Abstract

An analytic model for propagation of an optical pulse through an optically thick multiple-scattering medium such as a cloud is developed. Both a narrow collimated and a broad Gaussian beam are considered incident at the entrance of the medium, which is assumed to be a plane-parallel slab of infinite horizontal extent. Analytic expressions are derived for the radiance and power received by an on-axis receiver at any distance beyond the exit plane of the medium. Calculations with these expressions are compared with published Monte Carlo results for an infinite-plane, π/2-field-of-view receiver at the exit plane of the medium. Calculated and simulated results show excellent agreement for optical thickness beyond approximately 20.

© 1995 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, England, 1950).
  2. L. B. Stotts, “The radiance produced by laser radiation transversing a particulate multile-scattering medium,” J. Opt. Soc. Am. 67, 815–819 (1977).
    [CrossRef]
  3. E. A. Bucher, “Computer simulation of light pulse propagation for communication through thick clouds,” Appl. Opt. 12, 2391–2400 (1973).
    [CrossRef] [PubMed]
  4. R. F. Lutomirski, A. P. Ciervo, G. J. Hall, “Moments of multiple scattering,” Appl. Opt. 34, 7125–7136 (1995).
    [CrossRef] [PubMed]
  5. For simplicity, we define ∂/∂x = (∂/∂x, ∂/∂y, ∂/∂z)T and ∂2/∂x2 = (∂2/∂x2, ∂2/∂y2, ∂2/∂z2)T.
  6. W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1966), Vol. 2.
  7. P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  8. J. S. DeGroot, R. M. Frank, H. S. Stewart, “Effects of clouds on x-ray induced fluorescent light from a nuclear burst in space,” Rep. C73-65(u)11 (E. H. Plesset Associates, Santa Monica, Calif., 1965).
  9. S. Fritz, “Scattering and absorption of solar energy by clouds,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1953).
  10. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, 2d ed. (Clarendon, Oxford, England, 1959).

1995 (1)

1977 (1)

1973 (1)

Bucher, E. A.

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, 2d ed. (Clarendon, Oxford, England, 1959).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, England, 1950).

Ciervo, A. P.

DeGroot, J. S.

J. S. DeGroot, R. M. Frank, H. S. Stewart, “Effects of clouds on x-ray induced fluorescent light from a nuclear burst in space,” Rep. C73-65(u)11 (E. H. Plesset Associates, Santa Monica, Calif., 1965).

Feller, W.

W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1966), Vol. 2.

Feshbach, H.

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Frank, R. M.

J. S. DeGroot, R. M. Frank, H. S. Stewart, “Effects of clouds on x-ray induced fluorescent light from a nuclear burst in space,” Rep. C73-65(u)11 (E. H. Plesset Associates, Santa Monica, Calif., 1965).

Fritz, S.

S. Fritz, “Scattering and absorption of solar energy by clouds,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1953).

Hall, G. J.

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, 2d ed. (Clarendon, Oxford, England, 1959).

Lutomirski, R. F.

Morse, P.

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Stewart, H. S.

J. S. DeGroot, R. M. Frank, H. S. Stewart, “Effects of clouds on x-ray induced fluorescent light from a nuclear burst in space,” Rep. C73-65(u)11 (E. H. Plesset Associates, Santa Monica, Calif., 1965).

Stotts, L. B.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Other (7)

S. Chandrasekhar, Radiative Transfer (Clarendon, Oxford, England, 1950).

For simplicity, we define ∂/∂x = (∂/∂x, ∂/∂y, ∂/∂z)T and ∂2/∂x2 = (∂2/∂x2, ∂2/∂y2, ∂2/∂z2)T.

W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1966), Vol. 2.

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

J. S. DeGroot, R. M. Frank, H. S. Stewart, “Effects of clouds on x-ray induced fluorescent light from a nuclear burst in space,” Rep. C73-65(u)11 (E. H. Plesset Associates, Santa Monica, Calif., 1965).

S. Fritz, “Scattering and absorption of solar energy by clouds,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1953).

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, 2d ed. (Clarendon, Oxford, England, 1959).

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Figures (14)

Fig. 1
Fig. 1

Propagation path with model parameters.

Fig. 2
Fig. 2

Representation of photon density ρ z (z, ξ), ξ fixed, using the diffusion-theory approximation with an extended-medium boundary condition.

Fig. 3
Fig. 3

Comparison of simulated and calculated power pulses for τ = 30.

Fig. 4
Fig. 4

Comparison of simulated and calculated power pulses for τ = 80.

Fig. 5
Fig. 5

Comparison of simulated and calculated energy transition versus optical thickness for 〈cos θ〉 = 0.827.

