1. Introduction

Over the last few years, progress in solid-state laser technology has led to ultrafast systems that can provide up to a few kW average power and turn-key operation [1,2]. However, for many applications in ultrafast science and material processing [3–5], the few-hundred-fs pulses generated by these lasers are too long, and external pulse compression is required. Typically, this relies on optical nonlinearity to broaden the spectrum of laser pulses, followed by phase compensation. In optical systems with anomalous dispersion at the laser frequency, the interplay between nonlinearity and dispersion can cause laser pulses to shorten without need for additional phase compensation. This effect, known as soliton self-compression, was initially observed in conventional solid-core fibers and exploited to obtain pulses hundreds of fs in duration [6,7]. Since then, soliton self-compression has been used in a variety of different systems [8–12]. Among these, gas-based systems such as hollow-core waveguides and multi-pass cells, are particularly attractive for compression to durations of a few fs at high peak and average power [11–15]. Furthermore, the great flexibility of gas-filled hollow-core fibers is very convenient, as it allows few-fs pulses to be obtained over a wide spectral range (from the UV to the mid-IR) from pump pulse energies from sub-µJ up to several mJ [10,12,13,16–19]. This is because, in common with many phenomena in gas-based nonlinear optics [20], soliton dynamics in gas-filled hollow-core fibers is scale-invariant [8]. As a result, by using proper scaling relations, similar propagation patterns can be obtained for a plethora of different parameter sets. This flexibility comes, however, with the challenge of identifying optimal experimental parameters for high-quality compression. Soliton self-compression can be impaired in several different ways, depending on the system properties, pulse parameters, and laser repetition rate. It is therefore crucial to have a thorough understanding of these limitations and how optimal parameters can be identified.

Here we study, theoretically and experimentally, the scaling of soliton dynamics in noble gas-filled hollow-core photonic crystal fibers (HC-PCFs). We investigate the limits set by modulational instability, higher-order dispersion, self-focusing and photoionization, and identify an optimal parameter region for pulse compression. We then discuss practical constraints such as repetition rate scaling, and relate the theoretical quantities to experimental parameters. Finally, we present experimental results on soliton self-compression and the dependence on laser repetition rate. For this, we compress 250 fs, 1030 nm laser pulses to ∼5 fs duration in a gas-filled HC-PCF and vary the repetition rate between 1 and 10 MHz.

2. Theoretical discussion

The nonlinear propagation dynamics of ultrashort pulses in gas-filled hollow-core waveguides can be modelled by the generalized nonlinear Schrödinger equation (NLSE). In the case of a pure Kerr nonlinearity (as is the case for Raman-inactive noble gases), and ignoring for the moment the effects of self-focusing and ionization, the dynamics can be well described by three characteristic length-scales: the dispersion length (${L_\textrm{D}}$), the third-order dispersion (TOD) length (${L_{\textrm{TOD}}}$) and the nonlinear length (${L_{\textrm{NL}}}$) [8,21]. For a pulse of peak power ${P_0}$ and central angular frequency ${\omega _0}$, these lengths are given by [21]:

Conveniently, the characteristic length-scales can be reduced to two parameters: the soliton order $N = \sqrt {{L_\textrm{D}}/{L_{\textrm{NL}}}} $ and the zero dispersion (ZD) frequency ${\omega _{\textrm{ZD}}}$ [defined by ${\beta _2}({{\omega_{\textrm{ZD}}}})= 0$]. Once these parameters are fixed, similar (identical for the same ${\tau _0}$) nonlinear propagation dynamics can be reproduced for a wide range of different parameters [8,12]. Moreover, both N and ${\omega _{\textrm{ZD}}}$ can be continuously tuned by changing the gas species and density (both ${\beta _2}$ and the nonlinear coefficient $\gamma $ depend on the gas properties) [8,23], thus allowing precise control of the nonlinear propagation dynamics of ultrashort laser pulses. By selecting parameters such that ${\beta _2}({{\omega_0}} )< 0$ and $N > 1$, a laser pulse propagating in the fiber will undergo soliton self-compression, reaching the shortest duration at the compression length ${L_{\textrm{comp}}}$, which is essentially equal to the fission length ${L_{\textrm{fiss}}} \approx {L_\textrm{D}}/N$, provided ${L_{\textrm{comp}}} < \; {L_{\textrm{loss}}}$ [8,12,21]. Though widely used, this relation becomes increasingly inaccurate at low soliton orders. A more generally applicable expression takes the form [6]:

In Fig.  1 the compression length and compression factor are plotted against soliton order. The discrete datapoints (marked by symbols) were calculated by simulating the propagation of Gaussian pulses with FWHM durations 25 fs (dark blue), 100 fs (orange) and 250 fs (turquoise) along an Ar-filled HC-PCF using the unidirectional full-field nonlinear wave equation [25], including quantum noise [26] and neglecting fiber loss. Values of the nonlinear susceptibility ${\chi ^{(3 )}}\; $ were taken from [27] and photoionization was included via the Perelomov-Popov-Terent’ev (PPT) rate modified with the Ammosov-Delone-Krainov (ADK) coefficients [28]. The compression length predicted by Eq.  (2), plotted by the full lines in Fig.  1(a), agrees remarkably well with numerical simulations (the corresponding datapoints are denoted by the symbols and were obtained by selecting the position of highest peak power). The accuracy is observed to be typically well within a 15% margin even under the most extreme conditions and for low soliton orders. Additionally, we found that the expression ${Q_\textrm{c}} = 3.7/({N + 2.2} )$ gave a more accurate estimate of the quality factor (not shown), especially for low N. On the other hand, ${F_\textrm{c}}$ exhibits a linear dependence on N [dashed red line in Fig.  2(b)] only for small soliton orders. For short pulse durations (∼ 25 fs) the higher-order terms cause the compression factor to saturate. By analysing large sets of numerical data, we find that the compression factor is accurately described by the following fitting function that includes the TOD contribution:

For soliton orders greater than $N_{\textrm{ZD}}^{\textrm{max}}$, HOD becomes increasingly important and soliton self-compression is impaired through formation of pre- and post-pulses or pulse break-up. In hollow-core waveguides the influence of HOD can be reduced by shifting ${\omega _{\textrm{ZD}}}$ to frequencies much higher than ${\omega _0}$, thus allowing larger values of N to be chosen. Selecting large $\; {\mathrm{\Delta }_\omega }$ (e.g. by using a lighter filling-gas or decreasing the pressure) is, however, not always a good choice and in practice may not always be possible due to limited pulse energy and, more importantly, additional conditions on the maximum value of N set by SF and ION, which become relevant when ${\omega _{\textrm{ZD}}}$ is far away from ${\omega _0}$. Self-focusing causes energy transfer to higher-order modes, which distorts both the spatial and (since dispersion is mode-dependent) temporal profiles of the self-compressing pulse. This effect can be counter-balanced by the defocusing effect of free electrons resulting from gas photoionization [29]. While a certain amount of ionization is tolerable, it becomes detrimental if it distorts the temporal pulse profile. As we shall discuss later, ionization also becomes a constraint if too much energy is deposited in the gas, which can occur at repetition rates higher than a few hundred kHz [13,30,31].

 

