Seawater Dual Parametric Fiber Optic Temperature and Pressure Sensors Demodulated by Machine Learning Method

By employing polydimethylsiloxane (PDMS) characterized by elevated thermo-optical and elastic-optical coefficients for encapsulating an optical microfiber coupler integrated with a sagnac loop (OMCSL) structure, which exhibits large abrupt field characteristics, a fiber optic sensor capable of simultaneously measuring seawater temperature and pressure can be created. Nevertheless, the utilization of the traditional sensitivity matrix method (SMM) for demodulating the sensor led to unstable and considerably erroneous demodulation results. To enhance the accuracy of the demodulation process, this paper investigates and employs a machine learning method (MLM) for sensor demodulation. This paper centers on the investigation and application of MLM for sensor demodulation. The experimental results exhibit a significant decrease in demodulation error attained via MLM when contrasted with the traditional SMM.


Introduction
Ocean hydrographic conditions are profoundly influenced by seawater temperature and pressure.The real-time monitoring of these two parameters and their fluctuation patterns not only enhances meteorological, navigation, and hydroacoustic research but also holds crucial implications for various marine activities, including investigation [1,2], sea fishing [3], oceanographic exploration, and aquaculture [4].Consequently, the advancement and augmentation of marine temperature and pressure sensors, characterized by high performance, hold supreme significance.
Optical fiber sensors provide several advantages, including their cost-effectiveness, ease of deployment, adaptability for measuring multiple parameters, and the ability to perform large-scale region multiplexing.These sensors offer a cost-effective solution and can be easily integrated into existing systems.Their design flexibility allows for measuring various physical parameters, such as temperature, pressure, strain, and chemical composition.Additionally, optical fibers enable simultaneous sensing at multiple points, making them suitable for monitoring large-scale regions or structures.Overall, optical fiber sensors offer a range of benefits that make them highly desirable for different applications.In the contemporary period, there has been an escalating interest in the domain of optical fiber sensors, particularly those that employ Optical Microfiber Coupler (OMC) and Optical Microfiber (OM), for the purpose of monitoring applications within marine environments [1].These sensors are being explored due to their inherent advantages, including high sensitivity, rapid response, and versatile design capabilities.In 2015, Wang et al. introduced a The primary components of the OMCSL structure include a transition cone area, a uniform waist zone, and a sagnac loop, depicted in Figure 1(c).The transition cone area and uniform waist zone are both crucial in the coupling process of OMC.In cases where shifts in the coupling zone are gradual, the overall coupling performance of OMC can be considered as a sum of individual local couplings accumulated over the entire length.Due to the significant geometric dimensions of the transition cone area, the primary source of phase discrepancy within the coupling zone of OMC is typically the uniform waist zone.As a result, the length L of the uniform waist zone can be regarded as the effective length of the entire coupling area when applying the strong coupling theory to analyze the comprehensive coupling attributes of OMC [8].
When light P1 enters through port 1 and exits from port 2 after passing the coupling zone on two occasions, the power emanating from port 2, represented as P2, can be described as: Here, C(λ, n2, n3, r, z) [9] denotes the coupling coefficient at the position z.In this expression, λ signifies the wavelength of the incident wave, r represents the radius of the uniform waist zone, n2 and n3 are the refractive indices for the fiber cladding and PDMS, respectively, and φ corresponds to the phase difference generated within the coupling area.
The variations in seawater temperature lead to alterations in the parameters r, l, n2, and n3, primarily due to the thermo-optical effect and thermal expansion effect of PDMS.Due to the thermooptical and thermal expansion properties of PDMS, variations in seawater temperature lead to alterations in n2, n3, r and l.In a parallel manner, owing to PDMS's elasto-optical properties and the alteration in the OMC's structural parameters under applied pressure, variations in seawater pressure lead to modifications in n2, n3, r and l.As temperature and pressure charge, the reflection spectrum of OMCSL undergoes shifts.The expressions for OMCSL's temperature sensitivity ST and pressure sensitivity SP are given as [10]: (3)

