Nonlinear Fiber Optics - 4 ed. Agrawal

10.5. Polarization Effects 401
Brillouin gain discussed in Section 9.1. Second, the group-velocity mismatch among
the four pulses forces them to separate from each other. Both of these effects reduce
the effective length over which FWM can occur. For ultrashort pulses, one must also
include the GVD effects through Eq. (10.2.23).
10.5 Polarization Effects
The scalar theory of Section 10.2 is based on the assumption that all optical fields are
linearly polarized initially and maintain their SOP during propagation inside the fiber.
In practice, the SOP of the input pumps can be chosen, but the SOP of the input signal is
often arbitrary. The FWM process is highly polarization-dependent because it requires
conservation of angular momenta among the four interacting photons. As a result, the
gain spectrum of a FOPA depends on the relative SOPs of the input signal and pumps
and can change over a wide range depending on them. This section focuses on such
polarization effects.
It was shown as early as 1993 that polarization-independent operation of singlepump
FOPAs can be realized by employing a polarization-diversity loop, but the focus
was on wavelength conversion [101]. This technique was used in a 2002 experiment
to construct a FOPA whose gain was independent of the signal SOP to within 1 dB
[102]. In this scheme, the input pump beam is split by a polarizing beam splitter into
its orthogonally polarized components such that they have equal powers and counterpropagate
inside the loop containing the nonlinear fiber. The signal is also split by this
beam splitter into its orthogonally polarized components. Each of these components is
amplified inside the loop by the copolarized pump component. The two components
of the amplified signal are then recombined by the same polarizing beam splitter after
one round trip.
The dual-pump configuration of a FOPA provides an additional degree of freedom
that can be used to realize polarization-independent operation without a polarizationdiversity
loop [46]. It was shown in 1993 that, if the two pumps are launched with
linear but orthogonal SOPs, the FWM process can be made nearly independent of
the signal SOP [87]. This configuration is often used for dual-pump FOPAs, even
though its use reduces the signal and idler gains by a large factor because of a reduced
FWM efficiency. One may ask if the efficiency of the underlying FWM process can be
improved by choosing pump SOPs such that they are orthogonal but not linear [103].
To answer this question, it is necessary to develop a vector theory of FWM in which
the input SOPs of the two pumps and that of the signal are arbitrary [104].
10.5.1 Vector Theory of Four-Wave Mixing
A complete description of dual-pump FOPAs should include degenerate as well as
nondegenerate FWM processes. However, as discussed earlier, when two pumps are
located symmetrically around but relatively far from the zero-dispersion wavelength of
the fiber, the FOPA gain spectrum exhibits a central flat region having its origin in the
single nondegenerate FWM process (ω 1 + ω 2 → ω 3 + ω 4 ). As the flat portion of the
gain spectrum is used in practice, we may focus only on this nondegenerate process