Transverse impedance studies of 2D azimuthally symmetric devices of finite length

The accurate calculation of the beam coupling impedance for particle accelerators is necessary to carefully assess the machine stability against impedance-driven collective effects. A first order evaluation of the beam coupling impedance is often done by means of analytical formulas and/or 2D numerical codes. The infinite length approximation is often used to simplify the calculation of the beam coupling impedance of accelerator elements. This is expected to be a reasonable assumption for devices whose length is greater than the transverse dimension but may be a less accurate approximation for segmented devices. In this work, we present the application of the mode matching method to the calculation of the transverse dipolar impedance of a cylindrical cavity loaded with a toroidal insert. By choosing different insert electromagnetic properties (permittivity, permeability, and conductivity) and dimensions, the model can represent a beam pipe, a thin insert, a lossy cavity, or a collimator for which the effect of the finite length is investigated. The method is successfully benchmarked against available analytical formulas, field-matching codes, and 3D commercial solvers. The proposed model allows for performing wide parametric scans and reaching accurate results, therefore becoming an essential tool for the impedance evaluation of accelerator devices.