2. Basics of SBS

#### 2.1 Spontaneous and stimulated Brillouin scattering

Brillouin scattering, named after the French physicist Léon Brillouin, who theoretically predicted light scattering by a thermally excited acoustic wave (phonon) in 1922 [5], is one of the most prominent nonlinear effects in optical fibers [6]. For spontaneous Brillouin scattering (SpBS), an incident photon (pump wave) is transformed into a frequency-downshifted scattered photon (Stokes wave) and a phonon (acoustic wave). The angular distribution of the Stokes wave is governed by the laws of momentum and energy conservation, that is,

$$
\overrightarrow{k\_A} = \overrightarrow{k\_p} - \overrightarrow{k\_s} \qquad \alpha\_B = \alpha\_p - \alpha\_s \tag{1}
$$

Pth � Cth

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where gBð Þ ω<sup>B</sup> is the peak SBS gain, Leff ¼ ½ � 1 � exp ð Þ �αL =α is the effective length with L as the real fiber length, Aeff is the effective cross section of the fiber, and Pth is the estimated SBS threshold. Later on, this estimation has been revised with different theories and approximations, achieving also different Cth values [13–15]. However, instead of a constant, Cth has been recently found to be dependent on the fiber length and its parameters [16]. As a typical experimental example, Figure 1(a) illustrates the transmitted and the reflected power as a function of the incident power in a 20 km SMF. The dashed black line symbolizes the SBS threshold, beyond which the backscattered power rapidly increases and the output pump

The transferred energy from the pump to the Stokes wave can be regarded as an amplification when it is frequency downshifted to the pump wave by BFS. For typical SMF in the C-Band of optical telecommunications, the lifetime TB of the phonon involved in the SBS interaction is usually in the magnitude of �10 ns, which leads to a finite spectral distribution of the SBS gain. The complex SBS gain coefficient, which depicts the evolution of the probe wave as a function of frequency detuning ω between pump and probe wave, is approximated by a Lorentzian shape [17]:

<sup>g</sup>ð Þ¼ <sup>ω</sup> <sup>g</sup>0Pp

known as the Brillouin gain spectrum (BGS), and can be expressed as:

**-5 0 5 10 15 20 25**

**Reflection Transmission**

**Input power (dBm)**

1 � 2jð Þ ω � ω<sup>B</sup> =Γ<sup>B</sup>

<sup>B</sup> , Pp is the pump power, g<sup>0</sup> is related to the inherent material

**0.0 0.2 0.4 0.6 0.8 1.0**

**Normalized**

(a) The transition from SpBS to SBS in a 20 km SMF, from a distinct threshold (here Cth≈16), the reflected stokes power drastically increases, whereas the transmitted pump power stays almost constant; (b) simulated normalized Brillouin gain spectrum (BGS) and Brillouin phase spectrum (BPS) of a 20 km SMF.

 **Brillouin gain (a.u.)** <sup>2</sup> <sup>þ</sup> ð Þ <sup>Γ</sup>B=<sup>2</sup> <sup>2</sup> (5)

**1.0 (b)**

**-200-150-100 -50 0 50 100 150 200**

**Frequency detuning (MHz)**

Brillouin gain gp with g<sup>0</sup> ¼ gp=Aeff and Aeff is the effective cross section of the fiber, gp is in the range of 3 � <sup>10</sup>�<sup>11</sup> m/W to approximately 5 � <sup>10</sup>�<sup>11</sup> m/W at 1550 nm. The real part of Eq. (4) represents the power amplification of the probe wave, also

> gBð Þ¼ <sup>ω</sup> <sup>g</sup>0Ppð Þ <sup>Γ</sup>B=<sup>2</sup> <sup>2</sup> ð Þ ω � ω<sup>B</sup>

power stays almost constant.

2.2 The Brillouin gain

where <sup>Γ</sup><sup>B</sup> <sup>¼</sup> <sup>T</sup>�<sup>1</sup>

**-40 -30 -20 -10 0 10 20**

**(a)**

**SBS Threshold**

**Output power (dBm)**

Figure 1.

