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Abstract
To the best of our knowledge, a novel extensible multi-wavelength (EMW) method to interrogate arbitrary cavities in low-fineness fiber-optic multi-cavity Fabry-Pérot interferometric (LFMFPI) sensors is proposed and experimentally demonstrated. Based on the derived model of the LFMFPI sensor with any amount of cascaded cavities, theoretically, variation in each cavity of a LFMFPI sensor can be extracted simultaneously once the necessary parameters are acquired in advance. The feasibility of this method is successfully demonstrated in simulations and experiments utilizing LFMFPI sensors. In experiments with the LFMFPI sensor, optical path differences (OPD) of 78 nm and 2.95 µm introduced by temperature variation in two cavities, and the OPD induced by vibration with the amplitude from 5.891 nm to 38.116 nm were extracted, respectively. The EMW method is potential in multi-parameter sensing for pressure, vibration, and temperature.
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1. Introduction
Compared with traditional electrical sensors, the low-fineness fiber-optic Fabry-Pérot interferometric (LFFPI) sensor is more attractive in some aspects owing to its electromagnetic immunity, temperature endurance, compact size, high sensitivity, and fast response [1–3]. In many fields, such as exploration of oil, and aerospace, multi-parameter measurements are necessary because of the mixture of the target parameters. For instance, detecting temperature, pressure, or vibration under the oil well or in the aero-engine is critical to monitoring the working status. The low-fineness fiber-optic multi-cavity Fabry-Pérot interferometric (LFMFPI) sensor can reflect multiple parameters with different cavities, so this also makes the research of the LFMFPI sensor a hotspot [4–6]. With the development of the LFFPI sensor, the LFMFPI sensor has become a significant component. It is gradually verified to be a brilliant choice for in-situ parameters detection of combustors, turbines, and flight testing [7–9].
The difficulty in applying the LFMFPI sensor lies in the interrogation methods. The interrogation techniques for LFMFPI sensors can be categorized into three groups, spectra-based methods, Fizeau-based non-scanning cross-correlation methods, and multiple wavelengths phase demodulation techniques. The fast Fourier transform method is usually taken with different bandpass filtering on the LFMFPI sensor spectra to obtain separated spectra of different cavities, thus calculating the optical path difference (OPD) in each cavity [5,10–12]. The peak tracking technique is to trace position shifts of a certain peak or a pair of peaks on spectra, and the OPD of the corresponding cavity of the sensor can be calculated by the relation of phases [4,13–15]. The spectra correlation method constructs a virtual template function of the sensor spectra and determines the actual OPD of the cavity by seeking the extreme point of the cross-correlation operation between the function and the spectra [16–18]. Spectrometers are all needed for these spectra-based methods, whose cost limits the wide application. The Fizeau-based non-scanning cross-correlation method utilizes the optical wedge with the proper thickness range to achieve optical cross-correlation, and the un-target cavity can be excluded once the OPD is beyond the equivalent OPD of the wedge thickness, so the certain cavity in the LFMFPI sensor can be interrogated [19,20]. The demands on complex optical systems and high-precision optical wedges increase the difficulty of application. Dual-wavelength and three-wavelength demodulation techniques combine interferometric outputs at diverse wavelengths to construct a pair of quadrature signals, and the OPD of the shortest cavity can be obtained by converting the phase recovered from the quadrature signals [21,22]. Spectral equipment is not required in this technique, but the demodulation range is small, and only the certain cavity can be interrogated.
In this paper, a novel interrogation approach named the extensible multi-wavelength (EMW) method for the extraction of the OPD changes in several cascaded cavities from the LFMFPI sensor is proposed. The interferometric signal of the LFMFPI sensor with any number of cavities is modeled and analyzed. Based on the model, multiple parameters such as pressure or acceleration and temperature can be synchronously interrogated precisely by a simple system. The extensibility of the inverse matrix established shows potential for acquiring OPD changes from more different cavities, providing a reference for later work of obtaining signals from more cavities. The interrogation method has properties of high resolution, large interrogation range, cost-friendly, and multi-parameter interrogation.
