Ray propagation in nonuniform random lattices: part II

In this paper and its companion [J. Opt. Soc. Am. A. 23, 2251 (2006) ], the problem of ray propagation in nonuniform random half-plane lattices is considered. Cells can be independently occupied according to a density profile that depends on the lattice depth. An electromagnetic source external to the lattice radiates a monochromatic plane wave that undergoes specular reflections on the occupied sites. The probability of penetrating up to level k inside the lattice is analytically evaluated using two different approaches, the former applying the theory of Markov chains (Markov approach) and the latter using the theory of Martingale random processes (Martingale approach). The full theory concerned with the Martingale approach is presented here, along with an innovative modification that leads to some improved results. Numerical validation shows that it outperforms the Markov approach when dealing with ray propagation in dense lattices described by a slowly varying density profile.