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Subwavelength Localisation in Disordered Systems
(2025)SAM Research ReportWe consider disordered finite systems of subwavelength resonators where the disorder is in the spatial arrangement of the resonators and their material parameters are perturbed randomly. We elucidate the mechanism of wave localisation in such systems. To do so, we employ the capacitance matrix formalism, make precise the notions of bandgap and localised eigenmodes, study their stabilities with respect to fluctuations in the system, and ...Report -
Effective Medium Theory for Time-modulated Subwavelength Resonators
(2025)SAM Research ReportThis paper provides a general framework for deriving effective material properties of time-modulated systems of subwavelength resonators. It applies to subwavelength resonator systems with a general form of time-dependent parameters. We show that the resonators can be accurately described by a point-scattering formulation when the width of the resonators is small. In contrast to the static setting, where this point interaction approximation ...Report -
Nonlinear subwavelength resonances in three dimensions
(2024)SAM Research ReportIn this paper, we consider the resonance problem for the cubic nonlinear Helmholtz equation in the subwavelength regime. We derive a discrete model for approximating the subwavelength resonances of finite systems of high-contrast resonators with Kerr-type nonlinearities. Our discrete formulation is valid in both weak and strong nonlinear regimes. Compared to the linear formulation, it characterizes the soliton-like extra eigenmodes that ...Report -
Wave scattering with time-periodic coefficients: Energy estimates and harmonic formulations
(2024)SAM Research ReportThis paper investigates acoustic wave scattering from materials with periodic time-modulated material parameters. We consider the basic case of a single connected domain where absorbing or Neumann boundary conditions are enforced. Energy estimates limit the exponential growth of solutions to the initial value problem, thereby confining Floquet exponents to a complex half-space under absorbing boundary conditions and to a strip near the ...Report -
Truncated Floquet-Bloch transform for computing the spectral properties of large finite systems of resonators
(2024)SAM Research ReportThis paper aims at providing for the first time the mathematical foundations of the truncated Floquet-Bloch transform, which can be used to characterise the spectral properties of finite periodic and aperiodic large systems of resonators.Report -
Space-Time Wave Localisation in Systems of Subwavelength Resonators
(2024)SAM Research ReportIn 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 ...Report -
Applications of Chebyshev polynomials and Toeplitz theory to topological metamaterials
(2024)SAM Research ReportWe survey the use of Chebyshev polynomials and Toeplitz theory for studying topological metamaterials. We consider both Hermitian and non-Hermitian systems of subwavelength resonators and provide a mathematical framework to explain some spectacular properties of metamaterials.Report -
Wave packets propagation in the subwavelength regime near the Dirac point
(2024)SAM Research ReportIn [Ammari et al., SIAM J Math Anal., 52 (2020), pp. 5441–5466], the first author with collaborators proved the existence of Dirac dispersion cones at subwavelength scales in bubbly honeycomb phononic crystals. In this paper, we study the time-evolution of wave packets, which are spectrally concentrated near such conical points. We prove that the wave packets dynamics is governed by a time-dependent effective Dirac system, which still ...Report -
Generalised Brillouin Zone for Non-Reciprocal Systems
(2024)SAM Research ReportRecently, it has been observed that the Floquet-Bloch transform with real quasiperiodicities fails to capture the spectral properties of non reciprocal systems. The aim of this paper is to introduce the notion of a generalised Brillouin zone by allowing the quasiperiodicities to be complex in order to rectify this. It is proved that this shift of the Brillouin zone into the complex plane accounts for the unidirectional spatial decay of ...Report -
Tunable Localisation in Parity-Time-Symmetric Resonator Arrays with Imaginary Gauge Potentials
(2024)SAM Research ReportThe aim of this paper is to illustrate both analytically and numerically the interplay of two fundamentally distinct non-Hermitian mechanisms in a deep subwavelength regime. Considering a parity-time symmetric system of one-dimensional subwavelength resonators equipped with two kinds of non-Hermiticity — an imaginary gauge potential and on-site gain and loss — we prove that all but two eigenmodes of the system pass through exceptional ...Report