Edge-modes at interfaces between periodic media via reduced spatial dynamics near Dirac points

Abstract

We consider the Helmholtz operator in a d-dimensional waveguide, unbounded in x1-direction. The unperturbed waveguide has periodic coefficients in x1, the perturbations are different for x1 < 0 and x1 > 0. The perturbations are such that a band gap opens from a Dirac point. We show that an interface mode appears, corresponding to an eigenvalue in the band gap. Our proof uses the concept of reduced spatial dynamics and homogenization techniques. It is based on the analysis of the inhomogeneous problem for a right hand side that is a modulated eigenfunction of the unperturbed problem. We construct sequences of approximate solutions by solving ordinary differential equation; as these sequences are unbounded, we can conclude the existence of an eigenvalue.