Biharmonic split ring resonator metamaterial: Artificially dispersive effective density in thin periodically perforated plates

¹ Institute of Condensed Matter Theory and Solid State Optics, Abbe Center of Photonics, Friedrich-Schiller-Universität Jena - D-07743 Jena, Germany
² Division of Computer, Electrical, and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST) - Thuwal 23955-6900, Saudi Arabia
³ Aix-Marseille Université, CNRS, Centrale Marseille, Institut Fresnel, Campus universitaire de Saint-Jérôme 13013 Marseille, France

Received: 27 July 2014
Accepted: 4 August 2014

Abstract

We present in this paper a theoretical and numerical analysis of bending waves localized on the boundary of a platonic crystal whose building blocks are Split Ring Resonators (SRR). We first derive the homogenized parameters of the structured plate using a three-scale asymptotic expansion in the linearized biharmonic equation. In the limit when the wavelength of the bending wave is much larger than the typical heterogeneity size of the platonic crystal, we show that it behaves as an artificial plate with an anisotropic effective Young modulus and a dispersive effective mass density. We then analyze dispersion diagrams associated with bending waves propagating within an infinite array of SRR, for which eigen-solutions are sought in the form of Floquet-Bloch waves. We finally demonstrate that this structure displays the hallmarks of All-Angle Negative Refraction (AANR) and it leads to superlensing and ultrarefraction effects, interpreted thanks to our homogenization model as a consequence of negative and vanishing effective density, respectively.