Abstract

We use Monte Carlo simulations to study in detail the propagation of light in a plane-parallel medium containing scattering particles. In particular, we compute the forward and backward average path-length parameters (FAPP and BAPP, respectively) of four-flux radiative transfer models as functions of the optical depth. Strong dependence on the single scattering albedo and phase function asymmetry is found for both quantities. In general the values of the FAPP decrease with increasing absorption, whereas the opposite occurs for the BAPP. A similar effect is produced when changing from isotropic phase functions to phase functions with a large asymmetry in the forward direction. We present analytical results for the asymptotic values of the FAPP and BAPP as functions of albedo for the particular case of isotropic scattering. Our results differ markedly from the predictions obtained recently with two multiple-scattering models by Vargas and Niklasson [J. Opt. Soc. Am. A 14, 2243 (1997); Appl. Opt. 36, 3735 (1997)]. The differences found point out the intrinsic limitations of these models.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 1.
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1997), Chaps. 7–13.
  3. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).
  4. B. Maheu, J. N. Letoulouzan, G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 23, 3353–3362 (1984).
    [CrossRef]
  5. B. Maheu, G. Gouesbet, “Four-flux models to solve the scattering transfer equation: special cases,” Appl. Opt. 25, 1122–1128 (1986).
    [CrossRef] [PubMed]
  6. C. A. Arancibia-Bulnes, J. C. Ruiz-Suárez, “Spectral selectivity of Cermets with large metallic inclusions,” J. Appl. Phys. 83, 5421–5426 (1998).
    [CrossRef]
  7. T. Tesfamichael, W. E. Vargas, E. Wäcklegård, G. A. Niklasson, “Optical properties of silicon pigmented alumina films,” J. Appl. Phys. 82, 3508–3513 (1997).
    [CrossRef]
  8. W. E. Vargas, G. A. Niklasson, “Pigment mass density and refractive index determination from optical measurements,” J. Phys. Condens. Matter 9, 1661–1670 (1997).
    [CrossRef]
  9. A. P. Prishivalko, A. F. Sinyuk, “Studies of spectral characteristics of layers of ultradisperse silver particles taking into account the size dependence of optical constants and multiple light scattering,” Opt. Spectros. 79, 128–133 (1995).
  10. W. E. Vargas, “Generalized four-flux radiative transfer model,” Appl. Opt. 37, 2615–2623 (1998).
    [CrossRef]
  11. W. E. Vargas, G. A. Niklasson, “Generalized method for evaluating scattering parameters used in radiative transfer models,” J. Opt. Soc. Am. A 14, 2243–2252 (1997).
    [CrossRef]
  12. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3.
  13. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chaps. 3 and 4.
  14. W. E. Vargas, G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
    [CrossRef] [PubMed]
  15. W. E. Vargas, G. A. Niklasson, “Forward-scattering ratios and average path length parameter in radiative transfer models,” J. Phys. Condens. Matter 9, 9083–9096 (1997).
    [CrossRef]
  16. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 4.
  17. M. N. Özisik, Radiative Transfer and Interactions with Conduction and Convection (Wiley, New York, 1973), Chap. 10.
  18. W. E. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).
  19. W. E. Vargas, G. A. Niklasson, “Intensity of diffuse radiation in particulate media,” J. Opt. Soc. Am. A 14, 2253–2262 (1997).
    [CrossRef]
  20. G. N. Plass, G. W. Kattawar, F. E. Catchings, “Matrix operator theory of radiative transfer. 1. Rayleigh scattering,” Appl. Opt. 12, 314–329 (1973).
    [CrossRef] [PubMed]
  21. H.-W. Chang, T.-L. Wu, “Numerical solutions of matrix Riccati equations for radiative transfer in a plane-parallel geometry,” Waves Random Media 7, 147–168 (1997).
    [CrossRef]

1998

C. A. Arancibia-Bulnes, J. C. Ruiz-Suárez, “Spectral selectivity of Cermets with large metallic inclusions,” J. Appl. Phys. 83, 5421–5426 (1998).
[CrossRef]

W. E. Vargas, “Generalized four-flux radiative transfer model,” Appl. Opt. 37, 2615–2623 (1998).
[CrossRef]

1997

W. E. Vargas, G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
[CrossRef] [PubMed]

W. E. Vargas, G. A. Niklasson, “Forward-scattering ratios and average path length parameter in radiative transfer models,” J. Phys. Condens. Matter 9, 9083–9096 (1997).
[CrossRef]

T. Tesfamichael, W. E. Vargas, E. Wäcklegård, G. A. Niklasson, “Optical properties of silicon pigmented alumina films,” J. Appl. Phys. 82, 3508–3513 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Pigment mass density and refractive index determination from optical measurements,” J. Phys. Condens. Matter 9, 1661–1670 (1997).
[CrossRef]

