Abstract

A multiple-scattering approach is used to solve the radiative transfer equation in order to describe the angular and optical depth dependence of diffuse intensity propagating through an unsupported particulate slab perpendicularly illuminated with unpolarized electromagnetic radiation. Boundary reflections are taken into account for the unscattered radiation, as is each of the successive reflection and scattering orders, which are characterized by corresponding average path-length parameters and forward-scattering ratios. The optical depth dependence of the reflection- and scattering-order coefficients is displayed for given values of the particle size parameter, particle volume fraction, relative refractive index, and optical thickness of the slab. Diffuse intensity patterns at a given optical depth, and the angular dependence of the reflected and transmitted intensities, are also considered.

© 1999 Optical Society of America

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References

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  1. G. Zaccanti, “Monte Carlo study of light propagation in optically thick media: point source case,” Appl. Opt. 30, 2031–2041 (1991).
    [CrossRef] [PubMed]
  2. J. P. Briton, B. Maheu, G. Gréhan, G. Gouesbet, “Monte Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
    [CrossRef]
  3. M. H. Eddowes, T. N. Mills, D. T. Delpy, “Monte Carlo simulations of coherent backscatter for identification of the optical coefficients of biological tissues in vivo,” Appl. Opt. 34, 2261–2267 (1995).
    [CrossRef] [PubMed]
  4. N. T. Melamed, “Optical properties of powders. I. Optical absorption coefficients and the absolute value of the diffuse reflectance; II. Properties of luminescent powders,” J. Appl. Phys. 34, 560–570 (1963).
    [CrossRef]
  5. E. L. Simmons, “Diffuse reflectance spectroscopy: a comparison of the theories,” Appl. Opt. 14, 1380–1386 (1975).
    [CrossRef] [PubMed]
  6. A. Mandelis, F. Boroumand, H. Van den Bergh, “Quantitative diffuse reflectance spectroscopy of large powders: the Melamed model revisited,” Appl. Opt. 29, 2853–2860 (1990).
    [CrossRef] [PubMed]
  7. K. Stamnes, S. C. Tsay, W. Wiscombe, K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
    [CrossRef] [PubMed]
  8. J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [CrossRef]
  9. C. M. Lam, A. Ishimaru, “Mueller matrix calculation for a slab of random medium with both random rough surfaces and discrete particles,” IEEE Trans. Antennas Propag. 42, 145–156 (1994).
    [CrossRef]
  10. R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
    [CrossRef] [PubMed]
  11. A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
    [CrossRef]
  12. A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Scaling relationships for theories of anisotropic random walks applied to tissue optics,” Appl. Opt. 32, 504–516 (1993).
    [CrossRef] [PubMed]
  13. R. F. Lutomirski, A. P. Ciervo, G. J. Hall, “Moments of multiple scattering,” Appl. Opt. 34, 7125–7136 (1995).
    [CrossRef] [PubMed]
  14. V. G. Kolinko, F. F. M. de Mull, J. Greeve, A. V. Priezzhev, “Probabilistic model of multiple light scattering based on rigorous computations of the first and second moments of photon coordinates,” Appl. Opt. 35, 4541–4550 (1996).
    [CrossRef] [PubMed]
  15. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).
  16. J. Reichman, “Determination of absorption and scattering coefficients for nonhomogeneous media. 1. Theory,” Appl. Opt. 12, 1811–1815 (1973).
    [CrossRef] [PubMed]
  17. 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]
  18. C. M. Chu, S. W. Churchill, “Numerical solution of problems in multiple scattering of electromagnetic radiation,” Mult. Scatt. Electromag. Rad. 59, 855–863 (1955).
  19. P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology. II,” J. Colloid Interface Sci. 39, 551–567 (1972).
    [CrossRef]
  20. C. Whitney, “Efficient stream distributions in radiative transfer theory,” J. Quant. Spectrosc. Radiat. Transf. 14, 591–611 (1974).
    [CrossRef]
  21. W. E. Vargas, G. A. Niklasson, “Intensity of diffuse radiation in particulate media,” J. Opt. Soc. Am. A 14, 2253–2262 (1997).
    [CrossRef]
  22. 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]
  23. W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
    [CrossRef] [PubMed]
  24. W. E. Vargas, “Generalized four-flux radiative transfer model,” Appl. Opt. 37, 2615–2623 (1998).
    [CrossRef]
  25. W. Hartel, “Zur Theorie der Lichtstreuung durch trube Schichten besonders Trubglaser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).
  26. C. M. Chu, S. W. Churchill, “Representation of the angular distribution of radiation scattered by a spherical particle,” J. Opt. Soc. Am. 45, 958–962 (1955).
    [CrossRef]
  27. 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]
  28. W. E. Vargas, G. A. Niklasson, “Forward scattering ratios and average path-length parameters in radiative transfer models,” J. Phys.: Condens. Matter 9, 9083–9096 (1997).
  29. G. Gobel, J. Kuhn, J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5, 413–426 (1995).
    [CrossRef]
  30. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  31. H. G. Hecht, “The a priori calculation of the diffuse reflectance of a turbid medium,” Opt. Acta 30, 659–668 (1983).
    [CrossRef]
  32. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  33. P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
    [CrossRef]
  34. N. G. Khlebtsov, “Orientational averaging of light-scattering observables in the T-matrix approach,” Appl. Opt. 31, 5359–5365 (1992).
    [CrossRef] [PubMed]
  35. M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).
    [CrossRef]
  36. M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
    [CrossRef]

