Eigenvalue Problems of Discrete Curl Operators on Various Lattices for Simulating Three Dimensional Photonic Crystals

TL;DR

This paper extends Yee's scheme to all 14 Bravais lattices for simulating 3D photonic crystals, analyzes the eigenvalue problem of discrete curl operators, and improves computational efficiency using Fourier transforms and GPU acceleration.

Contribution

It introduces a generalized Yee's scheme for all Bravais lattices and derives eigen-decompositions of discrete curl operators, enabling efficient photonic crystal simulations.

Findings

Eigen-decomposition reduces to two types across lattices

Nullspace-free method improves convergence

GPU implementation accelerates calculations

Abstract

There are many numerical methods for simulate three-dimensional photonic crystals, after comparison, we choose Yee's scheme to be our discrete method. So far, this method can only be applied to simple cubic lattice and face-centered cubic lattice, but there are 14 Bravais lattices in three-dimensional space. In this paper we extended Yee's scheme to all of the 14 Bravais lattices. After discretization, we get a general eigenvalue problem, and we will analyze this general eigenvalue problem. The second important task of this paper is to find the eigen-decomposition of the discrete curl operator, after a series of complicated calculations, we find that all the lattices can be summed up into two kinds of decomposition. We are interested in finding the several few smallest real eigenvalues, but the large dimension of null space is 1/3 of all, this seriously affected the convergence of calculation, we use a technique which is called nullspace-free method to avoid this trouble. But this technique transforms the sparse matrix in our problem to a dense matrix, fortunately, the eigenvectors we found before are related to discrete Fourier transformation. The efficiency of calculation has been significantly improved by using the fast Fourier transformation. Finally, we calculate the band structures of photonic crystals on various lattices, and implement high performance calculations on the GPU.