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Abstract
For FPI sensor demodulation systems to be used in actual engineering measurement, they must have high performance, low cost, stability, and scalability. Excellent performance, however, necessitates expensive equipment and advanced algorithms. This research provides a new absolute demodulation system for FPI sensors that is high-performance and cost-effective. The reflected light from the sensor was demultiplexed into distinct channels using an array waveguide grating (AWG), with the interference spectrum features change translated as the variation of the transmitted intensity in each AWG channel. This data was fed into an end-to-end neural network model, which was utilized to interrogate multiple interference peaks’ absolute peak wavelengths simultaneously. This architecturally simple network model can achieve remarkable generalization capabilities without training large-scale datasets using an appropriate data augmentation strategy. Experiments show that in simultaneous multi-wavelength and cavity length interrogations, the proposed system has the precision of up to ± 14 pm and ± 0.07 µm, respectively. The interrogation resolution can theoretically reach the pm level benefit from the neural network method. Furthermore, the system’s outstanding demodulation repeatability and suitability were demonstrated. The system is expected to provide a high-performance and cost-effective, reliable solution for practical engineering applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Fiber-optics sensors have been widely applied in physics and chemistry due to their small size, lightweight, anti-electromagnetic interference, and corrosion resistance. Compared to Fiber Bragg grating (FBG) sensors [1], polarization type fiber sensors [2], Mach-Zender interferometer (MZI) fiber sensors [3], and Sagnac interferometric sensor [4], Fabry-Perot interferometric (FPI) sensors has significant advantages in measurement precision, resolution, stability and fabrication cost, and have been widely utilized in temperature [5,6], pressure [7,8], strain [9–11], biosensing [12,13] and dispersion [14] measurement applications.
The effective cavity length of FPI sensors will change in response to external perturbations (e.g., strain, temperature, pressure, etc.) in practical measurements, reliable and effective cavity length interrogation is required to get information about the measured target. FPI sensor interrogation techniques include laser interferometry [15,16] and white light interferometry (WLI) [17–21]. On the other hand, laser interferometry is limited to a small range of relative measurements of cavity length fluctuations, and the laser wavelength must be adjusted to keep the measurement stable at specific point. The system’s stability and precision are substantially harmed due to this.
WLI provides enriched sensing information by measuring the interferometer’s absolute optical path difference (OPD) and is suitable for monitoring static physical quantities. It is considered to be an excellent alternative for demodulating FPI sensors. Acquiring and analyzing the spectra of FPI sensors is the key to WLI. Nowadays, the spectra of FPI sensors are generally analyzed by fiber-optic Fabry-Perot tunable filter (FFP-TF) method [22], wavelength scanning method [23] and optical spectral analysis [24,25]. Among them, the FFP-TF method can achieve high-precision cavity length interrogation. Still, the nonlinear hysteresis behavior and poor repeatability can drastically degrade the measurement accuracy of the system. The wavelength scan method survives difficult to achieve engineering applications due to poor mechanical stability. The demodulation method of FPI sensors based on spectral detection and analysis is efficient. It does not require complex mechanical structures, but the spectrum’s peak positions are difficult to determine due to the sinusoidal distribution of the spectral profile. Numerous algorithms have been reported to reproted to identify the sensing peak of FPI sensor. Yu et al. [26] proposed a non-zero filled fast Fourier transform combined with a Buneman frequency estimation algorithm to achieve high-performance high-frequency real-time cavity length demodulation. Guo et al. [27] proposed a dual-frequency autocorrelation algorithm perform absolute demodulation of cavity length. Gui et al. [28] combined virtual reference interference (VRI) and minimum mean square error (MMSE) algorithms for absolute measurement of cavity length. For this reason, algorithms to extract sufficient information from the spectrum become important. However, these algorithms’ high complexity and low reproducibility seem to make them progressively less competitive, despite their excellent performance.
