## 2 Experimental Setup  
The proposed laser system, described on Figure 1, consists of a commercial 1 kHz CEP-stabilized amplifier (Femtopower Compact Pro CE Phase, Femtolasers GmbH) followed by an XPW contrast filter. The oscillator pulses (Rainbow) are CEP locked via pump-laser amplitude modulation and stretched in a 20 cm-long SF57 block before amplification up to 1.9 mJ in a ten-pass titanium-sapphire amplifier. The amplification stage is pumped by 11 W from a 1 kHz frequency-doubled Q-switched Nd:YLF laser (JADE, Thales). An AOPDF (low-jitter Dazzler HR800, Fastlite) is inserted between the fourth and fifth passes in the amplifier in order to optimize the spectral amplitude and phase. Finally, the 7 ps long amplified pulses are then compressed through a transmission-grating compressor (TGC) with an overall efficiency of 85%. The TGC consists of a pair of 1280 g/mm gratings used at the Littrow incidence angle (30.8\({}^{\circ}\)) and separated by 10 mm. We have already shown that this compression setup is fully compatible with CEP stabilization [21]. The compressed pulse duration is slightly below 30 fs and the Gaussian output beam has a diameter of 10 mm (1/e\({}^{2}\)). Self-referenced spectral interferometry (Wizzler device [22]) (Figure 2) enables the fine-tuning of the residual spectral phase of the compressed pulses, which is critical before they are sent into the XPW filtering stage [23]. Typical energy stability at this stage is around 1% RMS over hours.  
In our current setup, the XPW filter is an optimized version of that described in [24]. Thanks to the TGC providing pure polarization and thanks to little propagation between the TGC and the XPW filter, the use of an input polarizer is not necessary. The XPW setup consists of a 47 cm, 250 \(\mu\)m inner diameter hollow-core waveguide providing efficient (75% transmission) spatial filtering, followed by two nonlinear BaF\({}_{2}\) crystals, all under vacuum. Coupling of the beam into the hollow-core fiber is achieved using a 1.75 m focal length mirror. Slow drift of the input beam pointing is corrected using a home-built beam stabilization system, ensuring constant long-term fiber transmission. Two thin, 1.5 mm  
Figure 1: Experimental scheme of the 1 kHz CEP-stable CPA system and waveguide-XPW filter (TGC: Transmission Gratings Compressor).  
thick, BaF\({}_{2}\) crystals with holographic crystallographic orientation are employed for XPW conversion, situated 35 cm and 41 cm respectively after the fiber output. The position of the first crystal corresponds to a fundamental beam diameter of 1.7 mm (1/e\({}^{2}\)), adapted to the incident 1 mJ. This scheme does not correspond to the standard two-crystal scheme described in [25], where phase-matched XPW generation is ensured by Kerr focusing of the fundamental beam between the two crystals. In this scheme, the fundamental beam diverges out of the fiber and thus self-focusing in the first crystal mainly collimates the fundamental beam. However, we noticed that using two thin crystals instead of a thicker one leads to a slightly higher output energy and improved pulse-to-pulse stability. Because the XPW is driven closer into saturation, the pump spatial profile is depleted during XPW conversion and free-space propagation enables to optimize the fundamental spatial beam profile and improve the XPW conversion efficiency in the second crystal. A direct consequence of the improved efficiency XPW close to saturation is the enhanced pulse-to-pulse stability of the device. The advantage of this approach is that it is scalable to higher energies [24]. Here, the output beam is then expanded to go through the output Glan polarizer enabling XPW pulse selection. 300 \(\mu\)J pulses are routinely generated, corresponding to a XPW internal efficiency as high as 33% (fiber transmission and reflection losses on crystals deduced) and a global energy transmission of 22%.