Abstract

By use of the generalized von Kármán spectrum model that features both inner scale and outer scale parameters for non-Kolmogorov turbulence and the extended Rytov method that incorporates a modified amplitude spatial-frequency filter function under strong-fluctuation conditions, theoretical expressions are developed for the scintillation index of a horizontally propagating plane wave and spherical wave that are valid under moderate-to-strong irradiance fluctuations. Numerical results show that the obtained expressions also compare well with previous results in weak-fluctuation regimes. Based on these general models, the impacts of finite inner and outer scales on the scintillation index of an optical wave are examined under various non-Kolmogorov fluctuation conditions.

© 2012 OSA

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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. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283(7), 1229–1235 (2010).
    [CrossRef]
  3. H. G. Sandalidis, “Performance of a laser Earth-to-satellite link over turbulence and beam wander using the modulated gamma-gamma irradiance distribution,” Appl. Opt. 50(6), 952–961 (2011).
    [CrossRef] [PubMed]
  4. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE Optical Engineering Press, 2005).
    [CrossRef]
  5. 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]
  6. 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]
  7. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16(6), 1417–1429 (1999).
    [CrossRef]
  8. K. J. Mayer, Effect of Inner Scale Atmospheric Spectrum Models on Scintillation in All Optical Turbulence Regimes, Ph.D. dissertation, University of Central Florida, 2007.
  9. 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]
  10. 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]
  11. 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]
  12. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6457, 64570T–1–11 (2007).
  13. 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]
  14. 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).
    [CrossRef] [PubMed]
  15. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007).
    [CrossRef]
  16. P. Deng, X. H. Yuan, and D. X. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun., accepted for publication.
  17. J. Cang and X. Liu, “Average capacity of free-space optical systems for a partially coherent beam propagating through non-Kolmogorov turbulence,” Opt. Lett. 36(7), 3335–3337 (2011).
    [CrossRef] [PubMed]
  18. 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–1–12 (2007).
  19. 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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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  23. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 1–17 (1995).
  24. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, Bellingham, Wash., 1998).
  25. R. Mahon, C. I. Moore, H. R. Burris, W. S. Rabinovich, M. Stell, M. R. Suite, and L. M. Thomas, “Analysis of long-term measurements of laser propagation over the Chesapeake Bay,” Appl. Opt. 48(12), 2388–2400 (2009).
    [CrossRef] [PubMed]

2011 (5)

2010 (3)

2009 (2)

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]

R. Mahon, C. I. Moore, H. R. Burris, W. S. Rabinovich, M. Stell, M. R. Suite, and L. M. Thomas, “Analysis of long-term measurements of laser propagation over the Chesapeake Bay,” Appl. Opt. 48(12), 2388–2400 (2009).
[CrossRef] [PubMed]

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 (3)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6457, 64570T–1–11 (2007).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (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–1–12 (2007).

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]

1999 (1)

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 (1)

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 1–17 (1995).

1994 (2)

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]

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]

1993 (1)

Al-Habash, M. A.

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, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (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,” Proc. SPIE 6457, 64570T–1–11 (2007).

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–1–12 (2007).

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, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16(6), 1417–1429 (1999).
[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, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, Bellingham, Wash., 1998).

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

Bai, X. Z.

Belen’kii, M. S.

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]

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]

Burris, H. R.

Cang, J.

Cao, L.

Cao, X. G.

Cui, L. Y.

Deng, P.

P. Deng, X. H. Yuan, and D. X. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun., accepted for publication.

Dong, J. K.

Du, W.

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, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (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,” Proc. SPIE 6457, 64570T–1–11 (2007).

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–1–12 (2007).

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]

Golbraikh, E.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283(7), 1229–1235 (2010).
[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(1), 66–77 (2008).
[CrossRef]

Han, Q.

Hopen, C. Y.

Huang, D. X.

P. Deng, X. H. Yuan, and D. X. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun., accepted for publication.

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, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283(7), 1229–1235 (2010).
[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(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]

Liu, X.

Ma, J.

Mahon, R.

Mayer, K. J.

