"Introduction". In: Fiber-Optic Communication Systems

6.5. SYSTEM APPLICATIONS 269

uncontrolled. The tolerable value of the jitter can be estimated assuming the Gaussian
statistics forqso that

p(q)=( 2 πσt^2 )−^1 /^2 exp(−q^2 / 2 σt^2 ). (6.5.29)

The BER can be calculated following the method of Section 4.5. If we assume that
an error occurs whenever the pulse has moved out of the bit slot, we need to find the
accumulated probability for|q|to exceedTB/2, whereTB≡ 1 /Bis the bit slot. This
probability is found to be

BER= 2

∫∞

TB/ 2

p(q)dq=erfc

(

TB

2

√

2 σt

)

≈

4 σt

√

2 πTB

exp

(

−

TB^2

8 σt^2

)

, (6.5.30)

where erfc stands for the complimentary error function defined in Eq. (4.5.5). To re-
duce the BER below 10−^9 forσt/TBshould be less than 8% of the bit slot, resulting
in a tolerable value of the jitter of 8 ps for 10-Gb/s systems and only 2 ps for 40-Gb/s
systems. Clearly, the average dispersion of a fiber link should nearly vanish if the sys-
tem is designed not to be limited by the ASE-induced jitter. This can be accomplished
through dispersion management discussed in Chapter 7.

6.5.4 Accumulated Dispersive and Nonlinear Effects

Many single-channel experiments performed during the early 1990s demonstrated the
benefits of in-line amplifiers for increasing the transmission distance of point-to-point
fiber links [127]–[132]. These experiments showed that fiber dispersion becomes the
limiting factor in periodically amplified long-haul systems. Indeed, the experiments
were possible only because the system was operated close to the zero-dispersion wave-
length of the fiber link. Moreover, the zero-dispersion wavelength varied along the link
in such a way that the total dispersion over the entire link length was quite small at the
operating wavelength of 1.55μm. By 1992, the total system length could be increased
to beyond 10,000 km using such dispersion-management techniques. In a 1992 exper-
iment [130], a 2.5-Gb/s signal was transmitted over 10,073 km using 199 EDFAs. An
effective transmission distance of 21,000 km at 2.5 Gb/s and of 14,300 km at 5 Gb/s
was demonstrated using a recirculating fiber loop [133].
A crude estimate of dispersion-limitedLTcan be obtained if the input power is low
enough that one can neglect the nonlinear effects during signal transmission. Since
amplifiers compensate only for fiber losses, dispersion limitations discussed in Section
5.2.2 and shown in Fig. 5.4 apply for each channel of a WDM system ifLis replaced
byLT. From Eq. (5.2.3), the dispersion limit for systems making use of standard fibers
(β 2 ≈−20 ps^2 /km at 1.55μm) isB^2 LT<3000 (Gb/s)^2 -km: The distance is limited to
below 30 km at 10 Gb/s for such fibers. An increase by a factor of 20 can be realized by
using dispersion-shifted fibers. To extend the distance to beyond 5000 km at 10 Gb/s,
the average GVD along the link should be smaller thanβ ̄ 2 =− 0 .1ps^2 /km.
The preceding estimate is crude since it does not include the impact of the non-
linear effects. Even though power levels are relatively modest for each channel, the
nonlinear effects can become quite important because of their accumulation over long
distances [29]. Moreover, amplifier noise often forces one to increase the channel