Aberration coefficients of multi-element cylindrical electrostatic lens systems for charged particle beam applications

In order to analyze the imaging properties of an electrostatic lens system, it is necessary to know how various sources of aberration combine to increase the size of final image or spot. In this paper, we investigated the spherical and chromatic aberration coefficients of multi-element electrostatic lens systems as a function of the lens voltages and magnification, using the electron ray tracing simulation programs Simion and Lensys. These programs can be used to obtain electron optical aberration integrals which involve the axial potential distribution and its derivative, and two independent trajectories and their derivatives for the determination of the third- or fifth-order aberration coefficients of multi-element lenses. Optical simulation of the intensity distribution has quantitatively shown that the aberration in the crossover image causes an electron beam blur and a positioning error on the focus spot. If a high positive voltage with respect to the first element's potential is applied to the lens elements, the aberrations as well as the minimum beam divergence can be reduced. The reason, obtained from numerical simulation, is that a positive voltage increases the electron velocity, shortening the electron drift time across the region with aberrant field.