Analysis of Brillouin dynamic grating localized by intensity-modulated correlation-domain technique for distributed fiber sensing

Youhei Okawa; Rodrigo Kendy Yamashita; Masato Kishi; Kazuo Hotate

doi:10.1364/OE.385561

1. Introduction

Brillouin fiber sensing, which is a distribution measurement technique of temperature, strain, etc, using Brillouin scattering in optical fibers, has been actively researched as a new diagnostic technique for structures. Since the Brillouin gain spectrum (BGS) is linearly shifted with respect to the strain and temperature applied to the fiber, the strain and temperature can be measured from the frequency shift, called Brillouin frequency shift (BFS). For the distributed measurement, time-resolved measurement of Brillouin scattered light using short-time pulse was first developed [1,2], but this time-domain method, called Brillouin optical time-domain analysis (BOTDA), has the problem that the spatial resolution performance of the basic system is inherently limited by the acoustic phonon lifetime, which is typically on the order of one meter. In Brillouin optical correlation-domain analysis (BOCDA) devised to solve this problem [3,4], stimulated Brillouin scattering (SBS) occurs locally due to the control of optical coherence of a continuous-wave light source. Initially, a method of sinusoidal modulation of the light source frequency was used, but later a further localization method applying intensity modulation (called apodization [5,6]), and a method using phase modulation instead of frequency modulation [7,8] were developed. In addition, methods using incoherent light sources such as amplified spontaneous emission [9] and laser chaos [10] are also reported. Such a spontaneously incoherent light source requires a variable long delay to move the sensing point, called “correlation peak”, but can realize a high spatial resolution by utilizing its broadband spectrum.

When the counter-propagating lights satisfy the phase matching condition of SBS, an acoustic wave is driven by them through the electrostrictive effect. This acoustic wave can be regarded as a weak refractive index grating that flows at a speed of sound according to the transmission material, and is called a Brillouin dynamic grating (BDG) [11,12]. Since the acoustic wave is a longitudinal wave and interacts with lights regardless of its polarization, the acoustic wave can be interrogated by the light polarized orthogonal to the pump-probe light (we call “read” light in this paper) [11]. Unlike conventional fiber Bragg gratings, BDG can be rewritten at high speed, and many studies on distributed fiber sensing using its reflection spectrum (to be precise, diffraction spectrum) have been made [12,13]. In addition, it attracts attention as an all-optical signal processing device [14–16]. BDG reflection spectrum was first observed in polarization-maintaining fibers, but now there are a wide range of available platforms such as single-mode fibers [17], few-mode fibers [18], photonic crystal fibers [19], and photonic chips [20].

Localized BDG can be generated efficiently by the correlation-domain technique [7,21]. By using its reflection spectrum, we have demonstrated discriminative measurement of the distribution of temperature and strain, called BDG-BOCDA method [22–24]. Conventional Brillouin fiber sensing has a problem that it cannot distinguish changes in temperature and strain. For example in strain measurement of outdoor structures, temperature change is unavoidable according to sunshine hours, and then the discriminative measurement technology is demanded. In the simplest case, discriminative measurement may be possible by adding a loosely laid fiber for temperature measurement, which is not desirable from the viewpoint of costs and errors. A plurality of discrimination methods of distributed Brillouin sensing has been proposed so far [25–29]. In any method, in addition to the BFS, it is necessary to measure another quantity having different temperature/strain dependency. When the measured quantity is linearly converted with respect to changes in temperature and strain, the higher the regularity of the conversion matrix, the higher the separation accuracy. BDG-BOCDA method utilizes the fact that the birefringence of a general polarization-maintaining fiber is largely shifted by the temperature change from the difference in thermal expansion coefficient of the stress-induced material in a cladding. Therefore, the BDG-BOCDA method enables the separation measurement of temperature and strain with high accuracy [22].

Discriminative and distributed measurement using BDG-BOCDA method has already achieved a relatively high spatial resolution about 10 cm [23]. However, it is empirically known that the spatial resolution in BDG-BOCDA system is almost ten times worse than the BOCDA alone, and then the spatial resolution of the BDG reflection spectrum is a limiting factor in the discriminative measurement. The solution of this problem is the main purpose of this paper. In the correlation-domain technique, it is known that its performance can be improved by applying intensity modulation to the light source at the expense of brightness [5,6,30,31]. Although apodization, one of these methods, is also known to be effective for the spatial resolution of the reflection spectrum of BDG in the experiments [32,33], it has still no theoretical basis and quantitative evaluation.

In this paper, we numerically analyzed a reflection spectrum of BDG localized by intensity-modulated correlation-domain technique for the first time [34]. While providing a theoretical basis for the previous experimental results, it was additionally clarified that the improvement effect is expected to be limited when the sensing fiber length is fully long. Then, based on this analysis, we proposed a more effective intensity modulation method and showed that the spatial resolution can be greatly improved across the long fiber length. Our idea is based on the technique developed for dynamic range enhancement in optical reflectometry [35]. This proposed method will greatly improve the performance of distribution measurement using BDG generated by the correlation-domain method.

2. BDG-BOCDA system

In this section, the concept of BDG-BOCDA method is explained and the formulation used in later calculations is described. The analysis of intensity modulation effect and the explanation of the proposed method will be given in the next section.

