This appendix is based on the work of J. Skaar in his thesis work, Chapter 2 "Fiber Bragg grating model" [C-1].

C.1       Scalar wave approximation

The fiber is assumed lossless, single mode and weakly guiding (small refractive index difference between the claddings n_cladding and the fiber core n_core). The electromagnetic field is considered transverse to the fiber axis z and that the polarization state is conserved along the propagation (x-polarized). These hypotheses reduce the field description to the scalar wave equation [C-2]. A forward propagating wave with positive propagation constant b and pulsation w has a phase term e^i(bz-wt).

The fiber Bragg grating is treated as a perturbation of the fiber waveguide. The refractive index distribution of the fiber prior to the grating inscription is given by and the perturbed refractive index n(x,y,z) is z-dependant. The total electric field E_x is written as a superposition of the forward and backward propagating modes (b₊ and b_- respectively)

The coefficients b_± contain all the z-dependence of the modes when y describes the transverse dependence. The function y satisfies the scalar wave equation for the unperturbed fiber

where k = w/c₀ is the vacuum wavenumber (c₀ is the vacuum light speed) and b = n_effk (n_eff is the mode effective refractive index). The total electric field satisfies the scalar wave equation for the perturbed waveguide

From equations (C-1), (C-2) and (C-3) the following equation is obtained

This equation is multiplied by y and integrated over the fiber section and then