Introduction

High harmonic generation (HHG) in rare gases is a compact tabletop source of coherent extreme ultraviolet (XUV) radiation used in a large variety of research fields1,2,3. A very successful semiclassical microscopic description developed almost 30 years ago4 states that incoming laser radiation disrupts the atomic barrier potential, leading to ionization. The freed electron quivers along the linearly polarized laser field and recombines later with the parent ion within a half-cycle time period, emitting the excess energy as XUV radiation. Besides the effects influencing this microscopic phenomenon, the overall XUV signal strength is mainly affected by the phase-matching between the driving IR laser pulse and the generated high harmonic radiation5,6.

With the increasing peak power of laser systems currently available for driving HHG7,8, the f-number and the length of the generating medium can be increased following scaling laws9,10 to maintain the maximum conversion efficiency with a uniform generating medium11. Several approaches to control the phase-matching were examined, such as modifying the wavefront of the driving laser field12,13,14 or changing the properties of the generating medium by mixing multiple rare gases15. Other studies were devoted to guiding the beam and generating harmonics in a highly ionized medium16,17,18. In the case of short dense media, an overdriven regime uses effects of nonlinear laser propagation to shape the beam into favorable conditions of generation19,20. Not many studies, however, have examined the impact of nonlinear laser pulse propagation on phase-matching in long medium.

In this article, we first show that only limited phase-matching is reachable with the standard HHG scheme. This is due to the fact that in a long medium the laser pulse experiences an ionization-induced defocusing. Then, we introduce a controlled low pre-ionization of the medium as a method to overcome this bottleneck and verify it experimentally for what we believe is the first time. Additionally, an analytical model for defining the phase-matching conditions in a pre-ionized medium is derived to support the experimental results obtained with a weak capillary discharge. Moreover, we have performed numerical simulations of the whole process, modeling both microscopic and macroscopic aspects of HHG, to validate the method in a broader range of experimental conditions.

Phase-matching limitations due to defocusing in long medium

The impact of ionization-induced defocusing in long media is known to drastically drop the laser intensity along the direction of propagation once the entry intensity is higher than a critical value21,22. To illustrate the importance of such behavior in more detail, we have performed numerical simulations of an 800-nm laser pulse propagation in a 15 mm long gaseous krypton medium for different laser intensities and gas pressures. (Fig. 1a, see “Numerical simulations”). Similar to23, the results of our simulations show that the intensity drop (from 0 to 6 mm) is followed by a stable region with almost constant laser intensity. The value of stabilized intensity depends only on the medium type and density, and is practically independent of the laser intensity at the entrance of the medium. We can thus recognize two regions of generation in the long media: region I, where the pulse intensity is rapidly dropping during propagation and region II, the stable region.

Figure 1
figure 1

(a) Results of numerical simulations of a 30 fs pulse propagating in krypton demonstrating the stabilization of the on-axis intensity for different medium pressures and input laser intensities with emphasis on region I in light red, where intensity drops rapidly and region II in light green, where intensity stabilizes. (b) Evolution of phase-mismatch \(\Delta k\) between the 800-nm laser and the 17th harmonic order calculated for the same pulse propagation as in panel (a). (c) Phase-mismatch for the same laser pulse as in (b), but with the medium homogeneously pre-ionized to 8%. The medium length is set to 15 mm.

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Considering HHG in a long medium, the XUV reabsorption plays a critical role as soon as the absorption length of the generated harmonic is shorter than the total length of the medium. This is typically the case when generating harmonics below 100 eV with an 800 nm laser24. Therefore, achieving phase-matching in region II is crucial, as most of the detected XUV signal originates from this region.

