Waves in almost-periodic particle chains

Almost periodic particle chains exhibit peculiar propagation properties that are not observed in perfectly periodic ones. Furthermore, since they inherently support non-negligible long-range interactions and radiation through the surrounding free-space, nearest-neighbor approximations cannot be invoked. Hence the governing operator is fundamentally different than that used in traditional analysis of almost periodic structures, e.g. Harper's model and Almost-Mathieu difference equations. We present a mathematical framework for the analysis of almost periodic particle chains, and study their electrodynamic properties. We show that they support guided modes that exhibit a complex interaction mechanism with the light-cone. These modes possess a two-dimensional fractal-like structure in the frequency-wavenumber space, such that a modal phase-velocity cannot be uniquely defined. However, a well defined \emph{group velocity} is revealed due to the fractal's inner-structure.