Fig. 6
Fig. 6

Comparison of simulated and calculated mean radial distance of emerging photons versus optical thickness for 〈cos θ〉 = 0.827.

Fig. 7
Fig. 7

Comparison of simulated and calculated average multipath time delay versus optical thickness for 〈cos θ〉 = 0.827.

Fig. 8
Fig. 8

Received energy versus field of view, plotted with a generally applicable expression and one restricted to small-field-of-view receivers.

Fig. 9
Fig. 9

Received energy versus distance below the cloud (exit plane) for τ = 30 and various receiver fields of view.

Fig. 10
Fig. 10

Average multipath time delay versus distance below the cloud for τ = 30 and various receiver fields of view.

Fig. 11
Fig. 11

Received energy versus distance below the cloud for a 15° field of view and various cloud thicknesses.

Fig. 12
Fig. 12

Multipath time delay versus distance below the cloud for a 15° field of view and various cloud thicknesses.

Fig. 13
Fig. 13

Received energy versus optical thickness of a cloud for a 15° field of view and various receiver-to-cloud distances.

Fig. 14
Fig. 14

Multipath time delay versus optical thickness of a cloud for a 15° field of view and various receiver-to-cloud distances.

Equations (63)

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I ( x , α , ξ ) ξ + e α · x I = - I + ω ˜ 0 4 π 4 π I ( x , α ; ξ ) p ( α ; α ) d ω α + S 0 ( x , α ; ξ ) ,
ρ ( x , ξ ) ξ = [ D ( ξ ) 2 · ( 2 x 2 ) - a ( ξ ) · ( x ) - κ ] ρ ( x , ξ ) ,
ρ ( x , ξ ) = ( 2 π ) - 3 / 2 Λ - 1 × exp { - [ κ ξ + ( 2 Λ ) - 1 ( x - μ ) T ( x - μ ) ] } ,
μ = 0 ξ a ( ξ ) d ξ ,
Λ = 0 ξ D x ( ξ ) d ξ 0 ξ D y ( ξ ) d ξ 0 ξ D z ( ξ ) d ξ .
ρ ( x , ξ ) = exp ( - κ ξ ) ρ x ( x , ξ ) ρ y ( y , ξ ) ρ z ( z , ξ ) ,
a ( ξ ) x ¯ ( ξ ) ξ = [ 0 , 0 , z ¯ ( ξ ) ξ ] T ,
D ( ξ ) [ σ x 2 ( ξ ) ξ , σ y 2 ( ξ ) ξ , σ z 2 ( ξ ) ξ ] T ,
z ¯ ( ξ ) = 1 - exp ( - v ξ ) v ,
σ x 2 ( ξ ) = σ y 2 ( ξ ) = 2 3 w 2 [ exp ( - v ξ ) - 1 + v ξ ] - v 2 [ exp ( - w ξ ) - 1 + w ξ ] w v 2 ( w - v ) ,
σ z 2 ( ξ ) = 2 3 ( w 2 - 3 w v ) [ exp ( - v ξ ) - 1 + v ξ ] + 2 v 2 [ exp ( - w ξ ) - 1 + w ξ ] w v 2 ( w - v ) - [ 1 - exp ( - v ξ ) v ] 2 ,
ρ z ( z ) ± 0.71 ρ z z | z = 0 or z = T ,
ρ z ( z ) = ± 0.71 v ρ z z | z = 0 or z = T ,
ρ ( x , ξ ) = δ ( x ) δ ( y ) δ ( z ) ,             ξ = 0 ,
ρ ( x , ξ ) 0 ,             z = - 0.71 v ,             z = τ + 0.71 v ,
ρ ( x , ξ ) 0 ,             x = y = ± .
