A Fourier-Bessel solution of an Einzel-type electrostatic field with rotational symmetry

Abstract

A solution φ (r,z) is given of a rotationally symmetric boundary-value problem: φ (A,z) =0 for z⩽0 and z⩾L, and φ (A,z) =F (z) for 0⩾z⩾L. A and L are given constants and F (z) is a given function which is symmetrical with respect to the plane z=L/2. The solution is in the form of an infinite Fourier-Bessel series, the coefficients of which can be found without inverting matrices. It is shown that the given field can be approximated physically by means of two long equipotential cylinders and one or more central electrodes of curved cross sections. Since the electric intensity is also known at all points, the solution allows a precise determination of the electron-optical properties of a wide variety of electrostatic Einzel lenses with curved electrodes.