1998

1997

1996

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
[CrossRef]

V. G. Kolinko, F. F. M. de Mull, J. Greeve, A. V. Priezzhev, “Probabilistic model of multiple light scattering based on rigorous computations of the first and second moments of photon coordinates,” Appl. Opt. 35, 4541–4550 (1996).
[CrossRef] [PubMed]

1995

1994

C. M. Lam, A. Ishimaru, “Mueller matrix calculation for a slab of random medium with both random rough surfaces and discrete particles,” IEEE Trans. Antennas Propag. 42, 145–156 (1994).
[CrossRef]

1993

1992

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

J. P. Briton, B. Maheu, G. Gréhan, G. Gouesbet, “Monte Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

N. G. Khlebtsov, “Orientational averaging of light-scattering observables in the T-matrix approach,” Appl. Opt. 31, 5359–5365 (1992).
[CrossRef] [PubMed]

1991

1990

1988

1987

1984

1983

H. G. Hecht, “The a priori calculation of the diffuse reflectance of a turbid medium,” Opt. Acta 30, 659–668 (1983).
[CrossRef]

1979

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
[CrossRef]

1975

1974

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

C. Whitney, “Efficient stream distributions in radiative transfer theory,” J. Quant. Spectrosc. Radiat. Transf. 14, 591–611 (1974).
[CrossRef]

1973

1972

P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology. II,” J. Colloid Interface Sci. 39, 551–567 (1972).
[CrossRef]

1971

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

1963

N. T. Melamed, “Optical properties of powders. I. Optical absorption coefficients and the absolute value of the diffuse reflectance; II. Properties of luminescent powders,” J. Appl. Phys. 34, 560–570 (1963).
[CrossRef]

1955

C. M. Chu, S. W. Churchill, “Numerical solution of problems in multiple scattering of electromagnetic radiation,” Mult. Scatt. Electromag. Rad. 59, 855–863 (1955).

C. M. Chu, S. W. Churchill, “Representation of the angular distribution of radiation scattered by a spherical particle,” J. Opt. Soc. Am. 45, 958–962 (1955).
[CrossRef]

1940

W. Hartel, “Zur Theorie der Lichtstreuung durch trube Schichten besonders Trubglaser,” 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. (Leipzig) 12, 593–601 (1931).

Bonner, R. F.

Boroumand, F.

Briton, J. P.

J. P. Briton, B. Maheu, G. Gréhan, G. Gouesbet, “Monte Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

Chu, C. M.

C. M. Chu, S. W. Churchill, “Numerical solution of problems in multiple scattering of electromagnetic radiation,” Mult. Scatt. Electromag. Rad. 59, 855–863 (1955).

