**2. Brillouin scattering in optical fibers**

## **2.1. Principle**

Light scattering phenomena in optical fibers occur regardless of how intense the incident optical power is. They can be basically categorized into two groups, i.e. spontaneous scattering and stimulated scattering[1]. Spontaneous scattering refers to the process under conditions such that the material properties are unaffected by the presence of the incident optical fields.

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

For input optical fields of sufficient intensities spontaneous scattering becomes quite intense and stimulated scattering starts. The nature of the stimulated scattering process grossly modifies the optical properties of the material system and vice versa. Spontaneous and stimulated scattering in optical fibers are composed of Rayleigh, Raman, and Brillouin scattering processes. Each scattering process is always present in optical fibers since no fiber is free from microscopic defects or thermal fluctuations which originate the three processes. For a monochromatic incident lightwave of frequency *f*0 at *λ*0=~1550 nm (telecom wavelength), three processes are schematically described by the spectrum of the scattered light as shown in Fig. 1. The components, whose frequency is beyond *f*0, correspond to anti-Stokes while those below *f*0 correspond to Stokes.

Brillouin scattering is a "photon-phonon" interaction as annihilation of a pump photon creates a Stokes photon and a phonon simultaneously. The created phonon is the vibrational modes of atoms, also called a propagation density wave or an acoustic phonon/wave. In a silica-based optical fiber, Brillouin Stokes wave propagates dominantly backward [2] although very partially forward[3]. The frequency (~9-11 GHz) of Stokes photon at ~1550-nm wavelength is in quantity dramatically different from or smaller by three orders of magnitude than Raman scattering (see Fig. 1) and is dominantly down-shifted due to Doppler shift associated with the forward movement of created acoustic phonons. In a polymer optical fiber, the frequency is ~2-3 GHz due to the different phonon property[4].

**Figure 1.** Schematic spectrum of scattered light resulting from three scattering processes in optical fibers.

For input optical fields of sufficient intensities spontaneous scattering becomes quite intense and stimulated scattering starts. The nature of the stimulated scattering process grossly modifies the optical properties of the material system and vice versa. Spontaneous and stimulated scattering in optical fibers are composed of Rayleigh, Raman, and Brillouin scattering processes. Each scattering process is always present in optical fibers since no fiber is free from microscopic defects or thermal fluctuations which originate the three processes. For a monochromatic incident lightwave of frequency *f*0 at *λ*0=~1550 nm (telecom wavelength), three processes are schematically described by the spectrum of the scattered light as shown in Fig. 1. The components, whose frequency is beyond *f*0, correspond to anti-Stokes while those

4 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

Brillouin scattering is a "photon-phonon" interaction as annihilation of a pump photon creates a Stokes photon and a phonon simultaneously. The created phonon is the vibrational modes of atoms, also called a propagation density wave or an acoustic phonon/wave. In a silica-based optical fiber, Brillouin Stokes wave propagates dominantly backward [2] although very partially forward[3]. The frequency (~9-11 GHz) of Stokes photon at ~1550-nm wavelength is in quantity dramatically different from or smaller by three orders of magnitude than Raman scattering (see Fig. 1) and is dominantly down-shifted due to Doppler shift associated with the forward movement of created acoustic phonons. In a polymer optical fiber, the frequency is

Stokes components Anti-stokes components

*Brillouin Brillouin*

*Raman Raman*

*Rayleigh*

*~9-11 GHz*

*f0*

**Figure 1.** Schematic spectrum of scattered light resulting from three scattering processes in optical fibers.

*frequency*

below *f*0 correspond to Stokes.

~2-3 GHz due to the different phonon property[4].

*~13 THz*

**Figure 2.** Comparison of (a) spontaneous Brillouin scattering (SpBS) and (b) stimulated Brillouin scattering (SBS) in optical fibers.

Figure 2 illustrates the difference between spontaneous Brillouin scattering (SpBS) and stimulated Brillouin scattering (SBS) in optical fibers. In principle, the SpBS (see Fig. 2(a)) is started from a noise fluctuation and influences the pump wave (*E*p); the SBS (see Fig. 2(b)) occurs when the pump power for SpBS is beyond the so-called Brillouin threshold value (*P*th) or when two coherent waves with a frequency difference equivalent to the phonon's frequency are counter-propagated. Brillouin scattering dynamics in optical fibers are generally governed by the following coupling equations [5, 6]:

$$
\left(\frac{1}{\nu\_{\varrho}}\frac{\partial}{\partial t} + \frac{\partial}{\partial \varpi}\right)E\_{\rho} = -\frac{\alpha}{2}E\_{\rho} + i\kappa\_{1}\rho E\_{s},\tag{1}
$$

$$
\left(\frac{1}{\nu\_s}\frac{\partial}{\partial t} - \frac{\partial}{\partial z}\right)E\_s = -\frac{\alpha}{2}E\_s + i\kappa\_1\rho^"E\_s,\tag{2}
$$

$$\left( + \frac{\partial}{\partial t} + \frac{\Gamma\_B}{2} + 2\pi i \nu\_B \right) \rho = i\kappa\_2 E\_\rho E\_s \, ^\ast + N,\tag{3}$$

where *E*p and *E*s stand for the normalized slowly-varying fields of pump and Stokes (or probe) waves, respectively; and *P*p=*|E*p|2 and *P*s=*|E*s|2 correspond to their optical powers; *ρ* denotes the acoustic (or phonon) field in terms of the material density distribution; *N* represents the random fluctuation or white noise in position and time [5]; *v*g is the group light velocity in the fiber; α is the fiber's propagation loss; ΓB is the damping rate of the acoustic wave, which equals to the reciprocal of the phonon's lifetime (1/ΓB=τ*ρ*=~10 ns) and is related to the acoustic linewidth (∆ν*B=*ΓB/π) [7]; *κ*1 and *κ*2 are the coupling coefficients among *E*p, *E*s, and *ρ.* If SpBS is considered, it is reasonable to assume *E*<sup>s</sup> is sufficiently small so that the second term of *N* dominates in the right side of Eq. (3). In contrast, the first term of iκ2EpEs\* dominates for SBS.

