## 1 Introduction and Fundamental Considerations
The energy of an electron \(E_{\text{kin}}=m_{0}c^{2}(\gamma-1)\approx 0.511\,\text{MeV}\left(\gamma-1\right)\), and, e.g., for an electron with a kinetic energy of 0.5 MeV, the corresponding \(\gamma\approx 1.97\) and \(\beta\approx 0.54\), whereas, for an electron at 5 MeV, its Lorentz factor is \(\gamma\approx 10.87\) and \(\beta\approx 0.9957\), which is already close to the speed of light. This has an importance consequence, namely that, in LWFA, accelerated electrons will outrun the laser driver. They therefore move from the accelerating field phase in the back of the bubble forward into the decelerating field in the first part of the bubble, which limits the useful acceleration distance and hence energy gain. It is interesting to note that, as pointed out above, in LWFA, the wake excitation strength is independent of the plasma density, but the acceleration distance is, whereas in contrast in PWFA the wake excitation strength is dependent on the plasma density, but the acceleration distance is not. This is why PWFA, a sufficiently intense and energetic driver provided, can harness phase-constant, tens of GV/m-scale (or higher) accelerating fields over meter-scale distance, and hence realize tens of GeV energy gains in a single stage. Another important aspect that arises from the different wake velocities in LWFA and PWFA regards injection and dark current, respectively. While in PWFA, due to practically negligible dephasing, the wake velocity equals the driver velocity such that \(\gamma_{\text{wake}}\approx\gamma_{\text{p}}\approx 10^{4}\) for a 10 GeV driver beam, the wake velocity of the plasma wave is that of the driving laser beam such that \(\gamma_{\text{wake}}\approx\gamma_{p}=\left(1-\frac{\omega_{p}^{2}}{c^{2}} \right)^{-1/2}\approx\) 10-100 for typical plasma densities. A consequence of this is that electron injection and bunch generation via self-injection or other mechanisms is considerably easier to achieve in LWFA than in PWFA, while in turn it is easier to keep PWFA dark-current-free by excluding unwanted self-injection mechanisms. The general trapping condition for injected electrons is that the injected electrons have to catch up with the plasma wave such that \(\gamma_{\text{electron}}\geq\gamma_{\text{wake}}\approx\gamma_{p}\). The  
Figure 4: The electric fields of electron bunch PWFA drivers are unipolar, while electric fields of laser pulse LWFA drivers are oscillating. As a result, the peak electric fields, which are required to excite a similarly strong blowout/bubble, are three orders of magnitude lower for PWFA than for typical LWFA Ti:sapphire lasers. Hamiltonian is the sum of a kinetic energy term and the potential energy. If injected electrons are assumed to be initially at rest, this means the electrostatic wakefield potential energy in the wave frame needs to be larger than the required electron kinetic energy \(-e\Delta\phi^{\prime}\geq E^{\prime}_{\text{kin}}=E_{0}(\gamma_{p}-1)=m_{0}c^{2} (\gamma_{p}-1)\). The reduced threshold for injection due to lower \(\gamma_{p}\) in LWFA is helpful if aiming at producing electron bunches on the fly, but makes it more difficult to avoid dark current, and in turn the higher threshold for injection makes it is more difficult to produce electron bunches on the fly in PWFA, but on the other hand is advantageous in order to realize a robust, dark-current free system. As regards divergence and diffraction, both electron beams and the laser pulses expand hyperbolically when propagating in the forward direction \(z\). A Gaussian laser pulse focused to a spot size \(w_{0}\) exhibits diffraction following \(\omega(z)=\omega_{0}\sqrt{1+(z/Z_{R})^{2}}\). Similarly, for an electron beam focused to a spot size \(\sigma_{r0}\), the transverse evolution is \(\sigma_{r}=\sigma_{r0}\sqrt{1+(z/\beta^{*})^{2}}\).