Bloch Waves in High Contrast Electromagnetic Crystals

TL;DR

This paper develops analytic formulas and power series to describe the band structure of high contrast electromagnetic crystals, using resonance spectra of quasiperiodic modes to analyze wave behavior.

Contribution

It introduces a novel analytic framework utilizing resonance spectra to represent solutions and derive convergent power series for Bloch wave spectra in high contrast media.

Findings

Derived explicit conditions for the contrast ensuring convergence.

Provided lower bounds on the convergence radius of the power series.

Established criteria for spectral branch separation in dispersion relations.

Abstract

Analytic representation formulas and power series are developed describing the band structure inside non-magnetic periodic photonic three-dimensional crystals made from high dielectric contrast inclusions. Central to this approach is the identification and utilization of a resonance spectrum for quasiperiodic source-free modes. These modes are used to represent solution operators associated with electromagnetic and acoustic waves inside periodic high contrast media. A convergent power series for the Bloch wave spectrum is recovered from the representation formulas. Explicit conditions on the contrast are found that provide lower bounds on the convergence radius. These conditions are sufficient for the separation of spectral branches of the dispersion relation for any fixed quasi-momentum.