## 2 Materials and Methods
Combining the wave equation and Gaussian laser field distribution, the fifth-order expansion of the electric field component \(\mathbf{E}=\left\{E_{x},E_{y},E_{z}\right\}\) of the Gaussian laser field can be derived as follows:  
\[E_{x} =E_{0}\Big{\{}S_{0}+\varepsilon^{2}\Big{[}\alpha^{2}S_{2}-\frac{ \tau^{4}S_{3}}{4}\Big{]} \tag{6}\] \[\quad+\varepsilon^{4}\Big{[}\frac{S_{5}}{8}-\frac{\tau^{2}S_{3}} {4}-\frac{\tau^{2}\big{(}\tau^{2}-16\alpha^{2}\big{)}S_{4}}{16}-\frac{\tau^{4 }\big{(}\tau^{2}+2\alpha^{2}\big{)}S_{5}}{8}+\frac{\tau^{8}S_{6}}{32}\Big{]}\] \[\quad+\varepsilon^{2}C_{2}+\varepsilon^{4}\Big{[}r^{2}C_{4}- \frac{\tau^{4}C_{5}}{4}\Big{]}\Big{\}}\] \[\mathbf{E}=\sqrt{\epsilon}[(i/k)\
abla(\
abla\cdot\mathbf{A})+ik\mathbf{A}] \tag{1}\]0 (7) \[\mathbf{E}=\sqrt{\epsilon}[(i/k)\
abla(\
abla\cdot\mathbf{A})+ik\mathbf{A}] \tag{1}\]1 \[\mathbf{E}=\sqrt{\epsilon}[(i/k)\
abla(\
abla\cdot\mathbf{A})+ik\mathbf{A}] \tag{1}\]2 \[\mathbf{E}=\sqrt{\epsilon}[(i/k)\
abla(\
abla\cdot\mathbf{A})+ik\mathbf{A}] \tag{1}\]3 (8) \[\mathbf{E}=\sqrt{\epsilon}[(i/k)\
abla(\
abla\cdot\mathbf{A})+ik\mathbf{A}] \tag{1}\]4 \[\mathbf{E}=\sqrt{\epsilon}[(i/k)\
abla(\
abla\cdot\mathbf{A})+ik\mathbf{A}] \tag{1}\]5 \[\mathbf{E}=\sqrt{\epsilon}[(i/k)\
abla(\
abla\cdot\mathbf{A})+ik\mathbf{A}] \tag{1}\]6  
And the magnetic field components are described as follows:  
\[\mathbf{E}=\sqrt{\epsilon}[(i/k)\
abla(\
abla\cdot\mathbf{A})+ik\mathbf{A}] \tag{1}\]7  
\[\mathbf{E}=\sqrt{\epsilon}[(i/k)\
abla(\
abla\cdot\mathbf{A})+ik\mathbf{A}] \tag{1}\]8\[ \begin{split} B_{z}&=E_{0}\beta\Big{\{}eC_{1}+e^{3} \Big{[}\frac{C_{2}}{2}+\frac{r_{0}C_{3}}{2}-\frac{r_{0}C_{4}}{4}\Big{]}+e^{5} \Big{[}\frac{3C_{3}}{8}+\frac{3^{2}C_{4}}{8}+\frac{3^{2}C_{4}}{16}-\frac{r^{6} C_{4}}{4}+\frac{r^{8}C_{2}}{32}\Big{]}\Big{\}}\\ &\quad-E_{0}\alpha\Big{\{}eS_{1}+e^{3}\Big{[}\frac{S_{2}}{2}+ \frac{r_{0}S_{3}}{2}-\frac{r_{0}S_{4}}{4}\Big{]}+e^{5}\Big{[}\frac{3S_{3}}{8}+ \frac{3^{2}S_{4}}{8}+\frac{3^{4}S_{5}}{16}-\frac{r^{6}S_{4}}{4}+\frac{r^{8}S_{ 5}}{32}\Big{]}\Big{\}}\Big{\}}\end{split}  \tag{11}\]  
where \(\alpha=x/w_{0}\), \(\beta=y/w_{0}\), and \(r=\rho/w_{0}\). In the tightly focused laser (\(w_{0}<4\) um), terms expended to the order up to \(e^{5}\) are necessary and sufficient to ensure the accuracy of the description [20]. Among them, \(S_{m}\) and \(C_{m}\) are shown as follows:  
\[S_{m}=\Big{(}\frac{w_{0}}{w}\Big{)}^{m}sin(\psi+m\psi_{G}) \tag{12}\]  
\[C_{m}=\Big{(}\frac{w_{0}}{w}\Big{)}^{m}cos(\psi+m\psi_{G}) \tag{13}\]  
where \(m=0,1,2,\ldots\) The phase \(\psi=\psi_{0}+\eta-\psi_{R}+\psi_{C}\), in which the initial phase \(\psi_{0}\) is the phase of the plane wave. \(\psi_{R}=\rho^{2}/2R(z)\) is the phase related to the curvature of wave fronts where \(R(z)=z+z_{0}^{2}/z\) indicates the radius of curvature of a wave front intersecting the beam axis at the coordinate \(z\). \(\psi_{G}=\tan^{-1}(z/z_{0})\) is the Gouy phase, which means a total phase transition of \(\pi\) will be experienced by the Gaussian beam as \(z\) changes from \(-\infty\) to \(+\infty\). Considering a single electron with a rest mass \(m_{e}=9.1096\times 10^{-31}\) kg and charge \(-e=-1.6022\times 10^{-19}C\), the motion of the electron in an intense laser pulse can be determined by Lorentz and the energy equations given below:  
\[\frac{d\mathbf{p}}{dt_{e}}=-e(\mathbf{E}+\mathbf{u}\times\mathbf{B}) \tag{14}\]  
\[\frac{d\Gamma}{dt_{e}}-ec(\mathbf{u}\cdot\mathbf{E}) \tag{15}\]  
Here, \(\Gamma=\gamma_{0}m_{e}c^{2}\) represents the electron energy defined by equation \(\gamma_{0}^{-1}=\sqrt{1-\mathbf{u}^{2}}\) where \(\gamma_{0}\) is the Lorentz factor, which represents the normalized initial energy of the electron. The momentum of the electron \(\mathbf{p}=\Gamma\mathbf{v}/c^{2}\) where \(\mathbf{v}\) is the electron velocity and \(\mathbf{u}=\mathbf{v}/c\). In the rest frame of the observer, the time the electron moves through is represented by \(t_{e}\).