  Material resonances are fundamentally important in the field of
nano-photonics and optics. So it is of great interest to know what are the
limits to which they can be tuned. The bandwidth of the resonances in materials
is an important feature which is commonly characterized by using the quality
(Q) factor. We present bounds on the quality factor of two-phase quasi-static
metamaterial resonators evaluated at a given resonant frequency by introducing
an alternative definition for the Q-factor in terms of the complex effective
permittivity of the composite material. Optimal metamaterial microstrcuture
designs achieving points on these bounds are presented. The most interesting
optimal microstructure, is a limiting case of doubly coated ellipsoids that
attains points on the lower bound. We also obtain bounds on Q for three
dimensional, isotropic, and fixed volume fraction two-phase quasi-static
metamaterials. Some almost optimal isotropic microstructure geometries are
identified.