61

Aeff gBð Þ ω<sup>B</sup> Leff

(3)

(4)

**-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8**

**SBS phase shift (rad)**

where kA ! , kp ! , and ks ! are the wave vector of acoustic, pump, and Stokes wave, and ωB, ωp, and ω<sup>s</sup> are their corresponding angular frequencies. Considering the fiber geometry and provided that the phonon frequency is much smaller than that of both photons, an efficient Brillouin scattering only occurs in the backward direction. The Brillouin frequency shift (BFS) ν<sup>B</sup> ¼ ωB=2π is estimated to be �11 GHz for typical single mode fibers (SMF) and a pump wavelength in the C-Band of optical telecommunications (around 1550 nm) [7]. As has been shown very recently, forward SBS [8] can be used for sensing applications as well [9, 10]. However, the interaction is governed by transverse acoustic modes, and compared to backward SBS, the effect is rather weak. Since a description of this new field of SBS in sensing would go far beyond the scope of this book chapter, hereafter only backward SBS and its applications are discussed.

The basic origin of SBS is electrostriction, which tends to compress the material in the presence of an electrical field [11]. The superposition of the pump and the counterpropagating Stokes wave modulates the density and hence the refractive index of the optical fiber through electrostriction. Thus, a moving grating (an acoustic wave) is formed. If the velocity of this moving grating coincides with the speed of sound in the material, the effect is very efficient and it additionally reflects optical power from the pump wave. Due to the Doppler effect, the reflected pump wave is downshifted in frequency by the frequency difference between pump and probe and thus adds power to the Stokes wave. Therefore, a positive feedback loop is established. This transformation from SpBS to SBS can be quantitatively described by the differential equation system (DES) in the propagation direction z as:

$$\begin{split} \frac{dI\_p}{dz} &= -\mathbf{g}\_B(\boldsymbol{\alpha}) I\_p I\_s - a I\_p \\ -\frac{dI\_s}{dz} &= \mathbf{g}\_B(\boldsymbol{\alpha}) I\_p I\_s - a I\_s \end{split} \tag{2}$$

where Ip and Is are the intensity of pump and Stokes wave, α is the fiber attenuation, and gBð Þ ω is the SBS gain.

The threshold of SBS is defined by the critical power that characterizes the transformation from SpBS to SBS. However, its definition is rather controversial. Smith et al. first defined it as the input pump power at which the backscattered power equals to the transmitted power at the output [12] and Eq. (3) with the critical gain factor Cth≈21 gives an estimation of this critical value:

The State-of-the-Art of Brillouin Distributed Fiber Sensing DOI: http://dx.doi.org/10.5772/intechopen.84684

$$P\_{th} \equiv \mathcal{C}\_{th} \frac{A\_{\epsilon \mathcal{G}}}{\mathcal{g}\_B(\alpha\_B) L\_{\epsilon \mathcal{G}}} \tag{3}$$

where gBð Þ ω<sup>B</sup> is the peak SBS gain, Leff ¼ ½ � 1 � exp ð Þ �αL =α is the effective length with L as the real fiber length, Aeff is the effective cross section of the fiber, and Pth is the estimated SBS threshold. Later on, this estimation has been revised with different theories and approximations, achieving also different Cth values [13–15]. However, instead of a constant, Cth has been recently found to be dependent on the fiber length and its parameters [16]. As a typical experimental example, Figure 1(a) illustrates the transmitted and the reflected power as a function of the incident power in a 20 km SMF. The dashed black line symbolizes the SBS threshold, beyond which the backscattered power rapidly increases and the output pump power stays almost constant.

#### 2.2 The Brillouin gain

2. Basics of SBS

where kA ! , kp !

, and ks !

tion, and gBð Þ ω is the SBS gain.

60

2.1 Spontaneous and stimulated Brillouin scattering

Fiber Optic Sensing - Principle, Measurement and Applications

the laws of momentum and energy conservation, that is,

kA ! ¼ kp ! � ks !

backward SBS and its applications are discussed.

Brillouin scattering, named after the French physicist Léon Brillouin, who theoretically predicted light scattering by a thermally excited acoustic wave (phonon) in 1922 [5], is one of the most prominent nonlinear effects in optical fibers [6]. For spontaneous Brillouin scattering (SpBS), an incident photon (pump wave) is transformed into a frequency-downshifted scattered photon (Stokes wave) and a phonon (acoustic wave). The angular distribution of the Stokes wave is governed by

are the wave vector of acoustic, pump, and Stokes wave,

and ωB, ωp, and ω<sup>s</sup> are their corresponding angular frequencies. Considering the fiber geometry and provided that the phonon frequency is much smaller than that of both photons, an efficient Brillouin scattering only occurs in the backward direction. The Brillouin frequency shift (BFS) ν<sup>B</sup> ¼ ωB=2π is estimated to be �11 GHz for typical single mode fibers (SMF) and a pump wavelength in the C-Band of optical telecommunications (around 1550 nm) [7]. As has been shown very recently, forward SBS [8] can be used for sensing applications as well [9, 10]. However, the interaction is governed by transverse acoustic modes, and compared to backward SBS, the effect is rather weak. Since a description of this new field of SBS in sensing would go far beyond the scope of this book chapter, hereafter only