However, the intensity in region II is generally insufficient to ionize the medium by the driving laser itself to allow reaching the phase-matching condition. The central optical quantity to classify the phase-matching of two waves is their phase mismatch

$$\begin{array}{*{20}c} {\Delta k = k_{q} - qk_{1} , } \\ \end{array}$$
(1)

where \(k_{q}\) and \(k_{1}\) are the wavenumbers of the q-th harmonic and the fundamental, respectively. Ideal phase-matching corresponds to a situation where \(\left| {{\Delta }k} \right| = 0\). Figure 1b,c compare the mismatch for the 17th harmonic and 800 nm laser from a numerical experiment with and without the pre-ionization (see the Supplement S1 for detailed analysis). It demonstrates a significant reduction of the mismatch by homogeneously pre-ionizing the medium to ionization degree of 8%. In this case, the \(\left| {\Delta k} \right|\) is decreased by an order of magnitude in region II where an almost perfect phase-matching is achieved (Fig. 1c, note the different y-scale in Fig. 1b,c).

Analytical model

We now derive an analytical model that allows us to quantify the optimal pre-ionization condition of the medium and the subsequent XUV signal enhancement.

A macroscopic model of HHG in a medium of length \(L_{med}\) with the absorption of generated radiation characterized by absorption length \({ }L_{abs}\) was developed earlier25 providing an analytical estimate of the HHG gain in a homogeneous medium as

$$\begin{array}{*{20}c} {I_{q} = \frac{{4\rho^{2} A_{q}^{2} L_{abs}^{2} }}{{1 + 4\pi^{2} \left( {\frac{{L_{abs} }}{{L_{coh} }}} \right)^{2} }}\left[ {1 + \exp \left( { - \frac{{L_{med} }}{{L_{abs} }}} \right) - 2\cos \left( {\frac{{\pi L_{med} }}{{L_{coh} }}} \right)\exp \left( { - \frac{{L_{med} }}{{2L_{abs} }}} \right)} \right], } \\ \end{array}$$
(2)

where \(A_{q}\) is the dipole amplitude for the q-th harmonic (considered as constant in this model), \(\rho\) is the medium density, and \(L_{coh} = \pi /\left| {\Delta k} \right|\) is the coherence length, i.e. the distance along which all microscopic emitters contribute constructively. In the case of absorption-limited generation (\(L_{med} \gg L_{abs}\)), which is usually the condition of generation, the pre-factor in Eq. (2) dominates as the expression in square brackets approaches unity.

If the phase matching is achieved and the coherence length becomes indefinitely large, the denominator of the pre-factor decreases to unity, and we get the phase-matched absorption-limited gain which is independent of the medium density as \(L_{abs} = 1/\sigma \rho\), with \(\sigma\) being the absorption cross-section. With definite coherence length, it is the ratio of \(L_{coh}\) and \(L_{abs}\) that determines the gain of absorption-limited HHG. More generally, as the coherence length is inversely proportional to density, the absorption-limited gain becomes pressure independent.

The coherence length \(L_{coh}\) is defined by the mismatch of the wavenumbers of the driving laser \(k_{1}\), and the generated qth harmonic beam \(k_{q}\). In the following, we consider the dominant processes for each contribution to the wavenumbers, namely: the dispersion of neutrals for both fields26,27,28 and the effect of plasma on the medium susceptibility for the fundamental laser field. Effects of the dipole phase on the harmonic generation and the Gouy phase of the Gaussian beam are neglected. Using polarizabilities in the expression for refractive indices of neutrals and plasma, we get

$$\begin{array}{*{20}c} {\Delta k = \frac{{q\omega_{1} \rho }}{2c}\left( {{\Delta }\alpha - \eta \frac{{e^{2} }}{{\varepsilon_{0} m_{e} \omega_{1}^{2} }}} \right), } \\ \end{array}$$
(3)

where \({\Delta }\alpha = \alpha_{1n}^{\left( a \right)} - \alpha_{qn}^{\left( a \right)}\) is the difference of polarizabilities (at standard temperature and pressure) and \(\eta\) is the degree of ionization, i.e., the ratio of free electrons to neutral atoms and ions. Inserting this expression in the definition of the coherence length, we can express the degree of ionization \(\eta\) as a function of atomic density and coherence length:

$$\begin{array}{*{20}c} {\eta = \frac{{\omega_{1}^{2} \varepsilon_{0} m_{e} }}{{e^{2} }}\left( {{\Delta }\alpha \mp \frac{2\pi c}{{q\omega_{1} \rho L_{coh} }}} \right).} \\ \end{array}$$
(4)