Taken into account the SBS power transfer between *P*p and *P*s under the assistance of the acoustic wave and the so-called acousto-optic effect, Eqs. (1-3) can be rewritten as

#### 6 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

$$
\left(\frac{1}{\nu\_s}\frac{\partial}{\partial t} + \frac{\partial}{\partial \overline{z}} + \alpha\right)P\_p = -\mathbf{g}(\nu)P\_p P\_s,\tag{4}
$$

$$
\left(\frac{1}{\nu\_s}\frac{\partial}{\partial t} - \frac{\partial}{\partial z} + a\right)P\_s = +\mathbf{g}(\nu)P\_p P\_s,\tag{5}
$$

where the sign difference between the right hands of Eq. (4) and Eq. (5) means that the pump power is reduced or depleted but the probe (Stokes) power is increased or amplified.

g(ν) is called Brillouin gain spectrum (BGS), the key phraseology to represent Brillouin scattering in optical fibers. It denotes the spectral details of the light amplification from strong pump wave to weak counter-propagating probe/Stokes wave in SBS or those of the noiseinitialized scattered phonons in SpBS. The BGS is generally expressed by [8-10]

$$\mathbf{g}(\nu) = \sum\_{l} \mathbf{g}^{(l)}(\nu),\tag{6}$$

$$\mathbf{g}^{(l)}(\nu) = \left\{ \mathbf{g}\_{B0} \cdot \frac{\nu\_{\rm B} \cdot \Delta \nu\_{\rm B}}{\nu\_{\rm ac}^{(l)} \cdot \Delta \nu\_{\rm ac}^{(l)}} \right\} \cdot \left\{ \frac{\left(\Delta \nu\_{\rm ac}^{(l)} / \mathcal{D}\right)^{2}}{\left[\nu - \left(f\_{0} - \nu\_{\rm ac}^{(l)}\right)\right]^{2} + \left(\Delta \nu\_{\rm ac}^{(l)} / \mathcal{D}\right)^{2}} \right\} \cdot \left\{ \frac{1}{A\_{\rm ao}^{(l)}} \right\},\tag{7}$$

where Eq. (6) means that the BGS is the summation of all the longitudinal acoustic modes' gain spectra and Eq. (7) corresponds to the *l*th-order one assigned the subscript of "*l*". In Eq. (7), ∆ν*ac (l)* is the *l*th-order linewidth or full width at half magnitude (FWHM) which can be assumed to be approximately the same for all the acoustic modes; ν*ac (l)* is the effective acoustic velocity; *Aao (l)* (in μm2 ) is the so-called acousto-optic effective area of the *l*th-order one. ν*ac (l)* and *Aao (l)* are qualitatively different among all acoustic modes.

There are two basic methods to theoretically and numerically analyze the BGS in optical fibers [8, 11]. One method [11] is based on Bessel or modified Bessel functions for optical fibers with regular geometric and dopant distribution, such as step-index optical fibers. The other one [8] is called two-dimensional finite-element-method (2D-FEM) modal analysis of BGS for optical fibers with complicated or arbitrary distribution. The 2D-FEM modal analysis has been used to study a Panda-type polarization-maintaining optical fiber (PMF) [8], a SMF with arbitrary residual stress [12], a *w*-shaped triple-layer fiber [13], or optical fibers with non-uniform optical/acoustic profiles such as solid or microsctrucuted photonic crystal fibers (PCF) [14-21].

The contribution of the fundamental acoustic mode to the entire BGS is basically dominant, which has a Lorentzian feature as schematically depicted in Fig. 3(a). Besides, it modulates the refractive index of optical fiber and changes the group velocity of optical fields in a profile shown in the inset of Fig. 3(a), which has been adopted for Brillouin slow or fast light[22, 23]. There are three basic parameters of Brillouin frequency shift (BFS, νB), Brillouin gain peak (gB0), and Brillouin linwidth (∆νB) in the main-peak BGS (i.e. the fundamental acoustic mode or *l*=1 in Eq. (7)). νB is defined as

$$\nu\_{\rm av} \equiv \nu\_{\rm ac}^{(1)} = \frac{2}{\lambda\_0} \cdot n\_{\rm eff} \cdot V\_{\rm a},\tag{8}$$

where λ0 is the light wavelength (λ0=*c/f*0 with *c* the light speed in vacuum), *n*eff is the effective refractive index of the fiber, and *Va* is the effective acoustic velocity of the fundamental acoustic mode. *n*eff and *Va* in Eq. (8) as well as *Aao (l)* in Eq. (7) are all determined by the respective waveguide structures of the optical modes (n0 and n1) and those of the longitudinal acoustic modes (V*l1*, V*l0*), relative to the silica dopant materials and distributions in the cross section [24]. Figure 3(b) illustrates a simple example of step-index single-mode optical fiber (SMF), even for which the BGS comprises of several (typically, four) longitudinal acoustic modes due to the different contrast of optical and acoustic waveguides [8, 11]. The measured BGS of the SMF and a high-delta nonlinear optical fiber at 1550nm are depicted in Fig. 3(c)[13]. It is worth noting that ν<sup>B</sup> in optical fibers suffers strong influence from the residual elastic and inelastic strains induced by different draw tensions during fiber fabrication[12].

