Colloids and Surfaces, 20 (1986) 251-260 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

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Brief Note

A Derivation of the Gurevic Equation for Reflectance from Plane Bounded Scattering Media

L.F. GATE

English China Clays International, John Keay House, St. Austell, Cornwall PL25 405 (United Kingdom)

G.H. MEETEN

Department of Physics, Sir John Cass Faculty of Physical Sciences and Technology, City of London Polytechnic, 31 Jewry Street, London EC3N 2EY (United Kingdom)

(Received 26 July 1985; accepted in final form 17 June 1986)

INTRODUCTION

The Gurevic equation relates the diffuse reflectance R and transmittance T of a light-diffusing layer of arbitrary thickness to the reflectance R, of the same material in an infinitely-thick layer. It can be written

l-T2+R2 Rx++= R

x_

It was derived by Gurevic [l] from his theory of light propagation in diffusing layers. It was re-derived by Kubelka [ 21 using the two-flux multiple-scattering theory of Kubelka and Munk [3], and obtained by Orchard [4] from similar considerations. It has recently been re-derived by Scallan [ 51, using a multiple- scattering model proposed originally by Stokes in 1862. As noted by Scallan, the equation is useful in order to relate data measured for finitely-thick layers to theoretical values of R,, e.g. to the remission factor of the Kubelka-Munk theory. It has also been used by Gate [6] in an extrapolation procedure where sample reflectances were measured with backings of various known diffuse reflectance.

The derivation of the Gurevic equation appears to originate from multiple- scattering theories, and its validity therefore appears to rely on the assump- tions of such theories. We show below that the Gurevic equation can be very simply derived without reference to any particular multiple-scattering theory.

THEORY

We consider a diffusely-scattering layer of depth z. on the illuminated side of an infinitely-thick layer, see Fig. 1. The extent of the layer is assumed unlim-

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Fig. 1. The flux densities into and out of a layer of finite thickness z0 which is part of an infinitely thick layer.

ited in the x and y directions. Diffuse flux density E is incident on the plane z = 0, which at z = z. causes a forward-going diffuse flux density of i and a backward-going diffuse flux density of j. The diffusely-reflected flux density A4 at z = 0 is made up of ER by the reflectance of the layer of thickness .zo, and jT by its transmittance, thus

M=ER+jT (2)

The forward-going flux density is made up from ET by the transmittance of the layer of thickness z and jR by its reflectance, thus

i=ET+jR (3)

The backward-going flux density j is solely due to the reflectance of i by an infinite layer, i.e.

i = iR, (4)

On putting R, = M/E and eliminating i and j from Eqns (2)~( 4)) the Gurevic equation is obtained.

DISCUSSION

The validity of the Gurevic equation, Eqn (l), is not dependent on specific assumptions made in multiple-scattering theory. It follows logically from one assumption, which is that the characters of the forward- and backward-going fluxes remain unaltered as they propagate. Thus Eqn (1) is exact if the angle- distribution of the radiance I remains independent of position z within the layer, which is an implicit assumption in any two-flux description of multiple scattering [ 731. Otherwise, Eqn (1) will be a good approximation provided that the angle-distribution of I does not vary strongly with z. Using radiative trans- fer theory, Kottler [9] gives approximate but useful expressions for I(z,p) for weakly-absorbing layers, arccos p being the angle between the Poynting vector of I and the normal to the layer. These show that the angle-distribution of I is only strong when z < o-l, abeing the inverse scattering length in the radiative transfer theory, approximating to the scattering parameter S in the

Kubelka-Munk theory. When z > 20-l, I(zy) varies weakly with p and it follows that Eqn (1) is a good approximation for optically-thick layers of every- day importance.

Apart from its practical utility, the Gurevic equation may be used as a test of self-consistency in any multiple-scattering theory in which the angular dis- tribution of the radiance is constant. An example is the statistical theory of multiple scattering, where the layer is envisaged as a sequence of parallel lay- ers. This theory has been developed by a number of workers, and is described by Wendlandt and Hecht [lo]. The reflectance and transmittance for a sequence of m identical layers can be written

(5)

and

(6)

where a and b are functions of the material properties of the medium and are given in detail by Wendlandt and Hecht [lo]. From Eqn (5), R, = l/a when m is infinite, and the Gurevic equation is satisfied by Eqns (5) and (6).

We conclude that the Gurevic equation has greater validity than is generally assumed and that this does not rest on any particular light-scattering theory. It may be derived exactly on the assumption that the angle-distribution of the radiance is independent of position, and is a good approximation when the angle-distribution of the radiance only varies weakly with position as in opti- cally-thick layers. A numerical comparison of the Gurevic equation with radia- tive transport theory has been made [ 111. For isotropic scatter, this shows that the value of R, derived from the Gurevic equation varies by less than 0.008 as the optical thickenss Sz varies from 0.1 to infinity over the range 0 < K/S < 0.01, where S and Kare the scattering and absorption parameters, respectively, in the Kubelka-Munk theory.

REFERENCES

1 M. Gurevic, Physik. Z., 31 (1930) 753-763. 2 P. Kuhelka, J. Opt. Sot. Am., 38 (1948) 448-457. 3 P. Kuhelka and F. Munk, Z. Tech. Phys., 12 (1931) 593-601. 4 S.E. Orchard, J. Opt. Sot. Am., 59 (1969) 1584-1597. 5 A.M. Scallan, J. Pulp Paper Sci., 11 (1985) 80-84. 6 L.F. Gate, J. Opt. Sot. Am., 63 (1973) 312-317. 7 H.C. van de Hulst, Multiple Light Scattering, Vol. 2, Academic Press, New York, 1980. 8 A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1, Academic Press,

New York, 1978.

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9 F. Kottler, Prog. Opt., 3 (1964) 2-28. 10 W.W. Wendlandt and H.G. Hecht, Reflectance Spectroscopy, Interscience, New York, 1966. 11 L.F. Gate and G.H. Meeten, to be published.