## 3 Numerical Results and Discussion  
We take the He atom as an example to investigate the HH generation from atom in a circularly polarized laser field. Hence, the soft-core parameter \(a\) takes the value of 0.27 to simulate the ionization potential of the He atom (24.26 eV). For a circularly polarized laser field propagating along the z axis, the x and y components of the electric field are respectively given by:  
\[E_{x}(t)=E_{0}f(t)\sin(\omega t), \tag{9}\]  
\[E_{y}(t)=E_{0}f(t)\cos(\omega t) \tag{10}\]  
Here, \(E_{0}=\sqrt{\frac{2I_{0}}{c_{0}}}\) is the field amplitude corresponding to the intensity \(I_{0}\), \(f(t)\) is the envelope of the laser field. When \(f(t)=1\), the circularly polarized field has a constant amplitude. In Figure 1, the harmonic spectra for a circularly polarized field with a constant amplitude (panel a) and a sine-squared envelope (panel b) are presented. Three different wavelengths are considered, while the laser intensity is fixed at 1.0\(\times\)10\({}^{14}\) W/cm\({}^{2}\). The total interaction time is 10 optical cycles for all the cases. Interestingly, the harmonic spectra generated by the laser field with a constant amplitude and a sine-squared envelope possess quite different spectral structures. In the case of constant amplitude shown in Figure 1a, significant signals can only be observed below and near the ionization energy, as denoted by the black dash-dotted line. After that, the spectral intensity decreases very quickly. Moreover, the harmonic spectrum does not change too much when varying the wavelength of the driving laser. These features are quite different from those observed in the harmonic spectrum driven by a linearly polarized field, i.e. there is a plateau followed by a wavelength-dependent cutoff. The plateau and the dependence of the cutoff on the wavelength of the driving pulse are the typical features caused by the recollision mechanism. Hence, in the case of the laser field with constant amplitude, the harmonics caused by the recollision mechanism are significantly suppressed. They can't be observed in the spectrum. The significant suppression of the high-order harmonics implies that there is no sufficient returns of the electron. Whereas, in the case of laser field with a sine-squared envelope shown in Figure 1b, the HH spectra present the typical features caused by the recollision mechanism. In detail, there is an obvious plateau followed by a cutoff in each harmonic spectrum. The harmonic cutoff is extended when increasing the wavelength of the driving field. They are around 37.19 eV, 52.03 eV, and 72.76 eV for the wavelength of 800 nm, 1200 nm, and 1600 nm, respectively. If the ponderomotive energy of a free electron in a circularly polarized laser field is given by \(U_{p}=\frac{\left(E_{0}\right)^{2}}{4\omega^{2}}\), then the cutoff position satisfies \(E_{cutoff}\cong I_{p}+2U_{p}\), which is consistent with previous reports. The above harmonic spectral features indicate that sufficient electron recollision occurs to induce high-order harmonic generation in the case of sine-squared laser field. To explore the universality of the above phenomenon, various envelopes and laser intensities for the driving field are considered. The results are presented in Figure 2a-d for the cases of a constant amplitude (panel a), a trapezoidal envelope (panel b), a gaussian envelope (panel c), and a sine-squared envelope (panel d), respectively. Here, we consider two different pulse intensities of \(2.0\times 10^{14}\) W/cm\({}^{2}\) and \(1.0\times 10^{14}\) W/cm\({}^{2}\). The wavelength of the driving field is fixed at 800 nm. The total interaction time is 10 optical cycles for all the cases, where the rising and falling time of the trapezoidal pulse is two optical cycles. As shown in these figures, a similar phenomenon to that in Figure 1 is observed. Namely, in the case of constant amplitude shown in Figure 2a, there are no significant signals above the ionization energy and the spectrum is not influenced significantly by varying the laser intensity.