Nonlinear Fiber Optics - 4 ed. Agrawal

404 Chapter 10. Four-Wave Mixing
The same equations hold for the creation of signal photons if we exchange the subscripts
3 and 4.
The three terms on the right side of Eqs. (10.5.11) and (10.5.12) show how different
combinations of pump and signal photons produce idler photons and lead to the following
selection rules for the FWM process. The first term in these equations corresponds
to the case in which both pumps and the signal are copolarized and produce the idler
with the same SOP. Physically, if both pump photons are in the ↑ state with a total
angular momentum of 2¯h, the signal and idler photons must also be in the same state
to conserve the angular momentum. The last two terms in Eq. (10.5.11) correspond
to the case in which two pump photons are orthogonally polarized such that their total
angular momentum is zero. To conserve this value, the signal and idlers must also be
orthogonally polarized. As a result, a signal photon in the state ↑ 3 can only couple to
an idler photon with state ↓ 4 (and vice versa). This leads to two possible combinations,
↑ 3 + ↓ 4 and ↓ 3 + ↑ 4 , both of which are equally probable.
A pump with an arbitrary SOP consists of photons in both the ↑ and ↓ states with
different amplitudes and phases. All six terms in Eqs. (10.5.11) and (10.5.12) contribute
to FWM in this case. The polarization dependence of signal gain is a consequence
of the fact that FWM can occur through different paths, and one must add
probability amplitudes of these paths, as dictated by quantum mechanics. Such an addition
can lead to constructive or destructive interference. For example, if the two pumps
are right-circularly polarized, no FWM can occur for a signal that is left-circularly
polarized (and vice versa).
In the case of single-pump configuration, the two pump photons have the same SOP
as they are indistinguishable. If the pump is circularly polarized, it follows from Eqs.
(10.5.11) and (10.5.12) that only the first term can produce idler photons. In the case of
a linearly polarized pump, all terms can produce idler photons as long as the selection
rules are satisfied. However, it is easy to conclude that the signal gain for a singlepump
FOPA is always polarization-dependent. Physically speaking, it is impossible
to balance the FWM efficiency experienced by the ↑ and ↓ components of the signal,
unless a polarization-diversity loop is employed.
Often, one is interested in a FOPA configuration that yields the same signal gain
irrespective of the SOP of the input signal. Equations (10.5.11) and (10.5.12) show
that this situation can be realized by two orthogonally polarized pumps. More specifically,
if the two pumps are right- and left-circularly polarized, the terms containing
U 1 U 2 and D 1 D 2 vanish, and the FWM process becomes polarization-independent.
If the two pumps are linearly polarized with orthogonal SOPs, it turns out that
U 1 D 2 + D 1 U 2 = 0. The remaining two paths have the same efficiency, making the
FWM process polarization-independent. However, the gain is reduced considerably
for this configuration, as discussed next.
To study how the parametric gain depends on pump SOPs, we solve Eqs. (10.5.6)–
(10.5.9) for two pumps that are elliptically polarized with orthogonal SOPs. Noting
that these equations are invariant under rotations in the x–y plane, we choose the x
and y axes along the principal axes of the SOP ellipse associated with the pump at
frequency ω 1 . If we ignore the SPM and XPM terms in Eqs. (10.5.6) and (10.5.7), the