  To realise a planned high-luminosity and high-energy $e^+e^-$-collider, as
the ILC, a large amount of positrons have to be produced and the accelerated
particles have to be captured and matched according to the damping ring
acceptances. %There exist several technical possibilities. In this contribution
a new promising alternative method for capturing positrons will be presented,
the application of the plasma lens as an optical matching device. It will be
compared with the current matching device proposed for the ILC, namely the
quarter wave transformer. An advantage of the plasma lens is the different
magnetic field component, which focuses the divergent beam in a more effective
manner. Therefore it will be shown in this paper that the yield requirements
could be achieved more easily. The plasma lens can actually be a promising
alternative for focusing beams as soon as the technical feasibility has been
approved.\\ In the simulation, a tapered active plasma lens has been optimized
using the approximation of a homogeneous electric current density constant in
time. The optimization process led to a plasma lens design that improves on the
ILC's currently proposed optical matching device, namely the quarter wave
transformer, by approximately $50-100\%$. Furthermore the design has been shown
to guarantee a stable captured positron yield within $\pm1.5\%$ for single,
independent parameter deviations of about $\pm10\%$.
