I

As

Xa

b

a

ARA

KFSQoT

1

FoiietflayebmTtgtr

a

0h

ARTICLE IN PRESSG ModelJLEO-52240; No. of Pages 4

Optik xxx (2012) xxx–xxx

Contents lists available at SciVerse ScienceDirect

Optik

jo ur n al homepage: www.elsev ier .de / i j leo

hybrid method for designing fiber Bragg gratings with right-angled triangularpectrum in sensor applications

uelian Yua,b, Yong Yaoa,∗, Jiajun Tiana, Chao Liua

College of Electronic and Information Engineering, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, PR ChinaDepartment of Optics Information Science and Technology, Harbin University of Science and Technology, Harbin 150080, PR China

r t i c l e i n f o

rticle history:eceived 5 November 2011ccepted 18 March 2012

eywords:

a b s t r a c t

Based on the quantum-behaved particle swarm optimization (QPSO) algorithm and the discrete layerpeeling (DLP) approach, we propose a scheme to design the right-angled triangular spectrum fiber Bragggratings (RTS-FBGs). In this method, for a target reflectivity spectrum, the DLP approach is used to gener-ate an appropriate initial guess of the complex coupling coefficients, and the QPSO technique is applied

iber Bragg gratingynthesisuantum-behaved particle swarmptimizationriangular spectrum

to optimize the initial complex coupling coefficients. For the QPSO method, the whole complex solutionspace can be searched for the coupling coefficients. Using the proposed method, single- and multi-channelRTS-FBGs are designed. Simulation results demonstrate that the edge of the right-angled triangular spec-trums designed by our method is quite linear and the shape of the spectrum is also very consistent withthe target spectrum, in contrast to those obtained by existing method. The RTS-FBGs can find potentialapplications in wavelength interrogation in optical sensor system.

Crown Copyright © 2012 Published by Elsevier GmbH. All rights reserved.

. Introduction

Symmetrical triangular spectrum fiber Bragg gratings (STS-BGs) is a kind of useful wavelength-interrogation devices inptical sensor systems [1]. Compared with traditional wavelength-nterrogation devices such as edge filters [2], Mach–Zehndernterferometers [3], and Fabry–Perot filters [4], TS-FBGs is simple,conomical, highly flexible, and has the advantages of high sensi-ivity as well as immunity from the light source instability, poweructuations, and the uneven power distribution of source spectrumnd micro-bend attenuation [1]. Therefore, over the past severalears, design of STS-FBGs has attracted many interests and severalffective methods have been introduced [5–8]. These methods areasically optimization methods which are based on directly mini-izing the difference between the computed and target reflectivity.

he performances of these algorithms may be affected by the ini-ial guess value of the optimized parameters. An improper startinguess may lead to low efficiency optimizations and the termina-ion of iterations at local minimum, sometimes even with a false

Please cite this article in press as: X. Yu, et al., A hybrid method for desin sensor applications, Optik - Int. J. Light Electron Opt. (2012), http://

esult.In the design of STS-FBGs, two main concerns are linear edges

nd the large bandwidth of the reflectivity spectrum. Compared

∗ Corresponding author.E-mail addresses: yaoyong@hit.edu.cn, yxl-1216@sohu.com (Y. Yao).

030-4026/$ – see front matter. Crown Copyright © 2012 Published by Elsevier GmbH. Attp://dx.doi.org/10.1016/j.ijleo.2012.03.049

