Nonlinear Fiber Optics - 4 ed. Agrawal

334 Chapter 9. Stimulated Brillouin Scattering
second, for the same reason, α p ≈ α s ≡ α, i.e., fiber losses are nearly the same for the
pump and Stokes waves. With these changes, Eqs. (8.1.2) and (8.1.3) become
dI p
dz = −g BI p I s − αI p , (9.2.1)
− dI s
dz = g BI p I s − αI s . (9.2.2)
One can readily verify that in the absence of fiber losses (α = 0), d(I p − I s )/dz = 0,
and I p − I s remains constant along the fiber.
Equations (9.2.1) and (9.2.2) assume implicitly that the counterpropagating pump
and Stokes waves are linearly polarized along the same direction and maintain their
states of polarization along the fiber. This is the case when the two waves are polarized
along a principal axis of a polarization-maintaining fiber.
For the purpose of estimating the Brillouin threshold, pump depletion can be neglected.
Using I p (z)=I p (0)e −αz in Eq. (9.2.2) and integrating it over the fiber length
L, the Stokes intensity is found to grow exponentially in the backward direction as
I s (0)=I s (L)exp(g B P 0 L eff /A eff − αL), (9.2.3)
where P 0 = I p (0)A eff is the input pump power, A eff is the effective mode area, and the
effective fiber length is as defined in Eq. (4.1.6).
Equation (9.2.3) shows how a Stokes signal incident at z = L grows in the backward
direction because of Brillouin amplification occurring as a result of SBS. In practice,
no such signal is generally fed (unless the fiber is used as a Brillouin amplifier), and the
Stokes wave grows from noise provided by spontaneous Brillouin scattering occurring
throughout the fiber. Similar to the SRS case, the noise power is equivalent to injecting
a fictitious photon per mode at a distance where the gain exactly equals the fiber loss.
Following the method of Section 8.1.2, the Brillouin threshold is found to occur at a
critical pump power P cr such that [7]
g B (Ω B )P cr L eff /A eff ≈ 21, (9.2.4)
where the peak value of the Brillouin gain is given in Eq. (9.1.5). This threshold condition
should be compared with Eq. (8.1.13) obtained in the case of SRS.
To estimate the SBS threshold power, consider first a long fiber section (L > 50 km)
likely to be encountered in a typical optical communication system operating near
1.55 μm. Since αL ≫ 1 under such conditions, L eff = 1/α, and P cr = 21αA eff /g B .
If we use typical values for fiber parameters, A eff = 50 μm 2 , α = 0.2 dB/km, and
g B = 5 × 10 −11 m/W, Eq. (9.2.4) leads to P cr ≈ 1 mW. Such a low Brillouin threshold
is a consequence of the long fiber length. If the fiber section is relatively short, say
L = 1 m, the threshold power increases to 20 W. It can be increased even more by using
fibers with wider cores for which the effective mode area is larger.
9.2.2 Polarization Effects
The threshold level predicted by Eq. (9.2.4) is only approximate as several factors can
reduce the effective Brillouin gain in practice. An important role is played by the polarization
properties of the Brillouin gain. In contrast with the case of the Raman gain