Band-Gap Optimization for Wave Propagation in Periodic Media using Semi- Definite Programming.

摘要

Our research has focused on the optimal design of photonic crystals with large band-gaps, thereby enabling a wide variety of prescribed interaction with and control of mechanical and electromagnetic waves. We have developed computational methods to design two-dimensional photonic crystals with large absolute and relative band gaps. Our modeling methodology yields a series of finite-dimensional eigenvalue optimization problems that are large-scale and non-convex, with low regularity and non-differentiable objective. By restricting to appropriate eigen-subspaces, we have reduced the problem to a sequence of small-scale convex semidefinite programs (SDPs) for which modern SDP solvers can be efficiently applied. Among several illustrations we show that it is possible to design photonic crystals which exhibit multiple band gaps for the combined transverse electric and magnetic modes. The optimized crystals show complicated patterns which are far different from existing photonic crystal designs. We employ subspace approximation and mesh adaptivity to enhance computational efficiency.