Page 171 - Fiber Bragg Gratings

148 Chapter 4 Theory of Fiber Bragg Gratings

The solutions to the coupled-mode Eqs. (4.4.9) and (4.4.10) are found
by applying the boundary values as in the case of the reflection grating.
However, for the transmission grating, the input fields, R(—L/2) — 1 and
S(—L/2) = 0. The power couples from R to S so that the transmission in
the uncoupled state is

and the transmission in the coupled state (also known as the crossed
state) is

2 172
In Eqs. (4.4.11) and (4.4.12), a = (\K af + <5 ) , and

The difference between reflection as in contradirectional coupling and
codirectional mode coupling is immediately apparent according to Eqs.
(4.2.14) and (4.4.12). While the reflected signal continues to increase with
increasing aL, the forward-coupled mode recouples to the input mode at
aL > 77/2. Therefore, a codirectional coupler requires careful fabrication
for maximum coupling.
Figure 4.9 demonstrates the optimum coupling to the crossed state
with K acL = 77/2 (curve A) as the coupling length doubles, the transmission
band becomes narrower (C), while B shows the situation of K acL = IT,
when the light is coupled back to the input mode.

4.5 Polarization couplers: Rocking filters

Equations (4.4.9) and (4.4.10) also govern coupling of modes with orthogo-
nal polarization. An additional subscript is used to distinguish between
the laboratory frame polarizations. However, there are differences in the
detail of the coupling mechanism. In order to couple two orthogonally
polarized modes, the perturbation must break the symmetry of the waveg-
uide. This requires a source term, which can excite the coupled mode. In
perfectly circular fibers, any perturbation can change the state of the