2.1 Configuration

Figure 1 shows basic settings to generate BDG and interrogate its reflection spectrum [11]. The x-polarized pump $E_{1}$ and probe $E_{2}$ propagate oppositely through the polarization-maintaining fiber along the $z$-axis and interfere to drive the acoustic wave $Q$ through the electrostrictive effect. When y-polarized read light $E_{3}$ is incident, the diffracted light $E_{4}$ by the generated acoustic wave is grown. We define the electric field of each wave and the acoustic amplitude as follows:

(1)$$\begin{aligned}E_{1}(z,t)&=A_{1}(z,t)e^{-i\omega_{1}(t-\frac{n_{x}}{c}z)}+\textrm{c.c.},\hspace{3mm} E_{2}(z,t)=A_{2}(z,t)e^{-i\omega_{2}(t+\frac{n_{x}}{c}z)}+\textrm{c.c.},\\ E_{3}(z,t)&=A_{3}(z,t)e^{-i\omega_{3}(t-\frac{n_{y}}{c}z)}+\textrm{c.c.},\hspace{3mm} E_{4}(z,t)=A_{4}(z,t)e^{-i\omega_{4}(t+\frac{n_{y}}{c}z)}+\textrm{c.c.},\\ Q(z,t)&=\rho_{m}+[\rho(z,t)e^{-i\Omega_{B}(t-\frac{z}{V_{a}})}+\textrm{c.c.}]. \end{aligned}$$

Here, $n_{x,y}$ is the refractive index of each axis and its dispersion is neglected. We consider its distribution of small variation along the fiber, $n_{x,y}\rightarrow n_{x,y}+\Delta n_{x,y}(z)$, which is typically induced by strain and temperature change. We introduce constants that $c = 3\times 10^{8}$ m/s is the speed of light in vacuum, and $V_{a} = 5950$ m/s is the speed of sound in the fiber, and $\rho _m$ is the mean density, and $\Omega _B$ is the frequency of sound, typically $\sim 2\pi \times 11$ GHz. We also define $A_{i}$, $\rho$ as the slowly varying amplitude of each wave.

Fig. 1. Schematic of the theoretical model of BDG generation and readout process.

Download Full Size | PDF

Figure 2 illustrates schematic diagram of BDG-BOCDA system of our concern [24]. The light source is a narrow linewidth laser diode whose frequency is sinusoidally modulated by injection current modulation. This frequency modulated light source makes periodically the position called correlation peak, where the frequency difference between the pump and probe light is constant and then their mutual coherence is relatively high [4]. Since SBS efficiently occurs only at the correlation peak position, we can get the local BGS. We note that this frequency modulation is also needed for read light to obtain single peak spectrum because the wavenumber of generated acoustic wave is oscillating sinusoidally at the correlation peak [21]. Subsequent intensity modulator may be used for apodization, which further control the localization of the correlation peak by amplitude modulation. The light is divided into three waves, pump, probe, and read, using the fiber coupler. Local BGS at the correlation peak is measured by sweeping the probe frequency $\omega _2\rightarrow \omega _2+\Delta \omega _{2}$ in BOCDA technique [3,4]. After that, setting the probe frequency to maximize the Brillouin gain and then we measure the reflection spectrum of BDG by launching the read light and sweeping its frequency $\omega _3\rightarrow \omega _3+\Delta \omega _{3}$ [21]. For the distribution measurement, this measurement is conducted step by step at each position. The correlation peak position can be moved by changing the modulation frequency of the light source.

Fig. 2. Basic set up of BDG-BOCDA system. LD, laser diode; IM, intensity modulator; FM, frequency modulation; SSBM, single-sideband modulator; PBC, polarization beam combiner; PD, photodetector; FUT, fiber under test.

Download Full Size | PDF

2.2 Equations of BOCDA

We start with making an introduction of BOCDA theory [3], in order to provide clearly the entire structure of BDG-BOCDA system and describe the measurement protocol of BGS and BDG reflection in a unified manner. This makes it possible to discuss the measurement system while clarifying the difference in the roles of the refractive indices of the x and y axes. In this section, the expression of the acoustic wave making BDG generated by the x-polarized light is derived, and at the same time, the expression of the BOCDA output is derived in order to consider the intensity modulation effect in a later chapter. We note that our main concern is a standard BOCDA method using sinusoidally frequency modulated light source described above, but following equations can be applied to the other BOCDA methods.

We should consider the phase-matching conditions first. When pump frequency $\omega _{1}$ is given, following conservation relations of energy and momentum are satisfied in the SBS process:

(2)$$\omega_{1}=\omega_{2}+\Omega_{B}, \hspace{3mm} k_{1}=-k_{2}+q,$$

where $q$ is the wavenumber of acoustic phonon, and optical dispersion relation $k_{1}=n_{x}\omega _{1}/c$, $k_{2}=n_{x}\omega _{2}/c$ follows. When the probe frequency $\omega _{2}$ is scanned for the pump frequency $\omega _{1}$, SBS strongly occurs if the following phonon dispersion relation is satisfied,

(3)$$\Omega_{B}=V_{a}q.$$

From Eqs. (2), (3), we set the probe frequency $\omega _{2}$ for a strong SBS,

(4)$$\omega_{2}=\frac{c\,n_{x}^{-1}-V_{a}}{c\,n_{x}^{-1}+V_{a}}\,\omega_{1},$$

and the acoustic phonon parameters are,

(5)$$q\approx\frac{2n_{x}\omega_{0}}{c},\quad\Omega_{B}\approx\frac{2n_{x}V_{a}\omega_{0}}{c}.$$

Here we use $\omega _{1}\approx \omega _{2}=\omega _{0}$ for simplicity. $\Omega _{B}$ corresponds to BFS. If there is the refractive index change $\Delta n_{x}$, the variations are $\Delta q_{x}=2\Delta n_{x}\omega _{0}/c$, $\Delta \Omega _{B}=2\Delta n_{x}V_a\omega _{0}/c$.

