Diamagnetic properties of metamaterials: a magnetostatic analogy

The response of a metamaterial, consisting of a 3D lattice of lossy capacitively loaded metallic loops is studied theoretically when it is inserted into a homogeneous harmonically varying magnetic field. The current distribution is found by taking into account the magnetic coupling between any pair of loops in the approximation of no retardation. It is shown that in a frequency range above its resonant frequency the metamaterial behaves as a diamagnet expelling the applied magnetic field. As the resonant frequency is approached the magnetic field is shown to be expelled not only from the volume of the metamaterial but from a larger zone which in the vicinity of the resonant frequency takes the form of a sphere. In the lossless case the radius of this exclusion sphere tends to infinity. In the presence of losses the maximum radius is limited by the quality factor of the individual elements. The response of a single element is shown to be analogous to that of a sphere of magnetic material, an analogy that leads to an alternative definition of effective permeability.