# Frequency-resolved cross-correlation between XUV high harmonics and IR fundamental laser pulses by transient multiphoton absorption spectroscopy in gases

Juliette Dubois, Adrien Leblanc, Julien Gautier, Fabien Tissandier, Léonardo

Rico, Boris Vodungbo, Jérémie Caillat, Camille Lévêque, Richard Taïeb,
Guillaume Lambert

► To cite this version:

Juliette Dubois, Adrien Leblanc, Julien Gautier, Fabien Tissandier, Léonardo Rico, et al. Frequency-resolved cross-correlation between XUV high harmonics and IR fundamental laser pulses by transient multiphoton absorption spectroscopy in gases. Physical Review Research, 2025, 7, pp.023065. 10.1103/PhysRevResearch.7.023065. [hal-05038682]

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Distributed under a Creative Commons Attribution 4.0 International License
Frequency-resolved cross-correlation between XUV high harmonics and IR fundamental laser pulses by transient multiphoton absorption spectroscopy in gases

Juliette Dubois,<sup>1,2,*</sup> Adrien Leblanc,<sup>1</sup> Julien Gautier,<sup>1</sup> Fabien Tissandier,<sup>1</sup> Léonardo Rico<sup>●</sup>,<sup>2</sup> Boris Vodungbo<sup>●</sup>,<sup>2</sup> Jérémie Caillat<sup>●</sup>,<sup>2</sup> Camille Lévêque<sup>●</sup>,<sup>2</sup> Richard Taïeb,<sup>2</sup> and Guillaume Lambert<sup>●</sup><sup>1,†</sup>

<sup>1</sup>LOA, ENSTA Paris, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 828 Bd des Maréchaux, 91762 Palaiseau, France

<sup>2</sup>Sorbonne Université, CNRS, Laboratoire de Chimie Physique-Matière et Rayonnement, LCPMR, 75005 Paris, France

<figure><img src="image_2.png" /></figure>

(Received 26 September 2024; accepted 18 February 2025; published 17 April 2025)

Numerous methods have been developed to retrieve the full temporal properties of ultrafast infrared laser pulses, and are presently implemented on laser facilities all around the world. We present here a technique, developed for the XUV domain, suitable to characterize any femtosecond to picosecond pulse, that can be routinely produced on the high variety of existing sources of high harmonics, free electron lasers, and plasma-based XUV lasers. Especially, the versatility is such that no limitation on any property of any potential source exists. Our technique relies on a frequency-resolved cross-correlation measurement between an infrared pulse and an XUV pulse, combined with a transient multiphoton absorption in gases. It provides temporal information on the longest of the two pulses: intensity profile, pulse duration, and possible presence of pre- or postpulses. More importantly, the measurement of resolved XUV spectra and the use of iterative phase-retrieval algorithms give access to phase information. The simplicity of the technique makes its implementation on various XUV beamlines straightforward.

DOI: 10.1103/PhysRevResearch.7.023065

## I. INTRODUCTION

For the last decades, the development of IR femtosecond laser pulses opened new areas of research and applications in physics, chemistry, biology, and engineering. These new laser sources, delivering femtosecond pulse duration, allow to probe the dynamics of a quantum system on an unprecedented timescale. This breakthrough was possible thanks to the full characterization of these pulses—their frequency, envelope profile, duration, and carrier envelope phase, parameters that need to be known and controlled to enable applications. Consequently, a number of techniques have emerged in order to fully measure these IR pulses.

Historically, autocorrelation techniques were first used to characterize femtosecond pulses. A pulse is split in two, and the resulting twin pulses are combined through a physical phenomenon to create a correlation referred to as “auto,” since the pulses originate from a single source. The phenomenon used to create the correlation can be of a different nature, for instance, interferences with a Michelson interferometer [1], nonlinear effects in a second harmonic generation (SHG) crystal for an intensity autocorrelator [2], or a two-photon

absorption in a semiconductor [3]. The temporal measurement of the phenomenon duration gives access to the intensity profile of the laser pulse. Third-order intensity autocorrelations have also been developed using third-order nonlinear optical effects. Although more complex, they bring additional temporal information on the laser pulse, due to their asymmetry. An alternative exists when the two pulses are not related to each other; we talk about cross-correlation, and an “unknown” pulse is characterized thanks to a well-defined pulse, used as a reference. When the unknown pulse is much longer than the reference pulse, the cross-correlation gives a straightforward access to its intensity temporal profile without requiring knowledge on the reference temporal profile. Such characterization techniques were first developed in the IR range for femtosecond laser pulses, and then extended to the XUV range [4,5] together with other methods [6–15].

However, auto- and cross-correlation techniques do not provide access to the full temporal properties of laser pulses since the intensity informs us on the envelope profile and duration but not on the carrier envelope phase. The solution is to measure, in addition, the radiation spectrum, and use iterative phase-retrieval algorithms to fully reconstruct the pulse. Such techniques have been implemented, for instance, in SHG frequency resolved optical gating (FROG) [16], where two delayed twin pulses interact with a crystal, the second harmonic being acquired on a spectrometer. When one pulse is unknown, the term XFROG is used. One can also replace the spectrometer with a GRENOUILLE setup [17], composed of various optics that allow to create a relation between delay and wavelength with space coordinates. Finally, the SPIDER (spectral phase interferometry for direct electric-field reconstruction) technique is based on the interferometry of two spectrally and temporally shifted pulses [18]. These tech-

<footnote>*Also at Sorbonne Université, CNRS, Laboratoire de Chimie Physique-Matière et Rayonnement, LCPMR, 75005 Paris, France.</footnote>

<footnote>†Contact author: guillaume.lambert@ensta.fr</footnote>

<footnote>Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.</footnote>
niques have been widely used with success. More recently, several other techniques have also shown their capacity to fully characterize femtosecond pulses in the IR range, such as SRSI [19], Dscan [20], TIPTOE [21], FROSt [22] and TIGER [23], some of them down to the visible range.

Methods to fully characterize short UV and XUV laser pulses are particularly of interest with the development of various picosecond to femtosecond sources from UV to XUV over the last two decades. These sources include low-order harmonics from various crystals, high-order harmonics from gas (HHG) [24], molecules [25], plasma mirrors [26], plasma plumes [27], semiconductors [28], and liquids [29], but also free electron lasers (FEL) [30] and plasma-based XUV lasers (solid [31] and gas [32] targets). The zoo of above-mentioned characterization techniques cannot be extended in the XUV domain. Indeed, they require optics, crystals, or materials that are not adapted to this spectral range. The smallest wavelength that appears to have been reached is 260 nm, obtained by an interferometric autocorrelation through two-photon absorption (TPA) in a diamond pin-type photodiode [33] and also by induced-grating XFROG in ZnS [34]. This wavelength is still relatively far away to be considered for application in the XUV domain.

