5.3. LONG-HAUL SYSTEMS 195
different dispersion characteristics. Furthermore, mode mixing occurring at splices and
connectors tends to average out the propagation delay associated with different modes
of a multimode fiber. A statistical approach is often necessary to estimate the fiber
bandwidth and the corresponding rise time [22]–[25].
In a phenomenological approach,Tmodalcan be approximated by the time delay∆T
given by Eq. (2.1.5) in the absence of mode mixing, i.e.,
Tmodal≈(n 1 ∆/c)L, (5.2.13)wheren 1 ≈n 2 was used. For graded-index fibers, Eq. (2.1.10) is used in place of Eq.
(2.1.5), resulting inTmodal≈(n 1 ∆^2 / 8 c)L. In both cases, the effect of mode mixing is
included by changing the linear dependence onLby a sublinear dependenceLq, where
qhas a value in the range 0.5–1, depending on the extent of mode mixing. A reasonable
estimate based on the experimental data isq= 0 .7. The contributionTGVDcan also be
approximated by∆Tgiven by Eq. (2.3.4), so that
TGVD≈|D|L∆λ, (5.2.14)where∆λ is the spectral width of the optical source (taken as a full width at half
maximum). The dispersion parameterDmay change along the fiber link if different
sections have different dispersion characteristics; an average value should be used in
Eq. (5.2.14) in that case.
As an illustration of the rise-time budget, consider a 1.3-μm lightwave system de-
signed to operate at 1 Gb/s over a single-mode fiber with a repeater spacing of 50 km.
The rise times for the transmitter and the receiver have been specified asTtr= 0 .25 ns
andTrec= 0 .35 ns. The source spectral width is specified as∆λ=3 nm, whereas the
average value ofDis 2 ps/(km-nm) at the operating wavelength. From Eq. (5.2.14),
TGVD= 0 .3 ns for a link lengthL=50 km. Modal dispersion does not occur in single-
mode fibers. HenceTmodal=0 andTfiber= 0 .3 ns. The system rise time is estimated by
using Eq. (5.2.11) and is found to beTr= 0 .524 ns. The use of Eq. (5.2.10) indicates
that such a system cannot be operated at 1 Gb/s when the RZ format is employed for
the optical bit stream. However, it would operate properly if digital format is changed
to the NRZ format. If the use of RZ format is a prerequisite, the designer must choose
different transmitters and receivers to meet the rise-time budget requirement. The NRZ
format is often used as it permits a larger system rise time at the same bit rate.
5.3 Long-Haul Systems
With the advent of optical amplifiers, fiber losses can be compensated by inserting
amplifiers periodically along a long-haul fiber link (see Fig. 5.1). At the same time,
the effects of fiber dispersion (GVD) can be reduced by using dispersion management
(see Chapter 7). Since neither the fiber loss nor the GVD is then a limiting factor, one
may ask how many in-line amplifiers can be cascaded in series, and what limits the
total link length. This topic is covered in Chapter 6 in the context of erbium-doped
fiber amplifiers. Here we focus on the factors that limit the performance of amplified
fiber links and provide a few design guidelines. The section also outlines the progress