Page 150 - Fiber Bragg Gratings

4.2 Coupled-mode theory 127

Applying the orthogonality relationship of Eq. (4.1.15) directly results
in

Equation (4.2.10) is fundamentally the wave propagation equation,
which can be used to describe a variety of phenomena in the coupling
of modes. Equation (4.2.10) applies to a set of forward- and backward-
propagating modes; it is now easy to see how mode coupling occurs by
introducing forward- and backward-propagating modes. The total trans-
verse field may be described as a sum of both fields, not necessarily
composed of the same mode order:

Here the negative sign in the exponent signifies the forward- and the
positive sign the backward-propagating mode, respectively. The modes of
a waveguide form an orthogonal set, which in an ideal fiber will not couple
unless there is a perturbation. Using Eqs. (4.2.11) and (4.2.12) in Eq.
(4.2.10) leads to

4.2.1 Spatially periodic refractive index modulation
In a medium in which the dielectric constant varies periodically along
the wave-propagation direction, the total polarization can be defined with
the perturbed permittivity, Ae(z) and the applied field as