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Optimal design of short fiber Bragg gratings with triangular spectrum
Optics Communications 285 (2012) 631–637
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Optimal design of short fiber Bragg gratings with triangular spectrum Xuelian Yu a, b, Yong Yao a,⁎, Junjun Xiao a, Jiajun Tian a a b
College of Electronic and Information Engineering, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, 518055, China Department of optics information science and technology, Harbin University of Science and Technology, Harbin, 150080, China
a r t i c l e
i n f o
Article history: Received 6 June 2011 Received in revised form 9 September 2011 Accepted 5 November 2011 Available online 24 November 2011 Keywords: Optimization Fiber Bragg grating Quantum-behaved particle swarm optimization Particle swarm optimization
a b s t r a c t Based on the mutated adaptive quantum-behaved particle swarm optimization (MAQPSO) algorithm and the discrete layer peeling (DLP) algorithm, we propose a method to design triangular-spectrum fiber Bragg gratings (TS-FBGs). In this method, the DLP algorithm is used to generate an appropriate initial range of the index modulation, and the MAQPSO technique is applied to optimize the index modulation within this range. The quantum-behaved particle swarm optimization algorithm with mutation operation can improve the search performance of this algorithm. We purposely use a linear weighting factor for the fitness value function in view of the linear edge triangular-spectrum. This dramatically accelerates the convergence in the numerical implementation. Using the proposed method, a 0.2 nm bandwidth TS-FBG is designed and the performance is shown to be better than previously reported results in the literature. Then, a 2.5 nm bandwidth TS-FBG is reconstructed; it is indicated that a short grating length can be used to realize the TS-FBG with chirp-free structure so that complex phase modulation is avoided in fabrication. Finally, a triple-channel TS-FBG with sampled-free structure is successfully designed and optimized. We also present an analysis on the tolerance of the design method over possible fabrication error. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Fiber Bragg gratings (FBGs) have been widely employed in optical devices. Particularly, triangular-spectrum fiber Bragg gratings (TSFBGs) are recently applied to act as wavelength-interrogation devices in sensor systems [1]. The TS-FBGs are simple, economical, flexible, and have the advantages of higher sensitivity as well as immunity from the light source instability. Over the past decade, the synthesis of FBGs has attracted great interests from many researchers and basically two kinds of research effort are involved .The first is the inverse scattering (IS) algorithm [2–15]. In Ref. [15], several popular IS algorithms are critically compared and their relative speed and robustness are analyzed. However, the FBGs designed by IS algorithms generally have complicated index modulation profiles and long grating length. The second type of approach is based on the stochastic global optimization (SGO) algorithm [16–23]. In contrast to the IS algorithms, SGO algorithms can be practically implemented more easily. However, the performances of many SGO algorithms are affected by the initial value of the optimized parameters and the selected fitness value function. Gong et al. propose an approach employing both an IS algorithm and a SGO algorithm [24,25], which overcomes the shortcomings of the SGO algorithm and possess the advantages of the IS algorithm.
⁎ Corresponding author. E-mail address: [email protected] (Y. Yao). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.11.020
The purpose of this approach is to optimize the initial coupling coefficient (or the index modulation) obtained by the discrete layer peeling (DLP) algorithm [4]. However, it can not change the fact that FBGs designed by these methods usually have complex index modulation or long grating length although the maximum index modulation is minimized in these methods. Wu et al. show a simple and fast design technique for a triangular FBG filter which can give a straightforward solution to the design target by only solving a nonlinear function, but it has been based on a linearly chirped grating and need the sampling method to design the multiple-channel FBG [26–28]. In this paper, on the basis of the mutated adaptive quantumbehaved particle swarm optimization (MAQPSO) algorithm [29,30] and the DLP algorithm, another effective method is proposed to design TS-FBGs and a novel fitness value function corresponding to the linear edge characteristic of the TS-FBG spectrum is introduced. In our method, the DLP algorithm is only used to generate an appropriate initial range of the index modulation, and the MAQPSO algorithm is applied to optimize the initial index modulation within this range. The MAQPSO algorithm is an adaptive quantum-behaved particle swarm optimization (AQPSO) with mutation operation to improve the search performance of the AQPSO algorithm. The AQPSO algorithm is developed from particle swarm optimization (PSO) which has been successfully used to design FBGs [21]. The AQPSO algorithm has no velocity vector for the particles and the whole feasible solution space can be searched for the index modulation; only a parameter associated with the process of the iteration need to be adjusted. Moreover, the iterative equation of this algorithm is very different
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from that of PSO. Thus, the MAQPSO algorithm is much simple and easy to implement. To demonstrate the capability of the proposed method in designing TS-FBGs, we apply it to design a 0.2 nm bandwidth TS-FBG and a 2.5 nm bandwidth TS-FBG, and a triple-channel TS-FBG, respectively. We further discuss the success indicator and the tolerance to fabrication error of this method.
