Transverse Magnetic ENZ Resonators: Robustness and Optimal Shape Design

We study certain "geometric-invariant resonant cavitie"' introduced by Liberal et. al in a 2016 Nature Comm. paper, modeled using the transverse magnetic reduction of Maxwell's equations. The cross-section consists of a dielectric inclusion surrounded by an "epsilon-near-zero" (ENZ) shell. When the shell has the right area, its interaction with the inclusion produces a resonance. Mathematically, the resonance is a nontrivial solution of a 2D divergence-form Helmoltz equation $\nabla \cdot \left(\varepsilon^{-1}(x,\omega) \nabla u \right) + \omega^2 \mu u = 0$, where $\varepsilon(x,\omega)$ is the (complex-valued) dielectric permittivity, $\omega$ is the frequency, $\mu$ is the magnetic permeability, and a homogeneous Neumann condition is imposed at the outer boundary of the shell. This is a nonlinear eigenvalue problem, since $\varepsilon$ depends on $\omega$. Use of an ENZ material in the shell means that $\varepsilon(x,\omega)$ is nearly zero there, so the PDE is rather singular. Working with a Lorentz model for the dispersion of the ENZ material, we put the discussion of Liberal et.~al.~on a sound foundation by proving the existence of the anticipated resonance when the loss is sufficiently small. Our analysis is perturbative in character despite the apparently singular form of the PDE. While the existence of the resonance depends only on the area of the ENZ shell, the rate at which it decays depends on the shape of the shell. We consider an associated optimal design problem: what shape shell gives the slowest-decaying resonance? We prove that if the dielectric inclusion is a ball then the optimal shell is a concentric annulus. For an inclusion of any shape, we study a convex relaxation of the design problem using tools from convex duality, and discuss the conjecture that our relaxed problem amounts to considering homogenization-like limits of nearly optimal designs.