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Demodulation of the overlapping reflection spectrum of serial FBGs based on a weighted differential evolution algorithm

Weikang Liu, Wensong Zhou, and Hui Li

Open Access Open Access

Abstract

This study addresses the wavelength demodulation problem of the overlapping reflection spectrum of serial fiber Bragg gratings (FBGs) with nearly-identical wavelength. Specifically, a novel demodulation model for the overlapping reflection spectrum was presented based on spectrum similarity, and this model encodes FBGs through reflectivity. Subsequently, a weighted differential evolution algorithm was employed to calculate the FBG wavelengths. And the factors affecting the demodulation accuracy of the proposed method were simulated and analyzed. Finally, the proposed method was applied to demodulate the overlapping reflection spectra of serial FBGs. The experiment results indicate that the proposed method is suitable for completely overlapping, partially overlapping, and non-overlapping reflection spectra of serial FBGs. The wavelength demodulation accuracy demonstrated here in fully overlapping situations for two, three, and four FBGs was only 4.5, 14.9, and 24.6 pm, respectively.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Fiber Bragg gratings (FBGs) have the advantages of light weight, electromagnetic immunity, low cost, and corrosion resistance and can perform high-precision sensing of strain, temperature, vibration, pressure, and many other physical quantities [1,2]. They have been widely used in various fields of structural health monitoring, such as aerospace [3], civil engineering [4] bioengineering [5], power engineering [6], and railway engineering [7]. For an FBG sensor, demodulating the wavelength shift and converting it into the corresponding physical quantities is essential. The most commonly used FBG demodulation schemes are wavelength division multiplexing (WDM) [8] and time division multiplexing (TDM) [9]. In a WDM system, the operating wavelength range of each FBG must be independent, leading to inefficient utilization of light-source bandwidth resources and seriously limits the multiplexing number of FBGs. In a TDM system, the demodulation system is more complex because each FBG must be addressed, and a certain distance must be maintained between adjacent FBGs to meet the time difference requirement. In addition, the TDM system has the problems of spectral depression, shadow crosstalk, and multiple reflections, which limit the multiplexing number of the FBGs.

Scholars have proposed various spectral overlapping multiplexing methods to improve the multiplexing capacity of FBGs. Gong et al. [10] proposed a minimum variance shift (MVS) method for the demodulation of the overlapping reflection spectrum of two parallel FBGs. This method improves the multiplexing capability of WDM systems and has high detection accuracy. However, when the number of parallel FBGs is large, the solution set is also large owing to the traversal method, resulting in a low solution speed and difficulty in solving the problem. Hence, some scholars have used various optimization algorithms to improve the solution speed, such as the genetic algorithm (GA) [11], differential evolution (DE) [12], and particle swarm optimization (PSO) algorithm [13,14]. Compared with the traversal method for solving the MVS, these methods are more suitable for the static demodulation of FBGs. To ensure that the demodulation method of the overlapping reflection spectrum has dynamic demodulation performance, machine learning methods have been employed, such as deep neural networks [15–17] and extreme learning machines [18]. However, these methods are applicable only to specific FBGs. When the parameters of some FBGs in a particular array change, the dataset must be reobtained to train the neural network, which is a troublesome process. Furthermore, these methods have been proposed to demodulate the overlapping reflection spectrum of parallel FBGs. That is, they can expand the number of branches of an FBG array but not its sensing distance.

The demodulation method for the overlapping reflection spectrum of serial FBGs is very rare because of the lack of an overlapping reflection spectrum demodulation model for serial FBGs. In addition, the existing methods have several shortcomings. Jiang et al. [19] introduced an overlapping reflection spectrum demodulation model based on a transmission matrix model and realized the overlapping reflection spectrum demodulation of two serial FBGs. This model has many parameters and is complex; therefore, it is suitable only for modeling the overlapping reflection spectrum of a small number of serial FBGs. Guo et al. [14] proposed an overlapping reflection spectrum demodulation method for serial FBGs by using a modified particle swarm optimization algorithm. However, the demodulation error of this method is unsatisfactory and requires further investigation. Therefore, a novel overlapping reflection spectrum demodulation method suitable for serial FBGs that can series connect more FBGs with the nearly-identical wavelength is required. This is of great significance for long-distance and dense monitoring with FBGs, such as monitoring bridges, highways, and railways.

In this study, a novel demodulation method for the overlapping reflection spectrum of serial FBGs was proposed. Subsequently, an improved differential evolution algorithm called weighted differential evolution (WDE) was used to improve the solution efficiency of the proposed model.

In the remainder of this paper, the theory and principle of the proposed method are first presented. In addition, the principle of the demodulation method based on WDE is briefly described. Subsequently, the effects of spectral attenuation, shadow crosstalk and multiple reflections on the demodulation accuracy of the proposed method are simulated and analyzed. Finally, a series of experiments are performed to validate the proposed method, followed by the discussion and conclusions.

