## 3 Results
### Radiation Spectrum  
This section examines the properties of the radiation spectrum with respect to the laser amplitude. As depicted in Figure 7, as the laser amplitude increases continuously, the quantity of higher harmonics in the radiation spectrum keeps augmenting, and the cutoff  
Figure 6: The time distribution graphs of peak radiation power under different laser amplitudes and the magnified graphs of their main peaks.  
frequency rises from 1000 to 8000. This is due to the fact that an increase in laser amplitude intensifies the cross-collision interaction between the laser and electrons more vigorously, leading to a higher radiation intensity.  
Overall, the maximum intensity of the normalized radiation spectrum is not notably influenced by the laser amplitude. When the laser amplitude lies within the range of 1-3, the frequency of the radiation spectrum is predominantly distributed in the low-value region, and the frequency envelope is not distinct. The height of the main peak is conspicuously higher than that of the other secondary peaks, concentrating the majority of the energy. This is because at this juncture, the laser amplitude is relatively small; the emitted radiation intensity is low; and the output frequency is relatively low. When the laser amplitude is within the range of 4-6, the radiation spectrum gradually manifests an obvious multi-peak distribution. The heights of the main peak and the first secondary peak will exhibit a reciprocal relationship, equivalent to the maximum intensity of the radiation spectrum shifting to a higher frequency band. As the laser amplitude increases, the intensity of the interaction between the electron and the linearly polarized laser becomes higher, resulting in the output of radiation with a higher frequency. When the laser amplitude is within the range of 7-8, the height disparity between the main peak and the first peak of the radiation spectrum is narrowing, and both are significantly higher than the other secondary peaks. This implies that the energy is mainly concentrated on the main peak and the first peak in the medium- and low-frequency bands.  
As depicted in Figure 8, with the augmentation of the laser amplitude, the spectral bandwidth of the spectrum is conspicuously broadened. On the one hand, due to the relativistic nonlinear effect, the acceleration encompasses higher-order frequency components, leading to an increase in higher harmonics. On the other hand, as a result of the Doppler effect, a certain frequency shift is generated. In contrast to the time spectrum, in accordance with the principle of Fourier transform, the spectral width \(\Delta\omega\) of a short-time signal is proportional to \(1/\Delta t\). Hence, the shorter the pulse width, the broader the spectrum.  
Figure 7: Two-dimensional spectrum diagrams of the peak radiation power direction under various laser amplitudes.  
Through the aforementioned research, it can be discerned that when the laser amplitude is relatively strong, radiation with a rather concentrated frequency range can be output. Hence, by enhancing the intensity of the laser, namely increasing the laser amplitude, radiation of a specific frequency can be concentratedly emitted. This proves to be of definite assistance to the actual production research.  
For the sake of making the research more comprehensive, we also provide the three-dimensional spectrum related to \(\theta\). In Figure 9, it can be comprehended that as the laser amplitude keeps increasing, the three-dimensional radiation spectrum will constantly broaden in the \(\theta\) direction, and the cutoff frequency will also keep rising. When the laser amplitude is relatively high, it can be distinctly observed that the three-dimensional spectral diagram presents a pronounced \"valley-shaped\" image. Specifically, the distribution of the \(\theta\) polar angle is relatively broad at lower frequencies, and as the frequency increases, the radiation distribution in the \(\theta\) direction gradually becomes concentrated.