## Abstract

By numerically solving the time-dependent Schrödinger equation, we theoretically investigate the dynamics of the low-energy photoelectrons ionized by a single attosecond pulse in the presence of an infrared laser field. The obtained photoelectron momentum distributions exhibit complicated interference structures. With the semiclassical model, the originations for the different types of the interference structures are unambiguously identified. Moreover, by changing the time delay between the attosecond pulse and the infrared laser field, these interferences could be selectively enhanced or suppressed. This enables us to extract information about the ionization dynamics encoded in the interference structures. As an example, we show that the phase of the electron wave-packets ionized by the linearly and circularly polarized attosecond pulses can be extracted from the interference structures of the direct and the near-forward rescattering electrons.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The advent of attosecond pulses provides a powerful tool for investigating atomical and molecular dynamics on their natural time scale. Attosecond streaking, which is based on an attosecond pulse serving as the pump pulse and a phase controlled infrared (IR) field as the probe pulse, is an extensively used technique. It is initially proposed to characterize the attosecond pulses as well as the IR field [1–3]. This technique has been also demonstrated to be very useful for tracing and controlling the attosecond electron dynamics in atoms and molecules. In attosecond streaking scheme, the electrons are released by the attosecond pulses and accelerated by the moderate IR laser field. The photoelectron momentum distributions (PEMDs) are shifted by the amount of momentum transferred from the IR field to the continuum electron wave packets (EWPs). Thus, the time information is mapped onto the energy axis of photoelectrons with attosecond precision. The first proof-of-principle implementation is the measurement on the time delay of the atomic photoemission from Neon [4]. These observations have triggered a large number of theoretical investigations of time delay in atomic [5–12] and molecular photoionization [13–15], and photoemission from the solid surface [16–19]. In addition, the feasibility of observing the buildup of the Fano resonance in time domain with attosecond streaking technique has been proved [20–23]. Moreover, the streaking spectrum also encodes information of the electron wave-packets ionized by the attosecond pulses. The streaking technique has also been employed to retrieve the quantum phase information and the phase variation in atomic and molecular ionization [24,25].

In these attosecond streaking experiments, the photon-energy of the attosecond pulse is much higher than the electronic binding potential. The relatively weak IR field only leads to a momentum shift in the PEMDs along the laser polarization direction. For the ionization near threshold, the energy transfer of the low-energy electron is sensitive to the binding potential. Thus, by analyzing the low-energy photoelectron spectrum in the attosecond streaking, the effective binding potential can be extracted [26]. For the photoelectrons with further lower energy in attosecond streaking, they can even be driven back by the IR field and scatter with the parent ions. We should mention that in strong-field ionization by the IR field, electron-ion rescattering is an essential process and it could result in various interesting phenomena [27], such as high-order harmonic generation [28–32], high-order above-threshold ionization [33,34], nonsequential double ionization [35–37] and frustrated tunneling ionization [38,39]. The application of this rescattering process has been deeply explored. For example, the rescattering induced high-order harmonics could be used to generate attosecond pulses [40–42]. Moreover, the rescattering electrons and the resultant photons encode the structural and dynamical information of the parent ion, and thus it has been used to probe atomic and molecular structures and dynamics [43,44]. Fantastically, the rescattering electrons can interfere with the direct electrons in PEMDs, which is referred as strong-field photoelectron holography (SFPH) [45]. This SFPH is expected to be a powerful tool to probe the ultrafast electronic dynamics in atoms and molecules and it has attracted particular interests during the past years [46–49]. In our pervious works, we have demonstrated that the phase of the scattering amplitude can be extracted by the SFPH [50]. More interestingly, we have shown that the charge migration in molecules can be directly visualized by analyzing the SFPH structure [51]. Recently, we demonstrate that the ionization time [52] and the initial momentum of tunneling [53–55] can be probed using SFPH.