H.-W. Chang, T.-L. Wu, “Numerical solutions of matrix Riccati equations for radiative transfer in a plane-parallel geometry,” Waves Random Media 7, 147–168 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Generalized method for evaluating scattering parameters used in radiative transfer models,” J. Opt. Soc. Am. A 14, 2243–2252 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Intensity of diffuse radiation in particulate media,” J. Opt. Soc. Am. A 14, 2253–2262 (1997).
[CrossRef]

1995

A. P. Prishivalko, A. F. Sinyuk, “Studies of spectral characteristics of layers of ultradisperse silver particles taking into account the size dependence of optical constants and multiple light scattering,” Opt. Spectros. 79, 128–133 (1995).

1986

1984

1973

1940

W. E. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).

1931

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Arancibia-Bulnes, C. A.

C. A. Arancibia-Bulnes, J. C. Ruiz-Suárez, “Spectral selectivity of Cermets with large metallic inclusions,” J. Appl. Phys. 83, 5421–5426 (1998).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chaps. 3 and 4.

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 4.

Catchings, F. E.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 1.

Chang, H.-W.

H.-W. Chang, T.-L. Wu, “Numerical solutions of matrix Riccati equations for radiative transfer in a plane-parallel geometry,” Waves Random Media 7, 147–168 (1997).
[CrossRef]

Gouesbet, G.

Hartel, W. E.

W. E. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chaps. 3 and 4.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1997), Chaps. 7–13.

Kattawar, G. W.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3.

Kubelka, P.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Letoulouzan, J. N.

Maheu, B.

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Niklasson, G. A.

T. Tesfamichael, W. E. Vargas, E. Wäcklegård, G. A. Niklasson, “Optical properties of silicon pigmented alumina films,” J. Appl. Phys. 82, 3508–3513 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Generalized method for evaluating scattering parameters used in radiative transfer models,” J. Opt. Soc. Am. A 14, 2243–2252 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Forward-scattering ratios and average path length parameter in radiative transfer models,” J. Phys. Condens. Matter 9, 9083–9096 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Pigment mass density and refractive index determination from optical measurements,” J. Phys. Condens. Matter 9, 1661–1670 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Intensity of diffuse radiation in particulate media,” J. Opt. Soc. Am. A 14, 2253–2262 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
[CrossRef] [PubMed]

Özisik, M. N.

M. N. Özisik, Radiative Transfer and Interactions with Conduction and Convection (Wiley, New York, 1973), Chap. 10.

Plass, G. N.

Prishivalko, A. P.

A. P. Prishivalko, A. F. Sinyuk, “Studies of spectral characteristics of layers of ultradisperse silver particles taking into account the size dependence of optical constants and multiple light scattering,” Opt. Spectros. 79, 128–133 (1995).

Ruiz-Suárez, J. C.

C. A. Arancibia-Bulnes, J. C. Ruiz-Suárez, “Spectral selectivity of Cermets with large metallic inclusions,” J. Appl. Phys. 83, 5421–5426 (1998).
[CrossRef]

Sinyuk, A. F.

A. P. Prishivalko, A. F. Sinyuk, “Studies of spectral characteristics of layers of ultradisperse silver particles taking into account the size dependence of optical constants and multiple light scattering,” Opt. Spectros. 79, 128–133 (1995).

Tesfamichael, T.

T. Tesfamichael, W. E. Vargas, E. Wäcklegård, G. A. Niklasson, “Optical properties of silicon pigmented alumina films,” J. Appl. Phys. 82, 3508–3513 (1997).
[CrossRef]

Vargas, W. E.

W. E. Vargas, “Generalized four-flux radiative transfer model,” Appl. Opt. 37, 2615–2623 (1998).
[CrossRef]

T. Tesfamichael, W. E. Vargas, E. Wäcklegård, G. A. Niklasson, “Optical properties of silicon pigmented alumina films,” J. Appl. Phys. 82, 3508–3513 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Generalized method for evaluating scattering parameters used in radiative transfer models,” J. Opt. Soc. Am. A 14, 2243–2252 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Forward-scattering ratios and average path length parameter in radiative transfer models,” J. Phys. Condens. Matter 9, 9083–9096 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
[CrossRef] [PubMed]

W. E. Vargas, G. A. Niklasson, “Pigment mass density and refractive index determination from optical measurements,” J. Phys. Condens. Matter 9, 1661–1670 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Intensity of diffuse radiation in particulate media,” J. Opt. Soc. Am. A 14, 2253–2262 (1997).
[CrossRef]

Wäcklegård, E.

