**2. Spectral characteristics of SMBS gain, SRS scattering and FBG reflection contours**

#### **2.1. Spectral characteristics of SMBS gain and SRS scattering contours**

Two important nonlinear effects in optical fibers, known as SRS and SMBS, are related to vibration excitation modes of silica and fall in the category of stimulated inelastic scattering in which the optical field transfers part of its energy to the nonlinear medium [1]. Even though SRS and SMBS are very similar in their origin, the main difference between the two is that optical phonons participate in SRS while acoustic phonons participate in SMBS. In a simple quantum-mechanical picture applicable to both SRS and SMBS, a photon of the incident field (called the pump) is annihilated to create a photon at a lower frequency (belonging to the Stokes wave) and a phonon with the right energy and momentum, to conserve the energy and the momentum. Of course, a higher-energy photon at the so-called anti-Stokes frequency can also be created if a phonon of right energy and momentum is available. Different dispersion relations for acoustic and optical phonons lead to some basic differences between the two. The fundamental one is that SMBS in single-mode fibers occurs only in the backward direction whereas SRS can occur in both directions.

Both the Raman-gain spectrum *gR*(*ν*) and the Mandelstam-Brillouin-gain spectrum *gMB*(*ν*) have been measured experimentally for silica fibers. The Raman-gain spectrum is found to be very broad, extending up to 40 THz [17]. The peak gain *gR* ≈ 6×10−14 m/W at pump wavelength near 1.5 μm and occurs for a spectral shift of about 13.1 THz. In contrast, the Mandelstam-Brillouingain spectrum is extremely narrow and has a bandwidth of <100 MHz. The peak value of Mandelstam-Brillouin-gain occurs, for the Stokes shift of ~10 GHz, for pump wavelengths near 1.5 μm. The peak gain *gMB* ≈ 6×10−11 m/W for a narrow-bandwidth pump [18] and decreases by a factor of ΔνP/ΔνMB for a broad-bandwidth pump, where ΔνP is the pump bandwidth and ΔνMB is the Mandelstam-Brillouin-gain bandwidth. As the Mandelstam-Brillouin-gain coefficient *gMB* is larger by nearly three orders of magnitude compared with *gR*, typical values of SMBS threshold are ~1 mW and for SRS threshold are ~1 W.

Although a complete description of SRS *gR*(*ν*) in optical fibers is quite involved, the spectral characteristics for SMBS *gMB*(*ν*) can be described by a simple relation. Little reminding – SMBS is a result of scattering of light on acoustic waves (acoustic phonons), that are excited by thermal fields and produce periodic modulation of the refractive index of fiber [1]. As a result of Bragg diffraction the induced grating of refractive index scatters the pumping radiation. As the scattered light undergoes a Doppler effect, the frequency shift *νMB*, caused by SMBS, depends on acoustic velocity and is given by

$$\mathcal{V}\_{MB} = \frac{2nV\_A}{\mathcal{A}\_p},\tag{1}$$

where *VA* acoustic velocity in the fiber, *n* refractive index, *λ<sup>P</sup>* pump wavelength. The shape of the SMBS gain spectrum is determined by strong attenuation of sound waves in silica. Growth of Stokes wave's intensity is characterized by gain coefficient *gMB*(*ν*), maximum at *ν* =*νMB*. Because of exponential decay of the acoustic waves, the gain spectrum *gMB*(*ν*) will have the Lorentzian spectral profile

$$\mathbf{g}\_{\rm MB}(\nu) = \mathbf{g}\_0 \frac{\left(\Delta\nu\_{\rm MB}/2\right)^2}{\left(\nu - \nu\_{\rm MB}\right)^2 + \left(\Delta\nu\_{\rm MB}/2\right)^2},\tag{2}$$

where *ΔνMB* spectrum full width at half maximum. The maximum gain is given by

optical phonons participate in SRS while acoustic phonons participate in SMBS. In a simple quantum-mechanical picture applicable to both SRS and SMBS, a photon of the incident field (called the pump) is annihilated to create a photon at a lower frequency (belonging to the Stokes wave) and a phonon with the right energy and momentum, to conserve the energy and the momentum. Of course, a higher-energy photon at the so-called anti-Stokes frequency can also be created if a phonon of right energy and momentum is available. Different dispersion relations for acoustic and optical phonons lead to some basic differences between the two. The fundamental one is that SMBS in single-mode fibers occurs only in the backward direction