ρ ( x , ξ ) = exp ( - κ ξ ) ρ x ( x , ξ ) ρ y ( y , ξ ) ρ z ( z , ξ ) ,
ρ x ( x , ξ ) = 1 σ x 2 π exp [ - x 2 2 σ x 2 ( ξ ) ] ,
ρ y ( y , ξ ) = 1 σ y 2 π exp [ - y 2 2 σ y 2 ( ξ ) ] .
ρ z ξ = D z ( ξ ) 2 2 ρ z z 2 - a z ( ξ ) ρ z z ,
ρ z ( z , ξ ) = δ ( z ) ,             ξ = 0 , ρ z ( z , ξ ) 0 ,             z = - 0.71 v ,             z = τ + 0.71 v .
r z ( z , x ) = 1 s z ( x ) 2 p × k = - ( exp { - [ z - z ¯ ( x ) + 2 k ( t + 1.42 v ) ] 2 2 s z 2 ( x ) } - exp { [ z + z ¯ ( x ) + 2 k ( t + 1.42 v ) + 1.42 v ] 2 2 s z 2 ( x ) } ) ,
ρ ( x , ξ ) = exp ( - κ ξ ) ρ x ( x , ξ ) ρ y ( y , ξ ) ρ z ( z , ξ ) ,
κ = k a k s = 1 - ω ˜ 0 ω ˜ 0 ,
P { z ( ξ ) [ 0 , τ ¯ ] } = 1 - x = - y = - z = 0 τ exp ( κ ξ ) ρ ( x , ξ ) d V x = 1 - x = - y = - z = 0 τ ρ x ( x , ξ ) ρ y ( y , ξ ) ρ z ( z , ξ ) d x d y d z = 1 - 0 τ ρ z ( z , ξ ) d z .
ξ P { z ( ξ ) [ 0 , τ ¯ ] } = - 0 τ ρ z ξ d z = - 0 τ [ D z ( ξ ) 2 2 ρ z z 2 - a z ( ξ ) ρ z z ] d z = [ a z ( ξ ) ρ z - D z ( ξ ) 2 ρ z z ] z = τ - [ a z ( ξ ) ρ z - D z ( ξ ) 2 ρ z z ] z = 0 .
D z 2 z ρ z
A ( τ , ξ ) = [ exp ( - v ξ ) ρ z - D z ( ξ ) 2 ρ z ] z = τ = 1 σ z ( ξ ) 2 π k = - { exp [ - β 1 2 2 σ z 2 ( ξ ) ] × [ exp ( - v ξ ) + D z ( ξ ) 2 σ z 2 ( ξ ) β 1 ] - exp [ - β 2 2 2 σ z 2 ( ξ ) ] [ exp ( - v ξ ) + D z ( ξ ) 2 σ z 2 ( ξ ) β 2 ] } ,
β 1 = - z ¯ ( ξ ) + τ ( 1 + 2 k ) + 4 k ( 0.71 v ) , β 2 = z ¯ ( ξ ) + ( 1 + 2 k ) ( τ + 1.42 v ) ,
D z ( ξ ) = σ z 2 ( ξ ) ξ = 2 v × { ( w - 3 v ) [ 1 - exp ( - v ξ ) ] + 2 v [ 1 - exp ( - w ξ ) ] 3 ( w - v ) - exp ( - v ξ ) [ 1 - exp ( - v ξ ) ] } .
J ( r , τ ; ξ ) = E 0 exp ( - κ ξ ) ρ r ( r , ξ ) A ( τ , ξ ) ,
ρ r ( r , ξ ) = r σ x 2 ( ξ ) exp [ - r 2 2 σ x 2 ( ξ ) ] .
r = 0 J ( r , τ ; ξ ) d r = A ( τ , ξ ) 0 ρ r ( r , ξ ) d r A ( τ , ξ ) .
E τ = 0 A ( τ , ξ ) d ξ ,
r τ = 0 r h τ ( r ) d r ,
h τ ( r ) d r = P { r d r B } = P { r d r , B } / P { B } = ξ = 0 P { r d r , ξ d ξ , B } / P { B } = ξ = 0 P { r d r , B ξ } P { ξ d ξ , B } / P { B } .
P { r d r , B ξ } = r σ x 2 ( ξ ) exp [ - r 2 2 σ x 2 ] ,
P { B } = 0 A ( τ , ξ ) d ξ = E τ .
h τ ( r ) d r = E τ - 1 0 r σ x 2 ( ξ ) exp [ - r 2 2 σ x 2 ( ξ ) ] A ( τ , ξ ) d ξ ,
r τ = E τ - 1 r = 0 ξ = 0 r 2 σ x 2 ( ξ ) exp [ - r 2 2 σ x 2 ( ξ ) ] A ( τ , ξ ) d ξ d r .
r τ = E τ - 1 ξ = 0 { r = 0 r 2 σ x 3 ( ξ ) exp [ - r 2 2 σ x 2 ( ξ ) ] d r } × σ x ( ξ ) A ( τ , ξ ) d ξ = E τ - 1 π / 2 ξ = 0 σ x ( ξ ) A ( τ , ξ ) d ξ ,
Δ ξ ¯ = ξ = 0 ξ A ( τ , ξ ) d ξ ξ = 0 A ( τ , ξ ) d ξ - τ .
