Abstract

A 3-D transport code DIF3D, based on the diffusion approximation, is used to model the spatial distribution of radiant energy arising from volumetric isotropic sources. The limits of validity of the diffusion approximation are formulated quantitatively by comparing the results, in the case of a slab geometry, of the diffusion and transport theories. For 3-D geometry, the results are presented in the form of isosurface plots, which give the surfaces of constant energy density. It is shown that as the detector sensitivity decreases, individual sources cannot be spatially distinguished, thus leading to a discrimination problem. Applications of the diffusion approximation to imaging through a medium with isotropic scattering are described. For a periodic distribution of line sources, the image is considerably degraded if the optical depth of the scattering medium is 0.4 or larger.

© 1986 Optical Society of America

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References

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  1. J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).
  2. A. Zardecki, S. A. W. Gerstl, J. F. Embury, “Application of the 2-D Discrete-Ordinates Method to Multiple Scattering of Laser Radiation,” Appl. Opt. 22, 1346 (1983).
    [CrossRef] [PubMed]
  3. R. L. Fante, “Propagation of Electromagnetic Waves Through Turbulent Plasma Using Transport Theory,” IEEE Trans. Antennas Propag. AP-21, 750 (1973).
    [CrossRef]
  4. W. G. Tam, A. Zardecki, “Laser Beam Propagation in Particulate Media,” J. Opt. Soc. Am. 69, 68 (1979).
    [CrossRef]
  5. W. G. Tam, A. Zardecki, “Multiple Scattering Corrections to the Beer-Lambert Law. 1: Open Detector,” Appl. Opt. 21, 2405 (1982).
    [CrossRef] [PubMed]
  6. A. Zardecki, W. G. Tam, “Multiple Scattering Corrections to the Beer-Lambert Law. 2: Detector with a Variable Field of View,” Appl. Opt. 21, 2413 (1982).
    [CrossRef] [PubMed]
  7. A. Zardecki, A. Deepak, “Forward Multiple Scattering Corrections as a Function of the Detector Field of View,” Appl. Opt. 22, 2970 (1983).
    [CrossRef] [PubMed]
  8. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  9. A. Ishimaru, “Diffusion of a Pulse in Densely Distributed Scatterers,” J. Opt. Soc. Am. 68, 1045 (1978).
    [CrossRef]
  10. K. Furutsu, “Diffusion Equation Derived from Space-Time Transport Equation,” J. Opt. Soc. Am. 70, 360 (1980).
    [CrossRef]
  11. W. G. Tam, A. Zardecki, “Off-Axis Propagation of a Laser Beam in Low Visibility Weather Conditions,” Appl. Opt. 19, 2822 (1980).
    [CrossRef] [PubMed]
  12. A. Ishimaru, Y. Kuga, R. L.-T.- Cheung, K. Shimizu, “Scattering and Diffusion of a Beam Wave in Randomly Distributed Scatterers,” J. Opt. Soc. Am. 73, 131 (1983).
    [CrossRef]
  13. B. Friedman, Principles and Techniques of Applied Mathematics (Wiley, New York, 1956).
  14. S. A. W. Gerstl, A. Zardecki, “Discrete-Ordinates Finite-Element Method for Atmospheric Radiative Transfer and Remote Sensing,” Appl. Opt. 24, 81 (1985).
    [CrossRef] [PubMed]
  15. K. L. Derstine, “DIF3D: A Code to Solve One-, Two-, and Three-Dimensional Finite-Difference Diffusion Theory Problems,” Argonne National Laboratory Report ANL-82–64 (Apr.1984).
  16. K. D. Lathrop, “Remedies for Ray Effects,” Nucl. Sci. Eng. 45, 255 (1971).
  17. R. E. Hufnagel, N. R. Stanley, “Modulation Transfer Function Associated with Image Transmission through Turbulent Media,” J. Opt. Soc. Am. 54, 52 (1964).
    [CrossRef]
  18. A. Ishimaru, “Limitation on Image Resolution Imposed by a Random Medium,” Appl. Opt. 17, 348 (1978).
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  19. Y. Kuga, A. Ishimaru, “Modulation Transfer Function and Image Transmission through Randomly Distributed Spherical Particles,” J. Opt. Soc. Am. A 2, 2330 (1985).
    [CrossRef]
  20. N. S. Kopeika, “Spatial-Frequency Dependence of Scattered Background Light: The Atmospheric Modulation Transfer Function Resulting from Aerosols,” J. Opt. Soc. Am. 72, 548 (1982).
    [CrossRef]
  21. N. S. Kopeika, “Spatial Frequency and Wavelength-Dependent Effects of Aerosols on the Atmospheric Modulation Transfer Function,” J. Opt. Soc. Am. 72, 1092 (1982).
    [CrossRef]
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    [CrossRef] [PubMed]
  23. W. G. Tam, A. Zardecki, “Spatial Frequency Dependent Image Degradation in a Particulate Medium,” Int. J. Infrared Millimeter Waves 6, 249 (1985).
    [CrossRef]
  24. K. R. Barnes, The Optical Transfer Function (American-Elsevier, New York, 1972).

