1. Introduction

High harmonic generation (HHG) converts intense, short laser pulses to their harmonics and generates coherent radiation in the extreme ultraviolet (XUV) and soft X-ray spectral range. HHG is very flexible and able to fulfill the demand of different applications, viz. it can generate very short pulses with durations even of attoseconds [1,2], or very high harmonics with energies of few keV [3–6] for time resolved spectroscopy [7,8].

In order to tailor the spectral shape or the temporal profile of the HHG pulses and to improve the pulse energy according to the different demands, HHG is extensively studied. A very promising method consists in the illumination of the gas used for HHG with a vacuum ultraviolet (VUV) or XUV pulse together with a high-intensity infrared (IR) or near-infrared (NIR) short laser pulse. Such VUV/XUV pulse, generated by an independent source [9–11] or within a gas mixture [12], was used to enhance or synchronize the ionization of the gas atoms, and strong enhancement of HHG was reported.

Nonlinear parametric processes in HHG involving a short NIR laser pulse and a XUV pulse were reported in Ref [13]. Such nonlinear interactions, as X-ray parametric amplification (XPA), can cause amplification of the XUV pulse in the gas medium. Both nonlinear enhanced ionization and stimulated amplification have been theoretically studied, namely, a XUV seed pulse was shown to produce new harmonic lines [14], and it was shown to be amplified by backward scattering [15] or forward scattering [16–20]. Parametric amplification processes have been recently measured and described also by perturbative high-order parametric interaction [21] and by ab initio simulations in hydrogen molecular ions [22].

Here we show that the experimentally found amplification of coherent attosecond XUV pulses in He gas is fully supported by numerical simulations based on the quantum-mechanical description of HHG using the strong field approximation (SFA) [23]. Indeed, in Ref [18]. it was revealed that the amplification of coherent XUV attosecond pulses by strong-field induced stimulated forward scattering can be obtained by synchronizing a weak XUV pulse with a strong IR pulse. This theoretical prediction was soon corroborated by the experiments in Ref [25], which measured XUV attosecond pulse amplification in He gas at around 110 eV photon energies. Beyond independent theoretical [18] and experimental [25] demonstration, in the present work we show large agreement between theoretical and experimental observations concerning amplification by avalanche of a XUV attosecond pulse train in an He gas amplifier at the 110 eV region. The simulations show that the ionization potential of the gas [19] and the consequent dispersion caused by the free electrons in the amplifying medium are key factors to produce XUV amplification in a specific spectral region.

2. Experimental setup

In order to study parametric amplification of an XUV attosecond pulse train by stimulated forward scattering, experiments were performed using a Ti:sapphire laser system delivering 30 fs pulses with a central wavelength of 800 nm and 30 mJ of energy at 10 Hz repetition rate (see Fig. 1). The pulses were loosely focused to obtain an intensity of ~10¹⁵ W/cm². The HHG source consisted of two independent gas jets: the harmonics generated in the first gas jet in the form of an attosecond pulse train served as the XUV seed, and the second gas jet served as XUV amplifier. The beam profiles and the spectra of the harmonics after the second gas jet were measured at around 110 eV. The laser light and the low order harmonics were filtered out by thin metal foils of 200-nm-thick Zr and 200-nm-thick Ti for beam profile measurements, and two pieces of 300-nm-thick Zr foil were used for spectral measurements with 50 s integration time. The seed jet was filled with neon in order to produce a suitable intense seed beam for saturating the amplifier when it was necessary. The intensity of the seed beam was controlled by adjusting the Ne gas pressure. The gas medium in the amplifier jet was helium also with adjustable pressure. During the experiments, the backing pressure of Ne and He was adjusted up to 1.2 bar and 5 bar, respectively while the gas pressure in the interaction volume was about 4% of the backing pressure as given in the Method section of [25], where a more detailed description of the setup and the calibration of the measured XUV fluences can be also found.

 

Fig. 1 Experimental setup and the theoretical method. Both the experiments and the theory are based on an assembly of two jets for achieving and describing amplification of attosecond pulse trains. By decomposing the dipole matrix elements of the time-dependent dipole moment four different scattering processes can be identified, as indicated, of which x₄ is much smaller than the others and can be neglected (see text for details). At the bottom part, the physical interpretations of the four processes are presented.

