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Abstract
High-order harmonic generation (HHG) provides scalable sources of coherent extreme ultraviolet radiation with pulse duration down to the attosecond time scale. Efficient HHG requires the constructive interplay between microscopic and macroscopic effects in the generation volume, which can be achieved over a large range of experimental parameters from the driving field properties to those of the generating medium. Here, we present a systematic study of the harmonic yield as a function of gas pressure and medium length. Two regimes for optimum yield are identified, supporting the predictions of a recently proposed analytical model. Our observations are independent on the focusing geometry and, to a large extent, on the pulse duration and laser intensity, providing a versatile approach to HHG optimization.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High-order harmonic generation (HHG) in gases [1,2] is often used for producing pulses in the extreme ultraviolet (XUV) range and with ultrashort duration, down to the attosecond range. Strong efforts are being made to increase the generated photon flux, which is required for a range of applications including time-resolved ultrafast dynamics [3,4], as well as coherent imaging [5,6], metrology [7] and XUV nonlinear optics [8]. HHG is the result of an interplay between the response of a single atom to the strong driving laser field and the phase-matching between waves emitted by individual atoms in the macroscopic medium [9–11].
The single-atom response can be optimized to some extent. Short laser pulses [12,13] allow atoms to experience a higher intensity before being ionized, which leads to a higher efficiency, since the single atom response increases with intensity [14]. The laser wavelength [15] and atomic gas [16] should be chosen depending on the targeted photon energy range. Heavy atoms like xenon or krypton and/or short laser wavelength [17] lead to high efficiency over a reduced photon energy range. In contrast, neon or helium and/or long laser wavelength [18,19] produce long harmonic plateaus with poor efficiency. Finally, multicolor schemes [20] often lead to an increase of the conversion efficiency (CE), requiring, however, more complex optical setups.
From a macroscopic perspective, the wave propagation equation describing the generation of harmonics in gases is globally invariant under a scaling transformation [21–23], in which the longitudinal coordinate and the square of the radial coordinate scale inversely to the atomic density. Thus, a conceptually simple way to increase the generated harmonic pulse energy is to drive the process with highly-energetic laser pulses in a loose focusing geometry and in a long and low-density gas target. This has motivated the construction of long HHG setups, including the Intense XUV Beamline at the Lund Laser Centre [24,25] and the recently commissioned GHHG (Gas HHG) Sylos Long Beamline at the Extreme Light Infrastructure – Attosecond Light Pulse Source (ELI-ALPS) facility [26,27].
Optimizing the HHG process is a multiple-parameter problem. Besides the parameters affecting the single atom response, such as the laser wavelength and intensity, or the atomic system, phase-matching depends on the medium’s length and geometry, the atomic density, and the laser focusing geometry. In addition, HHG depends on the position of the gas medium relative to the laser focus [28,29]. Systematic HHG optimization studies are difficult and therefore rare. In the theoretical work of Weissenbilder et al. [30], a new analytic model supported by numerical simulations shows that the CE is optimized when the gas pressure and gas medium length are related by a hyperbolic equation [31], whose parameters depend on the absorption cross-section and polarizability at the generated frequency. In this work, we present experimental measurements performed both at the GHHG Sylos Long Beamline of ELI-ALPS and at the Intense XUV Beamline at the Lund Laser Centre. The harmonic yield obtained in argon using an $800\,\textrm {nm}$ femtosecond laser, is measured as a function of the pressure in the medium, for a number of gas cell lengths. We compare our results with numerical simulations and the predictions of the analytical model presented in [30]. We identify two phase-matching regimes corresponding to the vertical and horizontal branches of the hyperbole and study the properties of the radiation in the two cases.
