Abstract

The beam-wander contribution to the scintillation in a ground-to-satellite free-space optical link is one of major importance. An analytical model, based on the duality between beam wander and angle-of-arrival fluctuations, is proposed for the temporal statistics. The expression of the probability density function of the log-amplitude fluctuations is first obtained. Then, the expressions of the spatial and temporal autocovariances are also obtained. We present plots of the beam-wander contribution to the log-amplitude variance, as a function of the transmitter aperture size and the turbulence accumulated in the propagation path. We also present the angular fluctuation and log-amplitude scintillation spectrum plots for some selected cases.

© 2005 Optical Society of America

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References

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  1. F. Dios, J. A. Rubio, A. Rodriguez, A. Comeron, “Scintillation and beam-wander analysis in an optical ground station-satellite uplink,” Appl. Opt. 43, 3866–3873 (2004).
    [CrossRef] [PubMed]
  2. L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media, Vol. PM53 of SPIE Press Monograph Series (SPIE, Bellingham, Wash., 1998).
  3. J. H. Churnside, R. J. Lataitis, “Wander of an optical beam in the turbulent atmosphere,” Appl. Opt. 29, 926–930 (1990).
    [CrossRef] [PubMed]
  4. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
    [CrossRef]
  5. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence. Evaluation and Application of Mellin Transforms (Springer-Verlag, Berlin, 1994).
    [CrossRef]
  6. R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1998).
  7. R. E. Hufnagel, “Propagation through atmospheric turbulence,” in The Infrared Handbook,W. L. Wolfe, G. J. Zissis, eds. (Office of Naval Research, Washington, D.C., 1978).
  8. R. R. Beland, “Propagation through atmospheric optical turbulence,” in Atmospheric Propagation of Radiation, Vol. 2 of The Infrared & Electro-Optical Systems Handbook, F. G. Smith, ed. (SPIE, Bellingham, Wash., 1993).
  9. A. Ishimaru, “The beam wave case and remote sensing,” in Laser Beam Propagation in the Atmosphere,J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978).
    [CrossRef]
  10. A. Ishimaru, Wave Propagation and Scattering in Random Media, (Academic, New York, 1978), Vol. 2.
  11. H. T. Yura, “Short-term average optical-beam spread in a turbulent médium,” J. Opt. Soc. Am. 63, 567–572 (1973).
    [CrossRef]
  12. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972).
    [CrossRef] [PubMed]
  13. A. Papoulis, Probability, Random Variables and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984).
  14. D. L. Fried, “Statistics of laser beam fade induced by pointing jitter,” Appl. Opt. 12, 422–423 (1973).
    [CrossRef] [PubMed]
  15. J. N. Pierce, “Theoretical diversity improvement in frequency-shift keying,” Proc. IRE903–910 (1958).
    [CrossRef]
  16. Z. Sodnik, R. H. Czichy, “Design data summary of the ESA optical ground station for in-orbit check-out of laser communication payloads and for the observation and registration of space debris,” in ARTEMIS Laser Link for Atmospheric Turbulence Statistics (European Space Agency, 2001).
  17. “SP-8-1-1 ARTEMIS satellite performance specification,” Sigue 9, Appendix D, (European Space Agency, 1998).
  18. A. Comeron, J. A. Rubio, A. Belmonte, “ASTC inter-island measurement campaign. Final report. Data analysis. Atmospheric modelling. Fade and bit-error-rate statistics” (European Space Agency, July1996).
  19. T. Prud’homme, X. Calbet, E. Garcia, S. Hughes, “Optical ground station atmospheric seeing test campaign. Stellar seeing measurements,” (European Space Agency, April1996).
  20. G. J. Baker, R. S. Benson, “Gaussian beam scintillation on ground to space paths: the importance of beam wander,” in Free-Space Laser Communications IV, Proc. SPIE5550, 225–235 (2004).
    [CrossRef]
  21. I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, eds., Table of Integrals, Series and Products (Academic, San Diego, 1994).

2004 (1)

1990 (1)

1975 (1)

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

1973 (2)

1972 (1)

1958 (1)

J. N. Pierce, “Theoretical diversity improvement in frequency-shift keying,” Proc. IRE903–910 (1958).
[CrossRef]

Andrews, L. C.

