Wavelength-Dependent Bragg Grating Sensors Cascade an Interferometer Sensor to Enhance Sensing Capacity and Diversification through the Deep Belief Network

Abstract

Fiber-optic sensors, such as fiber Bragg grating (FBG) sensors and fiber-optic interferometers, have excellent sensing capabilities for industrial, chemical, and biomedical engineering applications. This paper used machine learning to enhance the number of fiber-optic sensing placement points and promote the cost-effectiveness and diversity of fiber-optic sensing applications. In this paper, the framework adopted is the FBG cascading an interferometer, and a deep belief network (DBN) is used to demodulate the wavelength of the sampled complex spectrum. As the capacity of the fiber-optic sensor arrangement is optimized, the peak spectra from FBGs undergoing strain or temperature changes may overlap. In addition, overlapping FBG spectra with interferometer spectra results in periodic modulation of the spectral intensity, making the spectral intensity variation more complex as a function of different strains or temperature levels. Therefore, it may not be possible to analyze the sensed results of FBGs with the naked eye, and it would be ideal to use machine learning to demodulate the sensed results of FBGs and the interferometer. Experimental results show that DBN can successfully interpret the wavelengths of individual FBG peaks, and peaks of the interferometer spectrum, from the overlapping spectrum of peak-overlapping FBGs and the interferometer.

1. Introduction

Fiber-optic sensing has been highly favored by researchers in recent years, and a large number of related studies have shown that fiber-optic sensing has excellent performance in various measurement applications due to its small size, sensitivity, and immunity to electromagnetic interference [1,2,3,4,5]. Fiber-optic sensors mainly use the position or power changes of specific wavelengths in the spectrum measured after the probing beam reaches the measurement position of the fiber-optic sensor to interrogate changes in various environmental physical parameters. FBGs based on the Bragg grating architecture are simple-to-manufacture fiber-optic sensors made by transversely exposing the core of single-mode fiber to intense ultraviolet light with a periodic pattern [6], which can be used to sense temperature [7], strain [8], pressure [9], vibration [10], acceleration [11], and more. Usually, the strain sensitivity is about a 1 pm wavelength shift with one microstrain (με) applied change, and the temperature sensitivity is about a 13 pm wavelength shift with a 1 °C change in environmental temperature for a standard FBG sensor around the 1550 nm wavelength [12]. On the other hand, the production process of a fiber-optic interferometer is more complicated than that of FBG, which usually requires the use of fiber fusion splicing equipment to splice different types of fibers, taper, side polishing, etching, etc. [13,14,15]. By changing the shape of these optical fibers, the path of light beam transmission is altered to form an optical path difference. This allows changes in different environmental physical parameters to shape different light beam paths, creating different interference states for sensing. Fiber-optic interferometers typically exhibit heightened sensitivity in measurements. Their accuracy for sensing temperature [16] and strain [17] is significantly greater—ranging from tens to hundreds of times higher—than that of standard FBGs.

In addition, standard FBGs are usually only capable of measuring either strain alike or temperature alone, requiring the use of another FBG for temperature compensation in the case of strain alike measurement [9]. Fiber-optic interferometers can overcome this problem by using the Vernier effect to measure temperature, strain, or other dual parameters simultaneously [18,19]. However, the main common disadvantages of fiber-optic interferometers are the high optical loss and the complexity of the optical spectrum, which make it difficult to cascade multiple fiber-optic interferometers to perform sensing tasks at the same time. In contrast, multiple FBGs can be cascaded in series to perform multi-point sensing tasks. Thus, if multiple FBGs and fiber-optic interferometers are cascaded in series, it is possible to meet the requirements of measurement conditions that require multiple measurement points, high sensitivity, and diversity of physical parameter measurement. In the previous instance, multiple FBGs cascaded an interferometer for sensing [20]. To increase the number of FBG sensing points, it is necessary to arrange the wavelengths of the FBGs very close to each other, and the overlapping of the wavelengths of the FBGs may occur during the sensing task, making it difficult to interpret the sensing results. In previous studies, it has been demonstrated that the overlapping of FBGs can be solved using machine learning to accurately interpret the sensed information [7,8]. However, FBGs cascade an interferometer framework for sensing tasks with the wavelength overlap of FBGs, which has not been explored.

