## 3 Evaluation of the Electron Bunch Shape by Using Electron Distribution Models  
### Pseudo-Voigt Distribution  
Although the form factor given by Equation (6) was evaluated from the measured CER spectrum in a limited frequency region, it contained information on the rough longitudinal shape of the electron bunch. Therefore, by simplifying the longitudinal structure of the electron bunch, we investigated the outline of the electron bunch. When the electron bunch shape is symmetric, \(S\) (\(z\)) is given with the square root of the form factor by the following equation [25]:  
\[S(z)=\frac{1}{\pi c}\int_{0}^{\infty}\sqrt{f(\omega)}\cos\left(\frac{\omega}{c }z\right)d\omega \tag{7}\]  
Here, the right side of Equation (7) is the inverse Fourier transform for the square root of the form factor. That is, when a Fourier transform for a function \(g\) is expressed as \(F[g]\), the following equation holds:  
\[S(z)=F^{-1}\left[\sqrt{f(\omega)}\right] \tag{8}\]  
Using the convolution theorem for the Fourier transform  
\[F^{-1}[g_{1}\cdot g_{2}]=\frac{1}{2\pi}F^{-1}[g_{1}]*F^{-1}[g_{2}] \tag{9}\]  
\[g_{1}(x)*g_{2}(x)\equiv\int g_{1}(x-y)g_{2}(y)dy \tag{10}\]  
\(S\) (\(z\)) can be expressed with Equation (6) as the following equation [26]:  
\[S(z)=\frac{e^{\frac{1}{2}c_{\Delta}}}{4\pi^{2}}F^{-1}\left[e^{\frac{1}{2}a_{ \Delta}\omega^{2}}\right]*F^{-1}\left[e^{\frac{1}{2}b_{\Delta}|\omega|}\right] \tag{11}\]  
If both \(a_{\Delta}\) and \(b_{\Delta}\) are negative, the righthand side in Equation (11) is a convolution of the Gaussian and Lorentz function. Therefore, under this assumption, \(S\) (\(z\)) can be described with a Voigt function \(V(z;\sigma,\gamma_{L})\) as the following equations:  
\[S(z)=\frac{e^{\frac{1}{2}c_{\Delta}}}{4\pi^{2}}V(z;\sigma,\gamma_{L}) \tag{12}\]  
\[\sigma=c\sqrt{-a_{\Delta}} \tag{13}\]  
\[\gamma_{L}=-\frac{1}{2}b_{\Delta}c \tag{14}\]  
As it is difficult for the Voigt function to capture the outline of the distribution, a pseudo-Voigt function consisted of elementary functions is often used instead of the Voigt function in spectroscopy [27; 28]. When the full width at half maximum of the Voigt function is defined as \(w_{v}\), \(S\) (\(z\)) can be approximated as the following equation [29; 30]:  
\[ \begin{array}{ll}S(z)&=\frac{e^{\frac{1}{2}c_{\Delta}}}{4\pi^{2}}V_{pV}(z;w_ {V},\eta)\\ &=\frac{e^{\frac{1}{2}c_{\Delta}}}{4\pi^{2}}\left[(1-\eta)\frac{2}{w_{V}}\sqrt {\frac{\ln 2}{\pi}}e^{-\frac{4\ln 2}{w_{V}^{2}}2}+\eta\frac{2w_{V}}{\pi(w_{V}^{2}+4 \varepsilon^{2})}\right]\end{array}  \tag{15}\]  
where \(\eta\) represents the ratio of the Lorentz component, and the pseudo-Voigt function is physically meaningful in the case of \(0\leq\eta\leq 1\). Nevertheless, herein, we consider the case in which \(\eta\) is negative. As the asymptotic behavior of the pseudo-Voigt function becomes negative in the case of \(\eta<0\), it is not usually treated as a distribution function. However, it is possible to avoid the negative distribution function by making a section in which the electrons are distributed finitely and by adding a positive constant term. When the behavior of the form factor in the low frequency region is ignored, it is possible to add a Gaussian distribution term with a large standard deviationinstead of the positive constant term. The physical meaning of the additional term is a background composed of low density electrons around the main components of the electron bunch, which has been actually observed in linear accelerators [31]. Figure 6 shows an electron distribution given by a pseudo-Voigt function with a negative \(\eta\) and its form factor. To avoid the negative electron distribution, a Gaussian distribution with a standard deviation of 5.0 mm, which is 100 times the standard deviation of the pseudo-Voigt function, is added as an additional term. As shown in Figure 6, there is a local maximum of the form factor at a frequency that is away from zero.