The effect of Coulomb field on laser-induced ultrafast imaging methods

TL;DR

This study reveals that the ionic Coulomb field significantly influences high-order above-threshold ionization spectra, especially near the cutoff, challenging previous assumptions in laser-induced electron diffraction methods and affecting imaging accuracy.

Contribution

It demonstrates the dominant role of 3rd-return-recollision trajectories in HATI spectra and highlights the need to consider Coulomb effects in laser-induced imaging techniques.

Findings

3rd-return-recollision trajectories dominate near the cutoff

Coulomb field alters electron beam characteristics

Implications for improving laser-induced imaging accuracy

Abstract

By performing a joint theoretical and experimental investigation on the high-order above-threshold ionization (HATI) spectrum, the dominant role of the 3rd-return-recollision trajectories in the region near the cutoff due to the ionic Coulomb field is identified. This invalidates the key assumption adopted in the conventional laser-induced electron diffraction (LIED) approach that the 1st-returnrecollision trajectories dominate the spectrum according to strong field approximation (SFA). Our results show that the incident (return) electron beams produced by the 1st and 3rd returns possess distinct characteristics of beam energy, beam diameter and temporal evolution law due to the influence of Coulomb field, and therefore the extracted results in the LIED will be altered if the significance of the 3rd-return-recollision trajectories is properly considered in the analysis. Such Coulomb field effect should be taken into account in all kinds of laser-induced imaging schemes based on recollision.

Full text

As one of the most important processes in strong field physics, recollision has provided an unprecedented insight into the inner working of atoms and molecules CorkumPT2011 ; JPB . In recollision picture, one electron is liberated from the target atom or molecule through tunneling ionization and then be accelerated in the field and pulled back by the field to collide with the parent ion. Most intriguing phenomena in strong field physics, such as high-order above-threshold ionization(HATI), high harmonics generation(HHG) and nonsequential double-ionization(NSDI), can be well understood based on the recollsion physics (see, e.g.,  Proto1997 ; BackerOP2002 ; KrauszPHYS2009 ; BackerPHYS2012 for reviews and references therein).

Since the products upon recollision carry information of the parent ion, they can be used to probe the structure of the parent ion. Various methods have been proposed to image molecules in intense laser fields based on the analysis of different products upon recollision, such as laser-induced electron diffraction (LIED) MmSCI2008 ; BlagaNAT2012 ; XuNC2014 ; PullenNC2016 ; MgNC2015 ; BwSCI2016 , molecular clockHnNAT2002 ; HnNAT2003 ; XTongPRL2003 ; AsPRL2004 , tomographic imaging of orbitals ItataniNAT2004 ; HawsslerPHYS2010 and laser-induced inelastic diffraction (LIID) QuanPRL2017 . Since these methods are self-imaging approaches based on coherent electron scattering, they can provide an unprecedented spatial-temporal resolution. In general, as long as the imaging method is based on electron scattering, the resolution of extracted results will depend upon the parameters of the incident electron beam, such as the beam energy, the beam diameter and the temporal evolution of the beam. Unlike the conventional electron diffraction (CED) method in which the information of the incident electron beam is well known, the incident (return) electron beam in the self-imaging method is produced by the laser-induced ionization of the target itself and its information is much more complicated. In LIED, for example, the temporal resolution relies on the knowledge of the temporal evolution of the beam, and more specifically, deconvolution of the exact recollision time. In the recollision process, however, the electron may miss the parent ion at the 1st return but collide with it at the subsequent returns as illustrated in Fig. 1 (a), which results in a large uncertainty in the recollision time. To complicate matters further, the incident energy and the impact parameter for different returns vary. In LIED, the aforementioned complexity is largely ignored by applying the strong field approximation (SFA) wherein the Coulomb interaction between the parent ion and the freed electron is ignored. According to SFA, the maximal kinetic energy at collision for 1st-return-recollision trajectories is much higher than that for multiple-return-recollision trajectories. Thus, the high energy part of the photoelectron momentum distribution selected for analysis in LIED is assumed to be produced mostly by the 1st-return-recollision trajectories. Furthermore, due to the spread of the electron wavepacket, the probability of the multiple-return-recollision trajectories is much smaller.