We consider ideal phase-matching as a situation where the coherence length is comparable or longer than the medium length. This is the case for a range of degrees of ionization. Figure 2a shows the boundaries of \(\eta\) as a function of pressure with coherence length equal to medium length for the 17th harmonic order generated by 800 nm laser in krypton. The case corresponding to an infinite coherence length is shown by the dashed line. As the laser ionizes the medium during generation, HHG is phase-matched as long as the ionization stays within the region shown in blue. This model explains the numerical observation presented in Fig. 1b,c, where the phase mismatch is strongly reduced in the region II when the medium is pre-ionized.

Figure 2
figure 2

(a) Range of optimal ionization degree with \(L_{med} = 15 mm\) for 17th harmonic order generated by a 800-nm laser in krypton as a function of pressure (cyan channel) with emphasis on ideal ionization corresponding to \(L_{coh} \to + \infty\) (black dashed line). (b) Describes evolution of the 17th harmonic signal based on Eq. (2), where the optimal signal is in the range \(L_{coh} > L_{med}\) (cyan dotted curves). Expected signal from the high-intensity laser in region II as shown in Fig. 1b is denoted by blue curve. By adding a constant pre-ionization of 8% to the same high-intensity laser, the expected signal moves into desired region (red curve). (c) The absorption-limited XUV signal as a function of the ionization degree \(\eta\) according to Eq. (5).

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Using the expression of the absorption and coherence lengths and inserting them into the pre-factor of Eq. (2), we obtain the absorption-limited harmonic signal

$$\begin{array}{*{20}c} {I_{q} \left( \eta \right) = \frac{{4A_{q}^{2} }}{{\sigma^{2} + \frac{{q^{2} \omega_{1}^{2} }}{{c^{2} }}\left( {{\Delta }\alpha - \eta \frac{{e^{2} }}{{\omega_{1}^{2} \varepsilon_{0} m_{e} }}} \right)^{2} }}.} \\ \end{array}$$
(5)

The most striking result is that the only variable is the degree of ionization \(\eta\). In particular, this expression is independent of the medium density. And it is also independent on the medium length (consequence of the assumption \(L_{med} \gg L_{abs}\)). The signal strength of 17th harmonic order generated in Kr given by Eq. (5) is shown in Fig. 2c as a function of degree of ionization \(\eta\). The maximum of the signal is reached for \(\eta_{{{\text{opt}}}} = {\Delta }\alpha \frac{{\omega_{0}^{2} \varepsilon_{0} m_{e} }}{{e^{2} }}\). The possible enhancement, \(I_{q} \left( {\eta_{{{\text{opt}}}} } \right)/I_{q} \left( 0 \right)\), turns out to be 8 for the 17th harmonic in krypton. Note that \(I_{q} \left( {\eta_{{{\text{opt}}}} } \right)/I_{q} \left( 0 \right)\) depends only on material constants and laser frequency, it thus provides a direct way to estimate the potential enhancement by pre-ionization in the absorption limited regime.

The effect of pre-ionization is demonstrated in Fig. 2b, where the pressure dependence of the 17th harmonic order signal generated in a 15-mm long krypton medium is calculated from Eq. (2). The XUV signal for the ideal generation is expected in the region \(L_{coh} > L_{med}\). These curves represent the best possible signals and show, that pressure up to \(\sim 35\) mbar is generally desired. We took the intensity from the region II of Fig. 1b and calculated the expected XUV signal based on corresponding ionization and coherent length \(L_{coh}\) using Eq. (2) (full blue curve of Fig. 2b). This XUV signal is well below the possible maximum. If we modify the medium by pre-ionizing it by 8%, the expected signal for the same laser pulse enhances significantly (red curve of Fig. 2b) and moves into the region of ideal absorption limited phase-matched generation, where the signal of 17th harmonic is 8 times stronger than its signal without the pre-ionization.

