Littrow resonators and the critical coupling concept

We describe a class of open resonators formed by highly corrugated broad waveguides. The contra-directional coupling induced by the corrugation can be seen as a Littrow diffraction event on a reflection grating. The presence of both boundaries still discretizes the modes. The modes most affected by this Littrow diffractions are the higher order modes of the broad waveguide. The new modes that emerge from their coupling may have collectively a low group velocity in a sizable part of the Brillouin zone. For this, a critical coupling value has to be found, minimizing the minibands and maximizing the "stripes of minigaps" between them. We give various implementations of this geometry, including an analogy with graphene nanoribbons. Furthermore, the analysis of the emerging field pattern of a toy-model shows phenomena previously identified in light localization studies, with large frequency intervals of "smooth" behaviour at "stripes of minigaps", separated by narrow frequency intervals of quickly evolving patterns.