Abstract

Formulation of the on-axis scintillation index of a focused Gaussian beam in weak oceanic turbulence is derived by using the Rytov method, and using this formulation, the average bit error rate (BER) is evaluated. The scintillation indices of collimated, focused Gaussian, plane, and spherical beams are compared. The scintillation index and BER versus the average signal-to-noise ratio is found by using the log-normal distributed intensity for the collimated and focused Gaussian beams, which are exhibited for various source sizes αs, focal lengths Fs, rates of dissipation of the mean squared temperature χT, and rates of dissipation of the turbulent kinetic energy per unit mass of fluid ε. Focused beams are found to have important advantages over collimated beams. For the focused beam, as the source size increases, the scintillation index and BER decrease. When the focal length is equal to the propagation length, the BER is found to possess the smallest value. The BER is proportional to χT, but inversely proportional to ε.

© 2014 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, 1978).
  3. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
  4. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
  5. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
    [CrossRef]
  6. L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave,” Waves Random Media 11, 271–291 (2001).
  7. R. K. Tyson, D. E. Canning, and J. S. Tharp, “Measurement of the bit-error rate of an adaptive optics, free-space laser communications system, part 1: tip-tilt configuration, diagnostics, and closed-loop results,” Opt. Eng. 44, 096002 (2005).
    [CrossRef]
  8. N. Namazi, R. J. Burris, and G. C. Gilbreath, “Analytical approach to the calculation of probability of bit error and optimum thresholds in free-space optical communication,” Opt. Eng. 46, 025007 (2007).
    [CrossRef]
  9. H. Gerçekcioğlu and Y. Baykal, “BER of annular and flat-topped beams in non-Kolmogorov weak turbulence,” Opt. Commun. 286, 30–33 (2013).
    [CrossRef]
  10. H. Gerçekcioğlu and Y. Baykal, “BER of annular and flat-topped beams in strong turbulence,” Opt. Commun. 298–299, 18–21 (2013).
    [CrossRef]
  11. S. A. Arpali, H. T. Eyyuboğlu, and Y. Baykal, “Bit error rates for general beams,” Appl. Opt. 47, 5971–5975 (2008).
    [CrossRef]
  12. H. Gerçekcioğlu, Y. Baykal, and H. T. Eyyuboğlu, “BER of annular beams in strong turbulence,” in Applications of Lasers for Sensing and Free Space Communications (Optical Society of America, 2010), paper LSTuA4.
  13. P. V. Kumar, S. S. K. Praneeth, and R. B. Narender, “Analysis of optical wireless communication for underwater wireless communication,” Int. J. Sci. Eng. Res.2 (2011).
  14. D. J. Bogucki, J. A. Domaradzki, D. Stramski, and J. R. Zaneveld, “Comparison of near-forward light scattering on oceanic turbulence and particles,” Appl. Opt. 37, 4669–4677 (1998).
    [CrossRef]
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    [CrossRef]
  16. W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A 8, 1052–1058 (2006).
    [CrossRef]
  17. N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285, 872–875 (2012).
    [CrossRef]
  18. M. L. Holohan and J. C. Dainty, “Low-order adaptive optics: a possible use in underwater imaging,” Opt. Laser Technol. 29, 51–55 (1997).
    [CrossRef]
  19. M. Tang and D. Zhao, “Propagation of radially polarized beams in the oceanic turbulence,” Appl. Phys. B 111, 665–670 (2013).
    [CrossRef]
  20. R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. 68, 1067–1072 (1978).
    [CrossRef]
  21. W. Fu and H. Zhang, “Propagation properties of partially coherent radially polarized doughnut beam in turbulent ocean,” Opt. Commun. 304, 11–18 (2013).
    [CrossRef]
  22. E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105, 415–420 (2011).
    [CrossRef]
  23. O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012).
    [CrossRef]
  24. Ç. Arpali, C. Yazıcıoğlu, H. T. Eyyuboğlu, S. A. Arpali, and Y. Baykal, “Simulator for general-type beam propagation in turbulent atmosphere,” Opt. Express 14, 8918–8928 (2006).
    [CrossRef]
  25. Y. Baykal, “Formulation of correlations for general-type beams in atmospheric turbulence,” J. Opt. Soc. Am. A 23, 889–893 (2006).
    [CrossRef]
  26. H. T. Eyyuboğlu, H. Gerçekcioğlu, and Y. Baykal, “Minimization of scintillation index against displacement parameters,” Opt. Commun. 281, 4224–4229 (2008).
    [CrossRef]

