7.6. FIBER BRAGG GRATINGS 293
alytically and is used to optimize the device design and performance [57]. In a 1994
implementation [58], a planar lightwave circuit with only five MZ interferometers pro-
vided a relative delay of 836 ps/nm. Such a device is only a few centimeters long,
but it is capable of compensating for 50 km of fiber dispersion. Its main limitations
are a relatively narrow bandwidth (∼10 GHz) and sensitivity to input polarization.
However, it acts as a programmable optical filter whose GVD as well as the operating
wavelength can be adjusted. In one device, the GVD could be varied from−1006 to
834 ps/nm [59].
7.6 Fiber Bragg Gratings
A fiber Bragg grating acts as an optical filter because of the existence of astop band,
the frequency region in which most of the incident light is reflected back [51]. The
stop band is centered at theBragg wavelengthλB=2 ̄nΛ, whereΛis the grating period
and ̄nis the average mode index. The periodic nature of index variations couples the
forward- and backward-propagating waves at wavelengths close to the Bragg wave-
length and, as a result, provides frequency-dependent reflectivity to the incident signal
over a bandwidth determined by the grating strength. In essence, a fiber grating acts as
a reflection filter. Although the use of such gratings for dispersion compensation was
proposed in the 1980s [60], it was only during the 1990s that fabrication technology
advanced enough to make their use practical.
7.6.1 Uniform-Period Gratings
We first consider the simplest type of grating in which the refractive index along the
length varies periodically asn(z)=n ̄+ngcos( 2 πz/Λ), wherengis the modulation
depth (typically∼ 10 −^4 ). Bragg gratings are analyzed using thecoupled-mode equa-
tionsthat describe the coupling between the forward- and backward-propagating waves
at a given frequencyωand are written as [51]
dAf/dz=iδAf+iκAb, (7.6.1)
dAb/dz=−iδAb−iκAf, (7.6.2)whereAfandAbare the spectral amplitudes of the two waves and
δ=2 π
λ 0−
2 π
λB, κ=πngΓ
λB. (7.6.3)
Hereδis the detuning from the Bragg wavelength,κis thecoupling coefficient, and
the confinement factorΓis defined as in Eq. (2.2.50).
The coupled-mode equations can be solved analytically owing to their linear nature.
The transfer function of the grating, acting as a reflective filter, is found to be [54]
H(ω)=r(ω)=Ab( 0 )
Af( 0 )=
iκsin(qLg)
qcos(qLg)−iδsin(qLg)