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Arbitrary strain distribution measurement using a genetic algorithm approach and two fiber Bragg grating intensity spectra
Optics Communications 239 (2004) 323–332 www.elsevier.com/locate/optcom
Arbitrary strain distribution measurement using a genetic algorithm approach and two fiber Bragg grating intensity spectra Hsu-Chih Cheng a, Yu-Lung Lo
b,*
a b
Department of Electrical Engineering, National Cheng Kung University, University Road, Tainan 701, Taiwan Department of Mechanical Engineering, National Cheng Kung University, University Road, Tainan 701, Taiwan Received 16 December 2003; received in revised form 21 May 2004; accepted 26 May 2004
Abstract This paper proposes and verifies a simple, convenient, and low cost method to inversely measure arbitrary strain distributions by applying a genetic algorithm approach to analyze the reflection intensity spectra of two fiber Bragg gratings (FBGs). The proposed method involves bonding one uniform FBG and one chirped FBG to the same location of the structure of interest such that they both encounter the same strain field. The arbitrary strain distribution within the fiber gratings is then determined inversely from the two Bragg intensity spectra by means of a genetic algorithm population-based optimization process. The proposed measurement method is suitable for many smart structure-monitoring applications. 2004 Elsevier B.V. All rights reserved. Keywords: Distributed sensing; Fiber Bragg gratings; Genetic algorithm
1. Introduction Many distributed strain-sensing techniques using FBGs and fiber Bragg grating synthesis method have been successfully developed in recent years [1–9]. One such method employs an interfer*
Corresponding author. Tel.: +886-275-7575x62123; fax: +886-235-2973. E-mail address: [email protected] (Y.-L. Lo).
ometric Fourier transform technique to measure arbitrary strain profiles within intra-core FBGs [1]. However, this technique is applicable only to a weak Bragg grating (i.e., reflectivity below 30%) and requires complex spectra (i.e., intensity and phase spectra). Techniques have also been proposed for reconstructing the grating period of an FBG from its corresponding complex reflection coefficient by means of time–frequency signal analysis [2] or by a layer-peeling method [3]. However,
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.05.054
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reconstructing the properties of the FBG [1–3] still requires its complex reflection coefficient to be known, and measuring the intensity and phase spectra of an FBG experimentally is both difficult and expensive [1]. Some researchers have obtained the strain distribution within an FBG by developing inverse solutions for the closed-form relationship between the strain and the reflection intensity spectrum [4] or the reflection phase spectrum [5]. However, these approaches are suitable only for monotonically increasing or decreasing strain profiles and measurement of the arbitrary strain distribution is difficult. Recently, Skaar and Risvik [6] proposed the use of genetic algorithms for the synthesis of FBGs. Specifically, these authors employed a binary genetic algorithm to design a fiber Bragg grating filter for optical communication applications. More recently still, Casagrande et al. [7] proposed the inverse measurement of distributed strain via the application of a genetic algorithm to the intensity spectrum of the FBG. However, this technique is valid only for monotonic strain profiles. Other optimization methods, including the simulated annealing and adaptive simulated annealing algorithms, can also be used for parameter synthesis and distributed strain sensing of fiber Bragg gratings [8,9]. However, these methods are only suitable when the inverse analysis involves a small number of parameters since they tend to become slower and more inaccurate when the number of parameters increases. The present study develops a novel approach for the inverse measurement of arbitrary strain distributions by applying a genetic algorithm population-based optimization technique to the analysis of two FBG intensity reflection spectra. Unlike the simulated annealing and adaptive simulated annealing algorithms, the genetic algorithm is ideally suited to the multiple-parameter synthesis problem addressed in the present study. A fundamental assumption of the proposed approach is that the two intra-core FBGs encounter the same strain field. It is noted that this assumption can be guaranteed by overwriting two Bragg gratings into the same intra-core fiber [10,11]. However, the present study demonstrates that the current approach that combine genetic algorithm and two
FBGs intensity reflection spectra to permits an arbitrary strain reconstruction in the noisy environment and overcomes the limitations of the methods presented previously [1–9]. The remainder of this paper is organized as follows. Section 2 introduces the basic concepts of the proposed distributed strain-sensing method using two FBGs. Section 3 describes the implementation of the novel FBG-based genetic algorithm strain measurement system using the T-matrix analysis method and the genetic algorithm. Section 4 presents the current numerical simulation results. Finally, Section 5 summarizes the principal conclusions of the present study.
2. Basic concepts of novel distributed strain-sensing method The majority of the distributed sensing techniques presented previously employed a single uniform or chirped fiber Bragg grating. However, the use of a single FBG reflection intensity spectrum prohibits the sensing of arbitrary strain distributions since, regardless of whether a uniform or a chirped FBG is employed, the reflection intensity spectrum associated with a positive linear gradient strain distribution is the same as that yielded by an FBG under a negative linear gradient strain distribution. In other words, when an arbitrary strain distribution is applied to either a single uniform FBG or to a single chirped FBG, there exists a specific strain distribution which induces reflection spectra of equal intensity but opposite phase. This study presents a method which overcomes this ambiguity by employing the novel sensing system illustrated schematically in Fig. 1. In the proposed arrangeChirped FBG
Loading
Loading Tunable Laser Uniform FBG
Photodetector Fig. 1. Schematic diagram of proposed two FBG arrangement.
H.-C. Cheng, Y.-L. Lo / Optics Communications 239 (2004) 323–332
Fig. 2. Different strain distributions.
ment, two FBGs (one uniform and one chirped) are glued onto a substrate in close proximity to one another. If the two FBGs are located sufficiently close to each other, it can be assumed that they both encounter the same applied strain field. In this way, two different reflection intensity spectra are obtained for the same strain distribution, and the arbitrary strain distribution can then be extracted inversely by means of a genetic algorithm. When an arbitrary strain distribution is applied to the two FBGs, two reflection intensity spectra are generated along the wavelength axis.
