## 2 Hybrid Combinations of LWFA and PWFA
### Wakefield-Induced Ionization--WII  
The wakefield-induced ionization (WII) injection method [22] exploits PWFAs operating at a high transformer ratio in order to induce ionization and trapping of high-quality electron bunches from and into the extreme accelerating fields of the plasma wake. These electrons originate from wakefield-induced ionization over an atomic species with appropriate ionization threshold, which is doped into the background plasma in a short axial region of the plasma target. Electrons from the dopant species not only need to be ionized by the accelerating wakefields, but they also need to be trapped by the wakefields before they reach the end of the blowout cavity. This establishes a necessary condition for trapping which can be expressed in terms of the difference in wakefield potential between the initial and final phase positions of the electrons within the plasma wake [35], i.e., \(\psi_{i}-\psi_{f}=1\), where \(\psi\) is the normalized wakefield potential, related to the longitudinal wakefields by \(E_{z}=(mc^{2}/e)\ \partial_{z}\psi\). Thus, the generated wakefields must be strong enough to generate a maximum difference in \(\psi\) greater than one. This imposes a constraint on the peak current of the drive beam, which needs to be at least 5 kA. This is a fundamental limit on the current of the drive beam required to generate a wakefield capable of trapping electrons originated from ionization [36]. In particular for WII injection, the drive beams need an even higher peak current (around 10 kA or higher), in order to generate a stronger wakefield capable of trapping within a shorter phase range. However, high-current drive beams could also induce ionization over the dopant species by meansof its space charge fields, if they are narrow enough. Therefore, in order to avoid ionization from the fields of the beam, it is necessary to start the WII injection process with a relatively wide drive beam. In this way, the trapped electrons will be originated by WII only, thereby constraining the final phase-space volume occupied by the witness beam, and, consequently, increasing its final quality. Remarkably, starting the plasma-wakefield generation process with a relatively wide drive beam also allows for largely improved stability of the PWFA system [37]. Figure 12 shows an example of WII injection performed with the PIC code OSIRIS [38] in 3D Cartesian geometry, and considering a drive beam with similar parameters to those attainable in the FLASHForward experiment [36]. In the simulation, a Gaussian drive beam with a peak current of \(I_{b}^{0}=10\) kA, a longitudinal (rms) size of \(\sigma_{z}=7\) um, a transverse (rms) size of \(\sigma_{x}=4\) um is initialized with an average energy of 1 GeV. The total charge of the beam is \(Q=575\) pC. In Figure 12, the drive beam is traversing a homogeneous plasma at the resonant density \(n_{0}=1.2\times 10^{18}\) cm\({}^{-3}\). The plasma is doped with neutral helium at 1% concentration. Near the end of the blowout cavity the accelerating wakefields are high enough to ionize completely the first electronic level of helium (Figure 12b). From the total phase-space volume of ionized electrons, only the ones closer to the axis can be trapped within the blowout, provided that a difference in potential \(\psi_{i}-\psi_{f}=1\) can be asymptotically reached at the same time that the acquired transverse oscillation amplitude is smaller than the radial extent of the blowout at the final phase position \(\zeta_{f}\). This constrains the initial phase-space volume of the trapped electrons to a thin disc centered on axis which extends up to a radial position \(r_{\text{max}}\). An upper estimate of the normalized transverse emittance of the so-injected witness bunch can be given in terms of the initial transverse extent of this disc \(r_{\text{max}}\)[39], i.e., \(k_{p}e_{n}^{\text{max}}=(k_{p}r_{\text{max}})^{2}/8\)[36].