K. J. Mayer, Effect of Inner Scale Atmospheric Spectrum Models on Scintillation in All Optical Turbulence Regimes, Ph.D. dissertation, University of Central Florida, 2007.

Miller, W. B.

Moore, C. I.

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, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (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,” Proc. SPIE 6457, 64570T–1–11 (2007).

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–1–12 (2007).

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, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16(6), 1417–1429 (1999).
[CrossRef]

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

Rabinovich, W. S.

Ricklin, J. C.

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 1–17 (1995).

Sandalidis, H. G.

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]

Stell, M.

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 1–17 (1995).

Suite, M. R.

Tan, L.

Thomas, L. M.

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, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (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,” Proc. SPIE 6457, 64570T–1–11 (2007).

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–1–12 (2007).

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.

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 1–17 (1995).

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. D.

Xue, W. F.

Yu, S.

Yuan, X. H.

P. Deng, X. H. Yuan, and D. X. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun., accepted for publication.

Zheng, S. L.

Zhou, F. G.

Zilberman, A.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283(7), 1229–1235 (2010).
[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(1), 66–77 (2008).
[CrossRef]

Appl. Opt. (2)

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. Opt. Soc. Am. A (4)

Opt. Commun. (1)

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283(7), 1229–1235 (2010).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Proc. SPIE (7)

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–1–12 (2007).

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 1–17 (1995).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (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,” Proc. SPIE 6457, 64570T–1–11 (2007).

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]

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]

Other (4)

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

K. J. Mayer, Effect of Inner Scale Atmospheric Spectrum Models on Scintillation in All Optical Turbulence Regimes, Ph.D. dissertation, University of Central Florida, 2007.

P. Deng, X. H. Yuan, and D. X. Huang, “Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence,” Opt. Commun., accepted for publication.

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

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Figures (3)

Fig. 1
Fig. 1

Scintillation index vs. path length, (a) plane wave model, (b) spherical wave model.

Fig. 3
Fig. 3

Spherical wave scintillation vs.L.

Fig. 2
Fig. 2

Plane wave scintillation vs.α.

Equations (65)

Equations on this page are rendered with MathJax. Learn more.