L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media, Vol. PM53 of SPIE Press Monograph Series (SPIE, Bellingham, Wash., 1998).

Baker, G. J.

G. J. Baker, R. S. Benson, “Gaussian beam scintillation on ground to space paths: the importance of beam wander,” in Free-Space Laser Communications IV, Proc. SPIE5550, 225–235 (2004).
[CrossRef]

Beland, R. R.

R. R. Beland, “Propagation through atmospheric optical turbulence,” in Atmospheric Propagation of Radiation, Vol. 2 of The Infrared & Electro-Optical Systems Handbook, F. G. Smith, ed. (SPIE, Bellingham, Wash., 1993).

Belmonte, A.

A. Comeron, J. A. Rubio, A. Belmonte, “ASTC inter-island measurement campaign. Final report. Data analysis. Atmospheric modelling. Fade and bit-error-rate statistics” (European Space Agency, July1996).

Benson, R. S.

G. J. Baker, R. S. Benson, “Gaussian beam scintillation on ground to space paths: the importance of beam wander,” in Free-Space Laser Communications IV, Proc. SPIE5550, 225–235 (2004).
[CrossRef]

Calbet, X.

T. Prud’homme, X. Calbet, E. Garcia, S. Hughes, “Optical ground station atmospheric seeing test campaign. Stellar seeing measurements,” (European Space Agency, April1996).

Churnside, J. H.

Comeron, A.

F. Dios, J. A. Rubio, A. Rodriguez, A. Comeron, “Scintillation and beam-wander analysis in an optical ground station-satellite uplink,” Appl. Opt. 43, 3866–3873 (2004).
[CrossRef] [PubMed]

A. Comeron, J. A. Rubio, A. Belmonte, “ASTC inter-island measurement campaign. Final report. Data analysis. Atmospheric modelling. Fade and bit-error-rate statistics” (European Space Agency, July1996).

Czichy, R. H.

Z. Sodnik, R. H. Czichy, “Design data summary of the ESA optical ground station for in-orbit check-out of laser communication payloads and for the observation and registration of space debris,” in ARTEMIS Laser Link for Atmospheric Turbulence Statistics (European Space Agency, 2001).

Dios, F.

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Fried, D. L.

Garcia, E.

T. Prud’homme, X. Calbet, E. Garcia, S. Hughes, “Optical ground station atmospheric seeing test campaign. Stellar seeing measurements,” (European Space Agency, April1996).

Hufnagel, R. E.

R. E. Hufnagel, “Propagation through atmospheric turbulence,” in The Infrared Handbook,W. L. Wolfe, G. J. Zissis, eds. (Office of Naval Research, Washington, D.C., 1978).

Hughes, S.

T. Prud’homme, X. Calbet, E. Garcia, S. Hughes, “Optical ground station atmospheric seeing test campaign. Stellar seeing measurements,” (European Space Agency, April1996).

Ishimaru, A.

A. Ishimaru, “The beam wave case and remote sensing,” in Laser Beam Propagation in the Atmosphere,J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media, (Academic, New York, 1978), Vol. 2.

Lataitis, R. J.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984).

Phillips, R. L.

L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media, Vol. PM53 of SPIE Press Monograph Series (SPIE, Bellingham, Wash., 1998).

Pierce, J. N.

J. N. Pierce, “Theoretical diversity improvement in frequency-shift keying,” Proc. IRE903–910 (1958).
[CrossRef]

Prud’homme, T.

T. Prud’homme, X. Calbet, E. Garcia, S. Hughes, “Optical ground station atmospheric seeing test campaign. Stellar seeing measurements,” (European Space Agency, April1996).

Rodriguez, A.

Rubio, J. A.

F. Dios, J. A. Rubio, A. Rodriguez, A. Comeron, “Scintillation and beam-wander analysis in an optical ground station-satellite uplink,” Appl. Opt. 43, 3866–3873 (2004).
[CrossRef] [PubMed]

A. Comeron, J. A. Rubio, A. Belmonte, “ASTC inter-island measurement campaign. Final report. Data analysis. Atmospheric modelling. Fade and bit-error-rate statistics” (European Space Agency, July1996).

