# Conceptual and Technical Design Aspects of Accelerators for External Injection in LWFA
## 3 Production of Ultra-Short e-Bunches
### Space Charge and CSR Effects
#### 3.3.1 RF Compression via Velocity Bunching  
The velocity bunching technique can be used to compress electron bunches only at low energies, because the method relies on the generation of a correlated spread of the velocities in the electron bunch. The bunch is injected into a RF cavity close to the 0 phase of the electric field, where the head of the bunch experiences lower field amplitude than the tail. By doing so, while the beam travels in the structure, it is compressed because of the induced velocity chirp. Moreover, since the average velocity of the bunch is smaller than the velocity of the RF field, the bunch slips along the traveling field wave, moving towards the crest. In this way it is possible to compress and accelerate an electron bunch at the same time [53]. This scheme also allows for the preservation of the transverse quality of the beam through the so-called emittance compensation scheme, which has been experimentally proven [54].  
The big advantage of the compression via velocity bunching is that there is no emission of CSR, because the beam always travels along a straight trajectory. The space charge effect is not negligible and it is dominant for the optimization of the working point.  
In velocity bunching compression the space charge couples the transverse focusing of the beam with its longitudinal compression. The longitudinal space charge field on axis in a drift region scales as [55]:  
\[E_{s}(r=0)=\frac{Q}{4\pi\epsilon_{0}}\frac{1}{\gamma^{2}s^{2}} \tag{12}\]  
where \(Q\) is the bunch charge, \(\epsilon_{0}\) is the free space permittivity, \(\gamma\) is the relativistic Lorentz factor describing the electron energy, and \(s\) is the longitudinal coordinate along the bunch. As we can observe, the space charge field vanishes at high energies.  
If the beam is transversely focused while its energy is still relatively low, the space charge effect limits the final minimum achievable bunch duration. On the other hand, if the bunch is focused too late, i.e., the compression takes place with negligible longitudinal space charge force, the longitudinal slices of the bunch \"cross-over\", provoking a deterioration of the transverse phase space [56], as shown in Figure 3. A numerical approach to control the bunch density versus the bunch energy gain in the RF-compressor has been proposed in [57].  
In this reference the following equation for the longitudinal RMS beam envelope is introduced:  
\[\sigma_{z}^{\prime\prime}+K_{z}\sigma_{z}+\frac{3\gamma^{\prime}\sigma_{z}^{ \prime}}{\beta^{2}\gamma}=\frac{Qc}{5\sqrt{5}I_{A}\beta^{2}\gamma^{4}\sigma_{z }\sigma_{t}}+\frac{\epsilon_{n}z^{2}}{\beta^{2}\gamma^{6}\sigma_{z}^{3}}. \tag{13}\]  
Figure 3: Example of compression with velocity bunching. Three cases are plotted corresponding to overcompression, maximum compression and undercompression respectively. It can be noted that as soon as the beam overcomes its maximum compression point (where we have slice-crossover) the transverse and longitudinal emittances start to rapidly increase. The simulations have been performed using the code ASTRA [32].  
The longitudinal laminarity parameter \(\rho_{z}=\frac{Qc(\gamma\sigma_{z})^{2}}{L_{0}\rho\epsilon_{z}}\) is defined, which allows for tuning the compression versus the acceleration in the linac while ensuring beam laminarity. More specifically when \(\rho_{z}>1\) the bunch maintains its longitudinal laminarity and \"cross-over\" is forbidden. In other words by maintaining \(\rho_{z}>1\) different longitudinal slices are not allowed to overlap or to cross-over each other, thus allowing the production of ultra-short electron bunches.  
At the maximum compression point it is possible to observe a characteristic _double-S shape_ of the bunch, due to the fact that each one of the longitudinal slices of the distribution start to bend. This feature corresponds to a narrow spike in the longitudinal projection of the beam (see plot B in Figure 4).