Wavelength tracking have been considered the best algorithm for extracting effective information from interferometric spectra because of its computational simplicity, high resolution, and independence from source fluctuations [29]. However, the presence of "stripe orientation blur" limits the dynamic range of the algorithm to half of the free spectral range (FSR), and the conventional wavelength tracking method only measures the relative wavelength shift of the interference peaks, which indirectly reduces the practical application of FPI sensors. Liu et al. [30] introduced a solution for calculating the OPD between two adjacent peak wavelengths in a spectrum. This significantly improves the dynamic range of the FPI sensor, while the resolution may not be satisfactory [31], and the lower resolution means that the measured physical quantities cannot be accurately calculated. Qi et al. [32] and Jiang et al. [22] introduced adjacent interference peaks to achieve high-resolution OPD calculations. Yet integer rounding errors in the operation may cause the results to deviate from a reasonable range [33]. Liu et al. [34]proposed to reduce the integer rounding errors by using two peaks far apart. Nevertheless, the method is complex and highly demands computational resources. In addition, commercial optical spectrum analyzers (OSA) are expensive and limited by tedious data processing, making them almost impossible to be applied in practical engineering scenarios. Therefore, developing an FPI sensor demodulation system with high performance, low cost, low complexity, and reusability is essential.
Artificial intelligence (AI) technology has made a splash in many domains, researchers are beginning to apply it to demodulation systems for fiber-optic sensors. Currently, machine learning (ML) algorithms are commonly utilized to improve the performance and flexibility of demodulation systems. For example, Chan et al. [35] used an adaptive linear network to improve wavelength detection accuracy for FBG sensors, Ren et al. [36] applied neural networks to FBG filters (FBGFs)-based FBG sensors demodulation systems to improve demodulation flexibility and accuracy, and Li et al. [37] applied Multilayer Perceptron Neural Network to achieve fast demodulation of FBG sensors. Our previous research used neural networks to dramatically lower the cost of FBG sensor demodulation devices while maintaining acceptable performance [38]. Unfortunately, no studies on machine learning techniques for the FPI sensor demodulation system have been reported. Attempting to demodulate an FPI sensor via peak tracking necessitates simultaneous probing of numerous peak wavelengths, which necessitates excellent model generalization performance. A high-performance model, on the other hand, must be trained on a large-scale dataset. These issues have hampered machine learning techniques in FPI sensor demodulation systems.
This paper presents a new, promising FPI sensor demodulation system based on a multi-peak wavelength tracking method to overcome the aforementioned problem, which combines array waveguide grating (AWG) and ML techniques. AWG demultiplexed the light reflected by the FPI sensor to its channels. The peak wavelength variation of each interference peak in the specified wavelength range was converted to the transmitted intensity variation under the corresponding AWG channel. The data generated by the FPI sensor under the continuous action of horizontal strain was utilized for training an end-to-end neural network that establishes a relationship between the transmitted intensity and the peak wavelength of multiple interference peaks. A data augmentation strategy was used during the training to permit the network to achieve robust generalization performance even with small-scale datasets. With simultaneous multi-wavelength and effective cavity length interrogation experiments, it is demonstrated that the proposed system has outstanding demodulation precision. In addition, experiments with several FPI sensors were also conducted to demonstrate the proposed demodulation system’s suitability. The proposed method can expand neural network techniques in the demodulation of FPI sensors, resulting in a cost-effective and customizable solution for engineering applications.
The remainder of this paper is organized as follows. Section 2 presents the theoretical analysis of the proposed demodulation system; Section 3 presents the artificial neural network model used for FPI sensor demodulation. Section 4 presents the experimental setup and performs the corresponding interrogation experiments. Section 5 discusses the related advantages of the system; Section 6 concludes this paper.
2. Demodulation scheme and theoretical analysis
2.1 Fabrication and interrogation theory of FPI sensor
2.1.1 Design and fabrication of FPI sensor
Figure 1 illustrates the fabrication process of the FPI sensor we utilized. First of all, the standard SMF (Corning SMF-28) and capillary glass tubes (Inner diameter 75 $\mathrm {\mu }$m, outer diameter 125 $\mathrm {\mu }$m) were placed in the left and right fiber holders of the commercial fusion splicer (Fujikura FSM-61S), as shown in Fig. 1(a). Secondly, as shown in Fig. 1(b), the left fiber end and the right fiber end were driven by the left and right motors of the fusion splicer, to move toward each other until the end surfaces of the two fiber end overlap in two dimensions. Where $d$ denotes the travel distance of the two fiber ends before they touch each other. Subsequently, the arc discharge was completed using the set parameters used to achieve the fusion of standard SMF, and the capillary glass tube ends in the fusion machine. As a result, the SMF and glass tube ends are spliced. Thirdly, using a precision fiber optic cleaver (Fujikura CT50) to cut the capillary glass tube over a certain length, the remaining capillary glass tube fused to the standard SMF was considered the F-P cavity component. Finally, the standard SMF with a glass tube structure and another standard SMF were placed on the left and right ends of the fusion splicer. The motor moves their ends to complete the two-dimensional overlap, followed by arc discharge using the set parameters to achieve the splicing of the two ends, completing the production of the FPI sensor we used.
