428 CHAPTER 9. SOLITON SYSTEMS
wherev=u/
√
p,d(ξ)=β 2 (ξ)/β 2 ( 0 ), andp(ξ)takes into account peak-power vari-
ations introduced by loss management. The distanceξis normalized to the dispersion
length,LD=T 02 /|β 2 ( 0 )|, defined using the GVD value at the fiber input.
Because of theξdependence of the second and third terms, Eq. (9.4.1) is not a
standard NLS equation. However, it can be reduced to one if we introduce a new
propagation variable as
ξ′=∫ξ0d(ξ)dξ. (9.4.2)This transformation renormalizes the distance scale to the local value of GVD. In terms
ofξ′, Eq. (9.4.1) becomes
i∂v
∂ξ′+
1
2
∂^2 v
∂τ^2+
p(ξ)
d(ξ)|v|^2 v= 0. (9.4.3)If the GVD profile is chosen such thatd(ξ)=p(ξ)≡exp(−Γξ), Eq. (9.4.3) reduces
the standard NLS equation obtained in the absence of fiber losses. As a result, fiber
losses have no effect on a soliton in spite of its reduced energy when DDFs are used.
Lumped amplifiers can be placed at any distance and are not limited by the condition
LALD.
The preceding analysis shows that fundamental solitons can be maintained in a
lossy fiber provided its GVD decreases exponentially as
|β 2 (z)|=|β 2 ( 0 )|exp(−αz). (9.4.4)This result can be understood qualitatively by noting that the soliton peak powerP 0
decreases exponentially in a lossy fiber in exactly the same fashion. It is easy to deduce
from Eq. (9.1.4) that the requirementN=1 can be maintained, in spite of power losses,
if both|β 2 |andγdecrease exponentially at the same rate. The fundamental soliton then
maintains its shape and width even in a lossy fiber.
Fibers with a nearly exponential GVD profile have been fabricated [76]. A practical
technique for making such DDFs consists of reducing the core diameter along the fiber
length in a controlled manner during the fiber-drawing process. Variations in the fiber
diameter change the waveguide contribution toβ 2 and reduce its magnitude. Typically,
GVD can be varied by a factor of 10 over a length of 20 to 40 km. The accuracy realized
by the use of this technique is estimated to be better than 0.1 ps^2 /km [77]. Propagation
of solitons in DDFs has been demonstrated in several experiments [77]–[79]. In a 40-
km DDF, solitons preserved their width and shape in spite of energy losses of more than
8 dB [78]. In a recirculating loop made using DDFs, a 6.5-ps soliton train at 10 Gb/s
could be transmitted over 300 km [79].
Fibers with continuously varying GVD are not readily available. As an alternative,
the exponential GVD profile of a DDF can be approximated with a staircase profile
by splicing together several constant-dispersion fibers with differentβ 2 values. This
approach was studied during the 1990s, and it was found that most of the benefits of
DDFs can be realized using as few as four fiber segments [80]–[84]. How should one
select the length and the GVD of each fiber used for emulating a DDF? The answer
is not obvious, and several methods have been proposed. In one approach, power