2. Principle
The interferometric model of reflective beams output by the LFMFPI sensor with arbitrary amounts of cavities is depicted in Fig. 1. If there are h parallel cascaded reflective surfaces in the sensor, for the $p$th surface, the reflective field ${E_{{r_p}}}$ can be conveyed as [23]
The normalized total interferometric light ${I_r}/{I_{in}}$ reflected by all the surfaces of the LFMFPI sensor with $h - 1$ cavities should be written as
LFMFPI sensors with three reflective surfaces are used to demonstrate the EMW interrogation technique. The schematic diagram illustrating a sensing system applied with the EMW method for the interrogation of LFMFPI sensors is shown in Fig. 2. The output light of amplified spontaneous emission (ASE) light source goes through a circulator to approach a LFMFPI sensor. Then the reflected light goes back through the circulator and it is divided into seven beams with different center wavelengths through a seven-channel wavelength division multiplexer (WDM) and received by photodetectors (PDs). All optical devices are connected by single-mode fiber. The converted electrical signals are acquired by a multi-channel data acquisition (DAQ) card and transmitted to a computer to finish data processing. If refractive indices of different optical medium in the sensor satisfy the relations that ${n_0} > {n_1},\textrm{ }{n_1} < {n_2}$ and ${n_2} > {n_3}$, then ${c_1} = {c_3} = 0,\textrm{ }{c_2} = 1$. Therefore, the normalized reflective light signal can be expressed as
Since the inverse matrix ${\mathbf{A}^{ - 1}}$ is employed to solve the phase change matrix, the reversibility of $\mathbf{A}$, which is related to the value of initial phase differences has to be discussed. If there are only some zero points in the determinant of $\mathbf{A}$, this method can still be utilized by selecting appropriate initial cavity lengths. In other words, the demodulation range is constrained by the determinant of $\mathbf{A}$, and this will also be investigated in the simulation section. For a determined sensor, the initial cavity length ${l_1}$ and ${l_2}$ can be obtained by the white-light interferometry (WLI) method before tests. Then the instant phase change can be derived by
3. Simulation
The feasibility of the EMW method was initially demonstrated by numerical simulation with a dual-cavity LFMFPI sensor. Main parameters used in simulations are listed in Table 1. The inverse matrix computation requires that the coefficient matrix is a non-singular matrix, from which condition the length of two cavities will be limited. To start with, the length of cavity 2 was set to 300 µm at first, and that in cavity 1 varied from 0 to 1 mm with a step of 1 µm, and then the length of cavity 1 was set to 80 µm and in cavity 2 increased from 0 to 1 mm with a step of 1 µm. The determinant value of $\mathbf{A}$ versus the lengths of cavity 1 and cavity 2 are portrayed in Fig. 4(a). It can be seen from Fig. 4(a) that the determinant values are not always zero, and the zero points in the curves denote the dead zones of the interrogation range. A 100 Hz sinewave with an amplitude of 3 µm was superimposed on cavity 1 for verification. The absolute errors of the interrogated cavity length in cavity 1 obtained with different initial cavity lengths are shown in Fig. 4(b). At cavity lengths corresponding to non-zero determinant values, the interrogation error is acceptable, and the points with significantly higher errors match well with those dead zones shown in Fig. 4(a).
Fig. 4. (a) The determinant values versus the length of cavity 1 and cavity 2, (b) the maximum absolute error versus diverse initial cavity length.
Table 1. Parameters list of simulations.
