Probability density functions for photon propagation in a binary (isotropic-Poisson) statistical mixture with unmatched positive and negative refractive indexes

The exact homogenized probability density function, for a photon making a step of length s has been analytically derived for a binary (isotropic-Poisson) statistical mixture with unmatched refractive indexes. The companions, exact, homogenized probability density functions for a photon to change direction ("scatter"), with polar ϑ and azimuthal φ angles, and the homogenized albedo, have also been obtained analytically. These functions also apply to negative refractive indexes and can reduce the number of Monte Carlo simulations needed for photon propagation in complex binary (isotropic-Poisson) statistical mixtures from hundreds to just one, for an equivalent homogeneous medium. Note, that this is not an approximate approach, but a mathematically equivalent and exact result. Additionally, some tutorial examples of homogenized Monte Carlo simulations are also given.