Band Structure Calculations of Dispersive Photonic Crystals in 3D Using Holomorphic Operator Functions

Abstract

We propose a finite element method to compute the band structures of dispersive photonic crystals in 3D. The nonlinear Maxwell’s eigenvalue problem is formulated as the eigenvalue problem of a holomorphic operator function. The Nédélec edge elements are employed to discretize the operators, where the divergence free condition for the electric field is realized by a mixed form using a Lagrange multiplier. The convergence of the eigenvalues is proved using the abstract approximation theory for holomorphic operator functions with the regular approximation of the edge elements. The spectral indicator method is then applied to compute the discrete eigenvalues. Numerical examples are presented demonstrating the effectiveness of the proposed method.