325
Fig. 2 presents three different strain distributions, while Fig. 3 indicates the corresponding reflection intensity spectra of the uniform and chirped FBGs. Comparing Fig. 3(a) and (b), which correspond to strain distributions (a) and (b), respectively, in Fig. 2, it is noted that the reflection intensity spectra of the uniform FBG are identical. Therefore, the use of the uniform FBG alone is insufficient to distinguish the nature of the strain distribution. However, if the reflection intensity spectra of the uniform and chirped FBGs are considered simultaneously, the respective strain distributions of the two cases can be distinguished. Similarly, in Fig. 3(a) and (c), the two reflection intensity spectra of the chirped FBG are identical, and the nature of the two different strain distributions can only be identified if the reflection intensity spectra of the uniform FBG are also taken into consideration. Therefore, although the application of two specific strains may cause either the uniform or the chirped FBG reflection intensity spectra to be equal, the reflection intensity spectrum of the second FBG will almost certainly be different. If the error function of the genetic algorithm is defined correctly, this methodology can be used to recover the corresponding strain distribution from the two reflection intensity spectra.
Fig. 3. Reflection intensity spectra of two FBGs subjected to the strain distributions of Fig. 2.
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3. Methodology and implementation S(1)
In most sensing applications, the strain along the fiber grating is uniform and can be determined from the reflective wavelength of the grating. However, if the grating is subjected to a strain gradient, its reflected intensity spectrum will not only be shifted, but will also be distorted due to nonuniform changes in both the physical pitch length and the refractive index of the grating. Gratings with non-uniform pitch length, refractive index modulation depth, and mean refractive index of the core can be analyzed in different ways. The present study adopts the T-matrix formalism [12,13], which offers a straightforward means of analyzing the reflection spectrum response of non-uniform grating structures. In the present methodology, two different reflective spectra are obtained from the uniform and chirped gratings when the substrate is subjected to a non-uniform strain. The principal effect of the spatial variation of K(z) and n(z) is that each section of the grating chiefly contributes to the reflection intensity spectrum at the wavelength of its local Bragg condition kB ¼ 2nðzÞKf ðzÞ ¼ 2n0 K0 ½1 þ SðzÞ;
ð1Þ
where n0 is the original effective refractive index of the fiber, K0 is the normal grating period, and S(z) is the strain profile along the fiber grating. The parameter Kf(z) is a constant along the z-axis direction in a uniform FBG, but is a linear function in a chirped FBG. The transverse stresses applied to the fiber are generally small, and it can be assumed that the radial strains in the fiber are determined by the Poisson contraction of the fiber. This assumption is particularly valid for the case of an FBG mounted on a substrate surface. Therefore, the optic-strain effect is manifested as a change in the optical refractive index, i.e.
where mf is the Poisson ratio, and p11 and p12 are the photo-elastic coefficients of the fiber core mate-
S(M-1)
S(M)
rial (assumed to be 0.113 and 0.252, respectively) [14]. In the present methodology, solutions of the distributed strain are encoded as a string of real values in a direct analogy to biological genes. The optic-strain effect is applied in the genetic algorithm model and the strain profile, S(z), is transformed into a sample string of real values, as shown in Fig. 4. The genetic algorithm involves three fundamental operators, namely selection, crossover, and mutation. The selection process is based on probability, i.e., solutions evaluated with lower error (higher fitness) values are most likely to be selected for the next generation, while those with high error values will most likely be rejected. Importantly, there is an element of randomness about those solutions which survive and those which do not, i.e., the survival process mirrors the survival of natural organisms. In the present application, this implies that the most suitable strain distributions have a greater probability of surviving to the next operator. Once the superior solutions have been selected, the crossover process randomly chooses two selected solutions (parents), exchanges their bit information, and then generates two further solutions (offspring). In this way, it is anticipated that the offspring generation will be improved by accumulating the exceptional bits information of the parentÕs generation. The present study employs the real valued crossover process [15], i.e. Get closer :
Pull away : ð2Þ
...
Fig. 4. Sample string of real values.
