Numerical studies on the ramped density plasma lens

We consider the so-called adiabatic plasma lens when the plasma density is ramped too quickly to be considered adiabatic. The lens length can be much shorter in such a case, but the final spot size is shown to be larger by a factor of √1+α2 than for a slowly ramped plasma lens with the same initial and final density (where α=-β'/2 is proportional to the plasma density gradient). We find that the final spot size is the same whether or not the Courant-Snyder parameters of the beam (α and β) are matched to the lens. However, matched beams allow the plasma density to be lower while unmatched beams allow the lens to be shorter (for the same α and for the same final to initial plasma density ratio). Finally, we find that a smaller spot size can be obtained for a given lens length and density ratio by starting at smaller α and increasing α along the lens