2013

H. Gerçekcioğlu and Y. Baykal, “BER of annular and flat-topped beams in non-Kolmogorov weak turbulence,” Opt. Commun. 286, 30–33 (2013).
[CrossRef]

H. Gerçekcioğlu and Y. Baykal, “BER of annular and flat-topped beams in strong turbulence,” Opt. Commun. 298–299, 18–21 (2013).
[CrossRef]

M. Tang and D. Zhao, “Propagation of radially polarized beams in the oceanic turbulence,” Appl. Phys. B 111, 665–670 (2013).
[CrossRef]

W. Fu and H. Zhang, “Propagation properties of partially coherent radially polarized doughnut beam in turbulent ocean,” Opt. Commun. 304, 11–18 (2013).
[CrossRef]

2012

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285, 872–875 (2012).
[CrossRef]

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012).
[CrossRef]

2011

E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105, 415–420 (2011).
[CrossRef]

2010

2008

S. A. Arpali, H. T. Eyyuboğlu, and Y. Baykal, “Bit error rates for general beams,” Appl. Opt. 47, 5971–5975 (2008).
[CrossRef]

H. T. Eyyuboğlu, H. Gerçekcioğlu, and Y. Baykal, “Minimization of scintillation index against displacement parameters,” Opt. Commun. 281, 4224–4229 (2008).
[CrossRef]

2007

N. Namazi, R. J. Burris, and G. C. Gilbreath, “Analytical approach to the calculation of probability of bit error and optimum thresholds in free-space optical communication,” Opt. Eng. 46, 025007 (2007).
[CrossRef]

2006

2005

R. K. Tyson, D. E. Canning, and J. S. Tharp, “Measurement of the bit-error rate of an adaptive optics, free-space laser communications system, part 1: tip-tilt configuration, diagnostics, and closed-loop results,” Opt. Eng. 44, 096002 (2005).
[CrossRef]

2001

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave,” Waves Random Media 11, 271–291 (2001).

1999

1998

1997

M. L. Holohan and J. C. Dainty, “Low-order adaptive optics: a possible use in underwater imaging,” Opt. Laser Technol. 29, 51–55 (1997).
[CrossRef]

1978

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,” Waves Random Media 11, 271–291 (2001).

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

Andrews, L. C.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave,” Waves Random Media 11, 271–291 (2001).

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

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

Arpali, Ç.

Arpali, S. A.

Baykal, Y.

H. Gerçekcioğlu and Y. Baykal, “BER of annular and flat-topped beams in non-Kolmogorov weak turbulence,” Opt. Commun. 286, 30–33 (2013).
[CrossRef]

H. Gerçekcioğlu and Y. Baykal, “BER of annular and flat-topped beams in strong turbulence,” Opt. Commun. 298–299, 18–21 (2013).
[CrossRef]

S. A. Arpali, H. T. Eyyuboğlu, and Y. Baykal, “Bit error rates for general beams,” Appl. Opt. 47, 5971–5975 (2008).
[CrossRef]

H. T. Eyyuboğlu, H. Gerçekcioğlu, and Y. Baykal, “Minimization of scintillation index against displacement parameters,” Opt. Commun. 281, 4224–4229 (2008).
[CrossRef]

Ç. Arpali, C. Yazıcıoğlu, H. T. Eyyuboğlu, S. A. Arpali, and Y. Baykal, “Simulator for general-type beam propagation in turbulent atmosphere,” Opt. Express 14, 8918–8928 (2006).
[CrossRef]

Y. Baykal, “Formulation of correlations for general-type beams in atmospheric turbulence,” J. Opt. Soc. Am. A 23, 889–893 (2006).
[CrossRef]

H. Gerçekcioğlu, Y. Baykal, and H. T. Eyyuboğlu, “BER of annular beams in strong turbulence,” in Applications of Lasers for Sensing and Free Space Communications (Optical Society of America, 2010), paper LSTuA4.

Bogucki, D. J.

Burris, R. J.

N. Namazi, R. J. Burris, and G. C. Gilbreath, “Analytical approach to the calculation of probability of bit error and optimum thresholds in free-space optical communication,” Opt. Eng. 46, 025007 (2007).
[CrossRef]

Canning, D. E.

R. K. Tyson, D. E. Canning, and J. S. Tharp, “Measurement of the bit-error rate of an adaptive optics, free-space laser communications system, part 1: tip-tilt configuration, diagnostics, and closed-loop results,” Opt. Eng. 44, 096002 (2005).
[CrossRef]

Dainty, J. C.

