Miniaturization and control of split ring structures from an analytic solution of their resonance

We derived simple polynomial equations to determine the entire resonance spectra of split ring structures. For double stacking split rings made with flat wires, we showed that the resonance frequency depends linearly on the ring-ring separation. In particular, we found that the wavelength of the lowest resonance mode can be made as large as the geometrical size of the ring for realistic experimental conditions, whereas for current systems this ratio is of the order of 10. Finite-difference-time-domain simulations on realistic structures verified the analytic predictions.