Tight bounds correlating peak absorption with Q-factor in composites and metallic clusters of particles

Resonances are fundamentally important in the field of nano-photonics and optics. Thus, 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 Q-factor. We present tight bounds correlating the peak absorption with the Q-factor of two-phase quasi-static metamaterials and plasmonic resonators evaluated at a given peak frequency by introducing an alternative definition for the Q-factor in terms of the complex effective permittivity of the composite material. This composite may consist of well-separated clusters of plasmonic particles, and, thus, we obtain bounds on the response of a single cluster as governed by the polarizability. Optimal metamaterial microstructure designs achieving points on the 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 and particle clusters with an isotropic polarizability. Some almost optimal isotropic microstructure geometries are identified.