Computations of Lossy Bloch Waves in Two-Dimensional Photonic Crystals

In this article we compute lossy Bloch waves in two-dimensional photonic crystals with dispersion and material loss. For given frequencies these waves are determined from non-linear eigenvalue problems in the wave vector. We applied two numerical methods to a demanding test case, a photonic crystal with embedded quantum dots that exhibits very strong and anamolous dispersion. The first method is based on the formulation with periodic boundary conditions leading to a quadratic eigenvalue problem. We discretize this problem by the finite element method (FEM), first of quadratic order and, second, of higher orders using curved cells (p-FEM). Second, we use the multiple-multipole method (MMP) with artificial sources and compute extrema in the field response determining the eigenvalues. Both MMP and FEM provide robust solutions for the investigated dispersive and lossy photonic crystal, and can approximate the Bloch waves to a high accuracy. Moreover, the MMP method and p-FEM show low computational effort for very accurate solutions.