Application of electric fields uniform in the Euler sense in electron spectrography

A new approach is developed for determining optimal electron-optics media in which the idea of the electron (ion) spectrograph can effectively be implemented. This approach is based on electrostatic fields with the Laplace potential obeying the Euler uniformity condition. The operation principle of the spectrograph with such coupling fields is associated with the special similarity property of constant-energy families of trajectories with different energies under uniformity conditions. For constructing the required fields, we propose a new method of analytic representation, in which the Donkin complex representation apparatus for conical fields plays the leading role. The algorithms developed here make it possible to easily construct the uniform potentials in a wide class in which classical solid spherical harmonics and spherical functions are just special cases. The Cauchy problem is formulated and solved in the general closed analytic form for uniform potentials with a symmetry plane. A number of specific structures of this class are considered.