Abstract

The scintillation statistics of a multiwavelength Gaussian optical beam are characterized when the beam is subjected to a turbulent optical channel. It is assumed that the level of turbulence in the atmosphere ensures a weak-turbulence scenario and that fluctuations in the signal intensity are due to variations in the refractive index of the medium, which in turn are caused by regional temperature variations due to atmospheric turbulence. Furthermore, it is assumed that the propagation path is nearly horizontal and that the heights of the transmitter and receiver justify a near-ground propagation assumption. The Rytov approximation is used to arrive at the desired results. Furthermore, it is assumed that the first- as well as second-order perturbation terms are present in modeling the impact of atmosphere-induced scintillation. Numerical results are presented to shed light on the performance of multiwavelength optical radiation in weak turbulence and to underscore the benefits of the proposed approach as compared with its single-wavelength counterpart in combating the effect of turbulence. Furthermore, it is shown that if the separation of wavelengths used is sufficiently large, wavelength separation affects the scintillation index in a measurable way.

© 2006 Optical Society of America

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References

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  1. K. Kiasaleh, 'Performance analysis of free-space on-off-keying optical communication systems impaired by turbulence,' in Free-Space Laser Communication Technologies XIV, S.Mecherle, ed., Proc. SPIE 4635, 150-161 (2002).
  2. K. Kiasaleh, 'Scintillation index of a multiwavelength beam in turbulent atmosphere,' J. Opt. Soc. Am. A 21, 1452-1454 (2004).
    [CrossRef]
  3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
  4. R. L. Fante, 'Electromagnetic beam propagation in turbulent media,' Proc. IEEE 63, 1669-1692 (1975).
    [CrossRef]
  5. D. L. Fried and J. B. Seidman, 'Laser beam scintillation in the atmosphere,' J. Opt. Soc. Am. 57, 181-185 (1967).
    [CrossRef]
  6. R. L. Fante, 'Two-position two-frequency mutual-coherence function in turbulence,' J. Opt. Soc. Am. 71, 1446-1461 (1981).
    [CrossRef]
  7. R. L. Fante, 'Wave propagation in random media: a systems approach,' in Progress in Optics XXII, E.Wolf, ed. (Elsevier, 1985).
    [CrossRef]
  8. H. T. Yura, C. C. Sung, S. F. Clifford, and R. J. Hill, 'Second-order Rytov approximation,' J. Opt. Soc. Am. 73, 500-502 (1983).
    [CrossRef]
  9. H. T. Yura and S. G. Hanson, 'Second-order statistics for wave propagation through complex optical systems,' J. Opt. Soc. Am. A 6, 564-575 (1989).
    [CrossRef]
  10. A. M. Obukhov, 'Effect of wave inhomogeneities in the atmosphere on sound and light propagation,' Izv. Akad. Nauk SSSR Ser. Geofiz. 2, 155-165 (1953).
  11. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961), translated by R. A. Silverman.
  12. A. Ishimura, Wave Propagation and Scattering in Random Media (IEEE, 1997).
  13. J. Strohbehn, ed., Laser Beam Propagation in the Atmosphere (Springer, 1978).
  14. W. B. Miller, J. C. Ricklin, and L. C. Andrews, 'Log-amplitude variance and wave structure function: a new presentation for Gaussian beams,' J. Opt. Soc. Am. A 11, 1653-1660 (1993).
  15. W. B. Miller, J. C. Ricklin, and 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]

2004

1994

1993

1989

1983

1981

1975

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

1967

1953

A. M. Obukhov, 'Effect of wave inhomogeneities in the atmosphere on sound and light propagation,' Izv. Akad. Nauk SSSR Ser. Geofiz. 2, 155-165 (1953).

Andrews, L. C.

Clifford, S. F.

Fante, R. L.

R. L. Fante, 'Two-position two-frequency mutual-coherence function in turbulence,' J. Opt. Soc. Am. 71, 1446-1461 (1981).
[CrossRef]

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

R. L. Fante, 'Wave propagation in random media: a systems approach,' in Progress in Optics XXII, E.Wolf, ed. (Elsevier, 1985).
[CrossRef]

Fried, D. L.

