Using Parabolic Equation for Calculation of Beam Impedance

In this paper we develop a new method of parabolic equation (PE) for calculation of both high-frequency and small-angle taper (or collimator) impedances. The applicability of PE in the high-frequency limit is based on the observation that in this case the contribution to impedance comes from the waves that catch up the beam far from the obstacle and propagate at small-angles to the axis of the pipe. One of the most important advantages of PE is that it eliminates the spatial scale of the small wavelength from the problem. As a result, the numerical solution of PE requires coarser spatial meshes. In the 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 parabolic equation