S 0 ( x , α ; ξ ) = E 0 δ ( x ) δ ( α ) δ ( ξ ) sin θ ,
S 0 ( x , α ; ξ ) = E 0 δ ( z ) δ ( α ) δ ( ξ ) 2 π σ B 2 sin θ exp [ - ( x 2 + y 2 ) 2 σ b 2 ] ,
ρ ( r , z , ξ ) = E 0 r σ B - 2 δ ( z ) δ ( ξ ) exp [ - r 2 2 σ B 2 ] ,
J ( r , τ ; ξ ) = E 0 0 J δ ( r - μ , τ ; ξ ) μ σ B 2 exp ( - μ 2 2 σ B 2 ) d μ = E 0 exp ( - κ ξ ) ρ r ( r , ξ ) A ( τ , ξ ) ,
ρ r ( r , ξ ) = r σ x 2 ( ξ ) + σ B 2 exp { - r 2 2 [ σ x 2 ( ξ ) + σ B 2 ] }
g ( α ) = 2 + 3 cos θ 7 π ,             0 θ π / 2 , 0 ϕ 2 π ,
J ( r , τ ; ξ ) = Ω I ( r , τ , α ; ξ ) cos θ d ω α ,
I ( r , τ , α ; ξ ) = J ( r , τ ; ξ ) g ( α ) Ω g ( α ) cos θ d ω α = E 0 4 π ex ( - κ ξ ) ( 2 + 3 cos θ ) ρ r ( r , ξ ) A ( τ , ξ ) ,             0 θ π / 2 ,
P τ ( ξ ) = r = 0 a θ = θ θ fov ϕ = 0 2 π I cos θ sin θ d θ d ϕ = E 0 4 exp ( - κ ξ ) ( 3 - cos 2 θ fov - 2 cos 3 θ fov ) × ( 1 - exp { - a 2 2 [ σ x 2 ( ξ ) + σ B 2 ] } ) A ( τ , ξ ) .
G ( r , τ + d , α , ξ ; r , τ , α , ξ ) = exp ( - κ d sec θ ) δ ( r - r - d tan θ n ) × δ ( α - α ) δ ( ξ - ξ - d sec θ ) .
I ( r , τ + d , α ; ξ ) = 0 d ξ R 2 d 2 r Ω d ω α , I ( r , τ , α ; ξ ) G = exp ( - κ d sec θ ) I ( r - d tan θ n , τ , α ; ξ - d sec θ ) ,
I ( r , τ + d , α ; ξ ) = E 0 4 π exp ( - κ ξ ) ( 2 + 3 cos θ ) ρ r ( r , ξ ) × A ( τ , ξ - d sec θ ) ,             0 θ π / 2 ,
ρ r ( r , ξ ) = 1 2 π [ σ x 2 ( ξ - d sec θ ) + σ B 2 ] × exp { - ( x - d tan θ cos ϕ ) 2 + ( y - d tan θ sin ϕ ) 2 2 [ σ x 2 ( ξ - d sec θ ) + σ B 2 ] } .
I ( r , τ + d , α ; ξ ) r = 0 = E 0 exp ( - κ ξ ) ( 2 + 3 cos θ ) A ( τ , ξ - d sec θ ) 8 π 2 [ σ x 2 ( ξ - d sec θ ) + σ B 2 ] × exp { - d 2 tan 2 θ 2 [ σ x 2 ( ξ - d sec θ ) + σ B 2 ] } .
a [ σ x 2 ( ξ ) + σ B 2 ] 1 / 2
P τ + d ( ξ ) r < a d 2 r θ = 0 θ fov d θ ϕ = 0 2 π d ϕ I r = 0 cos θ sin θ = E 0 a 2 4 exp ( - κ ξ ) θ = 0 θ f o v A ( τ , ξ - d sec θ ) σ x 2 ( ξ - d sec θ ) + σ B 2 × exp { - d 2 tan 2 θ 2 [ σ x 2 ( ξ - d sec θ ) + σ B 2 ] } × ( 2 + 3 cos θ ) cos θ sin θ d θ ,
P τ + d ( ξ ) = 5 a 2 E 0 4 d 2 exp ( - κ ξ ) A ( τ , ξ - d ) × [ 1 - exp { - d 2 θ fov 2 2 [ σ x 2 ( ξ - d ) + σ B 2 ] } ] .
lim d 0 P τ + d ( ξ ) = 5 a 2 E 0 θ fov 2 8 [ σ x 2 ( ξ ) + σ B 2 ] exp ( - κ ξ ) A ( τ , ξ ) ,
E τ + d ( a , θ fov ) = ξ = 0 P τ + d ( ξ ) d ξ
E τ + d ( a , θ fov ) = ξ = 0 P τ + d ( ξ ) d ξ ,
Δ ξ ¯ = E τ + d - 1 ( a , θ fov ) ξ = 0 ξ P τ + d ( ξ ) d ξ - ( τ + d ) .

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