The interaction between laser and matter is highly nonlinear under certain physical conditions, and many new physical phenomena are produced. Such as high-order above-threshold ionization (HATI) [1], high harmonic generation (HHG) [2-4], nonsequential double ionization(NSDI) [5-12], etc. These phenomena can be understood with a three-step recollision mode [13]. In this model, an electron is first emitted via tunneling from the atom or molecule when the laser electric field becomes comparable to the binding Coulomb field. Then, the emitted electron is accelerated and driven back by the oscillating laser electric field and recombines with the parent ion core to release high-energy photons or collide elastically or inelastically with the parent ion core, leading to HATI or NSDI. Due to the presence of the Coulomb field of the ion, the electron may also be captured into a Rydberg state after the end of the laser pulse, leading to excited atoms or molecules [14-21]. This is known as frustrated tunneling ionization (FTI). The capture of electrons into Rydberg states was also found when strong-field double ionization (DI) occurs [22-25], which can be referred to as frustrated double ionization (FDI).

FDI of atoms and molecules has received a lot of attention in recent years. For example, because the second released electron has a dominant contribution to frustrated double ionization in the sequential regime, the photoelectron momentum distribution of FDI changes from a broad double-hump to a narrow single-hump structure with the increase of laser intensity in the Ar atomic has been observed by Larimian et al [26]. In a circularly polarized laser field, the FDI prefers low ionization potentials and short laser wavelengths, as studied by Kang et al [27], due to the fact that recollision plays an essential role in FDI not only for the NSDI regime but also for the SDI regime. Compared with an atomic target, the FDI of molecules is more complicated due to additional degrees of freedom, such as electron orbits, molecular alignment, and internuclear distance [28]. For instance, in theoretical studies, Emmanouilidou et al found that the probability of the recaptured electron in molecular FDI attaching to different nuclei can be explained by the initial velocity of tunneling electrons [29]. They also demonstrated that the FDI of the double electron triatomic molecule D³⁺ is significantly enhanced under the drive of counter-rotating two-color circular (CRTC) laser fields [30]. The FDI of small molecules was also studied such as H₂ [23, 31], D₂ [32, 33], O₂ [34], and CO [35]. In these experimental studies, Manschwetus et al reported evidence of electron recapture during strong-field fragmentation of H₂—explained by using a frustrated tunneling ionization model; McKenna et al made a similar measurement of D^* in D₂ on Manschwetus's work, they examined the dependence of D^* generation on the pulse duration, intensity, ellipticity and angular distribution, it is also found that the features of D^* spectra are directly related to the D⁺ spectra; Zhang et al [33] experimentally tracked the stepwise dynamics of the dissociative FDI of D₂ by monitoring the KER spectrum of nuclear fragments and the momentum distribution of freed electrons as a function of the time delay, their results show that ionized electrons are more easily recaptured in the second ionization step of producing dissociated FDI channel; In addition, Zhang et al also studied the laser-induced dissociative frustrated multiple ionization of CO molecules, and found that the formation of C^* is more than that of O^*. Detailed studies have shown that molecular FDI can be identified by measuring the kinetic energies of the excited neutral fragments after dissociation, but what insight into the details of the electron emission dynamics underlying the molecular frustrated double ionization is still unclear.

In this paper, the alignment-dependence electron dynamics of molecular FDI driven by the strong laser fields are investigated with a three-dimensional classical model. The numerical results show that the FDI probability decreases as the wavelength increases. For the short wavelength regime, the contribution from the single-recollision and the multiple-recollision mechanisms to molecular FDI are equal. Contrast with the case in the long wavelength regime where the single-recollision channel dominates FDI. Back analysis reveals that the ionization-exit momentum of the recaptured electron is largely compensated by the vector potential and these results provide an insight into the complex dynamics of the molecular FDI.

