Dual-band semi-Dirac cones in two-dimensional photonic crystal and zero-index material

Abstract

<sec>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, and most of two-dimensional systems have only realized semi-Dirac cones at a single frequency. Therefore, we are to realize dual-band semi-Dirac cones in two-dimensional photonic crystals.</sec><sec>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 degeneracy at two different frequencies is achieved simultaneously in the center of the Brillouin zone. Using <inline-formula><tex-math id="M2">\begin{document}${\boldsymbol{k}} \cdot {\boldsymbol{p}}$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20240800_M2.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20240800_M2.png"/></alternatives></inline-formula> perturbation theory, the dispersion relations near the two degenerate points are proved to be nonlinear in one direction, and linear in other directions. 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 our designed photonic crystal. More interestingly, the dual-band semi-Dirac cones exhibit opposite linear and nonlinear dispersion relation along the major axis and the minor axis of the ellipse, respectively. 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 the frequencies of two semi-Dirac point, the designed Y-type waveguide can be used to realize frequency separation by leading out the plane waves of different frequencies along different ports. 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 regulating electromagnetic waves.</sec>