Transmission properties of time-dependent one-dimensional metamaterials

Abstract

We solve the wave equation with periodically time-modulated material parameters in a one-dimensional high-contrast resonator structure in the subwavelength regime exactly, for which we compute the subwavelength quasifrequencies numerically using Muller’s method. We prove a formula in the form of an ODE using a capacitance matrix approximation. Comparison of the exact results with the approximations reveals that the method of capacitance matrix approximation is accurate and significantly more efficient. We prove various transmission properties in the aforementioned structure and illustrate them with numerical simulations. In particular, we investigate the effect of time-modulated material parameters on the formation of degenerate points, band gaps and k-gaps.

Scattering from time-modulated subwavelength resonators

“…In order to allow for further numerical experiments involving the subwavelength quasifrequencies, we have proved a higher-order, discrete, capacitance matrix approximation for a system of resonators in theorem 3.1. It is worth emphasizing that our formula approximates the subwavelength quasi-frequencies up to order O(δ 3/2 ), as opposed to O(δ), formerly derived in [5,Theorem V.3]. Moreover, we derived a specific formulation of the non-zero subwavelength resonant frequency of a single-resonator system in corollary 3.3, which can be seen as a generalization of [10,Remark 3.4] to time-modulated systems.…”

“…Proof. The proof is similar to the proof of [5, Theorem III.4], but contains terms originating from the outgoing radiation condition, which is not present in [5]. Using the continuity condition on the boundaries of D i , we can establish the following relation for the exterior coefficients:…”

“…Under these assumptions, we know from [1,5] that (2.6) with u in = 0 admits 2N subwavelength resonant quasi-frequencies ω i , i = 1, . .…”

“…, N − 1. Moreover, the following lemma characterizing the solution to the Helmholtz equation inside the resonators has been proved in [5].…”

“…Time-modulated subwavelength metamaterials have recently emerged as a new paradigm for wave manipulation at subwavelength scales [1][2][3][4]. The exploration of time as a novel degree of freedom for metamaterials has led to a plethora of wave phenomena such as non-reciprocity, zero refractive index and wave amplification [1,[5][6][7]. The building blocks of subwavelength metamaterials are subwavelength resonators: objects exhibiting resonant phenomena in response to wavelengths much greater than their size.…”