Theoretical Investigation of Distributed Fiber Optic Sensing: Brillouin Stimulated Scattering

Abstract

The primary function of monitoring systems is to ensure the operational integrity and safety of the system being monitored. Continuous monitoring provides timely information on the state of the monitored system as well as early warning of anomalies that might lead to catastrophic failures. One of the leading sensing technologies that is gaining wide acceptance in the industry is fiber optic-based sensing technology. This technology enables the fiber optic cable to act as a sensor, providing continuous sensing with wide sensing coverage and advanced warning capabilities in real-time. When utilized in a distributed configuration, it detects changes in vibration, strain, and temperature along the monitored object. The strain on an offshore crude pipeline may indicate the presence of fractures; recognizing them early can prevent structural collapse, which might lead to a leak and, eventually, an oil spill, inflicting environmental damage. This chapter presents a theoretical analysis of fiber optic-based sensing and its practical applications.

1. Introduction

One of the most promising technologies that can provide ongoing monitoring is distributed fiber optic sensing. It can perform continuous sensing along the entire length of the monitored structure with great accuracy and precision. It enables the fiber optic cable to function as a sensor, providing greater sensing coverage and continuous real-time sensing capabilities. Moreover, it provides increased temperature accuracy and exceptional spatial resolution. When the optical laser propagates through the fiber, it gets scattered back into three different spectral forms with different frequencies and intensities. The spectral forms are called Rayleigh, Raman, and Brillouin scattering [1]. The scattering is created due to impurities or changes in composition and interaction of the laser light with the molecules of the fiber.

The scattering causes a frequency shift of the signals returning to the source and is dependent on the strain and temperature of the sensing fiber [2]. Thus, it can be used to reveal the change of temperature and strain that may take place along the fiber, or any object attached to the fiber cable.

Due to intrinsic impurities and microscopic flaws in manufacturing, scattering mechanisms are always present in the fiber. The scattering of light in an optical fiber is caused by variations in the fiber’s optical characteristics, such as density or refractive index. As mentioned above, scattering occurs in three different forms, Rayleigh, Raman, and Brillouin. Rayleigh scattering does not cause a frequency shift of the scattered light as it is an elastic process and is insensitive to changes in the sensing fiber material. Conversely, Raman and Brillouin scattering are inelastic processes and are sensitive to material properties.

Raman scattering is sensitive to temperature only and occurs in the forward and backward directions. While Brillouin scattering occurs in the backward scattering direction only and is sensitive to both temperature and strain. These two processes involve three waves that are of importance in the sensing regime. The three waves are the incident and the scattered waves as well as the thermal molecular vibration within the fiber.

There are two scattering schemes associated with this type of scattering, these are the spontaneous Brillouin and the stimulated Brillouin scattering [3]. When the incident light does not impose any changes on the optical material properties, we have a spontaneous scattering. On the other hand, when the incident light imposes a change, we have a stimulated scattering.

Along the way, a vibration of the fiber molecules takes place traveling at the acoustic speed which causes variation in density. Accordingly, the refractive index may change causing the light to scatter back to the source. This vibration is referred to as phonons. Basically, it is the scattering of light from sound waves in the fiber.

For distributed sensing, it is more attractive to use a Brillouin-based sensing technique rather than a Raman-based technique. This is because Brillouin scattering can perform both temperature and strain sensing. Moreover, Brillouin-based sensing outperforms Raman sensing in terms of sensing range, resolution and measuring time [1]. Therefore, the focus of this chapter is on the Brillouin scattering technique.

2. The stimulated Brillouin scattering

The Stimulated Brillouin Scattering (SBS) results from the interaction between two optical waves propagating in opposite directions, refer to Figure 1. As a result of this interaction, an acoustic wave will emerge that causes changes in the refractive index of the fiber resulting in coupling of the two counter-propagating optical waves.

Figure 1.

Illustration of a simplified Brillouin system.

