9.3. LOSS-MANAGED SOLITONS 419
Figure 9.9: Broadening of fundamental solitons in lossy fibers (Γ= 0 .07). The curve marked
“exact” shows numerical results. Dashed curve shows the behavior expected in the absence of
nonlinear effects. (After Ref. [55];©c1985 Elsevier; reprinted with permission.)
whereΓ=αLDrepresents fiber losses over one dispersion length. WhenΓ1, the last
term can be treated as a small perturbation [54]. The use of variational or perturbation
methods results in the following approximate solution of Eq. (9.3.1):
u(ξ,τ)≈e−Γξsech(τe−Γξ)exp[i( 1 −e−^2 Γξ)/ 4 Γ]. (9.3.2)The solution (9.3.2) shows that the soliton width increases exponentially because
of fiber losses as
T 1 (ξ)=T 0 exp(Γξ)=T 0 exp(αz). (9.3.3)Such an exponential increase in the soliton width cannot be expected to continue for
arbitrarily long distances. Numerical solutions of Eq. (9.3.1) indeed show a slower
increase forξ1 [55]. Figure 9.9 shows the broadening factorT 1 /T 0 as a function of
ξwhen a fundamental soliton is launched into a fiber withΓ= 0 .07. The perturbative
result is also shown for comparison; it is reasonably accurate up toΓξ=1. The dashed
line in Fig. 9.9 shows the broadening expected in the absence of nonlinear effects.
The important point to note is that soliton broadening is much less compared with the
linear case. Thus, the nonlinear effects can be beneficial even when solitons cannot be
maintained perfectly because of fiber losses. In a 1986 study, an increase in the repeater
spacing by more than a factor of 2 was predicted using higher-order solitons [56].
In modern long-haul lightwave systems, pulses are transmitted over long fiber
lengths without using electronic repeaters. To overcome the effect of fiber losses, soli-
tons should be amplified periodically using either lumped or distributed amplification
[57]–[60]. Figure 9.10 shows the two schemes schematically. The next two subsec-
tions focus on the design issues related to loss-managed solitons based on these two
amplification schemes.