Nonlinear Fiber Optics - 4 ed. Agrawal

278 Chapter 8. Stimulated Raman Scattering
integral in Eq. (8.1.8) can be evaluated approximately using the method of steepest descent
because the main contribution to the integral comes from a narrow region around
the gain peak. Using ω = ω s , we obtain
P s (L)=P eff
s0 exp[g R(Ω R )I 0 L eff − α s L], (8.1.9)
where the effective input power at z = 0isgivenby
( ) 2π 1/2 ∣ ∣∣∣
Ps0 eff = ¯hω ∂ 2 ∣
g R ∣∣∣
−1/2
sB eff , B eff =
I 0 L eff ∂ω 2 . (8.1.10)
ω=ω s
Physically, B eff is the effective bandwidth of the Stokes radiation centered near the
gain peak at Ω R = ω p − ω s . Although B eff depends on the pump intensity and the fiber
length, the spectral width of the dominant peak in Figure 8.2 provides an order-ofmagnitude
estimate for it.
The Raman threshold is defined as the input pump power at which the Stokes power
becomes equal to the pump power at the fiber output [16] or
P s (L)=P p (L) ≡ P 0 exp(−α p L), (8.1.11)
where P 0 = I 0 A eff is the input pump power and A eff is the effective core area as defined
in Section 2.3. Using Eq. (8.1.9) in Eq. (8.1.11) and assuming α s ≈ α p , the threshold
condition becomes
Ps0 eff exp(g RP 0 L eff /A eff )=P 0 , (8.1.12)
where Ps0
eff also depends on P 0 through Eq. (8.1.10). The solution of Eq. (8.1.12) provides
the critical pump power required to reach the Raman threshold. Assuming a
Lorentzian shape for the Raman-gain spectrum, the critical pump power, to a good
approximation, is given by [16]
g R P cr
0 L eff
A eff
≈ 16. (8.1.13)
A similar analysis can be carried out for the backward SRS. The threshold condition
in that case is still given by Eq. (8.1.13) but the numerical factor 16 is replaced
with 20. As the threshold for forward SRS is reached first at a given pump power,
backward SRS is generally not observed in optical fibers. Of course, the Raman gain
can be used to amplify a backward propagating signal. Note also that the derivation of
Eq. (8.1.13) assumes that the polarization of the pump and Stokes waves is maintained
along the fiber. If polarization is not preserved, the Raman threshold is increased by a
factor whose value lies between 1 and 2. In particular, if the polarization is completely
scrambled, it increases by a factor of two.
In spite of various approximations made in the derivation of Eq. (8.1.13), it is able
to predict the Raman threshold quite accurately. For long fibers such that α p L ≫ 1,
L eff ≈ 1/α p . At λ p = 1.55 μm, a wavelength near which the fiber loss is minimum
(about 0.2 dB/km), L eff ≈ 20 km. If we use a typical value A eff = 50 μm 2 , the predicted
Raman threshold is P0 cr ≈ 600 mW. Clearly, SRS is not likely to occur in an isolated
channel of an optical communication system because power levels are typically below