# Impact of oblique reflections on the generation of isolated attosecond pulses by polarization gating

C. Picot, D. Diaz Rivas, N. Ouahioune, D. Hoff, Lucas Perez, A.-K Raab, Stefan Skupin, C. Guo, P. Kumar Maroju, A. L'huillier, et al.

► To cite this version:

C. Picot, D. Diaz Rivas, N. Ouahioune, D. Hoff, Lucas Perez, et al. Impact of oblique reflections on the generation of isolated attosecond pulses by polarization gating. Optics Letters, 2025, 50 (21), pp.6738. 10.1364/OL.572691. hal-05336305

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# Impact of oblique reflections on the generation of isolated attosecond pulses by polarization gating

C. PICOT,<fnref n="1" /> D. DIAZ RIVAS,<fnref n="2" /> ID N. OUAHIOUNE,<fnref n="2" /> D. HOFF,<fnref n="2" /> ID L. PEREZ,<fnref n="1" /> S. SKUPIN,<fnref n="1" /> ID

A.-K RAAB,<fnref n="2" /> ID C. GUO,<fnref n="2" /> ID P. KUMAR MAROJU,<fnref n="2" /> A. L'HUILLIER,<fnref n="2" /> ID A.-L VIOTTI,<fnref n="2" /> ID

C. L. ARNOLD,<fnref n="2" /> ID M. GISSELBRECHT,<fnref n="2" /> AND E. CONSTANT<fnref n="1" />,*, ID

Received 4 July 2025; revised 25 August 2025; accepted 26 August 2025; posted 27 August 2025; published 28 October 2025

Isolated attosecond pulses are often generated via sub-optical-cycle polarization gating. This technique, based on the precise control of the time-dependent polarization of the infrared driver pulse, enables strong temporal confinement of the extreme ultraviolet (XUV) emission from high-order harmonic generation (HHG) in gases. In this work, we study the influence of pulse propagation between the position where the polarization gate is created and the nonlinear HHG medium on the generation of isolated attosecond pulses. Oblique reflections on mirrors, in particular, lead to a significant polarization-dependent carrier phase shift and group delay. In our experiments, seven reflections of the few-cycle infrared driver pulse on metallic mirrors at 45° lead to a delay of 2.8 fs, which is large enough to destroy the polarization gate. When these effects are precompensated, efficient XUV confinement via sub-cycle polarization gating is evidenced by continuous XUV spectra which strongly depend on the carrier-envelope phase (CEP). Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

https://doi.org/10.1364/OL.572691

high-order harmonic generation (HHG) [1,2] has emerged as a tabletop solution for generating short extreme ultraviolet (XUV) pulses [3]. However, isolating a single attosecond (1 as = $10^{-18}$ s) pulse [4–6] from the generated attosecond pulse train [7] remains a challenge. Many techniques have been developed for this task, such as time-dependent phase matching [8], short driving pulses [9], intensity gating [10], or polarization gating [5,11–15]. The latter relies on the temporal modulation of the polarization of the driver pulse from circular to linear to circular within a few femtoseconds (1 fs = $10^{-15}$ s). It confines the XUV emission within a temporal gate, where the driver's polarization is nearly linear [11–13]. Polarization manipulation is often performed in air, close to the laser system. The pulses are then transported into a vacuum chamber, where harmonics are generated in a gas medium. The propagation usually involves numerous reflections on optical elements (mirrors, beamsplit-

ters) at non-zero incidence angles, especially when pump–probe schemes are implemented.

In this article, we study the influence of this pulse propagation on the generation of isolated attosecond pulses. It is known from Fresnel coefficient that oblique reflections introduce a frequency-dependent dephasing between the *S* (vertical) and *P* (horizontal) polarization states [16]. As this dephasing changes with frequency, it also affects the position of the pulse envelope. This leads to polarization-dependent changes in dephasing, carrier-envelope phase (CEP), and group delay that can be very significant.

