Creating band gaps in periodic media

TL;DR

This paper develops a novel theoretical framework to identify explicit geometric and material conditions for creating band gaps in 2D photonic and 3D acoustic crystals using electro-static and quasi-periodic resonances.

Contribution

It introduces a new approach leveraging resonance spectra to systematically determine conditions for band gap formation in high contrast periodic media.

Findings

Explicit conditions for band gap creation derived

Spectral representation of solution operators established

Method applicable to design photonic and phononic crystals

Abstract

We identify explicit conditions on geometry and material contrast for creating band gaps in 2-d photonic and 3-d acoustic crystals. This approach is novel and makes use of the electro-static and quasi-periodic source free resonances of the crystal. The source free modes deliver a spectral representation for solution operators associated with propagation of electromagnetic and acoustic waves inside periodic high contrast media. An accurate characterization of the quasi-periodic and electrostatic resonance spectrum in terms of the shape and geometry of the scatters is possible. This information together with the Dirichlet and a Neumann like spectra associated with the inclusions deliver conditions sufficient for opening band gaps at finite contrast. The theory provides a systematic means for the identification of photonic and phononic band gaps within a specified frequency range.