Numerical Simulation of Transient and Steady State Nonlinear Beam-Cavity Dynamics in High Power Klystrons.

A 2-1over2 dimensional analysis of klystrons, assuming cylindrical space symmetry but retaining all velocity effects, is presented. The model handles all relativistic, electromagnetic, and non-linear effects. The klystron geometry is viewed as a conducting cylinder (the tube) to which are attached a number of resonant cavities at various locations. Within this tube, particle in cell plasma simulation techniques are used to model the electron beam dynamics and electromagnetic fields. The effects of the cavities are then "patched in" by the use of eigenmode expansions over the cavity geometries. These expansions reduce the effects of the cavity fields into relatively simple time varying boundary conditions on the tube. The plasma simulation in the tube is carried out using a number of "macroparticles," each representing many electrons and each obeying the equations of motion for an electron. A time centered cycloid fitting algorithm is used to move the particles. A time centered spectral method using Fourier series in z and Chebyschev polynomials in r is used to advance the fields. A non-uniform grid permits the use of FFT's to carry out the Chebyschev expansions. A method for finding time periodic solutions is also presented. This method will compute a qualitatively correct transient solution and yield a quantitatively correct steady state solution.