Semi-Dirac cones, a type of unique dispersion relation, always exhibit a series of interesting transport properties, such as electromagnetic topological transitions and anisotropic electromagnetic transmission. Recently, dual-band semi-Dirac cones have been found in three-dimensional photonic crystals, presenting great potential in electromagnetic wave regulation. However, to the best of our knowledge, there has been no report on dual-band semi-Dirac cones and their applications in two-dimensional photonic crystals, most two-dimensional systems have only realized semi-Dirac cones at a single frequency. Therefore, we aim to achieve dual-band semi-Dirac cones in two-dimensional photonic crystals.
In this work, a type of two-dimensional photonic crystal that comprises a square lattice of elliptical cylinders embedded in air is proposed. By rotating the elliptical cylinders and adjusting their sizes appropriately, accidental degeneracies at two different frequencies are achieved simultaneously in the center of the Brillouin zone. Using k · p perturbation theory, the dispersion relations near the two degenerate points are proved to be nonlinear in one direction, and linear in other directions, as shown in Figures (c) and (d). These results indicate that the double accidental degenerate points are two semi-Dirac points with different frequencies, and two different semi-Dirac cones, i.e., dual-band semi-Dirac cones, are realized simultaneously in the photonic crystal designed by us. More interestingly, the dual-band semi-Dirac cones exhibit opposite linear and nonlinear dispersion relations along the major axis and minor axis of the ellipse. And our photonic crystal can be equivalent to an impedance-matched double-zero index material in the direction of linear dispersion and a single-zero index material in the direction of nonlinear dispersion, which is demonstrated by the perfect transmission in the straight waveguide and wavefront shaping capabilities of electromagnetic waves. Based on the different properties of the equivalent zero-refractive-indices near two semi-Dirac points frequencies, a designed Y-type waveguide can be used to realize frequency separation by directing plane waves of different frequencies out along different ports, just as shown in Figures (e) and (f). We believe that our work is meaningful in broadening the exploration of the band structures of two-dimensional photonic crystals and providing greater convenience for the regulation of electromagnetic waves.