Effective Field Theory of Distorted-Photonic Crystal: Exact Solutions of the Geodesics Equation

Photonic crystals are periodic structure of dielectric materials that can control light propagations in the media because of their photonic-dispersion led by the well-ordered lattice-points arrangements. We here study the behavior of light propagation in distorted-photonic crystals (D-PCs), which possess the gradual spatial distortion of lattice-points positions, as the effective field theory in terms of the differential geometry. In order to investigate the trajectory of light ray in the D-PC, we derive the geodesics equation that is given from the principle of least action, with defining the metric tensor in terms of the lattice-positions distortion. The geodesics equation indicates that the trajectory can be bent by just introducing lattice-positions distortion. We show some exact solutions of the trajectory in the case of simple distortion and that those results well agree with the results of the Finite Difference Time Domain (FDTD) simulations.