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

TL;DR

This paper presents a novel approach to optimize the band gaps of two-dimensional photonic crystals by transforming a complex eigenvalue problem into a series of convex semidefinite programs, enabling efficient design of structures with unconventional patterns.

Contribution

It introduces a semidefinite programming-based method for photonic crystal design, handling large-scale, non-convex eigenvalue optimization problems with improved computational efficiency.

Findings

Optimized structures show unconventional patterns beyond typical intuition.

Method effectively handles TM and TE polarizations.

Numerical results demonstrate significant band gap improvements.

Abstract

In this paper, we consider the optimal design of photonic crystal band 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 problems 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 restricting 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 transverse 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.