Abstract

We analyze the ergodic capacity and e-outage capacity of coherent optical links through the turbulent atmosphere. 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 wavefront phase distortion. We study the effect of various parameters, including the ratio of receiver aperture diameter to wavefront coherence diameter, the strength of the scintillation index, and the number of modes compensated.

© 2009 Optical Society of America

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  1. C. E. Shannon, "A mathematical theory of communications," Bell Syst. Tech. J.27, 379-423 623-656 (1948).
  2. X. Zhu and J. Kahn, "Free space optical communication through atmospheric turbulence channels," IEEE Trans. Commun. 50, 1293-1300 (2002).
    [CrossRef]
  3. S. Haas and J. H. Shapiro, "Capacity of wireless optical communications," IEEE J. Sel. Areas Commun. 21, 1346-1357 (2003).
    [CrossRef]
  4. J. A. Anguita, I. B. Djordjevic, M. Neifeld, and B. V. Vasic, "Shannon capacities and error-correction codes for optical atmospheric turbulent channels," J. Opt. Netw. 4, 586-601 (2005).
    [CrossRef]
  5. A. Belmonte and J. M. Kahn, "Performance of synchronous optical receivers using atmospheric compensation techniques," Opt. Express 16, 14151-14162 (2008).
    [CrossRef] [PubMed]
  6. D. L. Fried, "Optical heterodyne detection of an atmospherically distorted signal wave front," Proc. IEEE 55, 57-67 (1967).
    [CrossRef]
  7. D. L. Fried, "Atmospheric modulation noise in an optical heterodyne receiver," IEEE J. Quantum Electron. QE-3, 213-221 (1967).
    [CrossRef]
  8. 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]
  9. J. H. Churnside and C. M. McIntyre, "Heterodyne receivers for atmospheric optical communications," Appl. Opt. 19, 582-590 (1980).
    [CrossRef] [PubMed]
  10. A. Winick, "Atmospheric turbulence-induced signal fades on optical heterodyne communication links," Appl. Opt. 25, 1817-1825 (1986).
    [CrossRef] [PubMed]
  11. J. Proakis and M. Salehi, Digital Communications (Mc Graw-Hill, 2007).
  12. J. W. Strohbehn, T. Wang, and J. P. Speck, "On the probability distribution of line-of-sight fluctuations of optical signals," Radio Sci. 10, 59-70 (1975).
    [CrossRef]
  13. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211 (1976).
    [CrossRef]
  14. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
  15. M. K. Simon, "A new twist on the Marcum Q-function and its applications," IEEE Commun. Lett. 2, 39-41 (1998).
    [CrossRef]

2008 (1)

2005 (1)

2003 (1)

S. Haas and J. H. Shapiro, "Capacity of wireless optical communications," IEEE J. Sel. Areas Commun. 21, 1346-1357 (2003).
[CrossRef]

2002 (1)

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

1998 (1)

M. K. Simon, "A new twist on the Marcum Q-function and its applications," IEEE Commun. Lett. 2, 39-41 (1998).
[CrossRef]

1986 (1)

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 Sci. 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]

Anguita, J. A.

Belmonte, A.

Churnside, J. H.

Djordjevic, I. B.

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]

Haas, S.

S. Haas and J. H. Shapiro, "Capacity of wireless optical communications," IEEE J. Sel. Areas Commun. 21, 1346-1357 (2003).
[CrossRef]

Kahn, J.

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

Kahn, J. M.

McIntyre, C. M.

Neifeld, M.

Noll, R. J.

Shapiro, J. H.

S. Haas and J. H. Shapiro, "Capacity of wireless optical communications," IEEE J. Sel. Areas Commun. 21, 1346-1357 (2003).
[CrossRef]

Simon, M. K.

M. K. Simon, "A new twist on the Marcum Q-function and its applications," IEEE Commun. Lett. 2, 39-41 (1998).
[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 Sci. 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 Sci. 10, 59-70 (1975).
[CrossRef]

Vasic, B. V.

Wang, T.

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

Winick, A.

Zhu, X.

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

Appl. Opt. (3)

IEEE Commun. Lett. (1)

M. K. Simon, "A new twist on the Marcum Q-function and its applications," IEEE Commun. Lett. 2, 39-41 (1998).
[CrossRef]

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 J. Sel. Areas Commun. (1)

S. Haas and J. H. Shapiro, "Capacity of wireless optical communications," IEEE J. Sel. Areas Commun. 21, 1346-1357 (2003).
[CrossRef]

IEEE Trans. Commun. (1)

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

J. Opt. Netw. (1)

J. Opt. Soc. Am. (1)

Opt. Express (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 Sci. (1)

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

Other (3)

C. E. Shannon, "A mathematical theory of communications," Bell Syst. Tech. J.27, 379-423 623-656 (1948).

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

J. Proakis and M. Salehi, Digital Communications (Mc Graw-Hill, 2007).

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Figures (5)

Fig. 1.
Fig. 1.

