## 3 Results and Discussions
The polarization of the probe pulse can influence the probing photoelectron imaging of the coherent electron dynamics. We next present the photoelectron imaging process by a circularly polarized probe laser pulse. Figure 4 shows results of photoelectron momentum \(\mathcal{J}(p_{x},p_{y})\) distributions by a left-handed \(\xi=-1\) circularly polarized \(\lambda_{pr}=5\) nm probe pulse. The corresponding photoelectron angular \(\mathcal{J}(\theta)\) distributions obtained from Equation (6) are also plotted (yellow curves). We use again a linearly polarized \(\lambda_{pu}=70\) nm pump pulse with its field vector polarized along the \(x\)-axis to excited the molecule H\({}_{2}^{+}\) aligned along the \(z\)-axis, the same as in Figures 2 and 3. We show the helicity effects of the probe pulse on the photoelectron angular and momentum distributions of the coherent excitation. \[\psi_{\pi_{u}}(\mathbf{r})=[\psi_{\pi_{u}}^{+}(\mathbf{r})+\psi_{\pi_{u}}^{-} (\mathbf{r})]/\sqrt{2}. \tag{8}\]0 Evolutions of photoelectron momentum \(\mathcal{J}(p_{x},p_{y})\) and angular \(\mathcal{J}(\theta)\) distributions as a function of the time delay \(\Delta\tau\) between the linearly polarized \(\lambda_{pu}=70\) nm pump and \(\lambda_{pr}=5\) nm probe pulses with their field vectors polarized along the \(x\)-axis. The time delays are: (**a**) \(\Delta\tau=4.75\tau_{pu}\); (**b**) \(5.0\tau_{pu}\); (**c**) \(5.25\tau_{pu}\); (**d**) \(5.5\tau_{pu}\); (**e**) \(5.75\tau_{pu}\); and (**f**) \(6.0\tau_{pu}\). The molecule H\({}_{2}^{+}\) is aligned along the \(z\)-axis. The pulse intensities \(l_{0}=1.0\times 10^{4}\) W/cm\({}^{2}\) (\(E_{0}=5.34\times 10^{-2}\) a.u.) and duration \(T_{pu}=10\tau_{pu}=2.32\) fs (1.16 fs FWHM) and \(T_{pr}=10\tau_{pr}=165.4\) as (83 as FWHM) are always fixed. The pulse intensities \(I_{0}=1.0\times 10^{14}\) W/cm\({}^{2}\) (\(E_{0}=5.34\times 10^{-2}\) a.u.) and duration \(T_{pu}=10\tau_{pu}=2.32\) fs (1.16 fs FWHM) and \(T_{pr}=10\tau_{pr}=165.4\) as (83 as FWHM) are always fixed while the time delay \(\Delta\tau\) between the pump-probe pulse varies from \(4.75\tau_{pu}\) to \(6.0\tau_{pu}\). Figure 4 shows that asymmetric photoelectron angular and momentum distributions are produced again which evolve periodically with the time delay \(\Delta\tau\). Comparing to the process by a linearly polarized probe pulse in Figure 3, the time resolved photoelectron distributions rotate periodically with an anti-clockwise direction around the molecular \(R\) or \(z\)-axis. As shown in Figure 4a at \(\Delta\tau=4.75\tau_{pu}\), the photoelectron distributions are mainly localized in the right half plane. As the time delay \(\Delta\tau\) increases, the distribution rotate to the upper half plane (\(p_{y}>0\)) and then to the left. At \(\Delta\tau=5.5\tau_{pu}\), the photoelectron distributions in the lower half plane (\(p_{y}<0\)) dominate. At \(\Delta\tau=5.75\tau_{pu}\) the distributions move back to the right. The evolution period of the photoelectron distributions is \(\tau_{pu}=2\pi/\Delta E=232\) as, as predicted in Equation (10) for the coherent electronic wave packets. The rotation of the probing photoelectron distributions with the time delay \(\Delta\tau\) corresponds to the helicity of the probe pulses. For the circularly polarized probe pulse, its field vector is isotropic in the \((x,y)\) polarization plane. Moreover, for the molecule H\({}_{2}^{+}\) aligned along the \(z\)-axis, the influence of the molecular potential on the high frequency photoelectron emission is negligible. As a result, the asymmetric photoelectron distributions are determined by the electric field vector of the probe pulse, leading to a rotation around the molecular axis. That differs from the linear probe process in Figure 3 where the photoelectron distributions are mainly localized along the laser polarization direction since the field-molecule interaction term follows the form \(\cos(\theta)\), i.e., \(\sim\mathbf{r}\cdot\mathbf{E}\).