Open Access
Add to BibSonomy
Abstract
Vernier effect enhances the sensitivity for interferometric fiber sensor, but indiscriminately amplifies cross-sensitivity to environmental parameters. Here, hybrid Vernier effect, a new theory based on the cascaded FPI, is proposed and demonstrated for cross-sensitivity elimination under the premise of sensitivity amplification. It combines traditional and high-order harmonic Vernier effects to measure two parameters simultaneously. The proposed sensor achieves strain sensitivity of 960.1 pm/µɛ, and temperature sensitivity of 1260.86 pm/°C. Stability experiments demonstrate excellent stability of envelope demodulation method, with minimum temperature resolution of 0.44 °C and minimum strain resolution of 0.58 µɛ. The proposed the hybrid Vernier effect can be achieved widely in common cascaded fiber FPI fiber sensor structure, making it good candidate for practical applications.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In the field of optical fiber sensors, high sensitivity is the foundation for high-precision detection and is also the development direction. Fiber sensor with ultra-high sensitivity plays an irreplaceable role in the fields of biochemistry research, pharmacy and biomedicine [1,2]. Among the numerous interferometric fiber sensor, Fabry–Perot (FP) Interferometer (FPI) is widely used in the measurement of chemical and physical parameters, such as air pressure [3,4], strain [5,6] and humidity [7] for its salient features of high sensitivity, compact structure, and easy fabrication. In addition to the use of high-correspondence materials, Vernier effect is another effective method in principle to amplify the sensitivity of FPI fiber sensor. A traditional Vernier effect based cascaded FPI was first reported in [8]. By matching two FP cavities with similar optical path length (OPL), two FP spectra with similar free spectral range (FSR) were superposed to generate a new spectrum with a larger FSR envelope, and the sensitivity was enhanced by tracing the drift of the envelope [9–18]. However, this type of Vernier effect sensor encounters challenges of two types of cross-sensitivity. Single FP cavities may have cross-sensitivity due to their sensitivity to multiple environmental parameters that vary in the environment, leading to a drift in the envelope. Meanwhile, in a cascaded structure with two FP cavities, the response of both cavities to the same. These effects jointly make Vernier effect-based sensor suffer huge cross-talk and powerless to simultaneous measurement of multiple parameters [9–17]. The improved Vernier effect sensor based on parallel FPI structure could effectively eliminate the crosstalk between two cavities, in which one FP cavity is used for sensing, while the other FP is placed in a controlled environment to provide steady reference spectrum. The cross-sensitivity between multiple parameters in the sensing FP was still amplified by this Vernier effect [18,19]. The latest development in the theory of the Vernier effect is that harmonic Vernier effect, which was achieved by two FPs with multiple level difference of OPLs [20,21]. This method still helps to improve sensitivity, but it is powerless to eliminate cross-talk. An attempt to measure salinity and temperature simultaneously using Vernier effect sensor based on parallel FPI was proposed [22]. However, the demodulation is limited by the wavelength range of spectrometer and broadband light source, leading to low magnification. And this work lacks a theoretical explanation for eliminating cross talk between two FPIs.
The effective method to eliminate the cross sensitivity of the normal cascaded FPI without Vernier effect is to restore the interferometric spectra to the individual FP spectrum by fast Fourier transform and the band-pass filtering method [23–28]. However, when the traditional Vernier effect was generated in the cascaded FPI, the spectral frequencies of two individual FP cavities are too close to be separated by using this method. Therefore, sensitivity amplification and cross-sensitivity elimination are mutually exclusive, that is, sensitivity amplification and information recovery in Vernier effect systems cannot be realized simultaneously.
As for theory, current Vernier effect theory limits the function improvement of cascaded FPI fiber sensor. Restricted to the traditional Vernier effect theory that two FP spectra superposition are required to generate Vernier effect [9], the number of spectra used for interference was simplified to two in theoretical analysis. This reduced analysis causes the information of the remaining FP cavity to be ignored [7,8], difficulties in interpretation of the phenomenon of excess spectrum peaks [14], or unknown source of envelope information [16], which further limits the function and performance improvement and practical application of Vernier effect based on FPI.
