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Abstract
We theoretically investigate the coherent control of strong-field high-harmonic generation in the presence of an isolated attosecond pulse. It is found that the rapid modulation of the controlled signal exhibits interference fringe structures in the delay-dependent spectra. By comparing the classical trajectory model with quantum mechanical calculation, it is demonstrated that the fringes are resulted from the interference between the photon- and the tunnelling-initiated recombination pathways. The relative recombination times for the two paths are reconstructed from the interference fringes, which provides a novel scheme for optical observation of the interplay of the photionization and tunneling ionization electron dynamics in attosecond resolution.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Attosecond spectral interferometry, which maps temporal dynamical information onto the final energy distributions, provides an ideal tool for observing the real-time processes in atomic and molecular systems on the electronic natural timescale [1], such as attosecond streaking [2,3], reconstruction of attosecond beating by interference of two-photon transitions (RABBIT) [4] and attosecond transient absorption [5,6]. Limited by the achievable fluences of isolated attosecond pulse (IAP), these attosecond metrology methods usually rely on photoionization processes initiated by attosecond pulses with the assistance of infrared-driving-laser pulses and have been applied to detect the ultrafast dynamical processes including the time delay of photoemission [7–9], electronic correlation dynamics [10,11] and strong-field-induced dipole response [12–14]. The infrared lasers involved in these techniques are generally expected to be strong enough to diffract the continuum electrons or perturb the highly excited states, but at the same time it is expected to be relative weak ($\leq 10^{14} \mathrm {W/cm}^2$) to avoid the induced complexity of ionization of the ground state by the laser field itself. If the intensity of the driving field increases to the strong field regime, the highly nonlinear dynamics involving tunnelling and rescattering processes [15,16] will dominate and complicate the observed results. It will be interesting yet challenging to decode the attosecond dynamical information from the strong-field-assisted spectra. In particular, it provides an opportunity to explore the nonlinear attosecond optics in time domain by combining intense laser pulses with attosecond pulses [17–19].
When the photoelectron is ejected to a strong laser field, the intense driving field can steer the photoelectron back to and recombine with the parent ion. In this recombination process, attosecond pulses are capable of initiating the emission of the bound electrons and leading to two kinds of field-induced recombination trajectories. According to the initial momentum, one is the photoelectron returning from the positive direction of the driving field, the other is the opposite case (shown in Fig. 1(a)). Both cases initiated by the attosecond pulse can be reflected by the recombination spectra [20]. Such attosecond-pulses-initiated recombination (APIR), similar to the tunnelling-initiated high harmonic generation (HHG) process, has been proposed and studied experimentally and theoretically aiming to enhance the yields of HHG [21–25]. In particular, the attosecond pulses precisely synchronized to the intense near-infrared (NIR) laser was found to enhance the intensity of the HHG once per half-cycle of the NIR field [25], implying that the ultrafast dynamical information of the APIR wavepacket can be resolved by the coherent radiation. Furthermore, such scheme is shown to be valuable when the inner-valence dynamics is involved in the HHG processes [26,27].
Fig. 1. (a)Sketch of the APIR trajectories. The ‘Pos’ and ‘Neg’ indicate that the photoelectrons recombine with the parent ion from positive and negative directions of the driving field, respectively. (b)The attosecond-controlled HHG spectra as a function of time delay (the intensity is in natural logarithmic scale). The classical recombination energies vs ejection times are shown in circle lines: green line is from tunnelling-initiated trajectories with zero initial momentum; black and blue are respectively with positive and negative initial momentum for the APIR processes indicated in upper panel. The isolated attosecond pulse is of peak intensity $10^{12} \textrm{W/cm}^2$ with FWHM $\tau _{E}=360$ as, and the intensity of the driving NIR laser is $10^{15} \textrm{W/cm}^2$.
