Abstract

Estimates of the scintillation index, fractional fade time, expected number of fades, and mean duration of fade time associated with a propagating Gaussian-beam wave are developed for uplink and downlink laser satellite-communication channels. Estimates for the spot size of the beam at the satellite or the ground or airborne receiver are also provided. Weak-fluctuation theory based on the log-normal model is applicable for intensity fluctuations near the optical axis of the beam provided that the zenith angle is not too large, generally not exceeding 60°. However, there is an increase in scintillations that occurs with increasing pointing error at any zenith angle, particularly for uplink channels. Large off-axis scintillations are of particular significance because they imply that small pointing errors can cause serious degradation in the communication-channel reliability. Off-axis scintillations increase more rapidly for larger-diameter beams and, in some cases, can lead to a radial saturation effect for pointing errors less than 1 μrad off the optical beam axis.

© 1995 Optical Society of America

Full Article  |  PDF Article

Errata

Larry C. Andrews, Ronald L. Phillips, and Peter T. Yu, "Optical scintillations and fade statistics for a satellite-communication system: errata," Appl. Opt. 36, 6068-6069 (1997)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-36-24-6068

References

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  1. P. O. Minott, “Scintillation in an earth-to-space propagation path,” J. Opt. Soc. Am. 62, 885–888 (1972).
    [CrossRef]
  2. J. L. Bufton, R. S. Iyler, L. S. Taylor, “Scintillation statistics caused by atmospheric turbulence and speckle in satellite laser ranging,” Appl. Opt. 16, 2408–2413 (1977).
    [CrossRef] [PubMed]
  3. J. L. Bufton, “Scintillation statistics measured in an earth-space-earth retroreflected link,” Appl. Opt. 16, 2654–2660 (1977).
    [CrossRef] [PubMed]
  4. H. T. Yura, W. G. McKinley, “Optical scintillation statistics for IR ground-to-space laser communication systems,” Appl. Opt. 22, 3353–3358 (1983).
    [CrossRef] [PubMed]
  5. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.
  6. W. B. Miller, J. C. Ricklin, L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10, 661–672 (1993).
    [CrossRef]
  7. W. B. Miller, J. C. Ricklin, L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11, 2719–2726 (1994).
    [CrossRef]
  8. P. A. Lightsey, “Scintillation in ground-to-space and retro-reflected laser beams,” Opt. Eng. 33, 2535–2543 (1994).
    [CrossRef]
  9. R. R. Beland, “Propagation through atmospheric optical turbulence,” in The Infrared and ElectroOptical Systems Handbook, F. G. Smith, ed. (SPIEOptical Engineering Press, Bellingham, Wash., 1993), Vol. 2, Chap. 2.
  10. L. C. Andrews, W. B. Miller, J. C. Ricklin, “Geometrical representation of Gaussian beams propagating through complex paraxial optical systems,” Appl. Opt. 32, 5918–5929 (1993).
    [CrossRef] [PubMed]
  11. L. C. Andrews, W. B. Miller, “Single-pass and double-pass propagation through complex paraxial optical systems,” J. Opt. Soc. Am. A 12, 137–150 (1995).
    [CrossRef]
  12. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (McGraw-Hill, New York, 1992).
  13. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
    [CrossRef]
  14. L. C. Andrews, W. B. Miller, J. C. Ricklin, “Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence,” J. Opt. Soc. Am. A 11, 1653–1660 (1994).
    [CrossRef]
  15. P. Beckman, Probability in Communication and Engineering (Harcourt, Brace & World, New York, 1967).
  16. R. J. Sasiela, J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1993).
    [CrossRef]

1995 (1)

1994 (3)

1993 (3)

1983 (1)

1977 (2)

1975 (1)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

1972 (1)

Andrews, L. C.

Beckman, P.

P. Beckman, Probability in Communication and Engineering (Harcourt, Brace & World, New York, 1967).

Beland, R. R.

R. R. Beland, “Propagation through atmospheric optical turbulence,” in The Infrared and ElectroOptical Systems Handbook, F. G. Smith, ed. (SPIEOptical Engineering Press, Bellingham, Wash., 1993), Vol. 2, Chap. 2.