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence. Evaluation and Application of Mellin Transforms (Springer-Verlag, Berlin, 1994).
[CrossRef]

Sodnik, Z.

Z. Sodnik, R. H. Czichy, “Design data summary of the ESA optical ground station for in-orbit check-out of laser communication payloads and for the observation and registration of space debris,” in ARTEMIS Laser Link for Atmospheric Turbulence Statistics (European Space Agency, 2001).

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1998).

Yura, H. T.

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Proc. IRE (1)

J. N. Pierce, “Theoretical diversity improvement in frequency-shift keying,” Proc. IRE903–910 (1958).
[CrossRef]

Other (14)

Z. Sodnik, R. H. Czichy, “Design data summary of the ESA optical ground station for in-orbit check-out of laser communication payloads and for the observation and registration of space debris,” in ARTEMIS Laser Link for Atmospheric Turbulence Statistics (European Space Agency, 2001).

“SP-8-1-1 ARTEMIS satellite performance specification,” Sigue 9, Appendix D, (European Space Agency, 1998).

A. Comeron, J. A. Rubio, A. Belmonte, “ASTC inter-island measurement campaign. Final report. Data analysis. Atmospheric modelling. Fade and bit-error-rate statistics” (European Space Agency, July1996).

T. Prud’homme, X. Calbet, E. Garcia, S. Hughes, “Optical ground station atmospheric seeing test campaign. Stellar seeing measurements,” (European Space Agency, April1996).

G. J. Baker, R. S. Benson, “Gaussian beam scintillation on ground to space paths: the importance of beam wander,” in Free-Space Laser Communications IV, Proc. SPIE5550, 225–235 (2004).
[CrossRef]

I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, eds., Table of Integrals, Series and Products (Academic, San Diego, 1994).

A. Papoulis, Probability, Random Variables and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984).

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence. Evaluation and Application of Mellin Transforms (Springer-Verlag, Berlin, 1994).
[CrossRef]

R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1998).

R. E. Hufnagel, “Propagation through atmospheric turbulence,” in The Infrared Handbook,W. L. Wolfe, G. J. Zissis, eds. (Office of Naval Research, Washington, D.C., 1978).

R. R. Beland, “Propagation through atmospheric optical turbulence,” in Atmospheric Propagation of Radiation, Vol. 2 of The Infrared & Electro-Optical Systems Handbook, F. G. Smith, ed. (SPIE, Bellingham, Wash., 1993).

A. Ishimaru, “The beam wave case and remote sensing,” in Laser Beam Propagation in the Atmosphere,J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media, (Academic, New York, 1978), Vol. 2.

L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media, Vol. PM53 of SPIE Press Monograph Series (SPIE, Bellingham, Wash., 1998).

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Figures (10)

Fig. 1
Fig. 1

Geometrical arrangement for the computation of the joint probability density of the angular fluctuations. The plane z = 0 corresponds to the location of the transmitting aperture. We use the classical approach of considering two apertures, separated by a distance ρ to calculate the spatial autocorrelation.

Fig. 2
Fig. 2

Structure constant of the fluctuations of the index of refraction as a function of height over sea level, using Ref. 18.

Fig. 3
Fig. 3

Short-exposure beam-spread mean value, as a function of the transmitter aperture size, for different path elevation angles.

Fig. 4
Fig. 4

Beam-wander contribution to scintillation variance at the spaceborne receiver, as a function of the transmitter aperture size, for different path elevation angles.

Fig. 5
Fig. 5

Angular fluctuation spectrum for a 1 cm diameter transmitter aperture, 5 m/s wind speed, and for different path elevation angles.

Fig. 6
Fig. 6

Angular fluctuation spectrum for a 10 cm diameter transmitter aperture, 5 m/s wind speed, and for different path elevation angles.

Fig. 7
Fig. 7

Angular fluctuation spectrum for a 1 m diameter transmitter aperture, 5 m/s wind speed, and for different path elevation angles.

