## 1.

## Introduction

Stimulated Brillouin scattering (SBS) in silica fiber is described as a nonlinear interaction between two counter-propagating optical waves with a frequency difference of Brillouin frequency shift (BFS) mediated through an acoustic wave, which not only transfers energy between these two optical waves but also changes their phases.^{1} Brillouin optical time-domain analysis (BOTDA) sensors based on SBS interaction have gained much attention over the world since 1989,^{2} owing to its superiority in distributed monitoring of temperature or strain in km-long fibers at a submeter resolution.^{3}4.5.6.^{–}^{7} The measurement of temperature or strain in conventional BOTDA sensor is based on the linear relationship between the BFS and temperature or strain, and the BFS is usually measured by scanning the gain or loss spectrum of SBS amplitude and locating the peak value of the spectrum. Unfortunately, the measurement accuracy in the amplitude spectrum-based system is limited by pump depletion and nonlocal effect.^{8}9.10.^{–}^{11} In recent years, however, the vector BOTDA (VBOTDA) system capable of measuring both the amplitude and phase-shift spectra of SBS response is introduced by Dossou et al.,^{12} and is used to feature the high-order acoustic resonances that are not clear on the SBS amplitude spectrum, by the SBS phase-shift spectrum. Since the temperature- or strain-induced BFS change results in a frequency shift of the SBS phase-shift spectrum, and the BFS can be measured by scanning the SBS phase-shift spectrum and locating its zero point, the SBS phase shift is also applied in BOTDA sensor to measure temperature or strain. A BOTDA sensor employing optical self-heterodyne detection and synchronous demodulation achieved a distributed phase-shift measurement over a 25-km long fiber and a 10.75-dB enhancement of signal-to-noise ratio (SNR).^{13} Since the SBS phase shift is independent of SBS amplitude, it is immune to nonlocal effect, which indicates a huge potential for enhanced long-haul distributed sensors.^{14} With the assistance of Raman amplification, VBOTDA-based sensor reduced significantly the relative intensity noise and showed very attractive application prospects in long-range distributed measurements.^{15} Xiaobo et al.^{16} successfully measured the Brillouin gain and phase-shift spectra by VBOTDA technique employing IQ demodulation and obtained the temperature dependence of Brillouin gain and phase-shift spectra.

To improve the system’s performance, unlike the existing VBOTDA system in which only a single sideband (SSB) is used, we propose a BFS measurement system for VBOTDA-based temperature sensor, in which double sideband (DSB) modulation is used and the SBS gain and loss processes work together to generate a superimposed SBS phase-shift spectrum and to achieve a double measurement accuracy. The temperature measurement error of the system is first evaluated mathematically, then a proof-of-concept experiment is performed by using a 100-m long standard single-mode fiber (SMF), and finally, the performances of the proposed DSB BFS measurement system and traditional SSB BFS measurement system for VBOTDA-based temperature sensor are compared.

## 2.

## Principle

The principle of the proposed DSB BFS measurement technique for VBOTDA-based temperature sensor is illustrated in Fig. 1. The technique relies on the SBS interaction between probe wave and two pump waves. The frequencies of the two pump waves are symmetrically up- and down-shifted from the frequency of probe wave by ${\nu}_{\mathrm{p}}$ that are tunable in the vicinity of the BFS ${\nu}_{\mathrm{B}}$. Pump wave 1 acting as the anti-Stokes wave induces an SBS gain response on the probe wave, whereas pump wave 2 acting as the Stokes wave induces an SBS loss response on the probe wave.

The heterodyne detection for measuring the SBS amplitude and phase-shift responses is realized by introducing a coherent local wave propagating together with the probe wave along the fiber in the same direction. The local wave is away from the probe wave by ${\nu}_{0}$ and outside the Brillouin scattering spectra of pump waves; therefore, it does not interact with the pump waves via SBS interaction. The modified steady-state SBS coupled wave equations are described as^{17}

