Distributed fiber optic vibration sensor based on polarization fading model for gas pipeline leakage testing experiment

Abstract

To design a distributed fiber optic vibration sensor for urban natural gas pipeline leak detection, the light polarization fading transmission model based on Jones matrix is built in this mixed Sagnac/Mach-Zehnder interferometer configuration. The interference light intensity of pipeline with leakage and without leakage cases is compared through spectrum analyzer. When fiber length ranges from 1.5 to 9.5 km, the light power intensity (central wavelength 1550 nm) reduces to −56 dB. In experiment, the leaky spectrum null frequencies are captured and the average relative error of leakage point location varies from 2.3 to 5.3%. When the sensing length is expanded to 9.5 km, the vibration sensor system sensitivity is 0.57 Hz/m.

Keywords

Fiber optic sensor vibration interferometer gas pipeline leakage polarization

Introduction

Detection of urban natural gas pipeline leakage requires attention due to the remoteness of gas locations and the sensitivity of the complex environmental conditions. The high pressure gas pipeline leakage sources of crack or slit caused broadband acoustic vibration signals.^1–3 Current monitoring methods include flow measurements at the beginning and end of the gas pipeline, offering an indication of the presence of a leak, but it is difficult to locate the leakage position. Recent developments of distributed optical fiber sensor promised to provide a cost-effective tool allowing monitoring long-distance pipeline.⁴ An optical fiber sensor embedded in or attached to gas pipelines expanded or contracted by small amounts according to strains on the structure and temperature variations, even vibration signals.^5,6 The distributed fiber optic sensing system integrated information collection and transmission in one optical fiber. Its advantages of high sensitivity, long-distance monitoring, and accurate location provided the availability of leakage location in the pipeline monitoring system.⁷

Distributed fiber optic sensing systems based on Brillouin and Raman scattering are used to detect the localized strain and/or temperature, allowing the monitoring of hundreds of kilometers along a structure with a single instrument with an accuracy of several meters.^8–10 The highly sensitive fiber optic liquid leak detection technique with an ultrasonic actuator and a fiber Bragg grating had been developed.¹¹ The backscattering fiber optic sensor can realize high positioning accuracy within several meters, but the system needs complex expensive apparatus and consuming much time for data processing. Therefore, interferometric systems owned high real-time performance and had been a well-known solution to fiber optic measurement configurations for pipeline leakages inspection.^12,13 The Sagnac interferometer owns merits of high sensitivity, large monitoring range, and passive sensing. The different complicated Sagnac configurations were proposed for gas pipeline leak detection. The core component of pipeline leaky point localization method is to search the null frequencies of the frequency spectrum. Such as distributed fiber optic sensing system based on Sagnac and Mach-Zehnder interferometers architecture was designed to monitor gas pipeline leaks signals using single mode fiber.¹³ A null frequency localization regression model based on 20 groups of leakage data was established through particle swarm optimization tuning of the support vector machine.¹⁴

Actually, polarization fading of many optic devices in the total optical paths configuration, such as mixed Sagnac and Mach-Zehnder interferometers, is not considered. The null frequency points are very difficult to be accurately captured owing to low signal-to-noise ratio (SNR) of leaky spectrum. Using low-birefringent single mode optical fiber, the distributed fiber optic sensor was subject to construct polarization-induced phase shift and polarization-induced fading, which will deteriorate the positioning accuracy of the fiber optic sensor.¹⁵ The unpredictable loss of interference efficiency arises because of random changes in the state of polarization of the two interfering beams caused by changes in the fiber birefringence.¹⁶ Polarization fading problem is also obvious in Sagnac interferometer and deteriorates the positioning accuracy. The interferometer phase spectrum exists many peaks, and it is not easy to discriminate first null frequency point. To eliminate and quantitatively evaluate the polarization fading in interferometric fiber optic sensor for gas pipeline leakage detecting is a challenge. With the analytic model deduced from Jones matrix, Choi and Jo¹⁷ had accurately evaluated the polarization characteristics in the integrated optic chip for interferometric fiber optic gyroscope. To find the null frequencies more accurately and efficiently, an improved algorithm twice fast Fourier transform (FFT) for multipoint intrusion location in distributed Sagnac sensing system was proposed.¹⁸ Furthermore, the Faraday rotation mirror (FRM) and polarization diversity receivers were used to remove the linear birefringence and the orientation birefringence in Sagnac and Michelson interferometer.¹⁹

