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Abstract
Vernier effect-based optical fiber sensors have been demonstrated for high-sensitivity measurements of a diverse array of physical and chemical parameters. The interrogation of a Vernier sensor typically needs a broadband source and an optical spectrum analyzer to measure amplitudes over a broad wavelength window with dense sampling points, facilitating accurate extraction of the Vernier modulation envelope for sensitivity-improved sensing. However, the stringent requirement on the interrogation system limits the dynamic sensing capability of Vernier sensors. In this work, the possibility of employing a light source with a small wavelength bandwidth (35 nm) and a coarsely resolved spectrometer (∼166 pm) for the interrogation of an optical fiber Vernier sensor is demonstrated with the assistance of a machine learning-based analysis technique. Dynamic sensing of the exponential decay process of a cantilever beam has been successfully implemented with the low-cost and intelligent Vernier sensor. This work represents a first step towards a simpler, faster, and cheaper way to characterize the response of optical fiber sensors based on the Vernier effect.
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1. Introduction
The possibility of employing the Vernier effect (VE) as a viable approach to enhancing the sensitivity of optical fiber interferometric sensors has been extensively explored in very recent years [1–4]. A Vernier sensor system involves two individual optical fiber interferometers, i.e., a sensing interferometer and a reference interferometer. The superposition of the interference fringes from the two interferometers generates a Vernier modulation envelope at the characteristic spectrum of the system. When subject to an external perturbation, the spectral shift of the Vernier envelope is significantly magnified, compared to the spectral shift of the single sensing interferometer. The sensitivity magnification factor is determined by the deviation of the optical path difference (OPD) between the two interferometers that constitute the system; the smaller the deviation, the larger the magnification factor. To date, the Vernier effect system has been implemented using various types of optical fiber interferometers for sensing different physical and chemical quantities, such as static strain [5–9], temperature [10–13], humidity [14,15], static pressure [16,17], refractive index [18–22], biomarker [23,24], etc. As an extension of the optical Vernier effect, the so-called microwave photonics Vernier effect has also been proposed [25–29], showing potential for high-sensitivity distributed optical fiber sensing.
Despite the great efforts that have been made in the development of Vernier effect-based optical fiber sensors (VE-OFSs), their practical applications remain hindered. As pointed out in a recent critical survey [1], one of the major limiting aspects of a VE-OFS system is the interrogation system. Typically, a broadband light source and a sophisticated benchtop optical spectrum analyzer (OSA) are needed to obtain the characteristic spectrum of a Vernier sensor with dense sampling points. Discrete fringe dips are determined, and then the Vernier envelope is extracted based on a nonlinear curve fit. The associated cumbersome signal-processing steps could introduce additional errors that might even deteriorate the overall performance of the Vernier sensor system, compared to a single interferometer system [30]. Meanwhile, the dynamic sensing capability of a VE-OFS is compromised due to the use of an OSA, although an attempt at acoustic sensing was reported [31]. Thus, this is an urgent need for the development of a simpler, cheaper, and faster system that enables accurate and economic characterization of VE-OFSs.
Machine learning (ML) has recently found successful applications in optics and photonics [32–35]. The use of ML for optical fiber sensors has also been explored [36,37]. Systems with expanded functionality, higher accuracy, and more intelligence have been made possible by combining ML with conventional fiber sensors. For example, a single one-dimensional sensor was proved able to sense three-dimensional space with ML as the tool for signal analysis [38,39]; a multimode fiber was employed for spatially distributed sensing in a cost-effective way by using ML to analyze the speckle patterns at the output [40,41]; accurate monitoring of temperature using a hybrid fiber structure device was proved even in the presence of strong environmental noise [42]. These functionalities would have been challenging, if not impossible, to realize based on conventional signal demodulation techniques, e.g., dip tracking and curve fitting methods.
Inspired by the recent success of ML in the field of sensing, in this work, the possibility of employing ML for the demodulation of VE-OFSs is further explored [43]. The distinct advantage is that the ML-based analysis uses the full information contained in the measured raw spectrum and maps the measurand of interest directly to the full spectrum, instead of only using a few principal features (e.g., typically dip wavelength). Thanks to the powerful one-to-one mapping, the strict requirements on the interrogation system (i.e., light source and OSA) can be remarkably released without sacrificing performance. Thus, an entirely new interrogation system assisted with ML analysis based on a C-band light source with a 35-nm bandwidth and a coarsely resolved spectrometer with a 166-pm resolution is demonstrated. Monitoring the exponential decay of a cantilever beam has been successfully implemented based on the interrogator with an update rate of 5 kHz (limited by the spectrometer) using a conventional VE-OFS that was reported previously [43], showing enhanced functionality.
