Abstract

A light beam propagating through a turbid medium (e.g., aerosol) can be severely attenuated by scattering losses and still retain coherence over distances comparable to particle diameters. An expression for the two-detector mutual-coherence function is rederived by means of approximations clarified by a physical model. Its spatial and temporal properties are further examined by means of a simplified physical aerosol model leading to tractable mathematical analysis.

© 1978 Optical Society of America

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References

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  1. R. L. Fante, J. Opt. Soc. Am. 64, 592 (1974).
    [CrossRef]
  2. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  3. K. M. Watson, J. Math. Phys. 10, 688 (1969).
    [CrossRef]
  4. K. M. Watson, Phys. Fluids 13, 2514 (1970).
    [CrossRef]
  5. A. Ishimaru, Radio Sci. 10, 45 (1975).
    [CrossRef]
  6. V. I. Tatarski, The Effect of the Turbulent Atmosphere on Wave Propagation (National Technical Information Services, Springfield, Va., 1971).
  7. V. Twersky in Proceedings of the American Mathematical Society Symposium on Stochastic Processes in Mathematical Physics and Engineering, Vol. 16 (American Mathematical Society, Providence, R.I., 1964), pp. 84–116.
    [CrossRef]
  8. H. M. Heggestad, J. Opt. Soc. Am. 61, 1293 (1971).
    [CrossRef]
  9. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  10. P. L. Fried, Proc. IEEE 55, 57 (1967).
    [CrossRef]

1975 (1)

A. Ishimaru, Radio Sci. 10, 45 (1975).
[CrossRef]

1974 (1)

1971 (1)

1970 (1)

K. M. Watson, Phys. Fluids 13, 2514 (1970).
[CrossRef]

1969 (1)

K. M. Watson, J. Math. Phys. 10, 688 (1969).
[CrossRef]

1967 (1)

P. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Fante, R. L.

Fried, P. L.

P. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

Heggestad, H. M.

Ishimaru, A.

A. Ishimaru, Radio Sci. 10, 45 (1975).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, The Effect of the Turbulent Atmosphere on Wave Propagation (National Technical Information Services, Springfield, Va., 1971).

Twersky, V.

V. Twersky in Proceedings of the American Mathematical Society Symposium on Stochastic Processes in Mathematical Physics and Engineering, Vol. 16 (American Mathematical Society, Providence, R.I., 1964), pp. 84–116.
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Watson, K. M.

K. M. Watson, Phys. Fluids 13, 2514 (1970).
[CrossRef]

K. M. Watson, J. Math. Phys. 10, 688 (1969).
[CrossRef]

J. Math. Phys. (1)

K. M. Watson, J. Math. Phys. 10, 688 (1969).
[CrossRef]

J. Opt. Soc. Am. (2)

Phys. Fluids (1)

K. M. Watson, Phys. Fluids 13, 2514 (1970).
[CrossRef]

Proc. IEEE (1)

P. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

Radio Sci. (1)

A. Ishimaru, Radio Sci. 10, 45 (1975).
[CrossRef]

Other (4)

V. I. Tatarski, The Effect of the Turbulent Atmosphere on Wave Propagation (National Technical Information Services, Springfield, Va., 1971).

V. Twersky in Proceedings of the American Mathematical Society Symposium on Stochastic Processes in Mathematical Physics and Engineering, Vol. 16 (American Mathematical Society, Providence, R.I., 1964), pp. 84–116.
[CrossRef]

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

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Figures (3)

Fig. 1
Fig. 1

Geometry of light beam incident upon cloud layers at altitude L.

Fig. 2
Fig. 2

Normalized temporal autocorrelation c(T) vs T = υTτ/2a for β = 10 dB (A), 20 dB (B), 30 dB (C), 40 dB (D), and 50 dB (E).

Fig. 3
Fig. 3

Integration domain of ρ in Eq. (A2) for υTτ ≤ 2a (A) and υTτ > 2a (B). Note ξ < 1 inside circles only.