Experimental results

Experimental verification was performed at the High Harmonic Beamline at the ELI Beamlines facility29 employing a Ti:Sapphire laser (Legend Elite Duo) with up to 40 fs pulses with central wavelength of 798 nm and energy of 10 mJ focused by a spherical mirror with the focal length of 5 m to drive the HHG. A 15 mm long glass capillary with a 1 mm inner diameter filled with Kr or Ar and hollow electrodes on both sides was placed in the laser focus for generation of a pre-ionized medium (see Fig. 3). Note that the inner diameter of such a short capillary was too large for the laser beam to be guided, so its propagation can be considered as free-space propagation in contrast to the work reported in16 and17 where the laser was guided in a narrower capillary with a highly ionized medium.

Figure 3
figure 3

Schematics of the gas-filled capillary with two hollow copper electrodes connected to the RLC circuit driving the capillary discharge to generate the pre-ionized medium for HHG.

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The electrical discharge was generated by a RLC circuit (Fig. 3)30, where energy is stored in a capacitor bank and then discharged through a 100 Ω resistor into the capillary. The current pulse had a 45 ns rising edge followed by an exponential decay with a characteristic time of 0.5 µs. The peak current was defined by charging voltages of 4 kV or 5 kV, corresponding to peak currents of 40 A or 50 A, respectively. The MHD simulations31 were used to find a current pulse that generates a discharge with the desired ionization degree. Considering this current pulse, the driving electrical circuit was designed accordingly. Electrical capillary discharge was selected as the most suitable pre-ionization method due to its overall homogeneity (confirmed by MHD simulations), its repeatability, and its simple construction allowing repetition rates up to 1 kHz.

The discharge was synchronized with the laser so that the laser pulse interacted with the pre-ionized target 250 ns after the breakdown of the discharge. The generated harmonic radiation was characterized by an imaging flat-field XUV spectrometer29. The pressure inside the capillary and the peak current of the discharge were varied to control the phase-matching of HHG by varying the pre-ionization of the medium while keeping the driving laser parameters unchanged.

The laser intensity during the experiment was set by aperturing the beam to reach harmonic the cut-off frequency corresponding to the 19th harmonic order in krypton and the 23rd harmonic order in Ar. The strength of the 17th harmonic signal measured in the experiment with Kr together with the estimate of the degree of ionization from the inverse of Eq. (5) (Fig. 4a dashed purple line) indicate that for low gas pressures (below 25 mbar), the ionization is too high for efficient HHG. For pressures higher than 25 mbar, the combined ionization from the discharge and the laser propagation falls exactly into the phase-matching region, leading to optimum phase-matched generation. In particular, the highest XUV signal of the 17th harmonic is seen at 35 mbar of krypton for the 17th harmonic generated in krypton.

Figure 4
figure 4

(a) Experimentally measured intensity of the 17th harmonic order as a function of pressure for HHG in Kr without pre-ionization (black curve) and in pre-ionized medium with discharge peak currents of 40 A (red diamonds). Red line (spline) is added to guide the eye. Estimate of degree of ionization in the capillary from the inverse of Eq. (5) with 35 mbar considered as maximum of the signal (purple dashed curve). (b) Enhancement factor for H15 (blue), H17 (orange), and H19 (green diamonds) in krypton with 40 A peak current. (c) Enhancement factor for H17 (blue), H19 (orange), and H21 (green squares) in argon with 50 A peak current. Each experimental point corresponds to the strongest shot from a set of ten laser shots.