C. M. Chu, S. W. Churchill, “Representation of the angular distribution of radiation scattered by a spherical particle,” J. Opt. Soc. Am. 45, 958–962 (1955).
[CrossRef]

Churchill, S. W.

C. M. Chu, S. W. Churchill, “Representation of the angular distribution of radiation scattered by a spherical particle,” J. Opt. Soc. Am. 45, 958–962 (1955).
[CrossRef]

C. M. Chu, S. W. Churchill, “Numerical solution of problems in multiple scattering of electromagnetic radiation,” Mult. Scatt. Electromag. Rad. 59, 855–863 (1955).

Ciervo, A. P.

de Mull, F. F. M.

Delpy, D. T.

Eddowes, M. H.

Fricke, J.

G. Gobel, J. Kuhn, J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5, 413–426 (1995).
[CrossRef]

Gandjbakhche, A. H.

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Scaling relationships for theories of anisotropic random walks applied to tissue optics,” Appl. Opt. 32, 504–516 (1993).
[CrossRef] [PubMed]

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

Gobel, G.

G. Gobel, J. Kuhn, J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5, 413–426 (1995).
[CrossRef]

Gouesbet, G.

J. P. Briton, B. Maheu, G. Gréhan, G. Gouesbet, “Monte Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

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]

Greeve, J.

Gréhan, G.

J. P. Briton, B. Maheu, G. Gréhan, G. Gouesbet, “Monte Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

Hall, G. J.

Hansen, J. E.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Hartel, W.

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

Havlin, S.

Hecht, H. G.

H. G. Hecht, “The a priori calculation of the diffuse reflectance of a turbid medium,” Opt. Acta 30, 659–668 (1983).
[CrossRef]

Ishimaru, A.

C. M. Lam, A. Ishimaru, “Mueller matrix calculation for a slab of random medium with both random rough surfaces and discrete particles,” IEEE Trans. Antennas Propag. 42, 145–156 (1994).
[CrossRef]

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

Jayaweera, K.

Khlebtsov, N. G.

Kolinko, V. G.

Kubelka, P.

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

Kuhn, J.

G. Gobel, J. Kuhn, J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5, 413–426 (1995).
[CrossRef]

Lam, C. M.

C. M. Lam, A. Ishimaru, “Mueller matrix calculation for a slab of random medium with both random rough surfaces and discrete particles,” IEEE Trans. Antennas Propag. 42, 145–156 (1994).
[CrossRef]

Letoulouzan, J. N.

Lutomirski, R. F.

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
[CrossRef]

Maheu, B.

J. P. Briton, B. Maheu, G. Gréhan, G. Gouesbet, “Monte Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

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]

Mandelis, A.

Melamed, N. T.

N. T. Melamed, “Optical properties of powders. I. Optical absorption coefficients and the absolute value of the diffuse reflectance; II. Properties of luminescent powders,” J. Appl. Phys. 34, 560–570 (1963).
[CrossRef]

Mills, T. N.

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
[CrossRef]

M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).
[CrossRef]

Mudgett, P. S.

P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology. II,” J. Colloid Interface Sci. 39, 551–567 (1972).
[CrossRef]

Munk, F.

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

Niklasson, G. A.

Nossal, R.

Priezzhev, A. V.

Reichman, J.

Richards, L. W.

P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology. II,” J. Colloid Interface Sci. 39, 551–567 (1972).
[CrossRef]

Simmons, E. L.

Stamnes, K.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
[CrossRef]

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Tsay, S. C.

Van den Bergh, H.

Vargas, W. E.

Waterman, P. C.

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
[CrossRef]

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Weiss, G. H.

Whitney, C.

C. Whitney, “Efficient stream distributions in radiative transfer theory,” J. Quant. Spectrosc. Radiat. Transf. 14, 591–611 (1974).
[CrossRef]

Wiscombe, W.

Zaccanti, G.

Appl. Opt.