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

Both the Raman-gain spectrum *gR*(*ν*) and the Mandelstam-Brillouin-gain spectrum *gMB*(*ν*) have been measured experimentally for silica fibers. The Raman-gain spectrum is found to be very broad, extending up to 40 THz [17]. The peak gain *gR* ≈ 6×10−14 m/W at pump wavelength near 1.5 μm and occurs for a spectral shift of about 13.1 THz. In contrast, the Mandelstam-Brillouingain spectrum is extremely narrow and has a bandwidth of <100 MHz. The peak value of Mandelstam-Brillouin-gain occurs, for the Stokes shift of ~10 GHz, for pump wavelengths near 1.5 μm. The peak gain *gMB* ≈ 6×10−11 m/W for a narrow-bandwidth pump [18] and decreases by a factor of ΔνP/ΔνMB for a broad-bandwidth pump, where ΔνP is the pump bandwidth and ΔνMB is the Mandelstam-Brillouin-gain bandwidth. As the Mandelstam-Brillouin-gain coefficient *gMB* is larger by nearly three orders of magnitude compared with *gR*, typical values

Although a complete description of SRS *gR*(*ν*) in optical fibers is quite involved, the spectral characteristics for SMBS *gMB*(*ν*) can be described by a simple relation. Little reminding – SMBS is a result of scattering of light on acoustic waves (acoustic phonons), that are excited by thermal fields and produce periodic modulation of the refractive index of fiber [1]. As a result of Bragg diffraction the induced grating of refractive index scatters the pumping radiation. As the scattered light undergoes a Doppler effect, the frequency shift *νMB*, caused by SMBS, depends

<sup>2</sup> , *<sup>A</sup>*

<sup>=</sup> (1)

*nV*

*P*

where *VA* acoustic velocity in the fiber, *n* refractive index, *λ<sup>P</sup>* pump wavelength. The shape of the SMBS gain spectrum is determined by strong attenuation of sound waves in silica. Growth of Stokes wave's intensity is characterized by gain coefficient *gMB*(*ν*), maximum at *ν* =*νMB*. Because of exponential decay of the acoustic waves, the gain spectrum *gMB*(*ν*) will have the

> 0 2 2 ( 2) ( ) , ( ) ( 2) *MB*

n

nn

*MB MB*

2

<sup>D</sup> <sup>=</sup> - +D (2)

 n

l

*MB*

n

whereas SRS can occur in both directions.

on acoustic velocity and is given by

Lorentzian spectral profile

*MB*

*g g*

n

of SMBS threshold are ~1 mW and for SRS threshold are ~1 W.

$$\mathbf{g}\_{\rm MB}(\nu\_{\rm MB}) = \mathbf{g}\_0 = \frac{2\pi n^{\gamma} p\_{12}^2}{c\lambda\_p^2 \rho\_0 V\_A \Delta\nu\_{\rm MB}},\tag{3}$$

where *p*12 longitudinal acousto-optic coefficient, *ρ*<sup>0</sup> density of material, *c* light speed in vacuum.

SMBS using allows to measure temperature (frequency shift of *νMB* about 1 MHz /°C) and strain (frequency shift of *νMB* about 493 MHz/%) of fiber so provide distributed technologies in sensor nets [1].

Let's return to the description of SRS *gR*(*ν*) in optical fibers [1]. A weak Stokes signal launched into a fiber with a stronger pump will be amplified due to SRS as discussed in this chapter. The amplification of the signal is described through the following equation

$$\mathbf{g}\_{\boldsymbol{\kappa}}(\boldsymbol{\nu}) = \sigma\_0 \left( \boldsymbol{\nu} \right) \frac{\boldsymbol{\mathcal{Z}}\_{\boldsymbol{s}}^{\boldsymbol{\beta}}}{c^2 \eta \boldsymbol{m}^2 \left( \boldsymbol{\nu} \right)},\tag{4}$$

where η is Planck's constant, *λ<sup>s</sup>* is the Stokes wavelength, *n*(*v*) is the refractive index, which is frequency dependent. The spontaneous Raman cross section *σ*<sup>0</sup> (*ν*) is defined as the ratio [1] of radiated power at the Stokes wavelengths to the pump power at temperature 0 °K, which can be obtained with the measured Raman cross-section *σ<sup>T</sup>* (*ν*) at temperature *T* °K, the thermal population factor *N* (*v*, *T*) and Boltzmann constant *κB*:

$$
\sigma\_0(\nu) = \sigma\_\tau(\nu) / \left( N(\Delta \nu, T) + 1 \right),
\tag{5}
$$

$$N\left(\Delta\nu, T\right) = \sqrt{\exp\left(\frac{\eta c \Delta\nu}{\kappa\_B T} - 1\right)}.\tag{6}$$