1985 (3)

1984 (1)

1983 (3)

1982 (4)

1980 (2)

1979 (1)

1978 (2)

1973 (1)

R. L. Fante, “Propagation of Electromagnetic Waves Through Turbulent Plasma Using Transport Theory,” IEEE Trans. Antennas Propag. AP-21, 750 (1973).
[CrossRef]

1971 (1)

K. D. Lathrop, “Remedies for Ray Effects,” Nucl. Sci. Eng. 45, 255 (1971).

1964 (1)

Barnes, K. R.

K. R. Barnes, The Optical Transfer Function (American-Elsevier, New York, 1972).

Cheung, R. L.-T.-

Deepak, A.

Derstine, K. L.

K. L. Derstine, “DIF3D: A Code to Solve One-, Two-, and Three-Dimensional Finite-Difference Diffusion Theory Problems,” Argonne National Laboratory Report ANL-82–64 (Apr.1984).

Duderstadt, J. J.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

Embury, J. F.

Fante, R. L.

R. L. Fante, “Propagation of Electromagnetic Waves Through Turbulent Plasma Using Transport Theory,” IEEE Trans. Antennas Propag. AP-21, 750 (1973).
[CrossRef]

Friedman, B.

B. Friedman, Principles and Techniques of Applied Mathematics (Wiley, New York, 1956).

Furutsu, K.

Gerstl, S. A. W.

Hufnagel, R. E.

Ishimaru, A.

Kopeika, N. S.

Kuga, Y.

Lathrop, K. D.

K. D. Lathrop, “Remedies for Ray Effects,” Nucl. Sci. Eng. 45, 255 (1971).

Martin, W. R.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

Shimizu, K.

Stanley, N. R.

Tam, W. G.

Zardecki, A.

Appl. Opt. (8)

IEEE Trans. Antennas Propag. (1)

R. L. Fante, “Propagation of Electromagnetic Waves Through Turbulent Plasma Using Transport Theory,” IEEE Trans. Antennas Propag. AP-21, 750 (1973).
[CrossRef]

Int. J. Infrared Millimeter Waves (1)

W. G. Tam, A. Zardecki, “Spatial Frequency Dependent Image Degradation in a Particulate Medium,” Int. J. Infrared Millimeter Waves 6, 249 (1985).
[CrossRef]

J. Opt. Soc. Am. (7)

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

Nucl. Sci. Eng. (1)

K. D. Lathrop, “Remedies for Ray Effects,” Nucl. Sci. Eng. 45, 255 (1971).

Other (5)

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

B. Friedman, Principles and Techniques of Applied Mathematics (Wiley, New York, 1956).

K. L. Derstine, “DIF3D: A Code to Solve One-, Two-, and Three-Dimensional Finite-Difference Diffusion Theory Problems,” Argonne National Laboratory Report ANL-82–64 (Apr.1984).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

K. R. Barnes, The Optical Transfer Function (American-Elsevier, New York, 1972).

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Figures (15)

Fig. 1
Fig. 1

Energy density as a function of distance computed from the diffusion code dif3d (upper figure) and from the transport code onetran (lower figure). Asymmetry parameter g = 0.1, optical depth τ = 2; the black dots indicate results of the analytic solution, Eq. (13).

