Development of a multi-point polarization-based vibration sensor

doi:10.1364/OE.21.005606

« Show journal navigation

Development of a multi-point polarization-based vibration sensor

Nicolas Linze, Pierre Tihon, Olivier Verlinden, Patrice Mégret, and Marc Wuilpart »View Author Affiliations

Optics Express, Vol. 21, Issue 5, pp. 5606-5624 (2013)
http://dx.doi.org/10.1364/OE.21.005606

View Full Text Article

Acrobat PDF (2744 KB)

Journal Search
Article Lookup

Article Tools

Citations

Alert me when this article is cited
Export Citation/Save

Full Text
Article Info
References (27)
Cited By
Figures (15)
Metrics

Abstract

In this paper we propose a novel kind of multi-point vibration sensor based on the polarization properties of light. Its principle relies on the combination of mechanical transducers with fiber Bragg gratings. When subject to vibrations, the mechanical transducers induce birefringence variations within the fiber and in turn modify the state of polarization, which appears as a power variation after going through a polarizer. The FBGs reflect light from different positions of the sensing fiber and provide wavelength multiplexing. We show that this sensor can provide the vibration frequencies in a quasi-distributed manner.

1. Introduction

Vibrations are of high importance as they are a health and ageing indicator of civil structures (bridges, buildings, dams for instance) and industrial machines (e.g. rotating machines). Changes in the vibration spectrum can be considered as alarm signals for problems within the structure. With vibration sensors it is thus possible to prevent damages and avoid serious consequences such as collapses and possible injuries. Many kinds of vibration sensors are already available and are based on mechanical sensors (piezoelectric and piezoresistive accelerometers, velocity and displacement sensors, for instance). For two decades there has been great research interest in optical vibration sensors. Compared to the mechanical sensors, they have several advantages, inherent to the use of the fiber: they can provide quasi-distributed (multi-point) and distributed (continuous) information along the optical fiber with only one interrogating element. This greatly reduces the wiring complexity since the classical methods involve as many wires as measurement points. Optical fiber sensors are also usable in harsh environments, in which conventional sensors cannot be used, such as EM-disturbed, humid, nuclear and high temperature media.

Optical vibration sensors are based on the modulation of a light property induced by the applied vibration. A majority of the vibrations sensors that have been developed is based on fiber Bragg grating (FBG) technology [1

X. Guo, Z. Yin, and N. Song, “Measuring vibration by using fiber Bragg grating and demodulating it by blazed grating,” Chinese Opt. Lett. 2, 393–395 (2004).

–3

Y. Weng, X. Qiao, T. Guo, M. Hu, Z. Feng, R. Wang, and J. Zhang, “A robust and compact fiber Bragg grating vibration sensor for seismic measurement,” IEEE Sens. J. 12, 800–804 (2011) [CrossRef] .

, 6

T. Guo, L. Shao, H-Y. Tam, P. A. Krug, and J. Albert, “Tilted fiber grating accelerometer incorporating an abrupt biconical taper for cladding to core recoupling,” Opt. Express 17, 20651–20660 (2009) [CrossRef] [PubMed] .

]. In these systems the measurement of the vibration is made possible by the modulation of the Bragg wavelength by the transduction. This transduction is mainly based on the bending of a structure on which an FBG is placed [1

X. Guo, Z. Yin, and N. Song, “Measuring vibration by using fiber Bragg grating and demodulating it by blazed grating,” Chinese Opt. Lett. 2, 393–395 (2004).

–3

] or on the movement of one or two flexural beams [4

Y. Zhu, P. Shum, C. Lu, B. M. Lacquet, P. L. Swart, and S. J. Spammer, “Temperature-insensitive fiber Bragg grating accelerometer,” IEEE Photon. Technol. Lett. 15, 1437–1439 (2003) [CrossRef] .

, 5

M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, “Flexural beam-based fiber Bragg grating accelerometers,” IEEE Photon. Technol. Lett. 10, 1605–1607 (1998) [CrossRef] .

]. Some systems are also based on the recoupling of cladding modes into the core [6

]. Another important family of vibration sensors is based on interferometry techniques [7

K. Wada, H. Narui, D. Yakamoto, T. Matsuyama, and H. Horinaka, “Balanced polarization maintaining fiber Sagnac interferometer vibration sensor,” Opt. Express 19, 21467–21474 (2011) [CrossRef] [PubMed] .

–12

Y. Lu, T. Zhu, L. Chen, and X. Bao, “Distributed vibration sensor based on coherent detection of phase-OTDR,” J. Lightwave Technol. 28, 3243–3249 (2010).

