## 3 Results
### Azimuth Distribution of NITS Temporal Intensity  
In this section, the joint distribution of NITS radiation in time and azimuth angle is explored, and the results are shown in Figure 9. The joint distribution presents a significant symmetry, and the changes of radiated pulses with time and azimuth angle \(\phi\) is continuous, i.e., the azimuth angle corresponding to the pulse main peak advances continuously as time progressing. With the increase in laser intensity, the speed of change in the pulse peak power with \(\phi\) becomes faster whether in pulse rise time or fall time, nevertheless, the change with time slows down, causing the increase in pulse interval between the main peak and secondary peaks, and also conspicuous contrast. It is worth mentioning that the subpeak power of the radiated pulse increases with laser field intensity \(a_{0}\), but the increase is far less than that of the main peak, so the subpeak of the radiated pulse is no longer obvious in comparison. Comparing each row in Figure 9, although the number of radiation pulses remains unchanged, the duration of the entire radiation process (note the time scale in Figure 9) is greatly compressed. In fact, this mechanism of compression is also what produces ultra-short pulses. Therefore, we took the same parameters as Figure 5i and analyzed the mechanism of producing ultrashort pulses through NITS in combination with the transverse acceleration of electron in Figure 10.  
In the process of nonlinear inverse Thomson scattering, the cause of ultrashort pulses is the nonlinear effect brought about by the ultra-intense laser field and high-energy electron. We know from Section 3.2 that a larger transverse ponderomotive force will bring a more intense pulse, which is why the main peak of pulses in Figure 10a corresponds to the maximum value of the combined transverse acceleration in Figure 10b. Nevertheless, the time scales of the electron motion process [Figure 10b,d] and the pulse radiation process [Figure 10a,c] are not the same because of the time-compression effect. Figure 9: The joint distribution of NITS radiation temporal power and azimuth angle \(\phi\), in which the electron’s normalized initial energy \(\gamma_{0}\) and the normalized laser field intensity \(a_{0}\) have been marked, and \(\theta\) is fixed as the direction of its radiation power peak. It is worth noting that the scales of different figures are different. In fact, when the radiation produced by the electron reaches the observer, the pulse train is not discrete but compressed together for two reasons. One is the relativistic/nonlinear effect caused by the high-energy electron. When the Lorentz factor \(\gamma_{0}=100\), the initial velocity of the electron \(v_{0}=0.9995c\); thus, the radiation pulses will be greatly compressed arising from the transformation from the rest frame of the electron to the observer's frame. The second reason is the time-compression effect of the electron radiation. The pulse generated first has a longer propagation time because it is farther from the observer, while the following pulses have shorter propagation time. The electron moves at a speed close to the light, so the pulses arrive at the observer almost at the same time. Thus, both the time duration of the whole generation of the radiation pulses and the pulse width of the individual pulses are further shortened. The degree of the time-compression effect is also different under different observation directions. As shown in Figure 7b, the radiation pulses are compressed to different degrees in different polar angles. This is because the electrons have a very high longitudinal velocity, which points in the direction of \(\theta=180^{\circ}\). When the polar angle of the observer deviates from \(\theta=180^{\circ}\), the velocity component of the electron in this direction becomes smaller, and the influence of the above two reasons is both weakened, so the degree of the pulses' compression is reduced.