# Multiple-Collision Free-Electron Laser Compton Backscattering for a High-Yield Gamma-Ray Source
## 2 Materials and Methods
### Description of the Multiple-Collision FEL-Compton Backscattering Gamma Rays  
We consider that \(m\) electron bunches per unit time collide head-on with \(n\) FEL pulses at collision points in the optical cavity. Using Equations (3)-(6), the total yield of the scattered gamma rays is given by  
\[N_{\text{CS}}=\sum_{j}^{m}N_{j}=\sum_{j=1}^{m}\sum_{i=1}^{n}\int_{0}^{2\pi}d \phi\int_{0}^{\theta_{i,j}}\frac{N_{e,j}N_{p,i}d\sigma_{i,j}(\theta,\phi)}{A _{i,j}}\sin\theta d\theta. \tag{7}\]  
For a storage ring FEL system in which the round-trip time of the FEL pulse in the optical cavity equals the revolution period of the electron bunch, this equation can be rewritten as  
Figure 1: Schematic diagram of the free-electron laser (FEL)-Compton backscattering in head-on collision.  
\[N_{\text{CS}}=f_{r}\sum_{j}^{m^{\prime}}N_{j}=f_{r}\sum_{j=1}^{m^{\prime}}\sum_{i= 1}^{n}\int_{0}^{2\pi}d\phi\int_{0}^{\theta_{ij}}\frac{N_{ej}N_{p,i}d\alpha_{ij} (\theta,\phi)}{A_{ij}}\sin\theta d\theta \tag{8}\]  
where \(f_{r}\) is the revolution frequency of the electron bunch in the case of the storage ring and \(m^{\prime}\) is the number of electron bunches with which one FEL pulse collides in the optical cavity. When the collision points are close as in linac FELs, Equation (8) shows that the total yield of the scattered gamma rays is approximately proportional to the product of the number of electron bunches per unit time and the number of FEL pulses which collide with the electron bunches in the optical cavity. Then, a high-yield gamma-ray source is expected by using backward Compton scattering between multiple electron bunches and multiple FEL pulses in an optical cavity.  
The spectrum of the scattered gamma rays in multiple-collision FEL Compton backscattering will be considered. When the incident light collides head-on with the electron beam in the backward Compton scattering process, the energy of the scattered gamma rays (\(E_{p}\)) is given by the following approximate equation using the energy of the incident light (\(E_{L}\)) [25; 28]:  
\[E_{p}=\frac{4\gamma^{2}E_{L}}{1+\xi^{2}+\frac{4\gamma E_{L}}{m_{e}c^{2}}}. \tag{9}\]  
As shown in this equation, the energy of the scattered gamma rays for a certain \(E_{L}\) is determined by the zenith angle regardless of the polarization of the FEL. Using the scattered gamma-ray energy normalized by its maximum energy \(E_{p,Max}\), which is the scattered gamma-ray energy at \(\theta=0\), the differential cross section of backward Compton scattering can be expressed by [16]  
\[\frac{d\sigma(\varepsilon)}{d\varepsilon}=4\gamma^{2}r_{e}^{2}\Big{(}2 \varepsilon^{2}-2\varepsilon+1\Big{)}, \tag{10}\]  
\[\varepsilon=\frac{E_{p}}{E_{p,Max}}. \tag{11}\]  
When the influence of the electron-beam emittance and energy spread on the backward Compton scattering is negligible, the lower limit of the scattered gamma-ray spectrum is determined by limiting the zenith angle by the collimator. Defining the energy spectrum of the scattered gamma rays which are generated by backward Compton scattering between electron bunch #\(j\) and FEL pulse #\(i\) and which pass through the collimator as \(S_{ij}(\varepsilon)\), the spectrum of the total gamma rays observed behind the collimator is given by the summation of the spectrum \(S_{i,j}(\varepsilon)\) from each collision point as given in the following equation:  
\[S(\varepsilon)=\sum_{j}^{m}S_{j}(\varepsilon)=\sum_{j}^{m}\sum_{i}^{n}S_{i,j}( \varepsilon). \tag{12}\]  
Considering the electron bunch length and temporal resolution of the general measurement system for the energy spectrum in the gamma-rays region, it is difficult to identify the spectrum \(S_{ij}(\varepsilon)\) of the FEL-Compton backscattering gamma rays which electron bunch #\(j\) generates at each collision point. The spectrum \(S_{j}(\varepsilon)\) of whole scattered gamma rays which the electron bunch #\(j\) generates with multiple FEL pulses is observed in actual experiments.