Application of the Donkin formula in the theory of electrostatic prisms

An electrostatic prism is an electron- and ion-optical device that transforms a parallel input beam of charged particles into a parallel output beam that is deflected by an angle dependent on the energy of a charged particle beam. The principle of similarity of trajectories for electric fields that are homogeneous in Euler terms provides perfect optical properties of electrostatic prisms when the fields with zero power of homogeneity are used. It is shown that the Donkin formula for 3D homogeneous harmonic functions makes it possible to employ analytical expressions using homogeneous electric potentials of a zero power. A few examples of electrostatic prisms are calculated using the Donkin formula.