Abstract

By using the rms beam width as the parameter to characterize the spreading property, the average spreading of Gaussian beam array propagates in non-Kolmogorov turbulence is studied. Numerical examples reveal that the beam width depends on the exponent value α, and the beam width in non-Kolmogorov turbulence is different from that in Kolmogorov turbulence with α=3.667. The beam width also depends on the distance between adjacent beamlets, the inner and outer scale parameters of the non-Kolmogorov turbulence. The beam array spreads more significantly with smaller inner scale and larger outer scale parameters.

© 2010 Optical Society of America

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[CrossRef]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, IEEE J. Sel. Top. Quantum Electron. 15, 248 (2009).
[CrossRef]

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, A. D. Sanchez, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, Proc. SPIE 7195, 71951M-1 (2009).

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, Appl. Phys. B: Photophys. Laser Chem. 98, 557 (2009).

2008 (3)

2007 (6)

S. J. Augst, J. Ranka, T. Y. Fan, and A. Sanchez, J. Opt. Soc. Am. B 24, 1707-1715 (2007).
[CrossRef]

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, Proc. SPIE 6457, 64570T-1 (2007).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, Proc. SPIE 6551, 65510E-1 (2007).

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, Appl. Phys. B: Photophys. Laser Chem. 88, 467 (2007).
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Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, Opt. Commun. 278, 157 (2007).
[CrossRef]

2005 (1)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
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C. Rao, W. Jiang, and N. Ling, J. Mod. Opt. 47, 175 (2000).
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A. E. Siegman, Proc. SPIE 1224, 2 (1990).
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1979 (1)

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G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
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I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, Proc. SPIE 6551, 65510E-1 (2007).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, Proc. SPIE 6457, 64570T-1 (2007).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
[CrossRef]

Asman, C. P.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

Augst, S. J.

Baker, J. T.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, A. D. Sanchez, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, Proc. SPIE 7195, 71951M-1 (2009).

Baykal, Y.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, Appl. Phys. B: Photophys. Laser Chem. 98, 557 (2009).

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, Appl. Phys. B: Photophys. Laser Chem. 88, 467 (2007).
[CrossRef]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, Opt. Commun. 278, 157 (2007).
[CrossRef]

Benjamin Weiss, S.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
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T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, A. D. Sanchez, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, Proc. SPIE 7195, 71951M-1 (2009).

Brosnan, S.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

Cai, Y.

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, Opt. Commun. 278, 157 (2007).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, Appl. Phys. B: Photophys. Laser Chem. 88, 467 (2007).
[CrossRef]

Chen, Y.

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, Appl. Phys. B: Photophys. Laser Chem. 88, 467 (2007).
[CrossRef]

Cheung, E. C.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

Chu, X.

P. Zhou, X. Xu, Z. Liu, and X. Chu, Opt. Laser Technol. 42, 392 (2010).
[CrossRef]

Du, W.

Du, X.

Eyyuboglu, H. T.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, Appl. Phys. B: Photophys. Laser Chem. 98, 557 (2009).

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, Appl. Phys. B: Photophys. Laser Chem. 88, 467 (2007).
[CrossRef]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, Opt. Commun. 278, 157 (2007).
[CrossRef]

Fan, T. Y.

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, Proc. SPIE 6551, 65510E-1 (2007).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, Proc. SPIE 6457, 64570T-1 (2007).

G, H.

Gallant, D.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, A. D. Sanchez, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, Proc. SPIE 7195, 71951M-1 (2009).

Golbraikh, E.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, Appl. Opt. 47, 6385 (2008).
[CrossRef] [PubMed]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, Atmos. Res. 88, 66 (2008).
[CrossRef]

Goodno, G. D.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

Guo, S.

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, IEEE J. Sel. Top. Quantum Electron. 15, 248 (2009).
[CrossRef]

Hammons, D.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

Han, Q.

Injeyan, H.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

Ji, X.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, Appl. Phys. B: Photophys. Laser Chem. 98, 557 (2009).