Experimental results
A temperature response test was conducted using the previously obtained PDMS-encapsulated OMCSL.  Figure 3(a) illustrates the sensor's reflection spectrum as the water temperature decreased from 28℃ to 22℃.With the decline in temperature, the refractive index of the PDMS increased, resulting in a red shift of the reflection spectrum.By tracking the feature wavelengths of four distinct dips (dip1, dip2, dip3, and dip4) in the range of 22℃ to 28℃, the authors determined the temperature sensitivities via linear regression, each result consistently presented R 2 exceeding 0.995.Figure 3(b) illustrates the sensitivities of dip1 to dip4 as -0.807 nm/℃, -0.911 nm/℃, -1.055 nm/℃, and -1.176 nm/℃, respectively.Figure 3(c) illustrates the sensor's reflection spectrum as the seawater pressure increased from 0 MPa to 3 MPa.As the pressure increased, the reflection spectrum exhibited a red shift, attributed to the elasto-optic effect and the deformation of PDMS under pressure.The wavelengths of the four dips, taken from 0 MPa to 3 MPa, were used to determine the pressure sensitivities through linear fitting, as depicted in Figure 3(d).The pressure sensitivities for the four dips were determined to be 0.767 MPa/nm, 0.838 MPa/nm, 1.053 MPa/nm, and 1.150 MPa/nm, All R² exceeded 0.995.

Sensitivity matrix demodulation method
For multi-parametric sensors like the one described in this paper, SMM is commonly applied for demodulation, as seen in references [5, 6,10].In this traditional demodulation approach, two distinct wavelengths from the spectra are typically selected for tracking and demodulation.
But for the sensor, is an ill-conditioned matrix under certain conditions [11] , which causes SMM to introduce large error to the demodulation.There is another reason why the sensor cannot be demodulated using SMM.As depicted in Figure 4, within the temperature range of 22℃ to 28℃, the wavelengths of dip2 and dip3 display strong linearity (R 2 > 0.996), suggesting their sensitivities remain consistent within this range.In examining the wider temperature span from 18℃ to 35℃, it's evident that the shifts in wavelength aren't linear.As the temperature rises, there's a decline in its sensitivity.The observed appearance can be attributed to the significant thermal elongation coefficient (TEC) of PDMS, which stands at 3 × 10 -4 /℃.As the temperature escalates, the inherent thermal expansion effect of PDMS kicks in, leading it to expand.This expansion, in turn, exerts a strain on the OMC's waist zone.From the pressure test with the sensor, it was evident that exerting strain on the sensor resulted in a red-shift of the dips, the PDMS's thermo-optical properties triggered a blue shift in the sensor's reflection spectrum at higher temperatures.This blue shift offset the red shift produced by thermal expansion, leading to decreased temperature sensitivity as temperatures rose.The experimental results showed that that the temperature sensitivity of the sensor's feature wavelengths are not constant across varying temperature intervals.It's essential to account for the non-linear phenomenon of the sensor's sensitivity.

Machine learning method
The Random Forest Regression (RFR) algorithm, which is a type of ensemble learning method based on the bagging technique, utilizes multiple decision trees to predict and analyze data [12].RFR capitalizes on the diversity of numerous decision trees to make more accurate and stable predictions.
Being able to address regression challenges involving continuous variables is one of its key strengths.
Given its versatility, it's not surprising that RFR has seen applications in diverse areas, including the prediction of stock market dynamics [13].Leveraging RFR for demodulating the sensor thus appears to be a promising approach, as it can adeptly handle the non-linearities and intricacies presented by the sensor's temperature sensitivity.
Below is an outline of the primary steps involved in utilizing RFR algorithm to predict sensor temperature and pressure demodulation.The schematic illustrating the principle of RFR can be referenced in Figure 5: 1) From the complete training set, k sub-training sets are randomly and repeatedly selected, each containing n samples.
2) To build each decision tree, RFR randomly selects m input features that affect the demodulation of temperature and pressure.The selected features are used as the candidate feature set for the decision tree's branch nodes.The branching criterion is used to determine the optimal feature for each node, and the decision tree is constructed using these criteria.
3) To predict temperature and pressure demodulation, RFR constructs k decision trees using k subtraining sets.It creates each decision tree independently of the others.The outputs from the k decision trees are then averaged to determine the predicted results for temperature and pressure.
Thus, RFR can predict temperature and pressure in the target environment from multiple perspectives.In RFR, dual random sampling processes are employed to maintain randomness and prevent model overfitting.In each individual regression tree within the random forest, m features are randomly selected from the entire set of input features.This results in variations in the selected features across different regression trees.As a result, RFR is capable of predicting temperature and pressure in the target environment from multiple perspectives.This approach enhances the overall robustness and generalization capability of the RFR model.The reflection spectrum samples of the sensor were organized and formatted into a dataset that is compatible with MLM.This dataset comprised 247 samples, with temperatures ranging from 18.5℃ to 35℃ and pressures from 0 to 2.88 MPa.Of the 247 distinct samples, approximately 85% (207 samples) were randomly allocated to a training set, while the remaining 15% (37 samples) were designated for a test set to validate the MLM demodulation outcomes.The performance of MLM is influenced by two key factors: the number of samples in the training set and the number of features selected for each sample.These factors can have a significant impact on the predictive power of the model.Therefore, it is important to carefully select an appropriate number of samples and features to optimize the performance of the model.We first examined the impact of the training set's sample size on the RFR model's prediction accuracy.The resulting temperature and pressure predictions are depicted in Figure 6  for training, the model can learn from a larger and more diverse set of data, improving its ability to make accurate predictions for temperature and pressure.Therefore, it is advisable to have an ample number of samples in the training set to optimize the performance of the RFR model.The authors examined the influence of the number of selected sample features on the RFR prediction outcomes.The experimental findings are depicted in Figure 7(a) and Figure 7(b).The three demodulation RFR models with different number of features selected: utilizing all 9 (100%) features, 6 (66%) selected features, and 3 (33%) selected features.For temperature predictions, the MAEs across the models were 0.43, 0.54, and 0.81; the MAPEs were 2.6%, 3.3%, and 4.9%; and the RMSEs were 0.63, 0.73, and 1.27.For pressure predictions, the corresponding MAEs were 0.22, 0.3, and 0.45; the MAPEs were 7.6%, 10.4%, and 15.6%; and the RMSEs stood at 0.32, 0.38, and 0.56.It can be concluded that increasing the number of selected features can reduce the prediction error of RFR.The results of the experiment are summarized in Table 1.

