On the Determination of the Electrodes Required to Produce a Given Electric Field Distribution along a Prescribed Curve

A method is derived for determining the potentials in a space-charge-free region, with given potentials and electric fields on a prescribed curve. The treatment is two-dimensional. The method used is to set up a complex analytic potential function along the trajectory, which may be continued analytically into an open region, and whose derivative is related to the electric field. The method is applied to two problems: that of the design of the electrodes for a crossed-field electron gun, and that of the design of the electrodes for Sturrock "leap-frog" focusing. It is shown that, owing both to the mathematical and the physical nature of the problems, care must be taken in the application of the method, in order to produce meaningful results. The electrode shapes required to produce exactly the beams in each of the two problems are given. Electrode systems which would be easier to fabricate and which adequately approximate the required fields at the edges of the beam are also described.