Nonlinear Fiber Optics - 4 ed. Agrawal

5.4. Perturbation of Solitons 149
where d(ξ )=|β 2 (ξ )/β 2 (0)| is the normalized local GVD. The distance ξ is normalized
to the dispersion length L D = T0 2/|β
2(0)|, defined using the GVD value at the input
end of the fiber.
If we rescale ξ using the transformation ξ ′ = ∫ ξ
0
p(ξ )dξ , Eq. (5.4.12) becomes
i ∂v
∂ξ ′ + 1 ∂ 2 v
2 ∂τ 2 + e−Γξ
d(ξ ) |v|2 v = 0, (5.4.13)
If the GVD profile is chosen such that d(ξ )=exp(−Γξ ), Eq. (5.4.13) reduces to the
standard NLS equation. Thus, fiber losses have no effect on soliton propagation if the
GVD of a fiber decreases exponentially along its length as
|β 2 (z)| = |β 2 (0)|exp(−αz). (5.4.14)
This result can be easily understood from Eq. (5.2.3). If the soliton peak power P 0
decreases exponentially with z, the requirement N = 1 can still be maintained at every
point along the fiber if |β 2 | were also to reduce exponentially.
Fibers with a nearly exponential GVD profile have been fabricated [150]. A practical
technique for making such DDFs consists of reducing the core diameter along
fiber length in a controlled manner during the fiber-drawing process. Variations in the
core diameter change the waveguide contribution to β 2 and reduce its magnitude. Typically,
GVD can be changed by a factor of 10 over a length of 20–40 km. The accuracy
realized by the use of this technique is estimated to be better than 0.1 ps 2 /km [151].
Since DDFs are not available commercially, fiber loss is commonly compensated by
amplifying solitons.
5.4.3 Soliton Amplification
As already discussed, fiber losses lead to broadening of solitons. Such loss-induced
broadening is unacceptable for many applications, especially when solitons are used
for optical communications. To overcome the effect of fiber losses, solitons need to
be amplified periodically so that their energy is restored to their initial value. Two
different approaches have been used for soliton amplification [129]–[136]. These are
known as lumped and distributed amplification schemes and are shown in Figure 5.13
schematically. In the lumped scheme [130], an optical amplifier boosts the soliton
energy to its input level after the soliton has propagated a certain distance. The soliton
then readjusts its parameters to their input values. However, it also sheds a part
of its energy as dispersive waves (continuum radiation) during this adjustment phase.
The dispersive part is undesirable and can accumulate to significant levels over a large
number of amplification stages.
This problem can be solved by reducing the spacing L A between amplifiers such
that L A ≪ L D . The reason is that the dispersion length L D sets the scale over which
a soliton responds to external perturbations. If the amplifier spacing is much smaller
than this length scale, soliton width is hardly affected over one amplifier spacing in
spite of energy variations. In practice, the condition L A ≪ L D restricts L A typically in
the range 20–40 km even when the dispersion length exceeds 100 km [130]. Moreover,