Open Access
Add to BibSonomy
Abstract
An optical refractive index sensor used for underway seawater salinity monitoring is proposed. Due to the empirical relation to salinity, refractive index measurement provides an alternative solution to obtain salinity of seawater. We developed a compact refractive index sensor based on total internal reflection (TIR) method. Through the repeatability and stability experiment and temperature correction, the performance of the sensor has been demonstrated experimentally. To evaluate the applicability of the sensor under real turbid sea conditions, field performance of the TIR sensor has been tested on an oceanographic cruise in the eastern of Yangtze Estuary in July 2017. The underway monitoring results show good correlation with the results provided by commercial CTD profiler.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Salinity is one of the most important parameters in oceanography. Measuring the salinity of seawater can reflect the state of the Marine environment. In 2009, the most recent definition of thermodynamic equations of seawater (TEOS-10) is proposed. The new definition emphasizes the assessment of absolute salinity (SA), defined as the ratio of the mass of dissolved material to the mass of seawater [1]. Until now, most of the salinity commercial sensors are based on conductivity ratio measurement. These sensors, such as the famous Sea-Bird CTD profiler (CTD stands for conductivity, temperature, depth), need a highly reliable pump to flush water through the conductivity cell at a constant rate and then can be applied to the underway monitoring. It is sensitive to the water velocity, and make the system complicated and difficult to miniaturize [2]. Besides, conductivity is 10 times more sensitive to temperature than to refractive index [3]. Also, according to requests of the TEOS-10, non-conductive part of dissolved material in seawater cannot be taken into account by the conductivity CTD [4].
Under this circumstance, refractive index (RI) measurement provides an advantageous alternative solution [5]. In oceanography, according to the relation of Lorentz-Lorenz, refractive index is the core parameter which is closely correlated with the density. Therefore it can be related to the absolute salinity [6]. Thus, accurate measurements of the seawater refractive index have provoked particular interest of ocean optics researchers. According to the past studies on the relation between refractive index and salinity, the significant features can be summarized as follow: (1) the rate of refractive index change with salinity is 2 × 10−4 refractive index unit (RIU) per 1‰ of salinity at constant temperature, where salinity is expressed in grams per kilogram of seawater, that is, in parts per thousand, (2) the refractive index decreases with the temperature under constant salinity and the refractive index varies with the temperature at the rate of 5 × 10−5 RIU per 1°C at 2–3°C and 1 × 10−4 RIU per 1°C at around 20°C irrespective of the salinity [7]. The equation for the refractive index of seawater is a function of pressure, temperature, salinity and wavelength of light used [8–10].
Multiple optical methods can achieve refractive index measurement. Only few of them can be implemented to sea trail environment. For example, the fiber interferometer structure has the benefit of high sensitivity. But it is inadaptable in harsh field measurement conditions and mostly used in laboratory [11]. Over the past years, based on the empirical relation between refractive index and salinity, many optical salinity sensors have been developed [12–15]. Zhao proposed an optical fiber sensor based on the detection of beam deviation, and tested the performance in laboratory environment [16]. A fiber-optic refractive index sensor based on surface plasma resonance is developed by N. Díazherrera et al, and the in situ measurement has been taken in the Baltic Sea [12]. Menn proposed a solution based on small beam deviation measurements by twin-prism refractometer and tested at sea to a depth of 2000m during an oceanographic cruise [17]. Most of these present methods are mainly based on the measurement of the beam deviation due to the refractive angle change. The devices developed by this method are designed as transmission optical path, which means, when the seawater in the optical path is turbid, the scattered light beam caused by the plankton and suspended sediment in turbid medium is likely to impact the measurement of light beam deviation, the measuring result may be obstructed [18–20]. Owing to river silt, the seawater is likely to be turbid in the estuary of river [21]. Thus, the traditional optical methods are susceptible to interference of turbidity in estuary area.
