1. Introduction

Advances in ultrafast laser technology [1–3] have made it possible to study and control the electronic properties of matter with unprecedented precision. At the heart of these developments lies the ability to generate low-noise trains of ultrashort high-intensity laser pulses, enabling access to strong-field phenomena and the generation of radiation in spectral regions that are otherwise hard to reach. For example, such pulses can be used to generate attosecond pulses at high-harmonics of their carrier wave frequency, in the extreme ultraviolet (UV) and soft-X-ray regions—essential for a plethora of experiments in ultrafast science, such as transient-absorption and photoelectron spectroscopy [4,5], laser-induced electron diffraction [6] and imaging of the bound electron response [7]. Intense ultrashort pulses are also crucial for spectroscopy in the mid-infrared region, via broad supercontinua [8] and frequency combs [9], and in material processing [10]. Pump lasers such as Ti:sapphire systems are restricted in average power, with the result that they can deliver mJ-level sub-100 fs pulses only at kHz-level repetition rates, although in many applications MHz repetition rates would be highly advantageous, permitting faster data acquisition, higher average powers and improved spectroscopic resolution [11,12].

Recent progress in ytterbium- and thulium-based solid-state lasers [13–15] has made possible highly stable and “turn-key” systems delivering pulses with tens or even hundreds of µJ energies at megahertz repetition rates and average powers above 100 W. Schemes for nonlinear temporal compression, involving multi-pass cells [16] or hollow-core fibers [17,18], are often used to reduce the pulse duration to a few femtoseconds. This is particularly important in the case of high-average-power ultrafast lasers, which typically produce 200 fs pulses (much longer than for Ti:sapphire lasers). Compression, which is often accomplished in several stages [19–21], becomes challenging when both the average power and the peak intensity are high, because even a small fraction of the pulse energy deposited in the nonlinear medium can cause thermal instabilities. In noble-gas-based compression systems, energy deposition is mainly caused by photoionization [16,22]. Despite the short free-electron lifetime (∼10 ns), photoionization can trigger transient refractive index changes in the gas that last for much longer times. Several previous studies of free-space laser filaments [23–26] have revealed that the energy deposited after recombination is concentrated in a small volume around the laser beam axis, causing strong heating and a pressure spike in the gas. The release of an acoustic pulse, propagating radially away from the axis, equalizes the pressure typically within few tens of µsec. In contrast, thermal diffusion is much slower—single-shot measurements of the post-recombination effects in an Ar-filled hollow-core photonic crystal fiber (HC-PCF), in which the gas is additionally spatially confined, have revealed a timescale of hundreds of µsec for the gas to fully relax [27,28]. Within this time interval the transverse gas-density distribution follows the inverse temperature profile, resulting in a density depression in the vicinity of the beam axis. If this duration exceeds the inverse laser repetition rate, transient index changes will affect the dynamics of successive pulses and may build up over many shots. Such effects have so far only been indirectly confirmed in gas-filled HC-PCF, when nonlinear pulse propagation is strongly affected [22,29], although reduced energy throughput and instabilities at high repetition rates frequently occur also in other high-power gas-based systems [30–33].

In this work we explore, both experimentally and numerically, the so far unexplored buildup of post-recombination refractive index changes in noble gases, generated at MHz repetition rates at the free-space focus of an Yb fiber laser beam. To study the dependence of the buildup dynamics on pulse repetition rate and peak intensity, we interferometrically monitor time-dependent refractive index changes in the ionized gas using a transversely propagating continuous-wave (CW) probe laser. The setup is designed to exclude effects of nonlinear pulse propagation on photoionization, making it possible to investigate pulse compression in different parameter ranges. To better understand the limits of repetition-rate scaling observed during compression to few- or single-cycle durations, we first explore the buildup of post-recombination effects caused by pulses tens-of-femtoseconds long. The results indicate that strong recombination-driven heating of the gas already occurs at repetition rates as low as 100 kHz. To suppress this we explore a simple yet effective mitigation strategy employing lighter gases to reduce ionization [22,31]. Pump pulses suitable for few-cycle compression at megahertz repetition rates can be produced in initial compression stages, typically relying on spectral broadening of hundreds-of-femtoseconds long laser pulses in gas-filled Heriott-type cells or hollow-core fibers. In a detailed study of the characteristic parameters we show that, despite the typically lower peak intensities at long pulse durations, a buildup of post-recombination refractive index changes in the gas cannot be neglected, particularly at higher pressures. The findings provide a deeper understanding of the effects of post-recombination thermal buildup and are important for optimizing the design of gas-based compression stages operating at high repetition rates.

