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

Zixuan Gao, Zhenli Xu, Zhiguo Yang

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.

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

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.
Paper Structure (16 sections, 6 theorems, 132 equations, 7 figures, 4 tables, 2 algorithms)

This paper contains 16 sections, 6 theorems, 132 equations, 7 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries on quasiperiodic functions
  3. Divergence-free projection method for quasiperiodic source problem
  4. Quasiperiodic divergence-free basis
  5. The divergence-free projection scheme
  6. Solution algorithm for quasiperiodic curl-curl source problems
  7. Reduced projection method for quasiperiodic Maxwell eigenvalue problem
  8. Decay rate of the divergence-free Fourier coefficients
  9. Divergence-free reduced projection method
  10. Numerical examples
  11. Example 1
  12. Example 2
  13. Example 3
  14. Example 4
  15. Example 5
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $m\in\mathbb Z, \;m>0$, suppose $F(\bm{x})\in H^m_{\rm per}(\mathbb [0,T]^n)$, then

Figures (7)

  • Figure 1: A schematic diagram of a quasicrystal. The red and blue colors represent two identical three-dimensional tetragonal lattices, each exhibiting periodicity in three dimensions. The blue lattice is obtained by rotating the red lattice around a fixed point (black dot) by a certain angle. The superposition of the two lattices forms the quasicrystal.
  • Figure 2: Convergence tests of the curl-curl problem with $\varepsilon=1$. (a) $L^2$-error versus $N$ with $\kappa=100$. (b) $L^{\infty}$-error versus $N$ with $\kappa=100$.
  • Figure 3: Convergence tests of the curl-curl problem with $\varepsilon$ defined in Eq. \ref{['eps0']}. (a) $L^2$-error versus $N$ with $\kappa=1,10,100,1000,10000$. (b) $L^{\infty}$-error versus $N$ with $\kappa=1,10,100,1000,10000$.
  • Figure 4: Absolute error versus the eigenvalue ID under (a) $N=12,M=10$ and (b) $N=8,M=6$.
  • Figure 5: Spectrum error $\varepsilon_1$ and $\varepsilon_2$. (a): Error as function of $N$ for different $M$. (b): Error as function of $M$ for different $N$.
  • ...and 2 more figures

Theorems & Definitions (21)

  • Definition 2.1
  • Remark 2.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.1
  • proof
  • proof
  • ...and 11 more