Energy scaling of carrier-envelope-phase-stable sub-two-cycle pulses at 1.76 µm from hollow-core-fiber compression to 1.9 mJ

: We present the energy scaling of a sub-two-cycle (10.4 fs) carrier-envelope-phase-stable light source centered at 1.76 µ m to 1.9 mJ pulse energy. The light source is based on an optimized spectral-broadening scheme in a hollow-core fiber and a consecutive pulse compression with bulk material. This is, to our knowledge, the highest pulse energy reported to date from this type of sources. We demonstrate the application of this improved source to the generation of bright water-window soft-X-ray high harmonics. Combined with the short pulse duration, this source paves the way to the attosecond time-resolved water-window spectroscopy of complex molecules in aqueous solutions.


Introduction
Ultrashort laser pulses have become an essential tool for the generation of isolated attosecond pulses [1][2][3], and for the study of electronic dynamics on their natural timescale [4][5][6][7][8][9].Among common schemes to generate millijoule-level, few-cycle pulses are techniques that rely on external compression in a hollow-core fiber (HCF) [10][11][12][13][14] and filamentation [15][16][17] subsequent to the laser system, so-called post-compression schemes.The HCF filled with noble gases has become an established pulse-compression technique paving the way to new applications in the field of extreme nonlinear light-matter interaction where the electric field of the laser pulse, rather than its intensity profile, is relevant.Thus, tailoring the electric field on electronic timescales requires the coherent superposition of electromagnetic waveforms with more than one octave.
In parallel to the above-mentioned compression techniques, optical-parametric chirped-pulseamplification (OPCPA) systems are commonly used: they enable high-power amplification to the millijoule level simultaneously with carrier-envelope-phase-stable (sub)-two-cycle pulses (two-cycle 0.74 mJ at 2.1 µm [18], 1.5-cycle 1.2 mJ at 2.1 µm [19], two-cycle 3 mJ at 1.7 µm [20]).Although such laser systems are tunable and deliver few-millijoule femtosecond pulses, they have several drawbacks: a difficulty in achieving phase-matching over an optical octave, a higher technical complexity and the crystal damage threshold, which ultimately limits the power scaling of OPCPAs.
Another alternative is the concept of frequency-domain optical parametric amplification (FOPA) where laser pulses are amplified not in a time-sequential manner, but in the frequency domain "slice by slice" (two-cycle 1.43 mJ pulses centered at 1.8 µm [21]).Such a technique allows to simultaneously upscale the pulse energy and increase the spectral bandwidth, i.e., amplification of sub-cycle pulses without increasing their pulse duration.Although this technique is very promising, especially for exploiting high-average-power pump lasers [22], it is not yet fully established.
An additional alternative is a parametric waveform synthesizer.This technique is able to tailor optical waveforms with scalable spectral bandwidth, energy, and average power.Giulio Rossi et al. [23] recently have shown that 0.5 mJ sub-cycle pulses can be obtained via the coherent combination of carrier-envelope-phase-stable pulses that emerge from different optical parametric amplifiers.They obtained a pulse spanning over 1.7 octaves with an FWHM pulse duration down to 0.6 optical cycles (2.8 fs) at a central wavelength of 1.4 µm.However, the pulse energy of 0.5 mJ is not sufficient for performing transient-absorption experiments, where this energy would be split into two parts for the generation of attosecond pulses and ionization of the sample.
Compared to the pulse compression techniques via filamentation and to the OPCPA systems, the gas-filled HCF compressors offer a high-quality output beam and beam-pointing stability thanks to their spatial-filtering characteristics [16,24].Eduardo Granados et al. [14] have shown that the high-quality operation of the HCF compressor is achieved when the quality factor, which depends on the central wavelength, pulse duration, and pump pulse energy, is more than 0.85.This limits the use of commercially available 30-fs chirped-pulse-amplification Ti:Sapphire lasers to less than 2 mJ at 800 nm (with the notable exception of Bohman et al. [25] who achieved pulse energies of 5 mJ with a pressure-gradient HCF) but allows for up to 5 mJ at longer wavelengths (up to 3.5 mJ at 1.8 µm and up to 2.6 mJ at 4 µm [26]).Here, we show that it is possible to compress 30-fs pulses with an energy of 2.5 mJ and a central wavelength at 1800 nm down to less than two optical cycles (10.4 fs) with a relatively high conversion efficiency of 76%.