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3. Theoretical model and single-atom simulations

Our theoretical model is based on an extension of the single-atom response calculated by solving the Schrödinger equation in the SFA in the nonadiabatic form, so that the full electric field of the laser pulse is used to calculate the nonlinear dipole moment [23]. In the present version of the theory the intense low-frequency IR field generating the high-harmonics is perturbed by a weak XUV electric field of which ionization probability is negligible compared to the ionization produced by the IR field, which is well suited since the SFA theory makes no explicit approximation on the frequency of the laser field. As it will be shown, this small perturbation allows us to describe the contribution of different nonlinear scattering processes induced by the weak XUV field, which are well understood and accurately described by our extended theory.

Following [23], after solving the Schrödinger equation and by considering the stationary values of the classical action in the frame of the saddle-point approximation for the canonical momentum, the time-dependent dipole moment can be written in the form of Eq. (13) in Ref [23] with the correction for the sign of the electron charge [24]:

3.1. Decomposition of the dipole moment

Let us further study the particular processes driving the time-dependent dipole moment [19]. By including $k=p_{st}^{}+A_{IR}+A_{XUV}$ into Eq. (4), we can write Eq. (6) as

The decomposition of the dipole matrix element written in Eq. (7) therefore provides four integrals x₁ - x₄ corresponding to the d₁ – d₄ terms in Eqs. (8)-(11), respectively, that we can compute separately. Figure 2 shows the contribution of these integrals to $x(t)$ for a single Gaussian XUV pulse [Figs. 2(a), 2(c) and 2(e)] and for a XUV pulse train [Figs. 2(b), 2(d) and 2(f)], both centered at $\approx$13 eV, in a single atom calculation. We observe that x₁ [blue solid line in Figs. 2(c) and 2(d)] accounts for the regular high-harmonic generation processes, i.e. the spectrum that would be obtained in the absence of the $E_{XUV}(t)$ field.

 

Fig. 2 Single atom calculations. The case of a single attosecond XUV pulse [(a), (c) and (e)] and a train of XUV pulses [(b), (d) and (f)] in He (I_p = 24.587 eV) was calculated. In (a) and (b) the IR and XUV fields are shown. The IR pulse is an 800 nm, 7 × 10¹⁴ W/cm² peak intensity pulse of 26 fs (FWHM) Gaussian temporal profile [black dotted lines in (a) and (b)]. The XUV field consists of Gaussian 200 as pulses, with peak intensity of 7 × 10⁴ W/cm² (i.e. 10⁻¹⁰ times the IR peak intensity). In (b) the pulse train has a super Gaussian envelope of 15 fs. The IR carrier envelope phase (CEP) is perfectly synchronized with the CEP of the XUV pulses, and the repetition rate in the XUV pulse train is half the period of the IR pulse. The different contribution from the time-dependent dipole moment factors to the spectra are shown in (c)-(f). The spectrum from the x₄ is negligible and not shown. Note that the vertical axis in (d) is in logarithmic scale. In (g) the absorption spectrum for a single attosecond XUV seed pulse is shown together with the normalized seed spectrum, which demonstrates IXPA (inset).

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The contribution from x₂ computes the probability of the release of the electron from the atom by both the presence of laser field $E_{IR}(t\text{'})$ and the XUV attosecond pulse field through its vector potential $A_{XUV}(t\text{'})$ at the time t’, the propagation to time t by the semiclassical action $S_{IR}^{st}(t,t\text{'})$ and the recombination at time t. Some of these processes concerning photon energies near the ionization potential of the gas medium have been extensively studied in the last years [9–12]. In these experiments, however, the first HHG source was optimized to produce intense low-order or bellow threshold harmonics to produce enhanced ionization (EI) processes to increase the HHG signal in the XUV.

XPA processes are readily contributed by x₃. The x₃ contribution can be read as the probability for an electron to be ionized by the laser field $E_{IR}(t\text{'})$ at time t’, propagated from t’ to t by the semiclassical action $S_{IR}^{st}(t,t\text{'})$, and recombined back to the ground state due to the presence of the attosecond $A_{XUV}(t)$ pulse at time t. Because $d_3(t)\propto A_{XUV}(t)$ [see Eq. (10)], XPA requires signal (seed) from the first gas jet at the same XUV photon energies as the amplified output spectrum, contrarily to the case of enhanced ionization [9–12].

