Abstract

We model the impact of atmospheric turbulence-induced phase and amplitude fluctuations on free-space optical links using synchronous detection. We derive exact expressions for the probability density function of the signal-to-noise ratio in the presence of turbulence. We consider the effects of log-normal amplitude fluctuations and Gaussian phase fluctuations, in addition to local oscillator shot noise, for both passive receivers and those employing active modal compensation of wave-front phase distortion. We compute error probabilities for M-ary phase-shift keying, and evaluate the impact of various parameters, including the ratio of receiver aperture diameter to the wave-front coherence diameter, and the number of modes compensated.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. L. Fried, "Optical heterodyne detection of an atmospherically distorted signal wave front," Proc. IEEE 55, 57-67 (1967).
    [CrossRef]
  2. D. L. Fried, "Atmospheric modulation noise in an optical heterodyne receiver," IEEE J. Quantum Electron. QE-3, 213-221 (1967).
    [CrossRef]
  3. J. H. Churnside and C. M. McIntyre, "Signal current probability distribution for optical heterodyne receivers in the turbulent atmosphere. 1: Theory," Appl. Opt. 17, 2141-2147 (1978).
    [CrossRef] [PubMed]
  4. J. H. Churnside and C. M. McIntyre, "Heterodyne receivers for atmospheric optical communications," Appl. Opt. 19, 582-590 (1980).
    [CrossRef] [PubMed]
  5. K. A. Winick, "Atmospheric turbulence-induced signal fades on optical heterodyne communication links," Appl. Opt. 25, 1817-1825 (1986).
    [CrossRef] [PubMed]
  6. N. Perlot, "Turbulence-induced fading probability in coherent optical communication through the atmosphere," Appl. Opt. 46, 7218-7226 (2007).
    [CrossRef] [PubMed]
  7. A. Belmonte, "Influence of atmospheric phase compensation on optical heterodyne power measurements," Opt. Express 16, 6756-6767 (2008).
    [CrossRef] [PubMed]
  8. E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, "Coherent detection in optical fiber systems," Opt. Express 16, 753-791 (2008).
    [CrossRef] [PubMed]
  9. M. P. Cagigal and V. F. Canales, "Speckle statistics in partially corrected wave fronts," Opt. Lett. 23, 1072-1074 (1998).
    [CrossRef]
  10. J. W. Goodman, Speckle Phenomena in Optics. Theory and Applications (Ben Roberts & Company, 2007).
  11. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211 (1976).
    [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999)
  13. J. W. Strohbehn, T. Wang, and J. P. Speck, "On the probability distribution of line-of-sight fluctuations of optical signals," Radio Science 10, 59-70 (1975).
    [CrossRef]
  14. R. F. Pawula, S. O. Rice, and J. H. Roberts, "Distribution of the phase angle between two vectors perturbed by Gaussian noise," IEEE Trans. Commun. COM-30, 1828-1841 (1982).
    [CrossRef]
  15. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
    [CrossRef]

2008 (2)

2007 (1)

1998 (1)

1986 (1)

1982 (1)

R. F. Pawula, S. O. Rice, and J. H. Roberts, "Distribution of the phase angle between two vectors perturbed by Gaussian noise," IEEE Trans. Commun. COM-30, 1828-1841 (1982).
[CrossRef]

1980 (1)

1978 (1)

1976 (1)

1975 (1)

J. W. Strohbehn, T. Wang, and J. P. Speck, "On the probability distribution of line-of-sight fluctuations of optical signals," Radio Science 10, 59-70 (1975).
[CrossRef]

1967 (2)

D. L. Fried, "Optical heterodyne detection of an atmospherically distorted signal wave front," Proc. IEEE 55, 57-67 (1967).
[CrossRef]

D. L. Fried, "Atmospheric modulation noise in an optical heterodyne receiver," IEEE J. Quantum Electron. QE-3, 213-221 (1967).
[CrossRef]

Barros, D. J. F.

Belmonte, A.

