A hybrid method for designing fiber bragg gratings with right angled triangular spectrum in sensor applications

Please cite this article in press as: X. Yu, et al., A hybrid method for designing fiber Bragg gratings with right-angled triangular spectrum
in sensor applications, Optik - Int. J. Light Electron Opt. (2012), http://dx.doi.org/10.1016/j.ijleo.2012.03.049
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IJLEO-52240; No.of Pages4
Optik xxx (2012) xxx–xxx
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A hybrid method for designing fiber Bragg gratings with right-angled triangular
spectrum in sensor applications
Xuelian Yua,b
, Yong Yaoa,∗
, Jiajun Tiana
, Chao Liua
a
College of Electronic and Information Engineering, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, PR China
b
Department of Optics Information Science and Technology, Harbin University of Science and Technology, Harbin 150080, PR China
a r t i c l e i n f o
Article history:
Received 5 November 2011
Accepted 18 March 2012
Keywords:
Fiber Bragg grating
Synthesis
Quantum-behaved particle swarm
optimization
Triangular spectrum
a b s t r a c t
Based on the quantum-behaved particle swarm optimization (QPSO) algorithm and the discrete layer
peeling (DLP) approach, we propose a scheme to design the right-angled triangular spectrum fiber Bragg
gratings (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
to optimize the initial complex coupling coefficients. For the QPSO method, the whole complex solution
space can be searched for the coupling coefficients. Using the proposed method, single- and multi-channel
RTS-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 with
the target spectrum, in contrast to those obtained by existing method. The RTS-FBGs can find potential
applications in wavelength interrogation in optical sensor system.
Crown Copyright © 2012 Published by Elsevier GmbH. All rights reserved.
1. Introduction
Symmetrical triangular spectrum fiber Bragg gratings (STS-
FBGs) is a kind of useful wavelength-interrogation devices in
optical sensor systems [1]. Compared with traditional wavelength-
interrogation devices such as edge filters [2], Mach–Zehnder
interferometers [3], and Fabry–Perot filters [4], TS-FBGs is simple,
economical, highly flexible, and has the advantages of high sensi-
tivity as well as immunity from the light source instability, power
fluctuations, and the uneven power distribution of source spectrum
and micro-bend attenuation [1]. Therefore, over the past several
years, design of STS-FBGs has attracted many interests and several
effective methods have been introduced [5–8]. These methods are
basically optimization methods which are based on directly mini-
mizing the difference between the computed and target reflectivity.
The performances of these algorithms may be affected by the ini-
tial guess value of the optimized parameters. An improper starting
guess may lead to low efficiency optimizations and the termina-
tion of iterations at local minimum, sometimes even with a false
result.
In the design of STS-FBGs, two main concerns are linear edges
and the large bandwidth of the reflectivity spectrum. Compared
∗ Corresponding author.
E-mail addresses: yaoyong@hit.edu.cn, yxl-1216@sohu.com (Y. Yao).
with STS-FBGs, right-angled triangular spectrum fiber Bragg grat-
ings (RTS-FBGs) have potential capability to have bigger effective
bandwidth so that they can be practically more useful when applied
to readout wavelength signal in sensor system. But it is difficult to
synthesize RTS-FBGs because their coupling coefficient is generally
complex value and quite complicated. In Refs. [7,8], RTS-FBGs are
considered and designed by combining the discrete layer peeling
(DLP) algorithm and an optimal method. However, the recon-
structed right-angled triangular spectrums have large deviation
although the left edge of triangular spectrum is pretty linear. In
Refs. [9,10], the DLP algorithm and the DLP algorithm incorporated
with a traditional Gerchberg–Saxton algorithm are introduced to
design the RTS-FBGs, respectively, but the edges of the triangular
spectrums have poorer linear. In this paper, by combining the DLP
method 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 design
general FBGs [13]. In contrast to PSO algorithm, the QPSO algorithm
has no velocity vector for the particles and the whole feasible solu-
tion space of complex number can be searched for the optimized
complex coupling coefficient, with only one adjustable param-
eter. Moreover, the iterative equation of this algorithm is very
different from that of PSO. That is to say, the QPSO algorithm
is very simple and easy to implement. We apply the proposed
method to design a single-channel and a multi-channel RTS-FBG,
respectively.
