Abstract

The anisotropic reflectance from turbid media predicted using the radiative transfer based DORT2002 model is experimentally verified through goniophotometric measurements. A set of paper samples with varying amounts of dye and thickness is prepared, and their angle resolved reflectance is measured. An alleged perfect diffusor is also included. The corresponding simulations are performed. A complete agreement between the measurements and model predictions is seen regarding the characteristics of the anisotropy. They show that relatively more light is reflected at large polar angles when the absorption or illumination angle is increased or when the medium thickness is decreased. This is due to the relative amount of near-surface bulk scattering increasing in these cases. This affects the application of the Kubelka–Munk model as well as standards for reflectance measurements and calibration routines.

© 2010 Optical Society of America

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References

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  1. A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J.  21, 1–22 (1905).
    [CrossRef]
  2. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  3. P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Z. Tech. Phys.  11a, 593–601 (1931).
  4. P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part I,” J. Opt. Soc. Am.  38, 330–335 (1948).
  5. P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part II,” J. Opt. Soc. Am.  44, 448–457 (1954).
    [CrossRef]
  6. 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).
  7. P. Edström, “A fast and stable solution method for the radiative transfer problem,” SIAM Rev.  47, 447–468 (2005).
    [CrossRef]
  8. J. H. Nobbs, “Kubelka-Munk theory and the prediction of reflectance,” Rev. Prog. Coloration  15, 66–75 (1985).
    [CrossRef]
  9. M. J. Leskelä, “Optical calculations for multilayer papers,” TAPPI  78, 167–172 (1995).
  10. 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]
  11. 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]
  12. M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. I. Theory,” J. Opt. Soc. Am. A  27, 1032–1039 (2010).
    [CrossRef]
  13. J. A. van den Akker, “Scattering and absorption of light in paper and other diffusing media,” TAPPI  32, 498–501 (1949).
  14. ISO 11664-1:2008(E)/CIE S 014-1/E:2006: Colorimetry—Part 1: CIE Standard Colorimetric Observers (Commission Internationale de l’Eclairage, 2008).
  15. 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).
  16. P. Edström, “A two-phase parameter estimation method for radiative transfer problems in paper industry applications,” J. Comput. Appl. Math.  16, 927–951 (2008).
  17. P. Edström, “Examination of the revised Kubelka-Munk theory: considerations of modeling strategies,” J. Opt. Soc. Am. A  24, 548–556 (2007).
    [CrossRef]
  18. P. Edström, “Numerical performance of stability enhancing and speed increasing steps in radiative transfer solution methods,” J. Comput. Appl. Math.  228, 104–114 (2009).
    [CrossRef]
  19. ISO 2469: Paper, Board and Pulps—Measurement of Diffuse Reflectance Factor (International Organization for Standardization, 1994).
  20. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J.  93, 70–83 (1941).
    [CrossRef]
  21. H. Granberg and M.-C. Béland, “Modelling the angle-dependent light scattering from sheets of pulp fibre fragments,” Nord. Pulp Pap. Res. J.  19, 54–359 (2004).
    [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. ISO 9416: Paper—Determination of Light Scattering and Absorption Coefficients (Using Kubelka-Munk Theory) (International Organization for Standardization, 1998).
    [PubMed]
  24. DIN 5033-4: Colorimetry; Spectrophotometric Method (Deutsches Institut Für Normung E. V., 1992).
    [PubMed]
  25. P. Edström, M. Neuman, S. Avramidis, and M. Andersson, “Geometry related inter-instrument differences in spectrophotometric measurements,” Nord. Pulp Pap. Res. J. (to be published).

2010 (1)

2009 (1)

P. Edström, “Numerical performance of stability enhancing and speed increasing steps in radiative transfer solution methods,” J. Comput. Appl. Math.  228, 104–114 (2009).
[CrossRef]

2008 (2)

ISO 11664-1:2008(E)/CIE S 014-1/E:2006: Colorimetry—Part 1: CIE Standard Colorimetric Observers (Commission Internationale de l’Eclairage, 2008).

P. Edström, “A two-phase parameter estimation method for radiative transfer problems in paper industry applications,” J. Comput. Appl. Math.  16, 927–951 (2008).

2007 (1)

2005 (1)

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

2004 (2)

H. Granberg and M.-C. Béland, “Modelling the angle-dependent light scattering from sheets of pulp fibre fragments,” Nord. Pulp Pap. Res. J.  19, 54–359 (2004).
[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]

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).

2000 (1)

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).

1998 (1)

ISO 9416: Paper—Determination of Light Scattering and Absorption Coefficients (Using Kubelka-Munk Theory) (International Organization for Standardization, 1998).
[PubMed]

1995 (1)

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

1994 (1)

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

1992 (1)

DIN 5033-4: Colorimetry; Spectrophotometric Method (Deutsches Institut Für Normung E. V., 1992).
[PubMed]

1987 (1)

1985 (1)

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

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).

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).

1941 (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J.  93, 70–83 (1941).
[CrossRef]

1931 (1)

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

1905 (1)

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

Andersson, M.

P. Edström, M. Neuman, S. Avramidis, and M. Andersson, “Geometry related inter-instrument differences in spectrophotometric measurements,” Nord. Pulp Pap. Res. J. (to be published).

Avramidis, S.

