Abstract

A spectral-projected gradient method and an extension of the Kubelka-Munk theory are applied to obtain the relevant parameters of the theory from measured diffuse reflectance spectra of pigmented samples illuminated with visible diffuse radiation. The initial estimate of the spectral dependence of the parameters, required by a recursive spectral-projected gradient method, was obtained by use of direct measurements and up-to-date theoretical estimates. We then tested the consistency of the Kubelka-Munk theory by repeating the procedure with samples of different thicknesses.

© 2002 Optical Society of America

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References

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  1. S. Fitzwater, J. W. Hook, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coatings Technol. 57, 39–47 (1985).
  2. 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]
  3. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
    [CrossRef]
  4. The reflection coefficients of diffuse radiation at the interfaces are defined as the ratio of the reflected diffuse flux over the (diffuse) incident flux.
  5. C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
    [CrossRef]
  6. P. Chylek, “Light scattering by small particles in an absorbing medium,” J. Opt. Soc. Am. 67, 561–563 (1977).
    [CrossRef]
  7. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).
  8. P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
    [CrossRef] [PubMed]
  9. P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part II: Nonhomogeneous layers,” J. Opt. Soc. Am. 44, 330–335 (1954).
    [CrossRef]
  10. G. Göbel, J. Kuhn, J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5, 413–426 (1995).
    [CrossRef]
  11. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  12. D. G. Phillips, F. W. Billmeyer, “Predicting reflectance and color of paint films by Kubelka-Munk analysis,” J. Coatings Technol. 48, 30–36 (1976).
  13. 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]
  14. W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka-Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
    [CrossRef] [PubMed]
  15. M. W. Ribarsky, “Titanium dioxide (TiO2) (rutile),” in Handbook of Optical Constants, E. D. Palik, ed. (Academic, New York, 1985), pp. 795–804.
    [CrossRef]
  16. W. E. Vargas, “Light scattering and absorption in pigmented coatings,” Ph.D. dissertation (Department of Materials Science, Uppsala University, Uppsala, Sweden, 1997).
  17. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, Ontario, Canada, 1985).
  18. Another possible source of absorption is the coating of the rutile crystallites added by the manufacturers of TiO2 to help in the dispersion of the pigment as well as to prevent the pigment to react with the resin.
  19. C. F. Bohren, “Applicability of effective-medium theories to problems of scattering and absorption by inhomogeneous atmospheric particles,” J. Atmos. Sci. 43, 468–475 (1986).
    [CrossRef]
  20. M. Athans, M. L. Dertouzos, R. N. Spann, S. J. Mason, Systems, Networks, and Computations: Multivariable Methods (McGraw-Hill, New York, 1974), pp. 132–143.
  21. R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
    [CrossRef]
  22. D. F. Shanno, “Conjugate gradient methods with inexact searches,” Math. Op. Res. 3, 244–256 (1978).
    [CrossRef]
  23. M. Raydan, “The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem,” SIAM J. Optim. 7, 26–33 (1997).
    [CrossRef]
  24. J. Barzilai, J. M. Borwein, “Two-point step size gradient methods,” IMA J. Numer. Anal. 8, 141–148 (1988).
    [CrossRef]
  25. L. Grippo, F. Lampariello, S. Lucidi, “A nonmonotone line search technique for Newton’s method,” SIAM J. Numer. Anal. 23, 707–716 (1986).
    [CrossRef]
  26. E. G. Birgin, J. M. Martinez, M. Raydan, “Nonmonotone spectral projected gradient methods on convex sets,” SIAM J. Optim. 10, 1196–1211 (2000).
    [CrossRef]
  27. E. G. Birgin, I. Chambouleyron, J. M. Martinez, “Estimation of the optical constants and the thickness of thin films using unconstrained optimization,” J. Comput. Phys. 151, 862–880 (1999).
    [CrossRef]
  28. W. E. Vargas, D. E. Azofeifa, N. Clark, “Retrieved optical properties of thin films on absorbing substrates from transmittance measurements” (submitted to Thin Solid Films).
  29. M. Quinten, J. Rostalski, “Lorenz-Mie theory for spheres immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13, 89–96 (1996).
    [CrossRef]
  30. W. E. Vargas, “Two-flux radiative transfer model under nonisotropic propagating diffuse radiation,” Appl. Opt. 38, 1077–1085 (1999).
    [CrossRef]

