## 2 Materials and Methods  
When a beam of light is incident on a transparent plate, an interference pattern is obtained due to multiple reflections from the plate's surfaces. In order to measure the profile of the transparent plate, it  
Figure 1: Diagram of the reflected rays forming the interferogram. is necessary to extract from the registered interference pattern information caused by each surface [5]. The intensity distribution in the interferogram formed by transparent plate can be expressed as:  
\[\mathrm{I}(x,y,k)= A_{0}+\sum_{n=1}^{\infty}A_{n}\cos(\varphi_{n}(x,y)+2\pi v_{n}k), \tag{1}\]  
where \(\mathrm{I}(x,y,k)\) is the intensity of the \(k\)th record image in the coordinate \((x,y)\), where the coordinate plane \(x,y\) is parallel to the measured interfaces. \(A_{0}\) is the intensity of the background, and \(A_{n}\) is a contrast associated with intensity of the \(n\)th harmonic. \(\varphi_{n}\) is the phase of the \(n\)th harmonic components, which corresponds to the path difference between the sample surface and the reference surface of the interferometer. The accuracy (peak-to-valley value) is less than one twentieth of the wavelength, \(\lambda/20\), generally. The modulation frequency is \(v_{n}\) and is proportional to the optical path difference (OPD) of each pair of interfering fringe. For the signal generated by the interference due to the reflection from the reference and front-and-rear surfaces, the frequency \(v\) is given by:  
\[v=\frac{2 OPD}{\lambda_{0}^{2}}\cdot\frac{d\lambda}{dk}, \tag{2}\]  
where \(\lambda_{0}\) is the central wavelength of the laser source, and \(\frac{d\lambda}{dk}\) is the wavelength-tuning rate per time unit. In Figure 1, each frequency in the measurement can be described as follows  
\[ \begin{array}{c}v_{1}=\frac{4n_{0}T}{\lambda_{0}^{2}}\cdot\frac{d\lambda}{dk }\\ v_{2}=\frac{4n_{0}T}{\lambda_{0}^{2}}\cdot\frac{d\lambda}{dk}\\ v_{3}=\frac{4(n_{0}T+H)}{\lambda_{0}^{2}}\cdot\frac{d\lambda}{dk}\end{array}  \tag{3}\]  
where \(n_{0}\) is the refractive index of the measured components when \(\lambda=\lambda_{0}\) and \(T\) is the thickness of the transparent plate. \(H\) is length of the air-gap (the distance between the reference and the front surface). Table 1 shows the relationship between the frequency information of the interference signals. We can detect the useful signals (front-and-rear surfaces and thickness information) from the superimposed interference patterns by setting the appropriate ratio \(M\) defined as \(M=H/(n_{0}T)\). The value of \(M\) determines the application for the measurement of the transparent plate of arbitrary thickness. It is possible to develop a spectral stretching approach for obtaining the appropriate \(M\) value for a transparent plate of any thickness in the actual measurement process using the Fizeau interferometer with wavelength tuning. \begin{table}
\begin{tabular}{c c c} \hline
**No.** & \(\mathbf{v_{n}}\)/\(\mathbf{v_{1}}\) & **Remark** \\ \hline \(v_{1}\) & 1 & thickness \\ \(v_{2}\) & \(M\) & Front interface \\ \(v_{3}\) & \(M+1\) & Rear interface \\ \hline \end{tabular}
\end{table} 
Table 1: Frequency information of each measuring surface. frequencies will be introduced to affect the extraction accuracy of the information to be tested in multi-surface measurements. In order to measure the highly reflective sample surface accurately, it is necessary to suppress the harmonic signals more effectively than the Hanning window function. The main factor for evaluating the validity of the window function is the amplitude of the sidelobe in the frequency domain. Therefore, the window function usually needs to be adjusted ulterior. In order to improve the restrain characteristics, the weighted values determined by the window function are subsequently processed. The processed results should satisfy the following equation [11]:  
\[