The achievement of sub-cycle polarization gating after several oblique reflections is thus challenging, as propagation-induced delays of a few femtoseconds are comparable to the delay introduced to create the polarization gate [13] with few-cycle pulses, and propagation can re-shape the polarization modulation [17]. Here, we investigate the polarization dependence of oblique reflections and propose a robust precompensation scheme for carrier phase shift and group delay. We perform polarization-gated HHG using 6.4 fs infrared (IR) pulses that undergo seven reflections on metallic mirrors at 45°. After proper precompensation, we generate CEP-sensitive continuous XUV spectra compatible with the generation of isolated attosecond pulses.

The experimental setup, shown in Fig. 1, consists of an optical parametric chirped pulse amplification (OPCPA) system operating at a repetition rate of 200 kHz, delivering linearly, horizontally polarized pulses centered at 850 nm and 15 µJ energy per pulse. CEP stabilization is ensured by active feedback to the oscillator using an f-2f interferometer and the stereo above-threshold-ionization technique (stereo-ATI) [18–20]. The CEP is tuned using two motorized SiO₂ wedges acting as a variable-thickness plate. The pulse duration, characterized by the dispersion-scan technique [21,22], is 6.4 fs full-width at half-maximum (FWHM) [23,24]. The polarization state of this pulse is controlled in two steps (see inset in Fig. 1). First, a narrow gate is created by two quartz waveplates [13]. The first plate (WP1) is a multi-order $\lambda/4$ waveplate (total retardance $7\lambda/4$) with optic axis at 45° relative to the horizontal. It splits the incoming linearly polarized pulse into two cross-polarized pulses delayed by 4.9 fs. The second plate, WP2, is a true zero-order $\lambda/4$ waveplate with optic axis oriented horizontally to

<footnote n="1">¹Université de Lyon 1 - CNRS, Institut Lumière Matière, UMR 5306, 69622 Villeurbanne, France</footnote>

<footnote n="2">²Department of Physics, Lund University, Box 118, SE-22100 Lund, Sweden</footnote>

<footnote>*eric constant@univ-lyon1.fr</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1. Schematic experimental setup. After the creation of the polarization gate with two quartz plates (WP1 and WP2 in inset), in air, the pulse passes through a polarization control system (wedges + WP3 and WP4 in inset), enters the vacuum chamber (dashed dotted), passes through a 1-mm-thick SiO₂ beamsplitter (BS, uncoated front side, AR-coated rear side) and is reflected by several metallic mirrors. It is then focused in an argon gas jet where harmonics are generated and analyzed using a flat field XUV spectrometer. Grey bars represent protected silver mirrors and the black bar the aluminum focusing mirror. ①, ②, and ③ represent the locations where a pick-up mirror is inserted to characterize the dephasing and delay between S and P polarizations. The inset shows the arrangement of quartz waveplates and wedges (blue) and fused silica wedges (grey) that are used to control the polarization state and precompensate for propagation effects.</figcaption></figure>

maximize the XUV confinement [13,25,26]. Second, additional birefringent plates are used to manipulate the relative phase and delay between the S and P polarization components. They consist of two movable identical quartz wedges (wedge angle of 1.1°) with horizontal optic axis, mimicking a birefringent parallel plate with a thickness adjustable between 1.3 mm and 2.6 mm, and two compensation quartz plates (WP3, WP4) with a total thickness of 1.872 mm and vertical optic axis. Compensating plates are necessary to achieve for small delays since the minimum wedge thickness is too large. The transverse wedge position, x, controls the effective thickness of the birefringent material and thereby the delay and dephasing between the S and P components. An additional pair of non-birefringent fused silica wedges moves opposite to the quartz wedges to ensure that the group delay dispersion is constant within ±5 fs² [27]. The pulses are then transported to the HHG medium, passing through the entrance window of the HHG chamber and a 45° incidence beam splitter (BS) and are reflected by six flat silver mirrors at 45° incidence and focused by a 45° incidence aluminum parabolic mirror. Harmonics are generated in an argon gas jet with a 40 µm nozzle diameter. Spatially resolved XUV spectra are observed after diffraction by a grating with a micro-channel plate (MCP)/Phosphor screen detector. This XUV spectrometer has no entrance slit, and the XUV spectra are obtained after spatial integration.

