## 1 Introduction and Fundamental Considerations
To excite the longitudinally trailing plasma wave, it is required to push plasma electrons mainly in the transverse direction as efficiently as possible. While a point charge (or an ensemble of point charges) has a radial Coulomb electric field distribution \(E_{r}=\frac{e}{4\pi\epsilon_{0}}\frac{1}{r^{2}}\), a Lorentz transformation of the electric fields yields \(E^{\prime}_{\perp}=\gamma E_{r}\gg E^{\prime}_{\parallel}=E_{r}\), such that the electric field (more accurately \(E^{\prime}_{\perp}=\gamma(\mathbf{E}+\mathbf{v}\times\mathbf{B})\)) of a relativistically moving electron is mainly transversal, scaling with the Lorentz factor \(\gamma=(1-v^{2}/c^{2})^{-1/2}\). For the magnetic field \(B_{0}=E_{0}/c\), we have \(B^{\prime}_{\parallel}=0\) and \(B^{\prime}_{\perp}=vE_{r}c^{-2}\). The case of an electron bunch interacting with plasma is much more complex, as magnetic self-fields, return currents, etc. have to be taken into account. Figure 3 visualizes the transverse field of an electron driver in the lab frame. The Coulomb force \(F_{\mathrm{Coulomb}}=e\mathbf{E}\) of an electron bunch on plasma electrons is unipolar and scales linear with the perceived electric field. This is highly suitable in order to expel electrons off axis and to set up a plasma wave. This is an important difference to laser drivers, where the electric field is not unipolar, but oscillates and puts the plasma electrons into a quiver motion. Hence, the plasma electron expulsion of electrons is achieved by means of the pondermotive force \(F_{\mathrm{Pond}}=-\frac{e^{2}}{4m\omega^{2}}\
abla\mathbf{E^{2}}\propto \
abla\mathbf{I}\), which scales with the gradient of the electric field squared or intensity \(I\). As a result, the electric fields which produce a similarly strong plasma wake excitation are orders of magnitude lower for an electron bunch PWFA driver when compared to a laser pulse LWFA driver, \(E_{\mathrm{r,PWFA}}\ll E_{\mathrm{r,LWFA}}\). This fundamental difference is visualized in Figure 4, showing that the electric fields of an electron bunch driver as used for Figure 1 are three orders of magnitude lower than for a correspondingly strong laser driver as used for Figure 1. This has important implications, in particular for the choice of media and injection mechanisms. Figure 3: The electric field of a point charge in the lab frame is mainly transverse, scaling with its Lorentz factor \(\gamma\). Figure 2: Linear, quasi-linear and non-linear regimes of PWFA (**a**) and LWFA (**b**), as indicated by the corresponding interaction value of \(\tilde{Q}\) and \(a_{0}\), respectively. Now, in a bunch of particles instead of only one electron, space charge forces will make the bunch diverge, and lengthen. However, as regards lengthening or \"dephasing\" \(\Delta L\) of a bunch over an acceleration distance \(L\) of a sufficiently relativistic and monoenergetic beam \(\left[\gamma_{i}\gamma+\Delta\gamma\right]\) such that \(\Delta\gamma\ll\gamma\), the lengthening \(\frac{\Delta L}{L}\approx\frac{1}{\gamma^{2}}\frac{\Delta\gamma}{\gamma}\) is typically very small. While bunch lengthening is typically negligible even over meter-scale distances in vacuum, energy transfer and thus deceleration of a drive beam in plasma leads to increasing \(\Delta L\) and therefore to significant lengthening, up to the point where drive beam electrons can slow down so much that they fall back and travel from the decelerating into the accelerating phase of the blowout [4]. For laser pulses drivers, we have a fundamentally different situation. While an electromagnetic wave propagates with the speed of light in vacuum, in plasma the group velocity of a laser pulse is \(v_{g}=c\left(1-\frac{\omega_{p}^{2}}{\omega_{0}^{2}}\right)^{1/2}\), thus is dependent on the plasma density \(n_{e}\) via \(\omega_{p}=\left(\frac{n_{e}c^{2}}{m_{e}\cdot c_{0}}\right)^{1/2}\). Figure 5 compares velocities of electrons and laser pulses in terms of corresponding \(\beta=v/c\).