Numerical calculation of electromagnetic toroidal resonators

Starting from the Lagrange function for electromagnetic fields, the eigenmodes and eigenfrequencies are calculated for an empty toroidal resonator with circular cross section. The calculation uses a Rayleigh-Ritz method, in which the vector potential of the electromagnetic field is represented in a space of vector trial functions. With this method it was possible to treat the general nonuniform case to compute the full set of E and H modes, where the forty lowest eigenvalues are approximated with an accuracy of approximately 0.0001.