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An angular displacement measurement sensor with high resolution for large range measurement is presented. The design concept of the proposed method is based on the birefringence effect and phase detection of heterodyne interferometry. High system symmetry and simple operation can be easily achieved by employing an innovative sandwich optical design for the angular sensor. To evaluate the feasibility and performance of the proposed method, several experiments were performed. The experimental results demonstrate that our angular displacement measurement sensor can achieve a measurement range greater than 26°. Considering the high-frequency noise, the measurement resolution of the system is approximately 1.2° × 10−4. Because of the common-path arrangement, our proposed method can provide superior immunity against environmental disturbances.
© 2016 Optical Society of America
1. Introduction
Precision measurement of angular displacement plays various crucial roles in modern manufacturing systems and equipment [1]. In particular, high-resolution measurement of large-range angular displacement is highly demanded [2]. Several optical angular sensors (ASs) are widely employed for angular displacement detection because they can provide high measurement sensitivity, high resolution, and high accuracy. The measurement technique in each optical AS has developed with its own merits. For example, the commercial autocollimator can be applied to dynamic or static small-angle detection, precision alignment, verification of angle standards, and various other processes [3,4]. Optical rotary encoders can provide the angle measurement with high resolutions and low uncertainties, they have been widely applied in science research and industry applications, such as imprinting fabrication [5] and 3D printing [6]. In recent years, numerous studies have investigated angular displacement sensing systems according to various effects, including the total internal reflection [7–9], surface plasmon resonance [9,10], two-beam interference [11,12], multibeam interference [13], and birefringence effects [14–16]. These sensing systems can convert the angular displacement information into the optical phase variation, which can be detected using the optical interferometer. The high sensitivity of the angular sensing system integrated with the interferometer can offer a high measurement resolution for the angular displacement. However, these techniques can only provide a small measurement range; a trade-off typically exists between a large measurement range and high sensitivity. The birefringence effect can reduce the need for this trade-off. The variation of the phase difference between the two orthogonal polarization states of the light beam is a function of the birefringence, propagation distance, and direction of the light beam in the birefringent crystal. Therefore, we can select a suitable crystal thickness and birefringence to build an angular sensing system with a large measurement range and high sensitivity.
According to the birefringence effect, an angular displacement sensor with high sensitivity and a large measurement range was developed in this study. The sensor consisted of two prisms and a calcite plate, and it was designed as a sandwich. The phase detection technique employed in our previous works, heterodyne interferometry [17,18], was used to detect the phase variation resulting from the AS. The angular displacement could be determined according to the birefringence characteristics of the sensor and the measured phase variation. On the basis of the experimental results, our system retained a maximum measurement range of 26°. Considering the high-frequency noise, the measurement resolution of the system was approximately 1.2° × 10−4.
2. Principle
In this section, we start by introducing the phase difference variation of an s- and p-polarization light beam, resulting from the birefringent crystal. According to the characteristics of the light beam phase difference variation, the design of the AS is explained. Finally, a heterodyne interferometry process for measuring the phase difference is briefly introduced.
2.1 Phase difference of s- and p-polarization in the birefringent crystal
As shown in Fig. 1(a), a linearly polarized light beam comprising s- and p-polarization states passes through the birefringent crystal with an incident angle of θi. Here, we assume that the optical axis of the birefringent crystal is set along the surface of the crystal. When the light beam enters the crystal, according to the theory of birefringence [19,20], the phase difference δ between the s- and p-polarization states can be expressed as
andwhere the suffixes x and y represent the optical axis along x- and y-axes, respectively. λ, d, no, ne, and np are the wavelength of the light beam, thickness of the crystal, ordinary refractive index, extraordinary refractive index, and surrounding refractive index, respectively.
Fig. 1 (a) A linearly polarized light beam passes through a birefringent crystal with an incident angle of θi. (b) Relationship between the phase difference (δx, δy) and the incident angle θi with different surrounding refractive indices np. The slope variation in the interval (−5° < θi < 5°) was approximately 30 °/°.
A procedure to determine the suitable parameters for the crystal for measuring the angular displacement should be performed. In this study, a 10.0-mm-thick calcite plate was employed to simulate the relationship between the phase difference and the incident angle. The ordinary and extraordinary refractive indices of the calcite crystal at a wavelength of 632.8 nm are 1.5427 and 1.5518, respectively. The simulation results for two group curves (δx and δy) are presented in Fig. 1(b).
