## Appendix A  
Here, consider an electron bunch that is accelerated near the crest of the RF field by accelerator tubes. The initial state before the acceleration is represented by the subscript \(i\), and the final state after acceleration is represented by the subscript \(f\). In addition, the subscript \(0\) is added to the center value of the electron distribution, and the slight change from the center value is represented by \(\delta\). As reported in [37; 38], the interaction between an electron and an electric wave of the accelerator tube can be described by a Hamiltonian  
\[H=\gamma-\sqrt{\gamma^{2}-1}-\alpha\cos\psi \tag{1}\]  
\[\alpha=\frac{eE_{z}}{km_{e}c^{2}} \tag{2}\]  
where \(\psi\) is the phase of the wave as seen by the electron, \(m_{e}\) is the rest mass of an electron, \(k\) is \(\pi\) the RF wave number, and \(E_{z}\) is the peak field of the longitudinal electric wave. The normalized energy of the electron \(\gamma\) is assumed to be sufficiently larger than 1; therefore, that the following approximation holds:  
\[\gamma-\sqrt{\gamma^{2}-1}\cong\frac{1}{2\gamma} \tag{10}\]  
As the Hamiltonian is conserved before and after the acceleration, the following relationship holds:  
\[\frac{1}{2\gamma_{i}}-\alpha\cos\psi_{i}=\frac{1}{2\gamma_{f}}-\alpha\cos\psi_ {f} \tag{11}\]  
Furthermore, the central energy of the final state, \(\gamma_{f0}\), is given by the following equation:  
\[\gamma_{f0}=\frac{1}{\frac{1}{\gamma_{i0}}+2\alpha\Big{(}\cos\psi_{f0}-\cos \psi_{i0}\Big{)}} \tag{12}\]  
First, we consider variation of energy in the final state caused by the variation of the phase in the initial state \(\delta\psi_{i}\). Assuming that \(\delta\psi_{i}\) does not influence the phase in the final state near the crest operation, the energy in the final state is given with variation of energy owing to the phase in the final state, \((\delta\gamma_{f})_{\psi}\), by  
\[\gamma_{f}=\gamma_{f0}+\Big{(}\delta\gamma_{f}\Big{)}_{\psi}\cong\frac{1}{ \frac{1}{\gamma_{i0}}+2\alpha\Big{[}\cos\psi_{f0}-\cos\psi_{i0}\Big{(}1-\frac {1}{2}\delta\psi_{i}^{2}\Big{)}\Big{]}} \tag{13}\]  
where, because \(\psi_{i}\) is almost zero near the crest operation, the following approximation is used:  
\[\cos\psi_{i}=\cos(\psi_{i0}+\delta\psi_{i})\cong\cos\psi_{i0}\Big{(}1-\frac{1 }{2}\delta\psi_{i}^{2}\Big{)} \tag{14}\]  
Furthermore, by expanding Equation (13) using the relationship \(\delta\psi_{i}<<1\), the energy in the final state is given by  
\[\gamma_{f}\cong\gamma_{f0}\Big{[}1-\gamma_{f0}\alpha\cos\psi_{i0}\delta\psi_ {i}^{2}\Big{]} \tag{15}\]  
By comparing Equations (13) and (15), the following relationship is derived:  
\[\Big{(}\delta\gamma_{f}\Big{)}_{\psi}\cong-\gamma_{f0}^{2}\alpha\cos\psi_{i0 }\delta\psi_{i}^{2} \tag{16}\]  
Next, we consider the variation of energy in the final state caused by the variation of the energy in the initial state \(\delta\gamma_{i}\). By replacing \(\gamma_{i0}\) with \(\gamma_{i}=\gamma_{i0}+\delta\gamma_{i}\) in Equation (12), the following equation is derived:  
\[\gamma_{f}=\gamma_{f0}+\Big{(}\delta\gamma_{f}\Big{)}_{\gamma}\cong\gamma_{ f0}\Big{(}1+\frac{\gamma_{f0}}{\gamma_{i0}}\delta\gamma_{i}\Big{)} \tag{17}\]  
Using Equations (16) and (17), the variation of energy in the final state is given by  
\[\alpha=\frac{eE_{z}}{km_{e}c^{2}} \tag{2}\]0  
As the second term in Equation (18) is a negative, the energy deviation \(\delta\gamma_{f}\) decreases in proportion to the square of the phase deviation of the initial state if \(\delta\gamma_{i}\) is positive. The electron distribution rotates \(90^{\circ}\) counterclockwise in the phase space by the \(180^{\circ}\) arc, therefore, the energy deviation is converted into the longitudinal position in the bunch. It was proved that the distance between the main and satellite pulses changes in proportion to the square of the RF phase difference between the electron gun and accelerator tube from the crest.