2. Simulations
Results of simulations are presented in Fig. 1 for three harmonics ($q=19,\; 23,\; 27$) generated in argon with a laser pulse of $800\,\textrm {nm}$ central wavelength and $20\,\textrm {fs}$ duration, at two different laser intensities. The laser focus is located at the center of the gas medium. The CE is plotted in colors as a function of gas pressure and medium length. The numerical calculations are based on the method presented in [9,30]. Briefly, it consists in solving the propagation equation for the fundamental and harmonic fields, within the slowly-varying and paraxial approximations. Tabulated single-atom data are obtained by solving the time-dependent three-dimensional Schrödinger equation, for a large number of peak intensity values (about five thousand) [32].
Fig. 1. Simulated conversion efficiency as a function of argon gas length and pressure. Driving wavelength centered at $800\,\textrm {nm}$, pulse duration $20\,\textrm {fs}$, laser intensity equal to ${1\times 10^{14}}$ W/cm$^2$ (a) and ${2\times 10^{14}}$ W/cm$^2$ (b). The dashed lines represent the phase-matching hyperbole equation calculated with $\varsigma =3$ (white) and $\varsigma =1$ (blue).
The simulations are performed for a certain focusing geometry. However, as shown in Heyl et al. [23], the propagation equation describing the generation of high-order harmonics in a gas medium is globally invariant under a transformation consisting in scaling the longitudinal coordinate by a factor $\eta$, the transverse coordinate by $\sqrt {\eta }$, and the pressure by $1/\eta$. The results presented in Fig. 1 become independent of the geometry, if the pressure $p$ is replaced by the product $p z_R$, $z_R$ denoting the Rayleigh length, and if the length of the medium is indicated in units of $z_R$.
The optimal conversion efficiency follows a hyperbolic behavior, as discussed in [30]. Figure 1 shows that the relative importance of the vertical and horizontal branches of the hyperbole depends on the laser intensity and process order. The vertical branch can be attributed to transient phase-matching, requiring high laser intensity, such that the influence of the free electrons during a short time interval compensates for the neutral atom dispersion. Since both contributions are proportional to the pressure, the conversion efficiency is only weakly pressure-dependent, thus leading to a vertical branch. Absorption leads to an optimal (short) interaction length.
The horizontal branch corresponds, on the other hand, to a phase-matching regime requiring a lower ionization degree, reached earlier in the pulse. In this case, the free electron contribution can be neglected and phase-matching is achieved at relatively low pressure, when the neutral atom dispersion approximately cancels the effect of the Gouy phase. Since the latter contribution does not depend on pressure and the neutral atom dispersion is proportional to it, phase-matching becomes strongly pressure-dependent, thus leading to a horizontal branch. As a consequence of the low pressure, a long medium length is needed to achieve high efficiency.
Interestingly, for the $27^{\textrm{th}}$ harmonic, besides the main hyperbolic behavior, interference fringes can be observed, which also follow a hyperbolic form. These can be interpreted as Maker’s fringes [22,33], corresponding to CE oscillations during the macroscopic build-up of the harmonic field, due to imperfect phase-matching. In addition, some weak vertical interference fringes can be observed for low-order harmonics.
The hyperbolic behavior can be parametrized using a one-dimensional phase-matching model described in detail in [30]. The equation of the hyperbole, represented in Fig. 1 as dashed lines, reads
where,- • $p_0$ is the minimum phase-matching pressure at zero ionization degree (see below).
- • $L$ is the medium length.
- • $\varsigma$ is a factor that depends on the ratio of the coherence length and the absorption length. The white dashed line is obtained with $\varsigma =3$, which means that the harmonic intensity reaches its maximum at $\sim$ 4 times the absorption length (see Fig. 1 in [10] and Fig. 5 in [30]). The blue dashed line corresponds to $\varsigma =1$, which describes better the behavior of the high-order harmonics for which the absorption length ($\propto 1/\sigma _{\textrm{abs}}$) is 10 times higher than for the low orders (see Table 1).
- • $k_B$ is the Boltzmann constant and $T$ the gas temperature (assumed to be room temperature).
- • $f_i$ is a small correction factor which takes into account the position of the laser focus in the generation medium. We use in general $f_i=1$, except for medium lengths larger than $z_R$, where we account for the fact that the coherent build-up will take place mostly at the end of the medium, i.e. after the laser focus. This leads to slightly decreasing (instead of horizontal) asymptotes.