Cagigal, M. P.

Canales, V. F.

Churnside, J. H.

Fried, D. L.

D. L. Fried, "Optical heterodyne detection of an atmospherically distorted signal wave front," Proc. IEEE 55, 57-67 (1967).
[CrossRef]

D. L. Fried, "Atmospheric modulation noise in an optical heterodyne receiver," IEEE J. Quantum Electron. QE-3, 213-221 (1967).
[CrossRef]

Ip, E.

Kahn, J. M.

Lau, A. P. T.

McIntyre, C. M.

Noll, R. J.

Pawula, R. F.

R. F. Pawula, S. O. Rice, and J. H. Roberts, "Distribution of the phase angle between two vectors perturbed by Gaussian noise," IEEE Trans. Commun. COM-30, 1828-1841 (1982).
[CrossRef]

Perlot, N.

Rice, S. O.

R. F. Pawula, S. O. Rice, and J. H. Roberts, "Distribution of the phase angle between two vectors perturbed by Gaussian noise," IEEE Trans. Commun. COM-30, 1828-1841 (1982).
[CrossRef]

Roberts, J. H.

R. F. Pawula, S. O. Rice, and J. H. Roberts, "Distribution of the phase angle between two vectors perturbed by Gaussian noise," IEEE Trans. Commun. COM-30, 1828-1841 (1982).
[CrossRef]

Speck, J. P.

J. W. Strohbehn, T. Wang, and J. P. Speck, "On the probability distribution of line-of-sight fluctuations of optical signals," Radio Science 10, 59-70 (1975).
[CrossRef]

Strohbehn, J. W.

J. W. Strohbehn, T. Wang, and J. P. Speck, "On the probability distribution of line-of-sight fluctuations of optical signals," Radio Science 10, 59-70 (1975).
[CrossRef]

Wang, T.

J. W. Strohbehn, T. Wang, and J. P. Speck, "On the probability distribution of line-of-sight fluctuations of optical signals," Radio Science 10, 59-70 (1975).
[CrossRef]

Winick, K. A.

Appl. Opt. (4)

IEEE J. Quantum Electron. (1)

D. L. Fried, "Atmospheric modulation noise in an optical heterodyne receiver," IEEE J. Quantum Electron. QE-3, 213-221 (1967).
[CrossRef]

IEEE Trans. Commun. (1)

R. F. Pawula, S. O. Rice, and J. H. Roberts, "Distribution of the phase angle between two vectors perturbed by Gaussian noise," IEEE Trans. Commun. COM-30, 1828-1841 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (2)

Opt. Lett. (1)

Proc. IEEE (1)

D. L. Fried, "Optical heterodyne detection of an atmospherically distorted signal wave front," Proc. IEEE 55, 57-67 (1967).
[CrossRef]

Radio Science (1)

J. W. Strohbehn, T. Wang, and J. P. Speck, "On the probability distribution of line-of-sight fluctuations of optical signals," Radio Science 10, 59-70 (1975).
[CrossRef]

Other (3)

J. W. Goodman, Speckle Phenomena in Optics. Theory and Applications (Ben Roberts & Company, 2007).

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999)

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1.

Symbol-error probability (SEP) vs. turbulence-free SNR per symbol γ0 for QPSK with coherent detection and additive white Gaussian noise (AWGN). Performance is shown for different values of: (a) the normalized receiver aperture diameter D/r 0, and (b) the number of modes J removed by adaptive optics. Amplitude fluctuations are neglected by assuming σ 2 β =0. Turbulence is characterized by the phase coherence length r 0. In (a), D/r 0 ranges from 0.1 (weak turbulence) to 10 (strong turbulence). In (b), the compensating phases are expansions up to tilt (J=3), astigmatism (J=6), and 5th-order aberrations (J=20). The no-correction case (J=0) is also considered. The no-turbulence case is indicated by black lines.

Fig. 2.
Fig. 2.