P. Edström, M. Neuman, S. Avramidis, and M. Andersson, “Geometry related inter-instrument differences in spectrophotometric measurements,” Nord. Pulp Pap. Res. J. (to be published).

Béland, M.-C.

H. Granberg and M.-C. Béland, “Modelling the angle-dependent light scattering from sheets of pulp fibre fragments,” Nord. Pulp Pap. Res. J.  19, 54–359 (2004).
[CrossRef]

Chandrasekhar, S.

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

Edström, P.

M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. I. Theory,” J. Opt. Soc. Am. A  27, 1032–1039 (2010).
[CrossRef]

P. Edström, “Numerical performance of stability enhancing and speed increasing steps in radiative transfer solution methods,” J. Comput. Appl. Math.  228, 104–114 (2009).
[CrossRef]

P. Edström, “A two-phase parameter estimation method for radiative transfer problems in paper industry applications,” J. Comput. Appl. Math.  16, 927–951 (2008).

P. Edström, “Examination of the revised Kubelka-Munk theory: considerations of modeling strategies,” J. Opt. Soc. Am. A  24, 548–556 (2007).
[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).

P. Edström, M. Neuman, S. Avramidis, and M. Andersson, “Geometry related inter-instrument differences in spectrophotometric measurements,” Nord. Pulp Pap. Res. J. (to be published).

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 M.-C. Béland, “Modelling the angle-dependent light scattering from sheets of pulp fibre fragments,” Nord. Pulp Pap. Res. J.  19, 54–359 (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).

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J.  93, 70–83 (1941).
[CrossRef]

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J.  93, 70–83 (1941).
[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).

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.  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).

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).

Latimer, P.

Leskelä, M. J.

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

Munk, F.

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

Neuman, M.

M. Neuman and P. Edström, “Anisotropic reflectance from turbid media. I. Theory,” J. Opt. Soc. Am. A  27, 1032–1039 (2010).
[CrossRef]

P. Edström, M. Neuman, S. Avramidis, and M. Andersson, “Geometry related inter-instrument differences in spectrophotometric measurements,” Nord. Pulp Pap. Res. J. (to be published).

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.

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).

Schuster, A.

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

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  32, 498–501 (1949).

Appl. Opt. (1)

Astrophys. J. (2)

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

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J.  93, 70–83 (1941).
[CrossRef]

J. Comput. Appl. Math. (2)

P. Edström, “A two-phase parameter estimation method for radiative transfer problems in paper industry applications,” J. Comput. Appl. Math.  16, 927–951 (2008).

P. Edström, “Numerical performance of stability enhancing and speed increasing steps in radiative transfer solution methods,” J. Comput. Appl. Math.  228, 104–114 (2009).
[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 (2)

J. Pulp Pap. Sci. (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).

Nord. Pulp Pap. Res. J. (2)

H. Granberg and M.-C. Béland, “Modelling the angle-dependent light scattering from sheets of pulp fibre fragments,” Nord. Pulp Pap. Res. J.  19, 54–359 (2004).
[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]

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 (2)

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

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

Z. Tech. Phys. (1)

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

Other (8)

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).

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

ISO 11664-1:2008(E)/CIE S 014-1/E:2006: Colorimetry—Part 1: CIE Standard Colorimetric Observers (Commission Internationale de l’Eclairage, 2008).

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

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).

ISO 9416: Paper—Determination of Light Scattering and Absorption Coefficients (Using Kubelka-Munk Theory) (International Organization for Standardization, 1998).
[PubMed]

DIN 5033-4: Colorimetry; Spectrophotometric Method (Deutsches Institut Für Normung E. V., 1992).
[PubMed]

P. Edström, M. Neuman, S. Avramidis, and M. Andersson, “Geometry related inter-instrument differences in spectrophotometric measurements,” Nord. Pulp Pap. Res. J. (to be published).

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Figures (3)

Fig. 1
Fig. 1

(a) Measured and (b) predicted BRDFs for increasing absorption. Absorption is increased in four steps corresponding to increasing amounts of dye. The BRDFs are scaled to coincide at polar angle θ = 0 ° . Lambertian (perfectly diffuse) reflectance is included for reference. It can be seen that the anisotropy increases with increasing absorption in both measurements and simulations in the same characteristic way where relatively more light is reflected at large polar angles.

Fig. 2
Fig. 2

(a) Measured and (b) predicted BRDFs of a 30 g m 2 paper, a 60 g m 2 paper, and an opaque pad of paper sheets. No papers contain dye. The BRDFs are scaled to coincide at polar angle θ = 0 ° . Lambertian reflectance is included for reference. It is seen that the relative reflectance in large polar angles increases as the medium thickness decreases. Thus the anisotropy is larger in this sense for thin media.

Fig. 3
Fig. 3

Measured BRDFs of (a) a barium sulphate diffusor and predicted BRDFs of (b) a non-absorbing and non-transmitting medium with varying illumination angle. Angles 0°, 30°, 45°, and 60° are used. It is seen that the relative reflectance at large polar angles increases as the illumination angle increases. The reflectance can thus be anisotropic both from an alleged perfect diffusor (a) and an ideal bulk scattering diffusor (b). Lambertian reflectance is included for reference.

Tables (1)

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Table 1 Overview of the Paper Handsheets Used in This Work

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