## 2 Hybrid Combinations of LWFA and PWFA  
The complementary features of laser beams, particle beams and LWFA and PWFA can be exploited. For example, using a laser pulse to pre-ionize the plasma for a PWFA stage is an obvious method to harness both the ability of laser pulses to tunnel ionize at comparably low intensities, and to make use of the long, dephasing-free acceleration distances achievable by PWFA. However, there are many other interesting hybrid permutations, and combinations of hybrid building blocks, which are more complicated than LWFA or PWFA alone, but allow achieving extremely high electron beam quality and tunability. A fundamental subset result of the above discussion and comparison of features of LWFA and PWFA is that:  
* High-current electron beams are ideal drivers for plasma waves. * Laser pulses are ideal to produce dense, high current electron bunches in compact setups. It is therefore a very attractive option to design and optimize an LWFA stage such that the electron output can be harnessed to drive an attached PWFA stage. This principle has been suggested to exploit purposefully in [4], where \"Monoenergetic Energy Doubling in a Hybrid Laser-Plasma Wakefield Accelerator\" was proposed. A more or less sharp transition from LWFA to PWFA may occur also in a single plasma stage [5; 6; 7] when dephasing is reached and/or when laser pulse power depletion and/or diffraction sets in. Figure 7 shows, based on PIC-simulations with OOPIC, how an electron double bunch from an LWFA stage is energy boosted in a driver-witness type PWFA stage. The key requirement for an electron bunch to drive a strong PWFA wakefield stage is its current. The energy spread of electron beam drivers is less important, because as described above for sufficiently relativistic electrons the phase slippage or bunch lengthening is small and not a primary concern. On the contrary, it is well-known that in a process called BNS damping [8] an energy spread can suppress instabilities such as the beam breakup (BBU) instability and hosing. While the energy transfer of the drive beam to the plasma will introduce an energy spread even if initially perfectly monoenergetic, it helps if the drive beam has a significant energy spread right from the beginning of the PWFA process [9]. This is a feature which LWFA can realize very well: produce inherently ultrashort, multi-kA electron bunches [10; 11; 12] with significant energy spreads in very compact setups. Significant energy spread of LWFA electron output is a drawback or even showstopper for example for demanding applications such as the free-electron laser, but in contrast, for PWFA, the significant energy spread can be even an asset. Experimentally, plasma beam dumps via collective deceleration [13]--a signature of energy transfer from driver beam to plasma and as such a first step in the direction of hybrid LWFA-PWFA--has been observed in a setup with two gas jets [14]. Passive plasma lensing has also been shown experimentally in a similar setup [15]. This is important, because one of the drawbacks of LWFA-generated electron bunches (in fact, of any plasma wakefield-accelerated electron bunch) is the typically large divergence with which they leave the plasma stage. This means that: (a) the divergence has to be reduced during the extraction process from the LWFA stage as much as possible; and (b) the electron beam needs to be captured by a transport line soon after the first stage. Long down-ramps which allow adiabatic extraction of the electron beam help with regard to Point (a), and with regard to Point (b) passive plasma lensing as in [15] can be exploited during the transition from LWFA to PWFA [16]--a highly attractive option, because at the same time it pinches the electron beam, which in turn increases its \(\tilde{Q}\) and hence strengthens the interaction with the plasma. Other options are active plasma lenses, or if possible, capture by strong permanent magnet quadrupoles.