The basic origin of SBS is electrostriction, which tends to compress the material in

the presence of an electrical field [11]. The superposition of the pump and the counterpropagating Stokes wave modulates the density and hence the refractive index of the optical fiber through electrostriction. Thus, a moving grating (an acoustic wave) is formed. If the velocity of this moving grating coincides with the speed of sound in the material, the effect is very efficient and it additionally reflects optical power from the pump wave. Due to the Doppler effect, the reflected pump wave is downshifted in frequency by the frequency difference between pump and probe and

thus adds power to the Stokes wave. Therefore, a positive feedback loop is

the differential equation system (DES) in the propagation direction z as:

dIp

� dIs

critical gain factor Cth≈21 gives an estimation of this critical value:

established. This transformation from SpBS to SBS can be quantitatively described by

dz ¼ �gBð Þ <sup>ω</sup> IpIs � <sup>α</sup>Ip

dz <sup>¼</sup> gBð Þ <sup>ω</sup> IpIs � <sup>α</sup>Is

where Ip and Is are the intensity of pump and Stokes wave, α is the fiber attenua-

The threshold of SBS is defined by the critical power that characterizes the transformation from SpBS to SBS. However, its definition is rather controversial. Smith et al. first defined it as the input pump power at which the backscattered power equals to the transmitted power at the output [12] and Eq. (3) with the

ω<sup>B</sup> ¼ ω<sup>p</sup> � ω<sup>s</sup> (1)

(2)

The transferred energy from the pump to the Stokes wave can be regarded as an amplification when it is frequency downshifted to the pump wave by BFS. For typical SMF in the C-Band of optical telecommunications, the lifetime TB of the phonon involved in the SBS interaction is usually in the magnitude of �10 ns, which leads to a finite spectral distribution of the SBS gain. The complex SBS gain coefficient, which depicts the evolution of the probe wave as a function of frequency detuning ω between pump and probe wave, is approximated by a Lorentzian shape [17]:

$$\lg(\alpha) = \frac{\lg\_0 P\_p}{1 - 2j(\alpha - \alpha\_B)/\Gamma\_B} \tag{4}$$

where <sup>Γ</sup><sup>B</sup> <sup>¼</sup> <sup>T</sup>�<sup>1</sup> <sup>B</sup> , Pp is the pump power, g<sup>0</sup> is related to the inherent material Brillouin gain gp with g<sup>0</sup> ¼ gp=Aeff and Aeff is the effective cross section of the fiber, gp is in the range of 3 � <sup>10</sup>�<sup>11</sup> m/W to approximately 5 � <sup>10</sup>�<sup>11</sup> m/W at 1550 nm. The real part of Eq. (4) represents the power amplification of the probe wave, also known as the Brillouin gain spectrum (BGS), and can be expressed as:

$$\text{g}\_B(\alpha) = \frac{\text{g}\_0 P\_p \left(\Gamma\_B/2\right)^2}{\left(\alpha - \alpha\_B\right)^2 + \left(\Gamma\_B/2\right)^2} \tag{5}$$

#### Figure 1.

(a) The transition from SpBS to SBS in a 20 km SMF, from a distinct threshold (here Cth≈16), the reflected stokes power drastically increases, whereas the transmitted pump power stays almost constant; (b) simulated normalized Brillouin gain spectrum (BGS) and Brillouin phase spectrum (BPS) of a 20 km SMF.

The full width at half maximum (FWHM) of the BGS Δν<sup>B</sup> ¼ ΓB=ð Þ 2π , also known as the natural Brillouin linewidth, is estimated to be several tens of MHz for SMF, as shown in Figure 1(b) for a 20 km SMF. The Brillouin linewidth can be engineered to several GHz with a broadened pump for applications such as SBSbased filters [18–20] and slow light [21–23] while it can also be narrowed to a record of 3.4 MHz with specific techniques [24–28].