X. Ji and X. Li, J. Opt. Soc. Am. A 26, 236 (2009).
[CrossRef]

Jiang, W.

C. Rao, W. Jiang, and N. Ling, J. Mod. Opt. 47, 175 (2000).
[CrossRef]

Komine, H.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

Kopeika, N. S.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, Atmos. Res. 88, 66 (2008).
[CrossRef]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, Appl. Opt. 47, 6385 (2008).
[CrossRef] [PubMed]

Kupershmidt, I.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, Atmos. Res. 88, 66 (2008).
[CrossRef]

Li, X.

X. Ji and X. Li, J. Opt. Soc. Am. A 26, 236 (2009).
[CrossRef]

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, Appl. Phys. B: Photophys. Laser Chem. 98, 557 (2009).

Lin, Q.

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, Opt. Commun. 278, 157 (2007).
[CrossRef]

Ling, N.

C. Rao, W. Jiang, and N. Ling, J. Mod. Opt. 47, 175 (2000).
[CrossRef]

Liu, Z.

P. Zhou, X. Xu, Z. Liu, and X. Chu, Opt. Laser Technol. 42, 392 (2010).
[CrossRef]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, IEEE J. Sel. Top. Quantum Electron. 15, 248 (2009).
[CrossRef]

Long, W. H.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

Lu, C. A.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, A. D. Sanchez, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, Proc. SPIE 7195, 71951M-1 (2009).

Lucero, A.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, A. D. Sanchez, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, Proc. SPIE 7195, 71951M-1 (2009).

Luo, B.

Ma, H.

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, IEEE J. Sel. Top. Quantum Electron. 15, 248 (2009).
[CrossRef]

Ma, J.

Ma, Y.

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, IEEE J. Sel. Top. Quantum Electron. 15, 248 (2009).
[CrossRef]

McClellan, M.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

McNaught, S. J.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, Proc. SPIE 6551, 65510E-1 (2007).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, Proc. SPIE 6457, 64570T-1 (2007).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
[CrossRef]

Plonus, M. A.

Pulford, B.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, A. D. Sanchez, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, Proc. SPIE 7195, 71951M-1 (2009).

Ranka, J.

Rao, C.

C. Rao, W. Jiang, and N. Ling, J. Mod. Opt. 47, 175 (2000).
[CrossRef]

Redmond, S.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

Robin, C. A.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, A. D. Sanchez, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, Proc. SPIE 7195, 71951M-1 (2009).

Sanchez, A.

Sanchez, A. D.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, A. D. Sanchez, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, Proc. SPIE 7195, 71951M-1 (2009).

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, A. D. Sanchez, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, Proc. SPIE 7195, 71951M-1 (2009).

Shay, T. M.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, A. D. Sanchez, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, Proc. SPIE 7195, 71951M-1 (2009).

Shtemler, Y.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, Atmos. Res. 88, 66 (2008).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Proc. SPIE 1224, 2 (1990).
[CrossRef]

Simpson, R.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

Sollee, J.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

Tan, L.

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, Proc. SPIE 6551, 65510E-1 (2007).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, Proc. SPIE 6457, 64570T-1 (2007).

Vergien, C. L.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, A. D. Sanchez, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, Proc. SPIE 7195, 71951M-1 (2009).

Virtser, A.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, Atmos. Res. 88, 66 (2008).
[CrossRef]

Wang, S. C. H.

Wang, X.

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, IEEE J. Sel. Top. Quantum Electron. 15, 248 (2009).
[CrossRef]

Weber, M.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

Wickham, M.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. Benjamin Weiss, and M. Wickham, IEEE J. Sel. Top. Quantum Electron. 13, 460 (2007).
[CrossRef]

Wu, G.

Xu, X.

P. Zhou, X. Xu, Z. Liu, and X. Chu, Opt. Laser Technol. 42, 392 (2010).
[CrossRef]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, IEEE J. Sel. Top. Quantum Electron. 15, 248 (2009).
[CrossRef]

Yu, S.

