Focusing laminar electron beams of high density with the trajectories of required configuration

Formation of high density electron beams is complex due to the very strong Coulomb interaction in the tightly focused beam. Therefore, a magnetic field is normally required, in addition to a properly configured electric field, in order to compress a stream of accelerated particles into the dense beam. Unfortunately, the magnetic field essentially complicates the internal beam structure due to heavy mixing of the electron trajectories that may be unacceptable in some applications. In addition, it is a complicated problem to calculate both the electric and magnetic fields in any combined electromagnetic system producing high-density electron beams (Forrester, 1988; Szilagyi, 1988; Hawkes and Kasper, 1989). In order to overcome the complications, another approach was developed based on considering and solving numerically a series of more general well-posed boundary value problems instead of a single problem arising due to the conventional approach. According to Yurchenko (Microwave and Opt. Tech. Lett. no. 10, pp. 104-108, 1995), some functions used for imposing the boundary conditions could be chosen arbitrarily so that a series of solutions for the electric and magnetic fields is obtained. In general, any of these could be used as the final solution as they all yield a solution to the original problem providing the required spatial structure of the beam. In this paper, we apply this approach to a special case of a beam with rapidly converging electron trajectories when an analytical solution to the part of the problem can be found (Yurchenko and Yurchenko, 2001). We compute some configurations of the electric and magnetic fields needed to prepare the dense laminar electron beams with the required trajectories