First-return statistics in bounded radiative transport: A Motzkin polynomial framework

A photon entering a scattering medium executes a three-dimensional random walk determined by the Henyey-Greenstein phase function. The photon either reaches the boundary for a first passage or is absorbed. Projecting the walk onto the axial direction produces a one-dimensional alternating process whose peaks and valleys correspond to changes in the sign of the projected step. This reduction preserves first-return and first-passage events and leads to a representation in terms of Motzkin-type polynomials. The analytical formulation is complete except for boundary-constrained return terms, which appear as high-order integrals. We treat these contributions with a single truncation factor determined from Monte Carlo simulations of first-return distributions over a wide range of anisotropy g and scattering steps ms. The resulting factor follows a Cauchy distribution. Incorporating it yields first-return probabilities in agreement with full three-dimensional Monte Carlo to within 2% for g<=0.7. The approach gives backscattering coefficients from phase-function integrals and provides an efficient alternative to full three-dimensional simulations for problems of radiative transport in semi-infinite media.