Abstract

The analytical formulas for the average intensity and the beam width of the elegant Laguerre–Gaussian beam (ELGB) in non-Kolmogorov turbulence have been derived based on the extended Huygens–Fresnel principle. Numerical results reveal that the ELGB converts to Gaussian form quicker for smaller values of beam order and for smaller wavelengths. The root-mean-square (rms) beam width of ELGB increases markedly with the propagation distance for higher beam order, smaller waist width, and larger wavelength. Furthermore, discussions of the influence of ELGB by the non-Kolmogorov turbulence reveal that the normalized intensity distribution of ELGB converts into Gaussian form more quickly and that the rms beam width of ELGB increases more rapidly in non-Kolmogorov turbulence with smaller parameter α,larger outer scale, smaller inner scale and larger structure constant.

© 2011 OSA

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2011 (4)

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011).
[CrossRef]

G. Wu, T. Zhao, J. Ren, J. Zhang, X. Zhang, and W. Li, “Beam propagation factor of partially coherent Hermite-Gassian beams through non-Kolmogorov turbulence,” Opt. Laser Technol. 43(7), 1225–1228 (2011).
[CrossRef]

X. Chu, C. Qiao, X. Feng, and R. Chen, “Propagation of Gaussian-Schell beam in turbulent atmosphere of three-layer altitude model,” Appl. Opt. 50(21), 3871–3878 (2011).
[CrossRef] [PubMed]

X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36(14), 2701–2703 (2011).
[CrossRef] [PubMed]

2010 (7)

2009 (3)

2008 (3)

2007 (1)

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–1­ 65510E-12 (2007).
[CrossRef]

2005 (3)

T. Shirai, “Polarization properties of a class of electromagnetic Gaussian Schell-model beams which have the same far-zone intensity distribution as a fully coherent laser beam,” Opt. Commun. 256(4-6), 197–209 (2005).
[CrossRef]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702, 598702-12 (2005).
[CrossRef]

H. T. Eyyubo?lu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44(6), 976–983 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (3)

2000 (1)

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[CrossRef]

1999 (1)

M. S. Belen’kii, S. J. Kars, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence,” Proc. SPIE 3749, 50–51 (1999).
[CrossRef]

1995 (2)

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375A, 1111–1126 (1995).

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 2471–22 (1995).

1993 (1)

K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 (1993).

1992 (1)

1986 (1)

1977 (1)

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[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–1­ 65510E-12 (2007).
[CrossRef]

Baykal, Y.

H. T. Eyyubo?lu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B 98(4), 857–863 (2010).
[CrossRef]

F. Wang, Y. Cai, H. T. Eyyubo?lu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and Elegant Laguerre-Gaussian beam in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–56 (2010).
[CrossRef]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyubo?lu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009).
[CrossRef] [PubMed]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyubo?lu, 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]

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

Y. Cai, H.T. Eyyubo?lu, and Y. Baykal, “Scintillation of astigmatic dark hollow beams in weak atmospheric turbulence,” J. Opt. Soc. Am. A 25(7), 1497–1503 (2008).
[CrossRef]

H. T. Eyyubo?lu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44(6), 976–983 (2005).
[CrossRef] [PubMed]

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

Beland, R. R.

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375A, 1111–1126 (1995).

Belen’kii, M. S.

M. S. Belen’kii, S. J. Kars, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence,” Proc. SPIE 3749, 50–51 (1999).
[CrossRef]

Cai, Y.

Cao, X. G.

Chen, R.

Chen, X.

D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite–Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. 5(5), 489–494 (2003).
[CrossRef]

Chu, X.

Cui, L. Y.

Cui, Z.

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011).
[CrossRef]

J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
[CrossRef]

Dayton, D.

Deng, D.

D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite–Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. 5(5), 489–494 (2003).
[CrossRef]

Dogariu, A.

Dong, J. K.

Eyyuboglu, H. T.

Eyyuboglu, H.T.

Felsen, L. B.

Feng, X.

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[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–1­ 65510E-12 (2007).
[CrossRef]

Golbraikh, E.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702, 598702-12 (2005).
[CrossRef]

Gonglewski, J.

Guo, H.

G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010).
[CrossRef] [PubMed]

D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite–Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. 5(5), 489–494 (2003).
[CrossRef]

Ji, X.