Both the x₂ and x₃ integrals also include processes that are produced in the time scale of the XUV pulse period. These processes have been here identified as XUV intrapulse parametric processes, this is, the accelerated electron absorbs part of the spectral contents of the XUV pulse and this is used by the other spectral components of the same XUV pulse for stimulated emission as the electron recombines to the ground state, what we name intrapulse X-ray parametric amplification (IXPA). IXPA is shown more effective when the XUV pulse coincides with the higher values of the IR field, contrarily to regular XPA that is produced at the recombination time of electrons that have gained energy by acceleration in a round trip trajectory in the continuum. The results reported in the present work hence confirm the calculations made in [18] (see in particular Figs. 1 and 2(b) in [18]). Furthermore, an analysis of the electron trajectories contributing to the amplification process, not shown here, reveals that for the parameters of interest in the present work (in particular considering XUV photon energies that are far from the He absorption region at about 20 eV), IXPA processes dictate the amplification at the single-atom level, as it will be further commented below. The relative contribution of other processes such as EI and XPA included in the x₂ and x₃ integrals during the propagation and amplification of the XUV signal has not been quantified yet and needs further investigation. In what follows, where necessary, we will differentiate IXPA₂ as the contribution coming from the x₂ term and IXPA₃ as the one coming from x₃. Finally, the contribution of factor x₄ describes a four-wave mixing between the XUV field and the fundamental laser field as l = n ± m ± 1, where l, m and n are harmonic line numbers. It is very small (more than 10-orders of magnitude smaller than x₁) and negligible in all cases of our study.

In Figs. 2(c)-2(f) the spectra are obtained from Eq. (7) by computing the complete dipole moment $x(t)\approx x_1(t)+x_2(t)+x_3(t)$ and the separated contributions from $x_1(t)$, $x_2(t)$ and $x_3(t)$, as indicated. As commented above, the contribution from term $x_4(t)$ is negligible. From the single atom calculations in Figs. 2(c)-2(f) we can therefore observe that the regular HHG spectrum is given by the $x_1(t)$ factor [blue solid line in (c) and (d)], and that the main contribution to the amplification is from the $x_3(t)$factor [black solid lines in (e) and (f)]. The amplification is completed by the contribution from the $x_2(t)$factor [orange dashed lines in (e) and (f)]. The higher XUV amplification obtained in the case of the XUV pulse train [Figs. 2(d) and 2(f)] compared to the amplification of the single attosecond XUV pulse [Figs. 2(c) and 2(e)] is obviously due to the larger XUV energy contained in the case of the seed train, since we take a super-Gaussian envelope for the train involving 11 subpulses, each with the same peak intensity as in the single-pulse case (7 × 10⁴ W/cm²), together with the spectral modulation due to the interference fringes of the seed train. Figures 2(e) and 2(f) show how the amplifications produced by the x₂ and x₃ integrals are centered at the same frequency as the seed. In Fig. 2(g) we show the XUV absorption spectra in the case of a single attosecond XUV pulse. The absorption signal is proportional to $\Im m\left\{E_{XUV}^{*}(\omega )x(\omega )\right\}$ [27], where $E_{XUV}^{}(\omega )$ and $x(\omega )$ are the Fourier transforms of the XUV field $E_{XUV}^{}(t)$ and the dipole moment $x(t)$. Since we are interested in the atomic response from processes different of the regular HHG we do not consider its contribution [$x_1(t)$], and therefore we compute $\Im m\left\{E_{XUV}^{*}(\omega )[x_2(\omega )+x_3(\omega )]\right\}$. The absorption spectra of $E_{XUV}^{}(t)$ detailed in Fig. 2(g) exhibits an absorption region in the low frequency spectral contents of the XUV pulse and an emission region at the higher frequency spectral components. This is characteristic of an intrapulse parametric process and, together with the frequency-time analysis of the amplified XUV signal [18], demonstrates the IXPA process. Indeed, as commented above, the ionized electron is accelerated in the presence of the strong IR field and this enables the absorption of XUV photons followed by stimulated emission in the recombination of the electron back to the ion. By analyzing the quantum electron trajectories involved in the IXPA process we have corroborated that it occurs within a few XUV periods and that it therefore corresponds to an XUV intrapulse effect.