]. In these systems the vibration has an impact on the phase of the optical signal and can be detected, among others, with interferometers based on a Sagnac [7

, 8

X. Fang, “Fiber-optic distributed sensing by two-loop Sagnac interferometer,” Opt. Lett. 21, 444–446 (1996) [CrossRef] [PubMed] .

], a Michelson [9

X. Hong, J. Wu, C. Zuo, F. Liu, H. Guo, and K. Xu, “Dual Michelson interferometers for distributed vibration detection,” Appl. Opt. 50, 4333–4338 (2011) [CrossRef] [PubMed] .

] or a Fabry-Perot configuration [10

R. M. Manuel, M. G. Shlyagin, and S. V. Miridonov, “Location of a time-varying disturbance using an array of identical fiber-optic interferometers interrogated by CW DFB laser,” Opt. Express 16, 20666–20675 (2008) [CrossRef] [PubMed] .

]. The vibration can also be detected with reflectometry techniques measuring the variation of the optical phase [11

J. C. Juarez and H. F. Taylor, “Field test of a distributed fiber-optic intrusion sensor system for long perimeters,” Appl. Opt. 46, 1968–1971 (2007) [CrossRef] [PubMed] .

, 12

Y. Lu, T. Zhu, L. Chen, and X. Bao, “Distributed vibration sensor based on coherent detection of phase-OTDR,” J. Lightwave Technol. 28, 3243–3249 (2010).

]. Current research also focuses on Brillouin sensors, in which the Brillouin frequency is modified by a dynamic longitudinal strain induced by the vibration. The main challenge in these systems is the conception of a fast measurement scheme [13

K. Hotate and S. S. L. Ong, “Distributed dynamic strain measurement using a correlation-based Brillouin sensing system,” IEEE Photon. Technol. Lett. 15, 272–274 (2003) [CrossRef] .

–15

R. Bernini, A. Minardo, and L. Zini, “Dynamic strain measurement in optical fibers by stimulated Brillouin scattering,” Opt. Lett. 34, 2613–2615 (2009) [CrossRef] [PubMed] .

]. Some other systems are based on optical power modulation by micro-bending [16

F. Qin, H. Li, W. Fan, and Q. Sheng, “Experimental study on vibration frequency response of micro-bend optic-fiber sensor,” Chinese Opt. Lett. 7, 556–559 (2009) [CrossRef] .

]. Finally some systems are based on the state of polarization (SOP) of light [17

Z. Zhang and X. Bao, “Continuous and damped vibraton detection based on fiber diversity detection sensor by Rayleigh backscattering,” J. Lightwave Technol. 26, 832–838 (2008) [CrossRef] .

–20

N. Linze, P. Tihon, O. Verlinden, P. Mégret, and M. Wuilpart, “Linearity considerations in polarization-based vibration sensors,” Appl. Opt. 51, 6997–7004 (2012) [CrossRef] [PubMed] .

], whose one approach is Polarization Optical Time Domain Reflectometry (POTDR) [21

M. Wuilpart, P. Mégret, M. Blondel, A. J. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett. 13, 836–838 (2001) [CrossRef] .

In the multi-point vibration sensor presented in this paper, the state of polarization is used as the sensitive optical parameter. The main advantages of this parameter are its ability to provide multi-point information and its sensitivity to external perturbations. Moreover the optical system has not to be as complex and expensive as interferometers (for instance, no need of a very narrow linewidth laser). Another advantage, principally compared to FBGs, relies on the transduction efficiency. In polarimetric systems the transduction can be made through the crushing of the fiber [20

N. Linze, P. Tihon, O. Verlinden, P. Mégret, and M. Wuilpart, “Linearity considerations in polarization-based vibration sensors,” Appl. Opt. 51, 6997–7004 (2012) [CrossRef] [PubMed] .

], perpendicularly to the propagation axis z. In the case of FBGs, the effect of a crushing is very weak because it has to generate an observable Bragg wavelength shift, which is the consequence of strain in the longitudinal axis z. With FBGs, transductions based on stretching or bending are preferably used but face practical implementation difficulties and are cumbersome.

The general principle of a polarimetric vibration sensor is as follows: a vibration modifies the birefringence properties of the fiber and induces an SOP modification. The use of a polarizer then transforms this SOP variation into a power modification, easily measurable on an oscilloscope. Different polarimetric vibration sensors have been previously developed [17

Z. Zhang and X. Bao, “Continuous and damped vibraton detection based on fiber diversity detection sensor by Rayleigh backscattering,” J. Lightwave Technol. 26, 832–838 (2008) [CrossRef] .