with STS-FBGs, right-angled triangular spectrum fiber Bragg grat-ings (RTS-FBGs) have potential capability to have bigger effectivebandwidth so that they can be practically more useful when appliedto readout wavelength signal in sensor system. But it is difficult tosynthesize RTS-FBGs because their coupling coefficient is generallycomplex value and quite complicated. In Refs. [7,8], RTS-FBGs areconsidered and designed by combining the discrete layer peeling(DLP) algorithm and an optimal method. However, the recon-structed right-angled triangular spectrums have large deviationalthough the left edge of triangular spectrum is pretty linear. InRefs. [9,10], the DLP algorithm and the DLP algorithm incorporatedwith a traditional Gerchberg–Saxton algorithm are introduced todesign the RTS-FBGs, respectively, but the edges of the triangularspectrums have poorer linear. In this paper, by combining the DLPmethod and the quantum-behaved particle swarm optimization(QPSO) algorithm [11], we propose a method to design RTS-FBGs.The QPSO algorithm is developed from the particle swarm opti-mization (PSO) [12] which has been successfully used to designgeneral FBGs [13]. In contrast to PSO algorithm, the QPSO algorithmhas no velocity vector for the particles and the whole feasible solu-tion space of complex number can be searched for the optimizedcomplex coupling coefficient, with only one adjustable param-eter. Moreover, the iterative equation of this algorithm is very

igning fiber Bragg gratings with right-angled triangular spectrumdx.doi.org/10.1016/j.ijleo.2012.03.049

different from that of PSO. That is to say, the QPSO algorithmis very simple and easy to implement. We apply the proposedmethod to design a single-channel and a multi-channel RTS-FBG,respectively.

ll rights reserved.

ING ModelI

2 ik xxx

2

2a

bttrp

T

w

�

agw

|

mct

�

r

wti

ttamttdmfgwrommgfdctdlTtLcbtQ

ARTICLEJLEO-52240; No. of Pages 4

X. Yu et al. / Opt

. Theory of the proposed method

.1. The principle of the discrete layer peeling algorithm andccuracy analysis

In the DLP method [14], the synthesis of the grating is facilitatedy discretizing the grating into D complex reflectors satisfying theime domain causality. The transfer matrix T of each discretiza-ion layer of thickness �l (�l = L/D, where L is the FBG length) isepresented by the product of the reflection matrix T�d and theropagation matrix T�l:

= T�l × T�d(1)

here the dth complex reflection coefficient �d is determined by

d = − tanh(|qd|�l)qd

∗

|qd| (2)

nd the complex coupling coefficient qd (and its complex conju-ate qd*) is related with the index modulation ınd and the Braggavelength �B by the following equation:

qd| = �ınd

�B(3)

The coupling coefficient at the front of the grating can be deter-ined only from the leading edge of the impulse response. The

omplex reflection coefficient �d is the Fourier transformation ofhe reflection rd:

d = F−1[rd(ı)]t=0 (4)

d+1(ı) = exp(−i2ı�l)rd(ı) − �d

1 − �d∗rd(ı)

(5)

here ı (ı = 2�neff/�B − 2�neff/�) is the wavelength detuning andhe detuning windows is ıω (ıω = �/�l), neff is the effective modalndex of the fiber core.

The above DLP algorithm is simple and effective to synthesizehe FBGs. However, its applicability is severely restricted becausehe synthesis problem by this method becomes ill-conditioned at

high level of noise and incorrect in conditions of deep indexodulation [15–18], At the same time, this method has also

he computational error which arises in the model discretiza-ion. The coupling coefficient has a comb shape inside a givenetuning window. In contrast, in the ordinary transfer matrixethod, the coupling coefficient is assumed to be piecewise uni-

orm, i.e., the coupling coefficient has a stair-like shape. For strongratings, because the Fourier relation breaks down and the piece-ise uniform approximation models more accurately the multiple

eflections inside each section. In the principal range the spectrumf the piecewise uniform approximation will be the exact spectrumultiplied by a (wide) sinc-function, and hence there will also beore inaccuracies there than the discretization model. In the weak

rating limit (Fourier), the situation is opposite. It is clear that theormer discretization model is most accurate in the detuning win-ow. This can be explained as the fact that provided the couplingoefficient is sampled with a sufficient number of points, accordingo the Nyquist theorem, the spectrum is represented exactly in theetuning window. But, for a given detuning window and grating

ength, the discretization layer can be expressed as �l = �/ıω = L/D.hat is to say, the number of the sampling points is identified andhe error may emerge, especially for the grating with shorter length. Therefore, in the paper, firstly an initial guess of the complex

