Open Access
Add to BibSonomy
Abstract
A Fabry-Perot (F-P) interferometric fiber-optic cantilever sensor is presented for simultaneous measurement of acoustic pressure and temperature, which are demodulated by a single high-speed spectrometer. The acoustic pressure wave pushes the cantilever to produce periodic deflection, while the temperature deforms the sensor and causes the F-P cavity length to change slowly. The absolute length of the F-P cavity of the fiber-optic cantilever sensor is calculated rapidly by using a spectral demodulation method. The acoustic pressure and temperature are obtained by high-pass filtering and averaging the continuously measured absolute cavity length value, respectively. The experimental results show that the acoustic pressure can be detected with an ultra-high sensitivity of 198.3 nm/Pa at 1 kHz. In addition, an increase in temperature reduces the resonant frequency of the acoustic response and increases the static F-P cavity length. The temperature coefficient of the resonance frequency shift and the temperature response of the sensor are -0.49 Hz/°C and 83 nm/°C, respectively. Furthermore, through temperature compensation, the measurement error of acoustic pressure reaches ± 3%. The proposed dual parameter measurement scheme greatly simplifies the system structure and reduces the system cost.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, a variety of fiber-optic sensors are designed to measure static and dynamic physical quantities such as temperature, pressure, vibration and gas concentration [1–4]. They have been increasingly used in many monitoring applications, such as structural health monitoring, fire alarming, power cable monitoring, down-hole monitoring and gas detection. This is due to their advantages of light weight, small size, high sensitivity, capability of remote sensing and tolerance to harsh environment.
To achieve highly sensitive detection of the acoustic pressure wave, various fiber-optic interferometric sensors have been proposed. The main types of interferometers used are Mach-Zehnder interferometer (MZI) [5], Michelson interferometer (MI) [6], Sagnac interferometer (SI) [7], Fizeau interferometer (FI) [8] and Fabry-Perot (F-P) interferometer (FPI) [9]. Due to the relatively large deformations that occur under sound waves, elastic diaphragms are commonly used. The FPI sensor is more suitable for measuring the deformation of a diaphragm because of its small size. Several types of elastic materials, such as silica glass [10], polymer [11,12], silver [13,14], titanium [15], stainless steel [16], silicon [17], graphene oxide [18] and graphene [19], have been used as the acoustic sensitive diaphragm. To further improve the sensitivity, we recently developed a fiber-optic cantilever microphone [20,21]. Compared with the diaphragm, the head of the cantilever beam is less constrained. Therefore, the deformation of the cantilever beam will be greater and have much higher acoustic sensitivity under the same sound pressure.
The demodulation method for the FPI sensor is critical to the sensitivity, dynamic range, and stability of the acoustic measurement. The tunable distributed feedback (DFB) laser based interference-intensity demodulation method is most commonly used for its low cost and easy to implement [22–24]. In order to achieve a good linear response, the central wavelength of the laser must be tracked and adjusted in real time to lock to the quadrature point. However, there is a problem that the quadrature point drifts with temperature change [25]. If factors such as sudden temperature changes make it impossible to lock to the quadrature point, large measurement distortion will occur. In addition, when the wavelength of the laser is ∼1550 nm, the dynamic range of acoustic pressure measurement is limited by the amount of change in F-P cavity length not exceeding ∼100 nm. In our recent work, we presented a stable and high-sensitivity fiber-optic F-P cantilever acoustic sensor based on a fast demodulated white-light interferometer. The cavity length is absolutely measured by realizing high-speed spectrum demodulation with a maximum measurement range of ∼1 mm, which is about 10000 times that of the interference-intensity demodulation [26,27]. However, these presented sensors are only used for the measurement of acoustic pressure.
Fiber-optic FPI sensors are also temperature sensitive [28,29]. Therefore, a single sensor can be used for simultaneous measurement of acoustic pressure and temperature. Morris et al. presented a polymer film based wideband fiber-tip F-P hydrophone [30]. The thickness changed as the temperature changed. The temperature rise was measured by tracking the quadrature point. However, the absolute temperature could not be measured with this method. Liu et al. designed a fiber-optic FPI sensor, which was formed by bonding two silicon wafers on both sides of a Pyrex glass [31]. The silicon F-P cavity and air F-P cavity were used for absolute temperature and acoustic pressure measurement, respectively. Acoustic wave deformed the top silicon diaphragm, which changed the air F-P cavity length. Meanwhile, the silicon F-P cavity had high thermal expansion coefficient and silicon thermo-optic coefficient, which could realize sensitive temperature measurement. A single-frequency laser combined with intensity demodulation method realized sound pressure measurement. Meanwhile, a broadband light source and a spectrometer were used in combination with the spectral demodulation method for temperature measurement. The measurement of the two parameters required two independent demodulation systems, which made the system structure complex.
