# Experimental and numerical study of direct sub-20fs multi-pass-cell post-compression of up to 4mJ, 80W Yb-based lasers

Jean-François Hergott, Fabrice Réau, Olivier Tcherbakoff, Fabien Lepetit, Nicolas Lericheux, Matthieu Guer, Hugo Marroux, Romain Géneaux, Akansha Tyagi, Romain Cazali, et al.

► To cite this version:

Jean-François Hergott, Fabrice Réau, Olivier Tcherbakoff, Fabien Lepetit, Nicolas Lericheux, et al. Experimental and numerical study of direct sub-20fs multi-pass-cell post-compression of up to 4mJ, 80W Yb-based lasers. Applied Physics B - Laser and Optics, 2025, 131 (3), pp.56. 10.1007/s00340-025-08415-0. cea-04979777

HAL Id: cea-04979777

https://cea.hal.science/cea-04979777v1

Submitted on 6 Mar 2025

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# Experimental and numerical study of direct sub-20fs multi-pass-cell post-compression of up to 4mJ, 80W Yb-based lasers

Jean-François Hergott<fnref n="1" /> · Fabrice Réau<fnref n="1" /> · Olivier Tcherbakoff<fnref n="1" /> · Fabien Lepetit<fnref n="1" /> · Nicolas Lericheux<fnref n="1" /> · Matthieu Guer<fnref n="1" /> · Hugo Marroux<fnref n="1" /> · Romain Géneaux<fnref n="1" />,<fnref n="2" /> · Akansha Tyagi<fnref n="1" /> · Romain Cazali<fnref n="1" /> · Pierre Nonnon<fnref n="1" />,<fnref n="2" /> · David Bresteau<fnref n="1" /> · Thierry Ruchon<fnref n="1" /> · Pascal Salières<fnref n="1" /> · Thierry Auguste<fnref n="1" />

Received: 2 December 2024 / Accepted: 6 February 2025
© The Author(s) 2025

## Abstract

In recent years, various methods for laser pulse post-compression have demonstrated their effectiveness. Nonlinear spectral broadening, achieved by coupling an ultrafast pulse into a gas-filled multi-pass cell (MPC), typically enables pulse compression factors exceeding 20, depending on the initial pulse duration. In this work, we report the direct post-compression of four Yb-based commercial laser systems to sub-20 fs durations, with energies up to 4 mJ and initial pulse durations ranging from 320 fs to 450 fs, using argon-filled MPCs. Each MPC was specifically designed to operate at a fixed optimal point for each laser, achieving the shortest possible pulse duration. Numerical simulations reproducing the experimental data highlight the critical influence of the driver pulse's temporal profile on the shape of the broadened spectra. These simulations are compared with the experimental results, demonstrating strong agreement.

## 1 Introduction

An increasing number of experiments demand lasers with high repetition rates (above 10 kHz), energy outputs of several mJ or more, high average powers approaching 100 W, and short pulse durations in the sub-30 fs range. These lasers benefit to a wide range of applications like intense ultra-broadband terahertz sources [1], high order harmonic generation and attosecond science [2–6], molecular research [7] time-angle resolved photoemission spectroscopy [8] or ultrashort electron pulse [9] and particle generation [10]. More and more laser systems used for such applications are ytterbium-doped (Yb) solid-state materials based lasers rather than other systems with lower repetition rate such as Ti: Sapphire based lasers. Thanks to their small quantum defect, Yb-based amplifiers can be efficiently pumped by high-power laser diodes, enabling repetition rates well above 10 kHz while minimizing heat generation. Among

Yb-based lasers, three main amplification schemes - fiber [11, 12], slab [13, 14], and thin-disk amplifiers [15–17] - ensure effective cooling and high performance.

In recent years, commercial Yb-based laser sources with high repetition rates (tens of kHz) and pulse energies in the mJ range have become widely available across these schemes. Despite their robustness and ease of use, these systems typically produce pulse durations of several hundred femtoseconds (fs) or even picoseconds (ps), which are unsuitable for high-intensity applications.

Various post-compression techniques exist to address this limitation, relying on nonlinear interactions during pulse propagation [18]. The most widely used approach involves coupling the pulse into a gas-filled stretched hollow-core fiber (HCF) [19–22], a technique initially developed for Ti: Sapphire lasers to achieve the few-cycle regime. More recently, the propagation of intense pulses through gas-filled or bulk multi-pass cells (MPCs), so-called Herriott cells [23], has been proposed for nonlinear interaction. This method is well suited to Yb lasers post-compression [24–29], offering higher transmission efficiencies, often exceeding 95%, depending on the spectral width of the pulses and the quality of mirror reflectivity. Post-compression of high average power Yb-based lasers using MPCs now presents a viable alternative to Ti: Sapphire systems, particularly in applications requiring high repetition rates above 10 kHz.