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 ,
σ ˜ R 2 = 8 π 2 A ( α ) Γ ( 1 α / 2 ) α 1 sin ( α π / 4 ) C ˜ n 2 k 3 α / 2 L α / 2
L / k ρ p l 2 ( α ) 1 , weak - fluctuation conditions , L / k ρ p l 2 ( α ) 1 , strong - fluctuation conditions .
D p l ( ρ , α ) = 8 π 2 k 2 L 0 κ Φ n ( κ , α ) [ 1 J 0 ( κ ρ ) ] d κ ,
D p l ( ρ , α ) = 8 π 2 A ( α ) C ˜ n 2 k 2 L n = 1 ( 1 ) n 1 ( ρ ) 2 n 2 2 n ( n ! ) 2 0 κ 2 n + 1 exp ( κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) α / 2 d κ .
D p l ( ρ , α ) = 4 π 2 A ( α ) C ˜ n 2 k 2 L κ 0 2 α n = 1 ( 1 ) n 1 n ! ( κ 0 ρ / 2 ) 2 n U ( n + 1 ; n + 2 α 2 ; κ 0 2 κ m 2 ) ,
D p l ( ρ , α ) π 2 A ( α ) C ˜ n 2 k 2 L κ 0 4 α U ( 2 ; 3 α 2 ; κ 0 2 κ m 2 ) ρ 2 , ρ l 0 .
D p l ( ρ , α ) π 2 A ( α ) Γ ( 2 α / 2 ) C ˜ n 2 k 2 L κ m 4 α ρ 2 , ρ l 0 .
ρ p l ( α ) = [ 0.5 π 2 A ( α ) Γ ( 2 α / 2 ) C ˜ n 2 k 2 L κ m 4 α ] 1 / 2 , ρ p l ( α ) l 0 .
D S , p l ( ρ , α ) = 4 π 2 k 2 L 0 1 0 κ Φ n ( κ , α ) [ 1 J 0 ( κ ρ ) ] [ 1 + cos ( L κ 2 ξ k ) ] d κ d ξ ,
D S , p l ( ρ , α ) = π 2 k 2 L ρ 2 0 1 0 κ 3 Φ n ( κ , α ) [ 1 + cos ( L κ 2 ξ k ) ] d κ d ξ .
D S , p l ( ρ , α ) = A ( α ) π 2 C ˜ n 2 k 2 L ρ 2 Re 0 1 0 κ 3 ( κ 2 + κ 0 2 ) α / 2 × { exp ( κ 2 κ m 2 ) + exp [ κ 2 ( 1 + i Q m ( α ) ξ ) κ m 2 ] } d κ d ξ ,
D S , p l ( ρ , α ) = 0.5 Γ ( 2 ) A ( α ) π 2 C ˜ n 2 k 2 L ρ 2 Re 0 1 { κ 0 4 α U ( 2 ; 3 α 2 ; κ 0 2 κ m 2 ) + κ 0 4 α U ( 2 ; 3 α 2 ; κ 0 2 ( 1 + i Q m ( α ) ξ ) κ m 2 ) } d ξ .
D S , p l ( ρ , α ) = 0.5 A ( α ) π 2 Γ ( 2 α 2 ) C ˜ n 2 k 2 L κ m 4 α ρ 2 Re 0 1 { 1 + ( 1 + i Q m ( α ) ξ ) α 2 2 } d ξ = A ( α ) π 2 Γ ( 2 α 2 ) C ˜ n 2 k 2 L κ m 4 α ρ 2 Re { 1 2 + [ 1 + i Q m ( α ) ] α / 2 1 1 i Q m ( α ) ( α 2 ) } .
D S , p l ( ρ , α ) = 0.5 Ξ A ( α ) π 2 Γ ( 2 α / 2 ) C ˜ n 2 k 2 L κ m 4 α ρ 2 , ρ l 0 ,
Ξ = 1 + 2 sin [ α 2 2 tan 1 Q m ( α ) ] ( α 2 ) [ Q m ( α ) ] 2 α / 2 { 1 + 1 [ Q m ( α ) ] 2 } α 2 4 .
σ I 2 ( α ) = 1 + 32 π 2 k 2 L 0 1 0 κ Φ n ( κ , α ) sin 2 [ L κ 2 ξ 2 k ] × exp { 0 1 D S , p l [ L κ k w ( τ , ξ ) , α ] d τ } d κ d ξ , L / k ρ p l 2 ( α ) 1 ,
w ( τ , ξ ) = { τ ( 1 Θ ¯ ξ ) , τ < ξ , ξ ( 1 Θ ¯ τ ) , τ > ξ ,
0 1 D S , p l [ L κ k w ( τ , ξ ) , α ] d τ = α Γ ( 2 α / 2 ) Ξ 16 Γ ( 1 α / 2 ) sin ( α π / 4 ) × σ ˜ R 2 [ Q m ( α ) ] 2 α / 2 ( L κ 2 k ) ξ 2 ( 1 2 3 ξ ) .
