Abstract

In this paper, the method of random wave vectors for simulation of atmospheric turbulence is extended to 2D×2D space to provide spatial degrees of freedom at both input and output planes. The modified technique can thus simultaneously simulate the turbulence-induced log-amplitude and phase distortions for optical systems with extended sources either implemented as a single large aperture or multiple apertures. The reliability of our simulation technique is validated in different conditions and its application is briefly investigated in a multibeam free-space optical communication scenario.

© 2012 Optical Society of America

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References

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  1. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  2. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
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    [CrossRef]
  4. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (Pergamon, 1981), Vol. 19, pp. 281–376.
  5. J. H. Shapiro, “Imaging and optical communication through atmospheric turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, 1978), Chap. 6.
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  7. X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
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  8. M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7, 5441–5449 (2008).
    [CrossRef]
  9. N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
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    [CrossRef]
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  16. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link,” Appl. Opt. 46, 6561–6571 (2007).
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  17. M. Safari and S. Hranilovic, “Diversity gain for near-field MISO atmospheric optical communications,” in Proceedings of IEEE International Conference on Communications (IEEE, 2012), p. 3167.
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    [CrossRef]
  23. M. Safari and M. Uysal, “Relay-assisted quantum-key distribution over long atmospheric channels,” J. Lightwave Technol. 27, 4508–4515 (2009).
    [CrossRef]

2009 (1)

2008 (1)

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7, 5441–5449 (2008).
[CrossRef]

2007 (1)

2004 (1)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330–341 (2004).
[CrossRef]

2003 (1)

J. H. Shapiro, “Near-field turbulence effects on quantum-key distribution,” Phys. Rev. A 67, 022309 (2003).
[CrossRef]

2002 (1)

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[CrossRef]

2000 (1)

1997 (1)

1996 (3)

1995 (1)

1992 (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

1990 (1)

N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

1973 (1)

1972 (1)

1967 (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–77 (1967).
[CrossRef]

Andrews, L. C.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330–341 (2004).
[CrossRef]

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

Anguita, J. A.

Cannon, R. C.

Carhart, G. W.

Dainty, J. C.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Eiss, R.

Erkmen, B.

B. Erkmen, “Performance analysis for near-field optical communications,” Master’s thesis (MIT, 2002).

Frehlich, R.

Fried, D. L.

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–77 (1967).
[CrossRef]

Glindemann, A.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Hopen, C. Y.

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

Hranilovic, S.

M. Safari and S. Hranilovic, “Diversity gain for near-field MISO atmospheric optical communications,” in Proceedings of IEEE International Conference on Communications (IEEE, 2012), p. 3167.

Jakobsson, H.

Kahn, J. M.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[CrossRef]

Kerr, J. R.

Korotkova, O.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330–341 (2004).
[CrossRef]

Kouznetsov, D.

Lane, R. G.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Montera, D.

Neifeld, M. A.

Ortega-Martinez, R.

Phillips, R. L.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330–341 (2004).
[CrossRef]

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

Pruidze, D. V.

Rhoadarmer, T. A.

Ricklin, J. C.

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (Pergamon, 1981), Vol. 19, pp. 281–376.

Roddier, N.

N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Roggermann, M. C.

Safari, M.

M. Safari and M. Uysal, “Relay-assisted quantum-key distribution over long atmospheric channels,” J. Lightwave Technol. 27, 4508–4515 (2009).
[CrossRef]

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7, 5441–5449 (2008).
[CrossRef]

M. Safari and S. Hranilovic, “Diversity gain for near-field MISO atmospheric optical communications,” in Proceedings of IEEE International Conference on Communications (IEEE, 2012), p. 3167.

Shapiro, J. H.

J. H. Shapiro, “Near-field turbulence effects on quantum-key distribution,” Phys. Rev. A 67, 022309 (2003).
[CrossRef]

J. H. Shapiro, “Imaging and optical communication through atmospheric turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, 1978), Chap. 6.

Tatarskii, V. I.

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

Titterton, P. J.

Uysal, M.

M. Safari and M. Uysal, “Relay-assisted quantum-key distribution over long atmospheric channels,” J. Lightwave Technol. 27, 4508–4515 (2009).
[CrossRef]

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7, 5441–5449 (2008).
[CrossRef]

Vasic, B. V.

Voelz, D. G.

Voitsekhovich, V. V.

Vorontsov, M. A.

Welsh, B. M.

Zhu, X.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[CrossRef]

Appl. Opt. (5)

IEEE Trans. Commun. (1)

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50, 1293–1300 (2002).
[CrossRef]

IEEE Trans. Wireless Commun. (1)

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7, 5441–5449 (2008).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Eng. (2)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330–341 (2004).
[CrossRef]

N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Phys. Rev. A (1)

J. H. Shapiro, “Near-field turbulence effects on quantum-key distribution,” Phys. Rev. A 67, 022309 (2003).
[CrossRef]

Proc. IEEE (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–77 (1967).
[CrossRef]

Waves Random Media (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Other (6)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (Pergamon, 1981), Vol. 19, pp. 281–376.

