Abstract

The revised Kubelka–Munk theory is examined theoretically and experimentally. Systems of dyed paper sheets are simulated, and the results are compared with other models. The results show that the revised Kubelka–Munk model yields significant errors in predicted dye-paper mixture reflectances, and is not self-consistent. The absorption is noticeably overestimated. Theoretical arguments show that properties in the revised Kubelka–Munk theory are inadequately derived. The main conclusion is that the revised Kubelka–Munk theory is wrong in the inclusion of the so-called scattering-induced-path-variation factor. Consequently, the theory should not be used for light scattering calculations. Instead, the original Kubelka–Munk theory should be used where its accuracy is sufficient, and a radiative transfer tool of higher resolution should be used where higher accuracy is needed.

© 2007 Optical Society of America

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  1. P. Kubelka and F. Munk, "Ein beitrag zur optik der farbanstriche," Z. Tech. Phys. (Leipzig) 11a, 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. P. Kubelka, "New contributions to the optics of intensely light-scattering materials. Part II," J. Opt. Soc. Am. 44, 330-335 (1954).
    [CrossRef]
  4. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  5. P. S. Mudgett and L. W. Richards, "Multiple scattering calculations for technology," Appl. Opt. 10, 1485-1502 (1971).
    [CrossRef] [PubMed]
  6. P. S. Mudgett and L. W. Richards, "Multiple scattering calculations for technology II," J. Colloid Interface Sci. 39, 551-567 (1972).
    [CrossRef]
  7. K. Stamnes, S.-C. Tsay, and I. Laszlo, "DISORT, a general-purpose Fortran program for discrete-ordinate-method radiative transfer in scattering and emitting layered media" (Goddard Space Flight Center, NASA, 2000).
  8. P. Edström, "A fast and stable solution method for the radiative transfer problem," SIAM Rev. 47, 447-468 (2005).
    [CrossRef]
  9. J. A. van den Akker, "Scattering and absorption of light in paper and other diffusing media," Tappi J. 32, 498-501 (1949).
  10. W. J. Foote, "An investigation of the fundamental scattering and absorption coefficients of dyed handsheets," Pap. Trade J. 109, 397-404 (1939).
  11. 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, 1996), Vol. 2, pp. 909-927.
  12. S. Moldenius, "Light absorption coefficient spectra of hydrogen peroxide bleached mechanical pulp," Pap. Puu 65, 747-756 (1983).
  13. 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).
  14. 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, 1968), pp. 27-46.
  15. 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).
  16. 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, F. Bolam, ed. (Technical Section, British Paper and Board Makers" Association, 1966), Vol. 2, pp. 948-950.
  17. 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).
  18. J. H. Nobbs, "Kubelka-Munk theory and the prediction of reflectance," Rev. Prog. Color. Relat. Top. 15, 66-75 (1985).
    [CrossRef]
  19. B. Philips-Invernizzi, D. Dupont, and C. Caz, "Bibliographical review for reflectance of diffusing media," Opt. Express 40, 1082-1092 (2001).
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    [CrossRef]
  21. L. Yang, B. Kruse, and S. J. Miklavcic, "Revised Kubelka-Munk theory. II. Unified framework for homogenous and inhomogenous optical media," J. Opt. Soc. Am. A 21, 1942-1952 (2004).
    [CrossRef]
  22. L. Yang and S. J. Miklavcic, "A theory of light propagation incorporating scattering and absorption in turbid media," Opt. Lett. 30, 792-794 (2005).
    [CrossRef] [PubMed]
  23. L. Yang and S. 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).
    [CrossRef]
  24. 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]
  25. G. H. Goedecke, "Radiative transfer in closely packed media," J. Opt. Soc. Am. 67, 1339-1348 (1977).
    [CrossRef]
  26. L. Coppel, H. Granberg, and M.-C. Béland, "Spectral reflectance prediction of fulltone offset printed LWC papers," in Proceedings of the 12th International Printing and Graphic Arts Conference (PAPTAC, 2004), pp. 235-239.
  27. L. Yang, "Characterization of inks and ink application for inkjet printing: model and simulation," J. Opt. Soc. Am. A 20, 1149-1154 (2003).
    [CrossRef]
  28. "ISO 2469: paper, board and pulps--measurement of diffuse reflectance factor" (International Organization for Standardization, Geneva, 1994).

2005 (3)

2004 (3)

2003 (2)

L. Yang, "Characterization of inks and ink application for inkjet printing: model and simulation," J. Opt. Soc. Am. A 20, 1149-1154 (2003).
[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).

2001 (1)

B. Philips-Invernizzi, D. Dupont, and C. Caz, "Bibliographical review for reflectance of diffusing media," Opt. Express 40, 1082-1092 (2001).

