## 1 Introduction and Fundamental Considerations
However, while the Rayleigh length \(Z_{R}=\pi\omega_{0}^{2}/\lambda\) of a typical high-power Ti:sapphire laser pulse with central wavelength \(\lambda\approx 0.8\,\upmu\)m is tunable only via the spot size, which in turn needs to be small in order to yield high intensity for a given laser power, the beta function \(\beta^{*}=\sigma_{r0}^{2}\gamma/\epsilon_{n}\) of an electron pulse can not only be tuned via the spot size \(\sigma_{r0}\), but also via the electron \(\gamma\) and normalized emittance \(\epsilon_{n}\). More importantly, it is also much larger for typical values of \(\gamma\) and normalized emittance \(\epsilon_{n}\). For example, for the same spot size of laser and electron beam \(w_{0}=\sigma_{r0}=10\,\upmu\)m, the Rayleigh range is \(Z_{R}\approx 400\,\upmu\)m for Ti:sapphire pulses, while \(\beta^{*}\approx 20\,\)cm for a typical electron beam energy of 1 GeV, which corresponds to \(\gamma\approx 2000\), and a typical normalized emittance of the drive beam of \(\epsilon_{n}=10^{-6}\) mrad. This dramatically different length over which an electron driver beam and laser pulse, respectively, stay compact, is sketched in Figure 6. This is another main reason PWFA can straightforwardly realize substantially longer acceleration distances, and hence energy gains in a single stage, than LWFA. Figure 5: The velocity of electrons in plasma is independent of plasma density, while the laser pulse group velocity in plasma decreases with increasing plasma density. Electron velocities in terms of corresponding \(\beta\) are plotted in black vs. electron energy, while the laser pulse group velocity is plotted in red vs. plasma density. It should be noted that self-focusing or active guiding, e.g., in a transversely parabolic density channel, can extend the effective Rayleigh range and hence the obtainable diffraction-limited acceleration distance of an LWFA driver pulse substantially. For an electron beam driver, there is a contribution to the divergence due to intra-bunch space charge forces. The radial Lorentz force of a particle inside the bunch is \(F_{r}=e(E_{r}+v\times B_{\theta})\) and the electric field within an (infinitely) long bunch with uniform density \(n_{b}\) is given by [PERSON] law as \(E_{r}=\frac{1}{2}en_{b}r/e_{0}\) and the magnetic field by [PERSON]'s law as \(B_{\theta}=\frac{1}{2}en_{b}\mu_{0}\sigma\). With \(c^{2}=1/(\epsilon_{0}\mu_{0})\), this yields \(F_{r}=eE_{r}\left(1-\frac{v^{2}}{c^{2}}\right)=eE_{r}\gamma^{-2}\). This means that transverse forces are limited, and also that for a witness bunch which is produced in a plasma wave (both driven by lasers as well as by particle beams) the transverse space charge forces quickly decrease during acceleration as \(\gamma^{-2}\). For a highly relativistic (\(\gamma\gg 1\)) bunch propagating in vacuum, the current-induced pinching magnetic field balances the Coulomb space charge repulsion. However, when propagating in plasma, the repulsive electron bunch space charge forces can be shielded by the plasma electrons, such that in total a net focusing force arises due to the magnetic field. To some extent, this \"plasma lens\" is the equivalent of relativistic self-focusing in LWFA. In some analogy to the classification of laser propagation through plasma, one differentiates between an underdense plasma lens, where the beam density \(n_{b}\gg n_{p}\), and an overdense plasma lens where the plasma density is so high that \(n_{b}\ll n_{p}\). The underdense plasma lens, a \(\tilde{Q}>1\) case in the nonlinear interaction regime, is straightforward because it simply means that the plasma electrons are expelled from the vicinity of the electron beam and thus a uniform ion background shields the space charge of the beam, and the focusing strength scales linearly with ambient plasma density \(n_{p}\) in principle without spherical aberrations. The head of the bunch, and depending on plasma skin depth and width and length of the bunch other parts of the bunch may not be seeing a fully uniform pure ion background.