In this paper, an F-P interferometric fiber-optic cantilever sensor is presented for simultaneous measurement of acoustic pressure and temperature, which are demodulated by a common spectrometer. The absolute length of the F-P cavity of the fiber-optic cantilever sensor is calculated rapidly by using the spectral demodulation method. The acoustic pressure and the temperature are obtained by high-pass filtering and averaging the continuously measured absolute cavity length value, respectively. Furthermore, the effect of temperature on the acoustic response of the cantilever beam has been studied through theoretical analysis and experiment.
2. Theory and sensing system design
2.1 FPI-based acoustic and temperature sensing
The acoustic pressure and temperature are simultaneously measured by a fiber-optic FPI sensor, which mainly composed of an optical fiber, a ceramic ferrule, a thin stainless steel cantilever beam and a cylindrical stainless steel shell [26]. The length, width and thickness of the cantilever are 1.8 mm, 1 mm and 10 µm, respectively. The acoustic pressure wave pushes the cantilever to produce periodic deflection, which is measured in an F-P resonator formed between the endface of the fiber and the inner surface of the cantilever. The temperature dependent frequency response of the cantilever can be expressed as [32]:
For 304 stainless steel, ρ = 7.93 g/cm3, E0 = 193${\times} {10^9}$ Pa and $\alpha $ = 17.3${\times} {10^{ - 6}}$/°C. According to Eqs. (1)–(3), the frequency response of the cantilever with temperatures of 0 °C, 20 °C, 40 °C, 60 °C and 80 °C was simulated, as shown in Fig. 1(a). According to the simulation result, the relationship between the resonant frequency and the temperature is plotted in Fig. 1(b). It indicates that the resonant frequency decreases linearly as the temperature increases. By using a linear fit, the temperature coefficient is obtained to be -0.435 Hz/°C. According to Fig. 1(a), temperature compensation is required for acoustic pressure measurement by simultaneously measuring the temperature.
Fig. 1. (a) Simulated frequency response of the cantilever with temperatures of 0 °C, 20 °C, 40 °C and 60 °C. (b) Simulated resonant frequency as a function of temperature.
The change in F-P cavity length is mainly due to the difference in thermal expansion coefficient between stainless steel and ceramic ferrule. We have simulated the temperature response of the fiber-optic cantilever sensor by using the finite element analysis method. The diameter and length of the cylindrical sensor are 13 mm and 12 mm, respectively. In order to improve the fringe visibility of the low-fineness F-P interference spectrum, the cavity length is set to ∼230 µm. Figure 2(a) shows the simulated temperature induced deformation of the sensor. By changing the temperature, the relationship between the cavity length and the temperature is obtained, as shown in Fig. 2(b). The temperature sensitivity of 80.9 nm/°C is obtained through linear fitting.
Fig. 2. (a) Simulated temperature induced deformation of the sensor. (b) Simulated F-P cavity length as a function of temperature.
