Research on Distributed Fiber Optical Strain Testing Technology Based on the Adam Algorithm

Distributed Optical Fiber Sensing Technology, with advantages such as long detection distance, resistance to electromagnetic interference, and easy maintenance, is widely applied in various fields including security monitoring, electrical safety, and aerospace. However, the current Distributed Optical Fiber Sensing Technology faces challenges such as short localization distance, limited range for stress measurement, low accuracy, and time-consuming processes. This study proposes a novel approach for extracting Brillouin frequency shift signals using an adaptive gradient descent algorithm (Adam Algorithm). A Brillouin strain testing system based on heterodyne coherent detection is also constructed. Experimental results show that the distributed strain testing system using the Adam Algorithm can achieve accurate and fast measurement of maximum strain up to 9500 µϵ within a range of 10 kilometers. The average strain measurement deviation is 32.88 µϵ, and the time required for frequency shift extraction is less than 18.5 ms. This method provides a theoretical and experimental basis for the application of BOTDR Distributed Optical Fiber Sensing Technology.

high-precision testing [5].In recent years, numerous methods for extracting Brillouin frequency shift have been proposed by domestic and international researchers.For instance, Y Zhang et al. utilized the LM algorithm to adjust the weights' radial direction in order to extract features of Brillouin scattering using neural networks.This approach enhances the accuracy of Brillouin scattering signal extraction, albeit at the cost of time-consuming and complex neural network training [6], [7].M.A. Farahani et al. proposed a Brillouin frequency shift extraction method based on cross-correlation convolution, which improves the signal-to-noise ratio.However, its accuracy is constrained by the degree to which the frequency sweep data adheres to the Lorentz distribution [8].W. Kang et al. optimized the generalized neural network using the differential evolution algorithm, resulting in reduced complexity for manual testing and better fitting for various signal-to-noise ratios and line widths.Nevertheless, it tends to produce larger errors when the signal-to-noise ratio is low [9].To address the need for a fast, accurate, and wide-ranging strain testing technique, this paper presents a distributed optical fiber strain test scheme based on the Adam Algorithm.The extraction method of Brillouin frequency shift is designed, and an experimental system for the BOTDR strain test with heterodyne coherence detection is constructed.The results demonstrate that the proposed distributed fiber strain testing technology, utilizing the Adam Algorithm, not only simplifies complexity and reduces operation time but also enables precise and rapid testing of long distances, large strains, and high precision.

A. Analysis of the Strain Sensing Mechanism Based on Brillouin Scattering
The phenomenon of Brillouin scattering entails non-elastic scattering of light [10], [11].The transmission characteristics of light in optical fibers dictate that the Brillouin gain spectrum conforms to a Lorentzian distribution.The power gain g(v) at a given frequency v can be expressed as: In Formula (1), the parameter v B refers to the peak frequency of the Brillouin gain spectrum, which is also known as the Brillouin frequency shift; g 0 and Δv B respectively denote the peak value of the Brillouin gain coefficient and half-width at half-maximum of the Brillouin spectrum.The maximum Brillouin gain occurs when v equals v B [12], as depicted in Fig. 1.
When the signal is influenced by external factors, v B can be expressed as: In Formula ( 2): V P is the Stokes optical frequency; V S is Pump frequency; Va is the speed of sound in the optical fiber; n is the refractive index of the medium; c is the speed of light; θ is the angle between the pump light and the probe light; w p is Pump wave frequency.
Due to the thermal-optic effect and elasto-optic effect, variations in stress (ε) and temperature (T) cause changes in the intrinsic parameters (n, V a ) of optical fibers.These parameters are functions of temperature (T) and stress (ε), denoted as n (T, ε), E (T, ε), k (T, ε), and ρ (T, ε) respectively.E is Young's modulus; k is Poisson's ratio; ρ is the density of the optical fiber medium.By substituting it into (2), the relationship between the Brillouin frequency shift and temperature as well as strain can be obtained.
(3) When the reference temperature is T 0 and there is a change (compression or tension) in fiber strain, substituting into (3) yields.
The relationship between the Brillouin frequency shift and temperature under zero strain is as follows: In the given equation: T denotes temperature; T 0 denotes reference temperature; C T denotes temperature coefficient; ε denotes stress; Cε,v denotes the stress coefficient for Brillouin frequency shift.

