Abstract

It is shown that the intensity of light reflected from plane-parallel turbid media is anisotropic in all situations encountered in practice. The anisotropy, in the form of higher intensity at large polar angles, increases when the amount of near-surface bulk scattering is increased, which dominates in optically thin and highly absorbing media. The only situation with isotropic intensity is when a non-absorbing infinitely thick medium is illuminated diffusely. This is the only case where the Kubelka–Munk model gives exact results and there exists an exact translation between Kubelka–Munk and general radiative transfer. This also means that a bulk scattering perfect diffusor does not exist. Angle-resolved models are thus crucial for a correct understanding of light scattering in turbid media. The results are derived using simulations and analytical calculations. It is also shown that there exists an optimal angle for directional detection that minimizes the error introduced when using the Kubelka–Munk model to interpret reflectance measurements with diffuse illumination.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J.  21, 1–22 (1905).
    [CrossRef]
  2. P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig)  11a, 593–601 (1931).
  3. P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part I,” J. Opt. Soc. Am.  38, 330–335 (1948).
  4. P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part II,” J. Opt. Soc. Am.  44, 448–457 (1954).
    [CrossRef]
  5. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  6. P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev.  47, 447–468 (2005).
    [CrossRef]
  7. J. H. Nobbs, “Kubelka-Munk theory and the prediction of reflectance,” Rev. Prog. Coloration  15, 66–75 (1985).
    [CrossRef]
  8. M. J. Leskelä, “Optical calculations for multilayer papers,” Tappi J.  78, 167–172 (1995).
  9. P. Latimer and S. J. Noh, “Light propagation in moderately dense particle systems: a reexamination of the Kubelka-Munk theory,” Appl. Opt.  26, 514–523 (1987).
    [CrossRef] [PubMed]
  10. S. H. Kong and J. D. Shore, “Evaluation of the telegrapher’s equation and multiple-flux theories for calculating the transmittance and reflectance of a diffuse absorbing slab,” J. Opt. Soc. Am. A  24, 702–710 (2007).
    [CrossRef]
  11. J. A. van den Akker, “Scattering and absorption of light in paper and other diffusing media,” TAPPI  32, 498–501 (1949).
  12. W. J. Foote, “An investigation of the fundamental scattering and absorption coefficients of dyed handsheets,” Pap. Trade J.  109, 333–340 (1939).
  13. L. Nordman, P. Aaltonen, and T. Makkonen, “Relationship between mechanical and optical properties of paper affected by web consolidation,” in Transactions of the Symposium on the Consolidation of the Paper Web, F.Bolam, ed. (Technical Section, British Paper and Board Makers' Association, 1966), Vol.  2, pp. 909–927.
  14. S. Moldenius, “Light absorption coefficient spectra of hydrogen peroxide bleached mechanical pulp,” Pap. Puu  65, 747–756 (1983).
  15. M. Rundlöf and 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, 220–223 (1997).
  16. J. A. van den Akker, “Theory of some of the discrepancies observed in application of the Kubelka-Munk equations to particulate systems,” in Modern Aspects of Reflectance Spectroscopy, W.W.Wendlandt, ed. (Plenum Press, 1968), pp. 27–46.
  17. A. A. Koukoulas and B. D. Jordan, “Effect of strong absorption on the Kubelka-Munk scattering coefficient,” J. Pulp Pap. Sci.  23, 224–232 (1997).
  18. J. A. van den Akker, “Discussion on ‘Relationships between mechanical and optical properties of paper affected by web consolidation’,” in Transactions of the Symposium on the Consolidation of the Paper Web,  Vol. 2, F.Bolam, ed. (Technical Section, British Board Makers' Association, 1966), pp. 948–950.
  19. H. Granberg and P. Edström, “Quantification of the intrinsic error of the Kubelka-Munk model caused by strong light absorption,” J. Pulp Pap. Sci.  29, 386–390 (2003).
  20. P. Edström, “Comparison of the DORT2002 radiative transfer solution method and the Kubelka-Munk model,” Nord. Pulp Pap. Res. J.  19, 397–403 (2004).
    [CrossRef]
  21. M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. II. Measurements,” J. Opt. Soc. Am. A  27, 1040–1045 (2010).
    [CrossRef]
  22. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Lamperis, “Geometrical Considerations and Nomenclature for Reflectance,” (National Bureau of Standards, 1977).
  23. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  24. M. Choulli and P. Stefanov, “An inverse boundary value problem for the stationary transport equation,” Osaka J. Math.  36, 87–104 (1998).
  25. P. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt.  10, 1485–1502 (1971).
    [CrossRef] [PubMed]
  26. P. Mudgett and L. W. Richards, “Multiple scattering calculations for technology II,” J. Colloid Interface Sci.  39, 551–567 (1972).
    [CrossRef]
  27. H. C. van de Hulst, Multiple Light Scattering. Tables, Formulas and Applications,  Vol. 2 (Academic, 1980).
  28. W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol.  33, 437–454 (1998).
    [CrossRef]
  29. S. N. Thennadil, “Relationship between the Kubelka-Munk scattering and radiative transfer coefficients,” J. Opt. Soc. Am. A  25, 1480–1485 (2008).
    [CrossRef]
  30. A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka-Munk theory,” J. Phys. D: Appl. Phys.  20, 2210–2216 (2007).
    [CrossRef]
  31. A. Kokhanovsky and I. Hopkinson, “Some analytical approximations to radiative transfer theory and their application for the analysis of reflectance data,” J. Opt. A, Pure Appl. Opt.  10, 1–8 (2008).
    [CrossRef]
  32. ISO 2469: Paper, Board and Pulps—Measurement of Diffuse Reflectance Factor (International Organization for Standardization, 1994).