In this paper, urban natural gas pipeline leak detection experiments are carried out based on distributed fiber optic vibration sensor. The birefringent path is represented as a general retarder and is characterized by Jones matrix in a birefringent interferometer. To evaluate the polarization fading in this mixed interferometer, the mathematical model deduced from Jones matrix is used to describe the optic devices polarization of the sensor system. Observing the light power spectrum of difference cases through high resolution spectrum analyzer, the influence of polarization fading on system localization performance was analyzed by experimental data. Finally, we discuss the positioning performance of different leakage points by distributed fiber optic vibration sensor.

Principle of operation

Measurement of gas pipeline leaky positions

Interferometric fiber optic acoustic sensors are based on measuring the phase change of light travelling in an optical fiber due to the strains, temperature, or vibration variations. Based on the basic optical paths structure proposed by Huang et al.,¹³ the configuration of this distributed fiber optic vibration sensor for urban gas pipeline leakage detecting is presented in Figure 1. The gas leakage point of the sensing fiber optic can be localized by Sagnac interferometer, and interferometric lights caused by outside vibration are processed by Mach-Zehnder interferometer. The FRM, polarization controller, and phase modulator are used to eliminate polarization-induced fading in light path. The piezoelectric ceramic transducer (KG-PM LiNbO₃ PZT) is applied as a phase modulator to generate a high frequency carrier. The distributed fiber optic sensor detects the vibration signal caused by natural gas pipeline leakage, and the details description is referred in Yue et al.¹⁴

Figure 1.

Schematic of fiber optic vibration sensor based on Sagnac and Mach-Zehnder interferometer.

The amplified spontaneous emission (ASE) light source transmits light through optical circulator (CIR) to the 2*2 coupler (DC1) terminal 1. Then the light is divided into two paths (through terminal 3 and 4), one is through PZT modulator and polarization controller, the other passes the delay optic fiber A. The lights transmit into the second coupler 1*2 (DC2). Finally, two optical paths pass the coupler to the sensing optic fiber and reflected by the FRM. As shown in Figure 2, there are two optical paths (path 3 and path 4) not equal to other paths, therefore these two paths cannot generate interference signal. Only the other two optical paths (path 1 and path 2) have equal optical paths and meet the zero optical path difference need of Sagnac interferometer.

Figure 2.

The optical path of the distributed fiber optic interferometer.

The natural gas pipeline crack or slit leakage consists of the transition of a pressurized gas from inside high pressure pipeline to the lower external pressure. The vibration signals are generated by high-speed jet flow.^20–22 Therefore, this gas pipeline section leakage point generates the acoustic field with frequency $ω_{s}$ and acts on the sensing fiber (L₁ and L₂). The optical path 1 and path 2 light beams pass the leakage point at different time in Figure 1, then the phase difference will be produced in this point. The phase difference of two optical paths detected by photo detector (PD) will vary with leakage point physical characteristics. The interferometer produces dynamic interference fringes variation. Not considering polarization of optical devices, the theoretic output light intensity of the interferometer can be expressed as¹³

I = cos {4 φ_{s} cos ω_{s} (t - \frac{τ_{t}}{2}) sin ω_{s} (\frac{τ_{A}}{2}) cos ω_{s} τ_{x} + 2 φ_{m} cos ω_{m} (t - \frac{τ_{t}}{2}) sin ω_{m} (\frac{τ_{t}}{2})]}

(1)

where

φ_{s}

is the amplitude of the leaky point phase signal,

ω_{s}

is the angular frequency of the leaky point phase signal,

τ_{A}

is the time consumption of the light transmit through delay fiber,

τ_{x}

is the time duration of the light transmit through leaky point to FRM,

φ_{m}

is the amplitude of the modulation carrier phase signal,

ω_{m}

is the angular frequency of the modulation carrier phase signal, and

τ_{t}

is the duration time of light passing the whole detecting system.

