Abstract

We present an analytical technique that solves exactly, and in closed form, for the first and second moments of the spatial and angular positions of photon distributions in a multiple-scattering medium. The analysis leads to simple analytic expressions for these moments, both conditioned on the number of scatterings and summed over all scattering events. The conventional results for small-angle forward scattering, and for the diffusion regime, are recovered in the appropriate limiting cases.

© 1995 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Oxford, New York, 1950).
  2. L. M. Milne-Thomson, Theoretical Hydrodynamics (Maximillan, London, 1938).
  3. S. Chapman, T. G. Cowling, Mathematical Theory of Non-Uniform Gases (Cambridge, New York, 1939).
  4. A. P. Ciervo, “Propagation through optically thick media,” Appl. Opt. 34, 7137–7148 (1995).
    [Crossref] [PubMed]
  5. E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962).
  6. L. B. Stotts, “The radiance produced by laser radiation transversing a particulate multiple-scattering medium,” J. Opt. Soc. Am. 67, 815–819 (1977).
    [Crossref]
  7. D. Arnush, “Underwater light-beam propagation in the small-angle-scattering approximation,” J. Opt. Soc. Am. 62, 1109–1111 (1972).
    [Crossref]
  8. G. C. Mooradian, M. Geller, L. B. Stotts, D. H. Stephens, R. A. Krantwald, “Blue-green pulsed propagation through fog,” Appl. Opt. 18, 429–441 (1979).
    [Crossref] [PubMed]
  9. E. A. Bucher, “Computer simulation of light pulse propagation for communication through thick clouds,” Appl. Opt. 12, 2391–2400 (1973).
    [Crossref] [PubMed]
  10. F. Reif, Statistical Thermal Physics (McGraw-Hill, New York, 1965).

1995 (1)

1979 (1)

1977 (1)

1973 (1)

1972 (1)

Arnush, D.

Bucher, E. A.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford, New York, 1950).

Chapman, S.

S. Chapman, T. G. Cowling, Mathematical Theory of Non-Uniform Gases (Cambridge, New York, 1939).

Ciervo, A. P.

Cowling, T. G.

S. Chapman, T. G. Cowling, Mathematical Theory of Non-Uniform Gases (Cambridge, New York, 1939).

Geller, M.

Krantwald, R. A.

Milne-Thomson, L. M.

L. M. Milne-Thomson, Theoretical Hydrodynamics (Maximillan, London, 1938).

Mooradian, G. C.

Parzen, E.

E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962).

Reif, F.

F. Reif, Statistical Thermal Physics (McGraw-Hill, New York, 1965).

Stephens, D. H.

Stotts, L. B.

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Figures (7)

Fig. 1
Fig. 1

Scattering geometry.

Fig. 2
Fig. 2

Mean photon penetration parameterized on 〈cos θ〉.

Fig. 3
Fig. 3

Spatial standard deviations for (a) cloud aerosols and (b) forward-scattering particles.

Fig. 4
Fig. 4

First two angular moments for 〈cos θ〉 = 〈cos2 θ〉 = 0.850.

Fig. 5
Fig. 5

Effects of small-angle assumption on rms beam spread. 〈cos θ〉 = 〈cos2 θ〉 = 0.85.

Fig. 6
Fig. 6

Average multipath time delay for 〈cos θ〉 = 0.827.

Fig. 7
Fig. 7

Conditional spatial moments for 〈cos θ〉 = 〈cos2 θ〉 = 0.850.

Equations (85)