Equations (40)

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Γ ( ρ , R , z ) = d 2 K Γ ( ρ + K ̂ z , K , 0 ) exp ( i K · R ) × exp [ σ ext z + σ sc ( ρ ) z σ sc ( 0 ) z ] ,
Γ ( ρ , R , z ) u ( r 1 , z ) u * ( r 2 , z ) ( ensemble average ) ,
Γ ( ρ , K , z ) 1 4 π 2 d 2 R Γ ( ρ , R , z ) exp ( i K · R ) , K ̂ = unit vector in direction of K ,
σ sc ( ρ ) N k 2 d 2 Ω | f ( Ω ) | 2 exp ( ik ρ · Ω ) , N = density of the medium ,
σ exp [ ( 4 π N ) / k 2 ] Re [ f ( 0 ) ] .
Γ ( ρ , z ) = Γ ( ρ , 0 ) exp [ σ ext z + σ sc ( ρ ) z σ sc ( 0 ) z ] .
u ( z ) exp [ ik ( L z ) ] .
m u ( z ) exp [ ik ( z m z ) ] exp ( ik R 1 m ) ik R 1 m f ( Ω 1 m ) = u ( z ) exp [ ik ( L z ) ] m exp [ ik ( R 1 m Z 1 m ) ] ik R 1 m f ( Ω 1 m ) ,
du ( z , r 1 ) = u ( z , r 1 ) × { m exp [ ik ( R 1 m Z 1 m ) ] ik R 1 m f ( Ω 1 m ) α sc exp [ ik ( L z ) ] dz + multiple scattering terms } .
d [ u ( z , r 1 ) u * ( z , r 2 ) ] = u ( z , r 1 ) u * ( z , r 2 ) × ( 2 Re { m exp [ ik ( R 1 m Z 1 m ) ] ik R 1 m f ( Ω 1 m ) } 2 α sc dz + m n exp [ ik ( R 1 m R 2 n ) ik ( Z 1 m Z 2 n ) ] k 2 R 1 m R 2 n f ( Ω 1 m ) f * ( Ω 2 n ) ) .
d ln Γ ( ρ , z ) = N ( z ) { 2 Re [ d 2 ρ i exp [ ik ( R i Z i ) ] ik R i f ( Ω i ) ] + d 2 ρ i exp ( ik R 1 i R 2 i ) k 2 R 1 i R 2 i f ( Ω 1 i ) f * ( Ω 2 i ) } dz 2 α sc dz ,
Γ ( ρ , z ) = Γ ( ρ , 0 ) exp 0 z dz 1 N ( z z ) × { 4 π Re [ f ( 0 ) ] k 2 + 1 k 2 d 2 Ω | f ( Ω ) | 2 exp ( ik Ω · ρ ) 2 α sc z } ,
δ u ( z , r 1 ) = i u ( z i , r 1 ) exp [ ik ( z i z ) ] × exp ( ik R is ) ik R is f ( Ω ) .
dI ( Ω ) = i | u ( z , r 1 ) | 2 | f ( Ω ) | 2 k 2 .
dI = | u ( z , r 1 ) | 2 d 2 Ω | f ( Ω ) | 2 k 2 N ( z ) dz ,
d [ u ( z , r 1 ) u * ( z , r 2 ) ] = u ( z , r 1 ) u * ( z , r 2 ) d 2 Ω | f ( Ω ) | 2 k 2 N ( z ) dz
2 α sc = N ( z ) d 2 Ω | f ( Ω ) | 2 k 2 ,
R 1 i R 2 i = Ω i · ( ρ + v T τ ) .
Γ ( ρ , z , τ ) = Γ ( ρ + v T τ , z , 0 ) ,
f ( Ω ) = ( ka ) 2 J 1 ( ka Ω ) / ( ka Ω ) ,
σ sc ( ρ , τ ) z 0 z dz 1 N ( z 1 ) Q ( ρ , τ ) , Q ( ρ , τ ) k 2 d 2 Ω | f ( Ω ) | 2 exp [ ik Ω · ( ρ + v T τ ) ] .
Q ( ρ , τ ) = π a 2 Q n ( ξ ) , Q n ( ξ ) = 2 π [ arccos ξ ξ ( 1 ξ 2 ) 1 / 2 ] , ξ 1 , = 0 , ξ > 1 ,
