6.3. RAMAN AMPLIFIERS 245
Consider first the case of small-signal amplification for which pump depletion can
be neglected [the last term in Eq. (6.3.3)]. SubstitutingPp(z)=Pp( 0 )exp(−αpz)in Eq.
(6.3.2), the signal power at the output of an amplifier of lengthLis given by
Ps(L)=Ps( 0 )exp(gRP 0 Leff/ap−αsL), (6.3.4)whereP 0 =Pp( 0 )is the input pump power andLeffis defined as
Leff=[ 1 −exp(−αpL)]/αp. (6.3.5)Because of fiber losses at the pump wavelength, the effective length of the amplifier is
less than the actual lengthL;Leff≈ 1 /αpforαpL1. SincePs(L)=Ps( 0 )exp(−αsL)
in the absence of Raman amplification, the amplifier gain is given by
GA=
Ps(L)
Ps( 0 )exp(−αsL)=exp(g 0 L), (6.3.6)where the small-signal gaing 0 is defined as
g 0 =gR(
P 0
ap)(
Leff
L)
≈
gRP 0
apαpL. (6.3.7)
The last relation holds forαpL1. The amplification factorGAbecomes length in-
dependent for large values ofαpL. Figure 6.12 shows variations ofGAwithP 0 for
several values of input signal powers for a 1.3-km-long Raman amplifier operating at
1.064μm and pumped at 1.017μm. The amplification factor increases exponentially
withP 0 initially but then starts to deviate forP 0 >1 W because of gain saturation. De-
viations become larger with an increase inPs( 0 )as gain saturation sets in earlier along
the amplifier length. The solid lines in Fig. 6.12 are obtained by solving Eqs. (6.3.2)
and (6.3.3) numerically to include pump depletion.
The origin of gain saturation in Raman amplifiers is quite different from SOAs.
Since the pump supplies energy for signal amplification, it begins to deplete as the
signal powerPsincreases. A decrease in the pump powerPpreduces the optical gain
as seen from Eq. (6.3.1). This reduction in gain is referred to as gain saturation. An
approximate expression for the saturated amplifier gainGscan be obtained assuming
αs=αpin Eqs. (6.3.2) and (6.3.3). The result is given by [29]
Gs=1 +r 0
r 0 +G−A(^1 +r^0 ), r 0 =ωp
ωsPs( 0 )
Pp( 0 ). (6.3.8)
Figure 6.13 shows the saturation characteristics by plottingGs/GAas a function of
GAr 0 for several values ofGA. The amplifier gain is reduced by 3 dB whenGAr 0 ≈1.
This condition is satisfied when the power of the amplified signal becomes comparable
to the input pump powerP 0. In fact,P 0 is a good measure of the saturation power.
Since typicallyP 0 ∼1 W, the saturation power of fiber Raman amplifiers is much larger
than that of SOAs. As typical channel powers in a WDM system are∼1 mW, Raman
amplifiers operate in the unsaturated or linear regime, and Eq. (6.3.7) can be used in
place of Eq. (6.3.8)