7.7. OPTICAL PHASE CONJUGATION 301
7.7.2 Compensation of Self-Phase Modulation
As discussed in Section 2.6, the nonlinear phenomenon of SPM leads to fiber-induced
chirping of the transmitted signal. Section 7.3 indicated that this chirp can be used to
advantage with a proper design. Optical solitons also use the SPM to their advantage
(see Chapter 9). However, in most lightwave systems, the SPM-induced nonlinear
effects degrade the signal quality, especially when the signal is propagated over long
distances using multiple optical amplifiers (see Section 6.5).
The OPC technique differs from all other dispersion-compensation schemes in one
important way: Under certain conditions, it can compensate simultaneously for both
the GVD and SPM. This feature of OPC was noted in the early 1980s [104] and has
been studied extensively after 1993 [97]. It is easy to show that both the GVD and
SPM are compensated perfectly in the absence of fiber losses. Pulse propagation in a
lossy fiber is governed by Eq. (5.3.1) or by
∂A
∂z+
iβ 2
2∂^2 A
∂t^2=iγ|A|^2 A−α
2A, (7.7.4)
where theβ 3 term is neglected andαaccounts for the fiber losses. Whenα=0,
A∗satisfies the same equation when we take the complex conjugate of Eq. (7.7.4)
and changezto−z. As a result, midspan OPC can compensate for SPM and GVD
simultaneously.
Fiber losses destroy this important property of midspan OPC. The reason is intu-
itively obvious if we note that the SPM-induced phase shift is power dependent. As a
result, much larger phase shifts are induced in the first-half of the link than the second
half, and OPC cannot compensate for the nonlinear effects. Equation (7.7.4) can be
used to study the impact of fiber losses. By making the substitution
A(z,t)=B(z,t)exp(−αz/ 2 ), (7.7.5)Eq. (7.7.4) can be written as
∂B
∂z+
iβ 2
2∂^2 B
∂t^2=iγ(z)|B|^2 B, (7.7.6)whereγ(z)=γexp(−αz). The effect of fiber losses is mathematically equivalent to
the loss-free case but with az-dependent nonlinear parameter. By taking the complex
conjugate of Eq. (7.7.6) and changingzto−z, it is easy to see that perfect SPM com-
pensation can occur only ifγ(z)=γ(L−z). This condition cannot be satisfied when
α =0.
One may think that the problem can be solved by amplifying the signal after OPC
so that the signal power becomes equal to the input power before it is launched in the
second-half section of the fiber link. Although such an approach reduces the impact
of SPM, it does not lead to perfect compensation of it. The reason can be understood
by noting that propagation of a phase-conjugated signal is equivalent to propagating
atime-reversedsignal [105]. Thus, perfect SPM compensation can occur only if the
power variations are symmetric around the midspan point where the OPC is performed