- • $\sigma _{\textrm{abs}}$ is the absorption cross-section, indicated in Table 1 for the harmonics investigated in the present work.
Table 1. Absorption cross-section $\sigma _{\textrm{abs}}$, polarizability difference $\alpha _0-\alpha _q$ and minimum phase-matching pressure $p_0z_R$ for several harmonic orders in argon.
3. Experimental setups
In order to test the universality of the model, we performed experiments on two different HHG beamlines, the GHHG Sylos Long Beamline at the ELI-ALPS facility and the Intense XUV Beamline at the Lund Laser Centre, over a wide range of gas medium lengths. The setups, sketched in Fig. 2, are both based on a loose focusing geometry in order to maximize the volume of interaction in the gas and drive the generation process with multi-mJ laser pulses.
Fig. 2. Schematic of the GHHG Sylos Long Beamline of ELI-ALPS (a) and the High-Intensity Attosecond Beamline at the Lund Laser Centre (b).
3.1 GHHG Sylos long beamline at ELI-ALPS
The results presented are those of the first user experiment of the Sylos GHHG Long Beamline [26,27] at ELI-ALPS, which is schematically depicted in Fig. 2(a). The beamline is designed to accommodate a very loose focusing geometry (up to $55\,\textrm {m}$ focal length), while allowing large flexibility in the XUV generation parameters. During the experiments, the generation process is driven by the Sylos Experiment Alignment (SEA) laser [34] providing $23\,\textrm {mJ}$ of pulse energy, $12\,\textrm {fs}$ pulse duration at $10\,\textrm {Hz}$ repetition rate. The central wavelength is $825\,\textrm {nm}$. The beam is focused by a deformable mirror (DM) into a variable-length cell filled with argon gas. The focal length used is equal to $19\,\textrm {m}$, leading to a measured focal spot size around $600\,\mathrm {\mu }\textrm {m}$ diameter (FWHM). The generated XUV radiation propagates through a $200\,\textrm {nm}$-thick Al filter and is analyzed by an XUV spectrometer.
The variable-length cell consists in two plates, one movable and one fixed, with pre-drilled pinholes for entrance and exit of the laser beam. The length of the cell can be varied continuously between $32.5\,\mathrm {cm}$ and $72.5\,\mathrm {cm}$. The measured Rayleigh length is approximately $z_R=$ $40\,\mathrm {cm}$, meaning that the gas cell length can be varied from 0.81 $z_R$ to 1.81 $z_R$. In this case, the gas pressure is directly measured in the cell. Additionally, four shorter cells were tested, with lengths $2.5\,\mathrm {cm}$ (corresponding to 0.06 $z_R$), $4\,\mathrm {cm}$ $\left (0.1\, z_R\right )$, $8\,\mathrm {cm}$ $\left (0.\, 2 z_R\right )$ and $16\,\mathrm {cm}$ $\left (0.4\, z_R\right )$. Here, the gas pressure in the cells cannot be directly measured but is estimated from the backing pressure (measured at the inlet gas tube) using a computational fluid dynamics package (Simcenter FloEFD) [35]. The focal spot is centered in the gas medium while using the short cells. In contrast, while increasing the length of the long cell the center of the medium moves away from the laser focal spot leading to a small offset between them. The center of the $1.3\, z_R$-long cell is estimated to be around $0.1\, z_R$ after the IR focus. For shorter lengths, this offset is smaller and can be neglected. We verify that a $0.1\, z_R$ offset does not substantially change the prediction of the simulations. Its effect is included in the hyperbolic equation through the parameter $f_i$.
