Imaging Properties and Aberration Analysis of Electrostatic Afocal-Zoom Lenses Using Computer Optimization

Abstract

The focusing properties of electrostatic lenses have been extensively studied both experimentally and theoretically due to their importance in practical applications in electron spectroscopy, surface science, electron microscopy, and mass spectrometry [1][2][3]. These applications require the optimization of the various lens parameters, such as a set of voltages applied to electrodes, to achieve minimum beam distortion, small aberrations and high transmission. It is well known that multi-element lenses are very useful in electron-optical systems due to their exceptional focusing capabilities. In most of the applications, it is suitable to vary the final-to-initial electron energy without altering the object and image positions, P and Q, respectively [4]. This condition can be obtained by using three-element "zoom" lenses [1]. Although the two voltage ratios are varied to keep the object and image positions constant as the overall voltage ratio is varied, the linear and/or angular magnification, M and M α , will change. Figure 1a shows schematically the focusing and the magnification of a zoom lens. In general, to keep n imaging properties constant while varying final-to-initial electron energy, a lens system with n+2 electrodes is needed. If one wants to control the image position together with one of the magnifications (M or M α ) for a given object position, an extra element must be introduced [5]. Lenses with more than four elements can have other advantages, such as lower aberration, a more extended range of overall voltage ratio or special lens modes like afocal lenses [6,7], i.e., parallel rays entering the lens will exit parallel, and the beam angle (θ b ) is zero. Here, the beam angle is a vital parameter to improve the angular spread of the beam, which is particularly important for high-resolution electron spectrometers [8][9][10][11]. This angle can be controlled well if the potentials are varied so as to make the beam angle zero at the image side (the beam is collimated). So, in order to control independently i) the image position Q, ii) the magnification M, and iii) the beam angle θ b while the final-to-initial electron energy is varied, a minimally five-element lens system is necessary. Operating the lens in "afocal-zoom" mode, i.e., a parallel input beam with respect to the optical axis leaves the lens parallel to the axis, and the beam at the object position is focused to the image position, as shown in Figure 1b, would be helpful to reduce the angular divergence of the beam prior to the electrostatic energy analyzers (see, for example, Ref.[12]).An electrostatic lens transports the electrons emanated from the source to the image point of the lens with certain acceleration/deceleration and linear magnification M. As it is well known, the following relation holds for imaging using such a lens system(the Liouville theorem)where α i and α o are the pencil angles of the electrons at the image and the object position, respectively, and r i and r o are the image and the original object sizes. The quanti...