Fig. 8
Fig. 8

Spectrum beam-wander contribution to scintillation for a 1 cm diameter transmitter aperture, 5 m/s wind speed, and for different path elevation angles.

Fig. 9
Fig. 9

Spectrum beam-wander contribution to scintillation for a 1 m diameter transmitter aperture, 5 m/s wind speed, and for different path elevation angles.

Fig. 10
Fig. 10

Spectrum beam-wander contribution to scintillation for a 10 cm diameter transmitter aperture, 5 m/s wind speed, and for different path elevation angles.

Equations (46)

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f α ( α ) = α σ α 2 e - α 2 2 σ α 2 .
σ α 2 6.13 k 2 W 0 1 / 3 r 0 5 / 3 ,
r 0 = [ 0.42 k 2 0 L C n 2 ( z ) d     z ] - 3 / 5 ,
I ( θ ) = I M e - θ 2 Δ 2 θ ,
Δ θ W ( L ) L ,
W ( L ) = [ W 0 2 + ( L     λ π     W 0 ) 2 + 2 ( L     λ π ρ s t ) 2 ] 1 / 2 .
ρ s t = ρ c [ 1 + 0.33 ( ρ c W 0 ) 1 / 3 ] .
ρ c = r 0 2 [ 1 + θ 0 2 2 ( 1 + 8 W 0 2 3 r 0 2 ) 1 + θ 0 2 6 ( 1 + 2 W 0 2 r 0 2 ) ] 1 / 2 ,
I = I M e - α 2 Δ 2 θ .
χ W = 1 2 ln I I M = - 1 2 α 2 Δ 2 θ = K     α 2 .
f χ W ( χ W ) = f α [ f - 1 ( χ W ) ] f [ f - 1 ( χ W ) ] .
f χ W ( χ W ) = ( Δ θ σ α ) 2 e ( Δ θ σ α ) χ W 2 ,     with     χ W 0.
C χ W ( ρ ) = [ χ W ( ρ + ρ ) - μ χ W ] [ χ W ( ρ ) - μ χ W ] = χ W ( ρ + ρ ) χ W ( ρ ) - μ χ W 2 ,
R χ W ( ρ ) = χ W ( ρ + ρ ) χ W ( ρ ) = f [ α ( ρ + ρ ) ] f [ α ( ρ ) ] ,
f [ α ( ρ + ρ ) ] f [ α ( ρ ) ] = - - f ( α 1 ) f ( α 2 ) f α α ( α 1 , α 2 ; ρ ) d α 1 d α 2 ,
α i = α i x 2 + α i y 2 ,     with i = 1 , 2.
F α 1 x α 1 y α 2 x α 2 y ( α 1 x α 1 y α 2 x α 2 y ; ρ ) = 1 ( 2 π ) 2 C 1 / 2 × exp ( - α T C - 1 α 2 ) ,
c i p j q ( ρ ) = α i p α j q ,
c 1 u 2 u ( ρ ) = 16 π 2 W 0 2 × 0.033 0 L C n 2 ( z ) 0 K - 8 / 3 × J 1 2 ( W 0 2 K ) J 0 ( ρ K ) d K d z ,
f α 1 x α 1 y α 2 x α 2 y ( α 1 x , α 1 y , α 2 x , α 2 y ; ρ ) = 1 ( 2 π ) 2 [ σ α 4 - c 1 u 2 u 2 ( ρ ) ] e - p ( α 1 x 2 + α 1 y 2 + α 2 x 2 + α 2 y 2 ) + 2 q ( α 1 x α 2 x + α 1 y α 2 y ) 2 ,
p = σ α 2 σ α 4 - c 1 u 2 u 2 ( ρ ) ,
q = - c 1 u 2 u ( ρ ) σ α 4 - c 1 u 2 u 2 ( ρ ) .