Evidently, the phase difference variation of δx was greater than that of δy. In addition, when the surrounding refractive index np was near the refractive index of the crystal (no or ne), the phase difference variation increased. Figure 1(b) also demonstrates that, as the incident angle increased, the slope (dδx/dθi or dδy/dθi) of the curve also increased. The high slope indicated that the sensitivity of the angular displacement was high.
2.2 Design of the AS
According to the calculations provided in the previous section, to obtain higher measurement sensitivity, the phase difference δx was detected, rather than δy, while the optical axis of the crystal was set along the x-axis. In addition, we used the prisms as the surrounding material to couple the light beam into the crystal, as displayed in Fig. 2. The refractive index (np) of the prisms should be close to the crystal, and a prism with a greater angle α can be chosen so that the light beam could enter the crystal with a higher incident angle (θi). According to Snell’s law, the incident angle (θi) at the interface between the prism and crystal can be expressed as
where θ represents the incident angle at the interface between the air and prism, regarded as the angular displacement in our experiments. When the light beam normally enters the prism (θ = 0), it passes through the crystal at an angle of α. A greater angle of α of the prism provides higher measurement sensitivity. However, considering the geometrical size of the crystal plate, a greater angle of the prism limits the measurement range of the angular displacement. Because the equilateral triangle prism is especially common, and the angle of the prism α = 60° can provide sufficiently high sensitivity, it was used in our experiment. After substituting the incident angle of θi (Eq [3].) into Eq. (1), the relationship between the phase difference (δx) and the incident angle (θ) can be obtained; it is expressed as follows:As displayed in Fig. 3, here we use the birefringence parameters of the calcite crystal to estimate the phase difference trends. Clearly, a higher surrounding refractive index provides a higher slope for the curve. Compared with the bare birefringence crystal, when the crystal integrated with the designed prisms, the measurement sensitivity increases by 1000 times around the condition of the normal incident. The technique of heterodyne interferometry was used to detect the phase difference variation (δx), and the angular displacement (θ) was subsequently determined by employing the curves shown in Fig. 3, clearly demonstrating the relationship between the phase difference and incident angle. The δx in Eq. (4) is the phase difference of the light resulting from the designed AS. The Jones matrix of the AS can be expressed as
Fig. 3 Relationship between the phase difference and incident angle (θ) with different refractive indices (np) of the equilateral triangle prism.
2.3 Heterodyne interferometry for phase detection
In our previous works, we have proposed several methods for measuring optical phase variation resulting from grating displacement [17,18] or surface plasmon resonance [21] through heterodyne interferometry. Here, the angular displacement measurement system was developed by combining a heterodyne detection technique with an innovative sandwich optical design concept. The schematic diagram of the proposed optical arrangement is displayed in Fig. 4; when a linearly polarized laser beam passes through the electro-optic modulator (EOM), which is modulated by a sawtooth signal with the half-wave voltage amplitude [22] at the angular frequency of ω, an optical frequency difference (ω) occurs between the p- and s-polarized components. This light beam is regarded as the heterodyne light source, and the Jones vector [18] can be written as
After passing through the beam splitter (BS), the heterodyne light beam is divided into the reflected and transmitted parts. The interference of the p- and s-polarized components of the reflected light occurs at the transmittance axis of the analyzer ANr and is received using a photodetector (Dr). The intensity (Ir) measured by (Dr) is regarded as the reference signal and can be given aswhere ANr (45°) is the Jones matrix of the analyzer with a transmission axis of 45°. The transmitted light from the BS passes through the AS, analyzer ANt (45°), and finally the photodetector (Dt). The intensity obtained using Dt is regarded as the measurement signal and can be written asThe reference and measurement signals are sent into a lock-in amplifier (LIA), and the phase difference (δx) can subsequently be obtained. The angular displacement (θ) can be determined by accounting for the measured phase difference and the relationship between the phase difference and angle in Eq. (4) and Fig. 3. The sandwich optical design of the AS promotes measurement system symmetry, and the photodetector (Dt) can receive the light beam without changing the position, even if the AS is rotating.3. Performance test
3.1 Experimental setup
The optical setup for measuring angular displacement in our experiment is shown in Fig. 4. The AS is also displayed in the photo. The AS consisted of a calcite plate with a diameter of 25.4 mm and thickness of 10.0 mm, and two equilateral triangle prisms composed of Schott F2 glass [23]. The refractive index of the prisms used in this study was 1.617. The AS was mounted on the precision rotation stage. Here, we define the counterclockwise rotation as a positive angle. A Ne–He laser with a wavelength of 632.8 nm was modulated using the EOM (Thorlabs EO-PM-NR-C1) with a frequency of 16 kHz to obtain a heterodyne light source. The reference and measurement signals from the photodetectors Dr and Dt were sent to the PC-based LIA (LabVIEW programing) for phase difference (δx) measurement. The angular displacement (θ) can be determined by the measured phase difference (δx).