J. H. Shapiro, “Imaging and optical communication through atmospheric turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, 1978), Chap. 6.

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

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

B. Erkmen, “Performance analysis for near-field optical communications,” Master’s thesis (MIT, 2002).

M. Safari and S. Hranilovic, “Diversity gain for near-field MISO atmospheric optical communications,” in Proceedings of IEEE International Conference on Communications (IEEE, 2012), p. 3167.

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Figures (7)

Fig. 1.
Fig. 1.

Geometry of atmospheric optical systems with extended sources: (a) large aperture and (b) multiple apertures.

Fig. 2.
Fig. 2.

Theoretical (solid curve) and simulated (1000 samples, dots; 10 5 samples, stars) log-amplitude structure. ρ and ρ are input and output displacement vectors and ϕ ρ is the angle between ρ and ρ .

Fig. 3.
Fig. 3.

Theoretical (solid curve) and simulated (1000 samples, dots; 10 5 samples, stars) phase structure function. ρ and ρ are input and output displacement vectors and ϕ ρ is the angle between ρ and ρ .

Fig. 4.
Fig. 4.

Theoretical (solid curve) and simulated (1000 samples, dots; 10 5 samples, stars) log-amplitude-phase structure function. ρ and ρ are input and output displacement vectors and ϕ ρ is the angle between ρ and ρ .

Fig. 5.
Fig. 5.

Amplitude and phase of a distorted optical field after propagation through the atmosphere over receive apertures with diameters D = 10 and D = 20 cm .

Fig. 6.
Fig. 6.

Simulated (dots) PDF with 10 5 samples and the log-normal fit (solid curve) of γ for the double-beam FSO system.

Fig. 7.
Fig. 7.

BER of the double-beam FSO system for different separations between transmit apertures ( d ).

Equations (28)