Please cite this article in press as: X. Yu, et al., A hybrid method for desin sensor applications, Optik - Int. J. Light Electron Opt. (2012), http://

oupling coefficient in the discretization model can be obtainedy the DLP algorithm. Then, in the piecewise uniform approxima-ion model, the complex coupling coefficients are optimized by thePSO algorithm

PRESS(2012) xxx–xxx

2.2. The optimization of the grating parameters usingquantum-behaved particle swarm optimization algorithm

The grating parameters from a target reflectivity can be viewedas an optimal problem of QPSO. In our method, for certain gratinglength L, the complex coupling coefficient q(z) · (z ∈ (0, L)) obtainedby the DLP algorithm is what should be optimized. For such anoptimal problem, it is very critical to select an appropriate objectivefunction which is used to represent how good a particular solutionis. In the grating synthesis problem it is customary to use the meansquared error between the computed and the target reflectivity asobjective function

F = 1M

M∑j=1

[Rcorrupted(�j) − Rtarget(�j)]2 (6)

where Rcorrupted and Rtarget are the computed and target reflectiv-ity, respectively, and �j is the jth discrete sampling wavelengthuniformly distributed over the total samples M of the evaluatedreflectivity.

Before we proceed, let us first outline the QPSO method. In theQPSO algorithm, the particles move according to the following iter-ative equations [11]

ped(n) = ϕd(n)Ped(n) + [1 − ϕd(n)]Pgd(n) (7)

Mbest(n) = 1S

S∑e=1

Ped(n) (8)

xid(n + 1) = pid(n) ± ˛(n)|Mbest(n) − xid(n)| ln[

1uid(n)

](9)

where ped is the attraction point of the particle swarm, Ped andPgd the best position of the eth the particle and the position ofthe best particle among all the particles in a D-dimensional space,respectively; Mbest is defined as the mean value of all particles’Ped, and ued, ϕd are random number distributed uniformly on [0, 1],respectively; S is the total particle number, and n denotes the itera-tive generation. The contraction-expansion coefficient ̨ is the onlyadjustable parameter used to control the convergence speed of thealgorithm and it decreases linearly from 0.3 to 0.1 as the iterationgrows.

Provided that ped in Eq. (7) falls into a local optimal solution, itwill lead to local convergence of the QPSO algorithm. In order toavoid that, we introduce the Gaussian mutation operation to thePed as

Ped(n) = Ped(n) + N(0, ς)(1 + i) (10)

where N(0, ς) is among a set of normal random numbers with mean0 and standard deviation ς, where ς denotes a positive constantless than 0.1.

The ways to update Ped and Pgd are identical to the correspondingones in a generic PSO [12], namely

Ped(n + 1) ={

xed(n + 1) (F[xed(n + 1)] < F[Ped(n)])Ped(n) (F[xed(n + 1)] ≥ F[Ped(n)])

(11)

Pgd(n + 1) = arg min1≤e≤S

{F[Ped(n)]} (12)

where F is the objective function used to evaluate each particle inthe swarm, as in Eq. (6).

igning fiber Bragg gratings with right-angled triangular spectrumdx.doi.org/10.1016/j.ijleo.2012.03.049

The QPSO algorithm is briefly summarized as follows:

Step 1: Initialize a particle population with random posi-tions xed(0) (xed(0) ∈ (q, q + N(0, ς)(1 + i))). Each particle’s

IN PRESSG ModelI

ik xxx (2012) xxx–xxx 3

S

hsit

0 0.2 0.4 0.6 0.8 1−40

−30

−20

−10

0

10

20

30

Grating position (cm)

Re.

q (

1/cm

)

0 0.2 0.4 0.6 0.8 1−20

−10

0

10

20

30

Grating position (cm)

Im.q

(1/

cm)

Reconstructed by the DLP methodOptimized by the proposed method

Reconstructed by the DLP methodOptimized by the proposed method

(a)

(b)

ARTICLEJLEO-52240; No. of Pages 4

X. Yu et al. / Opt

position xed(0) represents the dth complex coupling coef-ficients qd of the D-uniform sections of the FBG.

