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 $\pm10\%$.