Hanson, S. G.

Hill, R. J.

Ishimura, A.

A. Ishimura, Wave Propagation and Scattering in Random Media (IEEE, 1997).

Kiasaleh, K.

K. Kiasaleh, 'Scintillation index of a multiwavelength beam in turbulent atmosphere,' J. Opt. Soc. Am. A 21, 1452-1454 (2004).
[CrossRef]

K. Kiasaleh, 'Performance analysis of free-space on-off-keying optical communication systems impaired by turbulence,' in Free-Space Laser Communication Technologies XIV, S.Mecherle, ed., Proc. SPIE 4635, 150-161 (2002).

Miller, W. B.

Obukhov, A. M.

A. M. Obukhov, 'Effect of wave inhomogeneities in the atmosphere on sound and light propagation,' Izv. Akad. Nauk SSSR Ser. Geofiz. 2, 155-165 (1953).

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Ricklin, J. C.

Seidman, J. B.

Strohbehn, J.

J. Strohbehn, ed., Laser Beam Propagation in the Atmosphere (Springer, 1978).

Sung, C. C.

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961), translated by R. A. Silverman.

Yura, H. T.

Izv. Akad. Nauk SSSR Ser. Geofiz.

A. M. Obukhov, 'Effect of wave inhomogeneities in the atmosphere on sound and light propagation,' Izv. Akad. Nauk SSSR Ser. Geofiz. 2, 155-165 (1953).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. IEEE

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

Other

K. Kiasaleh, 'Performance analysis of free-space on-off-keying optical communication systems impaired by turbulence,' in Free-Space Laser Communication Technologies XIV, S.Mecherle, ed., Proc. SPIE 4635, 150-161 (2002).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

R. L. Fante, 'Wave propagation in random media: a systems approach,' in Progress in Optics XXII, E.Wolf, ed. (Elsevier, 1985).
[CrossRef]

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961), translated by R. A. Silverman.

A. Ishimura, Wave Propagation and Scattering in Random Media (IEEE, 1997).

J. Strohbehn, ed., Laser Beam Propagation in the Atmosphere (Springer, 1978).

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Figures (9)

Fig. 1
Fig. 1

Scintillation index as a function of propagation distance for N = 1 (squares), N = 2 (crosses), and N = 4 (diamonds) at r = 0 .

Fig. 2
Fig. 2

Normalized scintillation index as a function of the Fresnel ratio (for the case of N = 1 ) for N = 1 (squares), N = 2 (crosses), and N = 4 (diamonds) at the beam center.

Fig. 3
Fig. 3

Scintillation index as a function of propagation distance for N = 1 (squares), N = 2 (crosses), and N = 4 (diamonds) at r = W 0 .

Fig. 4
Fig. 4

Normalized scintillation index as a function of the Fresnel ratio (for the case of N = 1 ) for N = 1 (squares), N = 2 (crosses), and N = 4 (diamonds) at r = W 0 .

Fig. 5
Fig. 5

Normalized scintillation index as a function of W 0 for N = 1 (squares), N = 2 (crosses), and N = 4 (diamonds) at the beam center. In this plot, L = 1000 .

Fig. 6
Fig. 6

Normalized scintillation index as a function of W 0 for N = 1 (squares), N = 2 (crosses), and N = 4 (diamonds) at r = W 0 . In this plot, L = 1000 .

Fig. 7
Fig. 7

Scintillation index as a function of F 0 for N = 1 (squares), N = 2 (crosses), and N = 4 (diamonds) at the beam center. In this plot, L = 1000 and W 0 = 0.01 m .

Fig. 8
Fig. 8

Scintillation index as a function of F 0 for N = 1 (squares), N = 2 (crosses), and N = 4 (diamonds) at r = W 0 . In this plot, L = 1000 and W 0 = 0.01 m .