G. Zaccanti, “Monte Carlo study of light propagation in optically thick media: point source case,” Appl. Opt. 30, 2031–2041 (1991).
[CrossRef] [PubMed]

M. H. Eddowes, T. N. Mills, D. T. Delpy, “Monte Carlo simulations of coherent backscatter for identification of the optical coefficients of biological tissues in vivo,” Appl. Opt. 34, 2261–2267 (1995).
[CrossRef] [PubMed]

E. L. Simmons, “Diffuse reflectance spectroscopy: a comparison of the theories,” Appl. Opt. 14, 1380–1386 (1975).
[CrossRef] [PubMed]

A. Mandelis, F. Boroumand, H. Van den Bergh, “Quantitative diffuse reflectance spectroscopy of large powders: the Melamed model revisited,” Appl. Opt. 29, 2853–2860 (1990).
[CrossRef] [PubMed]

K. Stamnes, S. C. Tsay, W. Wiscombe, K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
[CrossRef] [PubMed]

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Scaling relationships for theories of anisotropic random walks applied to tissue optics,” Appl. Opt. 32, 504–516 (1993).
[CrossRef] [PubMed]

R. F. Lutomirski, A. P. Ciervo, G. J. Hall, “Moments of multiple scattering,” Appl. Opt. 34, 7125–7136 (1995).
[CrossRef] [PubMed]

V. G. Kolinko, F. F. M. de Mull, J. Greeve, A. V. Priezzhev, “Probabilistic model of multiple light scattering based on rigorous computations of the first and second moments of photon coordinates,” Appl. Opt. 35, 4541–4550 (1996).
[CrossRef] [PubMed]

J. Reichman, “Determination of absorption and scattering coefficients for nonhomogeneous media. 1. Theory,” Appl. Opt. 12, 1811–1815 (1973).
[CrossRef] [PubMed]

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]

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, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
[CrossRef] [PubMed]

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

N. G. Khlebtsov, “Orientational averaging of light-scattering observables in the T-matrix approach,” Appl. Opt. 31, 5359–5365 (1992).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propag.

C. M. Lam, A. Ishimaru, “Mueller matrix calculation for a slab of random medium with both random rough surfaces and discrete particles,” IEEE Trans. Antennas Propag. 42, 145–156 (1994).
[CrossRef]

J. Appl. Phys.

N. T. Melamed, “Optical properties of powders. I. Optical absorption coefficients and the absolute value of the diffuse reflectance; II. Properties of luminescent powders,” J. Appl. Phys. 34, 560–570 (1963).
[CrossRef]

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
[CrossRef]

J. Colloid Interface Sci.

P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology. II,” J. Colloid Interface Sci. 39, 551–567 (1972).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys.: Condens. Matter

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

J. Quant. Spectrosc. Radiat. Transf.

C. Whitney, “Efficient stream distributions in radiative transfer theory,” J. Quant. Spectrosc. Radiat. Transf. 14, 591–611 (1974).
[CrossRef]

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
[CrossRef]

J. Stat. Phys.

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

Licht

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

Mult. Scatt. Electromag. Rad.

C. M. Chu, S. W. Churchill, “Numerical solution of problems in multiple scattering of electromagnetic radiation,” Mult. Scatt. Electromag. Rad. 59, 855–863 (1955).

Opt. Acta

H. G. Hecht, “The a priori calculation of the diffuse reflectance of a turbid medium,” Opt. Acta 30, 659–668 (1983).
[CrossRef]

Part. Part. Syst. Charact.

J. P. Briton, B. Maheu, G. Gréhan, G. Gouesbet, “Monte Carlo simulation of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
[CrossRef]

Phys. Rev. D

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Space Sci. Rev.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Waves Random Media

G. Gobel, J. Kuhn, J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5, 413–426 (1995).
[CrossRef]

Z. Tech. Phys. (Leipzig)

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

Other

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

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Figures (5)

Fig. 1
Fig. 1

Reflected and transmitted diffuse intensities for a particulate slab perpendicularly illuminated with unpolarized collimated radiation. The forward and backward diffuse intensities are denoted I(τ, μ) and J(τ, μ), respectively, where τ is the optical depth, h is the geometrical thickness, and μ=cos θ.