2.2 Demodulation method
Due to the low effective reflectivity of both the endface of the optical fiber and the inner surface of the cantilever beam, the FPI sensing unit can be equivalent to a two-beam interferometer. The optical interference spectrum produced by the two reflected lights can be described by [35]:
where k is the wavenumber as an independent variable, k0 is the first wavenumber of the optical spectrum, I0(k + k0) is the light source intensity distribution with wavenumber, γ is the fringe visibility of the two-beam interferometric spectrum, φ0 is the initial phase, and d is the extrinsic F-P cavity length. When a sinusoidal acoustic pressure is applied to the cantilever beam, the length of the F-P cavity varies periodically. Meanwhile, the cavity length also varies slowly with temperature. Therefore, the cavity length d can be expressed as: where d0(T) is the temperature-dependent static cavity length, $\tilde{d}$(T) is the temperature-dependent dynamic change in cavity length caused by sound pressure. From Eq. (4) and (5), the acoustic pressure causes dynamic changes in the reflection spectrum of the extrinsic F-P cavity. The temperature and acoustic pressure can be demodulated by extracting DC and AC components of the measured cavity length, respectively.The absolute cavity length d can be calculated by a simple demodulation method of fast Fourier transform (FFT). However, due to the limitation of few sampling points in discrete spectrum, the resolution of the measured cavity length can only reach the order of 10 nm. To detect the acoustic pressure signal, the absolute cavity length of the fiber-optic FPI based cantilever sensing unit has been demodulated by a fast spectrum demodulation algorithm based phase demodulation method [36,37]. This algorithm mainly combines a non-zero-padded FFT and a Buneman frequency estimation. The absolute cavity length d can be obtained:
where k1 is the end wavenumber of the spectrum, and ξ is the real peak index, which falls between two maximal peak points of the FFT result. The peak index is often an integer for processing a discrete spectral signal. Therefore, the resolution of ξ is very low, which results in a low resolution of d. To further improve the resolution, the Buneman frequency estimation is used, and the peak index can be estimated by [36]:According to Eqs. (6) and (7), the cavity length d can be absolutely measured by the phase demodulation method. It can realize a large dynamic range from sub-nanometer level to millimeter level.
2.3 Sensing system
Figure 3 shows the schematic structure of the fast spectrum demodulation based fiber-optic detection system, which has been designed to realize high-sensitivity acoustic pressure detection and temperature measurement simultaneously. To suppress the influence of inter-mode interference effects, all fibers used in the system were single-mode fibers with a core/cladding diameter of 9/125 µm. The fiber-optic FPI based cantilever sensing unit was demodulated by a fiber-optic white-light interferometer. A superluminescent light emitting diode (SLED) (DL-CS5077, Denselight) was used as a probe light source with a central wavelength of ∼1550 nm and a power of ∼3 mW. Compared with the broadband amplified spontaneous emission (ASE) light source, the size of the SLED is much smaller. The probe light emitted into an optical fiber circulator and then propagated to the packaged FPI based cantilever sensor. The F-P interference spectrum was detected by a miniature spectrometer (FBGA-F-1510-1590, BaySpec). The spectrometer is composed of an optical dispersion element, an image detection unit and an electronic processing unit. A transmission volume phase grating (VPG) is used as the dispersion element that can void moving part and realize continuous spectrum measurement. Together with an InGaAs array image sensor, high-speed spectrum acquisition can be achieved. The maximum spectrum acquisition speed of the spectrometer specified by the manufacturer is 5 kHz. However, when the integral time of the image sensor is less than 20 µs, the actual highest spectral acquisition speed can reach 7.5 kHz, thereby increasing the detection bandwidth of the acoustic signal. Because the absolute F-P cavity length was measured by the spectrum demodulation method, the pressure signal and the temperature were obtained by high-pass filtering and averaging the continuously measured absolute cavity length value, respectively.
3. Experimental results and discussion
3.1 FPI cantilever based temperature sensing
The interference spectrums of the fiber-optic FPI cantilever were measured by changing the temperature of the sensing head with different temperatures of 0°C, 20°C and 40°C, as shown in Fig. 4. The integral time of the fiber-optic spectrometer was set to be 50 µs. As the temperature increased, the phase of the interference fringe changed, and the free spectral range became smaller.
Fig. 4. Interference spectrums of the fiber-optic FPI cantilever at different temperatures of 0°C, 20°C and 40°C.
Figure 5 shows the demodulated F-P cavity length as a function of temperature. It indicates that the cavity length becomes longer as the temperature increases. The FPI cantilever was proved to be a linear response to temperature by the calculated R2 of 0.9993. The temperature responsivity was estimated to be 83 nm/°C, which was in good agreement with the simulation the result of 80.9 nm/°C. To achieve temperature measurement, the temperature was calculated by the demodulated static cavity length (d0) with the equation of T=(d0-227.212)/0.083.
The long-term continuous measurement of the cavity length was carried out at a constant temperature of 20 °C, as shown in Fig. 6. Each point in the figure was obtained by averaging 50000 values of demodulated cavity length and then calculated according to the linear fitting relationship shown in Fig. 5. The calculated temperatures were in the range from 19.85 °C to 20.2 °C, which indicates that the FPI cantilever based temperature sensing had high accuracy and stability. In addition, through multiple measurements, we found that the temperature increased at the beginning of the test. We speculate that this is caused by the SLED emission light incident on the cantilever heating the sensor.