To characterize the dephasing and delay introduced by the mirrors, WP1 is set with the optic axis along the incoming linear polarization, and the λ/4 plate, WP2, is replaced by a zero-order λ/2 plate tuned to obtain a linearly polarized pulse at 45° with respect to the optic axis of the wedges. The pulses are characterized at different locations (indicated by numbers ①, ②, and ③ in Fig. 1) using a pick-up silver mirror followed by a linear polarizer oriented at 45° to the horizontal and a fiber-coupled spectrometer. The dephasing Δφ<sup>mir</sup>(ω) = φ<sub>S</sub>(ω) - φ<sub>P</sub>(ω) between S and P polarization is measured using the birefringent interferometer integrated in our setup. It consists of the quartz and fused silica wedges, the compensation plates WP3 and WP4, and a pick-up mirror followed by a polarizer. The quartz wedges

<figure><img src="image_3.png" /><figcaption>Fig. 2. (a) IR spectral interference map as a function of the position of the quartz wedges, x. The maximum of the spectrally integrated signal (yellow solid line) corresponds to the zero-delay position x<sub>d0</sub> = 19.8 mm (see text for details). (b) Normalized spectrally integrated signals for the three locations of the pick-up mirror indicated in Fig. 1: before the vacuum chamber (red solid line, x<sub>d0</sub> = 19.8 mm), after four additional protected silver mirrors (blue solid line, x<sub>d0</sub> = 18.6 mm), and after two additional mirrors (one Ag, one Al) near the position of the gas jet (black solid line, x<sub>d0</sub> = 17.3 mm). Curves in (b) are normalized and shifted vertically for clarity.</figcaption></figure>

and plates separate the S- and P-polarization components, while the polarizer allows them to interfere. This interferometer introduces a dephasing given by Δφ<sup>int</sup>(ω, x) = ω/cL(x)[n<sub>e</sub>(ω) - n<sub>o</sub>(ω)], where L(x) is the effective quartz thickness and n<sub>e</sub>(ω)(n<sub>o</sub>(ω)) denotes the extraordinary (ordinary) refractive index in quartz. The subsequent delay varies linearly as a function of the transverse position of the quartz wedges with a dynamic accuracy of σ = 30 as throughout a delay range of ±20 fs. The interferometer has a measured static delay stability of σ = 3 as over 2h.

Spectral interferences as a function of the wedge position are shown in Fig. 2 for three locations of the pick-up mirror. These interferences are determined by the total dephasing due to the mirrors and the interferometer, respectively

$$ \Delta\varphi^{\text{tot}}(\omega, x) = \Delta\varphi^{\text{mir}}(\omega) + \Delta\varphi^{\text{int}}(\omega, x). \quad (1) $$

The map in Fig. 2(a) is obtained by placing the pick-up mirror outside the vacuum chamber (position ① in Fig. 1), thus with two 45° reflections on silver mirrors between the interferometer and the polarizer. The oscillations give information on the wavelength-dependent dephasing between the S and P components. For example, the positions of the interference maxima x<sub>m</sub> at the center wavelength λ<sub>0</sub> = 2πc/ω<sub>0</sub> = 850 nm appear at Δφ<sup>tot</sup>(ω<sub>0</sub>, x<sub>m</sub>) = 2πm (m integer), that is, where the carrier shift induced by the interferometer compensates for the carrier shift induced by the mirrors. In contrast, at the position x<sub>d0</sub> where ∂<sub>ω</sub>Δφ<sup>tot</sup>(ω, x<sub>d0</sub>)|<sub>ω<sub>0</sub></sub> = 0, the group delay difference induced by the interferometer compensates for that due to the mirrors. This group delay difference is in general non-zero because ∂<sub>ω</sub>Δφ<sub>S</sub>(ω)|<sub>ω<sub>0</sub></sub> ≠ ∂<sub>ω</sub>Δφ<sub>P</sub>(ω)|<sub>ω<sub>0</sub></sub>
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Polarization-gated XUV spectra with precompensation (a) and (c) and without precompensation (b) and (d) of the oblique reflections effects. (a) and (b) XUV spectra as a function of the added SiO₂ thickness that controls the CEP of the polarization-modulated few-cycle driving pulse. (c) and (d) Three lineouts from (a) and (b) for three different CEPs. The generation of continuous spectra with complete absence of harmonic structures between 50 and 60 eV (cutoff region) for certain CEPs in (a) and (c) and broad harmonics after a π/2 CEP shift confirms a sub-cycle polarization gate when precompensation of the oblique mirror effects is applied. Curves in (c) and (d) are normalized and shifted vertically for clarity.</figcaption></figure>