Currently, accurate quantum simulations of a two-active-electron system in a strong laser field still present a great challenge. The classical ensemble model proposed by Eberly and co-workers [36-38] has been widely used to study the strong field double ionization (DI) [39-43] of atoms and molecules in intense laser fields. In this study, we use this model to investigate the FDI of molecules via a linearly polarized laser field. The evolution of the system in this model is governed by Newton's equations of motion (atomic units are used unless stated otherwise)where the subscript i is the label of the two electrons, r_i is the position of ith electron, ${\boldsymbol{E}}(t)=\hat{z}{E}_{0}f(t)\sin (\omega t)$ is a linearly polarized electric field along the $z$ axis. Here ${E}_{0}$ is the peak amplitude of the laser electric field and $f(t)$ is the pulse envelope with two cycles turning on, six cycles plateau, and two cycles turning off, the intensity I = 1 × 10¹⁴ W cm⁻². ${V}_{{\rm{ne}}}$ is the coulomb interaction potential between the electron and the nucleus, ${V}_{{\rm{ee}}}$ is the interaction potential between electron and electron. For N₂ molecules, they can be represented as: R is the internuclear distance, in this work, we chose R = 2 a.u. To avoid unphysical autoionization and numerical singularity, we set the softening parameters a to be 1.15 a.u. and b to be 0.05 a.u.

To get the initial calculation conditions, the two nuclei are fixed on the z axis, their coordinates are (0, 0, R/2) and (0, 0, −R/2), and the starting position of the electrons is randomly distributed around the two nuclei. The total kinetic energy of the two electrons is equal to the ground state energy of the molecular system (the first and second ionization potentials of the N₂ molecules) minus the potential energy. We chose the electron pairs that have positive total kinetic energy. The total kinetic energy is distributed between the two electrons randomly. Then the system is allowed to evolve for a sufficiently long time (100 a.u.) without the laser field for a stable position and momentum distribution. Once the initial state of the ensemble is obtained, the laser pulse is turned on. After the laser field is over, when the final energy of the two electrons is positive, a DI event is considered to have occurred. FDI events are identified when the two electrons both have positive energies at some time during the laser pulse while at the end of the laser pulse, one electron has positive final energy but the other has negative final energy. The energy of every electron includes the kinetic energy, the ion core-electron potential energy, and half of the electron-electron repulsion energy. By tracing these FDI trajectories, we find that there are many multiple-recollision trajectories in FDI, where the first electron returns to the parent ion many times and more than once significant energy exchange occurs during the returns. We define that a recollision occurs when the distance between two electrons is less than 3.5 a.u. For the multiple-recollision FDI (MRFDI) trajectories, the recollision electron visits this area multiple times, while for the single-recollision FDI (SRFDI) trajectories, the recollision electron visits this area only one time. In this work, MRFDI trajectories must have two recollisions.

The FDI yields are presented with regards to laser wavelengths ranging from 600 to 1800 nm in figure 1 at different molecular alignments, which illustrates the predicted probabilities of FDI for N₂ molecules as a function of wavelength. Φ is the angle between the molecular axis and the laser polarization direction. We see that for the different angles, as the wavelength increases, the FDI probability rapidly drops, indicating that FDI prefers shorter wavelengths, in line with earlier theoretical investigations of atoms [27, 44]. The yield of molecular FDI decreases as the angle between the molecular axis increases. In general, when Φ takes various angles, the probability curves of FDI display similar trends. The physical mechanism of parallel aligned molecular FDI, which is the major topic of our paper, is discussed in detail below.

For successful FDI events, figure 2 depicts the ionization-exit velocity $({V}_{z0})$ along the laser polarization direction versus the ionization time $({t}_{0})$ of the recaptured electron. The ionization-exit velocity in figure 2 are plotted in units of $\sqrt{{U}_{p}},$ where ${U}_{p}=I/4{\omega }^{2\,}$ is the ponderomotive energy. The ionization-exit velocity ${V}_{z0}$ is defined as the velocity of the recaptured electron after the recollision at the ionization time. The vector potential $A(t)=-\displaystyle {\int }_{-\infty }^{t}E(t){\rm{d}}t$ is represented by the red-dashed curve in each panel of figure 2. The vector potential is seen to compensate for the ionization-exit velocity in a molecular FDI event. The initial velocity must be equal to (or very close to) the vector potential at the time of emission so that the recaptured electron's final velocity ${P}_{z}\approx {V}_{z0}-A({t}_{0})$ at the end of the pulse (neglecting coulomb potential after emission) is equal or close to zero. Because of the existence of the negative ion core-electron potential energy, a small mismatch between ${V}_{z0}$ and $A({t}_{0})$ may be tolerated. That is, even if the kinetic energy is close to zero, the total energy can still be negative. Such a tolerance is plainly wavelength dependent, as evidenced by the distributions in figures 2(a) and (b), where the population becomes increasingly concentrated to the vector potential curve. This is because the shorter the wavelength, the closer the electron is to the ion core, and the more negative the ion core-electron potential energy. This is consistent with previous atoms work [44], indicating that the ionization-exit velocity in FDI is wavelength dependent.