Fig. 6. Long-term continuous measurement results of the cavity length at a constant temperature of 20°C.
3.2 FPI cantilever based acoustic sensing
The acoustic sensing performance of the FPI cantilever was evaluated by an acoustic test setup [26]. The designed sensor and a condenser microphone were placed side-by-side in an acoustic isolation box to suppress the interference of environmental sound noise. By applying different sound pressures with a frequency of 1000 Hz to the cantilever, the demodulated absolute cavity lengths were recorded at room temperature of ∼21 °C, as shown in Fig. 7(a). The cavity length was the superposition of the DC static cavity length and the sound pressure induced AC dynamic cavity length. Acoustic pressure signals were retrieved with high fidelity by utilizing a high-pass filter. Figure 7(b) shows the deflection of the cantilever as a function of acoustic pressure. It illustrates that the FPI cantilever has a linear response to acoustic pressure with an R-square value of 0.9994. In addition, the sensitivity of the cantilever can be obtained to be 198.3 nm/Pa at 1000 Hz by a linear fit.
Fig. 7. (a) Demodulation results of absolute cavity length with different sound pressure at the frequency of 1000 Hz. (b) Deflection of the cantilever as a function of acoustic pressure.
To evaluate the minimum detectable sound pressure level (MDSPL) of the FPI cantilever sensor, the spectrum of the measured 1-mPa acoustic pressure at the frequency of 1 kHz was analyzed, as shown in Fig. 8. The noise floor is about -1.54 dB with a 1-Hz resolution bandwidth, which is obtained by calculating twice the standard deviation of the noise around the frequency of 1 kHz. It indicates that the noise-limited MDSPL is -1.54 $\textrm{dB}/\sqrt {\textrm{Hz}} $ (16.7${\; {\mathrm{\mu}} {\textrm{Pa}}}/\sqrt {\textrm{Hz}} $) at 1 kHz. According to the acoustic sensitivity of 198.3 nm/Pa, the deflection noise density of the cantilever is calculated to be 3.3 $\textrm{pm}/\sqrt {\textrm{Hz}} $ at 1 kHz.
In order to investigate the influence of the temperature to the sensitivity of the FPI cantilever, the frequency responses were measured with different temperatures. The sensor head was wrapped by a polyimide heating film, and the temperature was adjusted by a thermostatic controller. The working frequency of the loudspeaker was scanned from 100 Hz to 2000 Hz, the effective values of the cavity length change were recorded at temperatures of 0 °C, 10 °C, 20 °C, 30 °C, 40 °C and 50 °C. The sensitivities were calculated by the ratio of the effective values to the sound pressure values calibrated by the condenser microphone, as shown in Fig. 9(a). The sensitivity below the resonant frequency is improved as the temperature increases, which is consistent with Eq. (1). The relationship of the resonant frequency and the temperature is plotted in Fig. 9(b). As the temperature increases, the resonant frequency of the FPI cantilever decreases. The temperature coefficient has been obtained to be -0.49 Hz/°C, which is close to the theoretical analysis value of -0.435 Hz/°C shown in Fig. 1(b).
Fig. 9. (a) Measured frequency response of the FPI cantilever with temperatures of 0 °C, 10 °C, 20 °C, 30 °C, 40 °C and 50 °C. (b) Measured resonant frequency as a function of temperature.
Because the acoustic pressure measurement was temperature sensitive, temperature compensation was needed. Firstly, according to the measured temperature, the sensitivity of different temperatures at the measured frequency in Fig. 9 (a) was calculated by spline interpolation to calculate the frequency response under the measured temperature. Then, another spline interpolation was performed on the obtained frequency response according to the frequency of the measurement signal to calculate the actual acoustic sensitivity. Finally, the sound pressure was calculated by the ratio of the measured deflection of the cantilever and the obtained acoustic sensitivity. The temperature compensation method for acoustic pressure measurement at a specific frequency was experimentally verified. Figure 10 shows the test results of acoustic pressure measurement after temperature compensation at different temperatures of 12 °C, 47 °C and 75 °C. Through temperature compensation and frequency-domain signal processing, the measurement error of sound pressure was within ± 3%.
