Transport mean free path tensor and anisotropy tensor in anisotropic diffusion equation for optical media

We derive the anisotropic diffusion equation for optical random media from the radiative transfer equation by correcting errors or eliminating simplifications in previous studies. The derived equation makes explicit the relation between microscopic (extinction cross-section and phase function) and macroscopic (transport mean free path tensor) parameters. We discover that transmittance and reflectance for a film made of an anisotropic material are determined from a dimensionless factor, which we define as anisotropy tensor, in addition to scattering/transport mean free paths and extrapolation length ratio. According to our derived equations, a direction component of the diagonalized transport mean free path tensor can be smaller than that of the diagonalized scattering mean free path tensor for forward scattering, which is physically impossible. We demonstrate that a direction component of the diagonalized inner product of transport mean free path tensor and the anisotropy tensor preserves the physical meaning of transport mean free path.