Existence of Gaps in the Spectrum of Periodic Dielectric Structures on a Lattice

We consider the nite-di erence counterpart, i. e. the true lattice analog of Maxwell's equations and equations that govern the propagation of acoustic waves in a medium with a periodic dielectric structure. In particular, the vector nature of electromagnetic waves is fully taken into account. The existence of true gaps for these lattice models is proved for a two-component medium for which the dielectric constant is everywhere real and positive, and the dielectric constant of the background is essentially larger than the one corresponding to the embedded component.