Analytic solution for the problem of gridded gap-electron flow interaction

The modelling of high power klystrons is discussed. One of the basic components of high power Klystron, the resonant cavities were studied. The formulation is not restricted to Klystron modelling, but is applicable to any problem involving the interaction of an electron beam with a resonant cavity. The hydrodynamic models of electron beams used to derive the small signal theories fail when particle trajectories cross each other. A Vlasov description of the electron beam to study the Klystron problem is emphasized. The Vlasov formulation follows the evolution of the electron distribution function in phase space. The general framework can naturally accommodate particle crossing, and the beam dynamics are accurately described even when the signals are large.