Photonic band gaps in quasiperiodic photonic crystals with negative refractive index

Abstract

We investigate the photonic band gaps in quasiperiodic photonic crystals made up of both positive $(\mathrm{Si}{\mathrm{O}}_{2})$ and negative refractive index materials using a theoretical model based on a transfer matrix treatment. The quasiperiodic structures are characterized by the nature of their Fourier spectrum, which can be dense pure point (Fibonacci sequences) or singular continuous (Thue-Morse and double-period sequences). These substitutional sequences are described in terms of a series of generations that obey peculiar recursion relations. We discussed the photonic band gap spectra for both the ideal cases, where the negative refractive index material can be approximated as a constant in the frequency range considered, as well as the more realistic case, taking into account the frequency-dependent electric permittivity $ϵ$ and magnetic permeability $\ensuremath{\mu}$. We also present a quantitative analysis of the results, pointing out the distribution of the allowed photonic bandwidths for high generations, which gives a good insight about their localization and power laws.