Nonlinear subwavelength resonances in three dimensions

TL;DR

The paper develops a discrete, Dirichlet-to-Neumann–based framework to approximate nonlinear subwavelength resonances in three-dimensional high-contrast resonator systems with Kerr-type nonlinearity. By extending the linear subwavelength theory through the generalized capacitance matrix to the nonlinear regime, it derives a semilinear variational formulation and asymptotic expansions that reveal nonlinear-induced eigenmodes and amplitude-dependent resonance shifts. The leading-order nonlinear eigenproblem ${\rm cap}^{gen}(\mathcal D)q_0-\omega_0^2\bigl(q_0-\beta c_r^2|q_0|^2q_0\bigr)=0$ generalizes the classical cap^{gen} framework and predicts multiple resonances, including a potential third branch for large amplitudes. Numerical experiments on a nonlinear dimer illustrate the emergence and interaction of nonlinear resonances under symmetry and amplitude variations, underscoring the rich nonlinear subwavelength physics and motivating further study of nonlinear soliton-like modes and topological transitions in structured media.

Abstract

In this paper, we consider the resonance problem for the cubic nonlinear Helmholtz equation in the subwavelength regime. We derive a discrete model for approximating the subwavelength resonances of finite systems of high-contrast resonators with Kerr-type nonlinearities. Our discrete formulation is valid in both weak and strong nonlinear regimes. Compared to the linear formulation, it characterizes the extra eigenmodes induced by the non-linearlity that have recently been experimentally observed.

Figures (2)

• Figure 5.1: In these plots the solution curves of equation \ref{['eq:char_resonance_num']} for different choices of $\mathop{\mathrm{\mathcal{D}}}\nolimits$ are plotted. From left to right, $\mathop{\mathrm{\mathcal{D}}}\nolimits = B_{0.2}\left(-\frac{1}{2}e_3\right) \cup B_{0.2}\left(\frac{1}{2}e_3\right)$, $\mathop{\mathrm{\mathcal{D}}}\nolimits = B_{0.2}\left(-\frac{1}{2}e_3\right) \cup B_{0.21}\left(\frac{1}{2}e_3\right)$ and $\mathop{\mathrm{\mathcal{D}}}\nolimits = B_{0.2}\left(-\frac{1}{2}e_3\right) \cup B_{0.22}\left(\frac{1}{2}e_3\right)$, respectively. The remaining parameters are given by $\beta = -0.1i\left|B_{0.2}\left(-\frac{1}{2}e_3\right)\right|^{-2}$ and ${c_r} = 1$. The colorful curves trace the magnitude of the entries of different solutions $q_0$ to equation \ref{['eq:char_resonance_num']} (the respective $\omega_0$ can be found in Figure \ref{['fig:resonancec_plot']}). The colors account for the complex valued solutions $q_0 \in \mathbb{C}^2$ and depict the phase of $(q_0)_1 / (q_0)_2$. The black dashed curves are the solutions to the respective linear problem when $\beta = 0$.
• Figure 5.2: In this figure the leading order asymptotics of the subwavelength resonances associated to the settings discussed in the caption of Figure \ref{['fig:solution_plot']} are displayed. The leading order asymptotics of the resonances are displayed as curves in the complex plane. The color of each curve is associated to the color of a solution curve $q_0$ depicted in the inlet plots.

Theorems & Definitions (35)

• Definition 1.1: Subwavelength resonance
• Theorem 1.2: Fundamental theorem of subwavelength physics review
• Lemma 2.1
• Definition 2.2
• Definition 2.3: Capacitance matrix
• Proposition 3.1
• proof
• Corollary 3.2
• Definition 3.3
• Lemma 3.4