Brillouin scattering is an effect caused by the χ(3) nonlinearity of a medium, specifically by that part of the nonlinearity which is the delayed nonlinear response related to acoustic phonons . An incident photon can be converted into a scattered photon of slightly lower energy, usually propagating in the backward direction, and a phonon. The coupling of optical fields and acoustic waves occurs via electrostriction.
Spontaneous and Stimulated Brillouin Scattering
Brillouin scattering can occur spontaneously even at low optical powers, then reflecting the thermally generated phonon field. Typically, only a very small amount of the incident light is scattered in that way.
For higher optical powers, there can be a stimulated effect, where the optical fields substantially contribute to the phonon population. Above a certain threshold power of a light beam in a medium, stimulated Brillouin scattering can reflect most of the power of an incident beam. This process involves a strong nonlinear optical gain for the back-reflected wave: an originally weak counterpropagating wave at the suitable optical frequency can be strongly amplified. Here, the two counter-propagating waves generate a traveling refractive index grating; the higher the reflected power, the stronger the index grating and the higher the effective reflectance.
Brillouin Frequency Shift and Bandwidth
The optical frequency of the reflected beam is slightly lower than that of the incident beam; the frequency difference νB corresponds to the frequency of emitted phonons. This so-called Brillouin frequency shift is set by a phase-matching requirement. For pure backward Brillouin scattering, the Brillouin shift can be calculated from the refractive index n, the acoustic velocity va, and the vacuum wavelength λ:
(For Brillouin scattering in fibers, the effective refractive index must be used.)
The Brillouin frequency shift depends on the material composition and to some extent the temperature and pressure of the medium. Such dependencies are exploited for fiber-optic sensors.
Brillouin scattering occurs only in a quite limited bandwidth, e.g. of the order of 100 MHz (which is a very tiny fraction of the optical frequency) in case of silica fibers. It depends on the damping time of the involved acoustic wave (the phonon lifetime), but can also be inhomogeneously increased, e.g. when the temperature of the active fiber in a fiber amplifier varies along the length.
Forward and Backward Brillouin Scattering
In optical fibers, Brillouin scattering occurs essentially only in backward direction. However, rather weak forward Brillouin scattering is also possible due to effects of the acoustic waveguide.
Optical Phase Conjugation
An important application of stimulated Brillouin scattering is optical phase conjugation. There are, for example, phase-conjugate mirrors for high-power Q-switched lasers which make it possible that the thermal distortions occurring in forward and backward direction in the laser crystal compensate each other.
Stimulated Brillouin Scattering in Optical Fibers
Stimulated Brillouin scattering (SBS) is frequently encountered when narrow-band optical signals (e.g. from a single-frequency laser) are amplified in a fiber amplifier, or just propagated through a passive fiber. While the material nonlinearity of e.g. silica is actually not very high, the typically small effective mode area and long propagation length strongly favor nonlinear effects.
Figure 1 shows what happens when a monochromatic light wave is injected into a 10 m long fiber. The counterpropagating Brillouin-shifted wave starts from quantum fluctuations with a very low optical power, but grows rapidly. Still, it stays far smaller than the input power of 1 W.
For a somewhat increased pump power of 1.8 W, the Brillouin gain (as measured in decibels) is nearly doubled, and the Brillouin wave becomes far stronger.
For a further increased pump power, the power of the Brillouin wave would become comparable to the pump power. In that case, substantial pump depletion occurs. For high SBS gain, that does not lead to a stable situation, but to chaotic fluctuations of the powers.
If the fiber is many kilometers long, milliwatt powers can be sufficient to cause substantial Brillouin scattering. However, one then has to take into account propagation losses, which are substantial of such fiber lengths. The affect both the pump wave and the Brillouin wave.
For silica fibers, the Brillouin frequency shift is of the order of 10–20 GHz, and the Brillouin gain has an intrinsic bandwidth of typically 50–100 MHz, which is determined by the strong acoustic absorption (short phonon lifetime of the order of 10 ns). However, the Brillouin gain spectrum may be strongly “smeared out” by various effects, such as transverse variations of the acoustic phase velocity [14, 19] or longitudinal temperature variations [11, 13]. Accordingly, the peak gain may be strongly reduced, leading to a substantially higher SBS threshold.
The Brillouin threshold of optical fibers for narrow-band continuous-wave light typically corresponds to a Brillouin gain of the order of 90 dB. (With additional laser gain in an active fiber, the threshold can be lower.) For trains of ultrashort pulses, the SBS threshold is determined not by a peak power, but rather by a power spectral density, as explained in a Spotlight article.
Note that for short laser pulses the spatial overlap of counterpropagating pulses is limited. Therefore, much less than the total fiber length may be relevant for calculating the Brillouin gain, and further increases of peak power by pulse shortening (i.e., for constant pulse energy) do not further increase the Brillouin gain.
SBS introduces the most stringent power limit for the amplification and the passive propagation of narrow-band optical signals in fibers. In order to raise the Brillouin threshold, various methods have been developed. For example, one may increase the bandwidth of the light beyond the Brillouin gain bandwidth, reduce the fiber length, concatenate fibers with slightly different Brillouin shift, or (in high-power active fiber devices) exploit the longitudinally varying temperature . There are also attempts to reduce the overlap of guided optical and acoustic waves, or to introduce significant propagation losses for the acoustic wave. To some extent, SBS problems can be reduced via basic amplifier design modifications, concerning e.g. the doping concentration, effective mode area and pump propagation direction.
On the other hand, the Brillouin gain can be used for operating a Brillouin fiber laser [5, 10, 18]. Such devices are often made as fiber ring lasers. Due to low resonator loss, they can have a relatively low pump threshold and a very small linewidth.
The temperature dependence of the Brillouin shift can be used for temperature and pressure sensing (→ fiber-optic sensors).
Numerical Simulation of Brillouin Scattering
It is possible to numerically simulate the propagation of light (e.g. in optical fibers) under the influence of Brillouin scattering. However, this is technically quite difficult, as it involves counterpropagating light waves and depends on their detailed frequency spectrum. Also, computation times can be fairly long.
However, in most practical cases it is not necessary to do full-blown numerical simulations for answering the relevant questions. The core question is usually whether or not Brillouin scattering will have substantial effects on life propagation, i.e., cause substantial nonlinear back-scattering – but not what exactly what would happen in that regime where it becomes substantial. That core question can usually be answered in a quite simple way. One simulates the propagation without taking into account Brillouin scattering and then calculates the resulting Brillouin gain. If that exceeds roughly 90 dB, substantial Brillouin scattering is to be expected. The mentioned threshold value somewhat depends on the circumstances, e.g. whether continuous-wave light or light pulses are considered. That simple approach may not be highly accurate, but note that for a high accuracy of full numerical simulations one would require a detailed knowledge of the involved optical inputs, which is often not available. For example, such details are usually not known for common types of seed lasers and would be rather difficult to measure. Further uncertainties may occur e.g. concerning the details of optical fibers such as their detailed Brillouin scattering and how that may change over the length of fiber.
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See also: Raman scattering, Kerr effect, delayed nonlinear response, fibers, fiber-optic sensors, The Photonics Spotlight 2007-09-01, The Photonics Spotlight 2010-03-03
and other articles in the categories fiber optics and waveguides, nonlinear optics