Nuclear Instruments and Methods in Physics Research A 573 (2007) 329–339 Aberration coefficients of multi-element cylindrical electrostatic lens systems for charged particle beam applications Omer Sise à , Melike Ulu, Mevlut Dogan Department of Physics, Science and Arts Faculty, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey Received 8 September 2006; received in revised form 1 November 2006; accepted 2 January 2007 Available online 12 January 2007 Abstract 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. r 2007 Elsevier B.V. All rights reserved. PACS: 41.85.Ne Keywords: Electrostatic lenses; Multi-element lenses; Electron optics; Zoom lens; Aberration coefficients; SIMION and LENSYS 1. Introduction In the development of charged particle optical instru- ments, such as electron or ion guns, lithography systems, energy analyzers, time of flight spectrometers, electron microscopes, it is necessary to determine the performance of the system and to fabricate lenses and apertures taking into account the aberration effects. It is well known that the effects of aberrations cause degradation of the focused beam spot and thereby severely restrict the spatial resolution of the system. The main part of this limitation arises from the spherical C s and chromatic C c aberrations, which are the principal aberrations and relatively large compared to optical lenses, that adversely affect the focusing properties of electrostatic lenses. Therefore, it is highly desirable to design multi-element electrostatic lenses with small spherical and chromatic aberration coefficients in order to obtain high resolution. The calculation of the third-order aberration coefficients using integrating functions of the axial potential distribu- tion is available in the literature. There are also many excellent books on charged particle optics in general as well as more specific books on aberration coefficients that cover much useful material [1–8]. Brunt and Read [9] have investigated the effects of the third-order spherical aberra- tions of electrostatic lenses for different operating condi- tions and pupil positions, and showed that the radius of disc of least confusion can be reduced by moving the Gaussian image plane to the fixed plane at a fixed voltage ratio. Renau and Heddle [10,11] developed a computer model to calculate the potential distribution of a two- element lens using a variational method, and described the theory of the model to calculate the third- and fifth-order aberration coefficients. The applicability of the Renau and Heddle approximation was investigated by Martinez and ARTICLE IN PRESS www.elsevier.com/locate/nima 0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.01.051 à Corresponding author. Tel.: +90 2722281311; fax: +90 2722281235. E-mail address: omersise@aku.edu.tr (O. Sise).