## 2 Simulation of the High Energy \(\gamma\) Photon Production
### Simulation Results  
In this section, we first demonstrate the 2D-QED-PIC simulation results of the photon production via the NCS process, and the photon polarization is discussed in the next section. We begin with a simulation of a laser pulse with a normalized intensity of \(a_{0}=192\) interacting with a 2 GeV electron bunch. Figure 1b,c show the distribution of the electron and produced \(\gamma\) photon number density. During the interaction, the dynamics of the electrons are highly affected by the laser field. The laser field is equivalent to a natural undulator, and most of the electrons oscillate perpendicularly, as illustrated in Figure 1b. The electrons absorb laser energy, accelerated higher energy than its initial energy and emit high-energy photons via the NCS process. Because the electron dynamics determine the photon emission and photons are emitted on the electron trajectories, the photons have an identical density distribution with the seed electron bunch, as illustrated in Figure 1c. Numerous photons are produced at the central region because of the Gaussian distribution of the electron beam density and laser intensity, corresponding to high laser intensity and energetic electrons at the central region. Figure 2 illustrates the energy spectrum of the electron spectrum after interaction and produced photons for different electron energies with a fixed laser intensity of \(a_{0}=192\). As the initial energy of the electron increases, high-energy photons can be produced. The energy of the \(\gamma\) beam is determined by that of the seed electron \(E_{\text{e}}=\gamma_{\text{e}}m_{\text{e}}c^{2}\) and is, at  
Figure 1: (**a**) Schematic of the laser electron bunch collision scheme. Density distribution of electron (**b**) and produced \(\gamma\) photons (**c**) following laser electron collision at time \(t=96\) fs. Laser normalized intensity and electron bunch initial energy are \(a_{0}=192\) (laser intensity of \(I_{0}=5\times 10^{22}\) W/cm\({}^{2}\)) and 2 GeV, respectively. Other laser electron bunch parameters are given in the text. most, radiated with an energy of \(E_{\gamma}\approx 0.44\eta_{e}E_{\text{e}}\) for ultra-relativistic electrons \(\gamma_{\text{e}}\gg 1\)[45]. Therefore, the \(\gamma\) photons cutoff energies are approximately \(E_{\gamma}\approx 1.7,~{}2.8,~{}5,~{}\text{and}~{}~{}10~{}\text{GeV}\) for initial electron energies of \(e_{e}=1,2,5,\) and \(10~{}\text{GeV}\), respectively, as shown in Figure 2. Notably, the \(\gamma\) photon cutoff energies were comparable with the electron energies for electrons with initial energies of \(5\) and \(10~{}\text{GeV}\). However, the photon cutoff energies exceeded electron energies for electrons with initial energies of \(1\) and \(2~{}\text{GeV}\). This can be explained as follows: because the laser exists during the interaction, some electrons accelerated to higher energy by laser electric field after laser-electron interaction and effective electron cutoff energy higher than incoming electron energy, as shown in Figure 2a. For example, with the low initial energy of the electron bunch (indicating low electron momentum) in our simulation, the electrons could stay in the high field region for a long time (indicating a long interaction time), and accelerated to energies higher than the initial incoming energy as shown in Figure 2a. This accounts for the high energy of the generated photons. Therefore, emitted photon final energy layer than electron beam initial energy but not exceed the electron final cut of energy after acceleration, this result is consistent with previous results [46]. However, for electrons with high initial energies (\(5\) and \(10~{}\text{GeV}\)), the momenta were also high, and they rapidly penetrated the high field region and pass through the laser (referring to a short interaction time) with less chance for additional acceleration in the laser field. Thus, the photon cutoff energy was comparable with the initial energy of the electron.