A numerical approach for the solution of Maxwell's equations is presented. Based on a finite difference Yee lattice the method transforms each of the four Maxwell equations into an equivalent matrix expression that can be subsequently treated by matrix mathematics and suitable numerical methods for solving matrix problems. The algorithm, although derived from integral equations, can be considered to be a special case of finite difference formalisms. A large variety of two and three-dimensional field problems can be solved by computer programs based on this approach: electro statics and magnetostatics, low-frequency eddy currents in solid and laminated iron cores, high-frequency modes in resonators, waves on dielectric or metallic waveguides, transient fields of antennas and waveguide transitions, transient fields of free-moving bunches of charged particles etc.

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Many state-of-art electron accelerator proposals incorporate TESLA-type superconducting radio-frequency (rf) cavities [1]. These standing wave rf cavities include rf input couplers and a pair of high order mode (HOM) couplers to absorb the energy associated to HOM field excited as the bunch passes through the cavity. In the present paper we investigate, using numerical simulations, the impact of the input and HOM couplers on the beam dynamics to zeroth and first order in initial position, and present parametric studies of the strength of these effects for various incoming beam energies. We finally study the impact of this asymmetric field on the beam dynamics, taking as an example the low energy section of the X-ray FEL injector.

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