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Abstract
We theoretically investigate the photoelectron momentum distribution of He atoms by numerically solving the time-dependent Schr$\ddot {\textrm {o}}$dinger equation (TDSE) in few-cycle ionization gating, which is synthesized by two linearly polarized laser pulses. When applying the TDSE, we can clearly see the spider-like structures in the photoelectron momentum spectra. We also find that the spider-like structures can be isolated by changing the relative phase. The directionality of the spider-like structure is changed from right-side to left-side and the ring-like interference structure gradually appears in the photoelectron momentum spectra when increasing the relative phase. The interference patterns observed in TDSE are recaptured well by the quantum-trajectory Monte Carlo (QTMC) model. We separate the ionization time window of the tunneling electron by analyzing the ionization rate. With the help of QTMC simulation, we illustrate the change of the interference structure and its directionality in the photoelectron momentum spectra. By changing the relative phase, the forward-backward asymmetry of the momentum distribution of the emitted electrons can also be controlled. Moreover, we find that the relative contribution of the nonrescattering and the rescattering trajectories can be controlled. These properties are beneficial for the application of photoelecron holography in probing atomic and molecular structures and dynamics.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The strong-field tunnel ionization of atoms is one of the fundamental events in the laser-atom interaction. And it gives rise to many non-linear phenomena, such as high-order harmonic generation [1], above-threshold ionization [2], and the nonsequential double ionization [3,4]. The electron can be ionized by absorbing several photons, and the number of the photons could more than the minimum number of photons required to free the electron, this process is called as the above-threshold ionization (ATI) process [2]. Based on the process of the ionization, many more complex phenomena can be generated. If the ionized electron is driven back to the parent ion, it can collide with its parent ion and generate several possible process. First, it may kick out a second electron and lead to the nonsequential double ionization (NSDI) [3,4], and the two ionized electrons are highly correlated. It may recombine with its parent ion and lead to the emission of a high energy photon, this process is called as high-order harmonic generation (HHG) [1].
When the atom is exposed to a strong laser pulse, the out-most electron may be ionized by tunneling through the barrier which formed by the Coulomb potential and the laser field [5]. The ionized electron wave packets (EWPs) which have the same final momentum and are emitted from different times will interference with each other [6]. Thus there are rich interference structures that could be generated in the final photoelectron momentum distributions (PMDs) [7–12].
The interference of the two EWPs ionized with exactly one optical cycle (o.c.) relative delay reaching the same final momentum gives rise to above-threshold ionization (ATI) rings, i.e., the intercycle interference, that are spaced by the energy of one photon in the photoelectrom spectrum [7,9,13]. Intra-cycle interference is another typical type of interference which is produced by the EWPs emitted within one cycle [12,14,15]. There are two normal types of the intra-cycle interference, one is the adjacent intra-cycle interference [16–18] that the interference fringes originate from EWPs emitted during adjacent quarter cycles of the optical wave, the other one is the nonadjacent intra-cycle interference [19] that the EWPs emitted during nonadjacent quarter cycles of the optical wave and interference with each other. Recently, it is reported that this intra-interference can be controlled by a phase-controlled two-color laser field. Photoelectron holography (PH) structure can be generated by the interference of direct and rescattering trajectories [20–23]. In PH, the rescattering electron acts as a signal wave, while the direct electron acts as a reference wave, the interference with these two kinds of trajectories give rise to the hologram. The spider-like pattern is a typical PH interference structure of the adjacent intra-cycle interference, which has been experimentally observed in the PMDs [10,11,20]. By analyzing the hologram patterns, the information of the ionization time can be reconstructed [24,25]. Furthermore, there is a different spider-like structure in the low-energy region which is called as inner-spider structure [21]. Recently, the researchers systematically introduce the development, the challenges and the future possibility about the PH in a review paper [23]. Usually, all of these interferences contribute together to the final PMDs and thus the interference patterns in PMDs are too complicated for people to extract useful information. So it is necessary to select some certain interferences to extract information.
The gating technique has been widely used in experimental measurements to control the ultrafast process. The polarization gating (PG) technique can provide a simple and reliable way to temporally shape the polarization of a pulse and generate a nearly-linearly polarized, single or sub-cycle laser pulse through combing two laser pulses of different polarization and frequencies. The normal way to generate the PG pulse is using a pair of counter-rotating circularly polarized pulses with a proper time delay. It has been used to measure the EWPs motion with attosecond resolution and to generate an isolated broadband attosecond XUV pulse. Recently, the PG pulse has been used to investigate the intra-cycle interferences of Ar atoms, and a unique interference structure has been observed which depend on the Coulomb potential [26]. The ionization gating can be achieved with a pair of linearly polarized pulse, which can be used to control the ionization process at the pulse peak by adjusting the second harmonic (SH) pulse as illustrated in Ref. [27]. It can be used to generate the isolate attosecond pulse and to enhance the harmonics and the attosecond XUV pulse. There are some other gating techniques which have been widely used to achieve the isolate attosecond pulse, such as amplitude gating [28] and double optical gating [29], etc.