<footnote>✉ Jean-François Hergott
jean-francois.hergott@cea.fr</footnote>

<footnote n="1">¹ Université Paris-Saclay, CEA, LIDYL, Gif-sur-Yvette
91191, France</footnote>

<footnote n="2">² CY Cergy Paris Université, CEA, LIDYL,
Gif-sur-Yvette 91191, France</footnote>
<table><caption>**Table 1** Main characteristics of the 4 different commercial Yb-based lasers from Light Conversion</caption><thead><tr><th>Laser</th><th>max. energy (mJ)</th><th>duration (fs)</th><th>max. rep. rate (kHz)</th><th>max. power (W)</th><th>M²(x, y)</th><th>diameter (mm)</th><th>CEP σ (mrad)</th></tr></thead><tbody><tr><td>Yb1</td><td>0.8 (0.4)</td><td>340</td><td>100 (200)</td><td>80</td><td>1.10×1.12</td><td>5.0</td><td>X</td></tr><tr><td>Yb2</td><td>2</td><td>320</td><td>40</td><td>80</td><td>1.12×1.09</td><td>5.2</td><td>200</td></tr><tr><td>Yb3</td><td>2</td><td>340</td><td>40</td><td>80</td><td>1.08×1.05</td><td>5.8</td><td>X</td></tr><tr><td>Yb4</td><td>4</td><td>450</td><td>5</td><td>20</td><td>1.08×1.11</td><td>7.0</td><td>180</td></tr></tbody></table>

<figure><img src="image_2.png" /><figcaption>**Fig. 1** Schematic of the post-compression setup composed of a mode-matching optical system with three lenses L1, L2 and L3, an argon-filled single-stage MPC, a convex lens L4 for collimation, an achromatic half wave plate HWP followed by a magnification telescope (L5 and L6) and a set of two chirped mirrors (CMS) for post-compression.</figcaption></figure>

# 2 Post-compression experimental results

## 2.1 Laser characteristics and experimental set-up

In this paper, we present the direct MPC-based post-compression of Yb lasers with up to 80 W average power, achieving pulse durations below 20 fs. To expand capabilities and replace the dual Ti: Sapphire system (2 mJ, 10 kHz and 15 mJ, 1 kHz beamlines providing 25 fs, CEP stabilized pulses) of the ATTOLab facility [30, 31], four new commercial Yb-doped laser systems have been purchased. These systems deliver pulses with durations between 320 fs and 450 fs (FWHM) at a central wavelength of 1030 nm. The output pulse energy can be easily tuned up to 2 mJ while maintaining an average power of 80 W for three systems and up to 4 mJ with an average power of 20 W for the fourth system. Notably, the 20 W laser and one of the 80 W systems are carrier envelop phase (CEP) stabilized.

After post-compression, each laser will be coupled to a high-harmonic generation chamber for high repetition rate XUV attosecond experimental lines as partially described in [31]. Table 1 provides an overview of the key characteristics of the four lasers—designated Yb1, Yb2, Yb3, and Yb4

C1 and C2 are cameras imaging the near field and far-field of the beam to ensure alignment in the MPC. Camera C3 images the transmitted pattern by a cavity mirror. Four different MPC based on two low-GDD cell mirrors set-ups are used depending on the input laser

(Carbide and Pharos systems from Light Conversion)—which were critical to the corresponding MPC design and study.

Given the diverse specifications of these lasers, we developed tailored MPC setups that accounted for the available laboratory space. The Yb-based lasers were installed in the laser room, with two MPCs positioned in each of the two experimental rooms. Four distinct MPC configurations—MPC1, MPC2, MPC3, and MPC4—were designed to correspond with lasers Yb1, Yb2, Yb3, and Yb4, respectively.

MPC 1 is designed to post-compress pulse energies ranging from 400 µJ and 800 µJ at maximum repetition rates of 200 kHz and 100 kHz respectively, with an initial pulse duration of 340 fs. MPC2 and MPC3 are configured for 2 mJ pulses with respectively 320 fs and 340 fs initial pulse duration, operating at a maximum repetition rate of 40 kHz. Due to space constraints, the design of MPC2 was optimized for compactness. Lastly, MPC4 is tailored for 4 mJ pulses with an initial pulse duration of 450 fs, running at a maximum repetition rate of 5 kHz. Across all laser systems, the measured output beam exhibits consistently high spatial quality, with a circularity of 98%, as shown in the near-field profile inset of Fig. 1. Beam diameters range from 5 mm to 7 mm depending on the laser with an $M^2_{x, y}$ value

<figure><img src="image_3.png" /></figure>
of approximately 1.1 in both x and y directions (see details in Table 1). All set-ups, including MPCs, are engineered to handle high average power, approaching 100 W.

The overall experimental setup (Fig. 1) includes a mode-matching lens telescope (L1, L2), a focusing lens L3 and a MPC placed in an argon-filled chamber capable of supporting pressures up to 5 bar. The MPC's dimensions depend on the incoming pulse energy and duration. Each MPC follows a similar design. Small rectangular dielectric mirrors (10 mm × 25.4 mm) are positioned at both ends of the cell for injection and extraction of the beam. The S-polarized pulse is injected at 45°, and the desired pass is extracted near 0°, guiding the beam out of the chamber. The Herriot cell comprises two broadband spherical mirrors with high laser-induced damage thresholds (LIDT), each 2 inches in diameter. The mirrors' radius of curvature (RoC) and separation distance (L) are selected to enable smooth spectral broadening during propagation through the argon-filled cavity. The setup allows for adjustable separation distances between the two chambers housing the cavity mirrors, ensuring compatibility with each MPC specification. The optics in the system have reflectivity exceeding 99.95% at 0° incidence and low group delay dispersion (GDD) of approximately 25 fs² within the 970–1100 nm wavelength range (Optoman). For long-term operation without optical damage, the optics' LIDT values are around 0.4 J/cm² at 100 kHz and 0.7 J/cm² at 10 kHz. These optics lack specific coatings for controlling spectral shape or width, as demonstrated in [32, 33], which could complicate the setup or reduce the LIDT.

