Group theoretical description of artificial magnetic metamaterials   utilized for negative index of refraction
  Group theoretical methods are used to determine the electromagnetic properties of artificial magnetic meta-materials, based solely upon the symmetries of the underlying constituent particles. Point groups for such materials are determined. From the transformation properties of an electromagnetic (EM) basis under symmetries of the particles, it is possible to determine, (i) the EM modes of the particles, (ii) the form of constitutive relations (iii) magneto-optical response of a meta-material or lack thereof. These methods are shown to be useful for determination of the isotropic or bi-anisotropic nature of artificial magnetic particles. The results for several artificial magnetic metamaterials are given. Based upon these methods, we predict an ideal planar artificial meta-material, which eliminates an undesirable electric resonance while still exhibiting a magnetic response. Further we determine the subset of point groups of which particles must belong to in order to yield an isotropic 3D magnetic response, and we show an example. 
W.J. Padilla