Geometrized theory of nontwisted relativistic electron beams

The second- and third-order approximation are constructed for the geometrized theory of hollow pencil relativistic electron beams in the absence of an external magnetic field. The formulas are useful for computing nonparaxial streams with large beam-compression factors. The regularization of a solution represented by a truncated Taylor series in the transverse coordinate is considered. For the near-cathode region, the third derivative of the emission-current density and the second derivative of the emitter curvature are calculated, the form of the expressions being determined from the initial data on the reference tube of current.