Spatial and spectral variations of high-order harmonics generated in noble gases

Generation of the coherent extreme ultraviolet (XUV) radiation through high-order harmonics generation (HHG) in atoms of noble gases is a frequently studied process [1–4]. The use of gas targets is important for the extension of the cut-off energy of generated XUV harmonics. The ionization potential of atoms plays a decisive role in extending the harmonic cut-off to larger energy values that [5] can be useful in studies that investigate water-containing biological systems.

Meanwhile, the restriction in harmonic conversion efficiency does not allow extending the practical applications of HHG [6]. Enhancement of HHG efficiency can be achieved using two-color laser fields, where the harmonics yield correlates with the enhanced probability of ionization [7, 8]. So far most of HHG experiments have been performed in the pure gas media using single-color pump (SCP) scheme. However, there is an increasing interest in the enhanced harmonics generation in gases using two-color scheme. HHG from mixed gases is an another attractive technique that allows for probing the dynamics of this process [9, 10], as well as the enhancement of the conversion efficiency of harmonics [11–14].

The influence of the beam shape on the spectral distribution of the HHG has been analyzed in earlier studies [15]. Generation of intense XUV optical vortices in laser-ablation plumes has been also reported by Singh et al [16]. Resonance-enhanced HHG in the field of structured Gaussian driving pulses (DPs) has been demonstrated. The application of the Gaussian and Gaussian–Laguerre beam distributions might open new routes to analyze nonlinear responses of target medium through the generation of HHG in gases and laser-ablated plasma plumes. Recently, the interest in XUV harmonics of different polarization states has increased due to their special phase and polarization characteristics. These features can be effective in probing magnetic materials, XUV lithography, ultrafast diffraction imaging, and in controlling the orbital angular momentum dichroism of materials etc [17–20]. Manipulating the parameters of the fundamental laser pulses using s-waveplates (SWPs) is a one of the options that can be employed to generate XUV harmonics with radial and azimuthal polarization states. Due to the non-perturbative nature of the HHG process, the generated XUV harmonics are sensitive to the polarization state of the laser field [5]. HHG process consists of three steps that start with tunnelling ionization of electron, its acceleration in the driving electric field, and finally its recombination with parent ion [21]. This scheme produces intense and high-cut-off harmonics compared to the time-structured DP because higher fraction of fundamental pulse energy is utilized for HHG. Additionally, this approach allows maintaining precise control over the spatial properties of the harmonics. The effect of spatially and temporally variable polarization on the intensity and polarization of the generated XUV harmonics was reported in references [22, 23]. It was demonstrated that circularly polarized even and odd harmonics can be produced in gas medium by varying the beam waist of the tightly focused, radially polarized driving beam. The tightly focused radially polarized DPs generated intense harmonics compared to the azimuthally polarized and spiral polarized beams. Moreover, harmonics with non-zero angular momentum (i.e., optical vortices) can be generated using non-uniform spatial beams. Hence, DPs with spatially variable polarization (i.e., vector beams) can be useful in generating and controlling harmonics of varying polarization representing elliptical, circular, vector, and vortex XUV beams.

In this work, we analyze the various dynamics in the spatial distribution of XUV harmonics using single- and two-color pump (TCP) schemes in Ar, Xe, and Kr gas targets. We investigate the influence of single- and two-color fields on the spatial distribution of the generated harmonic spectra. The spectral variation of DPs was controlled using SWP, which allowed the formation of radially- and azimuthally-oriented polarization components of the DPs. Application of the SWP in the TCP scheme of HHG enables the spatial and spectral analysis of the HH spectra generated in Ar and Xe gases, where free-electron-based phase-mismatch may lead to the deduction of the central parts of the harmonic distribution in the divergence-dependent spectrum of the HHG. Clear spectral splitting in the HHG is also observed in the case of TCP scheme of HHG with SWP.

