Formation of an ion beam in an elementary cell with inhomogeneous emission current density

A well-known Pierce solution that allows focusing a beam of charged particles using properly shaped electrodes outside the beam aperture is generalized to the case of an accelerating system with inhomogeneous emission current density. It is shown that the defocusing effect of the space charge can, in principle, be evenly compensated over the entire cross section of the beam. In contrast to the beam with a uniform emission current density, both the electric potential and the transverse electric field must be controlled along the beam boundary in order to eliminate the angular divergence. However, eliminating the angular spread evenly across the beam constitutes a mathematically ill-posed problem which needs to be solved with the use of one or another method of regularization. An alternative way of diminishing beam emittance is proposed for the beam where the emission current is uniform across the entire aperture except for a narrow beam edge layer and a simple formula for the Pierce electrodes is derived. Numerical simulation has proved the reasonable accuracy of our analytical theory.