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Ultrafast Science / 2022 / Article

Research Article | Open Access

Volume 2022 |Article ID 9834102 |

Mumta Hena Mustary, Liang Xu, Wanyang Wu, Nida Haram, Dane E. Laban, Han Xu, Feng He, R. T. Sang, Igor V. Litvinyuk, "Attosecond Delays of High-Harmonic Emissions from Hydrogen Isotopes Measured by XUV Interferometer", Ultrafast Science, vol. 2022, Article ID 9834102, 6 pages, 2022.

Attosecond Delays of High-Harmonic Emissions from Hydrogen Isotopes Measured by XUV Interferometer

Received28 Jun 2022
Accepted18 Sep 2022
Published09 Nov 2022


High-harmonic spectroscopy can access structural and dynamical information on molecular systems encoded in the amplitude and phase of high-harmonic generation (HHG) signals. However, measurement of the harmonic phase is a daunting task. Here, we present a precise measurement of HHG phase difference between two isotopes of molecular hydrogen using the advanced extreme-ultraviolet (XUV) Gouy phase interferometer. The measured phase difference is about 200 mrad, corresponding to ~3 attoseconds () time delay which is nearly independent of harmonic order. The measurements agree very well with numerical calculations of a four-dimensional time-dependent Schödinger equation. Numerical simulations also reveal the effects of molecular orientation and intramolecular two-center interference on the measured phase difference. This technique opens a new avenue for measuring the phase of harmonic emission for different atoms and molecules. Together with isomeric or isotopic comparisons, it also enables the observation of subtle effects of molecular structures and nuclear motion on electron dynamics in strong laser fields.

1. Introduction

The high-harmonic generation process is a sensitive probe of molecular dynamics and structures [14]. The hydrogen molecule, being the lightest, exhibits the fastest nuclear motion. Also, being the simplest neutral molecule, it allows accurate ab initio quantum mechanical simulations without resorting to severe approximations. Those simulations can be used to understand and validate the experimental results. Moreover, availability of different isotopes adds isotopic comparison as a valuable benchmarking and validation tool. A number of studies with such comparisons have already been performed, including effects of nuclear dynamics on relative HHG yields [5, 6] and tunnel ionization rates in and [7]. The strong field ionization of launches a free electron with in the 1 sσg state. The electron wave packet driven by the laser field accelerates away from the parent ion. At the same time, the nuclear wave packet evolves on the potential energy surface of (see Figure 1(a)). After a certain time delay, the electron wave packet returns to the parent ion upon the reversal of the driving laser field. Its consequent recombination induces an oscillating dipole, which imprints its amplitude and phase on the emitted HHG photons [8]. This ultrafast HHG process occurs in less than half of a laser oscillation period giving the potential to explore sensitive measurement on molecular electron structure and nuclear dynamics known as high-harmonic spectroscopy [913].

The investigation of the nuclear dynamics of hydrogen isotopes was envisioned theoretically [1416] and implemented experimentally [5, 6, 17] by comparing their high-harmonic yields. However, only two studies explored the isotope effects on the high-harmonic phase [18, 19]. The first one estimated the relative phase by measuring harmonic yields in a mixed gas cell with and [19]. The second study relied on the determination of group delay by using the reconstruction of attosecond beating by interference of two-photon transitions (RABITT) [18]. However, ensuring that both isotopes are at the same number density in the HHG interaction region is very challenging. Additionally, a sign ambiguity [19] and large experimental uncertainties of these measurements made it difficult to determine the phase difference and absolute delays accurately [18, 19]. Neither of those studies actually reproduced the measured phase difference theoretically.

To retrieve the phase information, the high-harmonic interferometers either split the driving laser beam into two paths and focus at different locations within a single gas jet that produce two phase-locked HHG photons [20, 21] or use a single driving laser beam in a gas mixture [2, 19]. The first method is limited to a single gas species, while the second has low resolution and issues with determining the sign of the relative harmonic phase. Recently, the phase difference between two atomic species was measured by an all-optical attosecond interferometric technique [22], where the delay between two extreme ultraviolet (XUV) pulses is controlled by a two-segment mirror which can provide the temporal resolution of ~6 attoseconds. Such an interferometer must maintain subcycle stability of its path difference (displacement between the two mirror segments) for XUV frequencies for the duration of the experiment, while this path difference is being scanned. This is an exceedingly difficult task which severely limits practical utility of the device.