Fig. 1. Scaling behavior of soliton self-compression and optimal parameter region for 1030 nm pulses. (a) Dependence of compression length on soliton order in argon for zero-dispersion wavelength $2\mathrm{\pi }\textrm{c}/{\omega _{\textrm{ZD}}} \approx 450\; \textrm{nm},$ with FWHM input pulse duration 25 fs (dark blue), 100 fs (orange) and 250 fs (turquoise). The symbols mark numerically calculated datapoints (position of highest peak power), and the full lines are solutions of Eq.  (2). For comparison, the fission length ${L_{\textrm{fiss}}} = {L_\textrm{D}}/N$ is also plotted (dashed lines). (b) Dependence of compression factor ${F_\textrm{c}}$ on soliton order for the same case as in (a). The symbols mark numerically calculated datapoints, and the full lines are solutions of Eq.  (3). The red dot-dashed line is a linear fit to ${F_\textrm{c}} = 4.4N - 4$. (c) Analytical plot of soliton order versus ${\tau _0}{\mathrm{\Delta }_\omega }/({2\pi } )$ for 100 fs input pulses with ${\mathrm{\Delta }_\omega } = |{{\omega_{ZD}} - {\omega_0}} |$. MI: modulational instability (turquoise dashed line); SF: self-focusing (dark blue dotted line); ION: photoionization (yellow to red solid lines); HOD: higher-order dispersion (blue dashed line). The size of the gray area, within which optimal compression occurs, is controlled by the ION limit, which depends on the gas species.

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Fig. 2. Ratio of peak compressed intensity I to peak launched intensity I₀ for 1030 nm pulses in Ar-filled HC-PCF, plotted against soliton order and ${\tau _0}{\mathrm{\Delta }_\omega }/({2\pi } )$, which depends on both core diameter and pressure. Initial pulse durations are (a) 250 fs, (b) 100 fs and (c) 25 fs. The optimal compression region is triangular in shape (yellow-white area) and shrinks with decreasing input pulse duration. At 1030 nm the maximum soliton order is limited mainly by HOD and ION for shorter pulses, whereas for longer pulses MI constrains the maximum soliton order to $N = 16$. The white dot in (a) marks the experimental parameters.

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Travers et al. [12] defined the SF limit on maximum soliton order by replacing the peak power in the expression for N with the critical power for self-focusing $P_{\textrm{SF}}^{\textrm{cr}} \approx 4\pi {\varepsilon _0}{c^3}/({\omega ^2}{\chi ^{(3 )}}{\rho _\textrm{r}})$ where ${\rho _\textrm{r}}$ is the gas density relative to standard conditions [32,33], and the ION limit by defining the ionization threshold power as ${P_{\textrm{ION}}} = \; {I_\textrm{c}}{A_{\textrm{eff}}}$, where ${I_\textrm{c}}$ is a critical intensity including an arbitrary factor [12]. In the work reported here we define ${I_\textrm{c}}$ as the clamping intensity, i.e., the intensity at which the ionization-induced phase shift equals the Kerr phase shift (see Appendix 1 for details). Expressing ${\beta _2}$ and $\gamma \; $ as functions of ${\omega _{\textrm{ZD}}}$, the following relations can then be obtained for the HE₁₁-like fundamental mode:

An additional constraint on self-compression of pulses hundreds of fs long is set by modulational instability (MI) [21,26], which causes the pulse to break up into a train of sub-pulses before the shortest duration is reached. Although under certain circumstances MI can be suppressed in HC-PCFs by the effects of spectral anti-crossings between the fundamental core mode and core-wall resonances [36], here we restrict the analysis to conventional MI, which is typically observed for N > 16 [26]. Using this as a limit, we can identify an optimal parameter range for soliton self-compression by plotting the maximum soliton order as a function of ${\tau _0}{\mathrm{\Delta }_\omega }/({2\pi } )$, a numerical quantity that is related to the fraction of the pulse bandwidth that lies within the anomalous dispersion region.