In previous, many researchers pay attention to the dynamics of the ionized electrons in long-cycle two-color parallel linearly polarized field [30–33]. Recently, the control of the ionization process and the asymmetry of PMDs by changing the relative phase in long-cycle parallel polarized field have been reported [30,32]. And we can also see that the ratio of the rescattering and nonrescattering trajectories could be regulated by the relative phase [33].
In this paper, we theoretically investigate the EWPs interference in PMDs with the ionization gating which achieved with a few-cycle laser pulse in combination with its second harmonic (SH), by numerically solving the time-dependent Schr$\ddot {\textrm {o}}$dinger equation (TDSE). The numerical results show that the interference patterns from only spider-like structure to both spider-like and ring-like structure, and the directionality of the spider-like patterns will be changed with increasing the relative phase. The quantum-trajectory Monte Carlo (QTMC) simulation can reproduce the interference pattern well. Adopting the QTMC analysis, we illustrate the change of the photoelectron momentum distribution and find that the forward-backward asymmetry of the momentum distribution of the emitted electrons can be controlled. And the spider-like interference structure can be isolated by changing the relative phase, in other words, the inter-cycle interference can be switched on and off by the relative phase with the ionization gating. In this way, the structure information of the target which is encoded in the hologram can be extract independently. Meanwhile, the relative contribution of the nonrescattering and the rescattering trajectories can also be controlled.
2. Methods
2.1 Time dependent Schr$\ddot {\textrm {o}}$dinger equation (TDSE) theory
In this paper, the photoelectron momentum distribution is investigated by solving the two-dimensional (2D) TDSE with single-active-electron (SAE) approximation with an ionization gating. The TDSE is expressed as:
where $\vec {r}$ denotes the electron position in the plane of polarization. The Hamiltonian $H(\vec {r},t)$ is given by where $V(r)$ is the effective soft-core potential: where $r=\sqrt {{x}^{2}+{y}^{2}}$, $x$ and $y$ are the position coordinates of the ionized electron. We set the soft-core parameters $b=1.5$ and $a=0.6$, so that the eigenvalue of ground state is ${{{I}_{p}}=0.9}$, which is equal to that of real He atoms [34]. $E(t)$ is the electric field of the laser pulse. The initial wave function is prepared by using the imaginary-time propagation method [35]. And we solve the 2D TDSE by fast Fourier transform technique combined with split-operator method [36]. The PMD is obtained by Fourier transforming the wave packet of the ionized part [37,38].2.2 Quantum-trajectory Monte Carlo (QTMC) theory
We also investigate the photoelectron momentum distribution of He atoms with the ionization gating by using the QTMC model. The detail of the QTMC model is described in [39]. Briefly, this model combined the traditional semi-classical model and Feynman’s path-integral approach. The EWPs in strong-field ionization are represented by a series of quantum trajectories. The tunneling ionization time and the gauss-like initial transverse momentum distribution of electron wave packets are given by the Ammosov-Delone-Krainov (ADK) theory [40,41]. The weight of each trajectory is given by
whereIn the QTMC model, we have included the phase for electron trajectory with ${{e}^{-i\Phi ({{t}_{0}},{{\vec {v}}_{0}})}}$. The phase $\Phi ({{t}_{0}},{{\vec {v}}_{0}})$ is given by integral along the trajectory. ${{\vec {v}}_{0}}$ is the initial velocity of the electron. Here, we use the full semi-classical phase, which is derived in detail in Ref. [42] and called semi-classical two-step (SCTS) model. It is given by
The probability of each asymptotic momentum is determined by
The ionization gating is achieved with few-cycle laser pulse in combination with its SH. The fundamental laser pulse and SH are both linearly polarized along the $\hat {x}$ direction and the electric field is given by
$f(t)$ is the pulse envelope which can be expressed by $f(t)={{\sin }^{2}}(\pi t/T)$. $T$ is the pulse duration. ${{E}_{0}}$ and ${{E}_{1}}$ are the amplitude, ${{\omega }_{0}}$ is the frequency of the fundamental laser and $\varphi$ is the relative phase. The intensity and the wavelength of the fundamental laser pulse are $6\times {{10}^{14}}\text {W/c}{{\text {m}}^{\text {2}}}$ and $800\text {nm}$; the intensity of the SH is $4\%$ of the fundamental field. Atomic units (a.u.) are used throughout the paper unless indicated otherwise.3. Results and discussions
