## 2 Hybrid Combinations of LWFA and PWFA
### Wakefield-Induced Ionization--WII
Since typically \(k_{p}r_{\text{max}}\) is smaller than one, a practical rule of thumb to scale the final normalized emittance of the WII-injected witness beams as a function of the plasma density can be given as \(\epsilon_{n}^{\text{max}}\approx 0.1/k_{p}=0.5\) um\(/\sqrt{n_{p}[10^{18}\) cm\({}^{-3}]}\). WII injection also allows for beam-loading optimization [40]. The magnitude of the current profile of the injected beams can be controlled by adjusting the concentration of the dopant species in a similar fashion to how it is done in LWFAs with ionization injection [12]. The magnitude of the current profile  
Figure 12: OSIRIS 3D simulation of a PWFA with WII injection: (**a**) electron density of the drive beam, the plasma and the ionized helium; and (**b**) ionization rate for He according to the ADK model. The contours in (**b**) show the wakefield-equipotential surfaces in steps of 0.2.  
required for a minimal time correlated energy spread along the witness bunch is higher for higher transformer ratio cases [36]. Therefore, since WII injection requires of a high-transformer ratio by design, the witness beams needs a high current (few tens of kiloamps) for an optimal beam-loading, and, consequently, for a low correlated energy spread [36]. In summary, WII injection allows for the production of high current, low energy spread and low emittance electron beams, which can be accelerated to double or thrice the initial energy of the drive beam, all in a conceptually particularly simple setup. Utilizing LWFA-generated electron beams to perform a PWFA with WII injection has unique advantages:  
* Electron beams from LWFAs can be of high current, reaching peaks of around 30 kA and beyond [12]. Thus, they fulfill the essential requirement to enable WII injection in the PWFA stage. Other typical features of LWFA beams such as a \(\sim\)10% relative energy spread or a relatively high normalized emittance of \(\sim\)5 um are even beneficial to use them as drivers, given the enhanced stabilization that they provide to the PWFA system [9; 37]. * WII injection is triggered by the wakefields themselves, and therefore, there is no need for a precise time synchronization of external components, such as lasers, to enable injection. Moreover, the WII injection method is insensitive to the jittering of the pointing angle of the electron beam emerging from the previous LWFA. * Electron beams from LWFAs are typically of short duration, 10-20 fs [12]. This means that the corresponding resonant density reaches values near 10\({}^{19}\) cm\({}^{-3}\). At these densities the produced witness beams feature normalized emittances values below 100 nm and sub-fs duration. * It is best for the performance of WII injection to have relatively wide drive beams entering the plasma stage in order to avoid injection from the drive beam fields and to provide improved stability [37]. This means that LWFA to PWFA staging with WII injection can tolerate a certain drift in vacuum of the highly divergent electron beam from LWFA. This simplifies the LWFA to PWFA setup as there is potentially no need of beam optics between stages to transport and refocus the LWFA beam into the PWFA stage.