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Abstract
Relative phase effect of nonsequential double ionization (NSDI) of aligned molecules by counter-rotating two-color circularly polarized (TCCP) fields is investigated with a three-dimensional classical ensemble model. Numerical results show that NSDI yield in counter-rotating TCCP fields sensitively depends on the relative phase of the two components, which exhibits a sin-like behavior with the period of π/2. NSDI yield achieves its maximum at the relative phase π/8 and minimum at 3π/8. Back analysis indicates the recollision time and the return angle of the electron strongly depend on the relative phase of the two components, which results in the dominant emission direction of the electrons, is different for different relative phases. This indicates that the recollision process can be steered by changing the relative phase of the two components in counter-rotating TCCP laser fields. Meantime, it provides an avenue to obtain information about the recollision time and the return angle in the recollision process from the electron momentum distribution.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The two electrons from nonsequential double ionization (NSDI) in strong laser fields are highly correlated by a recollision process between the return electron and the parent ion [1,2]. The correlation behavior attracted much attention and is deeply studied experimentally [3–11] and theoretically [12–21] in the past decades. Two typical ionization channels, direct recollision-impact ionization (RII) [5,22] and recollision-induced excitation with subsequent ionization (RESI) [22–27], have been well demonstrated.
In recent years, much attention is concentrated on two-color circularly polarized (TCCP) laser fields, which consists of two circularly polarized pulses with different frequencies. The wave shape of the combined electric field can flexibly be tailored by changing the parameters of those components constituting the combined electric fields [28–30]. It provides an effective means for steering the electron dynamics in two dimensional (2D) space. Recently, the TCCP laser field is used to generate high-brightness circularly polarized harmonics in the extreme ultraviolet and soft x-ray regions [31]. A subcycle interference in the light propagation direction [32] and photoelectron holographic interferences [33,34] have been shown for ionization of atoms by counter-rotating TCCP laser fields. An attoclock photoelectron interferometry for probing the phase and the amplitude of emitting wave packets [35] is achieved by co-rotating TCCP fields. The nonadiabatic offset of the initial electron momentum distribution in strong field tunneling ionization of argon [36] has been demonstrated by TCCP fields.
The studies of NSDI in TCCP laser fields indicate that NSDI yields and recollision trajectories strongly depend on the field amplitude ratios of two components [37–41]. Double-recollision trajectory and its intensity dependence are demonstrated in NSDI by counter-rotating TCCP laser fields [42,43]. Very recently, the angular distributions of the correlated electrons in NSDI of Mg in counter-rotating TCCP laser fields are discussed [44]. Electron recollision assisted enhanced ionization is also experimentally observed for N$_2$ molecule in counter-rotating TCCP laser fields [45]. In these studies on NSDI above, the phase difference between two components constituting the combined electric fields is fixed. As we know, a change of the relative phase between the two components can result in a corresponding rotation of the wave shape of the combined electric field in the 2D space, which will affect the directions of the emission and the returning of the electron. For atoms the change of the relative phase will not affect the NSDI yield and the electron momentum distributions because atoms are spherically symmetric. However, for aligned molecules, the spherical symmetry is broken. The change of the relative phase will significantly affect the ultrafast electron dynamics and the occurrence of NSDI of aligned molecules. Now the dependence of NSDI of aligned molecules on the relative phase in counter-rotating TCCP fields is still unclear.
In this paper, we focus on the relative phase effect of NSDI of aligned molecules by counter-rotating TCCP fields. Our numerical results show that NSDI yield in counter-rotating TCCP fields sensitively depends on the relative phase of the two components, which achieves its maximum at the relative phase $\pi$/8 and minimum at 3$\pi$/8. Back analysis indicates that the recollision time and the return angle of the electron strongly depend on the relative phase of the two components. It results in that the dominant emission direction of the electrons is different for different relative phases. This provides an avenue to obtain information about the recollision time and the return angle in the recolision process from the electron momentum distribution.