0030-4026/$ – see front matter. Crown Copyright © 2012 Published by Elsevier GmbH. All rights reserved.
http://dx.doi.org/10.1016/j.ijleo.2012.03.049

2 X. Yu et al. / Optik xxx (2012) xxx–xxx
2. Theory of the proposed method
2.1. The principle of the discrete layer peeling algorithm and
accuracy analysis
In the DLP method [14], the synthesis of the grating is facilitated
by discretizing the grating into D complex reflectors satisfying the
time domain causality. The transfer matrix T of each discretiza-
tion layer of thickness l ( l = L/D, where L is the FBG length) is
represented by the product of the reflection matrix T d and the
propagation matrix T l:
T = T l × T d
(1)
where the dth complex reflection coefficient d is determined by
d = − tanh(|qd| l)
qd
∗
|qd|
(2)
and the complex coupling coefficient qd (and its complex conju-
gate qd*) is related with the index modulation ınd and the Bragg
wavelength B by the following equation:
|qd| =
ınd
B
(3)
The coupling coefficient at the front of the grating can be deter-
mined only from the leading edge of the impulse response. The
complex reflection coefficient d is the Fourier transformation of
the reflection rd:
d = F−1
[rd(ı)]t=0 (4)
rd+1(ı) = exp(−i2ı l)
rd(ı) − d
1 − d
∗rd(ı)
(5)
where ı (ı = 2 neff/ B − 2 neff/ ) is the wavelength detuning and
the detuning windows is ıω (ıω = / l), neff is the effective modal
index of the fiber core.
The above DLP algorithm is simple and effective to synthesize
the FBGs. However, its applicability is severely restricted because
the synthesis problem by this method becomes ill-conditioned at
a high level of noise and incorrect in conditions of deep index
modulation [15–18], At the same time, this method has also
the computational error which arises in the model discretiza-
tion. The coupling coefficient has a comb shape inside a given
detuning window. In contrast, in the ordinary transfer matrix
method, the coupling coefficient is assumed to be piecewise uni-
form, i.e., the coupling coefficient has a stair-like shape. For strong
gratings, because the Fourier relation breaks down and the piece-
wise uniform approximation models more accurately the multiple
reflections inside each section. In the principal range the spectrum
of the piecewise uniform approximation will be the exact spectrum
multiplied by a (wide) sinc-function, and hence there will also be
more inaccuracies there than the discretization model. In the weak
grating limit (Fourier), the situation is opposite. It is clear that the
former discretization model is most accurate in the detuning win-
dow. This can be explained as the fact that provided the coupling
coefficient is sampled with a sufficient number of points, according
to the Nyquist theorem, the spectrum is represented exactly in the
detuning window. But, for a given detuning window and grating
length, the discretization layer can be expressed as l = /ıω = L/D.
That is to say, the number of the sampling points is identified and
the error may emerge, especially for the grating with shorter length
L. Therefore, in the paper, firstly an initial guess of the complex
coupling coefficient in the discretization model can be obtained
by the DLP algorithm. Then, in the piecewise uniform approxima-
tion model, the complex coupling coefficients are optimized by the
QPSO algorithm
2.2. The optimization of the grating parameters using
quantum-behaved particle swarm optimization algorithm
The grating parameters from a target reflectivity can be viewed
as an optimal problem of QPSO. In our method, for certain grating
length L, the complex coupling coefficient q(z) · (z ∈ (0, L)) obtained
by the DLP algorithm is what should be optimized. For such an
optimal problem, it is very critical to select an appropriate objective
function which is used to represent how good a particular solution
is. In the grating synthesis problem it is customary to use the mean
squared error between the computed and the target reflectivity as
objective function
F =
1
M
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 wavelength
uniformly distributed over the total samples M of the evaluated
reflectivity.