S h i2 1X f ¼ Rsynthesis λj −Rtarget λj W λj s j¼1
ð1Þ
8 1; λj ∈ stopband > > > BW < > =0:5BW; λj ∈ passband and λj ≤ λB ðj−jl Þ W λj ¼ S > > BW > > : ðj−ju Þ =0:5BW; λj ∈ passband and λj > λB S
ð2Þ
where S is the total sampling number of the evaluated reflection spectrum and λjis the j-th discrete sampling wavelength; Rsynthesis and Rtarget denote the synthesized reflectivity spectrum and target spectrum, respectively. The weighting factor W(λj) is introduced to define how the large errors in the spectrum are weighted with respect to the small errors; BW is the bandwidth of the TS-FBG; jl and ju are the lower and upper bound of the discrete wavelength number inside the bandwidth, respectively, and λB is the Bragg wavelength. Note that Eq. (1) could be a general choice for spectrum with other shape, by properly adjust the weighting factor W(λj) according to the spectrum profile. Before we proceed, let us first outline the MAQPSO method. In the AQPSO algorithm, the particles move according to the following iterative equations [29,30]:
M 1X P ðmÞ M e¼1 ed
xed ðm þ 1Þ ¼ ped ðmÞ α ðmÞjmbest−xed ðmÞj ln½1=ued ðmÞ
ð3Þ
ð4Þ ð5Þ
where ped is the attraction point of the particle swarm, Ped and Pgd the best position of the e-th the particle and the position of the best particle among all the particles in a D-dimensional space, respectively; mbestis defined as the mean value of all particles' Ped, and ued and φd are random number distributed uniformly on [0, 1], respectively; M is the total particle number, and m denotes the iterative generation. The contraction-expansion coefficient (CEC) α is the only adjustable parameter used to control the convergence speed of the algorithm. In order to improve the search performance of the AQPSO algorithm, we introduce the mutation operation to the Ped as follows: P ed ðmÞ ¼ P ed ðmÞ N 0; ςδn range
xed ðm þ 1Þ P ed ðmÞ
ðf ½xed ðm þ 1Þbf ½P ed ðmÞÞ ðf ½xed ðm þ 1Þ≥f ½P ed ðmÞÞ
1≤e≤M
Designing the grating parameters from a target reflectivity can be viewed as an optimal problem of MAQPSO. For a grating lengthL, the index modulation profile δn ðzÞ(z ∈ (0, L)) is what should be optimized. For such a problem, it is very critical to select an appropriate fitness value function. According to the linear edge of the triangular-spectrum, we propose a corresponding fitness value function with a linear weighting factor as follows:
mbest ðmÞ ¼
P ed ðm þ 1Þ ¼
P gd ðm þ 1Þ ¼ arg min ff ½P ed ðmÞg
2. Theory of the proposed method
ped ðmÞ ¼ φd ðmÞP ed ðmÞ þ ½1−φd ðmÞP gd ðmÞ
The ways to update Ped and Pgd are identical to the corresponding ones in a generic PSO [31], namely
ð6Þ
whereN 0; ςδn range is among a set of normal random numbers with mean 0 and standard deviation ςδn range where δn range denotes the range of the initial index modulation, andςis a mutated positive constant less than 1.
ð7Þ ð8Þ
where f is the fitness value function used to evaluate each particle in the swarm, as shown in Eq. (1). To accelerate the proposed method, the CEC parameter α is adaptive in the process of the each iteration. To do that, the following error function is used to identify how close the particle is to the global best position h i n h i F ðmÞ ¼ f e ðmÞ−f gbest ðmÞ = min abs½f e ðmÞ; abs f gbest ðmÞ
ð9Þ
where fe is the fitness value of the e-th particle and fgbest is the fitness value of the best particle, abs[fe] and abs[fgbest]denoting the absolute value of fe and fgbest, respectively. Furthermore, min{abs[fe], abs[fgbest]) refers to the smaller one of abs[fe] and abs[fgbest]. In this regard, the smaller the value of the error function F(m)for a certain particle, the closer it to the best particle. The particles that are far away from the best particle should be given a smaller α, whereas those close to the best particle should be given larger value of α. In other words, α is designed to be a nonlinear function α(log F, e, m). In our application, when changing log F from small value to large value, α(log F, e, m) are set from 0.3 to 0.01 which are different from that in Ref. [30]. This is because that the bigger αwill decrease the converging velocity of the MAQPSO algorithm. The MAQPSO algorithm is briefly summarized as follows: 1) Initialize a population of particles with random positions xed(0) inside the range of the initial index modulation δn range . Each particle's position xed(0) represents the d-th index modulation values δn ðdÞ of the D-uniform sections of the FBG; 2) Evaluate the fitness value of each particle by substituting its position into Eq. (1) and assign the particle's best position to Ped(0). Identify the best among Ped(0) as Pgd(0). According to Eq. (3), ped (0) is calculated, and mbest(0) is also obtained by Eq. (4); 3) The CEC parameterα is chosen according to Eq. (9); 4) Updating the position of the particle population xed(m) by Eq. (5); 5) Evaluate the fitness value of each particle by substituting its position into Eq. (1) and assign the particle's best position to Ped(m). Identify the best among Ped(m) as Pgd(m); 6) The particle's best position Ped(m) is operated with the mutation probability and its fitness value is compared with the one before the mutation operation. If its fitness value is better, then update
it and the fitness value. Namely, if f P ed ðmÞ þ N 0; ςδn range b f ½P ed ðmÞ, then P ed ðmÞ ¼ P ed ðmÞ þ N 0; ςδn range and f ½P ed ðmÞ ¼ f ½P ed ðmÞþ N 0; ςδn range ; 7) Calculate ped(m) and mbest(m) by Eqs. (3) and (4), respectively; 8) For each particle, the fitness value of the particle's current position is calculated. If it is better than the previous position, then update it and the fitness value. Namely, if f [xed(m + 1)] b f [Ped(m)], then Ped(m + 1) = xed(m + 1) and f [Ped(m + 1)] = f [xed(m + 1)]; 9) Identify the best among Ped(m) as the global best position of the current particle population Pgd(m + 1) and update the fitness value, namely P gd ðm þ 1Þ ¼ arg min1≤e≤M ff ½P ed ðmÞg and f [Pgd (m + 1)] = f [Pgd(m)]; 10) Compare the global best position of the current particle population with the previous global best position, if it is better than f [Pgd(m − 1)], update Pgd(m) and its fitness value; 11) Repeat steps 3–10 until the assigned fitness value function is achieved.
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Fig. 1. Flowchart of the proposed method.