Nevertheless, more and more studies have shown that the Coulomb field plays an important role in the ionization dynamics of atoms and molecules in intense laser fieldsBlagaNPH2009 ; QuanPRL2009 ; WuPRL2012 ; QuanREP2016 ; ShafirPRL2013 ; LiPRL2013 . Specially, the Coulomb focusing effect will significantly improve the contribution of the multiple-return-recollision trajectories BrabecPRA1996 ; YudinPRA2001 and in some cases it will even exceed that of the 1st return XTongPRL2003 ; HaoPRA2011 ; JiaPRA2013 in inelastic recollision process. Therefore, the basic assumption on the incident electron beam in the ultrafast imaging method may be in question, and, the impact of the Coulomb field on the imaging results has to be carefully accessed. In this paper, we theoretically and experimentally investigate the plateau in ATI spectrum and a distinct laser pulse duration dependence of the plateau is revealed. Analysis shows that this dependence is closely related to the increasing contribution of the 3rd-return-recollision trajectories due to the effect of Coulomb field. For multi-cycle laser field, this contribution becomes even dominant in the near-cutoff HATI spectrum. Since electron beams associated with the 1st and 3rd returns show distinct time evolution characteristic and impact parameter distribution, using the 3rd-return-recollision trajectories for analysis in LIED will change the imaging results.

In Fig. 1 we present the photoelectron energy spectra calculated with classical-trajectory Monte Carlo method (CTMC) (Fig. 1(b)) and numerical solution of time-dependent Schrödinger equation (TDSE) (Fig. 1(c)), and also the measured spectra (Fig. 1(d)), for Ar atoms exposed to intense laser pulses with different pulse durations at peak intensity of $1.25\times 10^{14}$W/cm2 and center wavelength of 800 nm. The details of the CTMC method QuanREP2016 ; HuPLA1997 ; ChenPRA2000 are presented in the supplementary material. The TDSE is solved using the freely available software QPROP QPROP . In both calculations, the hydrogen-like Coulomb potential is applied. The linearly polarized laser field has a sine-square pulse envelope in the form of $E\left(t\right)=E_{0}\sin^{2}\left(1.14t/\tau_{p}\right)\cos\left(\omega t+\varphi_{0}\right)$, where $\varphi_{0}$ is the carrier-envelope phase (CEP), $\tau_{p}$ is the duration of the pulse. Here the pulse duration is defined as the full width at half maximum (FWHM) of the intensity. The measured results are obtained with CEP unlocked, and correspondingly, in calculation of CTMC each trajectory is calculated with random CEP, and in TDSE the spectra are averaged over different CEPs with regular interval of $\pi/8$.

In our experiment (see supplementary material for details), the laser beam is introduced into the vacuum chamber of a homemade time-of-flight (TOF) photoelectron kinetic energy spectrometer QuanPRA2013 ; LaiJPB2018 with a limited detection angle of 0.026 sr (opening angle of $5{{}^{\circ}}$) in the direction of laser polarization. To compare directly with the measured spectra in Fig. 1(d), only trajectories with momentum direction in the corresponding range of $\theta_{p}<5{{}^{\circ}}$, where $\theta_{p}$ is the angle between the final momentum and the laser polarization, are considered to obtain the CTMC spectra (Fig. 1(b)). For TDSE, in Fig. 1(c) we present the spectra along the direction of the laser polarization, i.e. $\left(\mathbf{dw/dEd\Omega}\right)\left|{}_{\mathbf{\theta=\varphi=0}}\right.$. The ATI peaks in TDSE spectra are smoothed over by averaging adjacent data points to make the variation of the plateau more visible. All spectra in Fig. 1 are normalized to themselves by dividing the maximum of the individual spectrum. The simulated and measured entire spectra (insets in Figs. 1(b)-(d)) exhibit the well-documented ATI spectral features, i.e., a rapid decrease within $2U_{p}$ followed by a plateau extending to $10U_{p}$. If we focus on the dependence of the plateau on pulse duration, a qualitative agreement can be found between simulations (Figs. 1(b), (c)) and measurement (Fig. 1 (d)). The yield of the plateau first increases quickly when pulse duration increases from 5 fs to 11 fs, then only increases slightly until the pulse duration increases to 25 fs 50fs . In Figs. 1(e) and (f), we also present the angle-integrated spectra calculated by CTMC and TDSE to show that the pulse duration dependence is also prominent in the case of high-acceptance-angle with which LIED measurement is performed. This is not surprising, because the majority of photoelectrons will move along the polarization direction in linearly polarized laser field.