The light intensity I is demodulated by using phased generated carrier (PGC) technique to obtain the phase difference $Δ φ (t, L_{2})$ that is generated by the natural gas pipeline leakage. The phase difference between path 1 and path 2 caused by the leaky vibration signal is described as

Δ φ (t, L_{2}) = 4 φ_{s} cos ω_{s} (t - \frac{τ_{T}}{2}) sin ω_{s} (\frac{τ_{A}}{2}) cos ω_{s} (τ_{s})

(2)

In equation (2), $Δ φ (t, L_{2})$ is the phase difference of path 1 and path 2, which contains the leakage point information. The natural gas pipeline leakage signal is a wideband acoustic vibration signal. By applying FFT to $Δ φ (t, L_{2})$ , the frequency spectrum with obvious period feature can be seen. If the $Δ φ (t, L_{2})$ is constant to be zero, the equation for leak location can be derived based on the interference signal null frequency spectrum. Finally, the distance L₂ between leaky point and FRM is able to be calculated by null frequency spectrum¹⁴

L_{2} = \frac{c}{4 \cdot n \cdot f_{s}}

(3)

where f_s is the null frequency values, c is speed of light, and n is the refractive index of the fiber core.

Polarization fading analysis for Jones matrix

The light wave at any point in the circuit is represented by its Jones vector

E = (\begin{matrix} E_{x} \\ E_{y} \end{matrix})

(4)

Here $E_{x}$ , $E_{y}$ are electric field in x and y direction unit Jones vector, respectively. To apply the Jones calculus we need to identify the Jones matrix for each sequential element of the optical path.²³ The effect of each circuit element will then be represented by a $2 \times 2$ matrix operating on the Jones vector E.

In Figure 1, the distributed fiber optic vibration system of urban gas pipeline leakage includes many optical devices, such as single mode optic fiber, FRM reflection and couplers (DC1 and DC2), which contribute to polarization fading, interference noise, also reducing visibility of interference fringes in experiment. Jones matrix represents optical device polarization state by $2 \times 2$ matrix. Then the distributed fiber optic vibration sensing system is modeled by these optical devices' Jones matrix. The PZT phase modulator and optical circulator are polarization independent. The FRM is connected to the sensing fiber length (L₁ + L₂) to compensate its polarization variation. The delay fiber length (about 2 km) is far less than sensing fiber length (L₁ + L₂), and the delay fiber polarization effect can be ignored.

The ASE light source transmits the light through optical circulator, and the light is split into two mutually orthogonal beams by $2 \times 2$ coupler. To simplify the analysis, the propagation losses are neglected and the single mode directional coupler Jones matrix is given as²⁴

\begin{matrix} {\begin{matrix} K_{s} = [\begin{matrix} \sqrt{(1 - k)} 0 \\ 0 \sqrt{(1 - k)} \end{matrix}] \\ K_{A} = [\begin{matrix} j \sqrt{k} 0 \\ 0 j \sqrt{k} \end{matrix}] \end{matrix} \end{matrix}

(5)

In equation (5), $K_{s}$ and $K_{A}$ are directly coupled Jones matrix and cross coupled Jones matrix, k is coupling ratio.

