Page 302 - Fiber Bragg Gratings

6.7 In-coupler Bragg grating filters 279

Michelson/Mach-Zehnder interferometer with "zero-length" paths. This
will become clear as one compares the theoretical performance of the BRC
with that of the Michelson with detuned Bragg grating wavelengths. One
essential difference between the two devices are the single grating as well
as only one common path in the BRC, as opposed to two gratings and two
paths in the Michelson interferometer. Many of the features of the BRC are
easily understood by comparing it with the Michelson [79].

Theory of the BRC
The theory of the BRC has been worked out using coupled-mode analysis
[76,80,77]. We closely follow the nomenclature of Ref. [77]. Referring to
Fig. 6.41 (Hi), the fibers and A and B have unperturbed propagation
constants f3 a and fi b, respectively, and the grating with a coupling constant
of K ac exists in both fibers, evanescently coupled with a coupling constant
K. In region II, there are co- and counterpropagating modes, which are
coupled together. The presence of the grating introduces a detuning of
the propagation constants in each fiber, which are

Coupling between counterpropagating modes of different fibers (A- L/B 2
and A 2/Bi) is only significant if the fibers are very strongly coupled, so
that coupling occurs over a distance of a few wavelengths (when K is of
1
6
the order of 10 m~ ). Thus, these interactions can be ignored in a majority
of cases. The mode fields propagate with the modified propagation con-
stants and may be expressed as