2. Experimental setup and techniques

In the experiments, pulses with ∼250 fs duration (which we designate as “long”) at 1030 nm are directly delivered by an Yb fiber laser (Active Fiber Systems GmbH), capable of providing up to 100 W of average power. The laser is operated at repetition rates between 100 kHz and 1.92 MHz with a maximum pulse energy of ∼44 µJ, attaining a “vacuum” peak intensity (see Appendix 1 for definition) of up to ∼70 TW/cm² at the laser beam axis. These pulses can be additionally compressed to a duration of ∼26 fs in gas-filled HC-PCF [22], with a maximum energy of ∼19 µJ that corresponds to a (vacuum) peak intensity of ∼210 TW/cm². We designate these pulses as “short.”

An achromatic lens (f = 75 mm) focuses the laser pulses into a thick-walled silica capillary (660 µm outer diameter), mounted inside a gas cell and placed so that the focal spot (on the capillary axis) is located ∼1.2 mm away from the capillary input face. Neglecting nonlinear effects in the path of the ionizing beam, a collimated Gaussian waist of ∼4.2 mm was measured using a beam profiler, leading to a Rayleigh length of ∼0.5 mm and a 1/e² intensity diameter of ∼24 µm at the focus—much smaller than the bore of the capillary (D = 136 µm). As such, the ionizing pulses propagate within few Rayleigh lengths effectively in free-space, thus the main purpose of the capillary is to confine the gas in the vicinity of the laser focus. At that location we observe bright recombination luminescence for both “short” and “long” pulses. The photographs in Fig. 1(a), taken through the transparent lid of the gas cell, show the characteristic colors of the different gas species (gray for Kr, grayish-blue for Ar and orange for Ne). Luminescence provides only a weak channel for recombination and energy dissipation follows mainly from inter-atomic collisions, causing heating of the gas [34]. To investigate the buildup dynamics at high repetition rates, we placed a fast shutter (for average powers < 2W) or a chopper wheel (for higher average powers), both with 50% duty cycle, in the pump beam before the focusing lens.

lses as “short”.

 

Fig. 1. Detection of post-recombination effects in photoionized gases. (a) Recombination-driven luminescence of Kr, Ar and Ne, photoionized inside a glass capillary (top view). (b) Mach-Zehnder interferometer setup to probe refractive index changes in the gas. Components: PL, probe laser at 635 nm; GC, gas cell; CAP, glass capillary; RP1/2, Rochon prisms serving as beam splitters; λ/2, half-wave plate controlling the relative power in the interferometer arms; L, lens (f = 45 mm); POL, linear polarizer; PD, amplified photodiode (rise time ∼60 ns).

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2.1 Side-probe interferometer

Recombination-driven refractive index changes in the gas are probed transversely through the capillary walls using a novel free-space Mach-Zehnder interferometer [Fig. 1(b)]. Unlike its fiber-based counterpart in [27], this interferometer requires no active feedback stabilization, because the probe and reference beams follow almost identical paths, strongly suppressing differential path-length fluctuations.

Linearly polarized CW probe light at 635 nm is delivered from a narrow-linewidth diode laser (∼30 pm linewidth, stabilized by a volume Bragg grating). A Rochon prism (RP1) is used to split the collimated probe light into two orthogonally polarized beams, emerging at ∼1.4° relative angle. Placing RP1 in the back focal plane of a lens produces two parallel beams spaced by ∼1 mm in the y-direction. The lower beam is incident perpendicular to the capillary surface (to avoid deflections) and focused at the center of the capillary, acting as a probe of transient refractive index changes in the gas. We estimate a spot diameter (full width at half maximum, FWHM) of ∼5 µm. The upper beam is focused above the capillary and thus shielded from ionization effects, so serves as the reference beam. After re-collimation using an identical lens, the two beams are recombined at a second Rochon prism (RP2) and superimposed at a polarizer. In this way, phase changes between probe and reference beams are converted into variations in transmitted intensity, detected by a photodiode and followed over time on an oscilloscope. To optimize the phase sensitivity, a λ/2 plate is placed before RP1 to balance the probe power in the interferometer arms. The interferometer working point is adjusted to quadrature by shifting RP2 slightly along the x-axis.

Figures 2(a) and 2(b) show the temporal variation in probe phase for “short” and “long” pulses in Kr, both recorded close to the maximum available peak intensity. After unblocking the pump laser (at time t = 0) the gas in the focal region heats up abruptly and atoms are expelled in all directions, thus the gas density in the vicinity of the ionizing beam axis drops. This causes the optical path-length of the transverse probe beam (proportional to the integrated density change along its line of focus) to fall, as seen in the rapid negative buildup Δφ_B in probe phase. When the laser is blocked [after 260 ms in Fig. 2(a) and 60 ms in Fig. 2(b)] the gas cools down, the atoms diffuse back and the optical path-length recovers, causing a rapid recovery Δφ_R in probe phase. The measured “rise” and “fall” times [see lower panels of Fig. 2(a) and 2(b)] are longer than the actual response of buildup and recovery, because they are convolved with the slow shutter / chopper response (∼2.5 ms measured for 10 to 90% pump transmission). We note that since unblocking of the ionizing beam takes a finite time, the available peak intensity gradually builds up, causing a slow-down of the actual temporal heating dynamics during buildup. However, this does not affect either the post-buildup temperature or the Δφ_B values (see section 3.2 for measured temporal dynamics of the buildup). The shorter more intense pulses [Fig. 2(a)] produce more ionization and therefore a higher thermal load during recombination. The result is a slow positive drift in phase after the initial rapid buildup, reaching a maximum value of ∼0.15 rad after 250 ms. We attribute this to heating of the silica capillary walls: an average temperature rise of only ∼3 K would account for the observed phase change [35]. For the longer pulses, the ionization level and the thermal load are lower, the capillary heats up much less. As a result, the probe phase remains constant after the initial buildup [Fig. 2(b)], indicating that further heat accumulation is balanced by dissipation.

