Abstract

The analytical expression for the rms beam width of the radial Gaussian beam array propagating in non-Kolmogorov turbulence is derived, where the coherent combination is considered. The influences of the beam number, the generalized exponent, and the ring radius on the rms beam width are investigated. The results indicate that the rms beam width depends greatly on the generalized exponent and the beam number. Further, an optimum ring radius, which leads to a minimum beam width, is proved to exist within a certain traveling distance and the optimum ring radius increases when the beam number increases.

© 2011 Optical Society of America

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  1. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Validity of the Kolmogorov turbulence at higher elevations,” Proc. SPIE 5160, 397–405 (2004).
    [CrossRef]
  2. C. Rao, W. Jiang, and N. Ling, “Atmospheric characterization with Shack-Hartmann wave-front sensors for non-Kolmogorov turbulence,” Opt. Eng. 41, 534–541 (2002).
    [CrossRef]
  3. D. T. Kyrazis, J. B. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55(1994).
    [CrossRef]
  4. A. S. Gurvich and M. S. Be1en’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995).
    [CrossRef]
  5. A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77(2008).
    [CrossRef]
  6. D. Dayton, B. Pierson, B. Spielbusch, and J. Gonglewski, “Atmospheric structure function measurements with a Shack-Hartmann wave-front sensor,” Opt. Lett. 17, 1737–1739(1992).
    [CrossRef] [PubMed]
  7. M. S. Be1en’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
    [CrossRef]
  8. M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
    [CrossRef]
  9. L. J. Otten III, B. A. Jones, J. Lane, D. G. Black, and M. C. Roggemann, “High bandwidth atmosphere turbulence data collection platform,” Proc. SPIE 3866, 23–32 (1999).
    [CrossRef]
  10. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702 (2005).
    [CrossRef]
  11. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47, 6385–6391 (2008).
    [CrossRef] [PubMed]
  12. R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16(1995).
    [CrossRef]
  13. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
    [CrossRef]
  14. 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, 026003(2008).
    [CrossRef]
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    [CrossRef]
  17. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35, 1043–1045 (2010).
    [CrossRef] [PubMed]
  18. X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42, 604–609(2010).
    [CrossRef]
  19. X. Li, X. Ji, H. T. Eyyubog˘lu, and Y. Baykal, “Turbulence distance of radial GaussianSchell-model array beams,” Appl. Phys. B 98, 557–565 (2010).
    [CrossRef]
  20. 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] [PubMed]
  21. 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]

2010

2009

2008

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47, 6385–6391 (2008).
[CrossRef] [PubMed]

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, 026003(2008).
[CrossRef]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77(2008).
[CrossRef]

2007

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (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]

2005

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

2004

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Validity of the Kolmogorov turbulence at higher elevations,” Proc. SPIE 5160, 397–405 (2004).
[CrossRef]

2002

C. Rao, W. Jiang, and N. Ling, “Atmospheric characterization with Shack-Hartmann wave-front sensors for non-Kolmogorov turbulence,” Opt. Eng. 41, 534–541 (2002).
[CrossRef]

1999

M. S. Be1en’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

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

L. J. Otten III, B. A. Jones, J. Lane, D. G. Black, and M. C. Roggemann, “High bandwidth atmosphere turbulence data collection platform,” Proc. SPIE 3866, 23–32 (1999).
[CrossRef]

1995

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

A. S. Gurvich and M. S. Be1en’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995).
[CrossRef]

1994

D. T. Kyrazis, J. B. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55(1994).
[CrossRef]

1992

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, 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 (2007).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
[CrossRef]

Barchers, J. D.

M. S. Be1en’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of 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] [PubMed]

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

Be1en’kii, M. S.

M. S. Be1en’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

A. S. Gurvich and M. S. Be1en’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995).
[CrossRef]

Beland, R. R.

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

Belen’kii, M. S.

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

Bishop, K. P.

D. T. Kyrazis, J. B. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55(1994).
[CrossRef]

Black, D. G.

