Fourier-Bessel series solution for potential and intensity field of open electron optical systems with rotational symmetry, in terms of I0 Bessel functions

Abstract

A Fourier-Bessel series representation in terms of I0 Bessel functions is given for the potential distribution in certain open Einzel- or immersion-type electrostatic configurations with rotational symmetry described by φ (A,z) =F (z), 0<z<L; and φ (A,z) =0 for all z<0 and for all z≳L. F (z) is a given function and A and L are constants. The method is a direct one—no iterations or matrix inversions are required—and a superior rate of convergence in the paraxial region is achieved through the elimination of J0 Bessel functions. The precision reached is apparently limited by the computer word length only, programming is of a simple nature, and computer memory requirements are modest enough to allow implementation on small desk-top computers.