Fig. 2
Fig. 2

Same as Fig. 1, but g = 0.9, τ = 2.

Fig. 3
Fig. 3

Same as Fig. 1, but g = 0.1, τ = 16.

Fig. 4
Fig. 4

Same as Fig. 1, but g = 0.9, τ = 16.

Fig. 5
Fig. 5

Isosurface plot at 4% of maximum energy density. Radiation fields from two individual sources are well separated.

Fig. 6
Fig. 6

Isosurface plot at 2% of maximum energy density. Radiation fields from two individual sources overlap showing a discrimination problem.

Fig. 7
Fig. 7

Irradiance distribution in the plane z = 1.2 m for the wavelength λ = 0.70 μm.

Fig. 8
Fig. 8

Irradiance distribution in the plane z = 98.7 m for the wavelength λ = 0.70 μm.

Fig. 9
Fig. 9

Isosurface plot at 10% of maximum energy density; four distinct isotropic sources.

Fig. 10
Fig. 10

Isosurface plot for the configuration of four sources at 4% of maximum energy density.

Fig. 11
Fig. 11

Isosurface plot for the configuration of four sources at 2% of maximum energy density.

Fig. 12
Fig. 12

Irradiance due to a periodic object with spatial frequency f0 = 3.7 cycles/rad computed in the diffusion approximation. Slab with optical depth τ = 1.2. Image plane located at distance y = 0.4 (in units of optical depth) from the source plane.

Fig. 13
Fig. 13

Same as Fig. 12 but image plane located at y = 0.2 (in units of optical depth).

Fig. 14
Fig. 14

Same as Fig. 12, but f0 = 1.9 cycles/rad.

Fig. 15
Fig. 15

Same as Fig. 12, but f0 = 0.9 cycles/rad.

Equations (20)

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I ( r , Ω ) = 1 4 π [ ρ ( r ) + 3 Ω ) · J ( r ) ] ,
ρ ( r ) = I ( r , Ω ) d 2 Ω ,
J ( r ) = I ( r , Ω ) Ω d 2 Ω .
- · D ρ + ( σ - σ s ) ρ = q ,
J = - D ρ ,
μ = p ( Ω , Ω ) ( Ω · Ω ) d 2 Ω ,
σ t r = σ - σ s μ ,
D = 1 3 σ t r .
q ( z ) = q 0 δ ( z - z 0 ) .
Λ 2 = 3 σ 2 ( 1 - g ω ) ( 1 - ω ) ,
d 2 ρ d z 2 - Λ 2 ρ = - 3 σ ( 1 - ω g ) q 0 δ ( z - z 0 ) ,
ρ - 2 D d ρ d z = 0 a t z = 0 ,
ρ + 2 D d ρ d z = 0 a t z = d .
ρ ( z - z 0 ) = w 1 ( z ) w 2 ( z 0 ) H ( z 0 - z ) + w 2 ( z ) w 1 ( z 0 ) H ( z - z 0 ) K ( w 1 , w 2 ) .
w 1 ( z ) = ( 2 D Λ + 1 ) exp ( Λ z ) + ( 2 D Λ - 1 ) exp ( - Λ z ) ,
w 2 ( z ) = ( 2 D Λ - 1 ) exp [ Λ ( z - d ) ] + ( 2 D Λ + 1 ) exp [ - Λ ( z - d ) ] .
K ( w 2 , w 1 ) = w 1 w 2 - w 1 w 2 ,
n ( r ) = r α exp [ - ( r r c ) γ α γ ] .
F ( r ) = I ( r , Ω ) ( n ^ · Ω ) d 2 Ω ,
F ( x , y , z ) = 1 4 ρ ( x , y , z ) - 1 2 D ρ ( x , y , z ) z .

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