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On the validity of the tight-binding method for describing systems of subwavelength resonators
(2021)SAM Research ReportThe goal of this paper is to relate the capacitance matrix formalism to the tight-binding approximation. By doing so, we open the way to the use of mathematical techniques and tools from condensed matter theory in the mathematical and numerical analysis of metamaterials, in particular for the understanding of their topological properties. We firstly study how the capacitance matrix formalism, both when the material parameters are static ...Report -
Honeycomb-lattice Minnaert bubbles
(2018)SAM Research ReportThe aim of this paper is to rigorously show the existence of a Dirac dispersion cone in a bubbly honeycomb phononic crystal comprised of bubbles of arbitrary shape. The main result is an asymptotic formula for the quasi-periodic Minnaert resonance frequencies close to the symmetry points K in the Brilloun zone. This shows the linear dispersion relation of a Dirac cone. Our findings in this paper are illustrated in the case of circular ...Report -
Subwavelength resonances of encapsulated bubbles
(2018)SAM Research ReportThe aim of this paper is to derive a formula for the subwavelength resonance frequency of an encapsulated bubble with arbitrary shape in two dimensions. Using Gohberg-Sigal theory, we derive an asymptotic formula for this resonance frequency, as a perturbation away from the resonance of the uncoated bubble, in terms of the thickness of the coating. The formula is numerically verified in the case of circular bubbles, where the resonance ...Report -
Asymptotic Floquet theory for first order ODEs with finite Fourier series perturbation and its applications to Floquet metamaterials
(2021)SAM Research ReportOur aim in this paper is twofold. Firstly, we develop a new asymptotic theory for Floquet exponents. We consider a linear system of differential equations with a time-periodic coefficient matrix. Assuming that the coefficient matrix depends analytically on a small parameter, we derive a full asymptotic expansion of its Floquet exponents. Based on this, we prove that only the constant order Floquet exponents of multiplicity higher than one ...Report -
Bound states in the continuum and Fano resonances in subwavelength resonator arrays
(2021)SAM Research ReportWhen wave scattering systems are subject to certain symmetries, resonant states may decouple from the far-field continuum; they remain localized to the structure and cannot be excited by incident waves from the far field. In this work, we use layer-potential techniques to prove the existence of such states, known as bound states in the continuum, in systems of subwavelength resonators. When the symmetry is slightly broken, this resonant ...Report -
Functional analytic methods for discrete approximations of subwavelength resonator systems
(2021)SAM Research ReportWe survey functional analytic methods for studying subwavelength resonator systems. In particular, rigorous discrete approximations of Helmholtz scattering problems are derived in an asymptotic subwavelength regime. This is achieved by re-framing the Helmholtz equation as a non-linear eigenvalue problem in terms of integral operators. In the subwavelength limit, resonant states are described by the eigenstates of the generalized capacitance ...Report -
Wave interaction with subwavelength resonators
(2020)SAM Research ReportThe aim of this review is to cover recent developments in the mathematical analysis of subwavelength resonators. The use of sophisticated mathematics in the field of metamaterials is reported, which provides a mathematical framework for focusing, trapping, and guiding of waves at subwavelength scales. Throughout this review, the power of layer potential techniques combined with asymptotic analysis for solving challenging wave propagation ...Report -
Spectral convergence of defect modes in large finite resonator arrays
(2023)SAM Research ReportWe show that defect modes in infinite systems of resonators have corresponding modes in finite systems which converge as the size of the system increases. We study the generalized capacitance matrix as a model for three-dimensional coupled resonators with long-range interactions and consider defect modes that are induced by compact perturbations. If such a mode exists, then there are elements of the discrete spectrum of the corresponding ...Report -
Non-reciprocal wave propagation in space-time modulated media
(2021)SAM Research ReportWe prove the possibility of achieving non-reciprocal wave propagation in space-time modulated media and give an asymptotic analysis of the non-reciprocity property in terms of the amplitude of the time-modulation. Such modulation causes a folding of the band structure of the material, which may induce degenerate points. By breaking time-reversal symmetry, we show that these degeneracies may open into non-symmetric, unidirectional band ...Report -
Robust edge modes in dislocated systems of subwavelength resonators
(2020)SAM Research ReportRobustly manipulating waves on subwavelength scales can be achieved by, firstly, designing a structure with a subwavelength band gap and, secondly, introducing a defect so that localized modes fall within the band gap. We study a one-dimensional array of subwavelength resonators, prove that there is a subwavelength band gap, and show that by introducing a dislocation we can place localized modes at any point within the band gap. We ...Report