244 CHAPTER 6. OPTICAL AMPLIFIERS
Figure 6.11: Raman-gain spectra (ratiogR/ap) for standard (SMF), dispersion-shifted (DSF)
and dispersion-compensating (DCF) fibers. Normalized gain profiles are also shown. (After
Ref. [30];©c2001 IEEE; reprinted with permission.)
whereapis the cross-sectional area of the pump beam inside the fiber. Sinceapcan
vary considerably for different types of fibers, the ratiogR/apis a measure of the
Raman-gain efficiency [30]. This ratio is plotted in Fig. 6.11 for three different fibers.
A dispersion-compensating fiber (DCF) can be 8 times more efficient than a standard
silica fiber (SMF) because of its smaller core diameter. The frequency dependence of
the Raman gain is almost the same for the three kinds of fibers as evident from the
normalized gain spectra shown in Fig. 6.11. The gain peaks at a Stokes shift of about
13.2 THz. The gain bandwidth∆νgis about 6 THz if we define it as the FWHM of the
dominant peak in Fig. 6.11.
The large bandwidth of fiber Raman amplifiers makes them attractive for fiber-
optic communication applications. However, a relatively large pump power is required
to realize a large amplification factor. For example, if we use Eq. (6.1.7) by assuming
operation in the unsaturated region,gL≈ 6 .7 is required forG=30 dB. By using
gR= 6 × 10 −^14 m/W at the gain peak at 1.55μm andap= 50 μm^2 , the required pump
power is more than 5 W for 1-km-long fiber. The required power can be reduced for
longer fibers, but then fiber losses must be included. In the following section we discuss
the theory of Raman amplifiers including both fiber losses and pump depletion.
6.3.2 Amplifier Characteristics
It is necessary to include the effects of fiber losses because of a long fiber length re-
quired for Raman amplifiers. Variations in the pump and signal powers along the am-
plifier length can be studied by solving the two coupled equations given in Section
2.6.1. In the case of forward pumping, these equations take the form
dPs/dz=−αsPs+(gR/ap)PpPs, (6.3.2)
dPp/dz=−αpPp−(ωp/ωs)(gR/ap)PsPp, (6.3.3)whereαsandαprepresent fiber losses at the signal and pump frequenciesωsand
ωp, respectively. The factorωp/ωsresults from different energies of pump and signal
photons and disappears if these equations are written in terms of photon numbers.