  The problem of finding the optimal current distribution supported by small
radiators yielding the minimum quality (Q) factor is a fundamental problem in
electromagnetism. Q factor bounds constrain the maximum operational bandwidth
of devices including antennas, metamaterials, and nanoresonators, and have been
featured in seminal papers in the past decades. Here, we determine the lower
bounds of Q factors of small-size plasmonic and high-permittivity dielectric
resonators, which are characterized by quasi-electrostatic and
quasi-magnetostatic natural modes, respectively. We expand the induced current
density field in the resonator in terms of these modes, leading to closed-form
analytical expressions for the electric and magnetic polarizability tensors,
whose largest eigenvalue is directly linked to the minimum Q factor. Our
results allow also to determine in closed form the corresponding optimal
current density field. In particular, when the resonator exhibits two
orthogonal reflection symmetries the minimum Q factor can be simply obtained
from the Q factors of the single current modes with non-vanishing dipole
moments aligned along the major axis of the resonator. Overall, our results
open exciting opportunities in the context of nano-optics and metamaterials,
facilitating the analysis and design of optimally shaped resonators for
enhanced and tailored light-matter interactions.
