Determination of Electrode Shapes for Axially Symmetric Electron Guns

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Abstract
The determination of the electrode shapes for an electron gun involves solving Laplace's equation subject to specified boundary values of voltage and normal field on an open curve. Past attempts to solve this problem for the case of axial symmetry by mathematical methods have met with considerable difficulties because the problem is improperly set and leads to unstable solutions. Following Garabedian, we have reformulated the problem in such a manner that it becomes properly set, and have applied it to a curvilinear space-charge limited flow gun. First, a conformal transformation is made which maps the beam boundary into a coordinate axis. The second step, which constitutes the essence of the method, is accomplished by making an analytic continuation of Laplace's equation and its boundary values into a fictitious complex domain. Laplace's equation, which is elliptic in the real domain, is thereby converted into a set of hyperbolic equations. This leads to a stable scheme of computation by finite differences. This method should find particular application to curvilinear flow guns, where the use of analogs, such as the electrolytic tank, requires the use of involved experimental techniques. The method is very general, however, being applicable to any configuration where the boundary conditions are given through analytic functions. If one desires, these specifications for the boundary conditions may be given implicitly, as, for example, through a set of differential equations.