We used an Active Fiber laser system, which provided 37 fs pulses at 50 kHz repetition rate with an energy of 0.6 mJ at central wavelength of λ_ω = 1030 nm (figure 1). The DPs with Gaussian-like spatial distribution were focused into a gas jet (GJ) using the 400 mm focal length spherical lens (FL) to provide the SCP. The position of the GJ with respect to the focal plane of the focusing lens was controlled using a 3D motorized translating stage. The radius of the beam at the focal plane was equal to 50 μm. The maximal intensity of the fundamental beam inside the GJ at these focusing conditions was I_ω = 2 × 10¹⁴ W cm⁻². To modulate the spatial distribution of the DPs, an SWP was used. SWP (Radial/Azimuthal polarization converter, RPC-488-06-56 fabricated in the bulk of UVFS substrate of diameter 25 mm, and thickness of 3 mm, by Southampton, UK) allows for converting the linearly polarized Gaussian beam to radially or azimuthally polarized Gaussian–Laguerre beams. The SWP was installed between the focusing lens and nonlinear crystal. In the case of TCP scheme, the DPs propagated through the barium borate crystals (BBO, type I) of different thicknesses for second harmonic generation (λ_2ω = 515 nm). The SH conversion efficiencies in the case of 0.1, 0.4, and 1.5 mm thick BBO crystals were measured to be 2.8, 8 and 10%, respectively. The intensities of the fundamental and SH DPs in the TCP configuration were equal to be I_ω = 2 × 10¹⁴ W cm⁻² and I_2ω = 1 × 10¹³ W cm⁻². The efficiency for 1.5 mm crystal was almost the same as for the 0.4 mm crystal, since the crystals were inserted on the path of the focused beam. At these conditions, the application of a thicker crystal did not show notably larger conversion efficiency, which is ascribed to the growing influence of the group velocity dispersion in the thick nonlinear crystal.

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The gas pressure in the chamber during the experiment was ∼3.5 × 10⁻³ mbar, while the XUV spectrometer was maintained at ∼1 × 10⁻⁵ mbar. The effective length of GJ was equal to ∼1 mm. The harmonics were analyzed using the XUV spectrometer consisting of a flat field grating (G) and microchannel plate (MCP) detector coupled to a phosphor screen. The installation of gold-coated plane mirror or gold-coated cylindrical mirror (CM) inside the XUV spectrometer allowed for analyzing the divergence of the harmonics or collecting the focused images of harmonics, respectively. The signal of XUV harmonics was detected by MCP and imaged by charged coupled device (CCD camera).

Figure 2 shows the harmonics spectra with maximal cut-offs generated in gases at 1030 nm wavelength of the driving single-color laser field. The HHG process is based on the three-step model, which was developed by Corkum for hydrogen atom with single electron [5]. We observed difference in the cut-off energies of HHG spectra generated in three gases, which is attributed to the dependence of the maximal order of harmonics on the ionization potential of gas atoms. The dependence of maximal order of harmonics on the ionization potential of gas atoms and wavelength of DPs is presented by the equation: E_cutoff = I_p +3.17U_p, where I_p is the ionization potential of the target atom, and U_p (eV) = 9.337 × 10⁻¹⁴ I (W cm⁻²) (λ (μm))² is the electron's ponderomotive energy [24]. In the case of Ar gas, the maximal harmonic was 63rd order at the maximal used intensity of DPs. In the case of Kr and Xe gases, the maximal harmonics were 37th and 29th orders, respectively. All HHG spectra were taken at the same conditions. The gas pressure was equal to 50 torr, and the focusing geometry was similar for all experiments with the three gases.

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A dependence of maximal harmonic order on the intensity of the driving laser pulses was experimentally studied. Figure 3(a) shows the dependence of the harmonic cut-offs in rare gases on the energy of the DPs. At the low intensity of DP, we observed increase in the harmonic order corresponding to the three-step model of HHG. Further increase in the intensity of DP led to the decreasing maximal order of harmonics in Xe gases. This observation can be related to the lower ionization potential of Xe gas and the possibility of phase mismatch of interacting beams at the highly ionized area of the Xe GJ. As an evidence of decreased harmonic order, we present the spatial distributions of the harmonics generated in Xe gas for single-color and two-colors driving laser pulses (see figure 3(b)). The low ionization potential of Xe atoms results in the notable electron dispersion and ionization-induced laser defocusing (see upper panel of figure 3(b)). Here the driving laser pulse was linearly-polarized in cases of SCP and TCP schemes of HHG. The on-axis harmonics decrease due to the effect of propagation and the phase mismatching. Meanwhile, the advantage of Xe atom is a larger re-collision cross section leading to the generation of the bright harmonics in the range of the lowest orders. The application of TCP also allowed for demonstrating the decrease of on axis components of harmonics compared with the off-axis parts. The clear demonstration of the phase mismatching HHG in the case of the TCP also depends on the ionization rate of the gas atoms according to the calculation used by Ammosov–Delone–Krainov theory [7], which is equal to 30% for the orthogonally polarized laser pulses. The high ionization rate in the TCP of orthogonally polarized fields makes it difficult to maintain the phase matching conditions because, under such conditions, the driving field experiences severe self-phase modulation in the ionized medium and decrease of HHG conversion efficiency.

