Abstract

The gamma–gamma probability density function is commonly used to model the scintillation of a single laser beam propagating through atmospheric turbulence. One method proposed to reduce scintillation at the receiver plane involves the use of multiple channels propagating through independent paths, resulting in a sum of independent gamma–gamma random variables. Recently, a novel approach for an accurate, closed-form approximation for the sum of independent, identically distributed gamma–gamma random variables was introduced by Chatzidiamantis et al. [GLOBECOM 2009—2009 IEEE Global Telecommunications Conference (2009)]. Using this approximation, we present the first analytic model for the distribution of irradiance due to propagating multiple independent beams. This model compares favorably to wave-optics simulations.

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References

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  1. J. A. Louthain and J. D. Schmidt, Opt. Express 16, 10769 (2008).
    [CrossRef] [PubMed]
  2. J. A. Louthain and J. D. Schmidt, Opt. Express 18, 8948 (2010).
    [CrossRef] [PubMed]
  3. M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, Opt. Eng. 40, 1554 (2001).
    [CrossRef]
  4. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, Vol. PM99 of SPIE Press Monograph (SPIE Press, 2001).
    [CrossRef]
  5. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE Press, 2005).
    [CrossRef]
  6. N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, in GLOBECOM 2009—2009 IEEE Global Telecommunications Conference (IEEE, 2009), p. 16.
  7. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation With Examples in MATLAB (Press Monograph) (SPIE Press, 2010), pap/chrt ed.
  8. S. Coy, Proc. SPIE 5894, 589405 (2005)
    [CrossRef]

2010 (1)

2008 (1)

2005 (1)

S. Coy, Proc. SPIE 5894, 589405 (2005)
[CrossRef]

2001 (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, Opt. Eng. 40, 1554 (2001).
[CrossRef]

Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, Opt. Eng. 40, 1554 (2001).
[CrossRef]

Andrews, L. C.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, Opt. Eng. 40, 1554 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, Vol. PM99 of SPIE Press Monograph (SPIE Press, 2001).
[CrossRef]

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

Chatzidiamantis, N. D.

N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, in GLOBECOM 2009—2009 IEEE Global Telecommunications Conference (IEEE, 2009), p. 16.

Coy, S.

S. Coy, Proc. SPIE 5894, 589405 (2005)
[CrossRef]

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, Vol. PM99 of SPIE Press Monograph (SPIE Press, 2001).
[CrossRef]

Karagiannidis, G. K.

N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, in GLOBECOM 2009—2009 IEEE Global Telecommunications Conference (IEEE, 2009), p. 16.

Louthain, J. A.

Michalopoulos, D. S.

N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, in GLOBECOM 2009—2009 IEEE Global Telecommunications Conference (IEEE, 2009), p. 16.

Phillips, R. L.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, Opt. Eng. 40, 1554 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, Vol. PM99 of SPIE Press Monograph (SPIE Press, 2001).
[CrossRef]

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

Schmidt, J. D.

J. A. Louthain and J. D. Schmidt, Opt. Express 18, 8948 (2010).
[CrossRef] [PubMed]

J. A. Louthain and J. D. Schmidt, Opt. Express 16, 10769 (2008).
[CrossRef] [PubMed]

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation With Examples in MATLAB (Press Monograph) (SPIE Press, 2010), pap/chrt ed.

Opt. Eng. (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, Opt. Eng. 40, 1554 (2001).
[CrossRef]

Opt. Express (2)

Proc. SPIE (1)

S. Coy, Proc. SPIE 5894, 589405 (2005)
[CrossRef]

Other (4)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, Vol. PM99 of SPIE Press Monograph (SPIE Press, 2001).
[CrossRef]

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

N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, in GLOBECOM 2009—2009 IEEE Global Telecommunications Conference (IEEE, 2009), p. 16.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation With Examples in MATLAB (Press Monograph) (SPIE Press, 2010), pap/chrt ed.

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Figures (2)

Fig. 1
Fig. 1

Analytical and wave-optics generated CDF for an aperture-averaged, single-beam irradiance.

Fig. 2
Fig. 2

Analytical CDF (solid curves) and wave-optics results (dashed curves) plotted for one through seven beams.

Equations (19)

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p ( I ) = 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) I ( α + β ) / 2 1 × K α β ( 2 α β I ) , I > 0 ,
P ( I I T ) = 0 I T p ( I ) d I = π sin [ π ( α β ) ] Γ ( α ) Γ ( β ) × { ( α β I T ) β β Γ ( β 1 ) × F 1 2 ( β ; β + 1 , β 1 ; α β I T ) ( α β I T ) α α Γ ( α 1 ) × F 1 2 ( α ; α + 1 , α 1 ; α β I T ) } ,
σ ln x 2 ( D ) 0.49 σ 1 2 ( Ω G Λ 1 Ω G + Λ 1 ) 2 ( 1 3 1 2 Θ ¯ 1 + 1 5 Θ ¯ 1 2 ) × [ η x 1 + 0.40 η x ( 2 Θ ¯ 1 ) / ( Λ 1 + Ω G ) ] 7 / 6 ,
η x = ( 1 3 1 2 Θ ¯ 1 + 1 5 Θ ¯ 1 2 ) 6 / 7 ( σ B / σ 1 ) 12 / 7 ( 1 + 0.56 σ B 12 / 5 ) .
σ ln y 2 ( D ) 1.27 σ 1 2 η y 5 / 6 1 + 0.40 η y / ( Λ 1 + Ω G ) , η y 1 ,
η y = 3 ( σ 1 σ B ) 12 / 5 ( 1 + 0.69 σ B 12 / 5 ) .
σ B 2 3.86 σ 1 2 { 0.40 [ ( 1 + 2 Θ 1 ) 2 + 4 Λ 1 2 ] 5 / 12 ,
× cos [ 5 6 tan 1 ( 1 + 2 Θ 1 2 Λ 1 ) ] 11 16 Λ 1 5 / 6 } ,
Θ 1 = Θ 0 Θ 0 2 + Λ 0 2 ,
Λ 1 = Λ 0 Θ 0 2 + Λ 0 2 .
Θ 0 = 1 L F 0 , Λ 0 = 2 L k W 0 2 ,
I N = i = 1 N x i y i ,
I N = 1 N ( i = 1 N x i ) ( i = 1 N y i ) + 1 N i = 1 N 1 j = i + 1 N ( x i x j ) ( y i y j ) .
ϵ = 1 N i = 1 N 1 j = i + 1 N ( x i x j ) ( y i y j ) .
α N = N α + ϵ N ,
ϵ N = ( N 1 ) 0.127 0.95 α 0.0058 β 1 + 0.00124 α + 0.98 β .
P R = P T 0 2 π 0 D / 2 2 π W e exp ( 2 r 2 W e 2 ) r d r d θ = P T [ 1 exp ( D 2 2 W e 2 ) ] ,
W e W 1 [ 1 + 1.63 σ 12 / 5 Λ 1 ] 1 / 2 ,
W 1 = W 0 ( Θ 0 2 + Λ 0 2 ) 1 / 2 .

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