A mature method of measuring the refractive index can be applied to reduce the influence of turbid seawater. By getting the angular reflective curve from the total internal reflection (TIR), the refractive index can be carried out after analysis, which is called the TIR method. The device developed by the TIR method can be designed as an internal optical structure, which can reduce the influence of the turbid seawater and the algae in the marine environment. Besides, unlike the beam deviation measurement, it is conceivable to get more information of the seawater with the TIR method, such as absorption coefficient [22–27]. In addition, there are many researches reported improving the measurement of the TIR method. Guo, in our group, applied the differentiation algorithm in a two-reflection critical angle refractometer to improve the accuracy of RI measurement [27]. Augusto demonstrated an angle measuring method with a maximum sensitivity close to the diffraction limit [28]. Ye used the centroid method and Fourier analysis to expand the measurement range of the critical angle refractometer [29,30]. These are theoretical basis we applied the TIR method in seawater in situ measurement. The TIR method has the benefits of robustness in sea trail environment and potential of multi-parameter measurement. Hence, it is significant and necessary to study the measurement of seawater salinity by critical angle method.
In this paper, we applied the TIR method in a compact structure, developed a TIR salinity sensor. We tested the stability of the device in relatively constant conditions and verify the reliability of this TIR sensor. In its development, special attention has been given to the impact of temperature. And correction coefficient has been summarized. In order to evaluate the applicability and robustness of the sensor in the estuary area, the TIR sensor was deployed on an oceanographic cruise in the eastern of Yangtze Estuary in July 2017, showing good correlation with the results provide by a commercial CTD profiler. Based on these results, we achieved some meaningful progresses in the field of ocean field measurement. With the compact TIR sensor, we realized underway measurement of seawater and anti-interference measuring to the turbid environment.
2. Description of the instrument
2.1 The Transform formula
The empirical relation between salinity and the refractive index is well established. Several formulas have been proposed. Among them, considered the deployment environment, we used that of the Quan-Fry formula [9]:
Here, S is the salinity in ‰, T is the temperature in degrees Celsius and λ is the wavelength in nanometers. The coefficients have the following values: n0 = 1.31405, n1 = 1.779 × 10−4, n2 = −1.05 × 10−6, n3 = 1.6 × 10−8, n4 = −2.02 × 10−6, n5 = 15.868, n6 = 0.01155, n7 = −0.00423, n8 = −4382, n9 = 1.145 × 106. With this algorithm, we can convert salinity result into refractive index and vice versa. As the wavelength is constant, the relationships between refractive index, salinity and temperature at atmospheric pressure can be expressed as Fig. 1.
Fig. 1 Theoretical variations in refractive index versus: (a) salinity for different temperature, (b) temperature for different salinities.
2.2 TIR Method
Investigation of the total internal reflection (TIR) phenomenon is widely used in the detection of the RI [29,30]. By measuring the critical-angle of the TIR, the RI of the sample can be observed. The device developed by this method is called critical-angle refractometer, such as the Abbe refractometer. According to Snell law and the Fresnel reflective principle, the relation of the RI and the critical-angle can be expressed as Eq. (2),
where nglass is the RI of the prism and α is the critical-angle of the sample. To get the precise critical angle, the angular reflective curve is prerequisite. According to the Fresnel reflective law, the angular reflection at the interface between the prism and the liquid can be deemed as a function of the incident beam angle, which is the angular reflective curve. The critical angle can be summarized by differentiation of the curve [31].2.3 System setup
Based on the principle of total reflection and the hydrographic surveys environment, we designed the TIR sensor as shown in Fig. 2. In the optical system, an angular point laser source, a triangular prism, a field aperture and a charge-coupled device (CCD) were used. The angular point laser source is established by making the beam of a red laser diode with an emitting wave-length of 635nm coupling into a single mode fiber with a core diameter of 4μm. The divergence angle of the point laser source is 14.94°, as the numerical aperture of the single mode fiber is 0.13. The triangular prism, made of K9 glass (1.5150 RIU at 635nm), is designed to make the light beam incident on the dielectric interfaces at the angle of 60°. The field aperture is placed to limit the angle range of the reflected light. As shown in Fig. 2, the beam from the angular point laser source is incident on the bottom of the triangular prism with the incident angel range of 55.08° to 64.92°. The light rays with incident angle smaller than the critical angle cause part reflection and transmission into the liquid sample, whereas others cause total internal reflection based on Fresnel reflective law. The reflected light pass through the field aperture and the reflected angle range is limited to 61.28° to 62.8°. Afterward, the reflected beam is received by the CCD camera system. The resolution of the CCD is 1600*1200 (in pixels). The corresponding RI measuring range is from 1.328 to 1.346 RIU. The image is superimposed on line and divide by the system background (collected in air), and is summarized the reflectance curve. Differentiating the reflectance curve, the pixel position of the maximum differential value is the critical angle position we need. According to Eq. (2), the refractive index of the measuring sample can be obtained.
