Nonlinear Fiber Optics - 4 ed. Agrawal

10.1. Origin of Four-Wave Mixing 369
became available [6]–[26]. The main features of FWM can be understood from the
third-order polarization term in Eq. (1.3.1),
P NL = ε 0 χ (3) .EEE, (10.1.1)
where E is the electric field and P NL is the induced nonlinear polarization.
In general, FWM is polarization-dependent and one must develop a full vector theory
for it (presented in Section 10.5). However, considerable physical insight is gained
by first considering the scalar case in which all four fields are linearly polarized along a
principal axis of a birefringent fiber such that they maintain their state of polarization.
Consider four CW waves oscillating at frequencies ω 1 , ω 2 , ω 3 , and ω 4 and linearly
polarized along the same axis x. The total electric field can be written as
E = 1 2 ˆx 4
∑
j=1E j exp[i(β j z − ω j t)] + c.c., (10.1.2)
where the propagation constant β j = ñ j ω j /c,ñ j being the mode index. If we substitute
Eq. (10.1.2) in Eq. (10.1.1) and express P NL in the same form as E using
P NL = 1 2 ˆx 4
∑
j=1P j exp[i(β j z − ω j t)] + c.c., (10.1.3)
we find that P j ( j = 1 to 4) consists of a large number of terms involving the products
of three electric fields. For example, P 4 can be expressed as
P 4 = 3ε 0
4 χ(3) xxxx[
|E4 | 2 E 4 + 2(|E 1 | 2 + |E 2 | 2 + |E 3 | 2 )E 4
+ 2E 1 E 2 E 3 exp(iθ + )+2E 1 E 2 E ∗ 3 exp(iθ − )+···],
(10.1.4)
where θ + and θ − are defined as
θ + =(β 1 + β 2 + β 3 − β 4 )z − (ω 1 + ω 2 + ω 3 − ω 4 )t, (10.1.5)
θ − =(β 1 + β 2 − β 3 − β 4 )z − (ω 1 + ω 2 − ω 3 − ω 4 )t. (10.1.6)
The first four terms containing E 4 in Eq. (10.1.4) are responsible for the SPM and
XPM effects, but the remaining terms result from the frequency combinations (sum or
difference) of all four waves. How many of these are effective during a FWM process
depends on the phase mismatch between E 4 and P 4 governed by θ + , θ − , or a similar
quantity.
Significant FWM occurs only if the phase mismatch nearly vanishes. This requires
matching of the frequencies as well as of the wave vectors. The latter requirement
is often referred to as phase matching. In quantum-mechanical terms, FWM occurs
when photons from one or more waves are annihilated and new photons are created at
different frequencies such that the net energy and momentum are conserved during the
parametric interaction. The main difference between a FWM process and a stimulated
scattering process discussed in Chapters 8 and 9 is that the phase-matching condition