3.2 Intense XUV beamline at the Lund Laser Centre
The Intense XUV Beamline [24,36] is driven by a multiterawatt Ti:Sapphire laser system, delivering laser pulses with $40\,\textrm {fs}$ duration, $800\,\textrm {nm}$ central wavelength and $50\,\textrm {mJ}$ pulse energy at $10\,\textrm {Hz}$ repetition rate. As depicted in Fig. 2(b), the laser pulses are focused in a pulsed argon gas cell using a combination of a deformable mirror (DM) and a focusing mirror of focal length $8\,\mathrm {m}$. The DM allows for wavefront correction and for control of the laser focus position in the gas medium. A motorized iris can be used to optimize HHG. The XUV radiation is separated from the remaining IR by a 10 degrees grazing incidence reflection on a fused silica (FS) plate and through spectral filtering by a $200\,\textrm {nm}$ aluminum foil. Further, the XUV beam is focused using a Wolter-like assembly of two toroidal mirrors (Thales SESO), designed to image the generation position with minimal spatial aberrations [37]. The radiation further propagates after the focus and is analyzed by a homemade flat-field spectrometer, based on a Hitachi aberration-corrected concave grating, that images the XUV focus in the vertical direction (the dispersion direction) onto a microchannel plate (MCP), while the far field spatial profile is recorded along the horizontal axis. In the experiments presented here, the Rayleigh length is kept fixed at approximately $z_R=$ $30\,\mathrm {cm}$ for three different gas cell lengths: 1, 6 and $11\,\mathrm {cm}$, corresponding to $0.03\, z_R,\; 0.2\, z_R$ and $0.37\, z_R$. The gas density in the cell is estimated ex-situ with an interferometric setup by measuring the shift of interference fringes as a function of the opening voltage of the gas valve for a given opening time [38].
4. Results
In order to present the results independently of the specific focusing geometry, we show the quantity $pz_R$ in units of mbar cm (equivalent to Pa m) and we express the medium length in units of $z_R$. The data analysis focuses on the study of the harmonic yield’s dependency on gas length and pressure (section 4.1), harmonic order (section 4.2), laser intensity (section 4.3) and on the harmonic spatial profiles (section 4.4).
4.1 Dependence on gas length and pressure
In Fig. 3 (a), we show the normalized yield of harmonic 19, plotted as a function of the gas cell pressure for several medium lengths. The yield is obtained integrating the harmonic signal within the recorded spectrum and is averaged over multiple acquisitions (about 1000) for each gas cell pressure. As mentioned in Sec. 3.1, the focal spot is centered in the gas medium for short cell lengths, while there is a small offset ($< 0.1\, z_R$) for the long ones. For a given cell length, we vary the pressure and we normalize the results to the maximum yield, corresponding to the dark red color in the figure. Since the results for each length are obtained in different sets of measurements, often on different days, we do not compare the relative strengths between different lengths. The maximum yield follows the expected phase-matching hyperbole, indicated by the dashed orange line, obtained from Eq. (1) with the parameters reported in Table 1 and the phase-matching factor $\varsigma = 3$. Two different regimes are clearly distinguishable: a vertical branch for short cell lengths and a large gas pressure range in which efficient generation takes place and a horizontal branch for long cell lengths, with a narrow gas pressure range. Note that, here short and long cell lengths are always defined with respect to the specific Rayleigh length $z_R$ of the experimental setup. Figure 3 (b) compares the variation of the harmonic yield as a function of pressure, for medium lengths equal to $0.06 z_R$ (blue) and $0.85 z_R$ (violet). Short medium lengths lead to a much broader pressure distribution, with significant fluctuations. In contrast, the distribution obtained with a long medium length is much narrower and smoother. The fluctuations observed for the short medium length might be due to changes in the laser intensity in the case of the transient phase-matching regime (vertical branch). In contrast, the harmonic yields obtained in the regime corresponding to the horizontal branch are much less affected by laser fluctuations.