H. T. Eyyubo?lu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B 98(4), 857–863 (2010).
[CrossRef]

Jiang, W.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[CrossRef]

Kars, S. J.

M. S. Belen’kii, S. J. Kars, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence,” Proc. SPIE 3749, 50–51 (1999).
[CrossRef]

Kong, H.

D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite–Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. 5(5), 489–494 (2003).
[CrossRef]

Kopeika, N. S.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702, 598702-12 (2005).
[CrossRef]

Korotkova, O.

Li, W.

G. Wu, T. Zhao, J. Ren, J. Zhang, X. Zhang, and W. Li, “Beam propagation factor of partially coherent Hermite-Gassian beams through non-Kolmogorov turbulence,” Opt. Laser Technol. 43(7), 1225–1228 (2011).
[CrossRef]

Lin, Q.

Ling, N.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[CrossRef]

Liu, Z.

Luis, R. E.

K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 (1993).

Luo, B.

Ma, Y.

Osmon, C. L.

M. S. Belen’kii, S. J. Kars, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence,” Proc. SPIE 3749, 50–51 (1999).
[CrossRef]

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[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–1­ 65510E-12 (2007).
[CrossRef]

Pierson, B.

Qiao, C.

Qu, J.

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011).
[CrossRef]

J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
[CrossRef]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyubo?lu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009).
[CrossRef] [PubMed]

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyubo?lu, 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]

Rao, C.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[CrossRef]

Ren, J.

G. Wu, T. Zhao, J. Ren, J. Zhang, X. Zhang, and W. Li, “Beam propagation factor of partially coherent Hermite-Gassian beams through non-Kolmogorov turbulence,” Opt. Laser Technol. 43(7), 1225–1228 (2011).
[CrossRef]

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 2471–22 (1995).

Shchepakina, E.

Shi, J.

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011).
[CrossRef]

Shin, S. Y.

Shirai, T.

Sidoro, K.

K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 (1993).

Spielbusch, B.

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 2471–22 (1995).

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[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–1­ 65510E-12 (2007).
[CrossRef]

Wang, F.

F. Wang, Y. Cai, H. T. Eyyubo?lu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and Elegant Laguerre-Gaussian beam in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–56 (2010).
[CrossRef]

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
[CrossRef] [PubMed]

Wang, J. N.

Wang, X.

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 2471–22 (1995).

Wolf, E.

Wu, G.

G. Wu, T. Zhao, J. Ren, J. Zhang, X. Zhang, and W. Li, “Beam propagation factor of partially coherent Hermite-Gassian beams through non-Kolmogorov turbulence,” Opt. Laser Technol. 43(7), 1225–1228 (2011).
[CrossRef]

G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010).
[CrossRef] [PubMed]

Xue, B. D.

Yu, S.

Yuan, Y.

Zauderer, E.

Zhang, J.

G. Wu, T. Zhao, J. Ren, J. Zhang, X. Zhang, and W. Li, “Beam propagation factor of partially coherent Hermite-Gassian beams through non-Kolmogorov turbulence,” Opt. Laser Technol. 43(7), 1225–1228 (2011).
[CrossRef]

Zhang, X.

G. Wu, T. Zhao, J. Ren, J. Zhang, X. Zhang, and W. Li, “Beam propagation factor of partially coherent Hermite-Gassian beams through non-Kolmogorov turbulence,” Opt. Laser Technol. 43(7), 1225–1228 (2011).
[CrossRef]

Zhao, H.

Zhao, T.

G. Wu, T. Zhao, J. Ren, J. Zhang, X. Zhang, and W. Li, “Beam propagation factor of partially coherent Hermite-Gassian beams through non-Kolmogorov turbulence,” Opt. Laser Technol. 43(7), 1225–1228 (2011).
[CrossRef]

Zhong, Y.

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011).
[CrossRef]

J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
[CrossRef]

Zhou, P.

Zilberman, A.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702, 598702-12 (2005).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (2)

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011).
[CrossRef]

H. T. Eyyubo?lu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B 98(4), 857–863 (2010).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 (1993).

