Electric and magnetic resonances in broadside coupled split-ring resonators: An extended mode-expansion theory

We extend the previously established mode-expansion theory to study the eigenmodes of metallic ring systems made by thin wires possessing filmlike rectangular cross sections. Applications of the theory to a single split-ring resonator (SRR) yield essentially the same results with finite-difference-time-domain (FDTD) simulations on realistic structures, which justify some basic assumptions adopted in the theory. We then apply the theory to study the resonance properties of a broadside coupled split-ring resonator and show that such a planar resonator exhibits magnetic responses along all three dimensions under different conditions. FDTD simulations on realistic structures are performed to successfully verify the predictions based on the extended mode-expansion theory and reveal that mutual-SRR interactions in a periodic SRR array may lead to a significant change of the eigenmode properties, including a reversal of the frequency sequence for two resonance modes. We finally employ FDTD simulations to design a realistic layered metamaterial that exhibits magnetic responses along all three dimensions.