Aberration analysis of electron mirrors by a differential algebraic method

A differential algebraic (DA) method has been developed for the aberration analysis of electron mirrors. Since large ray slopes occur near the turning points in mirrors, the axial position is no longer suitable as the independent variable and the electron trajectory equation used in conventional lens theory is no longer feasible. A DA solution of the electron motion equation, wherein a single DA ray trace is performed on a non-standard extension of real number space called _nD_v, enables the aberrations of a mirror system to be obtained, in principle up to arbitrary order n, and with very high accuracy, due to the remarkable algebraic properties of _nD_v. With the DA method, the enormous effort to derive explicit formulae for the aberration coefficients of electron mirrors is avoided. A software package MIRROR_DA has been developed for the aberration analysis of electron mirrors, based on the DA method. Two examples of electron mirrors are presented. For the first example, for which the electrostatic and magnetic fields are represented by analytical models, the results computed with MIRROR_DA were shown to be in good agreement with those extracted by direct ray tracing, with relative deviations of less than 0.065% for all the primary aberration coefficients. The second example consists of a real magnetic lens and electrostatic mirror, with numerically computed fields, and from the results of MIRROR_DA, the spherical aberration coefficient C_s3 is almost cancelled out because of the correction effect of the mirror. The MIRROR_DA software is a novel, effective and precise tool for the aberration analysis of electron mirrors, capable of handling realistic and complicated systems of electron lenses and electron mirrors.