In the following we try to understand the pulse duration dependence of the plateau with CTMC approach by taking advantage of the transparent intermediate process that can be explored with this approach. If we employ the Yukawa potential instead of the Coulomb potential in the evolution of electrons after leaving the tunnel exit in CTMC, the spectra become independent of the pulse duration as shown in Fig. 1(g). The Yukawa potential is of the form $V\left(r\right)=-\left(Z^{\prime}/r\right)\exp\left(-r/a\right)$, with parameters $Z^{\prime}=4.547$ and $a=4$ which are chosen to retain the ground-state energy of Ar by TDSE. Therefore, the pulse duration dependence of the plateau comes from the effect of the ionic Coulomb potential.

Next we will show how the Coulomb field affects the plateau of the ATI spectrum. The pulse duration dependence of the plateau in Fig. 1 indicates that some specific multiple-return-recollision trajectories contribute significantly to the plateau. In the CTMC model, the different return recollision trajectories can be distinguished according to the travel time $t_{t}$ defined as the interval between the ionization time and the recollision time. For trajectories with $t_{t}$ in the interval [$(n/2)T,~{}((n+1)/2)T$] ($T$ is the optical cycle), we denote them as the $n$th-return-recollision trajectories HaoPRA2011 . The contributions of different returns to the spectra in Fig. 1(b) are presented in Fig. 2(a)-(c). In the case of the shortest duration of 5 fs (Fig. 2(a)), the 1st-return-recollision trajectories play a dominant role in the plateau, while the contributions of the multiple-return-recollision trajectories can be ignored. When the pulse duration increases to 11 fs (Fig. 2(b)), the contributions of the multiple-return-recollision trajectories increase significantly. The yields of the 2nd and 3rd returns even exceed that of the 1st return. Consequently, the 2nd-return-recollision trajectories becomes dominant in the low energy part of the plateau while the 3rd-return-recollision trajectories dominate the high energy part. When the pulse duration further increases to 25 fs (Fig. 2(c)), the contributions of the multiple returns continue to increase but much more slightly compared with that from 5 fs to 11 fs. The cutoffs of multiple-return-recollision trajectories, especially the 3rd and 4th return trajectories, also increase with pulse duration, which can be attributed to the variation of the pulse envelope. Since electrons are mostly probably ionized around the maximum of the envelope, the laser field of 5 fs pulse will decrease dramatically when electrons come back to collide with the ions due to steep gradient of the pulse envelope. So the energy cutoffs are much smaller than that in the plane-wave laser field BeckerPHYS1994 ; HaoCPL2016 . When the pulse duration increases, the cutoffs increase due to the smaller gradient of pulse envelope. At 25 fs, the cutoff of the 3rd return even becomes equal to the 1st return.