2000 (1)

E. G. Birgin, J. M. Martinez, M. Raydan, “Nonmonotone spectral projected gradient methods on convex sets,” SIAM J. Optim. 10, 1196–1211 (2000).
[CrossRef]

1999 (2)

E. G. Birgin, I. Chambouleyron, J. M. Martinez, “Estimation of the optical constants and the thickness of thin films using unconstrained optimization,” J. Comput. Phys. 151, 862–880 (1999).
[CrossRef]

W. E. Vargas, “Two-flux radiative transfer model under nonisotropic propagating diffuse radiation,” Appl. Opt. 38, 1077–1085 (1999).
[CrossRef]

1997 (2)

M. Raydan, “The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem,” SIAM J. Optim. 7, 26–33 (1997).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka-Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
[CrossRef] [PubMed]

1996 (1)

M. Quinten, J. Rostalski, “Lorenz-Mie theory for spheres immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13, 89–96 (1996).
[CrossRef]

1995 (2)

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]

G. Göbel, J. Kuhn, J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5, 413–426 (1995).
[CrossRef]

1994 (1)

1988 (1)

J. Barzilai, J. M. Borwein, “Two-point step size gradient methods,” IMA J. Numer. Anal. 8, 141–148 (1988).
[CrossRef]

1986 (2)

L. Grippo, F. Lampariello, S. Lucidi, “A nonmonotone line search technique for Newton’s method,” SIAM J. Numer. Anal. 23, 707–716 (1986).
[CrossRef]

C. F. Bohren, “Applicability of effective-medium theories to problems of scattering and absorption by inhomogeneous atmospheric particles,” J. Atmos. Sci. 43, 468–475 (1986).
[CrossRef]

1985 (1)

S. Fitzwater, J. W. Hook, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coatings Technol. 57, 39–47 (1985).

1979 (1)

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

1978 (1)

D. F. Shanno, “Conjugate gradient methods with inexact searches,” Math. Op. Res. 3, 244–256 (1978).
[CrossRef]

1977 (1)

1976 (1)

D. G. Phillips, F. W. Billmeyer, “Predicting reflectance and color of paint films by Kubelka-Munk analysis,” J. Coatings Technol. 48, 30–36 (1976).

1964 (1)

R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
[CrossRef]

1954 (1)

1948 (1)

1931 (1)

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

Athans, M.

M. Athans, M. L. Dertouzos, R. N. Spann, S. J. Mason, Systems, Networks, and Computations: Multivariable Methods (McGraw-Hill, New York, 1974), pp. 132–143.

Azofeifa, D. E.

W. E. Vargas, D. E. Azofeifa, N. Clark, “Retrieved optical properties of thin films on absorbing substrates from transmittance measurements” (submitted to Thin Solid Films).

Barzilai, J.

J. Barzilai, J. M. Borwein, “Two-point step size gradient methods,” IMA J. Numer. Anal. 8, 141–148 (1988).
[CrossRef]

Billmeyer, F. W.

D. G. Phillips, F. W. Billmeyer, “Predicting reflectance and color of paint films by Kubelka-Munk analysis,” J. Coatings Technol. 48, 30–36 (1976).

Birgin, E. G.

E. G. Birgin, J. M. Martinez, M. Raydan, “Nonmonotone spectral projected gradient methods on convex sets,” SIAM J. Optim. 10, 1196–1211 (2000).
[CrossRef]

E. G. Birgin, I. Chambouleyron, J. M. Martinez, “Estimation of the optical constants and the thickness of thin films using unconstrained optimization,” J. Comput. Phys. 151, 862–880 (1999).
[CrossRef]

Bohren, C. F.