In Fig.  1(c), we map out the optimal region for a ${\tau _{\textrm{FWHM}}} = \; 100$ fs pulse centered at ${\omega _0} = 2\pi \cdot 0.29\; $PHz (corresponding to a wavelength of 1030 nm). For the selected parameters, the region is mostly delimited by $N_{\textrm{ZD}}^{\textrm{max}}$ and $N_{\textrm{ION}}^{\textrm{max}}$, while self-focusing becomes an effective limit for pulses centered at shorter wavelengths ($N_{\textrm{SF}}^{\textrm{max}}$ depends inversely on $\sqrt {{\omega _0}} $). As seen in Fig.  1(c), a key limitation to optimal compression is ionization, which strongly depends on gas species. For lighter noble gases, the region of optimal compression widens provided $({N,{\tau_0}{\mathrm{\Delta }_\omega }/({2\pi } )} )$ is appropriately chosen. Both $N_{\textrm{ION}}^{\textrm{max}}$ and $N_{\textrm{SF}}^{\textrm{max}}$ are linear with ${\tau _\textrm{0}}\; $ and $N_{\textrm{ZD}}^{\textrm{max}}$ is approximately linear with it. As a result, the extension of the optimal region increases with ${\tau _0}$ and MI introduces an additional border to the region only for long initial pulse durations. For wide-bore capillaries, where the loss can be very significant, it is important to ensure that ${L_{\textrm{comp}}} < {L_{\textrm{loss}}} \approx {\omega ^2}{a^3}/({3{c^2}u_{01}^2} )$ (assuming a fundamental mode) [12]. In this case an additional constraint can be derived by solving the equation ${L_{\textrm{comp}}} = {L_{\textrm{loss}}}$ for N. The solution yields the following lower limit for the soliton order required for self-compression:

We verified the validity of these limits by numerically simulating the propagation of 1030 nm pulses in gas-filled HC fibers. Since at this wavelength self-focusing does not play a significant role, and the computational requirements scale quadratically with the number of modes [37], we only considered propagation of the HE₁₁-like mode. We used Gaussian pulses with initial FWHM durations 250 fs, 100 fs and 25 fs, and for each of these we ran a set of 256 simulations for a range of different soliton orders and zero dispersion frequencies. We considered a fiber with 15 µm core radius filled with argon, but we observed similar pulse propagation patterns using other noble gases and core radii (not shown). The results are shown in Fig.  2, where we plot, as function of N and ${\tau _0}{\mathrm{\Delta }_\omega }/({2\pi } ),$ the peak power of the pulses at the compression point normalized to their power at the fiber input. This quantity coincides with the product of the simulated compression and quality factors. As such it is a good measure for the quality of the self-compression. The curves are interpolated from simulated datapoints; since in each case N depends monotonically on ${\mathrm{\Delta }_\omega }$, the interpolations are very accurate.

For the input pulse durations considered, the simulations are in excellent agreement with the maximum soliton order predicted by Eqs.  (5–6). Self-compression of long pulses (250 fs) can enhance the peak power more than 20 times and is clearly limited by HOD and MI. In the vicinity of the latter limit, $I/{I_0}$ varies greatly with N and ${\tau _0}{\mathrm{\Delta }_\omega }$ thus indicating a pulse break up (we further verified this by analysing individual numerical simulations). For shorter durations, HOD and ION become dominant. The optimal compression region has a characteristic triangular shape, which shrinks as the initial pulse duration is decreased, while the peak power enhancement falls. The greatest enhancement is always located for $({N,{\tau_0}{\mathrm{\Delta }_\omega }/({2\pi } )} )$ values near the center of gravity of the triangle.

It is important to note that the limit set by ION appears to be more accurate for 25 fs input pulses. For much shorter pulse durations (e.g. < 10 fs) the limit is somewhat too strict, while for 250 fs pulses, $N_{\textrm{ION}}^{\textrm{max}}$/1.7 delimits the optimal region more precisely. This is because the definition of $N_{\textrm{ION}}^{\textrm{max}}$ neglects propagation effects and we calculate the clamping intensity in the multiphoton limit of the PPT rate (Appendix 2). When tunneling ionization becomes dominant, ${I_c}$ should be estimated in a different way, for example by fitting to the ADK rate [38].