α 1 x 2 + α 1 y 2 + α 2 x 2 + α 2 y 2 = α 1 2 + α 2 2 ,
α 1 x α 2 x + α 1 y α 2 y = α 1 α 2 cos ( ϕ - ϕ ) .
f α α ( α 1 , α 2 ; ρ ) d α 1 d α 2 = p { α 1 < α 1 < α 1 + d α 1 , α 2 < α 2 < α 2 + d α 2 } ,
f α α ( α 1 ,     α 2 ; ρ ) d α 1 d α 2 = 1 ( 2 π ) 2 [ σ α 4 - c 1 u 2 u 2 ( ρ ) α 1 α 2 e - p ( α 1 2 + α 2 2 ) 2 × 0 2 π 0 2 π e - q α 1 α 2 cos ( ϕ - ϕ ) d ϕ d ϕ d α 1 d α 2 .
0 2 π e - q α 1 α 2 cos ( ϕ - ϕ ) d ϕ = 2 π I o ( - q α 1 α 2 ) ,
f α α ( α 1 , α 2 ; ρ ) = 1 σ α 4 - c 1 u 2 u 2 ( ρ ) α 1 α 2 × exp { - σ α 2 ( α 1 2 + α 2 2 ) 2 [ σ α 4 - c 1 u 2 u 2 ( ρ ) ] } × I 0 [ c 1 u 2 u ( ρ ) σ α 4 - c 1 u 2 u 2 ( ρ ) α 1 α 2 ] ,
R χ W ( ρ ) = 1 4 Δ 4 θ [ σ α 4 - c 1 u 2 u 2 ( ρ ) ] × 0 0 α 1 3 α 2 3 exp { - σ α 2 ( α 1 2 + α 2 2 ) 2 [ σ α 4 - c 1 u 2 u 2 ( ρ ) ] } × I 0 [ c 1 u 2 u ( ρ ) σ α 4 - c 1 u 2 u 2 ( ρ ) α 1 α 2 ] d α 1 d α 2 ,
R χ W ( ρ ) = 1 Δ 4 θ [ σ α 4 + c 1 u 2 u 2 ( ρ ) ] .
C χ W ( ρ ) = c 1 u 2 u 2 ( ρ ) Δ 4 θ .
C χ W ( τ ) = c 1 u 2 u 2 ( τ ) Δ 4 θ ,
C n 2 ( h ) = { C n 0 2 ( h h s ) - 2 / 3 h h s C n 0 2 h s < h < h i , C n 0 2 e h i h r e - h h r + C n l 2 e - h h l + C n l 2 3 W 2 ( h h t ) 10 e - 10 h h t h > h i
s = σ α 4 - c 12 2 ( ρ ) .
0 α 2 3 exp ( - σ a 2 α 2 2 2 s ) I 0 ( c 12 s α 1 α 2 ) d α 2 ,
α 2 2 = x , d α 2 = 1 2 x - 1 / 2 d x .
1 2 0 x exp ( - σ α 2 x 2 s ) J 0 ( i c 12 s α 1 x 1 / 2 ) d x .
0 x n + 1 2 ν exp ( - α x ) J ν ( 2 β x ) d x = n ! β ν exp ( - β 2 α ) α - n - ν - 1 L n ν ( β 2 α ) ,
n = 1 , α = σ a 2 2 s , β = i     c 12 α 1 2 s ,     ν = 0 ,
L 1 0 ( z ) = 1 - z .
s 2 2 σ α 4 exp [ ( c 12 σ α ) 2 α 1 2 2 s ] [ 1 + ( c 12 σ α ) 2 α 1 2 2 s ] .
R χ W ( ρ ) = 2 K 2 s σ α 4 [ 0 α 1 3 exp ( - α 1 2 2 σ a 2 ) d α 1 + 1 2 s ( c 12 σ a ) 2 0 α 1 5 exp ( - α 1 2 2 σ a 2 ) d α 1 ] .
α 1 2 = x , d α 1 = 1 2 x - 1 / 2 d x ,
R χ W ( ρ ) = K 2 s σ α 4 [ 0 x exp ( - x 2 σ a 2 ) d x + 1 2 s ( c 12 σ α ) 2 × 0 x 2 exp ( - x 2 σ a 2 ) d x ] .
0 x n exp ( - μ x ) d x = n ! μ - n - 1 .
R χ W ( ρ ) = 1 Δ 4 θ [ σ α 4 + c 12 2 ( ρ ) ] .

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