2.6. NONLINEAR OPTICAL EFFECTS 63
the following set of two coupled equations [31]:
dIp
dz=−gRIpIs−αpIp, (2.6.8)dIs
dz=gRIpIs−αsIs, (2.6.9)wheregRis the SRS gain. In the case of backward SRS, a minus sign is added in front
of the derivative in Eq. (2.6.9), and this set of equations becomes identical to the SBS
case.
The spectrum of the Raman gain depends on the decay time associated with the
excited vibrational state. In the case of a molecular gas or liquid, the decay time is
relatively long (∼1 ns), resulting in a Raman-gain bandwidth of∼1 GHz. In the case
for optical fibers, the bandwidth exceeds 10 THz. Figure 2.18 shows the Raman-gain
spectrum of silica fibers. The broadband and multipeak nature of the spectrum is due
to the amorphous nature of glass. More specifically, vibrational energy levels of silica
molecules merge together to form a band. As a result, the Stokes frequencyωscan
differ from the pump frequencyωpover a wide range. The maximum gain occurs
when the Raman shiftΩR≡ωp−ωsis about 13 THz. Another major peak occurs
near 15 THz while minor peaks persist for values ofΩRas large as 35 THz. The peak
value of the Raman gaingRis about 1× 10 −^13 m/W at a wavelength of 1μm. This
value scales linearly withωp(or inversely with the pump wavelengthλp), resulting in
gR≈ 6 × 10 −^13 m/W at 1.55μm.
Similar to the case of SBS, the threshold powerPthis defined as the incident power
at which half of the pump power is transferred to the Stokes field at the output end of a
fiber of lengthL. It is estimated from [74]
gRPthLeff/Aeff≈ 16 , (2.6.10)wheregRis the peak value of the Raman gain. As before,Leffcan be approximated by
1 /α. If we replaceAeffbyπw^2 , wherewis the spot size,Pthfor SRS is given by
Pth≈ 16 α(πw^2 )/gR. (2.6.11)If we useπw^2 = 50 μm^2 andα= 0 .2 dB/km as the representative values,Pthis about
570 mW near 1.55μm. It is important to emphasize that Eq. (2.6.11) provides an
order-of-magnitude estimate only as many approximations are made in its derivation.
As channel powers in optical communication systems are typically below 10 mW, SRS
is not a limiting factor for single-channel lightwave systems. However, it affects the
performance of WDM systems considerably; this aspect is covered in Chapter 8.
Both SRS and SBS can be used to advantage while designing optical communi-
cation systems because they can amplify an optical signal by transferring energy to
it from a pump beam whose wavelength is suitably chosen. SRS is especially useful
because of its extremely large bandwidth. Indeed, the Raman gain is used routinely for
compensating fiber losses in modern lightwave systems (see Chapter 6).