Abstract

General solutions for a four-flux radiative transfer model, derived from the radiative transfer equation and based on Lorenz–Mie scattering and absorption parameters, have been obtained. Forward and backward average path-length parameters have been considered as well as forward-scattering ratios for diffuse anisotropic radiation going into the forward and the backward hemispheres. The reported solutions are generalizations of those obtained by Maheu et al. [Appl. Opt.23, 3353–3362 (1984)]. Compared with the generalized solutions, numerical calculations indicate that the δ-Eddington approximation and the standard four-flux model of Maheu et al. overestimate the collimated–diffuse reflectance of particulate coatings, whereas these models give similar results in the case of collimated–diffuse transmittance.

© 1998 Optical Society of America

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References

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  1. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).
  2. J. I. Frankel, “Computational attributes of the integral form of the equation of transfer,” J. Quant. Spectrosc. Radiat. Transfer 46, 329–342 (1991).
    [CrossRef]
  3. W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).
  4. A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
    [CrossRef]
  5. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).
  6. J. Reichman, “Determination of absorption and scattering coefficients for nonhomogeneous media. 1: Theory,” Appl. Opt. 12, 1811–1815 (1973).
    [CrossRef] [PubMed]
  7. P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
    [CrossRef] [PubMed]
  8. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  9. C. M. Chu, S. W. Churchill, “Numerical solution of problems in multiple scattering of electromagnetic radiation,” Multiple Scatt. Electromag. Radiation 59, 855–863 (1955).
  10. B. Maheu, J. N. Lotoulouzan, 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]
  11. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  12. J. K. Beasley, J. T. Atkins, F. W. Billmeyer, “Scattering and absorption of light in turbid media,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967), pp. 765–785.
  13. 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).
    [CrossRef]
  14. W. E. Vargas, G. A. Niklasson, “Average path-length parameter in radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
    [CrossRef] [PubMed]
  15. 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]
  16. W. E. Vargas, G. A. Niklasson, “Intensity of diffuse radiation in particulate media,” J. Opt. Soc. Am. A 14, 2253–2262 (1997).
    [CrossRef]
  17. K. Klier, “Absorption and scattering in plane parallel turbid media,” J. Opt. Soc. Am. 62, 882–885 (1972).
    [CrossRef]
  18. G. V. Efimov, W. von Waldenfels, R. Wehrse, “Analytical solution of the nondiscretized radiative transfer equation for a slab of finite optical depth,” J. Quant. Spectrosc. Radiat. Transfer 53, 59–74 (1995).
    [CrossRef]
  19. T. Kunitomo, Y. Tsuboi, S. Iwashita, H. M. Shafey, “Theoretical study on spectrally selective paint coatings,” in Proceedings of the Eighth Biennial Congress International Solar Energy Society, Vol. 3, S. V. Szokolay, ed. (Pergamon, Oxford, UK, 1983), pp. 1943–1947.
  20. S. Ito, “Optical wave propagation in discrete random media with large particles: a treatment of the phase function,” Appl. Opt. 32, 1652–1656 (1993).
    [CrossRef] [PubMed]
  21. F. B. Yurevich, L. A. Konyukh, “Radiation attenuation by disperse media,” Int. J. Heat Mass Transfer 18, 819–829 (1975).
    [CrossRef]
  22. E. P. Shettle, J. A. Weinman, “The transfer of solar irradiance through inhomogeneous turbid atmospheres evaluated by Eddington’s approximation,” J. Atmos. Sci. 27, 1048–1055 (1970).
    [CrossRef]
  23. J. H. Joseph, W. J. Wiscombe, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
    [CrossRef]
  24. F. Liu, J. Swithenbank, E. S. Garbett, “The boundary condition of the PN-approximation used to solve the radiative transfer equation,” Int. J. Heat Mass Transfer 35, 2043–2052 (1992).
    [CrossRef]
  25. S. Karanjai, M. Talukder, “Solution of the equation of transfer with general phase function by a modified spherical-harmonic method,” Astrophys. Space Sci. 197, 309–336 (1992).
    [CrossRef]
  26. W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980).
    [CrossRef]
  27. Harshvardhan, M. D. King, “Comparative accuracy of diffuse radiative properties computed using selected multiple scattering approximations,” J. Atmos. Sci. 50, 247–259 (1993).
    [CrossRef]
  28. 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. Grandqvist, C. M. Lampert, eds., Proc. SPIE1016, 89–99 (1988).
    [CrossRef]
  29. 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]
  30. L. Fukshansky, N. Fukshansky-Kazarinova, A. M. Remisowsky, “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]
  31. H. R. Wilson, W. Eck, “Transmission variation using scattering/transparent switching films,” Sol. Energy Mat. Sol. Cells 31, 197–214 (1993).
    [CrossRef]
  32. W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
    [CrossRef] [PubMed]
  33. B. Maheu, G. Gouesbet, “Four-flux models to solve the scattering transfer equation: special cases,” Appl. Opt. 25, 1122–1128 (1986).
    [CrossRef] [PubMed]