After selecting the desired pass and extracting the beam from the cavity, a collimating lens (L4) directs the beam through a 3 mm-thick fused silica exit window with a 28 mm clear aperture, similar to the entrance window. An achromatic half-wave plate (HWP) rotates the pulse polarization from S to P. A magnification telescope (L5, L6) then collimates the beam to a 16 mm diameter (1/e²). It is followed by a chirped-mirror compressor that compensates for the group delay dispersion (GDD) acquired during nonlinear propagation in the cavity and through various optics, achieving post-compression of the spectrally broadened pulses. The compressor consists of a pair of rectangular chirped mirrors (100 × 25.4 × 10 mm), with a second order spectral phase correction of –200 fs² per bounce over a wavelength range of 950–1120 nm at 10° angle of incidence (Laser-Optik GmbH). Reflectivity per bounce exceeds 99.8% for P-polarization. The final number of bounces is optimized based on the MPC set-up, gas pressure and temporal pulse profile requirements. Notably, third-order dispersion (TOD) is not compensated.

Each Yb laser and corresponding MPC set-up are equipped with two cameras for beam diagnostics and daily alignment (see Fig. 1). Camera C1 monitors the laser at the

exit window via leakage from the first steering mirror. A portion of this leakage is also used by camera C2 to image the focal spot in air, ensuring alignment with the cavity caustic by fitting the coupled beam. This setup facilitates the detection and correction of beam pointing variations. Despite the 3 m separation between the laser and the MPC, excellent long-term stability is achieved without active pointing control. Energy and pointing stabilities closely match those of the Yb lasers before MPC, with measured energy stability of 0.25% and pointing stability below 5 µrad over seven hours.

Additionally, the measured far-field profile provides the size of the focal spot obtained in air, which correlates to the beam size when coupled into the cavity. We capture the spot patterns resulting from the N bounces on one of the two cavity mirrors by imaging their leakage using camera C3. These spots correspond to either the odd or the even bounces on the cavity end mirror [29]. The evolution of the spot size is directly linked to the coupling efficiency and is critical for ensuring the safety of the mirrors by avoiding damage. This aspect will be discussed further in the next section.

## 2.2 Multi-pass-cells design and laser coupling

Four distinct MPC set-ups with various characteristics were developed to achieve optimal focused intensities that promote self-phase modulation (SPM) while minimizing ionization and self-steepening effects. The Herriot-type cells were dimensioned based on key parameters outlined in [29, 34], such as the waists at focus and on mirrors, peak intensities at focus, and fluences on mirrors. Argon gas was selected as the nonlinear medium for all setups. These combined constraints resulted in four unique configurations, each defined by a specific radius of curvature (RoC) and distance L between the spherical mirrors. The RoC-L pairs (in mm) are as follows: (250; 495), (300; 595), (500; 978), (750; 1480) for MPC1, MPC2, MPC3 and MPC4, respectively. The ratio $C=L/RoC$ remains between 1.96 and 1.98 in all cases, ensuring the cavity stability. These configurations allow increased intensity at focus while reducing fluence on the mirrors. Table 2 summarizes the relevant MPC parameters, including the extraction pass number, optimal argon pressure for spectral broadening, and estimated intensity reached at focus.

Nonlinear propagation, dependent on the gas pressure $p$, modifies waist size at focus and on the mirrors. This modifies the B-integral for each pass in the cavity and the fluence on the mirrors. A nonlinear propagation correction factor, $\sigma$, is defined in [34] as $\sigma = 1 - P_0/P_{crit}$ where $P_0$ is the beam's peak power and $P_{crit} = 3.77\lambda^2/(8\pi n_0 n_2 p)$ with $n_0$, $(n_2)$ being the gas' linear (nonlinear) index. The factor $\sigma$ adjusts the accumulated B-integral to account for nonlinear propagation effects. We used a modified version of the B-integral
<table><caption>**Table 2** Main characteristics of the 4 different specifically designed MPC for optimal energy coupling and post-compressed durations</caption><thead><tr><th>MPC</th><th>RoC (mm)</th><th>length L (mm)</th><th>extraction pass</th><th>mirror fluence Fm (mJ/cm²)</th><th>focus intensity I₀ (W/cm²)</th><th>Argon pressure p (mbar)</th></tr></thead><tbody><tr><td>MPC 1</td><td>250</td><td>495</td><td>27</td><td>75</td><td>2.0×10¹³</td><td>3500</td></tr><tr><td>MPC 2</td><td>300</td><td>595</td><td>25</td><td>130</td><td>5.3×10¹³</td><td>1300</td></tr><tr><td>MPC 3</td><td>500</td><td>978</td><td>19</td><td>160</td><td>2.0×10¹³</td><td>1200</td></tr><tr><td>MPC 4</td><td>750</td><td>1480</td><td>17</td><td>140</td><td>2.3×10¹³</td><td>1100</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 2** Calculated fluence on each cavity mirror (M1 in blue and M2 in red) as a function of the number of roundtrips in a MPC composed of two spherical mirrors with $RoC=500$ mm, $L=978$ mm filled with 2 bar of argon. The injected laser has a 2 mJ energy, 300 fs duration and a $M^2=1.05$ pulse. Open squares correspond to the optimized coupling conditions with linear propagation $\sigma=1$; Full squares to nonlinear propagation $\sigma \neq 1$ while keeping the previous coupling conditions; Full circles to the new optimized coupling conditions while considering nonlinear propagation $\sigma \neq 1$</figcaption></figure>

