## 4 Discussion  
We observed the CER spectra generated by electron beams passing through the downstream bending magnet of the undulator straight section at KU-FEL. It was observed that the CER spectrum had a local maximum on-crest operation of the RF field at the electron energy of 40 MeV. The frequency of the maximum depended on the RF phase difference between the electron gun and the accelerating tube. The frequency was maximized near the crest and decreased in proportion to the square of the RF phase difference from the crest. The form factor evaluated from the measured CER spectrum was proportional to a function whose exponent was given by the quadratic equation of the frequency. These experimental results suggest that the longitudinal electron distribution in the bunch can be approximated  
Figure 11: Fitting curves for the CER spectra at (**a**) \(\Delta=-5^{\circ}\) and (**b**) \(\Delta=+7^{\circ}\) (red curves). The measured spectra are indicated by black curves in the figures.  
by the pseudo-Voigt function in the real space. Although there are many phenomena in which the Voigt distribution appears in the frequency space as in the Raman spectroscopy, it is rare that a distribution in the real space is described by the pseudo-Voigt function.  
However, a negative electron distribution appears in the pseudo-Voigt function with a negative ratio \(\eta\). Therefore, we proposed a satellite pulse model consisting of two pulses with a large and small amount of charge as a practical model for the electron distribution. This model shows that the frequency at which the CER spectrum is maximized depends on the interval between the two pulses. It is preferable to evaluate the electron bunch shape using the Kramers-Kronig relation for more complex spectral structures [25; 33]. However, the accuracy of the measurement for the low frequency components of the CER spectrum is low due to the cutoff of the optical transport system and the sensitivity of the detector, and it is difficult to accurately reproduce the electron bunch shape. The satellite pulse model is effective in describing the outline of the electron distribution in the bunch despite its simplicity.  
As described above, it is possible to investigate the longitudinal distribution in the electron bunch by observing the CER spectrum. We improved the flat mirror so that the CER during FEL oscillations can be observed without damaging the FEL oscillation, and we observed that the FEL oscillations cause changes in the CER spectrum. Furthermore, in order to remove the influence of absorption by water vapor, we will consider placing the CER observation system under dry air. The measurements of the CER spectrum during FEL oscillations will elucidate the effect of the FEL interactions on the longitudinal electron bunch shape. The CER is the best tool to understand how the electron bunch is influenced by the FEL interaction.  
Conceptualization, N.S.; data curation, N.S. and H.Z.; formal analysis, N.S.; investigation, N.S. and H.O.; methodology, N.S. and H.Z.; writing--original draft, N.S.; writing--review and editing, N.S. and H.Z. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Japan Society for the Promotion of Science KAKENHI 19H0406 and has been carried out under the Visiting Researcher's Program of the Research Reactor Institute, Kyoto University, and ZE Research Program ZE2021B-22, Institute of Advanced Energy, Kyoto University.  
The authors would like to thank [PERSON] of Nihon University for useful discussions of the beam physics.  
The authors declare no conflict of interest.