3.2. Effect of the delay between the IR and XUV pulses

As it was reported first in Ref [18]. and corroborated experimentally in Ref [25], the synchronization of the XUV and IR pulses is essential for XUV amplification. We next show by single atom calculations the effect of the delay between the IR pulse and the XUV pulse on the amplification. The single atom calculations are the basis for understanding the more complicated effects included in the propagation of the coherent radiation in the gas medium, which will be treated later both experimentally and theoretically.

As well as in Fig. 2, we consider a driving laser field composed of a Gaussian temporal profile 26 fs (FWHM) 800 nm IR strong pulse of 7 × 10¹⁴ W/cm² peak intensity, carrier-envelope phase CEP = 0, which produces high-order harmonics in helium (I_p = 24.59 eV) with a photon energy cut-off at $\approx$150 - 160 eV, together with a super-Gaussian 15 fs envelope train of Gaussian XUV 200 as (FWHM) pulses of CEP = 0 and with central photon energy well in the plateau of the IR-generated HHG spectrum. The peak intensity of the XUV attosecond subpulses is in this case about 700 W/cm² (i.e. $\approx$10⁻¹² times the IR peak intensity), and the temporal separation between the subpulses in the train is half the IR pulse period [see Fig. 2(b)].

Figure 3(a) shows the amplification obtained by varying the delay of the XUV pulse train with respect to the IR pulse. It can be seen that the amplification is proportional to the IR field strength, therefore showing that IXPA processes are the dominant at the single-atom level for the parameters that we have considered, as already commented above. The simulations in Fig. 3(a) have been performed for different values of the central photon energy of the XUV pulse train, as indicated. The plotted enhancement factor is defined as the integrated HHG yield in the plateau obtained by using the combination of IR + XUV as input pulse divided by the integrated HHG yield obtained considering only the IR pulse. Note that because the integration has been performed over a wide spectral range, the enhancement factor is only somewhat larger than 1. Looking to one harmonic line however the enhancement is much larger. Furthermore, the enhancement in Fig. 3(a) is calculated for a single atom. When one considers propagation in the gas medium the calculation is repeated through several iteration steps and the enhancement becomes large, as it will be shown below.

 

Fig. 3 Delay dependent amplification. Single atom calculations for the case of a weak attosecond XUV pulse train interacting together with a strong IR pulse with He (I_p = 24.587 eV). The IR pulse is a 800 nm, 7 × 10¹⁴ W/cm² peak intensity pulse of 26 fs, CEP = 0. The XUV field consists on Gaussian 200 as pulses, CEP = 0, with peak intensity of $\approx$700 W/cm² (i.e. 10⁻¹² times the IR peak intensity). The XUV pulse train has a super Gaussian envelope of 15 fs width. The repetition rate of the train corresponds to half the period of the IR pulse [see Fig. 2(b)].(a) Integrated HHG yield enhancement for different values of the central photon energy of the XUV pulse train, as indicated, as a function of the delay between the IR pulse and the XUV pulse train. The XUV peak intensity has been chosen slightly increasing with the central photon energy for clarity of the figure. The delay is given in parts of half a cycle of the IR pulse, in radians, so that 2π rad ≡ λ_IR/c $\approx$2.66 fs. Note that for the weak XUV peak intensity used in the present calculation the single-atom enhancement is small. (c), (e) and (g) Spectra for the case that the XUV train peak is advanced 0.3325 fs (-π/4 rad) with respect to the IR pulse. (d), (f) and (h) Spectra for the case that the XUV train peak and the IR pulse peaks are perfectly synchronized. From up to down the XUV pulse trains centered at 100 eV, 120 eV, and 140 eV (green, purple and blue dashed curves, respectively) are shown, and (b) the black solid line show the HHG spectrum obtained with the IR laser pulse alone.

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The fast oscillations of the enhancement that can be observed in Fig. 3(a) are due to the interference between the harmonics generated by HHG and the amplified XUV train. The period of these fast oscillations coincides with the XUV period, as it was already reported in Ref [18]. Indeed, the amplified XUV is emitted at the time that the XUV pulse interacts with the medium, so that the fast oscillations are only present where the HHG and amplified XUV fields overlap in time. Due to this effect, the optimal delay between the IR pulse and the central peak of the XUV pulse train can be slightly shifted to positive delays in the case of low seed intensity [see the green and purple lines in Fig. 3(a)]. At higher values of the seed intensity such interferences become negligible [18] and the optimal delay is basically centered at 0 rad, as it is the case shown by the blue line in Fig. 3(a).