The FRM is designed to rotate the polarization of the input light by 45°. The light wave is input into the FRM with 45° deflection angle. Finally, the light wave is reflected by the mirror and returned back in the input direction. So FRM Jones matrix is expressed as

N = \begin{matrix} [\begin{matrix} - 10 \\ 0 - 1 \end{matrix}] \end{matrix}

(6)

According to Jones matrix equations (4) and (5), considering optical devices polarization variation, the electric fields of the lights passing through path 1 and path 2 are given by

E_{1} = \begin{matrix} (\begin{matrix} {E_{10}}_{x} \\ {E_{10}}_{y} \end{matrix}) K_{S} K_{S} BN B^{T} K_{A} K_{A} \end{matrix}

(7)

\begin{matrix} E_{2} = (\begin{matrix} {E_{20}}_{x} \\ {E_{20}}_{y} \end{matrix}) K_{A} K_{A} BN B^{T} K_{S} K_{S} \end{matrix}

(8)

where

E_{10}

and

E_{20}

are the amplitude of the electric field of the light beam propagating through path 1 and path 2, respectively. B is single mode fiber Jones matrix.

Therefore, the total intensity of electric field of two paths can be expressed as

I' = (E_{1} + E_{2}) (E_{1} + E_{2})^{*}

(9)

where * denotes the complex conjugate.

Considering the optic devices polarization variation, from equations (6) to (8), the actual interferometer output light intensity is acquired as

I' = exp [j (φ (t_{1}) + φ (t_{2}))] {K_{A} K_{A} BN B^{T} K_{S} K_{S} + exp [j Δ φ (t, L_{2})] K_{S} K_{S} BN B^{T} K_{A} K_{A}}

(10)

Comparing equation (10) with equation (1), $I'$ and I are actual interference light intensity and theoretic light intensity, respectively. $I'$ and I contain the phase difference information $Δ φ (t, L_{2})$ which is caused by gas pipeline leakage. The leakage point positioning method is based on null frequency spectrum of phase difference. Though phase difference $Δ φ (t, L_{2})$ is a fixed value, the polarization fading of fiber optic vibration sensor will cause actual interference light intensity signal $I'$ attenuation to some extent.

Experiments and data analysis

Measured interference light spectrum

In laboratory-scale experiment, the urban natural gas pipeline section leakage localization tests are conducted. The distributed fiber optic vibration sensor interferometric light spectrum under leaky case and no leaky case is compared by spectrum analyzer. The testing pipe section is a three-layer anticorrosion polyethylene (PE3) external coating metal pipe with 1.2 m length. The pipe inner diameter is 100 mm and thickness is 25 mm. In the experiment, the 2 mm diameter orifice in the pipeline section is used to simulate gas leakage. The pipe working pressure is 1.0 MPa. The sensing single model fiber is fixed on the outside surface of the pipe section, and its refractive index n is 1.458. The total distributed fiber optic vibration sensor system is shown in Figure 3. The ASE light source whose maximum output power is 15.8 dBm, the spectral range is 1528–1564 nm and wavelength λ 1250–1650 nm. The Yokogawa Corp AQ6370 spectrum analyzer was used to analyze the interferometer light spectrum. With the ability to provide high-speed accurate analysis of the short wavelength range between 600 and 1700 nm, this spectrum analyzer is well suited for a broad range of applications. In test, the selected wavelength resolution is 0.02 nm of spectrum analyzer.

Figure 3.

PE3 pipe section leakage testing system diagram. PE3: three-layer anticorrosion polyethylene.

When a leakage occurs in pipe section, the leaking gas percolates down through the outside PE coating material of pipe. The range of the sensing fiber span ranges from 1.5 to 9.5 km. The intensities of interferometric light of distributed fiber optic sensing system before and after leakage are compared using spectrum analyzer directly. The selected sensing fibers lengths (L₁+L₂) are 1.5 and 3.5 km, each case includes leakage signal and no leakage signals. The measured interference signals are analyzed by AQ6370 spectrum analyzer. The interferometer light intensity around the center wavelength 1550 nm under different working conditions is intercepted and presented in Figure 4.

Figure 4.

Interference light power spectrum. (a) Light power spectrum before leakage with 1.5 km optic fiber length, (b) light power spectrum after leakage with 1.5 km optic fiber length, (c) light power spectrum before leakage with 3.5 km optic fiber length, and (d) light power spectrum after leakage with 3.5 km optic fiber length.