Fig. 2
Fig. 2

First three (a), (b) scattering- and (c), (d) reflecting-order coefficients for a thick particulate slab of optical thickness τ=6.0, containing spherical particles characterized by a relative refractive index m=(2.75+i0.0)/1.50 and a size parameter x=0.10 with a particle volume fraction f=0.05. The index k specifies the order of the coefficients.

Fig. 3
Fig. 3

(a) Polar diagrams at the center (τ=τ/2) of particulate slabs characterized by three values of the optical thickness τ containing spherical particles with a relative refractive index m=(2.75+i0.0)/1.50 and a size parameter x=1.0. The particle volume fraction was set to f=0.05. The intensity values corresponding to τ=4.0 and τ=12.0 have been multiplied by 2 and 50 respectively, to display the intensity patterns more clearly. (b) Polar diagrams at the center of a thin slab [τ=0.50] containing nonabsorbing particles, as described in text. Solid curve, the intensity with both the reflecting- and the scattering-order contributions taken into account; dashed curve, the corresponding results when only the contributions from the scattering orders are considered.

Fig. 4
Fig. 4

Angular dependence of the scaled diffuse intensity of a rather thin slab (τ=1.0) containing (a) small or (b) large spherical particles at three optical depths, as indicated. The relative refractive index of the particles was set to m=(2.75+i0.0)/1.50; the particle volume fraction was set at f=0.05.

Fig. 5
Fig. 5

(a) Reflected and (b) transmitted diffuse intensity for a slab of optical thickness τ=2.0 perpendicularly illuminated with unpolarized collimated radiation. The slab contains spherical particles with size parameters x indicated. The relative refractive index of the particles was set to m=(2.75+i0.0)/1.50; the particle volume fraction was set to f=0.05.

Equations (77)

