MICROSCOPY Vol.46▶No.2 2011

â– Lectures

Generalization of Paraxial Trajectory Method for the Analysis of Non-Paraxial Rays: Simulation Program *G-optk* for Electron Gun Characterization

**Abstract:** The paraxial trajectory method has been generalized for the application to the cathode rays inside electron guns. The generalized method can handle rays that initially make a large angle with the optical axis. The key to success of the generalization is the adoption of the trigonometric function sine for the trajectory slope specification, instead of the conventional use of the tangent. It is possible to relate the ray conditions (position and slope of the ray at reference planes) on the cathode to those at the crossover plane using third-order polynomial functions, whose coefficients can be used as the optical parameters in the characterization of electron sources. Among them the electron gun focal length can be used for a quantitative estimate of both the crossover size and the angular current intensity. An electron gun simulation program *G-optk* has been developed based on the generalized paraxial trajectory theory. The program calculates the principal paraxial trajectories and the relevant optical parameters from axial potentials and fields. It gives a clear physical picture of the electron gun in a much more faster way than the conventional ray-tracing methods.

**Key words:** Ray tracing, numerical calculation, cathode lens, electron gun