Robust topology optimization of three-dimensional photonic-crystal band-gap structures

TL;DR

This paper presents a comprehensive 3D topology optimization method for photonic crystal structures to maximize band gaps, introducing a robust optimization approach to enhance fabrication tolerance.

Contribution

It develops a novel semidefinite-program formulation for nonconvex eigenvalue optimization and applies it to 3D photonic crystal band-gap design, including robust optimization under uncertainty.

Findings

Optimal gaps are close to previous designs, indicating near-optimality of current structures.

New structures with larger gaps than known designs were discovered.

Robust optimization increases tolerance to fabrication errors.

Abstract

We perform full 3D topology optimization (in which "every voxel" of the unit cell is a degree of freedom) of photonic-crystal structures in order to find optimal omnidirectional band gaps for various symmetry groups, including fcc (including diamond), bcc, and simple-cubic lattices. Even without imposing the constraints of any fabrication process, the resulting optimal gaps are only slightly larger than previous hand designs, suggesting that current photonic crystals are nearly optimal in this respect. However, optimization can discover new structures, e.g. a new fcc structure with the same symmetry but slightly larger gap than the well known inverse opal, which may offer new degrees of freedom to future fabrication technologies. Furthermore, our band-gap optimization is an illustration of a computational approach to 3D dispersion engineering which is applicable to many other problems in optics, based on a novel semidefinite-program formulation for nonconvex eigenvalue optimization combined with other techniques such as a simple approach to impose symmetry constraints. We also demonstrate a technique for \emph{robust} topology optimization, in which some uncertainty is included in each voxel and we optimize the worst-case gap, and we show that the resulting band gaps have increased robustness to systematic fabrication errors.