In this situation, equations describing dynamics of each wave are following [3,36]:

(6)$$\frac{\partial A_1}{\partial z}+\frac{n_{x}}{c}\frac{\partial A_1}{\partial t}=i \kappa \rho A_2e^{-i\Delta q_{x}z} $$

(7)$$\frac{\partial A_2}{\partial z}-\frac{n_{x}}{c}\frac{\partial A_2}{\partial t}=-i \kappa \rho^* A_1 e^{i\Delta q_{x}z} $$

(8)$$\frac{\partial \rho}{\partial t}+\left(\frac{\Gamma_{a}}{2}+i\Delta\Omega_{B}\right)\rho=i\Lambda A_1A_2^*\,e^{i\Delta q_{x}z}. $$

Here we neglect a phonon driving term due to the orthogonal lights $A_{3},A_{4}$ in Eq. (8), and then the acoustic phonons $\rho$ can be determined only by pump-probe light, $A_{1}, A_{2}$ [37]. We also neglect Kerr nonlinearity, fiber absorption and thermal fluctuation of the media density. It is noted that the phase-mismatching $\Delta q_{x}$ due to the refractive index change $\Delta n_{x}$ is explicitly shown, which is set to 0 at the correlation peak position in our scenario. Coupling constant $\kappa$ and $\Lambda$ are introduced, which are proportional to the electrostriction coefficient. The phonon decay rate $\Gamma _a$ is also introduced.

We consider the situation that the probe gain is sufficiently small and the depletion of the pump light can be ignored. In a first order approximation about electrostriction, we can calculate the acoustic wave $\rho$ from Eq. (8) such that

(9)$$\rho(z,t)=i\Lambda \int_{-\infty}^{t}d\tau\, e^{-\left(\frac{\Gamma_{a}}{2}+i\Delta\Omega_{B}\right)(t-\tau)} A_{1}(z,\tau)A^{*}_{2}(z,\tau)\,e^{i\Delta q_{x}z},$$

and its Fourier transform,

(10)$$\tilde{\rho}(z,\omega)=i\Lambda\tilde{D}(z,\omega)\tilde{B}(z,\omega)\,e^{i\Delta q_{x}z}, $$

(11)$$(\equiv\tilde{\rho}_{0}(z,\omega)e^{i\Delta q_{x}z} ) $$

where $\tilde {B}(z,\omega )=\mathcal {F}[A_{1}(z,t)A^{*}_{2}(z,t)],\ \tilde {D}(z,\omega )=\left (\frac {\Gamma _{a}}{2}-i(\omega -\Delta \Omega _{B}(z))\right )^{-1}$. $\mathcal {F}[:]$ stands for Fourier transformation about time, and tilde is used for indicating frequency component. We call $B(z,t)=A_{1}(z,t)A^{*}_{2}(z,t)$ beat signal, which describes optical coherence between pump and probe. $\tilde {D}$ is a complex Lorentzian function representing BGS. We also define their squared power as $S_{b}(z,\omega )\equiv |\tilde {B}(z,\omega )|^{2},\ g_{B}(z,\omega )\equiv |\tilde {D}(z,\omega )|^{2}$, which we call beat-power spectrum and intrinsic BGS, respectively.

We can show that by some calculations the variation of the probe amplitude due to Brillouin gain is expressed as an overlap integral of beat-power spectrum and intrinsic BGS in a second order approximation about electrostriction [3],

(12)$$\lim_{T\rightarrow \infty}\frac{1}{T}\int_{-T/2}^{T/2}dt\,\left(|A_{2}(0,t)|^{2}-|A_{2}(L,t)|^{2}\right) $$

(13)$$\approx\,\kappa\Lambda\int_{0}^{L}dz\int_{-\infty}^{\infty}d\omega\,g_{B}(z, \omega)\,S_{b}(z,\omega). $$

Finally, by sweeping the probe frequency $\omega _2\rightarrow \omega _2+\Delta \omega _{2}$, we can get BOCDA output as,

(14)$$\textrm{BGS}(\Delta\omega_{2})= \int_{0}^{L}dz\int_{-\infty}^{\infty}d\omega\,g_{B}(z, \omega)\,S_{b}(z,\Delta\omega_{2}-\omega).$$

We omit the unimportant constant. This equation shows that BOCDA output is a convolution of intrinsic BGS and beat-power spectrum. Note that when performing distribution measurement, the position $z$ is also convolved [3]. Therefore, if the beat-power spectrum is ideally a two-dimensional delta function, a local intrinsic BGS can be obtained correctly.

2.3 Equations of BDG reflection

Next, we describe the formula for calculating the reflection spectrum of BDG [38]. The formulation shown here is a more complete one than that in [38] in terms of indicating the difference in refractive index of each axis.

In the reading process of the BDG reflection, phonon state ($\Omega _{B}$, $q$) is given constant, and we scan the read frequency $\omega _{3}$. When the strong Brilluoin scattering induced by $x$-polarized pump-probe light occurs, similar relations to Eq. (2) are satisfied:

(15)$$\omega_{3}=\omega_{4}+\Omega_{B}, \hspace{3mm} k_{3}=-k_{4}+q$$

where we define $k_{3}=n_{y}\omega _{3}/c$, $k_{4}=n_{y}\omega _{4}/c$. From Eqs. (5), (15), we can get

(16)$$\omega_{3}\approx \left(1+\frac{n_{x}-n_{y}}{n_{y}}\right)\omega_{0}.$$

Then, equations of read light interaction with acoustic phonons are following [36,38];

(17)$$\frac{\partial A_3}{\partial z}+\frac{n_{y}}{c}\frac{\partial A_3}{\partial t}=i \kappa \rho_{0} A_4\,e^{i\Delta q_{xy}\,z} $$

