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A self-consistent nonlinear theory of the energy and current modulation in a relativistic electron beam propagating through a klystron amplifier is developed. A closed integro- differential equation for the beam current is obtained, assuming that the beam current is a function of time t and propagation distance z. Properties of the current and energy modulation are investigated from this integro- differential equation for a broad range of system parameters. Magnitudes of the energy and current modulation are determined in terms of the gap voltage, the microwave frequency, geometrical configuration, the beam intensity and initial kinetic energy of the beam. The modulation amplitude increases, reaches peak and decreases slowly, as the beam propagates through the amplifier. The theory is extended to a two-cavity klystron and it is shown that the theoretical results agree remarkably well with previous simulation data.
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