Resonance frequencies of conducting spheroids and the phase matching of surface waves

The complex resonance frequencies of conducting spheroids have been obtained by us for both transverse electric (TE) and transverse magnetic (TM) modes, using the
T
-matrix approach. The resonances originate from a phase matching of repeatedly circumnavigating surface waves, and we develop a quantitative model of this phenomenon. The propagation constants vary continuously during the passage of the waves over the curved surface along a geodesic, depending to first order on the local curvature along the path. This curvature dependence was obtained in a calculation of Franz and Galle in terms of a power series expansion in
k^{-2/3}
, comprising five terms of the series. Using this, we formulate the phase matching as an integral condition over the closed path, in the sense of Fermat's principle, and verify that the complex eigenfrequencies as obtained by us earlier indeed satisfy the phase matching condition. This indicates the correctness of both our physical phase-matching model, and of the
T
-matrix results for the eigenfrequencies.