Abstract

The probability density function (PDF) of aperture-averaged irradiance fluctuations is calculated from wave-optics simulations of a laser after propagating through atmospheric turbulence to investigate the evolution of the distribution as the aperture diameter is increased. The simulation data distribution is compared to theoretical gamma-gamma and lognormal PDF models under a variety of scintillation regimes from weak to strong. Results show that under weak scintillation conditions both the gamma- gamma and lognormal PDF models provide a good fit to the simulation data for all aperture sizes studied. Our results indicate that in moderate scintillation the gamma-gamma PDF provides a better fit to the simulation data than the lognormal PDF for all aperture sizes studied. In the strong scintillation regime, the simulation data distribution is gamma gamma for aperture sizes much smaller than the coherence radius ρ0 and lognormal for aperture sizes on the order of ρ0 and larger. Examples of how these results affect the bit-error rate of an on–off keyed free space optical communication link are presented.

© 2009 Optical Society of America

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References

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  1. S. M. Flatté, C. Bracher, and G. Wang, “Probability density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulation,” J. Opt. Soc. Am. A 11, 2080-2092 (1994).
    [CrossRef]
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    [CrossRef]
  3. J. H. Churnside and S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923-1930 (1987).
    [CrossRef]
  4. J. H. Churnside and R. G. Frehlich, “Experimental evaluation of log-normally modulated Rician and IK models of optical scintillation in the atmosphere,” J. Opt. Soc. Am. A 6, 1760-1766 (1989).
    [CrossRef]
  5. J. H. Churnside, “Joint probability-density function of irradiance scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 6, 1931-1940 (1989).
    [CrossRef]
  6. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum,” Waves Random Media 10, 53-70 (2000).
    [CrossRef]
  7. M. A. Al-Habush, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance PDF of a laser beam propagation through turbulent media,” Opt. Eng. 40, 1554-1562(2001).
    [CrossRef]
  8. M. A. Al-Habush, L. C. Andrews, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Proc. SPIE 4678, 91-101 (2002).
    [CrossRef]
  9. R. K. Tyson and D. E. Canning, “Indirect measurement of a laser communications bit-error-rate reduction with low-order adaptive optics,” Appl. Opt. 42, 4239-4243 (2003).
    [CrossRef] [PubMed]
  10. F. S. Vetelino, C. Young, L. C. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence,” Appl. Opt. 46, 2099-2108 (2007).
    [CrossRef] [PubMed]
  11. J. H. Churnside and R. J. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 727-733 (1987).
    [CrossRef]
  12. J. H. Churnside, “Aperture averaging of optical scintillations in the turbulent atmosphere,” Appl. Opt. 30, 1982-1994(1991).
    [CrossRef] [PubMed]
  13. Z. Azar, M. Loebenstein, G. Applebaum, E. Azoulay, U. Halavee, M. Tamir, and M. Tur, “Aperture averaging of the two-wavelength intensity covariance function in atmospheric turbulence,” Appl. Opt. 24, 2401-2407 (1985).
    [CrossRef] [PubMed]
  14. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
    [CrossRef]
  15. F. S. Vetelino, B. Clare, K. Corbett, C. Young, K. Grant, and L. C. Andrews, “Characterizing the propagation path in moderate to strong optical turbulence,” Appl. Opt. 45, 3534-3543 (2006).
    [CrossRef] [PubMed]
  16. F. S. Vetelino, C. Young, and L. C. Andrews, “Fade statistics and aperture averaging for Gaussian beam waves in moderate-to-strong turbulence,” Appl. Opt. 46, 3780-3789(2007).
    [CrossRef] [PubMed]
  17. M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC, 1996).
  18. R. Rao, “Statistics of the fractal structure and phase singularity of a plane light wave propagation in atmospheric turbulence,” Appl. Opt. 47, 269-276 (2008).
    [CrossRef] [PubMed]
  19. A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39, 5426-5445 (2000).
    [CrossRef]
  20. T. J. Brennan and P. H. Roberts, AOTools The Adaptive Toolbox Users Guide (The Optical Sciences Company, 2006), http://www.tosc.com/downloads/AOToolsUG.pdf.

2008 (1)

2007 (2)

2006 (1)

2003 (1)

2002 (1)

M. A. Al-Habush, L. C. Andrews, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Proc. SPIE 4678, 91-101 (2002).
[CrossRef]

2001 (1)

M. A. Al-Habush, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance PDF of a laser beam propagation through turbulent media,” Opt. Eng. 40, 1554-1562(2001).
[CrossRef]

2000 (2)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum,” Waves Random Media 10, 53-70 (2000).
[CrossRef]

A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39, 5426-5445 (2000).
[CrossRef]

1997 (1)

1994 (1)

1991 (1)

1989 (2)

1987 (2)

1985 (1)

Al-Habush, M. A.

M. A. Al-Habush, L. C. Andrews, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Proc. SPIE 4678, 91-101 (2002).
[CrossRef]

M. A. Al-Habush, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance PDF of a laser beam propagation through turbulent media,” Opt. Eng. 40, 1554-1562(2001).
[CrossRef]

Andrews, L. C.