Bufton, J. L.

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Gochelashvily, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Ishimaru, A.

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

Iyler, R. S.

Lightsey, P. A.

P. A. Lightsey, “Scintillation in ground-to-space and retro-reflected laser beams,” Opt. Eng. 33, 2535–2543 (1994).
[CrossRef]

McKinley, W. G.

Miller, W. B.

Minott, P. O.

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Ricklin, J. C.

Sasiela, R. J.

Shelton, J. D.

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Taylor, L. S.

Yura, H. T.

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Eng. (1)

P. A. Lightsey, “Scintillation in ground-to-space and retro-reflected laser beams,” Opt. Eng. 33, 2535–2543 (1994).
[CrossRef]

Proc. IEEE (1)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–809 (1975).
[CrossRef]

Other (4)

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (McGraw-Hill, New York, 1992).

P. Beckman, Probability in Communication and Engineering (Harcourt, Brace & World, New York, 1967).

R. R. Beland, “Propagation through atmospheric optical turbulence,” in The Infrared and ElectroOptical Systems Handbook, F. G. Smith, ed. (SPIEOptical Engineering Press, Bellingham, Wash., 1993), Vol. 2, Chap. 2.

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

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Figures (15)

Fig. 1
Fig. 1

Schematic of the uplink propagation channel, illustrating the Gaussian-beam profile of the mean intensity at the satellite and the relative pointing error. The height above the ground of the receiver–transmitter is h 0, the height of the satellite above the ground is H, and the slant path range L is given by L = (Hh 0)/cos(ζ), where ζ is the zenith angle.

Fig. 2
Fig. 2

C n 2(h) profile associated with the H–V day model as a function of altitude h. The values of A shown represent C n 2 near ground level, whereas ν denotes high-altitude wind speed.

Fig. 3
Fig. 3

Uplink effective beam radius at the receiver (in meters) as a function of beam radius for a collimated beam at the transmitter (in centimeters), zenith angles 0° and 60°, and λ = 1.5 μm. The atmospheric model is the H–V day model.

Fig. 4
Fig. 4

Uplink scintillation index on the optical beam axis (r = 0) for a point receiver at the satellite as a function of the transmitter beam radius of a collimated beam. The atmospheric model is the H–V day model.

Fig. 5
Fig. 5

Uplink scintillation index for a point receiver at the satellite as a function of angular receiver position from the optical axis for two collimated beam sizes at the transmitter and various zenith angles. The atmospheric model is the H–V day model.

Fig. 6
Fig. 6

Uplink fractional fade time or probability of a miss for a point receiver at the satellite as a function of angular receiver position and two zenith angles. The optical wave at the transmitter (h 0 = 0) is a collimated beam, the specified fade level at the receiver is 6 dB, and the atmospheric model is the H–V day model.

Fig. 7
Fig. 7

Uplink fractional fade time or probability of a miss for two zenith angles as a function of fade level F T below the mean on-axis intensity. The solid curves represent a point receiver on the optical axis, and the dotted curves represent a point receiver at 5-μrad offset from the optical axis as a result of pointing errors. The optical wave is a collimated beam (h 0 = 0), and the atmospheric model is the H–V day model.

Fig. 8
Fig. 8

Downlink fractional fade time or probability of miss for a point receiver on the ground as a function of angular receiver position and two zenith angles. The optical wave at the transmitter is a collimated beam, the specified fade level at the receiver is 6 dB, and the atmospheric model is the H–V day model.

Fig. 9
Fig. 9

Downlink fractional fade time or probability of a miss for two zenith angles as a function of fade level F T below the mean on-axis intensity. Both curves represent a point receiver on the optical axis. The optical wave is a collimated beam, and the atmospheric model is the H–V day model.

Fig. 10
Fig. 10

Uplink mean level crossing frequency or expected number of fades per second as a function of fade level F T below the mean on-axis intensity and two zenith angles. The solid curves represent a point receiver on the optical axis, and the dotted curves represent a point receiver at 5-μrad offset from the optical axis as a result of pointing errors. The optical wave is a collimated beam and the atmospheric model is the H–V day model.

