2.
Mxr2(α,z)=1+3T(α)zθx20+T(α)z3x20+3T2(α)z44x20θx20.
Mxr2[zMx(α)]=2.
zMx4(α)+4θx20zMx3(α)3T(α)+4x20zMx(α)T(α)4x20θx203T2(α)=0.
zMx(α)=θx203T(α)+12[4θx2029T2(α)+Q]1/2+12{8θx2029T2(α)Q+16θx203+216x20T2(α)27T3(α)[4θx2029T2(α)+Q]1/2}1/2,
Q=3223x20θx203T2(α)PP543,
P={16S+[217x203θx203T6(α)+256S2]12}13,
S=27x202T2(α)4θx203x20T4(α).
W(x1,x2;0)=i=(N1)/2(N1)/2j=(N1)/2(N1)/2Hm[2(x1ix0)w0]Hm[2(x2jx0)w0]exp((x1ix0)2+(x2jx0)2w02)exp([(x1x2)(ij)x0]22σ2),
x2=x20+θx20z2.
I(x,z)=k2πzW(x1,x2;0)exp{ik2z[(x12x22)2(x1x2)x}dx1dx2.
p=(x1+x2)/2(i+j)x0/2,
q=(x2x1)+(ij)x0,
I(x,z)=k2πzi=N12N12j=N12N12Hm[2(pq/2)w0]Hm[2(p+q/2)w0]×exp(2p2+q2/2w02)exp(q22σ02)exp{ikxz[q(ij)x0]}exp{ik2z[q(ij)x0][2p+(i+j)x0]}dpdq.
x2=x2I(x,z)dxI(x,z)dx=w02i=(N1)/2(N1)/2j=(N1)/2(N1)/2aij[Lm(O)2Lm(1)(O)+x02(i+j)2Lm(O)]4i=(N1)/2(N1)/2j=(N1)/2(N1)/2aijLm(O)+i=(N1)/2(N1)/2j=(N1)/2(N1)/2aij[Lm(O)2Lm(1)(O)+Lm(O)β2]aijO[Lm(O)4Lm(1)(O)+4Lm(2)(O)+Lm(O)β4+2Lm(O)β24Lm(1)(O)β2]k2w02i=(N1)/2(N1)/2j=(N1)/2(N1)/2aijLm(O)z2,
O=(ij)2x02,
aij=exp[x022(ij)2(β2+1)].
x20=w02i=(N1)/2(N1)/2j=(N1)/2(N1)/2aij[Lm(O)2Lm(1)(O)+x02(i+j)2Lm(O)]4i=(N1)/2(N1)/2j=(N1)/2(N1)/2aijLm(O),
θx20=i=(N1)/2(N1)/2j=(N1)/2(N1)/2aij[Lm(O)2Lm(1)(O)+Lm(O)β2]k2w02i=(N1)/2(N1)/2j=(N1)/2(N1)/2aijLm(O)i=(N1)/2(N1)/2j=(N1)/2(N1)/2aijO[Lm(O)4Lm(1)(O)+4Lm(2)(O)]k2w02i=(N1)/2(N1)/2j=(N1)/2(N1)/2aijLm(O)i=(N1)/2(N1)/2j=(N1)/2(N1)/2aijO[Lm(O)β4+2Lm(O)β24Lm(1)(O)β2]k2w02i=(N1)/2(N1)/2j=(N1)/2(N1)/2aijLm(O).
x20=w024(2m+1+N213x02),
θx20=λ24π2w02(2m+1+β2).

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