Theory of cathode trajectory characterization by canonical mapping transformation

The paraxial lens theory cannot directly be applied to the cathode trajectories inside the gun. This inconvenience makes the interpretation of cathode trajectories difficult since one cannot use the physical concepts familiar in the paraxial lens theory, such as focal length and magnification factor. We have proposed the canonical mapping transformation (CMT) to describe the electron trajectories inside the gun by relating the ray conditions on the cathode surface to those in the crossover plane. The method takes as variables the distance along the surface and the sine of the ray angle with respect to the surface normal to define ray conditions. It has been shown that the CMT can be characterized by a small number of optical parameters. One of the parameters is the 'electron gun focal length', an extension of the image side focal length in the paraxial lens theory. The crossover size of a triode gun can be calculated from the electron gun focal length and the initial transverse energy spread. The calculation predicts the dependence of the crossover size on the grid voltage due to the change in the electron gun focal length. The prediction is compared with the measurement and shows good agreement with it. Since the CMT optical parameters can be calculated from the representative trajectories only and as they predict practically all the necessary source properties of guns, the CMT can be used as a practical tool in the designing of various types of electron guns.