Page 304 - Fiber Bragg Gratings

6.7 In-coupler Bragg grating filters 281

with a point reflector (grating), the transmission through to the output
ports would follow the equation for the coupler [Eq. (6.3.la)], to route the
entire out-off-band transmitted light to B 2. However, because of the finite
length of the grating and the additional coupling that occurs in region II,
region III may no longer be equal to region I for optimum performance,
since K\(L l + L 2 + L 3) = mr.
As has been mentioned, a simple way of making such a device is to
draw two identical fibers together to form a coupler and subsequently
write a grating at the appropriate position, as shown in Fig. 6.41 (iv).
With two fibers and two gratings, there is always a possibility of a small
mismatch in the propagation constants after the fibers are polished and
the device assembled. The refractive index mismatch maybe typically be
5
~5 X 10~ , resulting in —95% coupling [38]. Gratings written into such
fibers may therefore need to be written carefully in order to match the
Bragg wavelengths. With a fused fiber coupler, the quality of the device can
be very good with coupling approaching 100%, indicating the uniformity of
the coupling region. Thus, assuming fibers with identical propagation
constants, A/3 ab = 0, and the detuning AySb = A/3, simplifying Eqs.
(6.7.9H6.7.12)

Note that despite using phase-synchronous fibers, in Eqs. (6.7.13)-
(6.7.16) the eigenvalues have been detuned from the exact phase-matching
condition by K.
To calculate the field at the input port 1 (return-loss) and the dropped
port 2, the boundary conditions are applied. The dropped "transmission"
in port 2 is

where the detuning, S l = — K — A/3, S 2 = K — A/6 and <f> = (\K\ + A/3)!/!.