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Space-Time Wave Localisation in Systems of Subwavelength Resonators

by H. Ammari and E.O. Hiltunen and L. Rueff

(Report number 2024-28)

Abstract
In this paper we study the dynamics of metamaterials composed of high-contrast subwavelength resonators and show the existence of localised modes in such a setting. A crucial assumption in this paper is time-modulated material parameters. We prove a so-called capacitance matrix approximation of the wave equation in the form of an ordinary differential equation. These formulas set the ground for the derivation of a first-principles characterisation of localised modes in terms of the generalised capacitance matrix. Furthermore, we provide numerical results supporting our analytical results showing for the first time the phenomenon of space-time localised waves in a perturbed time-modulated metamaterial. Such spatio-temporal localisation is only possible in the presence of subwavelength resonances in the unperturbed structure. We introduce the time-dependent degree of localisation to quantitatively determine the localised modes and provide a variety of numerical experiments to illustrate our formulations and results.

Keywords: space-time localisation, time-dependent degree of localisation, metamaterial, subwavelength resonance, time-modulation, wave equation, band gap, momentum gap, defect mode.

@Techreport{AHR24_1110,
  author = {H. Ammari and E.O. Hiltunen and L. Rueff},
  title = {Space-Time Wave Localisation in Systems of Subwavelength Resonators},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2024-28},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2024/2024-28.pdf },
  year = {2024}
}

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