Among the techniques used in the IR domain, the SPIDER technique has been adapted to XUV with relative success, for FEL [35,36] or for harmonics using a dazzler [37]. Nevertheless, the requirement of two spectrally shifted pulses in the XUV range is often impossible to fulfill for most sources, and the technique is limited to specific spectral and duration ranges.

In parallel, other techniques were developed for the HHG community to characterize XUV ultrashort pulses, such as the reconstruction of attosecond beating by interference of two-photon transitions (RABBIT) [38,39], the XUV-XFROG [40], and the FROG CRAB (for complete reconstruction of attosecond beatings) [41]. They rely on the interaction of an XUV pulse with a low-intensity IR dressing field through the ionization of a gas; photoelectrons are then collected thanks to a time-of-flight spectrometer. However, they are not adapted for XUV sources other than HHG sources composed of odd harmonics. To summarize, although quite powerful, such techniques that have been in reality developed essentially to measure attosecond pulses [42] are only devoted to classical high harmonic generation and cannot be easily implemented. One of the recent solutions is to associate FROG CRAB with ptychography and time-delay compensated monochromatization of an HHG source [43]. Using such combinations of techniques is highly challenging experimentally. Consequently, other techniques that work for all types of XUV femtosecond source, and that are easy to implement and use on a beamline, should be found.

Here, we present a technique that meets all the expected criteria. It allows to fully characterize any picosecond to femtosecond XUV pulses. We call it TIBETAN-FOX (or T-FOX), for multi-photon absorption spectroscopy in gases and a Frequency-resolved cross-correlation in Xuv. It relies on a cross-correlation between an IR pulse and an XUV pulse through the interaction with a gas and consists of measuring the transient absorption of this gas [44–47] with respect to the IR-XUV delay. This absorption presents a multiphoton aspect and takes place in a large spectral area for energies below the

ionization levels of gases. This technique actually requires the use of a simple millimeter-scale-long gas target that can be placed in an application chamber instead of a sample, an IR beam at low intensity, and a regular XUV spectrometer, which can be found on any typical pump-probe beamline.

The paper is organized as follows. After a brief description of the experiment, we present the physical phenomena used to characterize the pulses, namely, transient multiphoton absorption. Then, we apply our technique to the characterization of pulses in terms of temporal envelope shape and duration. Since the cross-correlation intensity can fully determine the temporal intensity profile of the longest pulse, we distinguish two different cases, for which either the IR or the XUV pulse is longer than the other. We also show that this technique can be used to measure the temporal contrast of such pulses. Finally, due to the simultaneous measurement of the spectrum, phase information is obtained using iterative phase-retrieval algorithms [48–50]. The full temporal information of any type of XUV source is then accessible.

## II. EXPERIMENTAL SETUP

We have set up an IR-XUV experiment at the “salle corail” of the Laboratoire d’Optique Appliquée, using a 5-kHz infrared laser system (Legend Elite Duo from Coherent) delivering pulses at a central wavelength of about 800 nm, with typical FWHM duration of about 40 fs and energy of 2.5 mJ. The experimental arrangement can be decomposed into five parts, the optical table located in air and four different vacuum chambers (see Appendix A). Briefly, on an XUV arm, a series of harmonics with odd and even orders [51–54] is generated from a first gas cell and refocused inside a second gas cell. In the latter, drilled by a focused IR laser coming from an IR arm, XUV and IR photons interact strongly with the gas medium (see Fig. 1).

The temporal delay between the IR and XUV pulses ($\tau$) is adjusted using an optical delay line, and the transmission of the harmonics signal is analyzed on a spectrometer. Two typical spectrograms are depicted in the inset of Fig. 1 for neon (25 mbar), for delays such that the XUV pulse does not overlap with the IR pulse of the IR arm in the gas sample (labeled as “No IR”) and when the IR and XUV pulses temporally overlap (labeled as “IR”). Comparing both spectrometer images, we see an absorption of some specific harmonics in the presence of IR pulses. Here, a decay of the transmission of the 13th harmonic ($H_{13}$) is seen while the other harmonics remain mainly unaffected. In the following we interpret this phenomenon and show how we can use it to characterize unknown light pulses.

In this paper, we present three different types of temporal profile related to the transmission of a given harmonic into a gas, the IR pulses, or the XUV pulses. These profiles are defined by their FWHM, denoted $\Delta_T$, $\Delta_{IR}$, and $\Delta_{XUV}$, respectively.

## III. RESULTS

### A. Transient multiphoton absorption in gas

We observe in Fig. 2 the variation of the transmission for some chosen harmonics as a function of the IR-XUV delay
<figure><img src="image_3.png" /><figcaption>FIG. 1. View from the top of the IR-XUV experimental setup scheme. SCP corresponds to silica cube position and the number refers to particular experimental cases detailed in the manuscript. Bottom right: Spectrometer image of an XUV signal on a CCD camera, showing a series of harmonics measured after passing through 25-mbar Ne gas cell and with or without the presence of an IR beam.</figcaption></figure>

$\tau$. The temporal delay $\tau = 0$ corresponds to the overlap in the gas sample of the maximum intensity of both pulses (see Appendix A). The value of the transmission for each harmonic in the presence of the IR laser has been normalized with the transmission measured without the IR laser, named for brevity $T_{\text{norm}}$. The delay-dependent variations of the transmission in helium (He) and neon (Ne), Figs. 2(a) and 2(b), show a similar pattern. Several hundred femtoseconds before $\tau = 0$ the IR pulse is not present, thus $T_{\text{norm}}$ is equal to 1. Then, close to $\tau = 0$, when the IR and XUV pulses temporally overlap, the transmission drops rapidly, indicating that a given harmonic is efficiently absorbed by the gas. Finally, for large values of $\tau$ the transmission goes back to its initial value, since the two pulses do not overlap anymore. Consequently, we are facing a transient absorption phenomenon with a quite symmetrical profile that is centered on the position corresponding to the temporal overlap between the two pulses.

Moreover, consecutive harmonic orders provide similar transmission signals in specific spectral ranges, as seen, for instance, for $H_{13}$ to $H_{15}$ in He and for $H_{11}$ to $H_{13}$ in Ne [Figs. 2(a) and 2(b), respectively]. It is yet visible, for the latter case, that the transmission of higher orders [$H_{14}$ to $H_{16}$ in Fig. 3(a)] remains constant. These observations enable us to discard the role of resonances since the absorption takes place on a large spectral area.

