## 1 Introduction  
Rotation angle measurement is essential in a variety of mechanical apparatuses. To achieve precision control for nanometer resolution, optical remote measurements have the advantage of being non-contact, and are easily performed using an optical interferometer. There are various optical methods of precision angle measurement, such as using surface plasmon resonance (SPR), total internal reflection (TIR), Fabry-Perot resonator, white-light interferometry, and imaging displacement sensing [1; 2; 3; 4; 5; 6; 7; 8; 9]. In optical metrology, the amplitude or phase of a reflected and refracted probe light is dependent on an incident angle in the optical transducer. In the SPR transducer, the amplitude and phase of the reflection \(p\)-wave (electric-field parallel to the incident plane) are abruptly changed at a resonance angle. In contrast, the amplitude and phase of the reflection s-wave (electric-field perpendicular to the incident plane) are only slightly changed. There are two schemes: intensity and phase interrogations, which have been proposed to explore angle measurement. The intensity interrogation approach is simple; however, the reflectance is minimal at the resonance condition. Only a small percentage of the incident photons are detected by the photodetector, which severely reduces the signal-to-noise ratio [10]. In addition, laser stability is also critical in obtaining precise measurements. In the phase interrogation approach, a common path polarization interferometry has been proposed to overcome the limited resolution resulting from the unstable intensity or weak signal of the probe light. The step-like measured signals can enhance measurement sensitivity, and require precise control of the initial incident angle [1; 5]. At the same time, the angle scanning range for maintaining a good linear relationship between the measured signal and the angle change is limited. Moreover, gold-film thickness and the probe wavelength are essential for attaining high sensitivity measurements [1; 2]. In contrast to the SPR transducers, the phase delay of an input with two orthogonal polarizations is dependent on the incident angle while over the critical angle during TIR conditions. The phase curves of the TIR transducers are smoother than those of the SPR measurements. The sensitivity is lower than the SPR, but with a wider range of scanning angles [4].  
It is well known that a birefringent plate (BP) has been widely applied for fabrication of wave plates and polarization splitters [11]. In addition, nonlinear birefringent crystals such as MgO-doped lithium niobate (MgO:LN) and potassium titanyl phosphate (KTP) are also important for the wavelength converters through nonlinear optics [12; 13]. Therein, MgO:LN is a uniaxial birefringent crystal with two different refractive indices. KTP is a biaxial birefringent with three different refractive indices. In this paper, we present a new optical angle sensor (OAS) for rotation angle measurement using a birefringent-refraction (BR) transducer performed in a common-path heterodyne interferometer. The OASs were made using the birefringent plates (MgO:LN and KTP) immersed in different liquids. The optical phase delay between two orthogonal polarizations (\(p\)- and \(s\)-wave) of the probe light is dependent on incident angle, the thickness of the birefringent plate and the refractive index (RI) of immersed liquids. In comparison with SPR and TIR transducers [1; 2; 3; 4; 5; 6], the proposed birefringent-type OAS does not require a precise setting of the specific angle. It can also provide sufficient sensitivity and wide dynamic measurements.