## 3 Results
### Radiation Spectrum  
This section explores the characteristics of the radiation spectrum concerning the beam waist radius \(b_{0}\) in the direction of maximum radiation. Under the circumstance of tight focusing, as \(b_{0}\) increases, the normalized maximum spectral intensity grows from 7000 to 40,000. When \(b_{0}\) increases to 5 and thereafter until the Gaussian pulse is in a non-tight focusing state, the normalized maximum spectral intensity remains essentially stable. This is because with the increase of the beam waist radius \(b_{0}\), the electron is exposed to a more complete period of the Gaussian laser pulse, and the longer the interaction time, the greater the spectral intensity. After \(b_{0}\) reaches 5, the effect of prolonging the interaction time between the electron and the pulsed laser diminishes, the interaction time remains largely unchanged, and thus the maximum spectral intensity does not increase significantly. From Figure 8, it can be discerned that as the waist radius increments continuously from 1 to 100, the beam transitions from tight focusing to non-tight focusing, and a marked reduction in the number of high-order harmonics within the spectrum can be observed. When \(b_{0}\) is within 7, the cut-off frequency is approximately \(1000\omega_{0}\). Subsequently, the cut-off frequency undergoes a marginal increase with the escalating waist radius. Irrespective of the variations in the waist radius, the width of the spectrum persists at \(8000\omega_{0}\). When \(b_{0}\) lies between 2 and 4, the spectrum manifests typical supercontinuum characteristics, corresponding to an ultrashort pulse in temporal terms. This is attributed to the fact that at this juncture, \(b_{0}\) is exceedingly small, and the interaction duration between the linearly polarized Gaussian pulse laser and the electron is extremely short, rendering the non-periodic nature of electron motion more pronounced, thereby giving rise to the traits of a supercontinuum. Figure 7: The linear expression and the logarithmic expression of the temporal when \(b_{0}=10\). A distinct transition mode emerges when \(b_{0}\) ranges from 5 to 6. During the process of the increase in the beam waist radius, the variation of the intensity distribution of the laser field in the direction perpendicular to the propagation becomes more gradual. The time evolution of the electron acceleration shifts from non-periodic oscillation to periodic oscillation, and the modulation effect of the relativistic effect on the electron radiation is intensified. Ultimately, this results in the transformation of the supercontinuum spectrum into a multi-peak state spectrum. When \(b_{0}\) falls within the range of 7 to 10 or even up to 100, a particular phenomenon emerges. A reciprocal relationship is witnessed between the intensities of the main peak and the secondary peak. The interchange of the primary and secondary peaks in the spectrum can be accounted for based on the time spectrum. As depicted in Figure 6, the height of the secondary peak evidently increases with the augmentation of \(b_{0}\), and the relative height with respect to the primary peak conspicuously decreases. This leads to the accumulation rate of the secondary peak value in the spectrum exceeding that of the primary peak value. Ultimately, the interchange of the primary and secondary peaks in the spectrum is realized when \(b_{0}=9\). The corresponding modification is indicated with a green shade in line 369 of the article. This implies that under the condition of tight focusing, appropriately increasing the waist radius \(b_{0}\) enables the attainment of rays with higher frequencies and greater energies. Correspondingly, appropriately reducing \(b_{0}\) facilitates the concentration of ray energy in the lower frequency segment. Currently, to enhance the radiation intensity of the rays, reliance is mainly placed on increasing \(a_{0}\) and \(\gamma\), which results in a certain degree of energy wastage.