2. Sensor system
A schematic diagram of the proposed fast and cost-effective system is given in Fig. 1(a). Instead of using an ultra-broadband source and an OSA (Yokogawa, AQ6370D), a C-band amplified spontaneous emission (ASE) source with a 35-nm bandwidth and a coarsely resolved spectrometer (BaySpec, FBGA) with an update rate of 5 kHz are used. The wavelength range of the spectrometer is ∼85 nm with 512 pixels, indicating a wavelength sampling interval of 166 pm. Considering the wavelength bandwidth of the ASE source, the effective sampling points of the spectrum obtained from the spectrometer are 209. A 3 dB-coupler assisted VE-OFS is employed in this work, as this type of configuration is easy to construct and stable [44]. Two air-gap Fabry-Perot interferometers (FPIs) are connected to the coupler in parallel. The construction of the FPI device is simple, where a short length of a capillary with an inner diameter of 50 µm and an outer diameter of 125 µm is spliced between two single-mode fibers. Note that the structural design of the sensing FPI can be further optimized using a microfiber offset-assisted configuration to improve strain sensitivity [44]. A conventional interrogation system based on a superluminescent diode (SLED) source and an OSA is also employed for comparison purposes.
Fig. 1. The proposed cost-effective, fast, and simple interrogation approach for VE-OFSs. (a) Experimental setup including the interrogation system and a 3-dB coupler assisted VE-OFS based on two separated air-gap FPIs. (b) Illustration of the ML-based signal analysis that directly maps the measured coarsely resolved spectrum to the measurand of interest (i.e., strain).
The sensitivity magnification factor for the VE-OFS can be calculated based on
where OPDsen and OPDref represent the OPD of the sensing interferometer and the reference interferometer, respectively. Thus, a smaller difference between the OPD of the reference interferometer and the sensing interferometer leads to a larger magnification factor. However, a larger magnification factor results in a Vernier envelope with a larger free spectral range (FSR) based on where FSRsen and FSRref denote the FSR of the sensing interferometer and the reference interferometer, respectively. Thus, the wavelength observation window prescribed by the bandwidth of the light source limits the largest magnification factor that can be physically realized; a larger magnification factor requires a light source with broader wavelength bandwidth.3. Results and discussion
3.1 System characterization
A comparison between the measured spectrum from the VE-OFS by the SLED-OSA interrogator and the ASE-spectrometer interrogator is shown in Fig. 2. The wavelength observation window limited by the hardware is 1450∼1610 nm and 1528∼1563 nm for the SLED-OSA and ASE-spectrometer system, respectively. The OSA was configured with 16001 sampling points. The cavity length of the sensing FPI and reference FPI was first determined to be 414.70 µm and 404.08 µm, respectively. According to Eq. (2), the FSR of the VE-OFS is expected to be approximately 110 nm. One can observe that the measured spectrum by the SLED-OSA interrogator revealed an FSR of ∼110 nm, as expected. Slight distortions of the Vernier envelope around ∼1510 nm can be observed, which is due to the anti-resonant guidance effect in the capillary [45]. For the case of the ASE-spectrometer, the wavelength observation window did not cover a complete FSR, making it challenging to extract the Vernier envelope that could be used for sensitivity-improved sensing. Meanwhile, the wavelength sampling of the spectrometer is much coarser than that of the OSA interrogator.
Fig. 2. Measured spectrum of the VE-OFS from (a) the SLED-OSA interrogator and (b) the ASE-spectrometer interrogator.
Tensile strains are incrementally applied to the sensing FPI by stretching the device using two translation stages (OMTOOLS, HFA-XYZ). The measured spectral responses of the VE-OFS are shown in Fig. 3. The evolution of the Vernier envelope is reliably quantified by the OSA-based interrogation, as can be observed in Fig. 3(a), whereas the spectral responses of the spectrometer-based interrogation shown in Fig. 3(b), are quite ambiguous due to the limited wavelength observation window. The spectral responses measured from the OSA can be analyzed based on the conventional method, where the Vernier envelope can be extracted first and then the dip wavelength in the Vernier envelope can be determined, which is then correlated to the tensile strain applied to the sensing FPI for sensitivity improved sensing.
Fig. 3. Measured spectra of the VE-OFS from (a) the SLED-OSA interrogator and (b) the ASE-spectrometer interrogator for different settings of tensile strains applied to the sensing FPI.
