Abstract

The description of optical properties of light-scattering materials has made extensive use of radiative transfer models. One of the most successful and simplest models is that of Kubelka and Munk (KM). With this model, optical properties of particulate films under diffuse illumination can be predicted from effective absorption and scattering coefficients of the material. We consider the applicability conditions of this kind of model. An extended KM model for the case of perpendicular collimated illumination is compared with results from a more general four-flux approach, and the differences between them are characterized in terms of a correction factor that depends on particle scattering and absorption, concentration of the scatterers, and film thickness. It is proved formally that the extended KM model under perpendicular illumination is a good approximation for the cases of optically thick films that contain weakly or nonabsorbing particles.

© 1997 Optical Society of America

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  1. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).
  2. P. Kubelka, “New contributions to the optics of intensely scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
    [CrossRef] [PubMed]
  3. D. C. Rich, “Computer-aided design and manufacturing of the color of decorative and protective coatings,” J. Coating Technol. 67, 53–60 (1995).
  4. A. B. Krewinghaus, “Infrared reflectance of paints,” Appl. Opt. 8, 807–812 (1969).
    [CrossRef] [PubMed]
  5. C. Rennel, M. Rigdahl, “Enhancement of the light-scattering ability of coatings by using hollow pigments,” Colloid Polym. Sci. 272, 1111–1117 (1994).
    [CrossRef]
  6. M. M. Beppu, E. C. de Oliveira Lima, F. Galembeck, “Aluminum phosphate particles containing closed pores: preparation, characterization, and use as a white pigment,” J. Colloid Interface Sci. 178, 93–103 (1996).
    [CrossRef]
  7. F. Thiébaud, F. K. Kneubuühl, “Infrared properties of quartz fibers and wool,” Infrared Phys. 23, 131–148 (1983).
    [CrossRef]
  8. J. Kuhn, S. Korder, M. C. Arduini-Schuster, R. Caps, J. Fricke, “Infrared-optical transmission and reflection measurements on loose powders,” Rev. Sci. Instrum. 64, 2523–2530 (1993).
    [CrossRef]
  9. N. Yamada, S. Fujimura, “Nondestructive measurement of chlorophyll pigment content in plant leaves from three-color reflectance and transmittance,” Appl. Opt. 30, 3964–3973 (1991).
    [CrossRef] [PubMed]
  10. L. Fukshansky, N. Fukshansky-Kazarinova, A. M. Remi-sowsky, “Estimation of optical parameters in a living tissue by solving the inverse problem of the multiflux radiative transfer,” Appl. Opt. 30, 3145–3153 (1991).
    [CrossRef] [PubMed]
  11. R. R. Anderson, J. A. Parrish, “The optics of human skin,” J. Invest. Dermatol. 77, 13–19 (1981).
    [CrossRef] [PubMed]
  12. J. D. Lindberg, R. E. Douglass, D. M. Garvey, “Absorption-coefficient-determination method for particulate materials,” Appl. Opt. 33, 4314–4319 (1994).
    [CrossRef] [PubMed]
  13. P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology II,” J. Colloid Interface Sci. 39, 551–567 (1972).
    [CrossRef]
  14. D. G. Phillips, F. W. Billmeyer, “Predicting reflectance and color of paint films by Kubelka-Munk analysis,” J. Coating Technol. 48, 30–36 (1976).
  15. L. F. Gate, “The determination of light absorption in diffusing materials by a photon diffusion model,” J. Phys. D 4, 1049–1056 (1971).
    [CrossRef]
  16. W. R. Blevin, W. J. Brown, “Total reflectance of opaque diffusers,” J. Opt. Soc. Am. 52, 1250–1255 (1962).
    [CrossRef]
  17. M. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Application of the Kubelka-Munk theory to thickness-dependent diffuse reflectance of black paints in the mid-IR,” Appl. Spectrosc. 49, 623–629 (1995).
    [CrossRef]
