## 2 Proposed Differential Quadrature Fabry-Perot Interferometer
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Item** & **Signal** & **Signal Processing** & **Problems** \\
**Fabry–Perot Interferometer** & & **Distribution** & **Method** & Not suitable for \\ \hline \multirow{2}{*}{Conventional structure} & \multirow{2}{*}{Reference 8} & Discontinuous & \multirow{2}{*}{Fringe counting} & Not suitable for \\  & & (high finesse) & & dynamic \\ \multirow{2}{*}{Conventional structure} & \multirow{2}{*}{Reference 8} & \multirow{2}{*}{(high finesse)} & \multirow{2}{*}{Fringe counting} & measurement in \\  & & & & large range \\ \hline \multirow{2}{*}{Polarized structure} & \multirow{2}{*}{Reference 11} & Continuous (low & Quadrature-based & \multirow{2}{*}{\
\begin{tabular}{} \end{tabular} } \\  & & & 2. Involved with \\ \multirow{2}{*}{Polarized structure} & \multirow{2}{*}{Reference 11} & \multirow{2}{*}{
\begin{tabular}{} \end{tabular} } \\  & & & photodiodes & the complex signal \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison between the previous Fabry–Perot interferometers. The Jones matrix of the incident beam and the relevant optical elements in this structure is as Equation (6) to Equation (11). \[\mathrm{E}_{(\Phi)}=\left( \begin{array}{c}\cos\Phi\\ \sin\Phi\end{array} \right) \tag{6}\]  
\[\mathrm{PF}=\mathrm{e}^{\mathrm{i}\cdot\delta}\mathrm{=e}^{\mathrm{i}\cdot \frac{\mathrm{i}\cdot\mathrm{strand}}{\lambda}} \tag{7}\]  
Figure 5: Proposed differential quadrature Fabry–Pérot interferometer: (**a**) Structure composition; (**b**) transmission of laser beams in two measurement types (**b**)1 plane mirrors (PM) type, (**b**)2 cube retro-reflector (CCR) type, (**c**) optical arrangement and the polarization direction of laser beams. PD, photodiode, PBS, polarizing beam splitter; BS, non-polarizing beam splitter. \[\text{BS}=\frac{1}{\sqrt{2}}\Bigg{(} \begin{array}{cc}1&0\\ 0&1\end{array} \Bigg{)} \tag{8}\]  
\[\text{QWP}(\theta)= \tag{9}\]  
\[\text{PBS}_{\text{o}}(\alpha)= \tag{10}\]  
\[\text{PBS}_{\text{e}}(\alpha)= \tag{11}\]  
Where E (\(\Phi\)) is the electric field of the linearly polarized laser beam after traveling the optical isolator, the angle of the polarization direction with respect to the \(y\)-axis is \(\Phi\). PF is defined as the phase factor (e\({}^{\text{i}\cdot\text{S}}\)) accounts for the optical phase difference (\(\delta\)) caused by the changing of the cavity distance (d). \(\delta\) equals 4\(\pi\)nd/\(\lambda\), where \(\lambda\) is the laser wavelength, and n is the refractive index. For the type of the plane mirror and the corner cube retro-reflector, the optical phase difference equals to 4\(\pi\)nd/\(\lambda\) (\(\delta\)) and 8\(\pi\)nd/\(\lambda\) (\(28\)), respectively. BS is the matrix of the non-polarizing beam splitter, and QWP (\(\theta\)) represents a matrix that corresponds to a quarter-wave plate whose fast axis is at an angle \(\theta\) with respect to the \(y\)-axis. PBS\({}_{\text{o}}\) (\(\alpha\)) and PBS\({}_{\text{e}}\) (\(\alpha\)) represent two separated beams rotated by an angle \(\alpha\) relative to the \(x\)-axis. The sum of the electric field can be express as Equation (12), where R and T are the reflectance and transmittance of the coated plane mirror, respectively, Q is the transmittance of the optical cavity, and N is the order number of the backward reflected beam. \[\text{E}_{\text{sum}}=\Bigg{[}\text{BS}^{3}\cdot\sqrt{\text{R}}+\text{BS}^{3} \big{(}\sqrt{\text{T}}\big{)}^{2}\cdot\sum_{\text{N}=2}^{\infty}\big{(}\sqrt {\text{R}}\big{)}^{2\text{N}-3}\cdot\text{Q}^{\text{N}-1}\cdot\text{PF}^{ \text{N}-1}\cdot\text{QWP}(\theta_{2})^{2\text{N}-2}\Bigg{]}\text{QWP}( \theta_{1})\cdot\text{E}(\Phi) \tag{12}\]  
For PD\({}_{1}\) to PD\({}_{4}\), the corresponding electric fields of each incident beam are shown from Equation (13) to Equation (16).