Zerinque, C.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, A. D. Sanchez, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, Proc. SPIE 7195, 71951M-1 (2009).

Zhao, D.

Zhou, P.

P. Zhou, X. Xu, Z. Liu, and X. Chu, Opt. Laser Technol. 42, 392 (2010).
[CrossRef]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, IEEE J. Sel. Top. Quantum Electron. 15, 248 (2009).
[CrossRef]

Zhu, Y.

Zilberman, A.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, Appl. Opt. 47, 6385 (2008).
[CrossRef] [PubMed]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, Atmos. Res. 88, 66 (2008).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B: Photophys. Laser Chem. (2)

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, Appl. Phys. B: Photophys. Laser Chem. 88, 467 (2007).
[CrossRef]

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, Appl. Phys. B: Photophys. Laser Chem. 98, 557 (2009).

Atmos. Res. (1)

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, Atmos. Res. 88, 66 (2008).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of the one-dimensional beam array.

Fig. 2
Fig. 2

Beam width ω ( z ) of the Gaussian beam array through non-Kolmogorov turbulence as a function of propagation distance with different α.

Fig. 3
Fig. 3

Dependence of ω ( z ) through non-Kolmogorov turbulence as a function of α with different x d .

Fig. 4
Fig. 4

Dependence of ω ( z ) through non-Kolmogorov turbulence as a function of propagation distance for different outer- and inner-scale parameters: (a) inner scale, (b) outer scale, (c) details of the subsection denoted in the squares in (b).

Equations (9)

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E ( x , 0 ) = j = N 1 2 N 1 2 exp [ ( x j x d ) 2 ω 0 2 ] ,
I ( p , z ) = k 2 ( 2 π z ) 2 E ( x , 0 ) E * ( ξ , 0 ) exp { i k 2 z [ ( p x ) 2 ( p ξ ) 2 ] } exp [ ψ ( x , p ) + ψ * ( ξ , p ) ] d x d ξ ,
exp [ ψ ( x , p ) + ψ * ( ξ , p ) ] = exp { 4 π 2 k 2 z 0 1 0 κ Φ n ( κ ) [ 1 J 0 ( κ η | x ξ | ) ] d κ d η } ,
ω 2 ( z ) = 4 p 2 I ( p , z ) d p I ( p , z ) d p .
ω 2 ( z ) = A + B z 2 + F z 3 ,
A = j 1 = ( N 1 ) 2 ( N 1 ) 2 j 2 = ( N 1 ) 2 ( N 1 ) 2 exp [ ( j 1 j 2 ) 2 x d 2 2 ω 0 2 ] [ ω 0 2 + ( j 1 + j 2 ) 2 x d 2 ] j 1 = ( N 1 ) 2 ( N 1 ) 2 j 2 = ( N 1 ) 2 ( N 1 ) 2 exp [ ( j 1 j 2 ) 2 x d 2 2 ω 0 2 ] ,
B = j 1 = ( N 1 ) 2 ( N 1 ) 2 j 2 = ( N 1 ) 2 ( N 1 ) 2 4 ω 0 2 κ 2 exp [ ( j 1 j 2 ) 2 x d 2 2 ω 0 2 ] [ 1 ( j 1 j 2 ) 2 x d 2 ω 0 2 ] j 1 = ( N 1 ) 2 ( N 1 ) 2 j 2 = ( N 1 ) 2 ( N 1 ) 2 exp [ ( j 1 j 2 ) 2 x d 2 2 ω 0 2 ] .
Φ n ( κ , α ) = H ( α ) C n 2 ̃ exp [ κ 2 κ m 2 ] ( κ 2 + κ 0 2 ) α 2 , 0 κ < , 3 < α < 4 ,
ω 2 ( z ) = A + B z 2 + [ 4 π 2 H ( α ) C n 2 ̃ κ m 2 α β exp ( κ 0 2 κ m 2 ) Γ ( 2 α 2 , κ 0 2 κ m 2 ) 4 κ 0 4 α 3 ( α 2 ) ] z 3 .

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