J. Mod. Opt. (1)

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[CrossRef]

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

D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite–Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. 5(5), 489–494 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

T. Shirai, “Polarization properties of a class of electromagnetic Gaussian Schell-model beams which have the same far-zone intensity distribution as a fully coherent laser beam,” Opt. Commun. 256(4-6), 197–209 (2005).
[CrossRef]

J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
[CrossRef]

Opt. Eng. (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
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Opt. Express (7)

E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010).
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L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010).
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H. T. Eyyubo?lu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(20), 4659–4674 (2004).
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Y. Cai, Q. Lin, H. T. Eyyubo?lu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
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Y. Yuan, Y. Cai, J. Qu, H. T. Eyyubo?lu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009).
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Y. Yuan, Y. Cai, J. Qu, H. T. Eyyubo?lu, 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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F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
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Opt. Laser Technol. (1)

G. Wu, T. Zhao, J. Ren, J. Zhang, X. Zhang, and W. Li, “Beam propagation factor of partially coherent Hermite-Gassian beams through non-Kolmogorov turbulence,” Opt. Laser Technol. 43(7), 1225–1228 (2011).
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Opt. Lett. (5)

Proc. SPIE (5)

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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–1­ 65510E-12 (2007).
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Prog. Electromagn. Res. (1)

F. Wang, Y. Cai, H. T. Eyyubo?lu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and Elegant Laguerre-Gaussian beam in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–56 (2010).
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Other (2)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, SPIE Press, Washington, 2001.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd edition SPIE Press, Bellington, (2005)

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Figures (6)

Fig. 1
Fig. 1

Normalized intensity distribution of ELGB through non-Kolmogorov turbulence for different propagation distances z with m=2, n=1, λ=850 nm, w 0 =0.02 m, and C ˜ n 2 =1× 10 14 m 3α .

Fig. 2
Fig. 2

Normalized intensity distribution of ELGB through non-Kolmogorov turbulence for different values of the wavelength and the beam order, respectively. (a) z=5 km, m=2 , and n=1. (b) z=6 km and λ=850 nm .

Fig. 3
Fig. 3

Normalized intensity distribution of ELGB through non-Kolmogorov turbulence at a fixed propagation distance z=5 km for different values of parameter α, outer scale L 0 , inner scale l 0 , and structure constant c ˜ n 2 .

Fig. 4
Fig. 4

Rms beam width of ELGB versus through non-Kolmogorov turbulence for different values of the beam order, waist size, and wavelength.

Fig. 6
Fig. 6

Rms beam width of ELGB versus through non-Kolmogorov turbulence for different parameter α, outer scale L 0 , inner scale l 0 , and structure constant C ˜ n 2 . The calculation parameters are m = 2, n = 1, λ = 850 nm, and w0 = 0.02 m. (a) L0 = 1 m, l0 = 0.01 m, and C ˜ n 2 = C ˜ n 2 10−14 m3-α. (b) α = 3.8, l0 = 0.01 m, and C ˜ n 2 = 10−14 m3-α. (c) α = 3.8, L0 = 1 m, and C ˜ n 2 = 10−14 m3-α. (d) α = 3.8, L0 = 1 m, and l0 = 0.01 m.

Fig. 5
Fig. 5

Rms beam width of ELGB through non-Kolmogorov turbulence as a function of α for different propagation distances. The calculation parameters are: m=2, n=1, λ=850 nm, , w 0 =0.02 m, , L 0 =1 m, , l 0 =0.01 m, , and C ˜ n 2 = 10 14 m 3α .

Equations (17)

