Honeycomb-Lattice Minnaert Bubbles
Habib Ammari, Brian Fitzpatrick, Erik Orvehed Hiltunen, Hyundae Lee, Sanghyeon Yu
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 [Ammari, Hiltunen, and Yu, Arch. Ration. Mech. Anal., 238 (2020), pp. 1559--1583], near the Dirac points, the use of honeycomb crystals of subwavelength resonators as near-zero materials has great potential. 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 resonance. Our first result is to derive an asymptotic formula for the quasi-periodic Minnaert resonance frequencies close to the symmetry points K in the Brilloun zone. Then we obtain the linear dispersion relation of a Dirac cone. Our findings in this paper are illustrated in the case of circular bubbles, where the multipole expansion method provides an efficient technique for computing the band structure.