Periodic Maxwell's operators with preassigned gaps in spectrum

We study the Maxwell operator in two dimensional dielectric medium with small heterogeneous inclusions that are periodically distributed with a small period $\varepsilon$. Media with such a structure are typical for photonic meta-materials being artificial composite materials with required electromagnetic properties. Spectrum of the Maxwell operator in such a medium is continuous and can have gaps, but their presence is not guarantied. On the other hand, it is an important application in radio-engineering to know the location of gaps in the spectrum. That is why our main purpose is to construct inclusions which provide existence of preassigned gaps in the spectrum.

We consider the traps-like inclusions that are the annuli of the completely conducting material with slim slits. We prove that for sufficiently small $\varepsilon$ the spectrum of the Maxwell operator is a finite gap with the edges converging to given numbers as $\varepsilon\to 0$. We establish a one-to-one correspondence between parameters of the traps-like inclusions and the edges of the limiting spectrum.