## 2 Materials and Methods
\begin{split} B_{x}=E_{L}\bigg{\{}C_{0}+\varepsilon^{2}\Big{[} \frac{\rho^{2}C_{2}}{2}-\frac{\rho^{4}C_{5}}{4}\Big{]}+\varepsilon^{4}\Bigg{[} -\frac{C_{2}}{8}+\frac{\rho^{2}C_{3}}{4}+\frac{5\rho^{4}C_{4}}{16}-\frac{\rho^{ 6}C_{5}}{4}+\frac{\rho^{8}C_{6}}{32}\Bigg{]}\bigg{\}}\end{split}  \tag{7}\]  
\[ \begin{split} B_{y}=E_{L}\bigg{\{}S_{0}+\varepsilon^{2}\Big{[} \frac{\rho^{2}S_{2}}{2}-\frac{\rho^{4}S_{3}}{4}\Big{]}+\varepsilon^{4}\Bigg{[} -\frac{S_{2}}{8}+\frac{\rho^{2}S_{3}}{4}+\frac{5\rho^{4}S_{4}}{16}-\frac{\rho^ {6}S_{5}}{4}+\frac{\rho^{8}S_{6}}{32}\Bigg{]}\bigg{\}}\end{split}  \tag{8}\]  
\[ \begin{split} B_{z}=E_{L}\beta\bigg{\{}\varepsilon C_{1}& +\varepsilon^{3}\Big{[}\frac{C_{2}}{2}+\frac{\rho^{2}C_{3}}{2}-\frac{\rho^{4} C_{4}}{4}\Big{]}\\ &+\varepsilon^{5}\Big{[}\frac{3C_{3}}{8}+\frac{3\rho^{2}C_{4}}{8 }+\frac{3\rho^{4}C_{5}}{16}-\frac{\rho^{6}C_{6}}{4}+\frac{\rho^{8}C_{7}}{32} \Big{]}\bigg{\}}\\ &-E_{L}\xi\bigg{\{}\varepsilon S_{1}+\varepsilon^{3}\Big{[}\frac{ S_{2}}{2}+\frac{\rho^{2}S_{3}}{2}-\frac{\rho^{4}S_{4}}{4}\Big{]}\\ &+\varepsilon^{5}\Big{[}\frac{3S_{3}}{8}+\frac{3\rho^{2}S_{4}}{8 }+\frac{3\rho^{4}S_{5}}{16}-\frac{\rho^{6}S_{6}}{4}+\frac{\rho^{8}S_{7}}{32} \Big{]}\bigg{\}}\end{split}  \tag{9}\]  
where \(\xi=x/\omega_{0}\), \(\beta=y/\omega_{0}\), \(\rho=r/\omega_{0}\), \(\omega=\omega_{0}\Big{(}1+\left(z/z_{R}\right)^{2}\Big{)}^{1/2}\), \(r=\sqrt{x^{2}+y^{2}}\) represent the vertical distance; \(\omega_{0}\) represents the minimum beam waist radius; and the diffraction angle is \(\varepsilon=\omega_{0}/z_{r}\). \(E_{L}\) can be represented by the following formula:  
\[E_{L}=a_{0}\frac{\omega_{0}}{\omega}exp\bigg{[}-\frac{\eta^{2}}{L^{2}}\bigg{]} exp\bigg{[}-\frac{\rho^{2}}{\omega^{2}}\bigg{]} \tag{10}\]where \(\eta=z-t\), \(\omega=\omega_{0}\sqrt{1+z^{2}/{z_{r}}^{2}}\) represent the waist radius at the z-axis; \(z_{r}=k{\omega_{0}}^{2}\) (\(k=\frac{2\pi}{\lambda}\)) represents the Rayleigh length; and \(a=\sqrt{{I{\lambda_{0}}^{2}}/{1.37}\times 10^{9}}\) represents the peak amplitude. \(S_{n}\) and \(C_{n}\) can be expressed by the following formula:  
\[S_{n}=\Big{(}\frac{\omega_{0}}{\omega}\Big{)}^{n}sin(\psi+n\psi_{G}) \tag{11}\]  
\[C_{n}=\Big{(}\frac{\omega_{0}}{\omega}\Big{)}^{n}cos(\psi+n\psi_{G}) \tag{12}\]  
where \(\psi=\phi_{0}+\phi_{P}-\phi_{R}+\phi_{G}\); \(\phi_{0}\) is the constant initial phase; \(\phi_{P}=\eta=\omega t-kz\) is the plane wave phase; \(\phi_{G}=\tan^{-1}(z/z_{r})\) is the Gouy phase, which is related to the total phase change of the Gaussian beam as it changes from \(-\infty\) to \(+\infty\) in the z-axis; \(\phi_{R}=kr^{2}/2R\) is related to the curvature of the wavefront; and \(R(z)=z+{z_{r}}^{2}/z\) is the radius of curvature at the intersection of the z-axis and the beam axis in the wavefront. The dynamics of a single relativistic electron, subject to the influence of an extremely intense laser pulse, can be accurately captured through the momentum and energy transfer equations outlined below:  
\[\frac{d\overrightarrow{P}}{dt}=-e\bigg{(}\overrightarrow{E}+\frac{ \overrightarrow{v}}{c}\times\overrightarrow{B}\bigg{)} \tag{13}\]  
\[\frac{d\gamma}{dt}=-\frac{e}{{m_{0}}c^{2}}\bigg{(}\overrightarrow{v}\cdot \overrightarrow{E}\bigg{)} \tag{14}\]  
where \(\overrightarrow{P}=\gamma m\overrightarrow{v}\) is the electron momentum, \(\gamma=1/\left({1-v^{2}/c^{2}}\right)^{1/2}\) is the relativistic factor, \(\overrightarrow{E}\) is the laser electric field, \(\overrightarrow{B}\) is the laser pulse magnetic field, and \(\overrightarrow{v}\) is the electron velocity. This research uses a normalized vector \(a=\left({I{\lambda^{2}}/{1.37}\times 10^{18}}\right)^{1/2}\) to represent the relativistic intensity of the laser pulse, where \(I\) is the laser intensity in physical units of W/cm\({}^{2}\). The subsequent mentions of laser intensity refer to the normalized relativistic intensity, which is a dimensionless quantity. \(\lambda\) is the wavelength. We take the laser pulse parameters \(\omega_{0}=2\lambda\) and \(L=1T\).