Before we proceed, let us first outline the QPSO method. In the
QPSO 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) =
1
S
S
e=1
Ped(n) (8)
xid(n + 1) = pid(n) ± ˛(n)|Mbest(n) − xid(n)| ln
1
uid(n)
(9)
where ped is the attraction point of the particle swarm, Ped and
Pgd the best position of the eth the particle and the position of
the 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 only
adjustable parameter used to control the convergence speed of the
algorithm and it decreases linearly from 0.3 to 0.1 as the iteration
grows.
Provided that ped in Eq. (7) falls into a local optimal solution, it
will lead to local convergence of the QPSO algorithm. In order to
avoid that, we introduce the Gaussian mutation operation to the
Ped as
Ped(n) = Ped(n) + N(0, ς)(1 + i) (10)
where N(0, ς) is among a set of normal random numbers with mean
0 and standard deviation ς, where ς denotes a positive constant
less than 0.1.
The ways to update Ped and Pgd are identical to the corresponding
ones 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 min
1≤e≤S
{F[Ped(n)]} (12)
where F is the objective function used to evaluate each particle in
the swarm, as in Eq. (6).
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

X. Yu et al. / Optik xxx (2012) xxx–xxx 3
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 by
substituting 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, and
Mbest(0) is also obtained by Eq. (8).
Step 3: Updating the position of the particle population xed(n) by
Eq. (9).
Step 4: Evaluate the objective function value of each particle and
assign the particle’s best position to Ped(n). Identify the
best among Ped(n) as Pgd(n).
Step 5: The particle’s best position Ped(n) is operated with the
mutation operation and its objective function value is
compared with the one before the mutation operation.
If its objective function value is better, then update it and
the 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 than
the previous position, then update it and the objective
function value.
Step 8: Identify the best among Ped(n) as the global best position
of the current particle population Pgd(n + 1) and update the
objective function value.
Step 9: Compare the global best position of the current particle
population with the previous global best position, if it is
better, update it and its objective function value.
Step 10: Repeat steps 3–9 until the assigned objective function
value is achieved.
Based on the above QPSO algorithm and the DLP method, a
hybrid method for the design of the RTS-FBGs is developed, as
sketched in Fig. 1. The transfer matrix method is applied to ver-
ify the final optimized complex coupling coefficients by calculating
the reflectivity spectrum [19].
d=1, is target reflectivity
Obtain from Eq.(4)
Criteria is satisfied?
( )dr δ
dρ
Obtain from Eq.(5)1dr +
d=d+1
if
Obtain from Eq.(2)dq
d D≤
Minimize objective function F by the
QPSO algorithm
End
The optimized coupling coefficent
True
False
TrueFalse
dq
Fig. 1. Flowchart of the proposed method.
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
Im.q(1/cm)
Reconstructed by the DLP method
Optimized by the proposed method
(a)
(b)
Fig. 2. The real parts and imaginary parts of the coupling coefficients reconstructed
by 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) bandwidth
of 4 nm, the Bragg wavelength is B = 1550 nm. The number of
sections was fixed to D = 100. The total samples of the evaluated
reflectivity are M = 100 and the effective modal index of the fiber
core is neff = 1.5. It is noteworthy that the initial complex cou-
pling coefficient obtained by the DLP method can be seen as an
inseparable variable and immediately optimized by the QPSO algo-
rithm. Instead, the real part and imaginary part of initial complex
coupling coefficient are optimized, respectively [7,8]. Fig. 2 shows
the complex coupling coefficients for a target spectrum by the pro-
posed method and the DLP method, respectively. As analyzed in
Section 2, for the DLP method, the computational error of the real
part and imaginary part of complex coupling coefficient arises in
the model discretization and is transferred by Eqs. (4) and (5).
1546 1548 1550 1552 1554
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Wavelength (nm)
Reflectivity
The 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.