Laser system
The laser system used in this work consists of a chirped-pulse amplifier (Coherent Legend Elite Duo HE+), a cryogenically-cooled power amplifier, and an optical parametric amplifier (Light Conversion HE-TOPAS-Prime), delivering passively carrier-envelope-phase stable 30 fs, 2.5 mJ idler pulses with a central wavelength of 1756 nm.This short-wave-infrared (SWIR) output is then spectrally-broadened in a gas-filled HCF and compressed with bulk material down to sub-two cycles pulses, as described in detail in the following sections.

Concept
The design criteria for a HCF compressor setup have already been discussed in several works [10,11,14,24,27,28].They include the gas type and pressure, the fiber length and its core radius, as well as the focusing geometry of the laser beam.These criteria are crucial to maximize the spectral broadening of laser pulses with a given energy, duration, and central wavelength while preserving the fiber throughput and the output beam quality.Thus, by appropriately selecting the gas type and fiber diameter, the ionization and resulting plasma effects can be reduced, and therefore, the spectral broadening can be driven mostly by the self-phase modulation (SPM) with a chirp close to linear.The design criteria for the present HCF compressor setup are the following (see Ref. [29] for a more detailed discussion).
When generating super-continua in the HCF at high pump pulse energies, two considerations set the energy limit.First, while a time-varying refractive index n 2 leads to SPM, the spatial dependence can result in self-focusing.This defines the critical power [11,30] that characterizes the strength of self-focusing.When the laser peak power P 0 is equal to the critical power P cr , self-focusing and diffraction are expected to cancel each other.At a peak power exceeding P cr , the self-focusing can no longer be stopped by diffraction, and a focus is formed after a finite propagation distance.Self-focusing remains negligible when the peak power is much smaller than P cr [30].In our case, a 30 fs, 2.5 mJ pulse reaches a peak power P 0 of about 78 GW (peak intensity of about 28.5•10 12 W/cm 2 ), while in the gas phase the critical power is of around P cr,Ar = 167 GW at 1 bar, hence leading to an increase in ionization of the gas.The optimal pressure was experimentally found in the range 0.260-0.400bar, leading to P cr = 416-640 GW resulting in (P 0 /P cr ) 2 = 0.015-0.035≪ 1.Thus, our HCF system satisfies this optimal working condition.
The second consideration is that the pump peak intensity should be smaller than the multiphoton ionization threshold at a given pulse duration [11].It means that for proper operation of the HCF compression, the variation of the refractive index ∆n Kerr induced by the Kerr effect should be much larger than the change of refractive index induced by the gas ionization ∆n p .Following the derivation based on the Ammosov-Delone-Krainov (ADK) ionization rates performed in [11] the minimal fiber core radius is given by: where T 0 is the pulse duration, E 0 the pulse energy, α and β constants with values 0.45 and 0.51 and A is a constant that in the case of argon amounts to A = 4.69 As a consequence of plasma-induced self-steepening, compression of the laser pulses with energies exceeding 1 mJ deals with the increased self-focusing and ionization effects when coupling into the HCF, which leads to poor transmission efficiency and a poor spectral broadening.In principle, this issue can be solved via an interplay of employing gases with higher ionization potentials (e.g., neon) and using fibers with larger inner diameters.Since the threshold for multi-photon ionization increases with decreasing pulse duration, this second consideration for our case is softened [27], but still applies a constraint on the fiber core radius and the type of gas medium.When the peak intensity is increased, ionization effects start to play an important role in the nonlinear interaction and, therefore, a lower limit in the core radius should be defined.For our case, with a central wavelength λ 0 of 1.8 µm, this leads to an optimal core radius of about 409 µm [Fig.1(B)].Additionally, the waveguide dispersion depends on the fiber core diameter.For small diameters, the negative dispersion is large, i.e., small-diameter fibers should yield greater compression.However, diameters that are too small can lead to large waveguide losses and, therefore, a poor transmission [Fig.1(A)].
One of the approaches to maximize the output energy of the fundamental mode and maintain its quality at the exit of the HCF is to maximize the fiber core radius because the attenuation constant of higher-order modes is inversely proportional to the fiber core radius [Fig.1(B)].On the other hand, the waveguide dispersion depends on the fiber core radius: for small radii, the negative dispersion is large, yielding a greater compression [13].Hence, the upper limit of the core radius is set close to the minimum core radius of a min = 409 µm, and the focusing geometry of the pump beam is tuned to fulfill this criterion.Thereby, by adjusting the ratio of beam waist to fiber core radius for best transmission through the fiber, it was found that a slightly smaller value of the core radius compared to the optimal value for coupling to the high-quality mode yielded higher output power.Hence, the optimum was found with a 1 m focusing CaF 2 lens and a fiber core radius of a = 350±20 µm.
Using equations (7-9) from [11] we obtain the broadening factor F as a function of the pressure ratio f = p/p max : with L eff = (1 − e −αL )α −1 and a eff = 0.45 • a.Using the parameters of our fiber: L=1 m, argon-filled fiber with radius a ≈ a min = 350 µm and input pulse energy 2.5 mJ the broadening factor is shown in Fig. 1(C).The maximum pressure is found to be p max = 0.64 bar, which differs from the 2.6 bar in [31].The maximum broadening factor obtained at the maximum pressure  decreases with the pulse energy, but increases with the pulse duration.Fixing the pressure ratio to f=1 in Eq. ( 2), the maximum broadening factor F as a function of the fiber core radius a, when its length is L =1 m is shown in Fig. 1(D).It should be mentioned that this calculation slightly underestimates the real broadening factor at a high pressure mainly because it does not include the self-steepening effect, which is quite important at high pressures and does not depend on the gas inside the fiber pulse energy, and the pulse duration.From Fig. 1(D), we find F max = 54 with a F max = 90 µm.For smaller core radius, the losses drastically limit the fiber's effective length L eff .Since the core radius for optimal broadening is significantly smaller than the minimal radius imposed by ionization (a min = 409 µm), a good compromise leading to optimal transmission and a good mode quality was found for a = 350 µm.For this core radius, a broadening factor F max = 8 is predicted from Fig. 1(C).