To overcome the limitations and theoretical shortcomings that prevent Vernier effect sensors from simultaneously measuring multiple parameters and amplifying sensitivity, we propose a hybrid Vernier effect theory based on a cascaded FPI fiber sensor. Our theory provides a clear interpretation of the multiple beam interference mechanism in cascaded FPI structures. The spectral superposition of the rear cavity and the hybrid cavity produces the traditional Vernier effect, which could effectively restore the sensor information of the front cavity in the system. For the first time, we have discovered and proved that the Vernier effect can amplify sensitivity and reconstruct information in the FP spectrum. Combining using of these two effects, cross-sensitivity could be eliminated and simultaneous measurement of multiple two parameters with the sensitivity amplification could be realized with the sensitivity amplification. The sensitivity of the sensor to strain is 960.1 ${\mathrm{pm}/\mathrm{\mu} \mathrm{\varepsilon}}$, with magnification of 99 compared to sensitive cavity, and to temperature 1260.86 pm/°C. Experimental results verified that hybrid Vernier effect has a good ability to eliminate cross sensitivity and has the functions of multi-parameter simultaneous measurement and sensitivity amplification. Such advantages of the proposed sensor make it good candidate for practical applications.
2. Sensor structure and operation principle
Figure 1 is a schematic of the hybrid Vernier effect based FPI sensor, which consists of two FPIs (FP1 and FP2) cascaded together. FP1 was constructed using a tube, while FP2 was constructed using a single mode fiber (SMF). ${n_1}$ and ${n_2}$ are refractive indices of FP1 and FP2, while ${L_1}$ and ${L_2}$ are cavity lengths of FP1 and FP2, respectively. ${L_2}{n_2} = i{L_1}{n_1} + \Delta $ (i > 3) is designed in our structure, where i depends on the ratio of the two FP OPLs and is expressed as [21]
where the mathematical symbol $\lfloor X \rfloor$ means rounding down of X. Considering the low reflectivity of the three mirrors (around 4% at 1550 nm), the multi-beam reflection model can be approximated as triple-beam interference. The interference light intensity can be simplified as [9]:
Fig. 2. Diagram of the spectra of FP1, FP2 and FP3. The phase of the envelope term of the interference spectrum of FP2 and FP3 is consistent with that of FP1, so the phase of the envelope term of the total spectrum is still ${\phi _1}$.(In the simulation, ${L_1}{n_1} = 40\textrm{ }\mu \textrm{m}$, ${L_2}{n_2} = 232\textrm{ }\mu \textrm{m}$ was set.)
Therefore, the strain sensitivity of upper envelope is equal to that of FP1, noted as K11:
Furthermore, these interferences exhibit a harmonic Vernier effect as well. With simulation through numerical analysis, the harmonic Vernier effect caused by interference of spectra of FP1 and FP2 would be investigated. In the simulation, ${L_1}{n_1} = 40\,{ \mathrm{\mu} \mathrm{m}}$, ${L_2}{n_2} = 232\,{ \mathrm{\mu} \mathrm{m}}$ was set, and ${L_2}{n_2} = 4{L_1}{n_1} + 32\,{ \mathrm{\mu} \mathrm{m}}$, in this case i = 5. Despite the unequal FSRs of FP1 and FP2 (FSR1 = 30 nm and FSR2 = 5.17 nm, respectively), a least common multiple exists between (6*FSR2) and FSR1. This is the basis of the harmonic Vernier effect [22]. Six FSR2s can be thought of as one large FSR, as shown by the red line in Fig. 3(a), and the peaks numbered as (1 + 6 k, k is the nature number) serves as endpoints of the new FSR. Thus, by selecting peaks numbered as (1 + 6 k) in the interference spectrum, an internal envelope can be fitted, indicated by the red line in Fig. 3(b). For another group of 6 FSR2s (depicted by the green line in Fig. 3(a)), this can be superimposed on FSR1 to produce another internal envelope curve (fitting the the peaks numbered as (2 + 6 k)), as indicated by the green line in Fig. 3(b). By repeating this process, 6 internal envelope curves can be generated. Based on the findings in Ref. [22], the FSR of the internal envelope can be derived as
Fig. 3. (a) Diagram of the spectra of FP1 and FP2; (b) interference spectra of FP1 and FP2 with fitted internal envelope.
