Space-charge waves of periodic-electrostatically focused electron beams

Space-charge wave propagation for periodic-electrostatically focused electron beams with nonuniform radial beam density distributions is the subject of this paper. The dispersion equation is obtained by solving a differential equation for the ac electric potential inside the electron beam and by matching the solution at the boundaries. An approximate analytic solution and a digital computer treatment are described. The reduced plasma frequency obtained is expressed in terms of the focusing parameters and is compared with that of a Brillouin beam. A study of the differential equation involved suggests that there are certain ranges of system parameters ("forbidden" zones) for which no wave propagation is possible, that is, no space-charge mode can be supported there.