## 4 Results and Discussions
The original RIs of liquids from the makers are used for the simulations. In the experiments, only the RIs of the immersed liquids are possibly different from the expected values due to quality change in stored period. Even if the traces of measurements and simulations are not completely alike, it is obvious that both are still close with good linearity. Actually, the measurements are repeatable in linearity and sensitivity. Especially, the variations of sensitivity are within 0.5% in each measurement under the same conditions. The achievable repeatability and linearity provide the sufficient evidence for the practical use. Figure 9 shows the measured results of the KTP plate immersed in the glycerin (\(n_{i}\) = 1.473) and the high refractive index oil (\(n_{i}\) = 1.7) using different incident angles. A good linearity is achieved only in the flattened region of the sensitivity curves. In the nonlinear region, the prompt correction factors shall be used to obtain the precision angle measurements based on the phase variation curves. The linearity error is defined as a ratio, half of maximum sensitivity difference versus mean value of sensitivity in the concern angle range. Therefore the linearity error is close to zero in the flattened region of the sensitivity curves. When the KTP plate is immersed in the glycerin, the values of measurement sensitivity are about 742 and 791 for the initial angles of 55\({}^{\circ}\) and 60\({}^{\circ}\), respectively. Since the system phase stability is around 0.01\({}^{\circ}\), the average sensitivity of around 766 can provide a measurement resolution of 1.3 \(\times\) 10\({}^{-5}\) with a wide range of 10\({}^{\circ}\) and linearity error of 0.032. By using the high refractive index oil, the values of measurement sensitivity are about 1439 and 1725 for the initial angles of 55\({}^{\circ}\) and 60\({}^{\circ}\), respectively. The measurement resolution can be further enhanced to 5.8 \(\times\) 10\({}^{-6}\) with an angle range of 5\({}^{\circ}\) and linearity error of 0.090. Figure 8: Comparisons of the MgO:LN and the KTP OASs under the different ranges of incident angle:  
(a) \(\theta i\) = 45\({}^{\circ}\)\(-\)50\({}^{\circ}\); and (b) \(\theta i\) = 50\({}^{\circ}\)\(-\)55\({}^{\circ}\). Figure 9: Comparisons of the KTP plate immersed in the different refractive indices of liquids under the different incident angles: (a) \(\theta i\) = 55\({}^{\circ}\)\(-\)60\({}^{\circ}\); and (b) \(\theta i\) = 60\({}^{\circ}\)\(-\)65\({}^{\circ}\). In the simulations, the phase variations only consider the optical path difference between the two orthogonal polarizations. As shown in Figure 7b, when the environmental RIs of over 1.473, the best incident angles for the highest sensitivity are larger than 65\({}^{\circ}\). In the experiments, the probing beam sizes of \(p\)- and \(s\)-waves are around 0.5 mm in diameter. The refracted angles of the \(p\)- and \(s\)-waves are different in the birefringent crystals. It will enhance walk off between the two orthogonal polarizations in the larger incident angles of over 65\({}^{\circ}\). The experimental results indicate that the reduced overlap between them can further decrease the sensitivity. Therefore, the incident angles smaller than 65\({}^{\circ}\) are used for repeatable measurements. Among the various measurement schemes [1; 2; 3; 4; 5; 6; 7; 8; 9], the measurement angle ranges and resolutions are all different, and most of the methods have a limited tradeoff relationship between them. It is desirable to have an accurate angle measurement resolution over a wide-angle measurement range. The measurement performance of this proposed birefringent-refraction (BR) methodology will be compared with other methods in the literature. The data of measurement resolution and linear angle range has been summarized and plotted on a performance map as shown in Figure 10. The horizontal and vertical axes are represented for the range and resolution, respectively.