Abstract

The relationship between the Kubelka–Munk (K–M) and the transport scattering coefficient is obtained through a semi-empirical approach. This approach gives the same result as that given by Gate [Appl. Opt. 13, 236 (1974)] when the incident beam is diffuse. This result and those given by Star et al. [Phys. Med. Biol. 33, 437 (1988)] and Brinkworth [Appl. Opt. 11, 1434 (1972)] are compared with the exact solution of the radiative transfer equation over a large range of optical properties. It is found that the latter expressions, which include an absorption component, do not give accurate results over the range considered. Using the semi-empirical approach, the relationship between the K–M and the transport scattering coefficient is derived for the case where the incident light is collimated. It is shown that although the K–M equation is derived based on diffuse incident light, it can also represent very well the reflectance from a slab of infinite thickness when the incident light is collimated. However, in this case the relationship between the coefficients has to include a function that is dependent on the anisotropy factor. Analysis indicates that the K–M transform achieves the objective of obtaining a measure that gives the ratio of absorption to scattering effects for both diffuse and collimated incident beams over a large range of optical properties.

© 2008 Optical Society of America

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References

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  1. G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, 1969).
  2. M. K. Gunde, J. K. Logar, Z. C. Orel, and B. Orel, “Application of the Kubelka-Munk theory to thickness-dependent diffuse reflectance of black paint in the mid-IR,” Appl. Spectrosc. 49, 623-629 (1995).
    [CrossRef]
  3. L. E. McNeil and R. H. French, “Light scattering from red pigment particles: multiple scattering in a strongly absorbing system,” J. Appl. Phys. 89, 283-293 (2001).
    [CrossRef]
  4. G. Dupuis and M. Menu, “Quantitative characterisation of pigment mixtures used in art by fibre-optics diffuse-reflectance spectroscopy,” Appl. Phys. A 83, 469-474 (2006).
    [CrossRef]
  5. J. Sirita, S. Phanichphant, and F. C. Meunier, “Quantitative analysis of adsorbate concentrations by diffuse reflectance FT-IR,” Anal. Chem. 79, 3912-3918 (2007).
    [CrossRef] [PubMed]
  6. P. Jeevanandam, R. S. Mulukutla, M. Phillips, S. Chaudhuri, L. E. Erickson, and K. J. Klabunde, “Near infrared reflectance properties of metal oxide particles,” J. Phys. Chem. 111, 1912-1918 (2007).
  7. B. J. Brinkworth, “Interpretation of the Kubelka-Munk coefficients in reflection theory,” Appl. Opt. 11, 1434-1435 (1972).
    [CrossRef] [PubMed]
  8. L. F. Gate, “Comparison of the photon diffusion model and Kubelka-Munk equation with the exact solution of the radiative transport equation,” Appl. Opt. 13, 236-238 (1974).
    [CrossRef] [PubMed]
  9. 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 (1988).
    [CrossRef] [PubMed]
  10. S. L. Jacques, “Reflectance spectroscopy with optical fiber devices, and transcutaneous bilirubinometers,” in Biomedical Optical Instrumentation and Laser-Assisted Biotechnology, Proceedings of the NATO Advanced Science Institute, A.M.Verga Scheggi, S.Martelluci, A.N.Chester, and R.Pratesi (Kluwer Academic, 1995).
  11. S. A. Prahl, “The adding-doubling method,” in Optical Thermal Response of Laser Irradiated Tissue, A.J.Welch and M.J. C.van Gemert, eds. (Plenum, 1995), pp. 101-129.
  12. M. A. Velazco-Roa and S. N. Thennadil, “Estimation of complex refractive index of polydisperse particulate systems from multiple-scattered ultraviolet-visible-near-infrared measurements,” Appl. Opt. 46, 3730-3735 (2007).
    [CrossRef] [PubMed]

2007 (3)

J. Sirita, S. Phanichphant, and F. C. Meunier, “Quantitative analysis of adsorbate concentrations by diffuse reflectance FT-IR,” Anal. Chem. 79, 3912-3918 (2007).
[CrossRef] [PubMed]

P. Jeevanandam, R. S. Mulukutla, M. Phillips, S. Chaudhuri, L. E. Erickson, and K. J. Klabunde, “Near infrared reflectance properties of metal oxide particles,” J. Phys. Chem. 111, 1912-1918 (2007).

M. A. Velazco-Roa and S. N. Thennadil, “Estimation of complex refractive index of polydisperse particulate systems from multiple-scattered ultraviolet-visible-near-infrared measurements,” Appl. Opt. 46, 3730-3735 (2007).
[CrossRef] [PubMed]

2006 (1)

G. Dupuis and M. Menu, “Quantitative characterisation of pigment mixtures used in art by fibre-optics diffuse-reflectance spectroscopy,” Appl. Phys. A 83, 469-474 (2006).
[CrossRef]

2001 (1)

L. E. McNeil and R. H. French, “Light scattering from red pigment particles: multiple scattering in a strongly absorbing system,” J. Appl. Phys. 89, 283-293 (2001).
[CrossRef]

1995 (3)

M. K. Gunde, J. K. Logar, Z. C. Orel, and B. Orel, “Application of the Kubelka-Munk theory to thickness-dependent diffuse reflectance of black paint in the mid-IR,” Appl. Spectrosc. 49, 623-629 (1995).
[CrossRef]

S. L. Jacques, “Reflectance spectroscopy with optical fiber devices, and transcutaneous bilirubinometers,” in Biomedical Optical Instrumentation and Laser-Assisted Biotechnology, Proceedings of the NATO Advanced Science Institute, A.M.Verga Scheggi, S.Martelluci, A.N.Chester, and R.Pratesi (Kluwer Academic, 1995).