For the optimal algorithm, the choice of the initial index modulation is very crucial. An improper initiation can lead to low-efficiency optimizations, although it doesn't lead to the local convergence and a wrong result in our MAQPSO algorithm. Therefore, an appropriate initial index modulation is the key to ensure fast convergence in the MAQPSO summarized above. As a matter of fact, we can obtain the range of the initial index modulation δn range by DLP algorithm [4]. In the DLP algorithm, the synthesis of the grating is facilitated by discretizing the grating into K complex reflectors satisfying the time domain causality. The transfer matrix T of each discretization layer of thickness Δ(Δ = L/K) is represented by the product of the reflection matrix Tρ and the propagation matrix TΔ: T ¼ T Δ T ρk
Based on the MAQPSO algorithm and the DLP algorithm, a new method is developed as shown in Fig. 1. For this method, it is noteworthy that the grating length selected is different when the range of the initial index modulation is obtained by the DLP algorithm and the initial index modulation is optimized by the MAQPSO algorithm. This is because that the DLP algorithm is only used to generate an appropriate initial range of the index modulation in the whole process,
ð10Þ
where the k-th complex reflection coefficient ρk is determined by ρk ¼ − tanhðjqk jΔÞ
qk jqk j
ð11Þ
And the real parts of the coupling coefficient qk is related with the index modulation δn k by qk ¼
πδn k λB
ð12Þ
Here Fourier transform analysis is reasonable. The coupling coefficient at the front of the grating can be determined only from the leading edge of the impulse response. The complex reflection coefficient ρk is the Fourier transformation of the reflection rk: ρk ¼ F
−1
½r k ðδÞt¼0
r kþ1 ðδÞ ¼ expð−i2δΔÞ
ð13Þ r k ðδÞ−ρk 1−ρk r k ðδÞ
ð14Þ
where δ is the wavelength detuning, δ = 2πneff/λB − 2πneff/λ(δw = π/ Δ, δw is the wavelength detuning window),neff is the effective modal index. By the DLP algorithm, the index modulation δn k can be obtained by Eq. (12), and we denote the depth of δn k as the range of the initial index modulation δn range , namely δn range ¼ δn max −δn min
ð15Þ
where δn max and δn min are the largest and least δn k , respectively .
Fig. 2. A TS-FBG with 0.2 nm bandwidth: (A) the optimized index modulation; (B) corresponding reflectivity of (A).
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To demonstrate the performance of the proposed method, three examples are considered. Firstly, as a benchmark example, we design
a TS-FBG with the parameters of BW = 0.2 nm, λB = 1550 nm and neff = 1.5. The target reflectivity spectrum is shown in Fig. 2(B) as the solid line. First, by the DLP algorithm, we obtain the initial range of the index modulation δn range ¼ 0:6 10−4 . Then, by the MAQPSO algorithm, the index modulation δn ðdÞ is optimized. In optimal process, the other parameters are chosen as L = 4 cm, D = 20 and M = 20. It is noteworthy that the MAQPSO algorithm is no velocity vector for the particles so that whole feasible solution space of real numbers can be searched for the index modulation, which is different from PSO algorithm and covariance matrix adapted evolution strategy (CMAES) algorithm [23]. For a random run, using the proposed method, we reconstructed the index modulation as shown in Fig. 2(A) with the mutated constant ς = 0.6 and the finial index modulation are positive (i.e. no phase shifts are needed). Therefore, the optimized index modulation is simple and can probably be fabricated using advanced point-by-point femotosecond laser writing FBG fabrications method [33]. To check the validity of the final index modulation in Fig. 2(A), the corresponding reflectivity spectrum obtained by the transfer matrix method is shown in Fig. 2(B) as the dashed line and the maximum reflectivity reaches 0.8727. From Fig. 2(B), we can see that the synthesized reflectivity spectrum is nearly consistent with the target reflectivity spectrum which has a triangular optical reflectivity and the wavelength changes linearly with the reflectivity. This agreement shows that our method is very effective. In this example the main program is implemented in Matlab and the most optimal results are obtained after several minutes. The random run is the same as one in which the Fig. (2) is obtained, with four different mutated constants (ς = 0, 0.4, 0.6, 0.8), Fig. 3 shows the convergence of the fitness value function in Eq. (5) as the iteration generation goes over 1000 and the fitness value is
Fig. 4. The influence of the weighting factor in the fitness value functions on the proposed method: (A) convergence of the fitness value function; (B) corresponding synthesized reflectivity spectrums.
Fig. 5. Comparison of the performance among the proposed method, the PSO method, and the CMAES methods: (A) the convergence of the fitness value function; (B) corresponding synthesized reflectivity spectrums.
Fig. 3. Convergence of the fitness value function f with four different mutated constant (ς = 0, 0.4, 0.6, 0.8).
which is different from Refs. [24,25]. The transfer matrix method is applied to verify the final optimized index modulation δn ðdÞ by calculating the reflectivity spectrum [32]. 3. Numerical examples and discussions
X. Yu et al. / Optics Communications 285 (2012) 631–637
least when the mutated constant ς is equal to 0.6. This indicates that the proposed method has a notable merit of fast convergence with the introduced fitness value function and the range of the initial index modulation obtained by DLP algorithm, and the introduction of the mutated constant can effectively improve the search performance of the proposed method. From the above analysis, it is also indicated that we can obtain the most optimal index modulation by adjusting the mutated constant for a random run. For a random run, Fig. 4 shows the influence of the weighting factor on the proposed method. As can be seen in Fig. 4(A), the convergence of the fitness value function with the weighting factor is faster than that of the fitness value function without the weighting factor (i.e., W(λj) ≡ 1), but the corresponding synthesized reflectivity spectra have no obvious difference [see Fig. 4(B)]. To demonstrate the effectiveness of the proposed method, we compare it with the PSO and CMAES [23] when the fitness value function of our method is equal to that used in those methods. By the proposed method, the best fitness value function of 20 individual runs is obtained which is shown in Fig. 5(A) where the PSO (solid line) and CMAES (dotted lines) results of Ref. 23 are reproduced. The results clearly indicate remarkable improvement in the performance of our method over the PSO and CMAES methods. Fig. 5(B) plots the corresponding reflectivity spectrums of the cases in Fig. 5(A). It is seen that the synthesis spectrum of our proposed method is visually closer to the target spectrum. To justify the success of our method, we further study several different grating lengths. Fig. 6(A) depicts that the fitness values
Fig. 6. The influence of the grating length on the synthesized reflectivity spectrum: (A) the fitness values for the different grating length change with the iterative generation; (B) the corresponding synthesized reflectivity spectrums of (A).
