Theories of Radiative Transfer

  • Georg Klein
Part of the Springer Series in Optical Sciences book series (SSOS, volume 154)


In this chapter, we ask the question as to how closer theoretical examinations could aid the calculation of the spectral values or rather color values which one would obtain experimentally. This is necessary in order to apply them to criteria such as strength of color, covering capacity, or most importantly to numerical recipe prediction. We will describe the optical radiative transfer in layers quantitatively with two different physical approaches: simple and multiple scattering. Although both formalisms achieve nearly the same results, for practical applications multiple scattering is preferred. In particular, for multiple scattering, the formalism is quite simpler, and the measurement geometry – which forms the basis of the experimental results – can be taken into account. In the following, we consider practical methods in order to obtain the interesting spectral and color values with a procedure as simply as possible.


Phase Function Diffuse Flux Corner Point Transmission Function Radiative Transfer Equation 
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  1. 1.
    Chandrasekhar, S: “Radiative Transfer” (1950), reprint, Dover, New York (1960)MATHGoogle Scholar
  2. 2.
    Born, M, Wolf, E: “Principles of Optics”, 7th ed, reprint, Cambridge Univ Press, Cambridge UK (2006)Google Scholar
  3. 3.
    Mie, G: “Beitraege zur Optik trueber Medien, speziell kolloidaler metalloesungen”, Annalen der Physik 25, 4th series (1908) 377ADSCrossRefMATHGoogle Scholar
  4. 4.
    Kerker, M: “The Scattering of Light and Other Electromagnetic Radiation”, 9th ed, Academic Press, San Diego (1990)Google Scholar
  5. 5.
    Bohren, CF, Huffmann, DR: “Absorption and Scattering of Light by Small Particles”, Wiley-VCH, Weinheim (2004)Google Scholar
  6. 6.
    Greiner, W: “Classical Electrodynamics”, Springer, New York (1998)CrossRefMATHGoogle Scholar
  7. 7.
    Hedinger, H, Pauli, H: “Messung optischer Konstanten an aufgedampften Pigmentschichten und daraus berechnete Farbe pigmentierter PVC-Folien”, XIXth FATIPEC-Congress Aachen, Vol II (1988) 503Google Scholar
  8. 8.
    Joshi, JJ, Vaidya, DB, Shah, HS: “Application of multi-flux theory based on Mie scattering to the problem of modeling the optical characteristics of colored pigmented paint films”, Col Res Appl 26 (2001) 234; Col Res Appl 28 (2003) 308CrossRefGoogle Scholar
  9. 9.
    Schuster, A: “Radiation through a foggy atmosphere”, Astrophys J 21 (1905) 1ADSCrossRefGoogle Scholar
  10. 10.
    Schwarzschild, K: “Ueber das Gleichgewicht der Sonnenstrahlung”, Goettinger Nachr (1906) p 41Google Scholar
  11. 11.
    Kubelka, P, Munk, F: “Ein Beitrag zur Optik der Farbanstriche”, Z Techn Physik 12 (1931) 593Google Scholar
  12. 12.
    Kubelka, P: “New contributions to the optics of intensely light-scattering materials part I”, J Opt Soc Am 38 (1947) 448ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Kortuem, G: “Reflexionsspektroskopie”, Springer, Berlin (1969)CrossRefGoogle Scholar
  14. 14.
    Voelz, HG: “Industrielle Farbpruefung”, 2nd ed, Wiley-VCH, Weinheim (2001)CrossRefGoogle Scholar
  15. 15.
