Abstract

Kolmogorov turbulence theory based models cannot be directly applied in non-Kolmogorov turbulence case, which has been reported recently by increasing experimental evidence and theoretical investigation. In this study, based on the generalized von Karman spectral model, the theoretical expression of the irradiance scintillation index is derived for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence with horizontal path. In the derivation, the expression is divided into two parts for physical analysis purpose and mathematical analysis convenience. This expression considers the influences of finite turbulence inner and outer scales and has a general spectral power law value in the range 3 to 4 instead of standard power law value of 11/3 (for Kolmogorov turbulence). Numerical simulations are conducted to investigate the influences.

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References

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  1. L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE 4489, 23–34 (2002).
    [CrossRef]
  2. X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009).
    [CrossRef]
  3. W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-20-17829 .
    [CrossRef] [PubMed]
  4. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express 18(6), 5356–5366 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-6-5356 .
    [CrossRef] [PubMed]
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    [CrossRef]
  7. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994).
    [CrossRef]
  8. J. D. Shelton, “Turbulence-induced scintillation on Gaussian-beam waves: theoretical predictions and observations from a laser-illuminated satellite,” J. Opt. Soc. Am. A 12(10), 2172–2181 (1995).
    [CrossRef]
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    [CrossRef]
  10. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005), Chap.8.
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    [CrossRef]
  12. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (trans.for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971).
  13. D. T. Kyrazis, J. B. Wissler, 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]
  14. 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]
  15. M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U, 63040U-12 (2006).
    [CrossRef]
  16. 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(1), 66–77 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  19. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
    [CrossRef]
  20. L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(2), 451–462 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-2-451 .
    [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, 65510E-12 (2007).
    [CrossRef]
  22. 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), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-20-21269 .
    [CrossRef] [PubMed]
  23. B. Xue, L. Cui, W. Xue, X. Bai, and F. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28(5), 912–916 (2011).
    [CrossRef] [PubMed]
  24. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 1998).
  25. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, 1998).

2011 (1)

2010 (3)

2009 (3)

X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009).
[CrossRef]

W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-20-17829 .
[CrossRef] [PubMed]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[CrossRef]

2008 (1)

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(1), 66–77 (2008).
[CrossRef]

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, 65510E-12 (2007).
[CrossRef]

2006 (1)

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U, 63040U-12 (2006).
[CrossRef]

2002 (1)

L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE 4489, 23–34 (2002).
[CrossRef]

2001 (1)

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media 11, 271–291 (2001). http://dx.doi.org/10.1080/13616670109409785 .
[CrossRef]

1997 (1)

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]

1995 (3)

1994 (2)

D. T. Kyrazis, J. B. Wissler, 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]

W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994).
[CrossRef]

1993 (1)

1992 (1)

L. C. Andrews, “An analytical model for the refractive index power dpectrum and its spplication to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992), http://dx.doi.org/10.1080/09500349214551931 .
[CrossRef]

Al-Habash, M. A.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media 11, 271–291 (2001). http://dx.doi.org/10.1080/13616670109409785 .
[CrossRef]

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[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, 65510E-12 (2007).
[CrossRef]

L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE 4489, 23–34 (2002).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media 11, 271–291 (2001). http://dx.doi.org/10.1080/13616670109409785 .
[CrossRef]

W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994).
[CrossRef]

W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10(4), 661–672 (1993).
[CrossRef]

L. C. Andrews, “An analytical model for the refractive index power dpectrum and its spplication to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992), http://dx.doi.org/10.1080/09500349214551931 .
[CrossRef]

Bai, X.

Belen’kii, M. S.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U, 63040U-12 (2006).
[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]

A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12(11), 2517–2522 (1995).
[CrossRef]

M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20(12), 1359–1361 (1995).
[CrossRef] [PubMed]

Bishop, K. P.

D. T. Kyrazis, J. B. Wissler, 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]

Brown, J. M.

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]

Cao, X. G.

Castillo-Vázquez, B.

Castillo-Vázquez, C.

Cheng, W.

Cuellar, E.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U, 63040U-12 (2006).
[CrossRef]

Cui, L.

Cui, L. Y.

Dong, J. K.

Du, W.

Feng, J.-W.

X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009).
[CrossRef]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[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, 65510E-12 (2007).
[CrossRef]

Fugate, R. Q.

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]

García-Zambrana, A.

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(1), 66–77 (2008).
[CrossRef]

Gurvich, A. S.

Han, Q.

Han, Y.

X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009).
[CrossRef]

Haus, J. W.

Hopen, C. Y.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media 11, 271–291 (2001). http://dx.doi.org/10.1080/13616670109409785 .
[CrossRef]

Hughes, K. A.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U, 63040U-12 (2006).
[CrossRef]

Karis, S. J.

