## 3 Evaluation of the Electron Bunch Shape by Using Electron Distribution Models
### Satellite Pulse Model for Electron Bunch  
The electron distribution expressed by a pseudo-Voigt function with a negative \(\eta\) could describe the observed CER spectrum by adding a term that made the electron distribution positive. However, the proportion of the electrons that form the main peak of the electron bunch is reduced by the additional component with a larger variance, and the CER power is significantly reduced although the total charge in the electron bunch remains unchanged. Figure 6: (**a**) Electron distribution based on a pseudo-Voigt function with \(w_{V}=90\) μm and \(\eta=-0.8\). The area and standard deviation of the additional Gaussian distribution are 5.4 and 100 times the Gaussian distributions that constitute the pseud-Voigt function, respectively. (**b**) Form factor spectrum calculated for the electron distribution in (**a**). The CER energy on-crest operation of the RF field was 0.12 mJ per macropulse, which was approximately 0.6 times the energy calculated with the CER theory. Most of the electrons contributed to the formation of the main pulse in the electron bunch, and a more realistic model of the electron distribution would be needed to describe the measured spectrum. As shown in Figure 6, a dip appears beside the main peak in the pseudo-Voigt function with a negative \(\eta\). As the absolute value of \(\eta\) is larger, the dip is deeper and the frequency at which the form factor is maximized is higher. It is noted that the dip of the electron distribution in the bunch concerns the local maximum appearing in the CER spectrum. In fact, electron distributions in the bunch with a dip structure have been reported at some linac facilities [33; 34; 35]. Therefore, as a simple model that can create a dip in a bunch shape, we consider a satellite pulse model in which a small pulse exists around a main pulse with most of the electrons. For simplicity, the electron distributions of the two pulses are assumed to be Gaussian. Furthermore, it is assumed that the RMS bunch length of the satellite pulse is larger than that of the main pulse and the charge of the satellite pulse is 0.3 times or less than that of the main pulse. Based on these assumptions, the contribution of the satellite pulse to the CER power is less than 1/10 that of the main pulse, and only the bunch shape of the main pulse contributes to the high frequency components of the CER spectrum. Figure 7 shows the RMS bunch length \(\sigma_{M}\) of the main pulse evaluated by fitting the measured CER spectrum to Equation (4) in the frequency region of 1-2 THz. The RMS bunch length evaluated from the CER spectrum behaved non-linearly with respect to the RF phase difference and became minimal near the crest. As the electrons in the bunch rotated 90\({}^{\circ}\) in the phase space by passing through the 180\({}^{\circ}\) arc, the energy distribution of the electron bunch after passing through the accelerator tube was considered to have the same dependence on the RF phase difference as the electron distribution in the undulator straight section. It was confirmed that the relationship between the RMS electron bunch length and the RF phase difference evaluated with the CER spectra was similar to the relationship between the RMS energy spread and the RF phase difference, which was reported at a linear accelerator facility [36]. Next, we consider the satellite pulse in the electron bunch. The satellite pulse has three independent parameters: the RMS bunch length \(\sigma_{S}\), the distance from the main pulse \(\Delta_{p}\), and the ratio of the charge to the main pulse \(r_{c}\). A deep dip is required to increase the frequency of the local maximum in the CER spectrum. When a constant \(d_{1}\) is expressed with \(\Delta_{p}\) as  
\[\Delta_{p}=d_{1}(\sigma_{S}+\sigma_{M}) \tag{16}\]  
Figure 7: Evaluated RMS bunch length of the main pulse for the measured CER spectrum at \(\Delta=+15^{\circ}\) in the satellite pulse model.