Space-Time Wave Localisation in Systems of Subwavelength Resonators

TL;DR

This paper addresses space-time localisation in a 1D metamaterial made of high-contrast subwavelength resonators with time-periodic parameters. It develops a first-principles capacitance-matrix framework that reduces the subwavelength dynamics to an ODE system $M^{\alpha}(t)\Psi(t)+\Psi''(t)=0$, with $M^{\alpha}(t)$ built from a generalised capacitance matrix $C^{\alpha}$ and time modulations. It proves a real-space capacitance formulation and a Toeplitz matrix approach to compute localized defect modes, and shows that simultaneous band and momentum gaps can be engineered by modulating $s(t)$ and $\kappa(t)$. Numerical experiments demonstrate space-time localised waves and introduce a time-dependent degree of localisation $d_*(t)$ that peaks at predicted times, validating the theory. The results provide a rigorous mathematical basis for subwavelength space-time control and suggest extensions to non-Hermitian and skin-effect phenomena in time-modulated metamaterials.

Abstract

In this paper we study the dynamics of metamaterials composed of high-contrast subwavelength resonators and show the existence of localised modes in such a setting. A crucial assumption in this paper is time-modulated material parameters. We prove a so-called capacitance matrix approximation of the wave equation in the form of an ordinary differential equation. These formulas set the ground for the derivation of a first-principles characterisation of localised modes in terms of the generalised capacitance matrix. Furthermore, we provide numerical results supporting our analytical results showing for the first time the phenomenon of space-time localised waves in a perturbed time-modulated metamaterial. Such spatio-temporal localisation is only possible in the presence of subwavelength resonances in the unperturbed structure. We introduce the time-dependent degree of localisation to quantitatively determine the localised modes and provide a variety of numerical experiments to illustrate our formulations and results.

Figures (6)

• Figure 2.1: An illustration of the setup of an infinite material with $N=2$ resonators in the unit cell $Y$.
• Figure 2.2: The band structure corresponding to $N=3$ resonators with the following parameter values: $\ell_1=\ell_2=\ell_3=1,\,\ell_{12}=\ell_{34}=1,\,\ell_{23}=2,\,v_0=v_{\mathrm{r}}=1,\,\Omega=0.034,\delta=0.0001,\,\varepsilon_{\kappa}=0.4,\,\varepsilon_{s}=0.2$. The blue line marks the real parts and the red dots the imaginary parts of the subwavelength resonant frequencies. The blue shaded area marks a momentum gap and the red shaded area marks a band gap.
• Figure 4.1: The spatial degree of localisation $d_i$ corresponding to $20$ reoccurring unit cells of $N=3$ resonators, where there is a defect in the wave speed inside $D^0_2$. Each resonator is of length $\ell=1$ and the spacing between two neighbouring resonators is $\ell_{ij}=2$.
• Figure 4.2: The eigenvalues $\left(\lambda_i^{\alpha}\right)_{i=1}^N$ corresponding to $20$ reoccurring unit cells of $N=3$ resonators, where there is a defect in the periodicity of $\kappa_2(t)$ inside $D^0_2$. Each resonator is of length $\ell=1$ and the spacing between two neighbouring resonators is $\ell_{ij}=2$. Note that in the static case $\lambda_i^{\alpha}\equiv1$, hence, it is impossible to achieve time localisation.
• Figure 4.3: The time-dependent degree of localisation $d_*(t)$ corresponding to $25$ recurrences of a unit cell with $N=3$ resonators of length $\ell=1$ and spacing $\ell_{12}=\ell_{34}=1,\,\ell_{23}=2$. The material parameters are chosen to be $v_0=v_{\mathrm{r}}=1,\,\Omega=0.034,\,\delta=0.0001,\,\varepsilon_{\kappa}=0.4,\,\varepsilon_{s}=0.2,\,\alpha=0.01$. The material has a spatial defect at $D_2^0$, where we set $v_{\mathrm{r}}=2$, and a temporal defect at $D_2^m$ of $c_2=1$ with $t_0=T=184.7996\,\mathrm{s}$.

Theorems & Definitions (31)

• Remark 2.1
• Definition 2.2: Operator $I_{\partial D}$
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• Lemma 2.4
• proof
• Lemma 2.5