Abstract

A diffusion model is developed for describing the propagation of optical pulses through dense scattering media (clouds). The model invokes both 1-D and 3-D diffusion and is valid even for clouds of only a few diffusion thicknesses. The predictions of the model are compared to experimental pulse shapes obtained by other workers for real clouds and scale model clouds and by Monte Carlo simulations. Comparisons indicate that the ratio of the 3-D to the 1-D component in the transmitted pulse increases with both the diffusion thickness of the cloud and the field of view of the receiver. Knowledge of the impulse response of a cloud of known length is sufficient to determine the diffusion length and hence the average scattering coefficient of the cloud.

© 1988 Optical Society of America

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References

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  1. S. Ito, K. Furutsu, “Theory of Light Pulse Propagation Through Thick Clouds,” J. Opt. Soc. Am. 70, 366 (1980).
    [CrossRef]
  2. A. Ishimaru, “Diffusion of a Pulse in Densely Distributed Scatterers,” J. Opt. Soc. Am. 68, 1045 (1978).
    [CrossRef]
  3. E. A. Bucher, “A Computer Simulation of Light Pulse Propagation for Communication Through Thick Clouds,” Appl. Opt. 12, 2391 (1973).
    [CrossRef] [PubMed]
  4. D. T. Gillespie, “Calculation of n-Scattered Lidar Returns for Large n in an Idealised Cloud,” J. Opt. Soc. Am. A 4, 455 (1987).
    [CrossRef]
  5. G. C. Mooradian, M. Geller, “Temporal and Angular Spreading of Blue-Green Pulses in Clouds,” Appl. Opt. 21, 1572 (1982).
    [CrossRef] [PubMed]
  6. E. A. Bucher, R. M. Lerner, “Experiments on Light Pulse Communication and Propagation Through Atmospheric Clouds,” Appl. Opt. 12, 2401 (1973).
    [CrossRef] [PubMed]
  7. R. A. Elliott, “Multiple Scattering of Optical Pulses in Scale Model Clouds,” Appl. Opt. 22, 2670 (1983).
    [CrossRef] [PubMed]
  8. R. A. Elliott, Oregon Graduate Center; unpublished.
  9. L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and its Applications (Applied Science Publishers, New York, 1981).
    [CrossRef]
  10. S. Rosseland, “Astrophysik und Atom—Theoretisches Grundlage” (Springer Verlag, Berlin, 1931) as cited in Ref. 9.
  11. L. F. Gate, “The Determination of Light Absorption in Diffusing Materials by a Photon Diffusion Model,” J. Phys. D 4, 1049 (1971).
    [CrossRef]
  12. S. Chandrashekar, Noise and Stochastic Processes (Dover, New York, 1954).
  13. R. M. Lerner, J. D. Summers, “Monte Carlo Description of Time- and Space-Resolved Multiple Forward Scatter in Natural Water,” Appl. Opt. 21, 861 (1982).
    [CrossRef] [PubMed]
  14. S. Simons, “On the Motion of Dust Particles in a Gas,” J. Appl. Phys. 59, 1415 (1986).
    [CrossRef]
  15. S. Ito, “Comparison of Diffusion Theories for Optical Pulse Waves Propagated in Discrete Random Media,” J. Opt. Soc. Am. A 1, 502 (1984).
    [CrossRef]
  16. A. Ishimaru, “Difference Between Ishimaru’s and Furutsu’s Theories on Pulse Propagation in Discrete Random Media,” J. Opt. Soc. Am. A 1, 506 (1984).
    [CrossRef]

1987 (1)

D. T. Gillespie, “Calculation of n-Scattered Lidar Returns for Large n in an Idealised Cloud,” J. Opt. Soc. Am. A 4, 455 (1987).
[CrossRef]

1986 (1)

S. Simons, “On the Motion of Dust Particles in a Gas,” J. Appl. Phys. 59, 1415 (1986).
[CrossRef]

1984 (2)

1983 (1)

1982 (2)

1980 (1)

S. Ito, K. Furutsu, “Theory of Light Pulse Propagation Through Thick Clouds,” J. Opt. Soc. Am. 70, 366 (1980).
[CrossRef]

1978 (1)

A. Ishimaru, “Diffusion of a Pulse in Densely Distributed Scatterers,” J. Opt. Soc. Am. 68, 1045 (1978).
[CrossRef]

1973 (2)

1971 (1)

L. F. Gate, “The Determination of Light Absorption in Diffusing Materials by a Photon Diffusion Model,” J. Phys. D 4, 1049 (1971).
[CrossRef]

Bayvel, L. P.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and its Applications (Applied Science Publishers, New York, 1981).
[CrossRef]

Bucher, E. A.

