Analog computer solutions for correction of ripple in magnetically-collimated electron beams

Analysis of the dynamics of an electron beam in a magnetic collimating field leads to formulas which can be processed conveniently in an analog computer to obtain individual solutions. This technique is applied to the problem of correcting the ripple which is generally imposed upon the envelope of the beam after passage through a spatial variation or reversal of the collimating field. However, a relatively general viewpoint is retained in developing methods and formulas for studying the motion of an outermost electron of a relatively uniform, round, straight electron beam. Fer this reason the results can be applied to beams from shielded or unshielded cathodes, passing through uniform, periodic, or nonperiodic reversing magnetic collimating fields. A simplified development of the mathematical theory of magnetically confined electron beams is presented, including the effects of spatial variations and reversals in the collimating magnetic fields. Formulas are developed for the equilibrium electron beam radius, and for the ripple frequency, axial wavelength, amplitude, and phase of the outermost electrons, in terms of the field parameters and the initial beam conditions. Analog computer solutions for an electron beam with various collimating magnetic fields serve to illustrate and confirm the developed formulas. From these computer solutions, suggestions are derived for guidance in shaping collimating magnetic fields so that electron beam ripple will be prevented or reduced.