## 3 Actively Phase-Stabilized Continuous Optical Delay Scan with Attosecond Timing Accuracy  
The uniqueness of the current ultrafast DUV/XUV beamline is the capability of continuous delay scanning with attosecond timing precision. For this purpose, as mentioned above, a double-layer interferometer (DLI) setup with an arm length of ~1.2 m has been constructed, as shown in the mid-panel of Figure 1a, in which the upper layer serves as a reference interferometer to scan and stabilize the DUV/XUV delay line on the lower layer. To achieve a high degree of phase stabilization, the optical breadboard of the delay line chamber is supported by vibration isolators (VIB320, Newport, Irvine, CA, USA), which greatly suppress the mechanical noise from various sources in the laboratory. In order to minimize the intensity drift of the upper-layer interferometer, which would cause a phase drift of the DUV/XUV delay, an intensity-stabilized HeNe laser is used to provide the phase reference in the upper layer. As shown in Figure 2a, custom-designed Al mirror mounts have been fabricated for the DLI setup to mechanically link the lower- and upper-layer interferometers. Since most mechanical drifts and vibrations in both layers are correlated with this design, the phase locking in the upper-layer interferometer automatically ensures the phase locking in the lower layer. Half-inch standard silver mirrors are used in the upper layer for the HeNe laser, whereas 1 inch dielectric mirrors and perforated dichroic beam separators are installed in the lower layer tailored for the DUV and XUV/IR pulses in the respective beam paths (see Figure 1a). For continuous scanning of the delay over a long range while actively stabilizing the phase in the DLI setup, we utilize the accumulated [PERSON]'s phase method [27] in the reference upper-layer interferometer. Figure 2b illustrates the active stabilization scheme in which an intensity-stabilized, linearly polarized (s-pol) HeNe laser beam passes through the upper-layer interferometer. After the first beamsplitter (BS1), the polarization in one of the arms is rotated by 90 degrees with respect to the other arm using a half-wave plate (HWP). The cross-polarized beams are then recombined at the second beamsplitter (BS2) and pass through a quarter-wave plate (QWP), which converts these crossed linear polarizations into circular polarizations of opposite helicity. The superposition of the two orthogonal circular polarizations results again in a linearly polarized beam whose polarization angle now only depends on the relative optical phase between the two circular polarization components. Effectively, this method converts a change in optical phase (or delay) into a change of the linear polarization angle, while in a simple interferometer an optical phase change results in an intensity change due to constructive or destructive interference. The advantage of this method is that the rotation of the polarization angle is directional and continuous with the optical delay, while the intensity variation due to interference in a normal interferometer changes direction after every half-cycle. This ambiguity of the interference amplitude as a function optical delay limits the scan range of a typical phase-locked interferometer to a half-cycle of the reference laser's wavelength. In contrast, using the [PERSON]'s phase method eliminates this ambiguity, since here a change of the optical delay in one direction results in a directional change of the polarization angle. Continuous tracking of the absolute angle (including the number of turns) allows for a unique mapping between optical delay and polarization angle over an arbitrary range. By analyzing the polarization angle with a polarizer (POL) and detecting the resulting HeNe laser intensity with a photodiode (PD), we can lock the optical delay by keeping the photodiode signal (and thus the polarization angle) at a certain value.