Propagation and excitation of perturbations in electron beams with velocity shear

A theoretical investigation of the influence of a spatially varying drift velocity on the perturbations in magnetically focused electron beams is presented. The wave propagation on such beams, focused by infinite magnetic fields, is studied under the small-wave assumption. The dispersion relation for slow waves is derived and solved for different transversal boundary conditions. Two sets of infinitely many propagating modes are found in a beam with small velocity shear. In a beam with sufficiently large velocity shear only two eigenfunctions exist. Since it is not possible to match the two eigenfunctions on arbitrary longitudinal boundary conditions, additional solutions (which cannot be written in form of a plane wave) must exist. The excitation of perturbations by ideal grids is solved by introducing the Laplace transform analysis. Additional solutions are ascertained which lead to a spatial decay of the perturbations according to a power law z^-α. This damping, arising from the spatially varying drift velocity, is similar to the Landau damping in electron beams with velocity distribution. Such damping effects are of great practical importance in conjunction with noise reduction in traveling-wave tubes.