## 4 Results and Discussion
### Functioning of the Mirau Configuration  
Different measurements are performed to illustrate the functioning of the MA. First, an aluminum mirror with a surface flatness less than \(\lambda/10\) is used as a measuring object to investigate optical aberrations and systematic deviations. As mentioned before, a beam splitter plate and a compensation plate of high optical quality is used. Nevertheless, tilting of these optical elements with respect to the optical axis, e.g., due to installation tolerances that may occur during assembly, can cause dispersion. A possibility to determine dispersion effects is given by the investigation of the measured interference signals. Therefore, the envelopes of signals of the custom-built 20x Mirau objective are compared with those of a commercial Mirau objective (Nikon 10x CF IC EPI Plan DI). Both objectives are directly applied in the same optical setup to keep the results the comparable. The signal envelopes for both MOs are obtained from the aluminum mirror using a white as well as a red LED for illumination. As shown in Figure 5a, significant difference between the envelopes (red curves) of both MOs can be observed for white light illumination. While the course of the signals obtained by the commercial objective is symmetrical, the course corresponding to the MA is asymmetric. Furthermore, the interference signal is broadened compared to the signal of the commercial system, which is illustrated by a full width at half maximum (FWHM) of 2.54 um for the MA and 1.52 um for the Nikon Mirau objective. With reference to the investigations of [PERSON] and [PERSON] [33] and under the assumption that the Mitutoyo MO is corrected to a high degree and only minor optical aberrations are present, the broadening of the envelope probably is caused mainly by dispersion. This assumption is confirmed comparing Figures 5c,d, where the same measurements are performed but using a red LED. Here, as a consequence of a narrower bandwidth the envelopes for both MOs are broadened compared to the signals obtained for a white LED. However, no significant difference is visible between the interference signals of both MOs. This is also corroborated by the difference of FWHMs obtained from the respective  
Figure 4: Schematic illustration of the correction algorithm. The CSI dataset is rearranged according to the real movement measured by the IDS (oriented to [24]). envelopes of the interference signals, which decrease from 1.02 \(\upmu\)m for the white LED to 0.02 \(\upmu\)m using the red LED as illumination source. The measurement of the aluminum mirror shows systematic deviations in the measured surface topography, which still remain when the measuring field is laterally shifted. In order to separate the systematic deviations from the topography of the measured object, nine measurements are performed on latterly shifted locations at the aluminum mirror used as measuring object. The pixel-wise averaging of the nine measured surface topographies leads to the systematic deviation of each pixel, similar to the procedure used in [34, 35, 36]. This deviation is also known as residual flatness and includes the systematic deviations due to the unevenness of the reference mirror. In Figure 6a, the illustrated residual flatness is measured using a white LED, whereas for the result depicted in Figure 6b, a red LED is used. Similarities in both topographies are apparent immediately. The topography in Figure 6a is slightly higher compared to that depicted in Figure 6b. This is also illustrated by calculating the standard deviations \(\sigma_{\text{tr}}=3.47\,\text{nm}\) for the surface obtained using the white LED and the red one with \(\sigma_{\text{r}}=3.25\,\text{nm}\). However, the specification of the standard deviation to represent the residual flatness is misleading, since the determined systematic deviations differ stronger at the edges compared to the center (see Figure 6).