Assuming steady state conditions, we will have two coupled wave equations that govern the scattering process. Assuming the fiber loss, α is the same for both waves, then the coupled wave equations can be expressed as [4]:

$$\frac{{\mathit{dI}}_P}{\mathit{d}\mathit{z}}=-g_B{I}_P{I}_{\mathit{CW}}-{\mathit{\alpha I}}_P$$E1

$$\frac{{\mathit{dI}}_{\mathit{CW}}}{\mathit{d}\mathit{z}}=-g_B{I}_P{I}_{\mathit{CW}}+{\mathit{\alpha I}}_{\mathit{CW}}$$E2

where I_P and I_CW are the input pump intensity and probe intensity, respectively, α is the fiber optic cable attenuation coefficient, g_B is the Brillouin gain and z is the distance.

2.1 Stimulated Brillouin gain

SBS is categorized into two forms, the Brillouin gain spectrum and the Brillouin loss spectrum. Brillouin gain operates by launching a pulsed laser ${\nu }_1$, the pump at one end of the fiber and a continuous wave (CW) probe laser ${\nu }_2$ into the other end of the fiber where pump frequency is greater than probe frequency. When the frequency of the pulsed laser (the pump laser) is higher than the frequency of the probe laser by the Brillouin frequency shift, ${\nu }_B$, then there will be a transfer of energy from the pump to the probe. In this case, the probe will experience gain [5]. Eq. (3) presents this relationship between the two waves as:

$${\nu }_1={\nu }_2+{\nu }_B$$E3

For the Brillouin loss, the frequencies are reversed (probe frequency > pump frequency). In this case, there is a transfer of energy from the probe to the pump and the probe experiences loss. Eq. (4) expresses this relationship as:

$${\nu }_2={\nu }_1+{\nu }_B$$E4

Based on the above discussion, the probe signal is amplified in the Brillouin gain process, and it is reduced in the Brillouin loss process. In the SBS configuration, we want to measure the signal coming toward the source. In this case, we want to know the intensity or the power level of the probe signal.

2.2 Stimulated Brillouin loss

The focus of this chapter is on Brillouin loss. Accordingly, the sign of the gain in Eqs. (1) and (2) should be reversed [6].

$$\frac{{\mathit{dI}}_P}{\mathit{dz}}=g_B{I}_P{I}_{\mathit{CW}}-{\mathit{\alpha I}}_P$$E5

$$\frac{{\mathit{dI}}_{\mathit{CW}}}{\mathit{dz}}=g_B{I}_P{I}_{\mathit{CW}}+{\mathit{\alpha I}}_{\mathit{CW}}$$E6

Integrating Eq. (5), we get:

$${\int }_0^z\frac{{\mathit{dI}}_P}{I_P}={\int }_0^z\left(g_B{I}_{\mathit{CW}}-\alpha \right)\mathit{dz}$$E7

At the time the pump laser is launched at point 0 (z = 0), there will be no gain or loss, then Eq. (7) becomes:

$$\mathit{ln}{I}_P\left(z\right)-\mathit{ln}{I}_P\left(0\right)=-\mathit{\alpha z}$$E8

$$\left(\frac{I_P\left(z\right)}{I_P\left(0\right)}\right)=\mathit{exp}\left(-\mathit{\alpha z}\right)$$E9

$$I_P\left(z\right)=I_P\left(0\right)\mathit{exp}\left(-\mathit{\alpha z}\right)$$E10

Eq. (10) calculates the intensity before the interaction between the two lasers.

Integrating Eq. (6) over the entire length (from z = L to z = 0), we get the intensity of the probe laser at distance z = 0:

$$\frac{{\mathit{dI}}_{\mathit{CW}}}{\mathit{dz}}=g_B{I}_P{I}_{\mathit{CW}}+{\mathit{\alpha I}}_{\mathit{CW}}={\int }_L^0\frac{{\mathit{dI}}_{\mathit{CW}}}{I_{\mathit{CW}}}={\int }_L^0\left(g_B{I}_P\left(z\right)+\alpha \right)\mathit{dz}$$E11

$$\frac{{\mathit{dI}}_{\mathit{CW}}}{\mathit{dz}}=g_B{I}_P{I}_{\mathit{CW}}+{\mathit{\alpha I}}_{\mathit{CW}}={\int }_L^0\frac{{\mathit{dI}}_{\mathit{CW}}}{I_{\mathit{CW}}}={\int }_L^0\left(g_B{I}_P\left(0\right)\mathit{exp}\left(-\mathit{\alpha z}\right)+\alpha \right)\mathit{dz}$$E12