2. Principles

2.1 Problem description

Figure 1 shows a FBG array with different nearly-identical wavelength units. This array is composed of multiple units, each of which uses FBGs with the nearly-identical wavelength. FBGs with the nearly-identical wavelength in each unit usually need to be demodulated by TDM, otherwise they overlap together in the demodulation module (DE). Due to the limitation of the pulse width of light source, a certain distance (usually 1 m) is required between adjacent FBGs with the nearly-identical wavelength. In addition, the wavelength demodulation accuracy in a TDM system is affected by spectral attenuation, shadow crosstalk and multiple reflections. This paper aims to solve these problems through an overlapping spectral model and the WDE algorithm.

 figure: Fig. 1.

Fig. 1. FBG array with different nearly-identical wavelength units (ASE: Amplified spontaneous emission, DM: Demodulation module).

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2.2 Demodulation model of serial FBGs

Take MU1 unit in Fig. 1 as an example, B1, B2, …, Bi, …, Bn represent FBGs with nearly-identical wavelength, and F(λ) is the initial incident-light signal produced by amplified spontaneous emission (ASE) light source. When the incident light signal passes through each FBG, the downward light signal can be expressed as [9]

$${F_{i - 1}}(\lambda ) = F(\lambda )\left[ {\prod\limits_{k = 1}^{i - 1} {({1 - {R_k}{S_k}({\lambda ,{\lambda_{{\textrm{B}_k}}}} )} )} } \right]\textrm{ ,}$$
where λ is the wavelength of the incident light, Rk is the reflectivity of the k-th FBG, Sk(λ, λBk) is the reflection spectrum function of the k-th FBG, and λBk is the Bragg wavelength of the k-th FBG.

When the downward light signal passes through the first FBG, the signal returned to the DM can be expressed as

$${G_1}({\lambda ,{\lambda_{{\textrm{B}_1}}}} )= F(\lambda ){R_1}{S_1}({\lambda ,{\lambda_{{\textrm{B}_1}}}} )\textrm{ ,}$$
When the downward light signal Fi-1(λ) passes through the i-th FBG (i = 2, 3, …, n), the signal reflected by Bi reaches the DM through Bi-1, …, B2, and B1, and the reflected signal of each FBG can be expressed as
$${G_i}({\lambda ,{\lambda_{{\textrm{B}_i}}}} )= {R_i}{S_i}({\lambda ,{\lambda_{{\textrm{B}_i}}}} ){F_{i - 1}}(\lambda )\left[ {\prod\limits_1^{k = i - 1} {({1 - {R_k}{S_k}({\lambda ,{\lambda_{{\textrm{B}_k}}}} )} )} } \right].$$
Substituting (1) into (3), the independent reflection spectrum of each FBG reaching the DM is obtained:
$${G_i}({\lambda ,{\lambda_{{\textrm{B}_i}}}} )= {R_i}{S_i}({\lambda ,{\lambda_{{\textrm{B}_i}}}} )F(\lambda ){\left[ {\prod\limits_{k = 1}^{i - 1} {({1 - {R_k}{S_k}({\lambda ,{\lambda_{{\textrm{B}_k}}}} )} )} } \right]^2}.$$
The overlapping reflection spectrum (RORS) collected by the DM without TDM can then be expressed as
$${R_{\textrm{ORS}}} = \sum\nolimits_{i = 1}^n {{G_i}(\lambda ,{\lambda _{{B_i}}})}.$$
To obtain the Bragg wavelength of each FBG from the overlapping reflection spectrum, an artificial spectrum (RAS) was constructed as shown below:
$${R_{\textrm{AS}}} = \sum\nolimits_{i = 1}^n {{G_i}(\lambda ,{x_{{B_i}}})}.$$
The similarity between RAS and RORS can be evaluated by the following equation, which can be expressed as
$$Var = \frac{{\sum\limits_{i = 1}^n {{{({{y_i} - {{\hat{y}}_i}} )}^2}} }}{{\sum\limits_{i = 1}^n {{{({{y_i} - \bar{y}} )}^2}} }}\textrm{ ,}$$
where yi represents the power amplitude at each point of RORS, ${\hat{y}_i}$ represents the power amplitude at each point of RAS, $\bar{y}$ represents the average power amplitude of RORS.

When Var is at its minimum, RORS and RAS are most similar. At this point, the actual wavelength of each FBG can be estimated with minimal error.

The response function Sk(λ, λBk) of the FBG in the above model can be approximately replaced by a Gaussian function expressed as

$${S_k}({\lambda ,{\lambda_{{\textrm{B}_k}}}} )= \exp \left[ { - 4\ln 2{{\left( {\frac{{\lambda - {\lambda_{{\textrm{B}_k}}}}}{{\Delta {\lambda_{{\textrm{B}_k}}}}}} \right)}^2}} \right],$$
where ΔλBk is the full width at half maximum (FWHM) of the k-th FBG.