**Figure 3.** (a) Brillouin gain spectrum (BGS) in optical fibers. The inset "i" denotes the change of group velocity of opti‐ cal fields. (b) Cross section of a step-index SMF. ∆=(n1-n0)/n0 is the relative index difference between the core (n1) and the cladding (n0). (c) Measured BGS of a step-index SMF (solid curve) and of a 17.0-mol% high-delta fiber (dashed curve). (Fig. 3(c) after Ref. [13]; © 2008 OSA.)

gB0 in Eq. (7) is determined by

1

6 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

*g*

1

*g*

n n

qualitatively different among all acoustic modes.

nn

*l B B ac*

to be approximately the same for all the acoustic modes; ν*ac*

( )

∆ν*ac*

*Aao*

*(l)* (in μm2

n

a

a

¶ ¶ è ø *<sup>s</sup> p s*

power is reduced or depleted but the probe (Stokes) power is increased or amplified.

initialized scattered phonons in SpBS. The BGS is generally expressed by [8-10]

n

 n

( ) ( ) ( ),

 <sup>=</sup> å *<sup>l</sup> l*

 n

0 () () () 2 () 2 () 0 ( / 2) <sup>1</sup> ( ) , [ ( )] ( / 2)

 n

*B l l ll l ac ac ac ac ao g g f A* (7)

where Eq. (6) means that the BGS is the summation of all the longitudinal acoustic modes' gain spectra and Eq. (7) corresponds to the *l*th-order one assigned the subscript of "*l*". In Eq. (7),

) is the so-called acousto-optic effective area of the *l*th-order one. ν*ac*

There are two basic methods to theoretically and numerically analyze the BGS in optical fibers [8, 11]. One method [11] is based on Bessel or modified Bessel functions for optical fibers with regular geometric and dopant distribution, such as step-index optical fibers. The other one [8] is called two-dimensional finite-element-method (2D-FEM) modal analysis of BGS for optical fibers with complicated or arbitrary distribution. The 2D-FEM modal analysis has been used to study a Panda-type polarization-maintaining optical fiber (PMF) [8], a SMF with arbitrary residual stress [12], a *w*-shaped triple-layer fiber [13], or optical fibers with non-uniform optical/acoustic profiles such as solid or microsctrucuted photonic crystal fibers (PCF) [14-21].

The contribution of the fundamental acoustic mode to the entire BGS is basically dominant, which has a Lorentzian feature as schematically depicted in Fig. 3(a). Besides, it modulates the refractive index of optical fiber and changes the group velocity of optical fields in a profile shown in the inset of Fig. 3(a), which has been adopted for Brillouin slow or fast light[22, 23].

*(l)* is the *l*th-order linewidth or full width at half magnitude (FWHM) which can be assumed

<sup>ì</sup> × D ü ì <sup>D</sup> üì ü =× × <sup>í</sup> ý í ýí ý <sup>×</sup> î ×D - - + D þ î þî þ

n

() 2

*l*

 n

where the sign difference between the right hands of Eq. (4) and Eq. (5) means that the pump

g(ν) is called Brillouin gain spectrum (BGS), the key phraseology to represent Brillouin scattering in optical fibers. It denotes the spectral details of the light amplification from strong pump wave to weak counter-propagating probe/Stokes wave in SBS or those of the noise-

¶ ¶ è ø *<sup>p</sup> p s*

æ ö ¶ ¶ ç ÷ + + =-

æ ö ¶ ¶ ç ÷ - + =+

 n() ,

 n() ,

*P g PP*

*vt z* (4)

*vt z* (5)

*g g* (6)

*(l)* is the effective acoustic velocity;

*(l)* and *Aao*

*(l)* are

*P g PP*

$$\mathbf{g}\_{B0} = \frac{4\pi n\_{\text{eff}}{\text{eff}}^{8} p\_{12}{}^{2}}{\lambda\_{0}^{3} \rho\_{0} \varepsilon \nu\_{B} \Delta \nu\_{B0}},\tag{9}$$

where *ρ*<sup>0</sup> is the density of silica glass (~2202 *kg/m3* ) and *p*12 the photo-elastic constant (~0.271). In most silica-based fibers, the peak gain value of *g*B0 lies in the range of 1.5~3× 10-11 m/W [25].

∆νB0 in silica optical fibers with a typical value of 30~40 MHz is characteristic of SpBS. However, in the SBS process, it was theoretically proved that ∆ν<sup>B</sup> strongly depends on the pump power, which is expressed as follows [5, 26]:

$$
\Delta \nu\_{\mathcal{B}} = \Delta \nu\_{\mathcal{B}0} \sqrt{\frac{\ln 2}{G\_s}},
\tag{10}
$$

where *Gs* is the single pass gain experienced by the weak probe wave from the strong pump wave, defined by

$$G\_s = \frac{\mathbf{g}\_{B0} P\_{\mu \text{sup}} L\_{\text{eff}}}{K \cdot A\_{\text{eff}}^{\text{ao}}},\tag{11}$$

where *K* (=1~2) is a polarization factor (=2 for a complete polarization scrambling process), *L*eff=[1-exp(-αL)]/α is the effective length of the fiber with α the optical loss (m-1) and *L* the fiber length, *Ppump* the pump power, and *A*effao=*Aao (1)* is the acoustic-optic effective area of the funda‐ mental acoustic mode. It is noted that the exponential (*Ge*) or logarithmic (*GdB, in dB*) gain of the weak probe power (*P*probe) are presented by

$$G\_s = \frac{\Delta P\_{probe}}{P\_{probe}} = \exp(G\_s) - \text{l},\tag{12}$$

$$G\_{\rm dB} = 10 \log\_{10}(G\_{\rm e}) \approx 4.342 G\_{\rm s}. \tag{13}$$

Eq. (13) is valid when *Ge*>>1.

