Nonlinear Fiber Optics - 4 ed. Agrawal

10.5. Polarization Effects 403
following set of four equations [104]:
d|A 1 〉
dz
d|A 2 〉
dz
d|A 3 〉
dz
d|A 4 〉
dz
= 2iγ (
)
〈A 1 |A 1 〉 + 〈A 2 |A 2 〉 + 1
3
2 |A∗ 1〉〈A ∗ 1| + |A 2 〉〈A 2 | + |A ∗ 2〉〈A ∗ 2| |A 1 〉, (10.5.6)
= 2iγ
3
= 2iγ
3
(
)
〈A 1 |A 1 〉 + 〈A 2 |A 2 〉 +
2 1|A∗ 2〉〈A ∗ 2| + |A 1 〉〈A 1 | + |A ∗ 1〉〈A ∗ 1| |A 2 〉, (10.5.7)
(
)
〈A 1 |A 1 〉 + |A 1 〉〈A 1 | + |A ∗ 1〉〈A ∗ 1| + 〈A 2 |A 2 〉 + |A 2 〉〈A 2 | + |A ∗ 2〉〈A ∗ 2| |A 3 〉
+ 2iγ (
)
|A 2 〉〈A ∗
3
1| + |A 1 〉〈A ∗ 2| + 〈A ∗ 1|A 2 〉 |A ∗ 4〉e −iΔkz , (10.5.8)
= 2iγ (
)
〈A 1 |A 1 〉 + |A 1 〉〈A 1 | + |A ∗
3
1〉〈A ∗ 1| + 〈A 2 |A 2 〉 + |A 2 〉〈A 2 | + |A ∗ 2〉〈A ∗ 2| |A 4 〉
+ 2iγ (
)
|A 2 〉〈A ∗
3
1| + |A 1 〉〈A ∗ 2| + 〈A ∗ 1|A 2 〉 |A ∗ 3〉e −iΔkz . (10.5.9)
In the case of single-pump configuration, only the first pump equation should be retained
because |A 2 〉 = 0. Also, one should replace |A 2 〉 with |A 1 〉 and 2γ with γ in the
signal and idler equations.
10.5.2 Polarization Dependence of Parametric Gain
The vector FWM equations, Eqs. (10.5.6) through (10.5.9), describe FWM in the general
case in which the two pumps and the signal are launched into an optical fiber with
arbitrary SOPs. Their complexity requires a numerical approach in general. To investigate
the relationship between the FWM efficiency and pump SOPs, we ignore the SPM
and XPM terms temporarily and focus on the selection rules that govern the creation
of idler photons. The SPM and XPM terms affect only the phase-matching condition
and their impact is considered later in this section.
Physically, the polarization dependence of FWM stems from the requirement of
angular momentum conservation among the four interacting photons in an isotropic
medium. This requirement can be described most simply in a basis in which ↑ and
↓ denote the left and right circular polarization states and represent photons with an
intrinsic angular momentum (spin) of +¯h and −¯h, respectively [105]. To describe
FWM among arbitrarily polarized optical fields, we decompose the Jones vector of
each field as
|A j 〉 = U j |↑〉+ D j |↓〉, (10.5.10)
where U j and D j represent the field amplitudes in the spin-up and spin-down states,
respectively, for the jth wave ( j = 1 to 4). Using this expansion, it follows from Eq.
(10.5.9) that the creation of idler photons in the two orthogonal states is governed by
the following two equations (assuming perfect phase matching):
dU 4
dz
dD 4
dz
= 4iγ
3 [U 1U 2 U ∗
3 +(U 1 D 2 + D 1 U 2 )D ∗ 3 ], (10.5.11)
= 4iγ
3 [D 1D 2 D ∗ 3 +(U 1 D 2 + D 1 U 2 )U ∗
3 ]. (10.5.12)