Rigorous calculation of spatially periodic electrostatic and magnetostatic fields

A rigorous solution is presented for the external (two-dimensional) Dirichlet problem for a Laplace equation with periodic boundary conditions. The solution obtained makes it possible to calculate both the electrostatic and the magnetostatic potential for a whole class of periodic structures using a single algorithm. A comparison is made between the theoretical and experimental values of the longitudinal and transverse components of magnetic induction in the channel of a periodic magnetic focusing system. The boundary paths of electron fluxes in the focusing system are calculated in the hydrodynamic approximation. The conditions of stable focusing are determined.