Chandrashekar, S.

S. Chandrashekar, Noise and Stochastic Processes (Dover, New York, 1954).

Elliott, R. A.

Furutsu, K.

S. Ito, K. Furutsu, “Theory of Light Pulse Propagation Through Thick Clouds,” J. Opt. Soc. Am. 70, 366 (1980).
[CrossRef]

Gate, L. F.

L. F. Gate, “The Determination of Light Absorption in Diffusing Materials by a Photon Diffusion Model,” J. Phys. D 4, 1049 (1971).
[CrossRef]

Geller, M.

G. C. Mooradian, M. Geller, “Temporal and Angular Spreading of Blue-Green Pulses in Clouds,” Appl. Opt. 21, 1572 (1982).
[CrossRef] [PubMed]

Gillespie, D. T.

D. T. Gillespie, “Calculation of n-Scattered Lidar Returns for Large n in an Idealised Cloud,” J. Opt. Soc. Am. A 4, 455 (1987).
[CrossRef]

Ishimaru, A.

Ito, S.

S. Ito, “Comparison of Diffusion Theories for Optical Pulse Waves Propagated in Discrete Random Media,” J. Opt. Soc. Am. A 1, 502 (1984).
[CrossRef]

S. Ito, K. Furutsu, “Theory of Light Pulse Propagation Through Thick Clouds,” J. Opt. Soc. Am. 70, 366 (1980).
[CrossRef]

Jones, A. R.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and its Applications (Applied Science Publishers, New York, 1981).
[CrossRef]

Lerner, R. M.

Mooradian, G. C.

G. C. Mooradian, M. Geller, “Temporal and Angular Spreading of Blue-Green Pulses in Clouds,” Appl. Opt. 21, 1572 (1982).
[CrossRef] [PubMed]

Rosseland, S.

S. Rosseland, “Astrophysik und Atom—Theoretisches Grundlage” (Springer Verlag, Berlin, 1931) as cited in Ref. 9.

Simons, S.

S. Simons, “On the Motion of Dust Particles in a Gas,” J. Appl. Phys. 59, 1415 (1986).
[CrossRef]

Summers, J. D.

Appl. Opt. (1)

G. C. Mooradian, M. Geller, “Temporal and Angular Spreading of Blue-Green Pulses in Clouds,” Appl. Opt. 21, 1572 (1982).
[CrossRef] [PubMed]

Appl. Opt. (4)

J. Appl. Phys. (1)

S. Simons, “On the Motion of Dust Particles in a Gas,” J. Appl. Phys. 59, 1415 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

S. Ito, K. Furutsu, “Theory of Light Pulse Propagation Through Thick Clouds,” J. Opt. Soc. Am. 70, 366 (1980).
[CrossRef]

A. Ishimaru, “Diffusion of a Pulse in Densely Distributed Scatterers,” J. Opt. Soc. Am. 68, 1045 (1978).
[CrossRef]

J. Opt. Soc. Am. A (1)

D. T. Gillespie, “Calculation of n-Scattered Lidar Returns for Large n in an Idealised Cloud,” J. Opt. Soc. Am. A 4, 455 (1987).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Phys. D (1)

L. F. Gate, “The Determination of Light Absorption in Diffusing Materials by a Photon Diffusion Model,” J. Phys. D 4, 1049 (1971).
[CrossRef]

Other (4)

S. Chandrashekar, Noise and Stochastic Processes (Dover, New York, 1954).

R. A. Elliott, Oregon Graduate Center; unpublished.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and its Applications (Applied Science Publishers, New York, 1981).
[CrossRef]

S. Rosseland, “Astrophysik und Atom—Theoretisches Grundlage” (Springer Verlag, Berlin, 1931) as cited in Ref. 9.

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Figures (10)

Fig. 1
Fig. 1

Discrete RC line made up of T sections. Each T section represents a length b′ of the cloud.

Fig. 2
Fig. 2

Electrical analogy for photon transport by 1-D and 3-D diffusion: (a) photon diffusion; (b) the electrical analog consists of two RC lines with (RC)1 = b′/c and (RC)3 = 3b′/c representing, respectively, 1-D and 3-D diffusion.