  18. 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] [PubMed]
  19. G. A. Niklasson, “Comparison between four-flux theory and multiple scattering theory,” Appl. Opt. 26, 4034–4036 (1987).
    [CrossRef] [PubMed]
  20. B. Maheu, J. P. Briton, G. Gouesbet, “Four-flux model and a Monte Carlo code: comparisons between two simple, complementary tools for multiple scattering calculations,” Appl. Opt. 28, 22–24 (1989).
    [CrossRef] [PubMed]
  21. J. P. Briton, B. Maheu, G. Gréhan, G. Gouesbet, “Monte Carlo simulations of multiple scattering in arbitrary 3-D geometry,” Part. Part. Syst. Charact. 9, 52–58 (1992).
    [CrossRef]
  22. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), pp. 191–196.
    [CrossRef]
  23. J. L. Saunderson, “Calculation of the color of pigmented plastics,” J. Opt. Soc. Am. 32, 727–736 (1942).
    [CrossRef]
  24. W. Theiss “wind: A radiation transfer program for simulating optical spectra of light scattering materials,” in Electromagnetic and Light Scattering: Theory and Applications, T. Wriedt, M. Quinten, K. Bauckhage, eds. (Universität Bremen, Bremen, Germany, 1996), pp. 79–82.
  25. P. Chylek, “Mie scattering into the backward hemisphere,” J. Opt. Soc. Am. 63, 1467–1471 (1973).
    [CrossRef]
  26. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–129.
  27. Y. P. Wang, Z. S. Wu, K. F. Ren, “Four-flux model with adjusted average crossing parameter to solve the scattering transfer equation,” Appl. Opt. 28, 24–26 (1989).
    [CrossRef] [PubMed]
  28. A. Ben-David, “Multiple-scattering transmission and an effective average photon path length of a plane–parallel beam in a homogeneous medium,” Appl. Opt. 34, 2802–2810 (1995).
    [CrossRef] [PubMed]
  29. 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]
  30. W. E. Vargas, G. A. Niklasson, “Generalized method for evaluating scattering parameters used in radiative transfer models,” J. Opt. Soc. Am. A 14 (September1997).
    [CrossRef]
  31. B. Maheu, G. Gouesbet, “Four-flux models to solve the scattering transfer equation: special cases,” Appl. Opt. 25, 1122–1128 (1986).
    [CrossRef] [PubMed]
  32. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 281.
  33. D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329–332 (1942).
    [CrossRef]
  34. C. F. Bohren, “Applicability of effective-medium theories to problems of scattering and absorption by nonhomogeneous atmospheric particles,” J. Atmos. Sci. 43, 468–475 (1986).
    [CrossRef]
  35. J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
    [CrossRef]
  36. G. A. Niklasson, T. S. Eriksson, “Radiative cooling with pigmented polyethylene foils,” in Optical Materials Technology for Energy Efficiency and Solar Energy Conversion VII, C. Granqvist, C. M. Lampert, eds., Proc. SPIE1016, 89–99 (1988).
    [CrossRef]
  37. T. M. J. Nilsson, G. A. Niklasson, “Optimization of optical properties of pigmented foils for radiative cooling applications: model calculations,” in Optical Materials Technology for Energy Efficiency and Solar Energy Conversion X, C. G. Granqvist, C. M. Lampert, eds., Proc. SPIE1536, 169–182 (1991).
    [CrossRef]
  38. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 151.
  39. M. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Optimum thickness determination to maximize the spectral selectivity of black pigmented coatings for solar collectors,” Thin Solid Films 277, 185–191 (1996).
    [CrossRef]
  40. N. P. Ryde, E. Matijevic, “Color effects of uniform colloidal particles of different morphologies packed into films,” Appl. Opt. 33, 7275–7281 (1994).
    [CrossRef] [PubMed]

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, “Generalized method for evaluating scattering parameters used in radiative transfer models,” J. Opt. Soc. Am. A 14 (September1997).
[CrossRef]