Fig. 9
Fig. 9

Normalized scintillation index (normalized to the first data point) at the beam center as a function of wavelength separation for N = 2 for W 0 = 10 cm , L = 1000 , and F 0 = . Wavelength pairs considered are [1330, 1400], [1330, 1470], and [ 1330 , 1550 ] nm .

Equations (66)

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E ( R , t ) = l = 1 N E l ( R ) exp ( j 2 π c λ l t + j ϕ l ( t ) ) ,
2 E ( R ) + k 2 n 2 ( R ) E ( R ) = 0 ,
2 E l ( R ) + k l 2 n 2 ( R ) E l ( R ) = 0 ; l = 1 , 2 , , N 1 , N ,
E ( r ) ( R , t ) = l = 1 N E l ( r ) ( R ) exp [ Ψ l ( R ) ] exp ( j 2 π c λ l t + j ϕ l ( t ) ) ,
I ( R ) = E ( r ) ( R , t ) 2 t = l = 1 N E l ( r ) ( R ) 2 exp [ Ψ l ( R ) + Ψ l * ( R ) ] ,
I ( R ) = l = 1 N A l ( R ) exp [ Ψ l ( R ) + Ψ l * ( R ) ] = l = 1 N A l ( R ) exp [ 2 ζ l ( R ) ] ,
σ sc 2 = I 2 ( R ) ¯ I ( R ) ¯ 2 1 ,
σ sc 2 = l 1 = 1 N l 2 = 1 N A l 1 ( R ) A l 2 ( R ) exp [ 2 ζ l 1 ( R ) + 2 ζ l 2 ( R ) ] ¯ I ( R ) ¯ 2 1 .
R l l ( R ; R ) = σ l 2 ( R ) = ζ l 2 ( R ) ¯ = 1 2 Re [ Ψ l ( R ) Ψ l * ( R ) ¯ ] + 1 2 Re [ Ψ l ( R ) Ψ l ( R ) ¯ ] ,
R l 1 l 2 ( R ; R ) = ζ l 1 ( R ) ζ l 2 ( R ) ¯ = 1 2 Re [ Ψ l 1 ( R ) Ψ l 2 * ( R ) ¯ ] + 1 2 Re [ Ψ l 1 ( R ) Ψ l 2 ( R ) ¯ ] ; l 1 l 2 .
exp ( U ) ¯ exp [ U ¯ + 1 2 ( U 2 ¯ U ¯ 2 ) ] .
γ l ( r , L ) = ζ 1 ( r , L ) ¯ = Re [ Ψ l ( r , L ) ] ¯
Γ l ( r , L ) = exp [ 2 ζ l ( r , L ) ] ¯ = exp [ 2 γ l ( r , L ) + 2 R l l ( r , L ; r , L ) ] = exp [ 2 ( γ l ( r , L ) + σ l 2 ( r , L ) ) ] ,
I ( r , L ) ¯ = l = 1 N A l ( r , L ) Γ l ,
Γ l 1 l 2 ( r , L ) = exp [ 2 ζ l 1 ( r , L ) + 2 ζ l 2 ( r , L ) ] ¯ = Γ l 1 Γ l 2 exp [ 4 R l 1 l 2 ( r , L ; r , L ) ] .
σ sc 2 ( r , L ) = l 1 = 1 N l 2 = 1 N A l 1 ( r , L ) A l 2 ( r , L ) Γ l 1 l 2 ( r , L ) [ l = 1 N A l ( r , L ) Γ l ( r , L ) ] 2 1 ,
n ( s , z ) = e j k s d η ( k , z ) ,
d η ( k 1 , z 1 ) d η * ( k 2 , z 2 ) ¯ = F n ( k 1 , z 1 z 2 ) δ ( k 1 k 2 ) d 2 k 1 d 2 k 2 ,
