## 1 Introduction and Fundamental Considerations  
Laser-driven wakefield accelerators (LWFA) and electron-driven plasma wakefield accelerators (PWFA) rely on similar concepts [1]. An intense laser or electron beam driver propagates through a plasma, and, based on its typically mainly transverse forces, plasma electrons are kicked out of the path of the driver beam, not unlike how a snowplough operates. However, unlike with a snowplough, these electrons then feel the re-attractive forces of the plasma ions, which have four orders of magnitude more mass than the electrons and therefore can be assumed quasistatic in first approximation. This transient charge separation follows the driver beam through the plasma and generates a co-moving plasma oscillation wave with more or less spherical geometry. The length of the arising bubble (LWFA) or blowout (PWFA) structure is determined by the plasma electron density \(n_{e}\) and the plasma wavelength \(\lambda_{p}=2\pi c[e_{0}m_{e}/(e^{2}n_{e})]^{1/2}\), which is solely dependent on \(n_{e}\) and where \(c\) is the speed of light, \(e_{0}\) is the vacuum permittivity, and \(m_{e}\) is the electron mass and \(e\) the elementary charge. In the ideal case, the driver beam length \(\sigma_{z}\) or laser pulse duration \(\tau\) fits into half of the plasma wavelength such that \(c\tau\approx\lambda_{p}/2\), because then, as with a harmonic oscillator such as a pendulum, the excitation is strongest because the driver excites the plasma wave just until the zero-crossing of the oscillatory movement. In terms of the plasma wave number \(k_{p}=2\pi/\lambda_{p}\), this desirable relation between driver length \(\sigma_{z}\) and plasma wave number \(k_{p}\) can be expressed as \(k_{p}\sigma_{z}<1\). Figure 1 visualizes (based on particle-in-cell simulations with the code VSim [2]) how the driver (green) excites a plasma blowout/bubble with large accelerating and decelerating fields up to tens of \(\mathrm{GV/m}\) (color-coded). Here, the plasma density is \(n_{p}=10^{17}\,\mathrm{cm^{-3}}\), the corresponding plasma wavelength \(\lambda_{p}\approx 106\,\mathrm{\SIUnitSymbolMicro m}\), and the plasma skin depth \(k_{p}^{-1}\approx 16\,\mathrm{\SIUnitSymbolMicro m}\). The width of the driver pulse is in both cases set to \(\sigma_{r}=10\,\mathrm{\SIUnitSymbolMicro m}\). The strength of the driver is characterized by the dimensionless electron beam charge \(\tilde{Q}\) and the dimensionless laser light amplitude \(a_{0}\), respectively. Here, the dimensionless light amplitude, defined by the ratio of electric field \(E\) and frequency \(\omega\), amounts to \(a_{0}=\frac{eE}{m_{0}\omega c}\approx 3\). The dimensionless beam charge [3], which is defined by the ratio of the beam electron density per cubic skin depth \(N_{b}k_{p}^{3}\) and the background plasma density \(\tilde{Q}=\frac{N_{b}k_{p}^{3}}{n_{p}}=4\pi k_{p}r_{e}N_{b}\), where \(r_{e}=4\pi e_{0}c^{2}m\) is the classical electron radius, which is also set to \(\tilde{Q}\approx 3\), since the values of \(a_{0}\) and \(\tilde{Q}\) define the interaction regime of LWFA and PWFA, respectively (see Figure 2). Figure 1: PIC-simulation visualization of electron-driven PWFA (**a**) and laser-driven LWFA (**b**) in the blowout/bubble regime, respectively. The driver beam is shown in green and propagates to the right, expels plasma electrons (not shown) and thus generates strong trailing electric decelerating/accelerating fields. Notably, the plasma density is not relevant for the type of LWFA interaction, just the laser frequency and intensity, whereas, for PWFA, \(\tilde{Q}\propto\sqrt{n_{e}}\). The background of this is that a particle beam has unipolar fields, while a quickly oscillating laser pulse can penetrate plasma up to the critical density \(n_{c}(\omega)=\epsilon_{0}m_{0}\omega^{2}/e^{2}\), which amounts to \(n_{c}\approx 1.7\times 10^{21}\,\mathrm{cm}^{-3}\) for a typical Ti:sapphire laser pulse.