1996

M. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Optimum thickness determination to maximize the spectral selectivity of black pigmented coatings for solar collectors,” Thin Solid Films 277, 185–191 (1996).
[CrossRef]

M. M. Beppu, E. C. de Oliveira Lima, F. Galembeck, “Aluminum phosphate particles containing closed pores: preparation, characterization, and use as a white pigment,” J. Colloid Interface Sci. 178, 93–103 (1996).
[CrossRef]

1995

1994

1993

J. Kuhn, S. Korder, M. C. Arduini-Schuster, R. Caps, J. Fricke, “Infrared-optical transmission and reflection measurements on loose powders,” Rev. Sci. Instrum. 64, 2523–2530 (1993).
[CrossRef]

1992

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

1991

1989

1987

1986

C. F. Bohren, “Applicability of effective-medium theories to problems of scattering and absorption by nonhomogeneous atmospheric particles,” J. Atmos. Sci. 43, 468–475 (1986).
[CrossRef]

B. Maheu, G. Gouesbet, “Four-flux models to solve the scattering transfer equation: special cases,” Appl. Opt. 25, 1122–1128 (1986).
[CrossRef] [PubMed]

1984

1983

F. Thiébaud, F. K. Kneubuühl, “Infrared properties of quartz fibers and wool,” Infrared Phys. 23, 131–148 (1983).
[CrossRef]

1981

R. R. Anderson, J. A. Parrish, “The optics of human skin,” J. Invest. Dermatol. 77, 13–19 (1981).
[CrossRef] [PubMed]

1976

D. G. Phillips, F. W. Billmeyer, “Predicting reflectance and color of paint films by Kubelka-Munk analysis,” J. Coating Technol. 48, 30–36 (1976).

1973

1972

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

1971

L. F. Gate, “The determination of light absorption in diffusing materials by a photon diffusion model,” J. Phys. D 4, 1049–1056 (1971).
[CrossRef]

1969

1962

1948

1942

J. L. Saunderson, “Calculation of the color of pigmented plastics,” J. Opt. Soc. Am. 32, 727–736 (1942).
[CrossRef]

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329–332 (1942).
[CrossRef]

1931

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

1904

J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
[CrossRef]

Anderson, R. R.

R. R. Anderson, J. A. Parrish, “The optics of human skin,” J. Invest. Dermatol. 77, 13–19 (1981).
[CrossRef] [PubMed]

Arduini-Schuster, M. C.

J. Kuhn, S. Korder, M. C. Arduini-Schuster, R. Caps, J. Fricke, “Infrared-optical transmission and reflection measurements on loose powders,” Rev. Sci. Instrum. 64, 2523–2530 (1993).
[CrossRef]

Ben-David, A.

Beppu, M. M.

M. M. Beppu, E. C. de Oliveira Lima, F. Galembeck, “Aluminum phosphate particles containing closed pores: preparation, characterization, and use as a white pigment,” J. Colloid Interface Sci. 178, 93–103 (1996).
[CrossRef]

Billmeyer, F. W.

D. G. Phillips, F. W. Billmeyer, “Predicting reflectance and color of paint films by Kubelka-Munk analysis,” J. Coating Technol. 48, 30–36 (1976).

Blevin, W. R.

Bohren, C. F.

C. F. Bohren, “Applicability of effective-medium theories to problems of scattering and absorption by nonhomogeneous atmospheric particles,” J. Atmos. Sci. 43, 468–475 (1986).
[CrossRef]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–129.

Briton, J. P.

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

B. Maheu, J. P. Briton, G. Gouesbet, “Four-flux model and a Monte Carlo code: comparisons between two simple, complementary tools for multiple scattering calculations,” Appl. Opt. 28, 22–24 (1989).
[CrossRef] [PubMed]

Brown, W. J.

Caps, R.