from [34] that includes the beam quality factor $M^2$ and energy losses at each bounce on the cavity spherical mirrors. Since injection and extraction mirrors share the same reflectivity as the cavity mirrors, there are 2 N bounces N roundtrips in the cavity. The estimated B-integral after N roundtrips in the cavity is given by the formula:

$$B_{\text{int}, N} = \sum_{n=1}^{2N} \frac{4\pi}{\lambda^2} n_2 p \frac{E}{\tau} R^n \frac{1}{\sqrt{\sigma M^2}} \arctan \left( \sqrt{\frac{L}{2RoC - L}} \right) \quad (1)$$

Here $R$ is the mirror reflectivity. The overall compression factor, $K \sim B_{\text{int}} N/\pi$ [29, 35], provides an estimate for the post-compressed Fourier transform limited (FTL) pulse duration. Our objective is to achieve post-compressed pulse durations below 25 fs, corresponding to a compression factor or B-integral between 14 and 18, depending on the initial laser pulse duration.

The $\sigma$ factor adjusts the B-integral by 10–20% depending on the case. Its impact is crucial when designing optical

setups for efficiently coupling the laser beam into the MPC. For pulse energies of 2 mJ and 4 mJ at optimal pressures, the waist size in the gas-filled MPC ranges from 20 µm to 50 µm, compared to optimal cavity coupling waists in vacuum of 110 µm and 160 µm, respectively. Propagation in the MPC was simulated using ABCD matrix analysis with nonlinear corrections for $\sigma$.

Figure 2 shows the calculated fluence on each cavity mirror as a function of the number of roundtrips in the cavity. Simulations use optimized focal lengths (L1; L2) and a specific coupling lens (L3). For a 2 mJ energy, 300 fs duration pulse propagating over 11 roundtrips in 2 bar of argon, the mean fluence with linear propagation ($\sigma=1$) is approximately 111 mJ/cm² with a small standard deviation (SD) of 4 mJ/cm². Incorporating the $\sigma$ factor without altering coupling parameters increases the fluence to 173 and 167 mJ/cm² on M1 and M2, respectively, with an SD of 51 mJ/cm². Maximum fluence reaches around 250 mJ/cm², exceeding the optics’ damage threshold. By modifying the magnification of lenses L1 and L2 to re-optimize coupling conditions, the mean fluence stabilizes at 170 mJ/cm² with an SD of 6 mJ/cm², reducing the risk of optical damage. Therefore, even though the mean value is close to the previous one, there is no risk to reach the optics LIDT. This example illustrates the importance of considering nonlinear propagation to establish proper coupling conditions, such as determining the beam magnification via focal lengths of L1 and L2 for a given lens L3. ABCD matrix calculations reveal a > 40% increase in fluence on mirrors when the focusing lens (2 m focal length) position shifts by $\pm 10$ mm or the magnification deviates by 10% from optimal conditions.

Coupling the nonlinear ABCD matrix calculations with a 1D analytical model enabled us to design optimal MPC setups, including lens focal lengths for efficient beam coupling, and estimate post-compressed pulse durations. These results were validated with 3D numerical simulations presented in Sect. 3.

Sections 2.3 and 2.4, present the results of spectral broadening and pulse post-compression experiments conducted with the different MPCs.

## 2.3 Spectral broadening and power transmission

For each MPC and laser system, we used the same experimental procedure. As discussed in the previous section, the
experimental setups have been installed with a high degree of precision, based on the ABCD matrix calculations. We first use the laser at maximal energy while reducing the repetition rate in order to stay around 8 W average power, thus limiting any possible heat load issue with the optics. Focusing optics are set-up for optimal coupling in the cavity for the optimal gas pressure. Under vacuum, the beam pattern thus shows some waist variations, as predicted. The MPC transmission is measured reaching 95% and 98% for MPC1 and MPC2, respectively, and close to the theoretical values of 98.5% and 99.2% for MPC3 and MPC4. Then, we increase progressively the argon static pressure up to its optimal value. The focus is adapted if necessary checking the beam pattern observed on C3. Finally, the repetition rate is set to maximum in order to deliver up to 80 W average power. After steering optics and chirped mirrors the overall power transmission is between 90% and 92% depending on the set-up.