In this paper, we focus on the rescattering process in the scheme of attosecond streaking. Here, the rescattering photoelectron is generated by the attosecond pulse through single-photon ionization and it is driven back to the parent ion by the infrared pulse. Different from the rescattering process in strong-field tunneling ionization by the IR field, the creation and the acceleration of the photoelectron are decoupled in attosecond streaking. Thus the ionization and the rescattering processes can be separately controlled. In the past decades, these rescattering electrons and their interference have drawn considerable attention. In the case of attosecond pulse trains, the laser-induced coherent electron scattering has been experimentally observed in the PEMDs [56]. This controlled and coherent scattering enables the time resolved measurements with very high spatial resolution. For the single attosecond pulse (SAP), the rescattering process has also been theoretically studied and its dependence on the strength and wavelength of the IR driven field has been explored [57,58]. Very recently, it has been shown that backward rescattering can be controlled by changing the time delay between the SAP and the IR pulse [59]. It is well known that these rescattered electrons deliver rich information of the atomic and molecular structures and dynamics. Thus, investigations on the rescattering electron in the attosecond streaking scheme, in particularly the interference structures originating from the rescattering electrons, could open a door for time-resolved holographic imaging of the ultrafast electronic dynamics in atoms and molecules.

In this work, we study the interference structures in the PEMD produced by the SAP in the presence of an IR field by numerically solving the time dependent Schrödinger equation. We observe six types of interference structures in the PEMDs. Using the semiclassical model, we show that these structures result from the interference of the direct electrons and different types of the rescattering electrons. The originations of each type of the interference structures are unambiguously identified. Furthermore, we demonstrate that the interference structures could be selectively enhanced or suppressed by changing the time delay between the SAP and the IR field. This enables us to extract information of electrons from the specific interference structures. As an example of the application of interference in attosecond streaking, we show that the interference of the direct and the near-forward rescattering electrons can be used to retrieve the phase of continuum EWPs ionized by the linearly and circularly polarized attosecond pulses.

## 2. Theoretical method

#### 2.1 Numerically solving TDSE

We obtain the PEMDs by numerically solving the two-dimensional (2D) TDSE [60,61]. The TDSE with the single-active-electron (SAE) approximation in the length gauge is given by (atomic unites are used unless otherwise stated)

The electric field $\textbf {E}$ is written as

In our simulation, we numerically solve TDSE using the split-operator spectral method [62] on a Cartesian grid ranging from -600 to 600 a.u.. The time step is fixed at $\Delta t=0.05$ a.u. and the spatial discretization is $\Delta x=\Delta y=0.2$ a.u.. The initial wavefunction is prepared by imaginary-time propagation [63].

#### 2.2 The semiclassical model

We employ the semiclassical model within the strong field approximation [59] to investigate the originations of the interference structures exhibited in the PEMDs. In the attosecond streaking scheme, the continuum EWPs is created by SAP and then its evolution is governed by the IR field. Neglecting the Coulomb interaction, the motion of the EWPs in the IR field is determined by Newtonian equation

In our calculation, the electron wave packet was launched by the SAP during the flat top of the trapezoidal IR pulse. Thus, the subsequent dynamic of the electron wave packet in the IR field could be described by considering the cos-like electric field: $\textbf {E}_{\textrm {IR}}(t)=E_\textrm {IR}\cos (\omega t)\textbf {e}_x$. The momentum $\textbf {p}(t)$ and the position $\textbf {r}(t)$ of the electron released at time $t_0$ are written as and where $\textbf {A}(t)=-\int ^t_{-\infty }\textbf {E}_\textrm {IR}(t')dt'=-E_{\textrm {IR}}/\omega _{1}\sin (\omega _{1} t)\textbf {e}_{\textbf {x}}$ is the vector potential of the IR field. $\textbf {p}_{0}$ and $\textbf {r}_{0}$ are the initial momentum and position of the electron, respectively. It has previously been pointed out that the choice of the initial position must match the asymptotic behavior of the exact Coulomb wavefunction [7]. Here, we select the initial position as the most probable position of the electron of the respective initial quantum state [12]. In our calculation, the initial state is 1S state, thus we choose the zero as the initial position. The initial momentum is determined by the binding potential and the frequency spectrum of the SAP. The ionization time $t_{0}$ is the center of the SAP.The low energy photoelectrons produced by the SAP could be driven back by the IR field and collide with the parent ion. Thus, there are two types of the electrons, i.e., the direct and the rescattering electrons. The direct electron reaches the detector without interaction with the parent ion after ionization. In our calculations, the vector potential of the IR field is zero as the IR field vanishes, $-\int _{-\infty }^{\infty } \textbf {E}_\textrm {IR}(t)dt=0$, and thus the final momentum of the direct electron is