2. Classical ensemble model
In this paper, we employ the three-dimensional (3D) classical ensemble model with a soft-core potential for the Coulomb interactions proposed by Eberly and coworkers [46,47], which has been widely recognized as a reliable and useful approach for interpretation and prediction of strong-field double ionization phenomena [16,17,41–44,48,49]. In this model, the evolution of the two-electron system is determined by the Newton’s equations of motion (atomic units are used throughout unless stated otherwise):
The electric field of the laser pulse is given by
To obtain the initial conditions for Eq. (1), the ensemble is populated starting from a classically allowed position for the energy of -1.67 a.u., corresponding to the sum of the first and second ionization potentials of H$_2$. The available kinetic energy is distributed between the two electrons randomly, and the directions of the momentum vectors of the two electrons are also randomly assigned. Then the two-electron system is allowed to evolve a sufficient long time (400 a.u.) in the absence of the laser field to obtain stable position and momentum distributions. Once the initial ensemble is obtained, the laser field is turned on and all trajectories are evolved in the combined Coulomb and laser fields. We check the energies of the two electrons at the end of the laser pulse, and a double ionization event is determined if both electrons achieve positive energies, where the energy of each electron contains the kinetic energy, potential energy of the electron-ion interaction, and half electron-electron repulsion energy.
3. Results and discussions
Figure shows NSDI probabilities as a function of the relative phase between the two components constituting the combined electric fields. The NSDI probability sensitively depends on the relative phase. It exhibits a sin-like behavior with the relative phase increasing. The period is $\pi$/2. The NSDI probability achieves its maximum at $\phi$=$\pi$/8 and 5$\pi$/8 and its minimum at $\phi$=3$\pi$/8 and 7$\pi$/8.
In order to understand the periodicity of the NSDI probability vs the relative phase, we check the variation of the space structure of the combined electric field with the relative phase. As shown in Fig. 2, the electric field and negative vector potential trace out a trefoil pattern and a triangle, respectively. A lobe of the electric field corresponds to a side of the triangle of the negative vector potential. Each lobe is marked by different colors. The negative vector potentials corresponding to the electric field maxima are located in the middle of each side of the triangle of the negative vector potential, which are indicated by the square blocks in Fig. 2. The arrows indicate the time evolution direction. The electric field recursively evolves from lobe 1 (blue), lobe 2 (red) to lobe 3 (green) and the negative vector potential evolves from side 1 (blue), side 2 (red) to side 3 (green). In the polarization plane the angle between any two lobes is 120$^{\circ }$. As the relative phase increases the trefoil-shape electric field and the triangle-shape negative vector potential both rotate clockwise. Moreover, for each $\pi$/8 increase of the relative phase, the trefoil-shape electric field rotates 15$^{\circ }$. From Fig. 2 one can see that when the relative phase increases $\pi$ the space structure of the combined field is completely the same. When the relative phase increases $\pi$/2, the wave shapes of the two combined fields are symmetrical with respect to the origin. In the two combined fields both angels the electron is emitted from and returns to the parent ion are the same with respect to the molecular axis. Thus NSDI probabilities are equal for those combined fields with the difference $\pi$/2 of the relative phase. This is the reason that the period of the NSDI probability vs the relative phase is $\pi$/2.
Fig. 2. The combined laser electric field E(t) (dashed) and the corresponding negative vector potential -A(t) (solid) for different relative phases. The arrows indicate the time evolution direction. The electric field and negative vector potential trace out a trefoil pattern and a triangle, respectively. A lobe of the electric field corresponds to a side of the triangle of the negative vector potential. The square blocks mark the field maxima and their negative vector potential. The black dumbbell represents the diatomic molecule aligned along x axis.
To obtain a deep understanding for ultrafast dynamics of NSDI in counter-rotating TCCP fields with different relative phases, we trace the classical NSDI trajectories. Figures 3(a)-3(c) show a sample trajectory from NSDI events at $\phi$=0. Figure 3(a) shows electron distances from the parent ion versus time. One electron is ionized at t=6.05T and then returns and collides with the parent ion at t=6.32T. The electron travels for 0.27T from single ionization to the recollision. It is obvious that prominent energy transfer from the returning electron to the bound electron takes place during the recollision, as shown in Fig. 3(b). The electrons’ 2D path in the field plane for the trajectory is shown in Fig. 3(c). The electron ionizes at lobe 3 and returns at lobe 1. So firstly the electron moves to -x direction and then returns from the third quadrant. We define the angle between the direction of the momentum of the return electron at the instant 3 a.u. before the recollision and the +x direction as the return angle. For the trajectory in Fig. 3(c), the return direction is opposite to the electric field vector and the return angle of the electron is 41$^{\circ }$. Furthermore, we do statistical analysis on the traveling time (the time interval between single ionization and the recollision) and find its distributions for different relative phases are similar. They all show a single-peak structure and the center of the peak is around 0.28T. The distribution of the traveling time for $\phi$=0 is shown in Fig. 3(d). It indicates that the recollision always occurs at the next electric field lobe after single ionization. Figures 3(e) and 3(f) show the electrons’ path in the field plane for another two typical sample NSDI trajectories for $\phi$=0. In Fig. 3(e) the electron is ionized at lobe 2 and returns at lobe 3 with the return angle of 160$^{\circ }$. For the trajectory in Fig. 3(f) the electron is ionized at lobe 1 and return at lobe 2 with the return angle of 283$^{\circ }$. This indicates that the electron can return to and collide with the parent ion from different angles via counter-rotating TCCP fields.