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]

Keating, D. B.

D. T. Kyrazis, J. B. Wissler, 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(1), 66–77 (2008).
[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(1), 66–77 (2008).
[CrossRef]

Kyrazis, D. T.

D. T. Kyrazis, J. B. Wissler, 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]

Ma, J.

Miller, W. B.

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[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, 65510E-12 (2007).
[CrossRef]

L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE 4489, 23–34 (2002).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media 11, 271–291 (2001). http://dx.doi.org/10.1080/13616670109409785 .
[CrossRef]

Preble, A. J.

D. T. Kyrazis, J. B. Wissler, 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]

Qiang, X.-W.

X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009).
[CrossRef]

Ricklin, J. C.

Rye, V. A.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U, 63040U-12 (2006).
[CrossRef]

Shelton, J. D.

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(1), 66–77 (2008).
[CrossRef]

Song, J.-P.

X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009).
[CrossRef]

Tan, L.

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[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, 65510E-12 (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(1), 66–77 (2008).
[CrossRef]

Wang, J. N.

Wissler, J. B.

D. T. Kyrazis, J. B. Wissler, 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]

Xue, B.

Xue, B. D.

Xue, W.

Yu, S.

Zhan, Q.

Zhou, F.

Zilberman, 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(1), 66–77 (2008).
[CrossRef]

Atmos. Res. (1)

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(1), 66–77 (2008).
[CrossRef]

IEEE Trans. Antenn. Propag. (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[CrossRef]

J. Mod. Opt. (1)

L. C. Andrews, “An analytical model for the refractive index power dpectrum and its spplication to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992), http://dx.doi.org/10.1080/09500349214551931 .
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Express (4)

Opt. Lett. (1)

Proc. SPIE (6)

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, 65510E-12 (2007).
[CrossRef]

D. T. Kyrazis, J. B. Wissler, 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]

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]

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U, 63040U-12 (2006).
[CrossRef]

L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE 4489, 23–34 (2002).
[CrossRef]

X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009).
[CrossRef]

Waves Random Complex Media (1)

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media 11, 271–291 (2001). http://dx.doi.org/10.1080/13616670109409785 .
[CrossRef]

Other (5)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005), Chap.8.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer, 1994).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (trans.for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 1998).

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, 1998).

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Figures (3)

Fig. 1
Fig. 1

Scaled irradiance scintillation index as a function of Λ 0 with different Q l values. (a): collimated beam ( Θ 0 = 1 ); (b): convergent beam ( Θ 0 = 0.7 < 1 ); (c): convergent beam ( Θ 0 = 0.1 < 1 ).

Fig. 2
Fig. 2

Scaled irradiance scintillation index as a function of Λ 0 with different outer scale values. (a): collimated beam ( Θ 0 = 1 ); (b): convergent beam ( Θ 0 = 0.7 < 1 ); (c): convergent beam ( Θ 0 = 0.1 < 1 ).

Fig. 3
Fig. 3

Scaled irradiance scintillation index as a function of Λ 0 with different α. (a): collimated beam ( Θ 0 = 1 ); (b): convergent beam ( Θ 0 = 0.7 < 1 ); (c): convergent beam ( Θ 0 = 0.1 < 1 ).

Equations (35)

