Chapter 3 : FBG simulation and reconstruction

Fig. 3-1 presents a synthetic view of the subjects treated in this chapter. A FBG can be described in three domains : space (z), frequency (n) and time (t). The methods used to go from one representation to the other are also indicated. The T-matrix method allows to calculate the complex spectral response r(n) when the complex coupling coefficient distribution q(z) is known [3-1]. The complex coupling coefficient amplitude is proportional to the refractive index modulation amplitude Dn_ac(z), while its phase represents the chirp function that mixes the average refractive index Dn_dc(z) and the period L(z). Inversely, q(z) can be retrieved from r(n) by the layer-peeling method [3-2]. This method is based on the coupled-mode formalism [3-3]. The impulse response h(t) can be obtained from the spectral response by Fourier transform.

Fig. 3-1 Different paths between the FBG representations

This chapter presents the T-matrix and the layer-peeling methods. The T-matrix is used to calculate the spectral and impulse response of homogeneous and non-homogenous gratings. From these responses, the layer-peeling method is studied and the optimal reconstruction parameters are presented. We also analyze the reconstruction limits observed for gratings with nearly 100 % reflectivity.

The T-matrix and the layer-peeling methods are only defined for lossless FBGs and for this reason we have adapted the two methods to takes account of loss effects.