Time-dependent high-contrast subwavelength resonators

Abstract

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. Based on this characterization, we numerically compute the dispersion relationship of several time-dependent structures exhibiting interesting wave manipulation properties.

On the validity of the tight-binding method for describing systems of subwavelength resonators

“…As shown in [11], the functions v * i,n (x) are asymptotically constant inside the resonators as δ → 0. In other words, we have v * i,n (x) = c i,n + o(δ).…”

“…We assume that D ⊂ R d is constituted by N disjoint bounded domains D i for i = 1, ..., N , each D i being connected and having boundary of Hölder class ∂D i ∈ C 1,s , 0 < s < 1. We will assume that the modulation only takes place inside the resonators, which was shown in [11] to be an effective way to obtain phenomena not achievable in the static case. The material parameters are hence discontinuous and of the form…”

“…We seek solutions to (2.1) under the modulation specified in (2.2). From the regularity of ρ and κ, we know that u is continuously differentiable in t (see [11]). Since e −iωt u(x, t) is a τ -periodic function of t, we have a Fourier series expansion as…”

“…Note that condition (ii) means that u only contains components which are either in the subwavelength regime (with frequencies of order of √ δ), or which are asymptotically small as δ → 0, see [11].…”

“…The capacitance matrix formulation is a discrete approximation of the wave equation. It has enabled the mathematical justification of many of the remarkable applications of metamaterials listed above [5,7,10,14,16], also in the case of time-dependent material parameters [11]. See also the recent review paper on the topic [4].…”