Figure 4 gives the shift of the envelope dip wavelength as a function of tensile strain. The responses from the sensing FPI without using the Vernier effect are also shown for the purpose of comparison. A sensitivity magnification factor of 37 is revealed, compared to the single sensing FPI, matching with the expected value from Eq. (1).
Fig. 4. Vernier envelope dip wavelength shift as a function of tensile strain determined from the OSA measurements. The response of the single FPI without the Vernier effect is shown for comparison.
The SLED-OSA interrogator is a widely used apparatus for quantifying the spectral responses of VE-OFSs. However, there are a few limitations to this approach. The system cost is quite high due to the involvement of the benchtop OSA. Meanwhile, the update rate of the OSA is restricted, due to the requirement of dense wavelength sampling over a broad range of wavelengths, hindering the application of VE-OFSs for dynamic sensing. Moreover, the signal processing is quite complicated and cumbersome, involving multi-step nonlinear approximation that could deteriorate the overall sensing performance of VE-OFSs. Thus, there is an urgent need for the development of a new approach for interrogating VE-OFSs. The ASE-spectrometer could be a potential solution. It should be noted that the responses measured from the spectrometer shown in Fig. 3(b) indeed contain the envelope information, but it is quite challenging to track due to the restricted wavelength window limited by the hardware (i.e., the light source). In such a context, it is rather the limitation imposed by the hardware (i.e., the ASE-spectrometer interrogator) than the software (i.e., signal demodulation). If there was an advanced signal analysis technique, the ASE-spectrometer-based interrogator could be employed for the VE-OFS, leading to a cheaper, faster, and simpler VE-OFS system. The fact that ML can address this issue will be shown.
3.2 ML-based analysis
To use ML for the demodulation of the VE-OFS, a certain amount of data needs to be acquired in the sensor calibration experiment, which can be used to train the ML model. It is worth mentioning that the fast update rate (up to 5 kHz) of the ASE-spectrometer interrogator facilitates the collection of a large amount of data, which would have been challenging for the SLED-OSA interrogator. Tensile strains were incrementally applied to the sensing FPI in steps of 50 µɛ within the range of 0∼1500 µɛ. Two hundred spectra were recorded at each setting of the tensile strain. A total of 6200 spectra were obtained in the calibration experiment. Thus, a 209-by-6200 matrix is formed.
As for the ML model, the Gaussian process regression (GPR) model was chosen, as it has been demonstrated efficient and effective in analyzing sensing signals from optical fiber sensors [46,47]. GPR is a supervised ML approach. Instead of choosing a particular function to fit (e.g., a polynomial function), GPR focuses on the relation between the input variables, and the similarity is expressed by a parameter called the kernel. One of the commonly used kernel functions is the squared-exponential one and is given by [47]
where β and θ are the hyperparameters that are optimized during the training process of the model; Xi and Xj represent two different spectra from the VE-OFS. Details regarding the optimization of the kernel can be found in [47]. To train the GPR model, a training data set needs to be generated first. The aforementioned 209-by-6200 matrix is considered the input variables, and the corresponding strains are the outputs. The root mean squared error (RMSE) is used to evaluate the performance of the model and is expressed as where ytrue_i and ypred_i are the ground truth and predicted value of the strain for the i-th input spectrum. Ten-fold cross-validation is employed to avoid over-fitting during the training.Figure 5 gives the results from the trained GPR model. The predicted values are plotted against the ground truth in Fig. 5(a), where a reference line y = x is shown to guide the eye. The discrete data points predicted by the GPR model are aligned very close to the reference line. The RMSE for the 6200 input spectra was determined to be 0.288 µɛ, demonstrating that reliable and accurate mapping between the input spectra and the output tensile strains is successfully realized by the GPR model. An error histogram for the 6200 input spectra is shown in Fig. 5(b). A typical normal distribution is revealed. The results demonstrate that the challenging issues encountered by the conventional demodulation method, when the spectrum for a VE-OFS is measured within a very restricted wavelength range, can be reliably and adequately addressed by the ML technique. Meanwhile, complicated and cumbersome data processing is avoided. The ambiguity of choosing appropriate curve fitting parameters in extracting the Vernier envelope that could directly affect the performance is avoided as well. Once the ML model is well trained, for an input spectrum, a tensile strain within the calibration range is given within a ms scale.
Fig. 5. Prediction results from the GPR model. (a) Scatter plot of the predicted strains vs. the ground truth. (b) Histogram showing the error distribution.