Using Helium atom as an example, the nonlinear coupling of the EUV photons and the NIR photons in the radiative recombination processes is investigated. It is shown that there exists a new type of quantum pathways initiated by attosecond pulses that interferes with tunneling initiated quantum paths of recombination leading to pronounced fringe structures. By varying the time-delay between the two pulses, the modulation of the harmonic yields is found in a time scale of tens of attoseconds and is well explained by the semiclassical trajectory model. The time difference between different recombination pathways are reconstructed from the interference fringes, which suggests a novel scheme for optical observation of the attosecond time-resolved ultrafast physical processes. In order to better distinguish the coherent pathways, the strong NIR field is chosen as short as possible. With the state-of-art techniques, sub-5fs NIR pulses can be readily generated [9,28–30]. In addition, by employing various gating techniques, the effective duration of HHG can be reduced further down to single cycle [30–33]. We will first focus on the dynamics in the combination of IAP and a single-cycle pulse and then compare to that in a three-cycle pulse. As recently shown, the combination of XUV pulse and NIR pulse can lead to the transient reshaping of the emission spectrum of the XUV light (T-REX) [34] and provides new means to control the space–time phase of the extreme ultraviolet emission [35,36]. This work complements the previous attsecond streaking techniques and can be readily employed. It is expected to stimulate more explorations on attosecond nonlinear optics.
2. HHG by an IAP and single-cycle pulses
The theoretical model that we consider is based on the single-atom response calculated by solving the time-dependent Schrödinger equation in the length gauge [37]:
In Fig. 1, the spectra of the single-atom response with the combined strong driving field and delayed attosecond pulse is shown, where the time delay is given by $t_d$ and the positive delay corresponds to the attosecond pulse arriving after the center of the NIR-dressing field. The spectra exhibits a substantial plateau with the cutoff photon energy around 70 eV, which agrees with the characteristics of tunnelling-recombination process induced by the driving field [16]. It is of interest that the delay-dependent spectra exhibit regular interference fringes, and the fringes vary rapidly within the attosecond time delay and over a wide range of energies. This unusual interference fringes, analogous to the multiphoton interference in the attosecond transient-absorption spectra [43], occur mainly in the overlap region of the two pulses, which indicates the existence of the “EUV+nNIR" coupling pathways, where n is the number of participating NIR photons.
To analyze the multiphoton-mediated processes, the classical picture of the APIR induced by the driving field is presented in Fig. 1(a). The attosecond pulses can promote the electron into the continuum with an initial energy $\omega _{E} - \textrm{Ip}$ (where Ip is the single ionization threshold of helium). Then the energy of the free electron is further increased by the NIR driving field, which leads to a possible pathway to radiative recombination. Depending on the initial momentum, the APIR trajectories are mainly classified into two kinds: one is that the electron is ejected with an initial momentum along the positive direction of the field and returns from the positive direction; the other is that the electron is ejected with a negative momentum and returns from the negative direction. The two distinct trajectory pathways are indicated by ‘Pos’ and ‘Neg’ in Fig. 1(a), which can be recorded by the delay-dependent HHG spectra (as shown in Fig. 1(b)) due to the interference with the conventional HHG pathways. The radiative recombination is related to the possible return trajectories.
The results of the classical recombination energies vs the ejection times are represented by the curves in Fig. 1(b). The black and blue curves are calculated for the cases of the electrons initiated with the positive and negative momentum respectively, and the green curve represents the case initiated with zero momentum corresponding to the tunnelling pathway induced by the strong driving field. According to the classical calculation, the returning processes of the electron with positive and negative initial momentum are separated in different initiated time which corresponds to the time delay, and the ionization electron from the Pos process obtains more return energy than that from the Neg process. The delay-dependent trace of the interference fringes in the spectra is found to be followed by the classical curves. As suggested by the classical calculations, the interference fringes in the region from -2 fs to -1 fs are dominated by the interference between the positive APIR pathway and the HHG pathway. When the time delay increases such that the attosecond pulse arrives within the first half cycle of the driving field, the recombination energy from the positive-momentum-initiated trajectory decreases, while that from the negative trajectory initiated by the attosecond pulses begins to rise, which reveals that the ‘Neg’ APIR trajectories contribute to the interference dominantly in the positive time delay region. From the classical picture, the ionized electron is driven back to the parent ion by the NIR field and gain energy during this process. The return energy is determined by initiated time, initial momentum and the waveform of the NIR field. For the negative time delay -0.75fs to -0.25fs, the work done by the driving field starts to vanish leading to less return energy, although the field itself is reaching maximum. At the same time, the tunneling-initiated pathway begins to contribute as shown in Fig. 1(b) and later the Neg APIR starts to dominate.