The rescattering electron is driven back by the IR pulse and rescatter with the parent ion. According to Eqs. (6) and (7), the rescattering time $t_s$ is determined byIn the semiclassical model, the interference pattern in the PEMDs could be estimated by considering the classical action along the trajectories [64,65]. Therefore, the classical actions of the direct and rescattering electrons with the final momentum $\textbf {p}_f=p_x\textbf {e}_x+p_y\textbf {e}_y$ are given by

Due to the broad frequency spectrum of the attosecond pulse, the atom could be prompted to high excited state by the SAP. The excited state is subsequently ionized by the weak IR pulse. The phase $S_{\textrm {d}}^{\textrm {IR}}$ of this type of electron is

## 3. Interference structures in PEMDs

#### 3.1 PEMDs obtained by solving TDSE

In Fig. 2, we show the PEMDs for the ionization of model He obtained by solving the 2D TDSE. The center frequencies of the SAP in Figs. 2(a) and 2(b) are 1.02 a.u. and 1.55 a.u., respectively. The 1600-nm IR pulse and SAP are both polarized along $x-$axis with the intensities of $6\times 10^{13}$ and $2\times 10^{14}$ W/cm$^2$, respectively. Here, we select He as an example to demonstrate the interference structure. The large binding potential of He ensures that the tunneling ionization from its ground state by the IR streaking field is negligibly small and could be safely neglected. The PEMDs reveal a two-lobe structure for both cases, which is the characteristic feature of the single-photon ionization from a ground state with S symmetry. The center of the two-lobe is zero due to the zero time delay (as shown in Fig. 1) between the SAP and IR pulse. Figures 2(a) and 2(b) exhibit the complex interference structures, as marked as I$-$VI.

The interference labeled as I displays a spiderlike structure in $p_x>0$ plane. This interference structure is similar to the holographic interference originating from the direct and the near-forward-rescattering electrons in strong-field tunneling ionization by IR or mid-infrared laser fields [45]. The interference pattern labeled as II has an arc-like structure in $p_x<0$ plane with a narrow width. Label III indicates the incoming semi-rings whose density decreases as $|p_x|$ increases, which is similar to the intracycle interference in strong field tunneling ionization [66]. The type IV fringes exhibit as outgoing semi-rings, opposed to type III. The width of the rings and the space between the rings increases as $|p_x|$ increases. This interference structure is similar to the interference between direct and the rescattering electrons launched on two successive quarters of the laser field in strong-field tunneling ionization [67]. In the relatively large momentum region, there is another type of structure labeled by V. This interference also displays as the curved rings structure, similar to type II. However, the width of the rings for type V is much larger than type II, indicating the different originations of types II and V. As the center frequency of the SAP increases, the interference patterns in PEMD become much simpler, as shown in Fig. 2(b). Only the semi-rings interference structure in $p_x>0$ plane is visible, as labeled by VI in Fig. 2(b). These interference structures are quite different from each other, indicating that they originate from different processes.

In pervious works, the central frequency of the SAP is usually far above the ionization threshold and thus the relatively weak IR field only leads to a momentum shift in the PEMDs. While for our situation, the photon energy of the attosecond pulse is close to the ionization threshold, the electrons can be driven back by the IR field and rescatter with the parent ions. In the next section, we will identify the responsible rescattering processes for these interference structures.