L. J. Otten III, B. A. Jones, J. Lane, D. G. Black, and M. C. Roggemann, “High bandwidth atmosphere turbulence data collection platform,” Proc. SPIE 3866, 23–32 (1999).
[CrossRef]

Brown, J. M.

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

M. S. Be1en’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

Dayton, D.

Eyyubog?lu, H. T.

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

Eyyuboglu, H. T.

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, 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 (2007).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
[CrossRef]

Fugate, R. Q.

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

M. S. Be1en’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

Golbraikh, E.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77(2008).
[CrossRef]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47, 6385–6391 (2008).
[CrossRef] [PubMed]

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

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Validity of the Kolmogorov turbulence at higher elevations,” Proc. SPIE 5160, 397–405 (2004).
[CrossRef]

Gonglewski, J.

Guo, H.

Gurvich, A. S.

Ji, X.

X. Li, X. Ji, H. T. Eyyubog˘lu, and Y. Baykal, “Turbulence distance of radial GaussianSchell-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] [PubMed]

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

X. Ji and X. Li, “Directionality of Gaussian array beams propagating in atmospheric turbulence,” J. Opt. Soc. Am. A 26, 236–243 (2009).
[CrossRef]

Jiang, W.

C. Rao, W. Jiang, and N. Ling, “Atmospheric characterization with Shack-Hartmann wave-front sensors for non-Kolmogorov turbulence,” Opt. Eng. 41, 534–541 (2002).
[CrossRef]

Jones, B. A.

L. J. Otten III, B. A. Jones, J. Lane, D. G. Black, and M. C. Roggemann, “High bandwidth atmosphere turbulence data collection platform,” Proc. SPIE 3866, 23–32 (1999).
[CrossRef]

Karis, S. J.

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

M. S. Be1en’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

Keating, D. D. B.

D. T. Kyrazis, J. B. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55(1994).
[CrossRef]

Kopeika, N. S.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77(2008).
[CrossRef]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47, 6385–6391 (2008).
[CrossRef] [PubMed]

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

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Validity of the Kolmogorov turbulence at higher elevations,” Proc. SPIE 5160, 397–405 (2004).
[CrossRef]

Kupershmidt, I.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77(2008).
[CrossRef]

Kyrazis, D. T.

D. T. Kyrazis, J. B. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55(1994).
[CrossRef]

Lane, J.

L. J. Otten III, B. A. Jones, J. Lane, D. G. Black, and M. C. Roggemann, “High bandwidth atmosphere turbulence data collection platform,” Proc. SPIE 3866, 23–32 (1999).
[CrossRef]

Li, X.

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

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

X. Ji and X. Li, “Directionality of Gaussian array beams propagating in atmospheric turbulence,” J. Opt. Soc. Am. A 26, 236–243 (2009).
[CrossRef]

Ling, N.

C. Rao, W. Jiang, and N. Ling, “Atmospheric characterization with Shack-Hartmann wave-front sensors for non-Kolmogorov turbulence,” Opt. Eng. 41, 534–541 (2002).
[CrossRef]

Liu, Z.

Luo, B.

Ma, Y.

Osmon, C. L.

M. S. Be1en’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[CrossRef]

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

Otten, L. J.

L. J. Otten III, B. A. Jones, J. Lane, D. G. Black, and M. C. Roggemann, “High bandwidth atmosphere turbulence data collection platform,” Proc. SPIE 3866, 23–32 (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, 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 (2007).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
[CrossRef]

Pierson, B.

Preble, A. J.

D. T. Kyrazis, J. B. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55(1994).
[CrossRef]

Rao, C.

C. Rao, W. Jiang, and N. Ling, “Atmospheric characterization with Shack-Hartmann wave-front sensors for non-Kolmogorov turbulence,” Opt. Eng. 41, 534–541 (2002).
[CrossRef]

Roggemann, M. C.

L. J. Otten III, B. A. Jones, J. Lane, D. G. Black, and M. C. Roggemann, “High bandwidth atmosphere turbulence data collection platform,” Proc. SPIE 3866, 23–32 (1999).
[CrossRef]

Shtemler, Y.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77(2008).
[CrossRef]

Spielbusch, B.