## (1)

$$\frac{d{I}_{\mathrm{p}1}}{dz}=-\alpha {I}_{\mathrm{p}1}-\frac{{g}_{0}\xb7\mathrm{\Delta}{\nu}_{\mathrm{B}}^{2}}{(4\mathrm{\Delta}{\nu}^{2}+\mathrm{\Delta}{\nu}_{\mathrm{B}}^{2})}{I}_{\mathrm{p}1}{I}_{\mathrm{p}\mathrm{r}},$$## (2)

$$\frac{d{I}_{\mathrm{p}2}}{dz}=-\alpha {I}_{\mathrm{p}2}+\frac{{g}_{0}\xb7\mathrm{\Delta}{\nu}_{\mathrm{B}}^{2}}{(4\mathrm{\Delta}{\nu}^{2}+\mathrm{\Delta}{\nu}_{\mathrm{B}}^{2})}{I}_{\mathrm{p}2}{I}_{\mathrm{p}\mathrm{r}},$$## (3)

$$\frac{d{I}_{\mathrm{p}\mathrm{r}}}{dz}=\alpha {I}_{\mathrm{p}\mathrm{r}}+\frac{{g}_{0}\xb7\mathrm{\Delta}{\nu}_{\mathrm{B}}^{2}}{(4\mathrm{\Delta}{\nu}^{2}+\mathrm{\Delta}{\nu}_{\mathrm{B}}^{2})}({I}_{\mathrm{p}2}-{I}_{\mathrm{p}1}){I}_{\mathrm{p}\mathrm{r}},$$## (4)

$$\frac{d{\varphi}_{\mathrm{p}\mathrm{r}}}{dz}=-\frac{2\xb7{g}_{0}\xb7\mathrm{\Delta}\nu \xb7\mathrm{\Delta}{\nu}_{\mathrm{B}}}{(4\mathrm{\Delta}{\nu}^{2}+\mathrm{\Delta}{\nu}_{\mathrm{B}}^{2})}({I}_{\mathrm{p}1}+{I}_{\mathrm{p}2}),$$In Eqs. (1)–(4), the probe wave is assumed to travel in the $+z$ direction, and the pump waves 1 and 2 in the $-z$ direction. Considering that for a practical BFS measurement system, a very small probe wave power is usually used to avoid pump depletion,^{11} therefore, to simplify the analysis, it is reasonable to assume that since the power of probe wave is much lower than those of pump waves 1 and 2, the power change of the two pump waves induced by SBS interaction is so small comparing to their initial level that the pump depletion is negligible and their powers remain approximately equal over the interaction length $\mathrm{\Delta}z$, defined as the spatial resolution of a sensing system. In this case, the second term on the right-hand side of Eq. (3) vanishes, which indicates that the probe power is not affected by the SBS processes due to the cancellation of SBS gain and loss on the amplitude of probe wave. However, the total phase shift ${\varphi}_{\mathrm{p}\mathrm{r}}$ experienced by the probe wave becomes the sum of the phase-shift responses induced by the two SBS interaction processes, which can be deduced by integrating Eq. (4) over the interaction length $\mathrm{\Delta}z$ of the fiber and expressed as Eq. (5), since the parameters concerned are independent of scattering location in the fiber under given conditions:

## (5)

$${\varphi}_{\mathrm{p}\mathrm{r}}({\nu}_{\mathrm{B}},T)=-\frac{2\xb7{g}_{0}\xb7\mathrm{\Delta}\nu \xb7\mathrm{\Delta}{\nu}_{\mathrm{B}}}{(4\mathrm{\Delta}{\nu}^{2}+\mathrm{\Delta}{\nu}_{\mathrm{B}}^{2})}\xb7\frac{2{P}_{\mathrm{p}}\xb7\mathrm{\Delta}z}{{A}_{\text{eff}}}=2{\varphi}_{\mathrm{p}\mathrm{r}}^{\prime}({\nu}_{\mathrm{B}},T),$$In the traditional SSB BFS measurement system, when a phase measurement instrument with an error of $\delta \varphi $ is used, the measurement error of BFS in the system can be given by

## (6)

$$\delta {\nu}_{\mathrm{B}}^{\mathrm{S}}=\frac{d{\nu}_{\mathrm{B}}}{d{\varphi}_{\mathrm{p}\mathrm{r}}^{\prime}}\delta \varphi .$$For the proposed DSB BFS measurement system, however, when the same measurement method and instrument are used, the measurement error of BFS in DSB system can be written as follows:

## (7)

$$\delta {\nu}_{\mathrm{B}}^{\mathrm{D}}=\frac{d{\nu}_{\mathrm{B}}}{d{\varphi}_{\mathrm{p}\mathrm{r}}}\delta \varphi .$$From Eqs. (5) and (6), Eq. (7) can be rewritten as follows:

## (8)

$$\delta {\nu}_{\mathrm{B}}^{\mathrm{D}}=\frac{1}{2}\xb7\frac{d{\nu}_{\mathrm{B}}}{d{\varphi}_{\mathrm{p}\mathrm{r}}^{\prime}}\delta \varphi =\frac{1}{2}\xb7\delta {\nu}_{\mathrm{B}}^{\mathrm{S}}.$$Again, since the temperature sensitivity of BFS in the two systems, ${C}_{\mathrm{\nu}\mathrm{T}}$, is only dependent on the fiber material, the measurement error of temperature in the proposed system can be expressed as