Fig. 2. (a) Microscope image of the fabricated FPI sensor. (b) Reflectance interference spectrum of the FPI sensor.
The FPI sensor fabricated above with capillary glass tubes as F-P cavity have tough enough structure to be applicable to strain experiments, which microscopic image is shown in Fig. 2(a). As presented in Fig. 2(b), interference pattern were observed in the reflection spectra of the FPI sensor by utilizing a 3 dB coupler, a broadband light source (ASELIGHTSOURCE) and an optical spectrum analyzer (OSA, YOKOGAWA AQ6370D).
2.1.2 Interrogation principles for FPI sensors
The FPI sensor can be considered as a multi-velocity interferometer, and the intensity of its reflected interferometric light (ignoring absorption) can be formulated as Eq. (1) [39]:
The absolute cavity length of FPI sensors can be calculated by counting the number of fringes over a specific wavelength range. Eq. (2) can be used to express the relationship between cavity length and optical wavelength [30].
where $\lambda _{1}$ and $\lambda _{2}$ are the wavelength that is $2m\pi$ out of phase, which can be considered as the peak wavelength of two adjacent interference peak, and $m$ is an integer. The OPD of two adjacent interference peaks can be defined as Eq. (3). In summary, target measurements can be achieved by monitoring the intensity of the interference signal, and absolute measurements of the OPD are essential.2.2 Design and theoretical analysis of FPI sensor demodulation system
Figure 3 depicts the proposed demodulation system. Two three-dimensional translation stages were used to mount the sensor. One has a signal generator-driven motorized translation table that applies strain to the sensor on a regular schedule. Reflected light from the FPI sensor flows through an optical circulator (OC) and into a beamsplitter, which feeds the light in a 50/50 ratio into the optical spectrum analyzer (OSA, YOKOGAWA AQ6370D) and the AWG (40 channels, 125GHz). During system operation, nine AWG channels were used simultaneously. The other was connected directly to an optical power meter, while the other eight were connected to an 8-channel MEMS optical switch (MEMS-FSW8-SM-A) with a controller. The intensity output of each was measured using an optical power meter. Sending an electrical signal to the optical switch to open and close the AWG channels, notably, allows simultaneous use of eight channels during the measurement process without the need for additional optical signal conversion sequences [Photodetector (PD) - Amplifier (AMP) - Data Acquisition (DAQ)] or more channels of optical power meters, which reduces system costs and data processing difficulties significantly. These data are finally received by the PC for further processing.
Fig. 3. Architecture of the FPI sensor demodulation system, which includes a broadband light source module, a strain applied module, an AWG, an 8-channel MEMS optical switch module with a controller, a dual-channel optical power meter, and a PC for signal processing, in addition to an OSA for obtaining reference in experiments.
The reflected light from the FPI sensor was demultiplexed into the nine channels by the AWG’s wave decomposition multiplexer architecture. The FPI sensor’s interference spectrum shifts as a result of external strain, and the transmitted intensity of its reflected light changes under each channel, where the transmitted intensity is defined as the area of the overlapping part of the sensor’s reflected spectrum and the AWG channel, as shown in Fig. 4. Under nine AWG channels, the system monitors the transmitted intensity of the FPI sensor’s reflected light and the peak fluctuations of three distinct interference peaks. It uses a neural network to construct a nonlinear relationship between transmitted intensity ($I$) and peak wavelengths ($\lambda$), which can be established as Eq. (4). The system can interrogate the peak wavelengths of the three independent interference peaks of the FPI sensor simultaneously without OSA to achieve high precision absolute demodulation.
Fig. 4. Schematic diagram of the demodulation principle of the system, where $d$ is the peak wavelength shift of the sensor interference peak.
Although the FPI sensor’s peak wavelength is unique for each interference peak, the cavity lengths calculated from different interferometric peaks may differ slightly. To minimize the difference mentioned above, the effective cavity length $d_{eff}$ computation for the FPI sensor in the system can be formulated as Eq. (5).
