Abstract

A two-flux model is considered as a particular case of a more-general four-flux approach to describing the properties of highly diffusing materials derived from the radiative transfer equation. Any degree of anisotropy is taken into account by means of average path-length parameters and forward-scattering ratios for diffuse radiation propagating in forward and backward directions. The conditions for applicability of the standard two-flux model of Kubelka and Munk are characterized in terms of particle size and refractive index as well as of optical thickness. Scattering and absorption coefficients are obtained in terms of the effective average path-length parameter and forward-scattering ratio of the propagating radiation as well as of the intrinsic scattering and absorption coefficients per unit length of the particulate medium.

© 1999 Optical Society of America

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References

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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. I,” J. Opt. Soc. Am. 38, 448–457 (1948).
    [CrossRef] [PubMed]
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  4. J. L. Saunderson, “Calculation of the color of pigmented plastics,” J. Opt. Soc. Am. 32, 727–736 (1942).
    [CrossRef]
  5. W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
    [CrossRef] [PubMed]
  6. M. K. Gunde, J. K. Logar, Z. C. Orel, B. Orel, “Optimum thickness determination to maximise the spectral selectivity of black pigmented coatings for solar collectors,” Thin Solid Films 277, 185–191 (1996).
    [CrossRef]
  7. B. Maheu, J. N. Letoulouzan, G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 26, 3353–3362 (1984).
    [CrossRef]
  8. 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. G. Granqvist, C. M. Lampert, eds., Proc. SPIE1016, 89–99 (1988).
    [CrossRef]
  9. 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. M. Lampert, C. G. Granqvist, eds., Proc. SPIE1536, 169–182 (1991).
    [CrossRef]
  10. 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]
  11. T. Tesfamichael, W. E. Vargas, E. Wackelgard, G. A. Niklasson, “Optical properties of silicon pigmented alumina films,” J. Appl. Phys. 82, 3508–3513 (1997).
    [CrossRef]
  12. W. E. Vargas, G. A. Niklasson, “Forward scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys. Condens. Matter. 9, 9083–9096 (1997).
    [CrossRef]
  13. 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]
  14. W. E. Vargas, “Generalized four-flux radiative transfer model,” Appl. Opt. 37, 2615–2623 (1998).
    [CrossRef]
  15. M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).
    [CrossRef]
  16. M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
    [CrossRef] [PubMed]
  17. M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
    [CrossRef]
  18. 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]
  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. W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).
  21. D. C. Rich, “Computer-aided design and manufacturing of the color of decorative and protective coatings,” J. Coatings Technol. 67, 53–60 (1995).
  22. B. J. Brinkworth, “Interpretation of the Kubelka–Munk coefficients in reflection theory,” Appl. Opt. 11, 1434 (1972).
    [CrossRef] [PubMed]
  23. K. Klier, “Absorption and scattering in plane parallel turbid media,” J. Opt. Soc. Am. 62, 882–885 (1972).
    [CrossRef]
  24. D. G. Phillips, F. W. Billmeyer, “Predicting reflectance and color of paint films by Kubelka–Munk analysis. IV. Kubelka–Munk scattering coefficient,” J. Coatings Technol. 48, 30–36 (1972).
  25. P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology. II,” J. Colloid Interface Sci. 39, 551–567 (1972).
    [CrossRef]
  26. B. L. Drolen, C. L. Tien, “Independent and dependent scattering in packed-sphere systems,” J. Thermophys. 1, 63–68 (1987).
    [CrossRef]
  27. S. A. El-Wakil, A. R. Degheidy, N. K. Radwan, “Light transfer problem in turbid media with surface reflectivity,” Waves Random Media 6, 101–118 (1996).
    [CrossRef]
  28. L. W. Richards, “The calculation of the optical performance of paint films,” J. Coatings Technol. 42, 276–286 (1970).
  29. W. L. Butler, “Absorption of light by turbid materials,” J. Opt. Soc. Am. 52, 292–299 (1962).
    [CrossRef]
  30. Y. Ma, V. K. Varadan, V. V. Varadan, “Enhanced absorption due to dependent scattering,” J. Heat Transfer 112, 402–407 (1990).
    [CrossRef]

1998 (1)

1997 (8)

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (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]

W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (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]

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

W. E. Vargas, G. A. Niklasson, “Forward scattering ratios and average pathlength 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]

1996 (3)

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

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

S. A. El-Wakil, A. R. Degheidy, N. K. Radwan, “Light transfer problem in turbid media with surface reflectivity,” Waves Random Media 6, 101–118 (1996).
[CrossRef]

1995 (1)

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

1991 (1)

1990 (1)

Y. Ma, V. K. Varadan, V. V. Varadan, “Enhanced absorption due to dependent scattering,” J. Heat Transfer 112, 402–407 (1990).
[CrossRef]