F. S. Vetelino, C. Young, L. C. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence,” Appl. Opt. 46, 2099-2108 (2007).
[CrossRef] [PubMed]

F. S. Vetelino, C. Young, and L. C. Andrews, “Fade statistics and aperture averaging for Gaussian beam waves in moderate-to-strong turbulence,” Appl. Opt. 46, 3780-3789(2007).
[CrossRef] [PubMed]

F. S. Vetelino, B. Clare, K. Corbett, C. Young, K. Grant, and L. C. Andrews, “Characterizing the propagation path in moderate to strong optical turbulence,” Appl. Opt. 45, 3534-3543 (2006).
[CrossRef] [PubMed]

M. A. Al-Habush, L. C. Andrews, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Proc. SPIE 4678, 91-101 (2002).
[CrossRef]

M. A. Al-Habush, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance PDF of a laser beam propagation through turbulent media,” Opt. Eng. 40, 1554-1562(2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum,” Waves Random Media 10, 53-70 (2000).
[CrossRef]

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

Applebaum, G.

Azar, Z.

Azoulay, E.

Belmonte, A.

Bracher, C.

Brennan, T. J.

T. J. Brennan and P. H. Roberts, AOTools The Adaptive Toolbox Users Guide (The Optical Sciences Company, 2006), http://www.tosc.com/downloads/AOToolsUG.pdf.

Canning, D. E.

Churnside, J. H.

Clare, B.

Clifford, S. F.

Corbett, K.

Flatté, S. M.

Frehlich, R. G.

Grant, K.

Halavee, U.

Hill, R. J.

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum,” Waves Random Media 10, 53-70 (2000).
[CrossRef]

Loebenstein, M.

Phillips, R. L.

M. A. Al-Habush, L. C. Andrews, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Proc. SPIE 4678, 91-101 (2002).
[CrossRef]

M. A. Al-Habush, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance PDF of a laser beam propagation through turbulent media,” Opt. Eng. 40, 1554-1562(2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum,” Waves Random Media 10, 53-70 (2000).
[CrossRef]

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

Rao, R.

Recolons, J.

Roberts, P. H.

T. J. Brennan and P. H. Roberts, AOTools The Adaptive Toolbox Users Guide (The Optical Sciences Company, 2006), http://www.tosc.com/downloads/AOToolsUG.pdf.

Roggemann, M. C.

M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC, 1996).

Tamir, M.

Tur, M.

Tyson, R. K.

Vetelino, F. S.

Wang, G.

Welsh, B. M.

M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC, 1996).

Young, C.

Appl. Opt. (8)

R. K. Tyson and D. E. Canning, “Indirect measurement of a laser communications bit-error-rate reduction with low-order adaptive optics,” Appl. Opt. 42, 4239-4243 (2003).
[CrossRef] [PubMed]

F. S. Vetelino, C. Young, L. C. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence,” Appl. Opt. 46, 2099-2108 (2007).
[CrossRef] [PubMed]

J. H. Churnside, “Aperture averaging of optical scintillations in the turbulent atmosphere,” Appl. Opt. 30, 1982-1994(1991).
[CrossRef] [PubMed]

Z. Azar, M. Loebenstein, G. Applebaum, E. Azoulay, U. Halavee, M. Tamir, and M. Tur, “Aperture averaging of the two-wavelength intensity covariance function in atmospheric turbulence,” Appl. Opt. 24, 2401-2407 (1985).
[CrossRef] [PubMed]

F. S. Vetelino, B. Clare, K. Corbett, C. Young, K. Grant, and L. C. Andrews, “Characterizing the propagation path in moderate to strong optical turbulence,” Appl. Opt. 45, 3534-3543 (2006).
[CrossRef] [PubMed]

F. S. Vetelino, C. Young, and L. C. Andrews, “Fade statistics and aperture averaging for Gaussian beam waves in moderate-to-strong turbulence,” Appl. Opt. 46, 3780-3789(2007).
[CrossRef] [PubMed]

R. Rao, “Statistics of the fractal structure and phase singularity of a plane light wave propagation in atmospheric turbulence,” Appl. Opt. 47, 269-276 (2008).
[CrossRef] [PubMed]

A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39, 5426-5445 (2000).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Eng. (1)

M. A. Al-Habush, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance PDF of a laser beam propagation through turbulent media,” Opt. Eng. 40, 1554-1562(2001).
[CrossRef]

Proc. SPIE (1)

M. A. Al-Habush, L. C. Andrews, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Proc. SPIE 4678, 91-101 (2002).
[CrossRef]

Waves Random Media (1)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum,” Waves Random Media 10, 53-70 (2000).
[CrossRef]

Other (3)

T. J. Brennan and P. H. Roberts, AOTools The Adaptive Toolbox Users Guide (The Optical Sciences Company, 2006), http://www.tosc.com/downloads/AOToolsUG.pdf.

M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC, 1996).