From Eq. (10), one could estimate that the linewidth goes gradually to zero for very high gain, which can be obtained by either increasing the pump power or the interaction length of the fiber (see Eq. (11)). A zero linewidth corresponds to acoustic oscillation with an infinite time. However, pump depletion always occurs when the single pass gain (*G*s) increases, which results in a limited effective single pass gain and in turn leads to a finite linewidth instead of zero one.

The experimental characterization of the phenomenon of the Brillouin linwidth's narrowing in three SMFs are depicted in Fig. 4 [27]. When the pump power is intensified to a high value of above ∼24 dBm (∼250 mW), the Brillouin main-peak linewidth of 180-m-long SMF becomes increasing. This is because the pump power depletes much faster than its contribution to the single-pass gain *Gs* since the probe wave experiences an amplification of more than *GdB*=20 dB Brillouin Scattering in Optical Fibers and Its Application to Distributed Sensors http://dx.doi.org/10.5772/59145 9

**Figure 4.** Measured Brillouin linewidth in three SMF varying with increase of Brillouin pump power. F-SMF denotes a SMF with pure-silica core and F-doped-silica cladding. (After Ref. [27]; © 2008 OSA.)

for a greater pump power than ∼24 dBm. From the other point of view, during the pumpprobe-based BGS measurement, which will be described later, the Brillouin probe wave with a down-shifted frequency of just νB feels more significantly the depletion of the pump power than the one with a downshifted frequency of a finite offset from νB. More details of the effect of pump depletion in SBS will be demonstrated in **Section 2.3**.

From Eq. (11), one can derivate the so-called pump threshold value of SBS originating from SpBS (also called Brillouin generator). It is given by

$$P\_{\rm th} = \kappa \frac{A\_{\rm eff}^{\rm ao}}{L\_{\rm eff}} \frac{K}{\mathbf{g}\_{\rm B0}},\tag{14}$$

where *κ* is a numerical factor (=~21) [28] that may change in terms of the fiber length [29]. If a long-enough SMF is considered, α=0.2 dB/km (or 0.046 /km) meaning *L*eff=21.7 km. *A*effao≈ *A*eff=100 μm2 and *g*B0=2× 10-11 m/W. For a perfectly-linearized pump wave, *P*th≈ 3.2 mW; for a completely polarization-scrambled pump wave, *P*th≈ 4.6 mW.

#### **2.2. Experimental characterization**

where *ρ*<sup>0</sup> is the density of silica glass (~2202 *kg/m3*

which is expressed as follows [5, 26]:

length, *Ppump* the pump power, and *A*effao=*Aao*

Eq. (13) is valid when *Ge*>>1.

zero one.

the weak probe power (*P*probe) are presented by

wave, defined by

In most silica-based fibers, the peak gain value of *g*B0 lies in the range of 1.5~3× 10-11 m/W [25].

∆νB0 in silica optical fibers with a typical value of 30~40 MHz is characteristic of SpBS. However, in the SBS process, it was theoretically proved that ∆ν<sup>B</sup> strongly depends on the pump power,

> 0 ln 2 D =D

where *Gs* is the single pass gain experienced by the weak probe wave from the strong pump

eff <sup>=</sup> , <sup>×</sup> *B pump eff s ao gP L <sup>G</sup>*

where *K* (=1~2) is a polarization factor (=2 for a complete polarization scrambling process), *L*eff=[1-exp(-αL)]/α is the effective length of the fiber with α the optical loss (m-1) and *L* the fiber

mental acoustic mode. It is noted that the exponential (*Ge*) or logarithmic (*GdB, in dB*) gain of

From Eq. (10), one could estimate that the linewidth goes gradually to zero for very high gain, which can be obtained by either increasing the pump power or the interaction length of the fiber (see Eq. (11)). A zero linewidth corresponds to acoustic oscillation with an infinite time. However, pump depletion always occurs when the single pass gain (*G*s) increases, which results in a limited effective single pass gain and in turn leads to a finite linewidth instead of

The experimental characterization of the phenomenon of the Brillouin linwidth's narrowing in three SMFs are depicted in Fig. 4 [27]. When the pump power is intensified to a high value of above ∼24 dBm (∼250 mW), the Brillouin main-peak linewidth of 180-m-long SMF becomes increasing. This is because the pump power depletes much faster than its contribution to the single-pass gain *Gs* since the probe wave experiences an amplification of more than *GdB*=20 dB

<sup>D</sup> = = - *probe e s probe P G G*

exp( ) 1,

 n, *B B Gs*

0

n

8 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

) and *p*12 the photo-elastic constant (~0.271).