1987 (1)

B. L. Drolen, C. L. Tien, “Independent and dependent scattering in packed-sphere systems,” J. Thermophys. 1, 63–68 (1987).
[CrossRef]

1984 (1)

B. Maheu, J. N. Letoulouzan, G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 26, 3353–3362 (1984).
[CrossRef]

1972 (4)

B. J. Brinkworth, “Interpretation of the Kubelka–Munk coefficients in reflection theory,” Appl. Opt. 11, 1434 (1972).
[CrossRef] [PubMed]

K. Klier, “Absorption and scattering in plane parallel turbid media,” J. Opt. Soc. Am. 62, 882–885 (1972).
[CrossRef]

D. G. Phillips, F. W. Billmeyer, “Predicting reflectance and color of paint films by Kubelka–Munk analysis. IV. Kubelka–Munk scattering coefficient,” J. Coatings Technol. 48, 30–36 (1972).

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

1970 (1)

L. W. Richards, “The calculation of the optical performance of paint films,” J. Coatings Technol. 42, 276–286 (1970).

1962 (1)

1948 (1)

1942 (1)

1940 (1)

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

1931 (1)

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

Billmeyer, F. W.

D. G. Phillips, F. W. Billmeyer, “Predicting reflectance and color of paint films by Kubelka–Munk analysis. IV. Kubelka–Munk scattering coefficient,” J. Coatings Technol. 48, 30–36 (1972).

Brinkworth, B. J.

Butler, W. L.

Degheidy, A. R.

S. A. El-Wakil, A. R. Degheidy, N. K. Radwan, “Light transfer problem in turbid media with surface reflectivity,” Waves Random Media 6, 101–118 (1996).
[CrossRef]

Drolen, B. L.

B. L. Drolen, C. L. Tien, “Independent and dependent scattering in packed-sphere systems,” J. Thermophys. 1, 63–68 (1987).
[CrossRef]

El-Wakil, S. A.

S. A. El-Wakil, A. R. Degheidy, N. K. Radwan, “Light transfer problem in turbid media with surface reflectivity,” Waves Random Media 6, 101–118 (1996).
[CrossRef]

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. G. Granqvist, C. M. Lampert, eds., Proc. SPIE1016, 89–99 (1988).
[CrossRef]

Gouesbet, G.

B. Maheu, J. N. Letoulouzan, G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 26, 3353–3362 (1984).
[CrossRef]

Gunde, M. K.

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

Hartel, W.

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

Ishimaru, A.

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

Kahn, R. A.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[CrossRef]

Klier, K.

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensely scattering materials. 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).

Letoulouzan, J. N.

B. Maheu, J. N. Letoulouzan, G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 26, 3353–3362 (1984).
[CrossRef]

Logar, J. K.

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

Ma, Y.

Y. Ma, V. K. Varadan, V. V. Varadan, “Enhanced absorption due to dependent scattering,” J. Heat Transfer 112, 402–407 (1990).
[CrossRef]

Macke, A.

Maheu, B.

B. Maheu, J. N. Letoulouzan, G. Gouesbet, “Four-flux models to solve the scattering transfer equation in terms of Lorenz–Mie parameters,” Appl. Opt. 26, 3353–3362 (1984).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

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. 12, 593–601 (1931).

Niklasson, G. A.

W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
[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]

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]

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

W. E. Vargas, G. A. Niklasson, “Forward scattering ratios and average pathlength parameter in radiative transfer models,” J. Phys. Condens. Matter. 9, 9083–9096 (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]

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. G. 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. M. Lampert, C. G. Granqvist, 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. M. Lampert, C. G. Granqvist, eds., Proc. SPIE1536, 169–182 (1991).
[CrossRef]

Orel, B.

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

Orel, Z. C.

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

Phillips, D. G.

D. G. Phillips, F. W. Billmeyer, “Predicting reflectance and color of paint films by Kubelka–Munk analysis. IV. Kubelka–Munk scattering coefficient,” J. Coatings Technol. 48, 30–36 (1972).

Radwan, N. K.

S. A. El-Wakil, A. R. Degheidy, N. K. Radwan, “Light transfer problem in turbid media with surface reflectivity,” Waves Random Media 6, 101–118 (1996).
[CrossRef]

Rich, D. C.

D. C. Rich, “Computer-aided design and manufacturing of the color of decorative and protective coatings,” J. Coatings 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]

L. W. Richards, “The calculation of the optical performance of paint films,” J. Coatings Technol. 42, 276–286 (1970).

Saunderson, J. L.

Tesfamichael, T.

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

Tien, C. L.