Figure 1(a)-(e) show the photoelectron momentum distribution of He atoms in the (${{P}_x}$, ${{P}_{y}}$) plane with the ionization gating at different relative phases by solving the TDSE. All photoelectron spectra show the spider-like structures which correspond to the PH interference, and the directionality of the spider-like structures is changed from the right to left and the ring-like interference structure gradually appears in the photoelectron momentum spectra with increasing the relative phase. While for the different ratio of the frequency, the direction of the spider-like structure has no change with the relative phases in the parallel linearly polarized laser field [31]. In Fig. 1(a) and (b), for $\varphi =0$ and $\varphi =0.25\pi$, the spectrum shows similar patterns which looks like the spider-leg. For $\varphi =0.5\pi$, the ring-like structure gradually appears which is shown in Fig. 1(c). With increasing the relative phase, clear ring-like structures centered at zero momentum and spider-like patterns appear in the PMDs as shown in Fig. 1(d) and (e) for $\varphi =0.75\pi$ and $\varphi =1.0\pi$. There is a little difference with that demonstrated in Ref. [30], where the ATI-ring structures accompany with the spider-like structure all the time for different relative phases in long-cycle parallel linearly polarized laser field.
Fig. 1. The photoelectron momentum distribution of He atoms with the ionization gating for different relative phases. (a)-(e) Simulation results of the TDSE method. (f)-(j) Simulation results of the QTMC model. (k)-(o) Illustration of the electric field of the laser pulse and the ionization rate for different relative phases, the gray dash-dotted curves marked the fundamental laser pulse, the blue solid line marked the SH laser pulse and the red solid line marked the ionization rate.
Figure 1(f)-(j) show the photoelectron momentum distribution of He atoms with the ionization gating at different relative phases calculated with QTMC model for (f) $\varphi =0$, (g) $\varphi =0.25\pi$, (h) $\varphi =0.5\pi$, (i) $\varphi =0.75\pi$ and (j) $\varphi =1.0\pi$, respectively. The simulation results exhibit a rather good agreement with the TDSE results. We can see that both classical and quantum momentum distributions exhibit a similar interference pattern, although there are some differences between these two methods for $\varphi =0.5\pi$. From Fig. 1(a)-(e) and (f)-(j), we can see that the spider-like patterns can be isolated by decreasing the relative phase, which means the inter-cycle interference can be switched on/off by the relative phase with the ionization gating. And we can see that the whole PMDs moves along the x-axis of the photoelectron momentum.
Figure 1(k)-(o) show the electric field of the laser pulse and the ionization rate for different relative phases, where the gray dash-dotted line marked the fundamental laser pulse, the blue solid line marked the SH laser pulse and the red solid line marked the ionization rate. In Fig. 1(k) and (l), we can see that the EWPs could be free from one time-window when the relative phase is $\varphi =0$ and $\varphi =0.25\pi$, which is marked by $\text {w}1$. With increasing the relative phase, the EWPs can emit from three or two time-windows at $\varphi =0.5\pi$, $\varphi =0.75\pi$ and $\varphi =1.0\pi$ as shown in Fig. 1(m)-(o) and the EWPs is marked by $\text {w}1$ $\text {w}2$ and $\text {w3}$. For $\varphi =0.5\pi$ and $\varphi =0.75\pi$, the ionization rate are different for $\text {w}2$ and $\text {w}3$ as shown in Fig. 1(m) and (n), this difference is caused by the asymmetry of the laser field. Because the dependence is exponential between the ionization rate and the field amplitude as Eq.(4)-(6), the oscillation of the curves of the ionization rate is obvious with a small change of the amplitude of the laser field. And for the symmetrical laser field with $\varphi =1.0\pi$, the tiny difference of the ionization rate between $\text {w}2$ and $\text {w}3$ caused by the precision of the calculation. We can see $\text {w}2$ and $\text {w}3$ are emitted around $2 \text {o}\text {.c}.$ and $3 \text {o}\text {.c}\text {.}$ respectively. The time gap of these two ionization windows is one optical cycle. The interference between the trajectories which come from these two time windows generate the ring-like interference pattern [7], which corresponds to Fig. 1(h)-(j).
