"Introduction". In: Fiber-Optic Communication Systems

9.3. LOSS-MANAGED SOLITONS 421

Because of rapid variations in the soliton energy introduced by periodic gain–loss
changes, it is useful to make the transformation

u(ξ,τ)=

√

p(ξ)v(ξ,τ), (9.3.5)

wherep(ξ)is a rapidly varying andv(ξ,τ)is a slowly varying function ofξ. Substi-
tuting Eq. (9.3.5) in Eq. (9.3.4),v(ξ,τ)is found to satisfy

i

∂v

∂ξ

+

1

2

∂^2 v

∂τ^2

+p(ξ)|v|^2 v= 0 , (9.3.6)

wherep(ξ)is obtained by solving the ordinary differential equation

dp

dξ

=[g(ξ)LD−Γ]p. (9.3.7)

The preceding equations can be solved analytically by noting that the amplifier gain is
just large enough thatp(ξ)is a periodic function; it decreases exponentially in each
period asp(ξ)=exp(−Γξ)but jumps to its initial valuep( 0 )=1 at the end of each
period. Physically,p(ξ)governs variations in the peak power (or the energy) of a
soliton between two amplifiers. For a fiber with losses of 0.2 dB/km,p(ξ)varies by a
factor of 100 whenLA=100 km.
In general, changes in soliton energy are accompanied by changes in the soliton
width. Large rapid variations inp(ξ)can destroy a soliton if its width changes rapidly
through emission of dispersive waves. The concept of the path-averaged or guiding-
center soliton makes use of the fact that solitons evolve little over a distance that is
short compared with the dispersion length (or soliton period). Thus, whenξA1,
the soliton width remains virtually unchanged even though its peak powerp(ξ)varies
considerably in each section between two neighboring amplifiers. In effect, we can
replacep(ξ)by its average value ̄pin Eq. (9.3.6) whenξA1. Introducingu=

√

pv ̄
as a new variable, this equation reduces to the standard NLS equation obtained for a
lossless fiber.
From a practical viewpoint, a fundamental soliton can be excited if the input peak
powerPs(or energy) of the path-averaged soliton is chosen to be larger by a factor 1/p ̄.

Introducing the amplifier gain asG=exp(ΓξA)and using ̄p=ξA−^1

∫ξA

0 e

−Γξdξ, the

energy enhancement factor for loss-managed (LM) solitons is given by

fLM=

Ps

P 0

=

1

p ̄

=

ΓξA

1 −exp(−ΓξA)

=

GlnG

G− 1

, (9.3.8)

whereP 0 is the peak power in lossless fibers. Thus, soliton evolution in lossy fibers
with periodic lumped amplification is identical to that in lossless fibers provided (i)
amplifiers are spaced such thatLALDand (ii) the launched peak power is larger by
a factorfLM. As an example,G=10 andfLM≈ 2 .56 for 50-km amplifier spacing and
fiber losses of 0.2 dB/km.
Figure 9.11 shows the evolution of a loss-managed soliton over a distance of 10 Mm
assuming that solitons are amplified every 50 km. When the input pulse width corre-
sponds to a dispersion length of 200 km, the soliton is preserved quite well even after