Figure 3(b) shows the spectra produced by the interaction of the strong IR pulse alone with the medium (black solid curve) and Figs. 3(c)-3(g) the spectra produced by the interaction of the combination of the strong IR and a weak XUV pulse train centered at 100 eV, 120 eV and 140 eV, as indicated. Clearly, the yield spectra obtained with the combination of IR + XUV input pulses show an enhancement in the spectral region around 100 eV, 120 eV and 140 eV (green, purple and blue dashed curves, respectively), as it was already shown in the case computed in Fig. 2. Here we show the dependence of this enhancement on the synchronization between the XUV and the IR pulses. In Figs. 3(c), 3(e) and 3(g) the XUV train is advanced in time by 0.3325 fs (π/4 rad) with respect to the peak of the IR pulse field; this geometry produces a weak amplification of the HHG yield around the corresponding spectral region (100 eV, 120 eV and 140 eV). When the XUV train is synchronized to the peak of the IR field strength, however, which is the case shown in Figs. 3(d), 3(f) and 3(h), the amplification is much larger. The dependence of the yield enhancement on the IR field strength is indeed expected from the theory considering the linear dependence of the time-dependent dipole moment factors $x_2(t)$ and $x_3(t)$ on the $E_{IR}$ field [see Eq. (7)].

4. Amplification of attosecond pulse trains

In this section, we demonstrate that a strong field driven gas behaves as an optical amplifier in the XUV regime by comparing experimental and theoretical results.

4.1. Simulation of the XUV pulse propagation in the amplifier jet

For the simulations of the experimental measurements, the calculations from Eq. (1) have been adapted to the particular experimental configuration. Specifically, a seed field is first produced by HHG from an intense IR pulse in Ne. This seed pulse combined at the optimal delay with the intense IR pulse is used as input for the interaction with a first numerical cell of He atoms. The HHG output from this first interaction together with the seed and IR pulses are propagated and used as input for a second interaction with a second cell of He atoms, and the process is repeated iteratively, so that propagation is described in 1D. In order to fully consider the macroscopic effects associated to propagation, we take into account the regular phase mismatch associated to neutral gas and dispersion from the free electrons together with the geometrical phase mismatch due to the shape of the driving pulse, which arises primarily in this case from the Gouy phase shift due to the focused driving laser beam. We give the detailed study of phase matching in section 4.4. We can estimate and have also checked numerically that pressure-induced phase matching can be produced for instance at $\approx$10 mbar with 7 × 10¹⁴ W/cm² IR peak intensity or at $\approx$5 mbar with 8 × 10¹⁴ W/cm², pressures that are much smaller than those used in the experiments (70-200 mbar). Furthermore we concluded that phase matching of the generated harmonics does not influence the amplification of the signal that we describe, being totally negligible in the parameters region that we are interested.

While we found that [Fig. 3(a)] synchronization of the seed XUV pulse train to the laser field in the He gas amplifier is a key issue for efficient amplification, we performed calculations to determine how propagation and phase matching affect this behavior. Figures 4(a)-4(c) show the calculated spectra as a function of the delay between the seed and the driving pulse, obtained after a propagation of 3 mm for different amplifying He pressures, as indicated. For the simulations in Fig. 4 we have considered an 800 nm driving laser field with a Gaussian temporal profile of 26 fs (FWHM) and 8 × 10¹⁴ W/cm² peak intensity. The delay has been varied along a period of the IR driving field. We observe that the regions where the maximum amplification is produced are depending on both the delay and the gas pressure.

 

Fig. 4 Delay dependent amplification for different amplifying gas pressures, as indicated. The simulations (a)-(c) show how the spectral region that is amplified depends on the delay and the gas pressure. The shape of the IR field is clearly reproduced. (d) The measurement where delay was produced by changing the distance between the gas jets reproduces the calculation in the corresponding delay range. The propagation distance was 3 mm for the calculation and measurement. The delay conversion is π rad ≈40 mm.

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The synchronization of the XUV pulse train with respect to the laser field remains a key parameter similarly to the single atom calculations of Fig. 3(a), however considering propagation it makes the picture more complex. While Fig. 3(a) shows an optimal delay near zero at every spectral range, the simulations including propagation show that zero delay is optimal only at around 100 eV [see Figs. 4(a)-4(c)] for the parameters used. At around 130 eV however, the optimal delay shifts to roughly ± π/8 rad, with four relative maxima per optical IR delay cycle, and it shifts to approximately -π/4 rad and + 3π/4 rad at 160 eV, where two maxima per optical IR delay cycle are produced.