ε-outage spectral efficiency vs. turbulence-free SNR per symbol γ0 for 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. The outage probability is fixed at ε=0.001. 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 AWGN Shannon limit is indicated by black lines.

Fig. 2.
Fig. 2.

ε-outage spectral efficiency vs. normalized receiver aperture diameter D/r 0 for coherent detection and AWGN. In (a), no phase compensation is employed, and performance is shown for different values of the outage probability ε. In (b), the outage probability is fixed at ε=0.001, 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), ε ranges from 0.1 (large outage probability) to 0.001 (small outage probability). In (b), the compensating phases are expansions up to tilt (J=3), astigmatism (J=6), and 5th-order aberrations (J=20). The AWGN Shannon limit is indicated by black lines. In (a), for ε=0.001, the dotted line considers the ε-outage spectral efficiency when the scintillation index is not neglected but fixed at σ2 β = 1

Fig. 3.
Fig. 3.

ε-outage fade margin vs. normalized receiver aperture diameter D/r 0 for coherent detection and AWGN. In (a), no phase compensation is employed, and performance is shown for different values of the outage probability ε. In (b), the outage probability is fixed at ε=0.001 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), ε ranges from 0.1 (large outage probability) to 0.001 (small outage probability). In (b), the compensating phases are expansions up to tilt (J=3), astigmatism (J=6), and 5th-order aberrations (J=20).

Fig. 4.
Fig. 4.

Ergodic spectral efficiency vs. turbulence-free SNR per symbol γ0 for 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 AWGN Shannon limit is indicated by black lines.

Fig. 5.
Fig. 5.

Ergodic spectral efficiency vs. normalized receiver aperture diameter D/r 0 for coherent detection and AWGN. In (a), no phase compensation is employed, and performance is shown for different values of the scintillation index σ2 β. In (b), the scintillation index is fixed at σ2 β = 1, 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), σ2 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 AWGN Shannon limit is indicated by black lines.

Equations (24)

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C ε = B log 2 ( 1 + γ R ) = B log 2 [ 1 + F c 1 ( 1 ε ) ] .
p γ ( γ ) = 1 + γ γ ̅ exp ( r ) exp [ ( 1 + r ) γ γ ̅ ] I 0 [ 2 ( 1 + r ) γ ̅ ] ,
α 2 ̅ = σ r 2 + σ i 2 + α ̅ r 2
1 r = σ r 2 + σ i 2 + α ̅ r 2 [ α ̅ r 4 + 2 α ̅ r 2 ( σ i 2 σ r 2 ) ( σ i 2 σ r 2 ) 2 ] 1 2 1 .
α ¯ r = exp ( 1 2 σ χ 2 ) exp ( 1 2 σ α 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 ) ] .
σ χ 2 = log e ( 1 + σ β 2 )
σ ϕ 2 = C J ( D r 0 ) 5 3
N = { 1.09 ( r 0 D ) 2 Γ [ 6 5 , 1.08 ( D r 0 ) 5 / 3 ] } 1 ,
F c ( γ R ) = 1 0 γ R p γ ( γ )
= Q ( 2 r , 2 ( 1 + r ) γ ̅ γ R )
F c ( γ R ) exp [ 1 2 ( 2 ( 1 + r ) γ ̅ γ R 2 r ) 2 ] .
F c 1 ( p ) = γ R γ ̅ 2 ( 1 + r ) ( 2 log e p + 2 r ) 2 .
C ε B log 2 [ 1 + γ ̅ 2 ( 1 + r ) ( 2 log e ( 1 ε ) + 2 r ) 2 ] .
C ε B log 2 [ 1 + γ ̅ 2 ( 1 + r ) ( 2 log e ( 1 ε ) 2 r ) 2 ]
P dB = 10 log 10 [ α ̅ 2 2 ( 1 + r ) ( 2 log e ( 1 ε ) + 2 r ) 2 ]
P dB = 10 log 10 [ α ̅ 2 1 + r ( ε + r ) 2 ] .
E [ C ] = B 0 log 2 ( 1 + γ ) p γ ( γ ) .
log 2 ( 1 + γ ) = log 2 ( 1 + γ ̅ ) + log 2 e m = 1 M ( 1 ) m 1 m ( 1 + γ ̅ ) m ( γ γ ̅ ) m + R M + 1 ,
R M + 1 = log 2 e ( 1 ) M ( M + 1 ) ξ M + 1 ( γ γ ̅ ) M + 1
E [ C ] = B log 2 ( 1 + γ ̅ ) + B log 2 e m = 2 M ( 1 ) m 1 m ( 1 + γ ̅ ) m ( γ γ ¯ ) m ¯ .
γ m ̅ = 0 γ m p γ ( γ ) = Γ ( 1 + m ) ( 1 + r ) m L m ( r ) ( γ ¯ ) m

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