A new method for calculating the axial potential due to space charge in electrostatic optics

It is well known that the spherical aberration is positive for an axisymmetric electrostatic lens that is empty and free from space charge. However, a space charge profile can be tailored to introduce negative spherical aberration in such a lens. This could be useful for modern high-resolution focused ion beam systems. A new technique is described for calculating potentials and their derivatives and is used to calculate the trajectories of charge particles propagating in regions in which space charge is present. The charge density and potential are expanded in powers of r and z and the Poisson equation is solved term by term. The method has been implemented numerically for axisymmetric configurations. An illustrative example is presented.