By using the QTMC model, we can simulate the interference pattern of EWPs which emit from specific time window and trace the trajectories for ionized electron. To disentangle the dynamic mechanism of the electrons, we can separate them to different time windows according to the ionization rate for $\varphi =0.5\pi$, $\varphi =0.75\pi$ and $\varphi =1.0\pi$. Fig. 2 shows the photoelectron momentum distribution of He for $\varphi =0.5\pi$ and $\varphi =0.75\pi$. Fig. 2(a) and (c) show the photoelectron momentum distribution for the case that electron emitted from w2 and w3 (correspond to Fig. 1(m) for $\varphi =0.5\pi$), respectively. Figure 2(b) shows the photoelectron momentum distribution for the case that the electron emitted from w1. From Fig. 2(a) and (c), we can see the obvious spider-like patterns which point to the left side. While in Fig. 2(b) the spider-like pattern point to the right side.
Fig. 2. The momentum distribution of He atoms which emit from the time window (a) w2, (b) w1 and (c) w3 for the relative phase $\varphi =0.5\pi$, and the red square marked the inner-spider structure.(d)-(f) The same as those in (a)-(c) for the relative phase $\varphi =0.75\pi$.
Fig. 3. The momentum distribution of He atoms which emit from different time windows which correspond to w2 and w3 for the relative phase $\varphi =1.0\pi$.
We do the same analysis for the case $\varphi =0.75\pi$, and the results are shown in Fig. 2(d)-(f). We can see the similar results with the case $\varphi =0.5\pi$. From the ionization rate which shown in Fig. 1(m) and (o), we can see that the ionization channel decreased from three to two, thus we separate two different ionization time windows ($\text {w}2$ and $\text {w}3$) for $\varphi =1.0\pi$ as shown in Fig. 3, which indicates that the spider-like pattern points to the left side.
From Fig. 1(k) and (l), we can see that only one ionization channel which marked by w1 for $\varphi =0$ and $\varphi =0.25\pi$ which correspond to the spider-like patterns point to the right side in the photoelectron momentum distribution as shown in Fig. 1(f) and (g). Through above analysis we can see that the electrons which emitted from w1 generate the spider-like structure pointed to the right side, while for the electrons which emitted from w2 and w3 generate the spider-like structure pointed to the left side.
In order to explore the underlying dynamics, we trace the trajectories of the ionized electrons. For instance, as shown in Fig. 4, we present the typical trajectories for the case $\varphi =0.5\pi$, which selected from the area marked by the red square in Fig. 2(a)-(c) and emitted from different time windows (correspond to w2, w1 and w3 in Fig. 1(m)). Fig. 4(a1)-(a3) and (b1)-(b3) indicate the nonrescattering trajectories and the rescattering trajectories, respectively [43]. To demonstrate the rescattering process more clearly, we set the range of the coordinate axis near the motion range of the ionized electron. Although the range of the coordinate axis is different, the features of these trajectories are similar with that demonstrated in Ref. [20,44].
Fig. 4. The typical trajectories emitted from different time windows which correspond to w2, w1 and w3 for the relative phase $\varphi =0.5\pi$. (a1)-(a3) the nonrescattering trajectories; (b1)-(b3) the rescattering trajectories.
For the electron which ionized from $\text {w2}$ [as shown in Fig. 4(a1) and (b1)] and $\text {w3}$ [as shown in Fig. 4(a3) and (b3)], we can see that the final positions of the trajectories locate in $X<0$ plane, while for $\text {w1}$ [as shown in Fig. 4(a2) and (b2)] the final positions of the trajectories mainly locate in $X>0$ plane. For different relative phases the phenomenon is similar. And it is agreement with the PMDs shown in Fig. 2 and Fig. 3. Specifically, for $\text {w1}$, the EWPs are emitted when the fundamental field is positive, which result that the spider-like structures point to the right-side in PMDs. For $\text {w2}$ and $\text {w3}$, the EWPs are emitted when the fundamental field is negative, which result that the spide-like structures point to the left-side in PMDs.In addition, from Fig. 2(a) and (c), we can see that the left-side spider-like structure can be generated by the interference of the electron which emitted from $\text {w2}$ and $\text {w3}$. However, the left-side spider-like structure is blur which is caused by the overlap of interference patterns, as shown in Fig. 1(h).
