Electromagnetic Eigenfrequencies in Concentric Spheroidal-Spherical Cavities

Abstract

The electromagnetic eigenfrequencies f_nsm in perfectly conducting concentric spheroidal-spherical cavities are determined analytically. Two types of cavities are examined, one with spheroidal outer and spherical inner boundary and inversely for the other. The problem is solved by two different methods. In the first, the electromagnetic field is expressed in terms of both spherical and spheroidal eigenvectors, connected with one another by well-known expansion formulas. In the second, a shape perturbation method, the field is expressed in terms of spherical eigenvectors only, while the equation of the spheroidal boundary is given in spherical coordinates. The analytical determination of the eigenfrequencies is possible for small values of h = d/(2R₂), (h « 1), with d the interfocal distance of the spheroidal boundary and 2R₂ the length of its rotation axis. In this case exact, closed-form expressions are obtained for the expansion coefficients 9(2)nsm and 9(4)nsm in the resulting relation f_nsm (h) = f_ns (0) [1 + h² 9(2)nsm + h⁴ 9(4)nsm + O(h⁶)] . Analogous expressions are obtained by using the parameter v = 1 - (R₂/R'₂)² (for lvl « 1) , with 2R'₂ the length of the other axis of the spheroidal boundary. Numerical results are given for various values of the parameters.