Abstract

To assess the coherent detection of an optical signal perturbed by atmospheric turbulence, the loss in the mean signal-to-noise ratio (SNR) is usually invoked although it constitutes a limited description of the signal fluctuations. To produce statistical distributions of the SNR, we generate random optical fields. A 5∕3-power law for the phase structure function is considered. The benefit of a wavefront tilt correction is assessed. Based on the 1%-probability fade, an optimum receiver size is found. For phase fluctuations only, a similarity between the signal distribution and the beta distribution is observed. Phase and amplitude are assumed independent, and the influence of amplitude perturbations is assessed with a scintillation index of 2. Turbulence impairments are compared for a coherent receiver and a direct-detection receiver.

© 2007 Optical Society of America

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References

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    [CrossRef]

2007 (1)

R. Lange and B. Smutny, "Homodyne BPSK-based optical intersatellite communication links," Proc. SPIE 6457, 645703 (2007).

2006 (3)

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, "Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects," Opt. Eng. 45, 076001 (2006).
[CrossRef]

R. Lange, B. Smutny, B. Wandernoth, R. Czichy, and D. Giggenbach, "142 km, 5.625 Gbps free-space optical link based on homodyne BPSK modulation," Proc. SPIE 6105, 61050A (2006).
[CrossRef]

N. Perlot, D. Giggenbach, H. Henniger, J. Horwath, M. Knapek, and K. Zettl, "Measurements of the beam-wave fluctuations over a 142 km atmospheric path," Proc. SPIE 6304, 63041O (2006).
[CrossRef]

2004 (1)

J. C. Ricklin, S. Bucaille, and F. M. Davidson, "Performance loss factors for optical communication through clear air turbulence," Proc. SPIE 5160, 1-12 (2004).
[CrossRef]

2002 (1)

1999 (1)

1998 (1)

1997 (2)

1995 (4)

J.-M. Conan, G. Rousset, and P.-Y. Madec, "Wave-front temporal spectra in high-resolution imaging through turbulence," J. Opt. Soc. Am. A 12, 1559-1570 (1995).
[CrossRef]

J. E. Kaufmann, "Performance limits of high-rate space-to-ground optical communications through the turbulent atmospheric channel," Proc. SPIE 2381, 171-182 (1995).
[CrossRef]

N. Shvartsman and I. Freund, "Speckle spots ride phase saddles sidesaddle," Opt. Commun. 117, 228-234 (1995).
[CrossRef]

G.-M. Dai, "Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen-Loeve functions," J. Opt. Soc. Am. A 12, 2182-2193 (1995).
[CrossRef]

1992 (1)

1989 (1)

1986 (1)

1978 (2)

1976 (1)

B. L. McGlamery, "Computer simulation studies of compensation of turbulence degraded images," Proc. SPIE 74, 225-233 (1976).

1968 (1)

J. W. Strohbehn, "Line of sight wave propagation through the turbulent atmosphere," Proc. IEEE 56, 1301-1318 (1968).
[CrossRef]

1967 (1)

D. L. Fried, "Optical heterodyne detection of an atmospherically distorted signal wave front," Proc. IEEE 55, 57-66 (1967).
[CrossRef]

Appl. Opt. (8)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

N. Shvartsman and I. Freund, "Speckle spots ride phase saddles sidesaddle," Opt. Commun. 117, 228-234 (1995).
[CrossRef]

Opt. Eng. (1)

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, "Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects," Opt. Eng. 45, 076001 (2006).
[CrossRef]

Proc. IEEE (2)

D. L. Fried, "Optical heterodyne detection of an atmospherically distorted signal wave front," Proc. IEEE 55, 57-66 (1967).
[CrossRef]

J. W. Strohbehn, "Line of sight wave propagation through the turbulent atmosphere," Proc. IEEE 56, 1301-1318 (1968).
[CrossRef]

Proc. SPIE (6)

J. E. Kaufmann, "Performance limits of high-rate space-to-ground optical communications through the turbulent atmospheric channel," Proc. SPIE 2381, 171-182 (1995).
[CrossRef]

R. Lange, B. Smutny, B. Wandernoth, R. Czichy, and D. Giggenbach, "142 km, 5.625 Gbps free-space optical link based on homodyne BPSK modulation," Proc. SPIE 6105, 61050A (2006).
[CrossRef]

R. Lange and B. Smutny, "Homodyne BPSK-based optical intersatellite communication links," Proc. SPIE 6457, 645703 (2007).

B. L. McGlamery, "Computer simulation studies of compensation of turbulence degraded images," Proc. SPIE 74, 225-233 (1976).

N. Perlot, D. Giggenbach, H. Henniger, J. Horwath, M. Knapek, and K. Zettl, "Measurements of the beam-wave fluctuations over a 142 km atmospheric path," Proc. SPIE 6304, 63041O (2006).
[CrossRef]

J. C. Ricklin, S. Bucaille, and F. M. Davidson, "Performance loss factors for optical communication through clear air turbulence," Proc. SPIE 5160, 1-12 (2004).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

A. Glindemann, "Relevant parameters for tip-tilt systems on large telescopes," Publ. Astron. Soc. Pac. 109, 682-687 (1997).
[CrossRef]

Other (5)

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Oxford Science Publications, 1998).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, 1971).