The interference signal optical power of pipeline without leakage is presented in Figure 4(a) and (c) when sensing fiber lengths are 1.5 and 3.5 km, respectively. In Figure 4(a), light intensity varies in the range from −16 to −25 dB with the sinusoidal wave shape. In Figure 4(c), interference light intensity ranges from −25 to −34 dB. When the sensing fiber length increases from 1.5 to 3.5 km, the light intensity attenuates about 9 dB. Comparing Figure 4(b) with (d), it is obvious that the leaky spectrum of different sensing fiber length owns different characteristics. In Figure 4(d), the attenuation of the light intensity varies in −25 to −37 dB with fluctuation sine waveform. It is clear that the wavelength of leaky signal spectrum is 1.2 times larger than original signal wavelength without leakage. Furthermore, interference light power spectrum crests and troughs exist many spiky noise. The period of the leaky signal spectrum curve is enlarged with the sensing fiber length increasing. So the light power spectrum variation is able to show whether the pipeline leakage accidents happened or not.

As illustrated in Figure 4, the light interference fringe state is stable and the light power spectrum sine wave shape is clear without leakage happened. The high pressure releasing gas causes friction with orifice and generates high frequency acoustic wave. The acoustic wave causes a distortion in the optical path length of the sensing section of the total optical fiber. Therefore, the distortion of optical paths will produce dynamic interference fringes. The spectrum has peak pulse interference around trough and crest. The period of the spectrum increases than before, and light power intensity degenerates obviously.

To validate this phenomenon, the optic fiber sensing length (L₁+L₂) is extended to 9500 m. In Figure 5, the light power intensity is reduced to −56 dB around center wavelength 1550 nm. The interference light intensity attenuates seriously, so leakage signal is not easy to be discriminated and covered by noise from outside in the measurement system.

Figure 5.

Interference light power spectrum after leakage with 9.5 km optic fiber length.

Leaky source localization

The interference light containing leaky point information will be converted into electricity through PD. Then the electricity signal is demodulated by PGC circuit (carrier frequency 300 kHz). Finally, the phase difference signal of leaky point was transformed into frequency domain and the first null frequency was captured. The leaky point position is able to be gained by equation (2). Figure 6(a) and (b) is the experimental localization results of null frequencies for various sensing fiber lengths (3500 and 9500 m, respectively) before denoising process. In the detection positioning spectrum diagram, although the downtrend is obvious, the null frequency points are covered by strong noises and the first null frequency point is obscure.

Figure 6.

The spectrum of null frequencies with difference leaky points. (a) Leakage point 3.5 km before denoising, (b) leakage point 9.5 km before denoising, (c) leakage point distance 3.5 km after denoising, and (d) leakage point distance 9.5 km after denoising.

The optic fiber sensor system SNR strongly depends on the distributed fiber optic sensor topology. In principle the noise sources belong to the light source, optical fibers, detector, electronic circuits, and environment. To study interferometric noise, we need to focus on the topology. As shown in Figure 1, the lights of path 1 and path 2 are all reflected by FRM. If the distance L₂ between leakage point and FRM is reduced, the interference signal intensity of interferometer will strengthen and SNR also be improved. Conversely, if the L₂ increased, the SNR will be decreased. The null frequency point is indistinguishable from the signal to be detected. After three-level Daubechies wavelet decomposition soft threshold denoising,^25–28 the SNR is obviously improved. The first null frequency is apparent and the identification of the null frequency was achieved in Figure 6(c) and (d). With the sensing fiber length increases from 3.5 to 9.5 km, the difference between the two null spectrum curves is obvious. It is clear that the polarization fading problem plays the negative effect on the positioning performance of distributed fiber optic vibration sensor.

