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A theory of nonrelativistic, laminar, elliptic-beam formation is presented. The beam is shown to have the one-dimensional Child-Langmuir flow form. An analytic expression for the electrostatic potential outside the beam is derived. Equipotentials corresponding to electrode surfaces are computed numerically. The effectiveness of the electric field formed by the electrodes in focusing and preserving the elliptic, 1D Child-Langmuir flow beam is verified via 3D ray-tracing simulations

Theory and simulation of nonrelativistic elliptic-beam formation with one-dimensional Child-Langmuir flow characteristics