## 2 Materials and Methods
\begin{array}{c}E_{z}=E_{0}\kappa\{\epsilon C_{1}+\epsilon^{3}[-\frac{C_{2}}{2}+r^ {2}C_{3}-\frac{r^{4}C_{4}}{4}]\\ +\epsilon^{5}[-\frac{3C_{4}}{8}-\frac{3r^{2}C_{4}}{8}+\frac{17r^{4}C_{5}}{16}- \frac{3r^{6}C_{5}}{16}-\frac{3r^{6}C_{5}}{8}-\frac{3r^{6}C_{7}}{32}]\}\\ -E_{0}\beta\{\epsilon S_{1}+\epsilon^{3}[-\frac{5_{2}}{2}+r^{2}S_{3}-\frac{r^{ 4}S_{4}}{4}]\] \[+\epsilon^{5}[-\frac{35_{2}}{8}-\frac{3r^{2}S_{4}}{8}+\frac{17r^{4 }S_{5}}{16}-\frac{3r^{6}S_{5}}{16}-\frac{3r^{6}S_{5}}{8}+\frac{r^{8}S_{7}}{32}]\} \end{array}  \tag{7}\]  
The components of a magnetic field are described as follows:  
\[ \begin{array}{c}B_{x}=E_{0}\{C_{0}+\epsilon^{2}[\frac{r^{2}C_{2}}{2}-\frac{r ^{4}C_{3}}{4}]+\epsilon^{4}[-\frac{C_{2}}{8}+\frac{r^{2}C_{3}}{4}+\frac{5r^{4} C_{4}}{16}-\frac{r^{6}C_{5}}{4}+\frac{r^{8}C_{6}}{32}]\}\end{array}  \tag{8}\]  
\[ \begin{array}{c}B_{y}=E_{0}\{S_{0}+\epsilon^{2}[\frac{r^{2}S_{2}}{4}-\frac{r ^{4}S_{3}}{4}]+\epsilon^{4}[-\frac{S_{2}}{8}+\frac{r^{2}S_{3}}{4}+\frac{5r^{4} S_{4}}{16}-\frac{r^{6}S_{5}}{4}+\frac{r^{8}S_{6}}{32}]\}\end{array}  \tag{9}\]  
\[ \begin{array}{c}B_{z}=E_{0}\beta\{\epsilon C_{1}+\epsilon^{3}[\frac{C_{2}}{ 2}+\frac{r_{2}C_{3}}{2}-\frac{r_{4}C_{4}}{4}]+\epsilon_{5}[\frac{3C_{3}}{8}+ \frac{3r^{2}C_{4}}{8}+\frac{3r^{4}C_{5}}{16}-\frac{r^{6}C_{4}}{4}+\frac{r^{8} C_{7}}{32}]\}\\ -E_{0}\alpha\Big{\{}\epsilon S_{1}+\epsilon^{3}[\frac{5_{2}}{2}+\frac{r_{2}S_ {3}}{2}-\frac{r_{4}S_{4}}{4}]+\epsilon_{5}[\frac{3C_{3}}{8}+\frac{3r^{2}S_{4} }{8}+\frac{3r^{4}S_{5}}{16}-\frac{r^{6}S_{6}}{4}+\frac{r^{8}S_{7}}{32}]\} \end{array}  \tag{10}\]  
where \(\alpha=x/w_{0}\), \(\beta=y/w_{0}\), \(r=\rho/w_{0}\). \(S_{m}\) and \(C_{m}\) are shown below:  
\[S_{m}=\Big{(}\frac{w_{0}}{w}\Big{)}^{m}\sin(\psi+m\psi_{G}) \tag{11}\]  
\[ \begin{array}{c}C_{m}=\Big{(}\frac{w_{0}}{w}\Big{)}^{m}\cos(\psi+m\psi_{G}) \end{array}  \tag{12}\]  
\[ \begin{array}{c}m=0,1,2,\ldots\end{array} \]  
It is worth noting that in the formula, the x and y components of E and B are expanded one order less than the z-component. This is because, in actual electromagnetic fields, the longitudinal components are an order of magnitude smaller than the transverse components. The phase is as follows: \(\psi=\psi_{0}+\eta-\psi_{R}+\psi_{G}+c_{0}\eta^{2}\), where \(c_{0}\eta^{2}\) is the phase shift caused by the chirp, and the initial phase \(\psi_{0}\) is the plane wave phase. \(\psi_{R}=\rho^{2}/2R(z)\) is the phase related to the curvature of the wavefront, and \(R(z)=z+z_{0}^{2}/z\) represents the z coordinate of the radius of curvature where the wavefront intersects the beam axis. \(\psi_{G}=tan^{-1}(z/z_{0})\) is an additional phase transition that a Gaussian beam undergoes, meaning that the Gaussian beam will undergo a total phase transition when changing from \(-\infty\) to \(+\infty\). Figure 1: Diagram of chirped pulse and electron interaction process. Considering the rest mass of a single electron \(m_{e}=9.1096\times 10^{-31}\) kg, charge \(-e=-1.6022\times 10^{-19}\)C, we use the following Lorentz equation and energy equation to determine and describe the motion state of the electron in the intense laser pulse  
\[\frac{d\mathbf{p}}{dt_{e}}=-e(\mathbf{E}+\mathbf{u}\times\mathbf{B}) \tag{13}\]  
\[\frac{d\Gamma}{dt_{e}}=-e(\mathbf{v}\cdot\mathbf{E}) \tag{14}\]  
where \(\Gamma=\gamma_{0}m_{e}c^{2}\) represents the electron energy, defined by the formula \(\gamma_{0}^{-1}=\sqrt{1-\mathbf{u}^{2}}\), where \(\gamma_{0}\) is the Lorentz factor and represents the normalized initial energy of the electron. Momentum of the electron \(\mathbf{p}=\Gamma\mathbf{v}/c^{2}\), where \(\mathbf{v}\) is the electron velocity and \(\mathbf{u}=\mathbf{v}/c\).