#### 3.2 Identify the originations of the interference structures

We employ the semiclassical model to analyze these interference structures in the PEMDs. As mentioned in Sec. 2.2, the EWPs created by the SAP could reach the detector through two different pathways, with (rescattering electron) and without (direct electron) a rescattering with the parent ion. Based on the scattering angles $\theta _{0}$, we classify rescattering electrons into the forward ($\theta _{0}\in [-\pi /2,\pi /2]$) and backward rescattering electrons ($\theta _{0}\in [\pi /2,3\pi /2]$). We calculate the phases of the direct, the forward and backward rescattering electrons with Eq. (12) and Eq. (13). The electrons could return to the parent ion many times and we calculate the electrons where rescattering occurs at the first returning in this case, as indicated by type C in Fig. 1. Figure 3(a) displays the interference between direct and the forward rescattering electrons. It exhibits the spiderlike structure and is located at the $p_x>0$ plane, which coincides with the structure labeled by I in in Fig. 2(a). Therefore, the interference structure labeled by I in Fig. 2(a) originates from the interference between the forward rescattering and the direct electrons. The interference between the backward-rescattering and the direct electrons exhibits the arc-like structure as shown in Fig. 3(b). This interference structure, which is located at the $p_x<0$ and has narrow width, has the same characteristic with the fringe labeled by II in Fig. 2(a), which means that the structure labeled by II originates from the interference between the direct and the backward rescattering electrons. These two types of interference structures are similar to the hologram in strong-field ionization. Thus, we refer to the interference structures labeled by I and II as forward and backward rescattering hologram, respectively. Note that the forward and backward rescattering electrons are classified by the scattering angle $\theta _{0}$, thus the forward-rescattering and the backward-rescattering hologram connect with each other at $\theta _{0}=\pi /2$, as shown in Fig. 2(a). This phenomenon is not easy to be observed in the ionization of an IR pulse due to the small differential scattering cross section for backward-rescattering electrons [67].

Due to the broad frequency spectrum of the SAP, when the central photon energy of the SAP is close to the ionization threshold, the atoms can be excited by the SAP. The weak IR field can ionize the atoms from the excited state. We calculate the phase of the electrons ionized by the IR field from the excited state with the Eq. (14). Figure 3(c) shows the interference between the electrons ionized by the IR pulse and the direct electrons ionized by the SAP. This incoming semi-rings structure, whose intensity decreases as $|p_x|$ increases, coincide with the fringes marked as III in Fig. 2(a), which means that the interference labeled by III originates from the interference between the direct electrons ionized by the IR pulse and the SAP. The interference between the electrons ionized by the IR pulse and the forward-rescattering electrons is displayed in Fig. 3(d). This interference structures exhibit the outgoing semi-rings and the space between the rings increases as $|p_x|$ increases. Comparing this semi-rings structures with the interference labeled by IV in Fig. 2(a) indicates that the type IV interference structures originate from the interference between the electrons ionized by the IR field and the forward-rescattering electrons ionized by SAP. In the pervious work, it was suggested that the interference between the electrons directly ionized from the ground state by a strong IR field and the electrons ionized by a SAP can be used to extract the carrier-envelope phase (CEP) of the attosecond pulses [68]. In these two types of interference structures, the creation and the acceleration of the EWPs are also coupled. It implies that these interference structures have potential to retrieve the CEP of the attosecond pulses.

In Fig. 4(a), we show the relationship between the final longitudinal momentum $p_x$ and the initial momentum $p_{0}$ for the rescattering occurring at the first returning (type C, Fig. 1). The blue and the red lines correspond to the forward-rescattering and backward-rescattering electrons, respectively. Here, we calculate the rescattering angle $\theta _{0}=0^\circ$ for forward-rescattering and $\theta _{0}=\pi$ for backward-rescattering. For the forward-rescattering electron, the final momentum is located at the $p_x>0$, and it is located at $p_x<0$ for the backward-rescattering electron. The final momentum of the forward-rescattering electron increases monotonously with the initial momentum. Whereas for the backward-rescattering electron, the final momentum decreases first and then increases as the initial momentum increases. There are two solutions of the initial longitudinal momentum for the same final momentum and we refer to these solutions as the short and long trajectories. The backward rescattering electrons from these two trajectories can interfere with each other in PEMDs. Figure 4(c) shows this interference structure obtained with the semiclassical model. The wide curved rings structures, which are located at the relatively large momentum region in $p_x<0$ plane, are similar to the interference structure labeled as V in Fig. 2(a), which means that the type V structures originate from the interference between the backward-rescattering electrons from short and long trajectories for the first rescattering time. The width of the rings is broader than other interference structures, which is quite different from the interference between the direct and the backward rescattering electrons as shown in Fig. 3(b).