The *gR*(*ν*) in (4) is the spontaneous Raman gain coefficient in bulk glass and is uniform in all directions. For the first time, the Gaussian decomposition technique was proposed earlier in [19] for a spontaneous Raman spectrum. It is known that the Raman gain profile differs from the spontaneous Raman spectrum, especially in the low frequency region [21]. So for SRS the Raman gain coefficient at the Stokes frequency *ν<sup>S</sup>* =*ν<sup>P</sup>* −*ν* with some assumptions can be written as follows [20]

$$\mathbf{g}\_{\kappa}(\nu) \approx \frac{\nu \tilde{A}}{\left(\nu\_{\mathcal{S}}^{2} - \nu^{2}\right)^{2} + \nu^{2} \tilde{A}^{2}},\tag{7}$$

where *νS* and *ν* are the resonance and phonon angular frequencies, respectively, Г is the phonon damping constant.

Model analysis of the Raman spectrum with Gaussian profiles is based on the following expression [21]:

$$\mathbf{g}\_{\mathcal{R}}\left(\boldsymbol{\nu}\right) = \mathbf{g}\_{\mathcal{R}} \sum\_{i=1}^{N\_n} A\_i \exp\left[-\left(\boldsymbol{\nu} - \boldsymbol{\nu}\_{\boldsymbol{\nu},i}\right)^2 / \boldsymbol{\Gamma}\_i^2\right],\tag{8}$$

where *Nm* is the number of modes used for decomposition, *v,i* is the central frequency of *i*-th Gaussian profile, *Г<sup>i</sup>* =FWHM*<sup>i</sup>* / (2 ln2) ≈0.6 FWHM*<sup>i</sup> ,* where FWHM*<sup>i</sup>* is the full width at the half maximum of *i-*th Gaussian profile. Amplitudes Г*<sup>i</sup>* together with *v,i* and Г*<sup>i</sup>* are used as param‐ eters in the nonlinear fitting procedure. There are two important aspects of the developed decomposition procedure. Firstly, this approach is seen as a possible method of subdividing the density of states according to specific contributions. Secondly, function *gR*(*ν*) that gives the best fit of the experimental Raman gain profile is constructed [21]. We can see SMBS gain spectra and its parameters on the fig. 1,a with the results of Lorentzian fitting [22], so as SRS gain spectra on the fig. 1,b with the results of triangle (linear) and Gaussian (nonlinear) fitting [23]. For the last one we can mark that Gaussian fitting is more applicable and accurate for the central lobe of spectra [24].

#### **2.2. Spectral characteristics of FBG reflection contours**

FBG couple the fundamental mode of an optical fiber with the same mode propagating in the opposite direction. This means that radiation propagating in the fiber at a certain wavelength (9) is reflected from the grating completely or partially. Central or resonant frequency FBG *λ*BG is defined by following expression [2]:

$$\mathcal{A}\_{\rm BG} = \mathcal{D}n\_{\rm eff} \Lambda,\tag{9}$$

where *n*eff is effective refraction factor of the basic mode, *Λ* is the period of its modulation.

The characteristics of this reflection depend on the grating parameters. It is possible to describe the envelope *R* of FBG reflection spectra defined by detunes *δ* as:

$$R = \sinh^2\left[\kappa L\sqrt{\mathbf{l} - \left(\delta\not\!/\kappa\right)^2}\right] \left\langle \left| \cosh^2\left[\kappa L\sqrt{\mathbf{l} - \left(\delta\not\!/\kappa\right)^2}\right] - \left(\delta\not\!/\kappa\right)^2\right| \approx \tanh\left[\kappa L\sqrt{\mathbf{l} - \left(\delta\not\!/\kappa\right)^2}\right],\tag{10}$$

Poly-harmonic Analysis of Raman and Mandelstam-Brillouin Scatterings and Bragg Reflection Spectra http://dx.doi.org/10.5772/59144 61

Figure 1. SMBS (a) and SRS (b) gain spectrums with the results of its fittings **Figure 1.** SMBS (a) and SRS (b) gain spectrums with the results of its fittings

described by the expression

10<sup>3</sup> *LL* (nm).

