Abstract

The effective scattering and absorption coefficients used to describe the optical properties of particulate materials in radiative transfer models are determined by the average path-length parameter of the diffuse radiation, as well as by the fraction of energy that each particle scatters into the forward and backward hemispheres relative to the direction of the impinging radiation. Until now, there were no well-established methods to calculate these parameters. We have devised an approach for evaluating average path-length parameters and forward-scattering ratios for both forward and backward diffuse radiation intensities. Single-scattering processes are described by Lorenz–Mie theory, and multiple-scattering effects have been taken into account by a generalization of Hartel theory. As a consequence of the formalism, the Kubelka–Munk scattering and absorption coefficients are explicitly related to average path-length parameters and forward-scattering ratios. These parameters display an optical depth dependence, characterized by values smoothly increasing or decreasing from the perpendicularly illuminated interface and saturation values at large optical depths.

© 1997 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. D. C. Rich, “Computer-aided design and manufacturing of the color of decorative and protective coatings,” J. Coat. Techn. 67, 53–60 (1995).
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    [CrossRef] [PubMed]
  7. A. A. Christy, O. M. Kvalheim, R. A. Velapoldi, “Quantitative analysis in diffuse reflectance spectrometry: a modified Kubelka–Munk equation,” Vibrational Spectrosc. 9, 19–27 (1995).
    [CrossRef]
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  35. W. E. Vargas, G. A. Niklasson, “Comparison between optimization criteria for light scattering from nonabsorbing spherical particles,” J. Colloid Interface Sci. 169, 497–499 (1995).
    [CrossRef]

1997 (2)

1996 (1)

1995 (5)

A. Ben David, “Multiple-scattering transmission and an effective average photon path length of a plane-parallel beam in a homogeneous medium,” Appl. Opt. 34, 2802–2810 (1995).
[CrossRef] [PubMed]

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

W. E. Vargas, G. A. Niklasson, “Comparison between optimization criteria for light scattering from nonabsorbing spherical particles,” J. Colloid Interface Sci. 169, 497–499 (1995).
[CrossRef]

D. C. Rich, “Computer-aided design and manufacturing of the color of decorative and protective coatings,” J. Coat. Techn. 67, 53–60 (1995).

A. A. Christy, O. M. Kvalheim, R. A. Velapoldi, “Quantitative analysis in diffuse reflectance spectrometry: a modified Kubelka–Munk equation,” Vibrational Spectrosc. 9, 19–27 (1995).
[CrossRef]

1994 (1)

1993 (1)

W. P. Hsu, R. Yu, E. Matijevic, “Paper whiteners I. Titania coated silica,” J. Colloid Interface Sci. 156, 56–65 (1993).
[CrossRef]

1992 (1)

1991 (1)

R. Koenigsdorff, F. Miller, R. Ziegler, “Calculation of scattering fractions for use in radiative flux models,” Int. J. Heat Mass Transfer 34, 2673–2676 (1991).
[CrossRef]

1989 (1)

1987 (1)

1986 (1)

1984 (1)

1983 (1)

H. G. Hecht, “The a priori calculation of the diffuse reflectance of a turbid medium,” Opt. Acta 30, 659–668 (1983).
[CrossRef]

1975 (1)

1973 (2)

1972 (1)

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

1971 (1)

1961 (1)

1955 (1)

1948 (1)

1943 (1)

1942 (1)

1940 (1)

W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).

1931 (2)

J. W. Ryde, “The scattering of light by turbid media. Part I,” Proc. R. Soc. London Ser. A 131, 451–464 (1931).
[CrossRef]

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

1927 (1)

L. Silberstein, “The transparency of turbid media,” Philos. Mag. 4, 1291–1296 (1927).

1905 (1)

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

Ben David, A.

Blevin, W. R.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Brown, W. J.

Christy, A. A.

A. A. Christy, O. M. Kvalheim, R. A. Velapoldi, “Quantitative analysis in diffuse reflectance spectrometry: a modified Kubelka–Munk equation,” Vibrational Spectrosc. 9, 19–27 (1995).
[CrossRef]

Chu, C. M.

Churchill, S. W.

Chylek, P.

Cohen, A.

de Mul, F. F. M.

Duntley, S. Q.

Gouesbet, G.

Graber, M.

Greve, J.

Grossman, J. P.

Gunde, M. K.

Hartel, W.

W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).

Hecht, H. G.

H. G. Hecht, “The a priori calculation of the diffuse reflectance of a turbid medium,” Opt. Acta 30, 659–668 (1983).
[CrossRef]

Hsu, W. P.

