Optimal Design of Photonic Crystals

Abstract

The present work considers the optimal design of photonic crystals. Convex optimization is formally used for the purpose of designing photonic crystal devices with desired eigenband structures. In particular, two-dimensional crystal structures with optimal band gap and three-dimensional fiber with optimal band width. The original infinite-dimensional eigenvalue problem is discretized using the finite element method into a series of finite-dimensional eigenvalue problems for appropriate values of the wave vector parameter. By restricting to appropriate sub-eigenspaces, and employing mesh adaptivity, we reduce the large-scale non-convex optimization problem via reparametrization to a sequence of small-scale convex semidefinite programs (SDPs) for which modern SDP solvers can be efficiently applied. The optimized structures exhibit patterns which go far beyond typical physical intuition on periodic media design.