 

Fig. 2. Temporal response of post-recombination thermal buildup in Kr. Time traces of interferometric phase change, obtained (a) with ∼26-fs-long ionizing pulses of ∼185 TW/cm² (vacuum) peak intensity (18.5 µJ energy) at 100 kHz repetition rate in 1 bar and (b) with 250-fs-long pulses of ∼66 TW/cm² peak intensity (38 µJ energy) at 505 kHz in 3 bar of Kr. The pump laser is blocked in the dark gray shaded regions. Upper panels show the time-evolution over a full shutter / chopper cycle, lower panels plot zoom-ins to the dynamics over the initial few ms. (c), (d) Zoom-ins to (a,b) showing single-shot effects.

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2.2 Estimation of mean gas temperature from single-pulse heating effects

The response time of the interferometer is sufficiently short to allow single-pulse dynamics to be monitored. Figure 2(c) plots 12-µs-long sections from the trace in Fig. 2(a) at two different times: t₁ = 0.69 ms, just after unblocking the laser has started, and t₂ = 5.29 ms, just after the initial negative buildup in phase. Each individual ionizing pulse heats the gas sufficiently to cause a rapid drop in phase, followed by a slower diffusion-driven recovery. We note that this effect can be only observed if the probe beam in the sample arm and the ionizing laser beam are focused at the same vertical (y) position [see Fig. 8(d)], i.e., when the alignment is good (the effects of a small vertical offset between the foci will be discussed in Appendix 2). Single-pulse heating is accompanied by fast oscillations in probe phase that are particularly pronounced in the vicinity of the phase minimum. These are caused by a recombination-driven acoustic pulse that is reflected multiple times at the gas-silica interface while decaying in strength. This phenomenon is not seen in free-space systems, but has been observed at low repetition rates in gas-filled HC-PCF [27,28]. While the single-shot response was clearly detectable for “short” pulses, for the “long” pulses it was much weaker, indeed below the noise limit (the standard deviation of the trace in Fig. 2(b) with the pump light blocked is ∼4 mrad). Consequently, it was only possible to uncover the single-shot response by averaging it over many [80 in Fig. 2(d)] successive pulses.

The time τ_ac taken for the acoustic pulse to travel across the bore of the capillary is observed to decrease both with increasing intensity in the quasi-steady state (for time t > 4 ms in Fig. 2) and with time during the initial negative phase buildup. For example, this “bounce” period τ_ac in Fig. 2(c) reduces from ∼496 ns at time t₁ to ∼416 ns at t₂. This effect stems directly the temperature dependence of the speed of sound, which for an ideal gas follows the relationship [36]:

For the measurement in Fig. 2(c) we obtain mean temperatures of ∼455 K close to the beginning and ∼645 K at the end of buildup. At higher repetition rates [e.g. 505 kHz in Fig. 2(d)] only few bounce periods can be detected between successive pulses, making it more difficult to obtain a reliable signal average and resulting in a less accurate estimate of τ_ac. Note that in a gas-filled HC-PCF pumped at 1 kHz repetition rate (too low to observe any buildup) the measured values of τ_ac agree well with calculations using the speed of sound at room temperature [27,28].

3. Results and discussion

3.1 Cumulative recombination heating in gases photoionized by short pulses

We explored the dependence of recombination heating on the vacuum peak intensity in a capillary filled with 1 bar of different gases and pumped with ∼26 fs pulses (such durations are often used for soliton self-compression in gas-filled HC-PCF [21,22] and hollow capillaries [18]). A combination of thin-film polarizer and a half-wave plate was used to reduce the peak intensity in equal steps from ∼200 to ∼40 TW/cm², at which point the detected phase change dropped below the noise level. We note that the highest peak intensities in the measurements are comparable to those reached by self-compressing pulses in gas-filled HC-PCF. However, spectral broadening [38–40] and self-compression [18] of tens-of-femtosecond-long pulses at millijoule energies in gas-filled hollow capillaries can produce even higher intensities.