In this paper, the main goal is to improve the sensing architecture of FBGs cascade with an interferometer by utilizing machine learning technology, specifically the DBN, to address the issue of FBGs’ wavelength overlap within this sensing architecture. The focus is on the power variations of different wavelengths caused by the interferometer spectrum drift, which results in different spectral profiles for the same overlapped FBG wavelengths through power modulation. Therefore, in order to interpret the center wavelength of FBGs in various situations based on the different interference spectra, it is essential to predict the central wavelength of FBGs through machine learning. DBN specializes in feature extraction, which is particularly useful for identifying different spectral shapes due to different interferometric and FBGs’ overlap patterns. The experimental results show that DBN can successfully decipher the overlap of different wavelengths of FBGs in different interference spectra. Therefore, machine learning will be beneficial for the sensing points’ capacity and sensing diversity in FBGs cascading an interferometer sensing architecture.

2. Experimental Setup

Figure 1 shows the experiment setup of wavelength-dependent Bragg grating sensors cascading an interferometer. In this scheme, two FBGs with the same initial wavelength are used for strain sensing, and the applied strain causes the FBG wavelengths to shift. Therefore, the center wavelengths of the two FBGs can overlap partially or completely. In addition, a tunable delay-line interferometer (TDI) is used to simulate the spectral changes of the fiber-optic interferometer during sensing. The TDI consists of two beam splitters placed at a positive and negative angle of 45 degrees, a reflector placed at a negative angle of 45 degrees, and a prism placed at a positive angle of 180 degrees [21]. The TDI splits a beam into two beams and creates a difference in the optical path, resulting in an interferometric spectrum. Furthermore, the TDI can cause a change in the optical path difference between the two beams by moving the position of the prism [21]. This shows that the optical principle of TDI is the same as that of the fiber-optic interferometer described in the previous section; only the medium of light transmission is different. Therefore, since the TDI has the same spectral properties as the fiber-optic interferometer and can adjust its overall spectral position arbitrarily, it is very easy to simulate the spectral shift caused by the fiber-optic interferometer during sensing.

Data acquisition consists of a broadband light source (BLS) emitting a beam that passes through port 1 of the circulator to port 2 of the circulator and senses the strain via the FBG. The reflectance spectra of the strain sensed on the FBG pass through port 2 of the circulator to port 3 of the circulator, where they then pass through a TDI and are then read by an optical spectrum analyzer (OSA). The collected spectra with specific wavelength reflections of FBG overlap interference patterns of the TDI are transmitted to a personal computer (PC) for machine learning model training and for machine learning models to verify the usefulness of the trained models. In addition, the strain application of the FBG is realized by the displacement variation of the linear stage (LS) to change the fixed position of the FBG. The brand models of the main equipment and components (BLS, Cir., FBG, LS, TDI, and OSA) are UNICE. Inc. NA0101 in Taoyuan, Taiwan (wavelength band: 1520 nm to 1570 nm), FOCI Inc. in Hsinchu, Taiwan (wavelength band: 1470 nm to 1610 nm), 3L Technologies Inc. in Miaoli County, Taiwan (standard FBG with 1 pm/με sensitivity), Onset Inc. CT02A in Taipei, Taiwan (0.01 mm moving resolution), Kylia Inc. WT-MINT in Paris, France, and Anritsu Inc. MS9740A in Atsugi, Japan (0.03 nm lowest resolution), respectively.