$$\mathit{ln}\left(\frac{I_{\mathit{CW}}\left(0\right)}{I_{\mathit{CW}}\left(L\right)}\right)=\left(\frac{g_B{I}_P\left(0\right)}{\alpha }\right)\left(\mathit{exp}\left(-\mathit{\alpha L}\right)-1\right)-\mathit{\alpha L}$$E13

$$I_{\mathit{CW}}\left(0\right)=I_{\mathit{CW}}\left(L\right)\mathit{exp}\left[\left(\frac{g_B{I}_P\left(0\right)}{\alpha }\right)\left(\mathit{exp}\left(-\mathit{\alpha L}\right)-1\right)-\mathit{\alpha L}\right]$$E14

Eq. (14) calculates the intensity of the probe laser at point 0 (z = 0) [6].

As the signal propagates along the fiber, the power of the signal drops due to the fiber optic cable attenuation. However, the power of the signal becomes constant at a specific length of the fiber, which is known as the effective length. This effective length can be expressed as [4]:

$$L_{\mathit{eff}}=\frac{1}{\alpha }\left(1-exp\left(-\mathit{\alpha L}\right)\right)$$E15

As an example, let us assume the attenuation, $\mathrm{\alpha }$ of a fiber optic cable is $\raisebox{1ex}{$0.2\mathit{dB}$}\!\left/ \!\raisebox{-1ex}{$\mathit{km}$}\right.$ and the length of the fiber optic cable is 20 km, then effective length is calculated to be 13 $\mathit{km}$. It must be noted that the first step in this example is to convert $\raisebox{1ex}{$\alpha \mathit{dB}$}\!\left/ \!\raisebox{-1ex}{$\mathit{km}$}\right.$ to $\alpha /\mathit{km}$ using Eq. (16) [4]. The derivation for this equation is presented in Appendix A.

$${\alpha }_{\raisebox{1ex}{$\mathit{dB}$}\!\left/ \!\raisebox{-1ex}{$\mathit{km}$}\right.}={4.343}_{\raisebox{1ex}{$\mathit{\alpha }$}\!\left/ \!\raisebox{-1ex}{$\mathit{km}$}\right.}$$E16

Using Eq. (15) in Eq. (14) yields:

$$I_{\mathit{CW}}\left(0\right)=I_{\mathit{CW}}\left(L\right)\mathit{exp}\left[-g_B{I}_P\left(0\right)L_{\mathit{eff}}-\mathit{\alpha L}\right]$$E17

The power of the signal is not equally distributed over the cross-sectional area of the fiber optic cable; hence, it is appropriate to use the effective area, $A_{\mathit{eff}}$ instead of the cross-sectional area. The effective area is the region where intensity of light does not exhibit nonlinear behaviors. It can be expressed in terms of the intensity, $I$, and power, $P$, as [4, 6]:

$$A_{\mathit{eff}}=\frac{P}{I},I=\frac{P}{A_{\mathit{eff}}}$$E18

Substituting Eq. (18) into Eq. (17) yields:

$$P_{\mathit{CW}}\left(0\right)=P_{\mathit{CW}}\left(L\right)\mathit{exp}\left[\left(-\frac{g_B{P}_P\left(0\right)L_{\mathit{eff}}}{A_{\mathit{eff}}}\right)-\mathit{\alpha L}\right]$$E19

$P_{\mathit{CW}}\left(0\right)$ is the power spectrum of the probe signal at z = 0. From Eq. (10), pump power at z can be expressed as:

$$P_P\left(z\right)=P_P\left(0\right)exp\left(-\mathit{\alpha z}\right)$$E20

2.3 Brillouin spectrum

The acoustic waves decay exponentially, and the Brillouin gain spectrum takes the shape of Lorentzian curve [6, 7]. It is defined as:

