New Contributions to the Optics of Intensely Light-Scattering Materials Part I

The system of differential equations of Kubelka-Munk, -di=-(S+K)idx+Sjdx, dj=-(S+K)jdx+Sidx(i, j⋯ intensities of the light traveling inside a plane-parallel light-scattering specimen towards its unilluminated and its illuminated surface; x⋯ distance from the unilluminated surface S, K⋯ constants), has been derived from a simplified model of traveling of light in the material. Now, without simplifying assumptions the following exact system is derived: -di=-12(S+K)uidx+12Svjdx,dj=-12(S+K)vjdx+12Suidx,u≡∫0π/2(∂i/i∂φ)(dφ/cosφ), v≡∫0π/2(∂j/j∂φ)(dφ/cosφ), φ≡angle from normal of the light). Both systems become identical when u=v=2, that is, for instance, when the material is perfectly dull and when the light, is perfectly diffused or if it is parallel and hits the specimen under an angle of 60° from normal. Consequently, the different formulas Kubelka-Munk got by integration of their differential equations are exact when these conditions are fulfilled. The Gurevic and Judd formulas, although derived in another way by their authors, may be got from the Kubelka-Munk differential equations too. Consequently, they are exact under the same conditions. The integrated equations may be adapted for practical use by introducing hyperbolic functions and the secondary constants a=12(1/R∞+R∞) and b=12(1/R∞-R∞), (R∞≡reflectivity). Reflectance R, for instance, is then represented by the formula R=1-Rg(a-b ctghbSX)a+b ctghbSX-Rg(Rg≡reflectance of the backing, X=thickness of the specimen) and transmittance T by the formula T=ba sinhbSX+b coshbSX.In many practical cases the exact formulas may be replaced by appropriated approximations.