We have also analyzed separately the contributions from neutral gas dispersion and free-electron dispersion. Clearly, and as it is expected, the dispersion of the free electrons is the dominant macroscopic effect. Importantly, for the amplification that we report, group velocity dispersion is the relevant parameter in propagation. Group velocity dispersion is responsible of the dynamic delay between the propagating spectral components in the attosecond HHG pulses and the peak of the IR pulse, and therefore it modifies their overlap with the IR field, which is the feed of the amplification. Consequently, the initial delay between seed and IR pulses together with free-electron dispersion determine the precise spectral region that is amplified. In this sense, it is important to stress that, as in the experiments, no spectral filter is applied to the HHG output from the first Ne gas jet, which is used as seed pulse for the second He jet by only scaling the value of its yield in order to match the experimental conditions, and therefore no particular spectral region is embedded in the seed pulse for amplification. The shape of the IR driving laser field is clearly reproduced by the amplified regions.

The maximum amplification is produced around 100 eV at 100 mbar and at 130 mbar, and it shifts to slightly higher photon energies in the case of 160 mbar. We compare this calculation with the measurement of Fig. 4(d) performed under similar conditions. In the measurement the delay was applied by scanning the distance between the two gas jets [25,28], so that in Fig. 4 a change of π rad corresponds to a scan of ≈40 mm. This method makes possible to scan only a short delay range around zero delay as indicated in Fig. 4(c). The measurement nicely reproduces the features of the He amplifier predicted by the calculation.

4.2. Experimental measurements and comparison with the calculations

The avalanche-like behavior of an amplifier means that the signal increases exponentially along the amplifier medium having length L, density n and σ gain cross-section, namely $J_{out}\approx J_{seed}{e}^{\sigma nL}$. It means further that the amplifier is linear, because the output fluence is linearly proportional to the seed fluence. For low seed fluence, the signal to noise ratio of the measurement was too small to extract reliable information for the linear amplification range directly; consequently we examined this behavior in another way, by changing the atomic density of the amplifier medium to observe the exponential dependence of the gain. The experimental arrangement was the same as presented in Fig. 1. The backing pressure of the neon gas in the seed jet was fixed at 0.8 bar yielding a measured seed fluence of ~5 × 10⁹ ph/cm². In the second gas jet, which served as an XUV amplifier, the backing pressure of the helium gas has been varied in a range up to 5 bar. A few measured spectra are plotted in the left column of Fig. 5. We have measured the spectrum of the seed beam (no gas in the amplifier jet, first row) and the spectra of the amplified beams (brown dashed curves) for different settings of the He gas backing pressures in the amplifier jet. Harmonics were also generated in the amplifier jet without the seed beam and we term this case as “unseeded” amplifier. Indeed, the generated harmonics in the amplifier jet are also amplified in the same gas medium, which acts as a self-seeded amplifier. These spectra are also plotted in Fig. 5 (black solid curves). When the amplifier is seeded by an independent external HHG source then we will term this amplifier simply as “seeded” amplifier.

 

Fig. 5 (a) Measured and (b) calculated spectra at increasing gas pressures and corresponding iteration numbers in the second gas jet, respectively. (a) In the He parametric amplifier, the generated high-harmonic spectra are altered when the amplifier is seeded (brown dashed curves) from an independent HHG source compared to the spontaneously generated spectra without seed (black solid curves). The spectra are normalized to the seed and so directly show the magnitude of the amplification. (b) The numerical simulations reproduce very well the behavior of the spectra of the seeded and unseeded amplifier.

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For the simulations shown in the right column of Fig. 5 we used a simple version of the “particle-in-cell” simulation. One cell contained ~3800 atoms and the effect of propagation and gas pressure was modeled by fitting the calculation of 30 iterations to the 5 bar gas backing pressure of the measurements. Despite the calculations describe the overall scaling of the experimental data very well, some differences between them are obvious, namely for higher pressure the calculated spectra are narrower than the measured ones, and the shift to higher photon energies observed on the amplified signal in the unseeded case with respect to the seed is larger in the experiment than in the simulations. This are probably the consequences of assuming a spatially uniform field distribution for the calculations i.e. supposing a plane wave, while in the experiments the profile of the laser beam was near Gaussian and the beam parameters changed somewhat by passing through the gas jet having finite length. Other parameters what are difficult to determine with precision in the experiment, as the laser peak intensity and accurate pressure at the interaction region, are also important and sensitive for the simulations.

