An inverse formulation for the identification of magnetic field profiles in plasma lenses

The knowledge and the control of the magnetic field profile is of major importance in the application of plasma discharges as magnetic lenses for charged particles beams, because of the complex dynamics of the plasma. Since conventional field measurements are quite difficult for high current plasma discharges, we propose an inverse formulation for the identification of the field profile directly from the focusing data. The problem is shown to be ill-posed, in general; therefore a regularisation is needed. This is obtained, first, by means of a proper restriction of the observation space; second, by using an over determined data set, leading to a pseudo-inversion procedure. The numerical solution is illustrated, showing the procedure to be robust, at least for the practical applications range. In particular it is shown how the: proposed method is effective in determining the amount of distortion of the field profile as compared to the reference linear one.