1997 (6)

1995 (1)

G. V. Efimov, W. von Waldenfels, R. Wehrse, “Analytical solution of the nondiscretized radiative transfer equation for a slab of finite optical depth,” J. Quant. Spectrosc. Radiat. Transfer 53, 59–74 (1995).
[CrossRef]

1993 (3)

H. R. Wilson, W. Eck, “Transmission variation using scattering/transparent switching films,” Sol. Energy Mat. Sol. Cells 31, 197–214 (1993).
[CrossRef]

Harshvardhan, M. D. King, “Comparative accuracy of diffuse radiative properties computed using selected multiple scattering approximations,” J. Atmos. Sci. 50, 247–259 (1993).
[CrossRef]

S. Ito, “Optical wave propagation in discrete random media with large particles: a treatment of the phase function,” Appl. Opt. 32, 1652–1656 (1993).
[CrossRef] [PubMed]

1992 (2)

F. Liu, J. Swithenbank, E. S. Garbett, “The boundary condition of the PN-approximation used to solve the radiative transfer equation,” Int. J. Heat Mass Transfer 35, 2043–2052 (1992).
[CrossRef]

S. Karanjai, M. Talukder, “Solution of the equation of transfer with general phase function by a modified spherical-harmonic method,” Astrophys. Space Sci. 197, 309–336 (1992).
[CrossRef]

1991 (2)

1986 (1)

1984 (1)

1980 (1)

W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980).
[CrossRef]

1976 (1)

J. H. Joseph, W. J. Wiscombe, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

1975 (1)

F. B. Yurevich, L. A. Konyukh, “Radiation attenuation by disperse media,” Int. J. Heat Mass Transfer 18, 819–829 (1975).
[CrossRef]

1973 (1)

1972 (1)

1971 (1)

1970 (1)

E. P. Shettle, J. A. Weinman, “The transfer of solar irradiance through inhomogeneous turbid atmospheres evaluated by Eddington’s approximation,” J. Atmos. Sci. 27, 1048–1055 (1970).
[CrossRef]

1955 (1)

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

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).

1905 (1)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Atkins, J. T.

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, “Scattering and absorption of light in turbid media,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967), pp. 765–785.

Beasley, J. K.

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, “Scattering and absorption of light in turbid media,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967), pp. 765–785.

Billmeyer, F. W.

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, “Scattering and absorption of light in turbid media,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967), pp. 765–785.

Bohren, C. F.

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

Chu, C. M.

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

Churchill, S. W.

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

Eck, W.

H. R. Wilson, W. Eck, “Transmission variation using scattering/transparent switching films,” Sol. Energy Mat. Sol. Cells 31, 197–214 (1993).
[CrossRef]

Efimov, G. V.

G. V. Efimov, W. von Waldenfels, R. Wehrse, “Analytical solution of the nondiscretized radiative transfer equation for a slab of finite optical depth,” J. Quant. Spectrosc. Radiat. Transfer 53, 59–74 (1995).
[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. Grandqvist, C. M. Lampert, eds., Proc. SPIE1016, 89–99 (1988).
[CrossRef]

Frankel, J. I.

