A discrete model of guided modes and anomalous scattering in periodic structures

We study a discrete prototype of anomalous scattering associated with the interaction of guided modes of a periodic scatterer and plane waves incident upon the scatterer. The transmission anomalies arise because of the non-robustness of a guided mode, a mode that exists only at a specific frequency and wave number pair. The simplicity of the discrete prototype allows one to make certain explicit calculations and proofs, and to examine details of important resonant phenomena of the open wave guides. The main results are (1) a formula for transmission anomalies near a non-robust guided mode with rigorous error estimates that extends the formula of Shipman and Venakides [28] to non-zero Bloch wave number and (2) rigorous analysis of a bifurcation that connects the anomaly for non-zero wave number to that of zero wave number