1 3 1 dn ¼ n0 D 2 2 n0 2;3 1 ¼ n30 ½ef ð1 mf Þp12 mf ef p11 ; 2
S(2)
x01 ¼ x1 þ rðx1 x2 Þ; x02 ¼ x2 rðx1 x2 Þ;
ð3Þ
x01 ¼ x1 þ rðx2 x1 Þ; x02 ¼ x2 rðx2 x1 Þ;
ð4Þ
where x1 and x2 are the parents, x01 and x02 are the new individuals (offspring), and r is a random and small real value. In the present application, x1 and x2 are taken from the sample string of real values illustrated
H.-C. Cheng, Y.-L. Lo / Optics Communications 239 (2004) 323–332
in Fig. 4. The aim of the genetic algorithm is to identify the most suitable solution (strain distribution) in each iteration. Within the genetic algorithm, an important process is that of mutation, which causes an existing individual to be modified, and hence introduces an additional variability into the population. This process is important since it can prevent the generation of a local optimal solution. In the mutation process, a species string is picked at random, a mutation point is randomly selected, and the bit information is then changed. The probability in the mutation process is controlled by the mutation probability. The current study employs the real valued mutation process, i.e. x00 ¼ x0 þ s random noise;
ð5Þ
where x00 is a new offspring after mutation, s is the control factor, and random_noise is a small value which is applied to change the original solution. The error function (fitness function) is a measurement of the distance between the calculated reflection intensity spectra and the target reflection intensity spectra. In the current methodology, there are two objective reflection intensity spectra, i.e., one associated with the uniform FBG under the distributed strain and another associated with the chirped FBG under the same distributed strain. In the multi-objective genetic algorithm, the objective is generally to identify one of the Pareto optimal solutions (a compromise solution) [15]. Such a multi-objective optimization problem requires the use of particular techniques, which are very different from the standard optimization techniques utilized for single objective optimization. Generally, the multi-objective functions are combined into a single overall objective function. The standard error function is defined as [15] F ðxÞ ¼
k X
wi fi ðxÞ;
the genetic algorithm finding the best solution with respect to the first objective and the worst solution with respect to the second objective is small, i.e., when each objective is given a different weight, the optimal solution will focus on the objective with the greatest weight. Hence, the error function in the present study can be defined as Error ¼ EU þ EC X X ¼ ðRU;Obj RU;Cal Þ2 þ ðRC;Obj RC;Cal Þ2 ; k
k
ð7Þ
where EU is the error function of a uniform FBG, EC is the error function of a chirped FBG, RU,Obj and RC,Obj represent the objective spectra from the uniform and chirped FBGs, respectively, under the same arbitrary distributed strain, and RU,Cal and RC,Cal represent the reflection intensity spectra of the two FBGs as calculated by the T-matrix analysis method. The objective of the inverse problem is to inversely derive the strain distribution which yields the two reflection intensity spectra most closely resembling the target spectra. Fig. 5 presents the major steps of the optimization algorithm. Initially, the genetic algorithm generates a group of random starting solutions (strain distributions) and then uses the T-matrix analysis method to calculate their corresponding reflection
Generate some solution randomly Calculate numerical intensity reflection spectra by T-matrix and Error values
Error values <0.0001 or Loops=10000
weights
327
Yes
i¼1
wi 2 ½0; . . . ; 1 and
k X
wi ¼ 1:
ð6Þ
No Selection
Solution S(z) (Strain distribution)
i¼1
In this paper, k = 2, w1 = 0.5, and w2 = 0.5, respectively. Since the selection process is based on the error function and each weight is 0.5 (the same weight) in the error function, the probability of
Crossover Mutatiom Fig. 5. Flow chart of genetic algorithm.
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intensity spectra. Subsequently, these spectra are substituted into Eq. (7) to determine their error values. If the objective spectra and the calculated spectra fall within the range of allowable error, the optimization program is terminated. However, if the error value is unacceptable, the selection and crossover operators are performed, followed by the mutation process, which changes one or several of the strain distribution sections of the sample strings shown in Fig. 4 to another value from the complete search space, hence preventing the generation of a local optimal solution. This iterative process is repeated until either the error value is less than 0.0001 or the specified number of iteration loops has been completed.
4. Numerical results In the present simulation, the grating length, L, is specified as 1 cm, and is divided into 20 uniform sections to give a spatial resolution of 0.5 mm. The wavelength ranges of the two gratings are 3 nm, and each range is sampled by 150 points. The effective refractive index of the fiber core is assumed to be neff = 1.457 and the index of the modulation depth is Dn = 2.5 · 104. From Eq. (1), the grating period of the uniform FBG is determined to be K0U = 0.5315 lm, while that of the chirped FBG is found to be K0C = 0.533 lm. Therefore, the variation in the chirped FBG grating period is defined as K(z) = K0C(1 + 0.0004z), where the z-coordinate is measured along the length of the fiber. As described above, the initial strain profile of the grating is generated randomly. In Eqs. (3) and (4), the r parameter of the genetic algorithm is a uniform distribution random variable and the region is set to ±0.1. The mutation probability in Eq. (5) is specified as 0.15 and the control factor, s, is set to 0.01. The terminal conditions are specified as an error value of less than 0.0001 or the completion of 10,000 iterations. To resolve the problem of ambiguity in the Section 2, two different FBG reflection intensity spectra generated by the same distributed strain must be implemented (as shown in Fig. 1), and a genetic algorithm utilized to inversely extract the arbitrary strain distribution. Figs. 6 and 7 present the results
Fig. 6. (a) Ideal (solid curve) and recovered (dashed curve) strain distribution. (b) Simulated (solid curve) and recovered (dashed curve) results from two FBGs subjected to the linear positive gradient strain distribution.
for two FBGs subjected to linear positive and linear negative gradient strains, respectively. Fig. 6(a) indicates that the simulated positive linear strain distribution (solid curve) and the recovered distributed strain (dashed curve) calculated by the genetic algorithm are in good agreement. The corresponding objective reflection intensity spectra (solid curves) and optimal reflection intensity spectra (dashed curves) are illustrated in Fig. 6(b). Similarly, Fig. 7 presents the case of two FBGs subjected to a negative linear distributed strain. It can be seen that the reflection intensity spectra from the chirped FBG subjected to linear positive and negative gradient strains are different. Therefore, using two FBG reflection intensity spectra to
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Fig. 7. (a) Ideal (solid curve) and recovered (dashed curve) strain distribution. (b) Simulated (solid curve) and recovered (dashed curve) results from two FBGs subjected to the linear negative gradient strain distribution.
Fig. 8. (a) Ideal (solid curve) and recovered (dotted curve) strain distribution. (b) Simulated (solid curve) and recovered (dotted curve) results from two FBGs subjected to the nonmonotonic strain distribution.
extract the arbitrary strain distribution is a feasible approach. Fig. 8 presents a more complicated simulated distributed strain (solid curve) along the grating length and the corresponding recovered strain. It is known that this distributed strain will produce a Febry–Perot-like effect in the reflection intensity spectrum [16], as illustrated in Fig. 8(b). After 10,000 iterations of the genetic algorithm, the error value given by Eq. (7) is 0.0024, which provides the acceptable result (dashed curve) shown in Fig. 8(a). The solid curves in Fig. 8(b) indicate the objective reflection intensity spectra of two FBGs under a non-uniform strain, while the dashed curves
show the calculated reflection intensity spectra for the optimal distributed strain, as determined from the genetic algorithm. It can be seen that the two sets of results are in good agreement. To the best of the current authorsÕ knowledge, two inverse methods have been proposed previously to solve a similar case [1,2]. The first method is a time–frequency analysis method, which requires a complex reflection spectrum. The second is a Fourier transform method, which not only requires the complex reflection spectrum, but is also restricted to reflection intensity spectra with less than 30% reflectivity. These limitations cause both methods to be difficult and expensive.