3. Machine learning algorithm for the demodulation system
3.1 Artificial neural network model
A back-propagation neural network was utilized to create the relationship between transmitted intensity and peak wavelengths to demodulate the FPI sensor. The architecture of the network is depicted in Fig. 5. It comprises nine independent neurons in the input layer representing the transmitted intensity of reflected light originating from the FPI sensor under each AWG channel and three neurons in the output layer that describe the peak wavelengths of three independent interference peaks interrogated. An intermediate hidden layer was implemented to further emulate the link between input and output for the multi-level calculation of the input features.
Fig. 5. The architecture of the neural network for peak wavelength interrogation consists of an input layer with nine neurons, a five-layer hidden layer with 100 neurons in each layer, and an output layer with three neurons.
3.2 Data pre-processing
The raw data must be pre-processed by normalization before being fed into the signal processing module to prevent inconvenience due to spatial inconsistencies in the data distribution and improve the model’s convergence speed, reliability, and accuracy. Min-Max normalization is utilized to linearly transform the raw data so that the results fall in $[0,1]$, and the transformation function can be formulated as Eq. (6).
3.3 Data augmentation
In order to enhance the generalization performance of the network model and prevent overfitting, based on the mixup [40], we introduced a data augmentation method during the training process, which expands the number of training set samples without affecting the original data. The principle can be formulated as Eq. (7). The schematic diagram is shown in Fig. 6.
4. Experiments
4.1 Experimental setup
To complete the experiment, we employed a previously fabricated FPI sensor (Fig. 2) based on the system (Fig. 3). The experiment’s ambient temperature was fixed at 26 $^\circ$C, and all of the system’s devices were preheated for thirty minutes before the operation.
In the experimental preparation stage, the two ends of the FPI sensor were fixed on a three-dimensional translation table of the same height, and the distance between the translation tables was set to 0.35 $m$. The sensor was tightened by fine-tuning the translation table horizontally, and this state was appointed as its initial state. The peak wavelength interval of the two adjacent channels of the AWG used in the experiment is 0.8 $nm$, and its full width at half maximum (FWHM) is 0.456 $nm$ after actual measurement. CH16, CH20, CH24, CH28, CH32, CH36, CH40, and CH48 are selected for demodulation of the FPI sensor, where CH16 was fixed to the optical power meter, and the rest of the channels were connected to the optical switch module then input to optical power meter. The selected channels are shown in Fig. 6(a). Their combined spectra with the sensor interferometric fringes are shown in Fig. 7(b), and they were used to perform absolute peak demodulation of three interferometric peaks in the range of 1530-1560 $nm$, which are noted as Interference Peak $\#$1, Interference Peak $\#$2 and Interference Peak $\#$3 in the order of left and right, and the corresponding initial peak wavelengths were measured by OSA ($\lambda _{1}$ = 1532.4 $nm$, $\lambda _{2}$ = 1544.9 $nm$ and $\lambda _{3}$ = 1557.1 $nm$).
Fig. 7. (a) Reflection spectra of the adopted AWG channels and their peak wavelengths. (b) Combined spectrum of FPI sensor and AWG channels, where $\lambda _{1}$-$\lambda _{3}$ are the peak wavelength of Interference Peak $\#$1, $\#$2 and $\#$3, respectively.
During the experiment, the motorized translation table was translated horizontally with a resolution of 5 $\mathrm {\mu }$m to control the stretching and relaxation of the FPI sensor. The transmitted intensity of each AWG channel was caught by the optical power meter and recorded by the PC after the translation stage was slightly shifted. The related interference peak’s peak wavelength from the OSA was also recorded. Figure 8(a) depicts the sensor’s spectral changes during stretching, with the interference fringe gradually shifting to the right as stretching progresses, and Fig. 8(b) depicts the relationship between the strain and the peak wavelength of each interference peak within 1500 $\mathrm {\mu } \varepsilon$. Figure 8(c) depicts the spectrum changes of the FPI sensor during relaxation, with the interference fringe gradually shifting to the left as relaxation progresses, and Fig. 8(d) depicts the relationship between the relaxation strain and the peak wavelength of each interference peak within 1500 $\mathrm {\mu } \varepsilon$.
Fig. 8. (a) and (c) is the schematic diagram of the shift of Interference Peak $\#$1, $\#$2, and $\#$3 during stretched horizontally and relaxing horizontally, respectively. (b) and (d) is the relationship between the strain and the peak wavelength shift of Interference Peak $\#$1, $\#$2, and $\#$3 within the 1500 $\mathrm {\mu } \varepsilon$ of the sensor in stretch and relaxation, respectively.