The results of our measurements (summarized in Figs. 3 and 4) directly reflect the changing free electron density, since all other parameters were kept constant. The on-axis free electron density N_e, calculated using Perelomov, Popov, Terent’ev (PPT) ionization rates [41], is plotted against peak intensity for three different noble gases in Fig. 3(a). In Kr, the gas with the lowest ionization potential (I_P), it varies over four orders of magnitude, from ∼2 × 10¹⁴ cm⁻³ to ∼5 × 10¹⁸ cm⁻³. Recombination-driven heating can be strongly reduced if gases with a higher ionization potential, such as Ar or Ne, are used [22,27,31]. In Fig. 3(a) we see that replacing Kr (I_P = 14.00 eV) with Ar (I_P = 15.76 eV) reduces N_e by a factor of 10. If Ne is used (I_P = 21.56 eV), N_e is reduced by more than four orders of magnitude. For comparability and to avoid heating-related instabilities at higher pressures (see Appendix 3), the pressure is kept at 1 bar (the effects of higher pressure will be studied in the next section).

 

Fig. 3. Cumulative post-recombination heating in different noble gases. Pump pulses ∼26 fs in duration were used to photoionize 1 bar of Kr, Ar and Ne at laser repetition rates between 100 and 505 kHz. (a) Room-temperature on-axis free electron density versus vacuum peak intensity in Kr (black), Ar (red) and Ne (blue), calculated with PPT ionization rates. (b) Mean temperature as a function of vacuum peak intensity in 1 bar of Kr (left: 100 kHz, right: 505 kHz). The data points measured at the beginning (open symbols) and end of buildup (filled symbols) are averaged over three consecutive measurements, error-bars plotting twice the standard deviation (99% confidence interval). Also shown are the results of simulations with (dashed lines) and without (solid lines) gas pre-heating at time t = 0 (to 360 K at 100 kHz and 380 K at 505 kHz). (c) Same as (b) but for 1 bar of Ar at three different repetition rates (left: 100 kHz, center: 253 kHz, right: 505 kHz). The pre-heating temperatures in this case were 360 K at 100 kHz and 253 kHz, and 340 K at 505 kHz.

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Fig. 4. Phase buildup in different noble gases photoionized by ∼26 fs long pulses, plotted against peak intensity for different repetition rates. Experimental results are shown for 1 bar of (a) Kr and (b) Ar. The filled symbols mark data averaged over three consecutive measurements (red circles, 100 kHz; purple crosses, 253 kHz; black squares, 505 kHz; green diamonds, 1.01 MHz; blue asterisks: 1.92 MHz), error bars showing twice the standard deviation. Also shown are numerical simulations with (dashed lines) and without (solid lines) gas pre-heating. The pre-heating temperatures are given in Fig. 3. The gray line and x's in (b) are simulations and measurements made in 1 bar of Ne at 1.92 MHz, when no buildup was detectable.

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We next used Eq. (2) to calculate the mean gas temperature from the measured values of τ_ac. The retrieved temperatures are plotted versus peak intensity in Fig. 3(b) at repetition rates of 100 kHz (left panel) and 505 kHz (right panel). They vary from ∼410 K to 650 K at 100 kHz and from 450 K to 1100 K at 505 kHz over the intensity range explored, and show stronger temperature rises at high intensity, as expected from the nonlinear trend of the free electron density [Fig. 3(a)]. Note that temperatures exactly on-axis will be even higher. Figure 3(c) plots the retrieved temperature rise against peak intensity for a capillary filled with 1 bar of Ar. Although the increase with repetition rate is similar to that seen in Kr, recombination heating is clearly weaker: at 100 kHz, a peak intensity of 185 TW/cm² yields ∼463 K in Ar, roughly 180 K lower than in Kr (∼645 K). The difference is even greater for 178 TW/cm² at 505 kHz: ∼534 K in Ar, compared to ∼860 K in Kr. At repetition rates above 505 kHz for a given pulse energy, when for both gases recombination heating is even stronger, single-shot effects could not be clearly resolved in the experiments, preventing calculation of $\bar{T}$ . The evolution of the gas density depression could however be tracked by measuring the dependence of the buildup phase Δφ_Β on repetition rate and peak intensity (Fig. 4). For Kr at 100 kHz [Fig. 4(a)] the magnitude |Δφ_Β| of the buildup amplitude increases monotonically with peak intensity, but interestingly it saturates at higher repetition rates (to a value of ∼0.2 rad). This is partly because at high temperature the gas density, and hence the refractive index, becomes less sensitive to temperature variations. In addition, the onset of defocusing at the anti-waveguide formed by the density depression, experimentally investigated in [29], acts to reduce the phase buildup by clamping the peak intensity as the pulse energy increases above a certain level [∼12 φJ in Fig. 4(a)]. These observations also apply when Kr is replaced with Ar at 1 bar [Fig. 4(b)], when the buildup amplitude |Δφ_Β| is again observed to increase both with peak intensity and laser repetition rate, showing signs (albeit weaker) of saturation. Note that the gas density changes seen by the probe beam in Kr and Ar cannot be compared simply by relating the measured |Δφ_Β| values because the linear susceptibility is species-dependent. Taking this into account, pulses of e.g. 185 TW/cm² peak intensity yield at 100 kHz repetition rate |Δφ_Β| = 164 mrad in Kr and 33 mrad in Ar, so that the integrated density change along the line of the probe focus is in Ar three times smaller than in Kr. These observations are also consistent with the lower post-buildup temperatures in Ar. When in contrast Ne is used as filling gas, no thermal buildup could be detected even at the highest repetition rate [1.92 MHz, Fig. 4(b)], as one would expect since the free electron densities are more than three orders of magnitude lower than in Ar [Fig. 3(a)].

