1317 ISSN 1054-660X, Laser Physics, 2007, Vol. 17, No. 11, pp. 1317–1322. © MAIK “Nauka / Interperiodica” (Russia), 2007. Original Text © Astro, Ltd., 2007. 1. INTRODUCTION Direct and inverse scattering problems for fiber Bragg gratings (FBG) are widely studied in optics [1, 2]. The direct problem here means the determination of the reflection spectrum using the spatial distribution of the glass refractive index. The inverse scattering prob- lem is the reconstruction of the refractive index from given complex reflection spectrum. However, the exist- ing numerical and analytical solutions describe the reg- ular profiles of FBG, while only a few works are devoted to the more realistic case of noisy reflection data [3, 4]. It was noticed that the inverse scattering problem becomes ill-conditioned at a high level of noise. Skaar and Feced [3] suggested the simplest regular- ization method in order to avoid incorrectness. If the reflection coefficient becomes greater than unity at some points due to random error, it is corrected by the regularization parameter μ > 1. Without this factor, the numerical calculation becomes unstable and cannot be completed, since the inverse problem becomes incor- rect. Parameter μ should be specially chosen to match the given noise level. Excessively large μ  1 makes the computation stable, but leads to a shift in the esti- mate. Parameter μ that is close to unity may be insuffi- cient to suppress the instability. Then, it is reasonable to make the regularization procedure dependent on the noise level. Calculations in the present paper are carried out for the purpose of developing the adaptive regularization. In Section 2, we present the general equations of the inverse scattering problem and the family of chirped profiles of grating that has been used for numerical modeling. Section 3 describes the numerical calcula- tions using the recently proposed TIB method [5]. The method demonstrates a stability for small noise. A high level of noise or a large reflectivity of the grating breaks down the numerical procedure. The adaptive regular- ization for Gaussian noise is described in Section 4. The procedure reconstructs the complex coupling coef- ficient and enables one to overcome the incorrectness at a high level of noise. The conclusions are summarized in Section 5. 2. BASIC EQUATION OF THE INVERSE SCATTERING PROBLEM The inverse scattering problem for coupled wave equations is reduced to the reconstruction of complex coupling coefficient q(x) from a given time dependence of the Fourier transform R(t) of the complex reflection coefficient r(ω). This can be done using the solution to the coupled Gelfand–Levitan–Marchenko equations [6]. In the representation [5], they have the form (1) where u, v are auxiliary functions of two variables. The coupling coefficient is given by the limit (2) Functions u, v are defined in the triangular region 0 ≤ s ≤ τ ≤ 2x ≤ 2L, where L is the length of the grating. Function R(t) is the impulse response of the grating, as follows from the causality principle that R(t) = 0 at t < 0. Notice that both equations involve difference ker- nels. This property is general and valid for the linear uxs , ( ) R * τ s – ( ) v x τ , ( )τ d s 2 x ∫ + 0, = v x τ , ( ) R τ s – ( ) uxs , ( ) s d 0 τ ∫ + R τ () , – = qx () v xy , ( ) . y 2 x 0 – → lim = FIBER OPTICS Reconstruction of High Reflectance Fiber Bragg Grating from Noisy Data O. V. Belai a , L. L. Frumin b , E. V. Podivilov a , and D. A. Shapiro a, * a Institute of Automation and Electrometry, Siberian Branch, Russian Academy of Sciences, pr. Akademika Koptyuga 1, Novosibirsk, 630090 Russia b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia *e-mail: shapiro@iae.nsk.su Received June 19, 2007 Abstract—The inverse scattering problem for fiber Bragg grating reconstruction becomes incorrect with an increasing level of noise in the input data or at a high reflection. The adaptive regularization procedure is pro- posed to restore the correctness and minimizing the reconstruction error. The proposed method is tested using numerical modeling with the Gaussian statistics of noise. PACS numbers: 42.65.-k, 42.55.Wd DOI: 10.1134/S1054660X07110084