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z ( T ) = V x Ω z f d V x d ω α .
σ z 2 ( T ) = V x Ω ( z - z ) 2 f d V x d ω α .
σ x 2 ( T ) = σ y 2 ( T ) = V x Ω x 2 f d V x d ω a .
cos θ ( T ) = V x Ω cos θ f d V x d ω α .
cos 2 θ ( T ) = V x Ω cos 2 θ f d V x d ω α .
z ( T ) = n = 0 P { n ( T ) = n } z n ( T ) ,
P { n ( T ) = n } = T n n ! exp ( - T ) ,             n 0.
( x n ( T ) y n ( T ) z n ( T ) ) = x n ( T ) = m = 0 n l m ( T ) μ m ,
μ m = ( k = 0 m B k ) ( 0 0 1 ) ,
B 0 = [ 1 0 0 0 1 0 0 0 1 ] ,
B k = [ ( cos ϕ k cos θ k ) ( - sin ϕ k ) ( cos ϕ k sin ϕ k ) ( sin ϕ k cos θ k ) ( cos ϕ k ) ( sin ϕ k sin θ k ) ( - sin θ k ) ( 0 ) ( cos θ k ) ] .
x n ( T ) = m = 0 n l m μ m .
f ( T 1 , T 2 , T n ) = n ! T n , 0 T 1 T 2 T n T = 0 , otherwise .
l k = λ k + 1 - λ k ,             k = 0 , 1 , , n .
λ k = n ! T n ( 0 T d T n 0 T n d T n - 1 0 T k + 2 d T k + 1 0 T k + 1 d T k T k × 0 T k d T k - 1 0 T 3 d T 2 0 T 2 d T 1 ) = k T n + 1 ,
l m = λ m + 1 - λ m = λ m + 1 - λ m = T n + 1 .
μ m = B 0 B 1 B 2 B m ( 0 0 1 ) = B m ( 0 0 1 ) ,
B = [ 0 0 0 0 0 0 - sin θ 0 cos θ ] .
μ m = ( 0 0 cos θ m ) .
x n ( T ) = y n ( T ) 0 ,
z n ( T ) = T n + 1 1 - cos θ n + 1 1 - cos θ .
x n 2 ( T ) = y n ( T ) = 2 T 2 3 w v 2 ( w - v ) × { w 2 [ ( 1 - v ) n + 2 - 1 + ( n + 2 ) v ] - v 2 [ ( 1 - w ) n + 2 - 1 + ( n + 2 ) w ] ( n + 1 ) ( n + 2 ) } ,
var [ x n ( T ) ] = x n 2 ( T ) ,             var [ y n ( T ) ] = y n 2 ( T ) .
var [ z n ( T ) ] = z n 2 ( T ) - z n ( T ) 2 ,
var [ z n ( T ) ] = 2 T 2 3 w v 2 ( w - v ) × { ( w 2 - 3 w v ) [ ( 1 - v ) n + 2 - 1 + ( n + 2 ) v ] + 2 v 2 [ ( 1 - w ) n + 2 - 1 + ( n + 2 ) w ] ( n + 1 ) ( n + 2 ) } - { T 2 ( n + 1 ) v [ 1 - ( 1 - v ) n + 1 ] } 2 .
v = 1 - cos θ ,
w = / 2 3 [ 1 - cos 2 θ ] .
cos θ ( T ) n = ( 1 - v ) n = cos θ n .
cos 2 θ ( T ) n = [ 1 + 2 ( 1 - w ) n ] ,
= [ 1 + 2 ( / 2 3 cos 2 θ - 1 ) n ] .
x ( T ) = n = 0 P { n ( T ) = n } x n ( T ) = 0 ,
y ( T ) = n = 0 P { n ( T ) = n } y n ( T ) = 0 ,
