Theory of cathode lenses for probe beams

In order to apply the trajectory equation to the cathode lens, the conventional paraxial and aberration theory and the K-S-Y theory are modified. A novel trajectory gradient, defined by the sine of the angle to the exis, is introduced, and the initial energy factor is added to the axial potential. This modification makes it possible to apply the trajectory equation to a gun whose cathode field is weak. Making use of the fact that the trajectory equation is suitable for calculation of the acceptance and for treatment of many trajectories, the current profiles of an SEM probe beam are calculated.