Adiabatic plasma lens designs for the final focus of TeV electrons*

The successful operation of future e+ e- linear colliders (LC) critically relies on the ability to tightly focus beams at the interaction point to achieve high luminosities. With spot sizes expected to reach the nanometer scale in TeV linear colliders, traditional beam delivery systems face challenges due to chromatic effects induced by synchrotron radiation at the final focusing lens. To evade the synchrotron-radiation limit, the concept of adiabatic plasma lenses has emerged as a potential solution. In an azimuthally symmetric passive adiabatic plasma lens, a plasma wave wake is excited by a particle beam and a trailing beam surfs on the wake, experiencing a linear focusing force of $momega_p^2 r/2$, which is proportional to the plasma density. An adiabatic plasma upramp is designed so that the focusing force on the trailing beam continuously and slowly (compared to the betatron wavelength) increases during propagation through the plasma ramp. However, for the beam parameters of a LC the tightly focused matched beam can induce ion motion within the beam, an aspect that has not been extensively investigated in previous studies of adiabatic lenses. We investigate the influence of ion motion on adiabatic plasma lens for high energy  electrons using advanced simulation tools, QuickPIC with adaptive mesh refinement and QPAD.  Furthermore, we will explore novel approaches to increase the focusing force slope through ion motion or asymmetric drive beams, potentially relaxing the stringent emittance requirements in linear colliders.