To better understand the observations and to obtain the spatiotemporal evolution of the gas temperature, we numerically modelled heat diffusion in a radially symmetric two-dimensional geometry (see Appendix 2 for details). The continuous curves in Fig. 3(b) represent the simulated post-buildup temperatures at different repetition rates in 1 bar of Kr when the gas at time t = 0 is at room temperature (294 K). At high peak intensities and 100 kHz repetition rate, the measured temperatures are only ∼50 K lower than in the simulations. While there is good qualitative agreement for both gases, the measured temperatures in Kr at 505 kHz and in Ar at all repetition rates are slightly higher than expected (Fig. 3). The numerical model also reproduces qualitatively the peak-intensity dependence of |Δφ_Β|, even though it markedly overestimates the values, particularly at higher repetition rates (Fig. 4). These discrepancies are predominantly due to pre-heating of the gas before time t = 0 (this is because data acquisition began after several skipped several chopper / shutter cycles during which the interferometer drifted.) To account for this effect in our numerical simulations, we estimated the upper limit of the pre-heating temperatures from the measured average single-shot response at the beginning of the buildup. The shutter- and chopper-related slow-down in temporal buildup dynamics facilitates this (see discussion in 2.1).

3.2 Effects of pulse duration and gas pressure on thermal buildup

Post-recombination effects are typically neglected in spectral broadening of pulses hundreds-of-femtoseconds long at relatively moderate peak intensities. When the repetition rate is high, however, this assumption is no longer valid. Simulations for a single pulse with a peak intensity of ∼70 TW/cm² in 1 bar of Kr [Fig. 5(a)] predict a free electron density of 1.2 × 10¹⁷ cm⁻³, close to 10 times higher than for ∼26 fs long pulses of identical peak intensity [Fig. 3(a)]. Avalanche ionization (not included in the model and more pronounced for longer pulse durations) is likely to result in even higher values. Significant recombination heating is detected in the setup at 505 kHz repetition rate for 250 fs pump pulses, despite the weak photoionization rate [Fig. 5(b)]. The measured buildup amplitude |Δφ_Β| is almost five times greater than for ∼26 fs pulses, despite the peak intensity being slightly lower [70 TW/cm² compared to 83 TW/cm², see Fig. 4(a) and 5(b)]. The mean gas temperature reached (455 K) is also marginally higher than for 26 fs pulses (439 K).

 

Fig. 5. Effects of cumulative recombination heating in Kr ionized by long laser pulses. (a) On-axis free electron density versus peak intensity for pressures of 1, 3, 5, 7.6 and 10.2 bar, calculated at room temperature for a single 250 fs long Gaussian pulse using PPT ionization rates. (b) Buildup Δφ_Β versus peak intensity for trains of pulses. The data points were measured at 1 bar and repetition rates of 505 kHz (red circles), 1.01 MHz (black squares), 1.92 MHz (blue asterisks). (c) Mean temperature rise as a function of peak intensity, measured at 1 bar and 505 kHz. The curves are numerically modelled for pre-heating gas temperatures (at t = 0, see Fig. 2) of 340 K (dashed) and 294 K (solid). Data points in (b,c) are averages of three consecutive measurements, error bars showing twice their standard deviation.

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Achieving similar spectral broadening in different gas species requires the third-order nonlinear coefficient n₂ to be comparable. For instance, at ambient conditions Ar requires a ∼2.7 times higher density than Kr. To explore the effects of increasing gas density, we carried out additional measurements at Kr pressures of 3, 5, 7.6 and 10.2 bar, pumping with 250 fs pulses at 505 kHz repetition rate. We note that typical peak intensities for spectral broadening of hundreds-of-femtoseconds long pulses in gas-filled hollow capillaries lie within our investigated (vacuum) peak intensity range (∼17 to 70 TW/cm²) [19,38–40], whereas in HC-PCF they are often lower (few TW/cm²) [21,22]. The experimental and numerical gas temperatures before (estimated) and after build-up follow the same overall trend [Fig. 6(a)]. Theory agrees much better with experiment when the gas is preheated to ∼350 K (dashed lines) rather than 294 K (solid lines). As is discussed in Appendix 2, the temperature increase directly after recombination for a single laser shot is almost independent of gas pressure. The slightly higher post-buildup temperatures at higher pressure, also predicted by the model, are caused by a drop in thermal diffusivity (inversely proportional to density), which in turn increases the time needed to reach the post-buildup state. We confirmed this in experiments at a constant peak intensity of ∼50 TW/cm² for a series of Kr pressures from 1 to 10.2 bar [Fig. 6(b)], over which pressure range the buildup time was observed to increase to almost 4 ms.