In Fig. 3a we show the spectral broadening as a function of pressure for MPC1 coupled to the Yb1 laser delivering pulses with 800 µJ energy. Spectra are recorded at the MPC output after the chirped mirror compressor. The 10 nm initial spectral width measured at $1/e^2$ broadens up to

<figure><img src="image_5.png" /><figcaption>Fig. 3 a Spectral broadening measured as a function of pressure for MPC1 with laser Yb1 delivering 800 µJ. b Spectral width measured at $1/e^2$ (black symbols) and corresponding FTL FWHM duration (red symbols) for MPC1 with laser Yb1 delivering 800 µJ (circles) and 400 µJ (squares) energies respectively</figcaption></figure>

150 nm at optimal pressure. The width of the spectra shows a linear increase with pressure up to 3.5 bar, as depicted by full circles in Fig. 3b. The FTL pulse duration decreases with increasing pressure as expected, reaching a minimum duration, measured at full width half maximum (FWHM), of 17 fs. This linear dependence of spectral width on pressure is a generic behavior observed for the four different MPC configurations. MPC1 could also be used with a lower input energy of 400 µJ and a repetition rate of 200 kHz. This would lead to a 23 fs pulse duration at 4.5 bar, which is the maximum pressure supported by the MPC chambers. Thus, it is possible to adjust the pulse duration by adapting the argon pressure for given energy and repetition rate.

## 2.4 Direct post-compression down to 18.6 fs pulse duration

Compression of the broadened spectra is achieved using chirped mirrors with second order dispersion. The number of bounces of each mirror must be adjusted depending on the pressure to achieve a pulse duration as close as possible to the FTL value. Each chirped mirror allows up to 6 reflections for beam diameters of 13 mm at $1/e^2$. The number of passes in the compressor is optimized for each set-up and pressure to produce sub-25 fs pulses. Depending on the MPC characteristics, the number of reflections required for post-compressing the pulses ranges from 8 to 12.

In Fig. 4a and b, we present the respective input and retrieved spectrograms obtained using a single shot SHG-FROG (Femtoeasy), after MPC1 and MPC4 with input pulse energies of 800 µJ and 4 mJ, respectively. These represent the extremes of the cases we have studied. The average retrieval FROG error is below $5 \times 10^{-3}$. Figure 4c shows the broadened spectrum obtained in each case (red and blue lines) at optimal pressure. While the profiles are quite similar, the bandwidth is larger in the lower-energy case. Beyond the respective optimal pressures of 3.0 and 1.15 bar, the pulse duration increases due to satellite peaks in both cases. As the spectrum broadens further, uncompensated third-order dispersion (TOD) significantly affects the pulse profile because only second-order dispersive mirrors are used. The FROG retrieved temporal profiles shown in Fig. 4d reveal FWHM pulse durations of 18.6 fs and 22.4 fs for input energies of 800 µJ and 4 mJ, respectively, while the corresponding FTL pulse durations are 18.3 fs and 20.5 fs. For the 4 mJ case, the pulse profile exhibits more pronounced residual second-order phase distortion due to the maximum allowable bounces on the chirped mirrors being reached. At 400 µJ energy, 200 kHz repetition rate, a pulse duration of 26 fs FWHM is achieved at 4.5 bar with MPC1, which represents the pressure limit of the chamber before leakage occurs. Shorter pulse duration in the 20 fs
<figure><img src="image_6.png" /><figcaption>**Fig. 4** Input and retrieved single shot SHG-FROG spectrograms for a MPC1 with 800 µJ pulse energy and b MPC4 with 4 mJ pulse energy coupled at 80 W and 20 W average power in the cells filled with 3.0 and 1.1 bar static pressure of argon, respectively. c Corresponding measured spectra (solid red and dashed blue lines). Solid and dashed</figcaption></figure>

green lines, depict the retrieved spectral phases. d Corresponding retrieved temporal profiles leading to respectively 18.4 fs and 22.4 fs FWHM pulse duration in low and high energy MPC. Solid and dashed green lines, depict the retrieved temporal phases

<figure><img src="image_7.png" /><figcaption>**Fig. 5** Broadened spectra obtained at optimal pressure at the output of a MPC3 and MPC4 b MPC1 and MPC2</figcaption></figure>

range can be obtained at higher pressure around 6 bar, but a continuous gas leakage emerges at these levels, necessitating the use of a chamber designed for higher pressures.

FROG traces and pulse durations measured after MPC2 and MPC3 are similar to those shown in Fig. 4. The post-compressed pulse durations at 2 mJ energy are respectively 19.2 fs and 20 fs for pressures of 1.1 and 1.2 bar, respectively. The FTL durations are 17.9 fs and 19.5 fs, respectively. The influence of TOD becomes more pronounced as

the spectra broaden, resulting in some energy being transferred into satellite peaks compared to ideal dispersion-compensated pulse compression. The final pulse duration represents a balance between residual group delay dispersion GDD and TOD.

