## 4 Results and Discussions  
Figure 5 shows the simulation results of phase change versus incident angle for the two crystal orientations and the different RIs of mediums with incident angles ranging from 45\({}^{\circ}\) to 65\({}^{\circ}\), according to Equations (1) and (2). The solid and empty symbols of the simulated curves represent the \(s\)-wave parallel and perpendicular to the \(n_{\mathrm{e}}\)-axis, respectively. The results show that in both cases phase changes are increased with a corresponding increase in the RI of the medium with the same crystal orientations. At the same RI, the phase change is higher when the \(s\)-wave is parallel to the \(n_{\mathrm{e}}\)-axis of MgO:LN plate. The phase difference between the two crystal orientations (Figure 1a,b) becomes evident as the RI of the environment is increased. Similar trends are obtained at the KTP plate (not shown), and having the \(s\)-wave parallel to the \(n_{\mathrm{z}}\)-axis can also result in higher phase variations. Therefore, it can be concluded that both crystal orientation and the RI of the environment are key parameters in enhancing angle measurement sensitivity. Figure 6a shows the relationships between phase variation and incident angle using various mediums with different RIs for the MgO:LN OAS. The quasi-linear regions of the phase curves are  
Figure 4: (**a**) MgO:LN plate immersed in glycerin medium (\(n_{i}\) = 1.473). (**b**) KTP plate immersed in glycerin medium. (**c**) KTP plate immersed in high refractive index oil (\(n_{i}\) = 1.7). Figure 5: Phase change vs. incident angle for the different orientations of the MgO:LN plate and the different refractive indices of mediums. dependent on the incident angle. This indicates that the good linearity between phase change and incident angle has a limited dynamic range. A slope (phase change vs. angle change) is defined as sensitivity, which gradually increases as the RI of mediums in the linear region are increased. Sensitivity versus angle curves are shown in Figure 6b. The sensitivity curves have maximum values at some specific angles, which in turn are dependent on the RI of mediums. The flattened region of the curves becomes smaller as the RI of mediums is increased. In comparison with the MgO:LN OAS, the phase variation and the sensitivity of the KTP OAS are greatly improved (see Figure 7). By using the high refractive index oil (\(n_{i}\) = 1.7), the maximum values of sensitivity are 525 and 3072 for the MgO:LN and KTP OASs, respectively. Figure 8 presents the measured results (solid symbol) of phase variation versus incident angle for the MgO:LN and the KTP plates immersed in the glycerin (\(n_{i}\) = 1.473) at different initial angles. By utilizing the MgO:LN plate, the values of measurement sensitivity are about 204 and 209 for the initial angles of 45\({}^{\circ}\) and 50\({}^{\circ}\), respectively. In the case of the KTP plate, the sensitivity values are about 557 and 676 for the initial angles of 45\({}^{\circ}\) and 50\({}^{\circ}\), respectively. In the simulations (empty symbol), the equations do not consider the cell parameters, including the curvature and refractive index of the cylindrical glass. There is a lens effect that causes the beam overlap difference between two orthogonal polarizations for the different RIs of the mediums. The lens effect is one of the reasons to cause the difference between the simulations and experiments. There are other possible reasons for this difference. According to the specifications provided by the makers, the errors of the thickness and RI of the BPs are less than 1%. These minor errors are not the key factors that cause huge difference. Figure 6: Simulations for MgO:LN OAS (optical angle sensor): (**a**) phase change vs. incident angle; and (**b**) sensitivity vs. angle for different refractive indices of environment. Figure 7: Simulations for KTP-OAS: (**a**) phase change vs. incident angle; and (**b**) sensitivity vs. angle for different refractive indices of environment.