 

Fig. 6. Effects of increasing Kr pressure on cumulative recombination heating caused by 250 fs pulses. (a) Effective temperatures versus vacuum peak intensity, measured at the beginning (open symbols) and end of buildup (filled symbols) for 505 kHz repetition rate and different pressures. (b) Temporal evolution of probe phase for different pressures at ∼50 TW/cm² peak intensity, 1.01 MHz repetition rate. Here, a higher chopping rate (40 Hz) resulted in a faster chopper response (∼0.5 ms). (c) Buildup Δφ_Β versus peak intensity at different Kr pressures and 505 kHz repetition rate (inset: Δφ_Β versus pressure for a fixed post-buildup effective temperature of ∼425 K). Data points in (a,c) were averaged over three consecutive measurements, error bars plot twice the standard deviation (red circles, 1 bar; pink crosses, 3 bar; black squares, 5 bar; green diamonds, 7.6 bar; blue asterisks, 10.2 bar). Lines in (a,c) show the results of numerical simulations with (dashed) and without (solid) pre-heating of the gas at time t = 0 (pre-heating temperatures: 3 bar, 355 K; 5 bar, 360 K; 7.6 bar, 340 K; 10.2 bar, 330 K).

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Unlike the post-buildup temperature, the buildup phase |Δφ_Β| was observed to increase strongly with gas pressure [Fig. 6(c)]. To interpret this we compared the values of |Δφ_Β| at different pressures in cases when the post-buildup temperature was ∼425 K. Under these circumstances numerical modelling predicts that the gas density profile across the capillary will have different peak values but very similar normalized shapes. Measurements of |Δφ_Β| [inset in Fig. 6(c)] indicate that, despite the unchanged normalized gas density profile, the drop in peak gas density and therefore in refractive index grows almost linearly with pressure.

Experiment and theory disagree increasingly at higher vacuum peak intensities and particularly at high pressures [Figs. 5(c) and 6] because larger absolute changes in gas density cause greater refractive index changes and stronger defocusing of the ionizing laser beam. This effect clamps the actual peak intensity and photoionization rate attainable in the experiments, resulting in the observed saturation in |Δφ_Β| and post-buildup temperature with pressure and (calculated) vacuum peak intensity. No such saturation is seen in the numerical simulations because defocusing was not included in the theoretical model.

4. Discussion and conclusions

We have shown that post-recombination thermal effects can be suppressed by using lighter noble gases with higher ionization energies. However, reproducing comparable pulse propagation dynamics would require a higher gas pressure, considerably slowing down thermal diffusion (see Appendix 2). To avoid this, H₂ is a good choice, offering similar nonlinearity and ionization potential to Ar, but a thermal conductivity that is almost ten times higher. Figure 7 compares the measured buildup response of Ar and H₂ for closely similar values of nonlinearity and dispersion. There is clearly a much smaller buildup for H₂, only weakly dependent on the repetition rate. We note that it is also possible to obtain simultaneously high nonlinearity along with high thermal conductivity by mixing light and heavy gases [42].

 

Fig. 7. Direct comparison of the buildup of thermal effects in Ar and H₂. Buildup Δφ_Β in relative probe phase as a function of vacuum peak intensity and pump-laser repetition rate in (a) 1 bar of Ar (I_P = 15.76 eV) and (b) 1 bar of hydrogen (I_P = 15.43 eV). From the measured values of |Δφ_Β| at 1.92 MHz repetition rate and ∼184 TW/cm² peak intensity, a five times lower integrated gas density change in H₂ compared to Ar is inferred. Data points are averaged over three consecutive measurements of the buildup, error bars showing twice the standard deviation.

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Fig. 8. Numerical modelling of recombination-driven periodic heating. The parameters used were: 136 µm capillary bore diameter, 3 bar of Kr, 250 fs FWHM pulse duration, 36 µJ pulse energy and 505 kHz repetition rate. (a) Full spatio-temporal evolution of the radially-dependent temperature, (b) zoomed in to the beginning (lower panel) and the end of buildup (upper panel), when a quasi-steady state is reached. (c) On-axis (in blue) and mean temperature (in red) as a function of time. (d) Relative probe phase versus time when the sample arm of the probe beam is vertically off-set from the capillary axis by Δy (simulated Δy values: 0 µm, blue; 4 µm, red; 8 µm, green).

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In conclusion, post-recombination thermal effects in a confined gas following femtosecond photoionization can become very strong at high repetition rates. In a gas-filled glass capillary, they can be studied by transverse interferometric probing, allowing measurements of temperature rise and density changes over ms time-scales with high temporal resolution. Such studies are fundamentally important for designing systems based on gas-filled hollow-core fibers that offer stable pulse compression and spectral broadening at high repetition rates.

