Abstract: This paper describes numerical techniques for computing axial field functions in focusing and deflection systems for charged particle beams. For systems with a straight optical axis, the computation of axial field functions using first- order and second-order finite element methods are compared, for electron lenses, deflectors and stigmators. The results show that the second-order finite element method gives greater accuracy for the high-order derivatives of the axial functions, which are needed for computing the fifth-order aberrations. The extension of these methods to computing perturbation fields caused by mechanical defects in the optical elements is also described. Methods for computing multipole field functions using a fully 3D field analysis are then described. These 3D methods are of quite general applicability to systems with fully 3D electrode and polepiece structures, and either straight or curved axes. Illustrative examples are shown of a multipole aberration corrector with a straight axis and a bending magnet with a curved optical axis.!17
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