## 3 Results  
### Radiation Angular Distribution  
In our numerical implementation, since the laser beam is tightly focused, in order to ensure that the intensity of the laser in the interaction region (corresponding to \(\alpha=5\)), that is \(I\sim 10^{19}\) w/cm\({}^{2}\), a linearly polarized laser pulse with a normalized peak intensity \(\alpha=5\) was employed, spanning a single optical cycle. Figure 2 displays representative electron trajectories for carrier-envelope phases (CEP) of \(\phi_{0}=0\), \(\phi_{0}=\pi/3\), \(\phi_{0}=\pi/2\), demonstrating clear alignment with the laser's linear polarization state through their oscillatory patterns. The two-dimensional plots of the motion trajectories show a symmetry about the polarization axis. This symmetry is an important feature of electron dynamics under specific conditions. As shown in the figure below, the electron moves from right to left along the negative direction of the z-axis. We can analyze the physical quantities associated with the radiation field by studying the motion trajectories of electrons under tightly focused linearly polarized laser pulses. Figure 2: The dynamics of counter-propagating electrons under a linearly polarized laser field (polarization plane: x-z) is demonstrated. The pulse parameters include normalized intensity \(a=5\), beam waist radius \(\omega_{0}=2\), and duration \(\mathrm{L}=1\). Electron trajectories are shown for CEP phases \(\phi_{0}=0\), \(\phi_{0}=\pi/3\), \(\phi_{0}=\pi/2\). Figure 3 displays the angular distribution of radiation produced by \(20\times 0.511\) MeV electrons in collision with a linearly polarized laser pulse, analyzed for multiple initial phase values (\(\phi_{0}=0,\ \pi/3,\pi/2,2\pi/3,\pi,4\pi/3,3\pi/2,5\pi/3\)). Our simulations reveal that in a head-on collision geometry, where relativistic electrons propagate along the \(-2\) direction while interacting with a linearly polarized laser beam advancing in the \(+2\) direction, the polar angle \(\theta_{m}\) splits at the peak radiation, and the radiant energy appears as a bimodal structure and exhibits a symmetric distribution about the laser polarization axis. The polar angle \(\theta\) of radiation is governed by the instantaneous angle between the electron's velocity vector \(\overset{\rightarrow}{v}\) and its acceleration vector \(\overset{\rightarrow}{\beta}\) as prescribed by Lienard-Wiechert potentials. For relativistic electrons (\(\gamma_{0}\gg 1\)), the radiative precession effect brings \(\theta_{m}\) the approach to \(\pi-c/\gamma_{0}\). With increasing initial phase \(\phi_{0}\), the azimuthal angle \(\psi\) remains locked to the laser polarization axis. For a linearly polarized laser with its electric field vector aligned along the direction of the x-axis, the radiation maintains two distinct peaks at \(\psi=0,\pi\), corresponding to the direction of the x-axis and the negative direction of the x-axis, respectively, independent of \(\phi_{0}\) variations. Furthermore, the analysis demonstrates that the azimuthal radiation pattern undergoes synchronous rotation with the laser polarization axis, maintaining a fixed angular relationship with the E-field vector orientation. It is worth noting that in Figure 3, we project the 3D bimodal distribution (\(\theta_{m}=\pm(\pi-c/\gamma_{0})\), \(\psi=0/\pi\)) to the \(\theta-\psi\) plane, and if we fix \(\psi=0\), then \(\theta=\theta^{\prime}\), there is no compression; if \(\psi\
eq 0\), then the polar angle appearing in the projection should be corrected to \(\theta^{\prime}=tan^{-1}(sin\theta cos\psi)<\theta\). This will lead to an underestimation of the polar angle. The effect arises from the regular oscillations of a linearly polarized field, which cause electrons to move alternately along the polarization axis, producing two dominant radiation directions.