To ensure that the observations presented above are not experimental artifacts, we compare the energy-dependent transmission of specific harmonic orders in Ne gas, with and without the presence of the IR pulses of the IR arm, with the reference transmission data from the CXRO website [55] [see Fig. 3(b)]. The latter (blue dashed line) is in perfect agreement with the experimental results (red triangle) for the energy range of interest. The abrupt drop of transmission for $H_{14}$ and higher harmonic orders is easily assigned to the ionization of

<figure><img src="image_4.png" /><figcaption>FIG. 2. Delay-dependent variation of transmission of specific harmonics through a gas medium. Temporal evolution of the strong and ultrafast absorption of high harmonics in (a) He and (b) Ne gases, observed when the latter are hit by IR pulses with maximum intensity at $\tau = 0$.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>FIG. 3. First threshold of first ionization of neon. (a) Delay-dependent variation of transmission for consecutive harmonics in 18 mbar of Ne.(b) Variation of transmission of harmonics in 50 mbar of Ne as a function of XUV energy, and with IR beam (for τ = 0) or not [Present (Exp.)]. The reference case corresponds to theoretical data obtained from the CXRO website [Reference (Th.)] [55]. (c) Schematic view of the energy levels involved (in eV) of atomic Ne together with possible single and multiphoton transitions (Appendix B). The short red arrows indicate the IR photons while the long colored arrows are the harmonics photons with different energies.</figcaption></figure>

Ne (ionization potential of 21.56 eV). We now compare the transmission in the presence of the IR pulse for a fixed delay τ = 0 (green rectangle). For harmonics with energies higher than the ionization potential of Ne, the results remain very similar to the case without the IR pulse, while at lower energies the transmission is strongly reduced. Equally, we observe that $H_{29}$ [see Fig. 4(b)] and $H_{31}$ are also subject to a strong transmission drop, while $H_{33}$ and higher order harmonics are not affected.

Based on these observations, we propose that the transient variation of the transmission results from two-color multiphoton ionization. More specifically, for harmonic orders below the ionization threshold of the gas used, the transmission is decreased due to the simultaneous absorption of one XUV photon and *n* IR photons.

Taking the specific cases of He in Fig. 2(a), $H_{15}$ to $H_{13}$ lie 1.45 to 4.55 eV below the first ionization threshold at 24.7 eV, respectively. The ionization can then take place by absorbing 1–3 IR-photons. As the ionization probability decreases with the number of IR pulses needed to be absorbed, the transmission increases for harmonics further away from the ionization threshold. This makes the phenomena dependent on the IR intensity ($I_{IR}$). Similarly for Ne, the first two thresholds of first ionization are 21.56 eV ($2s^2 2p^5$) and 48.5 eV ($2s^1 2p^6$) [56] [see Fig. 4(a)]. The number of IR photons required to lead to ionization for each harmonic is indicated by red arrows. For $H_{13}$ [Fig. 3(c)], lying just below the first two thresholds of first ionization, a unique IR photon is sufficient to ionize the atoms ($n = 1$), while for lower harmonic orders, larger values of *n* are necessary. This explains why $H_{12}$ transmission is decreased compared to the one of $H_{13}$ [see Fig. 2(b)]. However, the transmission of $H_{11}$ behaves differently, probably due

to a resonance with an excited state, positioned close to the spectral position of $H_{11}$, which enhances the expected probability. Similar behavior of the transmission of $H_{31}$ and $H_{29}$ is also observed due to the second threshold of first ionization, an innervalence ionization threshold of Ne [Fig. 4(a)], these harmonics needing to respectively absorb one and three IR photons.

To first experimentally confirm the role of the multiphoton IR absorption in our two-color scheme, the transmission of $H_{12}$ in Ne is measured in Fig. 5(a) for three different $I_{IR}$ ranging from $1.6 \pm 0.2 \times 10^{12}$ to $1.45 \pm 0.15 \times 10^{14}$ W cm$^{-2}$. In this case, simultaneous absorption of two IR photons leads to ionization, and the transmission should decrease with higher $I_{IR}$ as observed in Fig. 5(a). In addition, such intensity values (estimated carefully, see Appendix C) enable us to discard an effect solely due to a strong field ionization of the gas. Secondly, to ensure the single XUV absorption mechanism in our two-color scheme we measured the transmission of $H_{12}$ for three different XUV intensities ($I_{XUV}$), $2 \pm 0.2 \times 10^6$ to $1.6 \pm 0.15 \times 10^8$ W cm$^{-2}$ for $I_{IR} = 2 \pm 0.2 \times 10^{14}$ W cm$^{-2}$ [see Fig. 5(b)]. As expected for single-photon absorption, the normalized transmission is independent of $I_{XUV}$. Moreover, with a maximum value of $I_{XUV}$ for any harmonic order of about $10^8$ W cm$^{-2}$ (estimated carefully, see Appendix C), no multiphoton XUV ionization of the medium is likely to occur.

We have presented here a mechanism based on multiphoton absorption of the IR pulse that enables us to qualitatively explain all the experimental features that we observed. It is important to note that the mechanism is solely dependent on the ionization potential of the gas sample and not on specific resonances, even though the latter could have an impact on the transmission. Thus, the same mechanism should take place in
<figure><img src="image_6.png" /><figcaption>FIG. 4. Innervale ionization thresholds: Ne 2s¹2p⁶, Ar 3s¹3p⁶ and Kr 4s¹4p⁶. (a) Schematic view of the energy levels involved (in eV) of atomic Ne together with possible single and multiphoton transitions (Appendix B). (b) Delay-dependent variation of transmission for harmonics below the second ionization thresholds of Ne (yellow), Ar (pink), and Kr (purple).</figcaption></figure>

any atomic gas, and around any ionization threshold, as we confirmed for He [Fig. 2(a)], Ar and Kr [Fig. 4(b)], and also for a molecular gas such as N₂ (not shown here). It provides an important versatility of the method, further detailed, to characterize short laser pulses.

## B. Cross-correlation between IR and XUV pulses

### 1. Relation between pulses duration and the delay-dependent transmission

We established that the variation of the transmission results from transient multiphoton absorption involving one XUV photon and n IR photons. Considering that this mechanism is solely due to a temporal superposition of the IR and XUV pulses, thus the FWHM of the normalized transmission curve should correspond to the FWHM of the convolution of the two pulses at the single-atom level. Using an IR autocorrelator, we then measured $\Delta_{IR} = 40 \pm 5$ fs. Consequently, $\Delta_{XUV}$ could be estimated, as a first approximation, to $11.7 \pm 1.5$ fs using $\Delta_{IR}/\sqrt{N}$, with N the harmonic order [6]. This estimation holds in the perturbative regime (i.e., at low IR laser intensity)

and for a normally compressed Gaussian IR laser pulse and low harmonic orders. In our case, the intensity of the IR beam on the XUV arm reaches about $0.96 \pm 0.08 \times 10^{14}$ W cm⁻² (see Appendix C for details). The formula should be fairly accurate for H₉ and H₁₁ (the highest harmonic orders in the perturbative regime) and should also work for higher but near-harmonic orders, such as for H₁₅. Note that in Sec. III D, we have carefully estimated the value of $\Delta_{XUV}$, confirming the estimation used here. Thus, the cross-correlation should give $\sqrt{\Delta_{XUV}^2 + \Delta_{IR}^2} \approx 42$ fs. This value is weakly affected by the precision of the estimation of $\Delta_{XUV}$, as $\Delta_{IR}$ is significantly higher. However, from Fig. 6 we can extract $\Delta_{T} = 56 \pm 5$ fs, far from the expected result. This is due to the fact that we need to take into account the macroscopic propagation of the two pulses into the medium. To do that, we include in our ab initio simulations the Beer-Lambert law. At a single-atom level, the density of probability of absorption $S(\omega, \tau)$ [see Appendix F, Eq. (F1)] is obtained by solving the time-dependent Schrödinger equation,

