## 2 Simulation of the High Energy \(\gamma\) Photon Production  
### Model and Simulation Parameters  
The \(\gamma\) photon production process of an electron colliding with a laser field is referred to as NCS (\(n\gamma_{laser}+e^{-}\rightarrow\gamma+e^{-}\)). We focused on the production of a linearly polarized \(\gamma\) photon by an unpolarized electron bunch interacting with a linearly polarized laser pulse in the NCS process. Our physical model is shown in Figure 1a. The photon production process is tested by conducting a set of 2D-QED-PIC simulations using the code EPOCH [43]. The simulation size is set as \(x\times y=(40\times 40)~{}\upmu\)m, which corresponds to the grid cell number of \(2000\times 800\). We considered a linearly polarized laser pulse with a Gaussian intensity profile \(a=a_{0}\exp[-(y/y_{0})^{2}-(t/\tau)^{2}]\) entering the simulation region from the right boundary at \(t=0\). For the test simulation, the laser intensity is \(I_{0}=5\times 10^{22}~{}\text{W}/\text{cm}^{2}\), and the corresponding normalized field amplitude is \(a_{0}=eE_{0}/m_{e}c\omega_{0}=192\), where \(e\) and \(m_{e}\) are the electron charge and rest mass, respectively; \(c\) is the speed of light in a vacuum; and \(E_{0}\) and \(\omega_{0}\) are the laser electric field strength and laser frequency, respectively. The laser pulse duration, wavelength, and pulse spot size are \(\tau=30\) fs, \(\lambda_{0}=1~{}\upmu\)m, and \(y_{0}=6~{}\upmu\)m, respectively, in all simulations. To verify the effects of laser intensity on photon energy and polarization degree, the laser intensity was varied in the simulation as \(a_{0}=50,100,150,192,300,400,500,1000\).  
The electron beam enters the simulation region from the left boundary at \(t=0\). The electron density has a transversely Gaussian and longitudinally uniform distribution with a transverse size (FWHM) \(r_{0}=3~{}\upmu\)m, and the duration is \(\tau_{1}=30\) fs (beam length of \(9~{}\upmu\)m), which is represented by \(1\times 10^{7}\) of the total macroscopic particles and total charge of \(1\) pC. The energy spread of each electron beam is fixed to \(0.06\) in all calculations. The longitudinal and transverse initial temperatures of the electrons are set to \(1\) and \(0.01\) MeV, respectively. We set \(100\) macro electrons in each cell of the electron bunch. Periodic and simple-outflow boundary conditions are used in the transverse and longitudinal direction, respectively. To investigate the influence of the electron initial energy on the final photon production rate, energy, and polarization degree, the electron bunch energy changes from \(1\) GeV to \(10\) GeV. Such a high energy compact electron bunch can be obtained via laser wakefield acceleration experiments [9,42].  
In the EPOCH code, the emission of \(\gamma\) photons via the NCS process is controlled using the key quantum nonlinear parameter \(\eta_{\text{e}}=\sqrt{-(F_{\mu\
u}p^{\
u})^{2}}/(E_{s}m_{e}c)\). Here, \(F_{\mu\
u}=\partial_{\mu}A_{\
u}-\partial_{\
u}A_{\mu}\) is the electromagnetic field tensor, \(p^{\
u}=\tau_{\text{e}}m_{e}c(1,\beta)\) is the four-momentum of electron, \(\gamma_{\text{e}}\) is the electron relativistic factor, and \(E_{\text{s}}=m_{\text{e}}^{2}c^{3}/e\hbar=1.3 emission rate can efficiently be enhanced via the laser-electron colliding NCS process. We take in to account the radiation reaction in all simulations and calculations. In the Epoch code, we not implemented the electron spin polarization and photon polarization because of the complexity of the process. However, in Section 3, we introduced a simple and quick method to calculate the photon polarization by corporation of the PIC simulation and QED calculation. We theoretically calculate the photon polarization probability and polarization degree based on the data of the electron beam with acquired in the simulation [35].