Nonlinear Fiber Optics - 4 ed. Agrawal

8.1. Basic Concepts 277
set of two coupled equations:
dI s
dz = g RI p I s − α s I s , (8.1.2)
dI p
dz = −ω p
g R I p I s − α p I p ,
ω s
(8.1.3)
where α s and α p account for fiber losses at the Stokes and pump frequencies, respectively.
As shown in Section 8.1.3, these equations can be derived rigorously from the
theory of Section 2.3. They can also be written phenomenologically by considering
the processes through which photons appear in or disappear from each beam. One can
readily verify that in the absence of losses,
(
d Is
+ I )
p
= 0. (8.1.4)
dz ω s ω p
This equation merely states that the total number of photons in the pump and Stokes
beams remains constant during SRS.
Although pump depletion must be included for a complete description of SRS, it
can be neglected for the purpose of estimating the Raman threshold [16]. Equation
(8.1.3) is readily solved if we neglect the first term on its right side that is responsible
for pump depletion. If we substitute this solution in Eq. (8.1.2), we obtain
dI s /dz = g R I 0 exp(−α p z)I s − α s I s , (8.1.5)
where I 0 is the incident pump intensity at z = 0. Equation (8.1.5) can be easily solved,
and the result is
I s (L)=I s (0)exp(g R I 0 L eff − α s L), (8.1.6)
where L is the fiber length and
L eff =[1 − exp(−α p L)]/α p . (8.1.7)
The solution (8.1.6) shows that, because of fiber losses, the effective fiber length is
reduced from L to L eff .
The use of Eq. (8.1.6) requires an input intensity I s (0) at z = 0. In practice, SRS
builds up from spontaneous Raman scattering occurring throughout the fiber length.
It has been shown that this process is equivalent to injecting one fictitious photon per
mode at the input end of the fiber [16]. Thus, we can calculate the Stokes power by
considering amplification of each frequency component of energy ¯hω according to Eq.
(8.1.6) and then integrating over the whole range of the Raman-gain spectrum. The
result is given by
∫ ∞
P s (L)= ¯hω exp[g R (ω p − ω)I 0 L eff − α s L]dω, (8.1.8)
−∞
where the fiber is assumed to support a single mode. The frequency dependence of g R
is shown in Figure 8.2. Even though the functional form of g R (Ω) is not known, the