## (9)

$$\delta {T}^{\mathrm{D}}=\frac{\delta {\nu}_{\mathrm{B}}^{\mathrm{D}}}{{C}_{\mathrm{\nu}\mathrm{T}}}=\frac{1}{2}\xb7\frac{\delta {\nu}_{\mathrm{B}}^{\mathrm{S}}}{{C}_{\mathrm{\nu}\mathrm{T}}},$$To evaluate further the measurement error of BFS in the proposed system, the ratio of the phase-shift range to frequency range between the maximum and minimum of the phase-shift spectrum, namely the maximum frequency sensitivity of the phase shift, can be used. From the equation obtained by setting the derivative of Eq. (5) to be zero, when $\mathrm{\Delta}\nu =\mathrm{\Delta}{\nu}_{\mathrm{B}}/2$, the maximum of the phase-shift spectrum is obtained as

## (10)

$$\mathrm{max}({\varphi}_{\mathrm{p}\mathrm{r}})=\frac{{P}_{\mathrm{p}}}{{A}_{\text{eff}}}\xb7\mathrm{\Delta}z\xb7{g}_{0},$$## (11)

$$\mathrm{min}({\varphi}_{\mathrm{p}\mathrm{r}})=-\frac{{P}_{\mathrm{p}}}{{A}_{\text{eff}}}\xb7\mathrm{\Delta}z\xb7{g}_{0}.$$According to Eqs. (10) and (11), although the phase-shift range becomes twice that in the SSB system, the corresponding frequency range is kept to be $\mathrm{\Delta}{\nu}_{\mathrm{B}}$, which is the same as that in the SSB system. So, the maximum frequency sensitivity of phase shift can be given by

## (12)

$${S}_{\varphi \mathrm{f}}^{\mathrm{D}}=\frac{d{\varphi}_{\mathrm{p}\mathrm{r}}}{d{\nu}_{\mathrm{B}}}=\frac{2{P}_{\mathrm{p}}}{{A}_{\text{eff}}}\xb7\mathrm{\Delta}z\xb7{g}_{0}\xb7\frac{1}{\mathrm{\Delta}{\nu}_{\mathrm{B}}}=2\frac{d{\varphi}_{\mathrm{p}\mathrm{r}}^{\prime}}{d{\nu}_{\mathrm{B}}}=2{S}_{\varphi \mathrm{f}}^{\mathrm{S}},$$## (13)

$$\delta {T}^{\mathrm{D}}=\frac{\delta \varphi}{{S}_{\varphi \mathrm{f}}^{\mathrm{D}}\xb7{C}_{\mathrm{\nu}\mathrm{T}}}=\frac{1}{2}\xb7\frac{\delta \varphi}{{S}_{\varphi \mathrm{f}}^{\mathrm{S}}\xb7{C}_{\mathrm{\nu}\mathrm{T}}}.$$The heterodyne detection signal at frequency ${\nu}_{0}$ in the DSB system can be expressed as

## (14)

$$i({\nu}_{0})=R{E}_{\mathrm{p}\mathrm{r}}{E}_{\mathrm{L}}\text{\hspace{0.17em}}\mathrm{cos}(2\pi {\nu}_{0}t-{\varphi}_{\mathrm{p}\mathrm{r}}),$$## 3.

## Experimental Setup

The experimental setup of the proposed DSB BFS measurement technique is depicted in Fig. 2. The continuous wave output of a laser operating at a 1550.057-nm wavelength and with a 10-KHz linewidth is divided into two branches by a polarization-maintaining (PM) coupler. In the upper branch, the optical wave is launched into an electro-optic modulator (EOM) biased at null point and driven by a microwave generator, to produce a DSB suppressed carrier (SC) wave with 22-dB suppression. The two sidebands acting as pump waves 1 and 2 are amplified by an erbium-doped fiber amplifier (EDFA) and extracted from the EDFA output by an optical filter composed of a circulator and a Bragg grating with a center wavelength of 1550.055 nm and a bandwidth of 0.312 nm. The pump waves are passed through a polarization scrambler (PS) to reduce polarization-induced fluctuations and launched by a circulator into a length of standard SMF used as a test fiber. The lower branch is used as a probe wave and is input to a Mach–Zehnder interferometer (MZI) in which the lower arm is used to generate the local wave through a 200-MHz down-shifted acousto-optic frequency shifter (AOFS), and the upper arm is used to adjust the polarization of the probe wave through a polarization controller (PC) to ensure the maximum visibility of the beat signal of the probe wave and local wave. The probe wave and local wave are recombined by coupler 2, the upper output of which is directed to the test fiber through an optical isolator (ISO) that is used to ensure unidirectional transmission, and the lower output of which is detected by PD1 and is used as the reference signal of phase measurement. The probe wave, interacted with the pump waves through SBS interaction, and the local wave from port 3 of circulator 2 are detected by PD2 and used as the detection signal. Finally, the two heterodyne signals are captured by a digital oscilloscope (OSC). The reference signal, detection signal without SBS interaction, and detection signal with SBS interaction can be expressed, respectively, as