z ( T ) = n = 0 P { n ( T ) = n } z n ( T ) = n = 0 exp ( - T ) T n n ! T n + 1 1 - ( 1 - v ) n + 1 v = exp ( - T ) v { n = 0 T n + 1 ( n + 1 ) ! - n = 0 [ T ( 1 - v ) ] n + 1 ( n + 1 ) ! } = exp ( - T ) v ( exp ( T ) - 1 - { exp [ T ( 1 - v ) ] - 1 } ) = 1 - exp ( - v T ) v ,
σ x 2 ( T ) = σ y 2 ( T ) = { 2 3 w 2 [ exp ( - v T ) - 1 + v T ] - v 2 [ exp ( - w T ) - 1 + w T ] w v 2 ( w - v ) } ,
z 2 ( T ) = n = 0 P { n ( T ) = n } z n 2 ( T ) = n = 0 exp ( - T ) T n n ! 2 T 2 3 w v 2 ( w - v ) × { ( w 2 - 2 w v ) [ ( 1 - v ) n + 2 - 1 + ( n + 2 ) v ] + 2 v 2 [ ( 1 - w ) n + 2 - 1 + ( n + 2 ) w ] ( n + 1 ) ( n + 2 ) } = 2 3 ( w 3 - 3 w v ) [ exp ( - v T ) - 1 + v T ] + 2 v 2 [ exp ( - w T ) - 1 + w T ] w v 2 ( w - v ) ,
σ z 2 ( T ) = z 2 ( T ) - z ( T ) 2 ,
v = 1 - cos θ θ 2 + O θ 2 2 + ,
w = / 2 3 [ 1 - cos 2 θ ] / 2 3 θ 2 + O θ 2 2 + 3 v + 0 θ 2 2 + .
z n ( T ) T ( 1 - n θ 2 / 4 ) .
x n 2 ( T ) = y n 2 ( T ) / 9 1 n w T 2 n θ 2 T 2 ,
r n 2 ( T ) = x n 2 ( T ) + y n 2 ( T ) ~ n θ 2 T 2 ,
σ 2 [ z n ( T ) ] = z n 2 ( T ) - z n ( T ) 2 / 9 2 ( 3 v - w ) n T 2 / 12 1 θ 2 2 n T 2 .
z ( T ) T - v T 2 T - θ 2 T 2 .
σ x 2 = 2 w 3 ( w - v ) k = 3 ( - 1 ) k ( ν k - 2 - w k - 2 ) T k k ! ,
σ x 2 ( T ) w 9 T 3 .
r 2 = x 2 + y 2 = 2 σ x 2 ( T ) 2 w 9 T 3 = 1 3 θ 2 T 3
R 2 = S θ 2 Z 3 ,
σ 2 [ z ( T ) ] / 9 2 ( 3 v - w ) T 3 / 12 1 θ 2 2 T 3 .
lim T z ( T ) = 1 - cos θ .
lim T σ x 2 ( T ) = σ z 2 ( T ) = 2 3 T 1 - cos θ .
cos θ ( T ) = exp ( - v T ) ,
Var { cos [ θ ( T ) ] } = cos 2 θ ( T ) - cos θ ( T ) 2 = [ 1 + 2 exp ( - w T ) ] - exp ( - 2 v T ) ,
lim T Var ( cos θ ) = ,
T 1 v ln ( 1 1 - v ( z ) ) ,
Δ T 1 v ln ( 1 1 - v ( z ) ) - z ,
σ x ( n , T ) = [ x n 2 ( T ) ] 1 / 2 , σ z ( n , T ) = [ z n 2 ( T ) - z n ( T ) 2 ] 1 / 2
x n ( T ) x n tr ( T ) = [ m = 0 n l m ( T ) μ m ] [ k = 0 n l k ( T ) μ k ] tr = m = 0 n k = 0 n l m ( T ) l k ( T ) μ m μ k tr .
x n ( T ) x n tr ( T ) = m = 0 n k = 0 n l m l k μ m μ k tr = m = 0 n l m 2 μ m μ m tr + 2 m = 1 n k = 1 m - 1 l m l k μ m μ k tr .
l m 2 = ( λ m + 1 - λ m ) 2 ,
l m l k = ( λ m + 1 - λ m ) ( λ k + 1 - λ k ) .
λ k 2 = n ! T n ( 0 T d T n 0 T n d T n - 1 0 T k + 2 d T k + 1 0 T k + 1 d T k T k 2 0 T k d T k - 1 0 T 3 d T 2 0 T 2 d T 1 ) = k ( k + 1 ) T 2 ( n + 1 ) ( n + 2 ) ,
