Electrostatic resonances of a negative-permittivity hemisphere

This article studies the electric response of an electrically small hemispherical object with negative permittivity by computing its polarizability which is determined by two orthogonal components, the axial one and the transverse one. A certain range of negative permittivity values is found where the mathematical determination of the polarizability becomes impossible due to an unlimited number of singularities. These singularities are due to surface plasmons, also referred to as electrostatic resonances, caused by the sharp edge of the hemisphere. It is also found that the planar surface of the hemisphere may support resonant surface modes. Furthermore, there exists a dipolar resonance determined by the overall geometry. In addition, it is shown that the resonances can be smoothened by introducing losses and, even more importantly, rounding the edge.