9.4. DISPERSION-MANAGED SOLITONS 429
deviations are minimized in each section [80]. In another approach, fibers of different
GVD valuesDmand different lengthsLmare chosen such that the productDmLmis the
same for each section. In a third approach,DmandLmare selected to minimize shading
of dispersive waves [81].
9.4.2 Periodic Dispersion Maps
A disadvantage of the DDF is that the average dispersion along the link is often rel-
atively large. Generally speaking, operation of a soliton in the region of low average
GVD improves system performance. Dispersion maps consisting of alternating-GVD
fibers are attractive because their use lowers the average GVD of the entire link while
keeping the GVD of each section large enough that the four-wave mixing (FWM) and
TOD effects remain negligible.
The use of dispersion management forces each soliton to propagate in the normal-
dispersion regime of a fiber during each map period. At first sight, such a scheme
should not even work because the normal-GVD fibers do not support bright solitons
and lead to considerable broadening and chirping of the pulse. So, why should solitons
survive in a dispersion-managed fiber link? An intense theoretical effort devoted to
this issue since 1996 has yielded an answer with a few surprises [85]–[102]. Physically
speaking, if the map period is a fraction of the nonlinear length, the nonlinear effects
are relatively small, and the pulse evolves in a linear fashion over one map period. On
a longer length scale, solitons can still form if the SPM effects are balanced by the
average dispersion. As a result, solitons can survive in an average sense, even though
not only the peak power but also the width and shape of such solitons oscillate period-
ically. This section describes the properties of dispersion-managed (DM) solitons and
the advantages offered by them.
Consider a simple dispersion map consisting of two fibers with positive and nega-
tive values of the GVD parameterβ 2. Soliton evolution is still governed by Eq. (9.4.1)
used earlier for DDFs. However, we cannot useξandτas dimensionless parameters
because the pulse width and GVD both vary along the fiber. It is better to use the
physical units and write Eq. (9.4.1) as
i∂B
∂z−
β 2 (z)
2∂^2 B
∂t^2+γp(z)|B|^2 B= 0 , (9.4.5)whereB=A/
√
pandp(z)is the solution of Eq. (9.3.11). The GVD parameter takes
valuesβ 2 aandβ 2 nin the anomalous and normal sections of lengthslaandln, respec-
tively. The map periodLmap=la+lncan be different from the amplifier spacingLA.
As is evident, the properties of DM solitons will depend on several map parameters
even when only two types of fibers are used in each map period.
Equation (9.4.5) can be solved numerically using the split-step Fourier method.
Numerical simulations show that a nearly periodic solution can often be found by ad-
justing input pulse parameters (width, chirp, and peak power) even though these pa-
rameters vary considerably in each map period. The shape of such DM solitons is
typically closer to a Gaussian profile rather than the “sech” shape associated with stan-
dard solitons [86]–[88].