Discrete time propagation graphs for optical scattering

In this paper, we examine discrete time propagation graphs as a potential framework for thin scattering layer models. The particular class of interest are propagation graphs formed from Cayley graphs of Zd, as their underlying geometry and shift-invariance make them appealing as potential models of the desired physical process. We show that their shift-invariant nature makes it possible to use Fourier analysis and residue calculus to recover exact expressions for transfer functions on the graph when d = 1. Moreover, we reduce the propagation of Fourier modes on the graph in the general case to the d = 1 case, and provide exact solutions for propagation in a layer of the graph with a finite depth. We also consider direction-dependent propagation (e.g. Mie scattering) and extend both the models and methods developed to this new regime. The mathematical tools developed in this paper may provide a framework for the future development of a model of thin scattering layers using graph theory.