$$g_{B\left(\nu \right)}=g_{B0}\frac{{\left(\frac{{\mathrm{\Delta \nu }}_B}{2}\right)}^2}{{\left(\frac{{\mathrm{\Delta \nu }}_B}{2}\right)}^2+{\left(\nu -{\nu }_B\right)}^2}=\frac{g_{B0}}{1+\frac{4{\left(\nu -{\nu }_B\right)}^2}{{{\mathrm{\Delta \nu }}_B}^2}}$$E21

where $g_B$ is the Brillion gain, $\nu$ is the probe-pump frequency difference at location z along the fiber, g_Bo is the Brillouin gain that occurs when the frequency difference $\nu$ is equal to Brillouin frequency shift ${\nu }_B$_, and ${\mathrm{\Delta \nu }}_B$ is the Brillouin bandwidth. A typical Brillouin gain spectrum is shown in Figure 2 which indicates that the peak power or Brillouin gain occurs at the Brillouin frequency shift, ${\nu }_B$. As the figure illustrates, the shape of the spectrum may approximate Lorentzian distribution.

Figure 2.

Illustration of a typical Brillouin gain spectrum.

At the maximum gain, we can get the Brillouin frequency shift ${\nu }_B$, from which we can determine the temperature or strain at a particular position along the fiber optic cable or any object that is attached or bonded to the fiber optic cable.

3. Distributed Brillouin sensing techniques

Brillouin sensing techniques include Brillouin Optical Time Domain Reflectometry (BOTDR), Brillouin Optical Time Domain Analysis (BOTDA), Brillouin Optical Frequency Domain Analysis (BOFDA), Brillouin Optical Correlation-Domain Analysis (BOCDA), and Brillouin Echo Distributed Sensing (BEDS). The BOTDA technology is extensively utilized as a monitoring tool in the industry. It provides excellent temperature and strain-detecting capability [8].

3.1 Brillouin optical time domain analysis (BOTDA)

As previously explained, BODTA, when utilized in the stimulated Brillouin loss scheme, operates by firing lasers in two opposing directions, one pulsed and one continuous, as seen in Figure 1. The difference in frequency between the two lasers may be utilized to monitor strain and temperature along the fiber [1]. This frequency difference is frequently referred to as the Brillouin frequency shift because it happens in the reverse direction. It is the consequence of dispersed light, which is affected by material temperature and tension. This is one of the primary benefits of employing fiber optics as a distributed sensor to detect and quantify strain and temperature variations along the monitored structure.

The frequency shift is given by, [4]:

$${\nu }_B=\frac{2n{\mathrm{V}}_a\mathit{sin}\left(\frac{\theta }{2}\right)}{{\lambda }_i}$$E22

Since the scattering happens only in the backward direction, $\mathrm{\theta }={180}^0$, then the frequency shift will be:

$${\nu }_B=\frac{2n{\mathrm{V}}_a}{{\lambda }_i}$$E23

${\nu }_B$ is the Brillouin frequency shift, $V_a$ is the acoustic velocity of the phonons (approximately 5800 m/s), n is the refractive index of the fiber, ${\lambda }_i$ is the wavelength of the incident light. A photon is a particle of light while a phonon is a particle of sound, it is a unit of vibration energy. Both photons and phonons are the smallest amount of energy that a wave can carry. Assuming a wavelength of 1300 nm and a refractive index of 1.5, Eq. (23) yields a Brillouin frequency shift of 13,384.6 MHz.

3.2 Fiber optic distributed sensing based on Brillouin scattering

A linear relationship exists between the frequency shift and strain. Similarly, a linear relationship exists between the frequency shift and the temperature [5, 6, 9].

$${\nu }_B-{\nu }_0={\alpha }_{\epsilon }\mathrm{\Delta }\epsilon +{\alpha }_T\mathrm{\Delta }T$$E24

${\nu }_B$ is the Brillouin frequency shift, ${\nu }_0$ is the reference Brillouin frequency shift at no strain and ambient temperature in MHz, ${\alpha }_{\epsilon }$ is the strain coefficient expressed in MHz/με, ${\alpha }_T$ is the temperature coefficient expressed in MHz/Co, $∆T$, is the temperature change which is the difference between the measured temperature and the ambient temperature, $\mathrm{\Delta }\epsilon$ is the strain change.

