Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 19, Number 1, 2023
EOSAM 2022



Article Number  15  
Number of page(s)  5  
DOI  https://doi.org/10.1051/jeos/2023011  
Published online  07 April 2023 
Short Communication
Scalable subcycle pulse generation by soliton selfcompression in hollow capillary fibers with a decreasing pressure gradient
Grupo de Investigación en Aplicaciones del Láser y Fotónica, Departamento de Física Aplicada, Universidad de Salamanca, 37008, Salamanca, Spain
^{*} Corresponding author: marinafergal@usal.es
Received:
27
January
2023
Accepted:
13
March
2023
Advances in the generation of the shortest optical laser pulses down to the subcycle regime promise to break new ground in ultrafast science. In this work, we theoretically demonstrate the potential scaling capabilities of soliton selfcompression in hollow capillary fibers with a decreasing pressure gradient to generate nearinfrared subcycle pulses in very different dispersion and nonlinearity landscapes. Independently of input pulse, gas and fiber choices, we present a simple and general route to find the optimal selfcompression parameters which result in highquality pulses. The use of a decreasing pressure gradient naturally favors the selfcompression process, resulting in shorter and cleaner subcycle pulses, and an improvement in the robustness of the setup when compared to the traditional constant pressure approach.
Key words: Ultrafast nonlinear optics / Hollow capillary fibers / Solitonself compression / Subcycle pulses
© The Author(s), published by EDP Sciences, 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In a continuous effort to access the briefest and most fundamental phenomena in nature, intense ultrashort laser pulses have become indispensable tools for ultrafast science, breaking new ground in timeresolved spectroscopy and strongfield physics [1, 2]. At present, fewcycle femtosecond pulses in the optical spectral region are routinely generated by nonlinear postcompression in gasfilled hollow capillary fibers (HCFs) [3], which stand out among other compression schemes for their simplicity, highdamage threshold and the possibility of tuning their nonlinearity and dispersion by modifying the filling gas or its pressure. However, although HCF postcompression experiments have been greatly optimized [4], they are currently reaching their limit in terms of the shortest achievable pulse duration due to the complexity of dealing with uncompensated highorder dispersion in octavespanning spectra. Overcoming this problem, parametric lightfield synthesizers have succeeded in generating the shortest optical laser pulses well down into the subcycle regime, offering new opportunities for advancing realtime observation and precision control of electron dynamics at the atomic scale [5, 6]. As a promising alternative to these extremely complex systems, highenergy soliton dynamics in HCFs is attracting a great interest as a direct route to extreme pulse selfcompression down to the subcycle regime [7, 8]. As opposed to more conventional postcompression techniques, soliton selfcompression relies on the simultaneous nonlinear spectral broadening and phase compensation arising from the interplay between the negative groupvelocity dispersion (GVD) of the waveguide and selfphase modulation (SPM). Still, for a practical implementation, the complexity of this nonlinear interaction calls for theoretical investigations on scaling rules and design guidelines to identify the optimal experimental parameters which result in highquality selfcompression [9]. So far, extreme soliton selfcompression in HCFs in the nearinfrared (NIR) has only been demonstrated for precompressed (~10 fs) pump pulses [7, 10], or otherwise in configurations which ensure a strong anomalous response, i.e., working with longer wavelengths [8, 11], highorder modes [12], or smallcore photonic crystal fibers [9, 13]. However, the unexpected applicability of subcycle selfcompression to standard experimental setups driven by NIR multicycle pulses propagating in the fundamental mode of largecore HCFs has been recently demonstrated in negatively pumped fibers, i.e., HFCs filled with a decreasing pressure gradient [14]. Pressure gradients are routinely implemented by sealing the fiber into a gas cell at each end, which can be independently evacuated and filled with gas, yielding a longitudinal pressure distribution with a square roottype profile when the gas flows from highest to lowest pressure [10].