## 2.5 Specific spectral features

At respective optimal argon gas pressures all the broadened spectra exhibit a nearly identical spectral width at 1/e², ranging between 140 nm and 160 nm (Fig. 5a and b), resulting in sub-20 fs post-compressed pulse durations as described previously. Upon closer examination of the spectra, several interesting features emerge. First, as clearly observable in Fig. 4, there is an asymmetry between the blue and red wings of the spectra obtained with energies above 1 mJ. This asymmetry is a direct consequence of self-steepening caused by the nonlinear interaction of the pulse with the gas during the initial passes [36]. Notably, this effect occurs even with the relatively long pulses used in this study. Second, a distinct difference is evident in the central spectral region around 1030 nm. Specifically, in the spectra shown in Fig. 5b for the short RoC MPC setups, two sharp peaks appear around 1030 nm, whereas these features are absent in the spectra obtained from the long RoC MPC setups (Fig. 5a).

The intensity at focal point is twice as high in MPC2 compared to the other set-ups, which exhibit similar intensities (see Table 2). Notably, MPC1 and MPC2 share the same spectral behavior around the 1030 nm wavelength, despite the intensity in MPC2 being 2.6 times greater than that in MPC1 under equivalent cavity conditions. Consequently, these peaks cannot originate from nonlinearity or intensity-related effects associated with the shorter RoC of the cavity mirrors used in these setups.
An alternative explanation could be related to differences in the pulse profile or coherent contrast among the four Yb lasers. Indeed, autocorrelation traces performed by Light Conversion during the laser installation indicate larger pulse wings in Yb1 and Yb2 compared to the other two lasers. Unfortunately, our FROG device cannot measure pulse durations exceeding 50 fs and is therefore not suitable for precisely characterizing the Yb laser pulse profiles. A scanning FROG is currently under development to address this limitation. Contrast measurements will also be conducted using a third-order correlator, which is currently optimized for an 800 nm wavelength and requires modification to operate at 1030 nm.

In the absence of these specific experimental tools, we will rely on simulations to validate our interpretations. As previously noted, the MPC setups and their optimized operating conditions were designed using a combination of simplified and advanced models [29, 34]. The defined characteristics have been confirmed using a more sophisticated model, which will be described in detail in Sect. 3. Additionally, we will compare the numerical calculations with the experimental results to further corroborate our findings.

# 3 Numerical calculations

After a brief description of the equations used in the model, we will compare the calculated spectra and post-compressed pulse durations to the experimental results.

## 3.1 3D model description

The main equation of the model is the nonlinear envelope propagation equation, derived by Brabec and Krausz in the slowly evolving wave approximation (SEWA), valid for pulse durations as short as one optical cycle [37]. In the moving reference frame $\tau = t - \beta_1 z$ and $= z$, it is written, for a pulse propagating in the positive $\xi$ direction:

$$ \frac{\partial \mathbf{E}}{\partial \xi} = iD\mathbf{E} + \frac{i}{2\beta_0} T^{-1} \nabla_{\perp}^2 \mathbf{E} + i \frac{\beta_0}{2n_0^2} T \frac{\mathbf{P}_{\text{NL}}}{\epsilon_0} \quad (2) $$

In cylindrical geometry and in the general case, the field $\mathbf{E}$ depends on the three spatial coordinates $r, \xi$ and $\theta$, and on the retarded time $\tau$, i.e. $\mathbf{E} = \mathbf{E}(r, \xi, \theta, \tau)$, and $\beta_l = v^l$, where $v$ is an arbitrary reference frame velocity, generally taken as the group-velocity at the central pulse frequency $\omega_0$. We limit here our study to axis-symmetric beams and homogeneous media along the propagation direction. We then drop the polar angle $\theta$ and the coordinate $\xi$ in the following. In the first term on the right-hand side of Eq. 2, the dispersion operator linear in $\mathbf{E}$ is $D = \sum_{m=2}^{\infty} \frac{\beta_m}{m!} (i \frac{\partial}{\partial \tau})^m$. In the

present study, we limit the development to the third order: $D \approx -\frac{\beta_2}{2} \frac{\partial^2}{\partial \tau^2} - \frac{\beta_3}{6} \frac{\partial^3}{\partial \tau^3}$. We checked that the higher-order terms are negligible here. The second term describes the diffraction, and the third term is the nonlinear polarization. The operator $T = 1 + \frac{i}{\omega_0} \frac{\partial}{\partial t}$ in front of $\mathbf{P}_{\text{NL}}$ is responsible for the self-steepening of the pulse. $\beta_0 = \frac{n_0 \omega_0}{c}$ is the wavenumber, and $n_0 \equiv n(\omega_0)$ is the linear index of refraction at frequency $\omega_0$.

The nonlinear polarization is due to the Kerr effect, and to the optical-field ionization of the gas:

$$ \mathbf{P}_{\text{NL}}(r, \tau) = \mathbf{P}_{\text{Kerr}}(r, \tau) + \mathbf{P}_{\text{ion}}(r, \tau) \quad (3) $$

with:

$$ \mathbf{P}_{\text{Kerr}}(r, \tau) = 2\epsilon_0 n_0 n_2 I(r, \tau) \mathbf{E}(r, \tau) \quad (4) $$

where $I = \frac{1}{2} n_0 \epsilon_0 c |\mathbf{E}|^2$ is the laser intensity, $c$ is the speed of light, $\epsilon_0$ is the vacuum permittivity and $n_2$ is the nonlinear index of refraction of the medium $n_2 = (9.7 \pm 1.2) \times 10^{-20} \times \text{p cm}^2/\text{W}$, for argon gas [38]). Note that as we consider noble gases here, there is no Raman contribution to the Kerr effect.