2010 (1)

2008 (2)

S. N. Thennadil, “Relationship between the Kubelka-Munk scattering and radiative transfer coefficients,” J. Opt. Soc. Am. A  25, 1480–1485 (2008).
[CrossRef]

A. Kokhanovsky and I. Hopkinson, “Some analytical approximations to radiative transfer theory and their application for the analysis of reflectance data,” J. Opt. A, Pure Appl. Opt.  10, 1–8 (2008).
[CrossRef]

2007 (2)

2005 (1)

P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev.  47, 447–468 (2005).
[CrossRef]

2004 (1)

P. Edström, “Comparison of the DORT2002 radiative transfer solution method and the Kubelka-Munk model,” Nord. Pulp Pap. Res. J.  19, 397–403 (2004).
[CrossRef]

2003 (1)

H. Granberg and P. Edström, “Quantification of the intrinsic error of the Kubelka-Munk model caused by strong light absorption,” J. Pulp Pap. Sci.  29, 386–390 (2003).

1998 (2)

M. Choulli and P. Stefanov, “An inverse boundary value problem for the stationary transport equation,” Osaka J. Math.  36, 87–104 (1998).

W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol.  33, 437–454 (1998).
[CrossRef]

1997 (2)

M. Rundlöf and 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, 220–223 (1997).

A. A. Koukoulas and B. D. Jordan, “Effect of strong absorption on the Kubelka-Munk scattering coefficient,” J. Pulp Pap. Sci.  23, 224–232 (1997).

1995 (1)

M. J. Leskelä, “Optical calculations for multilayer papers,” Tappi J.  78, 167–172 (1995).

1994 (1)

ISO 2469: Paper, Board and Pulps—Measurement of Diffuse Reflectance Factor (International Organization for Standardization, 1994).

1987 (1)

1985 (1)

J. H. Nobbs, “Kubelka-Munk theory and the prediction of reflectance,” Rev. Prog. Coloration  15, 66–75 (1985).
[CrossRef]

1983 (1)

S. Moldenius, “Light absorption coefficient spectra of hydrogen peroxide bleached mechanical pulp,” Pap. Puu  65, 747–756 (1983).

1980 (1)

H. C. van de Hulst, Multiple Light Scattering. Tables, Formulas and Applications,  Vol. 2 (Academic, 1980).

1977 (1)

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Lamperis, “Geometrical Considerations and Nomenclature for Reflectance,” (National Bureau of Standards, 1977).

1972 (1)

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

1971 (1)

1968 (1)

J. A. van den Akker, “Theory of some of the discrepancies observed in application of the Kubelka-Munk equations to particulate systems,” in Modern Aspects of Reflectance Spectroscopy, W.W.Wendlandt, ed. (Plenum Press, 1968), pp. 27–46.

