JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 25, NO. 3, MARCH 2007 799

Optimization of Fiber Bragg Gratings Usinga Hybrid Optimization Algorithm

Nam Quoc Ngo, Rui Tao Zheng, J. H. Ng, S. C. Tjin, Member, IEEE, and L. N. Binh

Abstract—A new hybrid optimization algorithm is proposed forthe design of a fiber Bragg grating (FBG) with complex charac-teristics. The hybrid algorithm is a two-tier search that employsa global optimization algorithm (i.e., the Staged Continuous TabuSearch (SCTS) algorithm) and a local optimization method (i.e.,the Quasi-Newton method). First, the SCTS global optimizationalgorithm is used to find a “promising” FBG structure that hasa spectral response as close as possible to the targeted spectralresponse. Then, a local optimization method, namely, the Quasi-Newton method, is applied to further optimize the promisingFBG structure obtained from the SCTS algorithm to arrive at atargeted spectral response. To demonstrate the effectiveness of themethod, the design and fabrication of an optical bandpass filterare presented.

Index Terms—Fiber Bragg grating (FBG), hybrid optimization,Staged Continuous Tabu Search (SCTS).

I. INTRODUCTION

F IBER BRAGG gratings (FBGs) have been widelyemployed as key components in wavelength-division

multiplexed (WDM) systems. The complex spectrum of thegrating (i.e., spectral response and phase response) can be foundnumerically if the structure of the grating is given [1]. In someapplications, the structure of a grating must be found fromits complex spectrum. This problem of calculating the gratingprofile from its complex spectrum is usually referred to as aninverse scattering problem.

Several synthesis methods for solving the inverse scatteringproblems in FBGs have been demonstrated [2]–[10]. One so-lution is the differential inverse scattering method, which isalso referred to as the layer-peeling method [2]–[5]; it has beenapplied to the design of several types of FBGs [11], but thedesigned profiles of the FBGs (e.g., index modulation profile)are complex, making practical realization difficult. Moreover,when specifying ideal filter characteristics, it is desirable tohave a weighting mechanism that can be tailored to adjust thedifferent target requirements of the filter responses; however,the layer-peeling algorithm cannot support such a mechanismin a satisfactory way.

Manuscript received July 24, 2005; revised September 15, 2006.N. Q. Ngo, J. H. Ng, and S. C. Tjin are with the Photonics Research Cen-

ter, School of Electrical and Electronic Engineering, Nanyang TechnologicalUniversity, Singapore 639798 (e-mail: eqnngo@ntu.edu.sg).

R. T. Zheng is with the Fiber Optics Product Division, Avago Tech-nologies Manufacturing (Singapore) Pte. Ltd., Singapore 768923 (e-mail:ruitao@pmail.ntu.edu.sg).

L. N. Binh is with the Department of Electrical and Computer SystemsEngineering, Monash University, Melbourne, Vic. 3168, Australia.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JLT.2006.889703

To overcome these difficulties, optimization techniques areone of the promising solutions. When the optimization tech-niques were applied to the synthesis of FBGs [6]–[9], thesynthesis problems were formulated as nonlinear objectivefunctions. The optimized solution of the grating design was ob-tained by applying the optimization algorithm to find the globaloptimum of the objective function. Compared with the synthe-sis methods described earlier [2]–[5], optimization techniquescan facilitate the task of weighting different requirements tothe spectral responses of the filter [6]. Another advantage isthat the results obtained by the optimization method are morepractical because additional constraints can be imposed tosuit the fabrication conditions. Due to the multimodal and ill-conditioned character of those objective functions formulatedby different synthesis problems of FBGs, it is difficult to solvethese problems with traditional optimization algorithms. In thework of Plougmann and Kristensen [10], a local optimizationmethod, namely, the Levenberg–Marquardt algorithm, was em-ployed to solve the synthesis problem of grating. To improve theperformance of the Levenberg–Marquardt method, the Fouriertransform technique was used to obtain an initial solutionas the starting point for the Levenberg–Marquardt method.The Levenberg–Marquardt method converges relatively fast;however, because no constraints can be imposed on the indexmodulation depth (of the grating’s index profile to be opti-mized) during the Fourier transform process, the final optimizedgrating profile may not be easily realized in practice. In gen-eral, it is important to apply global optimization algorithms tothe synthesis of FBGs to ensure that a global optimum canbe obtained. However, convergence of a global optimizationalgorithm is normally not as good as that of the traditional localoptimization algorithm; thus, a global optimization algorithmmay not produce the best design when it is applied to thesynthesis of FBGs. To improve the convergence of a globaloptimization algorithm, a hybrid algorithm that employs aglobal optimization algorithm and a local optimization algo-rithm could be a better choice [13].

