The idea of exploiting the wakefield of a charged particles beam, formed when traversing a plasma, to focus on axis the beam itself was proposed on 1987 by P. Chen [ P. Chen, J. J. Su, T. Katsouleas, S. Wilks, and J. M. Dawson, IEEE Trans. Nucl. Sci. NS-15, 218 (1987) ]. Linear colliders luminosity benefits incredibly from a reduced spot size of the beams, being inversely proportional to the cross-sectional area.

We simulated a positron beam self-focusing traversing an under-dense plasma lens, which results to be non suitable as the focusing force acting upon the bunch of particles is inhomogeneous.
During the beam (in red) propagation through the plasma lens (in grey), the wake structure is formed due to the transverse motion of the plasma electrons which are drawn on axis, on the contrary of what happens with an electron bunch which repels the plasma electrons away from its path.

Since the plasma electrons are drawn on axis depending on the positive space charge distribution of the propagating beam, which varies longitudinally, the focusing wakefield has a longitudinal dependence and thus does not allow uniform focusing. The positron beam “explodes” as when a part of it is still in the focusing phase another part is already defocusing.