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ξ ( r ) = ξ ( r ) h ( r , r ) e a L / 2 d r ,
h ( r , r ) = e j k L + j k | r r | 2 / 2 L j λ L e χ ( r , r ) + j ϕ ( r , r ) ,
B χ ¯ ( ρ , ρ ) = χ ¯ ( r 1 , r 1 ) χ ¯ ( r 2 , r 2 ) = c 0 0 1 0 u 8 3 J 0 ( Δ u ) sin 2 ( u 2 z ( 1 z ) 2 k / L ) d u d z ,
B ϕ ¯ ( ρ , ρ ) = ϕ ¯ ( r 1 , r 1 ) ϕ ¯ ( r 2 , r 2 ) = c 0 0 1 0 u 8 3 J 0 ( Δ u ) cos 2 ( u 2 z ( 1 z ) 2 k / L ) d u d z ,
B χ ¯ ϕ ¯ ( ρ , ρ ) = χ ¯ ( r 1 , r 1 ) ϕ ¯ ( r 2 , r 2 ) = c 0 2 0 1 0 u 8 3 J 0 ( Δ u ) sin ( u 2 z ( 1 z ) k / L ) d u d z ,
B g ( ρ , ρ ) = 1 ( 2 π ) 2 0 1 f g ( u , z ) e j ( ρ · z u + ρ · ( 1 z ) u ) d u d z ,
f χ ¯ ( u , z ) = 2 π c 0 u 11 3 sin 2 ( u 2 z ( 1 z ) L / 2 k ) , f ϕ ¯ ( u , z ) = 2 π c 0 u 11 3 cos 2 ( u 2 z ( 1 z ) L / 2 k ) , f χ ¯ ϕ ¯ ( u , z ) = π c 0 u 11 3 sin ( u 2 z ( 1 z ) L / k ) .
W ( κ , κ ) = 1 ( 2 π ) 2 B ( ρ , ρ ) e j ( ρ · κ + ρ · κ ) d ρ d ρ
W g ( κ , κ ) = 0 1 f g ( u , z ) δ ( κ ( 1 z ) u ) δ ( κ z u ) d u d z .
W g ( κ , κ ) = { f g ( κ + κ , κ κ + κ ) κ ^ = κ ^ 0 otherwise ,
χ ¯ ( r , r ) = n = 1 M F ( x n , x n ) cos ( x n · r + x n · r + φ n ) ,
ϕ ¯ ( r , r ) = n = 1 M G ( x n , x n ) cos ( x n · r + x n · r + φ n + ψ n ) ,
B χ ¯ ( ρ , ρ ) = n , m = 1 M F n F m cos ( x n · r + x n · r + φ n ) cos ( x m · r + x m · r + φ m ) , B ϕ ¯ ( ρ , ρ ) = n , m = 1 M G n G m cos ( x n · r + x n · r + φ n + ψ n ) cos ( x m · r + x m · r + φ m + ψ m ) , B χ ¯ ϕ ¯ ( ρ , ρ ) = n , m = 1 M F n G m cos ( x n · r + x n · r + φ n ) cos ( x m · r + x m · r + φ m + ψ m ) ,
B χ ¯ ( ρ , ρ ) = 1 2 n = 1 M F n 2 ( x n , x n ) cos ( x n · ρ + x n · ρ ) , B ϕ ¯ ( ρ , ρ ) = 1 2 n = 1 M G n 2 ( x n , x n ) cos ( x n · ρ + x n · ρ ) , B χ ¯ ϕ ¯ ( ρ , ρ ) = 1 2 n = 1 M F n G n cos ( x n · ρ + x n · ρ ) cos ( ψ n ) ,
W χ ¯ ( κ , κ ) = π 2 n = 1 M F n 2 [ δ ( x n κ ) δ ( x n κ ) + δ ( x n + κ ) δ ( x n + κ ) ] , W ϕ ¯ ( κ , κ ) = π 2 n = 1 M G n 2 [ δ ( x n κ ) δ ( x n κ ) + δ ( x n + κ ) δ ( x n + κ ) ] , W χ ¯ ϕ ¯ ( κ , κ ) = π 2 n = 1 M F n G n [ δ ( x n κ ) δ ( x n κ ) + δ ( x n + κ ) δ ( x n + κ ) ] cos ( ψ n ) .
W χ ¯ ( κ κ ^ , κ κ ^ ) = π 2 n = 1 M d ζ n d 2 μ n Ω ( μ n , ζ n ) F n 2 [ δ ( ( 1 ζ n ) μ n κ κ ^ ) δ ( ζ n μ n κ κ ^ ) + δ ( ( 1 ζ n ) μ n + κ κ ^ ) δ ( ζ n μ n + κ κ ^ ) ] , W ϕ ¯ ( κ κ ^ , κ κ ^ ) = π 2 n = 1 M d ζ n d 2 μ n Ω ( μ n , ζ n ) G n 2 [ δ ( ( 1 ζ n ) μ n κ κ ^ ) δ ( ζ n μ n κ κ ^ ) + δ ( ( 1 ζ n ) μ n + κ κ ^ ) δ ( ζ n μ n + κ κ ^ ) ] , W χ ¯ ϕ ¯ ( κ κ ^ , κ κ ^ ) = π 2 n = 1 M d ζ n d 2 μ n d ψ n Ω ( μ n , ζ n ) η ( ψ n | μ n , ζ n ) F n G n cos ( ψ n ) [ δ ( ( 1 ζ n ) μ n κ κ ^ ) δ ( ζ n μ n κ κ ^ ) + δ ( ( 1 ζ n ) μ n + κ κ ^ ) δ ( ζ n μ n + κ κ ^ ) ] .
W χ ¯ ( κ κ ^ , κ κ ^ ) = 2 M π 2 Ω ( μ n = κ + κ , ζ n = κ κ + κ ) F ( κ , κ ) 2 ,
W ϕ ¯ ( κ κ ^ , κ κ ^ ) = 2 M π 2 Ω ( μ n = κ + κ , ζ n = κ κ + κ ) G ( κ , κ ) 2 ,
W χ ¯ ϕ ¯ ( κ κ ^ , κ κ ^ ) = 2 M π 2 Ω ( μ n = κ + κ , ζ n = κ κ + κ ) F ( κ , κ ) G ( κ , κ ) cos ( ψ ) η ( ψ | μ n , ζ n ) d ψ .
cos ( ψ n ) η ( ψ n | μ n , ζ n ) d ψ n = W χ ¯ ϕ ¯ ( κ κ ^ , κ κ ^ ) W χ ¯ ( κ κ ^ , κ κ ^ ) W ϕ ¯ ( κ κ ^ , κ κ ^ ) = { 1 sin ( κ κ L / k ) > 0 1 sin ( κ κ L / k ) < 0 ,
η ( ψ n | sin ( x n x n L / k ) > 0 ) = δ ( ψ n ) , η ( ψ n | sin ( x n x n L / k ) < 0 ) = δ ( ψ n π ) .
Ω ( μ n , ζ n ) = 1 2 π μ n 2 log ( K 2 / K 1 ) .
F ( κ , κ ) = 2 c 0 log ( K 2 / K 1 ) M ( κ + κ ) 5 6 | sin ( κ κ L / 2 k ) | , G ( κ , κ ) = 2 c 0 log ( K 2 / K 1 ) M ( κ + κ ) 5 6 | cos ( κ κ L / 2 k ) | .
D ( ρ , ρ ) = 2 [ B ( 0 , 0 ) B ( ρ , ρ ) ]
ξ 1 ( r ) = P 2 A δ ( r d 2 i ) , ξ 2 ( r ) = P 2 A δ ( r + d 2 i ) ,
γ = R | ξ 1 ( r ) + ξ 2 ( r ) | 2 d r = P A e a L 2 R | h ( d 2 i , r ) + h ( d 2 i , r ) | 2 d r ,
P e = 0 f γ ( γ ) Q ( R γ 2 σ n ) d γ ,
SNR = R 2 P 2 A 2 A 2 e 2 a L λ 4 L 4 σ n 2 .

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