In Figure 6(c), the first null frequency point is easy to be captured. While, in Figure 6(d), the null spectrum exhibits a large amount of noise and clutter when positioning, the null frequency curve is rough and irregular, and it is difficult to find the accurate null frequency point. Therefore, the positioning error of the system is unstable. In Figure 6(c), the first null frequency f_s is 14.37 kHz, the leaky distance from FRM (L₂ = c/(4nf_s)) is 3579.7 m, the leakage positioning error is 79.7 m, and relative error is about 2.3%. In Figure 6(d), there exist many spiky interference points around null frequency trough (marked by circle). Although the null frequency spectrum downtrend is obvious, the identification of the first null frequency is not easy. The measured first null frequency point f_s is 5.714 kHz, and the calculated leakage point L₂ is 9007.2 m. The leakage positioning absolute error is 492.8 m and localization relative error is about 5.2%.

Detection system sensitivity

The detection system sensitivity describes the frequency variation with range increment using $d f / d L_{x} = c / 4 {nL}_{x}^{2}$ . This distributed fiber optic vibration sensor based on mixed interferometers is inevitable to have polarization fading. So the polarization fading also has effect on the frequency–distance sensitivity. The sensitivity to the leakage is analyzed due to optic devices polarization. The 20 sets of data were obtained when the sensing fiber optic length ranges from 1.5 to 9.5 km. The results are presented in Figure 7. The null frequencies are inversely proportional to the leaky distance. As the sensing fiber length increases, the system sensitivity is reduced quickly, which is inversely proportional to $L_{2}^{2}$ . If sensing fiber length is expanded to 9.5 km, the sensitivity is 0.57 Hz/m.

Figure 7.

The relation between sensitivity and leakage location.

Conclusions

In conclusion, a sensitive fiber optic vibration sensor based on mixed interferometers for urban gas pipeline leak detection is discussed. Leakage detection experiments are performed to simulate actual urban gas pipeline leakage situation. To quantitatively analyze the polarization fading, the system polarization fading transmission model is built based on Jones matrix. Comparing the theoretical nonpolarization model with actual polarization model, we measure interference output intensity under pipeline with leakage case and without leakage case through high resolution spectrum analyzer.

The distributed fiber optic vibration sensor system allows a continuous monitoring and management of urban PE-3 coating pipeline leakage. But, in field work, the null frequency position location faces challenge with the optic fiber length increment. The interference light power spectrum is interfered by serious peak pulses around trough and crest in spectrum analyzer. When the span of fiber optic sensing length is from 1.5 to 9.5 km, the spectral intensity in 1500 nm center wavelength reduces from −16 to −56 dB. Furthermore, the average relative error of the experimental leakage point localization deteriorates from 2.3 to 5.3%.

Footnotes

Acknowledgments

Also I wish to thank Dr Thomas Schumacher for his valuable suggestions.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation of China under grant 51374188 and Zhejiang Provincial Natural Science Foundation of China under grant LR13E040001.

References

Wassef

Bassim

Houssny-Emam

, et al. Acoustic emission spectra due to leaks from circular holes and rectangular slits. J Acoust Soc Am 1985; 77: 916–923.

Zhong

Zhao

. Time-domain characterization of acoustic damping of a perforated liner with bias flow. J Acoust Soc Am 2012; 312: 271–281.

Zhao

Reyhanoglu

. Feedback control of flow disturbance transient growth in triggering thermoacoustic instability. J Sound Vib 2014; 333: 3639–3656.

Krohn

Macdougall

Mendez

. Fiber optic sensors: fundamentals and applications, 4th. Bellingham, WA: SPIE Press, 2014.

Maclean

Moran

Johnstone

, et al. Detection of hydrocarbon fuel spills using a distributed fibre optic sensor. Sens Actuat A Phys 2003; 109: 60–67.

Tanimola

Hill

. Distributed fiber optic sensors for pipeline protection. J Nat Gas Sci Eng 2009; 1: 134–143.

Thévenaz

Niklès

Fellay

, et al. Truly distributed strain and temperature sensing using embedded optical fibers. SPIE Conf Smart Struct Mater 1998; 3330: 301–314.