Note in [9] that the standard Tabu search process was em-ployed as the global optimization process. The standard Tabusearch algorithm [9] has several disadvantages, as described indetail in [12], and the disadvantages are summarized as follows:When the number of variables (number of parameters to be op-timized) increases, the efficiency (such as optimization speed)of the standard Tabu search algorithm will decrease, and it willbecome worse when solving problems with large dimensions orwith a large number of variables. To further improve the hybridmethod presented in [9], Zheng et al. [12] proposed a globaloptimization algorithm, namely the Staged Continuous Tabu

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800 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 25, NO. 3, MARCH 2007

Fig. 1. Schematic diagram of the proposed hybrid optimization method.

Search (SCTS) algorithm, and its efficiency and robustnesswere demonstrated using a set of test functions whose globaloptima are known. To further improve the SCTS algorithm[12], in this paper, a hybrid method is proposed for the syn-thesis of FBGs, and the method combines the SCTS algorithm[12] with another local optimization algorithm, namely theQuasi-Newton method. The proposed hybrid method has betterconvergence of the solution than the SCTS global optimiza-tion algorithm [12]. To demonstrate the effectiveness of theproposed hybrid method, an FBG-based bandpass filter wasdesigned and fabricated.

This paper is organized as follows: Section II presents detailsof the proposed hybrid method. Section III compares the FBG-based bandpass filter design using the hybrid method with thedesign obtained using the SCTS algorithm. Section IV presentsthe fabrication of the hybrid-optimized FBG bandpass filter.The main conclusions are described in Section V.

II. ALGORITHM DESCRIPTION

Fig. 1 shows the flowchart of the proposed hybrid optimiza-tion algorithm used for the synthesis of FBGs. It can be seenthat the first step involves the use of the global optimizationtechnique (the SCTS algorithm) to optimize the cascaded uni-form grating sections. Then, the second step makes use of alocal optimization algorithm (the Quasi-Newton method) tofurther optimize the index modulation profile obtained from thefirst step based on the cascading of the uniform FBGs.

Zheng et al. [12] have already presented the synthesis ofFBG-based bandpass filter using the SCTS algorithm. In thispaper, the first step of the hybrid method is the same as thatpresented in [12]. Before the synthesis process, the FBG lengthis divided into M points placed at equal intervals from one an-other, i.e., the index modulation function ∆nac(z) is sampled atthe discrete point zj , where j = 1, 2, . . . ,M . For every interval,the index modulation can be assumed to be constant. Thus, thetransfer matrix method can be applied to calculate the complexspectral response of the whole grating if the grating profile isknown. Thus, the error function for the index modulation of the

whole grating−−−→∆nac = [∆nac,1,∆nac,2, . . . ,∆nac,M ], can be

defined as

error(−−−→∆nac) =∑

j∈window

WRj ×

√∣∣∣Rj(−−−→∆nac) − Rtarget

j

∣∣∣(1)

where Rj(−−−→∆nac) is the calculated reflectivity of the jth wave-

length (the details involved in calculating the reflectivity canbe found in [1]), WR

j is the weight parameter of the jth

wavelength for the reflectivity, and Rtargetj is the target re-

flectivity of the jth wavelength. The SCTS algorithm is usedto produce a “promising” FBG structure by searching for theminimum of (1). Because of the global optimization propertyof the SCTS algorithm (which was demonstrated in [12]),the SCTS process can find the promising structure from arandomly selected initial FBG structure. After the promisingFBG structure is obtained from the SCTS process (the globaloptimization process), a local optimization algorithm is used tofurther optimize the FBG structure starting from this promisingindex modulation profile. Here, the Quasi-Newton method isemployed as a local optimization algorithm to further searchfor the minimum of (1). The Quasi-Newton method, which hashigh efficiency in solving multimode nonlinear optimizationproblems, has become a standard local optimization method,and its detailed description can be found in [14].

III. DESIGN OF BANDPASS OPTICAL FILTER

In this section, the hybrid optimization method is applied tothe design of a bandpass optical filter. The desired reflectivespectrum of an FBG-based bandpass optical filter with 0.2 nm(or 25 GHz in the 1550-nm window bandwidth) is given by

Rtargetλ =

{1; 1549.9 nm ≤ λ ≤ 1550.1 nm0; λ < 1549.9 nm and λ > 1550.1 nm

. (2)

In this bandpass filter design, the resolution of the wavelengthused is 0.01 nm, and the phase response of the filter is notconsidered here. The beam size of the UV exposure system isabout 0.5 mm, which corresponds to the minimum allowablesubgrating length in order to produce the FBG with an indexprofile that closely matches the desired one. Here, the totallength of the grating used is 20 mm, and the total number ofthe piecewise uniform FBGs is 40 (i.e., the subgrating lengthis 0.5 mm). The values of each element of

−−−→∆nac are set as[0, 0.0002].