Fig. 10. Test results of acoustic pressure measurement after temperature compensation at different temperatures of 12 °C, 47 °C and 75 °C.
It can also realize the temperature compensation of multi-frequency sound pressure signal measurement. Firstly, the frequency spectrum of the measured dynamic cavity length is obtained by Fourier transform. Then, the frequency spectrum is divided by the fitted frequency response of the sound pressure at this temperature in the frequency domain. Finally, the acoustic pressure signal in time domain is retrieved by inverse Fourier transform.
4. Conclusion
In conclusion, we have demonstrated a fiber-optic F-P interferometric sensing system for simultaneous measurement of acoustic pressure and temperature. The absolute length of the F-P cavity of the fiber-optic cantilever sensor was calculated rapidly by using spectral demodulation method. The acoustic pressure and the temperature were obtained by high-pass filtering and averaging the continuously measured absolute cavity length value, respectively. The acoustic sensitivity and the noise-limited MDSPL at 1 kHz were achieved to be 198.3 nm/Pa and -1.54 $\textrm{dB}/\sqrt {\textrm{Hz}} $ (16.7${\; {\mathrm{\mu}} {\textrm{Pa}}}/\sqrt {\textrm{Hz}} $), respectively. With the increase of temperature, the resonant frequency of the FPI cantilever decreases, while the acoustic sensitivity below the resonant frequency is improved. The experimentally obtained temperature coefficient was -0.49 Hz/°C, which was close to the theoretical analysis value of -0.435 Hz/°C. Meanwhile, the FPI cantilever had a linear response to temperature in the range of 0-80°C. The temperature responsivity was 83 nm/°C. The use of a single fiber-optic sensor and a single demodulation module to achieve multi-parameter measurement greatly simplifies the system structure and reduces costs. Furthermore, the measured temperature has been used to eliminate the influence of ambient temperature change on the acoustic response.
Funding
Natural Science Foundation of Liaoning Province (2019-MS-054); Fundamental Research Funds for the Central Universities (DUT18RC(4)040); National Natural Science Foundation of China (61727816, 61905034).
Disclosures
The authors declare no conflicts of interest.
References
1. T. Liu, J. Yin, J. Jiang, K. Liu, S. Wang, and S. Zou, “Differential-pressure-based fiber-optic temperature sensor using Fabry–Perot interferometry,” Opt. Lett. 40(6), 1049–1052 (2015). [CrossRef]
2. B. Xu, C. Wang, D. Wang, Y. Liu, and Y. Li, “Fiber-tip gas pressure sensor based on dual capillaries,” Opt. Express 23(18), 23484–23492 (2015). [CrossRef]
3. Y. Yang, E. Wang, K. Chen, Z. Yu, and Q. Yu, “Fiber-Optic Fabry–Perot Sensor for Simultaneous Measurement of Tilt Angle and Vibration Acceleration,” IEEE Sens. J. 19(6), 2162–2169 (2019). [CrossRef]
4. K. Chen, M. Guo, S. Liu, B. Zhang, H. Deng, Y. Zheng, Y. Chen, C. Luo, L. Tao, M. Lou, and Q. Yu, “Fiber-optic photoacoustic sensor for remote monitoring of gas micro-leakage,” Opt. Express 27(4), 4648–4659 (2019). [CrossRef]
5. D. Pawar, C. N. Rao, R. K. Choubey, and S. N. Kale, “Mach-Zehnder interferometric photonic crystal fiber for low acoustic frequency detections,” Appl. Phys. Lett. 108(4), 041912 (2016). [CrossRef]
