Radial focusing in a linear induction accelerator gap

The focusing properties of the electric field distribution in a linear induction accelerator gap are identical to those of a bipotential electrostatic lens if the particle gap-transit time is small compared with the time variation of the induced electric field. Bipotential lenses have been studied systematically only in the nonrelativistic case. In this paper we present the results of a theoretical study of the focusing properties of a symmetrical two-cylinder lens with identical radii and negligible separation for relativistic electron beams. Analytical formulas derived by using the thin-lens approximation are compared with the results of numerical integration of the relativistic paraxial ray equation and found to be sufficiently accurate for practical applications