d η ( k 1 , z 1 ) d η ( k 2 , z 2 ) ¯ = F n ( k 1 , z 1 z 2 ) δ ( k 1 + k 2 ) d 2 k 1 d 2 k 2 .
B n ( R 1 , R 2 ) = n ( R 1 ) n ( R 2 ) ¯ = B n ( R 1 R 2 ) .
Φ n ( κ x , κ y , κ z ) = 1 ( 2 π ) 3 B n ( R ) e j k R d r
Φ n ( κ x , κ y , κ z ) = 1 2 π F n ( k , μ ) cos ( μ κ z ) d μ ,
F n ( k , μ ) = 1 2 π Φ n ( k x , k y , k z ) cos ( μ κ z ) d κ z ,
F n ( k , μ ) d μ = 2 π Φ n ( k ) .
Ψ l ( r , L ) Ψ l , 1 ( r , L ) + Ψ l , 2 ( r , L ) ,
Ψ l , 1 ( r , L ) = j k l 0 L d η ( k , z ) exp [ j q l ( z ) k r j κ 2 q l ( z ) 2 k l ( L z ) ] d z ,
q l ( z ) = 1 + j α l z 1 + j α l L ,
Ψ l , 1 ( r , L ) ¯ = 0 .
γ l ( r , L ) = Re [ Ψ l , 2 ( r , L ) ¯ ] .
Γ l = exp { 2 Re [ Ψ l , 2 ( r , L ) ¯ ] + Ψ l , 1 ( r , L ) 2 ¯ + Re [ Ψ l , 1 ( r , L ) 2 ¯ ] } .
R l 1 l 2 ( r , L ; r , L ) = 1 2 Re [ Ψ l 1 , 1 ( r , L ) Ψ l 2 , 1 * ( r , L ) ¯ ] + 1 2 Re [ Ψ l 1 , 1 ( r , L ) Ψ l 2 , 1 ( r , L ) ¯ ] ; l 1 l 2 .
Ψ l , 1 ( r , L ) 2 ¯ = 4 π 2 k l 2 0 L 0 κ d κ d z Φ n ( κ , z ) I 0 ( 2 κ Im [ q l ( z ) ] r ) × exp { κ 2 Im [ q l ( z ) ] k l ( L z ) } ,
Ψ l , 1 2 ( r , L ) ¯ = 4 π 2 k l 2 0 L 0 κ d κ d z Φ n ( κ , z ) exp [ j κ 2 q l ( z ) k l ( L z ) ] .
σ l 2 ( r , L ) = 2 π 2 k l 2 0 L 0 κ d κ d z Φ n ( κ , z ) exp { κ 2 [ Im [ q l ( z ) ] k l ] ( L z ) } { I 0 ( 2 κ Im [ q l ( z ) ] r ) cos { κ 2 [ Re [ q l ( z ) ] k 1 ] ( L z ) } } .
σ l 2 ( L ) = 2 π 2 k l 2 0 L 0 κ d κ d z Φ n ( κ , z ) exp { κ 2 [ Im [ q l ( z ) ] k l ] ( L z ) } { 1 cos { κ 2 [ Re [ q l ( z ) ] k 1 ] ( L z ) } } ,
Ψ l 1 , 1 ( r , L ) Ψ l 2 , 1 * ( r , L ) ¯ = k l 1 k l 2 0 L 0 L d 2 k d z 1 d z 2 F n ( k , z 1 z 2 ) exp { j [ q l 1 ( z 1 ) q l 2 * ( z 2 ) ] k . r } exp [ j κ 2 q l 2 * ( z 2 ) ( L z 2 ) 2 k l 2 j κ 2 q l 1 ( z 1 ) ( L z 1 ) 2 k l 1 ] .
Ψ l 1 , 1 ( r , L ) Ψ l 2 , 1 * ( r , L ) ¯ = 2 π k l 1 k l 2 0 L 0 L 0 κ d κ d z 1 d z 2 F n ( κ , z 1 z 2 ) J 0 ( κ [ q l 1 ( z 1 ) q l 2 * ( z 2 ) ] r ) exp { j κ 2 q l 2 * ( z 2 ) ( L z 2 ) 2 k l 2 j κ 2 q l 1 ( z 1 ) ( L z 1 ) 2 k l 1 } ,
