Chapter Four - Solution of the Beam Formation Problem in Three Dimensions

Abstract

Abstract The urgency of studying the 3D problems in intense beam optics is directly conditioned by a number of practical problems, including, for example, the interaction of powerful electron beams with rectangular targets in laser physics and plasmochemistry, analysis of the fringe effects caused by the ribbon beams finiteness, and evaluation of the tolerances associated with small deviations of the beam geometry from planar and axial symmetry. In this section, consideration of the 3D problem is preceded by a review of the results of 2D formation theory. The algorithm to solve the 3D Cauchy problem for the Laplace equation is reduced to integral transformation with respect to the longitudinal coordinate and solution of the corresponding 2D problem using the Riemann method, which is modified as applied to the elliptical-type equations. The inverse integral transformation is then performed. Considered are cylindrical beams with an arbitrary smooth cross section, representing a “cut” from the planar diode in ρ-mode. The use of the Lipschitz‐Hankel integral allows a solution to be obtained in the form of a definite integral with the hypergeometric Gauss function. For an arbitrary cone representing a “cut” from the spherical diode, a solution for the Riemann function is constructed in the form of an asymptotic series. The same approach is used for an arbitrary toroid: It appears impossible to perform contour integration for the exact solution describing the Riemann function, which in the case under consideration, takes the form of a hypergeometric function. Discussed is the problem of ambiguity domains of the coordinate mesh arising from the analytical continuation of the cross-sectional contour in the formation problem for a cylindrical beam with a rectangular cross section. An algorithm to construct a conformal mapping with improved smoothness is formulated as applied to the beam with a near-rectangular cross section. Relevant examples are also discussed. Based on investigation of the forming electrodes in the emitter's vicinity, it is shown that the Pierce angle of 67.5° does not depend on the contour curvature for cylinders, cones, and toroids with an arbitrary cross section, just as this angle does not depend on the emitter curvature and the distribution of physical parameters on the emitter in the axisymmetric case (see Chapter 3 ). Local equations of the forming electrodes represent an effective and simple tool in 3D electron-optical system design.