W. P. Hsu, R. Yu, E. Matijevic, “Paper whiteners I. Titania coated silica,” J. Colloid Interface Sci. 156, 56–65 (1993).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Hulburt, E. O.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Koenigsdorff, R.

R. Koenigsdorff, F. Miller, R. Ziegler, “Calculation of scattering fractions for use in radiative flux models,” Int. J. Heat Mass Transfer 34, 2673–2676 (1991).
[CrossRef]

Kolinko, V. G.

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
[CrossRef] [PubMed]

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Kvalheim, O. M.

A. A. Christy, O. M. Kvalheim, R. A. Velapoldi, “Quantitative analysis in diffuse reflectance spectrometry: a modified Kubelka–Munk equation,” Vibrational Spectrosc. 9, 19–27 (1995).
[CrossRef]

Latimer, P.

Letoulouzan, J. N.

Logar, J. K.

Maheu, B.

Mandelis, A.

Matijevic, E.

N. P. Ryde, E. Matijevic, “Color effects of uniform colloidal particles of different morphologies packed into films,” Appl. Opt. 33, 7275–7281 (1994).
[CrossRef] [PubMed]

W. P. Hsu, R. Yu, E. Matijevic, “Paper whiteners I. Titania coated silica,” J. Colloid Interface Sci. 156, 56–65 (1993).
[CrossRef]

Miller, F.

R. Koenigsdorff, F. Miller, R. Ziegler, “Calculation of scattering fractions for use in radiative flux models,” Int. J. Heat Mass Transfer 34, 2673–2676 (1991).
[CrossRef]

Mudgett, P. S.

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

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

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Niklasson, G. A.

Noh, S. J.

Orel, B.

Orel, Z. C.

Priezzhev, A. V.

Reichman, J.

Ren, K. F.

Rich, D. C.

D. C. Rich, “Computer-aided design and manufacturing of the color of decorative and protective coatings,” J. Coat. Techn. 67, 53–60 (1995).

Richards, L. W.

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

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

Ryde, J. W.

J. W. Ryde, “The scattering of light by turbid media. Part I,” Proc. R. Soc. London Ser. A 131, 451–464 (1931).
[CrossRef]

Ryde, N. P.

Schuster, A.

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

Silberstein, L.

L. Silberstein, “The transparency of turbid media,” Philos. Mag. 4, 1291–1296 (1927).

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

Vargas, W. E.

Velapoldi, R. A.

A. A. Christy, O. M. Kvalheim, R. A. Velapoldi, “Quantitative analysis in diffuse reflectance spectrometry: a modified Kubelka–Munk equation,” Vibrational Spectrosc. 9, 19–27 (1995).
[CrossRef]

Wang, Y. P.

Yu, R.

W. P. Hsu, R. Yu, E. Matijevic, “Paper whiteners I. Titania coated silica,” J. Colloid Interface Sci. 156, 56–65 (1993).
[CrossRef]

Zheng, S. W.

Ziegler, R.

R. Koenigsdorff, F. Miller, R. Ziegler, “Calculation of scattering fractions for use in radiative flux models,” Int. J. Heat Mass Transfer 34, 2673–2676 (1991).
[CrossRef]

Appl. Opt. (10)

J. Reichman, “Determination of absorption and scattering coefficients for nonhomogeneous media. 1: theory,” Appl. Opt. 12, 1811–1815 (1973).
[CrossRef] [PubMed]

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]

B. Maheu, G. Gouesbet, “Four-flux models to solve the scattering transfer equation: special cases,” Appl. Opt. 25, 1122–1128 (1986).
[CrossRef] [PubMed]

P. Latimer, 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]

N. P. Ryde, E. Matijevic, “Color effects of uniform colloidal particles of different morphologies packed into films,” Appl. Opt. 33, 7275–7281 (1994).
[CrossRef] [PubMed]

W. E. Vargas, G. A. Niklasson, “Forward average path-length parameter in four-flux radiative transfer models,” Appl. Opt. 36, 3735–3738 (1997).
[CrossRef] [PubMed]

A. Ben David, “Multiple-scattering transmission and an effective average photon path length of a plane-parallel beam in a homogeneous medium,” Appl. Opt. 34, 2802–2810 (1995).
[CrossRef] [PubMed]

V. G. Kolinko, F. F. M. de Mul, J. Greve, A. V. Priezzhev, “Probabilistic model of multiple light scattering based on rigorous computation of the first and the second moments of photon coordinates,” Appl. Opt. 35, 4541–4550 (1996).
[CrossRef] [PubMed]

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

Y. P. Wang, S. W. Zheng, K. F. Ren, “Four-flux model with adjusted average crossing parameter to solve the scattering transfer equation,” Appl. Opt. 28, 24–26 (1989).
[CrossRef] [PubMed]

Appl. Spectrosc. (2)

Astrophys. J. (1)

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

Int. J. Heat Mass Transfer (1)

R. Koenigsdorff, F. Miller, R. Ziegler, “Calculation of scattering fractions for use in radiative flux models,” Int. J. Heat Mass Transfer 34, 2673–2676 (1991).
[CrossRef]

J. Coat. Techn. (1)

D. C. Rich, “Computer-aided design and manufacturing of the color of decorative and protective coatings,” J. Coat. Techn. 67, 53–60 (1995).