Fig. 11
Fig. 11

Downlink mean level crossing frequency or expected number of fades per second as a function of fade level F T below the mean on-axis intensity and two zenith angles. Both curves represent a point receiver on the optical axis. The optical wave is a collimated beam, and the atmospheric model is the H–V day model.

Fig. 12
Fig. 12

Uplink mean duration of a fade for a point receiver at the satellite as a function of angular receiver position and two zenith angles. The optical wave at the transmitter is a collimated beam, the specified fade level at the receiver is 6 dB, and the atmospheric model is the H–V day model.

Fig. 13
Fig. 13

Uplink mean duration of a fade for a point receiver at the satellite as a function of fade level F T below the mean on-axis intensity and two zenith angles. The solid curves represent a point receiver on the optical axis, and the dotted curves represent a point receiver at 5-μrad offset from the optical axis as a result of pointing errors. The optical wave is a collimated beam, and the atmospheric model is the H–V day model.

Fig. 14
Fig. 14

Downlink mean duration of a fade for a point receiver on the ground as a function of angular receiver position and two zenith angles. The optical wave at the transmitter is a collimated beam, the specified fade level at the receiver is 6 dB, and the atmospheric model is the H–V day model.

Fig. 15
Fig. 15

Downlink mean duration of a fade for a point receiver on the ground as a function of fade level F T below the mean on-axis intensity and two zenith angles. Both curves represent the point receiver on the optical axis. The optical wave is a collimated beam, and the atmospheric model is the H–V day model.

Equations (44)

