Electromagnetic coupling in a planar periodic configuration of resonators

We are studying arrays composed of a periodic arrangement of sub-wavelength resonators. An analytical model is developed inside an array of 4 by 4 multi-gap split ring resonators. To describe the frequency splitting of the single fundamental resonance, we propose a simple model based on the approximation of each resonator as an electrical dipole and a magnetic dipole that are driven by the same complex amplitude. We show that the relative strength of the two dipoles strongly depends on cell symmetry. With this approximation, the dispersion relation can be obtained for an infinite size array. A simple matrix diagonalization provides a powerful way to deduce the resonant frequencies for finite size array. These results are comforted by numerical simulations. Finally, an experimental demonstration of a tunable antenna based on this study is presented.