Electrostatic lenses are of great importance to electron lithography systems, electron microscopes, information display, and so forth. The canonical aberration theory in electron optics has an advantage over the conventional aberration theory in simplicity. In this paper, the primary canonical aberrations of electrostatic lenses are thoroughly investigated with Mathematica.
We start from the general Hamiltonian function to find the Gaussian value of its fourth-order approximation as well as the Gaussian value of the second-order approximation perturbed by chromatic effects. Furthermore, all the primary geometric and chromatic aberration coefficients are derived analytically.
Then, various typical geometric and chromatic aberration patterns are graphically displayed.
As an example, all the geometric and chromatic aberration coefficients of a given electrostatic lens are numerically calculated and the numerical values are compared with those calculated through another differential algebraic method. The result is very amazing, the computing accuracy reaching 10 significant figures or even better.
In conclusion, Mathematica is particularly suitable for exploring electron optical aberrations since it is a fully integrated computer algebra system containing symbolic computation, graphical display, and numerical calculations all in one.