J. I. Frankel, “Computational attributes of the integral form of the equation of transfer,” J. Quant. Spectrosc. Radiat. Transfer 46, 329–342 (1991).
[CrossRef]

Fukshansky, L.

Fukshansky-Kazarinova, N.

Garbett, E. S.

F. Liu, J. Swithenbank, E. S. Garbett, “The boundary condition of the PN-approximation used to solve the radiative transfer equation,” Int. J. Heat Mass Transfer 35, 2043–2052 (1992).
[CrossRef]

Gouesbet, G.

Harshvardhan,

Harshvardhan, M. D. King, “Comparative accuracy of diffuse radiative properties computed using selected multiple scattering approximations,” J. Atmos. Sci. 50, 247–259 (1993).
[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).

Huffman, D. R.

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

Ishimaru, A.

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

Ito, S.

Iwashita, S.

T. Kunitomo, Y. Tsuboi, S. Iwashita, H. M. Shafey, “Theoretical study on spectrally selective paint coatings,” in Proceedings of the Eighth Biennial Congress International Solar Energy Society, Vol. 3, S. V. Szokolay, ed. (Pergamon, Oxford, UK, 1983), pp. 1943–1947.

Joseph, J. H.

J. H. Joseph, W. J. Wiscombe, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Karanjai, S.

S. Karanjai, M. Talukder, “Solution of the equation of transfer with general phase function by a modified spherical-harmonic method,” Astrophys. Space Sci. 197, 309–336 (1992).
[CrossRef]

King, M. D.

Harshvardhan, M. D. King, “Comparative accuracy of diffuse radiative properties computed using selected multiple scattering approximations,” J. Atmos. Sci. 50, 247–259 (1993).
[CrossRef]

Klier, K.

Konyukh, L. A.

F. B. Yurevich, L. A. Konyukh, “Radiation attenuation by disperse media,” Int. J. Heat Mass Transfer 18, 819–829 (1975).
[CrossRef]

Kubelka, P.

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

Kunitomo, T.

T. Kunitomo, Y. Tsuboi, S. Iwashita, H. M. Shafey, “Theoretical study on spectrally selective paint coatings,” in Proceedings of the Eighth Biennial Congress International Solar Energy Society, Vol. 3, S. V. Szokolay, ed. (Pergamon, Oxford, UK, 1983), pp. 1943–1947.

Liu, F.

F. Liu, J. Swithenbank, E. S. Garbett, “The boundary condition of the PN-approximation used to solve the radiative transfer equation,” Int. J. Heat Mass Transfer 35, 2043–2052 (1992).
[CrossRef]

Lotoulouzan, J. N.

Maheu, B.

Meador, W. E.

W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980).
[CrossRef]

Mudgett, P. S.

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, “Forward scattering ratios and average path-length parameters in radiative transfer models,” J. Phys. Condens. Matter 9, 9083–9096 (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, “Average path-length parameter in 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, 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, “Generalized method for evaluating scattering parameters used in radiative transfer models,” J. Opt. Soc. Am. A 14, 2243–2252 (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. Grandqvist, C. M. Lampert, eds., Proc. SPIE1016, 89–99 (1988).
[CrossRef]

Reichman, J.

Remisowsky, A. M.

Richards, L. W.

Schuster, A.

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Shafey, H. M.

T. Kunitomo, Y. Tsuboi, S. Iwashita, H. M. Shafey, “Theoretical study on spectrally selective paint coatings,” in Proceedings of the Eighth Biennial Congress International Solar Energy Society, Vol. 3, S. V. Szokolay, ed. (Pergamon, Oxford, UK, 1983), pp. 1943–1947.

Shettle, E. P.

E. P. Shettle, J. A. Weinman, “The transfer of solar irradiance through inhomogeneous turbid atmospheres evaluated by Eddington’s approximation,” J. Atmos. Sci. 27, 1048–1055 (1970).
[CrossRef]

Swithenbank, J.

F. Liu, J. Swithenbank, E. S. Garbett, “The boundary condition of the PN-approximation used to solve the radiative transfer equation,” Int. J. Heat Mass Transfer 35, 2043–2052 (1992).
[CrossRef]

Talukder, M.