3.3 Dynamic sensing
So far, the GPR model has been demonstrated to be efficient and effective to achieve one-to-one mapping between an input spectrum of the VE-OFS measured from the ASE-spectrometer interrogator and a tensile strain applied to the sensing FPI. A comparison between the conventional SLED-OSA interrogation system and the proposed system is summarized in Table 1. It should be noted that one of the main drawbacks of the conventional OSA-based interrogator is the significantly restricted update rate of the OSA, thus limiting VE-OFSs for dynamic measurements. The fast update rate of the spectrometer and the reliable ML-based demodulation prompted us to employ the system for dynamic sensing, leading to the first demonstration of a VE-OFS for vibration sensing.
Table 1. Comparison between the conventional interrogation system and the proposed interrogator system.
As a proof of concept, a cantilever system was used. A stainless-steel ruler was employed as the beam, and the sensing FPI was glued to the upper surface of the ruler using epoxy resin at the starting and end point of the ruler, as shown in Fig. 6. One end of the beam was fixed, and the other end was pushed upwards and released. The free end of the beam vibrated and then stabilized. The strains experienced by the FPI were different as the beam deflected. Note that a pre-strain was applied to the sensing FPI to ensure that the whole dynamic behavior can be captured by the sensor. The spectrometer was configured at a 5-kHz update rate.
Fig. 6. Demonstration of dynamic sensing of the exponential decay process of a cantilever beam using the VE-OFS.
The measured raw spectra recorded from the spectrometer were directly fed into the trained GPR model, and the corresponding strains were obtained. Figure 7(a) shows the predicted strains experienced by the sensing FPI during the experiment. An overall exponential decay envelope was observed. Details of the whole process of the experiment were captured by the system. First, the free end of the cantilever was pushed upwards manually, representing a decrease in the tensile strain applied to the FPI. Figure 7(b) shows the first push process captured by the VE-OFS system. The predicted strain decreased as the beam was deflected upwards, as expected. The non-smooth curve was due to manual operation. After reaching a point, the free end of the cantilever was released and it started to vibrate. The vibrational behavior of the cantilever was also perfectly quantified by the strains measured from the sensing FPI, verifying the capability of the VE-OFS system for dynamic sensing. An enlarged view of the strains is given in Fig. 7(c). Finally, the cantilever tended to stabilize. The strains applied to the FPI recovered to the pre-strain value. The results demonstrate that the trained GPR model can accurately predict not only strain values that are included in the training dataset, but also arbitrary strains within the calibration range. Meanwhile, thanks to the fast update rate of the spectrometer and powerful ML analysis, the dynamic behavior of the cantilever is reliably captured by the VE-OFS. Moreover, as shown in the inset of Fig. 7(a) where the predicted strains within a time period of 0.4 s, the resolution of the VE-OFS is estimated to be higher than 0.5 µɛ, demonstrating the high resolution of the GPR-based demodulation method.
Fig. 7. Monitoring the exponential decay process of the cantilever beam using the VE-OFS system. (a) The overall decaying process. The inset gives an enlarged view of the predicted strains within a time period of 0.4 s, demonstrating the high resolution of the GPR-based demodulation. (b) The captured process of pushing the free end of the cantilever beam upwards. (c) Measured dynamic strains from the sensing FPI.
4. Conclusion
This work proposed a simple, fast, and cheap interrogation approach for VE-OFS by combining a coarsely resolved spectrometer, a C-band light source, and ML-based signal demodulation. It is shown that the ML model can accurately and reliably achieve direct one-to-one mapping between a measured raw spectrum within a very restricted wavelength range (even not including the Vernier envelope dip) and the measurand of interest (a tensile strain applied to the sensing interferometer). The proposed method can be applied to other VE-OFSs based on different types of interferometers for measuring different parameters, where new ML models also have to be trained accordingly. Thanks to the fast update rate provided by the spectrometer, dynamic sensing using the prototype VE-OFS for monitoring the exponential decay process of a cantilever with high resolution, enabled by ML-based demodulation, is demonstrated, for the first time.
In the proof of concept, a well-known VE-OFS based on two FPIs arranged in parallel is used to show that new functionality (i.e., dynamic sensing) can be introduced by the proposed interrogation approach, which would have been challenging for conventional methods. This work opens new avenues for the development of high-performance VE-OFS systems. It is envisioned that a simpler, faster interrogation method based on a few laser lines and photodetectors can be further developed based on the proposed technique, as expected in [1].
Funding
Research Initiation Project of Zhejiang Lab (2022ME0PI01); Researchers Supporting Project number (RSPD2023R654), King Saud University, Riyadh.
Acknowledgments
Research Initiation Project of Zhejiang Lab (2022ME0PI01) and Researchers Supporting Project number (RSPD2023R654), King Saud University, Riyadh, Saudi Arabia.
Disclosures
The authors declare no conflicts of interest.
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.
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