Generalization and synthesis of the Kino gun

A more general transformation is given for the cross-field gun problem, allowing for an arbitrary emission velocity; Kino's Solutions I and II then appear as particular cases. Close investigation of the region between the cathode and the front and rear focus electrodes (which must be separated to avoid thermal and migration problems) shows that all designs with nonzero emission velocity are theoretically flawed: imaginary ledges would be needed to connect the cathode surface to its supports. In addition, Kino's Solution II is shown to be unattainable because the slightest departure of the emission velocity from the prescribed value introduces singularities between the cathode and the accelerator electrode. Despite the theoretical flaws, useful designs can be generated in the neighborhood of Solution I, but matching realistic emission velocities on the order of 0.1 V. A synthesis procedure for this Solution III is given, which determines the electrodes in terms of the required exit-beam dimensions and current, the allowed cathode current density, and the electrode voltages.