Brillouin Scattering

Definition: a nonlinear scattering effect involving acoustic phonons

Categories: article belongs to category fiber optics and waveguides fiber optics and waveguides, article belongs to category nonlinear optics nonlinear optics

• optical effects
  • inelastic scattering
    • Rayleigh scattering
    • Brillouin scattering
      • spontaneous Brillouin scattering
      • stimulated Brillouin scattering
    • Raman scattering
    • hyper Raman scattering

• optical effects
  • nonlinear optical effects
    • nonlinear frequency conversion
    • self-phase modulation
    • cross-phase modulation
    • four-wave mixing
    • soliton propagation
    • modulational instability
    • parametric amplification
    • parametric fluorescence
    • nonlinear pulse distortion
    • self-focusing
    • nonlinear polarization rotation
    • self-steepening
    • stimulated Raman scattering
    • Brillouin scattering
      • spontaneous Brillouin scattering
      • stimulated Brillouin scattering
    • nonlinear absorption
    • thermal lensing
    • (more topics)

Related: Raman scattering Kerr effect delayed nonlinear response fibers fiber-optic sensors Stimulated Brillouin Scattering: Lower Peak Power, Stronger Effect? Thresholds for Nonlinear Effects in Fiber Amplifiers

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What is Brillouin Scattering?

Brillouin scattering is a nonlinear light scattering processes involving sound waves. The intricate physics are explained in detail in this article. The effect is mostly relevant in the following contexts:

• Brillouin gain is used for operating a Brillouin fiber lasers which can have an extremely small linewidth — well below what is typically possible with an ordinary fiber laser.
• Brillouin fiber amplifiers are also possible, but not common.
• Brillouin phase conjugation exploits special phase properties of a laser beam which is nonlinearly reflected by Brillouin scattering. It can be used for compensating optical aberrations in laser amplifiers, for example.
• Some fiber-optic temperature and pressure sensors (→ fiber-optic sensors ) exploit environmental dependencies of the Brillouin frequency shift [36].
• Brillouin scattering can be utilized for generating “slow light” [25], where long propagation time delays are achieved during a short passage.

In some cases, SBS has detrimental effects:

• Stimulated backward Brillouin scattering can introduce unwanted strong nonlinear reflections particularly in high-power fiber devices, which are often performance-limiting [3]. Ways to master such challenge are explained in the article.
• Stimulated forward Brillouin scattering can add amplifier noise and deteriorate the bit error rate (e.g., via limiting the allowable optical power) of optical fiber communications systems [12].

Fundamental Physics of Brillouin Scattering

Brillouin scattering, named after the French physicist Léon Brillouin, is a nonlinear scattering process:

• Light can be scattered at sound waves in a medium, generating a refractive index modulation. Since sound waves are moving, the optical frequency of light is modified upon reflection.
• Counterpropagating light beams with somewhat different optical frequencies can generate a moving interference pattern, which can interact with sound waves.

In most cases, the coupling of light to sound waves occurs due to electrostriction. In some cases, however, its arises from temperature variations due to absorption; this is called absorptive SBS.

Such processes can occur in a wide range of media — solids, liquids and gases.

Note that although scattering of light at a given sound wave can be regarded as linear scattering, the sound wave in the medium can be created or strongly modified by the interaction with intense light (often in the form of a laser beam or a guided wave in a fiber). Therefore, the interaction is actually nonlinear, i.e., strongly dependent on the involved optical intensity.

Brillouin scattering is related to the ($\chi^{(3)}$) nonlinearity of a medium, specifically by that part of the nonlinearity which is the delayed nonlinear response related to acoustic phonons [1]. An incident photon can be converted into a scattered photon of slightly lower energy, usually propagating in the backward direction, and a phonon.

One may also exploit interactions of light with sound waves which are generated in other ways, e.g. with piezoelectric transducers as used in acousto-optic modulators. The term Brillouin scattering, however, is restricted to cases without such sound inputs.

Spontaneous and Stimulated Brillouin Scattering; Nonlinear Amplification

Brillouin scattering can occur spontaneously even at relatively low optical intensities, when one injects a single light beam into a nonlinear medium, where it can interact with 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 is a stimulated effect, where the optical fields substantially contribute to the phonon population. Above a certain threshold power (see below) of a light beam in a medium, stimulated Brillouin scattering can even 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 (while two counter-propagating waves with identical frequency would produce a standing wave); the higher the reflected intensity, the stronger the index grating and the higher the effective reflectance.

