7.6. FIBER BRAGG GRATINGS 295
δ (cm−^1 )-20 -10 0 10 20βg 2
(ps2 /cm)-600-400-200020040060010 cm−^151Figure 7.9: Grating-induced GVD plotted as a function ofδfor several values of the coupling
coefficientκ.
signal transmitted over 100 km of standard fiber. The coupling coefficientκ(z)varied
smoothly from 0 to 6 cm−^1 over the grating length. Figure 7.10 shows the transmis-
sion characteristics of this grating, calculated by solving the coupled-mode equations
numerically. The solid curve shows the group delay related to the phase derivative
dφ/dωin Eq. (7.5.2). In a 0.1-nm-wide wavelength region near 1544.2 nm, the group
delay varies almost linearly at a rate of about 2000 ps/nm, indicating that the grating
can compensate for the GVD acquired over 100 km of standard fiber while providing
more than 50% transmission to the incident light. Indeed, such a grating compensated
GVD over 106 km for a 10-Gb/s signal with only a 2-dB power penalty at a bit-error
rate (BER) of 10−^9 [61]. In the absence of the grating, the penalty was infinitely large
because of the existence of a BER floor.
Tapering of the coupling coefficient along the grating length can also be used for
dispersion compensation when the signal wavelength lies within the stop band and the
grating acts as a reflection filter. Numerical solutions of the coupled-mode equations
for a uniform-period grating for whichκ(z)varies linearly from 0 to 12 cm−^1 over
the 12-cm length show that the V-shaped group-delay profile, centered at the Bragg
wavelength, can be used for dispersion compensation if the wavelength of the incident
signal is offset from the center of the stop band such that the signal spectrum sees a
linear variation of the group delay. Such a 8.1-cm-long grating was capable of com-
pensating the GVD acquired over 257 km of standard fiber by a 10-Gb/s signal [62].
Although uniform gratings have been used for dispersion compensation [61]–[64], they
suffer from a relatively narrow stop band (typically< 0 .1 nm) and cannot be used at
high bit rates.