1967 (1)

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

1966 (2)

J. A. van den Akker, “Discussion on ‘Relationships between mechanical and optical properties of paper affected by web consolidation’,” in Transactions of the Symposium on the Consolidation of the Paper Web,  Vol. 2, F.Bolam, ed. (Technical Section, British Board Makers' Association, 1966), pp. 948–950.

L. Nordman, P. Aaltonen, and T. Makkonen, “Relationship between mechanical and optical properties of paper affected by web consolidation,” in Transactions of the Symposium on the Consolidation of the Paper Web, F.Bolam, ed. (Technical Section, British Paper and Board Makers' Association, 1966), Vol.  2, pp. 909–927.

1960 (1)

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

1954 (1)

P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part II,” J. Opt. Soc. Am.  44, 448–457 (1954).
[CrossRef]

1949 (1)

J. A. van den Akker, “Scattering and absorption of light in paper and other diffusing media,” TAPPI  32, 498–501 (1949).

1948 (1)

P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part I,” J. Opt. Soc. Am.  38, 330–335 (1948).

1939 (1)

W. J. Foote, “An investigation of the fundamental scattering and absorption coefficients of dyed handsheets,” Pap. Trade J.  109, 333–340 (1939).

1931 (1)

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig)  11a, 593–601 (1931).

1905 (1)

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

Aaltonen, P.

L. Nordman, P. Aaltonen, and T. Makkonen, “Relationship between mechanical and optical properties of paper affected by web consolidation,” in Transactions of the Symposium on the Consolidation of the Paper Web, F.Bolam, ed. (Technical Section, British Paper and Board Makers' Association, 1966), Vol.  2, pp. 909–927.

Bristow, J. A.

M. Rundlöf and 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, 220–223 (1997).

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Choulli, M.

M. Choulli and P. Stefanov, “An inverse boundary value problem for the stationary transport equation,” Osaka J. Math.  36, 87–104 (1998).

Edström, P.

M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. II. Measurements,” J. Opt. Soc. Am. A  27, 1040–1045 (2010).
[CrossRef]

P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev.  47, 447–468 (2005).
[CrossRef]

P. Edström, “Comparison of the DORT2002 radiative transfer solution method and the Kubelka-Munk model,” Nord. Pulp Pap. Res. J.  19, 397–403 (2004).
[CrossRef]

H. Granberg and P. Edström, “Quantification of the intrinsic error of the Kubelka-Munk model caused by strong light absorption,” J. Pulp Pap. Sci.  29, 386–390 (2003).

Foote, W. J.

W. J. Foote, “An investigation of the fundamental scattering and absorption coefficients of dyed handsheets,” Pap. Trade J.  109, 333–340 (1939).

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Lamperis, “Geometrical Considerations and Nomenclature for Reflectance,” (National Bureau of Standards, 1977).

Granberg, H.

H. Granberg and P. Edström, “Quantification of the intrinsic error of the Kubelka-Munk model caused by strong light absorption,” J. Pulp Pap. Sci.  29, 386–390 (2003).

Hopkinson, I.

A. Kokhanovsky and I. Hopkinson, “Some analytical approximations to radiative transfer theory and their application for the analysis of reflectance data,” J. Opt. A, Pure Appl. Opt.  10, 1–8 (2008).
[CrossRef]

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Lamperis, “Geometrical Considerations and Nomenclature for Reflectance,” (National Bureau of Standards, 1977).

Jordan, B. D.

A. A. Koukoulas and B. D. Jordan, “Effect of strong absorption on the Kubelka-Munk scattering coefficient,” J. Pulp Pap. Sci.  23, 224–232 (1997).

Kokhanovsky, A.

A. Kokhanovsky and I. Hopkinson, “Some analytical approximations to radiative transfer theory and their application for the analysis of reflectance data,” J. Opt. A, Pure Appl. Opt.  10, 1–8 (2008).
[CrossRef]

Kokhanovsky, A. A.

A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka-Munk theory,” J. Phys. D: Appl. Phys.  20, 2210–2216 (2007).
[CrossRef]

Kong, S. H.

Koukoulas, A. A.