Step 2: Evaluate the objective function value of each particle bysubstituting its position into Eq. (6) and assign the parti-cle’s best position to Ped(0). Identify the best among Ped(0)as Pgd(0). According to Eq. (7), ped(0) is calculated, andMbest(0) is also obtained by Eq. (8).

Step 3: Updating the position of the particle population xed(n) byEq. (9).

Step 4: Evaluate the objective function value of each particle andassign the particle’s best position to Ped(n). Identify thebest among Ped(n) as Pgd(n).

Step 5: The particle’s best position Ped(n) is operated with themutation operation and its objective function value iscompared with the one before the mutation operation.If its objective function value is better, then update it andthe objective function value.

Step 6: Calculate Ped(n) and Mbest(n) by Eqs. (7) and (8), respec-tively.

Step 7: For each particle, the objective function value of the par-ticle’s current position is calculated. If it is better thanthe previous position, then update it and the objectivefunction value.

Step 8: Identify the best among Ped(n) as the global best positionof the current particle population Pgd(n + 1) and update theobjective function value.

Step 9: Compare the global best position of the current particlepopulation with the previous global best position, if it isbetter, update it and its objective function value.

tep 10: Repeat steps 3–9 until the assigned objective functionvalue is achieved.

Based on the above QPSO algorithm and the DLP method, aybrid method for the design of the RTS-FBGs is developed, as

Please cite this article in press as: X. Yu, et al., A hybrid method for desin sensor applications, Optik - Int. J. Light Electron Opt. (2012), http://

ketched in Fig. 1. The transfer matrix method is applied to ver-fy the final optimized complex coupling coefficients by calculatinghe reflectivity spectrum [19].

d=1, is tar get reflectivity

Obt ain from Eq.(4 )

Criteria is sa tisfie d?

( )dr δ

dρ

Obt ain from Eq.(5 )1dr +

d=d+1

if

Obtain from Eq.(2)dq

d D≤

Minimize objective function F by the QPSO algo rithm

End

The optimized coupling coeffice nt

TrueFalse

TrueFalse

dq

Fig. 1. Flowchart of the proposed method.

Fig. 2. The real parts and imaginary parts of the coupling coefficients reconstructedby the DLP method and optimized by the proposed method.

3. Numerical examples and discussion

Two designs of RTS-FBGs are investigated. The first design con-sists of a 1 cm long single-channel grating with a (broad) bandwidthof 4 nm, the Bragg wavelength is �B = 1550 nm. The number ofsections was fixed to D = 100. The total samples of the evaluatedreflectivity are M = 100 and the effective modal index of the fibercore is neff = 1.5. It is noteworthy that the initial complex cou-pling coefficient obtained by the DLP method can be seen as aninseparable variable and immediately optimized by the QPSO algo-rithm. Instead, the real part and imaginary part of initial complexcoupling coefficient are optimized, respectively [7,8]. Fig. 2 showsthe complex coupling coefficients for a target spectrum by the pro-

igning fiber Bragg gratings with right-angled triangular spectrumdx.doi.org/10.1016/j.ijleo.2012.03.049

posed method and the DLP method, respectively. As analyzed inSection 2, for the DLP method, the computational error of the realpart and imaginary part of complex coupling coefficient arises inthe model discretization and is transferred by Eqs. (4) and (5).

1546 1548 1550 1552 15540

0.2

0.4

0.6

0.8

1

1.2

1.4

Wavelength (nm)

Ref

lect

ivity

Reconstructed by the DLP methodOptimized by the proposed methodThe target reflectivity

Fig. 3. The reflectivity spectrums of the RTS-FBGs with 4 nm bandwidth recon-structed by the DLP method and optimized by the proposed method.