2.3 Model solution method based on the weighted differential evolution algorithm

The differential evolution (DE) algorithm has strong global search capacity and a high convergence rate [21,22]. However, DE is sensitive to the selection of control parameters and requires a time-consuming and difficult parameter-adjustment process for different problems. Recently, Civicioglu et al. [23] proposed a novel DE algorithm called the weighted differential evolution (WDE) algorithm. WDE does not require a time-consuming or difficult parameter-tuning process. The crossover and mutation processes of the WDE are simpler and much more efficient than those of the DE. In this study, the model proposed in Section 2.2 is combined with the WDE algorithm to realize the demodulation of the overlapping reflection spectrum of the serial FBGs. The principle of the WDE algorithm can be found in the literature [23] for interested readers. The key steps of the demodulation process are as follows:

Initialization: Randomly generate the initial population W. The formula for randomly obtaining the initial population W can be expressed as

$${W_{({i_0},{j_0})}} \sim {\mathbf U}({{w_{low}},{w_{up}}} )\textrm{ ,}$$
where wlow and wup are the lower and upper search limits of the wavelength, respectively. U(·) denotes a random array with a continuous uniform distribution. where i0 and j0 are the sizes of W, i0 = [1:2N], j0 = [1:D], N is the population size, and D is the number of FBGs. The fitness value of the initial population W is calculated by
$$fit{W_{({i_0})}} = F({{W_{({i_0})}}} )\textrm{ ,}$$
where F(·) and fitW(i0) denote the fitness evaluation function and fitness value of the initial population, respectively. In this study, the fitness evaluation function F(·) was obtained using Eqs. (1)–(7) in Section 2.2.

Selection: N individuals are randomly selected from W to obtain a subpopulation SubW in each iteration. SubW is defined as

$$SubW = \left.{W_{(k)}}\right| k = \left.{j_{(1:N)}}\right| j = permute(1:2N)\textrm{ ,}$$
where permute(·) denotes the permuting function.

The fitness value of SubW can be expressed as

$$fitSubW = F(SubW).$$
Mutation: The mutation process aims to generate new population individuals, that is, TempW, which is generated by
$$Temp{W_{(index\textrm{ })}} = \sum \left.{({w. \times {W_{(l)}}} )} \right| l = {j_{(N + 1:2N)}}\textrm{ ,}$$
where index = [1:N] and index∈Z + . w = rand(N,1).^3./sum(rand(N,1). ^3), “rand” denotes uniformly distributed pseudorandom numbers. j is defined in (13). The mutation process is then controlled by M(1:N,1:D) = 0 in each iteration, and M is updated by
$$M(index,J) = 1\textrm{ ,}$$
where J is defined as
$$\left. \begin{array}{l} J = V(1:[K \times D])\textrm{ }|\textrm{ }V = permute({{j_0}} )\\ \textrm{if }\alpha < \beta \textrm{ then }K\textrm{ = }{\kappa^\textrm{3}}\textrm{; else }K\textrm{ = 1 - }{\kappa^\textrm{3}} \end{array} \right\}\textrm{ ,}$$
where α, β, and κ are uniformly distributed pseudorandom numbers [0, 1].

In the WDE, the scale factor, F, is confirmed by the rule:

$$\left. \begin{array}{l} F = randn(1,D).^{\wedge} 3\textrm{ , }\alpha < \beta \\ F = randn(N,1).^{\wedge}3\textrm{, other} \end{array} \right\}\textrm{ ,}$$
where randn denotes the normally distributed pseudorandom number function. A trial vector obtained by mutation operation can be expressed as
$$T = SubW + F. \times M. \times ({TempW - Sub{W_{(m)}}} )\textrm{ ,}$$
where m is an integer between 1 and N generated by a random permutation function.

To ensure that the wavelength of the new individual is within the wavelength range of the FBG, the individual’s boundary must be constrained to T. The boundary control conditions are expressed as

$$\left. \begin{array}{l} {T_{(i,{j_0})}} = lo{w_{({j_0})}} + \kappa_{(1)}^3({u{p_{({j_0})}} - lo{w_{({j_0})}}} ), \textrm{ }{T_{(i,{j_0})}} < lo{w_{({j_0})}}\\ {T_{(i,{j_0})}} = u{p_{({j_0})}} + \kappa_{(1)}^3({lo{w_{({j_0})}} - u{p_{({j_0})}}} ), \textrm{ }{T_{(i,{j_0})}} > u{p_{({j_0})}} \end{array} \right\}.$$
The fitness value of T can be expressed as
$$fitT = F(T).$$
Crossover: Crossover operation is introduced to increase the diversity of interference parameter vectors. In WDE, T and fitT are used to update SubW and fitSubW, respectively, as per the greedy selection rule. The relevant process can be expressed as
$$[Sub{W_{({i^ \ast })}},fitSub{W_{({i^ \ast })}}] = [{T_{({i^ \ast })}},fit{T_{({i^ \ast })}}],\textrm{ }fit{T_{\textrm{(}{i^ \ast }\textrm{)}}}{\ < }fitSub{W_{({i^ \ast })}}\textrm{ ,}$$
where i*∈i.