From this frequency shift, sensing is possible, but the issue here is that the shift is dependent on both strain and temperature. From the frequency shift alone, you cannot tell which change has occurred. To solve this issue, make the fiber optic cable strain-free by placing it near the object that is being monitored. Under this arrangement, strain becomes zero which eliminates the first term in Eq. (24).

Therefore, the Brillouin frequency shift is calculated as indicated in Eq. (25):

$${\nu }_{B1}-{\nu }_{01}={\alpha }_T\mathrm{\Delta }T$$E25

${\nu }_{B1}$ is the measured Brillouin frequency shift, ${\nu }_{01}$ is the reference Brillouin frequency shift at the ambient temperature, in MHz. Figure 3 shows the linear relationship between the Brillouin frequency shift and the temperature change. The plot in the figure is based on an ambient temperature of 20°C, a reference Brillouin frequency shift of 16,650 MHz/C° and a temperature coefficient of 1.6 MHz/C°.

Figure 3.

Brillouin frequency shift – temperature change.

For strain measurement, it is recommended to use two fiber optic cables. One cable is attached to the object being monitored and another cable is made strain-free by placing it near the object being monitored.

The first cable which is attached to the object being monitored has these parameters: ${\nu }_{B1}$, ${\nu }_{01},{\alpha }_{\epsilon },{\mathit{\Delta \epsilon }}_1,{\alpha }_T,{\mathit{\Delta T}}_1$ and is used to monitor strain changes. The second cable, the strain-free cable, has these parameters: ${\nu }_{B2},{\nu }_{02},{\alpha }_{\epsilon },{\mathit{\Delta \epsilon }}_2,{\alpha }_T,{\mathit{\Delta T}}_2$ and is used to monitor the temperature changes.

Assuming the change in temperature in the bonded fiber cable equals the change in the free fiber, then the strain can be calculated using Eq. (26):

$${\mathrm{\Delta }\epsilon }_1=\frac{\left({\nu }_{B1}-{\nu }_{01}\right)-\left({\alpha }_T{\mathit{\Delta T}}_2\right)}{{\alpha }_{\epsilon }}$$E26

${\nu }_{B1}$ and ${\nu }_{01}$ are the measured Brillion frequency shift and the reference Brillouin frequency shift for the first fiber optic cable, and ${\mathrm{\Delta }T}_2$ is the temperature change. The strain measurement is known as micro-strain ($\mu \mathcal{E}$). Stretching a fiber optic cable from 1 m to 1.000010 m results in a strain of ten micro-strains (10$\mu \mathcal{E}$).

3.3 Location of the change

Calculating the time difference between the launched laser and the scattered laser, one can determine the location of temperature change or strain along the object being monitored using Eq. (27):

$$d=\frac{c\mathrm{\Delta }t}{2n}$$E27

c is the speed of light in vacuum, $\mathrm{\Delta }t$ is the difference between the laser launch time and arrival time (scattered laser), and n is the refractive index of the fiber. As an example, let us assume that a fiber optic cable is placed in the vicinity of an onshore crude oil pipeline, and it is given that the time difference between the launched and scattered lasers is 25 μs. For a fiber optic cable having a refractive index of 1.5 and a speed of light in vacuum of 300,000 km/s, the temperature change may be detected at 2.5 km away.

3.4 The lowest detectable temperature and strain

The threshold which is the lowest detectable change is calculated as a function of minimum Brillouin frequency shift. From this value, a threshold can be established. Exceeding the threshold will indicate the occurrence of a change in temperature or strain. The minimum detectable Brillouin shift ${\mathrm{\Delta }\nu }_B$ as a function of signal to noise ratio (SNR) is given by [2]:

$$\delta {\nu }_B=\frac{{\mathrm{\Delta }\nu }_B}{\sqrt{2}{\left(\mathit{SNR}\right)}^{0.25}}$$E28

${\mathrm{\Delta }\nu }_B$ is the Brillouin spectral width and the SNR is signal to noise ratio. Using Eqs. (29) and (30) one can determine the minimum detectable changes in temperature and strain respectively.

$$\mathit{\delta T}=\frac{{\mathrm{\Delta }\nu }_B}{\sqrt{2}{\alpha }_T{\left(\mathit{SNR}\right)}^{0.25}}$$E29

