Page 305 - Fiber Bragg Gratings

282 Chapter 6 Fiber Grating Band-pass Filters

Similarly, the back-reflected amplitude in port 1 is deduced as

We note that in Eq. (6.7.17) and (6.7.18), there are four terms of
interest. The two terms in brackets can be immediately recognized to be
identical to the reflectivity of two gratings at different Bragg wavelengths
[see Eq. (4.3.11)], given by equating Si and 8 2 to zero. Secondly, the phase
term, 2<£, has two components; the first is due to the difference in the
propagation constants of two modes propagating through a fiber of length,
LI, after being reflected by the grating. The second part, AcL 1? is simply
the accumulated phase change due to the coupling action of the coupler.
The equivalent reflectivity and phase factors of two gratings as in Eq.
(6.3.2) can replace the two terms within the square brackets of Eqs.
(6.7.17) and (6.7.18):

We note that the magnitude of the reflectivity pi = p 2, since it is the
same grating with identical parameters, only different detuning. The
dropped transmission in Eqs. (6.7.17) and (6.7.18) may be further simpli-
fied to

from which the power transmittance T 2 and back-reflectance R± are