Abstract

Pulse propagation in a random medium is studied through the calculation of the two-frequency mutual coherence function. An exact integral representation is formulated for the two-frequency mutual coherence function of a Gaussian beam pulse propagating in a weakly fluctuating random medium. Based on the modified von Karman spectrum for refractive-index fluctuations, an analytic approximation to the integral representation is presented and compared with exact numerical results.

© 1996 Optical Society of America

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References

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  1. S. T. Hong, A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci. 11, 551–559 (1976).
    [CrossRef]
  2. I. Sreenivasiah, A. Ishimaru, S.-T. Hong, “Two-frequency mutual coherence function and pulse propagation in a random medium: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
    [CrossRef]
  3. I. Sreenivasiah, A. Ishimaru, “Beam wave two-frequency mutual-coherence function and pulse propagation in random media: an analytic solution,” Appl. Opt. 18, 1613–1618 (1979).
    [CrossRef] [PubMed]
  4. V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, National Technical Information Service, Springfield, Va., 1971).
  5. H. T. Yura, C. C. Sung, S. F. Clifford, R. J. Hill, “Second-order Rytov approximation,” J. Opt. Soc. Am. 73, 500–502 (1983).
    [CrossRef]
  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. H. T. Yura, S. G. Hanson, “Second-order statistics for wave propagation through complex optical systems,” J. Opt. Soc. Am. A 6, 564–575 (1989).
    [CrossRef]
  8. 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]
  9. L. C. Andrews, “An analytic model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
    [CrossRef]

1995 (1)

1993 (1)

1992 (1)

L. C. Andrews, “An analytic model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

1989 (1)

1983 (1)

1979 (1)

1976 (2)

S. T. Hong, A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci. 11, 551–559 (1976).
[CrossRef]

I. Sreenivasiah, A. Ishimaru, S.-T. Hong, “Two-frequency mutual coherence function and pulse propagation in a random medium: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
[CrossRef]

Andrews, L. C.

Clifford, S. F.

Hanson, S. G.

Hill, R. J.

Hong, S. T.

S. T. Hong, A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci. 11, 551–559 (1976).
[CrossRef]

Hong, S.-T.

I. Sreenivasiah, A. Ishimaru, S.-T. Hong, “Two-frequency mutual coherence function and pulse propagation in a random medium: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
[CrossRef]

Ishimaru, A.

I. Sreenivasiah, A. Ishimaru, “Beam wave two-frequency mutual-coherence function and pulse propagation in random media: an analytic solution,” Appl. Opt. 18, 1613–1618 (1979).
[CrossRef] [PubMed]

I. Sreenivasiah, A. Ishimaru, S.-T. Hong, “Two-frequency mutual coherence function and pulse propagation in a random medium: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
[CrossRef]

S. T. Hong, A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci. 11, 551–559 (1976).
[CrossRef]

Miller, W. B.

Ricklin, J. C.

Sreenivasiah, I.

I. Sreenivasiah, A. Ishimaru, “Beam wave two-frequency mutual-coherence function and pulse propagation in random media: an analytic solution,” Appl. Opt. 18, 1613–1618 (1979).
[CrossRef] [PubMed]

I. Sreenivasiah, A. Ishimaru, S.-T. Hong, “Two-frequency mutual coherence function and pulse propagation in a random medium: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
[CrossRef]

Sung, C. C.

Tatarskii, V. I.

V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, National Technical Information Service, Springfield, Va., 1971).

Yura, H. T.

Appl. Opt. (1)

J. Mod. Opt. (1)

L. C. Andrews, “An analytic model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Radio Sci. (2)

S. T. Hong, A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci. 11, 551–559 (1976).
[CrossRef]

I. Sreenivasiah, A. Ishimaru, S.-T. Hong, “Two-frequency mutual coherence function and pulse propagation in a random medium: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
[CrossRef]

Other (1)

V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, National Technical Information Service, Springfield, Va., 1971).

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Figures (3)

Fig. 1
Fig. 1

Normalized MCF plotted versus separation distance ρ = |ρ 1ρ 2|. The parameter values are Ω0 = 1, k 1 = (2π)/0.5135 μm−1, k 2 =(2π)/0.5145 μm−1, z = 1 km, l = 0.2 m, 〈n 1 2〉 = 10−16, and ρ 2 = −ρ 1.

