Mathematical theory for the interface mode in a waveguide bifurcated from a Dirac point

Abstract

In this paper, we prove the existence of a bound state in a waveguide that consists of two semi-infinite periodic structures separated by an interface. The two periodic structures are perturbed from the same periodic medium with a Dirac point and they possess a common band gap enclosing the Dirac point. The bound state, which is called interface mode here, decays exponentially away from the interface with a frequency located in the common band gap and can be viewed as a bifurcation from the Dirac point. Using the layer potential technique and asymptotic analysis, we first characterize the band gap opening for the two perturbed periodic media and derive the asymptotics of the Bloch modes near the band gap edges. By formulating the eigenvalue problem for the waveguide with two semi-infinite structures using a boundary integral equation over the interface and analyzing the characteristic values of the associated boundary integral operator, we prove the existence of the interface mode for the waveguide when the perturbation of the periodic medium is small