A series of settings need to be specified before training the neural network, such as the learning rate (LR), the optimization algorithm, and the loss function. The LR was preset to 0.0001. Adam [41], Adagrad [42], RMSprop [43], and SGD [44] were chosen as the optimization algorithms to verify the model’s feasibility in simulating. Since the final goal of the network is to calculate of $\lambda _{1}$, $\lambda _{2}$ and $\lambda _{3}$, based on MSE, a multi-objective loss function was utilized, formulated as Eq. (8).
All models in this work were bulit on Pytorch, and run on the Intel Core i7-9750 (2.6GHz) and Geforce RTX2080 Max-Q GPU (8GB).
4.2 Horizontal stretch demodulation for FPI sensor
We tested the well-trained models training by the Adam, Adagrad, RMSprop, and SGD with data generated during the stretching process to illustrate the system’s favorable performance, noting that these data were never present in the training dataset and were scrambled. Figure 9 shows the interrogation errors of the models trained using the above algorithm for the three peak wavelengths in the stretching test dataset. For the three interference peaks, the wavelength interrogation error is within $\pm$ 0.06 $nm$ for all four models.
Fig. 9. Interrogation error of peak wavelength in stretching test dataset for the network trained by the Adam, Adagrad, RMSprop, and SGD.
MSE, RMSE (Root Mean Squared Error), $R^{2}$ and MAPE (Mean Absolute Percentage Error) were adopted as evaluation metrics to comprehensively evaluate the system, which can be expressed as Eq. (9)–Eq. (12), respectively.
Table 1 shows the statistical analysis of their demodulation performance using RMSE, MSE, $R^2$, MAPE, and the interrogation errors. The proposed neural network model can reach at least $\pm$ 26 $pm$ precision in wavelength interrogation, and models trained by all four algorithms can produce satisfactory interrogation performance. In a broad sense, the results trained with the RMSprop surpass the results trained with the other algorithms, which can achieved the best interrogation precision of $\pm$ 14 $pm$ when interrogating the three interference peaks simultaneously. This means that the model provides superior interrogation accuracy when performing multi-peak wavelength interrogation.
Table 1. Statistical analysis of performance evaluation metrics (MSE, RMSE, $R^2$, and MAPE) and interrogation error of models trained by Adam, Adagrad, RMSprop, and SGD, In the horizontal stretch test dataset.
Furthermore, in this test dataset, the absolute cavity length of the FPI sensor was estimated and compared to the measured values, with the results presented in Fig. 10. The difference between the cavity length interrogation values and the actual values derived from the models trained by each algorithm is minor. As indicated in Table 2, the errors of each algorithm in performing absolute cavity length interrogation were also evaluated. The neural network model has a cavity length interrogation precision of up to $\pm$ 80 $pm$ and at least $\pm$ 110 $nm$, meaning that the networks trained by the four algorithms above perform well in absolute cavity length interrogation.
Fig. 10. (a) Absolute cavity length interrogation the models trained by each algorithm in the stretching test dataset. (b), (c), (d), and (e) are the local schematics when the models trained by Adam, RMSprop, Adagrad, and SGD achieve the minimum error, respectively.
Table 2. Analysis of cavity length interrogation error of demodulation system in sensor stretching test dataset.
4.3 Horizontal relaxation demodulation for FPI sensor
Nonlinear hysteresis is a significant factor that affects the demodulation system’s effectiveness in actual measurements, and its presence reduces the system’s demodulation repeatability. To validate the system’s repeatability, test datasets derived from the FPI sensor relaxation procedure were applied to test the models trained by each algorithm. Their peak wavelength interrogation errors in the test dataset are presented in Fig. 11. The interrogation errors of these models for the peak wavelengths of the three interference peaks all fluctuate within $\pm$ 0.06 $nm$. Table 3 shows the statistical analysis of their demodulation performance using RMSE, MSE, $R^2$, MAPE, and the interrogation errors. The neural network obtains the wavelength interrogation precision of at least $\pm$ 24 $pm$ in the relaxation test dataset. The network trained by the RMSprop achieves the optimal precision of simultaneous interrogation of multiple peak wavelengths of at least $\pm$ 16 $pm$. The decreased MSE, RMSE, and MAPE reflect the system’s smoother wavelength interrogation error floating in the test dataset, confirming that demodulation operation is repeatable.