Building an interferometer in the XUV region is quite challenging for two reasons: firstly, it is challenging to control the delay of the XUV pulses precisely between the two arms with subcycle precision; secondly, the highly reflective XUV optics is yet to be developed. Our passively stabilized Gouy phase interferometer [23, 24], on the other hand, is an all-optical direct XUV interferometric technique. It does not require calibration of gas pressures to ensure the same number densities. Additionally, it does not require any XUV optics. The technique provides an elegant way to generate two coherent high-harmonic pulses without splitting the driving laser and XUV beams. The two mutually coherent XUV pulses are generated by exploiting the inherent properties (Gouy phase) of a single Gaussian focused laser beam. This method has an unprecedented resolution of ~300 μrad (~100 zeptoseconds) because unlike other interferometric techniques, it requires optical path length stability on the order of Rayleigh length of the driving infrared laser rather than the XUV wavelength.

We apply the technique to investigate the effect of nuclear dynamics on the electron motion in molecular hydrogen by precise measurement of high-harmonic phase difference (and corresponding HHG phase delays) produced in and . Since the ionization potentials of ( eV) and ( eV) are almost identical [5], the difference in the phase accumulated by electron in the continuum is considered to be negligible. However, due to the nuclear mass difference, the evolution of nuclear wave packet while electron propagates in the continuum and then recombines to the ground state may differ substantially. The harmonic intensity from the heavy isotope was shown to be higher compared to the lighter molecule as harmonic emission is sensitive to the nuclear motion [5] though for , the ionization probability is less than for [7]. The aim of this work is to measure a small phase difference of HHG signals and to gain an insight into the correlated electron—nuclear dynamics for the two isotopes of molecular hydrogen.

2. Experimental Methods

The high harmonics are generated by a linearly polarized, 800 nm, 9 fs pulses with a peak intensity of  W/cm2. To extract the relative phase difference between two different gases, our advanced Gouy phase interferometer consists of two spatially separated gas jets along the propagation direction in a single laser focus. One of the gas jets (bottom) is fixed in position at the center of laser focus, while the second jet (top) can move along the laser propagation direction before the focus of the laser beam, as shown in Figure 1(b) (see Supplementary Materials for details).

The Gouy phase modulates the carrier envelope phase (CEP) of a focused laser pulse which results in a phase shift of XUV pulses generated from the two gas jets given by where is the harmonic order and is the Gouy phase difference at two gas jet positions (separated by ). The diameter of the gas jets is 200 μm, and gas pressure in both jets is kept at 100 Torr even though optimal phase matching happens at a slightly higher pressure. The short interaction region and low gas density help to reduce the macroscopic phase matching effects and to minimize the reabsorption of XUV photons by the second gas jet. The gases are supplied from cylinders through individual gas lines. The pressures in each gas jets are controlled by two separate regulated valves and monitored with individual capacitance manometer pressure gauges. The pressures were kept constant throughout the experiments. The gases are switched in the jets by electronically actuated microvalves that allows faster measurement and thus minimizes the errors due to laser fluctuation over time and also compensates for a small difference of intensities (and associated phases) at the two jet positions.

The interference fringes of the high harmonics from H23 to H37 generated with H2 in both jets obtained by varying the separation between the jets are shown in Figure 2(a). To optimize the resolution for all the observable harmonic orders, the isotopic phase difference measurements were performed at jet separations of 0.63 mm and 1.27 mm where the fluctuation of laser intensities is only 0.2% and 0.8%, respectively.

To extract the relative phase difference between the harmonics generated from the two gases, for each of the two jet separations, measurements are performed for two configurations. In the first configuration, and are delivered from the top and bottom jets, respectively. The gases are swapped in the second configuration.

The HHG spectra are recorded for two different jet separations and for two gas configurations to account for systematic uncertainties. For each configuration, three HHG spectra are recorded: for both gas jets activated and for individual jet activated. The measured spectra are presented in Figure 3. The relative phase difference between and is calculated as (see Supplementary Materials for details) where and are the harmonic intensities from individual gas jets of and , respectively, and is the difference in the normalized intensity obtained from the two gas configurations. Here, is the normalized intensity when in top jet and in bottom jet, and is normalized intensity for the opposite gas configuration.

3. Results and Discussions with Theoretical Analysis

The results are presented in Figure 4(a). The left axis corresponds to the phase difference averaged over both separations and both gas configurations. The phase differences measured at gas jet separations of 0.63 mm and 1.27 mm are very similar. This indicates that mixing of gases from the two jets has a negligible effect on our measurements. It should be noted that our method measures both the value and sign of the phase difference, and the two different configurations of gases induce opposite signs. Their absolute values were averaged, and the shown phases correspond to emission from being delayed relative to as our measurements indicate (see Supplementary Materials for details). The corresponding phase delays of harmonics from relative to are depicted on the right axis of Figure 4(a) and are determined by where is the angular frequency of the harmonic order. The harmonics from are found to be ~3 attoseconds delayed in phase with respect to for all the observable harmonic orders.

