Creating and controlling band gaps in periodic media with small resonators

Abstract

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