As illustrated in Fig. 1(a)-(e) and (f)-(j), the distributions of the momentum move along $x$-axis by changing the relative phases. In order to quantify the breaking of inversion symmetry of the PMDs, Fig. 5(a) shows the momentum along $x$ direction as function of the relative phase with the ionization gating. From Fig. 5(a) we can see that the distribution of the momentum along polarization direction periodically changes with the relative phase, and the period is $2\pi$, which means that the momentum distributions are periodically moved along $x$ direction with the relative phase.
Fig. 5. (a)The photoelectron momentum distribution for Px as a function of the relative phase for He atoms; (b) The asymmetry parameter as a function of relative phase $\varphi$ for He atoms.
To illustrate this change more clearly, we also introduce an asymmetry parameter. The forward-backward asymmetry parameter $a$ is defined as $a(\varphi )=\frac {{{P}_{ionization}}(+)-{{P}_{ionization}}(-)}{{{P}_{ionization}}(+)+{{P}_{ionization}}(-)}$, where the ${{P}_{ionization}}(+)$ and ${{P}_{ionization}}(-)$ are the total emission probabilities in forward and backward directions, respectively [32]. Fig. 5(b) shows the asymmetry parameter as function of the relative phase, the red-cycle line and the blue-square line indicate the asymmetry parameter with and without inclusion the Coulomb potential, respectively. We can see that the asymmetry parameter strongly depends on the relative phase and changes periodically with the relative phase, and the period is $2.0\pi$. That are similar to the experimental results about Ar in a linearly polarized two-color laser field which reported in Ref. [32]. The tendency corresponds with Fig. 5(a). We can also see that the asymmetry parameter can be influenced by the Coulomb potential, such as $\varphi =0$, when the Coulomb potential is not considered the asymmetry corresponds to zero which means that the electron which emit to forward and backward directions are equal, while for considering the Coulomb potential the asymmetry corresponds to 0.2 which means the Coulomb potential can increase the probability of the electron which emit to the forward direction.
To shed more light on the influence of the relative phase, we demonstrate the emission angle. The emission angle is defined as $\text {tan}{{\text { }}^{-1}}(\frac {{{P}_{y}}}{{{P}_{x}}})$, which is the angle between the final momenta and the $x$ direction. Fig. 6 shows the photoelectron angular distribution with respect to the relative phase. It is clear that the emission angle mainly distribute around $90{}^\circ$ and $270{}^\circ$, while with increasing the relative phase the distribution of the electron which is around $90{}^\circ$ decreased for the relative phase around $0.5\pi$ to $0.75\pi$. And the electron which distribute around $270{}^\circ$ rarely changed with the relative phase. In brief, we find that the relative phase mainly influences the electron which emit around $90{}^\circ$. To further explore the mechanism, we will illustrate the ionization probability of the electrons which respect to the electron emission angle.
We separate the contributions of different trajectories to illustrate the rescattering effect on the asymmetry for different relative phases. As we known, for the nonrescattering trajectories, their final transverse momenta are smaller than the initial transverse momenta because of the Coulomb focusing effect. Thus the final and initial transverse momenta have the same sign [43]. For the rescattering trajectories, the finial transverse momenta have the opposite sign with the initial transverse momentum because of the recattering process [43]. Thus we can separate the contributions of those two kinds of trajectories by adopting the semi-classical QTMC model.
In Fig. 7, we show the ionization probability of the nonrescattering and rescattering trajectories which respect to the emission angle for different relative phases, the red-cycle line and the black-square line mark the rescattering and the nonrescattering cases, respectively. For the nonrescattering trajectories, the distribution of the emission angle demonstrates double-peak structure and the peaks locate around $90{}^\circ$ and $270{}^\circ$, although amplitude of the peak oscillate around $90{}^\circ$ for different relative phases. While for the rescattering trajectories, the distribution of the emission angle changes from single-peak to double-peak and then back to single peak with increasing the relative phase. As shown above, the emission angle of the rescattering and nonrescattering trajectories depends on the relative phase of the laser pulse. Thus, we can extract the rescattering or nonrescattering trajectories by tuning the relative phase. Comparing with Fig. 1(a)-(e), we can see that the role of the rescattering in the hologram in the final PMDs. In Fig. 7(a), (b), (d) and (e), the rescattering electron only distributed around one specific emission angle, which cause the spider-like structure point to the single direction. While for $\varphi =0.5\pi$, the emission angle of the rescattering angle are around $90{}^\circ$ and $270{}^\circ$, which make the spider-like structure point to both right and left side in the PMDs. There is a little difference with that demonstrated in Ref. [33], in which the rescatering and the nonrescattering trajectories both contribute to each emission angle for Xe atoms in long-cycle parallel two-color laser-pulse.