SEP vs. coherence diameter r 0 for QPSK with coherent detection and AWGN. In (a), no phase compensation is employed, and performance is shown for different values of the scintillation index σ 2 β. In (b), the scintillation index is fixed at σ 2 β=0.3, and performance is shown for different values of J, the number of modes corrected by adaptive optics. The severity of atmospheric turbulence increases when r 0 decreases. In all cases, we assume the turbulence-free SNR per symbol is γ0=10 dB, and the receiver aperture diameter is D=10 cm. In (a), the σ 2 β ranges from 0.3 (weaker turbulence) to 1 (stronger turbulence). In (b), the compensating phases are expansions through tilt (J=3), astigmatism (J=6), and 5th-order aberrations (J=20). The no-turbulence case is indicated by black lines.

Fig. 3.
Fig. 3.

SEP vs. normalized receiver aperture diameter D/r 0 for QPSK with coherent detection and AWGN. In (a), no phase compensation is employed, and performance is shown for different values of the scintillation index σ 2 β. In (b), the scintillation index is fixed at σ 2 β=0.3, and performance is shown for different values of J, the number of modes corrected by adaptive optics. In all cases, the turbulence-free SNR per symbol γ0 is proportional to the square of the aperture diameter D. For the smallest aperture considered, we assume γ0=10 dB. In (a), σ 2 β ranges from 0.3 (weaker turbulence) to 1 (stronger turbulence). In (b), the compensating phases are expansions up to tilt (J=3), astigmatism (J=6), and 5th-order aberrations (J=20). The no-turbulence case is indicated by black lines.

Equations (47)