In this communication we further investigate the scalability of soliton selfcompression down to the subcycle regime in HCFs filled with a decreasing pressure gradient. Varying the filling gas and choosing between atomic (Ne) and molecular (N_{2}) species, we study the compression process in two completely different dispersion and nonlinearity landscapes, which are of critical importance in soliton dynamics. Our results demonstrate that nearly identical selfcompression performance can be achieved in very distinct HCF scenarios, and provide a surprisingly simple universal route to find the optimal parameters for generating highquality NIR subcycle pulses.
Our work is based on onedimensional numerical simulations of nonlinear pulse propagation [15, 16], including the complete linear response of the gasfilled HCF [17], SPM, stimulated Raman scattering modeled in a damped harmonic oscillator approximation [18], and selfsteepening. This theorical model accurately describes ultrashort pulse propagation down to the singlecycle limit in a regime of moderate intensities [19].
In order to identify a route towards highquality selfcompression, we have followed the procedure detailed in [14]. In brief, we have systematically simulated the propagation of a transformlimited 30 fs gaussian pump pulse at 800 nm in the fundamental mode of a 3 m long, 100 μm core radius HCF, filled with either Ne or N_{2}, and with both constant gas pressure and a longitudinal decreasing pressure gradient ending in vacuum. The latter two situations are fairly compared by matching the integrated nonlinear phase shift acquired by the pulse peak during its propagation, which is often referred to as Bintegral. Neglecting the fiber losses, a system with a decreasing pressure gradient from p_{0} to vacuum can then be compared to that with a constant pressure p_{eq} simply if p_{0} = (3p_{eq})/2 [10, 12]. The main difference lies in that nonuniform pressure allows for a dynamic tuning of the dispersion and nonlinearity experienced by the pulse during its selfcompression, as the propagation constant and the nonlinear parameter scale linearly with the gas density and, thus, with pressure.
For the parameters considered in our study, Figure 1 shows the GVD and instantaneous nonlinear coefficient (related only to instantaneous Kerr effect or SPM [15]) of a HCF filled with Ne or N_{2} at different pressures. As we can see, a Nefilled HCF has a weaker instantaneous nonlinearity and displays anomalous dispersion (GVD < 0) over a larger pressure range than an identical fiber filled with N_{2}. An inner fiber radius of 100 μm was chosen because it offers a good balance between acceptable losses at 800 nm and a sufficiently strong anomalous response in N_{2}. Furthermore, a pulse propagating in N_{2} might experience a delayed molecular contribution to the optical Kerr effect which vanishes in noble gases like Ne. Therefore, owing to their very distinct linear and nonlinear nature, the optimal selfcompression in either Ne or N_{2} is expected to occur for different input pulse and fiber parameters.
Fig. 1 GVD (top) and instantaneous nonlinear coefficient (bottom) at 800 nm of the fundamental mode of a 100 μm core radius HCF filled with Ne or N_{2} as a function of gas pressure. Labels indicate the zerodispersion pressure (p_{ZD}) in each case. 