(18)$$\frac{\partial A_4}{\partial z}-\frac{n_{y}}{c}\frac{\partial A_4}{\partial t}=-i \kappa \rho_{0}^* A_3\,e^{-i\Delta q_{xy}\,z}. $$

Here we use the same coupling constant $\kappa$ as $x$-polarization, and introduce the birefringence induced wavenumber difference, $\Delta q_{xy}=2(\Delta n_{x}-\Delta n_{y})\omega _{0}/c$. Note that the change in the refractive index basically does not affect the wavenumber of acoustic wave itself, at least in our modeling. However, the writing wavenumber of pump-probe light varies by refractive index change $\Delta n_{x}$, so does the acoustic one. On the other hand, y-axis refractive index change $\Delta n_{y}$ does not affect the generated phonon wavenumber. In contrast, the wavenumber of read light varies according to y-axis refractive index change. Therefore, birefringence change becomes an issue. Differences in $\Delta n_x$ and $\Delta n_y$ dependence on temperature and strain change enable their discrimination [22].

In a first order approximation, we can neglect $A_{3}$ variation and integrate Eq. (18) directly,

(19)$$A_{4}(0,t) \approx i\kappa \int_{0}^{L}dz\ \rho^{*}_{0}\left(z,t-\frac{n_{y}z}{c}\right)A_{3}\left(z,t-\frac{n_{y}z}{c}\right)e^{-i\Delta q_{xy}\,z},$$

where we used the boundary condition $A_4(L,t)=0$. Then, the reflection power spectrum is obtained by time integration and sweeping read frequency $\omega _3\rightarrow \omega _3+\Delta \omega _3$, such as,

(20)$$\begin{aligned}&\lim_{T\rightarrow \infty}\frac{1}{T}\int_{-T/2}^{T/2}dt\,|A_{4}(0,t)|^{2}\\ \approx&\lim_{T\rightarrow \infty}\frac{1}{T}\int_{-T/2}^{T/2}dt\left|\,i\kappa\int_{0}^{L}dz\,\rho^{*}_{0}\left(z,t-\frac{n_{y}z}{c}\right)A_{3}\left(z,t-\frac{n_{y}z}{c}\right)e^{i\left(\frac{2n_y\Delta\omega_3}{c}-\Delta q_{xy}\right)\,z} \right|^{2}. \end{aligned}$$

By using Parseval’s theorem, we can get Fourier representation like BGS,

(21)$$\begin{aligned}&\textrm{BDG}(\Delta \omega_{3})\\ =&\int_{-\infty}^{\infty}d\omega\left|\int_{0}^{L}dz\,\left[\tilde{\rho}_{0}^{*}(z, \omega)\otimes_{\omega} \tilde{A}_3 (z, \omega) \right] \exp\left[ i\left(\frac{n_{y}(\omega+2\Delta\omega_{3})}{c}-\Delta q_{xy}\right)z\right] \right|^{2}. \end{aligned}$$

The simbol $\otimes _\omega$ is used for a convolution about $\omega$. We use this expression in later calculations of BDG reflection.

3. Analysis of intensity modulation effect

3.1 Three methods to compare

Here, we introduce three methods that will be compared. The first method is a standard technique of sinusoidally modulating the light source frequency without performing intensity modulation. The instantaneous frequency of the light source $f(t)$ is,

(22)$$f(t)=f_{0}+\Delta f \sin(2\pi f_{m}t),$$

where $f_{0}$ is a center frequency, $\Delta f$ is a modulation amplitude, and $f_{m}$ is a modulation frequency. As already mentioned, in this scheme there exists the position called correlation peak periodically along the fiber, where the frequency difference between the pump and probe light is constant. To evaluate the localization properties of the acoustic wave generated by this method, we introduce theoretical spatial resolution $\Delta z$ and measurable range $d$, known as [3],

(23)$$\Delta z= \left\{ \begin{array}{ll} \frac{c\Delta\nu_{B}}{2\pi n f_{m}\Delta f}\ & \Delta\nu_{B} \gg f_{m} \\ \frac{1.52c}{2\pi n \Delta f}\ & \Delta\nu_{B} \ll f_{m} \end{array}, \right.$$

(24)$$d = \frac{c}{2n f_{m}}.$$

We note that $\Delta z$ roughly corresponds to FWHM of phonon power distribution $|\rho (z)|^{2}$, and $d$ is the interval of the correlation peaks. We use the typical Brillouin gain width of $\Delta \nu _B$ = 30 MHz and the condition $f_m\ll \Delta \nu _B$ is typically satisfied in practical situations. Fig. 3(a) shows modulation waveform, light spectrum, and phonon power distribution in this technique. Although the acoustic phonon is well localized at correlation peak, this sinusoidal frequency modulation forms the spectrum with pointed edges at both ends, which constitute side lobes of coherence function [4,6]. This results in hindering phonon localization.

Fig. 3. Three methods to compare, (a) no apodized, (b) conventional apodized, and (c) proposed half-apodized method. Left column shows frequency modulation waveform of the light source and its temporal intensity is expressed in color strength. Middle column is time averaged power spectrum of the light source. Right column shows phonon power distribution generated by each of the modulated light source. Note that the ordinate is logarithmic.

Download Full Size | PDF

Next, we consider following intensity modulation $I(t)$ to each electric field of the light source $E_i$ ($i=1$, 2, 3), real and non-negative function:

(25)$$I (t)=\left( \frac{1+\cos(4\pi f_{m}t)}{2} \right)^{1/2}.$$

This modulation is used for a conventional apodization method [5,6,32,33] and its effects are illustrated in Fig. 3(b). The edges at both ends of the spectrum are rounded, and the phonon is more localized at the correlation peak position. However, we reveal that its power is restored at the position away from the correlation peak. As will be shown later, this constitutes a significant noise when measuring local BDG reflection spectrum along a long fiber.

