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arXiv:2604.19957

High-order harmonic generation in argon driven by short laser pulses: effects of post-pulse propagation and windowing

Aaron T. Bondy, Klaus Bartschat

physics.atom-phquant-phSubmitted Apr 21, 2026• Announced Apr 23, 2026
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Abstract

We present ab initio calculations using the $R$-matrix with time dependence (RMT) method for high-order harmonic generation (HHG) in argon in a short, intense pulse regime. The calculations employ a $6$-cycle $\sin^2$ pulse at $850$ nm with peak intensity $2.3\times 10^{14}$ W/cm$^2$ and, for comparison with the experiment by Guo et al. [J. Phys. B: At. Mol. Opt. Phys. 51, 034006 (2018)], a Gaussian pulse with the same frequency and peak intensity. Both pulse shapes yield the expected harmonic structure in the region above the ionization threshold (approximately $15.82$ eV in $LS$-coupling). The spectra exhibit strong carrier-envelope-phase (CEP) sensitivity. The energy region leading up to the ionization threshold contains spectral features arising from residual coherent dipole oscillations (free-induction decay) that strongly depend on spectral windowing and the post-pulse propagation time. We show that the HHG spectrum, particularly below the ionization threshold, is a defined quantity that depends on analysis choices rather than being a uniquely determined observable. Comparison between theoretical predictions and experimental observations in this energy regime, therefore, requires explicit specification of these parameters.