$$S(\omega, \tau) = C_1 e^{-(\tau/\alpha)^2}, \qquad (1)$$

with $C_1$ the proportionality coefficient between $S(\omega, \tau)$ and the convolution product of the envelopes of two Gaussian pulses. The parameter $\alpha$ is defined such that $2\alpha\sqrt{\log(2)}$ is the FWHM of the convolution product. We checked numerically that we recovered $S(\omega, \tau)$ from the convolution product of the two pulses for a given value of $\omega$. Nonetheless, the experimental transmission is macroscopic and we used the mathematical link between $S(\omega, \tau)$ and the absorption coefficient in the Beer-Lambert law (see Ref. [57]) to reproduce the experimental transmission. We start from Beer-Lambert law with the normalized transmission,

$$T_{\text{norm}} = e^{-\rho\epsilon z}, \qquad (2)$$

with $\rho$ the density of the medium (in cm⁻³), $\epsilon$ the absorption coefficient (cm²), and $z$ the length of the medium (cm). The microscopic $S(\omega, \tau)$ is proportional to $\epsilon$, as established in [57]

$$\epsilon = C_2 \times S(\omega, \tau) = C_2 \times C_1 e^{-(\tau/\alpha)^2}, \qquad (3)$$

with $C_2$ a proportionality coefficient between $S(\omega, \tau)$ and $\epsilon$. Then, by defining

$$C = \rho z C_2 \times C_1, \qquad (4)$$

we obtain the relation

$$T_{\text{norm}} = e^{-C e^{-(\tau/\alpha)^2}}. \qquad (5)$$

And finally, the FWHM of $T_{norm}$ is given by

$$\Delta_T = 2\alpha \sqrt{-\log\left[\log\left(\frac{2}{1+e^{-C}}\right)\right] + \log(C)}. \qquad (6)$$

Since an experimental determination of all parameters entering the Beer-Lambert law is cumbersome, we used Eq. (5) at $\tau = 0$, which corresponds to the minimum of transmission and leads directly to $\exp(-C)$. From Fig. 6, we extract the value of $\approx 0.15$ and thus $C \approx 1.9$. We can then evaluate $\alpha$ using Eq. (6), and we find $\alpha \approx 25.2$ fs, and thus a FWHM of the convolution product, $2\alpha\sqrt{\log(2)}$, of 42 fs, in very good agreement with our previous estimation of the cross-correlation.
<figure><img src="image_7.png" /><figcaption>FIG. 5. Effect of the IR intensity [$I_{IR}$ expressed in units of $10^{14}$ W cm$^{-2}$, with ±10% of error, (a)] and of the XUV intensity ($I_{XUV}$ expressed in units of $10^8$ W cm$^{-2}$, with ±10% of error, (b)] on the delay-dependent variation of transmission. (a) $H_{12}$ through 80 mbar of Ne. (b) $H_{13}$ through 70 mbar of Ne and for $I_{IR} = 2 \pm 0.2 \times 10^{14}$ W cm$^{-2}$.</figcaption></figure>

We now provide the recipe for correctly reading $\Delta_{IR}$ from the experimental transmission signal. First, considering a cross-correlation with two pulses of quite different durations, the transmission is largely dominated by the value of the duration of the longest pulse. In the present conditions this is the case since, as mentioned before, $\Delta_{IR} = 40 \pm 5$ fs and $\Delta_{XUV} = 11.7 \pm 1.5$ fs. Therefore, our scheme is expected to provide directly $\Delta_{IR}$. We recall that the transmission depends

on $I_{IR}$ due to the multiphoton process, as shown in Fig. 5(a). Indeed, even for $I_{IR}$ two orders of magnitude lower than the maximum achievable here, i.e., about $10^{12}$ W cm$^{-2}$, the IR pulse brings additional absorption. Thus, the transmission is affected not only at the maximum of the IR laser pulse but also for small temporal overlaps of the beginning or the ending tail of the IR pulse with the XUV pulse. This also explains why for a similar set of IR and XUV pulses the transmission for He and Ne are different, see Fig. 2. Indeed, Ne is easier to ionize than He, and as a consequence smaller IR intensities start to enlarge the transmission decay in Ne gas.

To take into account this effect, the transmission of the different harmonic orders was measured for different $I_{IR}$ for each gas at $\tau = 0$ (see Appendix C). We observed that the transmission as a function of $I_{IR}$ does not depend significantly on the harmonic order, while it behaves quite differently for different gases. This is in line with the ionization potential of the two gases and is also different from one ionization threshold to another.

We can now try to evaluate $\Delta_{IR}$ with the He gas target. Basically, when dividing $I_{IR}$ by a factor of 2, $T_{norm}$ increases by +0.2 (see Appendix D). As a consequence, in Fig. 6(a), $\Delta_{IR}$ corresponds to the temporal width of the normalized transmission curve for $T_{norm} = 0.35$ [0.15 (minimum) + 0.2]. It gives $40 \pm 5$ fs (green color), quite close to the evaluation of the calculated cross-correlation value in this case, i.e., 42 fs. Using the recipe for He and Ne and for harmonic energies below the first ionization levels of the respective gases, one can systematically retrieve $\Delta_{IR}$ values between 40 and 50 fs.

## 2. Measurements of temporal envelope shape and duration of IR pulses

We start with a more general characterization of the IR pulse, for which the IR autocorrelator measurements provide a reference. In our setup, we can modify the chirp of the 800-nm pulses, which are delivered directly at the exit of the laser chain, to access different pulse durations. This affects both $\Delta_{IR}$ and $\Delta_{XUV}$, the latter remaining much smaller than the former. In Fig. 7(a), we compare the results obtained for $\Delta_{IR}$ from the autocorrelator measurements and from the transmission of $H_{13}$ in He, for different chirp values, given