ARTICLE ING ModelIJLEO-52240; No. of Pages 4

4 X. Yu et al. / Optik xxx

0 0.5 1 1.5 2−10

−5

0

5

10

15

20

25

Grating position (cm)

Re.

q (1

/cm

)

0 0.5 1 1.5 2

−10

−5

0

5

10

Grating position (cm)

Im.q

(1/

cm)

1548 1549 1550 1551 15520

0.5

1

1.5

Wavelength (nm)

Ref

lect

iity

Optimized by the proposed methodThe target reflectivity

(c)

(b)

(a)

Fpc

Fotihlttae

RpsaaDapsrfmgt

4

mo

[

[

[

[

[

[

[

[

[18] A. Rosenthal, M. Horowitz, Inverse scattering algorithm for reconstructingstrongly reflecting fiber Bragg gratings, IEEE J. Quantum Electron. 39 (2003)1018–1026.

ig. 4. The triple-channel RTS-FBG. Real part (a) and imaginary part (b) of cou-ling coefficient are optimized by the proposed method, (c) the reflectivity spectrumorresponding with coupling coefficient.

or Fig. 2, Fig. 3 depicts their corresponding reflectivity spectrumsbtained by the transfer matrix method. From Fig. 3, it can be seenhat the computed reflectivity spectrum by the proposed methods nearly consistent with the target reflectivity spectrum whichas a triangular optical reflectivity and the wavelength changes

inearly with the reflectivity, while there is an obvious devia-ion for the results by the DLP method. The comparison of thewo computed reflectivity spectrums obtained by the DLP methodnd the proposed method demonstrates that our method is veryffective.

The second design consists of a 2 cm long triple-channelTS-FBGs usually used as a readout device in FBG-based multi-arameter sensor system. A triple-channel RTS-FBG is designeduch that each channel has a width 1 nm, channel spacing 1.5 nm,nd the three different central wavelengths are 1548.5, 1550,nd 1551.5 nm, respectively. The number of sections was fixed to

= 125. The total samples of the evaluated reflectivity are M = 300nd the effective modal index of the fiber core is neff = 1.5. By theroposed method, the complex coupling coefficients are recon-tructed as in Fig. 4(a) and (b), respectively. And the correspondingeflectivity spectrums are plotted in Fig. 4(c). As can be seenrom Fig. 4(c), the computed reflectivity spectrum by the proposed

ethod has good linear edge and is nearly consistent with the tar-et reflectivity spectrum. The excellent agreement demonstrateshe effectiveness of our proposed method.

. Conclusion

Please cite this article in press as: X. Yu, et al., A hybrid method for desin sensor applications, Optik - Int. J. Light Electron Opt. (2012), http://

In summary, we have proposed and demonstrated a newethod combining DLP with the QPSO algorithm for the design

f RTS-FBGs. A mutation operation is introduced to the QPSO

[

PRESS(2012) xxx–xxx

algorithm which can effectively prevent the local convergence.Compared with other optimal method, the whole feasible solu-tion space of complex number can be searched for the complexcoupling coefficient. By the proposed method, we design a single-channel and a multi-channel RTS-FBG, which can be used tomeasure single and multiple physical parameters. In the designedRTS-FBGs, the wavelength increases linearly with reflectivity.Thus the wavelength encoded signal in sensor system can beaccurately interrogated. In contrast to traditional wavelengthreadout devices, the proposed RTS-FBGs are simple and cost-effective, and will have important application in optical sensorsystem.

Acknowledgments

This work was supported in part by the NSFC (Grant Nos.60977034, 61107036 and 11004043) and the SMSTPR (Grant No.JC200903120167A, JC201005260185A), in part by China Postdoc-toral Science Foundation (Grant No. 20110491092). The authorsalso thank Professor Junjun Xiao of Harbin Institute of Tech-nology for assistance with manuscript preparation and helpfuldiscussions.

