Analytical model of the fundamental mode of 3D square split ring resonators

Abstract

An analytical model is developed for the charge and potential distributions of the fundamental mode of three-dimensional (3D) singly split ring resonators (SRRs) with square cross section, which nowadays can be conveniently fabricated by additive manufacturing techniques. This model allows the derivation of approximate formulas for the equivalent capacitance and the analysis of the resonant properties of this type of SRRs at any frequency. The total capacitance is expressed as the sum of the usual gap capacitance and a surface capacitance associated with the charges on the SRR walls, which are determined from the solution for the irrotational part of the electric field of a square split cylinder obtained by conformal mapping. The applicability of the proposed model for a broad range of SRR parameters is demonstrated by comparing the resonance frequency of sample SRR configurations found theoretically with the corresponding values obtained by numerical simulations and experiments. Furthermore, the functions describing the charge and current mode profiles provided in this work can be instrumental in estimating the near-field interaction and the coupling constants of an ensemble of resonant 3D square split rings and thus for tailoring the response of metamaterials and other devices formed by these elements.