The wavelengths involved are listed in Table 1, and the relative phase shift between ${\lambda _j}$ and ${\lambda _c}$ is ${{\Delta ({\Delta {\theta_{ij}}} )} / {\Delta {\theta _{ij}} = }}{{4\pi n\Delta l[{{{({{\lambda_j} - {\lambda_c}} )} / {{\lambda_j}{\lambda_c}}}} ]} / {({{{4\pi n\Delta l} / {{\lambda_j}}}} )}} \le {{{\lambda _7}} / {{\lambda _c}}} - 1 \approx 0.31\%$, where $\Delta l$ and $\Delta {\theta _{ij}}$ are the length change and the phase change of the cavity. $\Delta (\Delta {\theta _{ij}})$ is the variation of phase change, and n denotes the refractive index. In this circumstance, the approximation was employed, and the dual-cavity LFMFPI sensor described in Table 1 was used to investigate the error introduced by the approximation. A signal composed of a sinusoidal vibrating signal with an amplitude of 1.5 µm and a monotonically varied zero-point drift from 0 to 1 µm was imposed on cavity 1. At the same time, the length change of cavity 2 increased from 0 to 1 µm and stayed static after the variation reached 1 µm. The refractive index in cavity 2 increased by 0.0034 when the length of cavity 2 increased. The threshold of OPD correction was set to 194 nm. Considering cavity length and refractive index vary simultaneously, interrogation results are shown as OPD in Fig. 5. Figure 5(a) and (b) show that the interrogated OPD without correction is close to the true values, but the maximum errors increase with the growth of OPD change. That is because the real OPD of the cavity is gradually shifted from the initial value used in demodulation, and thus the error induced by the approximation progressively makes the prerequisite not valid. However, with the introduction of the OPD correction processing, which is explained in Fig. 3 and the last paragraph of the principle section, the EMW method greatly diminishes the effect of OPD offset, improving the interrogation accuracy. This is confirmed by the much smaller error after correction than the error without correction as shown in Figs. 6(a) and (b).
Fig. 5. (a) The true OPD and the interrogated OPD of cavity 1, (b) the true OPD and the interrogated OPD of cavity 2.
Fig. 6. (a) Relative error of the interrogation method, (b) absolute error of the interrogation method.
4. Experiment
To validate the practical performance of the EMW technique in measuring OPD changes in different cavities induced by vibration and temperature, a test system as shown in Fig. 7(a) was built. The test system consisted of the EMW interrogation system, a tube furnace and its temperature controller, and a vibration excitation system. The light emitted by the ASE source ranged from 1527 nm to 1567 nm. The center wavelengths of seven beams output by the WDM were 1545.32 nm, 1546.52 nm, 1548.11 nm, 1549.72 nm, 1551.32 nm, 1552.93 nm, and 1554.54 nm, with nearly the same bandwidth of 0.6 nm. In the experiment, a LFMFPI sensor with a structure similar to the design in the Ref. [2] was used. The sensor, as shown in Fig. 7(b), was formed by thrusting a single mode fiber into a sensing head, which was anodically bonded by the glass shell, cross beams with silicon central mass block, and the glass base. The fiber and sensor head were fixed by ceramic glue. The cross beam and mass block was fabricated by MEMS process. Cavity 1 of the sensor was an air cavity consisting of the parallel arranged fiber end face and the top surface of the mass block, while cavity 2 was composed of the upper and bottom surfaces of the silicon mass block. When the axial acceleration is applied to the sensor, the cross beams with central mass block will oscillate to introduce the length change of cavity 1. The initial lengths of cavity 1 and cavity 2 of the vibration sensor obtained by the FFT-based WLI method [26] at room temperature were 42.308 µm and 314.86 µm, respectively. The refractive indices of cavity 1 and cavity 2 were 1 and 3.476. The sensor was threaded to the top of a metal rod connected to the vibration exciter, and both were placed into the tube furnace with heat insulation cotton plugged into both ends of the tube furnace. The reference sensor was fixed on the bottom plate of the metal rod to provide the reference value of acceleration information.
To begin with, turn on the tube furnace to heat the temperature around the sensor from room temperature to 100 °C. The output of the sensor influenced by temperature change was sampled with a sampling rate of 1 kHz during this period, and the data was processed by computer. Besides, the spectra of the sensor were sampled when the temperature was 70 °C, and 100 °C. The spectra were calculated by the FFT-based WLI method to provide references for the interrogation results. It takes about 5200 seconds to raise the temperature from 25 °C to 100 °C, and the whole process was recorded as shown in Fig. 8(a). The interrogation results of this period of signals and the reference results and shown in Fig. 8(b). The OPD of both two cavities increases with temperature rise because of the thermal expansion and the variation of thermo-optic coefficient in the cavity compositions. It can be seen from the curves in Fig. 8(b) that the interrogated results of the OPD of both cavities show a similar trend to the reference OPD. The interrogated OPD of cavity 1 increases from 42.308 µm at 25 °C to 42.386 µm at 100 °C, and the reference OPD reaches 42.388 µm at 100 °C. The maximum relative error in cavity 1 is about 3.498%. The interrogated OPD of cavity 2 changes from 1094.45 µm at 25 °C to 1097.4 µm at 100 °C, while the reference OPD reaches 1097.441 µm at 100 °C. The maximum relative error in cavity 2 is about 1.651%.
