Page 337 - Fiber Bragg Gratings

314 Chapter 7 Chirped Fiber Bragg Gratings

The effect of the chirped grating is that it disperses light by introducing
a maximum delay of 2Lg/v g between the shortest and longest reflected
wavelengths. This dispersion is of importance since it can be used to
compensate for chromatic dispersion induced broadening in optical fiber
transmission systems. At 1550 nm, the group delay rin reflection is ~10
nsec/m. Therefore, a meter-long grating with a bandwidth of 1 nm will
have a dispersion of 10 nsec/nm.
An important feature of a dispersion-compensating device is the fig-
ure of merit. There are several parameters that affect the performance
of chirped fiber Bragg gratings for dispersion compensation. These are
the insertion loss (due to <100% reflectivity), dispersion, bandwidth, po-
larization mode-dispersion, and deviations from linearity of the group
delay and group delay ripple. Ignoring the first and the last two parame-
ters for the moment, we consider the performance of a chirped grating
with linear delay characteristics, over a bandwidth of AA chirp. Priest and
Giallorenzi [35] have proposed a figure of merit for coherent communica-
tions, but taking into account only the dispersion and the bandwidth of
the filter. This approach, while not entirely appropriate for chirped grat-
ings owing to the larger parameter set, is never the less a guide in as-
sessing the usefulness of the "ideal" chirped grating. It should be
remembered that chirped gratings have a limited bandwidth over which
the dispersion is useful, making them different from other truly broadband
compensating devices, such as dispersion-compensating fiber [36].
We consider the propagation of an optical pulse in normalized units,
in a frame of reference moving in the +z direction at a group velocity v g.
The pulse amplitude A(z, T), with a normalized amplitude U(z, r), following
Agrawal [37], is described as

in which a is the attenuation coefficient of the fiber, P in is the input power,
and the frame of reference normalized to the initial pulse width T 0 is

For a Gaussian pulse with a l/e- intensity half-width of T 0, the normal-
ized amplitude is [38]