$\mathit{\delta T}$ is the minimum detectable temperature change and α_T is the temperature coefficient expressed in MHz/C^o. As an example, let us assume that a system is designed to have a temperature coefficient = 1.52 MHz/C°, SNR = 35 dB, and Brillouin spectral width = 25 MHz, the minimum detectable temperature change becomes: 1.55°C. The reader is advised to convert the SNR to a linear scale when using the Equation.

$$\mathit{\delta \epsilon }=\frac{{\mathrm{\Delta }\nu }_B}{\sqrt{2}{\alpha }_{\epsilon }{\left(\mathit{SNR}\right)}^{0.25}}$$E30

δε is the minimum detectable strain change and α_ε is the strain coefficient expressed in MHz/με.

The spatial resolution is the minimum length of cable required to adequately detect a localized temperature change along the cable. Eq. (31) calculates the spatial resolution in terms of speed of light c in vacuum, the refractive index n of the fiber optic cable, and the pulse width $\tau$.

$$∆z=\frac{\mathit{c\tau }}{2n}$$E31

For a pulse width of 10 ns, fiber optic cable refractive index of 1.5, at a speed of light 300,000 km/s in vacuum, the spatial resolution becomes 1 meter.

4. Summary and concluding remarks

This chapter presented a theoretical study on fiberoptic-based sensing. The stimulated Brillouin gain and loss, Brillouin spectrum, distributed Brillouin sensing techniques, the smallest observable changes, and the location of the changes were all covered in detail. Additionally, equations associated with light scattering behaviors, as well as equations associated with the characteristics mentioned above were derived and presented.

It was pointed out that a linear relationship exists between the frequency shift and the changes in temperature and strain. Furthermore, it was indicated that the signal to noise ratio and the Brillouin spectral width are the two determining factors of the smallest measurable variations in temperature and strain. The time difference between launched and scattered lasers can be used to determine the location of potential temperature and strain changes.

Future research should include experimental investigation of the behaviors of scattered light along the optical fiber in different environments, underwater, near shorelines, and areas with varying weather conditions. This will enable researchers to gain a better understanding and insight of the impact of scattered light on the optical fiber in different situations and environments.

Appendix A: Conversion of ${\alpha }_{\mathit{dB}/\mathit{km}}$ to $\raisebox{1ex}{$\mathit{\alpha }$}\!\left/ \!\raisebox{-1ex}{$\mathrm{km}$}\right.$

As the signal propagates through the fiber, its power gets reduced due to the attenuation of the fiber optic cable. The output power through a fiber optic cable can be expressed in terms of the attenuation as [4]:

$$P_o=P_i{e}^{-\mathit{\alpha L}}$$E32

where $P_i$ is the input power, $\alpha$ is the attenuation factor in ${\mathit{km}}^{-1}$, and $L$ is the length of the fiber optic cable. $\alpha$ is usually provided with $\mathit{dB}/\mathit{km}$ unit which requires a conversion to the appropriate ${\mathit{km}}^{-1}$ unit through the following procedures. Recall that the equation for attenuation in dB is:

$${\alpha }_{\mathit{dB}}=-10{log}_{10}\left(\frac{P_o}{P_i}\right)$$E33

To convert Eq. (33) to $\mathit{dB}/\mathit{km}$, simply divide the equation by $L$:

$${\alpha }_{\mathit{dB}/\mathit{km}}=-\frac{10}{L}{log}_{10}\left(\frac{P_o}{P_i}\right)$$E34

Rearrange Eq. (32) to solve for $\frac{P_o}{P_i}$:

$$\frac{P_o}{P_i}=e^{-\mathit{\alpha L}}$$E35

Substitute Eq. (35) into Eq. (34):

$${\alpha }_{\mathit{dB}/\mathit{Km}}=-\frac{10}{L}{log}_{10}\left(e^{-\mathit{\alpha L}}\right)$$E36

After simplifying, we end up with the following equation:

$${\alpha }_{\mathit{dB}/\mathit{km}}=10\left({log}_{10}e\right)\alpha =4.343\alpha$$E37