4.2 Influence of harmonic order
A similar behavior is found for all the harmonics, as shown in Fig. 4. The data points in each plot indicate the gas pressure leading to the maximum harmonic yield for a specific medium length. Here, we show results obtained both at the GHHG Sylos Long beamline (circles) and at the Intense XUV Beamline of the Lund Laser Centre (triangles). Both data sets follow the hyperbolic trend given by Eq. (1). The orange (blue) dashed line is calculated assuming the phase-matching factor $\varsigma = 3$ ($\varsigma = 1$). The experimental data for the low order harmonics follow the orange hyperbole, while for harmonics 27 and 29 the data are in between the blue and orange hyperbole. The argon absorption cross-section changes abruptly between the $25^{\textrm{th}}$ and $29^{\textrm{th}}$ harmonic, as shown in Table 1. This leads to an abrupt change in the ratio between the absorption and coherence length, which explains why the experimental points are found to follow a hyperbole with lower eccentricity (between the orange and the blue hyperbole). This comparison confirms the universality of the proposed model and its independence on the specific focusing geometry and pulse duration.
Fig. 3. Integrated signal of harmonic $19^{\textrm{th}}$ measured as a function of the gas pressure for several medium lengths (a). The orange dashed line indicates the hyperbolic model. Comparison of gas pressure range for optimal signal between two different gas lengths (b). The data have been acquired at the GHHG Sylos Long beamline.
Fig. 4. Maximum harmonic yield conditions for orders $19^{\textrm{th}}-29^{\textrm{th}}$ generated in argon gas. Dataset from GHHG Sylos Long beamline (circles) and Intense XUV Beamline (triangles). The photon energies indicated are calculated for the GHHG Sylos Long beamline and it should be multiplied by 1.025 for the results obtained at the Intense XUV Beamline. Dashed lines indicate the hyperbolic model with $\varsigma = 3$ (orange) and $\varsigma = 1$ (blue).
4.3 Influence of laser intensity
A unique feature of our phase-matching model is the independence from, not only the focusing geometry, but also the laser intensity. In contrast, the high-order harmonic generation process, including the conversion efficiency, is highly dependent on the laser intensity (see Fig. 1). Figure 5 presents the maximum yield for harmonic 23 as a function of gas pressure and medium length for different driving pulse energy values: $8\,\textrm {mJ}$ (circles), $10\,\textrm {mJ}$ (squares) and $12\,\textrm {mJ}$ (diamonds). The corresponding intensities are estimated to be in the range from ${1\times 10^{14}}$ to ${3\times 10^{14}}$ W/cm$^{2}$. All other experimental parameters are kept constant. The figure shows that all three data sets are in good agreement with the expected behaviour given by Eq. (1) and plotted as orange dashed line, confirming the independence on the laser pulse energy and the universality of the model.
Fig. 5. Yield of harmonic 23rd at three different driving pulse energy values: $8\,\textrm {mJ}$ (circles), $10\,\textrm {mJ}$ (squares) and $12\,\textrm {mJ}$ (diamonds). Orange dashed line indicates the hyperbolic model. The data has been acquired at the GHHG Sylos Long beamline.
4.4 Spatial profile of harmonics
The different phase-matching regimes in the two branches of the hyperbole lead to different spatial profiles for the generated harmonics, with a stronger off-axis contribution for the vertical branch than for the horizontal branch possibly due to the contribution of long trajectories and off-axis phase matching [30].
Figure 6 shows spatial profiles in the far field for three different harmonic orders ($q = 19,\; 23,\; 27$) and two gas medium lengths: 0.06 $z_R$ (vertical branch) and 1.33 $z_R$ (horizontal branch). The top figures show simulations whereas the bottom ones present experimental results. For the simulations, a laser intensity value of $1.9\times 10^{14}$ W/cm$^{2}$ is used. The experimental spatial profiles are obtained from the spectrometer camera after integration along the spectral direction for each harmonic order. For both simulations and experimental results, the plotted profiles are obtained at the pressures giving the maximum harmonic yield for each medium length. The comparison shows a qualitative agreement between simulations and experimental results. In both cases, the spatial profiles are more irregular when the harmonics are generated on the vertical branch with clear off-axis contributions. In contrast, the radiation generated on the horizontal branch is more confined, with a smooth spatial profile, which is interesting for applications requiring high-quality spatial properties and/or high intensity after refocusing. The off-axis emission can be identified as a contribution from the long trajectory which becomes more divergent as the laser intensity increases. For the vertical branch, both short and long trajectories contribute, while for the horizontal branch only the short is phase-matched. This result is independent of the focusing geometry. Other details, such as the dependence of the divergence on the harmonic order, differ between simulations and experimental results. This might be due to differences in laser intensity and experimental artefacts, e.g. imposed by the XUV spectrometer. Previous work has also pointed out the importance of the exact generation position relative to the laser focus [29,39]. Further studies are required to investigate the origin of such differences.