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

p(μ, μ)=n=0ωnPn(μ)Pn(μ).
fk(cos θ)=14πn=0(2n+1)ωn/ω02n+1kPn(cos θ),
I(τ, μ)=N(+)k=1[Ik(+)(τ, μ)+Jk(+)(τ, μ)],
0<μ1,
J(τ, μ)=N(-)k=1[Ik(-)(τ, μ)+Jk(-)(τ, μ)],
-1μ<0,
μIk(±)τ=-Ik(±)(τ, μ)+12-10p(μ, μ)Ik-1(-)(τ, μ)dμ+01p(μ, μ)Ik-1(+)(τ, μ)dμ,
μJk(±)τ=-Jk(±)(τ, μ)+12-10p(μ, μ)Jk-1(-)(τ, μ)dμ+01p(μ, μ)Jk-1(+)(τ, μ)dμ,
dqk(+)dτ=-ξk(+)qk(+)+ω0{ξk-1(+)σk(+)qk-1(+)+ξk-1(-)[1-σk(-)]qk-1(-)},
-dqk(-)dτ=-ξk(-)qk(-)+ω0{ξk-1(+)[1-σk(+)]qk-1(+)+ξk-1(-)σk(-)qk-1(-)},
dvk(+)dτ=-ξk(-)vk(+)+ω0{ξk-1(-)σk(-)vk-1(+)+ξk-1(+)[1-σk(+)]vk-1(-)},
-dvk(-)dτ=-ξk(+)vk(-)+ω0{ξk-1(-)[1-σk(-)]vk-1(+)+ξk-1(+)σk(+)vk-1(-)},
ξ0(±)=1,w0(+)=2π01μδ(μ-1)dμ=2π,
w0(-)=-2π-10μδ(μ+1)dμ=2π,
q0(+)(τ)=F0(+) exp(-τ),q0(-)(τ)=F0(-) exp(τ),
F0(+)=1-rc1-[rc exp(-τ)]2,
F0(-)=(1-rc)rc exp(-2τ)1-[rc exp(-τ)]2
wk(+)=2π01μfk(μ)dμ,wk(-)=-2π-10μfk(μ)dμ.
01dμ01dμp1(μ, μ)fk-1(μ)
=-10dμ-10dμp1(μ, μ)fk-1(-μ)=ξk-1(+)wk-1(+)σk(+)/2π,
01dμ-10dμp1(μ, μ)fk-1(μ)
=-10dμ01dμp1(μ, μ)fk-1(-μ)=ξk-1(-)wk-1(-)[1-σk(-)]/2π,
01dμ01dμp1(μ, μ)fk-1(-μ)
=-10dμ-10dμp1(μ, μ)fk-1(μ)=ξk-1(-)wk-1(-)σk(-)/2π,
01dμ-10dμp1(μ, μ)fk-1(-μ)
=-10dμ01dμp1(μ, μ)fk-1(μ)=ξk-1(+)wk-1(+)[1-σk(+)]/2π,
qk(+)(τ)=Ck(+) exp[-ξk(+)τ]+i=0k-1{Fi,k(+) exp[-ξi(+)τ]+Gi,k(+) exp[ξi(-)τ]},
qk(-)(τ)=Ck(-) exp[ξk(-)τ]+i=0k-1{Fi,k(-) exp[-ξi(+)τ]+Gi,k(-) exp[ξi(-)τ]},
vk(+)(τ)=Ak(+) exp[-ξk(-)τ]+i=1k-1{Hi,k(+) exp[ξi(+)τ]+Li,k(+) exp[-ξi(-)τ]},
vk(-)(τ)=Ak(-) exp[ξk(+)τ]+i=1k-1{Hi,k(-) exp[ξi(+)τ]+Li,k(-) exp[-ξi(-)τ]},
qk(+)(τ=0)=0,qk(-)(τ=τ)=0,
vk(+)(τ=0)=rk(-)qk(-)(τ=0),
vk(-)(τ=τ)=rk(+)qk(+)(τ=τ).
rk(-)=-[2π/wk(-)]-10μfk(μ)r(μ)dμ,
rk(+)=[2π/wk(+)]01μfk(μ)r(μ)dμ,
Rd(-)=k=1rk(-)qk(-)(τ=0)k=1qk(-)(τ=0),
Rd(+)=k=1rk(+)qk(+)(τ=τ)k=1qk(+)(τ=τ).
Fk-1,k(+)=ω0ξk-1(+)σk(+)ξk(+)-ξk-1(+)Ck-1(+),
Gk-1,k(+)=ω0ξk-1(-)[1-σk(-)]ξk(+)+ξk-1(-)Ck-1(-),
Fi,k(+)=ω0ξk-1(+)σk(+)Fi,k-1(+)+ξk-1(-)[1-σk(-)]Fi,k-1(-)ξk(+)-ξi(+),
Gi,k(+)=ω0ξk-1(+)σk(+)Gi,k-1(+)+ξk-1(-)[1-σk(-)]Gi,k-1(-)ξk(+)+ξi(-),
Fk-1,k(-)=ω0ξk-1(+)[1-σk(+)]ξk(-)+ξk-1(+)Ck-1(+),