The updated SubW and fitSubW were used to update W(l) and fitW(l), respectively, and l is defined as in (15).

Optimal solution: The actual wavelength of each FBG can be obtained using the above operations. The obtained wavelength WOS can be expressed as

$$[{W_{\textrm{OS}}},fit{W_{OS}}] = [fit{W_{(\gamma )}},{W_{(\gamma )}}]\textrm{ |}fit{W_{\textrm{(}\gamma \textrm{)}}}\textrm{ = min(}fitW\textrm{), }\gamma \in i\textrm{ }\textrm{.}$$

3. Numerical analysis

3.1 Analysis of factors affecting the demodulation accuracy

The proposed model in this paper is based on the existing TDM model [9]. However, this model does not need to obtain the independent reflection spectrum of each FBG before wavelength demodulation. Thus, factors affecting the demodulation accuracy of TDM model, such as spectral attenuation, shadow crosstalk and multiple reflections, may have different effects on the model proposed in this paper. These influencing factors were analyzed below.

3.1.1 Analysis of spectral attenuation and shadow crosstalk

When the incident light signal propagates in a serial FBG array with the nearly-identical wavelength, the front FBG produces occlusion effect on the rear FBGs. This causes the spectra of FBGs arriving at the demodulation module to be distorted (Attenuation and shadow crosstalk). To explore the influence of spectral attenuation and shadow crosstalk on the demodulation accuracy of the proposed method, simulations were carried out, and the parameters of FBGs are shown in Table 1.

Tables Icon

Table 1. Simulation conditions of the serial FBG array with different reflectivity

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According to the parameters in Table 1, the overlapping spectra and reflection spectra of each FBG were obtained through Eqs. (1)–(5). Subsequently, the wavelength of each FBG in the overlapping spectrum was solved from the overlapped spectra using the method proposed in this paper. In the WDE algorithm, N and the maximum number of iterations were 200 and 5000. The wavelength search range was set to 1548–1552 nm. In addition, the conventional peak detection (CPD) methods [24], including centroid calculation (CC), polynomial curve fit (PCF), peak search (PS), Gaussian curve fit (GCF) and mean of half maxima (MHM), were used to solve the wavelength from the independent spectrum of each FBG. The spectra under different simulation condition and the demodulation error of the two demodulation methods are shown in Figs. 2 and 3.

 figure: Fig. 2.

Fig. 2. Influence of spectral attenuation: (a) Overlapping spectrum (OS) and independent spectrum of each FBG, (b) demodulation accuracy of different demodulation methods.

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 figure: Fig. 3.

Fig. 3. Influence of shadow crosstalk: (a) Overlapping spectrum (OS) and independent spectrum of each FBG, (b) a typical diagram of spectral distortion caused by shadow crosstalk, (c) demodulation accuracy of different demodulation methods.

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From Fig. 2(a), the spectral attenuation mainly causes the flat top and depression of the reflection spectra of rear FBGs. According to Fig. 2(b), except PS method, other methods have high demodulation accuracy. The reason for the large demodulation error of PS method is that it uses the wavelength corresponding to the maximum power value as the FBG wavelength. And the wavelength corresponding to the maximum power value in the reflection spectrum is not the actual wavelength of the FBG due to the flat top and depression.

From Fig. 3(a) and (b), the shadow crosstalk mainly causes the shift of the reflection spectra of rear FBGs. From Fig. 3(c), the demodulation error of CDP methods is large. The reason is that the spectra of the rear FBGs become asymmetrical due to the shadow crosstalk, which has a great impact on CPD methods [24]. In contrast, the demodulation accuracy of the proposed method (WDE) is high. The reason is that the proposed method does not need to obtain the independent reflection spectrum of each FBG, and the wavelength of each FBG is obtained through the similarity between the constructed spectrum and the actual overlapping spectrum.

3.1.2 Multiple reflections

Multiple reflections between the FBGs reduce the signal-to-noise ratio (SNR) of the signal received by the demodulation module, resulting in unrecognizable signals. The model proposed in Section 2.2 did not consider the influence of multiple reflections, so the deterioration of the signal-to-noise ratio must be prevented. A study [20] has shown that the relationship between the reflected signal of the FBG and the signal of multiple reflections in a serial FBG array with the same reflectivity and wavelength can be expressed as

$$\frac{P}{{P^{\prime}}} = \frac{{{{({1 - {r_0}} )}^2}}}{{r_0^2}}\textrm{ ,}$$
where P is the reflected signal power, $P^{\prime}$ is the multiple-reflection signal power, and r0 is the reflectivity of the FBGs.

The SNR of the reflected signal of the FBG can then be expressed as

$$SNR = 10\lg (\frac{P}{{P^{\prime}}})\textrm{ }\textrm{.}$$
Through Eq. (23), the relationship between the FBG reflectivity and the SNR under the influence of multiple reflections is shown in Fig. 4. Evidently, in a serial FBG array, the lower the reflectivity of the FBGs, the higher the signal-to-noise ratio of the FBGs. When the reflectivity of the FBGs is less than 20%, the SNR of the FBGs is higher than 10 dB.

 figure: Fig. 4.

