"Introduction". In: Fiber-Optic Communication Systems

268 CHAPTER 6. OPTICAL AMPLIFIERS

0 2000 4000 6000 8000 10000

Distance (km)

0

5

10

Timing Jitter (ps)

−0.3

−0.2

−0.1

−0.01 ps^2 /km

Figure 6.23: ASE-induced timing jitter as a function of system length for several values of the
average dispersionβ ̄ 2.

〈(δq)^2 〉=(Ssp/E 0 )Ti^2 , 〈δΩδq)〉=(Ssp/E 0 )Ci, (6.5.27)

whereE 0 is the input pulse energy andCiandTiare the chirp and width atz=zi. These
quantities can be calculated easily using the theory of Section 2.4. Note that the ratio
( 1 +C^2 i)/Ti^2 is related to the spectral width that does not change if the nonlinear effects
are negligible. It can be replaced byTm−^2 , whereTmis the minimum width occurring
when the pulse is unchirped.
Many lightwave systems employ the postcompensation technique in which a fiber
is placed at the end of the last amplifier to reduce the net accumulated dispersion (see
Section 7.4). Using Eqs. (6.5.24)–(6.5.27), the timing jitter for a CRZ system employ-
ing postcompensation is found to be [126]

σt^2 =(Ssp/E 0 )Tm^2

[

NA+NA(NAd+C 0 +df)^2

]

, (6.5.28)

whereC 0 is the input chirp,d=β ̄ 2 LA/Tm^2 , anddf=β 2 fLf/Tm^2 for a postcompensation
fiber of lengthLfand dispersionβ 2 f. Several points are noteworthy. First, if post-
compensation is not used (df=0), the dominant term in Eq. (6.5.28) varies asN^3 Ad^2.
This is the general feature of the ASE jitter resulting from frequency fluctuations [122].
Second, if the average dispersion of the fiber link is zero, the cubic term vanishes, and
the jitter increases only linearly withNA. Third, the smallest value of the jitter occurs
whenNAd+C 0 +df=0. This condition corresponds to zero net dispersion over the
entire link, including the fiber used to chirp the pulse initially.
The average dispersion of the fiber link can lead to considerable timing jitter in
CRZ systems when postcompensation is not used. Figure 6.23 shows the timing jitter
as a function of the total system lengthLT=NALAfor a 10-Gb/s system using four
values ofβ ̄ 2 withTm=30 ps,LA=50 km,C 0 = 0 .2, andSsp/E 0 = 10 −^4. The ASE-
induced jitter becomes a significant fraction of the pulse width for values of|β ̄ 2 |as
small as 0.2 ps^2 /km because of the cubic dependence ofσt^2 on the system lengthLT.
Such jitter would lead to large power penalties, as discussed in Section 4.6.3, if left