λ j λ k = n ! T n ( 0 T d T n 0 T n d T n - 1 0 T k + 2 d T k + 1 0 T k + 1 d T k T k 0 T k d T k - 1 0 T j + 2 d T j + 1 0 T j + 1 d T j T j 0 T j d T j - 1 0 T 3 d T 2 0 T 2 d T 1 ) = j ( k + 1 ) T 2 ( n + 1 ) ( n + 2 ) ,             for k > j ,
l m 2 = ( λ m + 1 - λ m ) 2 = λ m + 1 2 - 2 λ m λ m + 1 + λ m 2 = 2 T 2 ( n + 1 ) ( n + 2 )
l m l k = ( λ m + 1 - λ m ) ( λ k + 1 - λ k ) = λ m + 1 λ k + 1 - λ m + 1 λ k - λ m λ k + 1 + λ m λ k = T 2 ( n + 1 ) ( n + 2 ) = l m 2 2 ,             for m > k .
x n ( T ) x n ( T ) T = 2 T 2 ( n + 1 ) ( n + 2 ) ( [ 0 0 0 0 0 0 0 0 1 ] + m = 1 n μ m μ m tr + m = 1 n k = 0 m - 1 μ m μ k tr ) .
v m v k tr = B 1 B k B k + 1 B k + 2 B m ( 0 0 1 ) μ k tr = B 1 B k B k + 1 B k + 2 B m ( 0 0 1 ) μ k tr = B 1 B k B m - k ( 0 0 1 ) μ k tr = B 1 B k cos θ m - k ( 0 0 1 ) μ k tr = cos θ m - k μ k μ k tr .
x n ( T ) x n tr ( T ) = 2 T 2 ( n + 1 ) ( n + 2 ) ( [ 0 0 0 0 0 0 0 0 1 ] + m = 1 n μ m μ m tr + m = 1 n g m [ 0 0 0 0 0 0 0 0 1 ] + m = 2 n k = 1 m - 1 g m - k μ k μ k tr ) = 2 T 2 ( n + 1 ) ( n + 2 ) [ 1 - g n + 1 1 - g [ 0 0 0 0 0 0 0 0 0 ] + m = 1 n μ μ m tr + k = 1 n - 1 μ k μ k tr ( m = 1 n - k g m ) ] = 2 T 2 ( n + 1 ) ( n + 2 ) [ 1 - g n + 1 1 - g [ 0 0 0 0 0 0 0 0 1 ] + m = 1 n μ m μ m tr + g 1 - g k = 1 n - 1 μ k μ k tr - g n + 1 1 - g k = 1 n - 1 g - k μ k μ k tr ] .
μ k μ k tr = B 1 B 2 B k - 1 B k [ 0 0 0 0 0 0 0 0 1 ] B k T B k - 1 T B 2 T B 1 T = B 1 B 2 B k - 1 B k [ 0 0 0 0 0 0 0 0 1 ] B k T B k - 1 T B 2 T B 1 T = B 1 B 2 B k - 1 [ 1 2 sin 2 θ 0 0 0 1 2 sin 2 θ 0 0 0 1 2 sin 2 θ ] B k - 1 T B 2 T B 1 T = 1 3 [ 1 - ζ k 0 0 0 1 - ζ k 0 0 0 1 + 2 ζ k ] ,
ζ = 3 cos 2 θ - 1 2 .
m = 1 n ζ m = ζ ( 1 - ζ n ) 1 - ζ ,
x n ( T ) x n tr ( T ) = 2 T 2 ( n + 1 ) ( n + 2 ) { 1 - g n + 1 1 - g [ 0 0 0 0 0 0 0 0 1 ] + n 3 [ 0 0 0 0 0 0 0 0 1 ] + 1 3 ζ ( 1 - ζ n ) 1 - ζ [ 0 0 0 0 0 0 0 0 1 ] + n - 1 3 g 1 - g [ 0 0 0 0 0 0 0 0 0 ] + 1 3 g 1 - g ζ ( 1 - ζ n - 1 ) 1 - ζ [ - 1 0 0 0 - 1 0 0 0 2 ] - 1 3 g 1 - g g ( 1 - g n - 1 ) 1 - g [ 1 0 0 0 1 0 0 0 1 ] + 1 3 g n + 1 1 - g ( ζ / g ) [ 