Nonlinear Fiber Optics - 4 ed. Agrawal

282 Chapter 8. Stimulated Raman Scattering
been studied extensively [25]–[33]. This subsection describes the relevant features
qualitatively.
To understand how four-wave mixing can influence SRS, it is useful to reconsider
the physics behind the SRS process. As shown in Figure 8.1, Raman scattering can be
thought of as down-conversion of a pump photon into a lower-frequency photon and
a phonon associated with a vibrational mode of molecules. An up-conversion process
in which a phonon combines with the pump photon to generate a higher-frequency
photon is also possible, but occurs rarely because it requires the presence of a phonon
of right energy and momentum. The optical wave associated with higher-frequency
photons is called the anti-Stokes and is generated at a frequency ω a = ω p + Ω together
with the Stokes wave of frequency ω s = ω p − Ω, where ω p is the pump frequency.
Because 2ω p = ω a + ω s , four-wave mixing, a process where two pump photons annihilate
themselves to produce Stokes and anti-Stokes photons, can occur provided the
total momentum is conserved. The momentum-conservation requirement leads to the
phase-matching condition, Δk = 2k(ω p ) − k(ω a ) − k(ω s )=0, where k(ω) is the propagation
constant, that must be satisfied for FWM to occur (see Section 10.1).
The phase-matching condition in not easily satisfied in single-mode fibers for Ω ∼10
THz. For this reason, the anti-Stokes wave is rarely observed during SRS. As discussed
in Section 10.3, the phase-matching condition may be nearly satisfied when GVD is not
too large. In that case, Eqs. (8.1.20) and (8.1.21) should be supplemented with a third
equation that describes propagation of the anti-Stokes wave and its coupling to the
Stokes wave through four-wave mixing. The set of three equations can be solved approximately
when pump depletion is neglected [26]. The results show that the Raman
gain g R depends on the mismatch Δk, and may increase or decrease from its value in
Figure 8.2 depending on the value of Δk. In particular g R becomes small near Δk = 0,
indicating that four-wave mixing can suppress SRS under appropriate conditions. Partial
suppression of SRS was indeed observed in an experiment [26] in which the Raman
gain was reduced by a factor of 2 when the pump power P 0 was large enough to make
|Δk| < 3g R P 0 . A spectral component at the anti-Stokes frequency was also observed in
the experiment.
The effects of four-wave mixing on SRS were also observed in another experiment
[28] in which the spectrum of Raman pulses was found to exhibit a double-peak
structure corresponding to two peaks at 13.2 and 14.7 THz in Figure 8.2. At low pump
powers, the 13.2-THz peak dominated as the Raman gain is slightly larger (by about
1%) for this peak. However, as pump power was increased, the 14.7-THz peak began
to dominate the Raman-pulse spectrum. These results can be understood by noting
that the Raman-gain reduction induced by four-wave mixing is frequency dependent
such that the effective Raman gain becomes larger for the 14.7-THz peak for pump
intensities in excess of 1 GW/cm 2 .
The effects of fiber birefringence have been ignored in writing Eqs. (8.1.20) and
(8.1.21). Their inclusion complicates the SRS analysis considerably [29]. For example,
if a pump pulse is polarized at an angle with respect to a principal axis of the fiber so
that it excites both the slow and fast polarization modes of the fiber, each of them
can generate a Stokes pulse if its intensity exceeds the Raman threshold. These Stokes
waves interact with the two pump-pulse components and with each other through XPM.
To describe such an interaction, Eqs. (8.1.20) and (8.1.21) must be replaced by a set of