We also determine the photoelectron momentum distribution for FDI at two distinct wavelengths for ionized electrons. The momentum distribution of ionized electrons for 800 nm is depicted in figure 3 as having a double-hump pattern (blue curve). There is a four-hump structure for the 1064 nm case (red line). Our results show that for 1064 nm, the inner two humps correspond to instances when the ionization-exit velocity of an ionized electron is lower because it is closer to the vector potential at the time of emission. While the value of the ionization-exit velocity of an ionized electron is distant from that of the vector potential at the moment of emission for the outer two humps, it has a bigger velocity at the end of the laser pulse. Therefore, at a wavelength of 1064 nm, the final velocity ${P}_{z}\approx {V}_{z0}-A({t}_{0})$ of an ionized electron has a four-hump configuration.

We can unveil the microcosmic dynamics of molecular FDI by tracing the temporal evolution history of FDI trajectories. A classic trajectory perspective like this could provide an intuitive way to understand the complex dynamics of molecular FDI. FDI events in the current wavelength regime can thus be classified as SRFDI or MRFDI depending on the number of recollisions. In the SRFDI channel, one electron is first ionized, and then it is driven back by the laser field to recollide with the parent ions, resulting in the two electrons ionized after recollision. The second channel is the MRFDI channel, in which the second electron is released after two or three collisions with the first electron, which is driven back by the oscillating field. One of the two ionized electrons is recaptured for the FDI channels mentioned above before the laser is switched off.

SRFDI and MRFDI sample trajectories are displayed individually in figure 4 (left column and right column, respectively). According to the corresponding trajectories in the upper row of figure 4, the bottom row of the figure indicates the time evolution of the distance between each electron and parent ion. It is evident from the FDI trajectory in the left column that there is only one energy transfer taking place between the two electrons, and that the two electrons are promptly released after recollision (see figures 4(a) and (b)). In contrast, there are two energy transfers between the two electrons in the right column's trajectory, as depicted in figures 4(c) and (d), where the arrow is pointing. The first energy transfer results in one electron being ejected, the other electron being excited after the first recollision, and the second recollision releasing the excited electron.

Figure 5 displays, for an intensity of 1 × 10¹⁴ W cm⁻², the percentage of SRFDI and MRFDI trajectories in all FDI events as a function of wavelength. The rose red triangles line amplifies the point that the SRFDI channel's contribution rises as wavelength increases. The contribution of MRFDI (the green dots line) decreases as wavelength increases. According to figure 5, which compares the yields of SRFDI and MRFDI channels in the short wavelength regime, both the single-recollision and multiple-recollision processes contribute equally to molecular FDI in this regime. The single-recollision process, however, predominates FDI in the long wavelength regime as compared to the short wavelength regime.

We examine the time phase of single ionization (SI), recollision, and DI for 800 nm and 1064 nm, respectively, to provide a general knowledge of the wavelength-dependent evolution of the dynamics of the correlated electrons for the various FDI channels. We can easily determine the SI, recollision, and DI times through back analysis of the FDI trajectories, which can provide us insight into the molecular FDI sub-cycle dynamics. The recollision time (t_r) is defined as the moment when the two electrons are closest to each other. The SI time (t_SI) is defined as the time when the energy of the first electron is greater than zero. The DI time (t_DI) is the instant when the two electrons achieve positive energies, where the energy contains the kinetic energy, the ion-electron interaction, and half electron-electron interaction.

