Periodic electrostatic focusing of a ribbon beam

Abstract

Obtaining a solution of the equations of a beam satisfying certain conditions at the emitter is, as is known, only part of the problem. Any such solution determines the flow in an unbounded region, whereas actual beams have finite dimensions. To realize the flow described by the solution obtained, we must consider the problem of a system of focusing electrodes capable of producing a beam of given configuration. The solution of this problem reduces to the problem of analytic continuation of the potential given at the beam boundary together with its normal derivative into the charge-free region, i . e . , to the Cauchy problem for the Laplace equation. The problem was first formulated and solved in [1] in relation to a space-charge beam. In [2-4], the concepts of [1] were generalized to the case of plane curved trajectories. The mathematical bases of the method of e lectrostatic focusing were considered in [5] (problems of existence, uniqueness, and correctness). For a number of flows, a solution was obtained in terms of contour integrals which are very difficult to evaluate [6]. In [7], an analytic solution of the problem of the formation of arbitrary axially symmetric beams is given, Transition to ~he complex domain and transformation of the Laplace equation to hyperbolic form made it possible to give the solution in a form more convenient for obtaining final results. Only a few analytic solutions in elementary functions and closed form are known for the problem of focusing stationary flows [1, 4, 8-15] (plane diode [1, 13, 15], plane magnetron [4, 8, 9], hyperbolic [10] and elliptic [11, 12] beams, flow along circles and spirals in some nonuniform magnetic fields [14]). In [I6] electrodes were determined for several nonstationary beams.