Nonlinear Fiber Optics - 4 ed. Agrawal

148 Chapter 5. Optical Solitons
Figure 5.12: Variation of pulse width with distance in a lossy fiber for the fundamental soliton.
The prediction of perturbation theory is also shown. Dashed curve shows the behavior expected
in the absence of nonlinear effects. (After Ref. [147]; c○1985 Elsevier.)
Recalling that the amplitude and width of a soliton are related inversely, a decrease
in soliton amplitude leads to broadening of the soliton. Indeed, if we write η(τ − q) in
Eq. (5.4.2) as T /T 1 and use τ = T /T 0 , T 1 increases along the fiber exponentially as
T 1 (z)=T 0 exp(Γξ ) ≡ T 0 exp(αz). (5.4.11)
An exponential increase in the soliton width with z cannot be expected to continue
for arbitrarily large distances. This can be seen from Eq. (3.3.12), which predicts a linear
increase with z when the nonlinear effects become negligible. Numerical solutions
of Eq. (5.4.8) show that the perturbative solution is accurate only for values of z such
that αz ≪ 1 [147]. Figure 5.12 shows the broadening factor T 1 /T 0 as a function of ξ
when a fundamental soliton is launched into a fiber with Γ = 0.07. The perturbative
result is acceptable for up to Γξ ≈ 1. In the regime (ξ ≫ 1), pulse width increases
linearly with a rate slower than that of a linear medium [148]. Higher-order solitons
show a qualitatively similar asymptotic behavior. However, their pulse width oscillates
a few times before increasing monotonically [147]. The origin of such oscillations lies
in the periodic evolution of higher-order solitons.
How can a soliton survive inside lossy optical fibers? An interesting scheme restores
the balance between GVD and SPM in a lossy fiber by changing dispersive
properties of the fiber [149]. Such fibers are called dispersion-decreasing fibers (DDFs)
because their GVD must decrease in such a way that it compensates for the reduced
SPM experienced by the soliton, as its energy is reduced by fiber loss. To see which
GVD profile is needed, we modify Eq. (5.4.8) to allow for GVD variations along the
fiber length and eliminate the loss term using u = vexp(−Γξ /2), resulting in the following
equation:
i ∂v
∂ξ + d(ξ ) ∂ 2 v
2 ∂τ 2 + e−Γξ |v| 2 v = − i Γu, (5.4.12)
2