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Figure 4b summarizes the enhancement factor for the H15, H17, and H19 in krypton with 40 A current discharge as a function of pressure. Similarly, Fig. 4c represents the measurements with 50 A discharge in argon for the H17, H19, and H21. For krypton, the ideal phase-matching is achieved when the total ionization degree is 9.6%, 8.7%, and 8.0% for the presented orders. Similarly, ideal ionization values for these harmonics generated in argon are 7.3%, 6.3%, and 5.6%, respectively. The highest measured XUV signal enhancement in argon (Fig. 4c) is by a factor of 5, compared to a factor of 6 observed in krypton (Fig. 4b). The optimum pre-ionization is seen for all three harmonic orders at 35 mbar and 40 A current pulse in krypton and 40 mbar and 50 A current pulse in argon, demonstrating the improved phase-matching conditions for a broad spectral range. Such a broadband effect is expected because the intervals of degree of ionization for close-by harmonics do not change significantly.

Numerical simulations

To support the analytical model and experimental data we performed numerical simulations containing both microscopic and macroscopic aspects of HHG. First, we computed the propagation of the driving laser in the medium with a cylindrically symmetric unidirectional solver32. This model intrinsically contains the dispersion of the driving fields by neutrals, plasma generation, and propagation in the plasma and all geometrical effects as well. Once both plasma and intensity profiles in space and time are calculated (on-axis values of the intensity are shown in Fig. 1a), one can study the phase-matching conditions.

To estimate the coherence length in the whole medium, we add the microscopic dipole phase obtained by the Saddle point approximation as explained in33,34 and linear dispersion for a given harmonic order (see Supplement S1 for more details). The spatial profile of coherence length of the 17th harmonic generated in Kr at a gas pressure of 35 mbar and peak laser intensity of 0.9 × 1014 W/cm2 is shown in Fig. 5a (no pre-ionization) and Fig. 5b (with pre-ionization, note the different color scale). As seen in those figures, the coherence length exceeds 30 mm i.e., double the length of the medium, in a large area for the 8% pre-ionized case. On the contrary, without the pre-ionization, it stays always below 1.5 mm in the region II (second half of the medium). Therefore, the XUV is efficiently propagating, and the signal is added constructively from the extended region only in the pre-ionized case.

Figure 5
figure 5

Spatial distribution of the coherence length Lcoh at the peak of the laser pulse for the 17th harmonic generated in 35 mbar of Kr with vacuum intensity of 0.9·1014 W/cm2 and vacuum focus positioned at the center of the medium. Pre-ionization of the medium was set to 0% (a) and 8% (b). (c) The on-axis coherence lengths after 15 mm propagation at the peak of the laser pulse as a function of the input-laser peak intensity and medium pressure. \(L_{coh} = L_{med}\) (dotted line), \(L_{coh} = 2L_{med}\) (dashed line), and \(L_{coh} = 4L_{med}\) (full line).

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These simulations reveal as well that the major contribution to the phase-mismatch is provided by the polarizabilities of the medium (neutrals and plasma). Other effects, such as the geometrical phase of the laser beam and the intensity-dependent phase of the harmonic dipole become significant only in the optimum pre-ionization case, where the indices of refraction of the laser and the harmonic field become similar (see Supplement S1 for details). This fact justifies the application of the simple analytical model (Eq. 2), which is a plane-wave approximation of the signal built-up, in region II.

The on-axis coherence length at the end of the 15 mm-long krypton medium pre-ionized to 8% obtained from full numerical laser propagation code with various pressures and input laser intensities is shown in Fig. 5c. Let us recall also Fig. 1c, where the full on-axis profile of the associated phase mismatch \(\left| {{\Delta }k} \right|\) is shown for several points of Fig. 5c. As seen in the figure, the generation with this level of pre-ionization is optimum for a broad range of pressures and driving laser intensities and demonstrates the robustness of our phase-matching scheme.

Furthermore, we performed a set of numerical simulations of the whole process. The level of pre-ionization at various pressures was calculated by a one-dimensional dissipative magnetohydrodynamic (MHD) code utilizing a two-temperature model (ion and electron) and the following dissipative processes: electron thermal conductivities, Joule heating, Nernst and Ettinghausen effects, radiation losses, and viscosity35. In the next step, we have used a full numerical model describing the HHG experiment also including the exact quantum dipoles from 1D time-dependent Schrödinger equation36 and numerical XUV propagation to the far-field analogical to37. The resulting ionization and XUV signal as a function of the pressure is shown in Fig. 6a.