( )

nn

*S*

( ) ( )

*R Ri vi i*

1

*i g v g A vv* =

Gaussian profile, *Г<sup>i</sup>* =FWHM*<sup>i</sup>* / (2 ln2) ≈0.6 FWHM*<sup>i</sup>*

maximum of *i-*th Gaussian profile. Amplitudes Г*<sup>i</sup>*

**2.2. Spectral characteristics of FBG reflection contours**

*Nm*

*g*

damping constant.

expression [21]:

central lobe of spectra [24].

k

 dk

is defined by following expression [2]:

n

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

»

<sup>2</sup> 2 2 22 ( ) , *<sup>R</sup>*

 n *Ã*

<sup>2</sup> <sup>2</sup>

*,* where FWHM*<sup>i</sup>*

é ù = -- G å ë û (8)

,

exp / ,


is the full width at the half

together with *v,i* and Г*<sup>i</sup>* are used as param‐

= L 2 , *n* (9)

 k  dk

*Ã*

n

where *νS* and *ν* are the resonance and phonon angular frequencies, respectively, Г is the phonon

Model analysis of the Raman spectrum with Gaussian profiles is based on the following

where *Nm* is the number of modes used for decomposition, *v,i* is the central frequency of *i*-th

eters in the nonlinear fitting procedure. There are two important aspects of the developed decomposition procedure. Firstly, this approach is seen as a possible method of subdividing the density of states according to specific contributions. Secondly, function *gR*(*ν*) that gives the best fit of the experimental Raman gain profile is constructed [21]. We can see SMBS gain spectra and its parameters on the fig. 1,a with the results of Lorentzian fitting [22], so as SRS gain spectra on the fig. 1,b with the results of triangle (linear) and Gaussian (nonlinear) fitting [23]. For the last one we can mark that Gaussian fitting is more applicable and accurate for the

FBG couple the fundamental mode of an optical fiber with the same mode propagating in the opposite direction. This means that radiation propagating in the fiber at a certain wavelength (9) is reflected from the grating completely or partially. Central or resonant frequency FBG *λ*BG

BG eff

( ) { ( ) ( ) } ( ) 2 2 2 2 2 2 *R L* sinh 1

 dk

*<sup>L</sup>* tanh 1 , *<sup>L</sup>* é ùé ù é ù <sup>=</sup> - - -» - ê úê ú ê ú ë ûë û ë û (10)

 dk

k

where *n*eff is effective refraction factor of the basic mode, *Λ* is the period of its modulation.

The characteristics of this reflection depend on the grating parameters. It is possible to describe

l

the envelope *R* of FBG reflection spectra defined by detunes *δ* as:

cosh 1

where *L* – FBG length, *κ* – coupling factor of direct and return mode, (*δ* / *κ*) – relative detune. Detune of FBG with period *Λ* is equal to *δ* =*Ω* −(*π* / *Λ*), where *Ω* =2*πn*eff / *λ* [2]. The characteristics of this reflection depend on the grating parameters. It is possible to describe the envelope *R* of FBG reflection spectra defined by detunes as:

The spectral width of the resonance of a homogeneous FBG (fig. 2,*a*) measured between the first zeroes of its reflection spectrum is described by the expression 1sinh cosh <sup>1</sup> ,1tanh <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> *<sup>R</sup> <sup>L</sup> <sup>L</sup> <sup>L</sup>* (10) where *L* – FBG length, – coupling factor of direct and return mode, – relative detune. Detune of FBG with period is

$$
\Delta \mathcal{J}\_{\text{BG},0} = 2 \mathcal{J}\_{\text{BG}} \frac{\Lambda\_{\text{BG}}}{L} \left[ 1 + \left( \frac{\kappa L \sqrt{1 - (\delta / \kappa)}}{\pi} \right)^2 \right]^{1/2}. \tag{11}
$$

1

 

*BG* . (11)

*T*

 

BG,0 BG

2 1

*L*

   

 *L*

**Figure 2.** FBG reflection (a) and FBG-PS transmittance (b) calculated spectrums Figure 2. FBG reflection (a) and FBG-PS transmittance (b) calculated spectrums

The resonant FBG wavelength *λBG* depends on fiber temperature and from mechanical stretching or compressing pressure enclosed to it. This dependence is described by a following equation: The resonant FBG wavelength *BG* depends on fiber temperature and from mechanical stretching or compressing pressure enclosed to it. This dependence is described by a following equation: , <sup>11</sup> 12 <sup>12</sup> <sup>1211</sup> 2 ВРБ эфф *<sup>T</sup> n nT <sup>n</sup> PPP* (12)

where *T* temperature change, the enclosed pressure, the second item composed in a figure brace reflects photo elasticity factor. This parity gives typical values of BG shift depending on temperature ~0,01 nm/К from relative lengthening of a fiber ~