Fig. 8. (a) Signals sampled by DAQ, (b) reference OPD and interrogated OPD of two cavities corresponding to temperature from 25 °C to 100 °C.
After the furnace and the sensor cooled down to room temperature, the amplitude of vibration produced by the vibration exciter was set from 4 g to 20 g with a step of 4 g. The signal frequency was 200 Hz and the sensor was measured at 25 °C. Then the tube furnace started to heat the temperature to 100 °C again. When the temperature rose to 70 °C and 100 °C, the temperature would be kept for 40 minutes, and a set of the same vibration signals as before were sampled after a temperature holding time of 30 minutes. The vibration signals were sampled by the multi-channel DAQ with the same sampling rate of 100 kHz per channel. The test was conducted for 5 rounds. The sampled signals and the interrogation results are shown in Figs. 9 – 11. Figure 9(a) to (c) demonstrate the original vibration signals sampled by DAQ. To find out the actual response to the 200 Hz sinewave vibration signals with amplitude from 4 g to 20 g at 25 °C, 70 °C, and 100 °C, the interrogation results are filtered and depicted in Fig. 9(d) to (f). The FIR filter is designed as a Hamming-type low-pass filter, and the cutoff frequency is set as 3 kHz. The amplitudes of OPD change in cavity 1 introduced by the vibration signals vary from 5.891 to 27.404 nm at 25 °C, 6.929 to 33.807 nm at 70 °C, and 7.404 to 38.116 nm at 100 °C, respectively. As the temperature increases, the modulus of elasticity of the mass block (cavity 2) of the vibration sensor used in the experiment decreases, increasing the vibration sensitivity of the sensor. The linear fittings including error bars of standard deviation (SD) to 5 rounds of interrogated results of vibration signals are plotted in Fig. 10. ${R^2} > 0.999$ all the time, and the interrogated results are all included in ${\pm} 2\textrm{SD}$ around mean values. The nonlinearity error of the interrogation results is shown in Fig. 11(a). And the maximum nonlinearity error is 0.602% at 25 °C, 1.252% at 70 °C, and 1.015% at 100 °C. The power density spectrum of the interrogated results is presented in Fig. 11(b). The OPD changes of cavity 1 induced by vibration are successfully interrogated through the EMW method, and the length change of the corresponding cavity can be converted with OPD. In other words, the length change of the corresponding cavity can be obtained.
Fig. 9. Vibration signals with amplitude of 20 g sampled at (a) 25 °C, (b) 70 °C, (c) 100 °C, interrogation results of 200 Hz, 4 g to 20 g vibration signals at (d) 25 °C, (e) 70 °C, (f) 100 °C.
Fig. 11. (a) Nonlinearity of interrogation results of the vibration signals, (b) frequency spectrum of interrogated outputs at different temperatures.
The experimental results show that the EMW technique with seven wavelengths was used to successfully extract the OPD variation in a LFMFPI sensor with two Fabry-Pérot cavities. The feasibility of the interrogation technique is verified. On the basis of the derived model, it is considered that each cavity of the LFMFPI sensor with any number of cavities can be interrogated. If there are N cascaded Fabry-Pérot cavities in the LFMFPI sensor, the coefficient matrix order and the number of adopted wavelengths should be ${N^2} + N + 1$. For example, when interrogating a LFMFPI sensor containing four cascaded cavities, twenty-one wavelengths are needed. Due to the limitation of the experimental conditions and the performance of the hardware system, only a dual-cavity LFMFPI sensor and seven wavelengths were employed to validate the interrogation technique. However, it can be seen that the EMW technique is expected to be applied to LFMFPI sensors with arbitrary cavities and realize the real-time dynamic multi-parameter interrogation in the future.