Figure 6
figure 6

(a) Simulated XUV signal and ionization level as a function of pressure. Full black line shows the signal without pre-ionization. XUV signal with pre-ionization is denoted by diamonds. Red line (spline) is added to guide the eye. The dotted line shows the pre-ionization of the medium obtained from MHD simulations. The dashed line shows the estimate of the ionization on axis at the peak of the pulse in time and at the end of the medium. (b) Harmonic spectrum at 35 mbar in Kr comparing experimental data of optimal spectra (40 A peak current) with numerically computed spectrum for 8% pre-ionization.

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The simulated XUV signal as a function of pressure (Fig. 6a) shows a significant signal enhancement due to pre-ionization and agrees with the one-dimensional model (Eq. 1) and experimental observations (Fig. 4a). The highest amplification occurs around 8% of ionization, which agrees with the degree of ionization estimated from the experiment using an independent simplified model based on Eq. 5. There is a small shift of the maximum of the simulated signal in pressure as compared to the experimental results, caused by possible underestimation of the degree of ionization in the MHD simulations of the Kr discharge, which were used as the input for the complex simulations. The simulations also show that ionization by the laser field (in region II at the end of the medium) is small compared to the pre-ionization set by the discharge and alone cannot reach an optimal level of ionization.

Figure 6b compares the simulated and experimentally measured XUV spectra for Kr at a pressure of 35 mbar and the value of pre-ionization set to the optimal 8%. We found a good agreement between theory and experiment in the whole spectral region studied. In the experiment we reached an enhancement of the XUV signal by a factor of 6. The simulated spectrum shows an increase of a factor of up to 8, which is in excellent agreement with the value from the analytical model shown in Fig. 2c.

The pre-ionization levels considered here are too low to affect the optical field ionization rates. Therefore, the propagation of the IR laser pulse is not affected by the pre-ionization of the medium and its wavefront, as well as the wavefront of the harmonic beam, remain unchanged. A comparison of far-field intensity patterns of the simulated harmonic beams generated at various pre-ionization levels (not shown here) confirmed their shape-independence of the medium ionization degree considering low ionization levels relevant to our study.

Discussion and conclusion

We have shown experimentally and derived theoretically that a few percent of a medium pre-ionization leads to significant improvement of the phase matching and, consequently, an increase of the generated XUV signal. We have employed a weak capillary discharge to achieve a controllable pre-ionization of the generating medium and reached up to six-fold enhancement of the XUV signal.

To achieve the desired pre-ionization, other methods could also be used, such as radio-frequency discharge38, a sequence of ionizing laser pulses39, or mixing gas with low ionization potential acting as an electron donor on the rising edge of the laser pulse15. As long as the medium remains homogeneous, the weak uniform pre-ionization of the medium does not affect the propagation of the laser. Therefore, it does not affect the wavefront of the generated XUV beam, the divergence of the harmonic beam, and the position of the XUV source.

Controlling the phase-matching by pre-ionizing the generating medium. It increases the HHG conversion efficiency in the absorption-limited generation in long media. As the pre-ionization does not affect any other aspect of the generation, this mechanism turns out to be an independent tuning parameter of the experiment, paving the way for enabling photon-hungry applications with compact HHG sources.

Even though our method was verified for generation of low order harmonics in krypton and argon, it can be also applied for other gases, such as xenon for low harmonic orders, or neon, where we see a potential to significantly enhance the generation up to 70 eV. Moreover, the broadband nature of the method ensures that the temporal structure of the XUV pulse remains unchanged, which opens a way for the implementation of our method within the generation of single attosecond pulses. Another possible application might be HHG with picosecond laser pulses that suffers from fast over-ionization of the medium resulting in strong defocusing and overall low conversion efficiency. Interaction of lower intensity ps pulses within a properly pre-ionized medium could significantly improve the generation of spectrally very narrow harmonics.