IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 17, NO. 3, MARCH 2005 615 Particle Swarm Optimization for the Design of Low-Dispersion Fiber Bragg Gratings S. Baskar, R. T. Zheng, A. Alphones, Senior Member, IEEE, N. Q. Ngo, and P. N. Suganthan Abstract—We propose a novel formulation of the objective func- tion for the design of fiber Bragg grating (FBG)-based filters with respect to the given design specifications, instead of matching the desired magnitude and phase responses of the filter at each wave- length of the operating window that has commonly been used in previous works on FBG synthesis. The desired reflective spectrum and group delay characteristics of a filter are predefined using six design specifications. Particle swarm optimization (PSO) technique is employed here to find an optimum index modulation profile that meets the target design. To demonstrate the effectiveness of the PSO algorithm and the novel formulation of the objective function, an optimal design of a low-dispersion FBG-based filter with 0.2-nm bandwidth (or 25 GHz in the 1550-nm window) for three desired values of the maximum reflective power is presented. Index Terms—Fiber Bragg grating (FBG), filter design, low-dispersion, particle swarm optimization (PSO). I. INTRODUCTION F IBER BRAGG gratings (FBGs) have evolved as critical and essential components for a multitude of applications in wavelength-division-multiplexing-based fiber optics communi- cation systems [1]. The synthesis and fabrication of FBGs have recently attracted many researchers in the field of fiber optics. Several methods have been developed for the synthesis of FBG filters in the past [2]–[9]. A straightforward approach is to use Fourier transform techniques [2], but these methods can result in large design errors for strong gratings. Another method based on numerically solving the Gel’fand–Levitan–Marchenko equa- tions using iterative methods has been demonstrated for several grating designs [3]. A simple grating synthesis algorithm using standard coupled mode equations while simultaneously evaluating a simple inte- gral to obtain the grating strength has also been attempted in [4]. Most of the reported approaches for the design of low-dispersion FBGs are based on the layer peeling (LP) inverse scattering al- gorithm [5], [6]. Theoretically, FBGs with the required reflective and dispersion characteristics can be synthesized using the LP algorithm. However, in practice, the FBGs designed using the methods to achieve the best performance generally has compli- cated index modulation profiles and long grating lengths. Re- cently, an LP-based two-stage design approach has been pro- posed for the design of a low-dispersion FBG with a grating of 3 cm, a bandwidth (BW) of 0.4 nm, and a flat group-delay response [6]. However, all these methods do not provide the Manuscript received October 7, 2004; revised November 2, 2004. The authors are with the School of Electrical and Electronic Engi- neering, Nanyang Technological University, Singapore 639798, Singapore (e-mail: sbaskar@ntu.edu.sg; ealphones@ntu.edu.sg; eqnngo@ntu.edu.sg; epnsugan@ntu.edu.sg). Digital Object Identifier 10.1109/LPT.2004.840924 flexible means of changing the design conditions for different requirements. In recent years, different stochastic global optimization algo- rithms have been successfully applied to find the optimal index profiles to satisfy the prescribed filter specifications. Genetic algorithm (GA) [7], evolutionary programming [8], and Tabu search [9] techniques have been applied to solve the FBG syn- thesis problem. Compared to the inverse scattering method, the stochastic optimization approaches are, in general, potentially capable of obtaining an index profile that can be practically im- plemented more easily by imposing additional constraints on the solution. Recently, particle swarm optimization (PSO) has been pro- posed as a simple and alternative tool for solving optimization problems. PSO has been shown to be effective in optimizing mul- tidimensional discontinuous problems [10], [11]. PSO has also been successfully applied in electromagnetic design problems [12]. In this work, we propose a novel formulation of the objective function to search for design variables to satisfy the target design criteria specified in terms of 3-dB BW, sidelobe level (SLL), first null bandwidth (FNBW), in-band ripple (Rrip), maximum reflective power (Rmax), and in-band group delay ripple (Trip). To demonstrate the capability of the PSO algorithm and the effectiveness of the new formulation of the objective function, the design of a low-dispersion FBG filter with a BW of 0.2 nm for various desired values of the power maximum is presented. II. OBJECTIVE FUNCTION The synthesis problem of a low-dispersion FBG filter is to find an optimum index modulation profile for a desired reflec- tivity spectrum and a smooth in-band group delay response (i.e., low dispersion). In this letter, the grating is divided into piece- wise uniform sections [8]. If the index modulation profile is known for all the sections, then the transfer matrix for the en- tire grating can be obtained by chain multiplying the individual transfer matrices of each section [1]. In general, evolutionary algorithms use the concept of fit- ness to represent how well a particular solution satisfies the de- sign objectives. In the FBG synthesis problem, the sum of the weighted errors is normally used [7]–[9] as cost function. To ob- tain good performance, suitable weighing factor values at each wavelength are to be selected [9]. In practice, this method does not work well because the op- timization process attempts to maximize the nulls between ad- jacent sidelobe peaks while minimizing the sidelobe peaks in the reflective spectrum. In essence, this is a waste to optimize the nulls between the sidelobe peaks as they are of little impor- tance to the performance of the FBG filter. Instead, the desired 1041-1135/$20.00 © 2005 IEEE