635
convergence and Fig. 6(B) shows the corresponding synthesized reflectivity. It is seen that the final fitness values of the convergence are slightly different but the profiles of corresponding reflectivity spectra are nearly all consistent with the target reflectivity spectrum. That is to say, for certain range of short grating length, the target spectrum of the FBGs can be successfully synthesized by our method. To obtain a sensor based on the TS-FBG with a broad measurement range, it is better to ensure that the reflectivity spectrum of the TSFBG has linear edge with bandwidth as large as possible. Therefore, for the second example, we design a TS-FBG with the bandwidth BW = 2.5 nm, λB = 1550 nm and neff = 1.5. The target reflectivity spectrum (solid line) is shown in Fig. 7(B). First, by the DLP algorithm, we obtain the initial range of the index modulation δn range ¼ 1 10−3 . Then, by the MAQPSO algorithm, the index modulation δn ðdÞ is optimized. In optimal process, the other parameters are chosen as L = 0.3 cm,D = 20 and M = 20. The index modulation profile is shown in Fig. 7(A) and the corresponding reflectivity spectrum is presented as the dashed line in Fig. 7(B). Fig. 7(B) indicates that the proposed method can help to effectively design the TS-FBG with a larger bandwidth for a short grating, avoiding chirp structure. The capability of the proposed method in more complex situation is demonstrated with the analysis of a multi-channel TS-FBG, as illustrated in Fig. 8. The designed multi-channel TS-FBGs can be used for the purpose of measuring multiple physical parameters. As shown in Fig. 8(B), a triple-channel TS-FBG (solid line) is designed such that each channel has a width 1 nm, channel spacing 1.5 nm, and the three different central wavelengths are 1548.5, 1550, and 1551.5 nm, respectively. First, by the DLP algorithm, we obtain the
Fig. 7. A TS-FBG with 2.5 nm bandwidth: (A) the optimized index modulation; (B) corresponding reflectivity of (A).
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designed index modulation and their perturbations are plotted in the Figs. 9(A) and 10(A). Their corresponding reflectivity spectrum calculated by the transfer matrix method is shown in Figs. 9(B) and 10(B), respectively. These results indicate that a 10% UV irradiation error in fabrication poses no obvious influence on the reflectivity spectrum of the designed TS-FBGs.
5. Conclusions In summary, a new method combining the MAQPSO algorithm with the DLP algorithm is proposed for the design of TS-FBGs. Considering the linear edge characteristic of the target reflectivity, a novel fitness value function with linear weighting factor has been used. The employment of the DLP algorithm can improve the converging velocity of the MAQPSO algorithm. The mutated operation can rapidly adjust the converging velocity of the proposed method and can effectively improve its search performance. By the proposed method, we successfully design three TS-FBGs. The designed broad bandwidth TS-FBG has a chirp-free structure with a short grating length so that complex phase modulation can be avoided. With our method, multiple-channel TS-FBGs with sampled-free structure can also be designed and optimized. It is shown that for both single- and triplechannel TS-FBGs, a 10% error of the designed index modulation in fabrication does not obviously affect the reflectivity spectrum.
Fig. 8. A triple-channel TS-FBG with 1 nm bandwidth: (A) the optimized index modulation; (B) corresponding reflectivity of (A).
initial range of the index modulationδn range ¼ 1:6 10−3 . Then, the index modulation δn ðdÞ is optimized by the MAQPSO algorithm .In the optimal process, the other parameters are chosen as L = 0.5 cm, D = 30, and M = 20. The index modulation profile is reconstructed as in Fig. 8(A) and the corresponding reflectivity spectrum of it is plotted in Fig. 8(B) (dashed line). From above analysis, it is noteworthy that this triple-channel TS-FBG designed by our method is sampled-free structure, which is different from the method in Ref. [26]. 4. The tolerance of the proposed method to fabrication errors To evaluate possible fabrication error on the synthesis FBG, the influence of the designed index modulation perturbation on the reflectivity spectrum is investigated. To do this, a perturbation factor u(z) is added to the designed index modulation δn ðzÞ that was already optimized in Section 3, and the perturbation can be expressed as follows: 0
δn ðzÞ ¼ δn ðzÞ ð1 þ uðzÞÞ
ð16Þ
0
where δn ðzÞ and δn ðzÞ are the designed index modulation before the perturbation and after perturbation, respectively. We assume u(z) being normal distribution with a mean 0 and standard deviation 1 (e.g. | u(z)| b 10% means that the value of u(z) is generated randomly with a normal distribution with a mean 0 and deviation 1 in the range ±10%). For single-channel and triple-channel TS-FBGs, the
Fig. 9. The influence of the perturbation for the designed single-channel TS-FBG with 2.5 nm bandwidth: (A) perturbation on the optimized index modulation; (B) corresponding reflectivity of (A).
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References
Fig. 10. The influence of the perturbation for the designed triple-channel TS-FBG with 1 nm bandwidth: (A) perturbation on the optimized index modulation; (B) corresponding reflectivity of (A).
Acknowledgments This work was supported in part by the NSFC (Nos. 60977034, 61107036 and 11004043), the SMSTPR (Nos. JC200903120167A and JC201005260185A), and in part by the China Postdoctoral Science Foundation (No. 20110491092).
