## 4 Experiment Results and Discussion  
### Experimental Results  
A BK 7 plate (diameter: 10 cm; thickness: 5 mm) from the SCHOTT company with refractive index 1.515 at a wavelength of 633 nm was measured using the novel \(10N-9\) algorithm and wavelength-tuning  
Figure 7: RMS error as a function of phase shift miscalibration (\(r_{1}=4\%\), \(r_{2}=30\%\)).  
Fizeau interferometer. The measurement was taken under lab-temperature \(20.5\pm 0.1\)\({}^{\circ}\)C. The silicon wafer was aligned vertically on a mechanical stage. The air gap distance between the silicon wafer and reference surface was set to L = 10 mm. The interferometer was placed on a stainless-steel table to isolate the interferometer from external vibration. To eliminate the temperature gradient of the BK 7 plate, it was placed in the laboratory for one day. We set CCD frame rate as 0.5 fps, auto exposure time, and auto gain to get high quality interferograms. Figure 8 presents the observed raw interferogram.  
The necessary wavelength-tuning range \(\delta\lambda\) can be calculated as  
\[\delta\lambda=-\frac{\lambda^{2}}{4\pi L}\delta\varphi\approx 0.2024\,\mathrm{nm}, \tag{20}\]  
The wavelength was scanned linearly from 632.8524 to 633.0551 nm and 91 interferograms were acquired by CCD camera. The wavelength values from start to end were accurately measured by the wavelength meter shown in Figure 1 with an accuracy value of \(10^{-7}\) at a wavelength of 633 nm. For the whole wavelength-tuning process, the total phase was changed by \(20\pi\), and each phase step was \(\pi\)/5. The recording time was 3 min. Figure 9 shows five consecutive interference fringe patterns chosen arbitrarily from 91 interferograms captured by CCD.  
The phase \(\varphi\) can be calculated using the \(10N-9\) algorithm:  
Figure 8: Raw interferogram at a wavelength of 633 nm.  
Figure 9: Five consecutive interference fringe patterns captured by CCD.  
\[\varphi=\arctan\frac{\sum\limits_{k=1}^{91}w_{k}l_{k}\sin\frac{\pi k}{5}}{\sum \limits_{k=1}^{91}w_{k}l_{k}\cos\frac{\pi k}{5}}. \tag{21}\]  
where \(w_{k}\) is the polynomial window function, which is described in Appendix A, and \(l_{k}\) is the intensity of the \(k\)th image defined by Equation (1).  
The calculated surface profiles by phase unwrapping [51] are given in Figure 10, which presents the entire configuration of the silicon wafer. All 20 experiments were conducted successively within three days. To test the reliability of the proposed algorithm, the sample was measured using a FUJINON interferometer (G102, FUJIFILM Corporation). By using proposed new method, the obtained average peak-to-valley (PV) value of 20 experiments was 513.587 nm (Figure 10a), and the standard deviation of 20 experiments was 2.05 nm. By using FUJINON interferometer, the obtained average peak-to-valley value was 524.376 nm (Figure 10b), and the standard deviation was 2.33 nm. The error map of the surface shape is shown in Figure 11. Table 2 shows the peak-to-valley and RMS values of the \(10N-9\) method and the FUJINON interferometer, respectively.  
Figure 11: The error map of the surface shape.  
Figure 10: (**a**) Surface profile measured with new algorithm. (**b**) Surface profile measured with the FUJINON interferometer.