Optical and mechanical design
The optical layout of the pulse-compression setup is shown in Fig. 2. Our HCF compressor uses an input wavelength of 1.8 µm.The idler beam is focused by a lens (f =1 m) into 1 m long fused-silica HCF hosted in a vacuum-tight housing.The focal length has been chosen to fulfill the optimal coupling condition a 0 = 0.65 • a, where a 0 stands for the beam waist in focus.The HCF is placed in an under-pressure housing with an in-and out-FS window at the Brewster angle to minimize reflection losses.The housing is filled with argon at constant gas pressure.The fiber core diameter is 2a = 700 ± 20 µm.The fiber is statically filled with argon gas at about 0.6 bar under-pressure against the atmosphere.The gas pressure can be re-optimized during spectrum and pulse duration measurements as well as during soft-X-ray (SXR) generation.The input energy of 2.5 mJ results in sub-two-cycle pulses with below 1.95 mJ pulse energy.By properly mode-matching the input beam to the fundamental hybrid EH 11 mode of the HCF, an output coupling efficiency is expected to be about 92 % but is measured 76 %.This indicates that around 16 % is lost due to the core surface roughness, a possible bending of the fiber and focusing imperfections.The beam transmitted through the HCF is then collimated using a f =1 m concave silver mirror.Re-compression to the few-cycle regime in the current design is performed through the combined contributions of the output window of the fiber housing (it is chosen to be between 1 and 2 mm thin CaF 2 plate), the BS of the Mach-Zehnder interferometer (for the pump pulse), the entrance window to the vacuum chamber and a pair of FS wedges used for fine adjustment.Compression results finally into between 9 and 12 fs pulses with a typical output spectrum shown in Fig. 3(B) (orange curve), ranging from 1.1 to 2.2 µm, i.e. an octave-spanning spectrum.Fig. 3. XPW-SRSI pulse characterization of the few-cycle mid-IR pulse with a center-ofmass wavelength of 1.76 µm.(A) Single-shot reconstructed temporal profile of the 9.5 fs FWHM pulse (blue curve) on a linear scale.The Fourier transform limit is 9.4 fs FWHM (orange curve).(B) Normalized spectral intensities of the measured (orange curve) and the reconstructed (blue curve) compressed pulse.The green solid and dashed curves correspond to the retrieved and fitted spectral phase of the pulse, respectively.