Ψ l 1 , 1 ( r , L ) Ψ l 2 , 1 * ( r , L ) ¯ = 4 π 2 k l 1 k l 2 0 L 0 κ d κ d z Φ n ( κ , z ) J 0 ( κ [ q l 1 ( z ) q l 2 * ( z ) ] r ) exp { j κ 2 ( q l 1 ( z ) 2 k l 1 q l 2 * ( z ) 2 k l 2 ) ( L z ) } .
Ψ l 1 , 1 ( r , L ) Ψ l 2 , 1 ( r , L ) ¯ = 4 π 2 k l 1 k l 2 0 L κ d κ d z Φ n ( κ , z ) J 0 ( κ [ q l 1 ( z ) q l 2 ( z ) ] r ) exp { j κ 2 [ q l 1 ( z ) 2 k l 1 + q l 2 ( z ) 2 k l 2 ] ( L z ) } .
R l 1 l 2 ( r , L ; r , L ) = 2 π 2 k l 1 k l 2 0 L 0 κ d κ d z Φ n ( κ , z ) { Re { J 0 ( κ [ q l 1 ( z ) q l 2 * ( z ) ] r ) exp { κ 2 [ q l 1 ( z ) 2 k l 1 q l 2 * ( z ) 2 k l 2 ] ( L z ) } } Re { J 0 ( κ [ q l 1 ( z ) q l 2 ( z ) ] r ) exp { j κ 2 [ q l 1 ( z ) 2 k l 1 + q l 2 ( z ) 2 k l 2 ] ( L z ) } } } .
R l 1 l 2 ( 0 ) ( L ) = 2 π 2 k l 1 k l 2 0 L 0 κ d κ d z Φ n ( κ , z ) { Re { exp { j κ 2 [ q l 1 ( z ) 2 k l 1 q l 2 * ( z ) 2 k l 2 ] ( L z ) } } Re { exp { j κ 2 [ q l 1 ( z ) 2 k l 1 + q l 2 ( z ) 2 k l 2 ] ( L z ) } } } ,
Ψ l , 2 ( r , L ) ¯ + 1 2 Ψ l , 1 2 ( r , L ) ¯ = 2 π 2 k l 2 0 L 0 κ d κ d z Φ n ( κ , z )
γ l ( r , L ) = 2 π 2 k l 2 0 L 0 κ d κ d z Φ n ( κ , z ) { 1 Re { exp { j κ 2 [ q l ( z ) k l ] ( L z ) } } } .
Φ n ( κ , η ) = 0.033 C n 2 κ 11 3 ,
1 L 0 κ 1 l 0 , Kolmogorov ,
0.033 C n 2 κ 11 3 exp ( κ 2 κ m 2 ) ,
1 L 0 κ , κ m = 5.92 l 0 , Tatarskii ,
0.033 C n 2 ( κ 2 + κ 0 2 ) 11 6 exp ( κ 2 κ m 2 ) ,
κ 0 , κ 0 = 2 π L 0 , modified von Kármán ,
0.033 C n 2 [ 1 + 1.802 κ κ l 0.254 ( κ κ l ) 7 6 ] exp ( κ 2 κ l 2 ) ( κ 2 + κ 0 2 ) 11 6 ,
κ 0 , κ l = 3.3 l 0 , modified ,
R l 1 l 2 ( r , L ; r , L ) = 2 π 2 k l 1 k l 2 L 0 1 0 κ d κ d η Φ n ( κ ) × Re { J 0 ( κ [ q l 1 ( ( 1 η ) L ) q l 2 * ( ( 1 η ) L ) ] r ) exp { j κ 2 L [ q l 1 ( ( 1 η ) L ) 2 k l 1 q l 2 * ( ( 1 η ) L ) 2 k l 2 ] η } J 0 ( κ [ q l 1 ( ( 1 η ) L ) q l 2 ( ( 1 η ) L ) ] r ) exp { j κ 2 L [ q l 1 ( ( 1 η ) L ) 2 k l 1 + q l 2 ( ( 1 η ) L ) 2 k l 2 ] η } } .