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In accordance with Azana et al. [2], 10% and 20% of ‘‘normal’’ random noise were added to the intensity reflection spectra. The results of Fig. 9 indicate that the addition of 10% normal (Gaussian) distribution noise is acceptable. However, when the noise increases to a 20% noise variation, Fig. 10 reveals the presence of some ripples in the reconstructed strain distribution, which are due to the noisy intensity reflection spectra. The results of Table 1 indicate that the average error varies from 0.93% to 18.04% and that the maximum-error varies from 25.73% to 275.91%. The results of the positive and negative linear gradient cases indicate
Fig. 10. (a) Ideal (solid curve) and recovered (dashed curve) strain distribution. (b) Noisy (20% variation) intensity spectra (solid curve) and recovered (dashed curve) results from two FBGs subjected to the positive linear gradient strain distribution.
Fig. 9. (a) Ideal (solid curve) and recovered (dashed curve) strain distribution. (b) Noisy (10% variation) intensity spectra (solid curve) and recovered (dashed curve) results from two FBGs subjected to the positive linear gradient strain distribution.
that the average and maximum errors are less than 1.08% and 52.43%, respectively, in the noise-free intensity spectra. It is clear that the error induced by the non-monotonic strain profile exceeds those of the monotonic cases in the noise-free intensity spectra. It is noted that a greater error occurs at the edge of the strain profile. In the noisy-case, as the noise variation increases, the average error and maximum error both increase. Specifically, in an intensity spectrum with a 10% noise variation, the average and maximum errors are below 4.87% and 63.75%, respectively, while at a higher noise variation of 20%, the corresponding error values are 18.04% and 275.91%, respectively.
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Table 1 The strain distribution sensing results Strain profile
Maximum error (%)
Average error (%)
Mean of the error
Variance of the error
Correlation coefficient
Positive linear gradient (noise-free) Negative linear gradient (noise-free) Non-monotonic (noise-free) Positive linear gradient (noisy-10%) Positive linear gradient (noisy-20%)
25.73 4.87 52.43 63.76 275.91
1.08 0.93 3.26 4.87 18.04
6.65 · 107 1.77 · 107 1.29 · 106 6.44 · 106 1.75 · 105
4.27 · 1011 1.21 · 1010 5.06 · 1010 6.18 · 109 2.79 · 108
0.9998 0.9995 0.9974 0.9758 0.8828
The performance of the reconstructed method was assessed from the mean, variance and correlation coefficient of errors. The results are listed in Table 1. Generally, all of the noise-free cases indicated extremely high correlation coefficient (>0.99) and low mean and variance values. However, in 10% and 20% noisy cases, the correlation coefficients were 0.9758 and 0.8828. Obviously, the mean and variance of errors were higher and the correlation coefficient was lower than that in the noise-free cases. The results presented in Table 1 confirm that the proposed approach is also applicable to the measurement of noisy systems, but reveal that the quality of the reconstruction results depends on the degree of noise variation in the intensity spectra.
5. Conclusions This paper has proposed a novel arbitrary strain distribution sensing approach using a genetic algorithm to inversely trace the strain distribution from two FBG reflection intensity spectra. The effectiveness and feasibility of the proposed method have been demonstrated by solving the inverse problems of several strain sensing cases. The present simulation results have confirmed the genetic algorithm to be an effective means of optimally deriving the solution of a complicated strain distribution via the reflection intensity spectra of two FBGs. The current method eliminates the requirements for a monotonic axial strain field and a low reflectivity, and provides a high spatial resolution (0.5 mm). Furthermore, the proposed approach provides a simple and low cost method to measure arbitrary strain distributions. This technique has several limitations: (1) The two
FBGs must encounter the same strain field. (2) The increased number of divided sections and wavelength sample points can improve the spatial and wavelength resolution, but prolong the computation time. In this study, it takes approximately 1 hour to carry out 1000 loops in the simulation by using 2.4 GHz Pentium 4 PC. (3) The fiber Bragg gratingÕs characteristics, such as grating length, modulation of refractive index, apodise degree, etc., must be known in advance. (4) This technique only provides acceptable results for noise variations of 10% or less.
Acknowledgement The current authors acknowledge the financial support provided to this study by the National Science Council under Grant No. NSC 92-2622L011-001. Also, this work is partially supported by Ministry of Education Program for Promoting Academic Excellence of Universities under the Grant No. A-92-E-FA08-1-4.
References [1] M.M. Ohn, S.Y. Huang, R.M. Measures, J. Chwang, Electron. Lett. 33 (1997) 1242. [2] J. Azana, M.A. Muriel, L.R. Chen, P.W.E. Smith, J. Lightwave Technol. 19 (2001) 646. [3] R. Feced, M.N. Zervas, M.A. Muriel, IEEE J. Quantum Electron. 35 (1999) 1105. [4] M. LeBlanc, S.Y. Huamg, M. Ohn, R.M. Measures, Opt. Lett. 21 (1996) 1405. [5] S. Huang, M.M. Ohn, R.M. Measures, Appl. Opt. 35 (1996) 1135. [6] J. Skaar, K.M. Risvik, J. Lightwave Technol. 16 (1998) 1928.
