Modeling of Isotropic Backward-Wave Materials Composed of Resonant   Spheres
  A possibility to realize isotropic artificial backward-wave materials is theoretically analyzed. An improved mixing rule for the effective permittivity of a composite material consisting of two sets of resonant dielectric spheres in a homogeneous background is presented. The equations are validated using the Mie theory and numerical simulations. The effect of a statistical distribution of sphere sizes on the increasing of losses in the operating frequency band is discussed and some examples are shown. 
L. Jylha, I. Kolmakov, S. Maslovski, S. Tretyakov