Gun lens theory for nonparaxial trajectories by canonical mapping transformation: Characterization of general skewed rays inside electron guns by electron gun focal length

Abstract

The canonical mapping transformation (CMT) theory has been generalized to nonmeridional (skewed) rays. It has been derived that with minor mathematical modification a CMT optical parameter, “electron gun focal length,” can be applied to skewed rays. The generalization makes it possible to characterize the behavior of nonparaxial electron trajectories inside the gun by a small number of optical parameters derived in the CMT theory. Important gun properties such as the crossover size and the angular current intensity are readily available from the electron gun focal length. The relevant CMT optical parameters can be estimated either by ray tracing a few representative cathode trajectories or by use of the conventional paraxial trajectory method whose applicability is usually limited to paraxial trajectories. Interpretation of the parameters by the CMT method extends their validity to nonparaxial cases. It has been found that using the rotating coordinate system in the conventional paraxial trajectory theory is equivalent to adopting the canonical direction (direction based on the canonical momentum as opposed to the kinetic momentum) in specifying the ray condition in the CMT. The “vector potential shift” must be taken into consideration in the beam profile evaluation at the crossover when nonzero magnetic field exists on the cathode surface. The beams from the off-axis points on the cathode miss the optical axis and do not form a conventional crossover with consequent reduction in the average brightness of the electron beam when the beam current is increased by accepting the electrons from increasingly off-axis emission points on the cathode.