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Optimal design of short fiber Bragg gratings with triangular spectrum Xuelian Yu a, b, Yong Yao a,⁎, Junjun Xiao a, Jiajun Tian a a b
College of Electronic and Information Engineering, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, 518055, China Department of optics information science and technology, Harbin University of Science and Technology, Harbin, 150080, China
a r t i c l e
i n f o
Article history: Received 6 June 2011 Received in revised form 9 September 2011 Accepted 5 November 2011 Available online 24 November 2011 Keywords: Optimization Fiber Bragg grating Quantum-behaved particle swarm optimization Particle swarm optimization
a b s t r a c t Based on the mutated adaptive quantum-behaved particle swarm optimization (MAQPSO) algorithm and the discrete layer peeling (DLP) algorithm, we propose a method to design triangular-spectrum fiber Bragg gratings (TS-FBGs). In this method, the DLP algorithm is used to generate an appropriate initial range of the index modulation, and the MAQPSO technique is applied to optimize the index modulation within this range. The quantum-behaved particle swarm optimization algorithm with mutation operation can improve the search performance of this algorithm. We purposely use a linear weighting factor for the fitness value function in view of the linear edge triangular-spectrum. This dramatically accelerates the convergence in the numerical implementation. Using the proposed method, a 0.2 nm bandwidth TS-FBG is designed and the performance is shown to be better than previously reported results in the literature. Then, a 2.5 nm bandwidth TS-FBG is reconstructed; it is indicated that a short grating length can be used to realize the TS-FBG with chirp-free structure so that complex phase modulation is avoided in fabrication. Finally, a triple-channel TS-FBG with sampled-free structure is successfully designed and optimized. We also present an analysis on the tolerance of the design method over possible fabrication error. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Fiber Bragg gratings (FBGs) have been widely employed in optical devices. Particularly, triangular-spectrum fiber Bragg gratings (TSFBGs) are recently applied to act as wavelength-interrogation devices in sensor systems [1]. The TS-FBGs are simple, economical, flexible, and have the advantages of higher sensitivity as well as immunity from the light source instability. Over the past decade, the synthesis of FBGs has attracted great interests from many researchers and basically two kinds of research effort are involved .The first is the inverse scattering (IS) algorithm [2–15]. In Ref. [15], several popular IS algorithms are critically compared and their relative speed and robustness are analyzed. However, the FBGs designed by IS algorithms generally have complicated index modulation profiles and long grating length. The second type of approach is based on the stochastic global optimization (SGO) algorithm [16–23]. In contrast to the IS algorithms, SGO algorithms can be practically implemented more easily. However, the performances of many SGO algorithms are affected by the initial value of the optimized parameters and the selected fitness value function. Gong et al. propose an approach employing both an IS algorithm and a SGO algorithm [24,25], which overcomes the shortcomings of the SGO algorithm and possess the advantages of the IS algorithm.
⁎ Corresponding author. E-mail address: [email protected] (Y. Yao). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.11.020
The purpose of this approach is to optimize the initial coupling coefficient (or the index modulation) obtained by the discrete layer peeling (DLP) algorithm [4]. However, it can not change the fact that FBGs designed by these methods usually have complex index modulation or long grating length although the maximum index modulation is minimized in these methods. Wu et al. show a simple and fast design technique for a triangular FBG filter which can give a straightforward solution to the design target by only solving a nonlinear function, but it has been based on a linearly chirped grating and need the sampling method to design the multiple-channel FBG [26–28]. In this paper, on the basis of the mutated adaptive quantumbehaved particle swarm optimization (MAQPSO) algorithm [29,30] and the DLP algorithm, another effective method is proposed to design TS-FBGs and a novel fitness value function corresponding to the linear edge characteristic of the TS-FBG spectrum is introduced. In our method, the DLP algorithm is only used to generate an appropriate initial range of the index modulation, and the MAQPSO algorithm is applied to optimize the initial index modulation within this range. The MAQPSO algorithm is an adaptive quantum-behaved particle swarm optimization (AQPSO) with mutation operation to improve the search performance of the AQPSO algorithm. The AQPSO algorithm is developed from particle swarm optimization (PSO) which has been successfully used to design FBGs [21]. The AQPSO algorithm has no velocity vector for the particles and the whole feasible solution space can be searched for the index modulation; only a parameter associated with the process of the iteration need to be adjusted. Moreover, the iterative equation of this algorithm is very different
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from that of PSO. Thus, the MAQPSO algorithm is much simple and easy to implement. To demonstrate the capability of the proposed method in designing TS-FBGs, we apply it to design a 0.2 nm bandwidth TS-FBG and a 2.5 nm bandwidth TS-FBG, and a triple-channel TS-FBG, respectively. We further discuss the success indicator and the tolerance to fabrication error of this method.