    Pauli, H, Eitle, D: “Comparison of Different Theoretical Models of Multiple Scattering of Pigmented Media”, Colour 73, Adam Hilger, London (1973) 423Google Scholar
  16. 16.
    Reichl, LE: “A modern Course in Statistical Physics”, Wiley-VCH, Weinheim (2009)MATHGoogle Scholar
  17. 17.
    Peraiah, A: “An Introduction to Radiative Transfer”, Cambridge Univ Press, Cambridge UK (2002)MATHGoogle Scholar
  18. 18.
    Gerber, WH: “Messung und Charakterisierung von Metallic-Lacken”, Proc XXII. FATIPEC Congress, Budapest, Vol I, (1994) 263Google Scholar
  19. 19.
    Quateroni, A, Sacco, R, Saleri, F: “Numerical Mathematics”, Springer, Berlin, Heidelberg (2007)Google Scholar
  20. 20.
    van de Hulst, HC: “Multiple Light Scattering: Tables, Formulas and Applications”, Vol I & II, Academic Press, New York (1980)Google Scholar
  21. 21.
    Henyey, LG, Greenstein, JL: “Diffuse radiation galaxy”, Astrophys J 93 (1941) 70ADSCrossRefMATHGoogle Scholar
  22. 22.
    Gerber, WH: “Rezeptieren transluzenter Kunststoffe”, in: “Rationelles Verfahren zur Einfaerbung von Kunststoffen”, Deut Industrieforum Technologie, No 2140, Wuerzburg (1992) 1–24Google Scholar
  23. 23.
    Saunderson, JL: “Calculation of the color of pigmented plastics”, J Opt Soc Am 32 (1942) 727ADSCrossRefGoogle Scholar
  24. 24.
    Ryde, JW: “The scattering of light by turbid media. – Part I”, Proc Roy Soc Ser A, 131 (1931) 451ADSCrossRefMATHGoogle Scholar
  25. 25.
    Klein, GA: “Farbenphysik fuer industrielle Anwendungen”, Springer, Berlin, Heidelberg (2004)CrossRefGoogle Scholar
  26. 26.
    Rybicki, GB: “Radiative transfer”, J Astrophys Astr 17 (1996) 95ADSCrossRefGoogle Scholar
  27. 27.
    Richards, LW: “The calculation of the optical performance of paint films”, J Paint Techn 42 (1970) 276Google Scholar
  28. 28.
    Mudget, PS, Richards, LW: “Multiple scattering calculations for technology”, Appl Opt 10 (1971) 1485ADSCrossRefGoogle Scholar
  29. 29.
    Billmeyer, FW, Jr, Carter, EC: “Color and appearance of metallized paint films”, J Coat Techn, 48 (1976) 53Google Scholar
  30. 30.
    Ishimaru, A: “Wave Propagation and Scattering in Random Media”, IEEE Press, New York (2005)MATHGoogle Scholar
  31. 31.
    Martin, PA: “Multiple Scattering”, Cambridge Univ Press, Cambridge (2006)CrossRefGoogle Scholar
  32. 32.
    Lenoble, J, ed: “Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures”, Deepak Publishing, Hampton, VA (1985)Google Scholar
  33. 33.
    Dahlquist, G, Bjoerck, A: “Numerical Methods in Scientific Computing”, Soc Industrial and Applied Mathematics, Philadelphia, PA (2008)Google Scholar
  34. 34.
    Quateroni, A, Sacco, R, Saleri, F: “Numerical Mathematics”, Springer, Berlin, Heidelberg (2007)Google Scholar
  35. 35.
    Press, WH, Teukolsky, SA, Vetterling, WT, Flannery, BP: “Fortran Numerical Recipes”, Cambridge Univ Press, Cambridge (2008)MATHGoogle Scholar
  36. 36.
    Bronstein, IN, Semendjajew, KA, Musiol, G, Muehlig, H: “Handbook of Mathematics”, 9th ed, Springer, Berlin (2007)Google Scholar
  37. 37.
    Wolf, W: “High Performance Embedded Computing”, Elsevier, Amsterdam (2007)Google Scholar
  38. 38.
    Gerber, WH, Pauli, HKA: “Accurate Model for Metallic Paints” (1989), without referenceGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Georg Klein
    • 1
  1. 1.HerrenbergGermany

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