A. A. Koukoulas and B. D. Jordan, “Effect of strong absorption on the Kubelka-Munk scattering coefficient,” J. Pulp Pap. Sci.  23, 224–232 (1997).

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part II,” J. Opt. Soc. Am.  44, 448–457 (1954).
[CrossRef]

P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part I,” J. Opt. Soc. Am.  38, 330–335 (1948).

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig)  11a, 593–601 (1931).

Lamperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Lamperis, “Geometrical Considerations and Nomenclature for Reflectance,” (National Bureau of Standards, 1977).

Latimer, P.

Leskelä, M. J.

M. J. Leskelä, “Optical calculations for multilayer papers,” Tappi J.  78, 167–172 (1995).

Makkonen, T.

L. Nordman, P. Aaltonen, and T. Makkonen, “Relationship between mechanical and optical properties of paper affected by web consolidation,” in Transactions of the Symposium on the Consolidation of the Paper Web, F.Bolam, ed. (Technical Section, British Paper and Board Makers' Association, 1966), Vol.  2, pp. 909–927.

Marijnissen, J. P. A.

W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol.  33, 437–454 (1998).
[CrossRef]

Moldenius, S.

S. Moldenius, “Light absorption coefficient spectra of hydrogen peroxide bleached mechanical pulp,” Pap. Puu  65, 747–756 (1983).

Mudgett, P.

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

P. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt.  10, 1485–1502 (1971).
[CrossRef] [PubMed]

Munk, F.

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig)  11a, 593–601 (1931).

Neuman, M.

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Lamperis, “Geometrical Considerations and Nomenclature for Reflectance,” (National Bureau of Standards, 1977).

Nobbs, J. H.

J. H. Nobbs, “Kubelka-Munk theory and the prediction of reflectance,” Rev. Prog. Coloration  15, 66–75 (1985).
[CrossRef]

Noh, S. J.

Nordman, L.

L. Nordman, P. Aaltonen, and T. Makkonen, “Relationship between mechanical and optical properties of paper affected by web consolidation,” in Transactions of the Symposium on the Consolidation of the Paper Web, F.Bolam, ed. (Technical Section, British Paper and Board Makers' Association, 1966), Vol.  2, pp. 909–927.

Richards, L. W.

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

P. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt.  10, 1485–1502 (1971).
[CrossRef] [PubMed]

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Lamperis, “Geometrical Considerations and Nomenclature for Reflectance,” (National Bureau of Standards, 1977).

Rundlöf, M.

M. Rundlöf and 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, 220–223 (1997).

Schuster, A.

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

Shore, J. D.

Star, W. M.

W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol.  33, 437–454 (1998).
[CrossRef]

Stefanov, P.

M. Choulli and P. Stefanov, “An inverse boundary value problem for the stationary transport equation,” Osaka J. Math.  36, 87–104 (1998).

Thennadil, S. N.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering. Tables, Formulas and Applications,  Vol. 2 (Academic, 1980).

van den Akker, J. A.

J. A. van den Akker, “Theory of some of the discrepancies observed in application of the Kubelka-Munk equations to particulate systems,” in Modern Aspects of Reflectance Spectroscopy, W.W.Wendlandt, ed. (Plenum Press, 1968), pp. 27–46.

J. A. van den Akker, “Discussion on ‘Relationships between mechanical and optical properties of paper affected by web consolidation’,” in Transactions of the Symposium on the Consolidation of the Paper Web,  Vol. 2, F.Bolam, ed. (Technical Section, British Board Makers' Association, 1966), pp. 948–950.

J. A. van den Akker, “Scattering and absorption of light in paper and other diffusing media,” TAPPI  32, 498–501 (1949).

van Gemert, M. J. C.

W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol.  33, 437–454 (1998).
[CrossRef]

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Appl. Opt. (2)

Astrophys. J. (1)

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

J. Colloid Interface Sci. (1)

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

J. Opt. A, Pure Appl. Opt. (1)

A. Kokhanovsky and I. Hopkinson, “Some analytical approximations to radiative transfer theory and their application for the analysis of reflectance data,” J. Opt. A, Pure Appl. Opt.  10, 1–8 (2008).
[CrossRef]

J. Opt. Soc. Am. (2)

P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part I,” J. Opt. Soc. Am.  38, 330–335 (1948).

