3.1         FBG spectral response simulation in the coupled-mode formalism

3.1.4        T-matrix method

In the T-matrix method [3-2 to 3-4], the grating is divided in N sections of width D_j (j = 1, …, N), where the parameters Dn_ac, Dn_dc and L are constant. The grating is then defined by N sections with coupling coefficients q_j and physical thickness D_j (Fig. 3-2).

Fig. 3-2 FBG Slicing in sub-sections for the T-matrix method

The knowledge of the fields u_j and v_j at the entrance of section j allows to find the fields u_j+1 and v_j+1 at the layer output. This relation can be expressed in the form of a transfer matrix relation

where

where g_j² = |q _j |² - d². The fields u₁, v₁ and u_N+1, v_N+1 at the grating entrance and output respectively, are then related to each other by

The reflection coefficient amplitude r(d) is determined with the limit conditions u₁ = 1 and v_N+1 = 0 : r(d) = v₁ = -T₂₁/T₂₂. The transmission coefficient amplitude t(d) is found from the limit conditions u₁ = 0 and v_N+1 = 1 : t(d) = v₁ = 1/T₂₂.

The proposed T-matrix formulation takes into account the overlap integral h that is often neglected [3-1, 3-4] and the fringe visibility effect can be integrated in the definition of the refractive index distributions Dn_dc and Dn_ac.