Band-gap structure of the spectrum of periodic Maxwell operators

Abstract

We investigate the band-gap structure of some second-order differential operators associated with the propagation of waves in periodic two-component media. Particularly, the operator associated with the Maxwell equations with position-dependent dielectric constant ∈(x),xεR ³, is considered. The medium is assumed to consist of two components: the background, where ε(x) = ε _b , and the embedded component composed of periodically positioned disjoint cubes, where ε(x) = ε _a . We show that the spectrum of the relevant operator has gaps provided some reasonable conditions are imposed on the parameters of the medium. Particularly, we show that one can open up at least one gap in the spectrum at any preassigned point λ provided that the size of cubesL, the distancel=δL betwen them, and the contrast ε = ε _b /ε _a are chosen in such a way thatL ⁻²∼λ, and quantities ε^-1δ^-3/2 and εδ² are small enough. If these conditions are satisfied, the spectrum is located in a vicinity of widthw∼(εδ^3/2)^-1 of the set {π² L ^-2 k ²:k∈Z³}. This means, in particular, that any finite number of gaps between the elements of this discrete set can be opened simultaneously, and the corresponding bands of the spectrum can be made arbitrarily narrow. The method developed shows that if the embedded component consists of periodically positioned balls or other domains which cannot pack the space without overlapping, one should expect pseudogaps rather than real gaps in the spectrum.