However, according to SFA, trajectories with longer travel time will have smaller probability to collide with the ion due to wave packet spreading. So the multiple-return-recollision trajectories should have smaller contributions to the plateau, which is induced by backscattering, than the 1st-return-recollision trajectories. This can be clearly seen in the results of Yukawa potential in Fig. 2(d), in which the contribution of the 1st return is higher than that of multiple returns. In addition, the cutoff of the 3rd return is significantly lower than the 1st return for Yukawa potential. Therefore, the effect of Coulomb field increases not only the contribution of the 3rd return at the high energy part of the plateau but also the energy cutoff. This is the reason why the plateau exhibits distinct dependence on the pulse duration in Fig. 1.

The dominant contribution of the 3rd return to the spectrum near the cutoff as well as the increase of the cutoff energy can be attributed to modification of the incident (return) electron beam before recollision in the ionic Coulomb potential. The electron beam can be characterized by the temporal evolution of the beam (travel time $t_{t}$), the beam energy (recollision energy $E_{r}$) and the beam diameter (impact parameter $s$) as schematically illustrated in Fig. 3(a). In Fig. 3(c)-(d) we present the CTMC calculated 2D distributions of $E_{r}$ and travel time $t_{t}$ for Coulomb potential and Yukawa potential, respectively, at pulse duration of 25 fs. The four peaks in each plot correspond directly to the four returns, respectively. The distributions in Figs. 3(c) and (d) are similar except for some details. First, the relative contributions of multiple returns to the 1st return are much smaller in the case of Yukawa potential. Second, in Fig. 3(c) there is a peak on the top of the 3rd return (indicated by dashed line box), while it is absent in the distribution of Yukawa potential in Fig. 3(d). With this peak the recollision energy cutoff of the 3rd return even extends to a value very close to the 1st return as shown in Fig. 3(c). This causes the final energy cutoff of the 3rd return to be identical to the 1st return as shown in Fig. 2(c). After making analysis on the trajectories contributing to the peak on the top of the 3rd return, we find that more than $99\%$ of the trajectories always move on one side of the $z=0$ plane (laser field is polarized along $z$ axis) before recollision. The typical 3rd return trajectory is shown in Fig. 3(b). For such a trajectory, the Coulomb force on the electron is always in the same direction as the momentum before recollision, which results in an increase of the momentum at recollision. While for the 2nd return and 4th return trajectories, the electrons leave the tunnel exit on one side of the ions but will return to the ions from the other side, so they have to cross the $z=0$ plane to scatter hardly with the ions to produce high energy electrons. As a result, the Coulomb force will reverse its direction correspondingly, thus the net effect of Coulomb field is negligible. The typical 2nd-return-recollision trajectory is also shown in Fig. 3(b). After making statistics on trajectories contributing to the high energy part of the plateau ($E>5U_{p}$) in Fig. 2(c), we find that almost $100\%$ of the 2nd- and 4th-return-recollision trajectories cross the $z=0$ plane before recollision, while the proportion for the 3rd return is only $50\%$.

In Figs. 3(e) and 3(f) we plot the 2D distribution for the recollision energy $E_{r}$ and the impact parameter $s$ for the 1st and the 3rd return, respectively (see supplementary material for details of calculating $s$). The distributions are normalized to the maximum of the 3rd return to underline the relative contribution between the two returns. The distribution of the 1st return is more concentrated in the high energy region near the maximal return energy but distributes rather uniformly in the impact parameter direction. In contrast, the distribution of the 3rd return is rather uniform in the energy axis but concentrates in the small impact parameter regime. Therefore, the 3rd-return-recollision trajectories are more likely to experience backscattering to produce high energy electrons, and hence its contribution will dominate the ATI spectrum near the cutoff, although the integrated yield of the 1st return in the high recollision energy regime in Fig. 3(c) is considerably larger than that of the 3rd return.