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[7] F. Casagrande, P. Crespi, A.M. Grassi, A. Lulli, R.P. Kenny, M.P. Whelan, Appl. Opt. 41 (2002) 5238. [8] P. Dong, J. Azana, A.G. Kirk, Opt. Commun. 228 (2003) 303. [9] C.Z. Shi, N. Zeng, M. Zhang, Y.B. Liao, S.R. Lai, Opt. Commun. 226 (2003) 167. [10] J. Bao, X. Zhang, K. Chen, W. Zhou, Opt. Commun. 188 (2001) 31. [11] M.W. Rabin, W.C. Swann, S.L. Gilbert, Appl. Opt. 41 (2002) 7193.
[12] M. Yamada, K. Sakuda, Appl. Opt. 26 (1987) 3474. [13] S.Y. Huang, M. LeBlanc, M.M. Ohn, R.M. Measures, Appl. Opt. 34 (1995) 5003. [14] C.D. Butter, G.B. Hocker, Appl. Opt. 17 (1978) 2867. [15] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Springer-Verlag, New York, 1992. [16] G.E. Town, K. Sugden, J.A.R. Williams, I. Bennion, S.B. Poole, IEEE Photon. Technol. Lett. 7 (1995) 78.
Arbitrary strain distribution measurement using a genetic algorithm approach and two fiber Bragg grating intensity spectra Hsu-Chih Cheng a, Yu-Lung Lo
b,*
a b
Department of Electrical Engineering, National Cheng Kung University, University Road, Tainan 701, Taiwan Department of Mechanical Engineering, National Cheng Kung University, University Road, Tainan 701, Taiwan Received 16 December 2003; received in revised form 21 May 2004; accepted 26 May 2004
Abstract This paper proposes and verifies a simple, convenient, and low cost method to inversely measure arbitrary strain distributions by applying a genetic algorithm approach to analyze the reflection intensity spectra of two fiber Bragg gratings (FBGs). The proposed method involves bonding one uniform FBG and one chirped FBG to the same location of the structure of interest such that they both encounter the same strain field. The arbitrary strain distribution within the fiber gratings is then determined inversely from the two Bragg intensity spectra by means of a genetic algorithm population-based optimization process. The proposed measurement method is suitable for many smart structure-monitoring applications. 2004 Elsevier B.V. All rights reserved. Keywords: Distributed sensing; Fiber Bragg gratings; Genetic algorithm
1. Introduction Many distributed strain-sensing techniques using FBGs and fiber Bragg grating synthesis method have been successfully developed in recent years [1–9]. One such method employs an interfer*
Corresponding author. Tel.: +886-275-7575x62123; fax: +886-235-2973. E-mail address: [email protected] (Y.-L. Lo).
ometric Fourier transform technique to measure arbitrary strain profiles within intra-core FBGs [1]. However, this technique is applicable only to a weak Bragg grating (i.e., reflectivity below 30%) and requires complex spectra (i.e., intensity and phase spectra). Techniques have also been proposed for reconstructing the grating period of an FBG from its corresponding complex reflection coefficient by means of time–frequency signal analysis [2] or by a layer-peeling method [3]. However,
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.05.054
324
H.-C. Cheng, Y.-L. Lo / Optics Communications 239 (2004) 323–332
reconstructing the properties of the FBG [1–3] still requires its complex reflection coefficient to be known, and measuring the intensity and phase spectra of an FBG experimentally is both difficult and expensive [1]. Some researchers have obtained the strain distribution within an FBG by developing inverse solutions for the closed-form relationship between the strain and the reflection intensity spectrum [4] or the reflection phase spectrum [5]. However, these approaches are suitable only for monotonically increasing or decreasing strain profiles and measurement of the arbitrary strain distribution is difficult. Recently, Skaar and Risvik [6] proposed the use of genetic algorithms for the synthesis of FBGs. Specifically, these authors employed a binary genetic algorithm to design a fiber Bragg grating filter for optical communication applications. More recently still, Casagrande et al. [7] proposed the inverse measurement of distributed strain via the application of a genetic algorithm to the intensity spectrum of the FBG. However, this technique is valid only for monotonic strain profiles. Other optimization methods, including the simulated annealing and adaptive simulated annealing algorithms, can also be used for parameter synthesis and distributed strain sensing of fiber Bragg gratings [8,9]. However, these methods are only suitable when the inverse analysis involves a small number of parameters since they tend to become slower and more inaccurate when the number of parameters increases. The present study develops a novel approach for the inverse measurement of arbitrary strain distributions by applying a genetic algorithm population-based optimization technique to the analysis of two FBG intensity reflection spectra. Unlike the simulated annealing and adaptive simulated annealing algorithms, the genetic algorithm is ideally suited to the multiple-parameter synthesis problem addressed in the present study. A fundamental assumption of the proposed approach is that the two intra-core FBGs encounter the same strain field. It is noted that this assumption can be guaranteed by overwriting two Bragg gratings into the same intra-core fiber [10,11]. However, the present study demonstrates that the current approach that combine genetic algorithm and two
FBGs intensity reflection spectra to permits an arbitrary strain reconstruction in the noisy environment and overcomes the limitations of the methods presented previously [1–9]. The remainder of this paper is organized as follows. Section 2 introduces the basic concepts of the proposed distributed strain-sensing method using two FBGs. Section 3 describes the implementation of the novel FBG-based genetic algorithm strain measurement system using the T-matrix analysis method and the genetic algorithm. Section 4 presents the current numerical simulation results. Finally, Section 5 summarizes the principal conclusions of the present study.
