Exact Space-Charge Field Solutions for Cylindrically Symmetric Beam Currents in a Circular Conductor Pipe

We derive an exact formalism for calculating the space-charge fields of an electron beam inside of a perfectly conducting cylindrical pipe. The formalism utilizes a series expansion of the beam-charge and current densities in Bessel functions in conjunction with a time-dependent Green's function method to compute the exact space-charge fields due to the beam with the appropriate conductor boundary conditions. The formalism can be utilized for theoretical and numerical studies of high-space-charge beams, such as those found in high-power microwave sources. As a numerical example, we demonstrate how this formalism can be implemented numerically to compute the space-charge fields for a bunched beam which is undergoing radial oscillations.