P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part II,” J. Opt. Soc. Am.  44, 448–457 (1954).
[CrossRef]

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

J. Phys. D: Appl. Phys. (1)

A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka-Munk theory,” J. Phys. D: Appl. Phys.  20, 2210–2216 (2007).
[CrossRef]

J. Pulp Pap. Sci. (3)

H. Granberg and P. Edström, “Quantification of the intrinsic error of the Kubelka-Munk model caused by strong light absorption,” J. Pulp Pap. Sci.  29, 386–390 (2003).

M. Rundlöf and 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, 220–223 (1997).

A. A. Koukoulas and B. D. Jordan, “Effect of strong absorption on the Kubelka-Munk scattering coefficient,” J. Pulp Pap. Sci.  23, 224–232 (1997).

Nord. Pulp Pap. Res. J. (1)

P. Edström, “Comparison of the DORT2002 radiative transfer solution method and the Kubelka-Munk model,” Nord. Pulp Pap. Res. J.  19, 397–403 (2004).
[CrossRef]

Osaka J. Math. (1)

M. Choulli and P. Stefanov, “An inverse boundary value problem for the stationary transport equation,” Osaka J. Math.  36, 87–104 (1998).

Pap. Puu (1)

S. Moldenius, “Light absorption coefficient spectra of hydrogen peroxide bleached mechanical pulp,” Pap. Puu  65, 747–756 (1983).

Pap. Trade J. (1)

W. J. Foote, “An investigation of the fundamental scattering and absorption coefficients of dyed handsheets,” Pap. Trade J.  109, 333–340 (1939).

Phys. Med. Biol. (1)

W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol.  33, 437–454 (1998).
[CrossRef]

Rev. Prog. Coloration (1)

J. H. Nobbs, “Kubelka-Munk theory and the prediction of reflectance,” Rev. Prog. Coloration  15, 66–75 (1985).
[CrossRef]

SIAM Rev. (1)

P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev.  47, 447–468 (2005).
[CrossRef]

TAPPI (1)

J. A. van den Akker, “Scattering and absorption of light in paper and other diffusing media,” TAPPI  32, 498–501 (1949).

Tappi J. (1)

M. J. Leskelä, “Optical calculations for multilayer papers,” Tappi J.  78, 167–172 (1995).

Z. Tech. Phys. (Leipzig) (1)

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig)  11a, 593–601 (1931).

Other (8)

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

L. Nordman, P. Aaltonen, and T. Makkonen, “Relationship between mechanical and optical properties of paper affected by web consolidation,” in Transactions of the Symposium on the Consolidation of the Paper Web, F.Bolam, ed. (Technical Section, British Paper and Board Makers' Association, 1966), Vol.  2, pp. 909–927.

J. A. van den Akker, “Discussion on ‘Relationships between mechanical and optical properties of paper affected by web consolidation’,” in Transactions of the Symposium on the Consolidation of the Paper Web,  Vol. 2, F.Bolam, ed. (Technical Section, British Board Makers' Association, 1966), pp. 948–950.

J. A. van den Akker, “Theory of some of the discrepancies observed in application of the Kubelka-Munk equations to particulate systems,” in Modern Aspects of Reflectance Spectroscopy, W.W.Wendlandt, ed. (Plenum Press, 1968), pp. 27–46.

H. C. van de Hulst, Multiple Light Scattering. Tables, Formulas and Applications,  Vol. 2 (Academic, 1980).

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Lamperis, “Geometrical Considerations and Nomenclature for Reflectance,” (National Bureau of Standards, 1977).

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

ISO 2469: Paper, Board and Pulps—Measurement of Diffuse Reflectance Factor (International Organization for Standardization, 1994).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Three-dimensional BRDF for three media. The single scattering albedo is denoted by a and the optical thickness by τ. The supposed validity conditions of the KM model are used; that is, isotropic illumination and isotropic single scattering. A shaded half sphere is included for reference. The resulting intensity is anisotropic, since more light is reflected into larger polar angles for (b) thin and (c) highly absorbing media.