2. Basic concepts of novel distributed strain-sensing method The majority of the distributed sensing techniques presented previously employed a single uniform or chirped fiber Bragg grating. However, the use of a single FBG reflection intensity spectrum prohibits the sensing of arbitrary strain distributions since, regardless of whether a uniform or a chirped FBG is employed, the reflection intensity spectrum associated with a positive linear gradient strain distribution is the same as that yielded by an FBG under a negative linear gradient strain distribution. In other words, when an arbitrary strain distribution is applied to either a single uniform FBG or to a single chirped FBG, there exists a specific strain distribution which induces reflection spectra of equal intensity but opposite phase. This study presents a method which overcomes this ambiguity by employing the novel sensing system illustrated schematically in Fig. 1. In the proposed arrangeChirped FBG
Loading
Loading Tunable Laser Uniform FBG
Photodetector Fig. 1. Schematic diagram of proposed two FBG arrangement.
H.-C. Cheng, Y.-L. Lo / Optics Communications 239 (2004) 323–332
Fig. 2. Different strain distributions.
ment, two FBGs (one uniform and one chirped) are glued onto a substrate in close proximity to one another. If the two FBGs are located sufficiently close to each other, it can be assumed that they both encounter the same applied strain field. In this way, two different reflection intensity spectra are obtained for the same strain distribution, and the arbitrary strain distribution can then be extracted inversely by means of a genetic algorithm. When an arbitrary strain distribution is applied to the two FBGs, two reflection intensity spectra are generated along the wavelength axis.
325
Fig. 2 presents three different strain distributions, while Fig. 3 indicates the corresponding reflection intensity spectra of the uniform and chirped FBGs. Comparing Fig. 3(a) and (b), which correspond to strain distributions (a) and (b), respectively, in Fig. 2, it is noted that the reflection intensity spectra of the uniform FBG are identical. Therefore, the use of the uniform FBG alone is insufficient to distinguish the nature of the strain distribution. However, if the reflection intensity spectra of the uniform and chirped FBGs are considered simultaneously, the respective strain distributions of the two cases can be distinguished. Similarly, in Fig. 3(a) and (c), the two reflection intensity spectra of the chirped FBG are identical, and the nature of the two different strain distributions can only be identified if the reflection intensity spectra of the uniform FBG are also taken into consideration. Therefore, although the application of two specific strains may cause either the uniform or the chirped FBG reflection intensity spectra to be equal, the reflection intensity spectrum of the second FBG will almost certainly be different. If the error function of the genetic algorithm is defined correctly, this methodology can be used to recover the corresponding strain distribution from the two reflection intensity spectra.
Fig. 3. Reflection intensity spectra of two FBGs subjected to the strain distributions of Fig. 2.
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3. Methodology and implementation S(1)
In most sensing applications, the strain along the fiber grating is uniform and can be determined from the reflective wavelength of the grating. However, if the grating is subjected to a strain gradient, its reflected intensity spectrum will not only be shifted, but will also be distorted due to nonuniform changes in both the physical pitch length and the refractive index of the grating. Gratings with non-uniform pitch length, refractive index modulation depth, and mean refractive index of the core can be analyzed in different ways. The present study adopts the T-matrix formalism [12,13], which offers a straightforward means of analyzing the reflection spectrum response of non-uniform grating structures. In the present methodology, two different reflective spectra are obtained from the uniform and chirped gratings when the substrate is subjected to a non-uniform strain. The principal effect of the spatial variation of K(z) and n(z) is that each section of the grating chiefly contributes to the reflection intensity spectrum at the wavelength of its local Bragg condition kB ¼ 2nðzÞKf ðzÞ ¼ 2n0 K0 ½1 þ SðzÞ;
ð1Þ
where n0 is the original effective refractive index of the fiber, K0 is the normal grating period, and S(z) is the strain profile along the fiber grating. The parameter Kf(z) is a constant along the z-axis direction in a uniform FBG, but is a linear function in a chirped FBG. The transverse stresses applied to the fiber are generally small, and it can be assumed that the radial strains in the fiber are determined by the Poisson contraction of the fiber. This assumption is particularly valid for the case of an FBG mounted on a substrate surface. Therefore, the optic-strain effect is manifested as a change in the optical refractive index, i.e.
where mf is the Poisson ratio, and p11 and p12 are the photo-elastic coefficients of the fiber core mate-
S(M-1)
S(M)
rial (assumed to be 0.113 and 0.252, respectively) [14]. In the present methodology, solutions of the distributed strain are encoded as a string of real values in a direct analogy to biological genes. The optic-strain effect is applied in the genetic algorithm model and the strain profile, S(z), is transformed into a sample string of real values, as shown in Fig. 4. The genetic algorithm involves three fundamental operators, namely selection, crossover, and mutation. The selection process is based on probability, i.e., solutions evaluated with lower error (higher fitness) values are most likely to be selected for the next generation, while those with high error values will most likely be rejected. Importantly, there is an element of randomness about those solutions which survive and those which do not, i.e., the survival process mirrors the survival of natural organisms. In the present application, this implies that the most suitable strain distributions have a greater probability of surviving to the next operator. Once the superior solutions have been selected, the crossover process randomly chooses two selected solutions (parents), exchanges their bit information, and then generates two further solutions (offspring). In this way, it is anticipated that the offspring generation will be improved by accumulating the exceptional bits information of the parentÕs generation. The present study employs the real valued crossover process [15], i.e. Get closer :
Pull away : ð2Þ
...
Fig. 4. Sample string of real values.
