Finite slab first passage statistics of Henyey Greenstein scattering

Abstract

A photon entering a plane parallel scattering slab performs a random walk and eventually escapes through one of the two faces or is absorbed. The standard model employs a Henyey Greenstein phase function (HG) and an exponential step length distribution (Exp). Slab reflectance, transmittance, absorptance, and emergent angular distributions can be calculated in terms of random walk statistics. A central result is that the slab calculations factor into the order resolved first passage statistics of a half space combined with the a factor for the slab thickness. Absorptance is derived from order resolved walk statistics using the absorption rate. Two approaches are used. In the Monte Carlo (MC) approach, an extremely long random walk with many steps is efficiently generated without regard to any boundaries. The intersection of this walk with a large collection of target objects creates an ensemble of excursions of the objects. The MC approach relies explicitly on the memoryless property of Exp so that the portion of the first and last steps inside the object follow the same length distribution as the walk steps. The details of each excursion are recorded and any statistics can be extracted from the database of excursions. In particular, first passage statistics are extracted from this ensemble. In this work the objects are slabs with different positions and thicknesses. In the radiative transfer (RT) approach the slab is divided into thin layers with scattering treated to first order in each layer. The RT equations are then directly integrated over the slab to give the desired first passage statistics, reflectance, transmittance, and absorptance. The two methods agree to the Monte Carlo precision over the tested range of random walk parameters.