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, 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 (2007).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
[CrossRef]

Virtser, A.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77(2008).
[CrossRef]

Wang, X.

Wissler, J. B.

D. T. Kyrazis, J. B. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55(1994).
[CrossRef]

Wu, G.

Yang, F.

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

Yu, S.

Zhao, H.

Zhou, P.

Zilberman, A.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47, 6385–6391 (2008).
[CrossRef] [PubMed]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77(2008).
[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 (2005).
[CrossRef]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Validity of the Kolmogorov turbulence at higher elevations,” Proc. SPIE 5160, 397–405 (2004).
[CrossRef]

Appl. Opt.

Appl. Phys. B

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

Atmos. Res.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77(2008).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

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, 026003(2008).
[CrossRef]

C. Rao, W. Jiang, and N. Ling, “Atmospheric characterization with Shack-Hartmann wave-front sensors for non-Kolmogorov turbulence,” Opt. Eng. 41, 534–541 (2002).
[CrossRef]

Opt. Express

Opt. Laser Technol.

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

Opt. Lett.

Proc. SPIE

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]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Validity of the Kolmogorov turbulence at higher elevations,” Proc. SPIE 5160, 397–405 (2004).
[CrossRef]

D. T. Kyrazis, J. B. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55(1994).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the radial beam array.

Fig. 2
Fig. 2

(a)  w turb as the function of N with C ˜ n 2 = 1 × 10 15 m 3 α , α = 3.8 , and r 0 = 0.02 m . (b)  w turb as the function of r 0 for different N with C ˜ n 2 = 1 × 10 15 m 3 α and α = 3.8 .

Fig. 3
Fig. 3

A, B z 2 / k 2 , and w turb 2 as the function of r 0 with C ˜ n 2 = 1 × 10 15 m 3 α , α = 3.8 , and N = 10 .

Fig. 4
Fig. 4

z min and z max as the function of r 0 with N = 10 . (a)  C ˜ n 2 = 1 × 10 15 m 3 α , α = 3.8 . (b)  C ˜ n 2 = 1 × 10 14 m 3 α , α = 11 / 3 .

Fig. 5
Fig. 5

Optimum ring radius r 0 m as the function of N with C ˜ n 2 = 1 × 10 15 m 3 α and α = 3.8 .

Fig. 6
Fig. 6

w turb as the function of α with C ˜ n 2 = 1 × 10 15 m 3 α , N = 10 , and r 0 = 0.02 m .

Fig. 7
Fig. 7

w turb as the function of r 0 for different C ˜ n 2 with N = 10 . (a)  α = 3.3 . (b)  α = 11 / 3 . (c)  α = 3.8 .

Equations (20)

