## 1 Introduction and Fundamental Considerations
In the overdense plasma lens, the electron bunch space charge is only a small perturbation and the collective shielding by plasma electrons takes place locally and is proportional to the local bunch density. For example, in the case of a typical Gaussian electron bunch with density distribution \(n_{b}(x,y,z)=n_{b0}\exp\frac{-x^{2}}{2\sigma_{\pi}^{2}}\exp\frac{-y^{2}}{2 \sigma_{\pi}^{2}}\exp\frac{-z^{2}}{2\sigma_{\pi}^{2}}\), the peak bunch density in the center \(\hat{n}_{b,max}=N/((2\pi)^{3/2}\sigma_{\pi}^{2}\sigma_{z})\) and thus the focusing strength is much larger than farther outside, which results in spherical aberrations. It helps if the bunch is long compared to the skin depth of the plasma such that \(k_{p}\sigma_{z}\gg 1\), and if the beam is narrow such that \(k_{p}\sigma_{\pi}\ll 1\). In summary, the above considerations reveal a high degree of complementarity of LWFA and PWFA. Main features and considerations of employing laser pulses and electron beam pulses for plasma wakefield acceleration are:  
* Electron bunches drive plasma waves efficiently due to unidirectional fields, already at comparably low bunch self-field values such as few GV/m to tens of GV/m. * Laser pulses require large electric fields, typically in the TV/m range, to excite plasma waves via the ponderomotive force due to their oscillating field structure. * Laser pulses are very efficient to tunnel ionize matter and hence to provide plasma, due to the high peak electric laser pulse fields. * Electron bunches are not efficient for ionizing matter because of the low electric self-fields. Figure 6: An electron beam of typical energy and emittance stays focussed over much longer length than a typical laser pulse, focused to the same spot size. * Electron bunches propagate with approximately the speed of light even in plasma, whereas dephasing is a fundamental problem in LWFA. This allows much longer, and phase constant acceleration to be realized in PWFA. * Electron bunches are stiff and expand transversally much less than a laser pulse of typical parameters diffracts, which allows much longer acceleration distances in PWFA. * Laser pulses quickly diffract when focused strongly, which allows the generation of locally confined hot spots for tunneling ionization and electron release. * LWFA allow various injection mechanisms, supported by the low wake velocities due to the laser pulse group velocity, but are for the same reason prone to unwanted self-injection and dark current. * Injection thresholds for PWFA are comparably high, due to the high wake velocities of dephasing free systems, but, on the other hand, PWFA for the same reason allows realizing dark-current free systems. * LWFA can be realized in ultracompact, lab-scale setups, but not yet at highest repetition rates and stability. * Linac-driven PWFA requires large facilities, but can provide bunches with high stability at high repetition rate. * LWFA inherently generates beams with very high currents, but not extremely low energy spreads, both due to the small plasma cavity sizes and large field gradients. These features, advantages and disadvantages of LWFA and PWFA are highly complementary.