Creating and controlling band gaps in periodic media with small resonators

TL;DR

This paper studies how small resonators periodically placed in a medium can create and control spectral band gaps in the spectrum of the Neumann Laplacian, with implications for photonic crystal design.

Contribution

It demonstrates that the spectrum of the Neumann Laplacian on a domain with small resonators has at least as many gaps as resonator families, with controllable gap positions and sizes.

Findings

At least m spectral gaps are present for m resonator families.

The first m gaps converge to controllable intervals as resonator size shrinks.

Additional gaps, if any, tend to infinity.

Abstract

We investigate spectral properties of the Neumann Laplacian $\mathscr{A}_\varepsilon$ on a periodic unbounded domain $\Omega_\varepsilon$ depending on a small parameter $\varepsilon>0$. The domain $\Omega_\varepsilon$ is obtained by removing from $\mathbb{R}^n$ $m\in\mathbb{N}$ families of $\varepsilon$-periodically distributed small resonators. We prove that the spectrum of $\mathscr{A}_\varepsilon$ has at least $m$ gaps. The first $m$ gaps converge as $\varepsilon\to 0$ to some intervals whose location and lengths can be controlled by a suitable choice the resonators; other gaps (if any) go to infinity. An application to the theory of photonic crystals is discussed.