3. Experimental results

The limits discussed above are particularly useful for identifying the range of parameters for optimal soliton self-compression. Experimental implementation requires, however, consideration of additional constraints, such as core diameter, fiber length, and gas pressure handling capabilities. Furthermore, at high repetition rates, even weak gas photoionization can cause buildup of post-recombination refractive index changes, an effect which can occur already at hundreds of kHz repetition rates and impair the compression dynamics [30,31]. To minimize these effects, the interval between subsequent pulses should be longer than the time needed for the photoionization-induced heat to dissipate radially to the core-walls. This is described by the diffusion time, estimated by ${t_{\textrm{diff}}} = {a^2}/\alpha $, where a is the core radius and $\alpha = \; \kappa /({\varrho \; {C_\textrm{p}}} )$ is the thermal diffusivity, where $\varrho $ is the gas density, ${C_\textrm{p}}$ the heat capacity at constant pressure, and κ the thermal conductivity, which is larger for lighter gases. Matching the nonlinearity and ${\omega _{\textrm{ZD}}}$ of a heavier gas with a lighter gas requires much higher pressures, resulting in higher values of ${t_{\textrm{diff}}}$. To visualise this scaling, it is convenient to express the diffusion time as a function of the zero-dispersion frequency, derived from the ideal gas law using the relations in Appendix 1:

 

Fig. 3. Single-stage pulse compression to few-cycle durations. (a) Diffusion time as a function of the zero-dispersion frequency for different noble gases. The dashed gray line marks the experimental setpoint at $2\mathrm{\pi }\textrm{c}/{\omega _{\textrm{ZD}}} \approx 523\; \textrm{nm}$. The upper horizontal axis allows comparison with Fig.  1(c); however, note that the diffusion time is independent of the input pulse duration. (b) Scaling of core radius, gas pressure and compression length with pulse energy for 1030 nm, 250 fs pulses in argon at a fixed soliton order of 13.6. The dashed gray line marks the experimental parameters, where a core radius of 15 µm, a pressure of 5.6 bar and a fiber length of 0.8 m are required to compress pulses with an energy of 4 µJ. (c) Sketch of the experimental setup: λ/2: half-wave plate, TFP: thin-film polarizer, OAPM: off-axis parabolic mirror, W: wedge, CM: chirped-mirror. (d) Scanning electron micrograph of the cross-section (zoom) of the 80-cm-long, 8-capillary single-ring PCF with core diameter of 30 µm.

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We experimentally verified this prediction by compressing 250 fs pulses at ${\omega _0}\sim 2\pi \cdot 0.29\; \textrm{PHz}$ (1030 nm) to nearly single-cycle duration in a fiber with 15 µm core radius filled with either Ar or Ne. To achieve this ∼55-fold temporal compression with the highest efficiency, given the laser and fiber parameters we have access to, we used the numerical results in Fig.  2(a) and selected $N = 13.6$ and ${\tau _0}{\mathrm{\Delta }_\omega }/({2\pi } )\; \sim \; 40$ (yielding ${\lambda _{ZD}} = 2\pi c/{\omega _{\textrm{ZD}}} \approx $ 523 nm), values that are expected to yield the desired ${F_\textrm{c}}$ and compression quality.

To relate the optimal $({N,{\tau_0}{\mathrm{\Delta }_\omega }/({2\pi } )} )$-region with the physical parameters of gas-filled HC fibers, we express the core radius a and the gas pressure p as functions of the input pulse parameters and the zero-dispersion frequency. The core radius, given as:

Pump pulses (FWHM duration ∼250 fs) were generated by a commercial 1030 nm ytterbium fiber laser at repetition rates between 50 kHz and 19.2 MHz and up to 100 W average power. A half-wave plate and a thin-film polarizer were used for power control, and a second half-wave plate corrected for potential nonlinear polarization rotation along the fiber, to ensure horizontal polarization after the fiber, to meet the requirements of subsequent optics and diagnostics. The pulses were compressed in an 8-capillary single-ring PCF [Fig.  3(d)] with a length of 80 cm, which was mounted in a gas-filled cell. The core-wall thickness of ∼190 nm was chosen to ensure that the anti-crossings lie far from the pump wavelength, so as not to impair the compression [22]. Light leaving the cell through the uncoated magnesium fluoride window was collimated using a silver-coated off-axis parabolic mirror. Reflecting the beam at two glass wedges lowered the intensity to prevent damage to the diagnostics. The compressed pulses were characterized with a custom-built, all-reflective, dispersion-free, second-harmonic generation frequency-resolved optical gating device (SHG FROG). To achieve transform limited pulses in spite of the dispersion introduced by the output window of the gas cell and propagation in air, we used double-angle chirped mirrors (group delay dispersion of −80 fs²) and thin fused silica wedges, which were only adjusted once at the beginning to exactly compensate the beam path dispersion.

