  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 within a short
distance using a 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 drive beam and witness electron beam
co-propagate through the active plasma lens, interactions between the drive 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 for the AWAKE RUN 2
experiments. 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 RMS beam size, plasma density, and multiple
Coulomb scattering.
