A multislice approach to the spherical aberration theory in electron optics

Abstract

A multislice, wave-optical approach to the spherical aberration theory in electron optics is presented for the case of magnetic lenses. It is shown that, in order to remedy the inadequacy of a naive attempt carried out by taking the standard multislice equations and simply adding the fourth order terms in the transverse coordinates in the phase both of the transmission function and of the propagation kernel, it is necessary to approximate the Schrödinger equation by a wide-angle differential equation of the parabolic type and to solve it by means of a marching type algorithm. It turns out that the solution can be separated into two parts, describing, as in the multislice method, the interaction of the incoming spherically aberrated axial wavefunction with the field in a slice, and the propagation in a drift space. When the propagation is evaluated by means of the stationary phase method, a differential equation is obtained for the spherical aberration coefficient, whose solution is identical to that obtained by Glaser in his particle approach based on the eikonal.