Fig. 2
Fig. 2

Normalized MCF plotted versus separation distance ρ = |ρ 1ρ 2|. The parameter values are Ω0 = 0, Ω1 = 3.1623, k 1 = (2π)/0.5135 μm−1, k 2 = (2π)/0.5145 μm−1, z = 1 km, l = 0.2 m, 〈n 1 2〉 = 10−16, and ρ 2 = −ρ 1.

Fig. 3
Fig. 3

Degree of coherence plotted versus nondimensional quantity R = [(k 1ρ2)/z]1/2. The parameter values are σ1 2 = 0.02, k ˜ = 1.0194, Q 0 = 0, Q m = 100, and Ω0 = 1.

Equations (34)

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Γ 2 ( ρ 1 , k 1 , ρ 2 , k 2 , z ) = U ( ρ 1 , k 1 , z ) U * ( ρ 2 , k 2 , z ) ,
U ( ρ , k , z ) = U 0 ( ρ , k , z ) exp [ Ψ 1 ( ρ , k , z ) + Ψ 2 ( ρ , k , z ) ] ,
Γ 2 ( ρ 1 , k 1 , ρ 2 , k 2 , z ) = U 0 ( ρ 1 , k 1 , z ) U 0 * ( ρ 2 , k 2 , z ) M 2 ,
M 2 = exp [ Ψ 1 ( ρ 1 , k 1 , z ) + Ψ 1 * ( ρ 2 , k 2 , z ) + Ψ 2 ( ρ 1 , k 1 , z ) + Ψ 2 * ( ρ 2 , k 2 , z ) ] .
exp ( x ) = exp [ x + 1 2 ( x - x ) 2 ] .
M 2 = exp [ 2 E 1 ( 0 , k 1 , 0 , k 2 , z ) + E 2 ( ρ 1 , k 1 , ρ 2 , k 2 , z ) ] ,
2 E 1 ( 0 , k 1 , 0 , k 2 , z ) = Ψ 2 ( ρ 1 , k 1 , z ) + Ψ 2 * ( ρ 2 , k 2 , z ) + 1 2 Ψ 1 2 ( ρ 1 , k 1 , z ) + 1 2 Ψ 1 * 2 ( ρ 2 , k 2 , z ) ,
E 2 ( ρ 1 , k 1 , ρ 2 , k 2 , z ) = Ψ 1 ( ρ 1 , k 1 , z ) Ψ 1 * ( ρ 2 , k 2 , z ) .
U 0 ( ρ , k , 0 ) = exp [ - k ρ 2 2 ( 2 k W 0 2 + i 1 R 0 ) ] ,
U 0 ( ρ , k , z ) = ( Θ - i Λ ) exp ( i k z ) exp [ i k 2 z ( Θ ˜ + i Λ ) ρ 2 ] ,
Θ = Ω 0 Ω 0 2 + Ω 2 ,             Λ = Ω Ω 0 2 + Ω 2 ,             Θ ˜ = 1 - Θ ,
Ω 0 = 1 - z R 0 ,             Ω = 2 z k W 0 2 .
U 0 ( ρ 1 , k 1 , z ) = ( Θ 1 - i Λ 1 ) × exp ( i k 1 z ) exp [ i k 1 2 z ( Θ ˜ 1 + i Λ 1 ) ρ 1 2 ] ,
U 0 * ( ρ 2 , k 2 , z ) = ( Θ 2 + i Λ 2 ) exp ( - i k 2 z ) × exp [ - i k 2 2 z ( Θ ˜ 2 + i Λ 2 ) ρ 2 2 ] ,
Θ 1 = Ω 0 Ω 0 2 + Ω 1 2 ,             Λ 1 = Ω 1 Ω 0 2 + Ω 1 2 ,             Θ ˜ 1 = 1 - Θ 1 ,
Θ 2 = Ω 0 Ω 0 2 + Ω 2 2 ,             Λ 2 = Ω 2 Ω 0 2 + Ω 2 2 ,             Θ ˜ 2 = 1 - Θ 2 ,
Ω 0 = 1 - z R 0 ,             Ω 1 = 2 z k 1 W 0 2 ,             Ω 2 = 2 z k 2 W 0 2 .