Conclusion
In a nutshell, the authors first investigates the sensing performance of OMCSL sensor in PDMS packages.To decline demodulation error, the authors introduced MLM to demodulate the sensor using its reflection spectrum dataset.The paper introduced and analyzed RFR demodulation method due to its low error rate.The experimental results showed that the demodulation error in temperature and pressure sensors using SMM was significantly higher compared to the error in the RFR model.The temperature demodulation error was 4.87 times greater, and the pressure demodulation error was 2.79 times larger.This paper presents a methodology aimed at reducing demodulation errors in SMM-based sensors, providing an effective solution for more accurate temperature and pressure predictions.

Figure 1 .
Figure 1.(a) Microscopic picture of the waist zone of the OMC prior to encapsulation.(b) OMC packaging diagram.(c) OMCSL Structure Diagram.(d) Completed sensor.The primary components of the OMCSL structure include a transition cone area, a uniform waist zone, and a sagnac loop, depicted in Figure1(c).The transition cone area and uniform waist zone are both crucial in the coupling process of OMC.In cases where shifts in the coupling zone are gradual, the overall coupling performance of OMC can be considered as a sum of individual local couplings accumulated over the entire length.Due to the significant geometric dimensions of the transition cone area, the primary source of phase discrepancy within the coupling zone of OMC is typically the uniform waist zone.As a result, the length L of the uniform waist zone can be regarded as the effective length of the entire coupling area when applying the strong coupling theory to analyze the comprehensive coupling attributes of OMC[8].When light P1 enters through port 1 and exits from port 2 after passing the coupling zone on two occasions, the power emanating from port 2, represented as P2, can be described as:

Figure 2 (
a) displays the schematic diagram of the experimental setup for the temperature response.A temperature controlled water tank was used to change the temperature of the sensor's external environment.Following that, a pressure response experiment was carried out, with its schematic setup depicted in Figure2(b).The sensor was situated inside a water-filled pressure tank, with a digital pressure gauge attached to the pressurizing pump recording the tank's pressure in realtime.The pressure within the tank was incrementally increased from 0 MPa to 3 MPa in 0.5 MPa steps, and the sensor's reflection spectra was captured using the OSA.

Figure 2 .
Figure 2. (a) Schematic depiction of the experimental arrangement for the water temperature response analysis.(b)Schematic representation of the experimental configuration for investigating pressure response in seawater.

Figure 3 .
Figure 3. (a) Reflectance spectra of the sensor recorded at varying water temperatures.(b) The wavelengths of four dips as they vary with changes in temperature.(c) Reflectance spectra of the sensor under varying water pressure conditions.(d) The wavelengths of the four dips as they vary with changes in pressure.

Figure 4 .
Figure 4. Non-linear correlation between dips wavelengths and temperature.

Figure 5 .
Figure 5. Schematic illustration of the Random Forest Regression principle.

Figure 6 .
Figure 6.(a) Temperature demodulation outcomes for various RFR models based on varying sizes of the training set.(b) Pressure demodulation outcomes for different RFR models utilizing varying sizes of the training set.

Figure 7 .
Figure 7. (a) Temperature demodulation outcomes for different RFR models employing distinct feature sizes.(b) Pressure demodulation outcomes for different RFR models utilizing varying feature sizes.