J. Kuhn, S. Korder, M. C. Arduini-Schuster, R. Caps, J. Fricke, “Infrared-optical transmission and reflection measurements on loose powders,” Rev. Sci. Instrum. 64, 2523–2530 (1993).
[CrossRef]

Chylek, P.

de Oliveira Lima, E. C.

M. M. Beppu, E. C. de Oliveira Lima, F. Galembeck, “Aluminum phosphate particles containing closed pores: preparation, characterization, and use as a white pigment,” J. Colloid Interface Sci. 178, 93–103 (1996).
[CrossRef]

Douglass, R. E.

Eriksson, T. S.

G. A. Niklasson, T. S. Eriksson, “Radiative cooling with pigmented polyethylene foils,” in Optical Materials Technology for Energy Efficiency and Solar Energy Conversion VII, C. Granqvist, C. M. Lampert, eds., Proc. SPIE1016, 89–99 (1988).
[CrossRef]

Fricke, J.

J. Kuhn, S. Korder, M. C. Arduini-Schuster, R. Caps, J. Fricke, “Infrared-optical transmission and reflection measurements on loose powders,” Rev. Sci. Instrum. 64, 2523–2530 (1993).
[CrossRef]

Fujimura, S.

Fukshansky, L.

Fukshansky-Kazarinova, N.

Galembeck, F.

M. M. Beppu, E. C. de Oliveira Lima, F. Galembeck, “Aluminum phosphate particles containing closed pores: preparation, characterization, and use as a white pigment,” J. Colloid Interface Sci. 178, 93–103 (1996).
[CrossRef]

Garvey, D. M.

Gate, L. F.

L. F. Gate, “The determination of light absorption in diffusing materials by a photon diffusion model,” J. Phys. D 4, 1049–1056 (1971).
[CrossRef]

Gouesbet, G.

Gréhan, G.

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

Gunde, M. K.

M. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Optimum thickness determination to maximize the spectral selectivity of black pigmented coatings for solar collectors,” Thin Solid Films 277, 185–191 (1996).
[CrossRef]

M. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Application of the Kubelka-Munk theory to thickness-dependent diffuse reflectance of black paints in the mid-IR,” Appl. Spectrosc. 49, 623–629 (1995).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–129.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), pp. 191–196.
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 281.

Judd, D. B.

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329–332 (1942).
[CrossRef]

Kneubuühl, F. K.

F. Thiébaud, F. K. Kneubuühl, “Infrared properties of quartz fibers and wool,” Infrared Phys. 23, 131–148 (1983).
[CrossRef]

Korder, S.

J. Kuhn, S. Korder, M. C. Arduini-Schuster, R. Caps, J. Fricke, “Infrared-optical transmission and reflection measurements on loose powders,” Rev. Sci. Instrum. 64, 2523–2530 (1993).
[CrossRef]

Krewinghaus, A. B.

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensely scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
[CrossRef] [PubMed]

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

Kuhn, J.

J. Kuhn, S. Korder, M. C. Arduini-Schuster, R. Caps, J. Fricke, “Infrared-optical transmission and reflection measurements on loose powders,” Rev. Sci. Instrum. 64, 2523–2530 (1993).
[CrossRef]

Letoulouzan, J. N.

Lindberg, J. D.

Logar, J. K.

M. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Optimum thickness determination to maximize the spectral selectivity of black pigmented coatings for solar collectors,” Thin Solid Films 277, 185–191 (1996).
[CrossRef]

M. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Application of the Kubelka-Munk theory to thickness-dependent diffuse reflectance of black paints in the mid-IR,” Appl. Spectrosc. 49, 623–629 (1995).
[CrossRef]

Maheu, B.

Matijevic, E.

Maxwell Garnett, J. C.

J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
[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. 12, 593–601 (1931).

Niklasson, G. A.