For the SRFDI channel, the curve of the time delay between t_r and t_SI exhibits a single peak structure and the peak locates near 0.5 T (see figures 6(a) and (b)), which results are consistent with the prediction of the simple-man model [13]. While for the MRFDI channel, the distribution of the time delay between t_r and t_SI shows a series of peaks. These peaks are located near the odd multiple of 0.25 T. The first peak at 800 nm is higher than the others (see figure 6(a)), indicating that the probability of an MRFDI channel is primarily due to the contribution of the first peak. The fact that the second and third peaks are higher than the first peak at 1064 nm (figure 6(b)) indicates that the probability of an MRFDI channel is primarily due to the contribution of the second and third peaks. Figures 6(c) and (d) display the distribution of the time delay between t_DI and t_r for FDI events. The substantial probability distributions for time delays greater than 0.25 T indicate that FDI is closely related to the recollision excitation with subsequent ionization (RESI) mechanism, which is consistent with Shomsky et al [22]. It is clear that the narrow single peak of the distribution for the SRFDI channel is near 0.35 T (red solid line). For the MRFDI channel, the time delay curve between t_DI and t_r has a broad peak structure located around 0.65 T (green dotted line). The results show that the two electron dynamics of FDI are distinguishable.

We calculate the distribution of the ionization-exit velocity of recaptured electrons (figure 7) and the momentum distribution of ionized electrons (figure 8) to study the wavelength dependence of two different channels. For 800 nm, the ionization-exit velocity is largely compensated by the vector potential, whether SRFDI or MRFDI channels (see figures 7(a) and (b)). The distribution of V_z0 concentrates more and more tightly to the vector potential curve at the wavelength of 1064 nm (figures 7(c) and (d)), this confirms the conditions for the generation of molecular FDI in the above analysis and the dependence of the ionization-exit velocity of FDI on wavelength. Figures 8(a) and (b) display the momentum distribution of different FDI channels at different wavelengths. We notice that the momentum distribution of ionized electrons is wavelength dependent. At 800 nm, the momentum distribution of recaptured electrons in both SRFDI and MRFDI channels exhibits a double-hump structure. While the distribution of SRFDI and MRFDI channels at 1064 nm reveals a four-hump structure. This demonstrates that the channel has no effect on the ionization-exit velocity distribution of the recaptured electrons or the momentum distribution of the ionized electrons.

By tracing the kinetic energy evolution of the returning electron, the details of the recollision dynamics for the two correlated electron emissions in the long-wavelength regime can be better understood. The kinetic energy distribution of the returning electron for FDI is shown in figure 9. The white dotted line is plotted to compare energy change, the closer the distribution of kinetic energy is to the diagonal, the less energy is transferred. It is clearly seen that for the MRFDI channel of 800 nm (figure 9(c)), the distribution of E_k1 and E_k2 is closer to diagonal than SRFDI (figure 9(b)), indicating MRFDI transfers less kinetic energy than SRFDI, so the MRFDI happens more recollisions, the same as 1064 nm. When comparing 800 nm and 1064 nm, whether SRFDI or MRFDI mechanism, as the wavelength increases, the distribution of E_k1 and E_k2 moves closer to the diagonal, implying that the recollision efficiency (the ratio of the transferred kinetic energy to E_k1) decreases with the increase of wavelength. Confirming that the yields of FDI decrease with the increase of wavelength, which is shown in figure 1.

The laser intensity dependence of FDI is investigated by our classical model. The calculated FDI probabilities of N₂ molecules as functions of intensity are shown in figure 10 for 800 nm and 1064 nm, respectively. A similar ‘knee' shape can be seen in FDI results for the two wavelengths (figure 10(a)). In figure 10(b), we can see a crossing of the SRFDI and MRFDI probability curves around 8 × 10¹³ W cm⁻², below which MRFDI is more efficiently generated and above which SRFDI is more efficient. A similar crossing is not observed in the 1064 nm case (figure 10(c)), and the probabilities of SRFDI are always greater than those of MRFDI.

In conclusion, the alignment-dependence of FDI yields in N₂ molecules driven by linearly laser pulses is investigated and the results show that FDI yields decrease as wavelength increases. We demonstrated the intuitive classical FDI trajectories as well as their statistical characteristics. Tracing the classical trajectories reveals that the contributions of the SRFDI and MRFDI mechanisms are equal in the short wavelength regime, but the SRFDI channel is more important in the long wavelength regime. A specific requirement for producing molecular FDI events is that the vector potential must substantially offset the ionization-exit momentum of the recaptured electron at the time of ionization.