3. Data Collection Setup and Analysis

In the strain sensing experiment, the initial center wavelength of both FBGs was 1544.73 nm. The setup used was to fix the center wavelength of one of the FBGs at 1545.1 nm by applying strain. The other FBG was subjected to the strain from the initial center wavelength position of the FBG, and a total of 12 steps of strain application were performed. Finally, the center wavelength position of the FBG was located at 1545.45 nm. Each step was about 60 με of strain application with a 0.06 nm wavelength shift, since the wavelength shift of 1 pm corresponds to about 1 με of strain application. The 60 με strain was realized by shifting the LS by 20 μm. Since the distance between the fixed points on both sides of the FBG was 33 cm, the microstrain on the FBG was 20 μm divided by 33 cm, which is equal to 60 με. However, the moving operation of LS was via manual adjustment of the micrometer head, and with the limitations of the actual resolution of the OSA, there may have been an error of 5 με to 10 με in the application of each strain. From this setup, the spectrum of FBGs can be obtained with no overlap, partial overlap, and almost complete overlap. In order to verify that the overlapped spectra of FBG wavelengths can be successfully interpreted in different interference patterns obtained by fiber-optic interferometry, three different interferometric states of TDI were used to verify the success of the FBG spectra. Figure 2 shows the spectra of the FBG from 0 με to 720 με strain and the unstrained FBG in three different interferometric settings of TDI. The appearance of the interference spectrum is very similar to that of a sinusoidal wave, which shows periodic power modulation as the wavelength changes, and the displacement between the wavelength positions of the troughs is defined as the free spectral range (FSR). Here, the FSR of the three different interference states set by TDI is 2 nm. It can be observed that in the cases of Figure 2a–c, the wavelengths of the unstressed FBGs fall in the left, center, and right halves of one of the 2 nm wide lobes in the interference pattern. The FBG energy is higher in the case of Figure 2b and lower in the cases of Figure 2a,c because the interference power tunes the reflected power of the FBG. Therefore, it can be found that the power of the other FBG under strain will be modulated according to the wavelength position of the FBG and the corresponding power of the set interference.

From the above, it can be seen that even if the FBG wavelengths are at the same position, they will be affected by different interferences and thus show different spectra. Figure 3 shows this concept more clearly. Figure 3a–c show the FBG spectra of the first, eighth, and last steps of the strain in three different interferometric states, respectively. It can be seen that the spectra of the two FBGs, both superimposed and un-superimposed, are very different in the three different interference states. Furthermore, as mentioned before, optical sensing mainly utilizes specific wavelength variations for sensing, and sensing based on interferometric spectroscopy either utilizes the wavelength at a certain peak of the spectrum to obtain sensing information or utilizes the wavelength at a certain dip of the spectrum to obtain sensing information. Hence, in this scheme, the wavelength value at the peak of the interference’s lobe on the right side of the FBGs in Figure 3 is assumed to be the main wavelength for determining the sensed information during sensing. The peak wavelengths of the three selected lobe interferences are 1546.88 nm (interference 1), 1547.11 nm (interference 2), and 1547.39 nm (interference 3). Figure 4 clearly shows the fixed sensing wavelengths of the interferometric spectrum for different interference states at each strain step, the theoretical values of the two FBG wavelengths, and the overlapping areas of the two FBG wavelengths under different strain steps.

4. Deep Belief Network for Peak Wavelength Prediction

A DBN is a generative graphical model consisting of multiple hidden unit layers. Each pair of connected layers forms a restricted Boltzmann machine (RBM) [22,23,24,25,26], which serves as the core building block of a DBN. An RBM is a two-layer network capable of learning to represent complex data distributions [22,25]. It comprises a visible layer (v) for input data and a hidden layer (h) for learned features. The nodes in these layers are interconnected by undirected edges, with no connections within a layer, simplifying the learning process and rendering the RBM efficient for training. The energy of a configuration, including visible and hidden states in an RBM, is determined by [25]:

$$E\left(\mathrm{v},\mathrm{h}\right)=-{\sum }_i{a}_i{v}_i-{\sum }_j{b}_j{h}_j-{\sum }_{i,j}{v_{i}w}_{i,j}{h}_j$$

(1)

where v_i is visible units, h_j is hidden units, a_i is bias for visible units, b_j is bias for hidden units, and w_i,j is the weight matrix between visible and hidden units.