## (15)

$${I}_{\mathrm{R}}({\nu}_{0})={R}_{1}\sqrt{{P}_{\mathrm{p}\mathrm{r}}{P}_{\mathrm{L}}}\mathrm{cos}(2\pi {\nu}_{0}t),$$## (16)

$${I}^{\prime}({\nu}_{0})={R}_{2}\sqrt{{P}_{\mathrm{p}\mathrm{r}}{P}_{\mathrm{L}}}\mathrm{cos}(2\pi {\nu}_{0}t-{\varphi}_{\mathrm{f}}),$$## (17)

$$I({\nu}_{0})={R}_{2}\sqrt{{P}_{\mathrm{p}\mathrm{r}}{P}_{\mathrm{L}}}\mathrm{cos}(2\pi {\nu}_{0}t-{\varphi}_{\mathrm{f}}-{\varphi}_{\mathrm{p}\mathrm{r}}),$$When the center wavelength and bandwidth of Bragg grating in the optical filter are changed to 1549.967 and 0.296 nm, only the pump wave 1 can be extracted, and the SSB BFS measurement system based on SBS gain response can be achieved, which is expected to be used for performance comparison between the DSB system and the SSB system. The detection signal with SBS interaction in the SSB system can be approximately expressed as follows:^{13}

## 4.

## Results and Discussions

A proof-of-concept experiment of the proposed DSB BFS measurement technique is performed following the setup in Fig. 2, and the SBS phase shift and amplitude spectra of the DSB and SSB systems are compared. The powers of the probe wave and local wave are set to be $-5.23$ and $-3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{dBm}$, respectively; the power of pump waves 1 and 2, i.e., two sidebands of the DSB-SC wave, is fixed to 17.78 dBm, and the sampled heterodyne signals are averaged 1024 times to improve the SNR. A 100-m long standard SMF is used as the test fiber and immersed loosely in a temperature-controlled water tank with the water temperature increasing from 10°C to 80°C by a step of 10°C, and the frequency of the microwave generator is changed from 10.752 to 10.952 GHz with a step of 4 MHz. The frequency sweeping and data sampling are started after having been kept at each temperature for 10 min to ensure the temperature uniformity of the fiber. Three separate measurements for SBS phase shift and amplitude spectra are averaged and fitted by the minimum root mean square method, as shown in Fig. 3.

In Fig. 3, it is seen that the SBS phase shift and amplitude spectra move toward higher frequency when the water temperature increases. The variation of the SBS phase shift and amplitude peak is mainly owing to the power variation of the pump waves. Figure 3(a) shows clearly that the phase-shift range from the maximum to minimum of the SBS phase-shift spectrum in the DSB system is almost twice that in the SSB system at the same operational parameters, which indicates that the maximum frequency sensitivity of the phase shift in the vicinity of BFS can be improved by almost 100% in DSB system. Figure 3(b) illustrates clearly that compared with the SSB system, the probe power is almost unaffected by the SBS processes in the DSB system, which is consistent with the above theoretical analysis and indicates that the nonlocal effect induced by the energy accumulation on the probe wave transferred between the pump wave and probe wave through SBS interaction^{8}^{,}^{9} can be effectively reduced in the proposed system. The small but visible fluctuation of probe power in Fig. 3(b) is induced by the slight inconsistency between ${P}_{\mathrm{p}1}$ and ${P}_{\mathrm{p}2}$ due to equipment limitations.

Figure 4 shows the temperature dependencies of the BFSs obtained by fitting the SBS phase-shift spectra in DSB system and SSB system. It is seen that the measurements in both systems exhibit very good linearity and consistency, and the temperature coefficients are estimated to be 1.059 and $1.043\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{MHz}/\xb0\mathrm{C}$ through linear fitting, which demonstrates an excellent sensing performance of the proposed technique.