S. A. Prahl, “The adding-doubling method,” in Optical Thermal Response of Laser Irradiated Tissue, A.J.Welch and M.J. C.van Gemert, eds. (Plenum, 1995), pp. 101-129.

1988 (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 (1988).
[CrossRef] [PubMed]

1974 (1)

1972 (1)

1969 (1)

G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, 1969).

Brinkworth, B. J.

Chaudhuri, S.

P. Jeevanandam, R. S. Mulukutla, M. Phillips, S. Chaudhuri, L. E. Erickson, and K. J. Klabunde, “Near infrared reflectance properties of metal oxide particles,” J. Phys. Chem. 111, 1912-1918 (2007).

Dupuis, G.

G. Dupuis and M. Menu, “Quantitative characterisation of pigment mixtures used in art by fibre-optics diffuse-reflectance spectroscopy,” Appl. Phys. A 83, 469-474 (2006).
[CrossRef]

Erickson, L. E.

P. Jeevanandam, R. S. Mulukutla, M. Phillips, S. Chaudhuri, L. E. Erickson, and K. J. Klabunde, “Near infrared reflectance properties of metal oxide particles,” J. Phys. Chem. 111, 1912-1918 (2007).

French, R. H.

L. E. McNeil and R. H. French, “Light scattering from red pigment particles: multiple scattering in a strongly absorbing system,” J. Appl. Phys. 89, 283-293 (2001).
[CrossRef]

Gate, L. F.

Gunde, M. K.

Jacques, S. L.

S. L. Jacques, “Reflectance spectroscopy with optical fiber devices, and transcutaneous bilirubinometers,” in Biomedical Optical Instrumentation and Laser-Assisted Biotechnology, Proceedings of the NATO Advanced Science Institute, A.M.Verga Scheggi, S.Martelluci, A.N.Chester, and R.Pratesi (Kluwer Academic, 1995).

Jeevanandam, P.

P. Jeevanandam, R. S. Mulukutla, M. Phillips, S. Chaudhuri, L. E. Erickson, and K. J. Klabunde, “Near infrared reflectance properties of metal oxide particles,” J. Phys. Chem. 111, 1912-1918 (2007).

Klabunde, K. J.

P. Jeevanandam, R. S. Mulukutla, M. Phillips, S. Chaudhuri, L. E. Erickson, and K. J. Klabunde, “Near infrared reflectance properties of metal oxide particles,” J. Phys. Chem. 111, 1912-1918 (2007).

Kortüm, G.

G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, 1969).

Logar, J. K.

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 (1988).
[CrossRef] [PubMed]

McNeil, L. E.

L. E. McNeil and R. H. French, “Light scattering from red pigment particles: multiple scattering in a strongly absorbing system,” J. Appl. Phys. 89, 283-293 (2001).
[CrossRef]

Menu, M.

G. Dupuis and M. Menu, “Quantitative characterisation of pigment mixtures used in art by fibre-optics diffuse-reflectance spectroscopy,” Appl. Phys. A 83, 469-474 (2006).
[CrossRef]

Meunier, F. C.

J. Sirita, S. Phanichphant, and F. C. Meunier, “Quantitative analysis of adsorbate concentrations by diffuse reflectance FT-IR,” Anal. Chem. 79, 3912-3918 (2007).
[CrossRef] [PubMed]

Mulukutla, R. S.

P. Jeevanandam, R. S. Mulukutla, M. Phillips, S. Chaudhuri, L. E. Erickson, and K. J. Klabunde, “Near infrared reflectance properties of metal oxide particles,” J. Phys. Chem. 111, 1912-1918 (2007).

Orel, B.

Orel, Z. C.

Phanichphant, S.

J. Sirita, S. Phanichphant, and F. C. Meunier, “Quantitative analysis of adsorbate concentrations by diffuse reflectance FT-IR,” Anal. Chem. 79, 3912-3918 (2007).
[CrossRef] [PubMed]

Phillips, M.

P. Jeevanandam, R. S. Mulukutla, M. Phillips, S. Chaudhuri, L. E. Erickson, and K. J. Klabunde, “Near infrared reflectance properties of metal oxide particles,” J. Phys. Chem. 111, 1912-1918 (2007).

Prahl, S. A.

S. A. Prahl, “The adding-doubling method,” in Optical Thermal Response of Laser Irradiated Tissue, A.J.Welch and M.J. C.van Gemert, eds. (Plenum, 1995), pp. 101-129.

Sirita, J.