J. Colloid Interf. Sci. (1)

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

J. Colloid Interface Sci. (2)

W. P. Hsu, R. Yu, E. Matijevic, “Paper whiteners I. Titania coated silica,” J. Colloid Interface Sci. 156, 56–65 (1993).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Comparison between optimization criteria for light scattering from nonabsorbing spherical particles,” J. Colloid Interface Sci. 169, 497–499 (1995).
[CrossRef]

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (1)

Licht (1)

W. Hartel, “Zur Theorie der Lichtstreuung durch trübe Schichten besonders Trübgläser,” Licht 10, 141–143, 165, 190, 191, 214, 215, 232–234 (1940).

Opt. Acta (1)

H. G. Hecht, “The a priori calculation of the diffuse reflectance of a turbid medium,” Opt. Acta 30, 659–668 (1983).
[CrossRef]

Philos. Mag. (1)

L. Silberstein, “The transparency of turbid media,” Philos. Mag. 4, 1291–1296 (1927).

Proc. R. Soc. London Ser. A (1)

J. W. Ryde, “The scattering of light by turbid media. Part I,” Proc. R. Soc. London Ser. A 131, 451–464 (1931).
[CrossRef]

Vibrational Spectrosc. (1)

A. A. Christy, O. M. Kvalheim, R. A. Velapoldi, “Quantitative analysis in diffuse reflectance spectrometry: a modified Kubelka–Munk equation,” Vibrational Spectrosc. 9, 19–27 (1995).
[CrossRef]

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

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Other (3)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

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Figures (5)

Fig. 1
Fig. 1

Electromagnetic radiation fluxes inside an inhomogeneous film of thickness h, which is normally illuminated.  

Fig. 2
Fig. 2

Polar diagrams of generalized phase functions fk(cos θ) corresponding to different scattering orders k. The normally incident radiation is impinging from the left side of the figure. The size parameter and the particle volume fraction were set to x=2.50 and f=0.05, respectively. The particle refractive index and the free-space wavelength were set to m=2.0 and λ0=0.55 μm, respectively.  

Fig. 3
Fig. 3

Optical depth dependence of the forward- and backward-scattering-order coefficients Qk(+) and Qk(-) (in arbitrary units) for a size parameter x=0.10. The particle relative refractive index is m=(2.75+i0.0)/1.50, the free-space wavelength is λ0=0.55 μm, and the particle volume fraction was set to f=0.05.

Fig. 4
Fig. 4

Optical depth dependence of forward-scattering ratios in (a) forward and (b) backward directions, as well as average path-length parameters in (c) forward and (d) backward directions, in the case of films containing titanium dioxide particles hosted in polyethylene: m=(2.75+i0.0)/1.50. The particle volume fraction and the free-space wavelength were set to f=0.05 and λ0=0.55 μm, respectively. The values of α and β corresponding to different size parameters are given in Table 1.  

Fig. 5
Fig. 5

Optical depth dependence of the Kubelka–Munk scattering coefficient for a coating containing titanium dioxide particles embedded in polyethylene: m=(2.75+i0.0)/1.50. The particle volume fraction and the free-space wavelength were set to f=0.05 and λ0=0.55 μm, respectively.

Tables (1)

Tables Icon

Table 1 Values of the Scattering and Absorption Coefficients per Unit Length, α and β in μm-1 Units, for Different Materials and Size Parameters a

Equations (39)