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Φ n ( κ , α , l 0 , L 0 ) = A ( α ) C ^ n 2 ( κ 2 + κ 0 2 ) α / 2 exp ( κ 2 κ l 2 ) ( 0 κ < , 3 < α < 4 ) .
A ( α ) = Γ ( α 1 ) 4 π 2 cos [ α π 2 ] , c ( α ) = [ Γ ( 5 α 2 ) A ( α ) 2 3 π ] 1 α 5 .
σ I 2 ( ρ ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( Λ L κ 2 ξ 2 / k ) × { I 0 ( 2 Λ ρ κ ξ ) cos [ L κ 2 k ξ ( 1 Θ ˜ ξ ) ] } d κ d ξ .
Θ = 1 + L F = Θ 0 Θ 0 2 + Λ 0 2 , Λ = 2 L k W 2 = Λ 0 Θ 0 2 + Λ 0 2 ,
Θ 0 = 1 L F 0 , Λ 0 = 2 L k W 0 2 .
σ I 2 ( ρ ) = σ I , r 2 ( ρ ) + σ I , l 2 .
σ I , r 2 ( ρ ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( Λ L κ 2 ξ 2 / k ) { I 0 ( 2 Λ ρ κ ξ ) 1 } d κ d ξ ,
σ I , l 2 = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( Λ L κ 2 ξ 2 / k ) { 1 cos [ L κ 2 k ξ ( 1 Θ ˜ ξ ) ] } d κ d ξ .
σ I , r 2 ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ , α , l 0 , L 0 ) exp ( Λ L κ 2 ξ 2 / k ) { I 0 ( 2 Λ ρ κ ξ ) 1 } d κ d ξ .
I 0 ( x ) = J 0 ( i x ) = n = 0 ( x / 2 ) 2 n n ! Γ ( n + 1 ) ,
σ I , r 2 ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ( α ) C ^ n 2 L 0 1 { n = 1 ( Λ ρ ξ ) 2 n n ! Γ ( n + 1 ) 0 κ 2 n + 1 exp ( κ 2 / κ l 2 ) ( κ 2 + κ 0 2 ) α / 2 d κ } d ξ .
σ I , r 2 ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ( α ) C ^ n 2 L n = 1 ( Λ ρ ) 2 n n ! Γ ( n + 1 ) ( 2 n + 1 ) 0 κ 2 n + 1 exp ( κ 2 / κ l 2 ) ( κ 2 + κ 0 2 ) α / 2 d κ .
U ( a ; c ; z ) = 1 Γ ( a ) 0 e z t t a 1 ( 1 + t ) c a 1 d t ,
U ( a ; c ; z ) Γ ( 1 c ) Γ ( 1 + a c ) + Γ ( c 1 ) Γ ( a ) z 1 c , | z | 1 ,
σ I , r 2 ( ρ , α , l 0 , L 0 ) = 4 π 2 k 2 A ( α ) C ^ n 2 L n = 1 ( Λ ρ ) 2 n n ! ( 2 n + 1 ) × [ κ 0 α + 2 + 2 n Γ ( 1 n + α / 2 ) Γ ( α / 2 ) + Γ ( n α / 2 + 1 ) Γ ( n + 1 ) ( 1 κ l 2 + Λ L ξ 2 k ) 1 n + α / 2 ] .
σ I , r 2 ( ρ , α , l 0 , L 0 ) = σ I , r 1 2 ( ρ , α , l 0 , L 0 ) + σ I , r 2 2 ( ρ , α , l 0 , L 0 ) ,
σ I , r 1 2 ( ρ , α , l 0 , L 0 ) = 4 π 2 k 2 A ( α ) C ^ n 2 L n = 1 ( Λ ρ ) 2 n n ! ( 2 n + 1 ) [ κ 0 α + 2 + 2 n Γ ( 1 n + α / 2 ) Γ ( α / 2 ) ] ,
σ I , r 2 2 ( ρ , α , l 0 , L 0 ) = 4 π 2 k 2 A ( α ) C ^ n 2 L n = 1 ( Λ ρ ) 2 n n ! ( 2 n + 1 ) [ Γ ( n α / 2 + 1 ) Γ ( n + 1 ) ( 1 κ l 2 + Λ L ξ 2 k ) 1 n + α / 2 ] .
σ I , r 1 2 ( ρ , α , l 0 , L 0 ) = 4 π 2 k 2 A ( α ) C ^ n 2 L Λ 2 ρ 2 κ 0 4 α Γ ( 2 + α / 2 ) 3 Γ ( α / 2 ) ,
σ I , r 1 2 ( ρ , α , l 0 , L 0 ) = 8 Γ ( 2 + α / 2 ) 3 Γ ( α / 2 ) π 2 A ( α ) C ^ n 2 k 3 α / 2 L α / 2 Q 0 2 α / 2 Λ ρ 2 W 2 .
F 2 1 ( a , b ; c ; z ) = Γ ( c ) Γ ( b ) Γ ( c b ) 0 1 t b 1 ( 1 t ) c b 1 ( 1 t z ) a d t ,
