9.7. WDM SOLITON SYSTEMS 461
9.7.2 Effect of Lumped Amplification
The situation is worse in loss-managed soliton systems in which fiber losses are com-
pensated periodically through lumped amplifiers. The reason is that soliton collisions
are affected adversely by variations in the pulse energy. Physically, large energy vari-
ations occurring over a collision length destroy the symmetric nature of the collision.
Mathematically, theξdependence ofb(ξ)in Eq. (9.7.8) changes the frequency shift.
As a result, solitons do not recover their original frequency and velocity after the col-
lision is over. Equation (9.7.9) can be used to calculate theresidualfrequency shift
for a given form ofb(ξ). Figure 9.26(b) shows the residual shift as a function of the
ratioLcoll/Lpert, whereLpertis equal to the amplifier spacingLA[206]. Numerical sim-
ulations (circles) agree with the prediction of Eq. (9.7.9). The residual frequency shift
increases rapidly asLcollapproachesLAand can become∼ 0 .1 GHz. Such shifts are
not acceptable in practice since they accumulate over multiple collisions and produce
velocity changes large enough to move the soliton out of the bit slot.
WhenLcollis large enough that a collision lasts over several amplifier spacings,
effects of gain–loss variations begin to average out, and the residual frequency shift
decreases. As seen in Fig. 9.26, it virtually vanishes forLcoll> 2 LA(safe region).
SinceLcollis inversely related to the channel spacingΩch, this condition sets a limit on
the maximum separation between the two outermost channels of a WDM system. The
shortest collision length is obtained by replacingΩchin Eq. (9.7.4) withNchΩch. Using
Lcoll> 2 LA, the number of WDM channels is limited to
Nch<TsLD
T 0 ΩchLA. (9.7.13)
One may think that the number of channels can be increased by reducingΩch.How-
ever, its minimum value is limited to aboutΩch= 5 ∆ωs, where∆ωsis the spectral width
(FWHM) of solitons, because of interchannel crosstalk [207]. Using this condition in
Eq. (9.7.13), the number of WDM channels is limited such thatNch<LD/ 3 LA. Using
LD=T 02 /|β 2 |andB=( 2 q 0 T 0 )−^1 , this condition can be written as a simple design rule:
NchB^2 LA<( 12 q^20 |β 2 |)−^1. (9.7.14)For the typical valuesq 0 =5,|β 2 |= 0 .8ps^2 /km, andLA=40 km, the condition be-
comesB
√
Nch<10 Gb/s. The number of channels can be as large as 16 at a relatively
low bit rate of 2.5 Gb/s but only a single channel is allowed at 10 Gb/s. Clearly, inter-
channel collisions limit the usefulness of the WDM technique severely.
9.7.3 Timing Jitter
In addition to the sources of timing jitter discussed in Section 9.6 for a single isolated
channel, several other sources of jitter become important for WDM systems [208]–
[213]. First, each interchannel collision generates a temporal shift [see Eq. (9.7.12)]
of the same magnitude for both solitons but in opposite directions. Even though the
temporal shift scales asΩ−ch^2 and decreases rapidly with increasingΩch, the number
of collisions increases linearly withΩch. As a result, the total time shift scales as