Fig. 3. Planes (a), (b) and (c) show electron distances from the parent ion versus time, electron energies versus time and the electrons’ path in the field plane for a sample NSDI trajectory. The arrows indicate the time evolution direction. The dotted line marks the return direction of the free electron. (d) Distribution of the traveling time for $\phi$=0. Planes (e) and (f) show the electrons’ path in the field plane for another two sample NSDI trajectories with different return angle. These trajectories are from those NSDI events for $\phi$=0.
Based on the statistical analysis of those classical NSDI trajectories, we obtain the distributions of single ionization time t$_{SI}$ and the recollision time t$_R$, as shown in the first and second rows of Fig. 4 for $\phi$=0 (the first column), $\pi$/8 (the second column), $\pi$/4 (the third column) and 3$\pi$/8 (the fourth column). By comparing these distributions from different relative phases, one can see that the differences of NSDI yields triggered by the single ionization at lobe 1 and lobe 2 between different relative phases are small. The main difference of NSDI yields between different relative phases originates from those NSDI events induced by the single ionization at lobe 3. For the electric field with a specific amplitude, the smaller the angle between the electric field and the molecular axis is, the larger the probability of the single ionization is. From Figs. 2(a)–2(d), one can see that the angle between the lobe 3 and the molecular axis increases gradually with the relative phase increasing. So in the range of the relative phase (0, $\pi$/2) the probability of single ionization from lobe 3 decreases gradually as the relative phase increases. Those electrons ionized at lobe 3 return to and recollide with the parent ion at lobe 1. After recollision the occurrence of the double ionization is significantly affected by the height of the suppressed potential barrier by the lobe 1. Previous study has indicated that the smaller the angle between the electric field and molecular axis is, the lower the suppressed potential barrier is [50]. From Figs. 2(a)–2(d), one can see that the angle between the lobe 1 and molecular axis decreases gradually with the relative phase increasing. That is to say, in the range of the relative phase (0, $\pi$/2), as the relative phase increases the probability of single ionization from lobe 3 decreases and the suppressed potential barrier at the recollision lowers gradually. The former is unfavorable and the latter is favorable to occurrence of NSDI. The competition between the two factors results in that the NSDI yield achieves its maximum at $\phi$=$\pi$/8 and minimum at $\phi$=3$\pi$/8.
Fig. 4. Single ionization time (the first row), recollision time (the second row) and return angle distribution (the third row) of the electron are shown from up to down for $\phi$=0 (the first column), $\pi$/8 (the second column), $\pi$/4 (the third column) and 3$\pi$/8 (the fourth column). In order to more clearly show the laser phase at single ionization and recollision, the single ionization time and recollision time are transferred into the interval of [(k+$\frac {\pi -\phi }{6\pi }$)T, (k+$\frac {\pi -\phi }{6\pi }$+1)T], in which the combined electric field evolves from lobe 1 to lobe 3. The laser electric field in arbitrary units is shown in planes (a)-(h).
Figures 4(i)–4(l) show the return angle distributions of the free electron for $\phi$=0, $\pi$/8, $\pi$/4 and 3$\pi$/8 respectively. It is obvious that the free electron always returns to the parent ion from three directions for all relative phases, which correspond to the three single ionization bursts from three field lobes. The return direction of the electron is approximately opposite to the electric field direction at the recollision. So the specific return angle is strongly dependent on the relative phase. Similar to the trefoil-shape electric field, with the relative phase increasing the return angle distribution rotates clockwise. For $\phi$=0, $\pi$/8 and $\pi$/4, there is a return direction in which the return electrons result in more NSDI events. The particular return angles are 42$^{\circ }$, 25$^{\circ }$ and 11$^{\circ }$ respectively. For $\phi$=3$\pi$/8, those NSDI events returning from 0$^{\circ }$ and 240$^{\circ }$ are comparable. The result indicates that the return angle of the electron can be well controlled by changing the relative phase of the two components in counter-rotating TCCP fields.
