# Split-And-Delay Unit for FEL Interferometry in the XUV Spectral Range
## 4 Results and Discussion
### WLI Single-Shot Precision Determination  
To characterize the performance of the WLI system, we used the split-and-delay unit as a second independent interferometer. The SDU conveniently combines a beam splitter and two interferometer arms in a single device and requires only a light source and a detector to complete the setup. Hence, the second interferometric experiment can be performed simultaneously with the WLI measurement and used to gauge the precision of the latter. In the experiment, we used a small continuous wave (CW) diode laser to illuminate the SDU and imaged the reflected intensity in the focal plane with the second camera in synchronization with the WLI measurement. The interferogram of the laser recorded as a function of SDU displacement was then used to characterize the performance of the WLI system. The laser used in the experiment has a narrow spectrum with the central wavelength \(\lambda_{L}\,=\,638.4\) nm determined by a grating spectrometer. The laser beam was collimated and shaped with a \(1.5\times 7\) mm\({}^{2}\) rectangular aperture in order to restrict the illuminated area of the SDU to the structured region. After reflection from the SDU, the laser beam was focused by a toroidal mirror with an effective focal length \(f=317\) mm. The irradiance distribution in the focal plane was imaged by a camera placed behind the focus. The spatial distribution of the laser intensity for different OPDs between the gratings is shown in Figure 5. According to Equation (1), the separation between diffraction orders in the focal plane for our experimental geometry is \(\Delta r=801\)\(\upmu\)m and could be easily resolved by the camera. Figure 4: Vibrational spectrum of the SDU displacement. The spectrum was obtained by recording a movie with the WLI camera at a frame rate of 1380 Hz and then making the Fourier transform of the central fringe position. We scanned the OPD by translating the movable grating. The WLI and the laser cameras took images synchronously with equal exposure time of 1 ms at a rate of 8 Hz. The OPD \(s\) between the partial laser beams is connected to the displacement \(\Delta z\) between the gratings by the relation:  
\[s=2\Delta z\sin\alpha \tag{2}\]  
where is the angle of incidence on the SDU counted from its surface. We performed a 2130 nm long \(z\)-scan which corresponds to an OPD window of 1600 nm considering \(\alpha=22^{\circ}\) in our experiment. The actual displacement between the gratings for each exposure cycle was determined by the WLI according to the procedure described in Section 3. The laser intensity in the zeroth diffraction order as a function of derived OPD \(s\) is shown in Figure 6. The obtained interferogram is sinusoidal without any noticeable envelope as is expected for a highly monochromatic light source. The oscillation period of 640 nm agrees well with \(\lambda_{L}=638.4\) nm measured previously by the grating spectrometer. The deviation of the experimental data from the fit in the interferogram plot has two main contributions: (1) the noise of the camera imaging the focal plane and (2) the error in the OPD derived by the WLI. The latter error defines the precision of the WLI system and we will quantify it using its  
Figure 5: Irradiance distribution of the laser diode in the focal plane after reflection from the SDU for three different phase delays: (**a**) 0, (**b**) \(\pi/2\), (**c**) \(\pi\). The foci have line shapes because the laser had a \(1.5\times 7\) mm\({}^{2}\) rectangular aperture to match the beam size with the structured area of the SDU. Figure 6: Intensity of the diode laser in the zeroth diffraction order as a function of optical path difference between the gratings of the SDU derived from the WLI images. The red curve is a sine fit to the data. dependence on the interferogram shape in the following. The contribution of the WLI error to the standard deviation (SD) of the data is proportional to \(\mathrm{d}I/\mathrm{d}s\), where \(I\) is the interferogram intensity.