Using the parabolic equation for calculation of beam impedance

In this paper we develop a new method, using the parabolic equation (PE), for the calculation of both high-frequency and small-angle taper (or collimator) impedances. The applicability of the PE in the high-frequency limit is based on the observation that in this case the contribution to impedance comes from the electromagnetic waves that catch up with the beam far from the obstacle and propagate at small angles to the axis of the pipe. One of the most important advantages of the PE is that it eliminates the spatial scale of the small wavelength from the problem. As a result, the numerical solution of the PE requires coarser spatial meshes. In this paper we focus on the longitudinal impedance for an axisymmetric geometry and assume a perfect conductivity of the walls. We show how the known analytical results which include a small-angle collimator, step-in and step-out transitions, and a pillbox cavity, can be derived within the framework of the PE.