Light scattering as a Poisson process and first passage probability

The Kubelka-Munk equations describe one-dimensional transport with scattering and absorption. The reflectance for a semi-infinite slab is the Laplace transform of the distribution of the photon path length lambda. It is determined by the first passage probability of an alternating random walk after np peaks. The first-passage probability as a function of the number of peaks is a path-length distribution-free combinatoric expression involving Catalan numbers. The conditional probability as a function of lambda and np, is a Poisson process. We present a novel demonstration that the probability of first-passage of a random walk is step-length-distribution-free. These results are verified with two iterative calculations, one using the properties of Volterras composition products and the other via an exponential distribution. A third verification is based on fluctuation theory of sums of random variables. Particle trajectories with scattering and absorption on the real half-line are mapped into a random walk on the integer number line in a lattice model, therefore connecting to path combinatorics. Including a separate forward scattering Poisson process results in a combinatoric expression related to counting Motzkin paths.