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E nm ( r,θ,0 )= ( r m ) m L n m ( r 2 w 0 2 )exp( r 2 w 0 2 )exp( imθ ),
e imθ ρ m L n m ( ρ 2 )= ( 1 ) n 2 2n+m n! t=0 n s=0 m i s ( n t )( m s ) H 2t+ms ( x ) H 2n2t+s ( y ),
E( x,y,0 )= ( 1 ) n 2 2n+m n! t=0 n s=0 m i s ( n t )( m s ) H 2t+ms ( x w 0 )× H 2n2t+s ( y w 0 )exp[ x 2 + y 2 w 0 2 ].
I( p x , p y ,z ) = 1 ( λz ) 2 E( x 1 , y 1 ,0 ) E * ( x 2 , y 2 ,0 )exp[ ik 2z ( x 1 p x ) 2 ik 2z ( y 1 p y ) 2 ] ×exp[ ik 2z ( x 2 p x ) 2 + ik 2z ( y 2 p y ) 2 ]exp[ x 1 2 + y 1 2 w 0 2 ]exp[ x 2 2 + y 2 2 w 0 2 ] × exp[ ψ( x 1 , y 1 , p x , p y )+ ψ * ( x 2 , y 2 , p x , p y ) ] d x 1 d x 2 d y 1 d y 2 ,
exp[ ψ( ρ, ρ 1 ,z )+ ψ * ( ρ, ρ 2 ,z ) ] =exp{ 4 π 2 k 2 z 0 1 0 κ Φ n ( κ )[ 1 J 0 ( κξ| ρ 1 ρ 2 | ) ]dκdξ },
J 0 ( κξ| ρ 1 ρ 2 | )~1 1 4 ( κξ| ρ 1 ρ 2 | ) 2 ,
Φ n ( κ,α )=A( α ) C ˜ n 2 exp[ ( κ 2 / κ m 2 ) ] ( κ 2 + κ 0 2 ) ,0κ< , 3<α<4,
c( α )= [ Γ( 5 α 2 )A( α )2π/3 ] [ 1/( α5 ) ] .
A( α )=Γ( α1 )cos( απ/2 )/4 π 2 .
exp[ ψ( ρ, ρ 1 ,z )+ ψ * ( ρ, ρ 2 ,z ) ] =exp[ 1 3 π 2 k 2 z | ρ 1 ρ 2 | 2 Tdκ ],
T= 0 κ 3 Φ n ( κ )dκ = A( α ) C ˜ n 2 2 κ m 2α βexp( κ 0 2 κ m 2 )Γ( 2α/2, κ 0 2 κ m 2 )2 κ 0 4α α2 ,
I( p x , p y ,z ) = π 2 p 1 p 2 1 ( λz ) 2 ( 1 ) 2n 2 4n+2m ( n! ) 2 t 1 =0 n s 1 =0 m t 2 =0 n s 2 =0 m k 1 =0 [ 2 t 1 +m s 1 2 ] k 2 =0 [ 2n2 t 1 + s 1 2 ] v 1 =0 [ 2 t 2 +m s 2 2 ] × v 2 =0 [ 2n2 t 2 + s 2 2 ] u 1 =0 2 t 2 +m s 2 2 v 1 u 2 =0 2n2 t 2 + s 2 2 v 2 g 1 =0 [ 2 t 1 +2 t 2 +2m2 k 1 2 v 1 s 1 s 2 u 1 2 ] g 2 =0 [ 4n2 t 1 2 t 2 + s 1 + s 2 2 k 2 2 v 2 u 2 2 ] × i s 1 ( i ) s 2 ( n t 1 )( m s 1 )( n t 2 )( m s 2 )( 2 t 2 +m s 2 2 v 1 u 1 )( 2n2 t 2 + s 2 2 v 2 u 2 ) × ( 1 ) k 1 ( 2 t 1 +m s 1 )! k 1 !( 2 t 1 +m s 1 2 k 1 )! ( 1 ) k 2 ( 2n2 t 1 + s 1 )! k 2 !( 2n2 t 1 + s 1 2 k 2 )! ( 1 ) v 1 ( 2 t 2 +m s 2 )! v 1 !( 2 t 2 +m s 2 2 v 1 )! × ( 1 ) v 2 ( 2n2 t 2 + s 2 )! v 2 !