A self-consistent nonlinear theory of current modulation in relativistic klystron amplifiers

Abstract

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 integrodifferential 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 integrodifferential equation for a broad range of system parameters. Magnitudes of the energy and current modulation are determined in terms of the gap voltage, the cavity frequency, geometric 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. A simple expression of scaling law for maximum current modulation is obtained. This scaling law could be useful in the design and fabrication of a high-performance klystron. Nonlinear mode evolution in current profile is also investigated by Fourier decomposing the current modulation obtained from the integrodifferential equation. The mode evolution in a long-range propagation of an electron beam exhibits various interesting physical properties. For example, the maximum amplitude of the fundamental mode (l=1) occurs at the propagation distance, where all other modes vanish or have a very small amplitude. This property ensures a monochromatic frequency.