NONLINEAR-ANALYSIS OF THE 2-STREAM INSTABILITY FOR RELATIVISTIC ANNULAR ELECTRON-BEAMS

A nonlinear theory is developed for the two-stream interaction of two relativistic annular electron beams propagating through a grounded cylindrical conducting tube. The theory is based on the assumption that the beams experience energy modulation at a cavity before they enter the drift tube. Two coupled integrodifferential equations are derived which describe beam current modulation in terms of time t and propagation distance z. The evolution of the fundamental mode in the current modulation is investigated analytically by making use of these coupled equations. The amplitude of the fundamental mode, which is a function of the propagation distance, is expressed explicitly in terms of the initial energy modulation, the growth mte of the instability, and the beam intensity. It is found that self-field effects dominate two-stream effects at the beginning of the propagation. As the beams propagate further, the two-stream effects start to dominate and then the perturbations grow exponentially. The saturation mechanism is identified as multimode coupling. A theoretical model describing the mode coupling in the current modulation is also developed. The theory is verified by detailed comparisons between analytical and particle-in-cell simulation results.