"Introduction". In: Fiber-Optic Communication Systems

9.5. IMPACT OF AMPLIFIER NOISE 439

Notice thatT 0 is also equal toTmfor standard solitons which remain unchirped during
propagation and maintain their width all along the fiber. At first site, it appears that DM
solitons have a variance larger by a factor of 3 compared with the standard solitons.
However, this is not the case if we recall that the input pulse energyE 0 is enhanced for
DM solitons by a factor typically exceeding 3. As a result, the variance of frequency
fluctuations is expected to be smaller for DM solitons.
Frequency fluctuations do not affect a soliton system directly unless a coherent
detection scheme with frequency or phase modulation is employed (see Chapter 10).
Nevertheless, they play a significant indirect role by inducing timing jitter such that the
pulse in each 1 bit shifts from the center of its assigned bit slot in a random fashion.
We turn to this issue next.

9.5.3 Timing Jitter

If optical amplifiers compensate for fiber losses, one may ask what limits the total
transmission distance of a soliton link. The answer is provided by the timing jitter
induced by optical amplifiers [115]–[125]. The origin of timing jitter can be understood
by noting that a change in the soliton frequency byΩaffects the group velocity or
the speed at which the pulse propagates through the fiber. IfΩfluctuates because of
amplifier noise, soliton transit time through the fiber link also becomes random.
To calculate the variance of pulse-position fluctuations, we integrate Eq. (9.5.5)
over the fiber section between two amplifiers and obtain the recurrence relation

q(zn)=q(zn− 1 )+Ω(zn− 1 )

∫zn

zn− 1

β 2 (z)dz+δqn, (9.5.26)

whereq(zn)denotes the position at the output of thenth amplifier. This equation shows
that the pulse position changes between any two amplifiers for two reasons. First, the
cumulative frequency shiftΩ(zn− 1 )produces a temporal shift if the GVD is not zero
because of changes in the group velocity. Second, thenth amplifier shifts the position
randomly byδqn. It is easy to solve this recurrence relation for a cascaded chain ofNA
amplifiers to obtain the final position in the form

qf=

NA

∑

n= 1

δqn+β ̄ 2 LA

NA

∑

n= 1

n− 1

∑

i= 1

δΩi, (9.5.27)

whereβ ̄ 2 is the average value of the GVD and the double sum stems from the cumula-
tive frequency shift appearing in Eq. (9.5.26).
Timing jitter is calculated from this equation usingσt^2 =〈q^2 f〉−〈qf〉^2 together with
〈qf〉=0. As before, the average can be performed by noting that fluctuations at two
different amplifiers are not correlated. However, the timing jitter depends not only on
the variances of position and frequency fluctuations but also on the cross-correlation
function〈δqδΩ〉at the same amplifier. The result can be written as

σt^2 =

NA

∑

n= 1

〈(δq)^2 〉+β ̄ 2 LA

NA

∑

n= 1

(n− 1 )〈δqδΩ〉+(β ̄ 2 LA)^2

NA

∑

n= 1

(n− 1 )^2 〈(δΩ)^2 〉.(9.5.28)