## 2 Wavelength-Tuning Interferometry  
To show the optical path, the optical setup of the wavelength-tuning Fizeau interferometer, used for the measurement of the surface profile of highly reflective silicon wafers, is shown in Figure 1.  
A wavelength-tuning phase-shifting Fizeau interferometer has much better anti-air turbulence ability than other conventional interferometers. The light source is a tunable diode laser with a Littman external cavity [41] (Newport Inc., NewFocus TLB-6804, mode-hop-free fine-frequency tuning range \(>\) 120 GHz), which can scan the wavelength linearly from 632.5 to 640 nm. The light was transmitted out from the laser source and was then divided into two beams in the beam splitter: one was incident to the interferometer and the other was transmitted to a wavelength meter. In the interferometer, the collimated laser beam illuminated the object (silicon wafer) and reference surface. The beams reflected from silicon wafers and the reference surfaces combine and generate the interferograms on the screen and are acquired by a Charge Coupled Device (CCD) (The Imaging Source, DMK 33G445) camera and processed by an Analog to Digital Converter (ADC) (The Imaging Source, DFG/USB2 pro). The technical parameters of the CCD are shown in Table 1. The PC processor is an Intel(r) Core i7-6900K with an NVIDIA GeForce RTX 2080 Ti graphic card and 32 GB of RAM. In the wavelength-tuning Fizeau interferometry, significant interferences appear as a result of multi-reflections between the sample surface and the reference surface. Figure 2 shows the multi-reflections between the reference surface and the sample surface with reflectivities \(r_{1}\) and \(r_{2}\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Items** & **Technical Parameters** \\ \hline Dynamic range & 12 bit \\ Resolution & 1280 \(\times\) 960 pixels \\ Frame rate & 30 fps \\ Pixel size & Vertical/horizontal: 3.75 \(\upmu\)m \\ Sensor & SONY ICX 445 ALA \\ Shutter & 10 \(\upmu\)s–30 s \\ Gain & 0–30.39 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The technical parameters of the DMK 33G445. Figure 2: The reflections between surfaces. Figure 1: The setup of the wavelength-tuning Fizeau interferometer. PBS, Polarizing beam splitter; QWP, Quarter wave plate; CCD, Charge Coupled Device. The signal irradiance detected by a CCD camera is given by  
\[ \begin{array}{ll}I(\alpha_{k})&=\sum\limits_{k=1}^{\infty}A_{m}\cos(\varphi_{m} -\alpha_{k})\\ &=I_{0}+I_{0}\gamma_{1}\cos(\varphi_{1}-\alpha)+I_{0}\gamma_{2}\cos(\varphi_{2} -2\alpha)+\dots,\end{array}  \tag{1}\]  
where \(\alpha_{k}\) is the phase shift amount, \(I_{0}\) is the DC component, and \(A_{m}\) and \(\varphi_{m}\) are the amplitude and phase of the \(k\)th harmonic component, respectively. The parameters that are given by  
\[I_{0}=\frac{r_{1}+r_{2}-2r_{1}r_{2}}{1-r_{1}r_{2}}, \tag{2}\]  
\[\gamma_{1}=\frac{2(1-r_{1})(1-r_{2})}{r_{1}+r_{2}-2r_{1}r_{2}}\sqrt{r_{1}r_{2}}, \tag{3}\]  
\[\gamma_{2}=-\gamma_{1}\sqrt{r_{1}r_{2}}, \tag{4}\]  
\[\gamma_{3}=\gamma_{1}r_{1}r_{2}. \tag{5}\]  
where \(r_{1}\) is the reflectivity of the reference surface and \(r_{2}\) is the reflectivity of the sample surface. For an \(M\)-sample algorithm, the phase \(\varphi_{1}\) of the fundamental signal can be expressed as  
\[\varphi_{1}=\arctan\frac{\sum\limits_{k=1}^{M}b_{k}I(\alpha_{k})}{\sum\limits _{k=1}^{M}a_{k}I(\alpha_{k})} \tag{6}\]  
The reference phases can be divided into \((M-1)\) intervals. The phase shift value \(\alpha=2\pi/N\) (\(N\) is an integer), \(a_{k}\) and \(b_{k}\) are the \(k\)th sampling amplitude, and \(I_{k}\) is the \(k\)th sampled irradiance (\(I_{k}=I(\alpha_{k})\)) defined in Equation (1). The nonlinear relation between the \(\alpha_{k}\) value and the phase shift parameter is clearly present when the phase shift is nonlinear, which can be expressed as [42]  
\[