Honeycomb-lattice Minnaert bubbles

Abstract

The ability to manipulate the propagation of waves on subwavelength scales is important for many different physical applications. In this paper, we consider a honeycomb lattice of subwavelength resonators and prove, for the first time, the existence of a Dirac dispersion cone at subwavelength scales. As shown in [H. Ammari et al., A high-frequency homogenization approach near the Dirac points in bubbly honeycomb crystals, arXiv:1812.06178], near the Dirac points, honeycomb crystals of subwavelength resonators has a great potential to be used as near-zero materials. Here, we perform the analysis for the example of bubbly crystals, which is a classic example of subwavelength resonance, where the resonant frequency of a single bubble is known as the Minnaert resonant frequency. Our first result is an asymptotic formula for the quasi-periodic Minnaert resonant frequencies. We then prove the linear dispersion relation of a Dirac cone. Our findings in this paper are numerically illustrated in the case of circular bubbles, where the multipole expansion method provides an efficient technique for computing the band structure.

31 pages, 6 figures. Revised to include more details