This appendix is based on the work of
J. Skaar in his thesis work, Chapter 2 "Fiber Bragg grating model" [C-1].
The fiber is assumed lossless, single
mode and weakly guiding (small refractive index difference between the
claddings ncladding and the fiber core ncore). 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 ei(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 Ex is written as a
superposition of the forward and backward propagating modes (b+ and b-
respectively)
|
(C-1) |
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
|
(C-2) |
where k = w/c0
is the vacuum wavenumber (c0 is the vacuum light speed) and b = neffk
(neff is the mode effective refractive index). The total electric
field satisfies the scalar wave equation for the perturbed waveguide
|
(C-3) |
From equations (C-1), (C-2)
and (C-3) the following equation is obtained
|
(C-4) |
This equation is multiplied by y and integrated
over the fiber section and then
|
(C-5) |
|