W. E. Vargas, G. A. Niklasson, “Generalized method for evaluating scattering parameters used in radiative transfer models,” J. Opt. Soc. Am. A 14 (September1997).
[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]

G. A. Niklasson, “Comparison between four-flux theory and multiple scattering theory,” Appl. Opt. 26, 4034–4036 (1987).
[CrossRef] [PubMed]

G. A. Niklasson, T. S. Eriksson, “Radiative cooling with pigmented polyethylene foils,” in Optical Materials Technology for Energy Efficiency and Solar Energy Conversion VII, C. Granqvist, C. M. Lampert, eds., Proc. SPIE1016, 89–99 (1988).
[CrossRef]

T. M. J. Nilsson, G. A. Niklasson, “Optimization of optical properties of pigmented foils for radiative cooling applications: model calculations,” in Optical Materials Technology for Energy Efficiency and Solar Energy Conversion X, C. G. Granqvist, C. M. Lampert, eds., Proc. SPIE1536, 169–182 (1991).
[CrossRef]

Nilsson, T. M. J.

T. M. J. Nilsson, G. A. Niklasson, “Optimization of optical properties of pigmented foils for radiative cooling applications: model calculations,” in Optical Materials Technology for Energy Efficiency and Solar Energy Conversion X, C. G. Granqvist, C. M. Lampert, eds., Proc. SPIE1536, 169–182 (1991).
[CrossRef]

Orel, B.

M. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Optimum thickness determination to maximize the spectral selectivity of black pigmented coatings for solar collectors,” Thin Solid Films 277, 185–191 (1996).
[CrossRef]

M. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Application of the Kubelka-Munk theory to thickness-dependent diffuse reflectance of black paints in the mid-IR,” Appl. Spectrosc. 49, 623–629 (1995).
[CrossRef]

Orel, Z. C.

M. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Optimum thickness determination to maximize the spectral selectivity of black pigmented coatings for solar collectors,” Thin Solid Films 277, 185–191 (1996).
[CrossRef]

M. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Application of the Kubelka-Munk theory to thickness-dependent diffuse reflectance of black paints in the mid-IR,” Appl. Spectrosc. 49, 623–629 (1995).
[CrossRef]

Parrish, J. A.

R. R. Anderson, J. A. Parrish, “The optics of human skin,” J. Invest. Dermatol. 77, 13–19 (1981).
[CrossRef] [PubMed]

Phillips, D. G.

D. G. Phillips, F. W. Billmeyer, “Predicting reflectance and color of paint films by Kubelka-Munk analysis,” J. Coating Technol. 48, 30–36 (1976).

Remi-sowsky, A. M.

Ren, K. F.

Rennel, C.

C. Rennel, M. Rigdahl, “Enhancement of the light-scattering ability of coatings by using hollow pigments,” Colloid Polym. Sci. 272, 1111–1117 (1994).
[CrossRef]

Rich, D. C.

D. C. Rich, “Computer-aided design and manufacturing of the color of decorative and protective coatings,” J. Coating Technol. 67, 53–60 (1995).

Richards, L. W.

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

Rigdahl, M.

C. Rennel, M. Rigdahl, “Enhancement of the light-scattering ability of coatings by using hollow pigments,” Colloid Polym. Sci. 272, 1111–1117 (1994).
[CrossRef]

Ryde, N. P.

Saunderson, J. L.

Theiss, W.

W. Theiss “wind: A radiation transfer program for simulating optical spectra of light scattering materials,” in Electromagnetic and Light Scattering: Theory and Applications, T. Wriedt, M. Quinten, K. Bauckhage, eds. (Universität Bremen, Bremen, Germany, 1996), pp. 79–82.

Thiébaud, F.

F. Thiébaud, F. K. Kneubuühl, “Infrared properties of quartz fibers and wool,” Infrared Phys. 23, 131–148 (1983).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 151.

Vargas, W. E.

W. E. Vargas, G. A. Niklasson, “Generalized method for evaluating scattering parameters used in radiative transfer models,” J. Opt. Soc. Am. A 14 (September1997).
[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]

Wang, Y. P.

Wu, Z. S.

Yamada, N.

Appl. Opt.