( 2n2 t 2 + s 2 2 v 2 )! ( 2 t 1 +2 t 2 +2m2 k 1 2 v 1 s 1 s 2 u 1 )! ( 2 t 1 +2 t 2 +2m2 k 1 2 v 1 s 1 s 2 u 1 2 g 1 )! g 1 ! × ( 4n2 t 1 2 t 2 + s 1 + s 2 2 k 2 2 v 2 u 2 )! ( 4n2 t 1 2 t 2 + s 1 + s 2 2 k 2 2 v 2 u 2 2 g 2 )! g 2 ! ( 1 1 w 0 2 p 1 ) v 1 + v 2 × ( b 1 ) u 1 + u 2 × ( b 2 ) 4n+2m2 k 1 2 k 2 2 v 1 2 v 2 u 1 u 2 2 g 1 2 g 2 ( w 0 ) 4n2m+2 k 1 +2 k 2 +2 v 1 +2 v 2 × 2 4n+2m2 k 2 2 k 1 2 v 1 2 v 2 2 g 1 2 g 2 ( p 1 ) 2nm+2 v 1 +2 v 2 ( p 2 ) 4n2m+2 k 1 +2 k 2 +2 v 1 +2 v 2 + u 1 + u 2 + g 1 + g 2 × ( A i ) 2n+m2 v 1 2 v 2 u 1 u 2 ( p x ) 2 t 1 +2 t 2 +2m2 k 1 2 v 1 s 1 s 2 2 g 1 ( p y ) 4n2 t 1 2 t 2 + s 1 + s 2 2 k 2 2 v 2 2 g 2 ×exp[ ( b 1 2 p 1 + b 2 2 p 2 )( p x 2 + p y 2 ) ],
p 2 = A i A i 2 p 1 + ik 2z + 1 w 0 2 .
W ρ (z)= ρ 2 I( x,y,z ) dxdy I( x,y,z ) dxdy , ( ρ=x,y ).
W x ( z )= W y ( z )= P 1 ( z ) P 2 ( z ) .
P 1 ( z )= π 2 p 1 p 2 1 ( λz ) 2 ( 1 ) 2n 2 4n+2m ( n! ) 2 t 1 =0 n s 1 =0 m t 2 =0 n s 2 =0 m k 1 =0 [ 2 t 1 +m s 1 2 ] v 1 =0 [ 2 t 2 +m s 2 2 ] u 1 =0 2 t 2 +m s 2 2 v 1 × g 1 =0 [ 2 t 1 +2 t 2 +2m2 k 1 2 v 1 s 1 s 2 u 1 2 ] k 2 =0 [ 2n2 t 1 + s 1 2 ] v 2 =0 [ 2n2 t 2 + s 2 2 ] u 2 =0 2n2 t 2 + s 2 2 v 2 g 2 =0 [ 4n2 t 1 2 t 2 + s 1 + s 2 2 k 2 2 v 2 u 2 2 ] i s 1 ( i ) s 2 ×( n t 1 )( m s 1 )( n t 2 )( m s 2 )( 2 t 2 +m s 2 2 v 1 u 1 )( 2n2 t 2 + s 2 2 v 2 u 2 ) × ( 1 ) k 1 ( 2 t 1 +m s 1 )! k 1 !( 2 t 1 +m s 1 2 k 1 )! ( 1 ) k 2 ( 2n2 t 1 + s 1 )! k 2 !( 2n2 t 1 + s 1 2 k 2 )! ( 1 ) v 1 ( 2 t 2 +m s 2 )! v 1 !( 2 t 2 +m s 2 2 v 1 )! × ( 1 ) v 2 ( 2n2 t 2 + s 2 )! v 2 !( 2n2 t 2 + s 2 2 v 2 )! ( 2 t 1 +2 t 2 +2m2 k 1 2 v 1 s 1 s 2 u 1 )! ( 2 t 1 +2 t 2 +2m2 k 1 2 v 1 s 1 s 2 u 1 2 g 1 )! g 1 ! × ( 4n2 t 1 2 t 2 + s 1 + s 2 2 k 2 2 v 2 u 2 )! ( 4n2 t 1 2 t 2 + s 1 + s 2 2 k 2 2 v 2 u 2 2 g 2 )! g 2 ! × ( A i ) 2n+m2 v 1 2 v 2 u 1 u 2 × ( 1 1 w 0 2 p 1 ) v 1 + v 2 × ( w 0 ) 4n2m+2 k 1 +2 k 2 +2 v 1 +2 v 2 ( b 1 ) u 1 + u 2 ( b 2 ) 4n+2m2 k 1 2 k 2 2 v 1 2 v 2 u 1 u 2 2 g 1 2 g 2 × 2 4n+2m2 k 