Nonlinear Fiber Optics - 4 ed. Agrawal

374 Chapter 10. Four-Wave Mixing
Figure 10.1: Variation of parametric gain with phase mismatch Δk for several pump powers P 0 .
The shift of the gain peak from Δk = 0 is due to a combination of the SPM and XPM effects.
Significant FWM occurs if L < L coh . Even when this condition is satisfied, SRS can
influence the FWM process significantly when the frequency shift Ω s lies within the
Raman-gain bandwidth. The interplay between SRS and FWM has been studied extensively
[28]–[36]. The main effect in practice is that the Stokes component gets
amplified through SRS, resulting in an asymmetric sideband spectrum. This feature is
discussed further in the next section where experimental results are presented.
10.2.4 Ultrafast Four-Wave Mixing
The simplified analysis of this section is based on Eqs. (10.2.10) and (10.2.11) that
assume, among other things, a CW or quasi-CW regime so that group-velocity dispersion
(GVD) can be neglected. The effects of GVD can be included by allowing A j (z)
to be a slowly varying function of time and following the analysis of Section 2.3. If
polarization effects are neglected, assuming that all four waves are polarized along a
principal axis of a birefringent fiber, the inclusion of GVD effects and fiber losses in
Eqs. (10.2.1)–(10.2.4) amounts to replacing the derivative dA j /dz with
dA j
dz → ∂A j
∂z + β ∂A j
1 j + i ∂t 2 β ∂ 2 A j
2 j
∂t 2 + 1 2 α jA j (10.2.23)
for all four waves ( j = 1 to 4) in analogy with Eq. (2.3.28). The resulting four coupled
NLS equations describe FWM of picosecond optical pulses and include the effects of
GVD, SPM, and XPM. It is difficult to solve the four coupled NLS equations analytically,
and a numerical approach is used in practice. The group velocity of four pulses
participating in the FWM process can be quite different. As a result, efficient FWM
requires not only phase matching but also matching of the group velocities.