Equivalence of periodic magnetic field to uniform magnetic field in electron beam focusing

The general solution of the electron trajectory equation in a periodic magnetic field is derived in the form of series expansion, assuming laminar electron flow and small perturbation. It is concluded that if the cathode is not very heavily immersed in a magnetic field, beam focusing by a periodic magnetic field would be almost equivalent to that by a uniform magnetic field except near the unstable region. Under these restrictions, each focusing system would give almost the same beam ripple (in magnitude and phase) under the same injection conditions into the focusing system. It is also found that the first stable region of beam perturbation in the periodic magnetic field becomes narrower as the flux threading the cathode increases or the period of the magnetic field decreases, as far as linearization of the path equation is possible.