On Solving Maxwell Eigenvalue Problems for Accelerating Cavities

The common way to produce the accelerating electromagnetic fields in cyclic accelerators is to excite standing waves in accelerating cavities. The mathematical model for these high frequency electromagnetic fields is the eigenvalue problem solving the Maxwell equations in a bounded volume [1]. Usually, the eigenfield corresponding to the fundamental mode of the cavity is used as the accelerating field. A few modes of higher order have to be analyzed as well because these modes can be excited due to higher harmonic components contained in the RF (radio frequency) power fed into the cavity and through interactions between the accelerated particles and the electromagnetic field. The RF engineer designing such an accelerating cavity therefore needs a tool to compute the fundamental and about ten to twenty of the following eigenfrequencies together with the corresponding electromagnetic eigenfields. The most interesting quantities besides the eigenfrequencies are local maxima of the eigenmodes as well as the fields on the surface that induce heat in the metallic boundary and determine the power loss from surface currents. The finite element formulation for the solution of this