S. Karanjai, M. Talukder, “Solution of the equation of transfer with general phase function by a modified spherical-harmonic method,” Astrophys. Space Sci. 197, 309–336 (1992).
[CrossRef]

Tsuboi, Y.

T. Kunitomo, Y. Tsuboi, S. Iwashita, H. M. Shafey, “Theoretical study on spectrally selective paint coatings,” in Proceedings of the Eighth Biennial Congress International Solar Energy Society, Vol. 3, S. V. Szokolay, ed. (Pergamon, Oxford, UK, 1983), pp. 1943–1947.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

Vargas, W. E.

von Waldenfels, W.

G. V. Efimov, W. von Waldenfels, R. Wehrse, “Analytical solution of the nondiscretized radiative transfer equation for a slab of finite optical depth,” J. Quant. Spectrosc. Radiat. Transfer 53, 59–74 (1995).
[CrossRef]

Weaver, W. R.

W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980).
[CrossRef]

Wehrse, R.

G. V. Efimov, W. von Waldenfels, R. Wehrse, “Analytical solution of the nondiscretized radiative transfer equation for a slab of finite optical depth,” J. Quant. Spectrosc. Radiat. Transfer 53, 59–74 (1995).
[CrossRef]

Weinman, J. A.

E. P. Shettle, J. A. Weinman, “The transfer of solar irradiance through inhomogeneous turbid atmospheres evaluated by Eddington’s approximation,” J. Atmos. Sci. 27, 1048–1055 (1970).
[CrossRef]

Wilson, H. R.

H. R. Wilson, W. Eck, “Transmission variation using scattering/transparent switching films,” Sol. Energy Mat. Sol. Cells 31, 197–214 (1993).
[CrossRef]

Wiscombe, W. J.

J. H. Joseph, W. J. Wiscombe, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Yurevich, F. B.

F. B. Yurevich, L. A. Konyukh, “Radiation attenuation by disperse media,” Int. J. Heat Mass Transfer 18, 819–829 (1975).
[CrossRef]

Appl. Opt. (8)

Astrophys. J. (1)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Astrophys. Space Sci. (1)

S. Karanjai, M. Talukder, “Solution of the equation of transfer with general phase function by a modified spherical-harmonic method,” Astrophys. Space Sci. 197, 309–336 (1992).
[CrossRef]

Int. J. Heat Mass Transfer (2)

F. Liu, J. Swithenbank, E. S. Garbett, “The boundary condition of the PN-approximation used to solve the radiative transfer equation,” Int. J. Heat Mass Transfer 35, 2043–2052 (1992).
[CrossRef]

F. B. Yurevich, L. A. Konyukh, “Radiation attenuation by disperse media,” Int. J. Heat Mass Transfer 18, 819–829 (1975).
[CrossRef]

J. Atmos. Sci. (4)

E. P. Shettle, J. A. Weinman, “The transfer of solar irradiance through inhomogeneous turbid atmospheres evaluated by Eddington’s approximation,” J. Atmos. Sci. 27, 1048–1055 (1970).
[CrossRef]

J. H. Joseph, W. J. Wiscombe, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

W. E. Meador, W. R. Weaver, “Two-stream approximations to radiative transfer in planetary atmospheres: a unified description of existing methods and a new improvement,” J. Atmos. Sci. 37, 630–643 (1980).
[CrossRef]

Harshvardhan, M. D. King, “Comparative accuracy of diffuse radiative properties computed using selected multiple scattering approximations,” J. Atmos. Sci. 50, 247–259 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Phys. Condens. Matter (2)

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).
[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. Quant. Spectrosc. Radiat. Transfer (2)

G. V. Efimov, W. von Waldenfels, R. Wehrse, “Analytical solution of the nondiscretized radiative transfer equation for a slab of finite optical depth,” J. Quant. Spectrosc. Radiat. Transfer 53, 59–74 (1995).
[CrossRef]

J. I. Frankel, “Computational attributes of the integral form of the equation of transfer,” J. Quant. Spectrosc. Radiat. Transfer 46, 329–342 (1991).
[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).

