Systematic transformations of the asymptotic aberration coefficients of round electrostatic lenses (1)

Abstract

In previous work we formulated the third-order asymptotic aberration coefficients of round (axially symmetric) electrostatic lenses in a form independent of object and aperture positions, and expressions for the six quantities which are sufficient to specify completely the aberration properties of the lenses were derived in the form of integrals involving derivatives of the axial potential through the fourth order. Because actual calculations involved numerical differentiation of the axial potentials, integrations by parts were used to transform the integrals to two new forms with axial derivatives of lower degree. Many other forms of the aberration integrals can be obtained by further integrations by parts, but the transformations are laborious and it is not easy to predict the forms which are possible nor to determine the sequence of operations which will yield a desired result. However, using a method originally developed by Seman and extended by Hawkes, a completely general formula has been derived from which all of the possible forms of the asymptotic integrals can be obtained simply. A few of these possible forms are derived and discussed.