Page 148 - Fiber Bragg Gratings
P. 148
4.2 Coupled-mode theory 125
where e^ = 2 when /u, = 0 (fundamental mode) and 1 for /u, =£ 0. Matching
the fields at the core-cladding boundary results in the waveguide charac-
teristic eigenvalue equation, which may be solved to calculate the propaga-
tion constants of the modes:
4.2 Coupled-mode theory
The waveguide modes satisfy the unperturbed wave equation (4.1.13) and
have solutions described in Eqs. (4.1.17) through (4.1.20). In order to
derive the coupled mode equations, effects of perturbation have to be
included, assuming that the modes of the unperturbed waveguide remain
unchanged. We begin with the wave equation (4.1.11)
Assuming that wave propagation takes place in a perturbed system
with a dielectric grating, the total polarization response of the dielectric
medium described in Eq. (4.2.1) can be separated into two terms, unper-
turbed and the perturbed polarization, as
where
Equation (4.2.1) thus becomes,
where the subscripts refer to the transverse mode number JUL. For the
present, the nature of the perturbed polarization, which is driven by the
propagating electric field and is due to the presence of the grating, is a
detail which will be included later.