Multiple Scatt. Electromag. Radiation (1)

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

Sol. Energy Mat. Sol. Cells (1)

H. R. Wilson, W. Eck, “Transmission variation using scattering/transparent switching films,” Sol. Energy Mat. Sol. Cells 31, 197–214 (1993).
[CrossRef]

Z. Tech. Phys. (1)

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

Other (6)

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

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

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

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, “Scattering and absorption of light in turbid media,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967), pp. 765–785.

T. Kunitomo, Y. Tsuboi, S. Iwashita, H. M. Shafey, “Theoretical study on spectrally selective paint coatings,” in Proceedings of the Eighth Biennial Congress International Solar Energy Society, Vol. 3, S. V. Szokolay, ed. (Pergamon, Oxford, UK, 1983), pp. 1943–1947.

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

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Figures (3)

Fig. 1
Fig. 1

Collimated and diffuse radiation fluxes through a scattering and an absorbing coating on a thick substrate. The coating is perpendicularly illuminated with unpolarized collimated radiation.

Fig. 2
Fig. 2

Optical thickness dependence of the collimated–diffuse reflectance of an unsupported thick polyethylene film (h = 50 μm) containing titanium dioxide pigments at a particle volume fraction f = 0.05. The free-space wavelength was set to 0.55 μm. The refractive index of the pigments is 2.50 + i0.0, and the refractive index of the matrix was set to 1.50. MLG, values obtained from the four-flux model of Maheu, Letoulouzan, and Gouesbet,10 where we evaluate ξ and σ d by using the extended Hartel theory of Vargas and Niklasson13; GFF, values calculated from the generalized four-flux model described throughout the paper; EDD, reflectance values obtained from the integrated δ-Eddington approximation. Boundary reflections have been neglected.

Fig. 3
Fig. 3

Optical thickness dependence of the collimated–diffuse reflectance and transmittance of unsupported thick particulate slabs (h = 50 μm). The free-space wavelength was set to 0.55 μm, and the particle volume fraction is f = 0.05. The relative refractive index of the particles corresponds to (a) m = (4.08 + i0.04)/1.61 [silicon particles in an alumina matrix] and (b) m = (3.10 + i0.18)/1.56 (iron oxide particles in a KBr matrix). The MLG forward average path-length parameter ξ is calculated from the extended Hartel theory. Boundary reflections are taken into account.

Equations (58)

