Page 306 - Fiber Bragg Gratings

6.7 In-coupler Bragg grating filters 283

where

Interestingly, Eqs. (6.7.21) and (6.7.22) are exactly the same form as
Eqs. (6.3.7) and (6.3.8), which describe the transfer characteristics of the
Michelson interferometer with identical reflectivities but Bragg detuned
gratings. In the BRC, the detuning is implicit in the phase factors 0 X and
02 and calculated by equating S l and S 2 to zero, so that

where the sign in the denominator determines the perturbed Bragg wave-
length of the slow, symmetric (negative sign) and fast (positive sign)
antisymmetric supermodes. Note that for weak coupling, i.e., when \K\ —>
0, the splitting in the Bragg wavelength tends to zero. The detuning is
solely dependent on the coupling constant of the coupler. For a given
detuning, 2AA = A| ra ^ — A]j rag?, we can calculate the coupling constant
by solving Eq. (6.7.24) as

where A Bragg is the unperturbed Bragg wavelength.
Since the functional form of the properties of BRC band-pass filters
are almost identical to that for the Michelson interferometer, the detuning
that can be tolerated for low back reflection has been discussed in Section
6.3. For a back reflection of approximately -30 dB (requiring a detuning
1
of ~0.01 nm), we calculate the coupling constant K < 26 m" . This low
value of the coupling constant is necessary to suppress the reflections on
each side of zero detuning, as shown in Fig. 6.20. In a practical device,
there is additional "apodization" due to the variation in the coupling
constant in the tapered or curved region of the coupler, which will also
tend to reduce the back-reflected light. For minimum back reflection, we
note that T l in Eq. (6.7.20) is zero, so that the coupling length L x may
be calculated as