<figure><img src="image_8.png" /><figcaption>FIG. 6. Delay-dependent variation of transmission of $H_{15}$ in He (1s¹ first ionization threshold) for $I_{IR} = 1.5 \times 10^{14}$ W cm$^{-2}$. (a) Direct experimental evaluation of $\Delta_{IR}$. With $\Delta_{IR}$ much higher than $\Delta_{XUV}$, the transmission curve provides with a well-estimated value of $\Delta_{IR}$ (in green color), obtained after a recipe presented in Sec. 1 and Appendix D. Taking directly the FWHM of the transmission profile (red color) leads to a quite wrong estimation. (b) Comparison of the measurement with simulations using $\Delta_{IR} = 40$ fs (see Appendix F).</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>FIG. 7. Evaluations of $\Delta_{IR}$ in different conditions. (a) Comparison between an autocorrelator and the present method for $H_{13}$ in 40 mbar of He. $\Delta_{IR}$ is experimentally modified by tuning the laser chirp. The value -5 in the chirp scale corresponds to 40-fs FWHM (2). (b) Delay-dependent variation of the transmission of $H_{11}$ in 40 mbar of Ne and with [cyan, (4)] and without [yellow, (2)] a 3-cm-long silica cube positioned on the IR arm. Below (a) and (b) present a sketch of the IR (red) and XUV (blue) pulse envelopes at $\tau = 0$ for the different cases (1–4) and with estimations of $\Delta_{IR}$ or $\Delta_{XUV}$.</figcaption></figure>

in arbitrary units. Here, a chirp value of zero corresponds to the best HHG efficiency (in particular, the sign must not be interpreted in terms of pulse compression or stretching). We see that both measurements are in very good agreement, within 10% error, on all conditions considered for the chirp, resulting in pulse durations ranging from 50 to 200 fs.

We increased further $\Delta_{IR}$ using a 3-cm-long fused silica cube on the IR arm (SCP1, see Fig. 1), i.e., keeping unchanged the harmonic generating IR pulses. This allows to stretch $\Delta_{IR}$ up to $450 \pm 50$ fs. An accurate estimation of the pulse profile was obtained here by averaging several measurements performed with the IR autocorrelator, which revealed that its temporal intensity profile is almost flat-top (not shown). The transmission profile for $H_{11}$, shown in Fig. 7(b) (blue curve), turns to be even more flat-top than the measured IR profile. With our *in situ* recipe we can estimate $\Delta_{IR} = 457 \pm 20$ fs, deduced from the value of $T_{norm}$ corresponding to the minimum (0.47) + the Ne correction (0.11). This value is very close to the one obtained with the autocorrelator. For comparison, when the silica cube is removed, the minimum value of $T_{norm}$ is 0.41, which provides a consistent $\Delta_{IR}$ of $44 \pm 5$ fs [see the yellow curve in Fig. 7(b)].

The silica cube was also placed before the beam splitter (SCP2, see Fig. 1), modifying the shape and the duration of the XUV pulse while keeping it shorter than the IR pulse. We measured the same delay-dependent variation of transmission as a validating proof of the understanding of our technique.

We have shown that, using the TIBETAN-FOX technique, we are able to perfectly retrieve not only $\Delta_{IR}$ but also the temporal envelope of the IR pulse. When the two pulses are unknown, the technique can be used as we presented while one of the pulse has a duration much smaller than the second, since the cross-correlation is dominated by the longest pulse.

Note that the situation is different if one of the pulses is well known by other means, then another pulse of similar duration can be characterized.

### 3. Measurements of temporal envelope shape and duration of XUV pulses

Accurate techniques exist to characterize IR laser pulses, but for the XUV range, they remain elusive, as described above in the introduction. Here, we will establish the possibility to characterize directly the shape of any XUV pulse as long as the duration of the latter is significantly longer than the duration of any IR pulse used to realize the multiphoton absorption process.

To do that, we artificially increased $\Delta_{XUV}$ by introducing the 3-cm-long fused silica cube on the IR part of the XUV arm (SCP3, see Fig. 1). In that configuration, $\Delta_{IR} \approx 40$ fs is shorter than the estimated $\Delta_{XUV}$ for the harmonic orders considered. The delay-dependent variation of transmission is shown in Fig. 8(a) for $H_{14}$ in He. We can first observe that the shape of the transmission is not a flat-top, meaning that the original temporal profile is not conserved through the generation of harmonics. Using our *in situ* recipe in Fig. 8(a), we estimate $\Delta_{XUV} = 95$ fs $\pm 5$ fs, deduced from the minimum value of $T_{norm}$ (0.47) + the He correction (0.2). Our simulations, performed through convolution products and solving the time-dependent Schrödinger equation [Fig. 8(b)], fit well our experimental data for $\Delta_{XUV} = 95$ fs. We can conclude that even with the presence of a complex temporal shape in the IR profile, we can retrieve the temporal shape of the XUV profile.

It is also important to note that, if the IR pulse is known, it is no longer important to have a significantly longer pulse duration for the XUV radiation; one can retrieve a good
<figure><img src="image_10.png" /><figcaption>FIG. 8. Evaluation of $\Delta_{XUV}$. Delay-dependent variation of transmission of $H_{14}$ in 20 mbar of He with a 3-cm-long cube positioned on the IR part of the XUV arm and for $I_{IR} = 1.5 \times 10^{14}$ W cm$^{-2}$. (a) Measurement. (b) Comparison of the measurement with simulations using $\Delta_{XUV} = 95$ fs (see Appendix F). The insert presents a sketch of the IR (red) and XUV (blue) pulse envelopes at $\tau = 0$.</figcaption></figure>

estimation of $\Delta_{XUV}$ simply by using the cross-convolution formula given by Eq. (6).

Finally, this proof of principle, obtained thanks to the versatility of our setup, shows that T-FOX allows to measure the duration of any XUV sources. That is an important step forward to characterize such pulses since only a few methods exist and usually require involved experimental apparatus such as a magnetic bottle. In the following, we also show that this method is sensitive enough to address laser pulses with low intensity.

### C. Temporal contrast measurements

From Fig. 5(a), we know that our method allows to measure pulses with a relatively weak intensity, since we are able to observe clear transient absorption profiles for at least $I_{IR} = 1.6 \times 10^{12}$ W cm$^{-2}$. Consequently, T-FOX could be used to measure multiple pulses or substructures inside the main pulse, as one would obtain from a third-order cross-correlator [58]. While usual autocorrelation methods such as FROG and SPIDER use two copies of the same pulse, if any prepulse exists, the latter will be duplicated symmetrically. Our technique is not symmetric in essence and is able to distinguish any pre- or postpulses.

To demonstrate that our method is sensitive to several consecutive laser pulses, and especially at relatively high intensity for which possible plasma effects could happen in the absorbing gas, the IR beam on the IR arm is split into two parts. The intensity of each IR beam is estimated around $10^{14}$ W cm$^{-2}$,

with $\Delta_{IR} \approx 40$ fs and $\Delta_{XUV} \approx 11.7$ fs. By playing on the timing between these two IR beams (an additional optical delay line has been installed), from several picoseconds to hundreds of femtoseconds, we observed that the visibility of the second beam transmission is never affected by the first beam [see Fig. 9(a)]. When the delay becomes smaller than the temporal distance between the backward part of the first pulse and the forward part of the second pulse, the two responses add to each other. It shows that there is virtually no limitation on measurements caused by the change of the gas medium due to an anterior pulse.

