An Improved Method of Numerical Ray Tracing through Electron Lenses

An improved method is described for the tracing of electron trajectories through electron lenses by integrating the general ray equation with the help of Taylor's series, taking the series up to and including fourth-order terms. The paraxial trajectory is obtained by progressing from point P n on the trajectory to point P n +1 with the help of recurrence formulae r n +1 = Q 1 r n + Q 2 r ' n and r ' n +1 = Q 3 r ' n + Q 4 r n , where the coefficients Q 1 ... Q 4 are functions of the electrostatic or magnetic field distribution along the axis. The deviation of the first-order trajectory from the paraxial trajectory is built up with the help of similar recurrence formulae. The method is first developed for combined electrostatic and magnetic lenses, using the "equivalent potential" U = ϕ - eA 2 /2 m , and then specialized for pure magnetic or pure electrostatic lenses. The case of space charge within lenses is also considered. Comparison of the improved method with other methods shows that it is faster and capable of high accuracy. The factors influencing the accuracy are considered in detail and a short-cut method using extrapolation is given A fully calculated example is appended