1 3 1 dn ¼ n0 D 2 2 n0 2;3 1 ¼ n30 ½ef ð1 mf Þp12 mf ef p11 ; 2
S(2)
x01 ¼ x1 þ rðx1 x2 Þ; x02 ¼ x2 rðx1 x2 Þ;
ð3Þ
x01 ¼ x1 þ rðx2 x1 Þ; x02 ¼ x2 rðx2 x1 Þ;
ð4Þ
where x1 and x2 are the parents, x01 and x02 are the new individuals (offspring), and r is a random and small real value. In the present application, x1 and x2 are taken from the sample string of real values illustrated
H.-C. Cheng, Y.-L. Lo / Optics Communications 239 (2004) 323–332
in Fig. 4. The aim of the genetic algorithm is to identify the most suitable solution (strain distribution) in each iteration. Within the genetic algorithm, an important process is that of mutation, which causes an existing individual to be modified, and hence introduces an additional variability into the population. This process is important since it can prevent the generation of a local optimal solution. In the mutation process, a species string is picked at random, a mutation point is randomly selected, and the bit information is then changed. The probability in the mutation process is controlled by the mutation probability. The current study employs the real valued mutation process, i.e. x00 ¼ x0 þ s random noise;
ð5Þ
where x00 is a new offspring after mutation, s is the control factor, and random_noise is a small value which is applied to change the original solution. The error function (fitness function) is a measurement of the distance between the calculated reflection intensity spectra and the target reflection intensity spectra. In the current methodology, there are two objective reflection intensity spectra, i.e., one associated with the uniform FBG under the distributed strain and another associated with the chirped FBG under the same distributed strain. In the multi-objective genetic algorithm, the objective is generally to identify one of the Pareto optimal solutions (a compromise solution) [15]. Such a multi-objective optimization problem requires the use of particular techniques, which are very different from the standard optimization techniques utilized for single objective optimization. Generally, the multi-objective functions are combined into a single overall objective function. The standard error function is defined as [15] F ðxÞ ¼
k X
wi fi ðxÞ;
the genetic algorithm finding the best solution with respect to the first objective and the worst solution with respect to the second objective is small, i.e., when each objective is given a different weight, the optimal solution will focus on the objective with the greatest weight. Hence, the error function in the present study can be defined as Error ¼ EU þ EC X X ¼ ðRU;Obj RU;Cal Þ2 þ ðRC;Obj RC;Cal Þ2 ; k
k
ð7Þ
where EU is the error function of a uniform FBG, EC is the error function of a chirped FBG, RU,Obj and RC,Obj represent the objective spectra from the uniform and chirped FBGs, respectively, under the same arbitrary distributed strain, and RU,Cal and RC,Cal represent the reflection intensity spectra of the two FBGs as calculated by the T-matrix analysis method. The objective of the inverse problem is to inversely derive the strain distribution which yields the two reflection intensity spectra most closely resembling the target spectra. Fig. 5 presents the major steps of the optimization algorithm. Initially, the genetic algorithm generates a group of random starting solutions (strain distributions) and then uses the T-matrix analysis method to calculate their corresponding reflection
Generate some solution randomly Calculate numerical intensity reflection spectra by T-matrix and Error values
Error values <0.0001 or Loops=10000
weights
327
Yes
i¼1
wi 2 ½0; . . . ; 1 and
k X
wi ¼ 1:
ð6Þ
No Selection
Solution S(z) (Strain distribution)
i¼1
In this paper, k = 2, w1 = 0.5, and w2 = 0.5, respectively. Since the selection process is based on the error function and each weight is 0.5 (the same weight) in the error function, the probability of
Crossover Mutatiom Fig. 5. Flow chart of genetic algorithm.
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intensity spectra. Subsequently, these spectra are substituted into Eq. (7) to determine their error values. If the objective spectra and the calculated spectra fall within the range of allowable error, the optimization program is terminated. However, if the error value is unacceptable, the selection and crossover operators are performed, followed by the mutation process, which changes one or several of the strain distribution sections of the sample strings shown in Fig. 4 to another value from the complete search space, hence preventing the generation of a local optimal solution. This iterative process is repeated until either the error value is less than 0.0001 or the specified number of iteration loops has been completed.
4. Numerical results In the present simulation, the grating length, L, is specified as 1 cm, and is divided into 20 uniform sections to give a spatial resolution of 0.5 mm. The wavelength ranges of the two gratings are 3 nm, and each range is sampled by 150 points. The effective refractive index of the fiber core is assumed to be neff = 1.457 and the index of the modulation depth is Dn = 2.5 · 104. From Eq. (1), the grating period of the uniform FBG is determined to be K0U = 0.5315 lm, while that of the chirped FBG is found to be K0C = 0.533 lm. Therefore, the variation in the chirped FBG grating period is defined as K(z) = K0C(1 + 0.0004z), where the z-coordinate is measured along the length of the fiber. As described above, the initial strain profile of the grating is generated randomly. In Eqs. (3) and (4), the r parameter of the genetic algorithm is a uniform distribution random variable and the region is set to ±0.1. The mutation probability in Eq. (5) is specified as 0.15 and the control factor, s, is set to 0.01. The terminal conditions are specified as an error value of less than 0.0001 or the completion of 10,000 iterations. To resolve the problem of ambiguity in the Section 2, two different FBG reflection intensity spectra generated by the same distributed strain must be implemented (as shown in Fig. 1), and a genetic algorithm utilized to inversely extract the arbitrary strain distribution. Figs. 6 and 7 present the results
Fig. 6. (a) Ideal (solid curve) and recovered (dashed curve) strain distribution. (b) Simulated (solid curve) and recovered (dashed curve) results from two FBGs subjected to the linear positive gradient strain distribution.
for two FBGs subjected to linear positive and linear negative gradient strains, respectively. Fig. 6(a) indicates that the simulated positive linear strain distribution (solid curve) and the recovered distributed strain (dashed curve) calculated by the genetic algorithm are in good agreement. The corresponding objective reflection intensity spectra (solid curves) and optimal reflection intensity spectra (dashed curves) are illustrated in Fig. 6(b). Similarly, Fig. 7 presents the case of two FBGs subjected to a negative linear distributed strain. It can be seen that the reflection intensity spectra from the chirped FBG subjected to linear positive and negative gradient strains are different. Therefore, using two FBG reflection intensity spectra to
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Fig. 7. (a) Ideal (solid curve) and recovered (dashed curve) strain distribution. (b) Simulated (solid curve) and recovered (dashed curve) results from two FBGs subjected to the linear negative gradient strain distribution.