Fig. 11. Interrogation error of peak wavelength in relaxation test dataset for the network trained by the Adam, Adagrad, RMSprop, and SGD.
Table 3. Statistical analysis of performance evaluation metrics (MSE, RMSE, $R^2$, and MAPE) and interrogation error of models trained by Adam, Adagrad, RMSprop, and SGD, in the horizontal relaxation test dataset.
Similarly, the absolute cavity lengths of the FPI sensors in the test dataset were calculated and compared to the measured values, as shown in Fig. 12. The absolute demodulation errors of cavity lengths were also counted for each model, as shown in Table 4. The neural network can achieve a cavity length interrogation precision of up to $\pm$ 70 $nm$ and at least $\pm$ 110 $nm$ during the relaxation process of the FPI sensor, which is almost identical with the cavity length interrogation precision during the stretching process (Section 4.2). The cavity length interrogation precision can be achieved the best by the model trained by $RMSprop$, which indicates the excellent interrogation precision of the proposed system and reflects its excellent demodulation repeatability.
Fig. 12. (a) Absolute cavity length interrogation error of the model trained by each algorithm in the relaxation test dataset. (b), (c), (d), and (e) are the local schematics when the models trained by SGD, RMSprop, RMSprop, and Adagrad achieve the minimum error, respectively.
Table 4. Analysis of cavity length interrogation error in the FPI sensor relaxation test dataset.
4.4 Suitability assessment of the demodulation system
Suitability is critical for a demodulation system since it heavily influences the system’s technical application prospects. For additional experiments to evaluate the suitability of the proposed demodulation system, two novel FPI sensors were fabricated based on the procedure in Fig. 1. Sensor $\#$1 and Sensor $\#$2 were assigned to them, and their microstructures and reflectance spectra at 1530-1560 $nm$ are depicted in Fig. 13.
Fig. 13. (a) and (b) are microscope image and reflectance interference spectrum of Sensor $\#$1, respectively. (c) and (d) are microscope image and reflectance interference spectrum of Sensor $\#$2, respectively.
Following the previous setup (Section 4.1), we performed stretching and relaxation experiments on Sensor $\#$1 and Sensor $\#$2. Figure 14 depicts their reflection spectra during stretching and relaxation. RMSprop was regraded as the best algorithm for training the neural network upon combining the performance evaluation results of Section 4.2 and Section 4.3. Without any extra iterations, the datasets created by Sensor $\#$1 and Sensor $\#$2 were normalized and randomly scrambled before being directly fed into the model previously trained using the RMSprop for testing. The wavelength interrogation error distribution is illustrated in Fig. 15. The peak wavelength interrogation errors for Sensor $\#$1 and Sensor $\#$2 during stretching and relaxation, respectively, are shown in Fig. 15(a) and (b). During both stretching and relaxation, the interrogation errors of each interference peak converge within $\pm$ 0.06 $nm$, indicating that the system has a stable error floating range for interrogating novel sensors. Using the test dataset, the cavity lengths for Sensor $\#$1 and Sensor $\#$2 were computed and compared to the measured values, obtaining the results shown in Fig. 16. The system’s errors in the test dataset were also evaluated, as shown in Table 5. The proposed system’s interrogation precision is $\pm$ 21 $pm$, $\pm$ 0.20 $\mathrm {\mu }$m for Sensor $\#$1 and $\pm$ 20 $pm$, $\pm$ 0.17 $\mathrm {\mu }$m for Sensor $\#$2, in multi-wavelength and cavity length interrogation, respectively. This demonstrates that the proposed system can interrogate different FPI sensors with adequate precision, indicating that the system has suitability rather than a specific sensor structure.
Fig. 14. (a) and (b) are the schematic diagram of the shift of the interference peaks in 1530-1560 $nm$ during horizontal stretching and relaxation of Sensor $\#$1, $\#$2, respectively.
Fig. 15. (a) and (b) are the error distribution of peak wavelength interrogation for Sensor $\#$1 and Sensor $\#$2, respectively. Interference Peak $\#$1, $\#$2, and $\#$3 are the three interference peaks in 1530-1550 $nm$ from left to right.
Fig. 16. (a) and (b) are the comparison of the interrogation value and measured values of the cavity length of Sensor $\#$1, $\#$2 during stretching and relaxation, respectively.