The ionization contribution to the nonlinear polarization is written:

$$ \begin{aligned} \mathbf{P}_{\text{ion}}(r, \tau) = & n_0 \epsilon_0 c |I_P| \int_{-\infty}^{\tau} \frac{\partial \rho_e(r, \tau')}{\partial \tau'} \frac{\mathbf{E}(r, \tau')}{I(r, \tau')} d\tau' \\ & + \epsilon_0 \int_{-\infty}^{\tau} d\tau' \int_{-\infty}^{\tau'} [\omega_p(r, \tau'')]^2 \mathbf{E}(r, \tau'') d\tau'' \end{aligned} \quad (5) $$

where $|I_P|$ is the ionization potential of the gas, $\omega_p(r, \tau) = \sqrt{\frac{\rho_e(r, \tau)e^2}{m_e \epsilon_0}}$ is the plasma frequency, $\rho_e(r, \tau)$ is the spatio-temporal dependent electron density, and $e$ and $m_e$ are the charge and mass of the electron, respectively.

The time evolution of the electron density is given by:

$$ \frac{\partial \rho_e (r, \tau)}{\partial \tau} = [\rho_{at} - \rho_e (r, \tau)] W_{PPT} (|\mathbf{E}(r, \tau)|) \quad (6) $$

with $\rho_{at}$, the original density of neutral atoms, and $W_{PPT}$, the Coulomb-corrected Perelomov-Popov-Terent'ev (PPT) rate in linear polarization [39–41], valid in both multiphoton and tunpresent study, we limit the development to the third order regimes.

We solve Eq. 2 with auxiliary Eqs. 4, 5 and 6, using a symmetrized split-step method [42]. The linear part of the equation is solved by the finite-difference frequency-domain (FDFD) technique, while integration of the nonlinear part
is performed using a fifth-order Runge–Kutta method with Cash–Karp coefficients [43]. A Gaussian field profile is typically assumed in both space and time. Propagation of the pulse is performed over the N roundtrips in the cavity.

## 3.2 Comparison of numerical and experimental results

When using a perfect Gaussian temporal pulse profile for the calculations, we obtain a classical broadened spectrum with oscillations due to self-phase modulation (SPM), as shown in black in Fig. 6b. The calculation parameters match the experimental conditions for the MPC2 set-up and Yb2 laser parameters: a 2 mJ pulse energy, 320 fs pulse duration and a cavity composed of two concave spherical mirrors with a radius of curvature of 300 mm separated by L=595 mm, with 27 propagation passes.

The calculated spectrum is slightly broader than the experimental one and does not exhibit the sharp peaks around 1030 nm observed experimentally (blue line in Fig. 6c). These peaks appear in the calculation when using a slightly degraded temporal pulse with lower coherent contrast (red line Fig. 6a). Such a degraded profile may reflect an experimental situation where the pulse is affected by compression defects, such as residual third-order dispersion (TOD). Comparison between the calculated and experimental spectra (Fig. 6c) reveals excellent agreement, not only in the spectral width and oscillatory behavior but also in the two sharp peaks near 1030 nm. The experimental spectral width is reproduced by slightly adjusting the argon pressure in the calculations to 0.9 bar compared to 1.0 bar used in the experiment.

The overall agreement between the calculated and experimental spectra is remarkable, particularly in terms of the spectral shape and the width at $1/e^2$. It should also be noted that the calculations precisely reproduce the positions of the main oscillation maxima in the spectra, as well as the sharp peaks at the central wavelength. The asymmetry between the blue and red sides of the spectrum is also qualitatively replicated, even if the amplitude of the oscillations differs

slightly. This calculated asymmetry increases with pressure, as observed in the experiment. Finally, it can be stated that, under our conditions, ionization has no impact on the spectral shapes. When ionization and related absorption terms are excluded from the numerical calculations, no changes are observed in the output spectra.

The calculated temporal profile at the MPC output also aligns well with the measured profile. In Fig. 6d, the effect of spectral dispersion compensation is shown for the optimal MPC2 pressure (1.3 bar experimentally vs. 1.2 bar in the calculations). The pulse duration corresponds to an FTL of 17.9 fs full width at half maximum (FWHM), which increases slightly to 18.5 fs FWHM when –1200 fs² GDD and +2000fs³ TOD are applied. Notably, the FWHM remains unchanged when the residual TOD is compensated; only the energy in the satellite peaks is reduced. Experimentally, the shortest measured FWHM pulse duration is 19.4 fs for eight reflections on chirped mirrors (–200 fs² per bounce) at 1.3 bar, slightly larger than the prediction.

