Nonlinear Fiber Optics - 4 ed. Agrawal

10.6. Applications of Four-Wave Mixing 411
The two pump equations can be solved analytically if the pumps are assumed to
remain undepleted. The SPM and XPM terms in the remaining two equations can
be eliminated by introducing new Jones vectors |B 3 〉 and |B 4 〉 with a transformation
similar to that given in Eq. (10.2.12) for the scalar case. These new Jones vectors are
found to satisfy
d|B 3 〉
dz
d|B ∗ 4 〉
dz
= 8iγ (
)
|A 2 〉〈A ∗
9
1| + |A 1 〉〈A ∗ 2| |B ∗ 4〉e −iκz , (10.5.27)
= − 8iγ
9
(
)
|A 2 〉〈A ∗ 1| + |A 1 〉〈A ∗ 2| |B 3 〉e iκz . (10.5.28)
These two equations generalize Eqs. (10.2.13) and (10.2.14) to the vector case.
They can be solved for arbitrary SOPs of the two pumps and the signal because their
relative orientation does not change along the fiber. The signal amplification factor for
a dual-pump FOPA is found to be
G s = 1 2 [(G + + G − )+(G + − G − )cos(θ s )], (10.5.29)
where θ s is the input angle between the signal Stokes vector and the vector pointing
along ˆp 1 + ˆp 2 (on the Poincaré sphere). Here ˆp 1 and ˆp 2 are input Stokes vectors for the
two pumps. The two gains are defined as
G ± = |cosh(g ± L)+(iκ/2g ± )sinh(g ± L)| 2 , (10.5.30)
g 2 ± =(8γ/9) 2 P 1 P 2 [1 ± cos(θ p /2)] 2 − (κ/2) 2 , (10.5.31)
where θ p is the input angle between ˆp 1 and ˆp 2 .
Equation (10.5.29) provides the amplification factor of a dual-pump FOPA for arbitrary
input SOPs of the two pumps. Consider the perfect phase-matched case by
setting κ = 0. When the pumps are copolarized (θ p = 0), g − = 0 and G − = 1, while
G + = cosh 2 (g + L). It follows from Eq. (10.5.29), that G s varies with the signal SOP in
the range of 1 to G + , the minimum value of 1 occurring when the signal is polarized
orthogonal to the pumps. This SOP dependence of the signal gain can be avoided by
launching orthogonally polarized pumps (θ p = π). In this case, G + = G − , and G s becomes
independent of θ s , i.e., it does not change with the signal SOP. Of course, the
value of G s is reduced considerably because g + is smaller by a factor of 2. For example,
if the signal is amplified by 40 dB when all fields are copolarized, it is amplified
by only 20 dB when the pumps are orthogonally polarized.
10.6 Applications of Four-Wave Mixing
FWM in optical fibers can be both harmful and beneficial depending on the application.
It can induce crosstalk in WDM communication systems and limit the performance of
such systems. The FWM-induced crosstalk can be avoided in practice through dispersion
management, a technique in which the dispersion of each fiber section is made
large enough that the FWM process is not phase matched throughout the link length
[53]. At the same time, FWM is useful for a variety of applications [111]. As already