Fig. 2(a) shows the hybrid-optimized FBG index modulationprofile of the FBG-based bandpass filter obtained by the hybridoptimization method. For comparison, the index modulationprofile obtained in the first stage of the hybrid method (i.e.,the SCTS process) is illustrated in Fig. 2(b). As previouslymentioned, this profile shown in Fig. 2(b) is used as the promis-ing structure in the second stage of the hybrid method. As acomparison, the profile of a sine-apodized FBG with a gratinglength of 20 mm is also shown in Fig. 2(b). The correspondingreflective spectra of the three designs of the FBG-based band-pass filters illustrated in Fig. 2(a) and (b) are shown in Fig. 3.

NGO et al.: OPTIMIZATION OF FIBER BRAGG GRATINGS USING A HYBRID OPTIMIZATION ALGORITHM 801

Fig. 2. (a) Optimized FBG index modulation profile of an optical bandpassfilter designed by the hybrid optimization method. (b) Solid line is the promis-ing index modulation profile of an optical bandpass filter obtained by the firststage of hybrid method (i.e., the SCTS process), and the dashed line is the indexmodulation profile of a sine-apodized FBG (divided into 40 sections).

Fig. 3 shows that the performance of the SCTS-optimizedFBG (i.e., the FBG design obtained by the SCTS algorithm)is better than that of the standard sine-apodized FBG becausethe spectrum of the SCTS-optimized FBG (the dashed line inFig. 3) is more squared than the spectrum of the standard sine-apodized FBG (the dotted line in Fig. 3), and their sidelobesare nearly of the same level (about −25 dB). The hybrid-optimized FBG shows the best performance in the reflectivespectrum. The spectrum of the hybrid-optimized FBG (the solidline in Fig. 3) is the steepest compared to the two other spectra,and the sidelobes are suppressed to as low as −30 dB. It canthus be concluded that the two-stage hybrid algorithm is moreefficient than the single-stage SCTS process. The results alsodemonstrate that the hybrid method has better convergence thanthe single SCTS algorithm.

Fig. 3. Reflective spectra corresponding to the three index modulation profilesshown in Fig. 2(a) and (b). The solid line is the reflective spectrum of ahybrid-optimized FBG-based bandpass filter. The dashed line is the reflectivespectrum of an SCTS-optimized FBG-based bandpass filter (i.e., the first stageof the hybrid method). The dotted line is the reflective spectrum of a sine-apodized FBG-based bandpass filter with the same grating length as the twoother optimized filters.

Fig. 4. Measured reflective spectrum of a 20-mm-long hybrid-optimizedFBG-based bandpass filter (solid line) and measured reflective spectrum of auniform FBG-based bandpass filter with the same length (dashed line).

IV. FABRICATION OF THE OPTIMIZED GRATING

Asseh et al. [15] have proposed a technique for the fabrica-tion of long gratings with complex profiles, in which a largenumber of small subgratings were exposed in sequence by UVpulses. Each subgrating has a few hundred periods. Thus, thedepth of the index modulation of each subgrating can be tunedby adjusting the offset of the fiber dithering away from thephase mask. That is, if the offset of the fiber dithering is half ofthe grating period, the index modulation will be completely av-eraged out (i.e., no index modulation). Now, if there is no offsetof the fiber dithering from the phase mask, the index modulationwill have the maximum value. Using this method, one shouldbe able to fabricate an FBG with an optimized profile obtainedfrom the hybrid-optimized method, as shown in Fig. 2(a). Themeasured reflective spectrum is illustrated in Fig. 4. In Fig. 4,

802 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 25, NO. 3, MARCH 2007

the solid line is the measured spectrum of a fabricated 20-mm-long FBG-based bandpass filter with the optimized profile ob-tained using the hybrid method [see Fig. 2(a)]. As a comparison,a 20-mm-long uniform FBG was also fabricated and measured,and its reflective spectrum is shown as the dashed line in Fig. 4.Compared with the spectrum of the uniform FBG, the hybrid-optimized FBG has a more squared spectrum with sidelobesless than −20 dB. The sidelobe suppression level is higherthan the theoretical value of −30 dB (see Fig. 3), probablydue to fabrication errors. The possible fabrication errors arethe positioning error of the translation stage, the fluctuationof the UV laser power, and some possible dirty spots on thephase mask. The procedures that could be taken to improve thefabrication accuracy are listed as follows: 1) use more precisepositioning stage; 2) improve the stability of the UV laserpower; and (3) use a cleaner phase mask.

