Page 188 - Fiber Bragg Gratings
P. 188
4.7 Radiation mode couplers 165
if necessary but is ignored for now. For the backward phase-matching
condition, the radiation angle at resonance, <p L, is given by the A/fy = 0,
as has been seen in Section 4.2.5, so that
The last result is a longitudinal phase-matching condition, which is
exactly the same as normal Bragg reflection. It requires that the path
difference between light scattered from points that are both on a line
parallel to the optical axis of the fiber, and on adjacent fringes of the
grating, should be exactly A (Fig. 4.18).
Ignoring the forward scatter, we find the scattered counterpropagat-
ing power from Eqs. (4.7.11), (4.7.3), and (4.7.4) as
where the overlap integral over the profile of the grating, which we refer
to as the transverse phase-matching condition, is
Figure 4.18: Scattered light from the fringe planes of the gratings adds up
in phase when the resonance condition for longitudinal phase matching is met.
AB + BC = NA, at resonance.
