"Introduction". In: Fiber-Optic Communication Systems

438 CHAPTER 9. SOLITON SYSTEMS

whereEfis the output energy andE 0 is the input energy of the pulse. The energy
variance is calculated using Eq. (9.5.13) with〈δEn〉=Sspand is given by

σE^2 ≡〈E^2 f〉−〈Ef〉^2 = 2 NASspE 0. (9.5.20)

The optical SNR is obtained in a standard manner and is given by

SNR=E 0 /σE=(E 0 / 2 NASsp)^1 /^2. (9.5.21)

Two conclusions can be drawn from this equation. First, the SNR decreases as the
number of in-line amplifiers increases because of the accumulation of ASE along the
link. Second, even though Eq. (9.5.21) applies for both the standard and DM soli-
tons, the SNR is improved for DM solitons because of their higher energies. In fact,

the improvement factor is given byf
1 / 2
DM, wherefDMis the energy enhancement factor
associated with the DM solitons. As an example, the SNR is 14 dB after 100 am-
plifiers spaced 80-km apart for DM solitons with 0.1-pJ energy usingnsp= 1 .5 and
α= 0 .2 dB/km.
Frequency fluctuations induced by optical amplifiers are found by integrating Eq.
(9.5.4) over one amplifier spacing, resulting in the recurrence relation

Ω(zn)=Ω(zn− 1 )+δΩn, (9.5.22)

whereΩ(zn)denotes frequency shift at the output of thenth amplifier. As before,
the total frequency shiftΩffor a cascaded chain ofNAamplifiers is given byΩf=

∑
NA
n= 1 δΩn, where the initial frequency shift atz=0 is taken to be zero because the
soliton frequency equals the carrier frequency at the input end.
The variance of frequency fluctuations can be calculated using

σΩ^2 ≡〈Ω^2 f〉−〈Ωf〉^2 =

NA

∑

n= 1

NA

∑

m= 1

〈δΩnδΩm〉, (9.5.23)

where we used〈Ωf〉=0. The average in this equation can be performed by not-
ing that frequency fluctuations at two different amplifiers are not correlated. Using
〈δΩnδΩm〉=〈(δΩ)^2 〉δnmwith Eq. (9.5.13) and performing the double sum in Eq.
(9.5.23), the frequency variance for DM solitons is given by

σΩ^2 =NA(Ssp/E 0 )[( 1 +C^20 )/T 02 ]=NASsp/(E 0 Tm^2 ), (9.5.24)

whereTmis the minimum pulse width within the dispersion map at the location where
the pulse is transform-limited (no chirp). The variance increases linearly with the num-
ber of amplifiers. It also depends on the widthT 0 andC 0 of the input pulse. However,
the chirp parameter can be eliminated ifσΩ^2 is written in terms of the minimum pulse
width. In practice,Tmis also the width of the pulse at the optical transmitter before it
is prechirped.
In the case of standard solitons, we should use Eq. (9.5.17) while performing the
average in Eq. (9.5.23). The variance of frequency fluctuations in this case becomes

σΩ^2 =NASsp/( 3 E 0 T 02 ). (9.5.25)