Nonlinear Fiber Optics - 4 ed. Agrawal

8.2. Quasi-Continuous SRS 289
Figure 8.8: Variation of amplifier gain G A with pump power P 0 . Different symbols show the
experimental data for three values of input signal power. Solid curves show the theoretical prediction
using g R = 9.2 × 10 −14 m/W. (After Ref. [71]; c○1981 Elsevier.)
to gain saturation occurring because of pump depletion. The solid lines in Figure 8.8
are obtained by solving Eqs. (8.1.2) and (8.1.3) numerically to include pump depletion.
The numerical results are in excellent agreement with the data.
An approximate expression for the saturated gain G s in Raman amplifiers can be
obtained by solving Eqs. (8.1.2) and (8.1.3) analytically [17] with the assumption α s =
α p ≡ α. Making the transformation I j = ω j F j exp(−αz) with j = s or p, we obtain
two simple equations:
dF s
dz = ω pg R F p F s e −αz ,
dF p
dz = −ω pg R F p F s e −αz . (8.2.5)
Noting that F p (z)+F s (z) =C, where C is a constant, the differential equation for F s
can be integrated over the amplifier length, and the result is
G s = F ( )
s(L) C −
F s (0) = Fs (L)
exp(ω p g R CL eff − αL). (8.2.6)
C − F s (0)
Using C = F p (0)+F s (0) in this equation the saturated gain of the amplifier is given by
G s = (1 + r 0)e −αL
r 0 + G −(1+r , (8.2.7)
0)
A