# Conceptual and Technical Design Aspects of Accelerators for External Injection in LWFA
## 1 Introduction  
Particle accelerators have become of fundamental importance for many different applications such as high energy physics, radiation generation, material science, medical applications etc. The excellent performance and amazing scientific outcome achieved by km-scale accelerators have been possible thanks to the availability of vast resources (including large areas) required for their construction. As a result too few machines exist and the beam time available for users is very limited compared to the number of requests. Many applications therefore demand an improvement in the compactness and the cost efficiency of accelerators in order to allow, for example, compact accelerators to be built directly in university campuses and industry [1]. Figure 1 shows the scaling of the peak gradient versus the typical cell length in conventional RF (Radio-Frequency) accelerator cavities. It can be seen that by moving towards the left side of the plot, i.e., towards high frequency RF fields, it is in principle possible to increase at the same time the peak gradient and the compactness of the accelerator. Unfortunately moving towards the W-band is not trivial for two reasons: the lack of reliable sufficiently high power RF-sources, such as klystrons [2], to feed the structures and the ultimate deterioration of the metal constituting the structure in the presence of high gradients [3]. Also, these high frequencies require small apertures and thus wakefields driven by the beam become an issue. Three approaches have been developed in the last decades to overcome the power source limitation:  
* Use of wake-fields driven by charged particles as electromagnetic field driver;
* Use of wake-fields driven by laser pulses as electromagnetic field driver;
* Use of laser fields coupled to the beam through micro-structures. The material limitation has been overcome by replacing the metallic cavities with samples made of dielectrics or plasmas. Table 1 summarizes the principal parameters for characterizing the electron bunch size and the channel field for some laser-driven novel acceleration methods, namely: LWFA (Laser Wake-Field Accelerator), DLA (Dielectric Laser Accelerator) and TDA (THz Driven Accelerator). \begin{table}
\begin{tabular}{c c c c c} \hline
**Technique** & **Period Acc. Field [s]** & **Aperture Diameter [m]** & **Max Gradient [GV/m]** & **Rep. Rate [Hz]** \\ \hline LWFA & \textless{}10^{-12}\(n=10^{16}\) cm\({}^{-3}\) & \(\sim\)10\({}^{-6}\) (matching condition) & \textgreater{}100 [4] & 1–50 \\ DLA & \textless{}10\(\,\times\,10^{-15}\) & \textless{}10^{-6}\) & \(\sim\)1 & \(\sim\)10 \(\,\times\,10^{6}\) \\ TDA & 3\(\,\times\,10^{-12}\) & \textless{}10^{-3}\) & \(\sim\)0.4 & 1–100 \(\,\times\,10^{3}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Overview of the order of magnitude of typical beam and field parameters for several laser-driven novel acceleration techniques. The listed repetition rates are typical values mainly limited by the lasers and they are not fundamental limitations of the acceleration method. Figure 1: Scaling of the maximum achievable gradient versus the cell length in conventional accelerators. The gradients provided by novel acceleration techniques are superimposed to the plot and indicated with a star-symbol. For those items, also marked with an asterisk, the wavelength of the accelerating field has been plotted instead of the cell length. In the figure are used the following abbreviations: LWFA (Laser Wake-Field Accelerator), DLA (Dielectric Laser Accelerator) and TDA (THz Driven Accelerator). In LWFA an intense laser pulse propagates through a plasma target. The laser transversally pushes away the electrons by its ponderomotive force and creates an electron-depleted region. The resulting strong electrostatic field pulls back electrons on axis. The electrons oscillate and create a co-propagating wake-field. The electron bunch to be accelerated in LWFA can either be directly extracted from the plasma (through the so called self-injection process) or can be externally injected [5].