## 2 Materials and Methods
Based on these parameters, E is given by the following equation:  
\[E=E_{0}\frac{\omega}{\omega_{0}}exp[-\frac{r^{2}}{\omega^{2}}]exp[-(z-ct)^{2 }/L^{2}] \tag{10}\]  
\[\mathrm{B}=\sqrt{\epsilon}\bigg{[}\bigg{(}\frac{i}{k}\bigg{)}\
abla(\
abla \cdot A)+ikA\bigg{]} \tag{2}\]0 Interaction of tightly focused laser pulses with electrons in opposite directions. The hyperbolic wavefronts (green) represent the Gaussian beam moving rightward (+z), while electrons travel leftward (–z). Polarization is restricted to the x-z plane in this model. where, \(E_{0}\) is the peak amplitude, \(L\) is the pulse duration, \(\omega_{0}\) is the minimum spot size, \(S_{n}\), \(C_{n}\) are shown below. \[\mathrm{B}=\sqrt{\epsilon}\bigg{[}\bigg{(}\frac{i}{k}\bigg{)}\
abla(\
abla \cdot A)+ikA\bigg{]} \tag{2}\]1  
\[\mathrm{B}=\sqrt{\epsilon}\bigg{[}\bigg{(}\frac{i}{k}\bigg{)}\
abla(\
abla \cdot A)+ikA\bigg{]} \tag{2}\]2  
where, \(\psi=\psi_{0}+\psi_{P}-\psi_{R}+\psi_{G}\), the initial phase \(\psi_{0}\) is constant. \(\psi_{P}=\eta=\omega t-\mathrm{kz}\) and \(\psi_{G}=tan^{-1}(z/z_{R})\) represent the plane-wave phase and the Gouy phase, respectively. The latter is related to the characteristic that the total phase undergoes a change of \(\pi\) during the change of z from \(-\infty\) to \(+\infty\). \(\psi_{R}=kr^{2}/(2R)\) is the phase related to the curvature of the wavefront. \(\mathrm{R}_{(z)}=\mathrm{z}+\mathrm{z}_{R}^{2}/\mathrm{z}\) denotes the radius of curvature of the wavefront at z intersecting the beam axis. These definitions of phase are important for understanding and analyzing optical phenomena such as the propagation properties of Gaussian beams. The coupled energy-momentum evolution equations governing the interaction between an ultra-intense laser pulse and a relativistic electron are given by the following:  
\[\mathrm{B}=\sqrt{\epsilon}\bigg{[}\bigg{(}\frac{i}{k}\bigg{)}\
abla(\
abla \cdot A)+ikA\bigg{]} \tag{2}\]3  
\[\mathrm{B}=\sqrt{\epsilon}\bigg{[}\bigg{(}\frac{i}{k}\bigg{)}\
abla(\
abla \cdot A)+ikA\bigg{]} \tag{2}\]4  
Here, \(\overrightarrow{P}=\gamma m\overrightarrow{v}\) is the electron momentum \(\gamma=1/(1-\mathrm{v}^{2}/c^{2})^{2}\) is the relativistic factor, \(\overrightarrow{E}\) is the laser electric field, and \(\overrightarrow{B}\) is the laser magnetic field, and \(\overrightarrow{v}\) is the electron velocity. This work employs the normalized vector potential \(\alpha(=\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{ \mathrm{   \mathrm{     }}}}}}}}}}}}}}} 1\times 10^{18})^{1/2}\) to characterize the relativistic laser intensity, with the laser intensity I (W/cm\({}^{2}\)) and the laser wavelength \(\lambda\) (\(\mu m\)). The spatial and temporal scales are governed by the normalized wavelength \(\lambda\) and laser period T. The laser beam is tightly focused. The volume and intensity gradient of the interaction are controlled by \(\omega_{0}\), and at the same time, a significant diffraction phase is triggered when \(\omega_{0}<1\)\(\mu\)m. Therefore, we take the laser beam waist spot \(\omega_{0}=2\lambda=2\)\(\mu\)m (corresponding to the laser intensity). When the laser pulse length \(L<5T\) (pulse width), the electrons do not complete the acceleration cycle before the end of the pulse, the phase sensitivity is strong, and the energy spectrum will have a discrete structure; therefore, the laser pulse length is taken as \(L=1\)\(T\). In addition, when the numerical aperture \(NA>0.8\), it will lead to phase distortion. Therefore, the numerical aperture (medium focusing) is taken as \(NA=\lambda/(\pi\omega_{0})\approx 0.16<0.5\) (medium focus) when the phase is almost free of aberrations. Electrons with initial kinetic energy 11.4 MeV (\(v=0.999c\)) are injected from \(Z=20\lambda\) along the negative z-direction. In the absence of laser interaction, these electrons would naturally focus at the origin (0, 0, 0), corresponding to the laser beam center.