60 CHAPTER 2. OPTICAL FIBERS
viewed as scattering of the pump wave from this acoustic wave, resulting in creation
of a new wave at the pump frequencyΩs. The scattering process must conserve both
the energy and the momentum. The energy conservation requires that the Stokes shift
Ωequalsωp−ωs. The momentum conservation requires that the wave vectors sat-
isfykA=kp−ks. Using the dispersion relation|kA|=Ω/vA, wherevAis the acoustic
velocity, this condition determines the acoustic frequency as [31]
Ω=|kA|vA= 2 vA|kp|sin(θ/ 2 ), (2.6.1)where|kp|≈|ks|was used andθrepresents the angle between the pump and scattered
waves. Note thatΩvanishes in the forward direction (θ=0) and is maximum in the
backward direction (θ=π). In single-mode fibers, light can travel only in the forward
and backward directions. As a result, SBS occurs in the backward direction with a
frequency shiftΩB= 2 vA|kp|. Usingkp= 2 πn ̄/λp, whereλpis the pump wavelength,
theBrillouin shiftis given by
νB=ΩB/ 2 π=2 ̄nvA/λp, (2.6.2)where ̄nis the mode index. UsingvA= 5 .96 km/s and ̄n= 1 .45 as typical values for
silica fibers,νB= 11 .1 GHz atλp= 1. 55 μm. Equation (2.6.2) shows thatνBscales
inversely with the pump wavelength.
Once the scattered wave is generated spontaneously, it beats with the pump and
creates a frequency component at the beat frequencyωp−ωs, which is automatically
equal to the acoustic frequencyΩ. As a result, the beating term acts as source that
increases the amplitude of the sound wave, which in turn increases the amplitude of the
scattered wave, resulting in a positive feedback loop. SBS has its origin in this positive
feedback, which ultimately can transfer all power from the pump to the scattered wave.
The feedback process is governed by the following set of two coupled equations [73]:
dIp
dz=−gBIpIs−αpIp. (2.6.3)−
dIs
dz=+gBIpIs−αsIs, (2.6.4)whereIpandIsare the intensities of the pump and Stokes fields,gBis the SBS gain,
andαpandαpaccount for fiber losses.
The SBS gaingBis frequency dependent because of a finite damping timeTBof
acoustic waves (the lifetime of acoustic phonons). If the acoustic waves decay as
exp(−t/TB), the Brillouin gain has a Lorentzian spectral profile given by [77]
gB(Ω)=
gB(ΩB)
1 +(Ω−ΩB)^2 TB^2. (2.6.5)
Figure 2.17 shows the Brillouin gain spectra atλp= 1. 525 μm for three different kinds
of single-mode silica fibers. Both the Brillouin shiftνBand the gain bandwidth∆νB
can vary from fiber to fiber because of the guided nature of light and the presence
of dopants in the fiber core. The fiber labeled (a) in Fig. 2.17 has a core of nearly
pure silica (germania concentration of about 0.3% per mole). The measured Brillouin