  Active plasma lensing is a promising technology for compact focusing of
particle beams that has seen a recent surge of interest. While these lenses can
provide strong focusing gradients of order kT/m and focusing in both transverse
planes, there are limitations from nonlinear aberrations, causing emittance
growth in the beams being focused. One cause of such aberrations is beam-driven
plasma wakefields, present if the beam density is sufficiently high. We develop
simple, but powerful analytic formulas for the effective focusing gradient from
these wakefields, and use this to set limits on which parts of the beam and
plasma parameter space permits distortion-free use of active plasma lenses. It
is concluded that the application of active plasma lenses to conventional and
plasma-based linear colliders may prove very challenging, except perhaps in the
final focus system, unless the typical discharge currents used are dramatically
increased, and that in general these lenses are better suited for accelerator
applications with lower beam intensities.
