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 $g_i(x_i)= r^2\quad {\rm for} \quad 0\leq x_i<1, \hbox{and} g_i(x_i)= 0\quad {\rm for} \quad -1\leq x_i\leq 0$ 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.

Citation

(2011). On a class of three-phase checkerboards with unusual effective properties. Comptes Rendus Mécanique, 411-417. https://doi.org/10.1016/j.crme.2011.03.016

• Acceptance Date: Mar 11, 2011
• Publication Date: May 20, 2011
• Journal: Comptes Rendus Mécanique
• Print ISSN: 1631-0721
• Publisher: Elsevier
• Pages: 411-417
• DOI: https://doi.org/10.1016/j.crme.2011.03.016
• Keywords: waves, homogenization, negative refraction, acoustic band
• Publisher URL: http://dx.doi.org/10.1016/j.crme.2011.03.016