## 3 Results
### Frequency Spectrum Analysis  
In addition to the change in the time dimension, the power distribution of electrons also changes significantly in the frequency domain. As shown in Figure 5, it can be seen that after adding chirped parameters, a comparison is made under the same pulse width condition, the power peak and limit frequency of the three groups of spectrums are takenout, and the envelope lines are drawn for comparison. It can be seen that the envelope lines of the radiation peak and limit frequency under the three groups of pulse width conditions show a parabolic distribution.  
With the increase in the limit frequency, the frequency range of the power distribution expands, the number of sub-peaks generated at different frequencies increases, the gap between them and adjacent peaks decreases, and the power peaks in the high-frequency range become larger and larger. Under different pulse widths, the effect of the same chirp is not exactly the same, which is generally manifested as the larger the pulse width. Under the same chirped parameters, the frequency distribution tends to be uniform with an increasing frequency range. This is because under the action of a normal pulse, the frequency of the pulse is constant, so the response of the electron is also a single frequency, which will produce a sharp peak on the frequency spectrum because the chirped pulse has different frequencies at different stages of action, so the radiation frequency generated by the electron at different times and after the pulse action is different. Thus, the chirped electronic spectrum presents frequency continuity.  
At the same time, because the frequency of the chirped pulse is continuously changing, the action on the electron is a continuous process, and the spectrum change generated by the response is more continuous than the mutation in the non-chirped state, so the spectrum is manifested as energy at different frequencies. At the same time, due to the addition of the chirp, the frequency of the pulse action is denser and denser, and the frequency increases in the spectrum; the larger the chirp parameter, the wider the frequency range of the pulse is, so the peak point on the spectrum becomes more, and the power peak is also generated at a larger frequency. At the same time, we observe that when the chirped parameter is added, although the total amount increases, it is still mainly distributed within the low-frequency region. With an improvement in the chirped parameter, the power distribution tends to be evenly distributed. In the range of 0-6000 Hz, when the absolute value of the chirped parameter increases, the gap between the peaks of each frequency gradually decreases.  
However, the power peak of high-frequency peaks is lower than that of the low-frequency sub-peaks. Although the chirped pulse converts the equal frequency pulse into a pulse of different frequencies, the pulse action time gradually decreases as the frequency increases, and the pulse frequency changes before the electron fully interacts with the frequency pulse. Therefore, the low radiation value of high frequency is not because the energy of the high-frequency pulse itself is small but due to the result of the failure of the electron to fully interact with the pulse during the action.