Scanning Electron Microscopy Volume 3 Number 1 3rd Pfefferkorn Conference Article 6 1984 Recent Developments in Numerical Electron Optics Erwin K. Kasper Universitaet Auf der Morgenstelle Follow this and additional works at: https://digitalcommons.usu.edu/electron Part of the Biology Commons Recommended Citation Kasper, Erwin K. (1984) "Recent Developments in Numerical Electron Optics," Scanning Electron Microscopy: Vol. 3 : No. 1 , Article 6. Available at: https://digitalcommons.usu.edu/electron/vol3/iss1/6 This Article is brought to you for free and open access by the Western Dairy Center at DigitalCommons@USU. It has been accepted for inclusion in Scanning Electron Microscopy by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. Electron Optical Systems (pp. 63-73) 0-93 1288-34-7/84$ 1. 00+. 05 SEM In c., AMI' O'Hare (Chicago), IL 60666-0507, U.S.A. RECENT DEVELOPMENTS I N NUMERICAL ELECTRON OPT I CS Erwin K. Kasper In st itut fuer Angewandte Physik der Univers ita et Auf d er Mo rgenstelle 74 Tuebing e n, W. Germany Pho ne : 07071/296747 292429 Abstract Intr od uction The fa miliar met hods for the num e ri­ Due to the r apid advances in com ­ cal calcul at i on of fields in elect ron op ­ put er t ec hn o l ogy i t i s p oss ibl e to calcu­ t ic al devices are o utlin e d bri efly . for lat e now with reasonable effort a nd s u f­ the s ol uti o n of self - a dj o int ellipt ic fici ent acc u racy the pr ope rtie s o f elec ­ differential equations in orthogonal cur­ tron opt ic a l systems , an d this i s done vilinear coordinate systems a favou rabl e very frequent ly , since the a id of a com ­ ninepoint discr et iz at i o n is worked o u t put er facilitates essent i ally t h e d esign which can be applied favourably e .g. to of new devices . Generally such a co mpu­ spherical mes h grids . The field cal cul a ­ ter-aided design consists of fo u r subse­ ti o n in mag n et ic d eflect i on syste ms by qu e n t and partly in ter d e p e nd e nt ste p s . mearis of an int egral eq uati on metho d i s These are t he calculat i on of elect ri c and also highly advantageous . The methods for magnetic fie ld s from given source distri­ the field calc ul ation can be still more butions and boundary cond i tions , the improved by mea n s of s ui lable hybrid pr o ­ tracing of electron trajectories through cedures . t hese fields , the determination of p ar­ A second and shorter contribution is axial p ropert i es and of aberrations from concerned with r a y tracing and aber r a ­ the obta in ed trajectories and finally the tions . Some favourable num erically stable optimizat i on of the design in question. new fo rm s of the ray eq u at i on are derived The latter involves a repetition of the and thereafter a n ew simple method for preceding steps under suitable var i ations the d eterminat i on of aber r ations is out ­ of system parameters , until a relat i ve lin e d. minimum of the r esulting aberration is found. In al l of the above mentio ned fields some progress was made during t he last years, and it is the aim of this p aper to present a short r eview of this pr ogress an d to give some outlook on further developments . fi el d calculation Generally three diff ere nt met ho ds of f i eld calculation a re in wid espread us e , thes e are the fin it e - e lem e nt met hod (fEM) , t h e finit e -di fference me thod (fDM) an d the integral-equation met hod (IEM). Each of t h ese has s p ec ific advantages and also dis a d va ntag es , so that n one of them Key wo rds: field calculati o n, ray can b e ign o r ed compl ete ly o r de al t with tracing, aber ra t i ons , int egral equation, exclusively . hybrid metho d, Poisson ' s eq u at i on ,d iscre ­ Th e finite - ele me n t met h od tizati on . Th e fEM consists in the dissection of the d o ma in of so lution into su i tably ch osen volume e l e ments which must be ir­ regul a r in the general case in or d e r to fit to the bound ar i es . In the case o f tw o - dim e nsional pr o blems the mesh grid obtained in this way consists u s ually of irregular triangles with six ele ments joining together in eac h in te rn a l node. 63 E. K. Kasper It is familiar to derive the necessary An interesting