Finally, we have repeated measurements of the transmission of $H_{11}$ in Ne, by playing on the delay between the XUV beam and only one single IR beam on the IR arm, but on a larger time scale than before. We have also saturated as much as possible the absorption close to $\tau = 0$ [see Fig. 9(b)]. It consists of improving the spatial overlap between the IR and XUV beams, focusing at best the IR pulse to get the highest laser intensity, and also increasing the gas pressure at maximum. Surprisingly, we have observed two transmission decays that are positioned symmetrically about 700 fs before and after the main transmission drop. These signals correspond to weak IR pulses that probably come from the laser itself, since no optics on the optical table can introduce such a short delay. Thanks to our recipe, we estimate that the pulse energy is about $10^4$ lower than the main pulse, that is, $50 \pm 10$ nJ. These pulses could originate from the first part of our laser in the regenerative amplifier, in which several passes are performed and particularly through very thin optics that can introduce such a short delay.

### D. Full characterization of XUV pulses

All our measurements were performed directly on a spectrometer, and thus precisely resolved spectrograms can be obtained as for the SHG FROG technique. In the latter, the spectrograms represent laser intensity pictures as a function of spectral energies and are recorded for different delays between the two copies of the same pulse. Here, we use the variation of energy-resolved XUV transmission and the IR-XUV delay as inputs for an iterative phase-retrieval algorithm [16,22]. In Figs. 10(a)–10(e), such spectrograms are depicted for the transmission of $H_{15}$ in He for five different laser chirp values.

To retrieve the XUV pulse intensities and phases (Fig. 11) as a function of time [Figs. 11(a) and 11(c)] and spectral energy [Figs. 11(b) and 11(d)], we have not applied a FROG iterative phase-retrieval algorithm, but a specifically developed one for the frequency resolved optical switching (FROSt) [22] technique, since the latter had already been successfully used for transmission signals. Originally developed for IR to mid-IR femtosecond pulses, it has just been adapted for our special case of transmission in gases and in the XUV domain (see Appendix G). Note that only negative delays were used as input of the FROSt algorithm to obtain the results of Fig. 11. This choice is motivated to avoid transmissions that could be perturbed after the path of the IR beam at its maximal intensity close to $10^{14}$ W cm$^{-2}$.

We observe in Fig. 11(a) that $\Delta_{XUV}$, deduced from the normalized intensity profiles, remains rather constant for IR chirp values between 0 and +3, while for slight negative chirp values the duration increases. In this range of the chirp parameter, which corresponds to normal compression conditions,
<figure><img src="image_11.png" /><figcaption>FIG. 9. Delay-dependent variation of transmission with temporally shifted IR pulses. (a) Two delayed IR pulses response ($H_{12}$, 70 mbar of Ne). (b) Observation of pre- and postpulses coming from the laser system ($H_{11}$, 60 mbar of Ne) in the delay-dependent variation of the transmission profile.</figcaption></figure>

$\Delta_{XUV}$ is about $12.1 \pm 3$ fs, in good agreement with our starting estimate of $11.7 \pm 1.5$ fs as stated above. Figure 11 also presents the corresponding evolutions of the XUV phases in the time (a) and spectral energy (b) domain. For chirp values 0 and +1 the retrieved phases are rather flat, while other chirp values give a clear structure: for +3 we obtain a moderate XUV chirp, and for -1 and -3 a larger chirp is obtained with an opposite sign. To validate these results, the final spectrograms obtained from the algorithm, Figs. 10(f)–10(j), are compared to the measured ones, Figs. 10(a)–10(e), and present a very good agreement.

In conclusion, using T-FOX and a modified iterative phase-retrieval algorithm, we were able to follow the intensity and phase distributions of HHG femtosecond pulses in the XUV domain, in both the temporal and spectral domains. It

showed that for particular values of the IR laser chirp, one can minimize the XUV pulse duration and control the chirp of the resulting XUV pulses. However, the spectral resolution of our spectrometer is here limited, introducing for any harmonic a dilution of the real spectral response with the horizontal dimension onto the camera. Taking into account the quality of the results, this effect seems negligible. Anyway, one can solve such an issue relatively easily by using a more dispersive grating.

## IV. CONCLUSIONS

In summary, we have identified a multiphoton absorption process due to a temporal superposition of an IR pulse and an XUV pulse, for which energies should be below the ionization

<figure><img src="image_12.png" /><figcaption>FIG. 10. (a)–(j) Spectrograms of $H_{15}$ passing through He gas for five different laser chirps from -3 to +3 (arbitrary units). Variation of transmission ($T_{norm}$ in color scale) as a function of XUV spectral energy and IR-XUV delay. The intensity of the IR laser that generates the harmonics on the XUV arm is about $0.96 \pm 0.08 \times 10^{14} \text{ W.cm}^{-2}$. (a)–(e) obtained from experimental measurements. (f)–(j) retrieved from the algorithm. The arbitrary unit values used for the laser chirps are the one introduced in Fig. 7(a) and text above. (k)–(o) Temporal reconstructions of the intensity [$I(\text{norm.})$] and phase ($\phi$) profiles. For each chirp value, the thicker profile is the averaged of five independent reconstructions (presented with thinner profiles). Each reconstruction is based on an initial guess of the iterative algorithm, which is a random function and consequently different from one reconstruction to another [22].</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>FIG. 11. (a) Temporal and (b) spectral reconstructions of the intensity and phase profiles for the five different laser chirps corresponding to Figs. 10(a)–10(e) from -3 to +3 (arbitrary units). (a) corresponds to Figs. 10(k)–10(o).</figcaption></figure>

threshold of an arbitrary gas target. If the IR pulse duration is shorter than the XUV one, it becomes possible to obtain a cross-correlation trace, which provides the temporal intensity profile and pulse duration of any XUV source. The use of iterative phase-retrieval algorithms on experimental spectrograms showed the possibility of obtaining phase information of the XUV pulses. These results already give the TIBETAN-FOX technique a high degree of interest.

However, our technique is limited by the fact that to reconstruct an XUV pulse, one has to measure the transmission for several delays. As a consequence, the reconstructed pulse is a large average between several acquisitions. Here, in the case of Fig. 10(c), for example, the transmission has been evaluated for 80 temporal values. This problem could be fixed by inducing a relation between space and time for any of the beams. This could be done optically on the IR part, in order to introduce at the focus point of interaction a dependence of the spatial vertical position with the time delay. In the XUV spectrum, the temporal information would then be encoded vertically. Measuring XUV single shot pulses is still a quite long way to go, since each acquisition on the camera is an average of several pulses, here 4000 (in order to get a clear signal on the camera). This is due to the poor harmonic generation efficiency. Such single-shot measurements are possible with much more powerful lasers or with other XUV sources.