In the case of SPM-based spectral broadening the intensity, and thus the thermal buildup, will be highest at the input end of the fiber. Use of a low pressure or a positive pressure gradient along the fiber can reduce these effects at the launch end and avoid parasitic excitation of higher-order modes. In contrast, thermal effects in soliton self-compression stages are strongest in the vicinity of the temporal focus, where the optical modes will undergo continuous changes in shape and modal index [29] that affect the nonlinearity and, if non-adiabatic, cause excitation of higher-order and leaky modes. This may result in unstable pulse propagation, as has been seen in high-power fiber lasers [43], although in our case the refractive index changes are negative.

This study shows that the pressure and gas species must be carefully chosen if post-recombination effects are to be avoided, and that clean pulse compression in gas-filled hollow-core fibers is ultimately limited by post-recombination effects as the repetition rate increases. We note that the peak intensities and repetition rates studied in this work are predominantly relevant for HC-PCF-based setups. Nevertheless, we believe that cumulative post-recombination heating will no longer be negligible when gas-filled hollow capillaries are used to compress pulses from state-of-the-art high-energy femtosecond lasers operating at MHz- instead of kHz-level repetition rates [44]. Our numerical modeling suggests that a significant thermal buildup will appear when the repetition rate is increased to 200 kHz in the case of spectral broadening [45] and to only 50 kHz for soliton-effect self-compression [18].

Appendix 1: Definition of vacuum peak intensity

The “vacuum“ peak intensity of the ionizing pulses is calculated (for an unclipped pump laser beam) from energy-scaled temporal intensity profiles, measured for the short pulses via second harmonic-generation frequency resolved optical gating (SHG-FROG) and for the long pulses by SHG autocorrelation. Numerical forward-propagation to their focus inside the capillary accounts for the dispersion of gas and all optics in the beam path while disregarding spatial aberrations and losses due to nonlinear effects.

Appendix 2: Numerical model of thermal diffusion in a periodically heated gas

We numerically solve the differential equation governing heat diffusion in an axi-symmetric two-dimensional geometry in the absence of convective gas flow [46]:

(3)$$\frac{\partial }{{\partial t}}T({r,t} )= \frac{1}{r}\frac{\partial }{{\partial r}}\left[{r\alpha ({r,t} )\frac{\partial }{{\partial r}}T({r,t} )} \right] + s({r,t} ),$$

where T(r,t) is the time-dependent gas temperature, r the radial coordinate, $\alpha ({r,t} )= k(T )$${[{{C_\textrm{P}}{\rho_{\textrm{gas}}}({r,t} )} ]^{ - 1}}$ the thermal diffusivity of the gas, k(T) its temperature-dependent thermal conductivity (which scales with $\sqrt T $), C_P the heat capacity at constant pressure and ρ_gas the mass density (related to pressure and temperature via the ideal gas law), while the term s(r,t) models a heat source that depends both on time and radial position. We note that in Eq. (3) the effects of thermal diffusion along the capillary axis are expected to be small and so are neglected.

Next, we derive an expression for s(r,t) that accounts for cumulative recombination heating. For this, we calculate the radially-resolved free electron density N_e(r) generated by a single laser pulse at the pump focus using PPT ionization rates [41] (in Figs. 3(a) and 5(a) the on-axis value at r = 0 is plotted). Our implementation of the PPT rates includes the correction for the long-range Coulomb interaction, but disregards barrier-suppression ionization. We note that, although the barrier-suppression intensities [47] are slightly exceeded in section 3.1, measurements of the ion yield at comparable maximum intensities have confirmed that the unmodified PPT model remains a good approximation [48].

In case of ∼26 fs pulses the electric field measured by SHG-FROG was used, while for the 250 fs pulses (where SHG-FROG measurements were unreliable) a Gaussian shape with 250 fs full-width-half-maximum (from a fit to the measured SHG autocorrelation) was assumed. To estimate the energy freed by recombination we disregard acceleration of the electrons in the pump laser field, since simulations indicate that this causes less than 10% error for our experimental parameters. Furthermore, we neglect effects of avalanche ionization, collisional ionization and free-electron diffusion (which affects the radial shape of the heat source but modifies the post-buildup temperature distribution only weakly). Under these conditions, the energy density after complete recombination of the free electrons is given by I_P N_e(r,t), where I_P is the ionization potential of the gas (14.00 eV for Kr, 15.76 eV for Ar and 21.56 eV for Ne [49]). Assuming that the thermal diffusion is slow compared to the recombination time (typically a few ns) we can write the radially-dependent temperature change caused by recombination after a single ionization event as [26]:

(4)$$\Delta T(r )= {I_\textrm{p}}{N_\textrm{e}}({r,t} )/({{C_\textrm{V}}{\rho_{\textrm{gas}}}} ),$$

where C_V is the heat capacity at constant volume. As implied by Eq. (4), it is sufficient to calculate ΔT(r) only once, for the first ionization event: this is because N_e(r,t) scales with ρ_gas(r,t) [50] which at constant pressure is inversely proportional to the radial temperature profile T(r,t). Using Eq. (4), the source term in the heat diffusion equation [Eq. (3)] becomes

(5)$$s({r,t} )= \Delta T(r )\delta ({t - nf_{\textrm{rep}}^{ - 1}} ),$$

where f_rep is the repetition rate of the pump laser, n is an integer number, and δ(t) is the Dirac delta function.

