A divergence-free projection method for quasiperiodic photonic crystals in three dimensions

TL;DR

This paper introduces a novel divergence-free projection method for three-dimensional quasiperiodic photonic crystals, transforming Maxwell's equations into a higher-dimensional periodic form and employing spectral bases for efficient numerical solutions.

Contribution

It develops a divergence-free Fourier spectral basis and a reduced projection method for quasiperiodic Maxwell problems, improving computational efficiency and accuracy.

Findings

The method achieves high accuracy in numerical experiments.

It significantly reduces computational cost for eigenvalue problems.

The approach effectively handles divergence-free constraints in quasiperiodic settings.

Abstract

This paper presents a point-wise divergence-free projection method for numerical approximations of photonic quasicrystals problems. The original three-dimensional quasiperiodic Maxwell's system is transformed into a periodic one in higher dimensions through a variable substitution involving the projection matrix, such that periodic boundary condition can be readily applied. To deal with the intrinsic divergence-free constraint of the Maxwell's equations, we present a quasiperiodic de Rham complex and its associated commuting diagram, based on which a point-wise divergence-free quasiperiodic Fourier spectral basis is proposed. With the help of this basis, we then propose an efficient solution algorithm for the quasiperiodic source problem and conduct its rigorous error estimate. Moreover, by analyzing the decay rate of the Fourier coefficients of the eigenfunctions, we further propose a divergence-free reduced projection method for the quasiperiodic Maxwell eigenvalue problem, which significantly alleviates the computational cost. Several numerical experiments are presented to validate the efficiency and accuracy of the proposed method.