  An active plasma lens focuses the beam in both the horizontal and vertical
planes simultaneously using a magnetic field generated by a discharge current
through the plasma. A beam size of 5--10 $\mu$m can be achieved using an
focusing gradient on the order of 100 T/m. The active plasma lens is therefore
an attractive element for plasma wakefield acceleration, because an ultra-small
size of the witness electron beam is required for injection into the plasma
wakefield to minimize emittance growth and to enhance the capturing efficiency.
When the driving beam and witness electron beam co-propagate through the active
plasma lens, interactions between the driving and witness beams and the plasma
must be considered. In this paper, through particle-in-cell simulations, we
discuss the possibility of using an active plasma lens for the final focusing
of the electron beam in the presence of driving proton bunches. The beam
parameters for AWAKE Run 2 are taken as an example for this type of
application. It is confirmed that the amplitude of the plasma wakefield excited
by proton bunches remains the same even after propagation through the active
plasma lens. The emittance of the witness electron beam increases rapidly in
the plasma density ramp regions of the lens. Nevertheless, when the witness
electron beam has a charge of 100 pC, emittance of 10 mm mrad, and bunch length
of 60 $\mu$m, its emittance growth is not significant along the active plasma
lens. For small emittance, such as 2 mm mrad, the emittance growth is found to
be strongly dependent on the plasma density.