T. Tesfamichael, W. E. Vargas, E. Wäcklegård, G. A. Niklasson, “Optical properties of silicon pigmented alumina films,” J. Appl. Phys. 82, 3508–3513 (1997).
[CrossRef]

Wu, T.-L.

H.-W. Chang, T.-L. Wu, “Numerical solutions of matrix Riccati equations for radiative transfer in a plane-parallel geometry,” Waves Random Media 7, 147–168 (1997).
[CrossRef]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 4.

Appl. Opt.

J. Appl. Phys.

C. A. Arancibia-Bulnes, J. C. Ruiz-Suárez, “Spectral selectivity of Cermets with large metallic inclusions,” J. Appl. Phys. 83, 5421–5426 (1998).
[CrossRef]

T. Tesfamichael, W. E. Vargas, E. Wäcklegård, G. A. Niklasson, “Optical properties of silicon pigmented alumina films,” J. Appl. Phys. 82, 3508–3513 (1997).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. Condens. Matter

W. E. Vargas, G. A. Niklasson, “Pigment mass density and refractive index determination from optical measurements,” J. Phys. Condens. Matter 9, 1661–1670 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Forward-scattering ratios and average path length parameter in radiative transfer models,” J. Phys. Condens. Matter 9, 9083–9096 (1997).
[CrossRef]

Licht

W. E. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).

Opt. Spectros.

A. P. Prishivalko, A. F. Sinyuk, “Studies of spectral characteristics of layers of ultradisperse silver particles taking into account the size dependence of optical constants and multiple light scattering,” Opt. Spectros. 79, 128–133 (1995).

Waves Random Media

H.-W. Chang, T.-L. Wu, “Numerical solutions of matrix Riccati equations for radiative transfer in a plane-parallel geometry,” Waves Random Media 7, 147–168 (1997).
[CrossRef]

Z. Tech. Phys.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Other

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chaps. 3 and 4.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 4.

M. N. Özisik, Radiative Transfer and Interactions with Conduction and Convection (Wiley, New York, 1973), Chap. 10.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 1.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1997), Chaps. 7–13.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Analytical asymptotic values of ε (short-dashed curve) and η (long-dashed curve), from Eqs. (9) and (10), as functions of the albedo. The values of v 0 are also depicted (solid curve).

Fig. 2
Fig. 2

FAPP as a function of optical depth τ for Mie scattering with four size parameters. The refractive indices of the particles and the host medium are, respectively, n p = 2.75 and n m = 1.5.

Fig. 3
Fig. 3

FAPP for the HG phase function with three asymmetry parameters, μ̅ = 0, 0.75, and 0.9. ω0 = 1.

Fig. 4
Fig. 4

Same as Fig. 3 but for ω0 = 0.85.

Fig. 5
Fig. 5

Same as Fig. 3 but for ω0 = 0.65.

Fig. 6
Fig. 6

Comparison of the Monte Carlo calculations of the present study (points) and results calculated from previously published data (curves) for clouds of thickness τ. The results for the isotropic and HG phase functions were calculated from the data of Ref. 21, whereas the results for the Rayleigh phase function were calculated from Ref. 20.

Fig. 7
Fig. 7

BAPP as a function of τ for Mie scattering. The values of the parameters are the same as for Fig. 2.

Fig. 8
Fig. 8

BAPP for the HG phase function with the same parameters as in Fig. 3. The three upper curves correspond to an albedo of ω0 = 0.85; the lower curves correspond to ω0 = 1.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

dIcdz=k+sIc,
dJcdz=-k+sJc,
dIddz=εk+1-ζsId-ε1-ζsJd-1-ζsJc-ζsIc,
dJddz=-εk+1-ζsJd+ε1-ζsId+1-ζsIc+ζsJc.
ετ=01 Iτ, μdμ01 μIτ, μdμ,
ητ=--10 Iτ, μdμ-10 μIτ, μdμ,
Iμ, τ1v0-μ exp-τ/v0.
1-ω0v02 log1+1/v01-1/v0=0.
εa=1/log1-1/v0+v0-1,
ηa=1/log1+1/v0-v0-1.
pHGμ=ω01-g21+g2-2gμ-3/2,
Id=i=1 Qizfiμ.
fiμ=Ni-11 fi-1μpμ, μdμ,  fiμ=pμ,
Q1τ=ω0exp-τ-exp-τ/μ¯f1/μ¯f-1,
I1,Hartelμ, τ=ω0pμexp-τ-exp-τ/μ¯f1/μ¯f-1.
I1,Hartelμ, τ=pμω0τ;
I1,exactμ, τ=ω0pμexp-τ-exp-τ/μ1-μ,
I1,exactμ, τ=pμμ ω0τ
I1,exactμ, τ=pμ1-μ ω0 exp-τ

Metrics