B. L. Drolen, C. L. Tien, “Independent and dependent scattering in packed-sphere systems,” J. Thermophys. 1, 63–68 (1987).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

Varadan, V. K.

Y. Ma, V. K. Varadan, V. V. Varadan, “Enhanced absorption due to dependent scattering,” J. Heat Transfer 112, 402–407 (1990).
[CrossRef]

Varadan, V. V.

Y. Ma, V. K. Varadan, V. V. Varadan, “Enhanced absorption due to dependent scattering,” J. Heat Transfer 112, 402–407 (1990).
[CrossRef]

Vargas, W. E.

Wackelgard, E.

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

West, R. A.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[CrossRef]

Appl. Opt. (6)

J. Appl. Phys. (1)

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

J. Coatings Technol. (3)

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

D. G. Phillips, F. W. Billmeyer, “Predicting reflectance and color of paint films by Kubelka–Munk analysis. IV. Kubelka–Munk scattering coefficient,” J. Coatings Technol. 48, 30–36 (1972).

L. W. Richards, “The calculation of the optical performance of paint films,” J. Coatings Technol. 42, 276–286 (1970).

J. Colloid Interface Sci. (1)

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

J. Geophys. Res. (1)

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[CrossRef]

J. Heat Transfer (1)

Y. Ma, V. K. Varadan, V. V. Varadan, “Enhanced absorption due to dependent scattering,” J. Heat Transfer 112, 402–407 (1990).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (3)

J. Phys. Condens. Matter. (2)

W. E. Vargas, G. A. Niklasson, “Forward scattering ratios and average pathlength 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]

J. Thermophys. (1)

B. L. Drolen, C. L. Tien, “Independent and dependent scattering in packed-sphere systems,” J. Thermophys. 1, 63–68 (1987).
[CrossRef]

Licht (1)

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

Thin Solid Films (1)

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

Waves Random Media (1)

S. A. El-Wakil, A. R. Degheidy, N. K. Radwan, “Light transfer problem in turbid media with surface reflectivity,” Waves Random Media 6, 101–118 (1996).
[CrossRef]

Z. Tech. Phys. (1)

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

Other (3)

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

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. G. 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. M. Lampert, C. G. Granqvist, eds., Proc. SPIE1536, 169–182 (1991).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Dependence of optical thickness on the correction factor involved in collimated diffuse reflectance calculations [Eqs. (12) and (13)]. Two different refractive indices were considered, as were perfectly absorbing (R g = 0) and perfectly reflecting (R g = 1) substrates, for (a) small and (b) large particles.

Fig. 2
Fig. 2

Diffuse reflectance of a thick film (h = 100 µm) containing absorbing silicon particles in an alumina matrix [m = (4.08 + i0.04)/1.61] at a mid-visible wavelength (λ0 = 0.55 µm) calculated with the assumption of isotropy and symmetry (R KM) and of symmetry only (R MLG) and with anisotropy taken into account without the assumption of symmetry (R D). The particle volume fraction was set at f = 0.05; the film was deposited upon an aluminum substrate. The dashed curve depicts the difference between R MLG and R D , with ξ and σ d evaluated from the extended Hartel theory,12,13 whereas the dotted curve corresponds to the difference between R KM and R D .

Fig. 3
Fig. 3

Intrinsic (α) and effective (S) scattering coefficients per unit length of a thick coating (h = 100 µm) containing silicon particles hosted in an alumina matrix deposited upon an aluminum substrate. The particle volume fraction and the free-space wavelength were set at f = 0.05 and λ0 = 0.55 µm, respectively.

Fig. 4
Fig. 4

Intrinsic (β) and effective (K) absorption coefficients per unit length of a thick film (h = 100 µm) containing silicon particles hosted in an alumina matrix deposited upon an aluminum substrate. The particle volume fraction and the free-space wavelength were set at f = 0.05 and λ0 = 0.55 µm, respectively.

Fig. 5
Fig. 5

Effective average path-length parameter (ξ̅) and forward-scattering ratio (σ̅ d) corresponding to a thick coating (h = 100 µm) containing silicon particles in an alumina matrix supported by an aluminum substrate. The particle volume fraction and the free-space wavelength were set at f = 0.05 and λ0 = 0.55 µm, respectively. Circles denote values of ξ and σ d calculated from the extended Hartel theory.12,13

Equations (45)