*K A* (11)

*(1)* is the acoustic-optic effective area of the funda‐

*<sup>P</sup>* (12)

<sup>10</sup> *G GG dB* = 10log ( ) 4.342 . *<sup>e</sup>* » *<sup>s</sup>* (13)

(10)

The experimental characterization of BGS in optical fibers can be implemented by two individual ways that depend on which principle of SpBS or SBS is based on. The SpBS based configuration is illustrated in Fig. 5(a). A pump wave is amplified by an erbium doped fiber amplifier (EDFA) and its polarization is optimized by a polarization controller (PC), scrambled by a polarization scrambler (PS) or switched by a polarization switcher (PSW). It is launched through an optical coupler or circulator into the fiber under test (FUT). A weak Stokes wave with downshifted frequency around νB is backscattered towards the coupler/circulator and can be detected by three different schemes, which are depicted in Fig. 5(b). First, an optical filter (Etalon, Fabry-Perot filter or FBG) is inserted before a photo-detector (PD) so as to eliminate the influence of Rayleigh scattering on Stokes wave and spectrally analyzed by an electrical spectrum analyzer (ESA) [2, 30]. Second, Stokes wave is optically mixed with a part of the pump wave (serving as an optical oscillator) and then detected or heterodyne-detected by a high-speed PD or a high-speed balanced PD, which is directed to the ESA [31]. Third, heterodyne detection can be carried out at an intermediate frequency (IF) range by tuning the frequency of the optical oscillator or further using of a local microwave (RF) oscillator before ESA [32].

**Figure 5.** Experimental setup of BGS measurement based on spontaneous Brillouin scattering (SpBS). (a) Basic configu‐ ration. LD: laser diode; PC: polarization controller; PS: polarization scrambler; PSW: polarization switcher; EDFA: erbi‐ um-doped fiber amplifier; ESA: electrical spectrum analyzer. The dashed boxes of Function Generator or Pulse generator are used for distributed SpBS measurement. (b) Three different methods to detect the weak Stokes wave, corresponding to the dotted box "i" in (a) with two optical ports (port 1 and port 2) and one electrical port (port 3).

The pump-probe-based experimental configuration, depicted in Fig. 6(a), is more attractive to investigate the BGS in opticalfibers (especially with very short length) since SBS process occurs andhighBrillouin gain can beutilized.Two light waves froma laserunit are the optical sources: the one with larger optical frequency (*f*0) serves as SBS pump and the other with lower optical frequency (*f*0-ν) works as SBS probe wave. They are launched into the opposite ends of FUT so as to ensure their counter-propagation and generate intense SBS interaction. The pump wave transfers intense energy to the probe wave, called Brillouin gain; in contrast, the probe wave absorbs energy fromthepump wave, calledBrillouinloss.Themagnitude of bothBrillouingain and loss depends on the frequency offset (ν) between the pump and probe waves, deter‐ mined by the Lorentzian feature of BGS (see Eq. (7)). Subsequently, the BGS can be character‐ ized by monitoring the power of the probe wave at PD1 (i.e. Brillouin gain) [8] or that of the pump wave at PD2 (i.e. Brillouin loss) [33] as a function of ν (provided by the laser unit), respectively. Recently, a scheme based on a combination of Brillouin gain and loss was newly proposed to enhance the signal to noise ratio (SNR) of the BGS measurement [34]. It can be realized by periodic switching of the pump and probe wave and detected at either PD1 or PD2. Besides,thesimultaneousdetectionofPD1andPD2followedbyasubtractionmayworkequally.

with downshifted frequency around νB is backscattered towards the coupler/circulator and can be detected by three different schemes, which are depicted in Fig. 5(b). First, an optical filter (Etalon, Fabry-Perot filter or FBG) is inserted before a photo-detector (PD) so as to eliminate the influence of Rayleigh scattering on Stokes wave and spectrally analyzed by an electrical spectrum analyzer (ESA) [2, 30]. Second, Stokes wave is optically mixed with a part of the pump wave (serving as an optical oscillator) and then detected or heterodyne-detected by a high-speed PD or a high-speed balanced PD, which is directed to the ESA [31]. Third, heterodyne detection can be carried out at an intermediate frequency (IF) range by tuning the frequency of the optical oscillator or further using of a local microwave (RF) oscillator before

10 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

**Figure 5.** Experimental setup of BGS measurement based on spontaneous Brillouin scattering (SpBS). (a) Basic configu‐ ration. LD: laser diode; PC: polarization controller; PS: polarization scrambler; PSW: polarization switcher; EDFA: erbi‐ um-doped fiber amplifier; ESA: electrical spectrum analyzer. The dashed boxes of Function Generator or Pulse generator are used for distributed SpBS measurement. (b) Three different methods to detect the weak Stokes wave, corresponding to the dotted box "i" in (a) with two optical ports (port 1 and port 2) and one electrical port (port 3).

ESA [32].

**Figure 6.** Experimental setup of BGS measurement based on stimulated Brillouin scattering (SBS). (a) Basic configura‐ tion. (b) Three different methods of laser unit (dotted box "i" in (a)) with two optical ports and one electrical port. The dashed boxes of Function Generator or Pulse generator are used for distributed SBS measurement. (c) Three schemes of detection unit shown in dotted box "*j*" in (a).

