# Solving the radiative transfer equation in a plane-parallel geometry

Here, we consider the steady state radiative transfer equation (RTE) in a plane-parallel geometry for an infinitely extended and anisotropically scattering medium.

In general, a solution of the RTE is not straightforward and requires *a priori* knowledge of the optical properties of the medium, i.e. the absorption and scattering coefficients as well as the scattering phase function. In a simplifying setup, such as the plane-parallel geometry, a quite simple solution strategy can be devised: an expansion of all terms into a series of Legendre polynomials allows to replace the integro-differential form of the RTE by a much simpler set of ordinary differential equations. Its system matrix is found to be non-singular, symmetric and of tridiagonal symmetry, wherein the diagonal coefficients encode the characteristics of the underlying scattering phase function. Also, in this approach, the fluence for an equivalent spherically symmetric medium can be related to the fluence in the planar symmetry with no extra effort.

We implemented a solver for the plane-parallel geometry RTE that yield the fluence for scattering phase functions that might adequately be represented by a finite sequence of Legendre expansion coefficients. The resulting data is used for verification tests and benchmarking of Monte Carlo RTE solver, wherein the transfer of photons through media is modeled by a stochastic approach, tracing the paths of a large number of independent photon packets that interact with medium via absorption and scattering.

The figure below illustrates the good agreement of numerical simulations of the Legendre expansion of the radiative transfer equation (LE-RTE; dashed curves) and Mote Carlo estimates (MC-RTE, solid curves) in the limiting case where absorption and scattering coefficients are of the same order, i.e. a case outside the scope of the (otherwise popular) diffusion approximation, and for different values of the scattering anisotropy of the underlying Henyey-Greenstein scattering phase function.