Nonlinear Fiber Optics - 4 ed. Agrawal

264 Chapter 7. Cross-Phase Modulation
former. This feature can be verified experimentally by placing a polarizer before the
photodetector and noticing that the oscillatory structure changes as the polarizer is
rotated. When the pump pulse is elliptically polarized, the polarization dependence of
the probe is reduced as both components have some pump power in the same state of
polarization. In particular, when φ = 45 ◦ , the two probe components would evolve in
a nearly identical fashion.
7.7 XPM Effects in Birefringent Fibers
In a birefringent fiber, the orthogonally polarized components of each wave propagate
with different propagation constants because of a slight difference in their effective
mode indices. In addition, these two components are coupled through XPM, resulting
in nonlinear birefringence [106]–[108]. In this section we incorporate the effects of
linear birefringence in the Jones-matrix formalism developed in the preceding section.
7.7.1 Fibers with Low Birefringence
The induced nonlinear polarization can still be written in the form of Eq. (7.6.3) but
Eq. (7.6.4) for the optical field E j should be modified as
E j (r,t)=F j (x,y)[ ˆxA jx exp(iβ jx z)+ŷA jy exp(iβ jy z)], (7.7.1)
where β jx and β jy are the propagation constants for the two orthogonally polarized
components of the optical field with the carrier frequency ω j . It is useful to write Eq.
(7.7.1) in the form
E j (r,t)=F j (x,y)[ ˆxA jx exp(iδβ j z/2)+ŷA jy exp(−iδβ j z/2)]exp(i ¯β j z), (7.7.2)
where ¯β j = 1 2 (β jx + β jy ) is the average value of the propagation constants and δβ j is
their difference. One can now introduce the Jones vector |A j 〉 and write Eq. (7.7.2) in
the form of Eq. (7.6.4), provided |A j 〉 is defined as
( )
A
|A j 〉 = jx exp(iδβ j z/2)
. (7.7.3)
A jy exp(−iδβ j z/2)
At this point, we can follow the method outlined in Section 7.1.2 to obtain the
coupled vector NLS equations. It is important to keep in mind that δβ j (ω) itself is
frequency dependent and it should be expanded in a Taylor series together with ¯β j (ω).
However, it is sufficient in practice to approximate it as
δβ j (ω) ≈ δβ j0 + δβ j1 (ω − ω j ), (7.7.4)
where δβ j1 is evaluated at the carrier frequency ω j . The δβ j1 term is responsible for
different group velocities of the orthogonally polarized components. It should be retained
for fibers with high birefringence but can be neglected when linear birefringence
is relatively small. We assume that to be the case in this subsection.