The laser unit shown in Fig. 6(a) comprises three ports (two corresponding to optical fields and the other to the revealed value of ν or a microwave/RF input), which has three different schemes as illustrated in Fig. 6(b). First, one can utilize two individual lasers under frequency/ phase locking and frequency countering [33]. Second, one laser is divided into two parts. One part is amplified by an EDFA working as pump wave; the second part serving as probe wave is modulated by an electro-optic intensity modulator (EOM) to generate two sidebands working as probe wave[25]. An optical filter is inserted before launching into the FUT or laid after PD1 so as to cut off the influence of the frequency-upshifted sideband. Third, the second part can be also modulated by a single-sideband modulator [35] to get well-suppressed frequency-downshifted sideband directly serving as probe wave. There are also three schemes, depicted in Fig. 6(c), to realize the detection unit shown in Fig. 6(a). The first and simple scheme is related to the first scheme of the laser unit. A personal computer with a multi-channel data acquisition card (DAQ) can catch the value of ν and record the data of PD1 and/or PD2 so as to pick up the Brillouin signal (gain and/or loss) as a function of ν. The second and third schemes in Fig. 6(c) can be used for either the second and/or third laser unit in Fig. 6(b), respectively. For instance, a high-cost vector network analyzer provides a frequency-tuned RF signal to modulators and simultaneously detect the Brillouin signal [36]. Alternatively, the RF signal can be achieved from a microwave synthesizer and the data of PD1 and/or PD2 can be picked up by a DAQ with or without a lock-in amplifier (LIA). It is notable that the use of LIA for detection unit requires an intensity-chopping of the pump wave by an additional EOM [10] or periodic switching of upshifted or downshifted sideband at the SSBM [34], which is advantageous for characterization of very weak BGS or a short-length FUT due to its high SNR and accuracy[12, 34, 37].

Figure 7(a) depicts a high-accuracy experimental setup of pump-probe SBS-based BGS characterization by use of SSBM and LIA for a short-length FUT [37]. An EDFA is inserted after the SSBM to increase the probe power, which is aimed to reduce the impact of Rayleigh scattering or splicing/crack induced reflection of the pump wave in the FUT. The optical lights after a circulator include the following components:

$$P\_{\text{tot}} = P\_{\text{probe}}^0 + \delta P\_{\text{probe}} + P\_{\text{pump}}^R,\tag{15}$$

where *P*probe0 denotes the probe power experiencing no Brillouin amplification, ∆*P*probe the amplified probe power, and *P*pumpR the reflected pump power. Thanks to the lock-in detection, the component of *P*probe<sup>0</sup> is effectively cut off by the LIA since it has no relationship with the chopped pump power given by

$$P\_{pump}(t) = P\_{pump}^0 \cdot \cos(2\pi f\_{ch}t),\tag{16}$$

where *fch* is the chopping frequency. Simply assuming that there is no depletion for pump power and no optical propagation loss for either probe or pump light, ∆*P*probe can be expressed by

Brillouin Scattering in Optical Fibers and Its Application to Distributed Sensors http://dx.doi.org/10.5772/59145 13

The laser unit shown in Fig. 6(a) comprises three ports (two corresponding to optical fields and the other to the revealed value of ν or a microwave/RF input), which has three different schemes as illustrated in Fig. 6(b). First, one can utilize two individual lasers under frequency/ phase locking and frequency countering [33]. Second, one laser is divided into two parts. One part is amplified by an EDFA working as pump wave; the second part serving as probe wave is modulated by an electro-optic intensity modulator (EOM) to generate two sidebands working as probe wave[25]. An optical filter is inserted before launching into the FUT or laid after PD1 so as to cut off the influence of the frequency-upshifted sideband. Third, the second part can be also modulated by a single-sideband modulator [35] to get well-suppressed frequency-downshifted sideband directly serving as probe wave. There are also three schemes, depicted in Fig. 6(c), to realize the detection unit shown in Fig. 6(a). The first and simple scheme is related to the first scheme of the laser unit. A personal computer with a multi-channel data acquisition card (DAQ) can catch the value of ν and record the data of PD1 and/or PD2 so as to pick up the Brillouin signal (gain and/or loss) as a function of ν. The second and third schemes in Fig. 6(c) can be used for either the second and/or third laser unit in Fig. 6(b), respectively. For instance, a high-cost vector network analyzer provides a frequency-tuned RF signal to modulators and simultaneously detect the Brillouin signal [36]. Alternatively, the RF signal can be achieved from a microwave synthesizer and the data of PD1 and/or PD2 can be picked up by a DAQ with or without a lock-in amplifier (LIA). It is notable that the use of LIA for detection unit requires an intensity-chopping of the pump wave by an additional EOM [10] or periodic switching of upshifted or downshifted sideband at the SSBM [34], which is advantageous for characterization of very weak BGS or a short-length FUT due to its high SNR

12 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

Figure 7(a) depicts a high-accuracy experimental setup of pump-probe SBS-based BGS characterization by use of SSBM and LIA for a short-length FUT [37]. An EDFA is inserted after the SSBM to increase the probe power, which is aimed to reduce the impact of Rayleigh scattering or splicing/crack induced reflection of the pump wave in the FUT. The optical lights

amplified probe power, and *P*pumpR the reflected pump power. Thanks to the lock-in detection,

where *fch* is the chopping frequency. Simply assuming that there is no depletion for pump power and no optical propagation loss for either probe or pump light, ∆*P*probe can be expressed

, *<sup>R</sup> PP P P tot probe probe pump* (15)

*ch* (16)

is effectively cut off by the LIA since it has no relationship with the

denotes the probe power experiencing no Brillouin amplification, ∆*P*probe the

p

<sup>0</sup> =+ + d

<sup>0</sup> *P t P ft pump pump* ( ) cos(2 ), = ×

and accuracy[12, 34, 37].

where *P*probe0

by

the component of *P*probe<sup>0</sup>

chopped pump power given by

after a circulator include the following components:

**Figure 7.** High accuracy pump-probe-based BGS characterization. (a) Experimental setup. (b) Characterized BGS. (c) Measurement accuracy. ((a) and (b) after Ref. [37]; © 2007 OSA.)

$$
\delta \mathcal{P}\_{pwob} = \mathbf{g}(\nu) \cdot P\_{pwob}^{0} (f\_0 - \nu) \cdot P\_{punp}, \tag{17}
$$