To illustrate how the drift of FP1 and FP2 spectra affects the internal envelope, strain response was used as an example. Both FP1 and FP2 experience an increase in their respective OPLs when applying the strain to the sensor, so there is a simultaneous spectral red drift in their spectra. To simplify the derivation of the sensitivity equation, the simultaneous drift of the two spectra was treated as two distinct steps. In step1, illustrated in Fig. 4(a), it is assumed that FP2's spectra remains unchanged while FP1's spectra redshifts by 2 nm, resulting in a corresponding redshift of 64.1 nm in the interference spectrum. One of the multiple intersections between internal envelopes was identified with a marker for drift trace. It can be seen that the sensitivity of internal envelope is effectively amplified compared with FP1. The sensitivity magnification factor is defined as $M = \frac{{FS{R_{{\mathop{\rm int}} \textrm{ernal}}}}}{{FS{R_1}}}$ [21]. And the sensitivity of the internal envelope to the strain could be expressed as
In Step 2, as depicted in Fig. 4(b), a redshift of 0.5 nm is introduced in FP2’s spectrum. As a result, the intersection point of interference spectrum shifted towards the blue end by 14.4 nm. Similarly, the magnification of the internal envelope to FP2 could be expressed as $M = \frac{{FS{R_{{\mathop{\rm int}} \textrm{ernal}}}}}{{FS{R_2}}}$. Hence, the sensitivity value of the internal envelope is determined by calculating the difference between the two shift values, noted as K21, which can be expressed as
Based on (Eq. (11)), it can be inferred that the sensitivity of the internal envelope will remain linear, provided that the strain sensitivity of FP1 and FP2 is linear. Compared with the high-order harmonic Vernier effect reported [22], the sensitivity of hybrid Vernier effect in the cascaded FPI is associated with sensitivity of FP1 and FP2, which means it could be reduced (${\chi _\textrm{1}}$ and ${\chi _2}$ have the same signs) or be enhanced (${\chi _\textrm{1}}$ and ${\chi _2}$ have the opposite signs). Similarly, the sensitivity of the internal envelope to temperature is
Therefore, the internal envelope sensitivity with temperature and strain is
In general, there are two types of envelopes in this hybrid Vernier effect spectrum: the upper envelope of the sensor output signal is utilized to extract information from FP1, while the internal envelope serves as a means of sensitivity amplification. The sensitivities of the upper and internal envelopes to temperature and strain exhibit differences. Specifically, the sensitivity of the upper envelope relates solely to the FP1 cavity, whereas that of the internal envelope is jointly determined by FP1 and FP2, which incorporates information from both cavities. These sensitivities can be obtained by pre-testing the sensor responses to temperature and strain. The $\Delta T$ and $\varepsilon$ can be calculated with cross-sensitivity compensation by inserting the values of $\Delta {\lambda _{\textrm{upper}}}$ and $\Delta {\lambda _{{\mathop{\rm int}} \textrm{ernal}}}$ into into Eq. (13)
The fabrication process for the sensor is straightforward and involves only a few simple steps of cutting and fusing. Firstly, a section of silica tube (Polymicro TSP075150, inner diameter 75 µm, external diameter 125 µm after burning off the protective layer) was spliced with a piece of well-cleaved SMF using the commercial fiber fusion splicer (Fujikura 80S). Secondly, facilitated with a microscope and a translation stage (OMT-22071305DK), the silica tube was cleaved to a desired length by a fiber cleaver. Thirdly, a section of SMF was spliced with the cleaved silica tube using the automatic fusion mode. Finally, tail tube of 100 mm was fused to the right end of SMF to withstand the strain. And Fig. 5 shows microscopic image of the sensor and the measured spectrum of the sensor. According to $FSR = \frac{{{\lambda ^2}}}{{2nL}}$, the cavity length of FP1 can be deduced to be 62.5 µm theoretically, consistent with the microscopic measurement value of 60 µm in Fig. 5. For n1 = 1.03 and n2 = 1.456, ${L_2}{n_2} = 9{L_1}{n_1} + 55\,{ \mathrm{\mu} \mathrm{m}}$ in this case. Since i = 9 in our sensor, it would obtain a 9th order harmonic Vernier effect, where an internal envelope could be fitted within every 9 peak points. To facilitate monitoring of the shift, two internal envelopes were fitted and the intersection of two internal envelope curves served as a feature point for tracking, as shown in Fig. 5(b). Upper envelope was obtained by fitting all peak points of the spectrum. A larger value of i results in more peak points in the spectrum available for fitting the upper envelope. Typically, it is necessary to ensure i > 3 for accurate fitting. This is the reason why we specified i > 3 in the above section.