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Spatial and temporal variations of the driving laser beam might change the phase matching and focusing conditions, which can lead to reshaping the plasma responses [25]. The generation of over-dense plasma along the axis of the laser beam can cause defocusing, and phase mismatch conditions for the interacting waves in the high harmonic generation process; and hence reducing the HHG efficiency. Numerical studies of the HHG process in highly ionized Ar gases predicted a lighthouse effect that is correlated with the structure of the driving field in the temporal, spectral, and spatial domains during propagation [26]. An overdriven ionized medium also can be considered optimum medium for the generation of isolated attosecond pulses [27, 28]. Spatially and temporally modified driving laser beams can be used in high pressure and highly ionized gas medium without compromising the efficiency of the HHG. Whereas in the case of Gaussian beams, highly ionized gas medium may create phase mismatch conditions and hence reduce the efficiency. The spatiotemporal wavefront rotation of a driving laser can be optimized in the high pressure and highly ionized medium, which could lead to complete spatial separation of successive attosecond bursts in the far field [26]. Earlier, high intensity (2.3 × 10¹⁵ W cm⁻²) driving Gaussian laser pulses were used with Xe gas targets for spatiotemporal reshaping of the XUV harmonics distributions [29]. In our case and with using TCP scheme at even relatively low intensity (I_ω = 1.2 × 10¹⁴ W cm⁻² and I_2ω = 1 × 10¹³ W cm⁻²) of the DPs, reshaping of the spatiotemporal of XUV harmonics distributions could generate pulses both on-axis and off-axis of the laser beam. Moreover, macroscopic peculiarities related with gas HHG are shown in figure 3(b). One can see the unusual shapes of the divergence characteristics of the harmonics in those raw images of HHG spectra. The appearance of the off-axis components of harmonics is a manifestation of the worsened phase-matched conditions along the propagating axis. Similar findings were reported earlier in a pioneering work of vortex HHG in plasma [16]. Additionally, previous studies using an annular driving laser beam obtained by placing a circular block in the Gaussian laser beam path have shown that the generation of HOHs strongly peaked on the laser axis [30]. Hence, the double-lobe harmonic profile shown in figure 3(b) generated due to the presence of the fixed vertical slit in the path of the XUV-OV beam having a doughnut-shaped profile clearly indicated the generation of the vortex harmonics [31].

On the other hand, Gaussian–Laguerre beam shape in the case of SCP scheme of HHG generated a very weak signal of HHG in these three gases. In this case, generation of HH by Gaussian–Laguerre beam can be considered as a generation of harmonics from circularly polarized laser beams, where the intensity of harmonic depends on the angle of the polarization state of the DP pulses [32]. The application of circularly polarized laser pulses led to complete disappearance of harmonic emission, as it should be assuming the origin of HHG [5]. Here we present the ratio of the harmonic signal for Gaussian and structured laser beams. The spectral variation of HHG was studied by using two-color DPs in three gases. In the case of TCP, the temporal separation of two pulses occurred due to the group velocity dispersion in BBO. It is an important aspect for testing the influence of second field on the output of generated harmonics. We calculated the delay for λ_ω = 1030 nm pulses, with respect to the 515 nm radiation, in the 0.1 mm thick BBO crystal to be ∼12 fs. Correspondingly, the 0.4 mm thick BBO crystal produced 48 fs delay between these two pulses.

Figure 4 shows the harmonics spectra generated using TCP in the case of two crystals. The strong even and odd harmonics were generated in the case of all used gases. We tested the dependence of the spectral distribution of harmonics on the thicknesses (0.1 mm and 0.4 mm) of nonlinear crystals. Almost the same maximal order of the harmonics was generated from gases at the 12 and 48 fs delays between λ_ω = 1030 and λ_2ω = 515 nm pulses. Note that the higher intensities of XUV harmonics were demonstrated in the case of 0.4 mm thick BBO crystal.