C. F. Bohren, “Applicability of effective-medium theories to problems of scattering and absorption by inhomogeneous atmospheric particles,” J. Atmos. Sci. 43, 468–475 (1986).
[CrossRef]

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
[CrossRef]

Borwein, J. M.

J. Barzilai, J. M. Borwein, “Two-point step size gradient methods,” IMA J. Numer. Anal. 8, 141–148 (1988).
[CrossRef]

Chambouleyron, I.

E. G. Birgin, I. Chambouleyron, J. M. Martinez, “Estimation of the optical constants and the thickness of thin films using unconstrained optimization,” J. Comput. Phys. 151, 862–880 (1999).
[CrossRef]

Chylek, P.

Clark, N.

W. E. Vargas, D. E. Azofeifa, N. Clark, “Retrieved optical properties of thin films on absorbing substrates from transmittance measurements” (submitted to Thin Solid Films).

Dertouzos, M. L.

M. Athans, M. L. Dertouzos, R. N. Spann, S. J. Mason, Systems, Networks, and Computations: Multivariable Methods (McGraw-Hill, New York, 1974), pp. 132–143.

Fitzwater, S.

S. Fitzwater, J. W. Hook, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coatings Technol. 57, 39–47 (1985).

Fletcher, R.

R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
[CrossRef]

Fricke, J.

G. Göbel, J. Kuhn, J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5, 413–426 (1995).
[CrossRef]

Gilra, D. P.

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

Göbel, G.

G. Göbel, J. Kuhn, J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5, 413–426 (1995).
[CrossRef]

Grippo, L.

L. Grippo, F. Lampariello, S. Lucidi, “A nonmonotone line search technique for Newton’s method,” SIAM J. Numer. Anal. 23, 707–716 (1986).
[CrossRef]

Gunde, M. K.

Hook, J. W.

S. Fitzwater, J. W. Hook, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coatings Technol. 57, 39–47 (1985).

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
[CrossRef]

Ishimaru, A.

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

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, Ontario, Canada, 1985).

Kubelka, P.

Kuhn, J.

G. Göbel, J. Kuhn, J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5, 413–426 (1995).
[CrossRef]

Lampariello, F.

L. Grippo, F. Lampariello, S. Lucidi, “A nonmonotone line search technique for Newton’s method,” SIAM J. Numer. Anal. 23, 707–716 (1986).
[CrossRef]

Logar, J. K.

Lucidi, S.

L. Grippo, F. Lampariello, S. Lucidi, “A nonmonotone line search technique for Newton’s method,” SIAM J. Numer. Anal. 23, 707–716 (1986).
[CrossRef]

Martinez, J. M.

E. G. Birgin, J. M. Martinez, M. Raydan, “Nonmonotone spectral projected gradient methods on convex sets,” SIAM J. Optim. 10, 1196–1211 (2000).
[CrossRef]

E. G. Birgin, I. Chambouleyron, J. M. Martinez, “Estimation of the optical constants and the thickness of thin films using unconstrained optimization,” J. Comput. Phys. 151, 862–880 (1999).
[CrossRef]

Mason, S. J.

M. Athans, M. L. Dertouzos, R. N. Spann, S. J. Mason, Systems, Networks, and Computations: Multivariable Methods (McGraw-Hill, New York, 1974), pp. 132–143.

Matijevic, E.

Munk, F.

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

Niklasson, G. A.

Orel, B.

Orel, Z. C.

Phillips, D. G.

D. G. Phillips, F. W. Billmeyer, “Predicting reflectance and color of paint films by Kubelka-Munk analysis,” J. Coatings Technol. 48, 30–36 (1976).

Quinten, M.

M. Quinten, J. Rostalski, “Lorenz-Mie theory for spheres immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13, 89–96 (1996).
[CrossRef]

Raydan, M.

E. G. Birgin, J. M. Martinez, M. Raydan, “Nonmonotone spectral projected gradient methods on convex sets,” SIAM J. Optim. 10, 1196–1211 (2000).
[CrossRef]

M. Raydan, “The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem,” SIAM J. Optim. 7, 26–33 (1997).
[CrossRef]

Reeves, C. M.