Fig. 7. The ionization probability of the rescattering (the red-cycle line) and nonrescattering (the black-square line) trajectories with respect to the electron emission angle for different relative phases.
4. Conclusion
In summary, we have theoretically investigated the relative phase effect on the EWPs interference in photoelectron momentum distribution with the ionization gating. By solving the TDSE, we find that the interference patterns change from only spider-like structure to both spider-like structure and ring-like structure. And the directionality of the spider-like structures also change with increasing the relative phase. The direction of the spider-like structure changes from right-side to left-side in the photoelectron momentum distributions with increasing the relative phase. These features in the photoelectron momentum distribution can be reproduced by the QTMC model. By investigating the ionization rate and tracing the trajectories, we demonstrate the reason of the changes of the direction and the changes of the interference structures in the photoelectron momentum distribution. We also demonstrate that the forward-backward asymmetry of the doubly differential momentum distribution of the emitted electrons can be controlled and the PH interference structure can also be isolated by changing the relative phase. This is based on that the ionization channel of the specific time-window can be switched on and off by the relative phase in the ionization gating. Moreover, we find that the relative contribution of the nonrescattering and the rescattering trajectories can also be controlled. The structure and dynamic information of the He atoms are could be probed by these properties.
Funding
National Natural Science Foundation of China (12074142).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data that support the findings of this study are available from the corresponding author upon reasonable request.
References
1. X.-L. Ge, H. Du, J. Guo, and X.-S. Liu, “Quantum control of electron wave packet during high harmonic process of $\text {H}_{2}^{+}$ in a combination of a circularly polarized laser field and a terahertz field,” Opt. Express 23(7), 8837–8844 (2015). [CrossRef]
2. P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, “Free-free transitions following six-photon ionization of xenon atoms,” Phys. Rev. Lett. 42(17), 1127–1130 (1979). [CrossRef]
3. S. Ben, P.-Y. Guo, K.-L. Song, T.-T. Xu, W.-W. Yu, and X.-S. Liu, “Nonsequential double ionization of mg from a doubly excited complex driven by circularly polarized laser field,” Opt. Express 25(2), 1288–1295 (2017). [CrossRef]
4. T.-T. Xu, Q.-Y. Zhu, J.-H. Chen, S. Ben, J. Zhang, and X.-S. Liu, “Multiple recollisions in nonsequential double ionization by counter-rotating two-color circularly polarized laser fields,” Opt. Express 26(2), 1645–1654 (2018). [CrossRef]
5. L. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Soviet Physics JETP 20, 1307–1314 (1965).
6. W. Becker, F. Grasbon, R. Kopold, D. Milošević, G. Paulus, and H. Walther, “Above-threshold ionization:from classical features to quantum effects,” Advances in Atomic Molecular and Optical Physics 48, 35–98 (2002). [CrossRef]
7. W. Yang, H. Zhang, C. Lin, J. Xu, Z. Sheng, X. Song, S. Hu, and J. Chen, “Momentum mapping of continuum-electron wave-packet interference,” Phys. Rev. A 94(4), 043419 (2016). [CrossRef]
8. D. G. Arbó, K. L. Ishikawa, K. Schiessl, E. Persson, and J. Burgdörfer, “Intracycle and intercycle interference in above-threshold ionization: the time grating,” Phys. Rev. A 81(2), 021403 (2010). [CrossRef]
9. D. G. Arbó, K. L. Ishikawa, E. Persson, and J. Burgdörfer, “Doubly differential diffraction at a time in above-threshold ionization: intracycle and intercycle interference,” Nucl. Instruments Methods Phys. Res. Sect. B: Beam Interactions with Mater. Atoms 279, 24–30 (2012). [CrossRef]