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C n 2 ( h ) = 0 . 00594 ( ν / 27 ) 2 ( 10 5 h ) 10 exp ( h / 1000 ) + 2 . 7 × 10 16 exp ( h / 1500 ) + A exp ( h / 100 ) ,
Ω 0 = 1 L F 0 , Ω = 2 L k W 0 2 ,
θ = Ω 0 Ω 0 2 + Ω 2 , θ ˜ = 1 θ, Λ = 2 L k W 2 = Ω Ω 0 2 + Ω 2 ,
I 0 ( r , L ) = W 0 2 W 2 exp ( 2 r 2 / W 2 ) .
U ( r , L ) = U 0 ( r , L ) exp [ ψ ( r , L ) ] ,
ψ ( r , L ) = ψ 1 ( r , L ) + ψ 2 ( r , L ) + …,
E 1 ( 0 , 0 ) = ψ 2 ( r , L ) + 1 2 ψ 1 2 ( r , L ) = 2 π 2 k 2 sec ( ζ ) h 0 H 0 κ Φ n ( h , κ ) d κ d h ,
E 2 ( r 1 , r 2 ) = ψ 1 ( r 1 , L ) ψ 1 * ( r 2 , L ) = 4 π 2 k 2 sec ( ζ ) h 0 H 0 κ Φ n ( h , κ ) × exp ( Λ L κ 2 ξ 2 / k ) × J 0 [ κ | ( 1 θ ˜ ξ ) ρ 2 i Λ ξ r | ] d κ d h ,
E 3 ( r 1 , r 2 ) = ψ 1 ( r 1 , L ) ψ 1 ( r 2 , L ) = 4 π 2 k 2 sec ( ζ ) h 0 H 0 κ Φ n ( h , κ ) × exp ( Λ L κ 2 ξ 2 / k ) J 0 [ ( 1 θ ˜ ξ i Λ ξ ) κρ ] × exp [ i κ 2 L k ξ ( 1 θ ˜ ξ ) ] d κ d h ,
Φ n ( h , κ ) = 0 . 033 C n 2 ( h ) κ 11 / 3 .
I ( r , L ) = I 0 ( r , L ) exp [ 2 E 1 ( 0 , 0 ) + E 2 ( r , r ) ] ,
I ( r , L ) = W 0 2 W e 2 exp ( 2 r 2 / W e 2 ) ,
W e = W ( 1 + G u ) 1 / 2 ,
G u = 2 E 1 ( 0 , 0 ) E 2 ( 0 , 0 ) = 4 π 2 k 2 sec ( ζ ) h 0 H 0 κ Φ n ( h , κ ) × { 1 exp [ Λ L κ 2 k ( 1 h h 0 H h 0 ) 2 ] } d κ d h = 4 . 35 μ 1 Λ 5 / 6 k 7 / 6 ( H h 0 ) 5 / 6 sec 11 / 6 ( ζ ) ,
μ 1 = h 0 H C n 2 ( h ) ( 1 h h 0 H h 0 ) 5 / 3 d h .
G d = 4 π 2 k 2 sec ( ζ ) h 0 H 0 κ Φ n ( h , κ ) × { 1 exp [ Λ L κ 2 k ( h h 0 H h 0 ) 2 ] } d κ d h = 4 . 35 μ 2 Λ 5 / 6 k 7 / 6 ( H h 0 ) 5 / 6 sec 11 / 6 ( ζ ) ,
μ 2 = h 0 H C n 2 ( h ) ( h h 0 H h 0 ) 5 / 3 d h .
σ I 2 ( r , L ) σ L n I 2 ( r , L ) = 2 Re [ E 2 ( r , r ) + E 3 ( r , r ) ] ,
σ I 2 ( r , L ) = σ I 2 ( 0 , L ) + σ I , r 2 ( r , L ) .
σ I 2 ( 0 , L ) = 2 Re [ E 2 ( 0 , 0 ) + E 3 ( r , r ) ] = 8 π 2 k 2 sec ( ζ ) h 0 H 0 κ Φ n ( h , κ ) × exp [ Λ L κ 2 k ( 1 h H ) 2 ] × { 1 cos [ L κ 2 k ( 1 h H ) ( θ θ ˜ h H ) ] } d κ d h = 8 . 702 μ 3 k 7 / 6 ( H h 0 ) 5 / 6 sec 11 / 6 ( ζ ) ,
μ 3 = Re h 0 H C n 2 ( h ) { ( 1 h H ) 5 / 6 [ Λ ( 1 h H ) + i ( 1 + θ θ h H ) ] 5 / 6 Λ 5 / 6 ( 1 h H ) 5 / 3 } d h .
σ I , r 2 ( r , L ) = 2 Re [ E 2 ( r , r ) E 2 ( 0 , 0 ) ] = 8 π 2 k 2 sec ( ζ ) h 0 H 0 κ Φ n ( h , κ ) × exp [ Λ L κ 2 k ( 1 h H ) 2 ] × { I 0 [ 2 Λ κ r ( 1 h / H ) ] 1 } d κ d h = 14 . 508 μ 1 Λ 5 / 6 k 7 / 6 ( H h 0 ) 5 / 6 × sec 11 / 6 ( ζ ) ( r 2 W 2 ) , r W ,
