302 CHAPTER 7. DISPERSION MANAGEMENT
so thatγ(z)=γ(L−z)in Eq. (7.7.6). Optical amplification does not satisfy this prop-
erty. One can come close to SPM compensation if the signal is amplified often enough
that the power does not vary by a large amount during each amplification stage. This
approach is, however, not practical because it requires closely spaced amplifiers.
Perfect compensation of both GVD and SPM can be realized by usingdispersion-
decreasing fibersfor whichβ 2 decreases along the fiber length. To see how such a
scheme can be implemented, assume thatβ 2 in Eq. (7.7.6) is a function ofz.By
making the transformation
ξ=∫z0γ(z)dz, (7.7.7)Eq. (7.7.6) can be written as [97]
∂B
∂ξ+
i
2b(ξ)∂^2 B
∂t^2=i|B|^2 B, (7.7.8)whereb(ξ)=β 2 (ξ)/γ(ξ). Both GVD and SPM are compensated ifb(ξ)=b(ξL−ξ),
whereξLis the value ofξatz=L. This condition is automatically satisfied when
the dispersion decreases in exactly the same way asγ(z)so thatβ 2 (ξ)=γ(ξ)and
b(ξ)=1. Since fiber losses makeγ(z)to decrease exponentially as exp(−αz), both
GVD and SPM can be compensated exactly in a dispersion-decreasing fiber whose
GVD decreases as exp(−αz). This approach is quite general and applies even when
in-line amplifiers are used.
7.7.3 Phase-Conjugated Signal
The implementation of the midspan OPC technique requires a nonlinear optical ele-
ment that generates the phase-conjugated signal. The most commonly used method
makes use offour-wave mixing(FWM) in a nonlinear medium. Since the optical fiber
itself is a nonlinear medium, a simple approach is to use a few-kilometer-long fiber
especially designed to maximize the FWM efficiency.
The FWM phenomenon in optical fibers has been studied extensively [106]. Its
use requires injection of a pump beam at a frequencyωpthat is shifted from the sig-
nal frequencyωsby a small amount (∼ 0 .5 THz). The fiber nonlinearity generates
the phase-conjugated signal at the frequencyωc= 2 ωp−ωsprovided that thephase-
matching condition kc= 2 kp−ksis approximately satisfied, wherekj=n(ωj)ωc/c
is the wave number for the optical field of frequencyωj. The phase-matching con-
dition can be approximately satisfied if the zero-dispersion wavelength of the fiber is
chosen to coincide with the pump wavelength. This was the approach adopted in the
1993 experiments in which the potential of OPC for dispersion compensation was first
demonstrated. In one experiment [82], the 1546-nm signal was phase conjugated by
using FWM in a 23-km-long fiber with pumping at 1549 nm. The 6-Gb/s signal was
transmitted over 152 km of standard fiber in a coherent transmission experiment em-
ploying the FSK format. In another experiment [83], a 10-Gb/s signal was transmitted
over 360 km. The midspan OPC was performed in a 21-km-long fiber by using a pump
laser whose wavelength was tuned exactly to the zero-dispersion wavelength of the
fiber. The pump and signal wavelengths differed by 3.8 nm. Figure 7.14 shows the