S h i2 1X f ¼ Rsynthesis λj −Rtarget λj W λj s j¼1
ð1Þ
8 1; λj ∈ stopband > > > BW < > =0:5BW; λj ∈ passband and λj ≤ λB ðj−jl Þ W λj ¼ S > > BW > > : ðj−ju Þ =0:5BW; λj ∈ passband and λj > λB S
ð2Þ
where S is the total sampling number of the evaluated reflection spectrum and λjis the j-th discrete sampling wavelength; Rsynthesis and Rtarget denote the synthesized reflectivity spectrum and target spectrum, respectively. The weighting factor W(λj) is introduced to define how the large errors in the spectrum are weighted with respect to the small errors; BW is the bandwidth of the TS-FBG; jl and ju are the lower and upper bound of the discrete wavelength number inside the bandwidth, respectively, and λB is the Bragg wavelength. Note that Eq. (1) could be a general choice for spectrum with other shape, by properly adjust the weighting factor W(λj) according to the spectrum profile. Before we proceed, let us first outline the MAQPSO method. In the AQPSO algorithm, the particles move according to the following iterative equations [29,30]:
M 1X P ðmÞ M e¼1 ed
xed ðm þ 1Þ ¼ ped ðmÞ α ðmÞjmbest−xed ðmÞj ln½1=ued ðmÞ
ð3Þ
ð4Þ ð5Þ
where ped is the attraction point of the particle swarm, Ped and Pgd the best position of the e-th the particle and the position of the best particle among all the particles in a D-dimensional space, respectively; mbestis defined as the mean value of all particles' Ped, and ued and φd are random number distributed uniformly on [0, 1], respectively; M is the total particle number, and m denotes the iterative generation. The contraction-expansion coefficient (CEC) α is the only adjustable parameter used to control the convergence speed of the algorithm. In order to improve the search performance of the AQPSO algorithm, we introduce the mutation operation to the Ped as follows: P ed ðmÞ ¼ P ed ðmÞ N 0; ςδn range
xed ðm þ 1Þ P ed ðmÞ
ðf ½xed ðm þ 1Þbf ½P ed ðmÞÞ ðf ½xed ðm þ 1Þ≥f ½P ed ðmÞÞ
1≤e≤M
Designing the grating parameters from a target reflectivity can be viewed as an optimal problem of MAQPSO. For a grating lengthL, the index modulation profile δn ðzÞ(z ∈ (0, L)) is what should be optimized. For such a problem, it is very critical to select an appropriate fitness value function. According to the linear edge of the triangular-spectrum, we propose a corresponding fitness value function with a linear weighting factor as follows:
mbest ðmÞ ¼
P ed ðm þ 1Þ ¼
P gd ðm þ 1Þ ¼ arg min ff ½P ed ðmÞg
2. Theory of the proposed method
ped ðmÞ ¼ φd ðmÞP ed ðmÞ þ ½1−φd ðmÞP gd ðmÞ
The ways to update Ped and Pgd are identical to the corresponding ones in a generic PSO [31], namely
ð6Þ
whereN 0; ςδn range is among a set of normal random numbers with mean 0 and standard deviation ςδn range where δn range denotes the range of the initial index modulation, andςis a mutated positive constant less than 1.
ð7Þ ð8Þ
where f is the fitness value function used to evaluate each particle in the swarm, as shown in Eq. (1). To accelerate the proposed method, the CEC parameter α is adaptive in the process of the each iteration. To do that, the following error function is used to identify how close the particle is to the global best position h i n h i F ðmÞ ¼ f e ðmÞ−f gbest ðmÞ = min abs½f e ðmÞ; abs f gbest ðmÞ
ð9Þ
where fe is the fitness value of the e-th particle and fgbest is the fitness value of the best particle, abs[fe] and abs[fgbest]denoting the absolute value of fe and fgbest, respectively. Furthermore, min{abs[fe], abs[fgbest]) refers to the smaller one of abs[fe] and abs[fgbest]. In this regard, the smaller the value of the error function F(m)for a certain particle, the closer it to the best particle. The particles that are far away from the best particle should be given a smaller α, whereas those close to the best particle should be given larger value of α. In other words, α is designed to be a nonlinear function α(log F, e, m). In our application, when changing log F from small value to large value, α(log F, e, m) are set from 0.3 to 0.01 which are different from that in Ref. [30]. This is because that the bigger αwill decrease the converging velocity of the MAQPSO algorithm. The MAQPSO algorithm is briefly summarized as follows: 1) Initialize a population of particles with random positions xed(0) inside the range of the initial index modulation δn range . Each particle's position xed(0) represents the d-th index modulation values δn ðdÞ of the D-uniform sections of the FBG; 2) Evaluate the fitness value of each particle by substituting its position into Eq. (1) and assign the particle's best position to Ped(0). Identify the best among Ped(0) as Pgd(0). According to Eq. (3), ped (0) is calculated, and mbest(0) is also obtained by Eq. (4); 3) The CEC parameterα is chosen according to Eq. (9); 4) Updating the position of the particle population xed(m) by Eq. (5); 5) Evaluate the fitness value of each particle by substituting its position into Eq. (1) and assign the particle's best position to Ped(m). Identify the best among Ped(m) as Pgd(m); 6) The particle's best position Ped(m) is operated with the mutation probability and its fitness value is compared with the one before the mutation operation. If its fitness value is better, then update
it and the fitness value. Namely, if f P ed ðmÞ þ N 0; ςδn range b f ½P ed ðmÞ, then P ed ðmÞ ¼ P ed ðmÞ þ N 0; ςδn range and f ½P ed ðmÞ ¼ f ½P ed ðmÞþ N 0; ςδn range ; 7) Calculate ped(m) and mbest(m) by Eqs. (3) and (4), respectively; 8) For each particle, the fitness value of the particle's current position is calculated. If it is better than the previous position, then update it and the fitness value. Namely, if f [xed(m + 1)] b f [Ped(m)], then Ped(m + 1) = xed(m + 1) and f [Ped(m + 1)] = f [xed(m + 1)]; 9) Identify the best among Ped(m) as the global best position of the current particle population Pgd(m + 1) and update the fitness value, namely P gd ðm þ 1Þ ¼ arg min1≤e≤M ff ½P ed ðmÞg and f [Pgd (m + 1)] = f [Pgd(m)]; 10) Compare the global best position of the current particle population with the previous global best position, if it is better than f [Pgd(m − 1)], update Pgd(m) and its fitness value; 11) Repeat steps 3–10 until the assigned fitness value function is achieved.
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Fig. 1. Flowchart of the proposed method.