A. B. Krewinghaus, “Infrared reflectance of paints,” Appl. Opt. 8, 807–812 (1969).
[CrossRef] [PubMed]

N. Yamada, S. Fujimura, “Nondestructive measurement of chlorophyll pigment content in plant leaves from three-color reflectance and transmittance,” Appl. Opt. 30, 3964–3973 (1991).
[CrossRef] [PubMed]

L. Fukshansky, N. Fukshansky-Kazarinova, A. M. Remi-sowsky, “Estimation of optical parameters in a living tissue by solving the inverse problem of the multiflux radiative transfer,” Appl. Opt. 30, 3145–3153 (1991).
[CrossRef] [PubMed]

J. D. Lindberg, R. E. Douglass, D. M. Garvey, “Absorption-coefficient-determination method for particulate materials,” Appl. Opt. 33, 4314–4319 (1994).
[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] [PubMed]

G. A. Niklasson, “Comparison between four-flux theory and multiple scattering theory,” Appl. Opt. 26, 4034–4036 (1987).
[CrossRef] [PubMed]

B. Maheu, J. P. Briton, G. Gouesbet, “Four-flux model and a Monte Carlo code: comparisons between two simple, complementary tools for multiple scattering calculations,” Appl. Opt. 28, 22–24 (1989).
[CrossRef] [PubMed]

Y. P. Wang, Z. S. Wu, K. F. Ren, “Four-flux model with adjusted average crossing parameter to solve the scattering transfer equation,” Appl. Opt. 28, 24–26 (1989).
[CrossRef] [PubMed]

A. Ben-David, “Multiple-scattering transmission and an effective average photon path length of a plane–parallel beam in a homogeneous medium,” Appl. Opt. 34, 2802–2810 (1995).
[CrossRef] [PubMed]

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]

B. Maheu, G. Gouesbet, “Four-flux models to solve the scattering transfer equation: special cases,” Appl. Opt. 25, 1122–1128 (1986).
[CrossRef] [PubMed]

N. P. Ryde, E. Matijevic, “Color effects of uniform colloidal particles of different morphologies packed into films,” Appl. Opt. 33, 7275–7281 (1994).
[CrossRef] [PubMed]

Appl. Spectrosc.

Colloid Polym. Sci.

C. Rennel, M. Rigdahl, “Enhancement of the light-scattering ability of coatings by using hollow pigments,” Colloid Polym. Sci. 272, 1111–1117 (1994).
[CrossRef]

Infrared Phys.

F. Thiébaud, F. K. Kneubuühl, “Infrared properties of quartz fibers and wool,” Infrared Phys. 23, 131–148 (1983).
[CrossRef]

J. Atmos. Sci.

C. F. Bohren, “Applicability of effective-medium theories to problems of scattering and absorption by nonhomogeneous atmospheric particles,” J. Atmos. Sci. 43, 468–475 (1986).
[CrossRef]

J. Coating Technol.

D. C. Rich, “Computer-aided design and manufacturing of the color of decorative and protective coatings,” J. Coating Technol. 67, 53–60 (1995).

D. G. Phillips, F. W. Billmeyer, “Predicting reflectance and color of paint films by Kubelka-Munk analysis,” J. Coating Technol. 48, 30–36 (1976).

J. Colloid Interface Sci.

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Figures (4)

Fig. 1
Fig. 1

Geometry of the scattering problem. Reflection coefficients rc and rde for perpendicular collimated and semi-isotropic diffuse radiation incident on a supported film, respectively, and other reflection coefficients of film–air, film–substrate, and substrate–air interfaces are indicated in the figure.

Fig. 2
Fig. 2

Optical depth dependence of the correction factor. Films that contain nonabsorbing particles deposited on a perfectly reflecting (Rg = 1) or absorbing (Rg = 0) substrate are considered.

