Low-frequency Scattering Analysis and Homogenisation of Split-ring Elements

A key structure in so-called metamaterial mediums is the elementary split-ring resonator. We consider in this paper the lowfrequency electromagnetic scattering by a split-ring particle modelled as a perfectly conducting wire ring, furnished with a narrow gap, and derive analytical solutions for the electric and magnetic dipole moments for different kinds of incidence and polarisation in the quasistatic approximation. Through a vectorial homogenisation process, the expressions discovered for the dipole moments and the related polarisability dyadics are linked with the macroscopic constitutive equations for the medium. We further show that the condition for resonance of a medium consisting of simple split-rings cannot be achieved by means of the given quasi-static terms without violating the underlying assumptions of homogenisation. Nevertheless, the results are applicable for sparse medium of rings, and we derive numerical guidelines for the applicability with some examples of the effect of the considered split-ring medium on electromagnetic wave propagation.