Nonlinear Fiber Optics - 4 ed. Agrawal

402 Chapter 10. Four-Wave Mixing
in the following analysis. As before, we neglect pump depletion assuming that pump
powers are much larger than the signal and idler powers.
In general, one should consider the general form of the nonlinear polarization given
in Eq. (8.5.1) so that the Raman contribution is properly included. To simplify the following
discussion, we ignore the Raman effects and assume that the nonlinear polarization
is given by Eq. (10.1.1). The tensor χ (3) in this equation has the form of Eq.
(6.1.2) and can be written as
χ (3)
ijkl = 1 3 χ(3) xxxx(
δij δ kl + δ ik δ jl + δ il δ jk
)
. (10.5.1)
In the case of nondegenerate FWM, the total electric field and the nonlinear polarization
can be decomposed as
[ 4 ]
[ 4 ]
E = Re ∑ E j exp(−iω j t) , P NL = Re ∑ P j exp(−iω j t) , (10.5.2)
j=1
j=1
where Re stands for the real part and E j is the slowly varying amplitude at the frequency
ω j .
We now follow the procedure outlined in Section 8.5.1 and substitute Eq. (10.5.2)
in Eq. (10.1.1). Collecting the terms oscillating at ω 1 and ω 2 , the nonlinear polarization
at the pump frequencies is found to be
P j (ω j )= ε 0
4 χ(3) xxxx[
(E j · E j )E ∗ j + 2(E ∗ j · E j )E j
+ 2(E ∗ m · E m )E j + 2(E m · E j )E ∗ m + 2(E ∗ m · E j )E m
]
, (10.5.3)
where j,m = 1 or 2 with j ≠ m. Using the same process, the nonlinear polarization at
the signal and idler frequencies is found to be
P j (ω j )= ε 0
2 χ(3) xxxx[
(E ∗ 1 · E 1 )E j +(E 1 · E j )E ∗ 1 +(E ∗ 1 · E j )E 1
+(E ∗ 2 · E 2 )E j +(E 2 · E j )E ∗ 2 +(E ∗ 2 · E j )E 2
+(E ∗ m · E 1 )E 2 +(E ∗ m · E 2 )E 1 +(E 1 · E 2 )E ∗ m]
, (10.5.4)
where j,m = 3 or 4 with j ≠ m. In Eqs. (10.5.3) and (10.5.4), the SPM and XPM effects
induced by the two pumps are included but those induced by the signal and idler waves
are neglected because of their relatively low power levels.
To account for the polarization changes, we represent each field in terms of its Jones
vector |A j (z)〉 and use
E j (r)=F j (x,y)|A j 〉exp(iβ j z), (10.5.5)
where F j (x,y) represents the fiber-mode profile and β j is the propagation constant for
the field at frequency ω j . As in Section 10.2.1, we assume that mode profiles are
nearly the same for the four fields. This approximation amounts to assuming the same
effective mode area for the four waves.
Using Eqs. (10.5.3) through (10.5.5) in the wave equation (2.3.1) and following the
procedure of Section 2.3.1, the Jones vectors of the four fields are found to satisfy the