## 3 Results
### Frequency Spectrum  
Finally, we also delved into the spectral properties. In our configuration, electrons possessed 20 relativistic mass units (20 \(\times\) 0.511 MeV) of initial kinetic energy, and by analyzing it, we found that the time domain under linear polarization is correlated with the frequency domain, as shown in Figure 8a-c. When the initial phase \(\phi_{0}=0\), the time-domain bimodal peaks (spaced 0.67\(T_{0}\) apart) of the line-polarized radiation lead to the frequency-domain harmonic interference, and this results in sharp harmonic peaks, exclusively of odd-order harmonics, with highly concentrated energy; when the initial phase \(\phi_{0}=\pi/4\), the asymmetric bimodal peaks in the time domain lead to harmonic unfolding, with the harmonic peaks becoming wider, the splitting peaks appearing around them, the continuous spectrum component starts to appear; when the initial phase \(\phi_{0}=\pi/2\), the single-peak broad pulse in the time domain leads to the disappearance of the harmonics, and the continuous spectrum dominates. At the same time, the odd harmonic intensity fluctuates with the initial phase (carrier-envelope phase CEP), realizing a periodic modulation. Analysis reveals that the initial phase plays a crucial role in electron scattering dynamics within tightly focused, linearly polarized laser fields. In line-polarized light, the spectral properties are dynamically tunable through the initial phase, and this mechanism provides a key basis for phase-modulated attosecond pulses. In addition, as shown in Figure 8a, in the generation of high harmonics driven by linearly polarized laser light, we find that the radiation intensity exhibits a stepwise decay with the increase of the harmonic frequency, and the radiation intensity \(\propto n^{-3}\) (n is the harmonic order), which embodies the physical mechanism of nonlinear Thomson scattering, and this property provides a theoretical basis for the active modulation of high harmonics. It is worth noting that when we use elliptically polarized light, all harmonics appear in the radiation spectrum, and the harmonics are more densely spaced.  
Furthermore, electron energy spectra were analyzed for higher initial energies of \(40\times 0.511\) MeV (Figure 9b) and \(60\times 0.511\) MeV (Figure 9c). As can be seen from Figure 9a-c, the maximum radiation intensity corresponds to the values of 60,000, 240,000, and 530,000, respectively, and the threshold frequency (the frequency at which the first significant increase in radiation intensity occurs) corresponds to the values of 2000, 8000, and 18,000, respectively, in the horizontal coordinates. The electron energy increases quadratically from \(20\to 40\to 60\) (i.e., to a factor of 2 and 3), while the maximum radiation intensity increases from \(l_{a}\to l_{b}\to l_{c}\) satisfies: \(l_{b}/l_{a}\approx 4\), \(l_{c}/l_{a}\approx 9\), and the threshold frequency increases from \(\omega_{tha}\rightarrow\omega_{thb}\rightarrow\omega_{thc}\) satisfies: \(\omega_{thb}/\omega_{tha}\approx 4,\omega_{thc}/\omega_{tha}\approx 9\), both of which support quadratic increases. Thus, the threshold laser frequency corresponding to the maximum radiation intensity and peak radiation grows quadratically with increasing electron energy. Furthermore, we fit these three plots, and we find that the threshold frequency (vanishing radiation intensity) is linearly proportional to the electron energy.