1. Introduction
In Canada, Ken Hill discovered FBG in 1978 at a research communication center [
1]. The concept of FBG is based on the change in the refractive index of the core of FBG due to absorbing ultraviolet light [
2]. Since FBG was invented, FBG sensors have gained their potential in the optical sensing world due to their built-in pros such as their tiny size, low cost, immunity to high voltage, magnetic field, chemical, and electromagnetic interference, in addition to their accuracy and sensitivity [
3]. On top of that, they provide the distribution or multiplexed sensing capability, which covers long distances and wide surface area. On the other hand, the conventional sensors cannot offer these unique features of the FBG sensors [
4]. Therefore, FBG sensors have been utilized in various medical, civil, electrical, mechanical, aerospace, maritime, oil and gas, military applications, renewable energy and battery charging applications [
5]. In addition, they have shown standard sensing for most physical signals, such as temperature displacement, strain, stress, and pressure [
6,
7].
The FBG is constructed inside the fiber ultraviolet light by making a periodic perturbation in the refractive index in the fiber’s core due to high-intensity exposure [
8]. A brag wavelength has been yielded from the band rejection constructed by the grating zone’s constructive interference of the reflected wavelengths. That Bragg wavelength depends on the grating parameters, including grating length, refractive index, and the grating period. Therefore, changing these parameters due to exposure to physical signals such as pressure, strain, and temperature will change the yielded bragging wavelength, which is the key idea behind using FBG in sensing [
9]. Because FBG sensing performance depends on the grating parameters, and the reflected wave from FBG is characterized mainly by the reflectivity and bandwidth, optimizing grating parameters is crucial for optimizing the sensing accuracy and getting the desired bandwidth and reflectivity that fulfills the applications’ requirements. These requirements are not the same for all industrial applications. For example, most industrial applications require narrow bandwidth, while some other applications, such as oil and gas, require a wide range of bandwidth [
10]. In addition, the multiplexing feature of FBG enables measuring a wide-scale range of parameters, especially in the construction industry, oil and gas applications, and tracking the power in PV power systems [
11]. Therefore, the bandwidth and reflectivity characteristics of the reflected wave of FBG are critical factors in multiplexing more sensing points with different wavelengths [
12]. In partial shading PV application, temperature, and irradiance of an array of modules need to be monitored. So the multiplexing and distributing sensing will be beneficial for such applications [
13]. To sum up, bandwidth, reflectivity, and sidelobe are the most critical parameters that need to be optimized to enhance the performance of the sensing capabilities of the FBG and fulfill the different applications’ needs. In addition, some sidelobes are there with the reflected signal at the brag wavelength, which is undesirable in sensing applications [
14]. The problem is how to satisfy all applications’ needs when sometimes there is a conflict between these requirements. For instance, some require minimizing the bandwidth and maximizing the reflectivity, while other applications require maximizing the bandwidth and reflectivity. In conclusion, it is a multi-objective optimization problem in which the bandwidth and reflectivity are the optimized variables. Moreover, the grating parameters, including the grating length and modulation index, are the control variables considering the conflict between these objectives for some applications, as noted by surveying a couple of studies [
15,
16]. These studies reported a positive correlation between the reflectivity and the modulation index on one hand and grating length on the other hand. On the other hand, they showed a negative correlation between the bandwidth and grating length, while it is positively correlated with index modulation. To recap and highlight the problem for which this paper presents a solution, this paper aims to determine the optimum grating parameters, including the grating length, modulation index, and grating period, that realize the optimum bandwidth and reflectivity to suit every industrial application’s needs. For example, the optimization solution proposed by this research provides the industry with a tool to determine the optimum FBG grating parameters for their applications.