For the optimal algorithm, the choice of the initial index modulation is very crucial. An improper initiation can lead to low-efficiency optimizations, although it doesn't lead to the local convergence and a wrong result in our MAQPSO algorithm. Therefore, an appropriate initial index modulation is the key to ensure fast convergence in the MAQPSO summarized above. As a matter of fact, we can obtain the range of the initial index modulation δn range by DLP algorithm [4]. In the DLP algorithm, the synthesis of the grating is facilitated by discretizing the grating into K complex reflectors satisfying the time domain causality. The transfer matrix T of each discretization layer of thickness Δ(Δ = L/K) is represented by the product of the reflection matrix Tρ and the propagation matrix TΔ: T ¼ T Δ T ρk
Based on the MAQPSO algorithm and the DLP algorithm, a new method is developed as shown in Fig. 1. For this method, it is noteworthy that the grating length selected is different when the range of the initial index modulation is obtained by the DLP algorithm and the initial index modulation is optimized by the MAQPSO algorithm. This is because that the DLP algorithm is only used to generate an appropriate initial range of the index modulation in the whole process,
ð10Þ
where the k-th complex reflection coefficient ρk is determined by ρk ¼ − tanhðjqk jΔÞ
qk jqk j
ð11Þ
And the real parts of the coupling coefficient qk is related with the index modulation δn k by qk ¼
πδn k λB
ð12Þ
Here Fourier transform analysis is reasonable. The coupling coefficient at the front of the grating can be determined only from the leading edge of the impulse response. The complex reflection coefficient ρk is the Fourier transformation of the reflection rk: ρk ¼ F
−1
½r k ðδÞt¼0
r kþ1 ðδÞ ¼ expð−i2δΔÞ
ð13Þ r k ðδÞ−ρk 1−ρk r k ðδÞ
ð14Þ
where δ is the wavelength detuning, δ = 2πneff/λB − 2πneff/λ(δw = π/ Δ, δw is the wavelength detuning window),neff is the effective modal index. By the DLP algorithm, the index modulation δn k can be obtained by Eq. (12), and we denote the depth of δn k as the range of the initial index modulation δn range , namely δn range ¼ δn max −δn min
ð15Þ
where δn max and δn min are the largest and least δn k , respectively .
Fig. 2. A TS-FBG with 0.2 nm bandwidth: (A) the optimized index modulation; (B) corresponding reflectivity of (A).
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To demonstrate the performance of the proposed method, three examples are considered. Firstly, as a benchmark example, we design
a TS-FBG with the parameters of BW = 0.2 nm, λB = 1550 nm and neff = 1.5. The target reflectivity spectrum is shown in Fig. 2(B) as the solid line. First, by the DLP algorithm, we obtain the initial range of the index modulation δn range ¼ 0:6 10−4 . Then, by the MAQPSO algorithm, the index modulation δn ðdÞ is optimized. In optimal process, the other parameters are chosen as L = 4 cm, D = 20 and M = 20. It is noteworthy that the MAQPSO algorithm is no velocity vector for the particles so that whole feasible solution space of real numbers can be searched for the index modulation, which is different from PSO algorithm and covariance matrix adapted evolution strategy (CMAES) algorithm [23]. For a random run, using the proposed method, we reconstructed the index modulation as shown in Fig. 2(A) with the mutated constant ς = 0.6 and the finial index modulation are positive (i.e. no phase shifts are needed). Therefore, the optimized index modulation is simple and can probably be fabricated using advanced point-by-point femotosecond laser writing FBG fabrications method [33]. To check the validity of the final index modulation in Fig. 2(A), the corresponding reflectivity spectrum obtained by the transfer matrix method is shown in Fig. 2(B) as the dashed line and the maximum reflectivity reaches 0.8727. From Fig. 2(B), we can see that the synthesized reflectivity spectrum is nearly consistent with the target reflectivity spectrum which has a triangular optical reflectivity and the wavelength changes linearly with the reflectivity. This agreement shows that our method is very effective. In this example the main program is implemented in Matlab and the most optimal results are obtained after several minutes. The random run is the same as one in which the Fig. (2) is obtained, with four different mutated constants (ς = 0, 0.4, 0.6, 0.8), Fig. 3 shows the convergence of the fitness value function in Eq. (5) as the iteration generation goes over 1000 and the fitness value is
Fig. 4. The influence of the weighting factor in the fitness value functions on the proposed method: (A) convergence of the fitness value function; (B) corresponding synthesized reflectivity spectrums.
Fig. 5. Comparison of the performance among the proposed method, the PSO method, and the CMAES methods: (A) the convergence of the fitness value function; (B) corresponding synthesized reflectivity spectrums.
Fig. 3. Convergence of the fitness value function f with four different mutated constant (ς = 0, 0.4, 0.6, 0.8).
which is different from Refs. [24,25]. The transfer matrix method is applied to verify the final optimized index modulation δn ðdÞ by calculating the reflectivity spectrum [32]. 3. Numerical examples and discussions
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least when the mutated constant ς is equal to 0.6. This indicates that the proposed method has a notable merit of fast convergence with the introduced fitness value function and the range of the initial index modulation obtained by DLP algorithm, and the introduction of the mutated constant can effectively improve the search performance of the proposed method. From the above analysis, it is also indicated that we can obtain the most optimal index modulation by adjusting the mutated constant for a random run. For a random run, Fig. 4 shows the influence of the weighting factor on the proposed method. As can be seen in Fig. 4(A), the convergence of the fitness value function with the weighting factor is faster than that of the fitness value function without the weighting factor (i.e., W(λj) ≡ 1), but the corresponding synthesized reflectivity spectra have no obvious difference [see Fig. 4(B)]. To demonstrate the effectiveness of the proposed method, we compare it with the PSO and CMAES [23] when the fitness value function of our method is equal to that used in those methods. By the proposed method, the best fitness value function of 20 individual runs is obtained which is shown in Fig. 5(A) where the PSO (solid line) and CMAES (dotted lines) results of Ref. 23 are reproduced. The results clearly indicate remarkable improvement in the performance of our method over the PSO and CMAES methods. Fig. 5(B) plots the corresponding reflectivity spectrums of the cases in Fig. 5(A). It is seen that the synthesis spectrum of our proposed method is visually closer to the target spectrum. To justify the success of our method, we further study several different grating lengths. Fig. 6(A) depicts that the fitness values
Fig. 6. The influence of the grating length on the synthesized reflectivity spectrum: (A) the fitness values for the different grating length change with the iterative generation; (B) the corresponding synthesized reflectivity spectrums of (A).