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w ( 0 ) ( x 1 , y 1 , x 2 , y 2 , z = 0 ) = m = 0 N 1 n = 0 N 1 exp [ ( x 1 r 0 cos θ m ) 2 + ( y 1 r 0 sin θ m ) 2 w 0 2 ] × exp [ ( x 2 r 0 cos θ n ) 2 + ( y 2 r 0 sin θ n ) 2 w 0 2 ] ,
I ( x , y , z ) = ( k 2 π z ) 2 w ( 0 ) ( x 1 , y 1 , x 2 , y 2 , z = 0 ) × exp { i k 2 z [ ( x x 1 ) 2 + ( y y 1 ) 2 ( x x 2 ) 2 ( y y 2 ) 2 ] } exp [ ψ ( x , y , x 1 , y 1 ) + ψ * ( x , y , x 2 , y 2 ) ] d x 1 d y 1 d x 2 d y 2 ,
w turb = [ ( x 2 + y 2 ) I ( x , y , z ) d x d y I ( x , y , z ) d x d y ] 1 / 2 .
w turb = ( A + B z 2 / k 2 + 4 / 3 T z 3 ) 1 / 2 ,
A = 1 2 m = 0 N 1 n = 0 N 1 { w 0 2 + r 0 2 [ 1 + cos ( θ m θ n ) ] } S / m = 0 N 1 n = 0 N 1 S ,
B = 2 w 0 4 m = 0 N 1 n = 0 N 1 { w 0 2 r 0 2 [ 1 cos ( θ m θ n ) ] } S / m = 0 N 1 n = 0 N 1 S ,
S = exp { r 0 2 / w 0 2 [ 1 cos ( θ m θ n ) ] } ,
T = π 2 0 κ 3 Φ n ( κ , α ) d κ .
Φ n ( κ , α ) = A ( α ) C ˜ n 2 exp ( κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) α / 2 , 0 < κ < , 3 < α < 4 ,
T = π 2 A ( α ) C ˜ n 2 / [ 2 ( α 2 ) ] · [ β κ m 2 α exp ( κ 0 2 / κ m 2 ) Γ ( 2 α / 2 , κ 0 2 / κ m 2 ) 2 κ 0 4 α ] ,
z min = k A / B < z < 3 B / ( 4 T k 2 ) = z max .
I ( x , y , z ) = ( k 2 π z ) 2 m = 0 N 1 n = 0 N 1 d x 1 d y 1 d x 2 d y 2 exp [ 2 u x 2 + v x 2 / 2 2 ( a m + a n ) u x + ( a m a n ) v x + ( a m 2 + a n 2 ) w 0 2 ] × exp [ 2 u y 2 + v y 2 / 2 2 ( b m + b n ) u y + ( b m b n ) v y + ( b m 2 + b n 2 ) w 0 2 ] × exp { 4 π 2 k 2 z 0 1 0 κ Φ n ( κ , α ) [ 1 J 0 ( κ ξ v x 2 + v y 2 ) ] d κ d ξ } exp [ i k z ( x v x u x v x + y v y u y v y ) ] .
w turb = [ ( F 1 + F 2 ) / F 3 ] 1 / 2 ,
F 1 = x 2 I ( x , y , z ) d x d y ,
F 2 = y 2 I ( x , y , z ) d x d y ,
F 3 = I ( x , y , z ) d x dy .
F 1 = ( z k ) 2 m = 0 N 1 n = 0 N 1 d u x d v x d u y d v y exp [ 2 u x 2 + v x 2 / 2 2 ( a m + a n ) u x + ( a m a n ) v x + ( a m 2 + a n 2 ) w 0 2 ] × exp [ 2 u y 2 + v y 2 / 2 2 ( b m + b n ) u y + ( b m b n ) v y + ( b m 2 + b n 2 ) w 0 2 ] × exp { 4 π 2 k 2 z 0 1 0 κ Φ n ( κ , α ) [ 1 J 0 ( κ ξ v x 2 + v y 2 ) ] d κ d ξ } exp [ i k z ( u x v x + u y v y ) ] δ ( v x ) δ ( v y ) .
F 1 = π 2 w 0 m = 0 N 1 n = 0 N 1 exp [ 1 2 w 0 2 ( a m a n ) 2 ] × { 1 4 [ w 0 2 + ( a m + a n ) 2 ] + 1 k 2 w 0 2 [ 1 1 w 0 2 ( a m a n ) 2 ] z 2 + 2 3 π 2 z 3 0 κ 3 Φ n ( κ , α ) d κ } × π 2 w 0 exp [ 1 2 w 0 2 ( b m b n ) 2 ] ,
F 2 = π 2 w 0 m = 0 N 1 n = 0 N 1 exp [ 1 2 w 0 2 ( b m b n ) 2 ] × { 1 4 [ w 0 2 + ( b m + b n ) 2 ] + 1 k 2 w 0 2 [ 1 1 w 0 2 ( b m b n ) 2 ] z 2 + 2 3 π 2 z 3 0 κ 3 Φ n ( κ , α ) d κ } × π 2 w 0 exp [ 1 2 w 0 2 ( a m a n ) 2 ] ,
F 3 = π 2 w 0 2 m = 0 N 1 n = 0 N 1 exp { 1 2 w 0 2 [ ( a m a n ) 2 + ( b m b n ) 2 ] } .

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