## 2 Measurement Principles  
When the incident light with two orthogonal polarizations passes through the birefringent BPs, the phase delay between them can be measured in the common path polarization interferometry. Phase measurements based on angle scanning methods have been proposed to measure the retardation parameters of the multiple-order wave plate [14], and the liquid crystal cell parameters (pretilt angle and the cell gap) [15]. The results of studies measuring retardation parameters show that phase delay is deeply dependent on crystal orientations [14].  
To study the characteristics of the proposed OAS, crystal orientations and the incident polarizations (\(s\)-wave and \(p\)-wave) are schematically shown in Figures 1 and 2 for the MgO:LN and the KTP plates, respectively. The \(s\)-wave is parallel to the \(n_{\text{e}}\)-axis of MgO:LN and the \(n_{z}\)-axis of KTP, respectively, as shown in Figures 1a and 2a. The \(s\)-wave is perpendicular to the \(n_{\text{e}}\)-axis and the \(n_{z}\)-axis, as illustrated in Figures 1b and 2b.  
Figure 1: MgO:LN plate for optical angle sensors: (**a**) \(s\)-wave parallel to \(n_{\text{e}}\)-axis; and (**b**) \(s\)-wave perpendicular to \(n_{\text{e}}\)-axis.  
Figure 2: KTP (potassium titanyl phosphate) plate for optical angle sensors: (**a**) \(s\)-wave parallel to \(n_{z}\)-axis; and (**b**) \(s\)-wave perpendicular to \(n_{z}\)-axis.  
The corresponding relationships of phase variation and the incident angle are represented by Equations (1)-(4):  
\[\phi_{LN1}=\frac{2\pi}{\lambda}\cdot t\cdot\left(\sqrt{n_{\varepsilon}^{2}-n_{i}^ {2}\cdot\sin^{2}\!\theta_{i}}-\sqrt{n_{o}^{2}-n_{i}^{2}\cdot\sin^{2}\!\theta_{i }}\right) \tag{1}\]  
\[\phi_{LN2}=\frac{2\pi}{\lambda}\cdot t\cdot\left(\sqrt{n_{o}^{2}-n_{i}^{2} \cdot\sin^{2}\!\theta_{i}}-\sqrt{n_{\varepsilon}^{2}-n_{i}^{2}\cdot\frac{n_{ \varepsilon}^{2}}{n_{o}^{2}}\cdot\sin^{2}\!\theta_{i}}\right) \tag{2}\]  
\[\phi_{KTP1}=\frac{2\pi}{\lambda}\cdot t\cdot\left(\sqrt{n_{z}^{2}-n_{i}^{2} \cdot\sin^{2}\!\theta_{i}}-\sqrt{n_{y}^{2}-n_{i}^{2}\cdot\frac{n_{y}^{2}}{n_{ x}^{2}}\cdot\sin^{2}\!\theta_{i}}\right) \tag{3}\]  
\[\phi_{KTP2}=\frac{2\pi}{\lambda}\cdot t\cdot\left(\sqrt{n_{y}^{2}-n_{i}^{2} \cdot\sin^{2}\!\theta_{i}}-\sqrt{n_{z}^{2}-n_{i}^{2}\cdot\frac{n_{z}^{2}}{n_{ x}^{2}}\cdot\sin^{2}\!\theta_{i}}\right) \tag{4}\]  
where \(\lambda\) is a wavelength of the incident light, \(t\) is a thickness of the BP, \(n_{i}\) is a RI of the immersed medium and \(\theta i\) is an incident angle. The refractive indices of extraordinary and ordinary are represented by \(n_{e}\) and \(n_{o}\) (\(n_{e}\) = 2.203 and \(n_{o}\) = 2.286 at 632.8 nm) in the MgO:LN [16]. Three different refractive indices, \(n_{x}\), \(n_{y}\), and \(n_{z}\) (where \(n_{x}\) = 1.7619, \(n_{y}\) = 1.7712 and \(n_{z}\) = 1.8648 at 632.8 nm), are used for the KPT [17]. \(\phi_{\mathrm{LN1}}\) and \(\phi_{\mathrm{LN2}}\) are the phase variations of the \(s\)-wave applied parallel and perpendicular to the ne-axis of MgO:LN, respectively. \(\phi_{\mathrm{KTP1}}\) and \(\phi_{\mathrm{KTP2}}\) are the phase variations for the \(s\)-wave applied parallel and perpendicular to the \(n_{z}\)-axis of KTP, respectively.