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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 -
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 -
Exceptional points in parity-time-symmetric subwavelength metamaterials
(2020)SAM Research ReportWhen sources of energy gain and loss are introduced to a wave scattering system, the underlying mathematical formulation will be non-Hermitian. This paves the way for the existence of exceptional points, whereby eigenmodes are linearly dependent. The main goal of this work is to study the existence and consequences of exceptional points in the setting of high-contrast subwavelength metamaterials. We begin by studying a system of two ...Report -
Topologically protected edge modes in one-dimensional chains of subwavelength resonators
(2019)SAM Research ReportReport -
High-order exceptional points and enhanced sensing in subwavelength resonator arrays
(2020)SAM Research ReportSystems exhibiting degeneracies known as exceptional points have remarkable properties with powerful applications, particularly in sensor design. These degeneracies are formed when eigenstates coincide, and the remarkable effects are exaggerated by increasing the order of the exceptional point (that is, the number of coinciding eigenstates). In this work, we use asymptotic techniques to study PT-symmetric arrays of many subwavelength ...Report