σ l 2 ( r , L ) = 2 π 2 k l 2 L 0 1 0 κ d κ d η Φ n ( κ ) exp { κ 2 L Im [ q l 1 ( ( 1 η ) L ) ] k l η } { I 0 { 2 κ Im [ q l ( ( 1 η ) L ) r ] } cos { κ 2 L Re [ q l ( ( 1 η ) L ) ] k l η } } ,
γ l ( L ) = 2 π 2 k l 2 L 0 1 0 κ d κ d η Φ n ( κ ) { 1 Re { exp { j κ 2 L [ q l ( ( 1 η ) L ) k l ] η } } } .
R l 1 l 2 ( r , L ; r , L ) = R l 2 l 1 ( r , L ; r , L ) .
q l ( ( 1 η ) L ) = 1 + j α l ( 1 η ) L 1 + j α l L = 1 η j α l L 1 + j α l L = 1 η ( 1 ϴ l + j Λ l ) = ( 1 η ) + η ( ϴ l j Λ l ) ,
R l 1 l 2 ( r , L ; r , L ) = 2 π 2 k l 1 k l 2 L 0 1 0 κ d κ d η Φ n ( κ ) × Re { J 0 ( κ η [ ( ϴ l 1 ϴ l 2 ) j ( Λ l 1 + Λ l 2 ) ] r ) × exp { j κ 2 L [ ( 1 η ) + η ( ϴ l 1 j Λ l 1 ) 2 k l 1 ( 1 η ) + η ( ϴ l 2 + j Λ l 2 ) 2 k l 2 ] η } J 0 ( κ η [ ( ϴ l 1 ϴ l 2 ) j ( Λ l 1 Λ l 2 ) ] r ) exp { j κ 2 L [ ( 1 η ) + η ( ϴ l 1 j Λ l 2 ) 2 k l 1 + ( 1 η ) + η ( ϴ l 2 j Λ l 2 ) 2 k l 2 ] η } } .
σ l 2 ( r , L ) = 2 π 2 k l 2 0 1 0 κ d κ d η Φ n ( κ ) e κ 2 η 2 L Λ l k l { I 0 { 2 κ Λ l r } cos { L κ 2 k l η ( 1 η ϴ ̃ l ) } } ,
γ l ( L ) = 2 π 2 k l 2 L 0 1 0 κ d κ d η Φ n ( κ ) { 1 e κ 2 η 2 L Λ l k l cos { L κ 2 k l η ( 1 η ϴ ̃ l ) } } .
σ sc 2 ( r , L ) = 2 l 1 = 1 N l 2 = 1 l 1 1 A l 1 ( r , L ) A l 2 ( r , L ) Γ l 1 l 2 ( r , L ) + l = 1 N A l 2 ( r , L ) Γ l 2 ( r , L ) exp ( 4 σ l 2 ( r , L ) ) [ l = 1 N A l ( r , L ) Γ l ( r , L ) ] 2 1 .
σ 1 2 = 1.23 C n 2 ( 2 π λ ) 7 6 L 11 6 ,
2 E ( R , t ) μ 0 ϵ ( R ) 2 E ( R , t ) t 2 + 2 [ E ( R , t ) log n ( R ) ] = 0
2 E ( R , t ) μ 0 ϵ ( R ) 2 E ( R , t ) t 2 = 0 .
n ( R ) 1 + 77.6 × 10 6 ( 1 + 7.52 × 10 3 λ 2 ) P ( R ) T ( R ) ,
l = 1 N [ 2 E l ( R ) + l = 1 N k l 2 n 2 ( R ) E l ( R ) ] exp ( j 2 π c λ l t ) = 0 .
2 E l ( R ) + k l 2 n 2 ( R ) E l ( R ) = 0 ; l = 1 , 2 , , N 1 , N .

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