B. Extraction of Brillouin Frequency Shift Based on the Adam Algorithm
1) The Principles of the Adam Algorithm: Supposing the objective function is denoted as J(θ), and the conventional formula for updating in gradient descent is given by: In Formula ( 6): θ k represents the kth parameter; α is the learning rate, a constant value; Δθ k denotes the change in parameter θ k ; m represents the total number of parameters; ∂J(θ 0 , θ 1 , . . ., θ m )/∂θ k indicates the gradient of the objective function J(θ) in the direction of θ k [13], [14].
The traditional descent approach upholds a constant learning rate α to modify the weights of parameters θ k .The Adam algorithm employs the computation of first and second-moment estimates of the gradient descent to independently design adaptive learning rates for various parameters θ k .It also maintains an exponentially decaying average of previous gradients, ensuring that each iteration has a range for the learning rate, resulting in smaller fluctuations and promoting smoother parameter θ k .The formula for updating the parameters in the objective function is as follows: In Formula (7): η is the learning rate; The value of ε is 10 -8 ; β1, β2ϵ[01)represent the exponential decay rates estimated by the first and second-moment methods, respectively; v t is the non-central variance value of the second moment of the gradient; m t is the average value of the first moment of the gradient.
The iterative process of the Adam algorithm is illustrated in Fig. 2. First, set the parameters η, ε, β 1 , and β 2 .Then, initialize the parameter vectors for the first moment (m 0 ), the second moment (v 0 ), and the time step (t).When the parameter θ fails to converge, iterate through each component in a cyclic manner and update them accordingly until reaching the pre-defined number of iterations or achieving the desired accuracy in the sum of squared residuals.Stop the iteration and output the parameter θ.
2) Extraction and Simulation of Frequency Shift: In the strain sensing system based on BOTDR, the measured Brillouin gain spectrum (BGS) is represented as a set of discrete points (x i , y i ), where i = 0, 1, ••• N. N represents the number of points on the sensing system; x denotes the scanning frequency; y corresponds to the intensity of the Brillouin signal; g represents the ideal Lorentz curve.Utilizing function "g" to perform fitting on the measured scattered points (x i , y i ), Modify the parameters of function "g" to progressively approach the coordinates (x i , y i ).Use mean square error to quantify the degree of difference between two sets of data.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Utilize the iterative process flow illustrated in Fig. 2, to obtain the optimal solution for the objective function J (g 0 , Δv B , v B ).When the number of iterations or the root mean square error reaches its minimum, the fitting degree of function g to the discrete points (x i , y i ) is optimal.At this point, the frequency corresponding to the highest value g 0 of function g is defined as the Brillouin frequency shift.
To validate the performance of the Brillouin frequency shift extraction algorithm based on the Adam optimization algorithm in terms of long distance, large dynamic range, and fast measurement capabilities, a standard single-mode fiber (GB.653) with a total length of approximately 10 km was placed in a room temperature environment of 23°C, with one end connected to a circulator.The fiber was fixed on a tension meter at the tail end for stretching, with a length of approximately 1 m and a step size of 500 µε.Starting from 0 µε, the tension meter measured the stretching range of [0, 8000]µε, and data were collected.Three sets of discrete points were obtained using the Adam algorithm, Gauss-Newton algorithm, and L-M algorithm, respectively, and linear fitting analysis was performed.The results are shown in Fig. 3.
The correlation between the Brillouin frequency shift and strain variation is evident in Fig. 3.The Adam algorithm, as depicted in Fig. 3(a), exhibits a high degree of fitting with a value of 0.9989 (closest to the optimal value of 1) and a linear coefficient of 4.7086 MHz/100 µε.In contrast, the Gauss-Newton algorithm in Fig. 3(b) has a fitting degree of 0.9976 and a linear coefficient of 4.5511 MHz/100 µε.Similarly, the L-M algorithm in Fig. 3(c) shows a fitting degree of 0.9986 and a linear coefficient of 4.7279 MHz/100 µε.These findings suggest that the Adam algorithm provides a better fit than the other two algorithms.The simulation analysis of the Gauss-Newton algorithm, L-M algorithm, and Adam algorithm was conducted using MATLAB.Table I presents a comparison of the evaluation metrics for Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

