Aberrations in asymmetrical electron lenses

Starting from well established knowledge in light-optics we explore the question if electron-optical aberration can be improved in asymmetrical electron lenses. We show that spherical as well as chromatic aberration coefficients are reduced in asymmetric electrostatic einzel lenses when the center electrode is moved away from the center position towards the entrance electrode. Relative improvements up to 40% for both the chromatic and the spherical aberration coefficients can be obtained. We use analytical and numerical calculations to confirm this result for exemplary cases of a lens with fixed length and working distance. The agreement of the two calculation methods is very good. We then derive an estimate for the electron-optical aberration coefficients from light-optics. The derived expressions for chromatic and spherical aberrations are somewhat simpler than the ones derived from electron-optics as they involve integrals only over the electrostatic potential, not over the electron paths. The estimated formulas still agree well with the electron optical calculations. Overall, we are tempted to suggest that the enormous knowledge base of light optics can provide considerable guidance for electron-optical applications.