3.2 Phase–angle (δ–θ) characteristic curve
Equation (3) and Fig. 3 indicate that the relationship between the phase difference and the angular displacement was nonlinear. The algebraic or numerical method could be used to find the solution to the nonlinear equation. However, these methods are time-consuming in the context of a measurement system. Therefore, we used the fitting function method to determine the relationship between the angular displacement and the measured phase difference. First, we developed the polynomial function by fitting the characteristic curve of the measured phase difference with respect to the angular displacement. The rotation stage drove the AS to rotate from −23° to 3° with the angle of 0.02° per step while the phase difference was recorded. The input rotation angles and the measured phase differences were used to progress the fitting function and were subsequently coded in our PC-based program to determine the angular displacement. The characteristics of the phase difference and angular displacement are provided in Fig. 5. To demonstrate the feasibility and performance of our proposed method, experiments were conducted by assessing large angular displacement measurement and repeatability.
3.3 Experiment for large angular displacement measurement
To determine whether the proposed method could measure large angular displacement, the precision rotation stage (PI, M-660.45, rotation range: 360°; resolution: 4 μrad) was prepared to perform forward and backward rotation motions with various ranges. Three angular displacement motions for ± 5°, ± 10°, and ± 15° traveling ranges were executed. Figure 6 shows the angular displacement curves of the measurement results. The measurement curves obtained through our proposed method exhibited nearly the same behavior as the commanded angular displacements did. The relative deviations between the measured and commanded angular displacements were approximately 1%. We suspect that the discrepancy may have resulted from the stack-up error of the rotation stage. This problem was effectively improved by using our servo control technique, as demonstrated in the next experiment.
3.4 Experiment for repeatability of angular displacement measurement
Repeatability, a major performance index of any measurement system, is defined as the standard deviation of the measured position error between the starting position and the final position after the rotation stage has moved both forward and backward. A smaller deviation of the measured position error indicates that the measurement system possesses higher system repeatability. To test the repeatability of our proposed method, the rotation stage was operated in a closed-loop mode with the servo control technique to perform forward and backward movements with step angular displacements, subsequently returning to the starting point. As can be seen in Fig. 7, the experimental results obtained by our method and the commanded angular displacements are in agreement with each other. Moreover, the mean values of the starting and ending points obtained through our method both were approximately 1° × 10−4 (see the inset plot), respectively. This clearly demonstrated that our method retained high repeatability.
4. Discussion
4.1 Sensitivity, stability, and resolution
The measurement sensitivity (s) of our proposed method can be defined as the differential of Eq. (4) with an incident angle of θ, s = dδx/dθ. After substitution of the experimental parameters into the differential of Eq. (4), the measurement sensitivity could be obtained (Fig. 3). As displayed in Fig. 3, the phase difference (δx) and sensitivity (s) were both nonlinear. In the operation range (θ = −23° ~3°), the sensitivity was over 1000°; notably, at the normal incidence, it was greater than 3000°. According to the experimental setup, the minimum detectable phase resolution in our LabVIEW program was approximately dδx = 0.1°. Therefore, in the case of the conservation estimation, the measurement resolution of the angular displacement was approximately dθ = dδx/s = 1° × 10−4 in the operation range. Meanwhile, the measurement resolution will be approximately 1° × 10−5 around normal incidence.
