## 3 Photon Polarization
### Calculation Results  
Figure 4 presents the energy distribution of the scattered \(\gamma\) photons by the electron bunch with different initial energies while the laser pulse intensity is fixed to \(a_{0}=192\). With an increase in the initial energy of the electron bunch, the electron Lorenz factor \(\gamma_{e}=E_{e}/mc^{2}\) and QED parameter \(\eta_{e}\approx 2\gamma_{e}|E_{\perp}|/E_{s}\) increase. The production rate of the photon is proportional to the parameter \(\eta_{e}\), and thus the energy of the photon is given as \(E_{\gamma}=\eta_{e}E_{e}/2\). Therefore, the production rate and energy of photon increases with the initial electron energy, as shown in Figure 4. However, Figure 4 shows that most of the photons are linearly polarized in the \(y\)-direction (\(P_{1}>P_{2}\)) for different electron initial energies because the photon polarization mainly transfers from background field polarization for an unpolarized initial electron. In this study, the linearly polarized laser pulse is in the \(y\) direction and, therefore, the linearly polarized photons are parallel to the laser field. To validate the feasibility of our scheme, we compared the energy spectrum of the \(\gamma\) photons from the PIC simulations and spectrum from QED calculations shown in Figure 5 for different electron initial energies and fixed laser intensity. The results show that the energy spectra of the PIC simulation and QED calculation are consistent for low electron energies (for initial electron energies 1 and 2 GeV), whereas for high electron energies, a slight difference is observed at the tail of the photon spectrum, as illustrated in Figure 5. This indicates that our scheme and the theoretical calculation are consistent. To discuss the photon polarization degree, first, we consider different initial electron energies and fixed laser intensity, as shown in Figure 6. Here, for electron bunches with low initial energies (1 and 2 GeV), the polarization degree of the photon increases as the photon energy increases and reaches its maximum at approximately 63% and 60%, respectively, at a photon energy of approximately 700 MeV. However, for high electron bunch initial energies (5 and 10 GeV), the peak photon polarization degree is only approximately 55% at the same energy level (700 MeV) with the low initial energy cases. For photon energy exceeding 700 MeV, the photon polarization degree significantly decreases with a further increase in photon energy. This indicates that highly polarized photons are produced only at low photon energy regions, and high-electron initial energies only maintain the polarization unchanged in a broader photon energy range but do not increase the polarization. Notably, in our unpolarized initial electron bunch case, the photon polarization is transferred from the linearly polarized background field. Here, the polarization degree of the background field decreases with an increase in the energy absorbed by the scattered photon from the incident electron. The higher the photon energy, the lower the field polarization degree and polarization transfer efficiency to the photon. This results in a significant decrease in the photon polarization degree at high energy levels. For the initial electron energies of 5 and 10 GeV, the laser-electron interaction times may be shorter than those of the low electron initial energy cases. This results in the low photon polarization degree for the high initial electron energies, as illustrated in Figure 6.  
Figure 4: Energy distribution of the different polarized photons from QED calculations with different initial energies, (**a**) 1, (**b**) 2, (**c**) 5, and (**d**) 10 GeV. Solid dashed and dotted lines represent total y- and z-direction polarization probabilities of the photons, respectively. Other parameters are the same as presented in Figure 1.  
Finally, we discuss the effect of the laser field intensity on the photon spectrum and photon polarization degree.