Band Gap Optimization of Two-Dimensional Photonic Crystals Using Semidefinite Programming and Subspace Methods∗

In this paper, we consider the optimal design of photonic crystal structures for two-dimensional square lattices. The mathematical formulation of the band gap optimization problem leads to an infinite-dimensional Hermitian eigenvalue optimization problem parametrized by the dielectric material and the wave vector. To make the problem tractable, the original eigenvalue problem is discretized using the finite element method into a series of finite-dimensional eigenvalue prob-lems for multiple values of the wave vector parameter. The resulting optimization problem is large-scale and non-convex, with low regularity and non-differentiable objective. By restrict-ing to appropriate eigenspaces, 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. Numerical results are presented for both trans-verse magnetic (TM) and transverse electric (TE) polarizations at several frequency bands. The optimized structures exhibit patterns which go far beyond typical physical intuition on periodic media design.