"Introduction". In: Fiber-Optic Communication Systems

294 CHAPTER 7. DISPERSION MANAGEMENT

-10 -5 0 5 10

Detuning

0.0

0.2

0.4

0.6

0.8

1.0

Reflectivity

-10 -5 0 5 10

Detuning

-15

-10

-5

0

5

Phase

(a) (b)

Figure 7.8: (a) Magnitude and (b) phase of the reflectivity plotted as a function of detuningδLg
for a uniform fiber grating withκLg=2 (dashed curve) orκLg=3 (solid curve).

whereq^2 =δ^2 −κ^2 andLgis the grating length. Figure 7.8 shows the reflectivity
|H(ω)|^2 and the phase ofH(ω)forκLg=2 and 3. The grating reflectivity becomes
nearly 100% within the stop band forκLg=3. However, as the phase is nearly linear
in that region, the grating-induced dispersion exists only outside the stop band. Noting
that the propagation constantβ=βB±q, where the choice of sign depends on the
sign ofδ, and expandingβin a Taylor series as was done in Eq. (2.4.4) for fibers, the
dispersion parameters of a fiber grating are given by [54]

β 2 g=−

sgn(δ)κ^2 /v^2 g

(δ^2 −κ^2 )^3 /^2

, β 3 g=

3 |δ|κ^2 /v^3 g

(δ^2 −κ^2 )^5 /^2

, (7.6.5)

wherevgis the group velocity of the pulse with the carrier frequencyω 0 = 2 πc/λ 0.
Figure 7.9 shows howβ 2 gvaries with the detuning parameterδfor values ofκin
the range 1 to 10 cm−^1. The grating-induced GVD depends on the sign of detuningδ.
The GVD is anomalous on the high-frequency or “blue” side of the stop band where
δis positive and the carrier frequency exceeds the Bragg frequency. In contrast, GVD
becomes normal (β 2 g>0) on the low-frequency or “red” side of the stop band. The
red side can be used for compensating the anomalous GVD of standard fibers. Since
β 2 gcan exceed 1000 ps^2 /cm, a single 2-cm-long grating can be used for compensating
the GVD of 100-km fiber. However, the third-order dispersion of the grating, reduced
transmission, and rapid variations of|H(ω)|close to the bandgap make use of uniform
fiber gratings for dispersion compensation far from being practical.
The problem can be solved by using theapodization techniquein which the index
changengis made nonuniform across the grating, resulting inz-dependentκ. In prac-
tice, such an apodization occurs naturally when an ultraviolet Gaussian beam is used to
write the grating holographically [51]. For such gratings,κpeaks in the center and ta-
pers down to zero at both ends. A better approach consists of making a grating such that
κvaries linearly over the entire length of the fiber grating. In a 1996 experiment [61],
such an 11-cm-long grating was used to compensate the GVD acquired by a 10-Gb/s