Gk-1,k(-)=ω0ξk-1(-)σk(-)ξk(-)-ξk-1(-)Ck-1(-),
Fi,k(-)=ω0ξk-1(+)[1-σk(+)]Fi,k-1(+)+ξk-1(-)σk(-)Fi,k-1(-)ξk(-)+ξi(+),
Gi,k(-)=ω0ξk-1(+)[1-σk(+)]Gi,k-1(+)+ξk-1(-)σk(-)Gi,k-1(-)ξk(-)-ξi(-),
Hk-1,k(+)=ω0ξk-1(+)[1-σk(+)]ξk(-)+ξk-1(+)Ak-1(-),
Lk-1,k(+)=ω0ξk-1(-)σk(-)ξk(-)-ξk-1(-)Ak-1(+),
Hi,k(+)=ω0ξk-1(-)σk(-)Hi,k-1(+)+ξk-1(+)[1-σk(+)]Hi,k-1(-)ξk(-)+ξi(+),
Li,k(+)=ω0ξk-1(-)σk(-)Li,k-1(+)+ξk-1(+)[1-σk(+)]Li,k-1(-)ξk(-)-ξi(-),
Hk-1,k(-)=ω0ξk-1(+)σk(+)ξk(+)-ξk-1(+)Ak-1(-),
Lk-1,k(-)=ω0ξk-1(-)[1-σk(-)]ξk(+)+ξk-1(-)Ak-1(+),
Hi,k(-)=ω0ξk-1(-)[1-σk(-)]Hi,k-1(+)+ξk-1(+)σk(+)Hi,k-1(-)ξk(+)-ξi(+),
Li,k(-)=ω0ξk-1(-)[1-σk(-)]Li,k-1(+)+ξk-1(+)σk(+)Li,k-1(-)ξk(+)+ξi(-),
F0,1(+)=ω0σcF0(+)ξ1(+)-1,G0,1(+)=ω0(1-σc)F0(-)ξ1(+)+1,
F0,1(-)=ω0(1-σc)F0(+)ξ1(-)+1,G0,1(-)=ω0σcF0(-)ξ1(-)-1,
H1,2(+)
=ω0ξ1(+)[1-σ2(+)]r1(+)q1(+)(τ=τ)ξ2(-)+ξ1(+)exp[-ξ1(+)τ],
L1,2(+)
=ω0ξ1(-)σ2(-)r1(-)q1(-)(τ=0)ξ2(-)-ξ1(-),
H1,2(-)
=ω0ξ1(+)σ2(+)r1(+)q1(+)(τ=τ)ξ2(+)-ξ1(+)exp[-ξ1(+)τ],
L1,2(-)
=ω0ξ1(-)(1-σ2(-))r1(-)q1(-)(τ=0)ξ2(+)+ξ1(-),
v1(+)(τ)=[r1(-)q1(-)(τ=0)]exp[-ξ1(-)τ],
v1(-)(τ)=[r1(+)q1(+)(τ=τ)]exp[ξ1(+)(τ-τ)].
qk(+)(τ)=i=0k-1(Fi,k(+){exp[-ξi(+)τ]-exp[-ξk(+)τ]}+Gi,k(+){exp[ξi(-)τ]-exp[-ξk(+)τ]}),
qk(-)(τ)=i=0k-1{Fi,k(-) exp[-ξi(+)τ]+Gi,k(-) exp[ξi(-)τ]}-exp[-ξk(-)(τ-τ)]i=0k-1{Fi,k(-) exp[-ξi(+)τ]+Gi,k(-) exp[ξi(-)τ]},
vk(+)(τ)
=rk(-) exp[-ξk(-)(τ-τ)]i=1k-1(Fi,k(-){exp[ξk(-)τ]-exp[-ξi(+)τ]}+Gi,k(-){exp[ξk(-)τ]-exp[ξi(-)τ]})+i=1k-1(Hi,k(+){exp[ξi(+)τ]-exp[-ξk(-)τ]}+Li,k(+){exp[-ξi(-)τ]-exp[ξk(-)τ]}),
vk(-)(τ)
=rk(+) exp[-ξk(+)(τ-τ)]i=1k-1(Fi,k(+){exp[-ξi(+)τ]-exp[-ξk(+)τ]}+Gi,k(+){exp[ξi(-)τ]-exp[-ξk(+)τ]})+exp[-ξk(+)(τ-τ)]×i=1k-1{Hi,k(-) exp[ξi(+)τ]+Li,k(-) exp[-ξi(-)τ]}+i=1k-1{Hi,k(-) exp[ξi(+)τ]+Li,k(-) exp[-ξi(-)τ]},
Qk(+)(τ)=ω0Qk-1(+)(τ),τ0,Qk(+)(τ=0)=0,
Qk(-)(τ)=ω0Qk-1(-)(τ)ττ,Qk(+)(τ=τ)=0,
Vk(±)(τ)=ω0Vk-1(±)(τ).
I(τ, μ)=N(+)n=0cn(+)(τ)Pn(μ),
J(τ, μ)=N(-)n=0cn(-)(τ)Pn(μ),
cn(±)(τ)=(2n+1)4πk=1Qk(±)(τ)+(-1)nVk(±)k!ωn/ω02n+1k,

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