Transit-time effects on noise in planar crossed-field electron guns

A theory is developed for the noise in an unbounded parallel-plate crossed-field diode. The high-frequency regime is studied, where transit-time effects must be included. The diode is assumed to be operating in the space-charge-limited regime. Maxwell's equations and the Vlasov equation are expanded to first order. A key consideration in this expansion is the fluctuation in the boundaries of velocity space appearing in the integrals for the charge and current density. The first-order terms describe the linear response of the electric field and transmitted current resulting from the injection of a small additional cathode emission composed of electrons of a specified velocity. Using a statistical approach, these perturbations are combined to determine the noise factor. The special case of a parabolic average-potential distribution in space is considered.