Synthesis of electrostatic fields for transportation of charged particle beams

In this paper, an approach to creating corpuscular-optical devices for transportation and transformation of charged particle beams has been elucidated. These devices are able to optimize and create the most convenient configuration of ionic or electron paths. The approach relies upon the inverse dynamics problem formulated on the basis of the Hamilton-Jacobi equation. The motion in the symmetry plane of a three-dimensional (3D) field was considered. The problem was solved by analytical methods. An algorithm for constructing electric fields providing the particle motion on the desired trajectories was described. А key to this algorithm lies with a concept of conformal transformation from the theory of complex-valued function. This procedure was illustrated by examples. Quadratic potential was chosen as a basis. Three functions of conformal transformation were considered, providing the rotation of the focused charged particle beam at a fixed angle, the transformation of divergent flow to parallel one. The calculated two-dimensional potentials were extended into 3D-space by power series expansion on transverse coordinate. Device embodiments were suggested on the basis of the calculated field structures.