In Fig. 7a, we show the FTL pulse duration as a function of pressure for MPC2; black circles for the numerical results while red squares correspond to the experimental data. The calculated points are well fitted by a 1/p law, as shown by the solid line. We note here a close agreement between numerical and experimental results. Figure 7b shows the evolution as a function of pressure of the calculated and experimental pulse durations for different GDD compensations. In the calculations, the shortest pulse duration are achieved for GDD values between –2000 and –1000 fs², the amount of GDD depending on the pressure. The color symbols represent the experimental values, measured for different numbers of reflections on the chirped mirrors. Below 1 bar, the optimal pulse duration is obtained with 10 reflections on the chirped mirrors (GDD = -2000 fs²), consistent with the predictions (green diamonds). Between 1.0 and 1.3 bar, 8 reflections (GDD = -1600 fs²) yield the shortest pulse, though still longer than the FTL limit (blue triangles). Above 1.3 bar the TOD dispersion becomes too large increasing the energy in the satellite peaks that become bigger while keeping a central peak around 20 fs. At 1.4 bar the effect of TOD is so

<figure><img src="image_8.png" /><figcaption>**Fig. 6** a Initial 320 fs temporal profile used for the spectra simulations in the ideal case (black line) and with degraded coherent contrast (red line). b Respective calculated spectra obtained for a 2 mJ pulse propagating 27 passes in the MPC at 1 bar argon pressure. c Comparison</figcaption></figure>

between calculated (red line) and experimental spectrum (blue line). d Calculated FTL pulse profile (black), with GDD compensation of –1200 fs² (red) and with additional TOD compensation of +2000 fs³ (blue) at optimal argon pressure of 1.2 bar
<figure><img src="image_9.png" /><figcaption>**Fig. 7** FTL pulse duration as a function of pressure obtained in the calculations with its 1/p fit (black circles and line) and in the experiments (red squares). Pulse duration evolution with pressure of the calculated and experimental results for different GDD compensations: between –2000 and –1000 fs² for the calculations (black circles and line), –1600 fs² (blue triangles) and –2000 fs² (green diamonds) in the experiments</figcaption></figure>

huge that the satellite peaks are close to 50% distorting the FWHM measurement. Residual GDD and TOD are more prominent in experiments than in simulations, likely due to additional propagation in air and through optical components not included in the calculations. These residual dispersions increasingly affect pulse duration as spectral width grows. However, TOD has minimal impact on the FWHM duration as long as satellite peaks remain small, as demonstrated in Fig. 6d. Shorter pulse durations could be achieved experimentally with improved TOD management.

## 4 Conclusion

In summary, sub-20 fs post-compressed pulses were demonstrated at 80 W average power, starting with commercial Yb-based lasers delivering 340 fs pulses with energies between 0.8 and 2 mJ. Pulse durations close to 22 fs were also achieved at 20 W average power and 4 mJ energy, starting with 450 fs pulse durations. Four different multi-pass cells were designed to attain these short durations. The specific design enables versatile MPCs that can be easily

adjusted in size depending on the coupled pulse energy. The use of argon gas as a nonlinear medium was essential to reduce operational costs over extended periods.

To our knowledge, this is the first instance of achieving sub-20 fs pulse durations at this energy level through direct post-compression using gas-filled MPCs, without the need for dispersion engineered cavity mirrors or specially designed chirped mirrors for perfect dispersion control. Improved control of third-order dispersion (TOD) would refine the temporal pulse profile by limiting satellite peaks. However, as discussed, the full-width half-maximum (FWHM) duration would not change significantly for the same spectral width. Specifically designed chirped mirrors could potentially achieve a 17 fs Fourier-transform-limited (FTL) pulse duration.

Calculations performed using both a simple 1D model and a 3D propagation code were employed to design the various MPC setups and define the optimal conditions for achieving direct pulse duration post-compression down to sub-20 fs. These calculations are in quantitative agreement with experimental measurements across all MPC setups, regardless of the experimental conditions. The numerical results also emphasize the critical importance of the temporal profile of the injected pulse on the broadened spectra's shape. Experimental measurements will be conducted in the near future to validate these theoretical trends once the homemade devices become available.

**Supplementary Information** The online version contains supplementary material available at https://doi.org/10.1007/s00340-025-08415-0.

**Acknowledgements** We thank L. Daniault, R. Lopez-Martens and M. Hanna for fruitful discussions about the different MPC designs and for MPC injection/extraction mirror cutting.

**Author contributions** JF. H. made the substantial contribution to this work, in supervising the different experimental experiments, in defining the MPCs with 1D model as well as optimal laser coupling conditions with M. G. T. A. developed the 3D numerical code and performed the calculations and analysis with JF. H. F. R. dimensioned the MPC steps with JF. H. Experiments as well as collecting and analyzing the results have been performed by: JF. H., F. R., P. N., D. B., T. R. for MPC1, N. L., and H. M. for MPC2, O. T., F., L., F. R. for MPC3 and M. G., R. G., A. T., R. C. for MPC4. Drafting the manuscript has been done by JF. H. T. A., O. T. and P. S. All authors reviewed the manuscript.

**Funding** Open access funding provided by Commissariat à l'Énergie Atomique et aux Énergies Alternatives. Authors acknowledge fundings by the European Union (European Research Council Spinfield, Project No. 101041074 - European Research Council SATTOC, Project No. 101078595), and by Agence Nationale de la Recherche ANR-10-LABX-0039-PALM. This work benefited from state funding managed by the National Research Agency under the France 2030 investment plan bearing the reference ANR-22-EXLU-0002.
**Data availability** No datasets were generated or analysed during the current study.

## Declarations

**Competing interests** The authors declare no competing interests.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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