6. M. J. Murray, A. Davis, and B. Redding, “Fiber-Wrapped Mandrel Microphone for Low-Noise Acoustic Measurements,” J. Lightwave Technol. 36(16), 3205–3210 (2018). [CrossRef]
7. J. Ma, Y. Yu, and W. Jin, “Demodulation of diaphragm based acoustic sensor using Sagnac interferometer with stable phase bias,” Opt. Express 23(22), 29268–29278 (2015). [CrossRef]
8. C. Han, C. Zhao, H. Ding, and C. Chen, “Spherical microcavity-based membrane-free Fizeau interferometric acoustic sensor,” Opt. Lett. 44(15), 3677–3680 (2019). [CrossRef]
9. M. Islam, M. Ali, M. Lai, K. Lim, and H. Ahmad, “Chronology of Fabry-Perot interferometer fiber-optic sensors and their applications: a review,” Sensors 14(4), 7451–7488 (2014). [CrossRef]
10. J. Xu, X. Wang, K. L. Cooper, and A. Wang, “Miniature all-silica fiber optic pressure and acoustic sensors,” Opt. Lett. 30(24), 3269–3271 (2005). [CrossRef]
11. Q. Wang and Q. Yu, “Polymer diaphragm based sensitive fiber optic Fabry-Perot acoustic sensor,” Chinese Optics Letters,” Chin. Opt. Lett. 8(3), 266–269 (2010). [CrossRef]
12. Z. Gong, K. Chen, X. Zhou, Y. Yang, Z. Zhao, H. Zou, and Q. Yu, “High-sensitivity Fabry-Perot interferometric acoustic sensor for low-frequency acoustic pressure detections,” J. Lightwave Technol. 35(24), 5276–5279 (2017). [CrossRef]
13. F. Xu, J. Shi, K. Gong, H. Li, R. Hui, and B. Yu, “Fiber-optic acoustic pressure sensor based on large-area nanolayer silver diaphragm,” Opt. Lett. 39(10), 2838–2840 (2014). [CrossRef]
14. B. Liu, H. Zhou, L. Liu, X. Wang, M. Shan, P. Jin, and Z. Zhong, “An Optical Fiber Fabry-Perot Microphone Based on Corrugated Silver Diaphragm,” IEEE Trans. Instrum. Meas. 67(8), 1994–2000 (2018). [CrossRef]
15. B. Zhang, K. Chen, Y. Chen, B. Yang, M. Guo, H. Deng, F. Ma, F. Zhu, Z. Gong, W. Peng, and Q. Yu, “High-sensitivity photoacoustic gas detector by employing multi-pass cell and fiber-optic microphone,” Opt. Express 28(5), 6618–6630 (2020). [CrossRef]
16. K. Chen, B. Zhang, M. Guo, Y. Chen, H. Deng, B. Yang, S. Liu, F. Ma, F. Zhu, Z. Gong, and Q. Yu, “Photoacoustic trace gas detection of ethylene in high-concentration methane background based on dual light sources and fiber-optic microphone,” Sens. Actuators, B 310, 127825 (2020). [CrossRef]
17. X. Fu, P. Lu, J. Zhang, Z. Qu, W. Zhang, Y. Li, P. Hu, W. Yan, W. Ni, D. Liu, and J. Zhang, “Micromachined extrinsic Fabry-Pérot cavity for low-frequency acoustic wave sensing,” Opt. Express 27(17), 24300–24310 (2019). [CrossRef]
18. Y. Wu, C. Yu, F. Wu, C. Li, J. Zhou, Y. Gong, Y. Rao, I. E. E. E. Fellow, O. S. A. Fellow, and Y. Chen, “A highly sensitive fiber-optic microphone based on graphene oxide membrane,” J. Lightwave Technol. 35(19), 4344–4349 (2017). [CrossRef]
19. W. Ni, P. Lu, X. Fu, W. Zhang, P. Shum, H. Sun, C. Yang, D. Liu, and J. Zhang, “Ultrathin graphene diaphragm-based extrinsic Fabry-Perot interferometer for ultra-wideband fiber optic acoustic sensing,” Opt. Express 26(16), 20758–20767 (2018). [CrossRef]
20. K. Chen, Z. Gong, M. Guo, S. Yu, C. Qu, X. Zhou, and Q. Yu, “Fiber-optic Fabry-Perot interferometer based high sensitive cantilever microphone,” Sens. Actuators, A 279, 107–112 (2018). [CrossRef]
21. K. Chen, Q. Yu, Z. Gong, M. Guo, and C. Qu, “Ultra-high sensitive fiber-optic Fabry-Perot cantilever enhanced resonant photoacoustic spectroscopy,” Sens. Actuators, B 268, 205–209 (2018). [CrossRef]