2

$$
\Delta\mathcal{J}\_{\dot{\mathbf{A}}\dot{\mathbf{A}}} = 2n\_{\circ \dot{\mathbf{v}}\delta} \Lambda \left( \left\{ 1 - \left( \frac{\xi^2}{2} \right) \left[ P\_{12} - \nu \left( P\_{11} + P\_{12} \right) \right] \right\} \varepsilon + \left[ \frac{1}{\Lambda} \frac{\partial \Lambda}{\partial T} + \frac{1}{n} \frac{\partial n}{\partial T} \right] \Delta T \right), \tag{12}
$$

where *ΔT* is temperature change, *ε* is the enclosed pressure, the second item composed in a figure brace reflects photo elasticity factor. This parity gives typical values of *λ*BG shift depending on temperature ~0,01 nm/°К from relative lengthening of a fiber ~ 103 (*ΔL* / *L* ) (nm) [2].

The introduced phase shift (PS) [2] leads to the appearance of a narrow transmission band of width of a few tens of megahertz within the reflection band of FBG. Figure 2, b shows the calculated transmission spectrum of such FBG-PS grating. The phase shift in the grating can be introduced during the writing of the whole structure or later in the preliminary written grating. As the phase shift is increased (which is usually realized by writing two spatially separated gratings with the same FBG), the number of transmission regions in the reflection band increases, and such a structure is called, similarly to bulk optics, a Fabry-Perot interfer‐ ometer (or filter). FBG-PS becomes the grate instrument in telecommunication and sensor nets [25-27].

The FBG reflective spectrum line shape can be approximated with a Gaussian profile [28]

$$R(\lambda) = R\_B \exp[-4\ln 2] \mathbf{\tilde{f}} \left[ (\lambda - \lambda\_B) / \Delta \lambda\_B \mathbf{\tilde{f}}^2 \right]. \tag{13}$$

where λ is wavelength, λ<sup>B</sup> is the center wavelength or peak wavelength of FBG, Δλ<sup>B</sup> is the full width at half maximum, and *R*B is the maximum reflectivity.

As is known, the spectral dependence of the transmission band of FBG-PS has almost Lorent‐ zian profile [2, 29]. If we assume that the spectral line width of the laser emission lines is negligibly small (~ several KHz), the spectral dependence of the transmission band of FBG-PS can be represented as follows:

$$T\left(\lambda\right) = \frac{T\_B \left(\Delta\lambda\_B / 2\right)^2}{\left[ (\lambda - \lambda\_B)^2 \* (\Delta\lambda\_B / 2)^2 \right]}.\tag{14}$$

where *TB* is maximum transmittance on λB.

#### **2.3. Discussion of results**

Basics for the use of poly-harmonic probing methods of spectral characteristics for resonant circuits of arbitrary shape were described by us in a number of papers [4,9-10]. It is noted that their effective use (maximum slope of the measurement conversion) is possible at the location of equal amplitude symmetrical components of the probe radiation on the FWHM of contour with average frequency at the central (resonant) wavelength. Based on this requirement, the synthesis of the poly-harmonic (two-or four-frequency) probe radiation desired characteristics was carried out. The example results are shown in Tab. 1.

Poly-harmonic Analysis of Raman and Mandelstam-Brillouin Scatterings and Bragg Reflection Spectra http://dx.doi.org/10.5772/59144 63


**Table 1.** Spectral characteristics of Mandelstam-Brillouin gain contours, Raman scattering contours and FBG reflection spectra

We have show in [30-33] that poly-harmonic probing radiation can be forming by external electro-optic modulation of narrowband one-frequency laser one. For general estimations of requirements for electro-optic modulators we assume that 1 nm in wavelength band is 120 GHz in frequency bandwidth. Thus, the operating frequency of modulators, we need, must be equal from 10-20 MHz to 20-40 THz. Provision of the lower limit for the FBG-PS study causes no problems. To investigate the SMBS and FBG contours a wide range of modulators with a frequency range up to 100 GHz spacing are existed in LiNbO3, GaAs, InP realizations. Direct solutions to achieve 20-40 THz frequency does not exist today. The use of electro-optic modulators in the mode of frequency multiplication (in 12-16 times) is possible to obtain bandwidth in units of THz. However, given that we are interested in the scattering of the central part of the SRS spectrum allocated by a Gaussian filter, you can use a tunable laser to deliver carrier laser radiation to 13.1 THz. Thus, the implementation of methods for polyharmonic probing of Mandelstam-Brillouin gain, Raman scattering and FBG reflection contours spectral characteristics looks quite feasible.