J. Sirita, S. Phanichphant, and F. C. Meunier, “Quantitative analysis of adsorbate concentrations by diffuse reflectance FT-IR,” Anal. Chem. 79, 3912-3918 (2007).
[CrossRef] [PubMed]

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 (1988).
[CrossRef] [PubMed]

Thennadil, S. N.

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 (1988).
[CrossRef] [PubMed]

Velazco-Roa, M. A.

Anal. Chem. (1)

J. Sirita, S. Phanichphant, and F. C. Meunier, “Quantitative analysis of adsorbate concentrations by diffuse reflectance FT-IR,” Anal. Chem. 79, 3912-3918 (2007).
[CrossRef] [PubMed]

Appl. Opt. (3)

Appl. Phys. A (1)

G. Dupuis and M. Menu, “Quantitative characterisation of pigment mixtures used in art by fibre-optics diffuse-reflectance spectroscopy,” Appl. Phys. A 83, 469-474 (2006).
[CrossRef]

Appl. Spectrosc. (1)

J. Appl. Phys. (1)

L. E. McNeil and R. H. French, “Light scattering from red pigment particles: multiple scattering in a strongly absorbing system,” J. Appl. Phys. 89, 283-293 (2001).
[CrossRef]

J. Phys. Chem. (1)

P. Jeevanandam, R. S. Mulukutla, M. Phillips, S. Chaudhuri, L. E. Erickson, and K. J. Klabunde, “Near infrared reflectance properties of metal oxide particles,” J. Phys. Chem. 111, 1912-1918 (2007).

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 (1988).
[CrossRef] [PubMed]

Other (3)

S. L. Jacques, “Reflectance spectroscopy with optical fiber devices, and transcutaneous bilirubinometers,” in Biomedical Optical Instrumentation and Laser-Assisted Biotechnology, Proceedings of the NATO Advanced Science Institute, A.M.Verga Scheggi, S.Martelluci, A.N.Chester, and R.Pratesi (Kluwer Academic, 1995).

S. A. Prahl, “The adding-doubling method,” in Optical Thermal Response of Laser Irradiated Tissue, A.J.Welch and M.J. C.van Gemert, eds. (Plenum, 1995), pp. 101-129.

G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, 1969).

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Figures (4)

Fig. 1
Fig. 1

Reflectance from a slab of infinite thickness irradiated by diffuse incident beam as a function of μ a δ : (a) R over the whole range considered in this study. (b) Comparison of R calculated using Eq. (5) in the “linear” region. (c) Error in using Eq. (5) in the linear region.

Fig. 2
Fig. 2

Reflectance from a slab of infinite thickness irradiated by collimated incident beam as a function of μ a δ : (a) R over the whole range considered in this study. (b) R for two (low and high) values of g in the linear region. (c) C 0 as a function of g estimated by fitting Eq. (5) to exact values at different g.

Fig. 3
Fig. 3

Comparison of results for diffuse incident beam using K–M scattering coefficient S computed with relationships given by Star et al. [9], Brinkworth [7], and the present study. (a) R from the K–M equation using the various relationships is compared to the exact RTE. (b) Relative error of the K–M equation when Eq. (21) is used to related S with μ s .

Fig. 4
Fig. 4

Comparison of results for collimated incident beam using K–M scattering coefficient S computed with Eqs. (22, 24). (a) R when g = 0.1 ; (b) R when g = 0.9 ; (c) relative error when g = 0.1 ; and (d) relative error when g = 0.9 .

Equations (26)

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R , K - M = 1 + K S 2 ( K S ) 1 2 ( 1 + K 2 S ) 1 2 .
K S = ( 1 R ) 2 2 R .
K = 2 μ a .
S = y μ s x μ a ,
R = exp ( C 0 μ a δ ) .
δ = [ 3 μ a ( μ a + μ s ) ] 0.5 .
μ a μ s = 3 ( μ a δ ) 2 1 3 ( μ a δ ) 2 .
C 0 ( g ) = 4.8446 + 0.472 g 0.114 g 2 .
μ a δ = μ a μ s 3 ( 1 + μ a μ s ) .
μ a δ μ a μ s 3 .
R = exp ( C 0 3 μ a μ s ) .
R = 1 C 0 3 ( μ a μ s ) 1 2 + 1 2 ( C 0 3 ) 2 μ a μ s .
( 1 + K 2 S ) 1 2 = 1 + 1 2 K 2 S + .
R , K - M = 1 2 ( K S ) 1 2 + K S ,
K S = 2 μ a y μ s x μ a .
K S = 2 y μ a μ s .
R , K - M = 1 2 y ( μ a μ s ) 1 2 + 2 y μ a μ s .
C 0 3 = 2 y , C 0 2 6 = 2 y ,
y = 12 C 0 2 .
K S = C 0 2 6 μ a μ s .
S = 12 C 0 2 μ s .
S = 3 4 μ s ,
K S = 8 3 μ a μ s .
S = 12 ( 4.8446 + 0.472 g 0.114 g 2 ) 2 μ s .
K S = μ a μ s f ( g ) ,
f ( g ) = 6 C 0 2 ( g ) .

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