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

dIcdz=-(α+β)Ic,
dJcdz=(α+β)Jc,
dIddz=-ξ(+)βId-ξ(+)(1-σd(+))αId+ξ(-)(1-σd(-))αJd+σcαIc+(1-σc)αJc,
dJddz=ξ(-)βJd+ξ(-)(1-σd(-))αJd-ξ(+)(1-σd(+))αId-σcαJc-(1-σc)αIc,
p(μ)=n=0ωnPn(μ),
p(μ)=2x2Qext [|S1(μ)|2+|S2(μ)|2],
p(μ,μ)=n=0ωnPn(μ)Pn(μ).
fk(μ)=Nk-11dμfk-1(μ)p1(μ, μ)=14π n=0(2n+1)ωn/ω02n+1kPn(μ),
I(z, μ)=n=0cn(+)(z)Pn(μ)k=1Qk(+)(z)fk(μ),
J(z, μ)=n=0cn(-)(z)Pn(μ)k=1Qk(-)(z)fk(μ),
cn(±)(z)=2n+14π k=1 Qk(±)(z)k! ωn/ω02n+1k,
dQ1(+)dz=-ξ1(+)(α+β)Q1(+)+ξ1(+)αQ0,
dQk(+)dz=-ξk(+)(α+β)Qk(+)+a¯kξk-1(+)σk-1(+)αQk-1(+)+b¯kξk-1(-)[1-σk-1(-)]αQk-1(-),
-dQ1(-)dz=-ξ1(-)(α+β)Q1(-)+ξ1(-)αQ0,
-dQk(-)dz=-ξk(-)(α+β)Qk(-)+c¯kξk-1(+)[1-σk-1(+)]×αQk-1(+)+d¯kξk-1(-)σk-1(-)αQk-1(-),
ξk(+)=01fk(μ)dμ01μfk(μ)dμ,ξk(-)=--10fk(μ)dμ-10μfk(μ)dμ,
σ1(+)=σc,σk(+)=01dμ01dμfk-1(μ)p1(μ, μ)01dμ-11dμfk-1(μ)p1(μ, μ),
σ1(-)=σc,σk(-)=-10dμ-10dμfk-1(μ)p1(μ, μ)-10dμ-11dμfk-1(μ)p1(μ, μ),
a¯k=σk(+)wk-1(+)σk-1(+)wk(+),b¯k=[1-σk(-)]wk-1(-)[1-σk-1(-)]wk(+),
c¯k=[1-σk(+)]wk-1(+)[1-σk-1(+)]wk(-),d¯k=σk(-)wk-1(-)σk-1(-)wk(-),
wk(+)=01μfk(μ)dμ=18π 1+2ω13ω0k+2n=2(2n+1)hnωn/ω02n+1k,
wk(-)=--10μfk(μ)dμ=18π 1-2ω13ω0k+2n=2(2n+1)hnωn/ω02n+1k,
hn=01μPn(μ)dμ.
ξ(+)(z)=01I(z, μ)dμ01I(z, μ)μdμ.
ξ(-)(z)=--10J(z, μ)dμ-10J(z, μ)μdμ.
ξ(±)(z)=21±n=1 cn(±)(z)c0(±)(z) gn1±2c1(±)(z)3c0(±)(z)+2n=2 cn(±)(z)c0(±)(z) hn,
ξ(±)(z)=k=1qk(±)(z)ξk(±)k=1qk(±)(z),
qk(+)(z)=2πQk(+)(z)01μfk(μ)dμ,
qk(-)(z)=-2πQk(-)(z)-10μfk(μ)dμ
σd(+)(z)=01dμ01I(z, μ)p(μ, μ)dμ01dμ-11I(z, μ)p(μ, μ)dμ,
σd(-)(z)=-10dμ-10J(z, μ)p(μ, μ)dμ-10dμ-11J(z, μ)p(μ, μ)dμ.
σd(±)(z)=σd(i)±12 n=1 cn(±)(z)c0(±)(z) gn1+ωnχnnω0+12 n=1 ωnω0 gnm=2 cm(±)(z)c0(±)(z) χnm1±n=1 cn(±)(z)c0(±)(z) gn,
χnm=12n+m k=0(n/2)j=0(m/2) (-1)j+k[n+m+1-2(j+k)]×(2n-2k)!(2m-2j)!(n-k)!k!(n-2k)!(m-j)!j!(m-2j)!,
σd(i)=12ω0 ω0+n=1ωngn2.
S=α-10dμ01dμI(μ)p(μ, μ)-01dμ-10dμJ(μ)p(μ, μ)201μI(μ)dμ+-10μJ(μ)dμ.
S(z)=ω0ξ(+)(z)[1-σd(+)(z)]q(+)(z)-ξ(-)(z)[1-σd(-)(z)]q(-)(z)q(+)(z)-q(-)(z)α,
q(+)(z)=2π01μI(μ)dμ=2πc0(+)(z)2+c1(+)(z)3+n=2cn(+)(z)χn1
q(-)(z)=-2π-10μJ(μ)dμ=2πc0(-)(z)2-c1(-)(z)3+n=2cn(-)(z)χn1
K(+)(z)=ξ(+)(z)β,andK(-)(z)=ξ(-)(z)β.

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