To demonstrate the high-temperature measurement accuracy of the proposed technique experimentally, the maximum frequency sensitivity of the phase shift is obtained by taking the ratio of phase-shift range to the frequency range between the maximum and minimum of phase-shift spectrum, as shown in Fig. 5. From Fig. 5, the maximum frequency sensitivity of the phase shift increases with the increase of temperature, and the maximum frequency sensitivity of the phase shift in the DSB system is almost twice that in the SSB system, which shows that the measurement error of temperature in the proposed system can be reduced to only half that in SSB system, which is also well depicted in Eqs. (5)–(13).

To clarify the temperature dependence of the maximum frequency sensitivity of the phase shift, the relationships between frequency range $\mathrm{\Delta}{\nu}_{\mathrm{B}}$ and temperature in the two systems are also obtained by the measured phase-shift spectrum, as shown in Fig. 6. It is seen that the frequency range decreases with the increase of temperature in both systems, which is in good agreement with the results reported in the literature,^{18} thereby causing the increase of maximum frequency sensitivity with the increase of temperature, as depicted in Eq. (12). But the frequency range in the DSB system is slightly larger than that in the SSB system, which might be also induced by the slight inconsistency between ${P}_{\mathrm{p}1}$ and ${P}_{\mathrm{p}2}$ due to equipment limitations.

It is worth mentioning that since the BFS is proportional to fiber strain, the theoretical and experimental results about measurement error in this paper should be also applicable to fiber strain measurement by the proposed technique. In this case, the strain measurement error can be given as

## (19)

$$\delta {\u03f5}^{\mathrm{D}}=\frac{\delta \varphi}{{S}_{\varphi \mathrm{f}}^{\mathrm{D}}\xb7{C}_{\mathrm{\nu}\mathrm{\u03f5}}}=\frac{1}{2}\xb7\frac{\delta \varphi}{{S}_{\varphi \mathrm{f}}^{\mathrm{S}}\xb7{C}_{\mathrm{\nu}\mathrm{\u03f5}}},$$Although the proof-of-concept experiment for temperature sensing in this paper is conducted with continuous probe and pump waves injected into 100-m long fiber due to equipment limitations, the obtained results are equivalent to those from a $1\text{-}\mu \mathrm{s}$-width pulse pump injected VBOTDA-based temperature sensor; therefore, the spatial resolution of the proposed system can be estimated to be better than 100 m according to the relationship between system spatial resolution and sensing pulse width in a distributed optical fiber sensing system, and a DSB VBOTDA-based temperature sensing system with practical significance can be realized by inserting another EOM driven by a pulse generator between the microwave generator driven EOM and the EDFA, and by replacing the OSC with a data acquisition card and a computer in Fig. 2.

## 5.

## Conclusions

We have proposed and demonstrated a high-accuracy DSB SBS phase-shift based BFS measurement technique for VBOTDA-based temperature sensor. The measurement error of the technique has been evaluated mathematically and a proof-of-concept experiment has been implemented by using a 100-m long SMF. The theoretical and experimental results show that the temperature dependence of the BFS is of excellent linearity with a sensitivity of $1.059\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{MHz}/\xb0\mathrm{C}$, the measurement error of temperature in the proposed DSB system is only half that in traditional SSB system, and the probe wave is almost unaffected by the SBS interaction. The proposed technique can achieve high-accuracy temperature measurement and reduce nonlocal effect, which may lead to a performance-enhanced distributed sensor and new applications.

## Acknowledgments

This paper was supported by the National Natural Science Foundation of China (NSFC) (61377088); Natural Science Foundation of Hebei Province of China (F2014502098).

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## Biography

**Yongqian Li** received his BE degree in electronic instrument and measurement technology and his MS degree in communication and electronic systems from Tianjin University, China, in 1982 and 1988, respectively. He received his PhD degree at Gunma University, Japan, in 2003. Since 2004, he has been a professor in the Department of Electronics and Communication Engineering, North China Electric Power University. His research interests include optical communication and distributed optical fiber sensing.

**Qi An** received her BE degree in communication engineering from Hebei University of Science and Technology in 2009, and she is now a PhD student in North China Electric Power University with the major of electrical information technology. Her research interests include optical communication and distributed optical fiber sensing.

**Xiaojuan Li** received her BE degree in electronic and information engineering from Hebei University of Economic and Business in 2009, and she is now a PhD student in North China Electric Power University with the major of electrical information technology. Her research interests include optical communication and distributed optical fiber sensing.

**Lixin Zhang** received her BE degree in electronic information science and technology from North China Electric Power University in 2011, and she is now a PhD student in North China Electric Power University with the major of electrical information technology. Her research interests include optical communication and distributed optical fiber sensing.