M. L. Holohan and J. C. Dainty, “Low-order adaptive optics: a possible use in underwater imaging,” Opt. Laser Technol. 29, 51–55 (1997).
[CrossRef]

Domaradzki, J. A.

Eyyuboglu, H. T.

S. A. Arpali, H. T. Eyyuboğlu, and Y. Baykal, “Bit error rates for general beams,” Appl. Opt. 47, 5971–5975 (2008).
[CrossRef]

H. T. Eyyuboğlu, H. Gerçekcioğlu, and Y. Baykal, “Minimization of scintillation index against displacement parameters,” Opt. Commun. 281, 4224–4229 (2008).
[CrossRef]

Ç. Arpali, C. Yazıcıoğlu, H. T. Eyyuboğlu, S. A. Arpali, and Y. Baykal, “Simulator for general-type beam propagation in turbulent atmosphere,” Opt. Express 14, 8918–8928 (2006).
[CrossRef]

H. Gerçekcioğlu, Y. Baykal, and H. T. Eyyuboğlu, “BER of annular beams in strong turbulence,” in Applications of Lasers for Sensing and Free Space Communications (Optical Society of America, 2010), paper LSTuA4.

Farwell, N.

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285, 872–875 (2012).
[CrossRef]

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012).
[CrossRef]

E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105, 415–420 (2011).
[CrossRef]

Fu, W.

W. Fu and H. Zhang, “Propagation properties of partially coherent radially polarized doughnut beam in turbulent ocean,” Opt. Commun. 304, 11–18 (2013).
[CrossRef]

Gerçekcioglu, H.

H. Gerçekcioğlu and Y. Baykal, “BER of annular and flat-topped beams in strong turbulence,” Opt. Commun. 298–299, 18–21 (2013).
[CrossRef]

H. Gerçekcioğlu and Y. Baykal, “BER of annular and flat-topped beams in non-Kolmogorov weak turbulence,” Opt. Commun. 286, 30–33 (2013).
[CrossRef]

H. T. Eyyuboğlu, H. Gerçekcioğlu, and Y. Baykal, “Minimization of scintillation index against displacement parameters,” Opt. Commun. 281, 4224–4229 (2008).
[CrossRef]

H. Gerçekcioğlu, Y. Baykal, and H. T. Eyyuboğlu, “BER of annular beams in strong turbulence,” in Applications of Lasers for Sensing and Free Space Communications (Optical Society of America, 2010), paper LSTuA4.

Gilbreath, G. C.

N. Namazi, R. J. Burris, and G. C. Gilbreath, “Analytical approach to the calculation of probability of bit error and optimum thresholds in free-space optical communication,” Opt. Eng. 46, 025007 (2007).
[CrossRef]

Hanson, F.

Hill, R. J.

Holohan, M. L.

M. L. Holohan and J. C. Dainty, “Low-order adaptive optics: a possible use in underwater imaging,” Opt. Laser Technol. 29, 51–55 (1997).
[CrossRef]

Hopen, C. Y.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave,” Waves Random Media 11, 271–291 (2001).

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

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

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, 1978).

Korotkova, O.

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012).
[CrossRef]

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285, 872–875 (2012).
[CrossRef]

E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105, 415–420 (2011).
[CrossRef]

Kumar, P. V.

P. V. Kumar, S. S. K. Praneeth, and R. B. Narender, “Analysis of optical wireless communication for underwater wireless communication,” Int. J. Sci. Eng. Res.2 (2011).

Lasher, M.

Liu, L.

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A 8, 1052–1058 (2006).
[CrossRef]

Lu, W.

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A 8, 1052–1058 (2006).
[CrossRef]

Namazi, N.

N. Namazi, R. J. Burris, and G. C. Gilbreath, “Analytical approach to the calculation of probability of bit error and optimum thresholds in free-space optical communication,” Opt. Eng. 46, 025007 (2007).
[CrossRef]

Narender, R. B.

P. V. Kumar, S. S. K. Praneeth, and R. B. Narender, “Analysis of optical wireless communication for underwater wireless communication,” Int. J. Sci. Eng. Res.2 (2011).

Phillips, R. L.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave,” Waves Random Media 11, 271–291 (2001).

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

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

Praneeth, S. S. K.

P. V. Kumar, S. S. K. Praneeth, and R. B. Narender, “Analysis of optical wireless communication for underwater wireless communication,” Int. J. Sci. Eng. Res.2 (2011).

Shchepakina, E.