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This observation shows that the larger temporal mismatch between pumping pulses in this crystal was compensated by higher conversion efficiency of the second harmonic component. Correspondingly, the stronger λ_2ω = 515 nm field allowed higher conversion efficiency of the lower-order harmonics in the case of TCP of gases. We analyzed the HHG spectra for larger delay (180 fs) between two DPs in the case of second harmonic generation in the 1.5 mm-thick BBO crystal. In this case, when the overlap between λ_ω = 1030 nm and λ_2ω = 515 nm waves becomes lesser than in the case of application of the thinner crystals, the 2(2n + 1) orders of harmonics can be realized even at the significant delays between fundamental DPs and their second harmonics. In other words, those harmonics (H10, H14, H18, etc) can be considered as H5, H7, H9, etc generated from the second field, without the involvement of the fundamental wave. As an example, we present the HHG spectra in three gases using TCP configuration at the intensities I_ω = 1.2 × 10¹⁴ W cm⁻² and I_2ω = 1 × 10¹³ W cm⁻² of the fundamental and second harmonic pulses and at 180 fs delay between them (figure 5). The stronger even harmonics at 2(2n + 1) orders (H14, H18), where n is an integer number and H is the harmonic order, were observed in these gases using orthogonally-polarized two-color laser pulses. The stronger harmonic yield of 2(2n + 1) harmonics at the imperfect temporal overlap between the orthogonally-polarized two-color pulses after propagating through the 1.5 mm thick BBO crystal is attributed to the stronger yield of harmonics from the shorter-wavelength source. Meanwhile, even the small temporal overlap between the pump pulses became sufficient for the two-wave interaction in the GJ leading to the generation of other even harmonics (H12, H16, H20, H22, etc). Those harmonics became notably weaker than the 2(2n + 1) orders (H14, H18) (figure 5). It should be noted that we did not use Gaussian–Laguerre beam shape in the case of TCP with 1.5 mm thick BBO crystal. The reason was strong output of the HHG, which did not allow to see splitting of HHG spectra by recorded images from MCP detector.

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The variation of the spatial distribution of harmonics was examined by the modulation of the spatial distribution of driving fundamental pulses at the wavelength of λ_ω = 1030 nm. To analyze this process, we used a plane gold-coated mirror, which allowed for characterizing the divergence of harmonics generated in three gases like in the case shown in figure 3(b). To analyze the spectral splitting of the HHG spectra we used different distributions of the driving laser pulses by tuning SWP with respect to the polarization state of the driving laser pulses. Below we analyzed the difference in the spectra of HHG generated in Ar and Xe gases, with these two gases having high and low ionization potentials, respectively. Figure 6 shows the divergences of harmonics of the λ_ω = 1030 nm DPs with Gaussian and Gaussian–Laguerre distributions of laser beams. The shape of Gaussian driving beam was modified by using the SWP. The SWP was installed between focusing lens and nonlinear crystal, where we modified spatial distribution of the fundamental pulses. In the case of Gaussian beam (bottom panel of figure 6(a)), the generation of the H11 to 40th orders was obtained in the case of argon. We carried out the study of the divergence of odd and even harmonics at different spatial shapes and polarizations (azimuthally and radial polarized beams, upper and middle panels of figure 6(a)). The same spectra were obtained in the case of HHG in Xe (figure 6(b)).