Pulse characterization
After the HCF, the compressed pulse is around two optical cycles short, and therefore, it is of crucial importance to know its duration and phase.For the characterization of these pulses on a single-shot basis, a femtosecond shot-to-shot measurement device, the Wizzler from FASTLITE was used.This device is based on a Self-Referenced Spectral Interferometry (SRSI) [32][33][34] and a Cross-Polarized Wave generation (XPW) [35].
Before performing the measurement, the HCF output is split by a broadband BS into pump and probe pulses.One pair of ultra-thin FS wedges at Brewster angle is placed in the pump and probe path to achieve separate fine-tuning of the dispersion since the pump pulse is already traveling through the substrate of the BS.The total amount of FS is adjusted in each of the beam paths separately to obtain the best compression and, therefore, the shortest pulse duration.Figure 3(A) shows a reconstructed temporal profile of the few-cycle laser pulse with a center wavelength (calculated in the wavelength domain) of ≈ 1.76 µm; the spectrum of the laser beam directly measured by the spectrometer is shown in Fig. 3(B) (orange curve).The reconstructed spectrum (Fig. 3(B), blue curve) is close in shape to that of the directly measured spectrum (Fig. 3(B), orange curve).The corresponding reconstructed spectral phase is also shown (Fig. 3(B), green curve).The data shown in Fig. 3 correspond to one of the shortest pulses measured.The average pulse duration obtained over three independent measurements (quoted in the remained of this paper) amounts to 10.4±1.5 fs (Fourier-transform limit (FTL) of 9.8±1.8fs).The standard error is based on three independent measurements and is given with its 95 % confidence interval.

Carrier-envelope-phase stabilization
For a sub-two-cycle pulse, the electric-field amplitude can drastically change within half of an optical cycle, which significantly affects the generation process of a single isolated attosecond pulse, and therefore, the shot-to-shot reproducibility of the generated SXR spectra and their cutoff.For the transient-absorption measurements, the carrier-envelope-phase (CEP) stability is crucial for being able to observe the minor (of the order of 10 mOD [36][37][38]) absorption features.Thus, the stability of the carrier-envelope phase is essential to prevent cumulative errors in absorption measurements because of any shot-to-shot changes in the spectral shape and amplitude of the SXR beam: for example, to realize experiments in the near-edge or extended X-ray-absorption fine-structure (NEXAFS/EXAFS) energy regions, the absorption measurement needs to be performed at the pre-and post-edges over a range of a few tens of percent of the absorption-edge location (about 0.33 % at the carbon K-edge) and not just the edge itself.
In this work, a white-light seeded OPA is used, which has been shown to generate passively CEP-stable idler pulses [39,40].Under real conditions, the CEP is never perfectly stable.In order to monitor the CEP, a home-built f -to-2f interferometer was used.Figure 4(A) shows an in-loop measurement of the CEP jitter over 45 minutes (one data point is integrated over five laser shots).Under our laboratory conditions, we find a passive CEP stability of ∆ϕ CE, FWHM = 639 mrad over almost one hour.For a comparison, Seth Cousin et al. [41] have showed that with a passive CEP stability for a 1850 nm sub-two-cycle idler pulse, the single-shot CEP jitter was of ∆ϕ CE, FWHM = 315.5 mrad over 6 minutes, and of ∆ϕ CE, FWHM = 88.8 mrad over one hour when activating the electronic feedback loop of the f -to-2f interferometer.Cedric Schmidt et al. [17] have also reported the carrier-envelope-phase stability of their tabletop SXR light source.The passive free-running CEP stability resulted in ∆ϕ CE, RMS = 531 mrad over 30 minutes, while the actively-locked CEP stabilization reduced it down to ∆ϕ CE, RMS = 180 mrad.

