## 2 Materials and Methods
\[E_{x}=E\Bigg{\{}S_{0}+\epsilon^{2}\bigg{[}\xi^{2}S_{2}-\frac{\rho^{4}S_{3}}{4} \bigg{]}+\epsilon^{4}\bigg{[}\frac{S_{2}}{8}-\frac{\rho^{2}S_{3}}{4}-\frac{ \rho^{2}\big{(}\rho^{2}-16\xi^{2}\big{)}S_{4}}{16}-\frac{\rho^{4}\big{(}\rho^{2 }+2\xi^{2}\big{)}S_{5}}{8}+\frac{\rho^{8}S_{6}}{32}\bigg{]}\Bigg{\}} \tag{1}\]0  
In the equation, \(\xi_{0}\) indicates a constant phase, and it also shows that \(\xi=\xi_{0}+\xi_{p}-\xi_{R}+\xi_{G}\). \(\xi_{R}=k\pi/(2R)\) is expressed as the wavefront curvature phase, and \(\xi_{p}=\eta=\omega t-k\)z is expressed as the plane wave phase. When moving from \(-\infty\) to \(+\infty\), the Guoy phase associated with a Gaussian beam with a total phase shift in \(\pi\) is represented as \(\xi_{G}=tan^{-1}(z/z_{r})\), and the radius of curvature of the wavefront intersecting the beam axis is represented as \(R(z)=z+z^{2}/z\). The following definitions apply to several coefficients of the formula: \(k=\omega/c\), \(r^{2}=x^{2}+y^{2}\) and \(\rho=r/b_{0}\). These equations are created using the vector potential of amplitude \(A_{0}\) and frequency \(\omega\). Each field satisfies the Maxwell equation \(E\) plus the handwritten epsilon power of one-sixth of the order term. This work establishes a model that only designs one electron. Through the numerical solution of the equation, we analyzed the dynamics of the electron, mass \(m\), and charge \(\epsilon^{6}\). \[E_{x}=E\Bigg{\{}S_{0}+\epsilon^{2}\bigg{[}\xi^{2}S_{2}-\frac{\rho^{4}S_{3}}{4} \bigg{]}+\epsilon^{4}\bigg{[}\frac{S_{2}}{8}-\frac{\rho^{2}S_{3}}{4}-\frac{ \rho^{2}\big{(}\rho^{2}-16\xi^{2}\big{)}S_{4}}{16}-\frac{\rho^{4}\big{(}\rho^{2 }+2\xi^{2}\big{)}S_{5}}{8}+\frac{\rho^{8}S_{6}}{32}\bigg{]}\Bigg{\}} \tag{1}\]1  
The momentum \(p\) is equal to \(\gamma mc\cdot\mathbf{u}\), and the energy \(\gamma=\gamma_{0}\mathbf{m}\mathbf{c}^{2}\). The Lorentz factor \(\gamma_{0}=\left(\mathbf{1}-\mathbf{u}^{2}\right)^{-\frac{1}{2}}\mathbf{u}\) is the velocity, normalized by the speed of light \(c\) in a vacuum. The expression for the peak field strength \(\mathbf{I_{0}}\) is \(\mathbf{a_{0}}=\frac{eE_{0}}{m\omega}\), in which \(\mathbf{I_{0}}\mathbf{\lambda}^{2}\approx\mathbf{1.375}\times\mathbf{10}^{18}\mathbf{a_{0}}^{2} \left(\frac{\mathbf{\mathrm{W}}}{\mathbf{\mathrm{cm}}^{2}}\right)\left(\mu\mathbf{\mathrm{ m}}\right)^{2}\). This paper adopts \(\mathbf{\lambda_{0}}=\mathbf{1}(\mu\mathbf{\mathrm{m}})\). Therefore, \(\mathbf{I_{0}}\approx\mathbf{1.384}\times\mathbf{10}^{26}\left(\mathrm{W/cm^{2}}\right)\). It will not be repeated later. \(n\) represents the spatial coordinate system, and the radiation direction vector is expressed as \(\mathbf{n}=\left(\mathbf{sin}\mathbf{cos}\mathbf{,sin}\mathbf{sin}\mathbf{,cos}\mathbf{,}\mathbf{)}\right)\), which corresponds to the spherical coordinate representation. The azimuthal angle \(\varphi\) in the plane perpendicular to the origin describes the deflection orientation, while the polar angle \(\theta\) characterizes the angular deviation relative to the laser propagation direction. In this study, the single-electron radiation power and spectrum are of central importance. We therefore present the governing equations below:  
\[E_{x}=E\Bigg{\{}S_{0}+\epsilon^{2}\bigg{[}\xi^{2}S_{2}-\frac{\rho^{4}S_{3}}{4} \bigg{]}+\epsilon^{4}\bigg{[}\frac{S_{2}}{8}-\frac{\rho^{2}S_{3}}{4}-\frac{ \rho^{2}\big{(}\rho^{2}-16\xi^{2}\big{)}S_{4}}{16}-\frac{\rho^{4}\big{(}\rho^{2 }+2\xi^{2}\big{)}S_{5}}{8}+\frac{\rho^{8}S_{6}}{32}\bigg{]}\Bigg{\}} \tag{1}\]2  
\[E_{x}=E\Bigg{\{}S_{0}+\epsilon^{2}\bigg{[}\xi^{2}S_{2}-\frac{\rho^{4}S_{3}}{4} \bigg{]}+\epsilon^{4}\bigg{[}\frac{S_{2}}{8}-\frac{\rho^{2}S_{3}}{4}-\frac{ \rho^{2}\big{(}\rho^{2}-16\xi^{2}\big{)}S_{4}}{16}-\frac{\rho^{4}\big{(}\rho^{2 }+2\xi^{2}\big{)}S_{5}}{8}+\frac{\rho^{8}S_{6}}{32}\bigg{]}\Bigg{\}} \tag{1}\]3  
The radiation power at time t is normalized to \(e^{2}\omega^{2}/4\pi c\) per unit solid angle (\(\Omega\)).