# Conceptual and Technical Design Aspects of Accelerators for External Injection in LWFA
## 3 Production of Ultra-Short e-Bunches
### Space Charge and CSR Effects
#### 3.3.2 Magnetic Compression  
In magnetic compression the CSR effect also plays a significant role. In the CSR effect, the head of the beam is perturbed by the radiation emitted by the tail. Particles in the bunch radiate coherently at wavelengths much longer than the bunch length. This radiation produces a position-dependent energy modulation along the bunch, which will show up in the longitudinal phase space and distort it. CSR has also an important effect in degrading the transverse emittance of the beam. The absorption or emission of the photon changes the energy of an electron that it is located in a dispersive region. Different slices of the bunch are subject to different energy variations, therefore they start oscillations around different dispersive orbits, increasing the projection of the beam size and the angular divergence in the bending plane. This mechanism provokes an increase in the projected transverse emittance. CSR effects can also cause slice emittance growth due to the component which depends on the transverse position. In the case of very strong compression, electrons which end up in the same longitudinal slice experience different longitudinal CSR forces, resulting in high slice emittance values after a dispersive section [58].  
Figure 4: (**A**) longitudinal phase space of a beam at the RF gun exit. Different colors mark four longitudinal slices; (**B**) longitudinal phase space after the linac exit in the case of optimal bunch compression; (**C**) longitudinal phase space after the linac exit in the case of under-compression; (**D**) longitudinal phase space after the linac exit in the case of over-compression. The simulations have been performed using the code ASTRA [32].  
To get an idea about the scaling of the CSR effect with the most relevant beam parameters it is useful to have a look at the expression for the induced energy spread in a dipole in steady-state CSR emission for a Gaussian bunch [59]:  
\[\sigma_{\delta,CSR}=0.2459\frac{r_{e}Q\theta R^{1/3}}{e\gamma\sigma_{z}^{4/3}}, \tag{14}\]  
where \(\theta\) is the bending angle, \(R\) is the curvature radius in the dipole and \(r_{e}\) is the classical electron radius. Since the longitudinal charge distribution varies along the chicane, most of the CSR-induced energy spread arises between the end of the third dipole and the end of the fourth one, because the bunch reaches its shortest duration in this region. Also, the projected emittance increase caused by CSR can be analytically estimated by using a steady-state CSR model. It turns out that for a four-dipole chicane compressor bending in the x plane, the projected emittance growth inside the 4 th dipole of a chicane, assuming a constant bunch length, is [60; 61]:  
\[\epsilon_{x,n}=\epsilon_{xx0}\sqrt{1+\left(\frac{\theta\sigma_{\delta,CSR}}{ 2\sigma_{x0}}\right)^{2}}. \tag{15}\]  
To minimize the CSR effect some strategies are possible, such as using low \(R_{56}\) chicanes, and small bending angles, or squeezing the C-S \(\beta_{x}\) between the third and forth dipoles of a compressor [59].