## 3 Evaluation of the Electron Bunch Shape by Using Electron Distribution Models
### Satellite Pulse Model for Electron Bunch
the intensity of each pulse at the intersection of both the pulses is 5% of its peak value in the case of \(d_{1}=2.5\), and a dip becomes sufficiently deep. Figure 8 shows the CER spectra calculated in the cases of \(\sigma_{S}=2\sigma_{M}\), \(3\sigma_{M}\), and \(4\sigma_{M}\). The transmittance and diffraction loss of the optical transport system shown in Equation (1) are taken into consideration in these CER spectra calculated for some \(r_{c}\). As shown in Figure 8, it can be noted that the frequency of the maximum of the CER spectrum at \(d_{1}=3\) shifts to the high frequency side and exceeds 0.6 THz as \(r_{c}\) increases. Moreover, the modulation in the CER spectrum grows as \(r_{c}\) increases. The observed CER spectrum had a maximum on-crest operation at a frequency of approximately 0.6 THz, but there was no significant modulation in the CER spectrum. It is estimated that \(r_{c}\) was 0.15 from the CER spectrum calculated by changing \(r_{c}\). Therefore, the CER spectra are calculated by changing \(\Delta_{p}\) in conditions of \(\sigma_{S}=3\sigma_{M}\) and \(r_{c}=0.15\) as shown in Figure 9. It is noted that the coordinates of the maximum move counterclockwise in the graph as the distance between the main and satellite pulses increases from zero. The frequency of the maximum is maximized when the distance is \(10.5\sigma_{M}\). Therefore, we compare the calculated CER spectrum in the case of \((\sigma_{S},\Delta_{p},r_{c})=(3\sigma_{M},10.5\sigma_{M},0.15)\) to the measured CER spectrum near the crest, \(O_{-1}\), as shown in Figure 10a. The electron bunch shape in this condition is shown in Figure 10b. The coefficient of determination of the calculated CER spectrum for the set of frequencies used in Section 2.3 is as high as 0.983, it was found that the satellite pulse model can effectively reproduce the CER spectrum near the crest. As described in Section 2.2, the electron distribution in the bunch can be approximated by the Gaussian at the RF phase difference of +15 degrees. Figure 9 shows that the distance between the main and satellite pulses is almost zero at this RF phase difference. As proved in Appendix A, when the satellite pulse has a lower energy than the main pulse, the distance between the main and satellite pulses changes from the initial state on-crest operation by an amount proportional to the square of the RF phase difference. In fact, the frequency of the satellite pulses is almost zero at this RF phase difference. Figure 10: (**a**) Calculated CER spectrum in the satellite pulse model (red curve) and the measured CER spectrum for \(\Delta=-1^{\circ}\) (black curve). The parameters \(\sigma_{S}\), \(\Delta_{p}\), \(r_{c}\) are \(3\sigma_{M}\), \(10.5\sigma_{M}\), and \(0.15\), respectively. (**b**) Electron distribution at the aforementioned parameters in the satellite pulse model. Figure 9: Calculated CER spectra for several \(\Delta_{p}\) with \(\sigma_{S}=3\sigma_{M}\) and \(r_{c}=0.15\). the maximum of the form factor fitted by Equation (6) was given as a quadratic function of the RF phase difference. Therefore, the distance between the main and satellite pulses can be given by  
\[\Delta_{p}=-d_{2}\Delta^{2}+10.5\sigma_{M} \tag{17}\]  
where a constant \(d_{2}\) is determined so that \(\Delta_{p}\) becomes zero at \(\Delta=+15^{\circ}\). Figure 11 shows spectra fitted for \(\Delta=-5^{\circ}\) and \(+7^{\circ}\) with \(\Delta_{p}\) calculated by Equation (16) and \(\sigma_{M}\) shown in Figure 7. The ratio \(r_{c}\) is assumed to be 0.15 in these fittings. The coefficients of determination for these fitting curves are 0.988 for \(\Delta=-5^{\circ}\) and 0.991 \(\Delta=+7^{\circ}\), respectively. Although the satellite pulse model does not describe the complex structure of the measured CER spectrum, it can explain that there is a frequency fluctuating local maximum due to the RF phase difference in the CER spectrum. It was confirmed that this simplified model is a good representation of the outline of the electron bunch shape.