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

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

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Figures (10)

Fig. 1
Fig. 1

(Color online) Mean SNR 0 when the wavefront is not corrected and when C- and Z-tilt corrections are performed. Simulation results are compared with exact values.

Fig. 2
Fig. 2

(Color online) Normalized variances of i ϕ estimated from simulations and plotted along with their fitting curves.

Fig. 3
Fig. 3

(Color online) Comparison of the i ϕ distribution with the beta distribution. The beta fit improves as max ( F i ϕ F beta ) decreases.

Fig. 4
Fig. 4

(Color online) Cumulative density function of SNR 0 for the case D / r ϕ = 2 .

Fig. 5
Fig. 5

(Color online) Level of fades occurring with a probability of 1% for i = i ϕ .

Fig. 6
Fig. 6

(Color online) Ratio of the mean squares evaluated with and without the approximation i i A i ϕ .

Fig. 7
Fig. 7

(Color online) Same as Fig. 6 but for the ratio of the normalized variances.

Fig. 8
Fig. 8

(Color online) Same as Fig. 5 but with scintillation.

Fig. 9
Fig. 9

(Color online) Impairment caused by turbulence on the SNR. Results for coherent and direct detections are displayed (DD = direct detection).

Fig. 10
Fig. 10

(Color online) Aperture averaging factor a and detector diameter factor β det plotted on the same graph. As the receiver diameter D increases less scintillation is experienced due to aperture averaging. But the size of the detector in the focal plane must be adjusted to account for growing wavefront distortions.

Tables (2)

Tables Icon

Table 1 Parameter Values for Fitting Curves of σ i , 0 2

Tables Icon

Table 2 Limit Cases of the Photocurrent i

Equations (36)

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i = Re [ W ( r ) E s ( r ) E LO * ( r ) d r ] ,
i = | W ( r ) E s ( r ) d r | = | W ( r ) A ( r ) exp [ j ϕ ( r ) ] d r | ,
A 2 = I = 1 ,
W ( r ) = { 4 / ( π D 2 ) , if | r | D / 2 0 , if | r | > D / 2 .
D ϕ ( ρ ) = 6.88 ( ρ r ϕ ) 5 / 3 ,
i ϕ | W ( r ) exp [ j ϕ ( r ) ] d r | ,
i ϕ 2 = 2 π H W ( ρ ) R ϕ ( ρ ) ρ d ρ ,
H W ( ρ ) = W ( r ) W ( r + ρ ) d r .
R ϕ ( ρ ) = exp [ 0.5 D ϕ ( ρ ) ] .
SNR 0 ( D / r ϕ ) 2 i 2 .
SNR 0, u ( D / r ϕ ) 2 [ 1 + ( D / r ϕ ) 5 / 3 ] 6 / 5 .
SNR 0, C SNR 0, u ( 1 + ( 0.985 D / r ϕ ) 1.84 1 + ( 0.331 D / r ϕ ) 3.18 ) .
σ i , 0 2 i 2 i 2 1 .
  σ i , 0 2 0.273 ( ( a D / r ϕ ) c 1 + ( b D / r ϕ ) d + ( e D / r ϕ ) f 1 + ( e D / r ϕ ) f ) .
lim D / r ϕ σ i , 0 2 = 4 π 1 .
F beta ( x ) = B x ( α , β ) B ( α , β ) ,   0 x 1 ,
R A ( ρ ) = [ B I ( ρ ) + 1 σ I 2 + 1 ] 1 / 4 .
R A ( ρ ) = exp [ 0.5 D χ ( ρ ) ] .
A = ( σ I 2 + 1 ) 1 / 8 .
F I ( κ ) [ 1 κ I 2 κ 2 sin ( κ 2 κ I 2 ) ] κ 11 / 3 ,
χ ϕ ( ρ ) D χ ϕ ( ρ ) D χ ( ρ ) D ϕ ( ρ ) ,
i A W ( r ) A ( r ) d r .
i A = A ,
i A 2 = 2 π H W ( ρ ) R A ( ρ ) ρ d ρ .
i A 2 i ϕ 2 i 2 = 2 π H W ( ρ ) R A ( ρ ) ρ d ρ H W ( ρ ) R ϕ ( ρ ) ρ d ρ H W ( ρ ) R A ( ρ ) R ϕ ( ρ ) ρ d ρ .
lim D / r ϕ i A 2 i ϕ 2 i 2 = A 2 R ϕ ( ρ ) ρ d ρ R A ( ρ ) R ϕ ( ρ ) ρ d ρ ,
lim D / r ϕ σ A ϕ , 0 2 σ i , 0 2 = 1 ,
F i ( x ) = 0 1 f i ϕ ( y ) F i A ( x y ) d y ,
SNR D D , T N P 2 ,
P = W ( r ) A 2 ( r ) d r .
SNR D D , S N P .
PSF L T ( r ) exp [ K D 2 ( 1 + D / r ϕ ) 5 / 3 r 2 ] ,
PSF S T ( r ) exp [ K D 2 ( 1 + 0.28 D / r ϕ ) 5 / 3 r 2 ] .
β det = ( 1 + D / r ϕ ) 5 / 6 ,
β det = ( 1 + 0.28 D / r ϕ ) 5 / 6 ,
a = σ P 2 σ I 2 .

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