R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
[CrossRef]

Ribarsky, M. W.

M. W. Ribarsky, “Titanium dioxide (TiO2) (rutile),” in Handbook of Optical Constants, E. D. Palik, ed. (Academic, New York, 1985), pp. 795–804.
[CrossRef]

Rostalski, J.

M. Quinten, J. Rostalski, “Lorenz-Mie theory for spheres immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13, 89–96 (1996).
[CrossRef]

Ryde, N. P.

Shanno, D. F.

D. F. Shanno, “Conjugate gradient methods with inexact searches,” Math. Op. Res. 3, 244–256 (1978).
[CrossRef]

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, Ontario, Canada, 1985).

Spann, R. N.

M. Athans, M. L. Dertouzos, R. N. Spann, S. J. Mason, Systems, Networks, and Computations: Multivariable Methods (McGraw-Hill, New York, 1974), pp. 132–143.

Tsang, L.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, Ontario, Canada, 1985).

Vargas, W. E.

W. E. Vargas, “Two-flux radiative transfer model under nonisotropic propagating diffuse radiation,” Appl. Opt. 38, 1077–1085 (1999).
[CrossRef]

W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka-Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
[CrossRef] [PubMed]

W. E. Vargas, “Light scattering and absorption in pigmented coatings,” Ph.D. dissertation (Department of Materials Science, Uppsala University, Uppsala, Sweden, 1997).

W. E. Vargas, D. E. Azofeifa, N. Clark, “Retrieved optical properties of thin films on absorbing substrates from transmittance measurements” (submitted to Thin Solid Films).

Appl. Opt. (3)

Appl. Spectrosc. (1)

Comput. J. (1)

R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
[CrossRef]

IMA J. Numer. Anal. (1)

J. Barzilai, J. M. Borwein, “Two-point step size gradient methods,” IMA J. Numer. Anal. 8, 141–148 (1988).
[CrossRef]

J. Atmos. Sci. (1)

C. F. Bohren, “Applicability of effective-medium theories to problems of scattering and absorption by inhomogeneous atmospheric particles,” J. Atmos. Sci. 43, 468–475 (1986).
[CrossRef]

J. Coatings Technol. (2)

S. Fitzwater, J. W. Hook, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coatings Technol. 57, 39–47 (1985).

D. G. Phillips, F. W. Billmeyer, “Predicting reflectance and color of paint films by Kubelka-Munk analysis,” J. Coatings Technol. 48, 30–36 (1976).

J. Colloid Interface Sci. (1)

C. F. Bohren, D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72, 215–221 (1979).
[CrossRef]

J. Comput. Phys. (1)

E. G. Birgin, I. Chambouleyron, J. M. Martinez, “Estimation of the optical constants and the thickness of thin films using unconstrained optimization,” J. Comput. Phys. 151, 862–880 (1999).
[CrossRef]

J. Opt. Soc. Am. (3)

Math. Op. Res. (1)

D. F. Shanno, “Conjugate gradient methods with inexact searches,” Math. Op. Res. 3, 244–256 (1978).
[CrossRef]

Part. Part. Syst. Charact. (1)

M. Quinten, J. Rostalski, “Lorenz-Mie theory for spheres immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13, 89–96 (1996).
[CrossRef]

SIAM J. Numer. Anal. (1)

L. Grippo, F. Lampariello, S. Lucidi, “A nonmonotone line search technique for Newton’s method,” SIAM J. Numer. Anal. 23, 707–716 (1986).
[CrossRef]

SIAM J. Optim. (2)

E. G. Birgin, J. M. Martinez, M. Raydan, “Nonmonotone spectral projected gradient methods on convex sets,” SIAM J. Optim. 10, 1196–1211 (2000).
[CrossRef]

M. Raydan, “The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem,” SIAM J. Optim. 7, 26–33 (1997).
[CrossRef]

Waves Random Media (1)

G. Göbel, J. Kuhn, J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5, 413–426 (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 (9)

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
[CrossRef]

The reflection coefficients of diffuse radiation at the interfaces are defined as the ratio of the reflected diffuse flux over the (diffuse) incident flux.