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).

1985 (1)

J. H. Nobbs, "Kubelka-Munk theory and the prediction of reflectance," Rev. Prog. Color. Relat. Top. 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).

1977 (1)

1972 (1)

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

1971 (1)

1954 (1)

1949 (1)

J. A. van den Akker, "Scattering and absorption of light in paper and other diffusing media," Tappi J. 32, 498-501 (1949).

1948 (1)

1939 (1)

W. J. Foote, "An investigation of the fundamental scattering and absorption coefficients of dyed handsheets," Pap. Trade J. 109, 397-404 (1939).

1931 (1)

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

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, 1996), Vol. 2, pp. 909-927.

Béland, M.-C.

L. Coppel, H. Granberg, and M.-C. Béland, "Spectral reflectance prediction of fulltone offset printed LWC papers," in Proceedings of the 12th International Printing and Graphic Arts Conference (PAPTAC, 2004), pp. 235-239.

Bolam, F.

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, F. Bolam, ed. (Technical Section, British Paper and Board Makers" Association, 1966), Vol. 2, pp. 948-950.

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).

Caz, C.

B. Philips-Invernizzi, D. Dupont, and C. Caz, "Bibliographical review for reflectance of diffusing media," Opt. Express 40, 1082-1092 (2001).

Chandrasekhar, S.

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

Coppel, L.

L. Coppel, H. Granberg, and M.-C. Béland, "Spectral reflectance prediction of fulltone offset printed LWC papers," in Proceedings of the 12th International Printing and Graphic Arts Conference (PAPTAC, 2004), pp. 235-239.

Dupont, D.

B. Philips-Invernizzi, D. Dupont, and C. Caz, "Bibliographical review for reflectance of diffusing media," Opt. Express 40, 1082-1092 (2001).

Edström, P.

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, 397-404 (1939).

Goedecke, G. H.

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).

L. Coppel, H. Granberg, and M.-C. Béland, "Spectral reflectance prediction of fulltone offset printed LWC papers," in Proceedings of the 12th International Printing and Graphic Arts Conference (PAPTAC, 2004), pp. 235-239.

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).

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).

Kruse, B.

Kubelka, P

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

Kubelka, P.

Laszlo, I.

K. Stamnes, S.-C. Tsay, and I. Laszlo, "DISORT, a general-purpose Fortran program for discrete-ordinate-method radiative transfer in scattering and emitting layered media" (Goddard Space Flight Center, NASA, 2000).

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, 1996), Vol. 2, pp. 909-927.

Miklavcic, S. J.

Moldenius, S.

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

Mudgett, P. S.

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

P. S. 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).

Nobbs, J. H.

J. H. Nobbs, "Kubelka-Munk theory and the prediction of reflectance," Rev. Prog. Color. Relat. Top. 15, 66-75 (1985).
[CrossRef]

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, 1996), Vol. 2, pp. 909-927.

Philips-Invernizzi, B.

B. Philips-Invernizzi, D. Dupont, and C. Caz, "Bibliographical review for reflectance of diffusing media," Opt. Express 40, 1082-1092 (2001).

Richards, L. W.

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

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

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).

Stamnes, K.

K. Stamnes, S.-C. Tsay, and I. Laszlo, "DISORT, a general-purpose Fortran program for discrete-ordinate-method radiative transfer in scattering and emitting layered media" (Goddard Space Flight Center, NASA, 2000).

Tsay, S.-C.

K. Stamnes, S.-C. Tsay, and I. Laszlo, "DISORT, a general-purpose Fortran program for discrete-ordinate-method radiative transfer in scattering and emitting layered media" (Goddard Space Flight Center, NASA, 2000).

van den Akker, J. A.

J. A. van den Akker, "Scattering and absorption of light in paper and other diffusing media," Tappi J. 32, 498-501 (1949).

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, 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, F. Bolam, ed. (Technical Section, British Paper and Board Makers" Association, 1966), Vol. 2, pp. 948-950.

Wendlandt, W. W.

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, 1968), pp. 27-46.

Yang, L

Yang, L.

Appl. Opt. (1)

J. Colloid Interface Sci. (1)

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

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

J. Pulp Pap. Sci. (3)

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).

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).

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]

Opt. Express (1)

B. Philips-Invernizzi, D. Dupont, and C. Caz, "Bibliographical review for reflectance of diffusing media," Opt. Express 40, 1082-1092 (2001).

Opt. Lett. (1)

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, 397-404 (1939).

