Noise in planar crossed-field electron guns—I: Theory

A theory is developed for the shot effect in an unbounded parallel-plate crossed-field diode. The low-frequency régime is studied, where transit-time effects are negligible. It is further assumed that short-circuit conditions prevail, and that the diode is operating in the space-charge-limited régime. The theory begins with the derivation of general equations for the potential and the transmitted current. These are then subjected to a perturbation expansion. The zero-order terms in this expansion describe the steady-state potential and transmitted current. The first-order terms describe the linear response of the potential and transmitted current resulting from the injection of a small additional emission composed of electrons of a specified velocity. The theory shows how these perturbations are combined to determine the current noise power at the anode and thereby the noise factor. The special case where the average potential shows a parabolic distribution in space is considered.