# Multiple-Collision Free-Electron Laser Compton Backscattering for a High-Yield Gamma-Ray Source
## 4 Discussion
### Spatial Distribution of Multiple-Collision FEL-Compton Backscattering  
Figure 6 shows that the FEL-Compton backscattering gamma rays had greater dispersion in the vertical direction because the FELs were horizontally linearly polarized waves [40]. Substituting \(\varphi=1/2\) into Equation (2) and converting from the spherical coordinate system to the Cartesian coordinate system, the distribution function of the backward Compton scattering gamma rays on the vertical axis, \(N_{\text{CS}}(y)\), is given by the following approximate equation:  
\[N(y)=4\gamma^{2}r_{e}^{2}f_{r}\sum_{i}\sum_{j}\frac{N_{e,j}N_{p,j}}{A_{i,j}} \Bigg{[}1+\bigg{(}\frac{y}{l_{i,j}}\bigg{)}^{2}\Bigg{]}^{-2},\ i\
eq j \tag{13}\]  
Figure 6: (**a**) Spatial distribution of the FEL-Compton backscattering gamma rays on the imaging plate in the three-bunch operation, and (**b**) intensity distribution of the FEL-Compton backscattering gamma rays on the vertical axis indicated by the red broken line in (**a**). The calculated contributions of the collision points C\({}_{3}\), C\({}_{-2}\), and C\({}_{-3}\) to the total intensity distribution (red solid line) are indicated in (**b**) by the dotted line, the one-dot chain line, and the two-dot chain line, respectively.  
where the subscripts \(j\) and \(i\) denote the electron bunches (#0, #5, #10) and the FEL pulses (#0, #5, #10) for the storage ring NJJI-IV, respectively. The gamma-ray beam generated at collision point C\({}_{2}\) was deflected in the horizontal direction and had extra spread due to the electron bunch length. Thus, the contribution of collision point C\({}_{2}\) to the gamma-ray distribution on the vertical axis (the dotted line in Figure 6) is negligible. As shown in Figure 2, the numbers of collisions per revolution period at the collision points C\({}_{3}\), C\({}_{-2}\), and C\({}_{3}\) were two, one, and two, respectively. The distributions of the gamma-ray beams generated at each collision point can be evaluated on the vertical axis by taking the sum of \(j\) in Equation (13), as shown in Figure 6b. However, the calculated lines in this figure have a common arbitrary coefficient to match the measurement data. The sum of all contributions from each collision point is also indicated by the red solid line in Figure 6b. The coefficient of determination between the red solid line and the measured distribution of the total gamma-ray beam on the vertical axis is 0.98 in Figure 6b, and the calculation is thus in good agreement with the measured data. The High-Intensity Gamma-ray Source reported the profile of FEL-Compton backscattering gamma rays with nearly 100% linearly polarized FELs [41]. The agreement between the measured and calculated values in the gamma-ray profile indicates that the degree of linear polarization of the NJJI-IV FEL was also close to 100%. Moreover, this experimental result demonstrates that the distribution of the gamma-ray beam generated at the multiple collision points can be described as the superposition of the gamma-ray beams generated at each collision point. To increase the yield of monochromatic and wavelength-variable photon beams in the energy region above keV, multiple-collision FEL-Compton backscattering in an ERL facility will be a key technique [42; 43]. We will advance the development of an insertion device which is suitable for backward Compton scattering using ERL-FELs.