In the case of the rescattering occurring at the second returning (type D, Fig. 1), the relationships between the final momentum and the initial momentum for forward-rescattering $\theta _{0}=0^{\circ }$ and backward-rescattering $\theta _{0}=\pi$ are shown in Fig. 4(b). Different from the type C, the final momenta of backward and forward rescattering electrons are all in $p_x>0$ plane for type D. Same as type C, there are also two solutions of the initial momenta for the same final momentum which correspond to the short and long trajectories. The interference between the backward-rescattering electrons from these two different trajectories is shown in Fig. 4(d). It exhibits wide width semi-rings structure and is located at the $p_y>0$ plane. Comparing this wide semi-rings structure with the interference structures labeled by VI in Fig. 2(b) indicates that the type VI structures originate from the interference between the backward-rescattering electrons from long and short trajectories for type D.

## 4. Control the interference structures in attosecond streaking

In Fig. 2, different types of interference are mixed in the PEMDs, which hinders us from extracting the information from these interference structures. Therefore, it is necessary to selectively enhance the specific type of interference. Here, we control these interference structures by changing the time delay between the SAP and the IR pulses. Due to the binding potential of H atom is close to the current experiment condition, we choose the H atom as an example to show this method. Figure 5 shows the PEMDs for the ionization of model H atom for different time delays. The center frequency of the SAP is $0.54$ a.u. which is higher than the binding potential of target. The intensity of the IR pulse is $1\times 10^{13}$ W/cm$^2$ which is so weak that the ionization from the ground state by the IR field can be neglected. There are mainly two types of interference structures, i.e., the curved fringes located at the $p_x<0$ and the spiderlike structure located at the $p_x>0$. As analyzed in Sec. 3, the curved fringes are the backward-rescattering hologram and the spiderlike structure is the forward-rescattering hologram. Figures 5(a)–5(d) show that the backward-rescattering photoelectron hologram is suppressed and the forward-scattering holography is enhanced as the time delay $\tau$ between the SAP and the IR field increases. For instance, the backward-rescattering hologram is distinct at $\tau =0$, as shown in Fig. 5(a). Whereas, there is only the forward-rescattering hologram in Fig. 5(d) at $\tau =T_1/4$. This clear interference pattern can be used to retrieve the information of the atoms and electrons.

## 5. Retrieve the phase of the electron wave packets

Here, as an example of the application of the interference structures, we show that the phase of the EWPs ionized by the SAP can be extracted by these interference structures. In pervious works, the phase of the continuous EWPs ionized by the linearly polarized SAP has been extracted by the attosecond streaking technique [24,25]. In this section, we show the phase of the EWPs ionized by the linearly polarized as well as the circularly polarized SAP can be exacted by the forward-rescattering holography.

Figure 6 shows the PEMDs for ionization of model H atom by the linearly and circularly polarized SAP. For the linearly polarized SAP, the PEMDs reveal a two-lobe structure along the polarization direction of the SAP. For instance, when the angle between polarization direction and $x-$axis is $\pi /3$, the lobes distribute along the line of $\alpha =\pi /3$ as shown in Fig. 6(b). Here $\alpha$ is the angle between the direction of final momentum and $x-$axis. Figures 6(c) and 6(d) are the PEMDs for the right-hand and left-hand circularly polarized SAP, respectively. The PEMDs exhibit a ring structure with a momentum radius $p=\sqrt {p_x^2+p_y^2}=\sqrt {2(\omega _2-I_p)}$. These structures are the characteristic feature of single-photon ionization. The amplitude for a single photon ionization $M(\textbf {p})$ can be expressed in the dipole form as

To show the phase jump between the two lobes ionized by the linearly polarized SAP, we employ an IR field to induce interferences in the PEMD. The IR field is polarized along $x-$axis and we change the polarization direction of the SAP. Figure 8 shows the PEMDs for different polarization directions $\theta$ of the SAP. The near-forward rescattering holographic interference structure is clearly visible in the PEMDs. As explained in Sec. 2.2, the phase difference between the direct and the near forward rescattering electrons along the trajectories is $\frac {1}{2}p_y^2(t_s-t_{0})$ [45,65], where $t_{0}$ and $t_s$ are the ionization and rescattering times, respectively. This phase difference could qualitatively reproduce the overall shape of the interference pattern, while it fails in quantitative level. In our pervious study, we have demonstrate that the exact phase difference could be written as [51]