Fig. 4. Relationship between FBG reflectivity and signal-to-noise ratio.

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3.2 Summary of numerical results

According to the numerical results in Section 3.1, spectral attenuation and shadow crosstalk have no effect on the demodulation accuracy of the proposed method. However, when the spectral attenuation is serious, the reflected light of FBGs at the rear of the FBG array cannot reach the demodulation module. In addition, the deterioration of SNR caused by multiple reflections is also a factor limiting the proposed method, as described in Section 3.1.2. Thus, the proposed method is suitable for using multiple FBGs with different and lower reflectivity in a monitoring unit.

4. Experimental investigations

In this section, a series of experiments was designed and conducted. The proposed overlapping reflection spectrum demodulation method was applied to experimental data for verification. This paper aims to propose a novel wavelength demodulation method of FBGs with the nearly-identical wavelength. Thus, a unit with several FBGs in the schematic diagram of Fig. 1 was used for the test.

4.1 Experimental setup and procedure

4.1.1 Experimental setup

Figure 5(a) shows a photograph of the experimental setup. The optical path model and optical spectrum (OSA) were used to obtain the overlapping reflection spectrum of FBGs, and the principle of the demodulation system is shown in Fig. 5(b). The demodulation system was composed of an amplified spontaneous emission (ASE) light source, isolator, circulator, and OSA. The ASE light source was a C-band ASE broadband light source produced by Hoyatex Optoelectronics Co. Ltd., China. The OSA used was AQ6370D, produced by Yokogawa Electric Co. Ltd., Japan. The isolator and circulator were produced by Photonics Co., Ltd., USA.

 figure: Fig. 5.

Fig. 5. Experimental setup and schematic diagram of demodulation system: (a) Experimental setup, (b) schematic diagram of demodulation system.

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As shown in Fig. 5(a), a water bath (JOANLAB Co. Ltd., China) was used to prevent the disturbance of the FBG wavelength caused by the fluctuation of ambient temperature. The constant temperature resolution of the water bath is 0.1 °C, suggesting that the disturbance of the FBG wavelength caused by the fluctuation of ambient temperature can be approximately constrained within 1 pm. And different overlapping spectra were obtained by adjusting the deformation of each FBG in the monitoring unit. The deformation of the FBG was realized using a simple structure, as shown in Fig. 6. Before the test, the FBG was bonded to an elastic steel sheet, and the elastic steel sheet was placed on the outer skeleton. During the test, the deformation of the FBG was adjusted using a deformation-adjusting bolt. The reserved holes were used to prevent the FBG from breaking during the bolt tightening.

 figure: Fig. 6.

Fig. 6. Structure for FBG deformation simulation.

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The test scheme is presented in Table 2. Under each working condition, FBGs have different reflectivities, and the FBGs are arranged in descending order of reflectivity. The parameters for each FBG are listed in Table 3. Nine tests were performed under each working condition, and the nine tests included three reflection spectrum forms: completely overlapping, partially overlapping, and non-overlapping.

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Table 2. Simulation conditions

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Table 3. Nominal parameters of FBGs

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4.1.2 Experimental procedure

The experimental procedure was as follows: 1) The two ends of each FBG were connected with the FC/APC jumper. 2) The FBGs on the elastic steel sheets were fixed with adhesive and assembled with a deformation simulation structure. 3) FBGs were connected with jumpers according to different working conditions and placed into a water bath. 4) The temperature of the water bath was set to 30 °C and kept constant to eliminate the influence of ambient temperature on the wavelength of the FBGs. 5) The light source outlet of the optical path module was connected to the OSA and the spectrum of the incident light was tested and stored. 6) The FBGs were connected to the light source outlet of the optical path module, and the reflected light outlet of the optical path module was connected with the OSA. 7) The deformation-adjusting bolt was adjusted to move the reflection spectrum of each FBG away from the overlapping area, and the independent reflection spectrum of each FBG was collected through the OSA (The data obtained in this step were used for subsequent acquisition of the actual reflectivity and FWHM of the FBGs). 8) The overlapping reflection spectra of serial FBGs were obtained by randomly adjusting the deformation-adjusting bolt, and the overlapping reflection spectrum data were collected using the OSA. 9) The position of the deformation adjusting bolt must be maintained, each FBG at the jumper end was successively disconnected, and the independent reflection spectrum of each FBG was collected through the OSA (Different from the operation in step 7, the data obtained in this step were used for the extraction of the actual wavelengths of FBGs in the overlapping spectrum of step 8). 10) The above steps were repeated to obtain all the required data.

4.2 Results and discussions

The incident light spectra obtained under various working conditions from the tests are shown in Fig. 7. In this study, these spectra were used as the initial incident-light spectra of the proposed model described in Section 2.2.

 figure: Fig. 7.

Fig. 7. Incident light spectra.