As it is evident by comparing the experimental results with the simulations in Fig. 5, both the measured and calculated spectra show the same behavior. Without applying any seed, there is a continuous increase of the spectral intensity, which also can be seen by the black curves in Fig. 6. However, for the seeded amplifier, the spectra hardly change at low pressures and strong amplification can be observed at higher pressures. To study this behavior in more detail, we plotted the pressure dependence of the spectrally integrated intensity of a few harmonic lines (both measured and calculated) separately in Fig. 6.

 

Fig. 6 The avalanche effect in HHG can be observed during propagation (or increased He gas pressure) in the amplifier jet. In the different panels, the spectrally integrated intensities of few harmonic lines are plotted. Both the measurements (marks) and the calculations (solid/dashed lines) were performed for the seeded (brown dashed lines) and the unseeded (black solid lines) amplifier.

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For both the seeded (brown dashed) and unseeded (black solid) amplifier, the calculated curves fit very well to the measurement points for harmonics between 63 and 69, where the measured and calculated spectra were most intense. For every harmonics but especially for harmonics 65 and 67, the exponential increase of the harmonic signal in the case of the unseeded amplifier extends over three orders of magnitude. This exponential increase is the clear indication of the avalanche effect of the parametric amplification.

A closer inspection reveals a more complex behavior for the seeded amplifier that is true for the measurements and the calculations. First, in the calculations, the necessary seed energy is about 10-times smaller to obtain the same harmonic signal as in the experiment. This difference can be clearly seen at low gas pressure (below 1 bar) and supports the assumption that in the measurement probably only about 10% of harmonic beam generated in the first jet was used for seeding the amplifier. This observation is similar than reported in an earlier publication studying XPA at around 300 eV [13]. The difference can be explained by a partial overlap (spatially and temporally) in the amplifier medium. This mismatch is supported by the observation that only a small part of the seed beam was amplified. This observation requires a detailed theoretical study in the future.

4.3. Contribution of the $x_1$, $x_2$ and $x_3$integrals in the HHG signal

Another interesting feature of the seeded amplifier is the shoulder between 2 and 3 bar backing pressure. To explain this feature, we performed further calculations by following the $x_1$, $x_2$ and $x_3$ contributions separately during the propagation. The results can be seen on Fig. 7 for the most intense harmonic line of 65. The high harmonic part ($x_1$) remains very small in the full range of propagation or gas pressure. The contribution of EI + IXPA₂ ($x_2$) remains always bellow the XPA + IXPA₃ ($x_3$), however its rate increases as the XUV signal increases in the medium ($x_3/x_2$, pink). Comparing Fig. 7 with the same H65 of Fig. 6, it is clearly visible that for seeded amplifier the output signal is almost fully governed by the $x_3$ term (XPA + IXPA₃) alone and the EI + IXPA₂ and HHG gives only a small contribution. Consequently, both the small value of HHG and the modulation in $x_3$ are the consequence of the lack of phase matching. We can further observe and study in detail in chapter 4.4 that the periods and consequently the phase matching conditions for HHG and parametric amplification are different.

 

Fig. 7 Decomposed calculation of the seeded harmonic source. The three dominating contributions namely the HHG (x₁, blue), the EI + IXPA₂ (x₂, green) and the XPA + IXPA₃ (x₃, black) are plotted separately. The contribution from XPA + IXPA₃ dominates during the propagation (iteration number), however its dominance over the effect of IE + IXPA₂ (x₃/x₂, pink) decreases at higher XUV intensities.

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4.4. Contribution of dispersion and propagation effects

For a deeper understanding of the contribution of phase matching in the observed amplified signal, we performed calculations taking phase matching and the parametric amplification separately into account. Neutral gas dispersion and absorption was calculated from the scattering cross sections (f₁ and f₂) with data obtained from Ref [29]. As commented above, the plasma frequency of free electrons was obtained from the tunnel ionization rate in the ADK theory. The Gouy phase shift is given by arctan(z/z_R), with z being the distance from the beam focus along the axis of propagation and z_R the Rayleigh length (80 mm in the present experiment).

