## 3 Numerical Results and Discussion
Whereas, when the field is added a finite envelope, the typical features of the harmonic spectrum appear. For each harmonic spectrum shown in Figure 2b-d, there is a plateau above the ionization energy and followed by a cutoff. The cutoff positions are almost the same for the three different envelopes when the intensities of the driving pulses are the same, as indicated by the red dashed and blue solid arrows in Figure 2b-d. But the cutoff position is extended when increasing the pulse intensity. Moreover, the cutoff position satisfies \(E_{cutoff}\cong I_{p}+2U_{p}\) for all the cases. Hence, the high-order harmonics caused by the recollision mechanism are significantly suppressed, unless the driving field possesses an envelope. It is indicated that the time-varying amplitude of the driving field plays an important role in the electron return to the parent ion. Figure 1: High-order harmonic spectra generated by a laser field with a constant amplitude (**a**) and with a sine-squared envelope (**b**). Three different wavelengths are considered. The interaction time between He atom and the laser field is 10 optical cycles and the laser intensity is 1.0\(\times 10^{14}\) W/cm\({}^{2}\). To explain the above finding, we perform an analysis of the classical trajectory of the electron in circularly polarized laser fields. As examples, the laser fields with constant amplitude and a sine-squared envelope are considered. The intensity, wavelength, and pulse duration of the two driving laser field are respectively \(1.0\times 10^{14}\) W/cm\({}^{2}\), 800 nm, and 10 optical cycles. The corresponding harmonic spectra are presented by the red dotted curves in Figure 1. By solving the [PERSON]'s Equation of the freed electron in the two driving laser fields, the classical trajectories of the electron are obtained. For both two laser fields, the electron is set to start from the origin at zero velocity [18; 19]. The electron return is determined by the distance between the electron and the parent ion \(r\). When \(r\) is less than 4 a.u., then we treat this situation as an efficient return to the parent ion [33]. In Figures 3 and 4, we present a typical trajectory for the laser field with constant amplitude (Figure 3a-c) and four typical trajectories for the sine-squared laser field (Figure 3d-i) and Figure 4a-f). We denote the four typical trajectories of sine-squared laser field as trajectory \"a\" (Figure 3d-f)), trajectory \"b\" (Figure 3g-i), trajectory \"c\" (Figure 4a-c), and trajectory \"d\" (Figure 4d-f), respectively. The time-dependent \(r\), the two-dimensional displacement of the electron, and the evolution of the electron kinetic energy for each trajectory are presented in the first, second, and third columns of Figures 3 and 4, respectively. In Figure 3a,d,g, and Figure 4a,d, the x component of the electric field is presented by the pink dash-dotted curve. In addition, the electric field and the vector potential at the initial time for the four typical trajectories of sine-squared laser field are also depicted by the corresponding characters in Figure 4g,h, respectively. Figure 2: High-order harmonic spectra generated by a laser field with a constant amplitude (**a**), trapezoidal (**b**), gaussian (**c**), and sine-squared (**d**) envelopes. Two different intensities of the laser field (2.0\(\times 10^{14}\) W/cm\({}^{2}\) and \(1.0\times 10^{14}\) W/cm\({}^{2}\) ) are considered. Other parameters are the same as used in Figure 1.  
quite different from that of the electron trajectory driven by the laser field with constant amplitude. It is this oscillation that makes the electron return to the parent ion possible. As shown in Figure 3d-f for trajectory \"a\", the distance between the electron and the parent ion \(r\) appears less than 4 a.u. for five times. Hence, the corresponding electron has five chances to return to the parent ion. The return kinetic energy of the electron reaches the maximum value of 12.01 eV at the fourth time, as depicted in Figure 3f by the red dashed line.