## 3 Actively Phase-Stabilized Continuous Optical Delay Scan with Attosecond Timing Accuracy
This is achieved by using an analog proportional-integral-derivative (PID) controller (SIM960, Stanford Research Systems, Sunnyvale, USA) together with a piezoelectric transducer for the optical delay stage in a feedback loop with the set point set to roughly the middle of the range between the minimum and maximum photodiode signal after the polarizer as a function of optical delay. Delay scanning can now easily be achieved by slowly rotating the polarizer while maintaining the PID lock. Since the polarization change is continuous with the optical delay, the scan range is only limited by the reference laser's coherence length and the mechanical constraints of the optical delay stage. Turning the polarizer results in a unique corkscrew-like relationship between the absolute angle of the polarizer and the optical path difference between the interferometer arms as long as the PID lock is never lost. A full (2\(\pi\)) turn of the polarizer corresponds to an optical phase change of 4\(\pi\), or two wavelengths of the reference laser. To validate the performance of the DLI, for simplicity, the dielectric/dichroic mirrors in the lower layer have been replaced with Ag mirrors. The intensity-stabilized HeNe laser is now split in two parts. One part is used in the upper layer with the PID feedback loop to stabilize and scan the reference interferometer, referred to as in-loop (IL), while the second part is used for lower layer (out-of-loop, OOL) stability measurements. This second lower-layer (OOL) interferometer provides a direct performance assessment of the actively stabilized DLI, since it uses the same beam path as the DUV/XUV beams for pump-probe experiments. In addition, the HeNe intensity is referenced by another photodiode to eliminate intensity-related drift. Figure 3 shows the phase stability of the DLI measured over \(\sim\)2 h at a fixed optical delay. The signal from the upper- and lower-layer  
Figure 2: (a) Custom-designed coupled Al mirror mount for the double-layer interferometer setup. (b) Actively-stabilized continuous delay-scan scheme based on [PERSON]’s phase [24] in the upper-layer (reference) interferometer. BS: beam-splitter, PZT: piezoelectric transducer stage, PD: photodiode, POL: polarizer, PID: proportional-integral-derivative controller, RCP: right-handed circular polarization, LCP: left-handed circular polarization. photodiodes was sampled at a 6 kHz rate (well above the relevant acoustic frequencies) and converted into optical delay units by using the HeNe wavelength of 632.8 nm and the amplitude of both interference signals obtained from a short scan of the delay stage with the PID controller switched off. The power spectral density (PSD) of the timing drift signal of the upper- (IL) and lower-layer (OOL) interferometer as well as their corresponding integrated jitter (square-root of the integrated PSD, starting at high frequencies) are shown in Figure 3. In order to determine long-term stability, both the upper and lower layer stability were measured for two hours at a fixed optical delay position with and without active PID control. The analog PID controller was not bandwidth limited within the depicted frequency range. In Figure 3, upper layer measurements are shown in red, lower layer in blue, and free-running (PID control disabled) in green. It is evident from Figure 3a that the active stabilization notably reduces the timing drift in the lower-layer (OOL) interferometer over the entire frequency range below ~15 Hz. On the other hand, the active PID control introduces some noise on its own at higher frequencies due to a mechanical resonance of the piezoelectric stage around 30 Hz as well as electronic noise, which increases the overall lower-layer (OOL) timing jitter by about three as above 30 Hz (Figure 3b). Note that the upper layer (IL) measurement had a lower signal-to-noise ratio compared to the lower-layer signal due to a different amount  
Figure 3: Frequency-domain analysis of the optical delay timing jitter and drift.