Then, to circumvent this problem, we devise a technique to apply the additional intensity modulation, which we call half-apodization here:

(26)$$\chi (t)= \left\{ \begin{array}{ll} 1 \ & |t|\;<\;1/4f_{m} \\ 0 \ & 1/4f_{m}\;<\;|t|\;<\;1/2f_{m} \end{array}. \right.$$

See Fig. 3(c). This modulation is based on the following simple motive. At the position in the middle of the correlation peaks, SBS hardly occurs because the pump-probe frequency oscillates out of phase. However, in the conventional apodization method, the intensity of light at the timing when the frequency difference becomes equal to BFS is relatively large, so the phonon power is enhanced (see Fig. 5(a) also). On the other hand, in the proposed method, SBS hardly occurs because the pump-probe light does not exist simultaneously at the position away from the correlation peak by additionally applying rectangle intensity modulation. Although SBS is supposed to occur slightly just in the middle of the correlation peaks due to the imperfections of the rectangular waves in reality, it is ideally nothing at all. We note that this intensity modulation timing with respect to the frequency modulation is selected so as not to reduce the spectrum width of the light source.

As an example for the calculation, the external modulation effect on the light source amplitude $A_i$ in the half-apodized method is like that

(27)$$A_i (z,t)\rightarrow A_{i}(z,t)\,I\left(t\pm\frac{nz}{c}\right)\chi\left(t\pm\frac{nz}{c}\right),$$

where the signs are $\textrm {sgn}(-1)^i$. $A_i$ in right side represents frequency modulation effect $A_i(z,t)=\bar {A_i}\exp [2\pi i \int _0^tdt'f(t'\pm \frac {nz}{c})]$. From here, the coordinate system is set so that the center of the fiber is $z$ = 0.

3.2 Effects on BGS measurement

Fig. 4. Position distribution of Brillouin gain spectrum calculated in each technique, (a) no apodized, (b) apodized, (c) half-apodized method, and their back walls show position integrated one which corresponds to BOCDA output (see Eq. (14)). $\Delta f$ = 1 GHz, $f_{m}$ = 2 MHz.

Download Full Size | PDF

Fig. 5. Phonon dynamics generated by intensity-modulated correlation-domain technique. $|\rho (z,t)|^{2}$ is illustrated for the three cases, (a1) no apodized, (a2) conventional apodized, (a3) half-apodized method. (b) Phonon power at the correlation peak position depending on the modulation frequency $f_{m}$.

Download Full Size | PDF

Figure 4 shows position distribution of Brillouin gain spectrum in each technique, (a) no apodized, (b) apodized, (c) half-apodized case, and their back walls show position integrated one which corresponds to BOCDA output (see Eq. (14)). In our calculations, modulation parameter is set to the typical value, $\Delta f$ = 1 GHz, $f_{m}$ = 2 MHz (hence, $\Delta z = 0.5$ m, $d=52$ m). Position $z=0$ is where there is a correlation peak. Note that the cross-sectional view at frequency $\Delta \omega _2=0$ corresponds to the phonon distribution. In Fig. 4(a), since the intensity modulation is not performed, total gain by the frequency integration is equal at each position. On the other hand, it is not the case in Figs. 4(b) and 4(c) due to the intensity modulation. Especially in Fig. 4(c), the total gain equals to 0 at the both ends corresponding to the middle of the correlation peaks, which may have a merit in the signal-to-noise ratio although the total light amount is reduced. Fig. 4(c) also shows clear asymmetry in the frequency axis, but the spectrum shape of BOCDA output shown on the back wall is the same as conventional apodization method.

3.3 Effects on BDG measurement

First, phonon dynamics in each fiber position $|\rho (z,t)|^{2}$ is calculated, as shown in Fig. 5(a), in the case that $f_{m}$ = 2 MHz, then $f_{m}\ll \Delta \nu _{B}$ is satisfied. We note that their time-integrated one has been shown in the right column of Fig. 3. In Fig. 5(a1), phonon is generated by sinusoidal frequency modulated pump-probe light without intensity modulation. At the origin (z=0) where the correlation peak exists, the acoustic wave is always generated. However, at some moment, it is generated all over the fiber. In Fig. 5(a2), the conventional apodization technique is used. Although the acoustic waves are not always generated even at the correlation peak position, they are well localized in the vicinity. However, this is not the case at a position away from the correlation peak. In Fig. 5(a3), we can see that the half-apodized method successfully localizes acoustic wave over the whole area. In Fig. 5(b), relative phonon power at the correlation peak position is shown with respect to various modulation frequency (0.3 $\sim$ 20 MHz). When no intensity modulation is applied (black circle), the generated phonon power is constant. In the case of the apodized (blue triangle) and the half-apodized (red diamond) method, phonons cannot be generated efficiently, as the intensity modulation frequency approaches $\Delta \nu _B$ = 30 MHz. Therefore, a certain restriction is imposed on the frequency of the intensity modulation method as predicted.

Fig. 6. BDG reflection spectra of each method ((a), (b), and (c)). The strain equivalent to 1 GHz frequency shift is applied to the test fiber at the correlation peak position, and strained length is changed from 0 to 6$\Delta z$. The two peaks corresponding to the short strained portion and the remaining unstrained portion can be seen in the spectra (see Fig. (d)). Modulation parameters are $\Delta f$ = 1 GHz, $f_{m}$ = 2 MHz, and hence the theoretical spatial resolution for BOCDA $\Delta z$ = 0.5 m. Sensing fiber length is full measurable range, $L=d=52$ m.