Also, the TIR sensor has been containerized in order to make it usable for in situ measurement in oceanography. Based on the optical structure, the light path is completely inside the sensor, which effectively reduces the impacts of algal aggregation and turbidity of the seawater on the light path. Compared with the common optical salinity sensor, the TIR sensor is better adapted to the underway measurement environments.
3. Experimental results
3.1 Calibration results
Based on CAR theory and the optical structure, it’s easy to find out that there is a one-to-one correlation between the pixel position and the incident angle. With the Eq. (2), the critical Angle position-RI curve can be built directly. We summarized the relation between pixel position and the RI, shown in Fig. 3.
Fig. 3 Calibration curve: through calibration of 12 sets of data, a fitting curve with determination coefficient of 0.9975 is obtained.
In the experiment, an Abbe refractometer with the accuracy of 0.0002 RIU is used to calibrate the saline solution. From the calibration results, the resolution of the TIR sensor is 1.17 × 10−5RIU, which means one-pixel deviation of the critical-angle position represent the refractive index change of 1.17 × 10−5 RIU. And the coefficient of determination is 0.9975.
After the calibration, the results of RI have been obtained by locating the pixel position of the critical Angle. Finally, combining with the temperature data obtained by the temperature device inside the TIR sensor (Maxim, DS18B20), this profile has been expressed in salinity by inverting the Quan-Fry formula. And equivalent sensitivity in the salinity is nearly 0.064 g/kg. As we can see, range and precision of the device are highly depended on the geometrical factors of the sensing cell as well as of a resolution of the detection system. Obviously, the resolution can be greatly improved with a high precision camera system.
3.2 Repeatability experiment
After calibration of the measurement, repeatability experiments were carried out to examine the repeatability of the sensor. As shown in Fig. 4, three different seawater samples were measured under an invariant conditions, each sample was measured for 150 times continuously. The refractive index of each sample measured by Abbe refractometer is 1.3332 RIU, 1.3374 RIU and 1.3400 RIU, respectively.
The analysis results are shown in Table 1. From the results, the repeatability of the salinity sensor in refractive index unit is ± 6 × 10−5(RIU), which indicated a salinity repeatability of ± 0.3‰.
Table 1. Repeatability analysis results
3.3 Stabilization experiment
In order to examine the stabilization of the device, a long-time stabilization experiment was conducted. In case that salinity and temperature are invariable, the outputs of the measurement process are monitored. The temperature was kept in 23°C within ± 0.5°C. The salinity of the sample is 22.4 g/kg at 24.1°C. And the equivalent refractive index calculated with the Quan–Fry formula is 1.336675 RIU. The test result of the TIR sensor is shown in Fig. 5. The sensor measures and exports the result every 10 seconds. And the total monitoring time is 10.5 h.
Fig. 5 (a) Measuring results of the TIR sensor together with the temperature variation during the whole 10.5 hours. (b) Stability experiment results
From the monitoring results, we can see that the measuring results of RI change with the temperature at the first 5 hours. With the temperature slight arise from 22°C to 23.5°C, the refractive index drop from 1.336725 RIU to 1.33665 RIU, which shows a good response to the temperature stabilizing process. After the necessary time for stabilizing the temperature had passed, the temperature was kept stable at 23.5°C, the stabilization experiment was started. As shown in Fig. 5, a standard deviation of 4.78 × 10−6 RIU in measuring results is obtained after 5.5 hours long continuous operation. The main contribution to such deviation is believed to be due to the fluctuation of the light field, which can affect the definition of critical angle position.
3.4 Temperature variation experiment
As we know, changes in temperature do change the density and salinity, thereby changing the refractive index of seawater as well. Considered the refractometer behaving in real oceanic environments, the critical angle position is the impact of temperature changes. From the Eq. (2), the coefficient dn/dT can be determined as
Here, dnliquid/dT represents the variation of the measured liquid sample to the temperature. dθ/dT is the outcome of the differential calculus of the critical angle measuring result variation. As the refractive index of the glass also varies with the temperature, there exists a difference between the theoretical variation and the actual results detected by determining the critical angle. Thereby it’s necessary to determine the correction coefficient of the temperature variation.We tested the effect on the refractive index measurement in a thermostat water bath in the laboratory. The temperature variation ranged from 9°C to 26°C, which cover the main range of sea surface temperature during the summer. Four kinds of artificial sea salt solution, configured by artificial sea salt (Red sea coral pro salt) and deionized water, have been tested. The concentration of the solution is 5.05g/200ml, 6.0g/200ml, 7.03g/200ml and 8.02g/200ml, respectively. The results are shown in Fig. 6 together with the theoretical curve calculated by the Quan-Fry formula.
