# Split-And-Delay Unit for FEL Interferometry in the XUV Spectral Range
## 3 White Light Interferometer for Time Delay Diagnostics  
In our experimental setup we determine the time delay introduced by the SDU using a white light interferometer that monitors the topography of the split-and-delay unit. The typical task of a WLI system is to characterize the height profile \(h(x,y)\) of a sample surface placed in one of the interferometer arms by finding zero optical path difference (OPD) for each point of the surface. Usually an experiment is performed by mechanically scanning the length \(z\) of one arm and recording the light intensity \(I(x,y)\) for different \(z\) positions by a camera. The recorded intensity \(I(x,y;z)\) is a collection of \(z\)-dependent interferograms for all camera pixels. The optical path difference \(s\) between the two arms can be expressed as a sum of the surface heightmap \(h(x,y)\) relative to some reference plane and a translation \(z\) of this plane along the interferometer arm: \(s(x,y)=2(z+h(x,y))\). By scanning \(z\), the positions where \(s=0\) (interferogram maximum) are found for every pixel yielding thus the surface profile \(h(x,y)\).  
A mechanical \(z\)-scan requires the setup to be static on the timescale of at least few seconds. This condition is often difficult to fulfill. Equipment present in a typical laboratory environment (e.g., vacuum pumps or ventilation system) operates at frequencies from several tens to several hundred Hz and generates vibrations which may be transmitted to the experimental setup. For interferometric experiments in the XUV, even small vibration-induced displacements on a scale of tens of nm become crucial. With respect to an SDU, the important aspect to take into account is not the vibration of the system as the whole but the relative jitter of the two components of the device. These oscillations transform into the jitter of the time delay between the generated pulse replicas and thus have a direct impact on the outcome of an interferometric measurement. For example, a displacement of just 50 nm between the SDU reflectors translates into a delay which may comprise several optical cycles for XUV wavelengths. This makes interferometric, i.e., sub-cycle resolved, experiments impossible if the signal is accumulated over time without considering the exact topography of the SDU for each laser shot. Under such a scenario, a mechanical \(z\)-scan of the WLI is not applicable and one has to rely on a  
Figure 2: Layout of the experimental setup inside the vacuum chamber. As an illustration, the image on the position-sensitive detector (PSD) displays the spatial distribution of Xe\({}^{+}\) ions produced by 38 nm photons [22]. The image shows a characteristic triple foci structure generated by the SDU. The image is an accumulation of \(\sim\)5000 shots including many different pulse-replica delays.  
single-frame technique. Therefore, we used the interferometer in the so-called \"electronically scanned\" configuration [37; 38] with non-collinear beams. If one of the reflective surfaces is tilted to some small angle \(\beta\), the two beams intersect at the skew angle \(2\beta\). As a result, the OPD between the beams varies continuously along the intersection plane and a single camera image actually presents an OPD scan. In this way, the interferogram \(I(z)\) is mapped onto the axis perpendicular to the beam propagation: \(I(z)\to I(x)\) as shown in Figure 3. A shift of the interference pattern \(\Delta x\) along the horizontal axis of the image will correspond to some longitudinal displacement \(\Delta z\) of the reflective surface along the interferometer arm. The calibration factor \(\mathrm{d}z/\mathrm{d}x\) for a fixed tilt angle can be calculated from the spacing of interference fringes knowing the average wavelength of the white light. Using this factor, \(\Delta z\) can be derived for every single frame.