Higher-order dephasing $\partial_{\omega}^2 \Delta\varphi^{\text{tot}}(\omega, x)|_{\omega_0}$ can be neglected because the group delay dispersion in the interferometer is constant and $\partial_{\omega}^2 \Delta\varphi^{\text{mir}}(\omega)|_{\omega_0} \approx 0$. We use the position of the maximum of the envelope of the spectrally integrated signal, plotted as a solid yellow line in Fig. 2(a), as a measure of the zero-delay position, $x_{d0}$. Figure 2(b) shows that $x_{d0}$ changes with the number of reflections between the quartz interferometer and the polarizer, confirming that additional reflections on the mirrors induce a significant dephasing between the two polarizations. Due to space constraints in the setup, the measurement could only be performed at three locations indicated by ①, ②, and ③, in Fig. 1. At these locations, we find $x_{d0} = 19.8$ mm, 18.6 mm, and 17.3 mm, respectively. Calibration of the interferometer shows that the 2.5 mm shift between locations ① and ③ corresponds to a change in the delay of 2.8 fs near 850 nm, which is close to one optical cycle.

This result implies that the *S* and *P* envelopes of a pulse entering the vacuum chamber are further shifted by 2.8 fs with respect to each other at the position of the gas jet due to the seven mirror reflections and beam splitter transmission. This is far from negligible for few-cycle pulses. In addition, a non-zero carrier wavelength shift can alter the polarization state of the pulse, thus affecting HHG [17,28]. Our results also show that precompensation of the dephasing and delay caused by the mirrors is possible with a birefringent interferometer, as shown in Eq. (1). In the following, we perform polarization-gated HHG with and without precompensation.

In general, a polarization gate can be created by delaying two counter-rotating elliptically polarized pulses [13,26]. The duration of the gate is then given by $\tau_g = \epsilon_{thr} \tau^2 / [\delta \cos(2\beta) \ln(2)]$, where $\delta$ is the delay between the two pulses, $\beta$ is related to the ellipticity, $\tau$ is the pulse duration, and $\epsilon_{thr}$ is the threshold ellipticity, at which the harmonic yield is halved [25]. Sub-cycle

polarization gates can be achieved with CEP-controlled few-cycle pulses, leading to continuous XUV spectra when isolated attosecond pulses are emitted and broad harmonics when two attosecond pulses are emitted [5,6,14]. In our experiment, we create two delayed circularly polarized pulses with WP1 and WP2 to get a half-cycle gate with estimated duration $\tau_g = 1.4$ fs. We perform HHG with these pulses and measure the harmonic spectra as a function of the CEP. Figure 3 shows the XUV spectra in logarithmic scale with (a) and without (b) precompensation as a function of the CEP over a range of $3\pi$. In (a), the wedge position is $x = 16.95$ mm, while in (b) it is at the reference position $x = 19.8$ mm. Following our previous analysis (see Fig. 2), the position of the quartz wedges that ensures zero delay at the gas jet should be close to that obtained at ③ ($x = 17.3$ mm), which, however, includes an additional reflection at the pick-up mirror. The precompensation position $x = 16.95$ mm is obtained by optimizing the gated XUV signal close to $x = 17.3$ mm. The spectra in Fig. 3(a) vary periodically with added SiO₂ thickness, corresponding to a CEP periodicity of $\pi$, with the harmonic structure periodically disappearing and reappearing in the cutoff region. Figure 3(c) shows three lineouts with CEP differences of $\pi/2$. Continuous spectra between 50 eV and 60 eV (cutoff region) for certain CEP values (added SiO₂ thicknesses of 19.95 µm and 50.35 µm) are visible. Together with broad harmonic observation after a $\pi/2$ CEP shift, they are recognized as signatures of isolated attosecond pulse generation via sub-cycle polarization gating [5,6,26]. In contrast, the harmonic structure remains discrete across the lower-order plateau for all CEP values, as these harmonics are less temporally confined.