modificat i on of the equations for the values of the potential FDM has been proposed by Kang et al. in the nodes from a variational princip­ ( 1981, 1983). In order to overcome the le: in each of the finite elements the difficulties arising from the extre me stored electr ic or magnetic field energy differences of geometrical dimensions in must be minimized. In order to evaluate e l ectron guns with field emission catho­ this principle, some ass umptions about des, they proposed the use of an exponen­ the required solution must be made. The tially increasing spherical mesh grid simplest one is that the potential is a which they named SCWIM (spherical coor­ piecewise linear function. This version dinate with increasing mesh). Indeed, in of the FEM has been introduced by Munro this way a reasonable accuracy can be ( 19 71, 19 73) who applied it successfully achieved. This method can be further ge­ to a large variety of electron optical neralized and improved, as will be out ­ devices, most recently to magnetic and lined now. electric deflection systems (Munro and We start from a general self-adjoint Chu, 1982 a,b). A review of applications partial differential equation (PDE) of Munro's method to the design of un­ conventional magnetic lenses is given by Mulvey ( 19 82). While the version of the FEM, des­ cribed above is still frequently used, ( 1) great improvements have been made in the past decade, mainly outside electron op ­ tics . Better approximations than piecewise p(u,v), q(u,v) and s( u, v) being non-sin­ linear functions for the potential are gular analytic funct ions of the variables worked out (Silvester and Konrad, 1973), u and v , f ur ther p > 0 , o( ~-1, v .2: 0. more generally curvilinear and even in­ The exponent~ arises from a possible finite elements are employed (Lencova axial symmetry of the potential V(u,v) and Lene, 1982) and other than varia­ for instance o<- = 1 for rotationally tional formulations are proposed. A symmetric fields and o<. = 2m + 1 for review of recent improvements of the FEM multipole fields of order m. The product­ is given in a volume edited by Chari and relation Silvester ( 1980). Even with respect to V(u,v) U(u,v)/p(u,v) ( 2) the still most familiar case of planar triangular grids progress was made, as reduces ( 1 ) to He rmelin e (1982) prop o sed a new and re­ ,1 u -g(u,v) = q(u,v) - U - p. s ( 3) liable me thod for the automatic genera ­ c,L tion of such grids under given boundary with c o nditions and other constraints. In spite of all these improvements ( 4) the FEM has still severe disadvantages, the most serious one being the fact that and the interpolation in irregular mesh grids is too complicated in order to ensure the q(u,v) ( 5) continuity of the field strength and its derivatives along lines crossing orthogo­ For the discretization of (3) very nally the mesh lines of the grid. This accurate nine-point-formulae have been imposes strong restrictions on the ray derived (Kasper, 1976), but with respect tracing,as will be outlined in the "Ray to the transformation (2) a still more Tracing and Aberrations" section. favourable form of the discretization can The finite-difference method be found. If we introduce the coordinates In the applications of this method, u = ih, v = kh (i,k integers) of the the domain of solution is covered by a nodes and use corresponding subscripts regular, usually square-shaped mesh grid. (U,:*: = U(ih, kh)) then we can introduce Most frequently th i s grid does not fit to favourably a new field the boundaries, so that many irregular nod es are to be considered. Theoretically it is no problem to derive discretization w,;k:= l(,k + ;; Ji,k formulae for arbitrary irregular configu­ rations, but in practice the considera­ 6 tion of many different irregu lar situa­ = ~/ {~l+ J~l ( i.J V:;k+ S:;r)) ( l tions makes the method inconvenient. Ne­ vertheless , the FDM has the advantage and have now for k 3! 1 : that - besides exceptional cases in elec­ tron guns - the grid is regular within the domain of the electron beam, so that the necessary calculation of the field­ strength can be performed with sufficient ( 7) accuracy.A review of these standard tech­ niques has been given by Kasper (1982).