C. The Influence of Input Fiber Optic Power
Using an adjustable optical attenuator, the input optical power was increased in steps of 0.1 mW from 0.5 mW to 10 mW.The backward scattering light was measured using ESA and simulated using MATLAB.Fig. 5(a) shows the obtained Brillouin-Giant-Scattering (BGS) three-dimensional spectrum when the input fiber optic power is low, while Fig. 5(b) shows the spectrum when the input power is higher.As observed from these two charts, the Brillouin power increases linearly with the input power when it is low, but becomes narrower and Gaussian-like when the input power exceeds 3.5 mW.However, further increase in input power does not result in significant changes in the BGS scattering spectrum.
After fitting the Brillouin scattering spectrum in Fig. 5, a correlation between the input optical power and Brillouin scattering (BGS) spectral width for a fiber length of 10 km is established, as depicted in Fig. 6.As observed from Fig. 6, at a fiber length of 10 km, low input optical power does not induce self-induced Brillouin scattering within the sensing fiber.However, as the input optical power rises to 3.5 mW∼4 mW, the BGS spectral width increases significantly due to self-induced Brillouin scattering in the backward scattering within the fiber, and it decreases linearly.

A. The Establishment of an Experimental Platform
To evaluate the strain parameters of distributed optical fiber, a stress experimentation system using Brillouin optical timedomain reflectometry (BOTDR) has been set up.The detailed experimental setup diagram and physical configuration are depicted in Figs.7 and 8 respectively.The narrow linewidth laser generates an output power of 5 mW and emits continuous light with a central wavelength of 1550.12 nm and a linewidth of 10 kHz.The light is then split into two signals by coupler 1.Approximately 90% of the path consists of detection light, which is modulated by the electro-optic device into 30 ns detection pulse light.The repetition frequency is 8 kHz.Subsequently, the pulse light is amplified by the pulse light amplifier and passes through the grating filter before entering the distributed test optical fiber of 10 km, while being protected from backscattering by the optical isolator.The remaining 10% of the light serves as a reference light, which passes through a polarization scrambler to randomly vary its polarization state at a certain rate, aiming to match the polarization state of the returning signal light as closely as possible and reduce the amplitude fluctuation of the scattered signal.A high reflective mirror is added at the end of the fiber optic to minimize the Fresnel reflection.The amplified power from the erbium-doped fiber amplifier, after being coupled back into the optical fiber, enters the 3 dB coupler and interferes with the local reference light.Subsequently, it is converted into an electrical frequency signal by the photoelectric detector.The electrical signal is then amplified by a low-noise amplifier and the data is collected using an oscilloscope.The collected signal is processed using the Adam algorithm to extract the Brillouin frequency shift.

B. Calibration Device for Temperature and Strain Frequency Shift Coefficient
Utilize the differential coherent BOTDR experimental system depicted in Fig. 8 for the calibration of temperature and strain frequency shift coefficients.An experimental apparatus opted that performs the concurrent calibration of both temperature and strain frequency shift coefficients, as exemplified in Fig. 9. Choose an aluminum cylindrical conduit with a temperaturedependent expansion coefficient of 23 within the temperature range of 20 °C to 100 °C.A recess on the external facade of the conduit can be sculpted to appropriately house a conventional single-mode optical fiber.The breadth of the recess ought to match the diameter of the standard single-mode optical fiber, while its profundity should be equivalent to half the radius of said fiber.Envelop the undetermined optical fiber around the metallic tube, and secure the end with a 10 g mass to maintain tension.Subsequently, wrap the other end of the fiber around the bottom of a constant-temperature water bath, ensuring uniform heating of the fiber.This setup effectively addresses the issues of significant strain calibration errors, and low temperature and strain calibration efficiency.

A. The Calibrated Temperature and Strain Frequency Shift Coefficient
Set the temperature of the thermostatic water bath to T = [30,40,50,60,70,80,90] °C, the showcase of the Brillouin frequency displacement of the relaxed fiber with via iterative evaluations at distinct temperatures can be observed in Table II.
Based on the data provided in Table II, a fitting curve for the relaxation fiber measurements can be obtained by applying the method of linear regression, as shown in Fig. 10.The linearity of the regression is all above 0.98.Consequently, the temperature shift coefficient (C T ) for this particular fiber can be determined as 1.1483 MHz/°C.
The equation Δε(n) = 23 * ΔT(n) can be utilized to calculate the expansion amount of the metal tube at each temperature point.Then, by considering the expansion coefficient of the metal tube, the variation of the Brillouin frequency shift of the wound optical fiber at each temperature point can be determined, denoted as ΔV BS (n).Employing a linear fitting approach, the strain frequency shift coefficient of this optical fiber can be obtained as 4.7086 MHz/100 µε.