Download Full Size | PDF

Next, the reflection spectra of BDG generated by the intensity-modulated correlation-domain method were calculated by using Eq. (21). Fig. 6 shows BDG reflection spectra of a method, (a) that does not perform intensity modulation, (b) an apodized method (conventional), and (c) a half-apodized method (proposed). As shown in the Figs. 6(a)–6(c), a uniform elongation strain corresponding to a birefringence induced shift of 1 GHz is applied to the location on the center of the test fiber where the correlation peak exists, assuming strain dependency of each frequency shift as follows: BFS = 0.04 MHz/$\mu \varepsilon$, BDG = 0.9 MHz/$\mu \varepsilon$ [22]. In this case, the two peaks corresponding to the short strained portion and the remaining unstrained portion can be observed in the spectra. When the strained length is extended, the length in which the peak corresponding to the strained portion exceeds the unstrained one is the detectable length by the simple maximum value search. We use this length as an index of spatial resolution of the BDG reflection measurement. It can be seen that the spatial resolution is improved by the conventional apodization, and the half-apodization makes more improvement. Broadening of the reflection spectrum width reflects its localization effect. Interestingly, the spectrum in the half-apodized method become asymmetric. We are trying now to confirm this experimentally.

Shortest detectable length of the strained section depends on various factor, such as strain magnitude, sensing fiber length, and modulation parameter. Sensing fiber length $L$ has the particular importance among these for considering intensity modulation. Generally speaking, the shorter the overall fiber length, the better the spatial resolution can be realized. More importantly, the intensity modulation methods are greatly affected by the changes in fiber length. Fig. 7 illustrates these situations. When the fiber length is 0.3 and 1 times the measurable length $d$, the two-peak ratio in each method is compared by changing the strained length. A minimum length exceeding a peak ratio of 1 that is written with an orange dashed line indicates the spatial resolution. When the sensing fiber is relatively short as shown in Fig. 7(a), both the apodization method and the half-apodization method can improve the spatial resolution. The reason why the curve of the proposed method is not very smooth is presumed to be because the ratios of the maximum values of the narrow linewidth spectra are compared. It is in agreement with the experimental results that apodizing the three lights (pump, probe, and read) is more effective than apodizing two lights (pump and probe) alone [33]. The modest improvement effect over previous studies by three light apodization may be due to the poor extinction ratio in actual experiments. Next, when the fiber length is in the full range as shown in Fig. 7(b), the conventional apodization method greatly reduces the improvement effect, and may be worse than without the apodization. In contrast, the half-apodization method is greatly superior to these methods, and can detect a length of twice of the theoretical spatial resolution, even in the fully long sensing fiber.

Fig. 7. Two peak power ratio (strained / unstrained, see also Fig. 6) with respect to strained length. Sensing fiber length $L$ is set to (a) 0.3$d$ (b) $d$, where $d$ is measurable range (see Eq. (24)). Orange dashed line shows ratio equals to 1. The strained section with the length exceeding this line can be detected.

Download Full Size | PDF

Finally, we calculate distribution measurement results with half-apodized method as shown in Fig. 8. We simulate the situation where the reflection spectrum of BDG is measured by sweeping the correlation peak position for the strained section of 3$\Delta z$ length in the fully long fiber ($L=d$), which is not possible to detect by the other two methods. In this calculation, the scanning is performed by simply adding a delay to the probe light, instead of changing $f_m$. The result shows that the strained section is successfully detected using the half-apodized method, although the detected length is shorter than the actual one because it is close to the limit of its ability to resolve spatially. Some asymmetry about position is also observed due to its asymmetrical beat spectrum, which has never been found in the other methods. In fact, the reflection spectrum of BDG in half-apodized method slightly differs depending on the geometrical arrangement of the distortion section in front of or behind the correlation peak position. This effect may require us to make some corrections in deriving the actual position and length.

Fig. 8. Simulation of distribution measurement of BDG reflection spectrum in half-apodized method. We consider the situation where 3$\Delta z$ length of strain is applied to the center of test fiber with the full length $d$ and modulation parameters are the same as those in Fig. 7. (a) BDG reflection spectra at each fiber position. (b) Distribution of the peak frequency in BDG reflection spectra. Gray triangle shows the true frequency shift set in the simulation.

Download Full Size | PDF

4. Discussion and Conclusion

We analyzed BDG and its reflection spectra generated by the correlation-domain method taking into account the intensity modulation effect for the first time. Our calculations are in good agreement with the previous experimental studies [32,33], and the effect of spatial resolution improvement by intensity modulation was confirmed. We also developed a novel modulation method to effectively localize BDG at the target position. In our proposed method, the amount of light for half of the frequency modulation period is eliminated by the intensity modulator, so that the pump and the probe light do not exist simultaneously at a position away from the correlation peak. By adding this modulation to the conventional apodization method, it is possible to effectively localize the BDG across the entire measurement fiber. We also simulated the local reflection spectrum distribution of BDG and confirmed the effect of improving spatial resolution several times compared to no intensity modulation method. This technique will greatly improve the spatial resolution in distributed measurement of the reflection spectrum of BDG, which was a limiting factor in the temperature and strain discrimination distribution measurement by the BDG-BOCDA method.

Our modulation may also have a good effect on a BOCDA output in terms of getting the localized signal at the correlation peak position, as described in Sec. 3.2. Note that the effect is basically common to the case of BOCDR output, which is a similar method using spontaneous Brillouin scattering [39]. However, in actual distribution measurement by simple maximum searching, a negative influence on BOCDA (or BOCDR) may occur due to the asymmetry of the beat spectrum. The optimal intensity modulation waveform for BOCDA may be different from that for BDG. This effect could be mitigated by replacing and averaging the half-period selected by intensity modulator at certain time intervals. It should be added that this half-apodization is equivalent to a method in which the frequency is modulated only in one-way of sine wave and the measurement distance is doubled by the temporal gating method [30].