22. L. Liu, P. Lu, S. Wang, X. Fu, Y. Sun, D. Liu, J. Zhang, H. Xu, and Q. Yao, “UV adhesive diaphragm-based FPI sensor for very-low-frequency acoustic sensing,” IEEE Photonics J. 8(1), 1–9 (2016). [CrossRef]
23. Y. Zhao, M. Chen, F. Xia, and R. Lv, “Small in-fiber Fabry-Perot low-frequency acoustic pressure sensor with PDMS diaphragm embedded in hollow-core fiber,” Sens. Actuators, A 270, 162–169 (2018). [CrossRef]
24. Q. Liu, Z. Jing, A. Li, Y. Liu, Z. Huang, Y. Zhang, and W. Peng, “Common-path dual-wavelength quadrature phase demodulation of EFPI sensors using a broadly tunable MG-Y laser,” Opt. Express 27(20), 27873–27881 (2019). [CrossRef]
25. X. Mao, S. Yuan, P. Zheng, and X. Wang, “Stabilized fiber-optic Fabry–Perot acoustic sensor based on improved wavelength tuning technique,” J. Lightwave Technol. 35(11), 2311–2314 (2017). [CrossRef]
26. K. Chen, Z. Yu, Q. Yu, M. Guo, Z. Zhao, C. Qu, Z. Gong, and Y. Yang, “Fast demodulated white-light interferometry-based fiber-optic Fabry–Perot cantilever microphone,” Opt. Lett. 43(14), 3417–3420 (2018). [CrossRef]
27. K. Chen, Z. Yu, Z. Gong, and Q. Yu, “Lock-in white-light-interferometry-based all-optical photoacoustic spectrometer,” Opt. Lett. 43(20), 5038–5041 (2018). [CrossRef]
28. A. Zhou, B. Qin, Z. Zhu, Y. Zhang, Z. Liu, J. Yang, and L. Yuan, “Hybrid structured fiber-optic Fabry–Perot interferometer for simultaneous measurement of strain and temperature,” Opt. Lett. 39(18), 5267–5270 (2014). [CrossRef]
29. W. Ding, Y. Jiang, R. Gao, and Y. Liu, “High-temperature fiber-optic Fabry-Perot interferometric sensors,” Rev. Sci. Instrum. 86(5), 055001 (2015). [CrossRef]
30. P. Morris, A. Hurrell, A. Shaw, E. Zhang, and P. Beard, “A Fabry–Pérot fiber-optic ultrasonic hydrophone for the simultaneous measurement of temperature and acoustic pressure,” J. Acoust. Soc. Am. 125(6), 3611–3622 (2009). [CrossRef]
31. X. Liu, J. Jiang, S. Wang, K. Liu, L. Xue, X. Wang, Z. Ma, X. Zhang, Y. Zhang, and T. Liu, “A Compact Fiber Optic Fabry–Perot Sensor for Simultaneous Measurement of Acoustic and Temperature,” IEEE Photonics J. 11(6), 1–10 (2019). [CrossRef]
32. K. Chen, H. Deng, M. Guo, C. Luo, S. Liu, B. Zhang, F. Ma, F. Zhu, Z. Gong, W. Peng, and Q. Yu, “Tube-cantilever double resonance enhanced fiber-optic photoacoustic spectrometer,” Opt. Laser Technol. 123(2020), 105894 (2020). [CrossRef]
33. U. Gysin, S. Rast, P. Ruff, and E. Meyer, “Temperature dependence of the force sensitivity of silicon cantilevers,” Phys. Rev. B 69(4), 045403 (2004). [CrossRef]
34. L. Pan, S. Jiao, and X. Du, “The Relation Between Linear Expansion Coefficient and Yang’s Modulus of Allory Materials and Temperature at High Temperature,” Jour. Natur. Sci. Hunan. 23(2), 47–51 (2000).
35. K. Chen, X. Zhou, B. Yang, W. Peng, and Q. Yu, “A hybrid fiber-optic sensing system for down-hole pressure and distributed temperature measurements,” Opt. Laser Technol. 73, 82–87 (2015). [CrossRef]
36. Z. Yu and A. Wang, “Fast white light interferometry demodulation algorithm for low-finesse Fabry–Pérot sensors,” IEEE Photonics Technol. Lett. 27(8), 817–820 (2015). [CrossRef]
37. Y. Yang, D. Tian, K. Chen, X. Zhou, Z. Gong, and Q. Yu, “A fiber-optic displacement sensor using the spectral demodulation method,” J. Lightwave Technol. 36(17), 3666–3671 (2018). [CrossRef]
