An analytical approach to optical properties of plasma lenses with a non-linear magnetic field profile

The problem of evaluating the charged particle beam dynamics in plasma lenses with non-linear magnetic field profiles is considered. An analytical solution is given in terms of Jacobi functions for a proper class of field profiles. Such a solution is successfully compared with results obtained from a tracking code. From the analysis of the properties of the single-particle trajectory equation, deeper insight is gained about the focusing process. In particular, the phase space beam contour dynamic is evaluated. By analysing the beam envelope, it is demonstrated that for large emittance beams and in the case of a concave-type field profile the spot reduction ratio is larger than that obtained in the linear case and a considerable improvement in the focusing power is obtained.