Fig. 4. (a) Demonstration of the wavelength shift of the harmonic Vernier spectrum during Step1; (b) demonstration of the wavelength shift of the harmonic Vernier spectrum during Step2.
3. Experimental and results
3.1 Strain and temperature sensitivity
First, the strain response of proposed Vernier FPI was experimentally demonstrated. The lead-in fiber and tail tube were secured on the displacement platform ((OMT-22071305DK, the minimum single displacement is 0.2 µm) to apply strain to the sensor. The distance between the two fixed points is 25 cm, and the single displacement of the platform is 5 µm for 50 µɛ generation. The light from the broadband light source (BBS, FiberLake ASE, wavelength 1250-1650 nm) was incident into the sensor through optical path 2, and reflected light incident into the optical spectrum analyzer (OSA, YOKOGAMA AQ6370C) through the optical circulator optical, as shown in Fig. 6(a). The OSA records and analyzes the sensor's reflection spectrum, with the resolution of 20 pm. In order to improve the accuracy of spectrum shift tracing, the original spectral data was interpolated with 20 interpolations for every two intervals of data, so the resolution would be improved to 1 pm in theory.
Fig. 6. (a) The experimental setup for strain measurement; (b) the experimental setup for temperature measurement.
The sensitivities of the upper envelope and internal envelope to strain are shown in Fig. 7(a-c). After linear fitting, the strain sensitivity of the upper envelope was determined as K11 = 9.7 pm/µɛ. Compared with the strain sensitivity of air cavity in our previous work [26] of 2.97 pm/µɛ, sensitivity of FP1 is higher for shorter air cavity than that in case of [26]. The intersection point of the internal envelopes of the sensor was redshifted with the increase of strain, with a sensitivity of K21 = 960.1 pm/µɛ. The internal envelope sensitivity magnification was 99 times that of FP1.
Fig. 7. (a) The shift of upper envelope with 0-300 µɛ; (b) the measured shift of internal envelope with 0-300 µɛ; (c) the strain linear sensitivity of upper and internal envelope.
Then, the temperature response of proposed Vernier FPI was experimentally demonstrated. The sensor was placed in the oven (SIOM SG-XL, China) for the temperature test. The structure of the experimental system is shown in Fig. 6(b). The temperature ranged from 30 °C to 90 °C with a temperature interval of 10 °C. As shown in Fig. 8, the upper envelope redshifted as the temperature rises, with a sensitivity of K12 = 0.96 pm/°C, which is also in good agreement with the air cavity empirical sensitivity value of 0.86 pm/°C in the work [26]. The internal envelope blue shifted with the increase of temperature, with a sensitivity K22 = 1260.86 pm/°C. According to the work [26], the silica cavity sensitivity to temperature could be estimate as ${\tau _2}$=10.4 pm/°C. Inserting ${\tau _2}$=0.96 pm/°C and ${\tau _2}$=10.4 pm/°C into Eq. (12), the theoretical K22 = 1253.8 pm/°C could be calculated, which is good accordance with experiment value of K22 = 1260.86 pm/°C.
Fig. 8. (a) The shift of upper envelope with 0-90 °C; (b) the shift of internal envelope with 0-90 °C; (c) the temperature linear sensitivity of upper and internal envelope.