Fig. 3
Fig. 3

Size parameter dependence of the correction factor for films that contain (a), (c) weakly or (b), (d) highly scattering spherical particles with various absorptions, as indicated in the figures. The refractive index of the matrix was 1.40, the film thickness was 100 µm, and the particle volume fraction and free space wavelength were 0.05 and 0.55 µm, respectively. The cases of perfectly (a), (b) reflecting or (c), (d) absorbing substrate are considered.

Fig. 4
Fig. 4

Reflectance from films deposited on a perfectly reflecting substrate and containing (a), (b) weakly or (c), (d) highly scattering particles with various degrees of absorption, as indicated in the figures. Perpendicular illumination and boundary reflections are considered. The refractive index of the matrix was 1.40, the film thickness was 100 µm, and the particle volume fraction and free-space wavelength were 0.05 and 0.55 µm, respectively.

Equations (25)

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d I d d z = ( S + K ) I d + S   J d ,
d J d d z = ( S + K ) J d S   I d ,
R KM = 1 R g [ a b   coth   ( b S h ) ] a + b   coth   ( b S h ) R g ,
R g = r d + ( 1 r d r d ) r d   exp ( 2 τ s ) 1 r d r d   exp ( 2 τ s ) ,
R d d = r d e + ( 1 r d e ) ( 1 r d i ) R KM 1 r d i R KM ,
R c d = r c + ( 1 r c ) ( 1 r d i ) R KM 1 r d i R KM ,
R c d = r c + ( 1 r c ) ( 1 r d i ) R KM 1 r d i R KM F ,
F = c 0   exp [ ( s + k ) h ] + c 1 S [ ( s + k ) 2 ( b S ) 2 ] sinh ( b S h ) { 1 R g [ a b   coth ( b S h ) ] } ,
c 0 = A 1 ( A 3 A 2 R g ) ,
c 1 = A 1 ( A 3 A 2 R g ) cosh ( b S h ) + [ A 2 ( A 5 A 4 R g ) A 3 ( A 4 A 5 R g ) ] sinh ( b S h ) ,
A 1 = 2 { k [ k + 2 s ( 1 ζ ) ] } 1 / 2 = b S ,
A 2 = s [ 2 k ζ + 2 s ( 1 ζ ) + ( s + k ) ζ ] ,
A 3 = s ( 1 ζ ) ( s + k ) ,
A 4 = 2 [ k + s ( 1 ζ ) ] = K + S ,
A 5 = 2 s ( 1 ζ ) = S .
F = ( A 2 R g A 3 ) { 1 exp [ ( s + k ) h ] } + S h [ A 2 ( 1 a R g ) A 3 ( a R g ) ] [ ( s + k ) 2 ( b S ) 2 ] [ R g + S h ( 1 a R g ) ] .
F = 1 1 2 τ [ 1 exp ( τ ) ] ,
F = 1 exp ( τ ) .
r d = 0 π / 2 r ( θ ) sin ( 2 θ ) ,
r ( θ ) = 1 2 cos   θ - ( ε / μ ) 1 / 2 ( 1 sin 2   θ / εμ ) 1 / 2 cos   θ + ( ε / μ ) 1 / 2 ( 1 sin 2   θ / εμ ) 1 / 2 2 + cos   θ ( μ / ) 1 / 2 ( 1 sin 2   θ / εμ ) 1 / 2 cos   θ + ( μ / ε ) 1 / 2 ( 1 sin 2   θ / εμ ) 1 / 2 2 ,
ε eff = ε m { 1 + ( 3 i f / 2 x ¯ 3 ) [ S ( 0 ) + S ( π ) ] } ε m ( 1 + γ ) ,
μ eff = 1 + ( 3 i f / 2 x ¯ 3 ) [ S ( 0 ) S ( π ) ] 1 + δ,
ε MG = ε m 1 + ( 2 / 3 ) γ 1 γ / 3 ,
ε eff = ε m 1 + ( 2 / 3 ) γ 1 γ / 3 ,
μ eff = 1 + ( 2 / 3 ) δ 1 δ / 3 .

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