Rev. Prog. Color. Relat. Top. (1)

J. H. Nobbs, "Kubelka-Munk theory and the prediction of reflectance," Rev. Prog. Color. Relat. Top. 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 J. (1)

J. A. van den Akker, "Scattering and absorption of light in paper and other diffusing media," Tappi J. 32, 498-501 (1949).

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 (7)

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, 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, F. Bolam, ed. (Technical Section, British Paper and Board Makers" Association, 1966), Vol. 2, 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, 1996), Vol. 2, pp. 909-927.

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

K. Stamnes, S.-C. Tsay, and I. Laszlo, "DISORT, a general-purpose Fortran program for discrete-ordinate-method radiative transfer in scattering and emitting layered media" (Goddard Space Flight Center, NASA, 2000).

"ISO 2469: paper, board and pulps--measurement of diffuse reflectance factor" (International Organization for Standardization, Geneva, 1994).

L. Coppel, H. Granberg, and M.-C. Béland, "Spectral reflectance prediction of fulltone offset printed LWC papers," in Proceedings of the 12th International Printing and Graphic Arts Conference (PAPTAC, 2004), pp. 235-239.

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Figures (10)

Fig. 1
Fig. 1

Scattered photon path used in Rev KM to obtain the SIPV factor.

Fig. 2
Fig. 2

Longer path through a layer according to Rev KM.

Fig. 3
Fig. 3

In the calculation of the arc length of a curve, small line segments are approximated with straight lines. As the line segments are made smaller, the straight lines get closer to the curve, and in the limit the quotient between a true segment length and its straight-line approximation tends to unity (which is what must also happen to the SIPV factor μ).

Fig. 4
Fig. 4

Dye–paper mixture reflectances for (a) the Monte Carlo reference values, (b) Rev KM, (c) original KM, and (d) the DORT models. The paper grammage was 40 g m 2 , and the dye grammages were [ 0 , 0.005 , 0.01 , 0.02 , 0.05 , 0.1 , 0.2 ] g m 2 . The DORT models give results nearly identical to the reference, KM almost as well except for slight deviations in the absorption band of the dye, while Rev KM yields significant errors with clearly overestimated absorption.

Fig. 5
Fig. 5

Rev KM s and k dye–paper mixture parameters (a) as predicted from additivity, and (b) as calculated from dye–paper mixture reflectances. The paper grammage was 40 g m 2 , and the dye grammages were [ 0 , 0.005 , 0.01 , 0.02 , 0.05 , 0.1 , 0.2 ] g m 2 . Note the parameter dependencies (decrease in s with increased k) for predicted values. The model is clearly not self-consistent, as (a) and (b) are not at all similar neither in s nor in k. The statistical noise inherent in the Monte Carlo process is visible in the last pane, but that does not affect the conclusion.

Fig. 6
Fig. 6

Rev KM intrinsic σ s and σ a dye–paper mixture parameters (a) as predicted from additivity, and (b) as calculated from dye–paper mixture reflectances. The model is clearly not self-consistent, as (a) and (b) are not at all similar in σ a . (See the caption for Fig. 5 for grammages and comments on noise.)

Fig. 7
Fig. 7

Original KM s and k dye–paper mixture parameters (a) as predicted from additivity, and (b) as calculated from dye–paper mixture reflectances. The model is fairly self-consistent, as (a) and (b) are rather similar, but there are some deviations in the absorption band of the dye. Note that no parameter dependencies are present. (See the caption for Fig. 5 for grammages and comments on noise.)

Fig. 8
Fig. 8

DORT intrinsic σ s and σ a dye–paper mixture parameters (a) as predicted from additivity, and (b) as calculated from dye–paper mixture reflectances. The models are self-consistent, as (a) and (b) are very similar. (See the caption for Fig. 5 for grammages and comments on noise.)

Fig. 9
Fig. 9

Original KM s and k dye–paper mixture parameters as calculated from the dye–paper mixture reflectances with a higher dye amount. The paper grammage was still 40 g m 2 , but the dye grammages were increased to [ 0 , 0.02 , 0.1 , 0.2 , 1.0 , 1.5 , 2.0 ] g m 2 . Note that parameter dependencies (decrease in s with increased k) are now present.

Fig. 10
Fig. 10

Measured reflectances (curves) for handsheets with different amounts of dye, and Monte Carlo predictions (crosses). The good predictions confirm the relevance of the theoretical experiment.

Equations (10)

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

d i = ( s + k ) i d x + s j d x ,
d j = ( s + k ) j d x + s i d x
s = σ s ,
k = 2 σ a ,
s = 3 σ s 4 ,
k = 2 σ a ,
μ = α σ s D ,
μ = [ σ s 2 ( σ a 2 + σ a σ s ) ] 1 4 .
s = μ α σ s 2 ,
k = μ α σ a .

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