We now focus on the delay-dependent dynamics. Figure 2 shows the delay dependence of the spectral intensity at different energies. It is found that the intensity is modulated with the time delay in attosecond cycle, as can be seen in Fig. 2(b). The inclined lines are drawn to help compare the timing of the oscillations at different photon energies. The inclined lines indeed correspond to the interference fringes appearing in the spectra (Fig. 1). It can be seen that the change of slopes of the inclined lines coincides with the variation of the dominant APIR pathway which interferes coherently with the tunnelling-initiated pathway.
Fig. 2. (a) Fourier spectra of the controlled HHG intensities along the time delay interval from -2 fs to 1 fs. (b) The delay-dependent intensity signal (integrated over the range of 0.5 eV with the center photon energy) of the HHG spectra. The slope lines are drawn to indicate the interference fringes, and vertical line is drawn to divide the two kinds slope fringes from the ‘Pos’ and the ‘Neg’ pathways.
In order to analyze the modulation of the spectra, we take the Fourier transformation of the delay-dependent spectral intensity from -2 fs to 1 fs as shown in Fig. 2(a). The modulation of the specrtal intensity can be approximated by
where $\mathbf {A}$ is assumed as a slow delay-dependent variation and the intensity mainly relies on the coherence-phase difference $\omega _E t_d + \theta _{E} - \theta _{T}$. The spectral phase of the radiative recombination pathway involves the effects of the initial phase and the evolution phase. For the tunnelling-initiated pathway, the spectral phase $\theta _{T}$ is related to the action of the driving field, which is independent on the time delay. For the APIR pathways, the initial phase changes with the delayed spectral phase of the attosecond pulses, which is considered in the term $\omega _E t_d$. Then the rest delay-independent phase is contained in the term $\theta _{E}$. According to Eq. (4), the delay-independent phases from the two pathway are related to the modulation phase of the interference fringes. By taking the Fourier transformation of the attosecond-pulse-controlled HHG spectra along the time delay, the modulation phase $\phi _{\omega _{E}} \approx \theta _{E} - \theta _{T}$ can be extracted at depending frequency $\omega _{E}$.The coherent spectral phase $\theta _{E}- \theta _{T}$ reflects the synchronization of the strong-field-induced emission processes. It is known that a time-delay in photoemission can be defined as the group delay [44,45] from the derivative of the spectral phase. Here, the relative group delay is given by the derivative of $\phi _{\omega _E}(\omega )$:
which represents the photoemission time difference between the APIR and the tunneling-initiated pathways. Therefore the time-resolved dynamics induced by the strong field can be accessed from the interference fringes of the controlled HHG spectra. To further analyze the relative time-delay for the ‘Pos’ and the ‘Neg’ APIR pathways, we divide the delay-dependent spectra window into two regions with the bold red line (shown in Fig. 2(b)), which are separately analyzed by Fourier transformation within their time window. The obtained spectra are displayed in Figs. 3(a) and (b) and the corresponding relative group delay $\Delta G$ are shown in Fig. 3(c). It reveals that the photon emitted from the ‘Pos’ pathway is about 0.8 fs ahead that from the tunnelling pathway at the photon energy 45 eV, and the time difference is raised to 1.0 fs via a relatively negative chirp when the photon energy increases to 65 eV. Compared to the ‘Pos’ pathway, the photon emission from the ‘Neg’ pathway is occurring nearly at the same time with the tunnelling pathway, which is indicated by $\Delta G \approx 0$ fs over a broad photon energies. One should note that several spikes are caused by the minimum intensities of the HHG spectra, which mislead the achievements of the real values at these photon energies. For the ‘Pos’ APIR pathway, the electron is steered back a