Following [14], we have simulated the soliton selfcompression of the aforementioned 30 fs pulse while varying its initial energy and the equivalent gas pressure in the HCF. For each energypressure pair in the resulting bidimensional parameter space, we have plotted the intensity full width at halfmaximum (FWHM) duration and the ratio of output to input peak power of the selfcompressed pulses, as shown in Figure 2. In these plots, the optimal region for highquality selfcompression can be readily identified as the intersection between the areas of shortest output pulse duration and largest peak power enhancement. Surprisingly, the results for Ne and N_{2} show an identical behavior, which also follows that previously reported for Ar [14], except for the fact that they are displaced to different input energy and gas pressure ranges as mentioned earlier. In both cases, it is clear that, in the whole parameter ranges considered here, the selfcompression process is substantially enhanced when the fiber is negatively pumped rather than statically filled, resulting in the generation of selfcompressed pulses with extremely short durations well down into the subcycle regime (~1 fs) and high peak powers, which in turn implies a clean temporal profile. The most outstanding feature is that there is not just a single pair of input energy and gas pressure values that allow for a highquality compression, but there is a whole parameter region which yields similar results. This optimal region is found to always appear towards the same corner of the contour line in the energypressure map where the fixed fiber length (L) matches an average compression length (L_{av}), that we defined as:(1)where L_{sc} and L_{fiss} represent, respectively, the characteristic selfcompression and soliton fission lengths, which are given by [20]:(2)
Fig. 2 FWHM duration (top row) and ratio of output to input peak power (bottom row) of the selfcompressed pulses as a function of the input energy and the equivalent constant pressure (see text) in both a statically filled or a negatively pumped 3 m long, 100 μm core radius HCF filled with Ne (left) or N_{2} (right). The solid black lines represent the contour lines where L = L_{av}, which run along the optimal region for selfcompression in a decreasing pressure gradient. Note the one order of magnitude change in the pressure range from Ne to N_{2} owing to their different dispersion and nonlinearity. 
N = (L_{D}/L_{NL})^{1/2} being the soliton order, and the dispersion and the nonlinear lengths, which describe the characteristic length scales of GVD and SPM, respectively. Here T_{p} represents the intensity FWHM duration of the gaussian pump pulse, P_{0} is its input peak power and E_{0} its initial energy, β_{2} is the GVD coefficient of the HCF, and γ_{i} is the instantaneous nonlinear parameter as defined elsewhere [15]. The constraint L = L_{av} ensures that L_{sc} < L <L_{fiss} and, therefore, guarantees that the selfcompressing pulse reaches its maximum compression without entering in the soliton fission regime. In addition, the soliton order should be kept N < 15 to achieve a highquality compression [13], inevitably setting an upper limit to the achievable pulse energy. Independently of input pulse, gas and fiber parameters, the condition L = L_{av} always describes a contour line in the energypressure plane which, when falling inside the space with N < 15, can be used to identify the optimal region for highquality selfcompression in a universal way. A detailed inspection of the conditions L = L_{av} and N < 15 suggests that upscaling our results towards millijoule pump pulses should become possible, even in practical short fibers (~1 m), by pushing the central wavelength into the midinfrared spectral region.
As an example of the highquality subcycle waveforms that can be generated from the negatively pumped fiber, in Figure 3 we have plotted the selfcompressed pulses obtained for two different pairs of input pulse energy and equivalent gas pressure which lie towards the same area of the optimal selfcompression regions in Figure 2, corresponding to Ne and N_{2}, respectively. When compared to the output pulses in the equivalent constant pressure situations, it is clear that those generated with a decreasing gradient are much better, displaying shorter durations, higher peak powers, a cleaner temporal profile with a higher contrast, and a broader spectrum spanning from the NIR to the midultraviolet. The selfcompressed pulses from the negatively pumped HCF reach subcycle FWHM durations of 1.1 and 1.2 fs, and output peak powers of 8.8 and 10.7 GW in Ne and N_{2}, respectively. However, in the equivalent constant pressure situations, the output pulses where only 2.2 and 2.3 fs in duration, and 4.6 and 5.4 GW in peak power. The improvement with the decreasing pressure gradient has been attributed to an effective suppression of higherorder dispersion and selfsteepening in the last stages of the pulse compression, together with a continuous blueshift of the zerodispersion frequency at the same