There is not a strictly defined threshold power for Brillouin scattering. However, the nonlinear gain (in terms of optical power amplification factor for the counterpropagating wave) rises exponentially with the incoming power. Once it reaches e.g. 90 dB, the reflected power will be substantial even without any counterpropagating input beam, and steeply rises further with any small increase of input power.

One often uses a simple equation for estimating the magnitude of Brillouin gain in decibels (dB):

$$G_{\rm B} = 4.34 \: g_{\rm B} I_{\rm p} L$$

where ($g_{\rm B}$) is the Brillouin gain coefficient (e.g. 5 · 10⁻¹¹ m/W for silica), ($I_{\rm p}$) is the intensity of the pump wave (if necessary, spatially averaged) and ($L$) is the propagation length. The Brillouin gain coefficient depends on various material parameters, including the square of the elasto-optic coefficient, the Brillouin linewidth, the density, acoustic velocity and refractive index. Note that the equation holds only if various simplifying conditions are fulfilled; for example, it may not hold at all for transient phenomena involving short light pulses (see below).

Basic Conditions for Effective Brillouin Scattering

The optical frequency of the reflected beam is slightly lower than that of the incident beam; energy conservation requires that the frequency difference ($\nu_\textrm{B}$), called Brillouin frequency shift, corresponds to the frequency of generated phonons.

Further, there is a phase-matching requirement for efficient scattering over a substantial propagation length: The difference of the two optical wave vectors must correspond to the wavenumber ($q)$ of the phonon:

$$\vec{k}_{\rm incident} - \vec{k}_{\rm scattered} = \vec{q}$$

Both conditions together determine for which optical frequency differences Brillouin scattering can be substantial. This is generally the case only in a very narrow range of frequency differences.

Forward and Backward Brillouin Scattering

In principle, Brillouin scattering can occur in any direction. Typically considered cases are forward and backward scattering:

• Forward scattering means that the scattered light propagates in the same direction as the incoming light. In this case, the magnitude of ($\vec{q}$) is very small. For acoustic phonons, this implies a very small acoustic frequency (in the MHz region) and long acoustic wavelength.
• Backward scattering essentially means reflection of light. Here, the magnitude of ($\vec{q}$) is much larger — about twice the magnitude of the light's wavenumber. (Still, we are near the center of the Brillouin zone, since optical wavelengths are far longer than atomic lattice constants.)

Backward Brillouin scattering is most frequently the major concern, e.g. when designing fiber amplifiers for narrow-linewidth signals.

Backward Brillouin Frequency Shift and Bandwidth

For pure backward Brillouin scattering, the Brillouin shift can be easily calculated from the above explained phase-matching condition. As the wavelength of the scattered light is only marginally longer than that of the incident light, we can use a single vacuum wavelength ($\lambda$) in the formula. With the refractive index ($n$), the acoustic velocity ($v_\textrm{a}$), we obtain:

$${\nu _{\rm{B}}} = \frac{{2n{\upsilon _{\rm{a}}}}}{\lambda }$$

Note that for Brillouin scattering in fibers or other waveguides, the effective refractive index must be used instead of a normal refractive index.

The Brillouin frequency shift depends on the material composition (determining refractive index and sound velocity) and to some extent the temperature and pressure of the medium. Such dependencies are exploited for fiber-optic sensors.

Strong Brillouin scattering can occur in a quite limited bandwidth, e.g. of the order of 100 MHz (which is a very tiny fraction of the optical frequency) in the 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.

SBS With Light Pulses — Stationary vs. Transient Regime

In many cases, one considers SBS as a stationary process. For example, if long light pulses (with a pulse duration >100 ns, for example) interact with a solid where the phonon lifetime is far shorter (e.g. 10 ns), a steady state is approximately reached for every part of a pulse. The amplitude of the sound wave then essentially depends only on the optical intensities at the given time. Such situations can be described with much simplified equations.

More sophisticated models are required for transient SBS, occurring for shorter light pulses. Here, the sound amplitude also depends on optical intensities at earlier times. As the sound amplitude keeps building up during a pulse, not yet having reached its steady-state value, it stays weaker than if the same optical intensities would have occurred over a longer time.

Another important aspect for short light pulses is that counterpropagating light pulses can have a spatial overlap only over a limited length of a medium. This is particularly relevant for long media such as optical fibers (see below). Under such circumstances, it can easily happen that self-phase modulation becomes substantial before SBS becomes strong, and the resulting increase in bandwidth further raises the SBS threshold.

Both mentioned aspects can lead to far weaker Brillouin scattering effects than one might naively expect considering the high peak powers of short light pulses.

