3.1         FBG spectral response simulation in the coupled-mode formalism

3.1.5        Causal T-matrix method

For the section j of thickness D_j, the grating effect can be approximated by a single complex reflector of reflectivity r_j (Fig. 3-3). The complex reflectivity factor r_j is defined from the complex coupling coefficient q_j as

Fig. 3-3 Parameters for section j

In this case, the matrix T_j that represents the section j, can be formulated as the product of a pure propagation matrix T_D,j and of a transfer matrix T_r,j [3-5]

that is

The factor (1-|r_j|²)^-1/2 corresponds to the transmission amplitude. The matrix T_r,j can also be obtained from equation(3-8) by letting q_j ® µ and the matrix T_D,j by letting q_j ® 0 holding the factor q_jD_j constant.

From equations (3-11), the fields propagation can be expressed in a recursion form (instead of a matrix product)

This recursion formula allows calculating the reflectivity r_j+1 of the FBG constituted of the sections j to N. This propagation process is one of the required steps of the layer-peeling reconstruction method presented later in this chapter.

We have denominated this method "causal" as all reflections in the section are located in a single point. This method is similar to Rouard's method used in the simulation response of thin films [3-6]. The difference is in the thickness of the sections. For Rouard's method, each grating period would be divided in several sections (for a FBG this would lead to D_j of a few tens of nanometers) while in the causal T-matrix, only the necessary number of sections is used to represent the slowly varying coupling coefficient (thickness of tens of micrometers are possible). This method is the direct counterpart of the layer-peeling reconstruction method, which allows to recover the complex coupling coefficient for a given complex reflectivity response.