## 3 Numerical Results and Discussion
This maximum value turns out to be consistent with the harmonic cutoff around 37.19 eV. As for trajectory \"b\" shown in Figure 3g-i and trajectory \"c\" shown in Figure 4a-c, \(r\) reaches the values less than 4 a.u. for three and one times, respectively. Hence, these electrons can also return to the parent ion, but the return kinetic energy of the electron can't reach the maximum value of 12.01 eV, as shown in Figures 3i and 4c. Whereas for trajectory \"d\" shown in Figure 4d-f, although \(r\) presents the oscillation behavior, it still can't reach a value less than 4 a.u. This is because besides the oscillation, \(r\) also presents a behavior that is gradually increasing at a certain velocity, as shown in the first column of Figures 3 and 4. In other words, the electron also present a behavior of drifting away from the parent ion at a certain velocity, besides the circular motion, as shown in Figure 3e,h and Figure 4b,e. This drifting velocity is determined by the vector potential at the initial time of the trajectory, as shown in Figure 4h. The smaller the vector potential at the initial time is, the slower \(r\) increases. Because the oscillation magnitude of \(r\) is similar for all the trajectories, the trajectory with a smaller drifting velocity will have more chances to reach the value less than 4 a.u. This well explains the different return times of trajectory \"a\", \"b\",  
Figure 4: **(a–f)** Two typical trajectories for the sine-squared laser field. **(g)** The electric field and (h) the vector potential of the sine-squared laser field at the initial time for the four typical trajectories. **(i)** The return kinetic energy of the electron as a function of the initial time of the trajectory. The pink dash-dotted curves in panels **(a,d)** represent the x component of the driving field. The red dotted lines in panels **(c,f)** indicate the maximum return kinetic energy of the electron, which is consistent with the harmonic cutoff. The black dashed lines in panels **(a,c,d,f)** denote the position \(r\) = 4.0 a.u. and \"c\". When the vector potential becomes too large, such as trajectory \"d\", then \(r\) increases too fast that it can not be less than 4 a.u. during the oscillation. Hence, the corresponding electron can't return once being away from the parent ion. In Figure 4i, we present all the returning trajectories for the sine-squared laser field. The y axis stands for the kinetic energy at the return time plus the ionization energy of the atoms. The x axis stands for the initial time of the trajectory. From this figure, one can see that the returning kinetic energy of the electron generally reaches its maximum value at the third or fourth returns. This maximum value is consistent with the harmonic cutoff predicted by the TDSE simulation. From the above discussion, one can see that the oscillating behavior of \(r\) is caused by the gradually increasing amplitude of the electric field, i.e. the positive temporal gradient of the electric field amplitude. This oscillation behavior makes the electron's return to the parent ion possible. Hence, the positive temporal gradient of the electric field amplitude is a fundamental key factor for the electron return and finally for the high-order harmonic generation with a circularly polarized laser field. According to the classical trajectory analysis, the larger the temporal gradient of the electric field amplitude is, the larger magnitude of the oscillation of \(r\). This means that the electron will be pulled closer to the parent ion. We define the minimum value of the vector potential that leads to no return of the electron as the threshold vector potential. Then, the threshold vector potential will be larger for a laser field with a larger temporal gradient. As a result, more electron trajectories can lead to the electron's return to the parent ion. Correspondingly, the harmonic yield will be enhanced. This provides a new way of controlling the high-order harmonic generation from atoms driven by a circularly polarized laser field.