Persistent random walk in a honeycomb structure: light transport in foams.

We study light transport in a honeycomb structure as the simplest two-dimensional model foam. We apply geometrical optics to set up a persistent random walk for the photons. For three special injection angles of 30 degrees, 60 degrees, and 90 degrees relative to a hexagon's edge, we are able to demonstrate by analytical means the diffusive behavior of the photons and to derive their diffusion constants in terms of intensity reflectance, edge length, and velocity of light. Numerical simulations reveal an interesting dependence of the diffusion constant on the injection angle in contrast to the usual assumption that in the diffusive limit the photon has no memory for its initial conditions. Furthermore, for injection angles close to 30 degrees, the diffusion constant does not converge to the value at 30 degrees. We explain this observation in terms of a two-state model.