where the first term is the same as that given by the semiclassical model. The second term accounts for the phase due to the interaction of the rescattering electron with the parent ion during the rescattering process. This term has been revealed in our pervious paper [50]. The third term relates to the phases of the continuous EWPs ionize by SAP.In Figs. 8(a) and 8(b), the positions of the interference maxima and minima are exactly symmetric about $p_y=0$. In these cases, the direct and the forward-rescattering electrons are from the same lobe. As $\theta$ increases, the interference minima change to the interference maxima as shown in the black dotted box in Fig. 8(c). The interference maximum at $p_y=0$ changes into a minimum for $\theta =\pi /3$, as shown in Fig. 8(d). In this case, the direct and the forward-rescattering electrons originate from different lobes ionized by the linear polarization SAP. The change of the interference maxima to the minima of the interference structures means $\phi (0;t_{0})-\phi (p_y;t_{0})$ changing from $0$ to $\pi$, which indicates that there is a phase jump of $\pi$ between the two lobes.

Figure 9 shows the PEMD for the ionization of H atom by the right-hand circularly polarized SAP in a linearly polarized IR field. The positions of the interference minima and maxima in the hologram are obviously asymmetric with respect to $p_y=0$. Figure 10(a) shows the PEMD at $p_x=0.3$ a.u. and the results of the linearly polarized SAP are also presented for comparison. We fit the cuts by the function

Here, $f(p_y)$ describes the background contribution and is determined by fitting the logarithm of the PMD by a polynomial. Dividing the fitted function by $e^{-f(p_y)}$, we obtain an oscillating function with a smooth envelope $g(p_y)$, which is again fitted by a polynomial. By construction, the remaining factor $\cos \delta \phi$ is bounded between $-1$ and $+1$ which is shown in Fig. 10(b). More details of this phase extraction procedure were given in our pervious paper [51]. The shift of the positions of the minima and maxima of the circularly polarized SAP is visible. To retrieve the phase of continuous EWPs from the hologram, we extract the phase differences $\delta \phi _{\pm }$ from the PEMDs, as shown in Fig. 10(c). Here, $\delta \phi _{\pm }$ denote $\delta \phi (p_y)$ for $p_y>0$ and $p_y<0$, respectively. As discussed in our pervious work [52], the first and second terms in Eq. (18) for $p_y>0$ and $p_y<0$ exactly equal due to the symmetry of our system. Thus these terms can be eliminated by performing a subtracting procedure The result extracted from PEMDs is shown by the blue circle line in Fig. 10(d). For the atomic system studied here, this phase $\Delta \phi$ should be exactly the same as that in SFA, i.e, the one calculated from the corresponding spherical harmonics, which is shown by the red solid line in Fig. 10(d). It shows that the result extracted from the PEMDs agrees well with SFA, which indicates that the phase of the continuous EWPs ionized by the SAP can be extracted from the near-forward hologram. The small difference between the retrieved results and the phase of the spherical harmonic is due to our procedure of phase extraction. It can be improved in the future study on retrieving more meaningful phase of the continuous electron wave packets.## 6. Conclusions

In conclusion, we have studied the low energy interference structures in attosecond streaking scheme. Different from the high-energy attosecond streaking, the PEMDs calculated by numerically solving the TDSE contain rich interference structures. The originations of different types of interference structures are identified with the semiclassical model. Moreover, we show that these interference structures can be controlled by changing the time delay between the SAP and the IR pulse. As an example of the application of the interference structures in attosecond streaking, we show that the phase of the continuous EWPs ionized by the linearly and circularly polarized SAP can be retrieved from the forward-rescattering hologram. Though for the atomic system studied here, the retrieved phase is just the phase of the spherical harmonics, the concept of our scheme could be applied to retrieve more meaningful phase of the ionized electron wave packet in more complex systems. We should mention that the developed experiment technique, such as atto-COLTRIMS [69], is capable of detecting these interference structures of the low energy photoelectron streaking.

## Funding

National Natural Science Foundation of China (11604108, 11622431, 11627809, 11804233, 11874163); National Key Basic Research Program For Youth (2019YFA0308300).

## Acknowledgments

Numerical simulations presented in this paper were carried out using the High Performance Computing Center experimental testbed in SCTS/CGCL.

## Disclosures

The authors declare no conflicts of interest.

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