From Fig. 6, the difference is existing but not too significant in the two curves. It’s necessary to correct this deviation caused by the temperature variation in the subsequent processing. According to the test results of these four solutions, the relation between deviation and temperature can be approximately defined as a linear function. The correction formula can be written as:
Where nc is the corrected refractive index value, n0 is the measured value, α is the correction coefficient, which is 1.4 × 10−5 RIU/°C. T and T0 are the measured temperature value and the temperature of the calibration experiment, respectively. In this measurement, T0 is 20°C. After the correction, the measuring error caused by temperature is negligible compared to the general measurement error.4. Application for sea trial
We have tested the device in real in situ environment. The refractometer device was deployed from July 20th, 2017 on an oceanographic cruise (about 50 sites) in the eastern of Yangtze Estuary, where the seawater is turbid on account of the Yangtze River (see Fig. 8). The measuring campaign was held on the Runjiang2 oceanographic ship as a part of the NSFC project. Underway monitoring has been made during the cruise. For the underway monitoring, the device was settled on the water tank, as shown in Fig. 7(a). The details of the TIR sensor and the CTD profiler are demonstrated in Fig. 7(b). The tank was filled by the seawater pumped from the shaft in the middle of the ship. In order to validate the measuring result, a commercial CTD (Seabird SBE-37SI) was settled on the same water tank. The sampling rate of the CTD has been set to 0.1 Hz in accordance with the refractometer. The seawater pumped from the shaft in the middle of the ship and filled into the water tank, and measured by both TIR sensor and CTD. Finally the seawater got out through the outlet pipe. Hence these two data sets can be a one-to-one correspondence on the timeline to permit the comparison.
Fig. 7 (a) Photograph of settled environment of the TIR sensor. (b) Picture of the TIR sensor and CTD profiler.
Fig. 8 Map of the Yangtze Estuary area, the point and the line shows the path of the underway monitoring experiment.
We proposed the measuring data in Fig. 9, together with the voyage where the measurement was taken. For the whole voyage lasted thirteen days, we chose the typical data to propose. The measuring environment of Fig. 9(a)-9(c) is the most turbid area during the voyage. Compare with the commercial CTD, the measurement results are consistent, which indicate the robust ability to the turbid environment. In Fig. 9(a), our voyage started from Zhoushan port and sailed northwards. The result shows the salinity decreases gradually as we are getting close to the Yangtze Estuary. In Fig. 9(b), we enter the Yangtze Estuary, and the salinity decreases to zero and hold. The results shown in Fig. 9(b) reflect the veracity of measurement in low salinity area. Then we go out of the Yangtze River and continue north. The salinity experienced a change from low to high as shown in Fig. 9(c). Figure 9(d) is the salinity result of the day we went through the northernmost section. In this section, we can find the trough of salinity caused by the water mass.
From Fig. 9, the comparison of specific measurement value between refractometer and CTD is carried out. The measuring results observed by the refractometer are in reasonably good agreement with that observed by the commercial CTD. The coefficient of correlation between the refractometer data and the CTD data is 0.994, which shows good robust capability of the TIR sensor in turbid environment. Also, it is a reasonable result in field measurement.
To display our results vividly, we combined the salinity data with the GPS data and shown them in the map of the eastern of Yangtze Estuary in Fig. 10. Refractometer data and CTD data are displayed respectively, and a reasonably good match between the two data sets has been observed. From Fig. 10(a) and 10(b), the distribution of the sea surface salinity in the eastern of Yangtze Estuary is shown. In the Yangtze Estuary (the area I in Fig. 10), the surface water is entirely composed by the fresh water. The value of salinity is nearly zero. In the area (II), the surface water is a mixture of ocean water and the fresh water from the Yangtze River. And the salinity in this area presents an increasing trend from west to east. In area (III), the salinity of the middle of the section is obviously below the salinity of both ends. This measuring result shows that the water mass generated by the diluted water of the Yangtze River moved from Yangtze Estuary to the area III. From the underway monitoring result, the preliminary features of distribution and transformation of the East China Sea surface water mass can be carried out. It is evident from Fig. 10 that the differences of the salinity distribution are not too significant.