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

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Figures (10)

Fig. 1
Fig. 1

PDF of the normalized log irradiance ln ( I ) for the receiving apertures of diameter 1.8, 5, and 13 mm . Simulation results for two turbulence strengths C n 2 = 6.47 × 10 14 m 2 / 3 and C n 2 = 4.58 × 10 13 m 2 / 3 , respectively, are compared to gamma-gamma and lognormal PDFs. Results compare to Fig. 3 in [10].

Fig. 2
Fig. 2

PDF of the normalized log irradiance ln ( I ) for the receiving apertures of diameter (a), (c), (e)  5.6 mm and (b), (d), (f)  ρ 0 / 2 . Simulation results for turbulence strength C n 2 = 1 × 10 18 m 2 / 3 and propagation distance of 50, 75, and 100 km are compared to the gamma-gamma and lognormal PDF for each case.

Fig. 3
Fig. 3

PDF of the normalized log irradiance ln ( I ) for the receiving apertures of diameter (a), (c), (e)  ρ 0 and (b), (d), (f)  3 ρ 0 / 2 . Simulation results for turbulence strength C n 2 = 1 × 10 18 m 2 / 3 and propagation distance of 50, 75, and 100 km are compared to the gamma-gamma and lognormal PDF for each case.

Fig. 4
Fig. 4

PDF of the normalized log irradiance ln ( I ) for the receiving apertures of diameter (a), (c), (e)  5.6 mm and (b), (d), (f)  ρ 0 / 2 . Simulation results for turbulence strength C n 2 = 1 × 10 17 m 2 / 3 and propagation distance of 50, 75, and 100 km are compared to the gamma-gamma and lognormal PDF for each case.

Fig. 5
Fig. 5

PDF of the normalized log irradiance ln ( I ) for the receiving apertures of diameter (a), (c), (e)  ρ 0 and (b), (d), (f)  3 ρ 0 / 2 . Simulation results for turbulence strength C n 2 = 1 × 10 17 m 2 / 3 and propagation distance of 50, 75, and 100 km are compared to the gamma-gamma and lognormal PDF for each case.

Fig. 6
Fig. 6

PDF of the normalized log irradiance ln ( I ) for the receiving apertures of diameter (a), (c), (e)  5.6 mm and (b), (d), (f)  ρ 0 / 2 . Simulation results for turbulence strength C n 2 = 1 × 10 16 m 2 / 3 and propagation distance of 50, 75, and 100 km are compared to the gamma-gamma and lognormal PDF for each case.

Fig. 7
Fig. 7

PDF of the normalized log irradiance ln ( I ) for the receiving apertures of diameter (a), (c), (e)  ρ 0 and (b), (d), (f)  3 ρ 0 / 2 . Simulation results for turbulence strength C n 2 = 1 × 10 16 m 2 / 3 and propagation distance of 50, 75, and 100 km are compared to the gamma-gamma and lognormal PDF for each case.

Fig. 8
Fig. 8

PDF of the normalized log irradiance ln ( I ) for the receiving apertures of diameter (a), (c), (e)  5.6 mm and (b), (d), (f)  ρ 0 / 2 . Simulation results for turbulence strength C n 2 = 1 × 10 15 m 2 / 3 and propagation distance of 50, 75, and 100 km are compared to the gamma-gamma and lognormal PDF for each case.

Fig. 9
Fig. 9

PDF of the normalized log irradiance ln ( I ) for the receiving apertures of diameter (a), (c), (e)  ρ 0 and (b), (d), (f)  3 ρ 0 / 2 . Simulation results for turbulence strength C n 2 = 1 × 10 15 m 2 / 3 and propagation distance of 50, 75, and 100 km are compared to the gamma-gamma and lognormal PDF for each case.

Fig. 10
Fig. 10

BER of the laser communication channel for the pointlike and 3 ρ 0 / 2 diameter apertures for propagation distances of 75 and 100 km and for turbulence strengths of (a), (b)  C n 2 = 1 × 10 16 m 2 / 3 and (c), (d)  C n 2 = 1 × 10 15 m 2 / 3 .

Tables (1)

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Table 1 Numerical Simulation Parameters

Equations (9)

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p ( I ) = 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) I ( I ) ( α + β ) / 2 K α β ( 2 α β I ) , I > 0 ,
α = 1 σ X 2 , β = 1 σ Y 2 .
σ X 2 = exp ( σ ln X 2 ) 1 , σ Y 2 = exp ( σ ln Y 2 ) 1 ,
p ( I ) = 1 I 2 π σ ln I 2 exp { [ ln ( I ) + 1 2 σ ln I 2 ] 2 2 σ ln I 2 } , I > 0 ,
σ ln I 2 = ln ( σ I 2 + 1 ) ,
σ I 2 = I 2 I 2 I 2 ,
U ( x , y , z ) = A 0 W 0 W ( z ) exp [ ( x + y ) 2 W 2 ( z ) ] × P ( x , y ) ,
σ I 2 = ( 1 + 1 α ) ( 1 + 1 β ) 1.
BER ( OOK ) = 1 2 0 p ( I ) erfc ( SNR I 2 2 ) d s ,

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