Our calculations, of course, do not include the whole practical situations. Perhaps the most important phenomenon is the non-uniformity of the birefringence in a polarization-maintaining fiber [40]. Even if distortion due to the external environment is not applied, the birefringence of the fiber always varies due to instability of the manufacturing process. Since the strained section to detect is usually very short compared to the entire fiber, the reflection spectrum of the unstrained portion that makes up the noise will be more affected. Since the birefringence that constitutes the peak of the noise portion diffuses, it is estimated that the peak power of the noise portion is reduced and then the actual spatial resolution performance is better than the calculation. Nevertheless, this fluctuation also results in increased measurement uncertainty of the amount of strain and temperature.

Funding

Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number JP 19K14999.

Disclosures

The authors declare no conflicts of interest.

References

1. T. Horiguchi, T. Kurashima, and M. Tateda, “A technique to measure distributed strain in optical fibers,” IEEE Photonics Technol. Lett. 2(5), 352–354 (1990). [CrossRef]

2. T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995). [CrossRef]

3. K. Hotate and T. Hasegawa, “Measurement of Brillouin gain spectrum distribution along an optical fiber using a correlation-based technique–proposal, experiment and simulation,” IEICE Trans. Electron. E83-C, 405–412 (2000).

4. K. Hotate, “Brillouin optical correlation-domain technologies based on synthesis of optical coherence function as fiber optic nerve systems for structural health monitoring,” Appl. Sci. 9(1), 187 (2019). [CrossRef]

5. K. Y. Song, Z. He, and K. Hotate, “Optimization of Brillouin optical correlation domain analysis system based on intensity modulation scheme,” Opt. Express 14(10), 4256–4263 (2006). [CrossRef]

6. K. Y. Song, Z. He, and K. Hotate, “Effects of intensity modulation of light source on Brillouin optical correlation domain analysis,” J. Lightwave Technol. 25(5), 1238–1246 (2007). [CrossRef]

7. Y. Antman, N. Primerov, J. Sancho, L. Thevenaz, and A. Zadok, “Localized and stationary dynamic gratings via stimulated Brillouin scattering with phase modulated pumps,” Opt. Express 20(7), 7807–7821 (2012). [CrossRef]

8. A. Denisov, M. A. Soto, and L. Thévenaz, “Going beyond 1000000 resolved points in a Brillouin distributed fiber sensor: theoretical analysis and experimental demonstration,” Light: Sci. Appl. 5(5), e16074 (2016). [CrossRef]

9. R. Cohen, Y. London, Y. Antman, and A. Zadok, “Brillouin optical correlation domain analysis with 4 millimeter resolution based on amplified spontaneous emission,” Opt. Express 22(10), 12070–12078 (2014). [CrossRef]

10. J. Zhang, M. Zhang, M. Zhang, Y. Liu, C. Feng, Y. Wang, and Y. Wang, “Chaotic Brillouin optical correlation-domain analysis,” Opt. Lett. 43(8), 1722–1725 (2018). [CrossRef]

11. K. Y. Song, W. Zou, Z. He, and K. Hotate, “All-optical dynamic grating generation based on Brillouin scattering in polarization-maintaining fiber,” Opt. Lett. 33(9), 926–928 (2008). [CrossRef]

12. A. Bergman and M. Tur, “Brillouin dynamic gratings—a practical form of Brillouin enhanced four wave mixing in waveguides: The first decade and beyond,” Sensors 18(9), 2863 (2018). [CrossRef]

13. K. Y. Song, K. Hotate, W. Zou, and Z. He, “Applications of Brillouin dynamic grating to distributed fiber sensors,” J. Lightwave Technol. 35(16), 3268–3280 (2017). [CrossRef]

14. J. Sancho, N. Primerov, S. Chin, Y. Antman, A. Zadok, S. Sales, and L. Thévenaz, “Tunable and reconfigurable multi-tap microwave photonic filter based on dynamic Brillouin gratings in fibers,” Opt. Express 20(6), 6157–6162 (2012). [CrossRef]

15. S. Chin and L. Thévenaz, “Tunable photonic delay lines in optical fibers,” Laser Photonics Rev. 6(6), 724–738 (2012). [CrossRef]

16. M. Santagiustina, S. Chin, N. Primerov, L. Ursini, and L. Thévenaz, “All-optical signal processing using dynamic Brillouin gratings,” Sci. Rep. 3(1), 1594 (2013). [CrossRef]

17. K. Y. Song, “Operation of Brillouin dynamic grating in single-mode optical fibers,” Opt. Lett. 36(23), 4686–4688 (2011). [CrossRef]

18. S. Li, M.-J. Li, and R. S. Vodhanel, “All-optical Brillouin dynamic grating generation in few-mode optical fiber,” Opt. Lett. 37(22), 4660–4662 (2012). [CrossRef]

19. L. Teng, H. Zhang, Y. Dong, D. Zhou, T. Jiang, W. Gao, Z. Lu, L. Chen, and X. Bao, “Temperature-compensated distributed hydrostatic pressure sensor with a thin-diameter polarization-maintaining photonic crystal fiber based on Brillouin dynamic gratings,” Opt. Lett. 41(18), 4413–4416 (2016). [CrossRef]

20. R. Pant, E. Li, C. G. Poulton, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Observation of Brillouin dynamic grating in a photonic chip,” Opt. Lett. 38(3), 305–307 (2013). [CrossRef]