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

A = A S exp [ χ r j ϕ r ] ,
i S = η A 0 A S d x W r exp [ χ r ] cos [ 2 π Δ f t + Δ ϕ ϕ r ] ,
i S = η A 0 A S { cos [ 2 π Δ f t + Δ ϕ ] d r W r exp [ χ r ] cos [ ϕ r ] +
+ sin [ 2 π Δ f t + Δ ϕ ] d r W r exp [ χ r ] sin [ ϕ r ] } .
i S 2 = 1 2 ( η π 4 D 2 A 0 A S ) 2 [ α r 2 + α i 2 ] ,
α r = ( π 4 D 2 ) 1 d r W ( r ) exp [ χ ( r ) ] cos [ ϕ r ]
α i = ( π 4 D 2 ) 1 d r W ( r ) exp [ χ ( r ) ] sin [ ϕ r ] .
γ = 1 2 η e π 4 D 2 A S 2 α 2 .
γ = γ 0 α 2 .
P γ ( γ ) = α 2 ¯ γ ¯ P α 2 ( γ α 2 ¯ γ ¯ ) .
α r 1 N k = 1 N exp χ k cos ϕ k
α i 1 N k = 1 N exp χ k cos ϕ k ,
α r 1 N exp χ k = 1 N exp χ k χ cos ϕ k
α i 1 N exp χ k = 1 N exp ( χ k χ ) sin ϕ k .
p α r , α i α r α i = 1 2 π σ r σ i exp [ ( α r α ¯ r ) 2 σ r 2 2 ] exp [ ( α i α ¯ i ) 2 σ i 2 2 ] ,
α ¯ r = 1 2 exp χ ¯ exp ( χ k χ ¯ ) ¯ [ M ϕ ( 1 ) + M ϕ ( 1 ) ]
α ¯ i = j 2 exp χ ¯ exp ( χ k χ ¯ ) ¯ [ M ϕ ( 1 ) M ϕ ( 1 ) ] ,
σ r 2 = exp 2 χ exp 2 ( χ k χ ¯ ) ¯ 4 N [ 2 + M ϕ 2 + M ϕ 2 ]
exp 2 χ [ exp ( χ k χ ¯ ) ¯ ] 2 4 N [ 2 M ϕ ( 1 ) M ϕ 1 + M ϕ 2 1 + M ϕ 2 1 ] ,
σ i 2 = exp 2 χ exp 2 ( χ k χ ¯ ) ¯ 4 N [ 2 M ϕ 2 M ϕ 2 ]
exp 2 χ [ exp ( χ k χ ¯ ) ¯ ] 2 4 N [ 2 M ϕ ( 1 ) M ϕ ( 1 ) M ϕ 2 ( 1 ) M ϕ 2 ( 1 ) ] .
M ϕ ( ω ) = exp ( σ ϕ 2 ω 2 2 ) .
σ ϕ 2 = C J ( D r o ) 5 3 ,
χ ¯ = σ χ 2
exp ( χ k χ ¯ ) ¯ = exp ( 1 2 σ χ 2 ) .
α r ¯ = exp ( 1 2 σ χ 2 ) exp ( 1 2 σ ϕ 2 )
α i ¯ = 0
σ r 2 = 1 2 N [ 1 + exp ( 2 σ ϕ 2 ) 2 exp ( σ χ 2 ) exp ( σ ϕ 2 ) ]
σ i 2 = 1 2 N [ 1 exp ( 2 σ ϕ 2 ) ] .
N = [ 1 S d r W ( r ) C ( r ) ] 1 ,
C r = exp [ 1 2 6.88 ( r r 0 ) 5 3 ] .
N = [ 8 D 2 0 r dr exp [ 1 2 6.88 ( r r 0 ) 5 3 ] ] 1 .
N = [ 8 D 2 0 D 2 r dr exp [ 1 2 6.88 ( r r 0 ) 5 3 ] ] 1 ,
N = { 1.09 ( r 0 D ) 2 Γ [ 6 5 , 1.08 ( D r 0 ) 5 3 ] } 1 .
p α 2 ( α 2 ) = 1 4 π 1 σ r σ i π π d θ exp [ ( α cos θ α r ) 2 2 σ r 2 ] exp [ ( α sin θ ) 2 2 σ i 2 ] .
P α 2 ( α 2 ) = 1 2 σ 2 exp [ α 2 + a 2 2 σ 2 ] I 0 ( a α σ 2 ) ,
p α 2 ( α 2 ) = 1 + r α 2 ¯ exp ( r ) exp [ ( 1 + r ) α 2 α 2 ¯ ] I 0 2 α ( 1 + r ) r α 2 ¯ ,
α 2 ¯ = σ r 2 + σ i 2 + α ¯ r 2
α 2 ¯ ( 2 r + 1 ) / ( 1 + r ) 2 = 2 ( σ r 4 + σ i 4 ) + 4 σ r 2 α ¯ r 2 .
1 r = σ r 2 + σ i 2 + a r ¯ 2 [ a r ¯ 4 + 2 a r ¯ 2 ( σ i 2 σ r 2 ) ( σ i 2 σ r 2 ) 2 ] 1 2 1 ,
p γ ( γ ) = 1 + r γ exp ( r ) exp [ ( 1 + r ) γ γ ] I 0 [ 2 ( 1 + r ) r γ γ ] ,
p S ( E ) = 0 d γ p S ( E γ ) p γ ( γ ) .
p S ( E γ ) = π / 2 π / 2 π / M d θ exp [ γ sin 2 π M sin 2 θ ] .
p S ( E ) = 1 π π / 2 π / 2 π / M d θ ( 1 + r ) sin 2 θ ( 1 + r ) sin 2 θ γ sin 2 π M exp [ r γ sin 2 π M ( 1 + r ) sin 2 θ γ sin 2 π M ] .
p S ( E γ ) M 1 M exp ( γ sin 2 π M ) .
p S ( E ) M 1 M ( 1 + r ) ( 1 + r ) γ sin 2 π M exp [ r γ sin 2 π M ( 1 + r ) + γ sin 2 π M ] .
p S ( E ) = ( M 1 M ) { 1 γ sin 2 π M 1 + γ sin 2 π M ( M ( M 1 ) π ) [ π 2 + tan 1 ( γ sin 2 π M 1 + γ sin 2 π M cot π M ) ] } .

Metrics