The architecture of a DBN is constructed by stacking several RBMs on top of each other, as shown in Figure 5. The visible layer of the first RBM serves as the input layer of the DBN. The hidden layer of the first RBM becomes the visible layer for the next RBM in the stack, and this process continues for each subsequent layer. This hierarchical structure allows the DBN to learn increasingly abstract representations of the input data as it moves up the layers. The training of a DBN typically involves a two-step process: pre-training and fine-tuning. During the pre-training phase, each RBM is trained independently in a greedy, layer-by-layer manner, starting from the bottom-most RBM. This unsupervised pre-training step is crucial in effectively initializing the network’s parameters, like the network’s weights, in a way that captures the underlying structure of the data. The pre-training acts as a form of feature extraction, allowing the network to learn useful representations of the data [23].

Once all the layers are pre-trained, the entire DBN can be fine-tuned using supervised learning techniques if labeled data are available. This involves adding a final output layer and using backpropagation to adjust the weights across the entire network to minimize the error in predictions [24]. The fine-tuning process ensures that the learned representations are optimized for the specific task at hand, such as classification or regression. The combination of unsupervised pre-training and supervised fine-tuning makes DBNs powerful tools for learning from complex, high-dimensional data, providing a robust framework for capturing intricate patterns and structures in the data. Their ability to learn hierarchical representations of data makes them powerful for handling complex datasets. The hierarchical nature of DBNs means that higher layers capture more abstract features, enabling more accurate and efficient learning and prediction [23,24].

The process of utilizing a DBN for peak wavelength detection in FBGs and interferences involves several well-defined steps, each crucial for building a robust and accurate model. There were 3 different interferometric modes and 13 different FBG strained settings in the experiment, and the total number of interferometric spectral modes and FBG spectral modes of union was 39 different spectra. In order to predict the wavelength accurately in the final DBN model derived from the DBN model training, a total of 20 collections were made for each spectrum, and 780 spectral samples were finally obtained. From these samples, the dataset was split into training (80%) and test sets (20%). Once data were collected, the next phase was data preprocessing, which ensured the raw data were in a suitable form for model training. Thus, the training dataset (i.e., captured reflected spectrum) was preprocessed and normalized between 0 and 1. This involved normalizing the interference patterns to achieve consistency across the dataset. The most critical part of preprocessing is feature extraction, where relevant features indicative of peak wavelengths are extracted from the interference patterns. These features are essential, as they provide the DBN with the necessary information to learn and predict accurately.

The proposed DBN model was implemented using the TensorFlow framework, along with the Keras and Sklearn libraries. The training was conducted on a PC equipped with an Intel Core i7-4790 3.60 GHz CPU and 20.0 GB of RAM. Figure 5 illustrates the architecture and training process of the proposed DBN. The preprocessed reflection spectra of the FBGs with interferences were used as inputs to train the DBN algorithm, with the corresponding peak wavelengths of the FBGs and interferences serving as target values. Each layer of the DBN was pre-trained individually using RBMs. This was done in an unsupervised manner, utilizing contrastive divergence to initialize the network weights. This step is essential in capturing the underlying structure of the data without requiring labeled information. After pre-training, the RBMs were stacked to form the complete DBN. The entire network was then fine-tuned using supervised learning with a labeled dataset of known peak wavelengths. This fine-tuning adjusts the weights and biases across all layers to optimize the network for peak wavelength detection. The mean squared error (MSE), mean absolute error (MAE), and root mean square errors (RMSE) loss function were used for measuring the difference between predicted and actual wavelengths, and the optimizer Adam was used to minimize this loss, The MSE, MAE, and RMSE are expressed in the following equations:

$$MSE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{(y_i-y)}^2$$

(2)

$$MAE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n|y_i-y|$$

(3)

$$RMSE=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n{(y_i-y)}^2}{n}}}$$

(4)

where n is the number of predicted values, y_i is the actual value, and y is the predicted value, respectively. During the training of the DBN, various parameters such as the number of epochs, batch sizes, hidden layers, hidden units, and optimizer and activation functions were adjusted to achieve optimal values and to prevent overfitting. Tuning these parameters resulted in different training times, and MSE, MAE, and RMSE values.

