462 CHAPTER 9. SOLITON SYSTEMS
Ω−ch^1. Second, the number of collisions that two neighboring solitons in a given channel
undergo is slightly different. This difference arises because adjacent solitons in a given
channel interact with two different bit groups, shifted by one bit period. Since 1 and
0 bits occur in a random fashion, different solitons of the same channel are shifted by
different amounts. This source of timing jitter is unique to WDM systems because
of its dependence on the bit patterns of neighboring channels [211]. Third, collisions
involving more than two solitons can occur and should be considered. In the limit of a
large channel spacing (negligible overlap of soliton spectra), multisoliton interactions
are well described by pairwise collisions [210].
Two other mechanisms of timing jitter should be considered for realistic WDM
systems. As discussed earlier, energy variations due to gain–loss cycles make collisions
asymmetric whenLcollbecomes shorter than or comparable to the amplifier spacingLA.
Asymmetric collisions leave residual frequency shifts that affect a soliton all along the
fiber link because of a change in its group velocity. This mechanism can be made
ineffective by ensuring thatLcollexceeds 2LA. The second mechanism produces a
residual frequency shift when solitons from different channels overlap at the input of
the transmission link, resulting in an incomplete collision [208]. This situation occurs
in all WDM solitons for some bits. For instance, two solitons overlapping completely
at the input end of a fiber link will acquire a net frequency shift of 4/( 3 Ωch)since the
first half of the collision is absent. Such residual frequency shifts are generated only
over the first few amplification stages but pertain over the whole transmission length
and become an important source of timing jitter [209].
Similar to the case of single-channel systems, sliding-frequency filters can reduce
timing jitter in WDM systems [214]–[218]. Typically, Fabry–Perot filters are used
since their periodic transmission windows allow filtering of all channels simultane-
ously. For best operation, the mirror reflectivities are kept low (below 25%) to reduce
the finesse. Such low-contrast filters remove less energy from solitons but are as ef-
fective as filters with higher contrast. Their use allows channel spacing to be as little
as five times the spectral width of the solitons [218]. The physical mechanism remains
the same as for single-channel systems. More specifically, collision-induced frequency
shifts are reduced because the filter forces the soliton frequency to move toward its
transmission peak. The net result is that filters reduce the timing jitter considerably
even for WDM systems [215]. Filtering can also relax the condition in Eq. (9.7.13),
allowingLcollto approachLA, and thus helps to increase the number of channels in a
WDM system [217].
The technique of synchronous modulation can also be applied to WDM systems
for controlling timing jitter [219]. In a 1996 experiment involving four channels, each
operating at 10 Gb/s, transmission over transoceanic distances was achieved by us-
ing modulators every 500 km [220]. When modulators were inserted every 250 km,
three channels, each operating at 20 Gb/s, could be transmitted over transoceanic dis-
tances [221]. The main disadvantage of modulators is that demultiplexing of individual
channels is necessary. Moreover, they require a clock signal that is synchronized to the
bit stream. For this reason, the technique of synchronous modulation is rarely used in
practice.