Impedance studies of 2D azimuthally symmetric devices of finite length

In particle accelerators, the beam quality can be strongly affected by the interaction with self-induced electromagnetic fields excited by the beam in the passage through the elements of the accelerator. The beam coupling impedance quantifies this interaction and allows predicting the stability of the dynamics of high intensity, high brilliance beams. The coupling impedance can be evaluated with finite element methods or using analytical approaches, such as field matching or mode matching. In this paper we present an application of the mode matching technique for an azimuthally uniform structure of finite length: a cylindrical cavity loaded with a toroidal slab of lossy dielectric, connected with cylindrical beam pipes. In order to take into account the finite length of the structure, with respect to the infinite length approximation, we decompose the fields in the cavity into a set of orthonormal modes. We obtain a complete set of equations using the magnetic field matching and the nonuniform convergence of the electric field on the cavity boundaries. We present benchmarks done with CST Particle Studio simulations and existing analytical formulas and codes, pointing out the effect of different material conductivities, finite length, and nonultrarelativistic particle beam velocity.