The Kubelka-Munk Diffuse Reflectance Formula Revisited

Abstract We use an integral equation approach to finding the Kubelka-Munk (KM) diffuse reflectance formula and extend the result by finding the apparent path length and total intensity distribution inside an infinite, homogeneous, diffusely reflecting medium with isotropic scattering. We then expand the approach to three dimensions to show that the KM formula is correct for total diffuse reflectance when scattering, excitation, and detection are all isotropic. We obtain simple and exact results for the angular distribution of diffuse reflection and for the total diffuse reflectance when the incident light has an isotropic angular distribution, when it strikes at a single angle of elevation, and when it has the steady-state angular distribution. This work includes some results that employ Chandrasekhar's H function, so we also provide a program for the rapid evaluation of H. ?[Supplementary material is available for this article. Go to the publisher's online edition of Applied Spectroscopy Reviews for the following free supplemental resource: Program to compute the value of Chandrasekhar's H function] Abstract We use an integral equation approach to finding the Kubelka-Munk (KM) diffuse reflectance formula and extend the result by finding the apparent path length and total intensity distribution inside an infinite, homogeneous, diffusely reflecting medium with isotropic scattering. We then expand the approach to three dimensions to show that the KM formula is correct for total diffuse reflectance when scattering, excitation, and detection are all isotropic. We obtain simple and exact results for the angular distribution of diffuse reflection and for the total diffuse reflectance when the incident light has an isotropic angular distribution, when it strikes at a single angle of elevation, and when it has the steady-state angular distribution. This work includes some results that employ Chandrasekhar's H function, so we also provide a program for the rapid evaluation of H. ?[Supplementary material is available for this article. Go to the publisher's online edition of Applied Spectroscopy Reviews for the following free supplemental resource: Program to compute the value of Chandrasekhar's H function]