T h e optical properties and practical realization of ion and electron lenses are reviewed. Attention has chiefly been directed to electrostatic and magnetic electron lenses of rotational or planar symmetry, especially to those which seem to offer advantages in scientific instruments. Various competing methods of calculating the optical properties are analysed in some detail, with particular emphasis on the mathematical assumptions involved and their relevance to the real lens structure. Thus in the computation of magnetic electron lenses, methods are discussed that enable the whole of the magnetic circuit and the energizing coil to be taken into consideration. For magnetic lenses extensive data are provided concerning lens properties. T h e actual performance of contemporary magnetic lens designs, including flat helical lenses and superconducting lenses, is compared with the ultimate performance that is physically possible. A final section describes some of the important developments that have taken place in electrostatic lenses, both as separate imaging elements and as an integral part of ion sources and electron guns of extremely high brightness. T h e review concludes with an attempt to identify those computational methods that are most likely to be valuable in the future not only for the accurate computation of individual lenses but also for complete systems.

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Axial potential distributions in electrostatic slit-aperture lenses, consisting of plane and parallel electrodes, are derived using the method of conformal transformation. An improved iterative scheme is given which, for the first time, enables solutions to be obtained for multi-electrode configurations in which the aperture widths are larger than the electrode separations. No special conditions relating one aperture width to another, or relating one electrode spacing to another, were assumed. As an example, the axial potential distribution in the electron bombardment source used in AEI mass spectrometers was calculated; for this an analogue consisting of five apertured planes and an apertureless plane was used. Electrodes of finite thickness were each represented by two planes of infinitesimal thickness. The same problem required computer time greater by at least a factor of ten when solved by the method of numerical relaxation.