Fig. 2
Fig. 2

Schematic illustration of the probability for an exit direction θ from different scattering depths. The scattering event occurs at the point marked with an asterisk and the axes at the medium interface illustrate how the probability P varies with polar angle θ. The shape of the curve depends strongly on the distance to the surface. If near-surface scattering dominates in a medium, the relative reflectance at larger polar angles increases.

Fig. 3
Fig. 3

Schematic picture of the case treated analytically in Eq. (10). Light enters the medium (shaded gray) and scatters at the point marked with the asterisk. It then leaves the medium in the direction θ. The light must not scatter at the path marked with the dashed line.

Fig. 4
Fig. 4

Three-dimensional BRDF for an infinitely thick non-absorbing medium, with two different illuminations. A translucent half-unit sphere is included for reference. It can be seen that the diffuse illumination (a) gives a perfectly isotropic reflectance, while normally incident illumination (b) gives a lower reflectance for larger polar angles.

Fig. 5
Fig. 5

Relative error when assuming an isotropic intensity of the light reflected by a medium defined by the single scattering albedo a and optical thickness τ. The supposed validity conditions of the KM model are used, i.e., isotropic single scattering and diffuse illumination. The detection polar angle of I r chosen to represent the intensity in all directions is varied. The upper bound of the error is 20–40% in a large area and increases asymptotically for thin media, while approaching zero only for thick highly scattering media (a). When varying the polar angle of I r the characteristics of the error change (b)–(d). The smallest error is obtained when I r = I r ( 55 ° ) (d).

Fig. 6
Fig. 6

Mean relative error over the set τ [ 0.5 30 ] , a [ 0.5 0.999 ] introduced by the KM model. The supposed validity conditions of the KM model are used, i.e., isotropic single scattering and diffuse illumination. The polar angle θ of the intensity taken to represent the intensity in all directions, i.e., the angle of detection, is varied from 0° to 89°. It is seen that the KM model underestimates the reflectance when the intensity chosen is close to the normal direction, while it overestimates the reflectance when it is close to grazing angles. The error is zero when θ = θ * , and in an average sense this is thus the optimal angle when, for example, placing a detector.

Equations (22)

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

I = d E cos θ d a d ω d t .
d I = σ e I d s ,
σ e = σ a + σ s ,
a = σ s σ a + σ s
τ = σ e s .
S = a 4 π 4 π p ( cos Θ ) I d ω ,
d I d s = σ e ( I + S ) .
{ d i d x = ( s KM + k KM ) i + s KM j d j d x = ( s KM + k KM ) j + s KM i , ,
d u σ e exp ( σ e x ) d x = exp ( σ e d u ) ,
x 1 = 0 x 2 = x 1 u σ e exp ( σ e x 1 ) σ e exp ( σ e x 2 ) d x 1 d x 2 = 1 1 + 1 u .
I ( τ ) = I 0 exp [ ( 1 + a ) τ ] ,
d I d s = ( σ s + σ a ) I + σ s 4 π 4 π I d ω .
{ d I + d s = ( σ s + σ a ) I + + σ s 4 π 4 π ( I + + I ) d ω d I d s = ( σ s + σ a ) I + σ s 4 π 4 π ( I + + I ) d ω . }
{ u d I + d x = ( σ s + σ a ) I + + σ s 4 π 4 π ( I + + I ) d ω u d I d x = ( σ s + σ a ) I + σ s 4 π 4 π ( I + + I ) d ω .
{ 1 0 u d I + d x d u = 1 0 [ ( σ s + σ a ) I + + σ s 4 π 4 π ( I + + I ) d ω ] d u 0 1 u d I d x d u = 0 1 [ ( σ s + σ a ) I + σ s 4 π 4 π ( I + + I ) d ω ] d u .
{ 1 2 d I + d x = ( σ s + σ a ) I + + 1 2 σ s I + + 1 2 σ s I 1 2 d I d x = ( σ s + σ a ) I + 1 2 σ s I + + 1 2 σ s I .
{ d I + d x = ( σ s + 2 σ a ) I + + σ s I d I d x = ( σ s + 2 σ a ) I + σ s I + .
R ( ψ ) = ψ I r cos θ d ω ψ I r , d cos θ d ω .
R T = 2 π I r cos θ d ω .
R T , KM = I r π ,
I r , max = argmax I r ( | R T π I r | ) .

Metrics