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A Tight-binding Approach for Computing Subwavelength Guided Modes in Crystals with Line Defects

Habib Ammari, Erik Orvehed Hiltunen, Ping Liu, Borui Miao, Yi Zhu

TL;DR

The paper addresses computing subwavelength defect bands in crystals of high-contrast resonators with line defects by deriving a tight-binding approximation based on capacitance matrices. It proves exponential decay of off-diagonal capacitance elements, enabling truncation to nearest-neighbor interactions and reducing defect-band computations to eigenvalues of tridiagonal matrices. The authors validate the theory with numerical experiments, demonstrating accurate defect-mode computation and applicability to topological interface modes. This approach offers an efficient, scalable tool for designing subwavelength waveguides and exploring topological subwavelength phenomena.

Abstract

In this paper, we consider waveguide systems operating at subwavelength scales. A key feature of these systems is that they are high contrast periodic resonator systems with line defects, leading to resonant phenomena at subwavelength scales. Their spectral properties at the subwavelength scales can be approximated by using the capacitance matrix formulation. Our main objective is to investigate the exponential decay of the off-diagonal elements of the capacitance matrices associated with these waveguide systems. This decay property rigorously justifies a tight-binding approximation, which in turn enables a novel and efficient approach for computing the spectral properties of subwavelength resonators with non-compact defects. Various numerical experiments are provided to validate the theoretical results, including applications to topological interface modes.

A Tight-binding Approach for Computing Subwavelength Guided Modes in Crystals with Line Defects

TL;DR

The paper addresses computing subwavelength defect bands in crystals of high-contrast resonators with line defects by deriving a tight-binding approximation based on capacitance matrices. It proves exponential decay of off-diagonal capacitance elements, enabling truncation to nearest-neighbor interactions and reducing defect-band computations to eigenvalues of tridiagonal matrices. The authors validate the theory with numerical experiments, demonstrating accurate defect-mode computation and applicability to topological interface modes. This approach offers an efficient, scalable tool for designing subwavelength waveguides and exploring topological subwavelength phenomena.

Abstract

In this paper, we consider waveguide systems operating at subwavelength scales. A key feature of these systems is that they are high contrast periodic resonator systems with line defects, leading to resonant phenomena at subwavelength scales. Their spectral properties at the subwavelength scales can be approximated by using the capacitance matrix formulation. Our main objective is to investigate the exponential decay of the off-diagonal elements of the capacitance matrices associated with these waveguide systems. This decay property rigorously justifies a tight-binding approximation, which in turn enables a novel and efficient approach for computing the spectral properties of subwavelength resonators with non-compact defects. Various numerical experiments are provided to validate the theoretical results, including applications to topological interface modes.

Paper Structure

This paper contains 24 sections, 27 theorems, 198 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem setting and preliminaries
  3. Wave guiding in the subwavelength regime
  4. Sufficient conditions of the decay
  5. Proof of estimate \ref{['eqn:Decay_Condition1']}
  6. Truncation method
  7. Exponential decay of the integral on $I_{n}$
  8. Estimate on the boundary $\partial \widetilde{Y}$
  9. Proof of estimates \ref{['eqn:Decay_Condition2']}
  10. Truncation method on $\mathbb{R}$
  11. Exponential decay of the integral on $\widetilde{I}_{2,m}$
  12. Subwavelength problem formulation
  13. Numerical experiments
  14. Exponential decay of the off-diagonal elements
  15. Defect mode and truncation of the capacitance matrix
  16. ...and 9 more sections

Key Result

Theorem 1

For all $m,n,p,q\in \mathbb{Z}$, the entries of the full capacitance matrix satisfy the following estimate: for some positive constants $C,\rho$ with $\rho\in(0,1)$.

Figures (11)

  • Figure 1: Left panel: an example of a waveguide system (adapted from jems2021) in the case of a square lattice; right panel: illustration of a single cell $Y_{m,n}$.
  • Figure 2: Cell $Y_n$. The line $L_n$ divides the cell into two parts, denoted as $Y_{n,+}$ and $Y_{n,-}$.
  • Figure 3: Left panel: Illustration of $\mathcal{V}_n$ and the derivation of $h_n$. Right panel: Illustration of $\mathcal{W}_n$ and the derivation of $g_{n+1}$.
  • Figure 4: Left panel: Illustration of $\widetilde{\mathcal{V}}_n$ and the derivation of $h_n$ for $n<0$. Right panel: Illustration of $\widetilde{\mathcal{W}}_n$ and the derivation of $g_{n+1}$ for $n<0$.
  • Figure 5: Cell $Y_{m,n}$. The line $L_{2,m,n}$ divides the cell into two parts, denoted as $Y_{m,n,+}$ and $Y_{m,n,-}$.
  • ...and 6 more figures

Theorems & Definitions (53)

  • Theorem 1
  • Theorem 2
  • proof
  • Remark 1
  • Proposition 1
  • Lemma 1
  • proof
  • Remark 2
  • Theorem 3
  • proof
  • ...and 43 more