We found that the calculated core radius, gas pressure and fiber length are exceptionally accurate for predicting the real experimental behavior, despite the high soliton order of 13.6, which produces extreme nonlinear propagation dynamics, with strong spectral broadening and fast temporal compression. Using 5.6 bar of argon, we could compress the pulses to a duration of 4.6 fs at 1 MHz repetition rate with a pulse energy of 4.3 µJ. As we always optimized the input pulse energy to yield the broadest output spectrum, we found that the maximum spectral width and the required pulse energy decreased with increasing repetition rate: 3.8 µJ at 5 MHz and 3.6 µJ at 10 MHz. As the maximum spectral width decreased, the pulse duration increased and we obtained at 5 MHz trains of pulses with duration 4.9 fs, and 10 MHz trains with duration 5.4 fs. The measured spectra and retrieved temporal pulse shapes for different repetition rates are shown in Figs.  4(a) and 4(c), the corresponding measured and retrieved FROG traces at 1 MHz in Fig.  4(d). We attribute the decreasing spectral width to changes in the transverse refractive index distribution caused by cumulative post-recombination heating, which significantly alters the dispersion and nonlinearity, thus limiting the compression [30,39]. The power delivered through the gas-cell during pulse compression, including losses at the fiber and input and output windows, varied depending on the repetition rate, having similar values at 1 MHz (86.8%), 5 MHz (84.5%), and 10 MHz (84.7%). The maximum output power was ∼30 W at 10 MHz. In the experiment a quality factor of ${Q_{\textrm{exp}}}\sim 0.24$ was achieved, very close to the expected value of ${\sim} 0.23$ for $N = 13.6$ obtained from the numerical fit.

 

Fig. 4. Experimental results. Spectra measured at different repetition rates at the output of the 80-cm-long single-ring PCF for (a) 5.6 bar of argon and (b) 52 bar of neon. (c) Retrieved temporal shapes of the compressed pulses at different repetition rates in argon and (d) corresponding measured and retrieved SHG FROG traces at a repetition rate of 1 MHz.

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We next investigated the dynamics in the lighter noble gas, neon. For this, we selected a similar zero-dispersion frequency, corresponding to ${\lambda _{\textrm{ZD}}} = 2\pi c/{\omega _{\textrm{ZD}}} \approx $ 509 nm, by filling the fiber with 52 bar (the maximum pressure the output window of the gas cell could safely sustain). Since the resulting nonlinearity was marginally lower than for the Ar-filled fiber, we used a slightly higher pulse energy to match the soliton order. When launching 5.2 µJ pulses at 1 MHz into the fiber, we measured output spectra slightly broader than for Ar at the same repetition rate, and slightly shorter pulse durations (4.5 fs). At higher repetition rate we found that, in contrast to argon, compression in neon was much more unstable. Already at 5 MHz the output was too unstable for full characterization by SHG FROG and at 10 MHz no stable output could be achieved. This behavior is in accord with our predictions. The weaker ionization of neon compared to argon results in a slightly higher temporal compression at low repetition rates. As the repetition rate increases, however, the longer diffusion time seems to lead to a higher post-recombination heating in neon, affecting the propagation dynamics more strongly than in argon. We attribute this to a stronger non-uniformity in transverse refractive index profile, which affects the nonlinearity and dispersion during pulse compression. A full understanding of these phenomena will require, however, further careful investigations.