The experimental measurements are supported by numerical solutions of the four-dimensional time-dependent Schrödinger equation (4D-TDSE) within the single active electron (SAE) approximation [14] (see Supplementary Materials for details). In this model, the electronic and nuclear motions are both confined to a plane containing the molecular vibration and rotation. The simulated results (red crosses) are presented in Figure 4(a), and they agree well with the experimental data (blue stars). Additional simulations showed that is relatively insensitive to the laser intensity, so the laser focal-volume averaging was not included.

According to the well-known three-step model, the total HHG phase includes contributions from the ionization, propagation, and recombination processes. Thus, for molecules aligned along different angles towards the laser polarization direction, the dynamics of ionization and recombination should be different, resulting in the -dependent .

Numerically, by excluding molecular rotations and restricting the nuclear motion to vibrational degree of freedom, we reduced our 4D-TDSE model to a three-dimensional one. We used that 3D-TDSE model to obtain -dependent , presented in Figure 4(b). Only the molecules aligned at small angles (0-30 degrees) from the laser polarisation direction exhibit significant dependence of on harmonic order (energy of recolliding electrons) which could be attributed to two-center interference in the recombination step. Due to the different nuclear masses, the molecular bond elongation between the ionization and recombination steps is different for and . In the recombination step, the complex two-center destructive interference occurs at a specific internuclear distance when the recombining electron momentum meets the condition , and the HHG phase undergoes a sudden change at the corresponding frequency . It is the difference in two-center interference between and that is responsible for the angle and energy dependence of the curves in Figure 4(b). More simulations show that the molecular rotation induced by the laser field results in a phase difference not more than 0.03 radian. A more detailed analysis based on the Lewenstein model and the two-center interference as well as the rotation effect estimation can be found in the supplementary information.

4. Conclusions

In summary, we have demonstrated a novel all-optical Gouy phase interferometric technique for measuring HHG phase difference between two atomic or molecular species. We used this technique to measure the HHG phase difference between and to be about 0.2 radian, corresponding to 3 attoseconds phase delay for all the harmonics. We also simulated this phase difference by numerically solving the time-dependent Schrödinger equation. The theoretical results obtained at the highest level of simulations agree quite well with the experiment. The simulations reveal that depends on the molecular alignment, the bond stretching, and the two-center interference. The phase difference of harmonics from molecular isotopes can be used as a sensitive probe of ultrafast correlated electron-nuclear dynamics in molecules.

Data Availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this article.

Authors’ Contributions

D. L., I. L., and R. S. conceived, planned, and lead the project. M. M and D. L. set up the interferometer and the experiments. M. M. carried out the measurement, analyzed and interpreted experimental data, and improved the analysis and interpretation in discussion with D. L., I. L, and R. S. M. M wrote up experimental parts of this manuscript. L. X., F. H., and W. W. conducted numerical simulations. L. X and F. H wrote up the theoretical sections. N. H and H. X. contributed to important discussion on results and theoretical analysis. All authors contributed in discussions on the results and prepared the manuscript for submission.


The experimental research was supported by the Australian Research Council (LP140100813 and DP190101145) and Griffith University. M. M. was supported by a Griffith University International Postgraduate Research Scholarship (GUIPRS) and a Griffith University Postgraduate Research Scholarship (GUPRS). Theoretical work was supported by the National Key R&D Program of China (Nos. 2018YFA0404802 and 2018YFA0306303), Innovation Program of Shanghai Municipal Education Commission (2017-01-07-00-02-E00034), National Natural Science Foundation of China (NSFC) (Grant Nos. 11574205, 91850203, and 12204308), and Shanghai Science and Technology Commission (Grants No. 22ZR1444100).

Supplementary Materials

Experimental setup, derivation to extract relative HHG phase shift, data analysis, TDSE simulations, the calculation parameters, further analysis based on the Lewenstein model, and estimating molecular rotation effect. Figure S1: schematic of the experimental setup. Figure S2: the calculated HHG spectra for (black line) and (red line) from 4D-TDSE simulation. Figure S3: theoretical analysis for relative phase difference between and calculated by 3D-TDSE. Figure S4: the HHG phase differences between and . References [14, 2428]. (Supplementary Materials)


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Copyright © 2022 Mumta Hena Mustary et al. Exclusive Licensee Xi’an Institute of Optics and Precision Mechanics. Distributed under a Creative Commons Attribution License (CC BY 4.0).

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