9.7. WDM SOLITON SYSTEMS 463
9.7.4 Dispersion Management
As discussed in Section 8.3.6, FWM is the most limiting factor for WDM systems when
GVD is kept constant along the fiber link. The FWM problem virtually disappears
when the dispersion-management technique is used. In fact, dispersion management
is essential if a WDM soliton system is designed to transmit more than two or three
channels. Starting in 1996, dispersion management was used for WDM soliton systems
almost exclusively.
Dispersion-Decreasing Fibers
It is intuitively clear that DDFs with a continuously varying GVD profile should help a
WDM system. We can use Eq. (9.7.1) for finding the optimum GVD profile. By tailor-
ing the fiber dispersion asp(ξ)=exp(−Γξ), the same exponential profile encountered
in Section 8.4.1, the parameterbbecomes 1 all along the fiber link, resulting in an
unperturbed NLS equation. As a result, soliton collisions become symmetric despite
fiber losses, irrespective of the ratioLcoll/LA. Consequently, no residual frequency
shifts occur after a soliton collision for WDM systems making use of DDFs with an
exponentially decreasing GVD.
Lumped amplifiers introduce a new mechanism of FWM in WDM systems. In their
presence, soliton energy varies in a periodic fashion over each loss–amplification cycle.
Such periodic variations in the peak power of solitons create a nonlinear-index grating
that can nearly phase-match the FWM process [222]. The phase-matching condition
can be written as
|β 2 |(Ωch/T 0 )^2 = 2 πm/LA, (9.7.15)
wheremis an integer and the amplifier spacingLAis the period of the index grating.
As a result of such phase matching, a few percent of soliton energy can be transferred
to the FWM sidebands even when GVD is relatively large [222]. Moreover, FWM
occurring during simultaneous collision of three solitons leads to permanent frequency
shifts for the slowest- and fastest-moving solitons together with an energy exchange
among all three channels [223].
FWM phase-matched by the nonlinear-index grating can also be avoided by us-
ing DDFs with an exponential GVD profile. The reason is related to the symmetric
nature of soliton collisions in such systems. When collisions are symmetric, energy
transferred to the FWM sidebands during the first half of a collision is returned back
to the soliton during the second half of the same collision. Thus, the spectral side-
bands generated through FWM do not grow with propagation of solitons. In practice,
the staircase approximation for the exponential profile is used, employing multiple
constant-dispersion fibers between two amplifiers.
Figure 9.27 shows the residual energy remaining in a FWM sideband as a func-
tion of amplifier lengthLAwhen the exponential GVD profile is approximated using
m=2, 3, and 4 fiber sections chosen such that the productDmLmis the same for all
m[222]. HereDmis the dispersion parameter in themth section of lengthLm. The case
of constant-dispersion fibers is also shown for comparison. The average dispersion is
0.5 ps/(km-nm) in all cases. The double-peak nature of the curve in this case is due to
the phase-matching condition in Eq. (9.7.15), which can be satisfied for different values