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

μ   I τ ,   μ τ = - I τ ,   μ + 1 2 - 1 1   p μ ,   μ I τ ,   μ d μ ,
I τ ,   μ = k = 1   I k + τ ,   μ = k = 1   Q k + τ f k μ , 0 < μ 1 ,
J τ ,   μ = k = 1   I k ( - ) τ ,   μ = k = 1   Q k ( - ) τ f k μ , - 1 μ < 0 ,
μ f 1 μ d Q 1 ± d τ = - f 1 μ Q 1 ± + ω 0 Q 0 + 0 1   p 1 μ ,   μ × δ μ - 1 d μ + Q 0 ( - ) - 1 0   p 1 μ ,   μ × δ μ + 1 d μ ,
μ f k μ d Q k ± d τ = - f k μ Q k ± + ω 0 Q k - 1 + 0 1   p 1 μ ,   μ × f k - 1 μ d μ + Q k - 1 ( - ) - 1 0   p 1 μ ,   μ × f k - 1 μ d μ ,
Q 0 + τ = 1 - r c exp - τ 1 - r c exp - τ 2 , Q 0 ( - ) τ = 1 - r c r c exp - τ exp τ - τ 1 - r c exp - τ 2 ,
d Q k + d τ = - ξ k + Q k + + ω 0 a k σ k + ξ k - 1 + Q k - 1 + + b k 1 - σ k ( - ) ξ k - 1 ( - ) Q k - 1 ( - ) ,
- d Q k ( - ) d τ = - ξ k ( - ) Q k ( - ) + ω 0 c k 1 - σ k + ξ k - 1 + Q k - 1 + + d k σ k ( - ) ξ k - 1 ( - ) Q k - 1 ( - ) ,
w k + = 2 π   0 1   μ f k μ d μ ,     w k ( - ) = - 2 π   - 1 0   μ f k μ d μ ,
d q 1 + d τ = - ξ 1 + q 1 + + ω 0 σ c q 0 + + 1 - σ c q 0 ( - ) ,
d q k + d τ = - ξ k + q k + + ω 0 σ k + ξ k - 1 + q k - 1 + + 1 - σ k ( - ) × ξ k - 1 ( - ) q k - 1 ( - ) ,
- d q 1 ( - ) d τ = - ξ 1 ( - ) q 1 ( - ) + ω 0 1 - σ c q 0 + + σ c q 0 ( - ) ,
- d q k ( - ) d τ = - ξ k ( - ) q k ( - ) + ω 0 1 - σ k + ξ k - 1 + q k - 1 + + σ k ( - ) ξ k - 1 ( - ) q k - 1 ( - ) ,
d I d d τ = - ξ + 1 - ω 0 σ d + I d + ω 0 ξ ( - ) 1 - σ d ( - ) J d + ω 0 σ c I c + ω 0 1 - σ c J c ,
- d J d d τ = - ξ ( - ) 1 - ω 0 σ d ( - ) J d + ω 0 ξ + 1 - σ d + I d + ω 0 1 - σ c I c + ω 0 σ c J c ,
I d τ = k = 1   q k + τ ,   J d τ = k = 1   q k ( - ) τ , ξ ± τ = k = 1   ξ k ± q k ± τ k = 1   q k ± τ , σ d ± τ = k = 1   σ k + 1 ± ξ k ± q k ± τ k = 1   ξ k ± q k ± τ .
- d J d d τ = - γ 1 J d + γ 2 I d + π F ω 0 γ 3 exp - τ , d I d d τ = - γ 1 I d + γ 2 J d + π F ω 0 γ 4 exp - τ ,
d I c d z = α + β I c ,
d J c d z = - α + β J c ,
d I d d z = S α β + I d - S α ( - ) J d - σ c α I c - 1 - σ c α J c ,
d J d d z = S α + I d - S α β ( - ) J d + 1 - σ c α I c + σ c α J c ,
R cc = r c + 1 - r c 2 A 6 exp - 2 τ 1 - r c A 6 exp - 2 τ , T cc = G F 1 - r c exp - τ 1 - r c A 6 exp - 2 τ ,
T dd = G F 1 - r d e Γ 1 - Γ 2 D ,
T cd = G F 1 - r c exp - τ D 1 - r c A 6 exp - 2 τ × N 1 A 1 - δ 2 + B 0 δ + N 2 A 1 - δ 2 - B 0 δ ,
D = A 7 - Γ 2 1 - r d i Γ 1 exp r 1 h - A 7 - Γ 1 × 1 - r d i Γ 2 exp r 2 h ,
N 1 = Γ 1 - Γ 2 Ā 3 r d i - A 2 exp τ + Ā 3 - A 2 Γ 2 × 1 - r d i Γ 1 exp r 1 h - Ā 3 - A 2 Γ 1 × 1 - r d i Γ 2 exp r 2 h ,