sin 2 [ L κ 2 ξ 2 k ] L 2 κ 4 ξ 2 4 k 2 .
σ I , p l 2 ( α ) = 1 α Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 ( L k ) 3 α / 2 0 1 ξ 2 0 κ 5 ( κ 2 + κ 0 2 ) α / 2 × exp { κ 2 κ m 2 { 1 α Γ ( 2 α / 2 ) Ξ 16 Γ ( 1 α / 2 ) sin ( α π / 4 ) × σ ˜ R 2 [ Q m ( α ) ] 3 α 2 ξ 2 ( 1 2 3 ξ ) } } d κ d ξ .
σ I , p l 2 ( α ) = 1 α 2 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 ( L k ) 3 α / 2 0 1 ξ 2 κ 0 6 α Γ ( 3 ) × U ( 3 ; 4 α 2 ; κ 0 2 κ m 2 { 1 α Γ ( 2 α / 2 ) Ξ 16 Γ ( 1 α / 2 ) sin ( α π / 4 ) × σ ˜ R 2 [ Q m ( α ) ] 3 α 2 ξ 2 ( 1 2 3 ξ ) } ) d ξ .
σ I , p l 2 ( α ) = 1 α Γ ( 3 α / 2 ) 2 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 [ Q m ( α ) ] 3 α / 2 × 0 1 ξ 2 { 1 α Γ ( 2 α / 2 ) Ξ 16 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 [ Q m ( α ) ] 3 α 2 ξ 2 ( 1 2 3 ξ ) } 3 α / 2 d ξ .
σ I , p l 2 ( α ) = 1 + C p l ( α ) { σ ˜ R 2 [ Q m ( α ) ] 3 α / 2 } 2 α / 2 , α Γ ( 2 α / 2 ) σ ˜ R 2 [ Q m ( α ) ] 2 α / 2 16 Γ ( 1 α / 2 ) sin ( α π / 4 ) 1 ,
C p l ( α ) = α Γ ( 3 α / 2 ) 2 Γ ( 1 α / 2 ) sin ( α π / 4 ) × 0 1 ξ 2 { σ ˜ R 2 [ Q m ( α ) ] α / 2 3 α Γ ( 2 α / 2 ) Ξ 16 Γ ( 1 α / 2 ) sin ( α π / 4 ) ξ 2 ( 1 2 3 ξ ) } 3 α / 2 d ξ .
0 1 D S , p l [ L κ k w ( τ , ξ ) ] d τ = α Γ ( 2 α / 2 ) Ξ 48 Γ ( 1 α / 2 ) sin ( α π / 4 ) × σ ˜ R 2 [ Q m ( α ) ] 2 α / 2 ( L κ 2 k ) ξ 2 ( 1 ξ 2 ) ,
σ I , s p 2 ( α ) = 1 α Γ ( 3 α / 2 ) 2 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 [ Q m ( α ) ] 3 α / 2 × 0 1 ξ 2 ( 1 ξ ) 2 { 1 α Γ ( 2 α / 2 ) Ξ 48 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 [ Q m ( α ) ] 3 α 2 ξ 2 ( 1 ξ 2 ) } 3 α / 2 d ξ .
σ I , s p 2 ( α ) = 1 + C s p ( α ) { σ ˜ R 2 [ Q m ( α ) ] 3 α / 2 } 2 α / 2 , α Γ ( 2 α / 2 ) σ ˜ R 2 [ Q m ( α ) ] 2 α / 2 16 Γ ( 1 α / 2 ) sin ( α π / 4 ) 1 ,
C s p ( α ) = α Γ ( 3 α / 2 ) 2 Γ ( 1 α / 2 ) sin ( α π / 4 ) × 0 1 ξ 2 ( 1 ξ ) 2 { σ ˜ R 2 [ Q m ( α ) ] α / 2 3 α Γ ( 2 α / 2 ) Ξ 48 Γ ( 1 α / 2 ) sin ( α π / 4 ) ξ 2 ( 1 ξ ) 2 } 3 α / 2 d ξ .
Φ ˜ n , e ( κ , α ) = Φ ˜ n ( κ , α ) G ( κ , l 0 , L 0 , α ) = Φ ˜ n ( κ , α ) [ G X ( κ , l 0 , L 0 , α ) + G Y ( κ , α ) ] ,
G X ( κ , l 0 , L 0 , α ) = f ( κ , l 0 , α ) g ( κ L 0 ) exp ( κ 2 κ X 2 ) ,
G Y ( κ , α ) = κ α ( κ 2 + κ Y 2 ) α / 2 ,
