Hybrid computer aided design of thick electrostatic electron lenses

The synthesis of space charge free electron lenses can be approached by first assuming solutions for Laplace's partial differential equation in terms of cylindrical harmonics. The hybrid computer can quickly generate these assumed solutions and their gradient by working with the differential equations obtained from product separation of variables. The gradient is used in the integration for the trajectory to determine if the lens is useful. If so, either equipotential surfaces or the potentials along a desired surface are generated with the computer to guide the physical realization of the lens. Certain cases where space charge cannot be neglected are solved by truncation of rectilinear electron flow. Poisson's equation is solved in the computer to serve as a Cauchy boundary condition to Laplace's equation outside the electron beam. The hybrid computing technique and the experimental verification of some of the results in cathode-ray tube electron lenses are presented.