Fig. 10 Underway monitoring result of the Sea surface salinity in Yangtze Estuary Sea: (a) TIR sensor results, (b) CTD profiler results
5. Conclusions
We have demonstrated for the first time the application of the critical angle method for measuring salinity in actual sea trial conditions. Both the laboratory experiment and the sea trial test have been carried out. In laboratory, after the calibration of correction between the critical angle position and the refractive index, the device has been tested to prove the long term stability. It has been also tested in a thermostat water bath to study the effect of temperature on the RI result. The trials at sea have shown promising results. Data from device are consistent with lab test. Also, the feature of the sea surface salinity distribution has been obtained. The measuring results show good agreements with a commercial CTD.
Conclusively, these measurements have shown the feasibility of using TIR sensor for monitoring purposes in sea trials. With the compact TIR sensor we design, we realized underway measurement and anti-interference measuring in the turbid environment. These problems haven’t been reported to be realized in an actual sea trial from other researches. Thus, positive progress has been made in ocean field measurement during this work. In oceanographic, the TIR sensor can provide assistant measurement to the CTD profiler. Besides, in order to make it suitable for using in commercial devices, future studies are still necessary to improve the reliability and resolution capacities of the theoretical method.
Funding
National Natural Science Foundation of China (NSFC Grant No. 41406108); Project of “Scientific investigation of the Yangtze Estuary” founded by NSFC (No. 41649903).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
1. P. Ioc, “Th international thermodynamic equation of seawater-2010: Calculation and use of thermodynamic properties,” Ioc Paris (2010).
2. “CTD profilers”, http://www.seabird.com/products/profilers.htm.
3. P. Grosso, M. L. Menn, and Z. Y. Wu, Malard, andD. Eacute, “Practical versus absolute salinity measurements: New advances in high performance seawater salinity sensors,” Deep Sea Res. Part I Oceanogr. Res. Pap. 57(1), 151–156 (2010). [CrossRef]
4. T. J. Mcdougall, D. R. Jackett, F. J. Millero, R. Pawlowicz, and P. M. Barker, “A global algorithm for estimating Absolute Salinity,” Ocean Sci. Discussions 8(6), 1123–1134 (2012). [CrossRef]
5. A. P. Grosso, D. Malardé, M. L. Menn, and Z. Y. Wu, “Refractometer Resolution Limits For Measuring Seawater Refractive Index,” Opt. Eng. 49(10), 84 (2010). [CrossRef]
6. D. Malardé, Z. Y. Wu, P. Grosso, J. L. de Bougrenet de la Tocnaye, and M. Lemenn, “High-resolution and compact refractometer for salinity measurements,” Meas. Sci. Technol. 20(1), 015204 (2009). [CrossRef]
7. H. Minato, Y. Kakui, A. Nishimoto, and M. Nanjo, “Remote refractive index difference meter for salinity sensor,” IEEE Trans. Instrum. Meas. 38(2), 608–612 (1989). [CrossRef]
8. R. C. Millard and G. Seaver, “An index of refraction algorithm for seawater over temperature, pressure, salinity, density, and wavelength,” Deep-Sea Res. A, Oceanogr. Res. Pap. 37(12), 1909–1926 (1990). [CrossRef]
9. X. Quan and E. S. Fry, “Empirical equation for the index of refraction of seawater,” Appl. Opt. 34(18), 3477–3480 (1995). [CrossRef] [PubMed]
10. G. T. Mcneil, “Metrical Fundamentals of Underwater Lens System,” Opt. Eng. 16(2), 1079–1095 (1977). [CrossRef]
11. L. Li, L. Xia, Y. Wuang, Y. Ran, C. Yang, and D. Liu, “Novel NCF-FBG Interferometer for Simultaneous Measurement of Refractive Index and Temperature,” IEEE Photonics Technol. Lett. 24(24), 2268–2271 (2012). [CrossRef]
12. N. Díazherrera, O. Esteban, M. C. Navarrete, M. Lehaitre, and A. Gonzálezcano, “In situ salinity measurements in seawater with a fibre-optic probe,” Meas. Sci. Technol. 17(8), 2227–2232 (2006). [CrossRef]