2 2 k 1 2 v 1 2 v 2 2 g 1 2 g 2 ( p 1 ) 2nm+2 v 1 +2 v 2 ( p 2 ) 4n2m+2 k 1 +2 k 2 +2 v 1 +2 v 2 + u 1 + u 2 + g 1 + g 2 × Γ( 2 t 1 +2 t 2 +2m2 k 1 2 v 1 s 1 s 2 2 g 1 +3 2 ) ( b 1 2 p 1 b 2 2 p 2 ) 2 t 1 +2 t 2 +2m2 k 1 2 v 1 s 1 s 2 2 g 1 +3 × Γ( 4n2 t 1 2 t 2 + s 1 + s 2 2 k 2 2 v 2 2 g 2 +1 2 ) ( b 1 2 p 1 b 2 2 p 2 ) 4n2 t 1 2 t 2 + s 1 + s 2 2 k 2 2 v 2 2 g 2 +1 ,
P 2 ( z )= π 2 p 1 p 2 1 ( λz ) 2 ( 1 ) 2n 2 4n+2m ( n! ) 2 t 1 =0 n s 1 =0 m t 2 =0 n s 2 =0 m k 1 =0 [ 2 t 1 +m s 1 2 ] v 1 =0 [ 2 t 2 +m s 2 2 ] u 1 =0 2 t 2 +m s 2 2 v 1 × g 1 =0 [ 2 t 1 +2 t 2 +2m2 k 1 2 v 1 s 1 s 2 u 1 2 ] k 2 =0 [ 2n2 t 1 + s 1 2 ] v 2 =0 [ 2n2 t 2 + s 2 2 ] u 2 =0 2n2 t 2 + s 2 2 v 2 g 2 =0 [ 4n2 t 1 2 t 2 + s 1 + s 2 2 k 2 2 v 2 u 2 2 ] i s 1 ( i ) s 2 ×( n t 1 )( m s 1 )( n t 2 )( m s 2 )( 2 t 2 +m s 2 2 v 1 u 1 )( 2n2 t 2 + s 2 2 v 2 u 2 ) × ( 1 ) k 1 ( 2 t 1 +m s 1 )! k 1 !( 2 t 1 +m s 1 2 k 1 )! ( 1 ) k 2 ( 2n2 t 1 + s 1 )! k 2 !( 2n2 t 1 + s 1 2 k 2 )! ( 1 ) v 1 ( 2 t 2 +m s 2 )! v 1 !( 2 t 2 +m s 2 2 v 1 )! × ( 1 ) v 2 ( 2n2 t 2 + s 2 )! v 2 !( 2n2 t 2 + s 2 2 v 2 )! ( 2 t 1 +2 t 2 +2m2 k 1 2 v 1 s 1 s 2 u 1 )! ( 2 t 1 +2 t 2 +2m2 k 1 2 v 1 s 1 s 2 u 1 2 g 1 )! g 1 ! × ( 4n2 t 1 2 t 2 + s 1 + s 2 2 k 2 2 v 2 u 2 )! ( 4n2 t 1 2 t 2 + s 1 + s 2 2 k 2 2 v 2 u 2 2 g 2 )! g 2 ! × ( A i ) 2n+m2 v 1 2 v 2 u 1 u 2 × ( 1 1 w 0 2 p 1 ) v 1 + v 2 × ( w 0 ) 4n2m+2 k 1 +2 k 2 +2 v 1 +2 v 2 ( b 1 ) u 1 + u 2 ( b 2 ) 4n+2m2 k 1 2 k 2 2 v 1 2 v 2 u 1 u 2 2 g 1 2 g 2 × 2 4n+2m2 k 2 2 k 1 2 v 1 2 v 2 2 g 1 2 g 2 ( p 1 ) 2nm+2 v 1 +2 v 2 ( p 2 ) 4n2m+2 k 1 +2 k 2 +2 v 1 +2 v 2 + u 1 + u 2 + g 1 + g 2 × Γ( 2 t 1 +2 t 2 +2m2 k 1 2 v 1 s 1 s 2 2 g 1 +1 2 ) ( b 1 2 p 1 b 2 2 p 2 ) 2 t 1 +2 t 2 +2m2 k 1 2 v 1 s 1 s 2 2 g 1 +1 × Γ( 4n2 t 1 2 t 2 + s 1 + s 2 2 k 2 2 v 2 2 g 2 +1 2 ) ( b 1 2 p 1 b 2 2 p 2 ) 4n2 t 1 2 t 2 + s 1 + s 2 2 k 2 2 v 2 2 g 2 +1 .

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