Scattering from time-modulated subwavelength resonators

TL;DR

This work develops a mathematical framework for wave scattering by time-modulated, high-contrast subwavelength resonators in one dimension. It combines a pole-pencil decomposition of the scattered field with a higher-order discrete capacitance-matrix approximation to characterize subwavelength resonant quasifrequencies, including their imaginary parts, and derives a finite-dimensional linear system suitable for numerical computation. A novel energy notion for time-modulated systems is introduced, revealing that energy is generally not conserved when material parameters are modulated in time, with the total energy redistributed among Floquet modes and amplified near resonances. Numerical investigations across multiple resonator counts show energy gain or loss depending on modulation strength and operating frequency, highlighting the potential for controlled energy exchange via time modulation. The framework lays groundwork for future effective-medium theories in dilute limits and extensions to higher dimensions, with open-source code available for reproducibility.

Abstract

We consider wave scattering from a system of highly contrasting resonators with time-modulated material parameters. In this setting, the wave equation reduces to a system of coupled Helmholtz equations that models the scattering problem. We consider the one-dimensional setting. In order to understand the energy of the system, we prove a novel higher-order discrete, capacitance matrix approximation of the subwavelength resonant quasifrequencies. Further, we perform numerical experiments to support and illustrate our analytical results and show how periodically time-dependent material parameters affect the scattered wave field.

Figures (8)

• Figure 3.1: The imaginary parts of the subwavelength resonant frequencies $\omega_0$ and $\omega_1$ for fixed $\delta=0.0001,\,K=4$ and static material parameters, as a function of $\ell$. Note that the real parts of $\omega_0$ and $\omega_1$ are both zero. We use a logarithmic $x$-axis to present our numerical results.
• Figure 3.2: The imaginary parts of the non-zero quasifrequency $\omega_1$ for fixed $\delta=0.0001,\,v_{\mathrm{r}}=v_0=1,\,K=4,\,\Omega=0.03$ and time-modulated material parameters, as a function of $\ell$. Note that the quasifrequencies are purely imaginary. We use logarithmic axes to present our numerical results.
• Figure 3.3: The imaginary parts of the subwavelength resonant quasifrequencies as a function of $\varepsilon_{\kappa}$ for $N\in\{1,2,6\}$ resonators each of length $\ell=2$ and spacing $\ell_{i(i+1)}=10$ at time $t=0$ with fixed $\delta=0.0001,\,v_{\mathrm{r}}=v_0=1,\,K=4,\,\Omega=0.03$. The red dashed line in the left plot shows the approximation computed via (\ref{['eq:om1_Cepsk']}).
• Figure 4.1: The scattered wave field at time $t=0$ for $N=6$ resonators each of length $\ell=2$ with a spacing of $\ell_{i(i+1)}=10$ between two neighbouring resonators. The operating frequency is taken to be very close to the subwavelength quasifrequency of (\ref{['eq:new_CapApprox']}) with the largest real part. The corresponding parameters are: $\delta=0.0001,\,v_{\mathrm{r}}=v_0=1,\,\Omega=0.03$.
• Figure 4.2: The real and imaginary parts of the total wave field measured at several times $t$. These numerical results were obtained for $N=6$ resonators each of length $\ell=2$ and spacing $\ell_{i(i+1)}=10$. The operating frequency is taken to be very close to the subwavelength quasifrequency of (\ref{['eq:new_CapApprox']}) with the largest real part. The corresponding parameters are: $\delta=0.0001,\,v_{\mathrm{r}}=v_0=1,\,\Omega=0.03$.

Theorems & Definitions (16)

• Lemma 2.1
• Remark 2.1
• Theorem 3.1
• Remark 3.1
• proof
• Corollary 3.2
• proof
• Remark 3.2
• Lemma 4.1
• proof