5. Conclusion
The EMW technique for readout of OPD changes from different cavities of the LFMFPI sensor is verified through model analysis, simulations, and experiments. The variation of OPD in diverse cavities of the sensor is successfully extracted through calculation with the initial state inverse matrix of the sensor and the seven settled wavelengths. With temperature increased from 25 °C to 100 °C, OPD changes of 78 nm in cavity 1 and 2.95 µm in cavity 2 are extracted successfully. The maximum relative errors between the interrogated results and the reference results are 3.498% in cavity 1 and 1.651% in cavity 2. The recovered OPD change influenced by the temperature signal can be used to compensate accuracy of testing pressure or vibration signals in the future [27]. The 200 Hz sinusoidal OPD change with amplitude from 5.891 to 38.116 nm within 25 °C to 100 °C introduced by vibration signal is successfully interrogated from the LFMFPI sensor. The maximum nonlinearity error is 1.252% from 25 °C to 100 °C. The system can be miniatured through equal substitution of several DACs and integration of the sampling part, so the whole system can be compact. The interrogation technique has potential in multi-parameter measurements utilizing LFMFPI sensors.
Funding
National Natural Science Foundation of China (51935011, 52075505) and Innovative Research Group Project (51821003); National Major Science and Technology Projects of China (J2019-V-0015-0110); special Fund for Science and Technology Innovation Group of Shanxi Province (202204051001016); Shanxi Province Science Foundation for Youths (202303021212192).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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. Y. Cui, Y. Jiang, Y. Zhang, et al., “Sapphire optical fiber high-temperature vibration sensor,” Opt. Express 30(2), 1056–1065 (2022). [CrossRef]
2. J. Qian, P. Jia, H. Liu, et al., “A MEMS Fiber-Optic Fabry-Perot Vibration Sensor for High-Temperature Applications,” IEEE Access 10, 42908–42915 (2022). [CrossRef]
3. J. Liu, P. Jia, F. Feng, et al., “MgO Single Crystals MEMS-Based Fiber-Optic Fabry–Perot Pressure Sensor for Harsh Monitoring,” IEEE Sens. J. 21(4), 4272–4279 (2021). [CrossRef]
4. J. Zhu, P. Nan, K. Lim, et al., “Double F-P Interference Optical Fiber High Temperature Gas Pressure Sensor Based on Suspended Core Fiber,” IEEE Sens. J. 21(23), 26805–26813 (2021). [CrossRef]
5. J. Yin, T. Liu, J. Jiang, et al., “Batch-Producible Fiber-Optic Fabry–Pérot Sensor for Simultaneous Pressure and Temperature Sensing,” IEEE Photonics Technol. Lett. 26(20), 2070–2073 (2014). [CrossRef]
6. X. Yu, S. Wang, J. Jiang, et al., “Hybrid Sapphire Dual-Fabry—Perot-Cavities Sensor for High Temperature and Refractive Index Measurement,” J. Lightwave Technol. 39(12), 3911–3918 (2021). [CrossRef]
7. W. J. Pulliam, P. M. Russler, and R. S. Fielder, “High-temperature, high-bandwidth, fiber optic, MEMS pressure-sensor technology for turbine-engine component testing,” Proc. SPIE 4578, 229–238 (2002). [CrossRef]
8. M. Palmer, M. Davis, G. Engelbrecht, et al., “Un-Cooled Fiber-Optic Pressure Sensor for Turbine Engines, Operation to 1922°F and 500psig,” in 44th AIAA Aerospace Sciences Meeting and Exhibit (American Institute of Aeronautics and Astronautics, 2006). [CrossRef]
9. S. U. Kienitz, L. Lohr, M. J. Schmid, et al., “Static and Dynamic Pressure Measurement in Flight Test Application With Optical Fabry–Pérot Sensors,” IEEE Trans. Instrum. Meas. 70, 1–11 (2021). [CrossRef]
