Nonlinear Fiber Optics - 4 ed. Agrawal

330 Chapter 9. Stimulated Brillouin Scattering
9.1.1 Physical Process
The process of SBS can be described classically as a nonlinear interaction between the
pump and Stokes fields through an acoustic wave. The pump field generates an acoustic
wave through the process of electrostriction [10]. The acoustic wave in turn modulates
the refractive index of the medium. This pump-induced index grating scatters the pump
light through Bragg diffraction. Scattered light is downshifted in frequency because of
the Doppler shift associated with a grating moving at the acoustic velocity v A . The
same scattering process can be viewed quantum mechanically as if annihilation of a
pump photon creates a Stokes photon and an acoustic phonon simultaneously. As both
the energy and the momentum must be conserved during each scattering event, the
frequencies and the wave vectors of the three waves are related by
Ω B = ω p − ω s , k A = k p − k s , (9.1.1)
where ω p and ω s are the frequencies, and k p and k s are the wave vectors, of the pump
and Stokes waves, respectively.
The frequency Ω B and the wave vector k A of the acoustic wave satisfy the standard
dispersion relation
Ω B = v A |k A |≈2v A |k p |sin(θ/2), (9.1.2)
where θ is the angle between the pump and Stokes fields, and we used |k p |≈|k s | in
Eq. (9.1.1). Equation (9.1.2) shows that the frequency shift of the Stokes wave depends
on the scattering angle. In particular, Ω B is maximum in the backward direction
(θ = π) and vanishes in the forward direction (θ = 0). In a single-mode optical fiber,
only relevant directions are the forward and backward directions. For this reason, SBS
occurs only in the backward direction with the Brillouin shift given by
ν B = Ω B /2π = 2n p v A /λ p , (9.1.3)
where Eq. (9.1.2) was used with |k p | = 2πn p /λ p and n p is the effective mode index at
the pump wavelength λ p . Ifweusev A = 5.96 km/s and n p = 1.45, the values appropriate
for silica fibers, ν B ≈ 11.1 GHz at λ p = 1.55 μm.
Even though Eq. (9.1.2) predicts correctly that SBS should occur only in the backward
direction in single-mode fibers, spontaneous Brillouin scattering can occur in the
forward direction. This happens because the guided nature of acoustic waves leads to
a relaxation of the wave-vector selection rule. As a result, a small amount of Stokes
light is generated in the forward direction. This phenomenon is referred to as guidedacoustic-wave
Brillouin scattering [11]. In practice, the Stokes spectrum shows multiple
lines with frequency shifts ranging from 10 to 1000 MHz. Because of its extremely
weak character, this phenomenon is not considered further in this chapter.
9.1.2 Brillouin-Gain Spectrum
Similar to the case of SRS, the growth of the Stokes wave is characterized by the
Brillouin-gain spectrum g B (Ω) peaking at Ω = Ω B . However, in contrast to the SRS
case, the spectral width of the gain spectrum is very small (∼10 MHz in lieu of
∼10 THz) because it is related to the damping time of acoustic waves related to the