A High-Frequency Homogenization Approach Near the Dirac Points in Bubbly Honeycomb Crystals

Habib Ammari, Erik Orvehed Hiltunen, Sanghyeon Yu

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)


In (Ammari et al. in SIAM J Math Anal. arXiv:1811.03905), the existence of a Dirac dispersion cone in a bubbly honeycomb phononic crystal comprised of bubbles of arbitrary shape is shown. The aim of this paper is to prove that, near the Dirac points, the Bloch eigenfunctions is the sum of two eigenmodes. Each eigenmode can be decomposed into two components: one which is slowly varying and satisfies a homogenized equation, while the other is periodic across each elementary crystal cell and is highly oscillating. The slowly oscillating components of the eigenmodes satisfy a system of Dirac equations. Our results in this paper prove for the first time a near-zero effective refractive index near the Dirac points for the plane-wave envelopes of the Bloch eigenfunctions in a sub-wavelength metamaterial. They are illustrated by a variety of numerical examples. We also compare and contrast the behaviour of the Bloch eigenfunctions in the honeycomb crystal with that of their counterparts in a bubbly square crystal, near the corner of the Brillouin zone, where the maximum of the first Bloch eigenvalue is attained.

Original languageEnglish
Pages (from-to)1559-1583
Number of pages25
JournalArchive for Rational Mechanics and Analysis
Issue number3
Publication statusPublished - 2020 Dec 1

Bibliographical note

Publisher Copyright:
© 2020, The Author(s).

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering


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