References

[1] R. Huang, Y.W. Zhou, H.W. Cai, R.H. Qu, Z.J. Fang, A fiber Bragg grating withtriangular spectrum as wavelength readout in sensor systems, Opt. Commun.229 (2004) 197–201.

[2] S.M. Melle, K. Liu, R.M. Measures, A passive wavelength demodulation systemfor guided wave Bragg gratings sensors, IEEE Photon. Technol. Lett. 4 (1992)516–518.

[3] A.D. Kersey, T.A. Berkoff, W.W. Morey, Multiplexed fiber Bragg grating strain-sensor system with a fiber Fabry–Perot wavelength filter, Opt. Lett. 18 (1993)1370–1372.

[4] S.T. Lin, Y.R. Cheng, Wavelength shift determination using a dual-pathheterodyne Mach–Zehnder interferometer, Opt. Commun. 266 (2006)50–54.

[5] J.C.C. Carvalho, M.J. Sousa, C.S.S. Junior, J.C.W.A. Costa, C.R.L. Frances, M.E.V.Segatto, A new acceleration technique for the design of fibre gratings, Opt.Express 14 (2006) 10715–10725.

[6] Y. Gong, A. Lin, X. Hu, L. Wang, X. Liu, Optimal method of designing triangular-spectrum fiber Bragg gratings with low index modulation and chirp-freestructure, J. Opt. Soc. Am. B 26 (2009) 1042–1048.

[7] K. Aksnes, J. Skaar, Design of short fiber Bragg gratings by use of optimization,Appl. Opt. 43 (2004) 2226–2230.

[8] S. Baskar, P.N. Suganthan, N.Q. Ngo, A. Alphones, R.T. Zheng, Design of triangularFBG filter for sensor applications using covariance matrix adapted evolutionalgorithm, Opt. Commun. 260 (2006) 716–722.

[9] L.S. Li, X.Q. Feng, Synthesis of fiber Bragg grating with right-angled triangularspectrum, Chin. Phys. Lett. 27 (2010) 054210.

10] M. Li, J. Hayashi, H. Li, Advanced design of a complex fiber Bragg grating fora multichannel asymmetrical triangular filter, J. Opt. Soc. Am. B 26 (2009)228–234.

11] J. Sun, B. Feng, W.B. Xu, Particle swarm optimization with particles havingquantum behavior, in: IEEE Congress on Evolutionary Computation, Portland,Oregon, USA, 2004, pp. 325–331.

12] J. Kennedy, R.C. Eberhart, Particle swarm optimization, in: IEEE inter-national Conference on Neural Network, Perth, WA, Australia, 1995,pp. 1942–1948.

13] S. Baskar, R.T. Zheng, A. Alphones, N.Q. Ngo, P.N. Suganthan, Particle swarmoptimization for the design of low-dispersion fiber Bragg gratings, IEEE Photon.Technol. Lett. 17 (2005) 615–617.

14] J. Skaar, L. Wang, T. Erdogan, On the synthesis of fiber Bragg gratings by layerpeeling, IEEE J. Quantum Electron. 37 (2001) 165–173.

15] O.V. Belai, L.L. Frumin, E.V. Podivilov, D.A. Shapiro, Reconstruction of thehigh reflectance fiber Bragg grating from noisy data, Laser Phys. 17 (2007)1317–1322.

16] O.V. Belai, L.L. Frumin, E.V. Podivilov, D.A. Shapiro, Inverse scattering problemfor gratings with deep modulation, Laser Phys. 20 (2010) 318–324.

17] J. Skaar, R. Feced, Reconstruction of gratings from noisy reflection data, J. Opt.Soc. Am. A 19 (2002) 2229–2237.

igning fiber Bragg gratings with right-angled triangular spectrumdx.doi.org/10.1016/j.ijleo.2012.03.049

19] T. Erdogan, Fiber grating spectra, J. Lightwave Technol. 15 (1997)1277–1294.