Stimulated Brillouin Scattering in Optical Fibers

Strong SBS in Fibers

Optical fiber devices often exhibit strong stimulated Brillouin scattering:

• The effective mode area is usually rather small, and cannot be increased arbitrarily. That leads to high optical intensities even for moderate optical powers.
• In addition, the propagation length is often quite long — for example, several meters in a typical double-clad fiber amplifier.
• Note, however, that for short light 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. For ultrashort pulses, SBS is thus usually not a concern despite high peak powers.
• In some cases, light with very narrow bandwidth needs to be amplified for obtaining high temporal coherence of the output.

Backward scattering is usually the main concern, limiting the performance of high-power fiber devices [3]. However, forward scattering may also problems, for example related to amplifier noise.

Guided Acoustic Wave Brillouin Scattering

There are guided waves not only for light, but also for sound. Brillouin scattering involving guided acoustic waves is called GAWBS = Guided Acoustic Wave Brillouin Scattering.

A frequently considered case is that of an optical fiber, which is essentially a cylindrical rod. For a free-standing homogeneous rods, there are analytical methods for calculating various types of acoustic modes. (Substantial damping may be introduced by a fiber coating.) Note, however, that the acoustic properties are generally modified in the fiber core: The dopants used for modifying the refractive index also modify the acoustic velocity and the phonon damping. The consequences are very dependent on the type of scattering:

• For forward Brillouin scattering, where the acoustic wavelengths are long, the acoustic modification by the fiber core is not very relevant.
• For backward scattering, however, we have far shorter acoustic wavelengths — about half the optical wavelengths. As a result, there may be acoustic modes guided by the fiber core, having a similar mode area as the optical fields, and thus a strong spatial overlap with those. However, it is possible to engineer core structures for different acoustic properties, where the mode overlap may be substantially reduced, or one may even have acoustic anti-guidance.

Typical Values of Brillouin Shift and Bandwidth

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 [22, 28] or longitudinal temperature variations [15, 20, 27]. 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 RP Photonics Spotlight article.

Brillouin Fiber Lasers

Brillouin fiber lasers [5, 7, 14, 26] exploit narrow-band Brillouin gain generated with a narrow-band pump source, e.g. a distributed feedback fiber laser. Such devices are usually made as fiber ring lasers. Due to low resonator loss, they can have a low pump threshold, and their emission linewidth can be extremely small — far smaller than that of the used pump laser.

Raising the SBS Threshold

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 [47]:

• One should minimize the required fiber length, e.g. by maximizing the doping concentration of the laser-active dopant, although that can involve other trade-offs.
• Unfortunately, double-clad fibers as needed for high-power devices exhibit weak pump absorption, requiring longer fibers. Using triple-clad fibers with reduced pump cladding area can help.
• The effective mode area should be maximized. Several thousand μm² are possible, but eventually the guidance becomes weak. See the article on large mode area fibers.
• Backward pumping of a fiber amplifier can be advantageous, as it maximizes the amplifier gain in the last portion of fiber and thus reduces the integrated signal intensity.
• One may increase the optical bandwidth substantially beyond the Brillouin gain bandwidth, so that the SBS gain is effectively “smeared out” over a wider frequency range and gets accordingly weaker.
• Even if a narrow optical linewidth is needed, one may increase it before a fiber amplifier with a strongly driven electro-optic modulator and redo the phase changes with a second modulator after the fiber. A sinusoidal or pseudo-random phase modulation may be used.
• On can remove backward scattered light with fiber Bragg gratings before its power gets strong [21].
• Another option is to broaden the effective SBS gain spectrum through variations of Brillouin shift along the fiber:
  • One can concatenate fibers with slightly different Brillouin shift [33], or use a continuously tapered fiber [40].
  • One may wind the fiber on a tapered mandrel.
  • One can (in high-power active fiber devices) exploit the longitudinally varying temperature increase [31].
• Finally, one can optimize fiber designs:
  • There are attempts to reduce the overlap of guided optical and acoustic waves, or to achieve anti-guiding acoustic properties.
  • One may introduce significant propagation losses for the acoustic wave. Such damping increases the SBS bandwidth.

Numerical Examples of SBS in 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.

[画像:Brillouin scattering in an optical fiber]

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.

[画像:Brillouin scattering in an optical fiber]

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 in 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.

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 light 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.

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 [10, 18]. It is also possible to utilize this effect in coherent beam combining [16] and for pulse compression [6].

Phase-conjugate mirrors can be realized with solids, liquids and gases. Liquids allow for relatively low SBS thresholds with a simple cell design, and a liquid flow can remove damaged material. However, some applications favor solids or gases.

Bibliography

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