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convergence and Fig. 6(B) shows the corresponding synthesized reflectivity. It is seen that the final fitness values of the convergence are slightly different but the profiles of corresponding reflectivity spectra are nearly all consistent with the target reflectivity spectrum. That is to say, for certain range of short grating length, the target spectrum of the FBGs can be successfully synthesized by our method. To obtain a sensor based on the TS-FBG with a broad measurement range, it is better to ensure that the reflectivity spectrum of the TSFBG has linear edge with bandwidth as large as possible. Therefore, for the second example, we design a TS-FBG with the bandwidth BW = 2.5 nm, λB = 1550 nm and neff = 1.5. The target reflectivity spectrum (solid line) is shown in Fig. 7(B). First, by the DLP algorithm, we obtain the initial range of the index modulation δn range ¼ 1 10−3 . Then, by the MAQPSO algorithm, the index modulation δn ðdÞ is optimized. In optimal process, the other parameters are chosen as L = 0.3 cm,D = 20 and M = 20. The index modulation profile is shown in Fig. 7(A) and the corresponding reflectivity spectrum is presented as the dashed line in Fig. 7(B). Fig. 7(B) indicates that the proposed method can help to effectively design the TS-FBG with a larger bandwidth for a short grating, avoiding chirp structure. The capability of the proposed method in more complex situation is demonstrated with the analysis of a multi-channel TS-FBG, as illustrated in Fig. 8. The designed multi-channel TS-FBGs can be used for the purpose of measuring multiple physical parameters. As shown in Fig. 8(B), a triple-channel TS-FBG (solid line) is designed such that each channel has a width 1 nm, channel spacing 1.5 nm, and the three different central wavelengths are 1548.5, 1550, and 1551.5 nm, respectively. First, by the DLP algorithm, we obtain the
Fig. 7. A TS-FBG with 2.5 nm bandwidth: (A) the optimized index modulation; (B) corresponding reflectivity of (A).
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designed index modulation and their perturbations are plotted in the Figs. 9(A) and 10(A). Their corresponding reflectivity spectrum calculated by the transfer matrix method is shown in Figs. 9(B) and 10(B), respectively. These results indicate that a 10% UV irradiation error in fabrication poses no obvious influence on the reflectivity spectrum of the designed TS-FBGs.
5. Conclusions In summary, a new method combining the MAQPSO algorithm with the DLP algorithm is proposed for the design of TS-FBGs. Considering the linear edge characteristic of the target reflectivity, a novel fitness value function with linear weighting factor has been used. The employment of the DLP algorithm can improve the converging velocity of the MAQPSO algorithm. The mutated operation can rapidly adjust the converging velocity of the proposed method and can effectively improve its search performance. By the proposed method, we successfully design three TS-FBGs. The designed broad bandwidth TS-FBG has a chirp-free structure with a short grating length so that complex phase modulation can be avoided. With our method, multiple-channel TS-FBGs with sampled-free structure can also be designed and optimized. It is shown that for both single- and triplechannel TS-FBGs, a 10% error of the designed index modulation in fabrication does not obviously affect the reflectivity spectrum.
Fig. 8. A triple-channel TS-FBG with 1 nm bandwidth: (A) the optimized index modulation; (B) corresponding reflectivity of (A).
initial range of the index modulationδn range ¼ 1:6 10−3 . Then, the index modulation δn ðdÞ is optimized by the MAQPSO algorithm .In the optimal process, the other parameters are chosen as L = 0.5 cm, D = 30, and M = 20. The index modulation profile is reconstructed as in Fig. 8(A) and the corresponding reflectivity spectrum of it is plotted in Fig. 8(B) (dashed line). From above analysis, it is noteworthy that this triple-channel TS-FBG designed by our method is sampled-free structure, which is different from the method in Ref. [26]. 4. The tolerance of the proposed method to fabrication errors To evaluate possible fabrication error on the synthesis FBG, the influence of the designed index modulation perturbation on the reflectivity spectrum is investigated. To do this, a perturbation factor u(z) is added to the designed index modulation δn ðzÞ that was already optimized in Section 3, and the perturbation can be expressed as follows: 0
δn ðzÞ ¼ δn ðzÞ ð1 þ uðzÞÞ
ð16Þ
0
where δn ðzÞ and δn ðzÞ are the designed index modulation before the perturbation and after perturbation, respectively. We assume u(z) being normal distribution with a mean 0 and standard deviation 1 (e.g. | u(z)| b 10% means that the value of u(z) is generated randomly with a normal distribution with a mean 0 and deviation 1 in the range ±10%). For single-channel and triple-channel TS-FBGs, the
Fig. 9. The influence of the perturbation for the designed single-channel TS-FBG with 2.5 nm bandwidth: (A) perturbation on the optimized index modulation; (B) corresponding reflectivity of (A).
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References
Fig. 10. The influence of the perturbation for the designed triple-channel TS-FBG with 1 nm bandwidth: (A) perturbation on the optimized index modulation; (B) corresponding reflectivity of (A).
Acknowledgments This work was supported in part by the NSFC (Nos. 60977034, 61107036 and 11004043), the SMSTPR (Nos. JC200903120167A and JC201005260185A), and in part by the China Postdoctoral Science Foundation (No. 20110491092).
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