## 3 Applications  
The above mentioned schemes can allow production of electron beams with dramatically improved quality in compact hybrid setups. An improvement in electron beam quality, in particular normalized emittance \(\epsilon_{n}\), and corresponding output electron brightness \(B\propto I/\epsilon_{n}^{2}\) by orders of magnitude, and substantial decrease of relative energy spread down to the \(<\)0.01% level already at few GeV energies may be possible in a single stage. This has fundamental impact, for example on the feasibility of light sources such as free-electron lasers. Driving a free-electron-laser with electron output from plasma-based accelerators is one of the main goals of the plasma accelerator community. However, there are strict requirements on transverse phase space (emittance) and longitudinal phase space (energy and energy spread) of the FEL-driving electron beam in order to achieve lasing, which are so far prohibitive for realizing this goal with conventional LWFA-based approaches:  
* The emittance criterion: The emittance of the produced electron beam determines which resonant FEL wavelength \(\lambda_{r}\) may be realizable at given electron energy \(\gamma\) via \(\epsilon_{\mathrm{n}}<\lambda_{\mathrm{r}}\gamma/4\pi\) (also known as Pellegrini criterion). For a normalized emittance two orders of magnitude better than state-of-the-art, a hard X-ray FEL could be realized already at few GeV electron energies. For example, at \(\epsilon_{\mathrm{n}}<\) 50 nm, a hard X-ray lasing at \(\lambda_{\mathrm{r}}\approx\) 1.5  can be achieved already at an electron beam energy of 2 GeV.
* The energy spread criterion: The relative energy spread \(\sigma_{\gamma}/\gamma\) of the produced electron beam must be much smaller than the FEL Pierce parameter \(\rho\), which for a hard X-ray FEL means \(\sigma_{\gamma}/\gamma\) has to be better than 0.1%. Typical plasma accelerators produce beams with energy spread in the percent range, but, e.g., the plasma photocathode technique in an augmented version [17] will allow the production of beams with relative energy spread even down to the 0.01% range.  
* The FEL gain: The 1D FEL gain length scales with brightness \(B\) as \(L_{\text{g,1D}}\propto B_{\text{5D}}^{-1/3}\) and because \(B\propto\epsilon_{\text{n}}^{-2}\), e.g., a two orders of magnitude lower emittance and hence four orders of magnitude higher brightness means a much higher FEL gain, and shorter gain length, can be realized with such beams. This allows shrinking down the FEL undulator section to the ten-meter scale [21] instead of hundreds of meters as today, and may even allow for unprecedented ultrashort single spike, high brilliance coherent hard X-ray pulses.  
The features of the hybrid schemes described above therefore may allow transformative impact on future compact, high performance FELs on multiple levels via dramatically improved emittance, energy spread and brightness, and hence may allow hard X-ray FEL's to become ubiquitous, and to achieve higher performance. Other light sources such as inverse Compton scattering or betatron radiation/ion channel lasers would likewise profit from such enhanced electron beam quality. Figure 13 visualizes the principle setup of an FEL driven by a hybrid plasma wakefield accelerator (here, by a collinear Trojan Horse system).