Filling Effect on Constitutive Parameters of Metamaterials With Inductive Inclusions

A theoretical and experimental study of filling effect on the microwave constitutive parameters of a metamaterial with resonant inclusions is performed. The metamaterial is a planar anisotropic layer containing a regular array of coreless split ring inclusions and is considered to be a homogeneous medium. The electric and magnetic susceptibilities of the inclusions are measured and compared with predictions of the generalized Maxwell Garnett theory that accounts for the percolation threshold. The threshold for permittivity is calculated based on the structure of the sample and the sizes of the split rings. The electric interaction of the split rings is close to the dipole interaction of the metal disks. The magnetic interaction depends on the overlapping of ring-induced magnetic fields. When the overlap is negligible, the permeability is defined by the volume of a permeable prolate ellipsoid with magnetic moment equal to the moment induced by the ring. The flatter the split ring, the longer is the prolate ellipsoid and the higher is its volume compared to the volume of the ring inclusion itself. This approach allows the metamaterial constitutive parameters to be calculated based on the dimensions and filling factor of the inclusions, the same as in a powder-filled composite.