## 3 Results
### Temporal Distribution  
When the laser delay time is 5 fs, the temporal spectral characteristics of electron radiation are investigated in the observation direction with the maximum radiation power. As depicted in Figure 6, the position of the main peak is approximately at 51 fs, and a certain number of secondary peaks are symmetrically distributed on both sides. Generally, as the laser amplitude increases, the height of the main peak also increases rapidly, ranging from the \(10^{6}\) order of magnitude to the \(10^{9}\) order of magnitude, with a considerable span. This is because the increase in the laser amplitude enhances the effect of the linearly polarized Gaussian pulse on the high-energy electron, leading to a stronger radiation. As the laser amplitude increases, the position of the main peak will shift to the right correspondingly, and the coverage range of the main peak and the secondary peaks will expand. This suggests that the time for the electron to generate radiation in conjunction with the laser pulse is longer, and the time point for generating maximum radiation is retarded.  
Figure 5: The cross-sectional diagram of the angular distribution of the electron radiation power when \(a_{0}=5\).  
When the laser amplitude is relatively small, there are numerous secondary peaks on both sides of the main peak, and the height of the first secondary peak is highly proximate to that of the main peak. Nevertheless, as the laser amplitude escalates, the quantity of secondary peaks on both sides of the main peak diminishes, and ultimately, only two essentially symmetrical and approximately equal-height secondary peaks persist. Concurrently, the height of the first secondary peak steadily declines. This implies that augmenting the laser amplitude is conducive to energy concentration. It can be rationally conjectured that when the laser amplitude is intense to a certain degree, the main peak will be significantly higher than the secondary peaks and can be regarded as a solitary ultrashort attosecond pulse.  
It is noteworthy that although the multi-peak time spectrum generally assumes a symmetrical form, the heights of the symmetrical peaks exhibit differences and are not completely symmetrical. In the tightly focused context stipulated in this paper, the laser amplitude \(a_{0}\) serves as the coefficient in the field strength formula \(E=E_{0}\frac{b_{0}}{b}exp\left[-\frac{\eta^{2}}{L^{2}}-\frac{\rho^{2}}{P^{2}}\right]\). An increase in the laser amplitude \(a_{0}\) will lead to an augmentation in the gradient of the field strength formula, thereby causing an increase in the velocity difference of electrons at symmetrical positions and giving rise to a more pronounced asymmetry. Owing to the ponderomotive force of the laser along the z-axis on the electron, the z-axis velocity of a single electron will undergo certain alterations within the oscillation periods corresponding to the two symmetrical secondary peaks. The interaction between a single electron and the laser during these two periods cannot be considered entirely the same [12].  
It is rather intriguing that all the peaks, upon magnification, can be discerned to be constituted by two peaks with extremely minute time intervals. For the sake of convenient observation, taking the main peak as an example, we provided an enlarged diagram. After measurement, it is discovered that the distance between the closely adjacent double peaks of the main peak is constantly diminishing, and the intensity of the highest peak can reach the magnitude of \(10^{9}\).