G X ( κ , l 0 , L 0 , α ) = exp ( κ 2 κ m 2 ) [ exp ( κ 2 κ X 2 ) exp ( κ 2 κ X 0 2 ) ] ,
σ ln X 2 ( l 0 , L 0 , α ) = σ ln X 2 ( l 0 , α ) σ ln X 2 ( L 0 , α ) .
σ I 2 ( α ) = exp [ σ ln X 2 ( l 0 , α ) σ ln X 2 ( L 0 , α ) + σ ln Y 2 ( l 0 , α ) ] 1 ,
σ ln X , p l 2 ( l 0 , α ) = 8 π 2 k 2 L 0 1 0 κ Φ ˜ n ( κ , α ) exp ( κ 2 κ m 2 ) exp ( κ 2 κ X 2 ) × { 1 cos [ L κ 2 k ξ ] } d κ d ξ α 4 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 0 1 ξ 2 d ξ × 0 η 2 α / 2 exp ( η Q m ( α ) η η X ) d η ,
1 cos [ L κ 2 k ξ ] 1 2 [ L κ 2 k ξ ] 2 , κ κ X ,
σ ln X , p l 2 ( l 0 , α ) = α Γ ( 3 α / 2 ) 12 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 ( η X Q m ( α ) η X + Q m ( α ) ) 3 α / 2 .
σ ln X , p l 2 ( α ) = α Γ ( 3 α / 2 ) 12 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 ( η X ) 3 α / 2 ,
η X = 1 c 1 ( α ) + c 2 ( α ) L / k ρ p l 2 ( α ) { 1 / c 1 ( α ) , L / k ρ p l 2 ( α ) 1 , k ρ p l 2 ( α ) / c 2 ( α ) L L / k ρ p l 2 ( α ) 1 ,
σ ˜ P L 2 = σ ˜ R 2 sin ( α π / 4 ) { [ 1 + 1 Q m 2 ( α ) ] α / 4 sin [ α 2 tan 1 Q m ( α ) ] α 2 [ Q m ( α ) ] 1 α 2 } , σ ˜ R 2 < 1 ,
c 1 ( α ) = [ 5.88 Γ ( 1 α / 2 ) sin ( α π / 4 ) α Γ ( 3 α / 2 ) ] 2 α 6 .
σ ln X , p l 2 ( α ) C p l ( α ) 2 { σ ˜ R 2 [ Q m ( α ) ] 3 α / 2 } 2 α / 2 , α Γ ( 2 α / 2 ) σ ˜ R 2 [ Q m ( α ) ] 2 α / 2 16 Γ ( 1 α / 2 ) sin ( α π / 4 ) 1 .
c 2 ( α ) L k ρ p l 2 ( α ) = { C p l ( α ) 2 { σ ˜ R 2 [ Q m ( α ) ] 3 α / 2 } 2 α / 2 12 Γ ( 1 α / 2 ) sin ( α π / 4 ) α Γ ( 3 α / 2 ) σ ˜ R 2 } 2 α 6
η X = 1 / c 1 ( α ) 1 + [ 1.02 C p l ( α ) ] 2 α 6 σ ˜ R 2 [ Q m ( α ) ] 2 α 2 .
σ ln X , p l 2 ( l 0 , α ) = α Γ ( 3 α / 2 ) 12 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 × { Q m ( α ) 1 + c 1 ( α ) Q m ( α ) + c 1 ( α ) [ 1.02 C p l ( α ) ] 2 α 6 σ ˜ R 2 [ Q m ( α ) ] 3 α 2 } 3 α / 2
σ ln X , p l 2 ( L 0 , α ) = α Γ ( 3 α / 2 ) 12 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 ( η X 0 Q m ( α ) η X 0 + Q m ( α ) ) 3 α / 2 ,
η X 0 = η X Q 0 η X + Q 0 = Q 0 1 + c 1 ( α ) Q 0 + c 1 ( α ) [ 1.02 C p l ( α ) ] 2 α 6 σ ˜ R 2 Q 0 [ Q m ( α ) ] 2 α 2 ,
σ ln Y , p l 2 ( l 0 , α ) = α ( α 2 ) Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 η Y 1 α / 2 ,
η Y = c 3 ( α ) + c 4 ( α ) L / k ρ p l 2 ( α ) { c 3 ( α ) , L / k ρ p l 2 ( α ) 1 , c 4 ( α ) L / k ρ p l 2 ( α ) , L / k ρ p l 2 ( α ) 1 ,
