# Multiple-Collision Free-Electron Laser Compton Backscattering for a High-Yield Gamma-Ray Source
## 2 Materials and Methods  
### Backward Compton Scattering  
First, we consider the phenomenon in which one electron bunch collides head-on with a multi-pulse FEL beam in an optical cavity. In this paper, it is assumed that linearly polarized FEL pulses are incident light in the LCS process, and nonlinear effects due to the laser field of the FEL are negligible [24]. The optical cavity is assumed to be a Fabry-Perot resonator which is constructed using two high-reflectivity mirrors. As shown in Figure 1, the FEL-Compton backscattering gamma rays are observed in a screen which is separated from the collision point by a distance \(l\) and is perpendicular to an axis of the electron-beam orbit (\(z\) axis). The distance \(l\) is selected to be sufficiently larger than the beam sizes of the FEL pulse and the electron bunch at the collision point [12]. Representing the azimuthal and zenith angles in spherical coordinates as \(\theta\) and \(\varphi\), respectively, the differential cross section of the scattered gamma ray on the screen \(\mathcal{O}\) is given by  
\[\frac{d\sigma(\theta,\varphi)}{d\Omega}=r_{e}^{2}\bigg{[}1-\frac{\sin^{2} \theta}{\gamma^{2}(1-\beta\cos\theta)^{2}}\cos^{2}\varphi\bigg{]}\frac{1}{ \gamma^{2}(1-\beta\cos\theta)^{2}} \tag{1}\]  
where \(r_{e}\) is the classical electron radius, \(\beta\) is the electron speed normalized by the speed of light \(c\), and \(\gamma\) is the electron energy in units of its rest mass \(m_{e}c^{2}\), respectively [25]. When \(\gamma>>1\) and \(\theta<<1\) hold, Equation (1) can be approximated as  
\[\frac{d\sigma(\xi,\varphi)}{d\Omega}=4\gamma^{2}r_{e}^{2}\bigg{[}1-\frac{4\xi ^{2}}{(1+\xi^{2})^{2}}\cos^{2}\varphi\bigg{]}\frac{1}{(1+\xi^{2})^{2}} \tag{2}\]  
where \(\xi\) equals \(\gamma\theta\)[26]. The yield of gamma rays emitted by an electron bunch \(\#j\) colliding with \(n\) FEL pulses in a circle with radius \(r\) on the center of the screen \(\mathcal{O}\) is denoted as \(N_{j}\). The radius \(r\) is assumed to be sufficiently larger than the beam sizes of the FEL and the electron and to be sufficiently smaller than \(l\). Because the differential cross section of backward Compton scattering is very small, the probability that one electron in the bunch contributes to multiple processes of the backward Compton scattering is extremely low. It can be ignored that electrons lost energy in the backward Compton scattering collide with the FEL pulses again. Because the beam sizes of the electron bunch and the FEL pulses are sufficiently larger than the wavelengths of the scattered gamma rays and the electron bunch length is much longer than the wavelengths of the scattered gamma rays, spatial and time interference effects among the scattered gamma rays at the collision points can be ignored. Therefore, \(N_{j}\) is expressed via the following equations:  
\[N_{j}=\sum_{i}^{n}\int_{0}^{2\pi}dq\int_{0}^{\rho_{i,j}}d\sigma_{i,j}(\theta,q) L_{i,j}\sin\theta d\theta \tag{3}\]  
\[\theta_{i,j}=\arctan\left(\frac{r}{l_{i,j}}\right) \tag{4}\]  
where \(L_{i,j}\) is the luminosity between electron bunch #\(j\) and FEL pulse #\(i\) at the collision point [23]. Using the number of electrons in #\(j\), \(N_{e,j}\), and the number of photons in #\(i\), \(N_{p,i}\), the luminosity is given by  
\[L_{i,j}=\frac{N_{e,j}N_{p,i}}{A_{i,j}} \tag{5}\]  
\[A_{i,j}=2\pi\sqrt{\sigma_{x,j}^{2}+\sigma_{p,j}^{2}}\sqrt{\sigma_{y,j}^{2}+ \sigma_{p,j}^{2}} \tag{6}\]  
where \(A_{ij}\) denotes the effective overlap area between electron bunch #\(j\) and FEL pulse #\(i\)[16; 27]. The parameters \(\sigma_{x}\), \(\sigma_{y}\), and \(\sigma_{p}\) are the horizontal beam size of the electron bunch, the vertical beam size of the electron bunch, and the beam size of the FEL pulse at the collision points, respectively. When the electron bunch has slight transverse motion around the axis of the electron-beam orbit, phase factors are added to the angles \(\theta\) and \(\varphi\) in Equation (3). Here it is assumed that the phase factors are negligible.