New Contributions to the Optics of Intensely Light-Scattering Materials Part II: Nonhomogeneous Layers*

TLDR

The derived laws apply to layers whose scattering coefficient S and absorption coefficient K vary vertically with distance from the surface, requiring differential equations that treat these coefficients as functions of depth. In the case where the ratio K/S is constant, the authors introduce a variable p defined as the integral of the depth‑dependent function, reducing the nonhomogeneous problem to a previously solved homogeneous one. They derive explicit formulas for transmittance and reflectance of two non.

Abstract

The derived laws apply to layers whose scattering coefficient S and absorption coefficient K vary vertically to the surface of the layer. In the general case the differential equations of the preceding paper [ P. Kubelka , J. Opt. Soc. Am.38, 448 ( 1948)] must be used; the coefficients, however, hitherto constant, now are functions of the distance x from the surface. In the practically important case in which K/S is constant, one may introduce the variable p, such that p≡∫0x(x)dx. One reduces thereby the nonhomogeneous to the previously treated homogeneous case.Transmittance T1,2 and reflectance R1,2 of two nonhomogeneous sheets can be calculated by the following equations: T1,2=T1T21-R1R2, R1,2=R1+T12R21-R1R2,where T1, T2, R1, R2 are the transmittances and reflectances of the single sheets, and R1 represents the reflectance of the first sheet when illuminated in the inverse direction. Analogous formulas for more sheets and formulas relating transmittance, reflectance for specimens upon black, gray or white backing surfaces, and contrast ratio, are derived.It is shown by theory and experiment that reflectance and absorption of a nonhomogeneous specimen depend on the direction of illumination, whereas transmittance does not.