Reduced projection method for photonic moir\'e lattices

TL;DR

This paper introduces a reduced projection method for efficiently solving quasiperiodic Schr"{o}dinger eigenvalue problems in photonic moiré lattices, achieving high accuracy with fewer degrees of freedom.

Contribution

The paper develops a novel reduced projection approach with rigorous error estimates for photonic moiré lattices, significantly decreasing computational complexity.

Findings

Faster decay of Fourier coefficients along a fixed direction.

Reduced basis space maintains accuracy with fewer degrees of freedom.

Numerical examples confirm improved efficiency and precision.

Abstract

This paper presents a reduced projection method for the solution of quasiperiodic Schr\"{o}dinger eigenvalue problems for photonic moir\'e lattices. Using the properties of the Schr\"{o}dinger operator in higher-dimensional space via a projection matrix, we rigorously prove that the generalized Fourier coefficients of the eigenfunctions exhibit faster decay rate along a fixed direction associated with the projection matrix. An efficient reduction strategy of the basis space is then proposed to reduce the degrees of freedom significantly. Rigorous error estimates of the proposed reduced projection method are provided, indicating that a small portion of the degrees of freedom is sufficient to achieve the same level of accuracy as the classical projection method. We present numerical examples of photonic moir\'e lattices in one and two dimensions to demonstrate the accuracy and efficiency of our proposed method.