The phenomenological model of Kubelka and Munk (KM) for describing the reflection and transmission of diffuse radiation in turbid, plane parallel media is utilized in investigating the nature of inhomogeneous distributions (i.e., the ratio of the absorption and scattering coefficients, K/S, is not necessarily constant) which satisfy certain constraints in finitely thick slabs. The distributions are constrained to be continuous and to minimize the integral of K<sup>2</sup>/S<sup>2</sup> across the slab, while resulting in a specified reflectance when the slab rests upon a backing of specified reflectance. The form of the inhomogeneous distributions is obtained as the solution to the corresponding variational problem, and the associated Lagrange multiplier is found to be algebraically related to the transmittance. The sufficiency of the approach is justified a posteriori by direct comparison with the closed-form solutions of KM for homogeneous distributions. The qualitative nature of such optimal inhomogeneous distributions is discussed with regard to the effects of the boundary conditions and the scattering thickness and is found to be approximately exponential.
© 1977 the Optical Society of America
John Texter, "Continuous K/S minimizing distributions in Kubelka-Munk systems," J. Opt. Soc. Am. 67, 169-175 (1977)