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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 22, Iss. 9 — Sep. 1, 2005
  • pp: 1866–1873

Revised Kubelka–Munk theory. III. A general theory of light propagation in scattering and absorptive media

Li Yang and Stanley J. Miklavcic  »View Author Affiliations

JOSA A, Vol. 22, Issue 9, pp. 1866-1873 (2005)

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A generally applicable theoretical model describing light propagating through turbid media is proposed. The theory is a generalization of the revised Kubelka–Munk theory, extending its applicability to accommodate a wider range of absorption influences. A general expression for a factor taking into account the effect of scattering on the total photon path traversed in a turbid medium is derived. The extended model is applied to systems of ink-dyed paper sheets—mixtures of wood fibers with dyes—which represent examples of systems that have thus far eluded the original Kubelka–Munk model. The results of simulations of the spectral dependence of Kubelka–Munk coefficients of absorption and scattering show that they compare very well with those derived from experimental results.

© 2005 Optical Society of America

OCIS Codes
(290.4210) Scattering : Multiple scattering
(290.7050) Scattering : Turbid media
(300.1030) Spectroscopy : Absorption

Original Manuscript: January 18, 2005
Manuscript Accepted: March 9, 2005
Published: September 1, 2005

Li Yang and Stanley J. Miklavcic, "Revised Kubelka–Munk theory. III. A general theory of light propagation in scattering and absorptive media," J. Opt. Soc. Am. A 22, 1866-1873 (2005)

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  1. P. Kubelka, F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).
  2. P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948). [CrossRef] [PubMed]
  3. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  4. W. R. Blevin, W. J. Brown, “Light-scattering properties of pigment suspensions,” J. Opt. Soc. Am. 51, 1250–1256 (1962). [CrossRef]
  5. E. Allen, “Calculations for colorant formulations,” in Industrial Color Technology: Advances in Chemistry Series 107 (American Chemical Society, 1971), pp. 87–119.
  6. D. B. Judd, G. Wyszecki, “Physics and Psychophysics of colorant layers,” in Color in Business, Science and Industry, 3rd ed. (Wiley, 1975).
  7. W. F. Cheong, A. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990), and references therein. [CrossRef]
  8. J. H. Nobbs, “Kubelka–Munk theory and the prediction of reflectance,” Rev. Prog. Color. Relat. Top. 15, 66–75 (1985). [CrossRef]
  9. B. Philips-Invernizzi, D. Dupont, C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082–1092 (2001). [CrossRef]
  10. W. J. Foote, “An investigation of the fundamental scattering and absorption coefficients of dyed handsheets,” Pap. Trade J. 109, 333–340 (1939).
  11. J. A. Van den Akker, “Scattering and absorption of light in paper and other diffusing media,” Tappi J. 32, 498–501 (1949).
  12. L. Nordman, P. Aaltonen, T. Makkonen, Relationships between Mechanical and Optical Properties of Paper Affected by Web Consolidation (British Paper and Board Maker’s Association, 1966).
  13. J. A. Van den Akker, Theory of Some of the Discrepancies Observed in Application of the Kubelka–Munk Equations to Particulate Systems (Plenum, 1968).
  14. M. Rundlöf, J. A. Bristow, “A note concerning the interaction between light scattering and light absorption in the application of the Kubelka–Munk equations,” J. Pulp Pap. Sci. 23, J220–223 (1997).
  15. E. Allen, “Colorant formation and shading,” in Optical Radiation Measurements, Vol. 2, Color Measurements, F. Grum and C. J. Bartleson, eds. (Academic, 1980), pp. 290–336.
  16. W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997). [CrossRef] [PubMed]
  17. A. A. Koukoulas, B. D. Jordan, “Effect of strong absorption on the Kubelka–Munk scattering coefficient,” J. Pulp Pap. Sci. 23, J224–232 (1997).
  18. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998). [CrossRef]
  19. W. E. Vargas, “Inversion methods from Kubelka–Munk analysis,” J. Opt. A, Pure Appl. Opt. 4, 452–456 (2002). [CrossRef]
  20. J. F. Bloch, R. Sève, “About the theoretical aspect of multiple light scattering: Silvy’s theory,” Color Res. Appl. 28, 227–228 (2003). [CrossRef]
  21. 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]
  22. 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]
  23. L. Yang, B. Kruse, “Revised Kubelka–Munk theory. I. Theory and applications,” J. Opt. Soc. Am. A 21, 1933–1941 (2004). [CrossRef]
  24. L. Yang, B. Kruse, S. J. Miklavcic, “Revised Kubelka–Munk theory. II. A study of ink penetration in ink-jet printing,” J. Opt. Soc. Am. A 21, 1942–1952 (2004). [CrossRef]
  25. W. Wendlandt, H. Hecht, Reflectance Spectroscopy (Wiley Interscience, 1966), Chap. 3.
  26. L. Yang, S. J. Miklavcic, “A theory of light propagation incorporating scattering and absorption in turbid media,” Opt. Lett. 30, 792–794 (2005). [CrossRef] [PubMed]
  27. P. Emmel, “Modèles de Prédiction Couleur Appliqués à L’impression Jet D’encre,” Thèse No. 1857 (École Polytechnique Fédérale de Lausanne, Switzerland, 1998).
  28. L. Yang, B. Kruse, “Ink penetration and its effect on printing,” Proc. SPIE 3963, 365–375 (2000). [CrossRef]
  29. L. Yang, “Characterization of inks and ink application for inkjet printing: Model and simulation,” J. Opt. Soc. Am. A 20, 1149–1154 (2003). [CrossRef]
  30. N. Pauler, private communication. Similar experimental observations may be found in Ref. [14].
  31. T. Shakespear, J. Shakespear, “A fluorescent extension to the Kubelka–Munk model,” Color Res. Appl. 28, 4–14 (2003), Fig. 4. [CrossRef]

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