On the bifurcation of a Dirac point in a photonic waveguide without band gap openning

Abstract

Recent progress in topological insulators and topological phases of matter has motivated new methods for the localization of waves in photonic structures. Especially, it is established that a Dirac point of a periodic structure can bifurcate into in-gap eigenvalues if the periodic structure is perturbed differently on the two sides of an interface and if a common band gap can be opened for the two perturbed periodic structures near the Dirac point. The associated eigenmodes are localized near the interface and decay exponentially away from it. This paper addresses the less-known situation when the perturbation only lifts the degeneracy of the Dirac point without opening a band gap. Using a two-dimensional waveguide model, we constructed a wave mode bifurcated from a Dirac point of a periodic waveguide. We show that when the constructed mode couples to an analytically continued Floquet-Bloch mode near the Dirac energy, its eigenvalue acquires a strictly negative imaginary part, making the mode resonant. On the other hand, when the coupling vanishes, the imaginary part of the eigenvalue turns to zero, and the constructed mode becomes an interface mode that decays exponentially away from the interface. The developed method can be extended to other settings, thus providing a clear answer to the problem concerning the bifurcation of Dirac points.