3.2 Matrix verification experiment
In order to verify the accuracy of the matrix, a cross-experiment involving temperature and strain was performed. First, two sections of guide fiber connected to the sensor were fixed on the displacement platform without the application of any strain, while a heating platform was placed below the sensor with an initial temperature of 30 °C. The temperature was then increased by 10°C and the strain by 50 µɛ at each step, until the temperature reached 80°C and the strain reached 250 µɛ. Each spectrum was recorded at every step, and the internal and upper envelopes were obtained through fitting, which were shown in Fig. 9. After obtaining the drift values of these two envelopes at each step, the variations of temperature and strain could be obtained by inputting these data into the sensitivity matrix, which could be expressed as by
The recovered values for temperature and strain variations are presented in Table 1. It demonstrated that both temperature and strain variation values have been reproduced well, and the cross-sensitivity between temperature and strain has been effectively eliminated through our method. The measurement results of the sensitivity matrix show that the maximum errors for simultaneous measurement of temperature and strain are estimated to be around 2.7 °C and 2.8 µɛ, respectively. These deviations were most likely caused by using a heating platform instead of a temperature chamber, which generated errors in temperature readings.
Fig. 9. (a) The shift of upper envelope with temperature and strain increasing simultaneously; (b) the shift of internal envelope with temperature and strain increasing simultaneously.
Table 1. Experimental data of the sensitivity matrix validation
3.3 Stability and minimum resolution
Due to the sensitivity demodulation method of Vernier effect is based on tracking the fitted envelope, theoretically, its minimum resolution depends on the number of interpolations during envelope fitting. However, in practice, the envelope fitting is influenced by the stability/noise of the sensing system. To investigate the stability of the proposed sensor, the proposed sensor was set under specific temperature/strain conditions in 3 hours and the reflection spectra were recorded every 30 minutes. The spectral responses of the tracked internal envelopes during 3 hours are shown in Fig. 10(a) when the first measurement was conducted at a temperature of 40 °C and strain of 0 µɛ, respectively, and the second measurement was conducted at a temperature of 40 °C and strain of 200 µɛ, respectively. Figure 10(b) shows the corresponding fluctuation of the two wavelengths during the 3 hours. The largest fluctuation of the measured values in 3 hours were 0.55 nm and 0.59 nm, respectively, which shows that the tested wavelength shift of the proposed sensor in 3 hours is reasonable and acceptable. We define the maximum fluctuation of the as 0.59 nm. This also means that the internal envelope shift must be larger than largest fluctuation to differentiate minimum variations (minimum resolution) in strain/temperature, as shown in Fig. 10(b). By incorporating the sensitivity of temperature and strain into the calculation, the minimum resolution of temperature is determined to be 0.44 °C, and the minimum resolution of strain is 0.58 µɛ.
Fig. 10. (a) Spectra around 1292.4 nm and 1442.6 nm in 3 hours; (b) corresponding measured wavelength shifts.
4. Conclusion
In summary, we propose a hybrid Vernier effect based sensor for ultra-high sensitivity and simultaneous measurement of multiple parameters. The hybrid Vernier effect is generated through multi-cavity interference, where the interference from all cavities restores FP1 information, and the interference between FP2 and FP1 amplifies the sensitivity. The hybrid Vernier effect can effectively eliminate the cross-sensitivity under sensitivity amplification and realize the simultaneous measurement of multiple parameters as the experiment well proves. The internal envelope and upper envelope tracking method is simple to operate without complex filtering process. The minimum resolution of temperature is 0.44 °C, and the minimum resolution of strain is 0.58 µɛ. Since the hybrid Vernier effect widely exists in cascaded FPI, it could be leveraged to improve the performance of various sensors, and we predict that it will be highly competitive in practical applications.