Fig. 6. Simulated (a) and measured (b) normalized harmonic intensity (q = 19, 23, 27) along the spectrometer spatial axis for a gas medium length of ${0.06}\,z_R$ (dashed line) and ${1.33}\,z_R$ (solid line). The data have been acquired at the GHHG Sylos Long beamline. Laser intensity in simulations equals to $1.9\times 10^{14}$ W/cm$^{2}$.
5. Conclusions
We present in this work the first experimental validation of the hyperbolic phase-matching model, first published in [30]. The model allows us to identify the parameter space for phase-matching of high-order harmonics with a simple one-dimensional parametric curve. The analytic calculation required is simple and depends on only a few parameters, the gas medium, driving laser wavelength, and harmonic order. To test that the model has no explicit dependence on the specific focusing geometry, laser intensity and pulse duration, we show results of experiments performed at two different sources, the GHHG Sylos Long beamline at ELI-ALPS and the Intense XUV Beamline of the Lund Laser Centre, where these parameters are varied.
We demonstrate experimentally that the results agree well with the same hyperbolic equation. Two regimes of phase-matching can be identified, a vertical branch for medium lengths that are short with respect to the Rayleigh length of the focused driving field, and a horizontal branch for longer medium lengths. The vertical branch is characteristic of transient phase-matching, whereas the horizontal branch describes absorption-limited phase-matching taking place over a longer time. In the vertical branch, the spatial profiles exhibit strong off-axis contributions. In addition, laser intensity fluctuations have a strong effect on the harmonic yield. In contrast, the radiation generated in the horizontal branch is well collimated and relatively insensitive to laser intensity fluctuations. From this comparison, the horizontal branch seems to be more suitable for applications where good spatial profiles are required which could lead to higher refocused XUV intensity. However, simulations have shown somewhat higher conversion efficiencies on the vertical branch for high harmonic orders, as can be seen in Fig.1. In addition, the pulse duration is shorter since phase matching is possible only during a short time. In summary, the selection of the best generation regime depends on the particular experimental setup and application, making the phase-matching model [30], which is experimentally validated in the present work, a powerful tool for the design of future high-flux XUV sources.
Funding
Helmholtz-Lund International Graduate School (HELIOS, HIRS-0018); European Regional Development Fund (GINOP-2.3.6-15-2015- 00001); Wallenberg Center for Quantum Technology, Chalmers University of Technology; Knut och Alice Wallenbergs Stiftelse; European Research Council (QPAP, 884900); Vetenskapsrådet (2013-8185, 2016-04907, 2017-04106, 2018-03731, 2020-03315, 2020-0520, 2021-05992).
Acknowledgments
The authors acknowledge support from the Swedish Research Council (2013-8185, 2016-04907, 2017-04106, 2018-03731, 2020-0520, 2020-03315, 2021-05992), the European Research Council (advanced grant QPAP, 884900) and the Knut and Alice Wallenberg Foundation. AL is partly supported by the Wallenberg Center for Quantum Technology (WACQT) funded by the Knut and Alice Wallenberg foundation. MP and VP acknowledge the support of the Helmholtz Foundation through the Helmholtz-Lund International Graduate School (HELIOS, HIRS-0018). ELI-ALPS is supported by the European Union and co-financed by the European Regional Development Fund (GINOP-2.3.6-15-2015-00001).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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