N 2 = A 6 Γ 1 - Γ 2 Ā 2 r d i - A 3 exp - τ + Ā 2 - A 3 Γ 2 × 1 - r d i Γ 1 exp r 1 h - Ā 2 - A 3 Γ 1 × 1 - r d i Γ 2 exp r 2 h ,
R dd = r d e + 1 - r d e 1 - r d i D A 7 - Γ 2 Γ 1 exp r 1 h - A 7 - Γ 1 Γ 2 exp r 2 h ,
R cd = 1 - r d i 1 - r c exp - τ D 1 - r c A 6 exp - 2 τ × M 1 A 1 - δ 2 + B 0 δ + M 2 A 1 - δ 2 - B 0 δ ,
M 1 = A 7 - Γ 2 Ā 3 - Γ 1 A 2 exp r 1 h + τ - A 7 - Γ 1 Ā 3 - Γ 2 A 2 exp r 2 h + τ + Γ 2 - Γ 1 × Ā 3 - A 7 A 2 exp r 1 + r 2 h ,
M 2 = A 6 A 7 - Γ 2 Ā 2 - Γ 1 A 3 exp r 1 h - τ - A 7 - Γ 1 Ā 2 - Γ 2 A 3 exp r 2 h - τ + A 6 Γ 2 - Γ 1 × Ā 2 - A 7 A 3 exp r 1 + r 2 h .
T = exp - τ 1 - Ā 3 - A 2 Γ 1 exp r 2 h - Ā 3 - A 2 Γ 2 exp r 1 h - A 2 Γ 2 - Γ 1 exp τ A 1 - δ 2 + B 0 δ Γ 1 exp r 2 h - Γ 2 exp r 1 h ,
R = Γ 1 Ā 3 - A 2 Γ 2 exp r 2 h - Γ 2 Ā 3 - A 2 Γ 1 exp r 1 h + Ā 3 Γ 2 - Γ 1 exp r 1 + r 2 h - τ A 1 - δ 2 + B 0 δ Γ 1 exp r 2 h - Γ 2 exp r 1 h .
I c z = h = 1 - r c I c 0 + r c J c z = h , J c z = 0 = A 6 I c z = 0 ,
I d z = h = 1 - r d e I d 0 + r d i J d z = h , J d z = 0 = A 7 I d z = 0 ,
C 1 = 1 - r c I c 0 exp - τ 1 - r c A 6 exp - 2 τ ,   C 2 = 1 - r c A 6 I c 0 exp - τ 1 - r c A 6 exp - 2 τ ,
- d 2 I d d z 2 + B 0 d I d d z + A 1 I d = A 2 I c + A 3 J c ,
- d 2 J d d z 2 + B 0 d J d d z + A 1 J d = Ā 3 I c + Ā 2 J c ,
B 0 = S α β + - S α β ( - ) ,
A 1 = S α β + S α β ( - ) - S α + S α ( - ) 0 ,
A 2 = α S α β ( - ) σ c + S α ( - ) 1 - σ c + δ σ c ,
A 3 = α S α β ( - ) 1 - σ c + S α ( - ) σ c - δ 1 - σ c ,
Ā 2 = α S α β + σ c + S α + 1 - σ c + δ σ c ,
Ā 3 = α S α β + 1 - σ c + S α + σ c - δ 1 - σ c ,
I d z = C 3 exp r 1 z + C 4 exp r 2 z + C 5 exp δ z + C 6 exp - δ z ,
J d z = C 7 exp r 1 z + C 8 exp r 2 z + C 9 exp δ z + C 10 exp - δ z ,
r 1 = B 0 2 + A 1 + B 0 2 2 1 / 2 ,   r 2 = B 0 2 - A 1 + B 0 2 2 1 / 2 ,
C 5 = A 2 C 1 A 1 - δ 2 + B 0 δ ,     C 6 = A 3 C 2 A 1 - δ 2 - B 0 δ ,
C 9 = Ā 3 C 1 A 1 - δ 2 + B 0 δ ,     C 10 = Ā 2 C 2 A 1 - δ 2 - B 0 δ .
C 7 = Γ 1 C 3 ,     C 8 = Γ 2 C 4 ,
Γ 1 = A 4 - r 1 A 5 ,   Γ 2 = A 4 - r 2 A 5 ,   A 4 = S α β + ,   A 5 = S α ( - ) .
C 3 = A 13 A 10 - A 8 A 12 A 10 A 11 - A 9 A 12 ,     C 4 = A 8 A 11 - A 9 A 13 A 10 A 11 - A 9 A 12 ,
A 8 = Ā 3 - A 7 A 2 A 1 - δ 2 + B 0 δ   C 1 + Ā 2 - A 7 A 3 A 1 - δ 2 - B 0 δ   C 2 ,
A 9 = A 7 - Γ 1 ,
A 10 = A 7 - Γ 2 ,
A 11 = 1 - r d i Γ 1 exp r 1 h ,
A 12 = 1 - r d i Γ 2 exp r 2 h ,
A 13 = 1 - r d e I d 0 + Ā 3 r d i - A 2 exp τ A 1 - δ 2 + B 0 δ   C 1 + A 2 r d i - A 3 exp - τ A 1 - δ 2 - B 0 δ   C 2 .

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