A Semi-definite Optimization Method for Maximizing the Shared Band Gap of Topological Photonic Crystals

TL;DR

This paper introduces a semi-definite optimization approach to design topological photonic crystals with maximized shared band gaps, enhancing the bandwidth of robust edge modes for efficient electromagnetic energy transport.

Contribution

It develops a novel semi-definite programming framework for optimizing the dielectric structure of topological photonic crystals, incorporating topological invariants and symmetry constraints.

Findings

Optimized structures exhibit larger shared band gaps.

Enhanced robustness of edge modes demonstrated.

Efficient convex optimization algorithm developed.

Abstract

Topological photonic crystals (PCs) can support robust edge modes to transport electromagnetic energy in an efficient manner. Such edge modes are the eigenmodes of the PDE operator for a joint optical structure formed by connecting together two photonic crystals with distinct topological invariants, and the corresponding eigenfrequencies are located in the shared band gap of two individual photonic crystals. This work is concerned with maximizing the shared band gap of two photonic crystals with different topological features in order to increase the bandwidth of the edge modes. We develop a semi-definite optimization framework for the underlying optimal design problem, which enables efficient update of dielectric functions at each time step while respecting symmetry constraints and, when necessary, the constraints on topological invariants. At each iteration, we perform sensitivity analysis of the band gap function and the topological invariant constraint function to linearize the optimization problem and solve a convex semi-definite programming (SDP) problem efficiently. Numerical examples show that the proposed algorithm is superior in generating optimized optical structures with robust edge modes.