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4.2.1 Demodulation results and error analysis

In the WDE algorithm, N and the maximum number of iterations were 20 and 400, 100 and 500, and 200 and 500 for the two, three, and four serial FBG monitoring units, respectively. The wavelength search range was set to 1548–1554 nm. The overlapping reflection spectra of FBGs obtained from the testing under various working conditions and the comparison with the analytical results are shown in Fig. 8. It can be seen that the demodulation results of the algorithm are in good agreement with the overlapping reflection spectra of FBGs obtained from the experiments, except for some minor discrepancies at the peak. This might be because the actual reflection spectra in the tests are not completely consistent with the Gaussian function, leading to a light-intensity fitting error. Comparisons between the demodulation results of the algorithm and test results indicate that the proposed model can effectively fit the overlapping reflection spectrum of serial FBGs.

 figure: Fig. 8.

Fig. 8. Comparison between experimental and analysis results: (a) Condition C1, (b) Condition C2, and (c) Condition C3.

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To evaluate the performance of the algorithm, the root mean square (RMS) of the 20 demodulation results [19] was taken as the demodulation wavelength of each FBG, and the average value of the root mean square error (RMSE) of the demodulation results of each FBG was taken as the error. The observed value S and error E were calculated using the following equations.

$$S = \sqrt {\frac{{\sum\limits_{i = 1}^N {X_i^2} }}{N}} \textrm{ ,}$$
where Xi is the wavelength obtained from the i-th demodulation. N is the number of demodulations (N = 20 in this paper).
$$E = \sqrt {\frac{1}{{{N_F}}}(\sum\limits_{i = 1}^{{N_F}} {{{({\lambda _{Bi}} - {S_i})}^2}} )},$$
where λBi is the actual wavelength of the FBGi, Si is the algorithm demodulation wavelength corresponding to FBGi, and NF denotes the number of serial FBGs.

In this study, the mean of the half-maxima method [24] was employed to analyze the reflection spectrum of each FBG obtained in test step 9, and the obtained wavelength was considered as the actual wavelength of each FBG. The actual wavelengths of the FBGs and the demodulation results of the algorithm are shown in Figs. 9–11. Figure 12 shows the variation trend of the maximum RMSE of error under different overlapping conditions.

 figure: Fig. 9.

Fig. 9. Results of Condition C1: (a) Actual (A) and demodulation (D) wavelength of each FBG, (b) Absolute (Abs) error and RMSE (E) of error.

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 figure: Fig. 10.

Fig. 10. Results of Condition C2: (a) Actual (A) and demodulation (D) wavelength of each FBG, (b) Absolute (Abs) error and RMSE (E) of error.

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 figure: Fig. 11.

Fig. 11. Results of Condition C3: (a) Actual (A) and demodulation (D) wavelength of each FBG, (b) Absolute (Abs) error and RMSE (E) of error.

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 figure: Fig. 12.

Fig. 12. Variation trend of the maximum RMSE of error under different conditions and different overlapping conditions: C denotes completely overlapping, P denotes partially overlapping, and N denotes non-overlapping.

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Referring to Figs. 9–12, the error is different in each case, even in different test step. This may be because of the different complexities of the overlapping spectra and the stochastic properties of the WDE algorithm. The maximum RMSE of errors for the two, three, and four serial FBGs were 4.5, 14.9, and 24.6 pm, respectively, and the maximum error occurred when the reflection spectra completely overlapping. The maximum error increased with an increase in the number of serial FBGs. There are two main reasons for this: the complexity of the overlapping spectrum increases with an increase in the number of serial FBGs; as the number of serial FBGs increases, the dimension and quantity of the solution set matrix increases, and the WDE algorithm requires more iterations to converge.

For different overlapping cases, the maximum error is the largest when the reflection spectra of each FBG completely overlap; the error is smaller when there is partial overlap and no overlap. This can also be explained by the complexity of the overlapping reflection spectrum. The complexity of the reflection spectrum is lower when the reflection spectra are partially overlapping or non-overlapping and the complexity is the highest when the reflection spectra are completely overlapping.

A comparison between the maximum error of a single FBG under various working conditions and the results in the literature [14] is shown in Fig. 13. Compared with literature [14], the proposed method has a higher demodulation accuracy.

 figure: Fig. 13.

Fig. 13. Comparison of the maximum between the proposed algorithm and literature [14].

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4.2.2 Reflectivity difference analysis

In this study, the actual reflectivity of each FBG was calculated using (26). This calculation method is different from the existing FBG reflectivity calculation method but is applicable to the model proposed in Section 2.2.

$$R = \frac{{{P_{\textrm{peak}}}}}{{{P_{\textrm{light}}}}} \times 100\%,$$
where Ppeak denotes the peak power of the FBG reflection spectrum and Plight is the power of the incident light corresponding to the position of Ppeak.

