Habib Ammari, Brian Fitzpatrick, Erik Orvehed Hiltunen, Hyundae Lee, Sanghyeon Yu

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)5441-5466
Number of pages26
JournalSIAM Journal on Mathematical Analysis
Issue number6
Publication statusPublished - 2020

Bibliographical note

Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics


  • Bubble
  • Dirac cone
  • Honeycomb lattice
  • Minneart resonance
  • Subwavelength bandgap

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics


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