Numerical computation of topological properties of photonic crystals

Abstract

In recent years huge advances were made in the field of topological photonics. One area of interest are photonic crystals with broken time reversal symmetry resulting in gaps in their band structure that prevent light propagation within specific frequency ranges. These photonic crystals hold promising applications such as topological photonic insulators. In this context, Chern numbers play an important role in characterizing the optical properties of such components. Numerically calculating the Chern number for an energy band requires solving a certain number of resonance problems. The amount depends on the experimental setup and the chosen computation method. We apply the finite element method, in combination with a reduced basis approach, to efficiently obtain band structures of 2D photonic crystals. Furthermore, our approach allows us to consider problems with nonlinear frequency-dependent permittivities and permeabilities. Employing this method to compute resonance frequencies, we compare two ways of computing Chern numbers: the first principal calculation and the Wilson loop approach. All implementations are conducted using the high-performance multiphysics finite element software Netgen/NGSolve. We demonstrate that, even with significantly reduced dimensions of the system, accurate Chern numbers can be obtained. Additionally, we are able to calculate Chern numbers for photonic crystals with highly frequency dependent material parameters.