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

It is well 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. 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 from the bifurcation of a Dirac point of a periodic waveguide. We proved that when the constructed mode couples with the Floquet-Bloch modes of quasi-momentum away from the Dirac point, its associated eigenvalue has a negative imaginary part and the mode is a resonant mode that can radiate its energy into the bulk. 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. It is believed that the developed method can be extended to other settings, thus providing a clear answer to the problem concerned with the bifurcation of Dirac points.