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012).
[CrossRef]

E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105, 415–420 (2011).
[CrossRef]

Stramski, D.

Sun, J.

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A 8, 1052–1058 (2006).
[CrossRef]

Tang, M.

M. Tang and D. Zhao, “Propagation of radially polarized beams in the oceanic turbulence,” Appl. Phys. B 111, 665–670 (2013).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Tharp, J. S.

R. K. Tyson, D. E. Canning, and J. S. Tharp, “Measurement of the bit-error rate of an adaptive optics, free-space laser communications system, part 1: tip-tilt configuration, diagnostics, and closed-loop results,” Opt. Eng. 44, 096002 (2005).
[CrossRef]

Tyson, R. K.

R. K. Tyson, D. E. Canning, and J. S. Tharp, “Measurement of the bit-error rate of an adaptive optics, free-space laser communications system, part 1: tip-tilt configuration, diagnostics, and closed-loop results,” Opt. Eng. 44, 096002 (2005).
[CrossRef]

Yazicioglu, C.

Zaneveld, J. R.

Zhang, H.

W. Fu and H. Zhang, “Propagation properties of partially coherent radially polarized doughnut beam in turbulent ocean,” Opt. Commun. 304, 11–18 (2013).
[CrossRef]

Zhao, D.

M. Tang and D. Zhao, “Propagation of radially polarized beams in the oceanic turbulence,” Appl. Phys. B 111, 665–670 (2013).
[CrossRef]

Appl. Opt.

Appl. Phys. B

M. Tang and D. Zhao, “Propagation of radially polarized beams in the oceanic turbulence,” Appl. Phys. B 111, 665–670 (2013).
[CrossRef]

E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105, 415–420 (2011).
[CrossRef]

J. Opt. A

W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behavior of partially coherent beams in oceanic turbulence,” J. Opt. A 8, 1052–1058 (2006).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

H. T. Eyyuboğlu, H. Gerçekcioğlu, and Y. Baykal, “Minimization of scintillation index against displacement parameters,” Opt. Commun. 281, 4224–4229 (2008).
[CrossRef]

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285, 872–875 (2012).
[CrossRef]

W. Fu and H. Zhang, “Propagation properties of partially coherent radially polarized doughnut beam in turbulent ocean,” Opt. Commun. 304, 11–18 (2013).
[CrossRef]

H. Gerçekcioğlu and Y. Baykal, “BER of annular and flat-topped beams in non-Kolmogorov weak turbulence,” Opt. Commun. 286, 30–33 (2013).
[CrossRef]

H. Gerçekcioğlu and Y. Baykal, “BER of annular and flat-topped beams in strong turbulence,” Opt. Commun. 298–299, 18–21 (2013).
[CrossRef]

Opt. Eng.

R. K. Tyson, D. E. Canning, and J. S. Tharp, “Measurement of the bit-error rate of an adaptive optics, free-space laser communications system, part 1: tip-tilt configuration, diagnostics, and closed-loop results,” Opt. Eng. 44, 096002 (2005).
[CrossRef]

N. Namazi, R. J. Burris, and G. C. Gilbreath, “Analytical approach to the calculation of probability of bit error and optimum thresholds in free-space optical communication,” Opt. Eng. 46, 025007 (2007).
[CrossRef]

Opt. Express

Opt. Laser Technol.

M. L. Holohan and J. C. Dainty, “Low-order adaptive optics: a possible use in underwater imaging,” Opt. Laser Technol. 29, 51–55 (1997).
[CrossRef]

Waves Random Complex Media

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012).
[CrossRef]

Waves Random Media

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave,” Waves Random Media 11, 271–291 (2001).

Other

H. Gerçekcioğlu, Y. Baykal, and H. T. Eyyuboğlu, “BER of annular beams in strong turbulence,” in Applications of Lasers for Sensing and Free Space Communications (Optical Society of America, 2010), paper LSTuA4.

P. V. Kumar, S. S. K. Praneeth, and R. B. Narender, “Analysis of optical wireless communication for underwater wireless communication,” Int. J. Sci. Eng. Res.2 (2011).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, 1978).

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

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Figures (9)

Fig. 1.
Fig. 1.

Scintillation index versus propagation distance L for various source sizes αs.

Fig. 2.
Fig. 2.

Scintillation index versus propagation distance L for the focused Gaussian beams of various source sizes αs.

Fig. 3.
Fig. 3.

Scintillation index versus source size αs for various focal lengths Fs.

Fig. 4.
Fig. 4.

Scintillation index versus source size αs for various propagation distances L.