4 X. Yu et al. / Optik xxx (2012) xxx–xxx
0 0.5 1 1.5 2
−10
−5
0
5
10
15
20
25
Re.q(1/cm)
0 0.5 1 1.5 2
−10
−5
0
5
10
Im.q(1/cm)
1548 1549 1550 1551 1552
0
0.5
1
1.5
Wavelength (nm)
Reflectiity
The target reflectivity
(c)
(b)
(a)
Fig. 4. The triple-channel RTS-FBG. Real part (a) and imaginary part (b) of cou-
pling coefficient are optimized by the proposed method, (c) the reflectivity spectrum
corresponding with coupling coefficient.
For Fig. 2, Fig. 3 depicts their corresponding reflectivity spectrums
obtained by the transfer matrix method. From Fig. 3, it can be seen
that the computed reflectivity spectrum by the proposed method
is nearly consistent with the target reflectivity spectrum which
has a triangular optical reflectivity and the wavelength changes
linearly with the reflectivity, while there is an obvious devia-
tion for the results by the DLP method. The comparison of the
two computed reflectivity spectrums obtained by the DLP method
and the proposed method demonstrates that our method is very
effective.
The second design consists of a 2 cm long triple-channel
RTS-FBGs usually used as a readout device in FBG-based multi-
parameter sensor system. A triple-channel RTS-FBG 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. The number of sections was fixed to
D = 125. The total samples of the evaluated reflectivity are M = 300
and the effective modal index of the fiber core is neff = 1.5. By the
proposed method, the complex coupling coefficients are recon-
structed as in Fig. 4(a) and (b), respectively. And the corresponding
reflectivity spectrums are plotted in Fig. 4(c). As can be seen
from Fig. 4(c), the computed reflectivity spectrum by the proposed
method has good linear edge and is nearly consistent with the tar-
get reflectivity spectrum. The excellent agreement demonstrates
the effectiveness of our proposed method.
4. Conclusion
In summary, we have proposed and demonstrated a new
method combining DLP with the QPSO algorithm for the design
of RTS-FBGs. A mutation operation is introduced to the QPSO
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 complex
coupling coefficient. By the proposed method, we design a single-
channel and a multi-channel RTS-FBG, which can be used to
measure single and multiple physical parameters. In the designed
RTS-FBGs, the wavelength increases linearly with reflectivity.
Thus the wavelength encoded signal in sensor system can be
accurately interrogated. In contrast to traditional wavelength
readout devices, the proposed RTS-FBGs are simple and cost-
effective, and will have important application in optical sensor
system.
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 authors
also thank Professor Junjun Xiao of Harbin Institute of Tech-
nology for assistance with manuscript preparation and helpful
discussions.
References
[1] R. Huang, Y.W. Zhou, H.W. Cai, R.H. Qu, Z.J. Fang, A fiber Bragg grating with
triangular 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 system
for 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-path
heterodyne 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-free
structure, 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 triangular
FBG filter for sensor applications using covariance matrix adapted evolution
algorithm, Opt. Commun. 260 (2006) 716–722.
[9] L.S. Li, X.Q. Feng, Synthesis of fiber Bragg grating with right-angled triangular
spectrum, Chin. Phys. Lett. 27 (2010) 054210.
[10] M. Li, J. Hayashi, H. Li, Advanced design of a complex fiber Bragg grating for
a 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 having
quantum 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 swarm
optimization 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 layer
peeling, IEEE J. Quantum Electron. 37 (2001) 165–173.
[15] O.V. Belai, L.L. Frumin, E.V. Podivilov, D.A. Shapiro, Reconstruction of the
high 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 problem
for 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.
[18] A. Rosenthal, M. Horowitz, Inverse scattering algorithm for reconstructing
strongly reflecting fiber Bragg gratings, IEEE J. Quantum Electron. 39 (2003)
1018–1026.
[19] T. Erdogan, Fiber grating spectra, J. Lightwave Technol. 15 (1997)
1277–1294.

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