"Introduction". In: Fiber-Optic Communication Systems

9.5. IMPACT OF AMPLIFIER NOISE 435

transmission over 1000 km. Longer transmission distances were realized using a re-
circulating fiber loop [112] consisting of 102 km of standard fiber with anomalous
GVD (β 2 ≈−21 ps^2 /km) and 17.3 km of DCF with normal GVD (β 2 ≈160 ps^2 /km).
The map strengthSwas quite large in this experiment when 30-ps (FWHM) pulses
were launched into the loop. By 1999, 10-Gb/s DM solitons could be transmitted over
16 Mm of standard fiber when soliton interactions were minimized by choosing the
location of amplifiers appropriately [113].

9.5 Impact of Amplifier Noise

The use of in-line optical amplifiers affects the soliton evolution considerably. The
reason is that amplifiers, needed to restore the soliton energy, also add noise originating
fromamplified spontaneous emission(ASE). As discussed in Section 6.5, the spectral
density of ASE depends on the amplifier gainGitself and is given by Eq. (6.1.15). The
ASE-induced noise degrades the SNR through amplitude fluctuations and introduces
timing jitter through frequency fluctuations, both of which impact the performance of
soliton systems. Timing jitter for solitons has been studied since 1986 and is referred
to as the Gordon–Haus jitter [115]–[125]. The moment method is used in this section
for studying the effects of amplifier noise.

9.5.1 Moment Method

The moment method has been introduced in Section 6.5.2 in the context of nonsoliton
pulses. The same treatment can be extended for solitons [122]. In the case of Eq.
(9.4.5), the three moments providing energyE, frequency shiftΩ, and positionqof the
pulse are given by

E=

∫∞

−∞

|B|^2 dt, q=

1

E

∫∞

−∞

t|B|^2 dt, (9.5.1)

Ω=

i

2 E

∫∞

−∞

(

B∗

∂B

∂t

−B

∂B∗

∂t

)

dt. (9.5.2)

The three quantities depend onzand vary along the fiber as the pulse shape governed
by|B(z,t)|^2 evolves. DifferentiatingE,Ω, andqwith respect tozand using Eq. (9.4.5),
the three moments are found to evolve withzas [124]

dE

dz

=∑

n

δEnδ(z−zn), (9.5.3)

dΩ

dz

=∑

n

δΩnδ(z−zn), (9.5.4)

dq

dz

=β 2 Ω+∑

n

δqnδ(z−zn), (9.5.5)

whereδEn,δΩn, andδqnare random changes induced by ASE at thenth amplifier lo-
cated atzn. The sum in these equations extends over the total numberNAof amplifiers.