66 CHAPTER 2. OPTICAL FIBERS
worst case in which all channels simultaneously carry 1 bits and all pulses overlap is
given by
φNLj =(γ/α)( 2 M− 1 )Pj. (2.6.17)
It is difficult to estimate the impact of XPM on the performance of multichannel
lightwave systems. The reason is that the preceding discussion has implicitly assumed
that XPM acts in isolation without dispersive effects and is valid only for CW opti-
cal beams. In practice, pulses in different channels travel at different speeds. The
XPM-induced phase shift can occur only when two pulses overlap in time. For widely
separated channels they overlap for such a short time that XPM effects are virtually
negligible. On the other hand, pulses in neighboring channels will overlap long enough
for XPM effects to accumulate. These arguments show that Eq. (2.6.17) cannot be used
to estimate the limiting input power.
A common method for studying the impact of SPM and XPM uses a numerical
approach. Equation (2.4.9) can be generalized to include the SPM and XPM effects by
adding a nonlinear term. The resulting equation is known as the nonlinear Schr ̈odinger
equation and has the form [31]
∂A
∂z+
iβ 2
2∂^2 A
∂t^2=−
α
2A+iγ|A|^2 A, (2.6.18)where we neglected the third-order dispersion and added the term containingαto ac-
count for fiber losses. This equation is quite useful for designing lightwave systems
and will be used in later chapters.
Since the nonlinear parameterγdepends inversely on the effective core area, the
impact of fiber nonlinearities can be reduced considerably by enlargingAeff. As seen in
Table 2.1,Aeffis about 80μm^2 for standard fibers but reduces to 50μm^2 for dispersion-
shifted fibers. A new kind of fiber known as large effective-area fiber (LEAF) has been
developed for reducing the impact of fiber nonlinearities. The nonlinear effects are not
always detrimental for lightwave systems. Numerical solutions of Eq. (2.6.18) show
that dispersion-induced broadening of optical pulses is considerably reduced in the case
of anomalous dispersion [81]. In fact, an optical pulse can propagate without distortion
if the peak power of the pulse is chosen to correspond to a fundamental soliton. Solitons
and their use for communication systems are discussed in Chapter 9.
2.6.3 Four-Wave Mixing
The power dependence of the refractive index seen in Eq. (2.6.12) has its origin in the
third-order nonlinear susceptibility denoted byχ(^3 )[73]. The nonlinear phenomenon,
known asfour-wave mixing(FWM), also originates fromχ(^3 ). If three optical fields
with carrier frequenciesω 1 ,ω 2 , andω 3 copropagate inside the fiber simultaneously,
χ(^3 )generates a fourth field whose frequencyω 4 is related to other frequencies by a
relationω 4 =ω 1 ±ω 2 ±ω 3. Several frequencies corresponding to different plus and
minus sign combinations are possible in principle. In practice, most of these com-
binations do not build up because of a phase-matching requirement [31]. Frequency
combinations of the formω 4 =ω 1 +ω 2 −ω 3 are often troublesome for multichannel
communication systems since they can become nearly phase-matched when channel