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Exponentially localised interface eigenmodes in finite chains of resonators

Habib Ammari, Silvio Barandun, Bryn Davies, Erik Orvehed Hiltunen, Thea Kosche, Ping Liu

Abstract

This paper studies wave localisation in chains of finitely many resonators. There is an extensive theory predicting the existence of localised modes induced by defects in infinitely periodic systems. This work extends these principles to finite-sized systems. We consider finite systems of subwavelength resonators arranged in dimers that have a geometric defect in the structure. This is a classical wave analogue of the Su-Schrieffer-Heeger model. We prove the existence of a spectral gap for defectless finite dimer structures and find a direct relationship between eigenvalues being within the spectral gap and the localisation of their associated eigenmode. Then we show the existence and uniqueness of an eigenvalue in the gap in the defect structure, proving the existence of a unique localised interface mode. To the best of our knowledge, our method, based on Chebyshev polynomials, is the first to characterise quantitatively the localised interface modes in systems of finitely many resonators.

Exponentially localised interface eigenmodes in finite chains of resonators

Abstract

This paper studies wave localisation in chains of finitely many resonators. There is an extensive theory predicting the existence of localised modes induced by defects in infinitely periodic systems. This work extends these principles to finite-sized systems. We consider finite systems of subwavelength resonators arranged in dimers that have a geometric defect in the structure. This is a classical wave analogue of the Su-Schrieffer-Heeger model. We prove the existence of a spectral gap for defectless finite dimer structures and find a direct relationship between eigenvalues being within the spectral gap and the localisation of their associated eigenmode. Then we show the existence and uniqueness of an eigenvalue in the gap in the defect structure, proving the existence of a unique localised interface mode. To the best of our knowledge, our method, based on Chebyshev polynomials, is the first to characterise quantitatively the localised interface modes in systems of finitely many resonators.
Paper Structure (17 sections, 14 theorems, 89 equations, 7 figures)

This paper contains 17 sections, 14 theorems, 89 equations, 7 figures.

Table of Contents

  1. Introduction
  2. One-dimensional subwavelength resonator systems
  3. Dimer chains with a defect
  4. Perturbed tridiagonal $2$-Toeplitz matrices
  5. Eigenvalues
  6. Eigenvectors
  7. Asymptotic spectral gap and localised interface modes
  8. Asymptotic spectral gap
  9. Localised interface modes
  10. Existence, uniqueness, and convergence of the eigenvalue in the gap
  11. Existence
  12. Uniqueness
  13. Convergence
  14. Stability analysis
  15. Concluding remarks
  16. ...and 2 more sections

Key Result

Proposition 2.2

Consider a system of $N$ subwavelength resonators with size $\ell$ and spacings $s_i$ for $1\leqslant i \leqslant N-1$. Assume that the eigenvalues of $\mathcal{C}$ are simple. Then, the $N$ subwavelength resonant frequencies $\omega_i$ of eq: system of coupled equations satisfy to the first order where $(\lambda_i)_{1\leqslant i\leqslant N}$ are the eigenvalues of the eigenvalue problem Further

Figures (7)

  • Figure 1: A chain of $N$ resonators, with lengths $(\ell_i)_{1\leqslant i\leqslant N}$ and spacings $(s_{i})_{1\leqslant i\leqslant N-1}$ (which will be chosen to alternate between two distinct values, as depicted).
  • Figure 2: Dimer structure with a defect.
  • Figure 3: Eigenvalues and eigenvectors of the capacitance matrix \ref{['eq: strucutre capacitance matrix']} for $N=41, s_1=1$ and $s_2=3$.
  • Figure 4: Eigenvector behaviour based on the location of eigenvalue. Computation performed with $N=81, s_1=1, s_2=3$.
  • Figure 5: Convergence of the eigenvalue in the gap ($y$-axis in log scale). We display the left-hand side of \ref{['eq: error estimate convergence frequency in gap']} for a structure with $s_1=1$ and $s_2=2$.
  • ...and 2 more figures

Theorems & Definitions (19)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 3.1: Eigenvalues
  • Proposition 3.2
  • Proposition 3.3
  • Definition 4.1: Spectral bulk and gaps
  • Proposition 4.2
  • Definition 4.3: Localised interface mode
  • Proposition 4.4: Eigenvectors of $\mathcal{C}$
  • Proposition 5.1
  • ...and 9 more