Stimulated Brillouin scattering is described as the nonlinear interaction between a pump wave, a Stokes wave and an acoustic wave [AGRAWAL 2006]. The power levels at which SBS occurs in an optical fiber can be much lower than those needed for other
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nonlinear phenomena such as Stimulated Raman Scattering. The physical process underneath stimulated Brillouin scattering is depicted in figure 1.1. The pump and the Stokes wave propagate in opposite directions in the fiber, inducing an acoustic wave because of electrostriction effect. This acoustic wave generates a density variation of the material as it propagates in the fiber. Due to the dependence of refractive index with the density of the material, a moving and periodic disturbance of the refractive index in the fiber is generated. As a consequence of the interaction with periodic disturbance of the refractive index, a fraction of light is transferred through Bragg diffraction from the pump to the Stokes wave. The frequency of the transferred light is downshifted due to the Doppler effect associated with the velocity of the acoustic wave. Therefore a power transfer between pump and probe power occurs which simultaneously strengths the acoustic wave, stimulating the described process consequently. Finally, the interaction ends up in an amplification of the Stokes wave and the depletion of the pump wave as they travel through the fiber. From the quantum mechanics point of view, SBS consist in the destruction of a pump photon to create simultaneously a Stokes photon and an acoustic phonon. The three waves must fulfill the energy conservation condition relating their frequencies:
S P
A
(1.1)
and the conservation of the momentum:
S P
A k k
k (1.2)
where the sub-indexes P, S and A are referred to the pump, the Stokes and the acoustic waves respectively. These two relationships have the following consequences:
the Stokes wave only experiments gain when counter-propagating with the pump wave, and the frequency shift between pump and stokes is given by:
c nv
nv a
p p B a
2
2
(1.3)
where n is the effective refractive index of the fiber, va the acoustic velocity in the fiber, and P the wavelength of the pump wave. In telecom grade single mode fibers, this frequency shift is around 11GHz when operating in C-Band. The Stokes wave can be injected to the fiber or can be generated from the reflection of the pump wave with thermally excited phonons [CUMMINS 1972]. In the latter case, the effect is called Spontaneous Brillouin scattering.
Introduction to distributed optical fiber sensors based on stimulated Brillouin Scattering
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Figure 1.1: Diagram of the SBS interaction in an optical fiber.
The coupled equations that describe the interaction model are the following respectively, gB the Brillouin gain spectrum, c the speed of light, n the refractive index of the fiber, and =1+j2(P-B) with 1=1/(2a), where a~6ns is the phonon lifetime for silica fibers. The Brillouin gain spectrum, gB, characterizes the growth of the Stokes wave, and it is related to the damping time of the acoustic waves or the phonon lifetime. It has a Lorentzian profile, and it is given by:
where gmax is the maximum Brillouin gain, B is the Brillouin frequency shift, B is the Brillouin bandwidth and P is the pump frequency. The maximum gain is given by:
B
where p12 is the longitudinal elasto-optic coefficient of the fiber, 0 the density of the fiber, P the pump wavelength and a factor that depends on the relative polarization orientation between pump and probe waves. This last factor reaches its maximum
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Stimulated Brillouin scattering (SBS)
Occurs when a pump and an stokes wave counterpropagate in a medium
Is the nonlinear interaction between the Pump and Stokes fields
though an acoustic wave
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value equal to one when the two waves have the same polarization during SBS interaction.
The nature of SBS is very appropriate to be exploited as a sensing mechanism. This is because there is a linear dependence between the Brillouin frequency shift and the temperature, T, or the strain difference in the medium, , an optical fiber in this case.
This relationship can be described with the following expression:
)]
where T0 is the reference temperature and the constants that relate temperature and strain variations are A and B respectively, and are given by [ZOU 2008]:
where E1 is the second order nonlinearity of Young’s modulus of the fiber. Apart from this dependence, Brillouin gain and line-width have slight dependences with strain and temperature too. The Brillouin line-width decreases with temperature, due to a lower absorption of the acoustic phonon while temperature increases [NIKLES 1997]. The Brillouin gain also varies, while the product of both, gB·B is kept constant. The increase of strain causes a smaller Brillouin gain. This has been related to the variations in density suffered by the material, which is proportional to Brillouin gain as expressed in (1.6) [NIKLES 1997].
Brillouin distributed sensors take advantage of these relationships in order to monitor the temperature at which the optical fiber is or the strain that has been applied to the fiber. The measurements are usually based on the characterization of the Brillouin spectrum along the fiber, and specifically, is the Brillouin frequency shift the one parameter taken as a reference. This is because it is the parameter measured with the highest resolution. Brillouin gain or line-width measurements are usually neglected since they are more sensible to noise.