## 3 Photon Polarization
### Theory  
When an unpolarized electron bunch collides with a linearly polarized laser pulse, the photon polarization is determined by the laser field. For the intense laser field in this study, as the radiation formation length is significantly smaller than the laser wavelength, the  
Figure 3: Energy angular distribution of the produced \(\gamma\) photons for different electron initial energy with a fixed laser intensity of \(a_{0}=192\), the (**a–d**) represents the results for electron initial energy of 1, 2, 5, and 10 GeV, respectively. The other parameters are the same as in Figure 1.  
laser field can be considered constant during the photon formation interval. Therefore, we use an LCFA [41] and Nikishov-Ritus method for the QED calculations, photon production probability, and polarization degree. The detailed derivation of the basic formulas is given in Refs. [32; 33], so we will briefly introduce the ideas and key points below. The probability of emitting a photon in the polarization state \(\epsilon_{j}\) with a momentum \(k\) via the NCS process can be expressed as [32; 33]  
\[P_{1}=-\frac{\alpha}{2}\int d^{3}p\rho_{e}(\mathbf{p})\frac{1}{\eta_{p}}\int_{0}^{ 1}ds\int d\phi\Bigg{[}\bigg{(}1+\frac{s^{2}}{t}+2\frac{\xi_{y}^{2}}{|\xi|^{2}} \bigg{)}\frac{1}{x}\text{Ai}^{\prime}(x)+\text{Ai}_{1}(x)\Bigg{]}, \tag{1}\]  
\[P_{2}=-\frac{\alpha}{2}\int d^{3}p\rho_{e}(\mathbf{p})\frac{1}{\eta_{p}}\int_{0}^{ 1}ds\int d\phi\bigg{[}\bigg{(}1+\frac{s^{2}}{t}+2\frac{\xi_{z}^{2}}{|\xi|^{2}} \bigg{)}\frac{1}{x}\text{Ai}^{\prime}(x)+\text{Ai}_{1}(x)\bigg{]}, \tag{2}\]  
corresponding to the photon scattered by an unpolarized electron onto the polarization state \(\epsilon_{1}\) and \(\epsilon_{2}\):  
\[\epsilon_{1}^{\mu}=\epsilon_{y}^{\mu}-\frac{\ell\cdot\epsilon_{y}^{\mu}}{k \cdot\ell}k^{\mu},\ \ \ \ \ \epsilon_{2}^{\mu}=\epsilon_{z}^{\mu}-\frac{\ell\cdot\epsilon_{z}^{\mu}}{k \cdot\ell}k^{\mu}, \tag{3}\]  
where \(\ell^{\mu}\) is the photon momentum, \(k^{\mu}=\omega_{0}(1,-1,0,0)\) is the laser wave vector, \(\epsilon_{y}^{\mu}=(0,0,1,0)\), and \(\epsilon_{z}^{\mu}=(0,0,0,1)\). The Airy functions are defined as follows:  
\[\text{Ai}_{1}(x) =\int_{x}^{\infty}dk\ \text{Ai}(\textbf{k}),\ \ \text{Ai}(x)\] \[=1/\pi\int_{x}^{\infty}dk\cos(kx+k^{3}/3), \tag{4}\]  
where \(\text{Ai}^{\prime}(x)\) is the derivative of the Airy function Ai(x). For small-angle collision \(\ell^{\mu}\approx\ell^{0}(1,1,0,0)\), \(\epsilon_{1}^{\mu}\) (\(\epsilon_{2}^{\mu}\)) indicates polarization in the \(y\)-direction (\(z\)-direction). The total yield is  
\[P =P_{1}+P_{2}\] \[=-\alpha\int d^{3}p\rho_{e}(\mathbf{p})\frac{1}{\eta_{p}}\int_{0}^{1} ds\int d\phi\bigg{[}\frac{1+t^{2}}{t}\frac{1}{x}\text{Ai}^{\prime}(x)+\text{Ai}_{1}(x) \bigg{]}, \tag{5}\]  
where \(\alpha\) is the fine-structure constant, \(\eta_{p}=\frac{k\cdot p}{m^{2}}\), \(s=\frac{k\cdot\ell}{k\cdot p}\) is the light-front momentum fraction of the scattered photon, \(t=1-s\), \(x=\left(\frac{s}{\chi_{p}t}\right)^{2/3}\), \(\chi_{p}=|\xi(\phi)|\eta_{p}\), and \(\xi(\phi)\) is the field intensity parameter (\(\phi\) is the laser pulse phase). \(\rho_{e}(\mathbf{p})\) is the momentum distribution of the electron bunch and satisfies the normalization condition \(\int d^{3}p\rho_{e}(\mathbf{p})=1\). The polarization degree can be determined by the probability change (\(\Delta P=P_{1}-P_{2}\)) and total probability (\(P=P_{1}+P_{2}\)) of photon scattering to a linearly polarized state as  
\[\Gamma_{1}=\frac{P_{1}-P_{2}}{P_{1}+P_{2}}. \tag{6}\]  
Notably, for the intense laser field in this study, as the radiation formation length is significantly smaller than the laser wavelength, the laser field can be considered constant during the photon formation interval. Therefore, we use an LCFA [41] for the QED calculations, photon production probability, and polarization degree.