Table 5. Analysis of wavelength interrogation errors of Sensors $\#$1, $\#$2 during their respective stretching and relaxation processes.
This suitability, however, does not apply to all sizes of FPI sensors. Only sensors having three consecutive interference peaks in the range of 1530-1560 $nm$ are accepted by this system, which is limited by the architecture of the proposed demodulation system. If higher interrogation precision is desired, transfer learning (TL) on top of current models is an effective option.
5. Discussion
Except for its high performance, low cost, stability, and scalability, the proposed FPI sensor demodulation system promises cost-effectiveness in engineering measurements. These properties will be described below down.
- • High performance This has been demonstrated in the experimental test section. The proposed demodulation system can simultaneously interrogate the peak wavelengths of three interference peaks with the precision of up to $\pm$ 14 $pm$, and absolute cavity length interrogation precision of up to $\pm$ 0.07 $\mathrm {\mu }$m. In addition, the wavelength interrogation resolution can theoretically reach the $pm$ level, but this is attributed to the neural network algorithm rather than the hardware construction of the system.
- • Low cost This can be reflected by hardware and the demodulation algorithm. To complete the demodulation of the FPI sensor on the hardware side, we use nine AWG channels simultaneously. These nine channels only require a two-channel optical power meter (equal to two PD-AMP-DAQ sequences). This is due to an eight-channel optical switch, which eliminates the need for too many PD-AMP-DAQ sequences, lowering the system’s hardware cost and simplifying data processing by eliminating the requirement to process and fuse numerous channels at the same time. In terms of algorithms, the neural network model’s simplistic architecture requires few parameters and is easy to utilize. In addition, the proposed model does not need to be trained for a long time and does not require excessive resource consumption, as shown in Table 6.
- • Stability The AWG is suitable for use as a demultiplexer in demodulation systems, along with its components’ resilience, compact size, passive, low loss, and flat spectrum [45,46]. In addition, the whole system’s demodulation repeatability and suitability have been proved, and easy to integrate and package.
- • Scalability The AWG demultiplexes the reflected light from the FPI sensor in the various channels, and the range of interferometric peaks that the system may probe is determined by the AWG channel selected. In theory, the system may be customized to interrogate many interferometric peaks over a wide wavelength range by changing the AWG channels and the neural network model’s input and output. Furthermore, the system’s performance may be improved by employing a complicated network structure, although this would require additional processing resources. Due to hardware restrictions, we’ve only tested three interference peaks within 1530-1560 $nm$.
Table 6. Training / Testing (mean value in stretch and relaxing) time and training-time resource consumption of the proposed model under each algorithm.
6. Conclusion
This paper presents a new absolute demodulation method for FPI sensors that is both high-performing and cost-effective. Using AWG, the system translates the peak fluctuations of the FPI sensor’s interferometric peaks into transmitted intensity variations in the channel. It uses a neural network model to simultaneously interrogate the peak wavelengths of multiple interferometric peaks, achieving absolute indirect interrogation of their effective cavity lengths. It is demonstrated that the proposed system has high precision and demodulation repeatability, with multi-peak interrogation precision of up to $\pm$ 14 $pm$ and cavity length interrogation precision of up to $\pm$ 0.07 $\mathrm {\mu }$m, and its maximal interrogation resolution can reach $pm$ level attributable to the neural network algorithm. Furthermore, adequate suitability was demonstrated on novel sensors, with wavelength interrogation precision of at least $\pm$ 21 $pm$ and cavity length interrogation precision of at least $\pm$ 0.20 $\mathrm {\mu }$m. This research utilized neural network algorithms for FPI sensor demodulation systems, resulting in a high-performance, cost-effective, stable, and scalable absolute demodulation solution, promising a novel and reliable platform for practical engineering measurements.
Funding
National Natural Science Foundation of China (61762033, 61865005, 62175054); Natural Science Foundation of Hainan Province (2019CXTD400, 617079, 620RC554); Major Science and Technology Program of Haikou City (2021-002); Wuhan National Laboratory for Optoelectronics (2020WNLOKF001); National Key Technology Support Program (2015BAH55F01, 2015BAH55F04); Major Science and Technology Project of Hainan Province (ZDKJ2016015); Scientific Research Starting Foundation of Hainan University (KYQD(ZR)1882).
Acknowledgments
We are appreciative of the help from M. Z.
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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