  Recently, a theoretical investigation of the collective and nonlocal quantum
effects has been carried out within the framework of a quantum approach to the
relativistic charged particle beam travelling in a cold, collisionless,
strongly magnetized plasma. This has been done taking into account both the
plasma wake field excitation and the quantum paraxial approximation. On the
basis of this theory, here we carry out a preliminary study of the transverse
effects experienced by a cold relativistic beam through a thin plasma slab
(plasma lens). In the strongly nonlocal regime, in which the beam experiences a
very strong focusing effect, the scheme of plasma lens is reviewed in terms of
the wave description provided by the above quantum theory.