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We present ab initio calculations using the RRR-matrix with time dependence (RMT) method for high-order harmonic generation (HHG) in argon in a short, intense pulse regime. The calculations employ a 666-cycle sin⁡2\sin^2sin2 pulse at 850850850 nm with peak intensity 2.3×10142.3\times 10^{14}2.3×1014 W/cm2^22 and, for comparison with the experiment by Guo et al. [J. Phys. B: At. Mol. Opt. Phys. 51, 034006 (2018)], a Gaussian pulse with the same frequency and peak intensity. Both pulse shapes yield the expected harmonic structure in the region above the ionization threshold (approximately 15.8215.8215.82 eV in LSLSLS-coupling). The spectra exhibit strong carrier-envelope-phase (CEP) sensitivity. The energy region leading up to the ionization threshold contains spectral features arising from residual coherent dipole oscillations (free-induction decay) that strongly depend on spectral windowing and the post-pulse propagation time. We show that the HHG spectrum, particularly below the ionization threshold, is a defined quantity that depends on analysis choices rather than being a uniquely determined observable. Comparison between theoretical predictions and experimental observations in this energy regime, therefore, requires explicit specification of these parameters.
The amplification of ultrashort extreme-ultraviolet (XUV) pulses in argon in high-order harmonic generation processes is studied by using the time-dependent Schr\"odinger equation in the spin-free one-active-electron and single-atom approximation. We consider a neutral argon atom initially in the valence 3p state and a sufficiently intense two-cycle driving infrared (IR) pulse for the atom to be mainly ionized after the first laser cycle. The correlated dynamics and transitions from the valence 3p to a virtual subvalence 3s state and in the ionized regions are examined by synchronizing a 1.5-fs XUV pulse to the IR pulse. The calculated single-atom gain spectrum (26--45 eV) agrees with recent experimental measurements. We discuss different channels that can be present in the gain process as a function of pulse parameters and through an analysis of the dynamics of the populations in terms of field-free eigenstates. When the XUV pulse is considered at the end of the driving IR field the amplification is due to contributions of stimulated recombination from excited Rydberg and low energy continuum states to the 3s and 3p states of argon. In regions where the IR field is intense, high energy and angular momentum states are populated and the medium can interact with the pulses through bound and continuum states involving parametric transitions, which is further confirmed by studying classical electron trajectories. We discuss how these parametric interactions might be suitable for amplification of photon energies far from the ionization threshold as observed in the experiments.
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We demonstrate Rabi oscillations in nonlinear ionization of argon by an intense femtosecond extreme ultraviolet (XUV) laser field produced by high-harmonic generation. We monitor the formation of Ar2+\mathrm{A}{\mathrm{r}}^{2+}Ar2+ as a function of the time delay between the XUV pulse and an additional near-infrared (NIR) femtosecond laser pulse, and show that the population of an Ar+∗\mathrm{A}{\mathrm{r}}^{+*}Ar+∗ intermediate resonance exhibits strong modulations both due to an NIR laser-induced Stark shift and XUV-induced Rabi cycling between the ground state of Ar+\mathrm{A}{\mathrm{r}}^{+}Ar+ and the Ar+∗\mathrm{A}{\mathrm{r}}^{+*}Ar+∗ excited state. Our experiment represents a direct experimental observation of a Rabi-cycling process in the XUV regime.
The influence of ultrashort low-frequency laser pulses of different intensities and pulse lengths on the dynamics of a resonantly driven atomic system is investigated theoretically. We demonstrate the periodical dependence of the total ionization rate on the appearance time of a strong few-cycle laser pulse. By depleting the excited state, the intense short pulse not only resets the Rabi oscillations, but also introduces a time shift that is proportional to the pulse length. This time shift can be employed to determine the ultrashort pulse length. In the case of ultrashort pulses with lower intensities, we observe side-bands in the two-photon ionization spectrum, whose intensities vary periodically with the appearance time of the pulse. This can be utilized to measure the period of Rabi oscillations. We have developed an S-matrix theory of two-colour ionization which can explain some of the observations of our numerical calculations. Furthermore, we have studied the high harmonic generation spectra as well as the attosecond pulses generated from the high harmonics. We show that the Rabi oscillation can be used to alter the ionization rate within laser sub-cycles and therefore select different quantum trajectories in the high harmonics generation.
We perform detailed calculations of harmonic conversion in a 15-Torr jet of xenon into which a 1064-nm-wavelength 36-ps-pulse-width laser has been tightly focused, so that the peak intensity ranges from 5\ifmmode\times\else\texttimes\fi{}1012{10}^{12}1012 to 5\ifmmode\times\else\texttimes\fi{}1013{10}^{13}1013 W ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}2}$. The single-atom emission rates are obtained by integrating the time-dependent Schr\"odinger equation. We employ an improved atomic model which includes excitation and ionization through states with both low-lying ionic cores. The propagation equations are solved using a general and efficient finite-difference technique. Excellent agreement with experimental data is obtained. We consider the effect of the defocusing of the pump beam by the free electrons in the saturation regime, and find it to be small for the conditions studied here. The spatial, temporal, and spectral profiles of the harmonic emission are presented. Significant blueshifts of the harmonics' spectral line shapes are observed in the saturation regime. Finally, using model polarizations, we discuss how the harmonic fields build up in the nonlinear medium, through a series of interferences. It allows us to understand why efficient phase matching can be achieved in a strong-field regime for the laser-atom interaction.
Two-electron processes can generate high harmonics beyond the conventional single-active-electron cutoff. Motivated by recent experimental evidence of an extended secondary plateau in the helium high-harmonic spectrum [S. Wang et al, Optica, (2023); S. Wang et al, In Print in Nature Photon., (2026)], we present a two-electron generalisation of the strong-field approximation. We analyse the resulting expressions using the saddle-point method and determine the extended cutoff. We find good agreement with classical predictions of cutoff scalings of 4.74.74.7 and 5.55.55.5 times the ponderomotive energy, which significantly exceed the established single-electron scaling of 3.17. We calculate high-harmonic spectra generated via a two-electron process in helium atoms driven by an intense few-cycle infrared laser pulse. Our results demonstrate that the harmonic spectrum extends far beyond the water window, reaching photon energies up to ≈1.2 keV\approx 1.2\,\mathrm{keV}≈1.2keV in the soft x-ray region. The large spectral bandwidth can support the generation of sub-attosecond soft x-ray pulses, which are of particular interest for probing ultrafast dynamics across matter, including applications in core-level spectroscopy and biological imaging.