456 CHAPTER 9. SOLITON SYSTEMS
ever, as solitons are periodically amplified, their state of polarization becomes random
because of the ASE added at every amplifier. Such polarization fluctuations lead to
timing jitter in the arrival time of individual solitons through fiber birefringence be-
cause the two orthogonally polarized components travel with slightly different group
velocities. This timing jitter, introduced by the combination of ASE and PMD, can be
written as [194]
σpol=(πSspfLM/ 16 EsLA)^1 /^2 DpL. (9.6.13)
As a rough estimate,σpol≈ 0 .4 ps for a standard-soliton system designed with
α= 0 .2 dB/km,LA=50 km,Dp= 0 .1 ps/
√
km, andL=10 Mm. Such low values
ofσpolare unlikely to affect 10-Gb/s soliton systems for which the bit slot is 100 ps
wide. However, for fibers with larger values of the PMD parameter (Dp>1 ps/
√
km),
the PMD-induced timing jitter becomes important enough that it should be considered
together with other sources of the timing jitter. If we assume that each source of timing
jitter is statistically independent, total timing jitter is obtained using
σtot^2 =σt^2 +σacou^2 +σpol^2. (9.6.14)Bothσacouandσpolincrease linearly with transmission distanceL. The ASE-induced
jitter normally dominates because of its superlinear dependence onLbut the situation
changes when in-line filters are used to suppress it.
An important issue is related topolarization-dependentloss and gain. If a light-
wave system contains multiple elements that amplify or attenuate the two polarization
components of a pulse differently, the polarization state is easily altered. In the worst
situation in which the polarization is oriented at 45◦from the low-loss (or high-gain)
direction, the state of polarization rotates by 45◦and gets aligned with the low-loss
direction only after 30–40 amplifiers for a gain–loss anisotropy of< 0 .2 dB [195].
Even though the axes of polarization-dependent gain or loss are likely to be evenly
distributed along any soliton link, such effects may still become an important source of
timing jitter.
Interaction-Induced Jitter
In the preceding discussion of the timing jitter, individual pulses are assumed to be
sufficiently far apart that their position is not affected by the phenomenon of soliton–
soliton interaction. This is not always the case in practice. As seen in Section 9.2.2,
even in the absence of amplifier noise, solitons may shift their position because of
the attractive or repulsive forces experienced by them. As the interaction force be-
tween two solitons is strongly dependent on their separation and relative phase, both
of which fluctuate because of amplifier noise, soliton–soliton interactions modify the
ASE-induced timing jitter. By considering noise-induced fluctuations of the relative
phase of neighboring solitons, timing jitter of interacting solitons is generally found to
be enhanced by amplifier noise [196]. In contrast, when the input phase difference is
close toπbetween neighboring solitons, phase randomization leads to reduction in the
timing jitter.
An important consequence of soliton–soliton interaction is that the probability den-
sity of the timing jitter deviates considerably from the Gaussian statistics expected for