4. Conclusion and outlook

In conclusion, an optimal region for soliton self-compression in hollow-core fibers can be identified in $({N,{\omega_{\textrm{ZD}}},{\tau_0}} )$ space. When the optical loss is negligible, e.g., in anti-resonant guiding PCF far from spectral anti-crossings with resonances in the core walls, the optimal region is enclosed within limits set by modulational instability, higher-order dispersion, self-focusing and photoionization. Of these, only photoionization depends on the choice of filling gas, becoming especially important when scaling to high repetition rates. The analytically derived limits are in excellent agreement with numerical simulations and reveal a characteristic triangular shape for the optimal compression region that shrinks with decreasing input pulse duration. Furthermore, we find that increasing the photoionization limit by employing lighter gases broadens the optimal region. However, matching ${\omega _{\textrm{ZD}}}$ and N of a heavier gas (e.g. argon) using a lighter gas (e.g. neon) results in longer diffusion times, leading to the thermal buildup of refractive index changes and impairing pulse compression.

When the core radius, gas pressure and fiber length are matched to the calculated values, good agreement is obtained between experiment and theory. Stable compression of 250 fs pulses at 1030 nm to almost single-cycle durations at repetition rates up to 10 MHz can be achieved experimentally in argon. As expected, the performance in neon is worse, stable compression being possible only at 1 MHz repetition rate.

We expect these results to be helpful in the design of high-performance soliton-compression systems. Proper selection of experimental parameters makes soliton self-compression highly convenient for compressing even long (∼250 fs) few-µJ pulses in a single compression stage down to few-cycle duration. The compression efficiency is similar to that achieved in two-stage systems based on gas-filled capillaries, though these require much higher pulse energies. The much lower pulse energy requirements allow operation at MHz instead of kHz repetition rates, allowing mode-locked pump lasers to be used.

Appendix 1

According to the capillary model [8], the propagation constant $\beta (\omega )$ of the HE_nm modes in a gas-filled HC waveguide of radius a is given by:

(11)$$\begin{array}{{c}} {\beta (\omega )= \frac{\omega }{c}{n_{\textrm{eff}}} \approx \frac{\omega }{c}\left[ {1 + \frac{{{\rho_r}{\chi^{(1 )}}}}{2} - \; \frac{1}{2}{{\left( {\frac{{{u_{n - 1,m}}c}}{{a\omega }}} \right)}^2}} \right],} \end{array}\; $$

where ${\rho _\textrm{r}}$ is the gas density relative to standard conditions, ${\chi ^{(1 )}}(\omega )$ the linear susceptibility of the gas, and ${u_{\textrm{qm}}}$ is the m-th zero of the q-th order Bessel function of the first kind. The group velocity dispersion and TOD follow from this expression as:

(12)$$\begin{array}{{c}} {{\beta _2}(\omega )\approx \; \frac{{{\rho _\textrm{r}}}}{c}f(\omega )- \frac{{u_{n - 1,m}^2c}}{{{a^2}{\omega ^3}}},\; \; \; \; {\beta _3} \approx \frac{{{\rho _\textrm{r}}}}{c}{\partial _\omega }f(\omega )+ \frac{{3u_{n - 1,m}^2c}}{{{a^2}{\omega ^4}}},} \end{array}$$

$$\textrm{with}\; f(\omega )= {\partial _\omega }{\chi ^{(1 )}} + \; \frac{\omega }{2}\partial _\omega ^2{\chi ^{(1 )}}$$

where ${\partial _\omega }$ the derivative with respect to the frequency $\omega .$ If we consider only the HE₁₁-like fundamental mode, we can relate the relative density to the zero dispersion frequency via ${\rho _\textrm{r}} = \; u_{01}^2{c^2}/[{{a^2}\omega_{\textrm{ZD}}^3f({{\omega_{\textrm{ZD}}}} )} ]$ with a obtained from Eq.  (9) and thus express ${\beta _2}$ and ${\beta _3}$ as functions of ${\omega _{\textrm{ZD}}}$:

(13)$$\begin{array}{{c}} {{\beta _2}({\omega ,{\omega_{\textrm{ZD}}},\; a} )= \frac{{u_{01}^2c}}{{{a^2}}}\; \left( {\frac{{f(\omega )}}{{f({{\omega_{\textrm{ZD}}}} )}}\frac{1}{{\omega_{\textrm{ZD}}^3}} - \frac{1}{{{\omega^3}}}} \right) = \frac{{\delta ({\omega ,{\omega_{\textrm{ZD}}}} )}}{{{a^2}}}\; ,} \end{array}$$

$$\; {\beta _3}({\omega ,{\omega_{\textrm{ZD}}},\; a} )= \frac{{\; {\partial _\omega }\; \delta ({\omega ,{\omega_{\textrm{ZD}}}} )}}{{{a^2}}}.\; $$

Similarly, the nonlinear coefficient can also be expressed as function of ${\omega _{ZD}}$:

(14)$$\begin{array}{{c}} {\gamma = \; \frac{{3{\omega _0}{\chi ^{(3 )}}{\rho _\textrm{r}}}}{{4{\varepsilon _0}{c^2}n_{\textrm{eff}}^2{A_{\textrm{eff}}}}} \approx \frac{{{\omega _0}{\chi ^{(3 )}}}}{{2{\varepsilon _0}{a^4}}}\frac{{u_{01}^2}}{{\omega _{\textrm{ZD}}^3f({{\omega_{\textrm{ZD}}}} )}},\; } \end{array}\; $$

where ${\varepsilon _0}$ is the vacuum permittivity, ${\chi ^{(3 )}}$ the nonlinear susceptibility, and we have used the approximation $n_{\textrm{eff}}^2 \approx 1$ and ${A_{\textrm{eff}}} \approx 1.5{a^2}$ [8].

Appendix 2

The clamping intensity is defined as the intensity ${I_\textrm{c}}$ at which the nonlinear phase shift induced by ionization equals the Kerr phase shift. This can be expressed as [28]:

(15)$$\begin{array}{{c}} {{n_2}{I_\textrm{c}} = \; \frac{{{N_\textrm{e}}({{I_\textrm{c}}} )}}{{2{N_\textrm{c}}}},} \end{array}$$

where ${N_\textrm{c}}\; $ is the critical density and ${N_\textrm{e}}$ is the free-electron density, which can be calculated from the ionization rate. To obtain an analytical expression for ${I_\textrm{c}}$, we consider only multiphoton ionization (MPI), thus ${N_\textrm{e}}({{I_\textrm{c}}} )\approx {\sigma _\textrm{K}}I_\textrm{c}^K{\tau _0}{N_0}$, where ${N_0}$ is the density of neutral atoms, and ${\sigma _\textrm{K}}\; is\; $ the MPI cross-section for multi-photon absorption by K photons. The last two parameters can be derived from the PPT rate in the multi-photon limit [28]. Table  1 lists the MPI coefficients calculated at a wavelength of 1030 nm for the noble gases.

Table 1. MPI cross-sections for the noble gases and for a wavelength of 1030 nm

View Table

By solving Eq.  (15), we obtain the following expression for the clamping intensity:

(16)$$\begin{array}{{c}} {{I_c} = \; {{\left( {\frac{{2{n_2}{N_\textrm{c}}}}{{{\sigma_\textrm{K}}{\tau_0}{N_0}}}} \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {K - 1}}} \right.}\!\lower0.7ex\hbox{${K - 1}$}}}} = \; {{\left( {\frac{{3{\chi^{(3 )}}{N_\textrm{c}}{\rho_r}}}{{2{\varepsilon_0}cn_{\textrm{eff}}^2{\sigma_\textrm{K}}{\tau_0}{N_0}}}} \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {K - 1}}} \right.}\!\lower0.7ex\hbox{${K - 1}$}}}}.} \end{array}$$

Noting that ${N_0} \propto {\rho _\textrm{r}},\; $ this expression is only weakly pressure-dependent, via n_eff.

Funding

Max-Planck-Gesellschaft.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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