Before designing the proposed solution, literature has been searched and reviewed to fetch the research contributions for solving the problem above. As a result, some studies worked on determining the grating parameters for obtaining the maximum reflectivity and the minimum sidelobes intensity. Moreover, other studies determined the relation between bandwidth and reflectivity and the grating parameters [
15,
16]. These studies contributed some simulation and experimental work for reporting the bandwidth, sidelobe, and reflectivity at different values of grating length and index modulation for determining the optimum bandwidth and reflectivity. However, these research works did not contribute to determining the optimum values for different requirements of industrial applications. They consider a specific range of grating length from 0 to 10 mm and index modulation from 0.0005 to 0.002. Nevertheless, some other studies can be used to validate the results obtained by the proposed optimization method in the present study. Moving to the proposed solution in this study, this paper presents an optimization method based on the nondominated sorting genetic algorithm II (NSGA-II) for solving the current multi-objective optimization problem that has some contradictions in its objectives. This method has been selected because it can apply the Pareto optimization concept and determine the Pareto front that can fetch superior solutions compared with other solutions in the search space. At the same time, these solutions are non-dominant to other objectives that conflict with them. The concept of nondominated sorting is that Pareto dominance is utilized to sort the population. The process commences by moving the first-ranked individuals of the non-dominating members from the initial population to the first front. Then the remaining individuals are then ordered based on the non-dominating sorting procedure to determine the second front. The process continues, and all populations are sorted and categorized based on their ranks on their equivalent fronts [
17]. Surveying literature on the contributions of the NSGA-II method to solve multi-objective problems, it is found that there are research works that utilize NSGA-II. For example, the assignment problem has been solved by using NSGA-II [
18], in which N tasks are optimally assigned to N agents to maximize the performance of the task and minimize the total cost of achieving the tasks. In addition, the allocation problem has been optimized using NSGA-II, in which activities are optimally allocated between resources to maximize activities that can be handled by limited resources [
19]. Furthermore, the most known traveling salesman problem has been optimized by using NSGA-II to find out the shortest path the salesman can take to visit a city and return to the starting point [
20]. Moreover, the scheduling problem has been optimized using NSGA-II to determine the optimum sequence of processing of N jobs by M machines aiming to optimize the total flow time, waiting time, service time, and make span [
21]. Therefore, in the end, it comes up with the Pareto front that offers the optimum values of grating parameters for the manufacturing decision makers to select the grating length, grating period, and index modulation by which they can obtain the bandwidth and reflectivity that suit their applications’ requirements. The obtained results are compared to those other studies that yielded the same input parameters for verifying the proposed methods [
15,
16]. In addition, Thereby, the proposed method’s performance has been validated, the same scenarios are tested using another standard optimization method, such as a genetic algorithm, and a comparison has been conducted. The paper is organized as follows:
Section 2 presents the background and the parameters of FBG sensors.
Section 3 describes the methodology of the proposed optimization method.
Section 4 demonstrates the results. Ultimately,
Section 5 concludes the work.
2. Background
Fulfilling the aim of this research, which is controlling the bandwidth and the power of the reflected wave of FBG by fetching the optimum grating parameters, it is important from the beginning to show the correlation between the control variables and the prospected optimized variables. Therefore,
Figure 1 represents data obtained from one study in the literature showing the resulting bandwidth and reflected power of five FBG sensors with different grating parameters [
22]. It depicts the effects of changing the grating parameters on the reflectivity and bandwidth. The variable R, the percentage of the reflected power, is positively proportional to the grating length. In contrast, the bandwidth represented by the full width at half maximum (FWHM) is negatively correlated. In addition, it is obviously shown how the base wavelength of the reflected wave is shifted by changing the grating length.
The reflected power and bandwidth of the reflected signal wave be mathematically represented as a function of the grating parameters as a relation between inputs and outputs. The inputs in this context are the grating length, grating period, and refractive modulation index, and the outputs are the reflected power and the bandwidth of the reflected wave. The reflection occurs at and only at the grating condition realized as in Equation (1) tells that the base wavelength of the reflected wave depends on the changing in the grating period and the refraction index.
where
: the Bragg base wavelength
the grating period
: the refractive index
For deriving a relation between bandwidth, reflectivity, sidelobe and grating length, and the change in refractive index. The bandwidth can be determined by Equation (2), which shows the bandwidth depends on the grating parameters, such as the variation in the refractive index [
22].
where
: the variation in the refractive index
: the fraction of power in the core
In addition, the reflected and transmitted power can be determined by Equations (3) and (4) that also show the reflectivity depends on the grating parameters as follows [
22]:
where
the grating periods
V: the fringe visibility
: the grating length
Using the above mathematical equations makes it reasonable to analyze the correlation between these input and output parameters.
