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Exponentially localised interface eigenmodes in finite chains of resonators

by H. Ammari and S. Barandun and B. Davies and E.O. Hiltunen and T. Kosche and P. Liu

(Report number 2024-01)

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 then show the existence of an eigenvalue in the gap of the defect structure. We find a direct relationship between an eigenvalue being within the spectral gap and the localisation of its associated eigenmode, which we show is exponentially localised. 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.

Keywords: Finite Hermitian systems, subwavelength resonators, capacitance matrix, topological protection, Chebyshev polynomials, wave localisation

@Techreport{ABDHKL24_1083,
  author = {H. Ammari and S. Barandun and B. Davies and E.O. Hiltunen and T. Kosche and P. Liu},
  title = {Exponentially localised interface eigenmodes in finite chains of resonators},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2024-01},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2024/2024-01.pdf },
  year = {2024}
}

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