To test the stability of our method, the variation of the phase difference was detected when the rotation stage was held stationary. The measured variation of the phase difference for the stability test is shown in Fig. 8; as evidenced, the detected low phase drift was approximately 2°, and the high-frequency noise was approximately 0.4°. As can be found from the result, the environment actually affects the stability of the system. Not only the refractive indexes of the material (calcite plate, glass and air), but also the mechanical configuration drifted with the environmental temperature. According to the dispersion of the dielectric materials and the thermal expansion coefficients of the mechanical configuration, we believe that the main source of measurement instability is slow drift of the mechanical configuration, such as the thermal drift or the mechanical vibration. For example, the relative position or the angular displacement between the laser beam point and the optical components is slow drifting in the test period. This problem can be suppressed in an ideal experimental environment. On the other hand, the phase measurement resolution and accuracy of the lock-in amplifier is much more stable. However, the high-frequency noises generated inside the system components, such as the laser source, electronic noise, photodetector, and DAQ card, were inevitable. Considering only high-frequency noises, the measurement resolution of our system was approximately 1.2° × 10−4.
4.2 Limitation of measurement speed
Measurement speed is among the most crucial features of any measurement technique. In our current system, the limitation of measurement speed was determined according to the performance of the PC-based LIA. Even if the wrapped phase appeared in the heterodyne signal, the PC-based program could successfully determine the continuous optical phase variation. The high-rotation speed AS resulted in the high variation rate of the optical phase (dδx/dt). When the wrapped phase rate is higher than the output data rate (fODR) of the PC-based program, missed data cause an error in the measured phase, affecting the measurement results of angular displacement. To prevent the data from missing, the variation rate of the optical phase should be smaller than the output data rate, that is dδx/dt = s × dθ/dt<π × fODR. The fODR is about 26 Hz in our system [24]. If considering the average of the measurement sensitivity 2000 (°/°), the theoretical measurement speed is about dθ/dt ~π fODR/s ~2.3°/sec. Therefore, to test the limitation of the measurement speed of our system, we drove the rotation stage forward and backward by using four speeds (0.88°/s, 1.34°/s, 1.80°/s, and 2.30°/s) and simultaneously measured the phase variation of the AS. The experimental results are provided in Fig. 9. The traveling range of the angular displacement was approximately 1.4°, and the phase variations were approximately 4320°. However, the missed data clearly resulted in faults in the phase unwrapping process when the rotation speed reached 2.30°/s. Therefore, the maximum measurement speed we could achieve was approximately 2°/s.
4.3 Measurement error analysis
In our measurement system, the light beam was parallel to the optical table, and the pitch angle of the AS was 0°. However, when there is a misalignment angle (incorrect pitch angle) between the AS and light beam, the measurement result is influenced. This type of misalignment is equivalent to that of the optical axis of the crystal deviating from the x-axis (rotation axis). Assuming the deviation angle is ϕ, the phase difference in Eq. (4) can be modified as
With identical parameters for the AS and the different deviation angle ϕ, the relationship between the relative deviation of the phase difference [δ (ϕ) − δ (0°)]/δ (0°) and the incident angle θ is illustrated in Fig. 10. This figure indicates that the relative deviation of the phase difference increased with the deviation angle ϕ. In our current setup, the deviation angle ϕ could be adjusted to under 0.1° by careful alignment so that the relative deviation of the phase difference could be smaller than 2 × 10−5. This relative error was attributable to system error and could be improved through a calibration procedure.Notably, owing to the effects of the polarization and frequency mixing of the interference beams and the misalignment of polarization components, the heterodyne detection technique was also subject to the nonlinear periodic error. Our previous work indicated that the typical periodic optical phase error resulting from the polarization and frequency mixing effects was approximately 1.5° [25]. The relative error of the angular displacement resulting from the nonlinearity periodic optical phase error was the same as in the pitch angle deviation of the AS.
5. Conclusion
An angular displacement measurement based on the birefringence effect and the heterodyne interferometer was presented in this study. The optical phase variation resulting from the rotation of the AS was detected using the optical heterodyne technique. The angular displacement could then be determined from the measured optical phase variation. Compared with the bare birefringence crystal, when the crystal integrated with the designed prisms, the measurement sensitivity increases by 1000 times around the condition of the normal incident. The experimental results demonstrate that our proposed method can be used to measure long-range angular displacement with high resolution while maintaining high system stability. Furthermore, these results show that the measurement resolution and speed can reach 2° × 10−4 and 2°/s, respectively.
Acknowledgments
This study was supported by Ministry of Science and Technology (MOST), Taiwan, under the Grant Nos. MOST 103-2221-E-008 −066 -MY3 and MOST 104-2221-E-011-116.
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