10. M. Haertelt, X.-B. Bian, M. Spanner, A. Staudte, and P. B. Corkum, “Probing molecular dynamics by laser-induced backscattering holography,” Phys. Rev. Lett. 116(13), 133001 (2016). [CrossRef]
11. S. G. Walt, N. B. Ram, M. Atala, N. I. Shvetsov-Shilovski, A. V. Conta, D. Baykusheva, M. Lein, and H. J. WöRner, “Dynamics of valence-shell electrons and nuclei probed by strong-field holography and rescattering,” Nat. Commun. 8(1), 15651 (2017). [CrossRef]
12. J. Tan, Y. Zhou, Q. Ke, M. He, J. Liang, Y. Li, M. Li, and P. Lu, “Resolving strong-field tunneling ionization with a temporal double-slit interferometer,” Phys. Rev. A 101(1), 013407 (2020). [CrossRef]
13. M. Li, P. Zhang, S. Luo, Y. Zhou, Q. Zhang, P. Lan, and P. Lu, “Selective enhancement of resonant multiphoton ionization with strong laser fields,” Phys. Rev. A 92(6), 063404 (2015). [CrossRef]
14. X.-B. Bian, Y. Huismans, O. Smirnova, K.-J. Yuan, M. Vrakking, and A. D. Bandrauk, “Subcycle interference dynamics of time-resolved photoelectron holography with midinfrared laser pulses,” Phys. Rev. A 84(4), 043420 (2011). [CrossRef]
15. Y. Li, Y. Zhou, M. He, M. Li, and P. Lu, “Identifying backward-rescattering photoelectron hologram with orthogonal two-color laser fields,” Opt. Express 24(21), 23697–23706 (2016). [CrossRef]
16. X. Xie, “Two-dimensional attosecond electron wave-packet interferometry,” Phys. Rev. Lett. 114(17), 173003 (2015). [CrossRef]
17. D. G. Arbó, E. Persson, and Burgdörfer, “Time double-slit interferences in strong-field tunneling ionization,” Phys. Rev. A 74(6), 063407 (2006). [CrossRef]
18. R. Gopal, K. Simeonidis, R. Moshammer, T. Ergler, M. Dürr, M. Kurka, K.-U. Kühnel, S. Tschuch, C.-D. Schröter, D. Bauer, J. Ullrich, A. Rudenko, O. Herrwerth, T. Uphues, M. Schultze, E. Goulielmarkis, M. Uiberacker, M. Lezius, and M. F. Kling, “Three-dimensional momentum imaging of electron wave packet interference in few-cycle laser pulses,” Phys. Rev. Lett. 103(5), 053001 (2009). [CrossRef]
19. X. Xie, T. Wang, S. Yu, X. Lai, S. Roither, D. Kartashov, A. Baltuška, X. Liu, A. Staudte, and M. Kitzler, “Disentangling intracycle interference in photoelectron momentum distributions unsing orthogonal two-color laser fields,” Phys. Rev. Lett. 119(24), 243201 (2017). [CrossRef]
20. Y. Huismans, A. Rouzée, A. Gijsbertsen, J. H. Jungmann, A. S. Smolkowska, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, H. G. Muller, W. Vermin, K. J. Schafer, M. Spanner, M. Y. Ivanov, O. Smirnova, D. Bauer, S. V. Popruzhenko, and M. J. J. Vrakking, “Time-resolved holography with photoelectrons,” Science 331(6013), 61–64 (2011). [CrossRef]
21. D. D. Hickstein, P. Ranitovic, S. Witte, X.-M. Tong, Y. Huismans, P. Arpin, X. Zhou, K. E. Keister, C. W. Hogle, B. Zhang, C. Ding, P. Johnsson, N. Toshima, M. J. J. Vrakking, M. M. Murnane, and H. C. Kapteyn, “Direct visualization of laser-driven electron multiple scattering and tunneling distance in strong-field ionization,” Phys. Rev. Lett. 109(7), 073004 (2012). [CrossRef]
22. P. A. Korneev, S. V. Popruzhenko, S. P. Goreslavski, W. Becker, G. G. Paulus, B. Fetić, and D. B. Milošević, “Interference structure of above-threshold ionization versus above-threshold detachment,” New J. Phys. 14(5), 055019 (2012). [CrossRef]
23. C. F. de Morisson Faria and A. S. Maxwell, “It is all about phases: ultrafast holographic photoelectron imaging,” Rep. Prog. Phys. 83(3), 034401 (2020). [CrossRef]
24. S. Skruszewicz, J. Tiggesbäumker, K.-H. Meiwes-Broer, M. Arbeiter, T. Fennel, and D. Bauer, “Two-color strong-field photoelectron spectroscopy and the phase of the phase,” Phys. Rev. Lett. 115(4), 043001 (2015). [CrossRef]
25. G. Porat, G. Alon, S. Rozen, O. Pedatzur, M. Krüger, D. Azoury, A. Natan, G. Orenstein, B. D. Bruner, and M. Vrakking, “Attosecond time-resolved photoelectron holography,” Nat. Commun. 9(1), 2805 (2018). [CrossRef]