Fig. 3
Fig. 3

Normalized impulse response of the RC line for different numbers τD of RC sections. Normalized time t′ = t/(RC)n; (RC)n = nb′/c, n = 1,3 for the 1-D and 3-D diffusive modes.

Fig. 4
Fig. 4

Effect of τD on the normalized instant of peaking t p n [ = t p n / ( b / c ) ;     n = 1 , 3 ] of the impulse response. The 1-D and 3-D values bracket the values obtained from scale model and real cloud experiments and Monte Carlo simulations.3 It can be seen that for τD > 4, t p n 0.1 n τ D 2 .

Fig. 5
Fig. 5

Comparison of the present model with experimental pulse shapes obtained7,8 from scale model clouds. The waveshapes I1 and I3 for the 1-D and 3-D diffuse modes are calculated using actual cloud parameters. Only the weights P1 and P3 for the two modes are adjusted. (a)–(d) DIM/water emulsion cloud; (e)–(f) oil/water emulsion cloud. Note that in (f) τD = 3.1, but P3 = 0 (i.e., the 3-D component is insignificant), SF (a sourcelike component not predicted by the model) appears.

Fig. 6
Fig. 6

Effect of diffusion thickness τD on the relative magnitude (P3/P1) of the 3-D and 1-D components for Elliott’s7 scale model clouds. The 3-D component becomes dominant with increasing τD: ●, code A; ■, code B; ×, code C; ⊙, code D; △, code E; + code F; ◇, code G.

Fig. 7
Fig. 7

Comparison of present model with real cloud data5: (a) narrow FOV (4°); the 1-D component dominates the response, P3/P1 = 0.17; (b) FOV = (17°), the 1-D and 3-D components are comparable, P3/P1 = 0.68; (c) large FOV (34°), the 3-D component dominates, P3/P1 = 4.95, I1 and I3 remain unchanged for all the cases (a)–(c).

Fig. 8
Fig. 8

Pulse shape predicted by the present model compared with Bucher’s Monte Carlo simulation3: (a) τD = 5.2; (b) τD = 13.8. The differences may be because Bucher uses a point source.

Fig. 9
Fig. 9

Value of 〈t〉 of the present model compared with those obtained by Monte Carlo simulation.3 Note that 〈t〉 for the 3-D impulse response is almost the same as for the Monte Carlo simulation. Thus the major contribution to 〈t〉 is made by the 3-D diffusion mode.

Fig. 10
Fig. 10

Standard deviation σt obtained by the present model compared with the Monte Carlo simulation.3 The value of σt for the simulation and the 3-D mode are almost equal indicating that the major contribution to σt is from the 3-D diffusion mode.

Tables (2)

Tables Icon

Table II Scale Model Cloud Parameters b ( 1 - cos θ ¯ ) and L as given by Elliott7,8 and the Parameters of our Model τD and (RC)1 for These Samples

Equations (20)

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J = - D 3 N ,
D 3 2 N = N / t .
D n 2 N = N / t ,
D n = b c / n ,
J n = - D n N ;             n = 1 , 3.
D n 2 N / x 2 = N / t ,
J n = - D n N / x ;             n = 1 , 3.
2 J n / x 2 + / t ( N / x ) = 0.
D n 2 J n / x 2 = J n / t ,
1 R C 2 I x 2 = I t ,
( Δ x ) 2 RC Δ 2 I ( Δ x ) 2 = Δ I Δ t ,
D 1 = ( Δ x ) 2 ( RC ) 1 ;             D 3 = ( Δ x ) 2 ( RC ) 3 .
( RC ) 1 = b / c ;             ( RC ) 3 = 3 b / c .
I n ( t ) = S ( t ) * h n ( t ) ;             n = 1 , 3.
I n ( t ) = P 1 I 1 ( t ) + P 3 I 3 ( t ) ,
t p n = 0.1 n τ D 2 b c - 1
= 0.1 n τ D L c - 1
= 0.1 n ( b ) - 1 c - 1 L 2 .
l = 0.5 b c - 1 τ D 1.96 ; σ l = 0.64 b c - 1 τ D 1.81 .
t = 0.55 b c - 1 τ D 1.9 , σ t = 0.43 b c - 1 τ D 1.8 .

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