High-harmonic generation
The HCF output is coupled to a Mach-Zehnder interferometer for attosecond transient-absorption spectroscopy measurements in the soft-X-ray energy region.This apparatus is described in more detail in [42].The sub-two-cycle mid-IR pump (30%) is used to trigger dynamics in the samples under investigation, and the sub-two-cycle mid-IR probe (70%) is converted into the attosecond SXR pulse.The water-window spectrum is generated by focusing the post-compressed pulses with a 250 mm focal length off-axis parabolic mirror into a static high-pressure helium gas target.The approximate f-number of the focusing condition is f =31 and changes slightly from day to day.The typical backing pressure used for the HHG in helium is around 4 bar.The phase-matching conditions for the HHG process are optimized via the helium gas pressure, focus position, iris, and by fine-tuning the pulse compression.The HHG radiation is filtered from the driving mid-IR pulses by pinholes and a thin metallic filter.The SXR beam is then refocused with a toroidal mirror to the gas sample.Right before the sample, the attosecond probe pulse is recombined with the sub-two-cycle pump pulse using a perforated silver mirror.Finally, the pulse transmitted through the sample is analyzed in a flat-field SXR spectrometer.The SXR spectrum shown in Fig. 5 was acquired with a spectrometer based on a microchannel plate and Shimadzu grating.The spectrum was accumulated over 400 ms, averaged over 10 images, and corrected by the Jacobian transformation and for the grating efficiency.However, for time-resolved measurements, where the energy resolution is crucial, a spectrometer based on a vacuum charge-coupled-device (CCD) camera and Hitachi grating is used.With a CCD camera, data are typically collected with the integration time of 0.5 s, a CCD-camera gain of two, a full vertical binning, and 50 images per step.The SXR light source developed in the present work has the harmonic cutoff energy above 450 eV and thus covers the carbon and nitrogen K-edges.This light source is therefore ideally suited to perform attosecond spectroscopy experiments in the water window.The first attosecond time-resolved absorption measurements at the carbon K-edge with such sub-two-cycle laser pulses were performed in ethylene, which lead to the simultaneous observations of multidimensional structural dynamics and the fastest electronic relaxation dynamics via a conical intersection observed to date [9].It was shown that the electronic relaxation from the electronic first excited state to the ground state of the ethylene cation takes place with a short time constant of 6.8 ± 0.2 fs.

