On a class of three-phase checkerboards with unusual effective properties

TL;DR

This paper investigates the spectral properties of a class of three-phase periodic checkerboards, revealing unusual effective behaviors and band gaps that challenge traditional homogenization theories.

Contribution

It introduces a detailed analysis of the band spectrum for a specific class of three-phase checkerboards, highlighting cases with unconventional effective properties and limitations of classical homogenization.

Findings

For r^2 > -1, the lowest frequency branch is linear at the origin.

At r^2 = -1, the frequency behaves like sqrt(k) near zero, an unusual behavior.

When r^2 < -1, a zero-frequency band gap appears, and effective medium theory fails.

Abstract

We examine the band spectrum, and associated Floquet-Bloch eigensolutions, arising in a class of three-phase periodic checkerboards. On a periodic cell $[-1,1[^2$, the refractive index is defined by $n^2= 1+ g_1(x_1)+g_2(x_2)$ with $g_i(x_i)= r^2\quad {\rm for} \quad 0\leq x_i<1, \hbox{and} g_i(x_i)= 0\quad {\rm for} \quad -1\leq x_i\leq 0$ where $r^2$ is constant. We find that for $r^2>-1$ the lowest frequency branch goes through origin with linear behaviour, which leads to effective properties encountered in most periodic structures. However, the case whereby $r^2=-1$ is very unusual, as the frequency $\lambda$ behaves like $\sqrt{k}$ near the origin, where $k$ is the wavenumber. Finally, when $r^2<-1$, the lowest branch does not pass through the origin and a zero-frequency band gap opens up. In the last two cases, effective medium theory breaks down even in the quasi-static limit, while the high-frequency homogenization [Craster et al., Proc. Roy. Soc. Lond. A 466, 2341-2362, 2010] neatly captures the detailed features of band diagrams.