Γ ( ρ , z , τ ) = Γ ( ρ , 0 , 0 ) exp { π a 2 dzN ( z ) [ 3 Q n ( ξ ) ] } .
Γ ( ρ , z , 0 ) = Γ ( 0 , z , 0 ) exp ( π a 2 Nz ) ,
I ( t ) = η 2 A 0 d 2 ρ U ( ρ ) u ( ρ , z , t ) exp ( i δ ω t ) ,
C I ( τ ) = d 2 ρ 1 d 2 ρ 2 U ( ρ 1 ) U ( ρ 2 ) Γ ( ρ 1 ρ 2 , z , τ ) ,
Q n ( ξ ) = 1 ξ for ξ 1 , = 0 for ξ > 0 .
β = σ ext z = π a 2 dzN ( z ) .
C I ( τ ) = exp ( 2 β ) d 2 ρ 1 d 2 ρ 2 U ( ρ 1 ) U ( ρ 2 ) Γ ( ρ , 0 , 0 ) exp ( β ξ ) , ξ 1 , = exp ( 2 β ) d 2 ρ 1 d 2 ρ 2 U ( ρ 1 ) U ( ρ 2 ) Γ ( ρ , 0 , 0 ) exp ( β ) , ξ > 1 , ρ = ρ 1 ρ 2 .
C I ( τ ) = C I ( 0 ) + δ C I ( τ ) , C I ( 0 ) = exp ( 2 β ) Γ 0 ( π D 2 / 8 ) 2 exp ( β ) , δ C I ( τ ) exp ( 2 β ) Γ 0 d 2 ρ 1 d 2 ρ 2 U ( ρ 1 ) U ( ρ 2 ) × { exp [ β ξ ( τ ) ] exp ( β ) } , ξ ( τ ) = min ( | ρ 1 ρ 2 + v T τ | / 2 a , 1 ) .
C I ( τ ) = exp ( 2 β ) Γ 0 π D 2 8 [ π D 2 8 exp ( β ) + 8 π a 2 β 2 c ( T ) ] , c ( T ) ( 1 + β T ) exp ( β T ) [ 1 + β + 1 2 β 2 ( 1 T 2 ) ] exp ( β ) for T 1 , 0 for T > 1 ,
a D = exp ( β / 2 ) 8 β ,
δ C I ( τ ) Γ 0 d 2 ρ 1 | U ( ρ 1 ) | 2 d 2 ρ [ exp ( β ξ ) exp ( β ) ] Γ 0 ( π D 2 / 8 ) δ c ( T ) for T 1 ,
δ c ( T ) 4 a 2 d 2 t [ exp ( β ξ ( t , T ) ) exp ( β ) ] , ξ ( t , T ) = ( t 2 + 2 tT cos ϕ + T 2 ) 1 / 2 , ξ 1 .
δ c ( T ) = 4 a 2 π π d ϕ T 1 d ξ ξ [ 1 T cos ϕ ( ξ 2 T 2 sin 2 ϕ ) 1 / 2 ] × [ exp ( β ξ ) exp ( β ) ] .
δ c ( T ) = 8 π a 2 T 1 d ξ ξ [ exp ( β ξ ) exp ( β ) ] = 8 π a 2 β 2 { ( 1 + β T ) exp ( β T ) [ 1 + β + β 2 ( 1 T 2 ) / 2 ] exp ( β ) } , for T 1 .
δ C I ( τ ) Γ 0 d 2 ρ 1 U ( ρ 1 ) U ( ρ 1 + v T τ ) δ c ( T ) for T > 1 ,
δ c ( T ) = 4 a 2 ϕ 2 ϕ 1 d ϕ t 1 t 2 dtt [ exp ( β ξ ) exp ( β ) ] , T > 1 , t 1 = T cos ϕ + ( 1 T 2 sin 2 ϕ ) 1 / 2 , t 2 = T cos ϕ ( 1 T 2 sin 2 ϕ ) 1 / 2 , ϕ 1 = π arcsin ( 1 / T ) , ϕ 2 = π + arcsin ( 1 / T ) ,
δ c ( T ) = 16 a 2 0 1 du 0 ( 1 u 2 ) 1 / 2 d υ exp [ β ( u 2 + υ 2 ) 1 / 2 ] .
δ C I ( τ ) = Γ 0 π D 2 8 · 8 π a 2 β 2 { ( 1 + β T ) exp ( β T ) [ 1 + β + β 2 ( 1 T 2 ) / 2 ] exp ( β ) } for T 1 = 0 for T > 1 .

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