E 1 ( 0 , k 1 , 0 , k 2 , z ) = - π 2 ( k 1 2 + k 2 2 ) z 0 κ Φ n ( κ ) d κ ,
E 2 ( ρ 1 , k 1 , ρ 2 , k 2 , z ) = 4 π 2 k 1 k 2 z 0 1 0 × Φ n ( κ ) J 0 ( κ γ 1 ρ 1 - γ 2 * ρ 2 ) × exp [ - i κ 2 2 z ξ ( γ 1 k 1 - γ 2 * k 2 ) ] κ d κ d ξ ,
γ 1 = 1 - ( Θ ˜ 1 + i Λ 1 ) ξ ,
γ 1 * = 1 - ( Θ ˜ 2 + i Λ 2 ) ξ .
Φ n ( κ ) = n 1 2 l 3 8 π π exp ( - 1 4 κ 2 l 2 ) ,
Φ n ( κ ) = 0.033 C n 2 exp [ - ( κ 2 / κ m 2 ) ] ( κ 2 + κ 0 2 ) 11 / 6 ,
E 1 ( 0 , k 1 , 0 , k 2 , z ) = - 0.3177 2 σ 1 2 ( 1 + k ˜ - 2 ) × ( Q 0 - 5 / 6 - 5.5833 Q m - 5 / 6 ) ,
E 2 ( ρ 1 , k 1 , - ρ 1 , k 2 , z ) = 0.6355 σ 1 2 Q 0 - 5 / 6 k ˜ × [ 1 + 0.5 Q 0 R 2 ( 1 + Θ + Θ 2 ) ] - 3.54 σ 1 2 Q m - 5 / 6 k ˜ × [ ( 1 + 0.321 Λ Q m ) 5 / 6 + 0.0694 Q m R 2 Θ ˜ d 1 ] ,
E 2 ( ρ 1 , k 1 , ρ 1 , k 1 , z ) = 0.6355 σ 1 2 Q 0 - 5 / 6 × ( 1 - 0.5 Q 0 R 2 Λ 1 2 ) - 3.54 σ 1 2 Q m - 5 / 6 × [ ( 1 + 0.321 Λ 1 Q m ) 5 / 6 + 0.0694 Q m R 2 Λ 1 2 d 2 ] ,
d 1 = ( 1 + 0.033 R 2 Q m + 0.211 Λ Q m ) - 1 / 6 - Θ 3 × ( 1 + 0.033 R 2 Θ 2 Q m + 0.211 Λ Q m ) - 1 / 6 ,
d 2 = ( 1 - 0.033 Q m R 2 Λ 1 2 + 0.547 Λ 1 Q m ) - 1 / 6 ,
Λ = Λ 1 + k ˜ Λ 2 2 ,             Θ = Θ 1 + Θ 2 2 ,             Θ ˜ = 1 - Θ ,
σ 1 2 = 1.23 C n 2 k 1 7 / 6 z 11 / 6 ,             k ˜ = k 1 k 2 ,             ρ = ρ 1 - ρ 2 ,
Q 0 = z κ 0 2 k 1 ,             Q m = z κ m 2 k 1 ,             R = ( k 1 ρ 2 z ) 1 / 2 .
M 2 = exp ( 0.6355 σ 1 2 Q 0 - 5 / 6 { - ( 1 + k ˜ - 2 ) 2 + 1 k ˜ × [ 1 + 0.5 Q 0 R 2 Θ ˜ ( 1 - Θ 3 ) ] } + 3.54 σ 1 2 Q m - 5 / 6 { ( 1 + k ˜ - 2 ) 2 - 1 k ˜ × [ ( 1 + 0.321 Λ Q m ) 5 / 6 + 0.0694 Q m R 2 Θ ˜ d 1 ] } ) .
ϒ ( ρ 1 , k 1 , ρ 2 , k 2 , z ) = Γ 2 ( ρ 1 , k 1 , ρ 2 , k 2 , z ) [ Γ 2 ( ρ 1 , k 1 , ρ 1 , k 1 z ) Γ 2 ( ρ 2 , k 2 , ρ 2 , k 2 , z ) ] 1 / 2 .
ϒ ( ρ 1 , k 1 , ρ 2 , k 2 , z ) = exp ( Re { E 2 ( ρ 1 , k 1 , ρ 2 , k 2 , z ) - 1 2 [ E 2 ( ρ 1 , k 1 , ρ 1 , k 1 , z ) + E 2 ( ρ 2 , k 2 , ρ 2 , k 2 , z ) ] } ) .

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