It is noteworthy that laser fields with various wavelengths from infrared to mid-infrared are applied in the LIED scheme to extract time-resolved dynamics of molecules MmSCI2008 ; BlagaNAT2012 ; XuNC2014 ; PullenNC2016 ; MgNC2015 ; BwSCI2016 . In order to give a more complete assessment of the Coulomb effect in the LIED approach, in Fig. 4 we present the CTMC simulated results for Ar at wavelength of 3100 nm and intensity of $1.25\times 10^{14}$W/cm2, which is another kind of typical laser pulse applied in LIED BwSCI2016 . Overall, the distributions at 3100 nm in Fig. 4 are similar to the results at 800 nm. As shown in Fig. 4(a), the 3rd-return-recollision trajectories dominate the high energy part of the ATI spectrum. In Figs. 4(c) and (d), similar to the case of 800 nm, the 2D distribution of $s$ and $E_{r}$ for the 1st return distributes uniformly in the impact parameter direction, while the distribution for the 3rd return concentrates in the small impact parameter regime. However, upon a careful inspection, there are some differences between the 800 nm and 3100 nm cases. First, the peak on the top of the 3rd return in Fig. 3(c) disappears in Fig. 4(b). This causes the ATI spectrum cutoff of the 3rd return obviously smaller than the 1st return. As a consequence, a two-step, i.e., two-cutoff structure arises on the spectrum in Fig. 4(a). It should be noted that this two-cutoff structure may not be observed in experiment for two reasons. (i) The signal of the second cutoff is too low for experimental observation; (ii) The realistic cutoff in a quantum system would not be as sharp as in the classical simulations, which makes the two cutoffs hard to be distinguished. Second, the difference between contributions from the 1st return and the 3rd return to the spectrum at 3100 nm is much greater than that at 800 nm, which can be clearly seen by comparing Fig. 4(a) with Fig. 2(c). These differences can be attributed to the weaker effect of Coulomb field at 3100 nm. When wavelength increases, the quiver distance increases quickly but the effective radius of the Coulomb field keeps unchanged, thus the piece of the trajectory affected by the Coulomb field decreases. In addition, the electron momentum also increases and becomes harder to be disturbed by the Coulomb field. Since the peak on the top of the 3rd return in Fig. 3(c) is induced by Coulomb field effect, its absence at 3100 nm is not surprising. The difference of relative contributions between the 1st and 3rd returns can be interpreted with the help of the so-called defocusing effect KBM2016 ; WangPRA2017 , namely, stronger focusing leads to stronger deflection of the electron at the 1st return, which causes the electron hard to come back to the core at the subsequent returns. Since the Coulomb focusing effect is more prominent for 800 nm, the contribution of the 3rd return is suppressed due to the accompanying enhanced defocusing effect.

Based on the above, we believe that the 3rd-return-recollision trajectories, instead of the 1st ones, should be used for analysis in the LIED scheme. In this case, the imaging results will be affected since the travel time for the 3rd-return-recollision trajectories is about one optical cycle longer than that of the 1st return. Then the extracted result is actually the molecular structure at the time about $7/4$ T instead of $3/4$ T after ionization, which is crucial if LIED is applied to investigate the time evolution of the molecular structure after ionization.

In conclusion, we perform a joint investigation on the dependence of ATI spectrum on the pulse duration theoretically and experimentally. The results indicate that the 3rd-return-recollision trajectories dominate the high energy part of the ATI spectrum due to the ionic Coulomb field effect, which invalidates the key assumption of SFA applied in LIED. The incident (return) electron beams associated with different returns show distinct characteristics in the presence of the Coulomb field. Compared with the 1st return, the 3rd return generates an electron beam with a much smaller diameter and a much higher intensity, thus providing more high energy photoelectrons. Moreover, it is found that the Coulomb field will also increase the cutoff energy of the 3rd-return-recollision trajectories, although this effect will become weaker with increasing wavelength. The above Coulomb effect will change the results extracted from the LIED approach and should be taken into account in current imaging schemes based on recollision physics.

This work is supported by the National Key program for S&T Research and Development (No. 2019YFA0307700 and No. 2016YFA0401100), the National Natural Science Foundation of China (Nos. 11504215, 11774387, 11874246, 11834015, 11974383), the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDB21010400), and the Science and Technology Department of Hubei Province (No. 2019CFA035).