Numerical Ray Tracing in Electron Lenses

Liebmann's method of tracing paraxial rays is critically reviewed, corrections being made to the higher terms of his formulae. A new and very accurate method of ray tracing is described, which is a development of that used by Burfoot; the predictor is one used by Goddard and the corrector is the double integration formula on which the method of Stormer is based. Like Goddard's, this method is applicable to marginal as well as to paraxial rays. The results of numerical tests of various methods are compared. The methods of Gautier and Laudet are discussed. The use of the constancy of the Wronskian determinant as a criterion of accuracy of paraxial ray tracing is considered, examples being given of instances in which it is and is not reliable.