It is also worth to mention that the complex SBS gain changes not only the probe wave in amplitude, but also in phase by its imaginary part. As shown in Figure 1(b), the SBS phase shift has a linear dependence on the frequency detuning in the vicinity of the BFS and zero-phase shift directly at the BFS, that is, the peak Brillouin gain. These properties are important for applications such as SBS-based microwave photonic filters [29] and distributed Brillouin dynamic sensing [30].

#### 2.3 The polarization effect of SBS

Since the state of polarization (SOP) of the probe and pump waves hover randomly in the fiber due to the weak birefringence of SMF and only an averaged interaction can be detected at the output, the dependence of the SBS gain on the SOP has been neglected for long. However, with the rise of the polarization maintaining technique [31], the polarization effect of SBS has been investigated, and it has been found that the SBS efficiency of the pump-probe interaction with arbitrary polarization states is governed by [7]:

$$\eta\_{\rm SBS} = \frac{1}{2}(\mathbf{1} + \hat{\mathbf{s}} \cdot \hat{\mathbf{p}}) = \frac{1}{2}(\mathbf{1} + \mathfrak{s}\_1 p\_1 + \mathfrak{s}\_2 p\_2 - \mathfrak{s}\_3 p\_3) \tag{6}$$

where νBð Þ T; ε represents the BFS at a temperature T and strain ε, CT and C<sup>ε</sup> are the temperature and strain coefficients. Although both temperature and strain contribute to the BFS shift, the physical difficulty in discriminating the response from these two factors can be solved with specific strategies [43]. For standard SMF, CT and C<sup>ε</sup> are measured to be 1.081 MHz/°C and 42.93 kHz/με, respectively [42]. The slope of the linearity has also been studied intensively and optimized with

BFS dependence on (a) temperature and (b) strain in a SMF for a pump wavelength of 1550 nm [42].

**10.75**

**Brillouin frequency**

 **shift (MHz)**

**10.80**

**10.85**

**10.90**

**0 100 200 300**

**Microstrain ( )**

Since the first demonstration of the most widely used distributed Brillouin sensing scheme in time domain [4], which is now called Brillouin optical time domain analyzer (BOTDA), several different schemes have been proposed and

The principle of BOTDA is based on the Brillouin interaction between a pulsed pump (or probe) wave and a counterpropagating CW probe (or pump) wave. The acoustic wave is generated locally at the point where the pump pulse and the probe CW meet. The energy transfer via the acoustic wave at each position of the fiber under test (FUT) is determined by the frequency detuning between the two signals in comparison to the phonon frequency, that is, the probe wave is amplified when ν<sup>p</sup> � ν<sup>s</sup> ¼ ν<sup>a</sup> and depleted when ν<sup>s</sup> � ν<sup>p</sup> ¼ ν<sup>a</sup> where νp, νs, and ν<sup>a</sup> are the pump, probe, and phonon frequency, respectively. As shown in Figure 3(a) as a typical example, local Brillouin gain (or loss) can be translated from the time-dependent to the distance-dependent information according to the round trip relation z ¼ ct=ð Þ 2n where z is the fiber position where pump and probe wave interact, t is the propagation time of the pulse, c is the velocity of light in vacuum, and n is the refractive

In order to derive the local BFS at each position of the fiber, the reconstruction of the BGS, as illustrated in Figure 3(b), should be carried out by a frequency sweep with every frequency detuning around the phonon frequency. The accurate BFS at each fiber section can be achieved by fitting every measured BGS with the theoretical profile (either Voigt [45] or Lorentzian, dependent on pulse width) as

different doping concentrations [44].

3.2.1 BOTDA

**-50 -40 -30 -20 -10 0 10**

**Brillouin frequency**

Figure 2.

 **shift (MHz)**

index.

63

illustrated in Figure 4(a).

3.2 Overview of SBS sensing techniques

**10 20 30 40 50 60**

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**(a) (b)**

**Temperature (°C)**

developed with their own advantages and disadvantages.

where ^s ¼ ð Þ s1; s2; s<sup>3</sup> and p^ ¼ p1; p2; p<sup>3</sup> represent the unit vectors of Stokes and pump waves in the Poincaré sphere. As shown in Eq. (6), the SBS efficiencies of a parallel and orthogonal SOP are 1 � s 2 <sup>3</sup> and s 2 3. The SBS efficiency reaches zero when both of the pump and Stokes are orthogonally linear polarized (s<sup>3</sup> ¼ 0). This behavior can be used for many different applications from filters [19, 32, 33] via highresolution spectrum analyzers [34–36] to the generation of THz waves [37, 38].