26. Y. Wang, S. Yu, X. Lai, H. Kang, S. Xu, R. Sun, W. Quan, and X. Liu, “Separating intracycle interferences in photoelectron momentum distributions by a polarization-gated pulse,” Phys. Rev. A 98(4), 043422 (2018). [CrossRef]
27. P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Attosecond ionization gating for isolated attosecond electron wave pocket and bradband attosecond xuv pulses,” Phys. Rev. A 76(5), 051801 (2007). [CrossRef]
28. M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001). [CrossRef]
29. H. Mashiko, S. Gilbertson, M. Chini, X. Feng, C. Yun, H. Wang, S. D. Khan, S. Chen, and Z. Chang, “Extreme ultraviolet supercontinua supporting pulse durations of less one atomic unit of time,” Opt. Lett. 34(21), 3337–3339 (2009). [CrossRef]
30. X. Zheng, M.-M. Liu, H. Xie, P. Ge, M. Li, and Y. Liu, “Control of photoelectron interference in asymmetric momentum distributions using two-color laser fields,” Phys. Rev. A 92(5), 053422 (2015). [CrossRef]
31. M. Mandrysz, M. Kübel, J. Zakrzewski, and J. S. Prauzner-Bechcicki, “Rescattering effects in streaking experiments of strong-field ionization,” Phys. Rev. A 100(6), 063410 (2019). [CrossRef]
32. D. Arbó, C. Lemell, S. Nagele, N. Camus, L. Fechner, A. Krupp, T. Pfeifer, S. D. López, R. Moshammer, and J. Burgdörfer, “Ionization of argon by two-color laser pulses with coherent phase control,” Phys. Rev. A 92(2), 023402 (2015). [CrossRef]
33. S. Luo, M. Li, H. Xie, P. Zhang, S. Xu, Y. Li, Y. Zhou, P. Lan, and P. Lu, “Angular-dependent asymmetries of above-threshold ionization in a two-color laser field,” Phys. Rev. A 96(2), 023417 (2017). [CrossRef]
34. C.-L. Xia and X.-S. Liu, “Quantum path control and isolated attosecond pulse generation with the combination of two circularly polarized laser pulses,” Phys. Rev. A 87(4), 043406 (2013). [CrossRef]
35. M. Protopapas, C. H. Keitel, and P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys. 60(4), 389–486 (1997). [CrossRef]
36. M. Feit, J. Fleck Jr, and A. Steiger, “Solution of the schrödinger equation by a spectral method,” J. Comput. Phys. 47(3), 412–433 (1982). [CrossRef]
37. H.-D. Zhang, S. Ben, T.-T. Xu, K.-L. Song, Y.-R. Tian, Q.-Y. Xu, S.-Q. Zhang, J. Guo, and X.-S. Liu, “Photoelectron-momentum distribution of $\text {H}_{2}^{+}$ molecules at different internuclear distances in an elliptically polarized laser field,” Phys. Rev. A 98(1), 013422 (2018). [CrossRef]
38. Q. Zhen, S. Q. Zhang, Z. J. Yang, and X. S. Liu, “Cross-channel contributions to vortex patterns in photoionization by a pair of counter-rotating circularly polarized attosecond pulses,” Europhys. Lett. 133(3), 33001 (2021). [CrossRef]
39. M. Li, J.-W. Geng, H. Liu, Y. Deng, C. Wu, L.-Y. Peng, Q. Gong, and Y. Liu, “Classical-quantum correspondence for above-threshold ionization,” Phys. Rev. Lett. 112(11), 113002 (2014). [CrossRef]
40. M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms anf of atomic ions in an alternating electromagnetic field,” Soviet Physics JETP 664, 138–141 (1986). [CrossRef]
41. N. Delone and V. P. Krainov, “Energy and angular electron spectra for the tunnel ionization of atoms by strong loe-frequency radiation,” J. Opt. Soc. Am. B 8(6), 1207–1211 (1991). [CrossRef]
42. N. Shvetsov-Shilovski, M. Lein, L. Madsen, E. Räsänen, C. Lemell, J. Burgdörfer, D. Arbó, and K. Tökési, “Semiclassical two-step model for strong-field ionization,” Phys. Rev. A 94(1), 013415 (2016). [CrossRef]
43. S. D. López and D. G. Arbó, “Holographic interference in atomic photoionization from a semiclassical standpoint,” Phys. Rev. A 100(2), 023419 (2019). [CrossRef]
44. A. S. Maxwell and C. F. de Morisson Faria, “Coulomb-free and coulomb-distorted recolliding quantum orbits in photoelectron holography,” J. Phys. B: At., Mol. Opt. Phys. 51(12), 124001 (2018). [CrossRef]
