In the field of metamaterials, many intriguing phenomena arise from having a structure which is periodic in space. In time-dependent structures, conceptually similar properties can arise, which nevertheless have fundamentally different physical implications. In this work, we study time-dependent systems in the context of subwavelength metamaterials. The main result is a capacitance matrix characterization of the band structure, which generalizes previous recent work on static subwavelength metamaterials. This characterization provides both theoretical insight and efficient numerical methods to compute the dispersion relationship of time-dependent structures. We exemplify this in several structures exhibiting interesting wave manipulation properties.
The 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 problems at subwavelength scales is demonstrated.
Halide perovskites are promising materials with many significant applications in photovoltaics and optoelectronics. In this paper, we use integral methods to quantify the resonant properties of halide perovskite nano-particles. We prove that, for arbitrarily small particles, the subwavelength resonant frequencies can be expressed in terms of the eigenvalues of the Newtonian potential associated with its shape. We also characterize the hybridized subwavelength resonant frequencies of a dimer of two halide perovskite particles. Finally, we examine the specific case of spherical resonators and de…
In this paper, we study the behaviour of the coupled subwavelength resonant modes when two high-contrast acoustic resonators are brought close together. We consider the case of spherical resonators and use bispherical coordinates to derive explicit representations for the capacitance coefficients which, we show, capture the system's resonant behaviour at leading order. We prove that the pair of resonators has two subwavelength resonant modes whose frequencies have different leading-order asymptotic behaviour. We, also, derive estimates for the rate at which the gradient of the scattered pressu…
Robustly 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 eigenfrequencies fall within the band gap. Such frequencies are well known to correspond to localized modes. We study a one-dimensional array of subwavelength resonators, proving that there is a subwavelength band gap, and showing that by introducing a dislocation we can place localized modes at any point within the band gap. We complement this analysis by studying the stability properties of the corresponding finite ar…
When 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 state can be excited from the far field. Remarkably, this may create asymmetric (Fano-type) scattering behaviour where the transmission is fundamentally differ…
We present a design for an acoustic metamaterial that mimics the behaviour of the active cochlea. This material is composed of a size-graded array of cylindrical subwavelength resonators, has similar dimensions to the cochlea and is able to reproduce the frequency separation of audible frequencies. Non-linear amplification is introduced to the model in order to replicate the behaviour of the cochlear amplifier. This formulation takes the form of a fluid-coupled array of Hopf resonators. We seek solutions based on a modal decomposition, so as to retain the physically-derived coupling between re…
The goal of this paper is to advance the development of wave-guiding subwavelength crystals by developing designs whose properties are stable with respect to imperfections in their construction. In particular, we make use of a locally resonant subwavelength structure, composed of a chain of high-contrast resonators, to trap waves at deep subwavelength scales. We first study an infinite chain of subwavelength resonator dimers and define topological quantities that capture the structure's wave transmission properties. Using this for guidance, we design a finite crystal that is shown to have wave…
The 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 and modulated, can be posed in a Hamiltonian form. Then, we use this result to compare this formalism to the tight-binding approximation. We prove that the cor…
Wave scattering structures with amplification and dissipation can be modelled by non-Hermitian systems, opening new ways to control waves at small length scales. In this work, we study the phenomenon of topologically protected edge states in acoustic systems with gain and loss. We demonstrate that localized edge modes appear in a periodic structure of subwavelength resonators with a defect in the gain/loss distribution, and explicitly compute the corresponding frequency and decay length. Similarly to the Hermitian case, these edge modes can be attributed to the winding of the eigenmodes. In th…
Systems 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 resonators and search for high-order asymptotic exceptional points. This analysis reveals the range of different configurations that can give rise to high-order e…
The aim of this work is to derive precise formulas which describe how the properties of subwavelength devices are changed by the introduction of errors and imperfections. As a demonstrative example, we study a class of cochlea-inspired rainbow sensors. These are devices based on a graded array of subwavelength resonators which have been designed to mimic the frequency separation performed by the cochlea. We show that the device's properties (including its role as a signal filtering device) are stable with respect to small imperfections in the positions and sizes of the resonators. Additionally…
We investigate the Fano resonance in grating structures by using coupled resonators. The grating consists of a perfectly conducting slab with periodically arranged subwavelength slit holes, where inside each period, a pair of slits sit very close to each other. The slit holes act as resonators and are strongly coupled. It is shown rigorously that there exist two groups of resonances corresponding to poles of the scattering problem. One sequence of resonances have imaginary part on the order of $\varepsilon$, where $\varepsilon$ is the size of the slit aperture, while the other sequence have im…
When 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, where eigenmodes are linearly dependent. The primary goal of this work is to study the existence of exceptional points in high-contrast subwavelength metamaterials. We begin by studying a parity-time-symmetric pair of subwavelength resonators and prove that this system supports asymptotic exceptional points. These are points at which the subwavelength eigenvalues and eigenvectors coincide at l…
The aim of this paper is to understand the behaviour of a large number of coupled subwavelength resonators. We use layer potential techniques in combination with numerical computations to study the acoustic pressure field due to scattering by a graded array of subwavelength resonators. Using this method, we study a graded-resonance model for the cochlea. We compute the resonant modes of the system and explore the model's ability to decompose incoming signals. We are able to offer mathematical explanations for the cochlea's so-called "travelling wave" behaviour and tonotopic frequency map.
Mathematical Analysis of Subwavelength Resonances and Gradient Blow-up for Two Close-to-Touching Inclusions within the Two-Dimensional Elasticity
Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心，清华大学）；Academy for Multidisciplinary Studies, Capital Normal University（首都师范大学多学科研究院）；Qiuzhen College, Tsinghua University（清华大学求真学院）
The ability to manipulate the propagation of waves on subwavelength scales is important for many different physical applications. In this paper, we consider a honeycomb lattice of subwavelength resonators and prove, for the first time, the existence of a Dirac dispersion cone at subwavelength scales. As shown in [H. Ammari et al., A high-frequency homogenization approach near the Dirac points in bubbly honeycomb crystals, arXiv:1812.06178], near the Dirac points, honeycomb crystals of subwavelength resonators has a great potential to be used as near-zero materials. Here, we perform the analysi…
Our 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 will be perturbed linearly. The required multiplicity can be achieved via folding of the system through certain choices of the periodicity of the coefficient…
Frequency-dependent capacitance matrix formulation for Fabry-Perot resonances in two and three dimensional systems
ETH Zürich（苏黎世联邦理工学院）；Hong Kong Institute for Advanced Study, City University of Hong Kong（香港城市大学高等研究院）；City University of Hong Kong（香港城市大学）；Institute of Fundamental and Transdisciplinary Research, Zhejiang University（浙江大学基础与交叉研究院）；School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）