Figure 2 depicts a conceptual scenario showing how the FBG temperature sensor works. When an optical source indicated on the left of the figure is applied to the FBG sensor with a structure and grating shown in the middle of the figure that changes by increasing the applied temperature based on the coupled mode theory, which says the wavelength of Bragg depends on the effective refractive index and the grating period and represented mathematically in Equations (1)–(4) [
22]. Therefore, for every change that occurred to the measured physical signal, there was an equivalent change that happened to the refractive index of the core of FBG, which, in turn, caused a shift in the base wavelength of the reflected wave.
Figure 2 shows on the right the change in the base wavelength from
to
due to changing the temperature from
to
.
In addition, optimizing a problem with multi-objective functions requires a suitable algorithm for determining the grating parameters that lead to optimum bandwidth and reflectivity of the reflected wave that fulfills applications’ needs. The following section presents the methodology and logic flow of the proposed optimization method to solve this multi-objective problem.
3. Materials and Methods
The problem for whom this paper proposed a novel solution has multi-objective functions, which are maximizing the reflectivity, minimizing or maximizing the bandwidth based on the applications’ needs, and minimizing the sidelobe. Moreover, the control variables are the grating length, grating period, and the change in refractive index. In general, multi-objective functions can be formulated by determining
k decisions that satisfy
n objective functions and comply with m constraints as follows [
23]:
Subject to the following constraints
The above equations formulate the optimization problem of multi-criterion decision making. In Equation (5), Equation (5) represents the set of solutions for minimizing or maximizing n objective functions 6ns above represents the set of restrictions or constraints. In addition, Equation (7) presents the set of the determined decisions,
X. In addition, Equation (9) represents the space of objective
Y. In addition, Equation (9) shows an example of inequality constraints. In contrast, Equation (10) presents the equality constraints. Solving this optimization problem yields Pareto-optimal solution that represents the most suitable region A where no region B is more feasible so that it could minimize or reduce some objectives and maximize or increase at least one other goal [
23,
24]. By projecting that concept on the current problem for whom this paper presents a solution, the objective functions might conflict, given that increasing the bandwidth does not go with increasing the reflectivity. Therefore, the appropriate solution for such a problem should apply Pareto optimization concepts, which is the mathematical optimization of two or more objective functions simultaneously without degrading the other functions [
24]. The mathematical representation of a multi-objective criteria problem can be represented as follows:
With
is the minimum number of the objectives, and set
X is the feasible set of vectors of decisions. In addition, the vector of the objective function can be formulated as follows:
is a feasible solution or decision and is called an objective.
In Pareto optimization, the target is to find a set of Pareto-optimal solutions that satisfy the condition that, for all points in the variable space, there is no other point that can give a less than or equal and at the same time it can give at least one value less than that is given by the Pareto-optimal point. That can be formulated mathematically as follows:
The following section presents the proposed optimization method’s design by projecting the Pareto optimization on the problem of this study by using NSGA-II methods to determine the grating parameters for optimum bandwidth and reflectivity.
6. Validation and Comparison with Literature
Many studies were found in the literature that contributed to determining the relation between the grating parameters such as the grating length and the change in reflective index and the reflectivity and bandwidth [
15,
16]. One of these studies has experimented with varying the grating length from 1 to 10 nm to determine a wider bandwidth to fulfill the requirements of oil and gas applications [
15]. In that study, the authors kept varying the grating length and monitoring the change in bandwidth and reflectivity, as shown in
Figure 9 and
Figure 10. They offer a positive correlation between the reflectivity and grating length, while the correlation between the bandwidth and grating length is negative. That shows the problem for which this study proposed a solution. In addition, this method does not offer an optimization; instead, it determines the suitable value based on varying the grating length and remarking the equivalent bandwidth and reflectivity. By applying the same parameters to the proposed optimization methods, higher reflectivity has been obtained and can handle the two contradictory test cases.
Similarly, another study tried to determine the least bandwidth and the highest reflectivity by changing the grating length and remarking on the bandwidth and reflectivity [
16]. The study showed a negative correlation between the bandwidth and the grating length, as shown in
Figure 11. Moreover, it shows a positive correlation between the grating length and the reflectivity, as depicted in
Figure 12. However, the proposed method showed a complete optimization in that c in the contradicting test case. In addition, the current method shows higher reflectivity and narrower bandwidth.