The XUV spectra shown in Fig. 3(b) (without precompensation) are very different and much less sensitive to the CEP. We observe a slight CEP-dependent spectral shift of the low-order plateau harmonics [23]. This shift is a fraction of the harmonic width and is $\pi$ periodic. No XUV continuum is observed,
independently of the CEP. Without precompensation of the oblique reflections, the sub-cycle polarization gate is not effective in the gas jet. Another striking difference between Figs. 3(a) and 3(b) is the behavior of the spectra close to the Cooper minimum, around 47.5 eV. The minimum is more pronounced with gated HHG in good agreement with strong temporal confinement, where intensity smearing is reduced.

The key result of our work is that multiple oblique reflections on metallic mirrors have a large impact on any short pulse that is not linearly polarized along the horizontal or vertical direction by inducing significant group delays and carrier shifts between the two polarization components. The dephasing due to reflections on metallic surfaces can, in principle, be computed from the phases $\phi_S(\omega)$ and $\phi_P(\omega)$ of the complex-valued Fresnel coefficients $r_S$ and $r_P$ [16], and the group delay difference by subsequent differentiation of these phases. The metallic mirrors used in the present work have, however, a thin protective layer. Using tabulated values for the refractive indices [29] and assuming an $SiO_2$ protective layer, the delay for a 28° reflection at an $SiO_2$-coated silver surface (45° incidence angle at the vacuum-SiO$_2$ interface) at $\lambda_0 = 850$ nm is estimated to -0.068 fs for each reflection. The measured value is about five times higher, and we attribute this discrepancy to the unknown protective dielectric coating and/or to the partial oxidation of the mirrors that can significantly change the refractive index. Additional measurements performed with a white light source confirm that mirrors with identical commercial specifications can lead to delays/dephasing that differ greatly in amplitude and even in sign and that estimates from the Fresnel coefficient are not sufficient. It also confirmed that transmission through a thin fused silica plate, mimicking the beamsplitter appearing in Fig. 1, has little impact on the dephasing between S and P polarizations. It is therefore important to characterize the effects of oblique reflections in a given experimental setup and compensate for them if necessary. Interestingly, our birefringent interferometer allows simultaneous compensation of the group delay and the carrier shift induced by the mirrors, as evidenced by the fact that the maximum of the oscillating signal almost coincides with the peak of its envelope in Fig. 2(b). Finally, tuning the interferometer around the estimated precompensation delay allows us to study various properties of the polarization gate. For example, the true zero-order waveplate, WP2, can be changed to $-\lambda/4$ or $\pm 3\lambda/4$, leading to a similar result. However, for larger plate orders, the width of the gate is too large, and sub-cycle gating is not achieved.

To conclude, we studied the impact of oblique reflections on metallic mirrors on polarization-gated HHG with CEP-stable few-cycle pulses. Large delays, on the order of 2.8 fs after seven reflections, can be induced between the S- and P-polarization components of the electric field. These delays can be prec compensated with suitable quartz plates to restore a sub-cycle polarization gate. Then, the measured XUV spectra, which are strongly CEP-dependent, become continuous at high photon energy for some values of the CEP. This is compatible with the generation of high contrast isolated attosecond pulses. The polarization dependence of oblique reflections has so far been largely overlooked and underestimated in the context of polarization gating. We believe that our results will have an important practical impact on the future design of experiments in ultrafast optics and attosecond science.

**Funding.** Laserlab-Europe (654148); Vetenskapsrådet (2021-04691, 2022-03519, 2023-0460); European Research Council (Advanced grant QPAP); Knut och Alice Wallenbergs Stiftelse; Wallenberg Centre for Quantum Technology (WACQT); Crafoord Foundation.

**Acknowledgment.** This work is supported by Laserlab Europe, grant agreement 654148. The authors acknowledge financial support from the Swedish Research Council (Grant Nos. 2021-04691, 2022-03519, and 2023-0460), the European Research Council (Advanced grant QPAP), the Knut and Alice Wallenberg Foundation, the Wallenberg Centre for Quantum Technology (WACQT), and the Crafoord Foundation.

**Disclosures.** The authors declare no conflicts of interest.

**Data availability.** No data were generated or analyzed in the presented research.

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