In addition, in the near future some advances could also be brought on our technique. On one hand, the direct measurements of the XUV pulse durations rely on the fact that the IR pulse duration needs to be significantly smaller, in order to directly get both the envelope shape and pulse duration of the XUV pulses. To demonstrate the interest of our technique, we had artificially obtained an HHG-based XUV pulse duration higher than the IR duration, by introducing a large cube on the XUV arm. But after all, one can already benefit from XUV sources with pulses of several hundred femtoseconds duration, such as in plasma-based XUV lasers [31,32]. On the contrary, another possibility would be to shorten the IR pulse duration down to several femtoseconds with postcompression systems, opening measurements to most of the existing XUV sources.

On the other hand, with the T-FOX technique applied to XUV-resolved spectrograms and using algorithms, whatever the durations of the IR and/or XUV pulses are, one can always retrieve any XUV temporal characteristic.

Nonetheless, T-FOX presents many advantages. First, there is no constraint on the polarization properties of both IR and XUV pulses (not a topic discussed here), contrary to many techniques due to the nonlinear effects on which they are based. Second, there is also no limitation on the repetition rate of the laser or the XUV source, since consecutive pulses can be measured with a few tens of femtoseconds separation. Third, even with XUV pulses with quite large spectral bandwidth, the process is working. Actually, we are able to reconstruct pulses simultaneously from consecutive harmonics, as long as the latter have an energy below the targeted ionization threshold of the gas that is used.

Furthermore, T-FOX allows, thanks to its very high sensitivity and its nonsymmetric temporal behavior, high temporal resolution, high contrast, and to distinguish pre- from post-pulses. Such additional information can be obtained on the IR temporal profiles, as seen in this paper, but this is also expected for XUV pulses, as long as the IR pulse duration is smaller than the XUV one. This could be of huge interest, notably to measure fine and complex physical structures that can be found in some XUV laser sources, such as Rabi oscillations.

## ACKNOWLEDGMENTS

We acknowledge Sergey Stremoukhov, from the Lomonosov Moscow State University, for fruitful discussions. This research received the financial support of the French National Research Agency through Grants No. ANR-20-CE30-0007-DECAP, No. ANR-21-CE30-0036-ATTOCOM, and No. ANR-21-CE09-0042-HYPNOSE.

## APPENDIX A: SETUP

On the optical table, the laser beam (25 mm aperture, s polarized) is separated into two parts via a beam splitter (Fig. 1). On the IR arm, an optical delay line is installed to control the timing between the IR and the XUV pulses. A half wave plate is inserted to control the IR pulse polarization. On the XUV arm, the laser is just passing through three elements, another half wave plate positioned to polarize linearly the laser beam in *p* state, an iris that allows to optimize the harmonic efficiency and stabilize it, and a 1.5-m-long focal lens that focuses the laser in the generating gas cell. In the first vacuum chamber, in order to generate both odd and even harmonics a 200-microns-long BBO (barium beta borate) crystal can be placed 1.15 m after the focal lens on the fundamental (*ω*, 800 nm) beam path, to obtain a strong 2*ω* signal (400 nm). Then, both orthogonally polarized *ω* and 2*ω* beams are spatially recombined and focused inside a 1-cm-long gas cell. The latter can be filled continuously with various rare gases, allowing generating a harmonic comb signal optimized in the different spectral ranges requested for this experiment. Most of the time xenon or krypton is used. In the second vacuum chamber, four XUV optical elements are placed at 4° grazing incidence. The main optic is a toroidal gold-covered
mirror that properly focuses the harmonics 1.5 m away in a second gas cell. In addition, three flat mirrors in SiO₂, plus an aluminum filter (300 nm thickness), canceled the residual part of the laser beams ω and 2ω. At the opposite these optics highly reflect the various harmonic orders in XUV. In the third chamber, the IR beam, coming from the IR arm, is deviated horizontally by less than 0.5° out of the axis of the harmonic path and is refocused, using a 50-cm-long focal lens inside the second gas cell. The latter is about 5 mm long and is constantly filled with gas at pressures up to 100 mbar. Spatial and temporal overlaps between IR and XUV beams are performed in two steps. Firstly, we optimize the contrast of fringes obtained from IR pulses coming from the two arms interfering in a plan that is placed in the middle of the second gas cell (both beams being perfectly spatially superimposed). One can have then a sub-10-fs precision. In the XUV arm, even if XUV and IR pulses are theoretically perfectly temporally overlapped at the generation point, for particular generation conditions, pulses can have very slightly different trajectories (even from one harmonic to another). Consequently, secondly, a very slight spatial realignment of the XUV beam is performed (with picomotors placed on mirrors) in order to pass through the gas cell drilled by the IR pulses of the IR arm, constituting the finest spatial overlap. In the fourth vacuum chamber, an XUV spectrometer is set up with a second aluminum filter to suppress any IR laser residual energy. The spectrometer is composed of a grazing incidence spherical mirror and a gold mesh (1000 lines per millimeter) grating that focus and separate the different harmonic orders onto a CCD camera (Princeton PIXIS). Images acquired on the chip of the camera are calibrated precisely in order to perfectly know the order and wavelength of the involved harmonics.

## APPENDIX B: SPECTROMETER CALIBRATION

The calibration of the spectrometer is performed in several steps. It first relies on the basic formula of diffraction through a transmission grating, for which the number of lines per millimeter is known. It easily allows to determine the different orders of the harmonics. Then, we have used three different sharp spectral steps located inside the spectral range of our targeted harmonics. For high energies, the cut of the aluminum corresponds to 72.6 eV, while for lower energies the first ionization threshold of Ne and He are, respectively, 21.7 and 24.7 eV. Finally, it leads to a laser central wavelength of 795 nm, giving, for example, 17.19 eV (72 nm) for $H_{11}$ and 45.32 eV (27 nm) for $H_{29}$.

## APPENDIX C: ESTIMATION OF IR AND XUV INTENSITIES

To carefully interpret all the results of the manuscript, we have estimated the IR intensities at the focusing positions in the two gas cells and the XUV intensity in the second gas cell.

First, the principle of the calibration of the maximal IR intensity achievable on the IR arm relies on the measurement of the harmonics that can be generated in the second gas cell using different gases. The formula of the cutoff energy ($E_C$) in the harmonics comb spectrum that refers to the laser intensity

is

$$I_{IR}(10^{14} \text{ W cm}^{-2}) = \frac{E_C(\text{eV}) - I_p(\text{eV})}{29.6\lambda(\mu\text{m})^2}, \quad (\text{C1})$$

where $\lambda$ is the laser wavelength and $I_p$ is ionization potential of the considered gas. Measurements have been made using nitrogen ($E_C = 52$ eV, $I_p = 15.56$ eV) and neon ($E_C = 60$ eV, $I_p = 21.56$ eV) gases, and for which an iris aperture of 16 mm has been placed on the IR path, leading to an energy per pulse of about 0.7 mJ. They give, respectively, 1.86 and 1.96 ± 0.15 × 10¹⁴ W cm⁻². Consequently, an experimental value of 1.9 ± 0.2 × 10¹⁴ W cm⁻² is considered. We can scale from that the highest intensity achievable on our setup when the iris is fully open to about 2.5 × 10¹⁴ W cm⁻².

