Normal Modes of AN Axisymmetric Spherical Cavity Resonator.

The problem of the normal modes of electromagnetic oscillations in a spherical cavity resonator with axisymmetric interior and ideally conducting walls is solved. The method involves the construction of a complete set of solutions of the axisymmetric wave equation in spherical co-ordinates, a co-ordinate system in which the equation is not separable. Fitting the boundary conditions at the surface of the sphere results in an equation for the normal modes in the form of the roots of an infinite dimensional determinant. The determinant is evaluated by the method of successive truncations. Numerical results are presented for the normal modes as a function of the dielectric asymmetry. The field lines for some of the lowest modes and selected choices of parameters are drawn. Furthermore, the effect of a hole in the cavity's wall (necessary for the excitation of the modes) is investigated. It is shown that the existence of the hole perturbs the normal frequencies and this perturbation is calculated. The method of solution is based on the Rayleigh-Schrodinger perturbation theory. It is similar to time-independent degenerate perturbation theory in non-relativistic Quantum Mechanics. It is found that the frequency shift depends on the value of the perturbed electric field at the hole. This field is calculated using the quasistatic approximation, which involves the solution of a mixed boundary value problem for the potential. An expression for the frequency shift and broadening is obtained. A preliminary comparison between the theory and experiment is made.