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Iz, μ=k=1 Ikz, μ=k=1 Qk(+)zfkμn=0 cn(+)zPnμ,
Jz, μ=k=1 Jkz, μ=k=1 Qk(-)zfkμn=0 cn(-)zPnμ,
fkμ=14πn=02n+1ω¯n2n+1kPnμ,
dQ1(+)dz=-ξ1(+)α+βQ1(+)+ξ1(+)αQo,
dQk(+)dz=-ξk(+)α+βQk(+)+akξk-1(+)σk-1(+)αQk-1(+)+bkξk-1(-)1-σk-1(-)αQk-1(-),
-dQ1(-)dz=-ξ1(-)α+βQ1(-)+ξ1(-)αQo,
-dQk(-)dz=-ξk(-)α+βQk(-)+ckξk-1(+)1-σk-1(+)×αQk-1(+)+dkξk-1(-)σk-1(-)αQk-1(-),
Q1(+)τ=ω0exp-τ-exp-ξ1(+)τξ1(+)ξ1(+)-1,
Qk(+)τ=Fk(+)exp-τ-exp-ξk(+)τ+i=1k-1 Gi,k(+)×exp-ξk(+)τ-exp-ξi(+)τ,
Q1(-)τ=ω0ξ1(-) exp-τξ1(-)+1,
Qk(-)τ=Fk(-) exp-τ-i=1k-1 Gi,k(-) exp-ξi(+)τ,
ξ(±)=21±n=1 gnc¯n(±)1±2c¯1(±)/3+2 n=2 hnc¯n(±),
σd(±)=σdi±12n=1 gnc¯n(±)1+χnnω¯n+12n=1 gnω¯nm=2 χnmc¯m(±)1±n=1 gnc¯n(±),
cn(±)z=2n+14πk=1Qk(±)zk!ω¯n2n+1k,  χnm=01 PnμPmμdμ,
dIcdz=α+βIc,
dJcdz=-α+βJc,
dIddz=Sαβ(+)Id-Sα(-)Jd-σcαIc-1-σcαJc,
dJddz=Sα(+)Id-Sαβ(-)Jd+1-σcαIc+σcαJc,
Rdd=rde+1-rde1-rdiDRg-Γ2Γ1 expr1h-Rg-Γ1Γ2 expr2h,
Rcc=rc+1-rc2rg exp-2δh1-rcrg exp-2δh,
Rcd=1-rdi1-rcexp-δh1-rcrg exp-2δhDM1A1-δ2+Boδ+M2A1-δ2-Boδ,
D=Rg-Γ21-rdiΓ1expr1h-Rg-Γ1×1-rdiΓ2expr2h,
M1=Rg-Γ2A¯3-Γ1A2expr1+δh-Rg-Γ1×A¯3-Γ2A2expr2+δh+Γ2-Γ1×A¯3-RgA2expr1+r2h,
M2=rgRg-Γ2A¯2-Γ1A3expr1-δh-Rg-Γ1A¯2-Γ2A3expr2-δh+rgΓ2-Γ1×A¯2-RgA3expr1+r2h,
A1=Sαβ(+)Sαβ(-)-Sα(+)Sα(-)0,
A2=αSαβ(-)σc+Sα(-)1-σc+δσc,
A3=αSαβ(-)1-σc+Sα(-)σc-δ1-σc,
A¯2=αSαβ(+)σc+Sα(+)1-σc+δσc,
A¯3=αSαβ(+)1-σc+Sα(+)σc-δ1-σc,
Bo=Sαβ(+)-Sαβ(-)
-d2Iddz2+BodIddz+A1Id=A2Ic+A3Jc,
-d2Jddz2+BodJddz+A1Jd=A¯3Ic+A¯2Jc.
r1=Bo2+A1+Bo/221/2,  r2=Bo2-A1+Bo/221/2,
Γ1=Sαβ(+)-r1Sα(-),  Γ2=Sαβ(+)-r2Sα(-).
Rdd=rde+1-rde1-rdiRD1-rdiRD,
RD=Rg-Γ2Γ1 expγh-Rg-Γ1Γ2 exp-γhRg-Γ2expγh-Rg-Γ1exp-γh,
Rc=rc+1-rc1-rdiRD1-rdiRD F,
F=Rg-Γ2A¯3-Γ1A2expr1h-Rg-Γ1A¯3-Γ2A2expr2h+Γ2-Γ1A¯3-RgA2expr1+r2-δhA1-δ2+BoδRg-Γ2Γ1 expr1h-Rg-Γ1Γ2 expr2h.
R¯D=1-Rga-b cothbSha+b cothbSh-Rg,
Rs=rc+1-rc1-rdiRD1-rdiRD.
dIddz=S+KId-SJd,  -dJddz=S+KJd-SId.
-d2Iddz2+S+K2-S2Id=0,  -d2Jddz2+S+K2-S2Jd=0,
-d2Iddz2+BodIddz+A1Id=0,  -d2Jddz2+BodJddz+A1Jd=0,
S=ξ¯α1-σ¯d,  K=ξ¯β,
ξ¯=ξ(+)ξ(-),  σ¯d=1/2σd(+)+σd(-).

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