## 3 Results
### Temporal Spectrum  
We also investigated the temporal spectrum of electron radiation. Specifically, when the initial electron energy was set to \(20\times 0.511\) MeV, we examined the temporal spectrum of the initial phase, varying from \(0\) to \(2\pi\), with increments of \(\pi/18\). Our investigation revealed that the initial phase of the tightly focused circularly polarized laser had a minimal impact on the temporal spectrum. During the initial phase changes, the peak value of the laser pulse remained at a value of \(1.0812\times 10^{11}\) arb.units, and the temporal spectra across different phases exhibited remarkable similarity. However, the peak value of the high-order harmonic radiation generated under linearly polarized laser pulses underwent notable periodic variations in accordance with the initial phase, demonstrating distinctive phase-sensitive behavior. Furthermore, these periodic changes exhibited symmetry, reflecting the inherent properties of linearly polarized light.  
Subsequently, we intensified the pulse and observed a distinct single-peak pattern in the wave crest of the circularly polarized laser pulse, as depicted in Figure 5a. As the initial phase shifted, the pulse width of the wave crest experienced slight variations. Figure 5b illustrates the correlation between the pulse width and the initial phase. A thorough analysis of the figure, coupled with detailed calculation data, revealed the periodic nature of the pulse width concerning the initial phase, exhibiting a period-like characteristic of \(\pi\). The pulse width fluctuated within a range of \(0.0409\) to \(0.0436\) as.  
The observed phenomenon can be attributed to the diverse motion trajectories of electrons across different phases, as depicted in Figure 2. When the electron trajectory aligns with the phase of the laser field, the scattering process is significantly augmented, leading to more focused radiation energy and a narrower pulse width within the radiation time spectrum. Conversely, if the electron trajectory misaligns with the polarization direction or phase of the laser field, the scattering process is suppressed, causing a reduction in radiation energy and the broadening of the pulse width within the radiation time spectrum. This periodic variation in the pulse width concerning the initial phase underscores the dynamic interplay between the phase matching of the electron trajectory and the laser field.  
We further obtained the laser pulse diagram at the initial energy of \(40\times 0.511\) MeV and \(60\times 0.511\) MeV, respectively, and compared it with that when the initial energy of the electron was \(20\times 0.511\) MeV. As Figure 6 illustrates, an increase in the initial electron energy results in a corresponding increase in the peak value of the laser pulse and a concurrent decrease in the pulse width. Specifically, for an initial electron energy of \(40\times 0.511\) MeV, the pulse width measures approximately \(0.0104\) as, whereas, for an initial electron energy of \(60\times 0.511\) MeV, the pulse width narrows to around \(0.0050\) as.  
Figure 5: The time spectrum and the relationship between the crest pulse width and the initial phase of \(gm_{0}=20\). (**a**) A time spectrum; (**b**) the relationship between the peak pulse width and initial phase.  
The observed phenomenon can be attributed to the findings presented in Figure 4. As the initial energy of the electron increases, the resulting radiation energy becomes more prominent, leading to a more focused radiation ability. Consequently, the pulse width of the radiation time spectrum narrows correspondingly.