F 2 1 ( a , b ; c ; z ) = ( 1 + z ) a F 2 1 ( a , c b ; c ; z 1 + z ) ,
σ I , r 2 2 ( ρ , α , l 0 , L 0 ) = 8 Γ ( 2 α / 2 ) 3 π 2 A ( α ) C ^ n 2 k 3 α / 2 L α / 2 × Q l 2 α / 2 F 2 1 ( 2 α 2 , 3 2 ; 5 2 ; Λ Q l ) Λ ρ 2 W 2 .
σ I , r 2 ( ρ , α , l 0 , L 0 ) = α 3 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ I _ p l 2 ( α ) Λ α / 2 1 ρ 2 W 2 [ Γ ( 2 α 2 ) × ( Λ Q l ) 2 α / 2 F 2 1 ( 2 α 2 , 3 2 ; 5 2 ; Λ Q l ) + Γ ( 2 + α / 2 ) Γ ( α / 2 ) ( Λ Q 0 ) 2 α / 2 ] ,
σ I _ p l 2 ( α ) = 8 Γ ( 1 α / 2 ) α sin ( α π 4 ) π 2 A ( α ) C ^ n 2 k 3 α / 2 L α / 2 .
σ I , l 2 ( α , l 0 , L 0 ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ , α , l 0 , L 0 ) × exp ( Λ L κ 2 ξ 2 / k ) { 1 cos [ L κ 2 k ξ ( 1 Θ ˜ ξ ) ] } d κ d ξ .
σ I , l 2 ( α , l 0 , L 0 ) = 8 π 2 k 2 L 0 1 0 κ ( κ 2 + κ 0 2 ) α / 2 { exp [ ( Λ L ξ 2 k + 1 κ l 2 ) κ 2 ] Re { exp [ ( Λ L ξ 2 k + 1 κ l 2 + i L k ξ ( 1 Θ ˜ ξ ) ) κ 2 ] } } d κ d ξ .
σ I , l 2 ( α , l 0 , L 0 ) = 4 Γ ( 1 α 2 ) π 2 A ( α ) C ^ n 2 k 3 α 2 L α 2 Q l 1 α 2 { F 2 1 ( 1 α 2 , 1 2 ; 3 2 ; Λ Q l ) Re [ 0 1 ( 1 + Λ Q l ξ 2 + i Q l ξ ( 1 Θ ˜ ξ ) ) 1 + α / 2 d ξ ] } .
σ I , l 2 ( α , l 0 , L 0 ) = 4 Γ ( 1 α 2 ) π 2 A ( α ) C ^ n 2 k 3 α 2 L α 2 Q l 1 α 2 × { F 2 1 ( 1 α 2 , 1 2 ; 3 2 ; Λ Q l ) Re [ [ 1 + 2 3 Λ Q l + i Q l ( 1 2 3 Θ ˜ ) ] α / 2 1 α 2 Q l [ 2 3 Λ + i ( 1 2 3 Θ ˜ ) ] ] } ,
σ I , l 2 ( α , l 0 , L 0 ) = 4 Γ ( 1 α 2 ) π 2 A ( α ) C ^ n 2 k 3 α 2 L α 2 Q l 1 α 2 { F 2 1 ( 1 α 2 , 1 2 ; 3 2 ; Λ Q l ) 2 α Re ( { 1 + Q l [ 2 Λ / 3 + i ( 1 + 2 Θ ) / 3 ] } α / 2 1 Q l [ 2 Λ / 3 + i ( 1 + 2 Θ ) / 3 ] ) } ,
Re ( { 1 + Q l [ 2 Λ / 3 + i ( 1 + 2 Θ ) / 3 ] } α / 2 1 Q l [ 2 Λ / 3 + i ( 1 + 2 Θ ) / 3 ] ) = Q l α / 2 1 [ ( 1 + 2 Θ ) 2 + ( 2 Λ + 3 / Q l ) ] α / 4 3 α / 2 1 [ ( 1 + 2 Θ ) 2 + 4 Λ 2 ] 1 / 2 sin ( α 2 φ 1 + φ 2 ) 6 Λ Q l [ ( 1 + 2 Θ ) 2 + 4 Λ 2 ] ,
φ 1 = tan 1 [ ( 1 + 2 Θ ) Q l 3 + 2 Λ Q l ] , φ 2 = tan 1 [ 2 Λ 1 + 2 Θ ] ,
σ I , l 2 ( α , l 0 , L 0 ) = σ I _ p l 2 ( α ) sin ( α π / 4 ) { [ ( 1 + 2 Θ ) 2 + ( 2 Λ + 3 / Q l ) 2 ] α / 4 3 α / 2 1 [ ( 1 + 2 Θ ) 2 + 4 Λ 2 ] 1 / 2 sin ( α 2 φ 1 + φ 2 ) 6 Λ Q l α / 2 [ ( 1 + 2 Θ ) 2 + 4 Λ 2 ] α 2 Q l 1 α 2 F 2 1 ( 1 α 2 , 1 2 ; 3 2 ; Λ Q l ) } .
σ I 2 ( ρ , α , l 0 , L 0 ) = σ I , r 2 ( ρ , α , l 0 , L 0 ) + σ I , l 2 ( α , l 0 , L 0 ) .
σ I 2 ( ρ , α , l 0 , L 0 ) = σ I _ p l 2 ( α ) sin ( α π / 4 ) { [ ( 1 + 2 Θ ) 2 + ( 3 / Q l ) 2 ] α / 4 3 α / 2 1 ( 1 + 2 Θ ) sin ( α 2 φ 1 + φ 2 ) α 2 Q l 1 α 2 } .

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