We observe from our simulations that the coherence length of the harmonics (about 100 - 300 μm) is much shorter than the propagation that we consider (3 mm). Further, taking all processes into account, we have carefully checked that the phase matching of the generated harmonics does not influence the amplification of the seed signal that we describe, being totally negligible in the parameters range that we are interested. Figure 8(a) shows the calculation results at two gas pressures of 30 mbar and 100 mbar and at the harmonic lines of H63 and H69 (97 eV and 107 eV, respectively). At 30 mbar (red curves), the intensity of the particular harmonics reaches a maximum at about the half of the gas jet and the intensity decreases after it. At 100 mbar (black curves), the intensity of the H63 reaches a maximum later and H69 does not reach its maximum within the jet. This behavior of the harmonic source is clearly contrary to what one can expect for phase mismatched or absorption limited HHG where maximum is reached earlier at higher pressures. On the other hand, we find that pressure-induced phase matching due to the geometric Gouy phase shift is also negligible, since it applies to much smaller pressures (approx. 5 mbar) than the pressures that we consider and were applied in the experiments (30-200 mbar). Figure 8(b) shows the results obtained at the phase matching pressure of 5 mbar. Calculating with only the x₁ term (HHG), pressure-induced phase matching is clearly reproduced (dashed lines) for the detailed harmonics of H47 and H53, showing the expected parabolic increase of the harmonic signal. At this small pressure, which was not considered in the experiments, the amplification induced by x₂ + x₃ gives a maximum at about 1 mm, and it is afterwards suppressed by group velocity dispersion, which removes the optimal overlap between the propagating XUV pulse train and the driving IR laser pulse. The strong modulation of the signal in Fig. 8(b) has the same origin as the modulations shown in Fig. 3(a) and also reported in [18]. In this case the low pressure that we consider (5 mbar), makes the amplified signal to be weak enough to interfere with the generated harmonics. This modulation almost disappears for a stronger amplified signal, as it is the case in Fig. 8(a).

 

Fig. 8 Simulation performed with 8 × 10¹⁴ W/cm² IR peak intensity. (a) The figure shows how the regular phase mismatch induced by neutral gas and free electron dispersion is negligible at the region of the experimental parameters. (b) Calculations performed at 5 mbar where pressure-induced phase matching is achieved. The dashed lines correspond to the calculations neglecting the parametric amplification.

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5. Discussion and conclusions

We theoretically studied XUV generation in a two-gas-jet arrangement and compared the results with experimental measurements. We have demonstrated that coherent attosecond XUV pulses can be amplified in He gas in the context of HHG by carefully adjusting the delay between the intense IR laser pulse and the XUV seed pulses. The numerical simulations show that free-electron dispersion in the amplifying medium is a key factor to produce XUV amplification in a specific spectral region and that this amplified spectral region can be fine-tuned by adjusting the delay. We have investigated in detail the characteristics of an optical amplifier, namely the avalanche-type increase of the generated harmonics during propagation, and have found that numerical simulations based on the SFA fully support and reproduce the experimental measurements and describe the He gas medium as an amplifier of the XUV coherent light pulses. To look into the phenomenon, we distinguished three contributions from the theoretical description of the process, namely HHG, enhanced ionization and parametric amplification. We find that parametric amplification dominates over the other processes and determines the main characteristics of the XUV source. Specifically, we have identified a new type of XPA process (IXPA) that is produced during the short interaction of the attosecond XUV pulses with the ionized electron and is enabled by the presence of the strong IR field. We find that IXPA is the dominant parametric amplification effect at the single-atom response. Our results indicate the optimal conditions and the interpretation of HHG in two-jet experimental geometries in a broad spectral region considering propagation. In [19], it was showed that XUV parametric amplification can be most efficient by using atoms or ions with a high ionization potential and that the nonlinear amplification is robust at high photon energies where HHG is not efficient, such as in the water-window spectral region. A high ionization potential is also most optimal for amplification because a small free-electron density in the medium allows group velocity dispersion effects to be minimized. In this direction, recent simulations in the higher spectral region (300 - 400 eV) using ions with a high ionization potential, such as Li⁺ or Ba⁺⁺, indicate that amplification of the X-ray signals might be substantially boosted and only limited by saturation. The present research hence settles know-how for the generation of intense XUV and X-ray coherent ultrashort light pulses at high repetition rates in typical university laboratories.