half cycle sooner than the tunnelling pathway due to the reversed driving field. For the ‘Neg’ APIR pathway, the initial momentum $\sqrt {\omega _{E}-\textrm{Ip}}$ of the photoelectron released by attosecond pulse photoionization is not much higher than the momentum of the tunnelling electron at the tunnelling exit, which makes the difference of their recombination times close to zero at the same photon energy. The results of the relative recombination time give the direct attosecond time-resolved observation of the recombination dynamics via the pump-probe method.Fig. 3. Fourier spectra of the delay-dependent high harmonic yields for the ‘Pos’ part (a) and the ‘Neg’ part (b), which correspond to the coherent pathways of the positive and the negative APIR trajectories respectively. (c) Group delays $\Delta G_{P}$ and $\Delta G_{N}$ as functions of photon energies, extracted from the respective Fourier spectra of the ‘Pos’ and the ‘Neg’ parts. The relative recombination time differences between the APIR pathways and the tunnelling-initiated pathways are calculated from the classical model and indicated by the solid lines.
To verify the analysis from the interference fringe spectra, the recombination times of the ionization electrons with different initial momentums are calculated by the classical three-step model [16,46]. The tunnelling-initiated trajectory with initial momentum zero corresponds to the tunnelling pathway, whose recombination time is indicated by $t_{r}$. The recombination times of the APIR trajectories with positive and negative momentums are indicated by $t_{P}$ and $t_{N}$ respectively. The time differences between the different pathways are then given by $\Delta T_{P}=t_{P}-t_{r}$ and $\Delta T_{N}=t_{N}-t_{r}$, which are displayed in Fig. 3(c). The results of the classical model are found to agree well with the relative recombination time reconstructed from the interference fringe spectra. It demonstrates that the attosecond dynamical process induced by the strong field can be decoded from the delay-dependent HHG spectra.
3. HHG by an IAP and few-cycle pulses
We now considering the effect of the laser waveform on the HHG processes. The delay-dependent HHG spectra with 3-cycle driving field is calculated, whose waveform is given by Eq. (3) with $\tau _{N} = 3T_{N}$ and laser intensity of $3\times 10^{15} \textrm{W/cm}^2$. The multiple recombination events occur in the few-cycle regime and vary with the CEOP [47,48]. It can be seen from the spectra shown in Fig. 4 that distinguishable interference fringes appear in multiple time-delays. The simulations of the classical recombination energy with the three-cycle NIR field are also given in Fig. 4, which exhibit different curve features for changing the NIR waveform. The classical curves in the panel make the interference fringes correspond to different coherent pathways intuitively.
Fig. 4. (a)Same as in Fig. 1 for the three-cycle NIR field with carrier-envelope offset phases (a) $\theta _C = 0$ and (b) $\theta _C = 0.5\pi$. (c) Waveform of the three-cycle NIR fields.
The relative group delays are extracted from the fringes with different slopes as shown in Fig. 5. For $\theta _C = 0$, the relative group delay $\Delta G$ from the ‘Pos’ APIR pathway presents about half NIR-cycle earlier than the one from the ‘Neg’ pathway in the photon energy range from 50 eV to 75 eV, where two tunnelling-recombination events occur in time sequence (as indicated with green lines in Fig. 4). By comparison with the relative recombination times $\Delta T$, the relative group delays are found to be agreement with the relative recombination time between the APIR pathways and the second tunnelling-recombination pathway, as shown in Fig. 5. It suggests that the tunnelling-initiated pathway by the second recombination processes is the dominant contribution to the interference. The possible cause is the intensity dependence of the tunnelling probability, which leads to a larger HHG contribution of the second recombination process than that from the first.