Second, on the XUV arm, the IR energy is estimated in the same way. For most of the experimental cases, harmonics have been generated through krypton. Through the analysis of the spectra cutoff, we can estimate the intensity to be about 0.96 ± 0.08 × 10¹⁴ W cm⁻².

Concerning the XUV intensity, one has first to take into account several calibrations of filters, optics and camera, that were done in the past on our laser beamline. Second, one has to estimate the waist of the harmonics at the interaction point. Because the focal length of the lens used to focus the IR pulses is much shorter than the equivalent one for the toroidal mirror, the waists of IR and XUV beams are quite different. The IR pulses drill the second gas cell, while the harmonics are highly filtered by the hole created at the entrance. Consequently, our estimation shows, under no circumstances, that the intensity of any harmonic order exceeds 10⁸ W cm⁻².

## APPENDIX D: RECIPE FOR FWHM PULSE DURATION EVALUATION

When the IR pulse duration is higher than the HHG pulse duration, one can estimate the IR pulse duration at FWHM using T-FOX. For that, one should measure the transmission response. The latter depends mainly on the gas type, and more particularly on the ionization state. We then measured the transmission response for various IR intensities and analyzed the curves in Fig. 12. The variations are quite linear, with a rather different slope between He, Fig. 12(a), and Ne, Fig. 12(b), and very different from one ionization threshold to another (not shown here). Nevertheless, from one harmonic to another, while probing the same gas and the same ionization state, the fit parameter is similar. With the fitted curve, we deduced the transmission shift that corresponds to a factor of two lower compared to the maximal value of $I_{IR}$ in our experimental range of $I_{IR}$ values. One has to apply for the 1s¹ state in He + 0.2 ± 0.05, and for the 2s² 2p⁵ in Ne + 0.11 ± 0.02. Then, one has to add to the value corresponding to the minimum of transmission the specific shift of transmission. For this new value of absorption, we can measure the FWHM pulse duration of the cross-correlation, and consequently of the laser.
<figure><img src="image_14.png" /><figcaption>FIG. 12. Minimum value of the variation of transmission (corresponding to $\tau = 0$) (a) of $H_{14}$ and $H_{15}$ in helium and (b) of $H_{10}$ and $H_{12}$ in neon, as a function of the IR intensity. The plain lines indicate the linear fits that allow to get the correction value to apply for the recipe.</figcaption></figure>

## APPENDIX E: FWHM PULSE DURATION EVALUATION PRECISION

The temporal resolution depends mainly on the shortest pulse duration. Nevertheless, with the choice of an even shorter temporal sampling between each delay point and when a large number of data points are acquired, one can improve the temporal resolution. For instance, in the general case of an IR pulse duration of ~40 fs and an XUV pulse duration of ~11.7 fs, with a sampling of ~2 fs, we can even see in the absorption profile clear small substructures shorter than the XUV pulse. The resolution probably reaches ~5 fs in these conditions. In addition, in the case for which the cube was inserted in the XUV part, the resolution was expected to be theoretically limited by the IR pulse duration, i.e., to about 40 fs, since the XUV pulse duration was estimated to ~95 fs. With a sampling of ~10 fs, one can estimate about ~30 fs resolution by analyzing repeated measurements done with the same conditions, and that show equivalent typical transmission shapes.

## APPENDIX F: SIMULATIONS

The simulations were done by solving the time-dependent Schrödinger equation in two dimensions of space, using the split-operator method [59] with the FFTW library [60]. We ensured convergence of our computations by varying the integration time step, the number of grid points in every dimension, and the size of the simulation box. The results presented here are obtained using a time step of $2.69 \times 10^{-2}$ atomic units (a.u.), a regular grid spacing of 0.25 a.u. in both dimensions extending from -128 to 128 a.u.. The helium atom is modeled using a short range potential $V(x, y) = -2.71 \times e^{-x^2}$, in a.u., providing an ionization potential of 0.907 a.u.. The absorption, and thus the transmission, is computed in the

dipole approximation and in the velocity gauge, using a potential vector $\vec{A}$ with a sine squared envelope. The frequency of the IR laser is fixed to 1.55 eV with $I_{IR} = 1.5 \times 10^{14}$ W cm$^{-2}$ and the XUV pulse solely includes the 15th harmonic with $I_{XUV} = 3.5 \times 10^{10}$ W cm$^{-2}$. Perpendicular polarizations for the two pulses are used to avoid harmonic generation due to the strongly intense IR pulse. Experimentally, this is avoided by a small angle between the propagation axes of the two pulses. All simulations used $\Delta_{IR} = 40$ fs, and $\Delta_{XUV} = 11.7$ fs in Fig. 6 and $\Delta_{XUV} = 95$ fs in Fig. 8. The considered observable is the density of probability of absorption $S(\omega, \tau)$ for a given delay $\tau$, defined as

$$S(\omega, \tau) = 2\mathrm{Im}[\tilde{d}(\omega, \tau)\tilde{\epsilon}(\omega, \tau)]. \quad (\mathrm{F1})$$

Here, $\tilde{\epsilon}$ is the Fourier transform of the electric field and $\tilde{d}$ is the Fourier transform of the time-dependent dipole, $\langle\Psi(t)|x|\Psi(t)\rangle$, with $|\Psi(t)\rangle$ the time-dependent wave function within the single active electron approximation.

## APPENDIX G: RECONSTRUCTION PROCEDURE

The spectrogram reconstruction was performed with a phase retrieval algorithm of the so-called temporal ptychography, initially developed to reconstruct spectrograms obtained by the FROSt technique [22], which has very similar features. Algorithms used for temporal ptychography or to reconstruct spectrograms obtained by the FROG technique are both based on the same hypothesis consisting of assuming the spectrogram to be described by the following equation:

$$I(\omega, \tau) = \left| \int_{t} P(t)G(t - \tau_{G})e^{i\omega t} dt \right|^{2}, \quad (\mathrm{G1})$$

where $I$ is the measured spectral intensity of the spectrogram, i.e., the norm squared of the Fourier transform of the temporal amplitude, in intensity and phase, of the pulse $P$ to be characterized multiplied by the temporal amplitude of a gate
$G$ for different delays $\tau_G$ between the pulse and the gate. The gate function can be either another pulse in the case of FROG measurements or a temporal switch in the case of FROSt measurements. In both cases, the main assumption is that both temporal functions are independent. For our study, we assume that the spectrograms, as presented in Figs. 10(a)–10(e), for

instance, can be described by this equation, with $P$ being the temporal amplitude, in intensity and phase, of $H_{15}$ and $G$ being the variation of transmission. Considering this assumption, we can reconstruct the spectrograms with the ptychographic phase retrieval algorithm used in Ref. [22] and deduce the temporal intensity and phase profiles of $H_{15}$.

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