Fig. 5.
Fig. 5.

BER versus SNR.

Fig. 6.
Fig. 6.

BER versus source size αs for various focal lengths Fs.

Fig. 7.
Fig. 7.

BER versus SNR for collimated and focused beams at various rates of dissipation of turbulent kinetic energy per unit mass of fluid ε.

Fig. 8.
Fig. 8.

BER versus SNR for collimated and focused beams at various rates of dissipation of the mean squared refractive index fluctuation χn.

Fig. 9.
Fig. 9.

BER versus SNR at various source sizes αs.

Equations (12)

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

Φn(κ)=(4π)1βχnε1/3κ11/3[1+Q(κηs)2/3][w2θ+1w(1+θ)]1×(w2θexp{βPrT1[23(κηs)4/3+Q(κηs)2]}+exp{βPrS1[23(κηs)4/3+Q(κηs)2]}w(1+θ)exp{0.5β(PrT+PrS)PrTPrS[23(κηs)4/3+Q(κηs)2]}),
m2=4πRe{0Ldη0κdκ02πdϕ[GA1(η,κ,ϕ)+GA2(η,κ,ϕ)]Φn(κ)},
GA1(η,κ,ϕ)=k2T(L)2A2(1+iαL)2exp[i(Lη)k1(1+iαη)(1+iαL)1κ2],
GA2(η,κ,ϕ)=k2|T(L)|2AA*(1+iαL)1(1iα*L)1×exp{0.5i(Lη)k1[(1+iαη)(1+iαL)1(1iα*η)(1iα*L)1]κ2},
m2=8π2k2Re{|T(L)|2AA*(1+iαL)1(1iα*L)10Ldη(AA1+BB1)T(L)2A2(1+iαL)20Ldη(AA2+BB2)},
AA1=(8π)1Kk1{w2θk=0(1)kAT1kΓ(2k/35/6)k![AT2+i(Lη)2k(1+iαη1+iαL1iα*η1iα*L)]2k/3+5/6w(1+θ)k=0(1)kATS1kΓ(2k/35/6)k![ATS2+i(Lη)2k(1+iαη1+iαL1iα*η1iα*L)]2k/3+5/6+k=0(1)kAS1kΓ(2k/35/6)k![AS2+i(Lη)2k(1+iαη1+iαL1iα*η1iα*L)]2k/3+5/6},
BB1=(8π)1Kk2{w2θk=0(1)kAT1kΓ(2k/31/2)k![AT2+i(Lη)2k(1+iαη1+iαL1iα*η1iα*L)]2k/3+1/2w(1+θ)k=0(1)kATS1kΓ(2k/31/2)k![ATS2+i(Lη)2k(1+iαη1+iαL1iα*η1iα*L)]2k/3+1/2+k=0(1)kAS1kΓ(2k/31/2)k![AS2+i(Lη)2k(1+iαη1+iαL1iα*η1iα*L)]2k/3+1/2},
AA2=(8π)1Kk1{w2θk=0(1)kAT1kΓ(2k/35/6)k![AT2+i(Lη)k1+iαη1+iαL]2k/3+5/6+k=0(1)kΓ(2k/35/6)k!AS1k[AS2+i(Lη)2k1+iαη1+iαL]2k/3+5/6w(1+θ)k=0(1)kATS1kΓ(2k/35/6)k![ATS2+i(Lη)k1+iαη1+iαL]2k/3+5/6},
BB2=(8π)1Kk2{w2θk=0(1)kk!AT1kΓ(2k/31/2)[AT2+i(Lη)k1+iαη1+iαL]2k/3+1/2+k=0(1)kAS1kΓ(2k/31/2)k![AS2+i(Lη)k1+iαη1+iαL]2k/3+1/2w(1+θ)k=0(1)kATS1Γ(2k/31/2)kk![ATS2+i(Lη)k1+iαη1+iαL]2k/3+1/2}.
Kk1=βχnε1/3[w2θ+1w(1+θ)]1,Kk2=βχnε1/3Qηs2/3[w2θ+1w(1+θ)]1,AT1=23ηs4/3βPrT1,AT2=βPrT1Qηs2,AS1=23ηs4/3βPrS1,AS2=βPrS1Qηs2,ATS1=13ηs4/3β(PrT+PrS)PrT1PrS1,andATS2=12Qηs2β(PrT+PrS)PrT1PrS1.
pI(u)=1m2πuexp{0.5m2[ln(u)+12m2]2}.
BER=0.50pI(u)erfc(SNR22u)du,

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