Funding
National Natural Science Foundation of China (61675055); Shenzhen Municipal Science and Technology Innovation Council (JCYJ20190806143818818, GXWD20201230155427003-20200731103843002).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. H. Farahani, R. Wagiran, and M. N. Hamidon, “Humidity Sensors Principle, Mechanism, and Fabrication Technologies: A Comprehensive Review,” Sensors 14(5), 7881–7939 (2014). [CrossRef]
2. Y. S. Shi, B. Polat, Q. Huang, and D. J. Sirbuly, “Nanoscale Fiber-optic Force Sensors for Mechanical Probing at The Molecular and Cellular Level,” Nat. Protoc. 13(11), 2714–2739 (2018). [CrossRef]
3. J. Eom, C. J. Park, B. H. Lee, J. H. Lee, I. B. Kown, and E. Chung, “Fiber Optic Fabry–Perot Pressure Sensor Based on Lensed Fiber and Polymeric Diaphragm,” Sensors and Actuators A: Physical 225, 25–32 (2015). [CrossRef]
4. C. Liao, S. Liu, L. Xu, C. Wang, Y. P. Li, Q. Wang, and D. N. Wang, “Sub-micron Silica Diaphragm-based Fiber-tip Fabry–Perot Interferometer for Pressure Measurement,” Opt. Lett. 39(10), 2827–2830 (2014). [CrossRef]
5. S. Pevec and D. Donlagic, “Miniature All-fiber Fabry–Perot Sensor for Simultaneous Measurement of Pressure and Temperature,” Appl. Opt. 51(19), 4536–4541 (2012). [CrossRef]
6. C. X. Liu, X. T. Yang, H. Zhang, L. G. Hou, and J. R. Yang, “Micro-Open-Cavity Interferometer for Highly Sensitive Axial-Strain Measurement via Bias-Taper and Vernier Effect,” IEEE Sens. J. 22(22), 21676–21683 (2022). [CrossRef]
7. A. M. Shrivastav, D. S. Gunawardena, Z. Y. Liu, and H. Y. Tam, “Microstructured Optical Fiber based Fabry-Pérot Interferometer as A Humidity Sensor Utilizing Chitosan Polymeric Matrix for Breath Monitoring,” Sci. Rep. 10(1), 6002 (2020). [CrossRef]
8. P. Zhang, M. Tang, F. Gao, B. P. Zhu, S. N. Fu, J. Ouyang, P. P. Shum, and D. Liu, “Cascaded Fiber-optic Fabry-Perot Interferometers with Vernier Effect for Highly Sensitive Measurement of Axial strain and Magnetic field,” Opt. Express 22(16), 19581–19588 (2014). [CrossRef]
9. M. Quan, J. J. Tian, and Y. Yao, “Ultra-high Sensitivity Fabry–Perot Interferometer Gas Refractive Index Fiber Sensor Based on Photonic Crystal Fiber and Vernier Effect,” Opt. Lett. 40(21), 4891–4894 (2015). [CrossRef]
10. P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified Hollow-Core Fiber-Based Fabry–Perot Interferometer With Modified Vernier Effect for Highly Sensitive High-Temperature Measurement,” IEEE Photonics J. 7(1), 1–10 (2015). [CrossRef]
11. G. Zhang, X. Wu, W. Zhang, S. Li, J. Shi, C. Zuo, S. Fang, and B. Yu, “High Temperature Vernier Probe Utilizing Photonic Crystal Fiber-based Fabry-Perot Interferometers,” Opt. Express 27(26), 37308–37317 (2019). [CrossRef]
12. Y. Yang, Y. Wang, J. Jiang, Y. Zhao, X. He, and L. Li, “High-Sensitive All-fiber Fabry-Perot Interferometer Gas Refractive Index Sensor Based on Lateral Offset Splicing and Vernier effect,” Optik 196, 163181 (2019). [CrossRef]
13. Y. Zhao, R. Tong, M. Chen, and Y Peng, “Relative Humidity Sensor Based on Vernier effect with GQDs-PVA Un-fully Filled in Hollow Core Fiber,” Sensors and Actuators A: Physical 285, 329–337 (2019). [CrossRef]
14. T. Nan, B. Liu, Y. F. Wu, J. F. Wang, Y. Y. Mao, L. L. Zhao, T. T. Sun, and J. Wang, “Ultrasensitive Strain Sensor Based on Vernier-effect Improved Parallel Structured Fiber-optic Fabry-Perot Interferometer,” Opt. Express 27(12), 17239–17250 (2019). [CrossRef]