Next, we solve Eq. (3) by linearizing the derivatives on a mesh equidistant in space and time and applying the Crank-Nicolson scheme to the resulting system of equations [46] while assuming that the temperature at the boundary between gas and capillary remains constant (i.e., ${|{\Delta T} |_{D/2}} = 0$). We note that this will somewhat underestimate heating in contiguous gas regions.

Figure 8(a) plots the spatiotemporal evolution of the gas temperature in a capillary with 136 µm bore diameter when 3 bar of Kr is photoionized and recombination-heated at 505 kHz repetition rate by 250 fs long Gaussian pulses of energy 36 µJ (corresponding to a peak intensity of ∼63 TW/cm²). A uniform gas temperature of 294 K (room temperature) is assumed before the first single-shot temperature increase. As shown in Fig. 8(b), the gas temperature initially increases from shot to shot and reaches a quasi-steady state after ∼0.6 ms (corresponding to ∼300 pulses), when any further heating is balanced by dissipation enforced through the boundary conditions. These simulations yield a peak on-axis temperature of 794 K (590 K time-averaged) and a (time-averaged) mean temperature $\bar{T} = 392\textrm{ K}$ Interestingly, the model predicts a temporal variation in mean temperature of only ∼5 K, even though single-pulse heating causes strong on-axis temperature changes of more than 250 K [Fig. 8(c)]. This is because the ionized gas region is only ∼6 µm in (FWHM) diameter (similar values were calculated for all measurements in this work) and contributes only weakly to the mean temperature computed across the entire capillary diameter. The numerical model also allows investigation of the effect of a small vertical offset Δy between the foci of the ionizing beam (at the capillary axis) and the sample arm of the probe beam, which can occur as a result of misalignment. Figure 8(d) suggests that an increase in Δy from 0 (exactly co-aligned) to 8 µm will reduce the buildup phase Δφ_B by only ∼10%, even though the single-shot phase change is expected to weaken much more strongly. We expect misalignment to have an even smaller effect on the measurement of the acoustic period τ_ac since this relies on two-dimensional averaging across the capillary bore and is therefore insensitive to the exact probing position.

Appendix 3: Observation of spontaneous instabilities in probe phase

Spontaneous temporal instabilities in the measured relative probe phase were observed throughout all the measurements for both pulse durations at Kr pressures higher than 1 bar. For example, when 250 fs pulses were focused at 1.01 MHz repetition rate into a capillary filled with 5, 7.6 and 10 bar, instabilities started to appear at a pressure-independent threshold energy of ∼39 µJ (corresponding to a peak intensity of ∼68 TW/cm²). Typical temporal traces at 5 and 7.6 bar are shown in Fig. 9(a), showing pulsations in probe phase that begin shortly before completion of the buildup. We note that, while no instabilities were seen at 505 kHz, the threshold was markedly lower at higher repetition rates (∼34 µJ energy or 59 TW/cm² peak intensity at 1.92 MHz). Focusing 26 fs pulses of peak intensity ∼184 TW/cm² at only 100 kHz repetition rate into 2 bar of Kr also resulted in periodic pulsations, however with spikes ∼0.3 ms long repeating every ∼0.7 ms [upper panel in Fig. 9(b)]. A zoom-in to the temporal trace at the top of one of these narrow peaks [lower right-hand panel in Fig. 9(b)] shows that recombination heating by a single laser pulse and acoustic effects are stronger than directly before the onset of instabilities (lower left-hand panel). Interestingly, on top of the peaks in probe phase, the bouncing to-and-fro of the recombination-induced acoustic pulse is unable to fully decay before the arrival of the next ionizing pulse. The physics underlying these instabilities is currently unclear, although the observed ms timescales suggest a thermal origin.

 

Fig. 9. Spontaneous instabilities observed for intense short and long pulses in Kr. (a) Periodic (∼6 ms, upper) and random pulsations (lower) in relative probe phase for 5 and 7.6 bar of Kr, observed when the vacuum peak intensity of ∼250 fs long pulses at 1.01 MHz repetition rate exceeded a threshold. (b) (Upper) Strong periodic pulsation in 2 bar of Kr, pumped by ∼26 fs long pulses at 100 kHz repetition rate, above a threshold peak intensity of ∼160 TW/cm². (Lower) Zoom-ins of the single-shot phase changes before the onset of instability (left) and at the peak of the first observed spike (right).

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Disclosures

The authors declare no conflicts of interest.

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