1 - ( ζ / g ) n - 1 ] 1 - ζ / g [ - 1 0 0 0 - 1 0 0 0 2 ] } = 2 T 2 2 ( n + 1 ) ( n + 2 ) { 3 [ 0 0 0 0 0 0 0 0 1 ] ( 1 - g n + 1 1 - g ) + [ 1 0 0 0 1 0 0 0 1 ] [ n 1 - g - g ( 1 - g n ) ( 1 - g ) 2 ] + [ - 1 0 0 0 - 1 0 0 0 2 ] [ ζ ( 1 - ζ n ) 1 - ζ + g ζ ( 1 - ζ n - 1 ) ( 1 - g ) ( 1 - ζ ) - g 2 ζ ( g n - 1 - ζ n - 1 ) ( 1 - g ) ( g - ζ ) ] } .
x n ( T ) x n tr ( T ) = [ x n t ( T ) x n ( T ) y n ( T ) x n ( T ) z n ( T ) y n ( T ) x n ( T ) y n 2 ( T ) y n ( T ) z n ( T ) z n ( T ) x n ( T ) y n 2 ( T ) ] .
B H = B 0 B 1 B n .
B H = B n = [ 0 0 0 0 0 0 - sin θ cos θ n - 1 0 cos θ n ] .
B ¯ H = [ 0 0 0 0 0 0 - sin θ n 0 cos θ n ] .
cos θ n = cos θ n = ( 1 - v ) n .
μ n μ n tr = B H [ 0 0 0 0 0 0 0 0 1 ] B H tr = [ 1 2 sin 2 θ n 0 0 0 1 2 sin 2 θ n 0 0 0 cos 2 θ n ] .
cos 2 θ n = ~ ( 1 + 2 ζ n ) = ~ [ 1 + 2 ( 1 - w ) n ] .
σ x 2 ( T ) = x 2 ( T ) - x ( T ) 2 = n = 0 P { n ( T ) = n } x n 2 ( T ) = n = 0 exp ( - T ) T n n ! 2 T 2 3 ( n + 1 ) ( n + 2 ) × { w 2 [ ( 1 - v ) n + 2 - 1 + ( n + 2 ) v ] - v 2 [ ( 1 - w ) n + 2 - 1 + ( n + 2 ) w ] w v 2 ( w - v ) }
= 2 exp ( - T ) 3 w v 2 ( w - v ) ( w 2 { n = 0 [ ( 1 - v ) T ] n + 2 ( n + 2 ) ! - n = 0 T n + 2 ( n + 2 ) ! + v T n = 0 T n + 1 ( n + 1 ) ! } - v 2 { n = 0 [ ( 1 - w ) T ] n + 2 ( n + 2 ) ! - n = 0 T n + 2 ( n + 2 ) ! + w T T n + 1 ( n + 1 ) ! } ) .
n = 0 T n + 1 ( n + 1 ) ! = exp ( T ) - 1 ,
n = 0 T n + 2 ( n + 2 ) ! = exp ( T ) - 1 - T ,
σ x 2 ( T ) = σ y 2 ( T ) = 2 3 w 2 [ exp ( - v T ) - 1 + v T ] - v 2 [ exp ( - w T ) - 1 + w T ] w v 2 ( w - v ) .
z 2 ( T ) = n = 0 P { n ( T ) = n } z n 2 ( T ) = n = 0 exp ( - T ) T n n ! 2 T 2 3 ( n + 1 ) ( n + 2 ) × { ( w 2 - 2 w v ) [ ( 1 - v ) n + 2 - 1 + ( n + 2 ) v ] + 2 v 2 [ ( 1 - w ) n + 2 - 1 + ( n + 2 ) w ] w v 2 ( w - v ) } = 2 3 ( w 2 - 3 w v ) [ exp ( - v T ) - 1 + v T ] + 2 v 2 [ exp ( - w T ) - 1 + w T ] w v 2 ( w - v ) .
σ z 2 ( T ) = z 2 ( T ) - z ( T ) 2 = 2 3 ( w 2 - 3 w v ) [ exp ( - v T ) - 1 + v T ] + 2 v 2 [ exp ( - w T ) - 1 + w T ] w v 2 ( w - v ) - [ 1 - exp ( - v T ) v ] 2 .

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