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We used three different (linear, radial, and azimuthally) polarization states of DPs. The examination of harmonics spectra generated in gases show their different spatial, spectral distributions and conversion efficiencies. In case of radially polarized DPs, the shift of spectral components of harmonics (right panels of figures 6(a) and (b)) was observed. This is ascribed to Gouy phase shift of the different polarized beams. The comparison analysis of HHG using a Gaussian and optical vortex driving laser fields in laser induced plasma on the surface of graphite target was performed by Singh et al [16], driven by an amplified Ti:sapphire laser. It was shown that the intense HHG outputs in these two Gaussian and optical vortex beams can be explained by existing three-step model of HHG. In our case we observed spectral variation of the HHG as a function of the driving beam's shape. Moreover narrower divergence angle of HHG in the cases of the radially and azimuthally polarized laser beams might also be attributed to the Gouy phase shift, which plays an important role in phase matching condition of the beam during nonlinear processes in gases [33]. Based on numerical solution of the three-dimensional macroscopic propagation equation, the gradual splitting of the spectral and spatial profiles of harmonics generated in Ar gas was demonstrated by Zhang et al [34]. This splitting was attributed mainly to the distortion that the driving field suffers during propagation, which consequently affects the phase matching conditions. A scheme for correcting the spectral fluctuations of high-harmonic radiation via spectral splitting of HHG has been developed by Volkov et al [35]. Intensity-dependent double-peak spectral splitting of high harmonics was clearly observed due to the propagation effects in rapidly ionizing medium [36]. Systematic examination of the interference fringe pattern with increasing the energy of the DP and with different phase-matching conditions has been utilized to explain the spectral splitting effect of HHG [37]. The observed high-contrast fine interference fringes in the harmonic spectra near the propagation axis were attributed to the interference between long and short quantum paths. The influence of the long and short trajectories on the spatial, temporal, and spectral structures of the HHG spectrum has been studied with Ti:sapphire laser sources at near-800 nm wavelength [38]. The divergence of the HHG in different beam shapes also varied due to the focusing conditions of the laser driving beams in the GJs. Two other factors could also play important role in the evolution of the intensity profile of the HHG. The first one is the focusing effect of the harmonic field which contributes to the phase shift similar to the Gouy phase shift; and the second is the effect on the interference conditions due to the short- and long-trajectories emissions [39]. In our case we have observed that the spectral split is occurring by using TCP with Gaussian–Laguerre beam shape at much lower intensity of the driving laser pulses contrary to high intensity requirement of Gaussian beam shape of the DPs reported by [26]. We have not observed any spectral splitting of HHG spectrum in the case of azimuthal and linear polarized beams in the two gases.

Our study demonstrates the efficacy of spatially modified driving laser beams in optimizing the efficiency of HHG, and in effectively controlling the phase-matching conditions for HHG based on macroscopic processes. In traditional HHG experiments, the application of Gaussian beam has some advantages in generating high harmonics at the macroscopic level, because of its favourable short-trajectory phase-matching conditions, which is a few millimeters after the focus point, where the source gas cell/jet is usually placed. That helps in producing harmonics of high quality [40]. But, when the propagation distance of Gaussian beam is greater than the Rayleigh distance, the intensity of the beam sharply decreases, which leads to the generation of only few-orders of harmonics. To overcome this constraint, adjustments to the focus area and phase-matching conditions by changing the spatial distribution of the driving beam can be more effective. It is also well noted that Bessel–Gaussian distributed driving beams can be considered a solution for optimizing of the generation of high-order harmonics in gases [41, 42]. Impressive results were recorded using a two-color Bessel–Gaussian beam as DPs in XUV harmonics generation [43]. These results demonstrate the positive influence of using spatially-controlled beams on the generation of high harmonics at the macroscopic level.

In conclusion, we presented the results of HHG studies in rare gases using 50 kHz repetition rate femtosecond laser pulses centered at around 1030 nm wavelength. A decrease in the cut-off energy of HHG based on the ionization rate of the used rare gases during the generation of high harmonics of 1030 nm DPs was demonstrated. In the case of Ar gas, the maximal harmonic was 63rd order (with photon energy 75 eV) at the maximal used intensity of DPs. In the case of Kr and Xe gases, the maximal harmonics were 37th (with photon energy 45 eV) and 31st orders (with photon energy 37 eV), respectively. Moreover, SWP was used to change the shape and polarization of the Gaussian beam. A prominent splitting of the spectral components of the harmonics was observed for radially-polarized DPs. This splitting is explained by Gouy phase shift of the radial polarized beam. However in the case of azimuthal and linearly polarized beams, the splitting of the spectral components of the harmonics was not observed. When TCP scheme is applied with Xe gas, significant changes in the on-axis spatial distribution of the harmonics with E > 16 eV was observed. These changes in the harmonics distribution are attributed to the appearance of large amount of free electrons causing phase-mismatch. Due to the difference in the phase-matching conditions of Gaussian–Laguerre beam, spectral splitting of harmonics spectra was obtained.

The work in this paper was supported by the FRG Grant # FRG19-L-S61 from the American University of Sharjah. GSB acknowledges support from the Uzbek-Belarus Project # MRB-2021-543. RAG acknowledges a support from the European Regional Development Fund (1.1.1.5/19/A/003).

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