21. W. Zou, Z. He, K.-Y. Song, and K. Hotate, “Correlation-based distributed measurement of a dynamic grating spectrum generated in stimulated Brillouin scattering in a polarization-maintaining optical fiber,” Opt. Lett. 34(7), 1126–1128 (2009). [CrossRef]

22. W. Zou, Z. He, and K. Hotate, “Complete discrimination of strain and temperature using Brillouin frequency shift and birefringence in a polarization-maintaining fiber,” Opt. Express 17(3), 1248–1255 (2009). [CrossRef]

23. W. Zou, Z. He, and K. Hotate, “Demonstration of Brillouin distributed discrimination of strain and temperature using a polarization-maintaining optical fiber,” IEEE Photonics Technol. Lett. 22(8), 526–528 (2010). [CrossRef]

24. W. Zou, Z. He, and K. Hotate, “One-laser-based generation/detection of Brillouin dynamic grating and its application to distributed discrimination of strain and temperature,” Opt. Express 19(3), 2363–2370 (2011). [CrossRef]

25. T. R. Parker, M. Farhadiroushan, R. Feced, V. A. Handerek, and A. J. Rogers, “Simultaneous distributed measurement of strain and temperature from noise-initiated Brillouin scattering in optical fibers,” IEEE J. Quantum Electron. 34(4), 645–659 (1998). [CrossRef]

26. M. N. Alahbabi, Y. T. Cho, and T. P. Newson, “Simultaneous temperature and strain measurement with combined spontaneous Raman and Brillouin scattering,” Opt. Lett. 30(11), 1276–1278 (2005). [CrossRef]

27. X. Lu, M. A. Soto, and L. Thévenaz, “Temperature-strain discrimination in distributed optical fiber sensing using phase-sensitive optical time-domain reflectometry,” Opt. Express 25(14), 16059–16071 (2017). [CrossRef]

28. M. A. S. Zaghloul, M. Wang, G. Milione, M.-J. Li, S. Li, Y.-K. Huang, T. Wang, and K. P. Chen, “Discrimination of temperature and strain in Brillouin optical time domain analysis using a multicore optical fiber,” Sensors 18(4), 1176 (2018). [CrossRef]

29. Z. Li, L. Yan, X. Zhang, and W. Pan, “Temperature and strain discrimination in BOTDA fiber sensor by utilizing dispersion compensating fiber,” IEEE Sens. J. 18(17), 7100–7105 (2018). [CrossRef]

30. K. Hotate, H. Arai, and K. Y. Song, “Range-enlargement of simplified Brillouin optical correlation domain analysis based on a temporal gating scheme,” SICE J. Control. Meas. Syst. Integration 1(4), 271–274 (2008). [CrossRef]

31. R. K. Yamashita and K. Hotate, “Numerical simulation for optimization of temporal gatings in Brillouin optical correlation domain analysis,” in Fifth European Workshop on Optical Fibre Sensors, vol. 8794L. R. Jaroszewicz, ed., International Society for Optics and Photonics (SPIE, 2013), pp. 522–525.

32. R. K. Yamashita, Z. He, and K. Hotate, “Spatial resolution improvement in correlation domain distributed measurement of Brillouin grating,” IEEE Photonics Technol. Lett. 26(5), 473–476 (2014). [CrossRef]

33. T. Sasai, M. Kishi, and K. Hotate, “Enhancement of spatial resolution in distributed measurement of Brillouin dynamic grating spectrum by optical correlation domain analysis,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2016), p. SM3P.6.

34. Y. Okawa, R. K. Yamashita, M. Kishi, and K. Hotate, “Spatial resolution improvement of discriminative and distributed measurement of temperature/strain with pmf using intensity-modulated correlation-domain technique,” in Optical Sensors and Sensing Congress, (Optical Society of America, 2019), p. STh4A.3.

35. K. Kajiwara, Z. He, and K. Hotate, “Dynamic range enhancement in reflectometry by synthesis of optical coherence function with half-wave intensity modulation,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2011, (Optical Society of America, 2011), p. OTuL3.

36. R. Boyd, Nonlinear Optics (Academic Press, 2008), 3rd ed.

37. D. P. Zhou, Y. Dong, L. Chen, and X. Bao, “Four-wave mixing analysis of Brillouin dynamic grating in a polarization-maintaining fiber: theory and experiment,” Opt. Express 19(21), 20785–20798 (2011). [CrossRef]

38. R. K. Yamashita, M. Kishi, and K. Hotate, “Theoretical evaluation of Brillouin dynamic grating length localized by optical correlation domain technique through reflection spectrum simulation,” Appl. Phys. Express 10(4), 042501 (2017). [CrossRef]

39. Y. Mizuno, W. Zou, Z. He, and K. Hotate, “Proposal of Brillouin optical correlation-domain reflectometry (BOCDR),” Opt. Express 16(16), 12148–12153 (2008). [CrossRef]

40. Y. H. Kim and K. Y. Song, “Characterization of nonlinear temperature dependence of Brillouin dynamic grating spectra in polarization-maintaining fibers,” J. Lightwave Technol. 33(23), 4922–4927 (2015). [CrossRef]

Analysis of Brillouin dynamic grating localized by intensity-modulated correlation-domain technique for distributed fiber sensing

Abstract

1. Introduction

2. BDG-BOCDA system

2.1 Configuration

2.2 Equations of BOCDA

2.3 Equations of BDG reflection

3. Analysis of intensity modulation effect

3.1 Three methods to compare

3.2 Effects on BGS measurement

3.3 Effects on BDG measurement

4. Discussion and Conclusion

Funding

Disclosures

References

Cited By

Figures (8)

Equations (27)

Optics Express