Figure 5 shows the momentum distributions of the two electrons in the field plane for eight relative phases. The corresponding negative vector potentials -A(t) are also shown. According to simple-man model, where the initial momentum at the ionization instant and the effect of the Coulomb potential on the ionized electron are ignored, the final momentum of the ionized electron is equal to the negative vector potential -A(t) at the ionization instant. So the electrons mainly distribute along the negative vector potential curve. But these electrons do not uniformly distribute on the three sides of the negative vector potential curve. For $\phi$=0, $\pi$/8, $\pi$/4, more electrons cluster on the side 1 of the negative vector potential triangle. For $\phi$=3$\pi$/8, the electrons on the side 1 and side 2 are comparable and more than those on the side 3. Because the negative vector potential curve rotates clockwise with the relative phase increasing, the emission direction of the electrons changes with the relative phase. In addition, the electron momentum distributions for $\phi$=0, $\pi$/8, $\pi$/4, $\phi$=3$\pi$/8 and $\phi$=$\pi$/2, 5$\pi$/8, 3$\pi$/4, $\phi$=7$\pi$/8 are symmetrical with respect to the origin. It is because that their electric fields are symmetrical with respect to the origin.
Fig. 5. Electron momentum distributions in the field plane for $\phi$=0 (a), $\pi$/8 (b), $\pi$/4 (c), 3$\pi$/8 (d), $\pi$/2 (e), 5$\pi$/8 (f), 3$\pi$/4 (g) and 7$\pi$/8 (h). Both electrons are included. The negative vector potentials -A(t) are shown.
Based on the final ionization order after recollision the two electrons are defined as the first and the second electron. We separately present the momentum distributions of the first electrons (the first row) and the second electrons (the second row) for $\phi$=0, $\pi$/8, $\pi$/4 and $\phi$=3$\pi$/8 in Fig. 6. For all relative phases, the distributions of the second electrons on the three sides of the negative vector potential are similar. The distributions of the first electrons determine the shape of the total momentum distribution of two electrons on the three sides. Experimentally, it is not possible to distinguish the two electrons involved in NSDI as we do. But the results above indicate that the measured total momentum distribution of two electrons in experiment can reflect the dominant emission direction of the first electrons. For the laser parameters considered in this work, after recollision the first electron is freed immediately. The second electrons are ionized with a prominent time delay with respect to the recollision as shown in Fig. 7. It indicates that most of NSDI proceed by RESI channel. If the initial momentum at the ionization instant and the effect of the Coulomb potential on the ionized electron are ignored, finally the first electron is most likely emitted with the momentum -A(t$_R$). This establishes a connection between the final momentum of the first electron and the recollision time. From Figs. 4(e), 4(f) and 4(g), one can see that more electrons return to and recollide with the parent ion at lobe 1 for $\phi$=0, $\pi$/8 and $\pi$/4. After recollision the first electron is ionized immediately and thus more first electrons distribute on the side 1. For $\phi$=3$\pi$/8, those NSDI events recolliding at lobe 1 and lobe 2 are comparable. Thus the first electrons mostly distribute on side 1 and side 2 of the negative vector potential triangle. So for different relative phases, the dominant NSDI events originate from different recollision times, different return angles and finally the electrons are emitted to different directions. Conversely, the electron momentum distribution can be regarded as the map of the recollision time and the return angle. The information about the recollision time and the return angle can be derived from the electron momentum distribution. In addition, our results also demonstrate that the recollision process can be steered by changing the relative phase of the two components in counter-rotating TCCP laser fields.
Fig. 6. Momentum distributions of the first and second electrons in the field plane for $\phi$=0, $\pi$/8, $\pi$/4 and 3$\pi$/8.
Fig. 7. Distributions of the time separation between the double ionization and the recollision for $\phi$=0 (blue), $\pi$/8 (red), $\pi$/4 (green) and 3$\pi$/8 (magenta).
4. Conclusion
In conclusion, we have investigated NSDI of aligned molecules by counter-rotating TCCP fields for different relative phases. Numerical results indicate that NSDI yield in counter-rotating TCCP fields sensitively depends on the relative phase of the two components, which achieves its maximum at the relative phase $\pi$/8 and minimum at 3$\pi$/8. Back analysis shows that the recollision time and the return angle of the electron can be well steered by changing the relative phase of the two components in counter-rotating TCCP laser fields. Correspondingly, it also results in the difference of the dominant emission direction of the electrons between different relative phases. The information about the recollision time and the return angle can be derived from the electron momentum distribution. This provides an avenue to access the subcycle dynamics of the recollision process.
Funding
National Natural Science Foundation of China (11504302, 61475127); Fundamental Research Funds for the Central Universities (XDJK2019C110).
Disclosures
The authors declare no conflicts of interest.
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