Effects of parameter variations on negative effective constitutive parameters of non-metallic metamaterials

Analytical expressions describing the variability of effective constitutive parameters of non-metallic metamaterials, as a function of the constituent geometric and material parameters and their variations, have been developed from the total differential of Clausius-Mossotti expressions (using Mie dipole polarizabilities) for the effective (bulk) constitutive parameters of the metamaterial. In practice, these expressions are important for estimating the performance of a metamaterial with particular variations in the parameters of its constituents that arise during the fabrication process, and can be used to guard against extinction of desired double negative (DNG) behavior. With the derived expressions, the effects of parameter variations on effective constitutive parameters of non-metallic metamaterials have been analyzed for three types of metamaterials: (i) cubic arrays of identical magnetodielectric spheres; (ii) cubic arrays of dielectric spheres with equal radius but two different permittivities; and (iii) cubic arrays of dielectric spheres with equal permittivity but two different radii. These effects are evaluated in terms of the calculated variations in values of the effective constitutive parameters of the metamaterial in the vicinity of the DNG or single negative (SNG) band for particular geometric and material parameters and their variations. Results show that variation in the following parameters impacts DNG bandwidth. Listed in order from greatest to least influence: (i) sphere radius; (ii) sphere permittivity and permeability; (iii) lattice constant of the array; and (iv) the constitutive parameters of the array medium, all impact the width of the achievable DNG band. For particular cases studied here, results also show that the DNG behavior may be extinguished if there are 0.78%, 0.016%, and 0.016% variations in all parameters of metamaterial types (i), (ii), and (iii), respectively, as defined above. For the design of non-metallic metamaterials with inclusions, having arbitrary material parameters, in either periodic or random arrangement, the presented results can give a qualitative guide on the level of fabrication tolerances that should be achieved in order to observe the predicted SNG or DNG behavior experimentally.