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Negative refraction, surface modes, and superlensing effect via homogenization near resonances for a finite array of split-ring resonators

M. Farhat1, S. Guenneau2, S. Enoch1, and A. B. Movchan2

  • 1Institut Fresnel, CNRS, Aix-Marseille Université, Campus Universitaire de Saint-Jérôme, 13013 Marseille, France
  • 2Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 3BX, United Kingdom

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Phys. Rev. E 80, 046309 – Published 12 October, 2009

DOI: https://doi.org/10.1103/PhysRevE.80.046309

Abstract

We present a theoretical and numerical analysis of liquid surface waves (LSWs) localized at the boundary of a phononic crystal consisting of split-ring resonators (SRRs). We first derive the homogenized parameters of the fluid-filled structure using a three-scale asymptotic expansion in the linearized Navier-Stokes equations. In the limit when the wavelength of the LSW is much larger than the typical heterogeneity size of the phononic crystal, we show that it behaves as an artificial fluid with an anisotropic effective shear modulus and a dispersive effective-mass density. We then analyze dispersion diagrams associated with LSW propagating within an infinite array of SRR, for which eigensolutions are sought in the form of Floquet-Bloch waves. The main emphasis is given to the study of localized modes within such a periodic fluid-filled structure and to the control of low-frequency stop bands associated with resonances of SRRs. Considering a macrocell, we are able to compute the dispersion of LSW supported by a semi-infinite phononic crystal of SRRs. We find that the dispersion of this evanescent mode nearly sits within the first stop band of the doubly periodic structure. We further discover that it is linked to the frequency at which the effective-mass density of the homogenized phononic crystal becomes negative. We demonstrate that this surface mode displays the hallmarks of all-angle negative refraction and it leads to a superlensing effect. Last, we note that our homogenization results for the velocity potential can be applied mutatis mutandis to designs of electromagnetic and acoustic superlenses for transverse electric waves propagating in arrays of infinite conducting SRRs and antiplane shear waves in arrays of cracks shaped as SRRs.

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Phys. Rev. E 80, 046309– Published 12 October, 2009
  • Received 10 March 2009

© 2009 American Physical Society

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