Page 273 - Fiber Bragg Gratings

250 Chapter 6 Fiber Grating Band-pass Filters

pass filter. Substituting Eq. (6.3.2) into Eq. (6.3.3), and remembering that
ReCR, <S) = 1/2CR, S + R*, <S*), the transmitted power in each output port
of the filter is the product of the complex field with their conjugate. By
simple expansion and algebraic manipulation of the equations, the power
transmittance at ports 1 and 2 can be shown to be

where the phase difference d between the reflections from the two gratings
is

Equations (6.3.4) and (6.3.5) describe how the transmitted power at
the output depends on the path-length difference, the reflectivities, and
the Bragg wavelengths of the two gratings in the arms of the Michelson
interferometer filter. For a 50:50 coupler, /cL c = 77/4; ignoring loss and
polarization effects, Eqs. (6.3.5) simplify to

Note that the power transfers is cyclic between the two ports, de-
pending on the phase difference S and which is of paramount importance
for the proper operation of the filter. This cyclic behavior is well known for
unbalanced broad band interferometers, but in this device it is restricted to
the bandwidth of the gratings. The choice of the gratings determines the
wavelength at which the interference will occur. With the phase difference
d — 2N7T (where N > 0 is an integer), all the power is routed to port
1. This phase can to be adjusted mechanically [40], thermally [41], or
permanently by optical "trimming" of the path using UV radiation [53].
Detuning of the interferometer is an important issue for the accept-
able performance of the band-pass filter. As such, there are two parame-
ters, which are variables in a filter of this type. Assuming that the