10. H. Gao, Y. Jiang, Y. Cui, et al., “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]
11. D. Zhao, Y. Wu, and J. Wu, “Pressure and Temperature Sensor Based on Fiber-Optic Fabry-Perot Interferometer by Phase Demodulation,” IEEE Access 7, 179532–179537 (2019). [CrossRef]
12. Y. Cui, Y. Jiang, T. Liu, et al., “A Dual-Cavity Fabry–Perot Interferometric Fiber-Optic Sensor for the Simultaneous Measurement of High-Temperature and High-Gas-Pressure,” IEEE Access 8, 80582–80587 (2020). [CrossRef]
13. 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]
14. J. Wang, L. Li, S. Liu, et al., “Investigation of Composite Structure with Dual Fabry–Perot Cavities for Temperature and Pressure Sensing,” Photonics 8(5), 138 (2021). [CrossRef]
15. G. Liu, W. Hou, and M. Han, “Unambiguous Peak Recognition for a Silicon Fabry–Pérot Interferometric Temperature Sensor,” J. Lightwave Technol. 36(10), 1970–1978 (2018). [CrossRef]
16. H. Chen, J. Liu, X. Zhang, et al., “High-Order Harmonic-Frequency Cross-Correlation Algorithm for Absolute Cavity Length Interrogation of White-Light Fiber-Optic Fabry-Perot Sensors,” J. Lightwave Technol. 38(4), 953–960 (2020). [CrossRef]
17. G. Zhang, Z. Xiong, M. Guo, et al., “Low-Frequency Optical Fiber Fabry-Perot Acoustic Sensor Based on All-Phase Cross-Correlation Demodulation,” J. Lightwave Technol. 40(22), 7431–7438 (2022). [CrossRef]
18. J. Kang, H. Chen, X. Zhang, et al., “Valley-positioning-assisted discrete cross-correlation algorithm for fast cavity length interrogation of fiber-optic Fabry–Perot sensors,” Measurement 198, 111411 (2022). [CrossRef]
19. Z. Guo, W. Lv, W. Wang, et al., “Absolute Single Cavity Length Interrogation of Fiber-Optic Compound Fabry–Perot Pressure Sensors Through a White Light Non-Scanning Correlation Method,” Sensors 19(7), 1628 (2019). [CrossRef]
20. H. Chen, Q. Chen, W. Wang, et al., “Fiber-optic, extrinsic Fabry–Perot interferometric dual-cavity sensor interrogated by a dual-segment, low-coherence Fizeau interferometer for simultaneous measurements of pressure and temperature,” Opt. Express 27(26), 38744–38758 (2019). [CrossRef]
21. Q. Ren, P. Jia, G. An, et al., “Dual-wavelength demodulation technique for interrogating a shortest cavity in multi-cavity fiber-optic Fabry–Pérot sensors,” Opt. Express 29(20), 32658–32669 (2021). [CrossRef]
22. Q. Ren, P. Jia, J. Qian, et al., “Three-wavelength Dynamic Demodulation Technique for Multi-cavity Fiber Fabry-Perot Sensor,” Acta Photonica Sinica 51(9), 0906005 (2022). [CrossRef]
23. M. Han and A. Wang, “Exact analysis of low-finesse multimode fiber extrinsic Fabry-Perot interferometers,” Appl. Opt. 43(24), 4659–4666 (2004). [CrossRef]
24. Y. Rao, M. Deng, D. Duan, et al., “In-line fiber Fabry-Perot refractive-index tip sensor based on endlessly photonic crystal fiber,” Sens. Actuators, A 148(1), 33–38 (2008). [CrossRef]
25. Q. Ren, P. Jia, G. An, et al., “Self-compensation three-wavelength demodulation method for the large phase extraction of extrinsic Fabry–Pérot interferometric sensors,” Opt. Laser. Eng. 164, 107535 (2023). [CrossRef]
26. Y. Jiang, “Fourier Transform White-Light Interferometry for the Measurement of Fiber-Optic Extrinsic Fabry–PÉrot Interferometric Sensors,” IEEE Photonics Technol. Lett. 20(2), 75–77 (2008). [CrossRef]
27. P. Jia, D. Wang, G. Yuan, et al., “An Active Temperature Compensated Fiber-Optic Fabry–Perot Accelerometer System for Simultaneous Measurement of Vibration and Temperature,” IEEE Sens. J. 13(6), 2334–2340 (2013). [CrossRef]