σ I 2 ( r , L ) = 8 . 702 k 7 / 6 ( H h 0 ) 5 / 6 sec 11 / 6 ( ζ ) × [ μ 3 + 1 . 667 μ 1 Λ 5 / 6 α 2 ( H h 0 ) 2 sec 2 ( ζ ) W 0 2 ( Ω 0 2 + Ω 2 ) ] , α W / L ,
σ I 2 ( 0 , L ) = 8 . 702 μ 4 k 7 / 6 ( H h 0 ) 5 / 6 sec 11 / 6 ( ζ ) ,
μ 4 = Re h 0 H C n 2 ( h ) ( ( h H ) 5 / 6 { Λ ( h H ) + i [ 1 θ ˜ ( h H ) ] } 5 / 6 Λ 6 / 5 ( h H ) 5 / 3 ) d h .
σ I , r 2 ( r , L ) = 14 . 508 μ 2 k 7 / 6 Λ 5 / 6 ( H h 0 ) 5 / 6 × sec 11 / 6 ( ζ ) ( r W ) 2 , r W ,
σ I 2 ( r , L ) = 8 . 702 k 7 / 6 ( H h 0 ) 5 / 6 sec 11 / 6 ( ζ ) × [ μ 4 + 1 . 667 μ 2 Λ 5 / 6 α 2 ( H h 0 ) 2 sec 2 ( ζ ) W 0 2 ( Ω 0 2 + Ω 2 ) ] , α W / L .
b I ( ρ , L ) = B I ( ρ, L ) B I ( 0 , L ) = Re [ E 2 ( r , r ) + E 3 ( r , r ) ] Re [ E 2 ( 0 , 0 ) + E 3 ( 0 , 0 ) ] ,
p ( I ) = 1 I σ I ( r , L ) 2 π × exp ( { ln [ I I ( r , L ) ] + 1 2 σ I 2 ( r , L ) } 2 2 σ I 2 ( r , L ) ) ,
I ( r , L ) = W 0 2 W e 2 exp ( 2 r 2 / W e 2 ) = I ( 0 , L ) exp ( 2 r 2 / W e 2 ) .
p ( I ) = 1 I σ I ( r , L ) 2 π × exp ( { ln [ I I ( 0 , L ) ] + 2 r 2 W e 2 + 1 2 σ I 2 ( r , L ) } 2 2 σ I 2 ( r , L ) ) , I > 0 .
F T = 10 log 10 [ I ( 0 , L ) I T ] .
P ( I I T ) = 1 I σ I ( r , L ) 2 π 0 I T exp ( { ln [ I I ( 0 , L ) ] + 2 r 2 W e 2 + 1 2 σ I 2 ( r , L ) } 2 2 σ I 2 ( r , L ) ) d I = 1 2 { 1 + erf [ 1 2 σ I 2 ( r , L ) + 2 r 2 W e 2 0 . 23 F T 2 σ I ( r , L ) ] } ,
ln [ I T I ( 0 , L ) ] = 0 . 23 F T .
n ( I T ) = ν 0 exp { [ 1 2 σ I 2 ( r , L ) + 2 r 2 W e 2 0 . 23 F T ] 2 2 σ I 2 ( r , L ) } ,
ν 0 = 1 2 π [ B I ( 0 ; r , L ) B I ( 0 ; r , L ) ] 1 / 2 .
B I ( ν τ ; r , L ) = 8 π 2 k 2 ζ h 0 H 0 κ Φ n ( h , κ ) J 0 ( κ ν τ ) × exp ( Λ L κ 2 χ 2 / k ) { I 0 ( 2 Λ κ r ξ ) cos [ L κ 2 k ξ ( 1 θ ˜ ξ ) ] } d κ d h .
B I ( 0 ; r , L ) = σ I 2 ( r , L ) .
B I ( 0 ; r , L ) = 3 . 627 k 7 / 6 ( H h 0 ) 5 / 6 sec 11 / 6 ( ζ ) × ( k ν 2 L ) ( μ 5 + 1 3 μ 6 Λ 1 / 6 r 2 W 2 ) ,
μ 5 = Re h 0 H C n 2 ( h ) { Λ 1 / 6 ( 1 h H ) ( 1 h H ) 1 / 6 × [ Λ ( 1 h H ) + i ( 1 + θ θ h H ) ] 1 / 6 } d h ,
μ 6 = h 0 H C n 2 ( h ) ( 1 h h 0 H h 0 ) 1 / 3 d h .
B I ( 0 ; L ) 3 . 627 k 7 / 6 μ 7 ( H h 0 ) 5 / 6 sec 11 / 6 ( ζ ) ( k ν 2 L ) ,
μ 7 = h 0 H C n 2 ( h ) ( H h 0 h h 0 ) 1 / 6 d h .
t ( F T ) = P I ( I I T ) n ( I T ) = 1 2 ν 0 exp { [ 1 2 σ I 2 ( r , L ) + 2 r 2 W e 2 0 . 23 F T ] 2 2 σ I 2 ( r , L ) } × ( 1 + erf { [ 1 2 σ I 2 ( r , L ) + 2 r 2 W e 2 0 . 23 F T ] 2 σ I ( r , L ) } ) ,

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