IMPROVEMENTS OF METHODS FOR ELECTRON OPTICAL FIELD CALCULATIONS.

The finite-difference method is improved by the derivation of new nine-point discretization formulas valid for fairly general two-dimensional self-adjoint elliptic differential equations to be solved in a regular square-shaped grid in orthogonal coordinates. This discretization can be applied to many electron optical systems, as shown in some examples. Next, the integral-equation method is applied to electron optical multipole fields. By means of azimuthal Fourier series expansions a sequence of uncoupled one-dimensional Fredholm equations are obtained for which an improved solution technique is derived. Finally two hybrid methods are proposed, the first one combining the finite-difference method with the integral-equation method, while the second one couples the latter procedure to the familiar finite-element method. In this way it is possible to make use of the advantages of the pure methods and to circumvent their disadvantages.