A theorem on high-order geometric spherical aberrations for Hutter`s electrostatic lens
Abstract: In this paper third-, fifth-, seventh-, and ninth-order geometric spherical aberrations for Hutter's electrostatic lens have been studied analytically and calculated numerically by means of Mathematica. Therefore, a theorem on high-order geometric spherical aberrations has been established. In addition, in the present work we use the Hamiltonian function expansion method in stead of the generalized integration transformation method because it is more concise and simpler. The numerical results show that up to ninth-order spherical aberrations always keep the positive sign for Hutter's lenses at the condition of infinite magnification. This theoretical conclusion would be regarded as a complement of Scherzer's theorem on the third-order spherical aberration and may be helpful for estimating the effect of high-order spherical aberrations in electrostatic immersion lenses.