A SIMPLE NUMERICAL-METHOD FOR A MODIFIED ABEL INVERSION IN WHICH THE DENSITY CAN BE APPROXIMATED BY ELLIPTIC SYMMETRY

In electron beam or plasma diagnostics, the standard Abel inversion to obtain the corresponding densities cannot be used if the density does not have circular symmetry, that is, if it is not a symmetric radial function, F(r), of the radius in the beam cross section. We introduce a type of asymmetry not previously treated in the literature in which the density is constant on an ellipse in the beam cross section with e ccentricity epsilon. From the equation for an ellipse in polar coordin ates, we introduce a variable eta = r[1 + cos2 thetaepsilon2/(1 - epsi lon2)]1/2 such that eta = a0 is the equation for an ellipse with semi- major axis a0. The corresponding density is represented by G(eta, epsi lon). This density can be found from the probe current I(x) by multipl ying the standard Abel inversion of I(x) by the constant [1 + epsilon2 /(1 - epsilon2)]1/2. To determine whether a density can be approximate d by elliptical symmetry, it is necessary to use two probe measurement s separated by an aperture. The aperture acts as an analyser, both to test for circular symmetry and, if not present, to analyse for ellipti cal symmetry as described in this paper. This technique is applied to the output of a Van de Graaff generator whose electron flux density is found to be approximated by elliptical symmetry.