13. Z. Qiu, “A simple optical model to estimate suspended particulate matter in Yellow River Estuary,” Opt. Express 21(23), 27891–27904 (2013). [CrossRef] [PubMed]
14. R. Röttgers, D. McKee, and C. Utschig, “Temperature and salinity correction coefficients for light absorption by water in the visible to infrared spectral region,” Opt. Express 22(21), 25093–25108 (2014). [CrossRef] [PubMed]
15. C. P. Artlett and H. M. Pask, “New approach to remote sensing of temperature and salinity in natural water samples,” Opt. Express 25(3), 2840–2851 (2017). [CrossRef] [PubMed]
16. Y. Zhao and Y. Liao, “Novel optical fiber sensor for simultaneous measurement of temperature and salinity,” Sens. Actuators B Chem. 86(1), 63–67 (2002). [CrossRef]
17. M. L. Menn, J. L. de Bougrenet de la Tocnaye, P. Grosso, L. Delauney, C. Podeur, P. Brault, and O. Guillerme, “Advances in measuring ocean salinity with an optical sensor,” Meas. Sci. Technol. 22(11), 115202 (2011). [CrossRef]
18. W. R. Calhoun, H. Maeta, A. Combs, L. M. Bali, and S. Bali, “Measurement of the refractive index of highly turbid media,” Opt. Lett. 35(8), 1224–1226 (2010). [CrossRef] [PubMed]
19. B. Hou, “Turbidimeter based on a refractometer using a charge-coupled device,” Opt. Eng. 51(2), 160–168 (2012). [CrossRef]
20. B. Hou, P. Grosso, J.-L. de Bougrenet de la Tocnaye, and M. Le Menn, “Principle and implementations of a refracto-nephelo-turbidimeter for seawater measurements,” Opt. Eng. 52(4), 044402 (2013). [CrossRef]
21. J. H. Liu, S. L. Yang, Q. Zhu, and J. Zhang, “Controls on suspended sediment concentration profiles in the shallow and turbid Yangtze Estuary,” Cont. Shelf Res. 90, 96–108 (2014). [CrossRef]
22. G. H. Meeten and A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6(2), 214–221 (1995). [CrossRef]
23. H. Liu, J. Ye, K. Yang, M. Xia, W. Guo, and W. Li, “Real part of refractive index measurement approach for absorbing liquid,” Appl. Opt. 54(19), 6046–6052 (2015). [CrossRef] [PubMed]
24. H. Contrerastello, R. Márquezislas, O. Vázquezestrada, C. Sánchezpérez, and A. Garcíavalenzuela, “Understanding the performance of Abbe-type refractometers with optically absorbing fluids,” Meas. Sci. Technol. 25(7), 075201 (2014). [CrossRef]
25. H. Sobral and M. Peña-Gomar, “Determination of the refractive index of glucose-ethanol-water mixtures using spectroscopic refractometry near the critical angle,” Appl. Opt. 54(28), 8453–8458 (2015). [CrossRef] [PubMed]
26. G. Morales-Luna, H. Contreras-Tello, A. García-Valenzuela, and R. G. Barrera, “Experimental Test of Reflectivity Formulas for Turbid Colloids: Beyond the Fresnel Reflection Amplitudes,” J. Phys. Chem. B 120(3), 583–595 (2016). [CrossRef] [PubMed]
27. W. Guo, M. Xia, W. Li, J. Dai, X. Zhang, and K. Yang, “A local curve-fitting method for the complex refractive index measurement of turbid media,” Meas. Sci. Technol. 23(4), 47001–47004 (2012). [CrossRef]
28. A. García-Valenzuela, G. E. Sandoval-Romero, and C. Sánchez-Pérez, “High-resolution optical angle sensors: approaching the diffraction limit to the sensitivity,” Appl. Opt. 43(22), 4311–4321 (2004). [CrossRef] [PubMed]
29. J. Ye, M. Xia, H. Liu, W. Li, W. Guo, C. Xiong, and K. Yang, “Expanding the measurement range of a critical-angle refractometer through Fourier analysis,” EPL 104(2), 20001 (2013). [CrossRef]
30. J. Ye, K. Yang, H. Liu, J. Dai, W. Guo, W. Li, and M. Xia, “Expand the measurement range of a critical angle refractometer by a centroid method for transparent fluids,” Opt. Laser Technol. 65, 175–179 (2015). [CrossRef]
31. P. S. Huang and J. Ni, “Angle measurement based on the internal-reflection effect using elongated critical-angle prisms,” Appl. Opt. 35(13), 2239–2241 (1996). [CrossRef] [PubMed]
