## 3 Results
After unwrapping and compensating for the tilt phase aberration of the phase map obtained from the both surfaces of the transparent plate, the profiles of both surfaces of the studied sample can be obtained according to the following relationship:  
\[\delta=\frac{4\pi d(x,y)}{\lambda}, \tag{12}\]  
where \(d\) describes the optical path difference (OPD). For the three phase maps generated from the reference surface and the front-rear surfaces of the tested transparent plate, \(\phi_{f}(x,y)\), \(\phi_{r}(x,y)\) and \(\phi_{f\cdot r}(x,y)\) can be expressed as follows:  
\[\phi_{f}(x,y)=\frac{4\pi[H(x,y)+W_{t}(x,y)]}{\lambda} \tag{13}\] \[\phi_{r}(x,y)=\frac{4\pi[H(x,y)+W_{t}(x,y)+n_{0}(W_{t}(x,y)-W_{t}( x,y))+n_{0}T(x,y)]}{\lambda}\] \[\phi_{f\cdot r}(x,y)=\frac{4\pi[n_{0}T(x,y)+n_{0}(W_{t}(x,y)-W_{t} (x,y))]}{\lambda}\]  
Figure 5: Distributions of the registered and processed signals obtained in the experiment. (**a**) The fringe pattern from both surfaces of the transparent plate at the initial wavelength; (**b**) wrapped phase map of the front surface after the windowed Fourier transform (WFT) filter was applied; (**c**) wrapped phase map of the rear surface after the WFT filter was applied; (**d**) wrapped phase map of variation in the thickness after the WFT filter was applied. where \(H\) represents the distance of the air-gap and \(W_{\text{f}}\), \(W_{\text{r}}\) are front and rear surface profiles of the sample, \(n_{0}\) and \(T\) are the reflective index of the sample, which were assumed to have a uniform distribution, and the thickness of the sample, respectively. The measured shape of the front surface, rear surface and variation in the thickness are shown in Figure 6a-c, respectively. The mean peak-to-valley (PV) values of the front-and-rear surfaces of the tested transparent plate and its thickness variation were found as 1.38 \(\upmu\)m, 0.89 \(\upmu\)m and 0.51 \(\upmu\)m, respectively, by carrying out 10 repeated measurements at the same position. The repeatability of the 36-step algorithm is confirmed by a pair of measurements taken successively within three days. According to the obtained results by 10 repeated measurements, the root mean square errors (RMSE) for determining both surfaces did not exceed 1.5 nm. The detailed data of the measurements are shown in Table 2. To test the reliability of the 36-step algorithm, the sample was measured by using a ZYGO interferometer (Zygo Corporate, Middlefield, USA), which is considered as a reliable testing instrument in industrial applications. However, it cannot measure the front surface and rear surface profiles simultaneously. Therefore, only one surface was measured, whereas the other surface was covered with Vaseline to suppress its reflection. Then, the other surface was measured while Vaseline covered the first surface. Figure 6b,d,f show the measured front-and-rear surface profiles and variation in the thickness respectively, using a ZYGO interferometer. The results indicate that the shapes measured with the proposed 36-step algorithm were consistent with the ZYGO interferometer. Quantificationally, the PV of front-and-rear surface profiles and variation in the thickness were 1.40 \(\upmu\)m, 0.88 \(\upmu\)m and 0.53 \(\upmu\)m, respectively. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & & \multicolumn{8}{c}{**Measurement Number**} & \multicolumn{3}{c}{**Average**} \\ \cline{2-13}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **Value** & **RMSE** \\ \hline front surface & 1.3815 & 1.3809 & 1.3814 & 1.3824 & 1.3815 & 1.3803 & 1.3814 & 1.3838 & 1.3810 & 1.3802 & 1.3814 & 0.0010 \\ rear surface & 0.8932 & 0.8923 & 0.8943 & 0.8915 & 0.8924 & 0.8911 & 0.8932 & 0.8927 & 0.8900 & 0.8911 & 0.8922 & 0.0013 \\ variation of thickness & 0.5114 & 0.5126 & 0.5015 & 0.5127 & 0.5113 & 0.5112 & 0.5103 & 0.5139 & 0.5117 & 0.5108 & 0.5116 & 0.0011 \\ \hline \hline \end{tabular}
\end{table}