If *R* denotes the reflectivity of pump power arising from both Rayleigh scattering and some reflection points, the reflected pump power can be expressed:

$$P\_{pump}^{\mathcal{R}} = \left[P\_{pump}^{0} - P\_{loss}\right] \cdot R,\tag{18}$$

where *Ploss* is the so-called Brillouin loss of pump power during Brillouin interaction which is approximately equal to ∆*P*probe as

$$P\_{\rm loss} = \delta P\_{\rm probe} = \mathbf{g}(\nu) \cdot P\_{\rm probe}^0 \cdot P\_{\rm pump}^0 \cdot \cos(2\pi f\_{\rm ch} t),\tag{19}$$

The demodulated electric amplitude via a LIA at *f*ch is given by

$$P\_{dc} = \left[P\_{probo}^0 \cdot \mathbf{g}(\nu) \cdot (1 - R) + R\right] \cdot P\_{punp}^0,\tag{20}$$

where the part of P probe <sup>0</sup> ⋅ g(ν) ⋅ (1− R) ⋅ P pump <sup>0</sup> is the signal to be detected by the LIA and the rest part of R ⋅ P pump<sup>0</sup> is the noise level. From it, one can deduce the signal-to-noise ratio (SNR):

$$LSNR = \frac{P\_{\text{probi}}^0 \cdot \text{g}(\nu) \cdot (\text{l} - R)}{R},\tag{21}$$

which is independent on the pump level, but just determined by the probe power of *P*probe0 and the reflection rate of *R* as well as the BGS of *g*(ν). In other words, an increase of probe power can drastically enhance SNR and also improve the system accuracy. As an example, Figure 7(b) shows a characterized fundamental-order or higher-order resonance BGS in a *w*-shaped high-delta fiber with fluorine inner cladding (F-HDF) [37]. The measurement system has a high accuracy of 0.13-MHz standard deviation at laboratory condition or 0.05 MHz for welltemperature-controlled condition, as illustrated in Fig. 7(c).

#### **2.3. Pump depletion effect**

As mentioned above (see Fig. 4), pump depletion effect influences the linewidth of pumpprobe-based BGS. Early in 2000 [38], it was first observed that the spectrum broadening and hole burning occurs in a SBS generator (i.e. noise-started spontaneous Brillouin scattering). The reason was thought as the waveguide interaction among different angular components of the pump and backscattered Stokes signals. Besides, during the application of SBS-based amplifier, two coherent optical waves with precise frequency difference equal to Brillouin frequency shift are launched into optical fibers; then a frequency-scanned weak signal could suffer non-uniform amplification if the two waves' powers are too high [39, 40]. In SBS-based distributed fiber optical sensor, which will be introduced in **Section 3**, two coherent waves (pulse and/or continuous wave (CW)) are injected into the two opposite ends of the sensing fiber. Recently, it was found that pump depletion of pump-probe-based system configuration could induce a significant measurement error of the local Brillouin frequency shift in the far end of the probe (Stokes) wave [41].

Assuming that CW probe wave, *Ps*(0), is injected at the near end of the fiber (*z=*0) while CW pump wave, *Pp*(*l*), is launched at the far end of the fiber (*z=l* with *l* the fiber length). Considering the steady-state condition and neglecting the transmission loss of the fiber, the coupling equations of Eq. (4) and Eq. (5), describing the SBS interaction, can be modified to the dimen‐ sionless equations [42]:

$$\frac{dQ\_{\rho}}{d\mathbf{x}} = kQ\_{\rho}Q\_{s},\tag{22}$$

Brillouin Scattering in Optical Fibers and Its Application to Distributed Sensors http://dx.doi.org/10.5772/59145 15

$$\frac{d\underline{Q}\_s}{d\mathbf{x}} = k\underline{Q}\_\rho \underline{Q}\_s,\tag{23}$$

where *Qp*=*Pp/Ps*(0) and *Qs*=*Ps/Ps*(0) represent the normalized pump and probe waves with respect to the injected probe wave of *Ps*(0), *x=z*/*l* is the normalized position, and *k=G∙l∙Ps*(0) is the normalized Brillouin gain with *G*=*g*(ν)/*A*eff. Figure 8(a) illustrates the normalized BGS at different positions of an arbitrary-length SMF, which are numerically calculated according to Eqs. (22) and (23). It is found that BGS gradually gets broadened, saturated, and hole-burned when the position moves from the far end (*z=l*) towards the near end of the fiber (*z=*0). We define the BGS saturation as the critical condition of spectral hole burning phenomenon.

The demodulated electric amplitude via a LIA at *f*ch is given by

14 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

temperature-controlled condition, as illustrated in Fig. 7(c).

where the part of

R ⋅ P pump

**2.3. Pump depletion effect**

end of the probe (Stokes) wave [41].

sionless equations [42]:

rest part of

P probe

<sup>0</sup> ⋅ g(ν) ⋅ (1− R) ⋅ P pump

0 0 = × ×- + × é ù ( ) (1 ) , n

<sup>0</sup> ( ) (1 ) , × × n<sup>=</sup> *Pg R probe SNR*

which is independent on the pump level, but just determined by the probe power of *P*probe0 and the reflection rate of *R* as well as the BGS of *g*(ν). In other words, an increase of probe power can drastically enhance SNR and also improve the system accuracy. As an example, Figure 7(b) shows a characterized fundamental-order or higher-order resonance BGS in a *w*-shaped high-delta fiber with fluorine inner cladding (F-HDF) [37]. The measurement system has a high accuracy of 0.13-MHz standard deviation at laboratory condition or 0.05 MHz for well-