B. Strain Experiment Results and Analysis
The application of strain is exerted at the position of the fiber optic tail end as depicted in Fig. 11.The initial section of the fiber spans a length of 9990 meters, while the subsequent section measures approximately 10 meters in length.
Take a certain length of the last 10 meters and fix it on the adjustable tensiometer for stretching, applying a strain of 600 µε.Take one of the measurement results and the waveform captured by the oscilloscope is shown in Fig. 12.
The initial rise in the backscattering curve is attributed to the phenomenon of end-face reflection in the optical fiber.The later rise can be attributed to the variations in the refractive index and speed of sound within the fiber caused by applying strain, which in turn modifies the Brillouin frequency shift.After photoelectric conversion, these changes are characterized as alterations in voltage.
The operating mode of the spectrometer is set to "zero-span".The measurement is centered around a frequency of 1.2 GHz, with a scanning range from 1.226 GHz to 1.386 GHz, and an average of 5000 times per measurement.The Brillouin frequency shift plot, as shown in Fig. 13, will be generated using MATLAB.
From Fig. 13, it can be observed that the direction of the Brillouin frequency shift is towards increasing values.Specifically, at a distance of 9990 m to 10000 m, the change in frequency shift is approximately 0.03 GHz.By utilizing the previously calibrated strain frequency shift coefficient of 4.7089 MHz/100 µε, the strain can be determined to be 637.1 µε, which deviates from the initially applied strain by 37.1 µε.
Using a tension gauge with adjustable pulling force, the fiber optic is subjected to stretching, while a strain detector is employed to measure the applied strain values.Applying strains of 3000 µε and 6000 µε separately, the fiber optic cable is stretched at distances of 2 km, 4 km, 6 km, and 8 km.The measurement results are illustrated in Figs. 14 and 15.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Based on the analysis described above, an error analysis graph depicting varying strains at different distances is presented in Fig. 16.The maximum error values for both sets of measurements fall within the range of ±35.76.Additionally, it can be observed that, under stable conditions for all other factors, the detected error of the system increases as the distance of the fiber optic sensor increases, aligning with theoretical evidence.In terms of absolute error magnitude, when two different strains are applied at the same location, the error increases proportionally with the magnitude of the strain.
To assess the maximum measurement range of distributed optical fibers, the stretching stress range was varied incrementally by 500 µε, from 500 µε to 10,000 µε.The measurement results for the range of 500 µε to 10,000 µε, as depicted in Fig. 1, were analyzed for errors.
It can be observed from Fig. 17 that the maximum measurement error exhibits an irregular distribution and is consistently below 30 µε.When stretched to 10000 µε, the strain measurement range of the system is limited due to the bandwidth limitations of the balance detector used.Therefore, the maximum strain measurement range of the system is 9500 µε, indicating that the system can achieve accurate measurements within a range of 10 kilometers.

V. CONCLUSION
This article presents an analysis of the strain sensing mechanism based on Brillouin scattering and proposes a Brillouin frequency shift extraction method using the Adam algorithm.The Adam algorithm is simulated and analyzed based on the extraction model.Additionally, a BOTDR stress experimental measurement system based on heterodyne coherent detection is constructed.The Adam algorithm is used to perform strain testing on the distributed optical fiber, and error analysis is conducted.The experimental results demonstrate that this method achieves accurate and rapid measurement of a distributed optical fiber spanning 10 km with a maximum strain measurement range of 9500 µε.The research findings in this article provide theoretical and experimental evidence for the application of distributed Brillouin optical fiber sensing technology.

Fig. 3 .
Fig. 3. Fitting curve of Brillouin frequency shift versus strain change.(a) Fitting points and strains for Adam's algorithm.(b) Fitting points and strains for Gauss-Newton algorithm.(c) Fitting points and strains for L-M algorithm.

Fig. 4 .Fig. 5 .
Fig. 4. Frequency shift error of the three algorithms under different strains.(a) Frequency shift error of the fitting curve of Adam algorithm.(b) Frequency shift error of fitting curve of Gauss-Newton algorithm.(c) Frequency shift error of the fitting curve of L-M algorithm.

Fig. 6 .
Fig.6.Fitting curves between the input power and BGS spectral width.

Fig. 9 .
Fig. 9. Device for the simultaneous calibration of temperature and strain frequency shift coefficients.

Fig. 10 .
Fig. 10.Data fitting curves for multiple measurements of relaxed fibers.(a) The first measurement.(b) The second measurement.(c) The third measurement.(d) The fourth measurement.

TABLE II FREQUENCY
SHIFT OF RELAXED OPTICAL FIBER AT DIFFERENT TEMPERATURES