Finite-Slab Reflectance and Transmittance for Henyey-Greenstein Scattering via a First-Passage Transfer Operator

Abstract

We compute the reflectance R and transmittance T of a plane-parallel slab of optical thickness tau for Henyey-Greenstein (HG) scattering with asymmetry parameter g, single-scattering albedo a, and incidence angle theta_0. The method is a Monte-Carlo-free first-passage transfer operator on the depth-direction state (z, mu): exact in formulation, with no physical approximation beyond the transport model, and evaluated by a numerically convergent discretization. A free flight in depth at fixed direction cosine is followed by the azimuthally averaged HG angular redistribution. Confining the operator to (0, tau) with absorbing boundaries yields, in a single evaluation, the order-resolved reflection and transmission laws P_R(n) and P_T(n), from which R and T follow at every albedo through the weighting sum_n P(n) a^n, together with the emergent angular distributions R(mu), T(mu) at no extra cost. Reflectance obeys the factorization R_tau = sum_n P_inf(n) S(n, tau) a^n. Beyond serving as a forward model, the central result is structural: the finite slab recovers the half-space first-return law P_inf(n) order by order as tau -> infinity, placing slab reflectance and the half-space return statistics in one framework. The operator reproduces full three-dimensional Monte Carlo in both channels to <= 1.6 x 10^-3 (absolute) across g in [0, 0.95], tau in [0.5, 16], albedo a in [0.5, 1], and normal-to-oblique incidence, with energy conservation R + T = 1 recovered to < 10^-4 at a = 1.