η Y = [ α 0.51 ( α 2 ) Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 σ ˜ P L 2 ] 2 α 2 ( 1 + 0.73 2 α 2 σ ˜ P L 4 α 2 ) .
σ ln Y , p l 2 ( l 0 , α ) = 0.51 σ ˜ P L 2 ( 1 + 0.73 2 α 2 σ ˜ P L 4 α 2 ) 1 α / 2 .
σ I , p l 2 ( α ) = exp [ σ ln X , p l 2 ( l 0 , α ) σ ln X , p l 2 ( L 0 , α ) + σ ln Y , p l 2 ( l 0 , α ) ] 1 , 0 σ ˜ R 2 < .
σ ˜ S P 2 = σ ˜ R 2 sin ( α π / 4 ) { 3 1 α 2 [ 1 + 9 Q m 2 ( α ) ] α / 4 sin [ α 2 tan 1 ( Q m ( α ) 3 ) ] α 2 [ Q m ( α ) ] 1 α 2 } , σ ˜ R 2 < 1 .
σ ln X , s p 2 ( l 0 , α ) = 8 π 2 k 2 L 0 1 0 κ Φ ˜ n ( κ , α ) exp ( κ 2 κ m 2 ) exp ( κ 2 κ X 2 ) × { 1 cos [ L κ 2 k ξ ( 1 ξ ) ] } d κ d ξ α 4 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 0 1 ξ 2 ( 1 ξ 2 ) d ξ × 0 η 2 α / 2 exp ( η Q m ( α ) η η X ) d η = α Γ ( 3 α / 2 ) 120 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 ( η X Q m ( α ) η X + Q m ( α ) ) 3 α / 2 .
σ ln X , s p 2 ( α ) = α Γ ( 3 α / 2 ) 120 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 ( η X ) 3 α / 2 .
σ ln X , s p 2 ( α ) { 0.49 β ˜ 0 2 , σ ˜ R 2 < 1 , C s p ( α ) 2 { σ ˜ R 2 [ Q m ( α ) ] 3 α / 2 } 2 α / 2 , α Γ ( 2 α / 2 ) σ ˜ R 2 [ Q m ( α ) ] 2 α / 2 16 Γ ( 1 α / 2 ) sin ( α π / 4 ) 1 ,
η X = 1 / d 1 ( α ) 1 + Δ ( α ) ,
d 1 ( α ) = [ 58.8 Γ ( 1 α / 2 ) sin ( α π / 4 ) Γ ( α / 2 ) Γ ( 1 + α / 2 ) α Γ ( 3 α / 2 ) Γ ( α ) ] 2 α 6 ,
Δ ( α ) = [ 1.02 Γ ( α ) C s p ( α ) Γ ( α / 2 ) Γ ( 1 + α / 2 ) ] 2 α 6 σ ˜ R 2 [ Q m ( α ) ] 2 α 2 .
σ ln X , s p 2 ( l 0 , α ) = α Γ ( 3 α / 2 ) 120 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 × ( Q m ( α ) 1 + d 1 ( α ) Q m ( α ) + d 1 ( α ) Δ ( α ) Q m ( α ) ) 3 α / 2 .
σ ln X , s p 2 ( L 0 , α ) = α Γ ( 3 α / 2 ) 120 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 × { Q 0 Q m ( α ) Q 0 + Q m ( α ) + d 1 ( α ) Q 0 Q m ( α ) [ 1 + Δ ( α ) ] } 3 α / 2 .
η Y = [ α 0.51 ( α 2 ) Γ ( 1 α / 2 ) sin ( α π / 4 ) σ ˜ R 2 σ ˜ S P 2 ] 2 α 2 ( 1 + 0.73 2 α 2 σ ˜ S P 4 α 2 ) .
σ ln Y , s p 2 ( l 0 , α ) 0.51 σ ˜ S P 2 ( 1 + 0.73 2 α 2 σ ˜ S P 4 α 2 ) 1 α / 2 .
σ I , s p 2 ( α ) = exp [ σ ln X , s p 2 ( l 0 , α ) σ ln X , s p 2 ( L 0 , α ) + σ ln Y , s p 2 ( l 0 , α ) ] 1 , 0 σ ˜ R 2 < .

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