Homogenization of the scattered wave and scattering resonances for periodic high-contrast subwavelength resonators

TL;DR

This work addresses time-harmonic scattering by a periodic array of high-contrast subwavelength resonators confined to a bounded domain, and derives a frequency-dependent homogenized model as the period $\varepsilon\to 0$ for the heterogeneous operator $L_\varepsilon$ with $A_\varepsilon = \varepsilon^2\mathbf{1}_{D_\varepsilon} + \mathbf{1}_{\mathbb{R}^d\setminus D_\varepsilon}$. The authors combine two-scale expansions with boundary-layer analysis to obtain quantitative convergence of the scattered field and to identify the limiting set of scattering resonances, described by $\Sigma_{\lim} = \Sigma_{hom} \cup \Sigma_{D,0}$, along with explicit convergence rates for resonances and the far-field pattern. They prove $L^2$ and weighted $H^1$ convergence rates (e.g., $O(\varepsilon^{1/2})$ in general and $O(\varepsilon)$ under higher regularity), characterize the frequency-dependent effective parameters $A_0$ and $\mu_0^k$, and establish robust resolvent estimates via a meromorphic continuation framework for the outgoing problem. The results provide rigorous error control for effective-medium descriptions of resonant metamaterials and lay groundwork for extensions to more complex or random media and for improved numerical simulations near subwavelength resonances.

Abstract

We study time-harmonic scattering by a periodic array of penetrable, high-contrast obstacles with small period, confined to a bounded Lipschitz domain. The strong contrast between the obstacles and the background induces subwavelength resonances. We derive a frequency-dependent effective model in the vanishing-period limit and prove quantitative convergence of the heterogeneous scattered wave to the effective scattered wave. We also identify the limiting set of scattering resonances and establish convergence rates. Finally, we establish convergence rates for the far-field pattern of the heterogeneous problem to that of the effective model.

Figures (4)

• Figure 1: An illustration for the scattering by a periodic array of small inclusions.
• Figure 2: The curves represent the function $k\mapsto k^2\mu_0^k$. On a bounded domain $\Omega$, the limiting spectrum of $-L_{\varepsilon}$ with Dirichlet boundary conditions consists of the spectrum of $-\Delta_D$ together with the red crosses, where each $\lambda_j$ is an element of $\Sigma_{D,1}$ and each $\sigma_j$ is a Dirichlet eigenvalue of $-L_0$ on $\Omega$. It is clear that $\Sigma_{D,1}$ is the essential spectrum, since the red crosses in each interval of $(\lambda_{j-1},\lambda_j)$ converge to each $\lambda_j$.
• Figure 3: An illustration of light propagation: (a) $\mu_0^k<0$, light cannot transmit through the medium; (b) $\mu_0^k \approx 0$, light propagates with a uniform phase inside the medium; (c) $\mu_0^k \rightarrow \infty$, the medium looks iridescent as light passes through, due to strong dispersive effects.
• Figure 4: An illustration of the resonances distribution: the blue crosses denote the eigenvalues in $\Sigma_{D,0}$, the red crosses denote the eigenvalues in $\Sigma_{D,1}$, the green crosses denote the resonances in $\Sigma_{\mathrm{hom}}$, and the black crosses denote the resonances of $L_{\varepsilon}$.

Theorems & Definitions (89)

• Theorem 2.1
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Theorem 2.2
• Remark 6
• Theorem 2.3
• Definition 2.4