16. Photonic Dirac Cones and Relevant Physics

In this chapter, we derive a necessary and sufficient condition for materializing the photonic Dirac cone, which is an isotropic linear dispersion relation, on the \(\varGamma \) point of periodic metamaterials and photonic crystals by the \(\mathbf{k}\cdot \mathbf{p}\) perturbation theory and the group theory. We analyze the coupling between the Dirac-cone modes in metamaterials/photonic crystal slabs and the free-space modes by the Green function method, and prove that the propagation direction of the Dirac-cone modes in the slab can be controlled by the polarization of the incident wave. We further analyze the shapes of dispersion curves of the slab modes in the presence of diffraction loss and show that the group velocity of the slab modes exceeds the light velocity in free space. This problem of superluminal propagation is often found for non-Hermitian systems. Finally, we extend our discussion to electronic waves and prove that we can also materialize the Dirac cone on the \(\varGamma \) point of periodically modulated quantum wells.