Fig. 8. (a) Ideal (solid curve) and recovered (dotted curve) strain distribution. (b) Simulated (solid curve) and recovered (dotted curve) results from two FBGs subjected to the nonmonotonic strain distribution.
extract the arbitrary strain distribution is a feasible approach. Fig. 8 presents a more complicated simulated distributed strain (solid curve) along the grating length and the corresponding recovered strain. It is known that this distributed strain will produce a Febry–Perot-like effect in the reflection intensity spectrum [16], as illustrated in Fig. 8(b). After 10,000 iterations of the genetic algorithm, the error value given by Eq. (7) is 0.0024, which provides the acceptable result (dashed curve) shown in Fig. 8(a). The solid curves in Fig. 8(b) indicate the objective reflection intensity spectra of two FBGs under a non-uniform strain, while the dashed curves
show the calculated reflection intensity spectra for the optimal distributed strain, as determined from the genetic algorithm. It can be seen that the two sets of results are in good agreement. To the best of the current authorsÕ knowledge, two inverse methods have been proposed previously to solve a similar case [1,2]. The first method is a time–frequency analysis method, which requires a complex reflection spectrum. The second is a Fourier transform method, which not only requires the complex reflection spectrum, but is also restricted to reflection intensity spectra with less than 30% reflectivity. These limitations cause both methods to be difficult and expensive.
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In accordance with Azana et al. [2], 10% and 20% of ‘‘normal’’ random noise were added to the intensity reflection spectra. The results of Fig. 9 indicate that the addition of 10% normal (Gaussian) distribution noise is acceptable. However, when the noise increases to a 20% noise variation, Fig. 10 reveals the presence of some ripples in the reconstructed strain distribution, which are due to the noisy intensity reflection spectra. The results of Table 1 indicate that the average error varies from 0.93% to 18.04% and that the maximum-error varies from 25.73% to 275.91%. The results of the positive and negative linear gradient cases indicate
Fig. 10. (a) Ideal (solid curve) and recovered (dashed curve) strain distribution. (b) Noisy (20% variation) intensity spectra (solid curve) and recovered (dashed curve) results from two FBGs subjected to the positive linear gradient strain distribution.
Fig. 9. (a) Ideal (solid curve) and recovered (dashed curve) strain distribution. (b) Noisy (10% variation) intensity spectra (solid curve) and recovered (dashed curve) results from two FBGs subjected to the positive linear gradient strain distribution.
that the average and maximum errors are less than 1.08% and 52.43%, respectively, in the noise-free intensity spectra. It is clear that the error induced by the non-monotonic strain profile exceeds those of the monotonic cases in the noise-free intensity spectra. It is noted that a greater error occurs at the edge of the strain profile. In the noisy-case, as the noise variation increases, the average error and maximum error both increase. Specifically, in an intensity spectrum with a 10% noise variation, the average and maximum errors are below 4.87% and 63.75%, respectively, while at a higher noise variation of 20%, the corresponding error values are 18.04% and 275.91%, respectively.
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Table 1 The strain distribution sensing results Strain profile
Maximum error (%)
Average error (%)
Mean of the error
Variance of the error
Correlation coefficient
Positive linear gradient (noise-free) Negative linear gradient (noise-free) Non-monotonic (noise-free) Positive linear gradient (noisy-10%) Positive linear gradient (noisy-20%)
25.73 4.87 52.43 63.76 275.91
1.08 0.93 3.26 4.87 18.04
6.65 · 107 1.77 · 107 1.29 · 106 6.44 · 106 1.75 · 105
4.27 · 1011 1.21 · 1010 5.06 · 1010 6.18 · 109 2.79 · 108
0.9998 0.9995 0.9974 0.9758 0.8828
The performance of the reconstructed method was assessed from the mean, variance and correlation coefficient of errors. The results are listed in Table 1. Generally, all of the noise-free cases indicated extremely high correlation coefficient (>0.99) and low mean and variance values. However, in 10% and 20% noisy cases, the correlation coefficients were 0.9758 and 0.8828. Obviously, the mean and variance of errors were higher and the correlation coefficient was lower than that in the noise-free cases. The results presented in Table 1 confirm that the proposed approach is also applicable to the measurement of noisy systems, but reveal that the quality of the reconstruction results depends on the degree of noise variation in the intensity spectra.
5. Conclusions This paper has proposed a novel arbitrary strain distribution sensing approach using a genetic algorithm to inversely trace the strain distribution from two FBG reflection intensity spectra. The effectiveness and feasibility of the proposed method have been demonstrated by solving the inverse problems of several strain sensing cases. The present simulation results have confirmed the genetic algorithm to be an effective means of optimally deriving the solution of a complicated strain distribution via the reflection intensity spectra of two FBGs. The current method eliminates the requirements for a monotonic axial strain field and a low reflectivity, and provides a high spatial resolution (0.5 mm). Furthermore, the proposed approach provides a simple and low cost method to measure arbitrary strain distributions. This technique has several limitations: (1) The two
FBGs must encounter the same strain field. (2) The increased number of divided sections and wavelength sample points can improve the spatial and wavelength resolution, but prolong the computation time. In this study, it takes approximately 1 hour to carry out 1000 loops in the simulation by using 2.4 GHz Pentium 4 PC. (3) The fiber Bragg gratingÕs characteristics, such as grating length, modulation of refractive index, apodise degree, etc., must be known in advance. (4) This technique only provides acceptable results for noise variations of 10% or less.
Acknowledgement The current authors acknowledge the financial support provided to this study by the National Science Council under Grant No. NSC 92-2622L011-001. Also, this work is partially supported by Ministry of Education Program for Promoting Academic Excellence of Universities under the Grant No. A-92-E-FA08-1-4.
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