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

M. Athans, M. L. Dertouzos, R. N. Spann, S. J. Mason, Systems, Networks, and Computations: Multivariable Methods (McGraw-Hill, New York, 1974), pp. 132–143.

M. W. Ribarsky, “Titanium dioxide (TiO2) (rutile),” in Handbook of Optical Constants, E. D. Palik, ed. (Academic, New York, 1985), pp. 795–804.
[CrossRef]

W. E. Vargas, “Light scattering and absorption in pigmented coatings,” Ph.D. dissertation (Department of Materials Science, Uppsala University, Uppsala, Sweden, 1997).

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, Ontario, Canada, 1985).

Another possible source of absorption is the coating of the rutile crystallites added by the manufacturers of TiO2 to help in the dispersion of the pigment as well as to prevent the pigment to react with the resin.

W. E. Vargas, D. E. Azofeifa, N. Clark, “Retrieved optical properties of thin films on absorbing substrates from transmittance measurements” (submitted to Thin Solid Films).

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Figures (6)

Fig. 1
Fig. 1

Diagram of a pigmented coating on a thick substrate, illuminated with semi-isotropic diffuse radiation. Partial reflection of the external (internal backward) diffuse radiation is described by the reflection coefficient R d e (R d i ). R g describes the partial reflection of forward diffuse radiation at the backcoated interface.

Fig. 2
Fig. 2

Experimental, first approximation, and retrieved diffuse reflectance spectra R of a 48-µm-thick titanium dioxide (rutile-) pigmented polymer coating illuminated with visible diffuse radiation. The mean pigment radius a is equal to 0.15 µm and the filling fraction is equal to 0.20. The retrieved values were obtained after 1612 iterations.

Fig. 3
Fig. 3

Spectral dependence of the approximated and the retrieved values of (a) scattering coefficient S, (b) absorption coefficient K, (c) external and internal diffuse reflection coefficients R d e and R d i , respectively, for the film whose diffuse reflectance spectrum is displayed in Fig. 2.

Fig. 4
Fig. 4

Diffuse reflectance spectra of three pigmented coatings at thicknesses listed in the figure. The dashed curve corresponds to the first approximation of the diffuse reflectance for the thicker coating.

Fig. 5
Fig. 5

Visible spectral dependence of (a) scattering coefficients S, (b) absorption coefficients K, (c) internal reflection coefficient R d i for three TiO2-pigmented coatings at thicknesses listed in the figure, as obtained with the SPGM.

Fig. 6
Fig. 6

Spectral dependence of (a) mean isotropy factor ξ that characterizes the angular distribution of the diffuse radiation intensity that propagates through rutile TiO2-pigmented polymer coatings at thicknesses listed in the figure, and (b) extinction coefficient k M of the matrix at the spectral region in which the pigments do not absorb. The thin solid curve in Fig. 5(b) corresponds to the initial estimated value.

Tables (1)

Tables Icon

Table 1 Performance of the SPGM on Four Rutile-Pigmented Coatings

Equations (9)

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

R=Rde+1-Rdi1-RdeRKM1-RdiRKM,
RKM=1-Rg1+KS-b cothbSh1+KS+b cothbSh-Rg.
F=i=1NRexpλi-R Sλi, Kλi, Rdeλi, Rdiλi, λi, h2,
Zi= Sλi,i=1, 2,, NKλi,i=N+1, N+2,, 2NRdeλi,i=2N+1, 2N+2,, 3NRdiλi,i=3N+1, 3N+2,, 4Nh,i=4N+1.
αm=Zm-Zm-1tZm-Zm-1Zm-Zm-1tgm-gm-1,
FZm+1FZm-j+γgZmt · Zm+1-Zm
S S=ξ 1-σs, K K=ξk,
ξ2SSi,
kM= 2ξKKikMi,

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