Figure 14 shows the actual reflectivities of the FBGs under different working conditions. The actual reflectivity of each FBG differs significantly from the nominal reflectivity, as shown in Table 3. There are two main reasons for this discrepancy. First, the optical devices are connected by jumpers (FC/APC) in the test, which leads to a large loss of light energy. Second, the nominal parameters of the FBGs were measured by the FBG manufacturer, and the test conditions and devices were different from those in this study. The difference in reflectivity of each FBG at different points in time was due to the random insertion loss caused by the insertion and extraction of jumpers in each test.

 figure: Fig. 14.

Fig. 14. Actual reflectivity of FBGs under different working conditions: (a) Condition C1, (b) Condition C2, and (c) Condition C3.

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Figure 15 shows the reflectivity difference between adjacent FBGs under various working conditions. For two, three, and four serial FBGs, the minimum reflectivity differences in each test were 1.1%, 0.8%, and 0.1%, respectively. This reflectivity difference is smaller than that reported in the existing literature [14,17]. That is, the overlapping reflection spectrum demodulation method proposed in this study has lower requirements for the difference in the reflectivity of the adjacent FBG in a serial FBG unit.

 figure: Fig. 15.

Fig. 15. Reflectivity difference between adjacent FBGs under various working conditions: (a) Condition C1, (b) Condition C2, and (c) Condition C3.

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4.2.3 Comparison and analysis

In order to illustrate the advantages of the proposed method, the demodulation results were compared with those of the standard DE algorithm. In the DE algorithm, N, the maximum number of iterations and the wavelength search range were the same as those in WDE. DE and WDE algorithms were run on a computer with average performance. The processor of this computer is Intel Core i7–8700 @ 3.2 GHz six core, and the memory is 16 GB. The overlapping spectra corresponding to 9 data in Figs. 9–11 were used for comparative analysis. The demodulation error and time of the two algorithms are shown in Fig. 16. The error (E) was still calculated by Eqs. (24) and (25). The average time of 20 demodulation was taken as the single demodulation time (STD).

 figure: Fig. 16.

Fig. 16. Comparison of demodulation accuracy and time between WDE and DE: (a) Demodualtion accuracy (RMSE of errors, E), (b) single demodulation time (STD).

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It can be seen from Fig. 16, the demodulation accuracy of the proposed method is basically the same as that of the standard DE algorithm (Fig. 16(a)), but the single demodulation time is shorter (Fig. 16(b)). For the demodulation of two, three, and four serial FBG overlapping reflection spectra, the demodulation time of the proposed method is approximately 5.2%, 23.7% and 43.3% of the demodulation time of standard DE, respectively. More importantly, the demodulation speed of the proposed method can be further improved by using high-performance computer or parallelization.

5. Conclusions

This paper proposed a novel demodulation method for the overlapping reflection spectrum of serial FBGs. This method includes a new demodulation model of the overlapping reflection spectrum of serial FBGs and the WDE optimization method. The effectiveness of the method was verified by applying it to the actual overlapping reflection spectra of serial FBGs. The following conclusions were drawn:

  • (1) Spectral attenuation and shadow crosstalk do not affect the demodulation accuracy of the proposed method.
  • (2) The proposed method is applicable to the overlapping reflection spectrum demodulation of two, three, and four serial FBGs. This applicability indicates that the proposed method can increase the capacity of the FBG wavelength division multiplexing without upgrading hardware.
  • (3) For the demodulation of two, three, and four serial FBG overlapping reflection spectra, the demodulation accuracy (RMSE of error) reached 4.5, 14.9, and 24.6 pm, respectively.
  • (4) Compared with the existing research, the demodulation accuracy of the proposed method for completely overlapping spectrum is improved. However, it still needs further improvement.
  • (5) The proposed method distinguishes and locates the FBGs by the reflectivity difference of each FBG, and the minimum reflectivity difference requirement of adjacent FBGs is less than that in existing studies.
  • (6) The demodulation speed of the proposed method is significantly improved compared with the standard DE algorithm.

Funding

National Key Research and Development Program of China (2021YFB3704402); National Natural Science Foundation of China (51978217).

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.

References

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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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Figures (16)
Fig. 1.
Fig. 1. FBG array with different nearly-identical wavelength units (ASE: Amplified spontaneous emission, DM: Demodulation module).

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Fig. 2.
Fig. 2. Influence of spectral attenuation: (a) Overlapping spectrum (OS) and independent spectrum of each FBG, (b) demodulation accuracy of different demodulation methods.

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Fig. 3.
Fig. 3. Influence of shadow crosstalk: (a) Overlapping spectrum (OS) and independent spectrum of each FBG, (b) a typical diagram of spectral distortion caused by shadow crosstalk, (c) demodulation accuracy of different demodulation methods.

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Fig. 4.
Fig. 4. Relationship between FBG reflectivity and signal-to-noise ratio.

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Fig. 5.
Fig. 5. Experimental setup and schematic diagram of demodulation system: (a) Experimental setup, (b) schematic diagram of demodulation system.

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Fig. 6.
Fig. 6. Structure for FBG deformation simulation.

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Fig. 7.
Fig. 7. Incident light spectra.

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Fig. 8.
Fig. 8. Comparison between experimental and analysis results: (a) Condition C1, (b) Condition C2, and (c) Condition C3.

