Abstract

The change in reflectance of a diffuser, due to a known change in the ratio of its absorption and scattering coefficients, can be predicted from the simple Kubelka–Munk equations or from Giovanelli’s more detailed theory. These predictions may differ by up to about 0.01 in reflectance. Differences of this magnitude have also been found between reflectance changes predicted by Giovanelli’s theory and measured changes. There is evidence that diffuser theory cannot give greater accuracy unless based on a scattering distribution which is more realistic than isotropic scattering. In the absence of such a theory, the Kubelka–Munk theory is as reliable as any for practical reflectance problems with opaque diffusers.

© 1962 Optical Society of America

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References

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  1. P. Kubelka and F. Munk, Z. tech. Physik 12, 593 (1931).
  2. R. G. Giovanelli, Optica Acta (Paris) 2, 153 (1955).
    [Crossref]
  3. S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, England, 1950).
  4. J. W. T. Walsh, D.S.I.R. (G.B.) Illum. Research Comm. Tech. Paper 2, 10 (1926).
  5. V. Kourganoff, Basic Methods in Transfer Problems (Clarendon Press, Oxford, England, 1952), Chap. III.
  6. D. R. Duncan, J. Oil and Colour Chemists’ Assoc. 32, 296 (1949).
  7. J. L. Saunderson, J. Opt. Soc. Am. 32, 727 (1942).
    [Crossref]
  8. W. R. Blevin and W. J. Brown, J. Opt. Soc. Am. 51, 129 (1961).
    [Crossref]
  9. W. E. K. Middleton and C. L. Sanders, J. Opt. Soc. Am. 41, 419 (1951).
    [Crossref] [PubMed]
  10. W. R. Blevin and W. J. Brown, J. Opt. Soc. Am. 51, 975 (1961), Fig. 1.
    [Crossref]

1961 (2)

1955 (1)

R. G. Giovanelli, Optica Acta (Paris) 2, 153 (1955).
[Crossref]

1951 (1)

1949 (1)

D. R. Duncan, J. Oil and Colour Chemists’ Assoc. 32, 296 (1949).

1942 (1)

1931 (1)

P. Kubelka and F. Munk, Z. tech. Physik 12, 593 (1931).

1926 (1)

J. W. T. Walsh, D.S.I.R. (G.B.) Illum. Research Comm. Tech. Paper 2, 10 (1926).

Blevin, W. R.

Brown, W. J.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, England, 1950).

Duncan, D. R.

D. R. Duncan, J. Oil and Colour Chemists’ Assoc. 32, 296 (1949).

Giovanelli, R. G.

R. G. Giovanelli, Optica Acta (Paris) 2, 153 (1955).
[Crossref]

Kourganoff, V.

V. Kourganoff, Basic Methods in Transfer Problems (Clarendon Press, Oxford, England, 1952), Chap. III.

Kubelka, P.

P. Kubelka and F. Munk, Z. tech. Physik 12, 593 (1931).

Middleton, W. E. K.

Munk, F.

P. Kubelka and F. Munk, Z. tech. Physik 12, 593 (1931).

Sanders, C. L.

Saunderson, J. L.

Walsh, J. W. T.

J. W. T. Walsh, D.S.I.R. (G.B.) Illum. Research Comm. Tech. Paper 2, 10 (1926).

D.S.I.R. (G.B.) Illum. Research Comm. Tech. Paper (1)

J. W. T. Walsh, D.S.I.R. (G.B.) Illum. Research Comm. Tech. Paper 2, 10 (1926).

J. Oil and Colour Chemists’ Assoc. (1)

D. R. Duncan, J. Oil and Colour Chemists’ Assoc. 32, 296 (1949).

J. Opt. Soc. Am. (4)

Optica Acta (Paris) (1)

R. G. Giovanelli, Optica Acta (Paris) 2, 153 (1955).
[Crossref]

Z. tech. Physik (1)

P. Kubelka and F. Munk, Z. tech. Physik 12, 593 (1931).

Other (2)

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, England, 1950).

V. Kourganoff, Basic Methods in Transfer Problems (Clarendon Press, Oxford, England, 1952), Chap. III.

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Figures (2)

Fig. 1
Fig. 1

This figure compares theoretical data on the total reflectance for normally incident light of a semi-infinite diffuser whose matrix is of refractive index 1.0. The upper graph superimposes curves showing (i) reflectances from Giovanelli’s results for isotropic scattering (our Table I) plotted against the ratio α/σ of absorption and scattering coefficients, and (ii) reflectances from the Kubelka–Munk equation [our Eq. (1)] plotted against the ratio K/S of absorption and scattering coefficients. The lower graph shows the differences in reflectance plotted against α/σ and K/S.

Fig. 2
Fig. 2

This figure compares theoretical data on the total reflectance for normally incident light of a semi-infinite diffuser whose matrix is of refractive index 1.5. The upper graph superimposes curves showing (i) reflectances from Giovanelli’s results for isotropic scattering (our Table I) plotted against the ratio α/σ of absorption and scattering coefficients, and (ii) reflectances from the Kubelka–Munk equations [our Eqs. (1) and (2)] plotted against the ratio K/S of absorption and scattering coefficients. The lower graph shows the differences in reflectance plotted against α/σ and K/S.

Tables (2)

Tables Icon

Table I Theoretical values (after Giovanelli) for the total reflectance of a semi-infinite diffuser, for diffuser matrices of refractive indices n of 1, 1.333, 1.46, and 1.50. Isotropic scattering, optically smooth surface, normally incident light.

Tables Icon

Table II Measured values of the total spectral reflectance, relative to a perfect diffuser, of opaque pigment suspensions mixed with various amounts of black pigment suspension. The differences between the measured reflectances and reflectances predicted from Giovanelli’s theoretical results are also tabulated: occurrence of the letter p in a difference column signifies that the corresponding measured value of reflectance was used as a basis for the theoretical predictions.

Equations (3)

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R 1 = ( 1 + K / S ) - ( 2 K / S + K 2 / S 2 ) 1 / 2 .
R n = r + t n - 2 R 1 ( 1 - r ) / [ 1 - R 1 ( 1 - t n - 2 ) ] ,
ω ˜ 0 = σ / ( σ + α ) .