Stability of Periodic-Field Beam Focusing

Abstract

By taking into consideration both the effects of space-charge and large-amplitude scalloping, an analytic solution for periodic-field beam focusing is obtained through use of successive approximation and Fourier series expansion. The stability limit is then determined by studying a perturbation of the obtained solution. The perturbation results in a Hill's differential equation for the incremental beam radius. This equation yields a stability criterion for periodic-field beam focusing. It is found that instability starts at some degree of beam scalloping and ends at a higher one, and thus forms an unstable band.