15. Y. Zhao, P. Wang, R. Lv, and X. Liu, “Highly Sensitive Airflow Sensor Based on Fabry–Perot Interferometer and Vernier Effect,” J. Lightwave Technol. 34(23), 5351–5356 (2016). [CrossRef]
16. J. J. Tian, Z. G. Li, Y. X. Sun, and Y. Yao, “High-sensitivity Fiber-optic Strain Sensor Based on The Vernier Effect and Separated Fabry–Perot Interferomer,” J. Lightwave Technol. 37(21), 5609–5618 (2019). [CrossRef]
17. J Zhang, H. Liao, P. Liu, X. Y. Jiang, X. Fu, W. J. Ni, D. M. Liu, and J. S. Zhang, “Ultrasensitive Temperature Sensor With Cascaded Fiber Optic Fabry–Perot Interferometers Based on Vernier Effect,” IEEE Photonics J. 10(6), 1–9 (2018). [CrossRef]
18. J. Deng and D. N. Wang, “Ultra-Sensitive Strain Sensor Based on Femtosecond Laser Inscribed In-Fiber Reflection Mirrors and Vernier Effect,” J. Lightwave Technol. 37(19), 4935–4939 (2019). [CrossRef]
19. T. Paixao, F. Araujo, and P. Antunes, “Highly Sensitive Fiber Optic Temperature and Strain Sensor Based on An Intrinsic Fabry–Perot Interferometer Fabricated by A Femtosecond Laser,” Opt. Lett. 44(19), 4833–4836 (2019). [CrossRef]
20. A. D. Gomes, M. Ferreira, J. Bierlich, J. Kobelke, M. Rothhardt, H. Bartelt, and O. Frazão, “Optical Harmonic Vernier Effect: A New Tool for High Performance Interferometric Fibre Sensors,” Sensors 19(24), 5431 (2019). [CrossRef]
21. A. D. Gomes, H. Bartelt, O. J. L. Frazo, and P. Review, “Optical Vernier Effect: Recent Advances and Developments,” Laser Photonics Rev. 15(7), 2000588 (2021). [CrossRef]
22. R. Flores, R. Janeiro, and J. Viegas, “Optical Fibre Fabry-Pérot Interferometer Based on Inline Microcavities for Salinity and Temperature Sensing,” Sci. Rep. 9(1), 9556 (2019). [CrossRef]
23. Z G. Li, J. J. Tian, Y. Z. Jiao, Y. X. Sun, and Y. Yao, “Simultaneous Measurement of Air Pressure and Temperature Using Fiber-Optic Cascaded Fabry-Perot Interferometer,” IEEE Photonics J. 11(1), 1–10 (2019). [CrossRef]
24. Y. F. Wu, Y. D. Zhang, J. Wu, and P. Yuan, “Simultaneous Measurement of Transverse Load and Temperature Using Hybrid Structured Fiber-optic Fabry–Perot Interferometer,” Sci. Rep. 7(1), 10736 (2017). [CrossRef]
25. H. C. Gao, Y. Jiang, Y. Cui, L. C. Zhang, J. S. Jia, and J. Hu, “Dual-Cavity Fabry-Perot Interferometric Sensors for the Simultaneous Measurement of High Temperature and High Pressure,” IEEE Sens. J. 18(24), 10028–10033 (2018). [CrossRef]
26. J. J. Tian, Y. Z. Jiao, S. B. Ji, X. L. Dong, and Y. Yao, “Cascaded-cavity Fabry–Perot Interferometer for Simultaneous Measurement of Temperature and Strain with Cross-sensitivity Compensation,” Opt. Commun. 412, 121–126 (2018). [CrossRef]
27. A. Zhou, B. Qin, Z. Zhu, Y. Zhang, Z. H. Liu, J. Yang, and L. B. Yuan, “Hybrid Structured Fiber-optic Fabry–Perot Interferometer for Simultaneous Measurement of Strain and Temperature,” Opt. Lett. 39(18), 5267–5270 (2014). [CrossRef]
28. H. Sun, X. Zhang, L. Yuan, L. Zhou, X. Qiao, and M. Hu, “An Optical Fiber Fabry–Perot Interferometer Sensor for Simultaneous Measurement of Relative Humidity and Temperature,” IEEE Photonics J. 15(5), 1 (2014). [CrossRef]
29. C. Zhou, Q. Zhou, J. J. Tian, and Y. Yao, “High-sensitivity relative humidity fiber-optic sensor based on an internal–external Fabry–Perot cavity Vernier effect,” Opt. Express 29(8), 11854–11868 (2021). [CrossRef]
