Lower Bounds to Quality Factor of Small Radiators through Quasistatic Scattering Modes

The problem of finding the optimal current distribution supported by a small radiator 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 open optical resonators. In this manuscript, a representation of the optimal current distribution in terms of quasistatic scattering modes is introduced. Quasi-electrostatic and quasi-magnetostatic modes describe the resonances of small plasmonic and highpermittivity particles, respectively. The introduced representation leads to analytical and closed form expressions of the electric and magnetic polarizability tensors of arbitrarily shaped objects, whose eigenvalues are known to be linked to the minimum Q. Hence, the minimum Q and the corresponding optimal current are determined from the sole knowledge of the eigenvalues and the dipole moments associated with the quasistatic modes. It is found that, when the radiator exhibits two orthogonal reflection symmetries, its minimum Q factor can be simply obtained from the Q factors of its quasistatic modes, through a parallel formula. When an electric type radiator supports a spatially uniform quasistatic resonance mode, or when a magnetic type resonator supports a mode of curl type, then these modes are guaranteed to have the minimum Q factor.