As mentioned above (see Fig. 4), pump depletion effect influences the linewidth of pumpprobe-based BGS. Early in 2000 [38], it was first observed that the spectrum broadening and hole burning occurs in a SBS generator (i.e. noise-started spontaneous Brillouin scattering). The reason was thought as the waveguide interaction among different angular components of the pump and backscattered Stokes signals. Besides, during the application of SBS-based amplifier, two coherent optical waves with precise frequency difference equal to Brillouin frequency shift are launched into optical fibers; then a frequency-scanned weak signal could suffer non-uniform amplification if the two waves' powers are too high [39, 40]. In SBS-based distributed fiber optical sensor, which will be introduced in **Section 3**, two coherent waves (pulse and/or continuous wave (CW)) are injected into the two opposite ends of the sensing fiber. Recently, it was found that pump depletion of pump-probe-based system configuration could induce a significant measurement error of the local Brillouin frequency shift in the far

Assuming that CW probe wave, *Ps*(0), is injected at the near end of the fiber (*z=*0) while CW pump wave, *Pp*(*l*), is launched at the far end of the fiber (*z=l* with *l* the fiber length). Considering the steady-state condition and neglecting the transmission loss of the fiber, the coupling equations of Eq. (4) and Eq. (5), describing the SBS interaction, can be modified to the dimen‐

> = , *<sup>p</sup> p s*

*dQ kQ Q dx* (22)

*P P g R RP de probe* ë û *pump* (20)

<sup>0</sup> is the noise level. From it, one can deduce the signal-to-noise ratio (SNR):

<sup>0</sup> is the signal to be detected by the LIA and the

*<sup>R</sup>* (21)

Further introduce the injected power ratio between pump and probe waves, defined as *γ*=*Qp*(*x*=1)=*Pp*(*l*)*/Ps*(0). Since *dQp/dx=dQs/dx*, the difference between *Qp* and *Qs* maintains a constant (*A*), i.e. *A=Qp*-*Qs*, which is determined by *k* and *γ*. Consequently, the analytical solutions to Eqs. (22) and (23) are derived as

$$\mathcal{Q}\_{\rho} = \frac{A\left(A+\mathbf{l}\right)e^{-k\mathbf{l}\cdot\mathbf{x}}}{\left(A+\mathbf{l}\right)e^{-k\mathbf{l}\cdot\mathbf{x}}-\mathbf{l}},\tag{24}$$

$$
\underline{Q}\_s = \frac{A}{(A+\mathbf{l})e^{-k\mathbf{l}\cdot\mathbf{x}} - \mathbf{l}}.\tag{25}
$$

The critical condition of the spectral hole burning phenomenon can be theoretically expressed by:

$$\frac{d\underline{Q}\_s}{d\nu} = 0.\tag{26}$$

By numerically solving Eqs. (24)-(26), one can interpret the critical condition by two different ways: (1) the critical position *xc* for the fixed pump and probe power; (2) the critical powers for a specific position of the fiber. Figure 8(b) depicts the calculated relation of *xc* to *k* and *γ*, which indicates that *xc* moves towards the fiber far end when *k* and *γ* reach higher values (i.e. the fiber gets longer or the injected powers are stronger). It means that the pump depletion gets worse since much longer segments in the fiber suffer spectral hole burning. It is notable to address that the physical nature of the critical powers is essentially the same as that of the critical position because they can be also deduced by the contour (i.e. *k*-*γ* curve) at a fixed position *x* in Fig. 8(b):

$$P\_s\left(0\right) = k \mid GL,\tag{27}$$

$$P\_p\left(L\right) = \gamma k / GL.\tag{28}$$

**Figure 8.** (a) Simulated BGS at different positions in the fiber. (b) Critical position *xc*, determined by the normalized gain *k* and injected power ratio γ. The critical positions divide the fiber into two parts with and without the spectral hole burning phenomenon. Measured BGS in the middle (c) or at the end (d) of a 50-m-long fiber. (after Ref. [42]; © 2014 JJAP.)

Figure 8(c) and 8(d) illustrates the measured BGS under different pump power for two different positions (in the middle and at the far end, respectively) of a 50-m-long dispersion compen‐ sated fiber (DCF). The probe power is fixed at 9.8 dBm. At the far end, the BGS [see Fig. 8(d)] rises with the power increased but always preserves the Lorentz shape. While in the middle, the experimental result [see Fig. 8(c)] is in a qualitative accordance with the numerical analysis [see Fig. 8(a)]. The Brillouin gain keeps rising with the increase of optical power, while the peak at the local Brillouin frequency shift seems to be saturated gradually and a hollow starts appearing when it reaches ~20 dBm, which is just the spectral hole burning phenomenon. The hollow in the BGS may introduce great errors to pump-probe-based Brillouin distributed sensors since it disables the peak-searching of the Brillouin frequency shift.

The pump power leading to the BGS saturation is approximately characterized as the critical pump power (for instance, 21.3 dBm at z=30 m). The measured critical powers for two positions (*z=*20 m or 30 m) of 50-m-long DCF are depicted in Fig. 9, where the simulated critical powers are compared. It illustrates that the critical pump powers approximately measured for several probe powers (open symbols) have very similar trend as the theoretical analysis (curves). It is clear that the position of bigger *z* requires greater critical powers. This is because the pump depletion is weaker and the distortion of the BGS is less serious if the position is much closer to the fiber far end.

**Figure 9.** Critical powers for two different positions in a 50-m-long DCF. Solid and dashed curves, simulation; open symbols: experiment. (after Ref. [42]; © 2014 JJAP.)