Fig. 5. Group delay same as in Fig. 3 for the three-cycle NIR field with (a) $\theta _C = 0$ and (b) $\theta _C = 0.5\pi$. The corresponding relative recombination times are also shown in solid lines, where the relative recombination time between the ‘Neg’ APIR pathway and the third tunnelling-initiated pathway (as shown in Fig. 5(b) from 0 fs to 0.5 fs) is indicated by $\Delta T_{N1}$.
In contrast to the case with of $\theta _C = 0$, the two recombination events become one single in the photon energy range from 70 eV to 90 eV for CEOP $\theta _C$ changing to $0.5\pi$, as shown in Fig. 4(b). The isolated tunnelling-initiated pathway makes the interference fringes plain in the energy range, where the relative group delays are close to the relative recombination times $\Delta T_N$ (in Fig. 5(b)). Besides, the delay line also displays a jump-like change below 60 eV, where the relative group delays are advanced by half NIR-cycle. This phenomenon reflects the change of the dominant pathways. In the return energy range below 60 eV, three tunnelling-initiated pathways begin to participate in the interaction according to the classical analysis indicated with green lines in Fig. 4(b). However it can be seen that the third one is the dominant pathway, since the tunnelling process is induced by the strongest part in the central half cycle of the driving pulse. Comparing with the results of the classical analysis in Fig. 4(b), the jump change is shown in accordance with the relative recombination time between the ‘Neg’ APIR pathway and the third tunnelling-initiated pathway.
By comparison of the cases of the different CEOPs, the interference fringes in the delay-dependent HHG spectra with the few-cycle driving field are determined by the dominant interference pathways. The variation of the CEOP may strongly control the recombination processes and create the identifiable pathways. Besides, although using a few-cycle pulse allows to retrieve the interference information as in a single-cycle pulse, but the interference fringes will be blurred by the multiple recombinations. It suggests the “EUV+nNIR" coupling pathway is an unavoidable factor for the rapid sub-cycle modulation of HHG intensity [25], and to study the rapid modulation phenomenon requires the extremely short pulse-shaping techniques [29,30]. Note that the current simulations are restricted to single-atom response and therefore the coherent excitation processes of the medium such as free-induction decay [34,36] are ignored.
The interference fringes, including the phase information about the APIR pathway, have temporal correlation feature with the nonlinear process in attosecond, which reflects the time-resolved characters of the attosecond pulse and the recombination process. The methods can be applied to analyse of the waveform of the isolated attosecond pulse and to coherently control HHG emission process [35,36].
4. Conclusion
In conclusion, we have theoretically studied the strong-field-induced recombination of photoelectron initiated by a time-delayed isolated attosecond pulse. With the time delay varying, the strong-field-induced HHG spectra are coherently controlled by the transient action of the isolated attosecond pulse. It is found that this kind of control contributes to a rapid delay-dependent modulation of the spectral intensity on a timescale of tens attoseconds, which exhibits an interference fringe structure in the delay-dependent HHG spectra. Our study provides more insight into the mechanism of the interference fringe, and the results show that the delay-dependent modulation can be attributed to the interference between the APIR pathways and the HHG pathways. Furthermore, the interference fringes structure will be distinguishable when the isolated recombination process can be prepared by the tailored ultrashort pulse. It provides a means to reconstruct the information of the recombination time difference between the photon- and the tunneling-initiated pathways from the delay-dependent HHG spectra.
Funding
National Key Research and Development Program of China (2019YFA0307703); Major Research Plan (91850201); National Natural Science Foundation of China (11774415).
Disclosures
The authors declare no conflicts of interest.
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