Extension of the One-dimensional (Klystron) Solution to Finite Gaps†

Rigorous analytical solution is derived for the quasi-linear partial differential equation system, governing the motion of the electrons in the buncher gap and in the adjoining drift space. The method closely follows that described by Solymar (1961 a) and so the validity of the solution breaks down after crossover. The nth harmonic of current (n =l, 2, 3, 4, 8) is plotted for a number of gap angles, space-charge coefficients (ω/ ωp) and modulation coefficients (α). It is shown that high harmonic currents can be obtained (as predicted by previous approximations) but beyond a certain critical density the approximations are no longer valid, and the harmonic content decreases drastically The solution contains the zero space-charge (ballistic) and infinitesimal gap theories as limiting cases.