V. CONCLUSION

A novel synthesis method of FBGs using a hybrid optimiza-tion algorithm has been presented. The effectiveness of theproposed method has been demonstrated by synthesizing andfabricating a bandpass optical filter with 25-GHz bandwidth.Note that the grating phase can be easily included in the list ofparameters to be optimized. For example, the index modulationvalues [which are denoted as

−−−→∆nac; see (1)] to be optimized canalso take negative values, where a negative index modulationrepresents a π-phase shift inserted into a grating section. Thus,the proposed hybrid synthesis method of FBGs can be furtherdeveloped into a powerful toolbox for a variety of fiber gratingdesigns.

ACKNOWLEDGMENT

The authors would like to thank the reviewers for theirconstructive criticism and useful suggestions for improving themanuscript.

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[5] A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for recon-structing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Elec-tron., vol. 39, no. 8, pp. 1018–1026, Aug. 2003.

[6] J. Skaar and K. M. Risvik, “A genetic algorithm for the inverse problemin synthesis of fiber gratings,” J. Lightw. Technol., vol. 16, no. 10,pp. 1928–1932, Oct. 1998.

[7] N. Q. Ngo, R. T. Zheng, P. Shum, Y. H. Lee, S. C. Tjin, and S. Y. Li, “Tabusearch synthesis of cascaded fiber Bragg gratings for linear phase filters,”Opt. Commun., vol. 241, no. 1–3, pp. 79–85, Nov. 2004.

[8] S. Baskar, R. T. Zheng, A. Alphones, N. Q. Ngo, and P. N. Suganthan,“Particle swarm optimization for the design of low-dispersion fiber Bragggratings,” IEEE Photon. Technol. Lett., vol. 17, no. 3, pp. 615–617,Mar. 2005.

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J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 21, no. 12, pp. 2399–2405,Dec. 2004.

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[12] R. T. Zheng, N. Q. Ngo, P. Shum, S. C. Tjin, and L. N. Binh, “A stagedcontinuous Tabu search algorithm for the global optimization and itsapplications to the design of fiber Bragg gratings,” Comput. Optim. Appl.,vol. 30, no. 3, pp. 319–335, Mar. 2005.

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[15] A. Asseh, H. Storoy, B. E. Sahlgren, S. Sandgren, and R. Stubbe, “Awriting technique for long fiber Bragg gratings with complex reflectivityprofiles,” J. Lightw. Technol., vol. 15, no. 8, pp. 1419–1423, Aug. 1997.

Nam Quoc Ngo received the B.Eng. and Ph.D. de-grees in electrical and computer systems engineeringfrom Monash University, Melbourne, Vic., Australia,in 1992 and 1998, respectively. His doctoral workinvolved the use of digital signal processing tech-niques for the design of several novel fiber-opticsignal processors and integrated optical devices forapplication in optical computing and optical commu-nication systems.

From July 1997 to July 2000, he was a Lecturer inthe School of Microelectronic Engineering, Griffith

University, Brisbane, Qld., Australia, where he was the founding Director ofthe Optical Communications Research Laboratory. Since July 2000, he has beenwith the Photonics Research Center, School of Electrical and Electronic Engi-neering, Nanyang Technological University, Singapore, where he is currentlyan Associate Professor. He has published more than 90 international journalpapers and more than 40 conference papers in these areas. His current researchinterests include the design and development of waveguide-based devices forcommunication systems and fiber-based and fiber-Bragg-grating-based devicesfor communication systems, sensor networks, and RF photonics networks.

Prof. Ngo is a Reviewer for a number of international journals. He was arecipient of two awards for outstanding contributions for his Ph.D. dissertation.

Rui Tao Zheng received the B.Eng. degree from Tianjian University, Tianjin,China, in 1995, the M.Eng. degree from Shanghai Institute of Optics andFine Mechanics, Shanghai, China, in 1998, and the Ph.D. degree in photonicsfrom Nanyang Technological University, Singapore, in 2004. His doctoralwork involved the development of new optimization techniques and the use ofoptimization techniques for the design and development of several novel fiber-grating-based optical filters for application in optical communication systemsand optical fiber sensor systems.

From October 2004 to October 2005, he was a Research Associate withthe School of Electrical and Electronic Engineering, Nanyang TechnologicalUniversity. Since March 2006, he has been with the Fiber Optics ProductDivision, Avago Technologies Manufacturing (Singapore) Pte. Ltd., Singapore.His current research interests include the design and development of fiber-basedand fiber-Bragg-grating-based devices for communication systems, sensor net-works, nonimage optics, and optical thin films.

J. H. Ng, photograph and biography not available at the time of publication.

S. C. Tjin (M’03), photograph and biography not available at the time ofpublication.

L. N. Binh, photograph and biography not available at the time of publication.