Convergent Power Series for Fields in Positive or Negative High-Contrast Periodic Media

TL;DR

This paper develops convergent power series methods to represent Bloch waves in high-contrast periodic media, accommodating both positive and negative material coefficients, and analyzing their dispersion relations.

Contribution

It introduces a novel convergent power series approach for Bloch waves in high-contrast media with positive or negative coefficients, extending previous asymptotic methods.

Findings

Positive coefficients yield infinite dispersion branches with convergent power series.

Negative coefficients result in a single dispersion branch.

Power series solutions are explicitly constructed from cell problem solutions.

Abstract

We obtain convergent power series representations for Bloch waves in periodic high-contrast media. The material coefficient in the inclusions can be positive or negative. The small expansion parameter is the ratio of period cell width to wavelength, and the coefficient functions are solutions of the cell problems arising from formal asymptotic expansion. In the case of positive coefficient, the dispersion relation has an infinite sequence of branches, each represented by a convergent even power series whose leading term is a branch of the dispersion relation for the homogenized medium. In the negative case, there is a single branch.