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 nonnegligible long-range interactions and radiation through the surrounding free space, nearest-neighbor approximations cannot be invoked. Hence the governing operator is fundamentally different from 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-wave number space, such that a modal phase velocity cannot be uniquely defined. However, a well-defined group velocity is revealed due to the fractal's inner structure.