5. Results and Discussions

The different parameters, such as epochs, hidden layers, batch sizes, and optimizer and activation functions, were adjusted until an optimal or well-trained model was obtained. The optimal parameters for the well-trained DBN model are crucial in achieving high accuracy in peak wavelength detection. Thus, the optimal parameters used for training the proposed DBN included 3 hidden layers, 32 batch sizes, 150 epochs, the Adam optimizer, and ReLU activation functions in the hidden layers that introduced non-linearity, improving the network’s ability to model complex relationships in the data. Figure 6a shows the DBN’s training performance evaluated through its training and validation losses, as well as training and validation accuracy using these optimal parameters. The training loss and validation loss curves provide insights into the model’s performance during training and on unseen data, respectively. The training loss quickly declined and stabilized near zero at epoch 150 on the bottom x-axis, indicating effective learning of training data patterns. The validation loss also decreased significantly early on and remains stable, showing good generalization to unseen data without overfitting. The training accuracy reached nearly 1.0 (100%) by epoch 150 at the top x-axis, and the validation accuracy also approached 1.0 (100%), though with some variability. In general, the DBN demonstrated strong performance, effectively learning and generalizing from the data, with high accuracy and low loss metrics.

Figure 6b shows that the performance of the DBN for peak wavelength detection of FBG strain sensors was evaluated in terms of prediction error, measured as MSE, MAE, and RMSE, and training time, measured in seconds per epoch. As shown in the figure, the training process showed stabilization of training time after initial fluctuations, indicating computational efficiency. Key error metrics, including MSE, MAE, and RMSE, at epoch 0 were relatively high. However, as training progressed, all significantly decreased, remaining close to zero at 150 epochs, which signifies that the model effectively minimized prediction errors and indicates the DBN’s ability to accurately learn and predict peak wavelengths, demonstrating its reliability and effectiveness for this application. Furthermore, to determine the best optimizer, the model was trained and tested with different optimizers, and their performance was compared based on MSE, MAE, and RMSE. According to Figure 7, Adam achieved lower MSE, MAE, and RMSE values of 0.0012, 0.015, and 0.0187 compared to other optimizers. This was due to its computational efficiency and noise minimization. As a result, the proposed DBN model was trained and tested using the Adam optimizer.

Figure 8 shows the predicted peak wavelengths of the strained FBGs, fixed FBGs, and three different interferences. In Figure 8a, the x-axis represents the peak wavelength of the strained FBGs in reality, while the y-axis represents the peak wavelength of the strained FBGs predicted by the DBN model. The yellow symbols on the graph show the ideal values, indicating where the predicted values should be if they exactly match the ideal values. Along this line, 15 randomly selected prediction data points are positioned to demonstrate the accuracy of the DBN model. The closer these points are to yellow symbols, the more accurate the prediction becomes, suggesting that the DBN model is very effective in accurately predicting the peak wavelength of the strained FBG in the case of overlap with the FBG and interferences. The x-axis of Figure 8b is the marked peak wavelength of fixed FBG and three different interferences, which are 1545.1 nm, 1546.88 nm (interference 1), 1547.11 nm (interference 2), and 1547.39 nm (interference 3), respectively. The y-axis corresponds to the predicted wavelength peak of the DBN model. From the results, it can be seen that the DBN model can also accurately predict the marked peak wavelengths of fixed FBG and three different interferences.

6. Conclusions

This research centers on the application of DBN to predict sensing outcomes in a flexible sensing design that combines FBG sensors with a simulated fiber-optic interferometer sensor. It is known that the coexistence of FBG sensors and fiber-optic interferometer sensors in the sensing architecture can perform diverse sensing tasks due to the high sensitivity of fiber-optic interferometers and the advantage that multiple FBGs can be arranged in an array. However, in practice, to maximize the number of FBG sensing points, the problem of FBG spectral overlap will inevitably occur. The overlap of the overlapping FBG spectrum and the spectrum sensed by the fiber interferometer will inevitably cause the entire spectrum power to be modulated, making it challenging to accurately extract sensing information. In this case, the use of machine learning, specifically the DBN model, offers an optimal solution for accurately interpreting the sensed information. The DBN model can effectively predict all sensed information for various levels of overlapping FBG spectra under different spectrums sensed by the interferometer, ultimately enhancing the cost-effectiveness, flexibility, and accuracy of the sensing system.