Motil

Bergman

Tur

. State of the art of Brillouin fiber-optic distributed sensing. Optics Laser Technol 2016; 78: 81–103.

Mirzaei

Bahrampour

Taraz

, et al. Transient response of buried oil pipelines fiber optic leak detector based on the distributed temperature measurement. Int J Heat Mass Transfer 2013; 65: 110–122.

10.

Glisic

Yao

. Fiber optic method for health assessment of pipelines subjected to earthquake-induced ground movement. Struct Health Monit 2012; 11: 696–711.

11.

Lee

Tsuda

. Fiber optic liquid leak detection technique with an ultrasonic actuator and a fiber Bragg grating. Optics Lett 2005; 30: 3293–3295.

12.

Hang

. Novel distributed optical fiber acoustic sensor array for leak detection. Opt Eng 2008; 47: 525–534.

13.

Huang

Lin

Tsai

. Fiber optic in-line distributed sensor for detection and localization of the pipeline leaks. Sens Actuat A Phys 2007; 135: 570–579.

14.

Huang

Wang

Shi

, et al. Underwater gas pipeline leakage source localization by distributed fiber-optic sensing based on particle swarm optimization tuning of the support vector machine. Appl Optics 2016; 55: 242–247.

15.

Chen

Liu

, et al. An elimination method of polarization-induced phase shift and fading in dual Mach–Zehnder interferometry disturbance sensing system. J Lightwave Technol 2013; 31: 3135–3141.

16.

Keysey

Marrone

M-J

Dandridge

. Experimental investigation of polarization- induced fading in interferometric fibre sensor arrays. Electron Lett 1991; 27: 562–563.

17.

Choi

W-S

M-S

. Accurate evaluation of polarization characteristics in the integrated optic chip for interferometric fiber optic gyroscope based on path-matched interferometry. J Opt Soc Korea 2009; 13: 439–444.

18.

Wang

Sun

, et al. Improved location algorithm for multiple intrusions in distributed Sagnac fiber sensing system. Optics Express 2014; 22: 7587–7597.

19.

Ming

Yang

Xiong

, et al. Investigation of polarization-induced fading in fiber-optic interferometers with polarizer-based polarization diversity receivers. Appl Optics 2006; 45: 2387–2390.

20.

Alam

Tokhi

. Hybrid fuzzy logic control with genetic optimisation of a single-link flexible manipulator. Eng Appl Artif Intell 2008; 21: 858–873.

21.

Zhao

. Lattice Boltzmann investigation of acoustic damping mechanism and performance of an in-duct circular orifice. J Acoust Soc Am 2014; 135: 3243–3251.

22.

Zhao

. Two-dimensional lattice Boltzmann investigation of sound absorption of perforated orifices with different geometric shapes. Aerosp Sci Technol 2014; 39: 40–47.

23.

Birks

Morkel

. Jones calculus analysis of single-mode fiber Sagnac reflector. Appl Opt 1988; 27: 3107–3113.

24.

Yanbiao

. Polarization optics, Beijing: Beijing Science Press, 2003, pp. 167–198.

25.

Abad

MRAA

Moosavian

Khazaee

. Wavelet transform and least square support vector machine for mechanical fault detection of an alternator using vibration signal. J Low Freq Noise V A 2016; 35: 52–63.

26.

Kane

Andhare

. Application of psychoacoustics for gear fault diagnosis using artificial neural network. J Low Freq Noise V A 2016; 35: 207–220.

27.

Jiaqiang

Qian

Liu

, et al. Design of the H∞ robust control for the piezoelectric actuator based on chaos optimization algorithm. Aerosp Sci Technol 2015; 47: 238–246.

28.

Liu

C-C

Ren

C-B

Liu

, et al. Optimal control of nonlinear resonances for vehicle suspension using linear and nonlinear control. J Low Freq Noise V A 2013; 32: 335–345.