time as the pulse spectrum broadens by SPM [12, 14]. When propagating in the anomalous dispersion regime, it is straightforward to understand from the trends shown in Figure 1 that a decreasing pressure gradient is the most natural way to emphasize and favor the characteristic dynamics of the selfcompression process [11]. In short, at the fiber entrance the higher pressure enhances the accumulation of nonlinear phase shift and the spectral broadening of the input pulse by SPM. In later stages, the larger spectral extent combined with an increase in the magnitude of the anomalous GVD and a reduction of thirdorder dispersion due to the drop in pressure, assist the phase compensation for pulse selfcompression and delay the fission process beyond the maximum compression point that can be reached with constant pressure. Altogether, the decreasing pressure gradient enables unprecedent compression ratios (≿25) which had remained out of reach due to detrimental highorder effects [13]. Furthermore, the great similarities between the pulses in Figure 3, generated with different gases, energies and pressures, demonstrates the promising scaling capabilities of HCF selfcompression down to the subcycle regime in different configurations. Another interesting point is that the optimal selfcompressed pulse is accompanied by the onset of resonant dispersive wave (RDW) emission, when the strongly nonlinear selfcompressing soliton transfers its excess energy to a linear wave propagating in the normal dispersion regime [7, 8, 10, 16]. This is manifested by the isolated peak around 200 nm in the output spectra of the lower panels in Figure 3. At this point, RDW emission has just started and the energy transfer to the ultraviolet is still low, resulting in conversion efficiencies quite below saturation. This fact could be used to experimentally predict the best subcycle pulse parameters based on RDW spectral content at the fiber output.
Fig. 3 Temporal intensity profile (top row) and spectrum (bottom row) of the selfcompressed subcycle pulses obtained after propagation through a HCF filled with Ne (left) or N_{2} (right), at both constant or decreasing pressure, for two different pairs of input pulse energy and equivalent gas pressure which lie towards the same area of the optimal selfcompression regions in Figure 2. 
In summary, we have demonstrated that broadly similar highquality NIR subcycle pulses can be generated by extreme soliton selfcompression in negatively pumped HCFs in different configurations. Independently of input pulse, gas and fiber choices, the optimal selfcompression parameters can always be found by matching the fiber length to and average compression length, providing a simple design guideline for experiments. Furthermore, the decreasing pressure gradient can help to improve the robustness of HCF selfcompression and the quality of the generated subcycle pulses when compared to the equivalent constant pressure situations, also preventing the onset of undesirable highorder effects. We believe that these findings will pave the way towards a new generation of ultrafast experiments which might benefit from the availably of tailored subcycle waveforms, especially those which are carried out in vacuum chambers, like the synthesis of highfrequency isolated attosecond pulses through highorder harmonic generation.
Conflict of interest
The authors declare no conflict of interest.
Acknowledgments
Authors acknowledge financial support from Ministerio de Ciencia e Innovación under grant PID2019106910GBI00 funded by MCIN/AEI/ 10.13039/501100011033. M.F.G. acknowledges support from Ministerio de Universidades under grant FPU21/02916.
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All Figures
Fig. 1 GVD (top) and instantaneous nonlinear coefficient (bottom) at 800 nm of the fundamental mode of a 100 μm core radius HCF filled with Ne or N_{2} as a function of gas pressure. Labels indicate the zerodispersion pressure (p_{ZD}) in each case. 

In the text 
Fig. 2 FWHM duration (top row) and ratio of output to input peak power (bottom row) of the selfcompressed pulses as a function of the input energy and the equivalent constant pressure (see text) in both a statically filled or a negatively pumped 3 m long, 100 μm core radius HCF filled with Ne (left) or N_{2} (right). The solid black lines represent the contour lines where L = L_{av}, which run along the optimal region for selfcompression in a decreasing pressure gradient. Note the one order of magnitude change in the pressure range from Ne to N_{2} owing to their different dispersion and nonlinearity. 

In the text 
Fig. 3 Temporal intensity profile (top row) and spectrum (bottom row) of the selfcompressed subcycle pulses obtained after propagation through a HCF filled with Ne (left) or N_{2} (right), at both constant or decreasing pressure, for two different pairs of input pulse energy and equivalent gas pressure which lie towards the same area of the optimal selfcompression regions in Figure 2. 

In the text 
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