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Fig. 9.
Fig. 9. Results of Condition C1: (a) Actual (A) and demodulation (D) wavelength of each FBG, (b) Absolute (Abs) error and RMSE (E) of error.

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Fig. 10.
Fig. 10. Results of Condition C2: (a) Actual (A) and demodulation (D) wavelength of each FBG, (b) Absolute (Abs) error and RMSE (E) of error.

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Fig. 11.
Fig. 11. Results of Condition C3: (a) Actual (A) and demodulation (D) wavelength of each FBG, (b) Absolute (Abs) error and RMSE (E) of error.

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Fig. 12.
Fig. 12. Variation trend of the maximum RMSE of error under different conditions and different overlapping conditions: C denotes completely overlapping, P denotes partially overlapping, and N denotes non-overlapping.

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Fig. 13.
Fig. 13. Comparison of the maximum between the proposed algorithm and literature [14].

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Fig. 14.
Fig. 14. Actual reflectivity of FBGs under different working conditions: (a) Condition C1, (b) Condition C2, and (c) Condition C3.

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Fig. 15.
Fig. 15. Reflectivity difference between adjacent FBGs under various working conditions: (a) Condition C1, (b) Condition C2, and (c) Condition C3.

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Fig. 16.
Fig. 16. Comparison of demodulation accuracy and time between WDE and DE: (a) Demodualtion accuracy (RMSE of errors, E), (b) single demodulation time (STD).

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Tables (3)

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Table 1. Simulation conditions of the serial FBG array with different reflectivity

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Table 2. Simulation conditions

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Table 3. Nominal parameters of FBGs

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Equations (26)

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F i − 1 ( λ ) = F ( λ ) [ ∏ k = 1 i − 1 ( 1 − R k S k ( λ , λ B k ) ) ]  ,
G 1 ( λ , λ B 1 ) = F ( λ ) R 1 S 1 ( λ , λ B 1 )  ,
G i ( λ , λ B i ) = R i S i ( λ , λ B i ) F i − 1 ( λ ) [ ∏ 1 k = i − 1 ( 1 − R k S k ( λ , λ B k ) ) ] .
G i ( λ , λ B i ) = R i S i ( λ , λ B i ) F ( λ ) [ ∏ k = 1 i − 1 ( 1 − R k S k ( λ , λ B k ) ) ] 2 .
R ORS = ∑ i = 1 n G i ( λ , λ B i ) .
R AS = ∑ i = 1 n G i ( λ , x B i ) .
V a r = ∑ i = 1 n ( y i − y ^ i ) 2 ∑ i = 1 n ( y i − y ¯ ) 2  ,
S k ( λ , λ B k ) = exp ⁡ [ − 4 ln ⁡ 2 ( λ − λ B k Δ λ B k ) 2 ] ,
W ( i 0 , j 0 ) ∼ U ( w l o w , w u p )  ,
f i t W ( i 0 ) = F ( W ( i 0 ) )  ,
S u b W = W ( k ) | k = j ( 1 : N ) | j = p e r m u t e ( 1 : 2 N )  ,
f i t S u b W = F ( S u b W ) .
T e m p W ( i n d e x   ) = ∑ ( w . × W ( l ) ) | l = j ( N + 1 : 2 N )  ,
M ( i n d e x , J ) = 1  ,
J = V ( 1 : [ K × D ] )   |   V = p e r m u t e ( j 0 ) if  α < β  then  K  =  κ 3 ; else  K  = 1 -  κ 3 }  ,
F = r a n d n ( 1 , D ) . ∧ 3  ,  α < β F = r a n d n ( N , 1 ) . ∧ 3 , other }  ,
T = S u b W + F . × M . × ( T e m p W − S u b W ( m ) )  ,
T ( i , j 0 ) = l o w ( j 0 ) + κ ( 1 ) 3 ( u p ( j 0 ) − l o w ( j 0 ) ) ,   T ( i , j 0 ) < l o w ( j 0 ) T ( i , j 0 ) = u p ( j 0 ) + κ ( 1 ) 3 ( l o w ( j 0 ) − u p ( j 0 ) ) ,   T ( i , j 0 ) > u p ( j 0 ) } .
f i t T = F ( T ) .
[ S u b W ( i ∗ ) , f i t S u b W ( i ∗ ) ] = [ T ( i ∗ ) , f i t T ( i ∗ ) ] ,   f i t T ( i ∗ )   < f i t S u b W ( i ∗ )  ,
[ W OS , f i t W O S ] = [ f i t W ( γ ) , W ( γ ) ]  | f i t W ( γ )  = min( f i t W ),  γ ∈ i   .
P P ′ = ( 1 − r 0 ) 2 r 0 2  ,
S N R = 10 lg ⁡ ( P P ′ )   .
S = ∑ i = 1 N X i 2 N  ,
E = 1 N F ( ∑ i = 1 N F ( λ B i − S i ) 2 ) ,
R = P peak P light × 100 % ,
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