Discussion
Bruno Schmidt et al. [12] have demonstrated that with the pulse compression by spectral broadening in a HCF with linear propagation in FS, it is possible to generate 1.6-cycle laser pulses at 1800 µm at a repetition rate of 1 kHz.These pulses were obtained with the same type of white-light seeded OPA as ours besides the fact that they modified the commercial OPA such that the idler rather than the signal pulse is amplified in a collinear with the pump configuration in the first nonlinear crystal.The generated 0.24 mJ quasi single-cycle pulses resulted in FWHM pulse duration of 10.4 fs.The summary, comparing their compression with ours, is presented in Table 1, showing that the high-intensity pulses can still be compressed with the HCFs, and thus, preserve their spatio-temporal quality.Several research groups have demonstrated spectral broadening of mid-IR pulses based on other techniques than the HCF, i.e. optical filamentation [16,17,24].In filamentation, ultrashort amplified pulses are focused in a long gas cell in a regime where the self-focusing and the plasma generation dominate the propagation dynamics: the laser beam is guided by an interplay between the subsequent self-focusing due to the Kerr effect, the plasma generation caused by an intense beam, and the plasma-defocusing.The defocusing leads to a beam quality that may not be as good as that from a waveguide.Similar to the HCF technique, the Kerr effect leads to the broadening of the pulse spectrum during propagation in the filament through SPM.After the filament, the continuum output can be compressed with dispersion compensation methods like a bulk material.
The filamentation approach is an attractive alternative to the compression in the HCF for a couple of reasons: its experimental setup is simpler than the one with a HCF, and its throughput is higher because of the absence of a waveguide and its associated coupling and transmission losses (84% [17] vs. 76% in this work).On the other hand, the beam-pointing stability of the optical filamentation is lower compared to the HCF.Lukas Gallmann et al. [24] have shown that the filamentation and the HCF techniques yield comparable pointing stability and spectral bandwidths at their optimum gas pressure.Nevertheless, the performance of filamentation critically depends on gas pressure, while the performance of the HCF strongly depends on the alignment.A comparative overview of the relevant parameters is given in Table 1.
Regarding the spatial chirp along the horizontal axis of the laser beam, in the case of filamentation, it changes dramatically at higher wavelengths, whereas the structure of the spatial chirp in the case of the HCF is relatively evenly distributed.Additionally, in the case of filamentation, the broadest spectrum is obtained in the beam center while the broadening considerably decreases towards the lower-intensity wings.It implies the use of only the center part of the beam to obtain the shortest output pulses.In the case of the HCF, the transform limit remains almost constant for the center part and slightly decreases towards the lower intensity wings of the beam [24].Thus, in our application, we want to preserve the short pulse duration over the whole beam profile and its intensity.Hence, the HCF approach is a considerably better choice than the optical filamentation approach in order to deliver the required performance for our application.
Lukas Gallmann et al. [24] have also shown that the spectra generated with optical filamentation possess a long tail towards short wavelengths due to interaction with the plasma, whereas the HCF possesses a tail towards long wavelengths given by dominant SPM.Thus, since in our case, the goal is to obtain an HHG cutoff as high as possible (E cutoff ∝ λ 2 [43]), the pulse-compression approach with the HCF remains the most advantageous.

Conclusion
In conclusion, 10.4 fs pulses with energy up to 1.9 mJ with a central wavelength of 1757 nm have been achieved with the compression technique based on spectral broadening in a hollow-core fiber statically filled with argon gas and post-compression in bulk material.Up to date, this is the highest pulse energy achieved with this technique in the short-wave-infrared region.This source is directly applicable to attosecond transient absorption spectroscopy in the water window, which will facilitate the measurement of attosecond dynamics of complex biomolecules in their natural, aqueous environment.

Fig. 1 .
Fig. 1. (A)Transmission versus fiber length of the EH 11 (blue) and EH 12 (orange) modes for 350 µm (solid) and 150 µm (dashed) fiber core radii.(B) Minimum fiber radius at 780 nm and 1800 nm for different pump pulse energies and durations.(C) Broadening factor as a function of the pressure ratio, f = p/p max in a 1 m fiber with a radius of 350 µm and filled with argon for input pulse energy of 2.5 mJ.(D) Broadening factor as a function of the fiber core radius filled with gas at pressure p = p max .Data are calculated at λ 0 =1.8 µm.In this figure, FS stands for fused silica.

Fig. 2 .
Fig.2.Schematic layout of the pulse-compression setup based on a one-meter long HCF and bulk material.Beam-pointing stabilization at the fiber input is implemented via two sets of a four-quadrant photodiode and a piezo-driven mirror.BB stands for the broadband.

Fig. 4 .
Fig. 4. Passive carrier-envelope phase stability.(A) Evolution of the interference fringes in the f -to-2f interferometer over a period of 45 min.The idler-based system is free running, that is, without any feedback loop to correct for CEP fluctuations.Each measurement is integrated over five laser shots.(B) Stability of the input pulses delivered by the HCF that results in a relative phase of ∆ϕ CE, FWHM = 639 mrad or ∆ϕ CE, RMS = 272 mrad.The orange line is a convolution of the presented data with a boxcar of 50 data points.

Fig. 5 .
Fig.5.SXR spectra generated with sub-two-cycle 1.2 mJ mid-IR pulses.The harmonic cutoff energy is above 450 eV, covering the carbon and nitrogen K-edges.The photon spectrum is transmitted through a 100-nm thick Sn filter.