IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 14, NO. 11, NOVEMBER 2002 1557 Evolutionary Programming Synthesis of Optimal Long-Period Fiber Grating Filters for EDFA Gain Flattening C. L. Lee and Y. Lai Abstract—An innovative long-period fiber grating (LPG) syn- thesis method based on the stochastic evolutionary programming (EP) algorithms is demonstrated to be effective for designing op- timal LPG filters. Synthesis of a single-stage LPG filter for the en- tire C-band erbium-doped fiber amplifier (EDFA) gain flattening is used as an example to show the feasibility and the effectiveness of the proposed algorithm. To the best of our knowledge, this is the first demonstration that good EDFA gain flattening for the entire band can be achieved with a single-stage LPG filter that is prop- erly designed. Index Terms—Erbium-doped fiber amplifier (EDFA) gain flat- tening, evolutionary programming (EP), long-period fiber gratings (LPGs). I. INTRODUCTION F IBER gratings are photoinduced fiber devices that have found many applications in optical communication and optical sensing. Among various fiber grating devices, the long- period fiber gratings (LPGs), in which the guided core mode is coupled to one or several forward propagating cladding modes, have been demonstrated to be useful in applications like band-rejection filters, high sensitivity sensors, mode con- verters, and especially erbium-doped fiber amplifier (EDFA) gain flattening filters [1]. In recent years, several synthesis or inverse design methods to determine the required fiber grating index modulation profile corresponding to a given reflection spectrum have been developed quite successfully for fiber Bragg gratings (FBGs) [2]–[4]. Among these methods, the layer-peeling implementation of the inverse scattering method can effectively calculate the required grating index profile from the targeted reflection spectrum and is, thus, widely used in designing special FBG devices for fiber communication appli- cations. For designing transmission-type fiber grating filters like LPGs, the inverse scattering method is still applicable [5]–[7] and a solution can be uniquely determined if an addi- tional assumption about the filter properties (the under-coupled assumption) is used [6]. In the literature, another class of syn- thesis methods for FBGs are the optimization methods based Manuscript received May 20, 2002; revised August 1, 2002. This work is sup- ported in part by the National Science Council of the Republic of China under Contract NSC 91-2215-E-009-026, the Ministry of Education of the Republic of China, and the Lee Center at National Chiao-Tung Univervisity. C. L. Lee is with the Institute of Electric-Optical Engineering, National Chiao-Tung University, Hsinchu, Taiwan, R.O.C. and also with the Department of Electro-Optical Engineering, National Lien-Ho Institute of Technology, Miaoli, Taiwan, R.O.C. Y. Lai is with the Institute of Electric-Optical Engineering, National Chiao- Tung University, Hsinchu, Taiwan, R.O.C. (e-mail: yclai@mail.nctu.edu.tw). Digital Object Identifier 10.1109/LPT.2002.803910. on variational or genetic algorithms (GA) [2]. A combinational use of the inverse scattering and genetic optimization methods for designing binary LPGs have also been reported recently [8]. Compared to the inverse scattering method, the optimization approaches have the potential capability of obtaining an index profile that can be more practically implemented by imposing additional constrains on the solution to be found. In this letter, a novel approach to the problems of synthesizing long-period fiber gratings is proposed. The new method is based on the evolutionary programming (EP) algorithm which employs the population-based optimization mechanism. EP and GA are two important branches in the family of evolutionary algorithms (EA), which are a class of probabilistic search and optimization algorithms gleaned from the organic evolution process. Com- pared to the genetic algorithms for fiber grating synthesis that have been reported in the literature, we find our EP algorithm has a higher convergence velocity as well as higher reliability. This may be due to the fact that we only use the mutation process of continuous variables in the EP algorithm and without using the binary crossover process. To verify the effectiveness of the pro- posed approach, the EP-based synthesis algorithm together with the transfer-matrix model based on the couple-mode theory for calculating the transmission spectra of LPGs are implemented by using the Matlab software package. The performance of the al- gorithm is tested by designing a single-stage LPG filter for the entire C-Band EDFA gain flattening. It is worth to note that the coupling coefficients obtained from the proposed algorithm are normally less complex and easy to implement in practice. II. SYNTHESIS OF LPGS USING EP Evolutionary programming (EP), an important branch in the field of EA, was originally developed by Fogel et al. [9] in the 1960s. At that time they used a quite simple model based on the finite-state machine concept and used mutation as the only operator. The EP algorithm used in this letter can be briefly de- scribed as follows. Starting from a population of “individ- uals” (parents), a new set of individuals is generated through some selection rules and then a set of offsprings is generated through a mutation process. This set of offsprings is then used as the parents for next iteration. Since the main objective of the synthesis is to find a coupling coefficient profile that pro- duces a transmission spectrum as close as possible to the target spectrum, an “individual” is thus a particular coupling coeffi- cient profile function , which will be descretized later and represented by a complex vector . To complete the algorithm, one then needs to define the error function and the fitness func- 1041-1135/02$17.00 © 2002 IEEE