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

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 $g_i(x_i)=r^2\mathrm{for}0\le x_i<1,and{g}_i(x_i)=0\mathrm{for}-1\le x_i\le 0$ 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.

8 pages, 3 figures

References in corpus (1)

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