, 18

Z. Zhang and X. Bao, “Distributed optical fiber vibration sensor based on spectrum analysis of polarization-OTDR system,” Opt. Express 16, 10240–10247 (2008) [CrossRef] [PubMed] .

]. The limitation in these papers is that the birefringence modifications do not correspond to practical perturbations: the SOP modifications are indeed simulated either by rotating a polarization controller or as the stretching of a fiber section wounded around a piezoelectrical crystal (PZC).

In this paper, vibrations are induced via the crushing of an optical fiber on a 3 mm length, which makes the developed configuration easy to install in practice, comparatively to the sensors described in [17

Z. Zhang and X. Bao, “Continuous and damped vibraton detection based on fiber diversity detection sensor by Rayleigh backscattering,” J. Lightwave Technol. 26, 832–838 (2008) [CrossRef] .

, 18

Z. Zhang and X. Bao, “Distributed optical fiber vibration sensor based on spectrum analysis of polarization-OTDR system,” Opt. Express 16, 10240–10247 (2008) [CrossRef] [PubMed] .

]. With these sensors, accelerations as low as 1 m/s² can be detected. Moreover, a multiplexing interrogating scheme allows to interrogate several sensors along the fiber and to provide multi-point information, while the sensor described in [19

G. Rajan, M. Ramakrishnan, Y. Semenova, A. Domanski, A. Boczkowska, T. Wolinski, and G. Farrell, “Analysis of vibration measurements in a composite material using an embedded PM-PCF polarimetric sensor and an FBG sensor,” IEEE Sens. J. 12, 1365–1371 (2012) [CrossRef] .

] only consists of point measurements. The paper is divided as follows : in the first section the working principle of the proposed sensor is given. In the second section the proposed vibration sensor is simulated so as to assess the experimental results, which are given in the third section. The fourth section gives discussions about these experimental results. Finally the last section gives performance values.

2. Working principle of the proposed vibration sensor

The proposed multi-point vibration sensor is depicted in Fig. 1. Light is emitted from a broadband source (here, an Amplified Spontaneous Emission source, ASE, with a range extending from 1515 nm to 1575 nm) and is launched into the 2 meter long fiber under test (FUT) through a circulator and a linear polarizer which fixes the input SOP. The polarizer extinction ratio is higher than 25 dB. We here consider the case of three measurement points (sensors S_i, i=1,2,3).

Fig. 1 Experimental setup (plain lines: optical links; dashed lines: electrical links, FUT: fiber under test; MT: mechanical transducer; S_i: sensor i; FBG: fiber Bragg grating.

During its propagation along the fiber, the lightwave is successively reflected by 3 FBGs (which are not used here as sensors but as reflecting devices only) with different Bragg wavelengths λ_i (i=1,2,3, depending on which sensor is analyzed), each of these being separated in our experiments by 20 nm to avoid crosstalk between the different FBGs (λ₁=1571.5 nm, λ₂=1550.6 nm and λ₃= 1530.9 nm). As the proposed sensor is based on wavelength multiplexing, the FBGs can be highly reflective and the sensors can be as spatially close as possible to each other. As shown in Fig. 1, each of these FBGs is placed after one mechanical transducer (MT), which is depicted in Fig. 2.

Fig. 2 Mechanical transducer used to induce birefringence into the fiber.

This MT is made of a rigid base and a vibrating beam. The base is screwed on the vibration source (shaker, in our case) while the vibrating beam is free to move at one of its ends (the other extremity is screwed on the base). A second screw, whose edge is covered with rubber and is in contact with the optical fiber, is placed at the free extremity. Note that the direction of this second screw is transversal to the fiber propagation axis z. When the vibration source vibrates perpendicularly to the fiber propagation axis z, the inertia of the upper beam, via the presence of the second screw, induces a crushing of the optical fiber over a 3 mm length and modifies the birefringence. The transduction generates a linear birefringence δβ equal to [22

S. C. Rashleigh, “Origins and control of polarization effects in single-mode fibers,” J. Lightwave Technol. 1, 312–331 (1983) [CrossRef] .

δ β = 2 k n_{0}^{3} (p_{11} - p_{12}) (1 + ν_{P}) \frac{f}{π b E}

(1)

The phase retardance δ, i.e. the accumulated phase delay δβ along the crushed length L, is equal to:

δ = 2 k n_{0}^{3} (p_{11} - p_{12}) (1 + ν_{P}) \frac{f}{π b E} L = K f

(2)

In Eqs. 1 and 2 k = 2π/λ, λ is the light wavelength in vacuum, n₀ is the effective refractive index (about 1.45), p₁₁ and p₁₂ are the elasto-optic coefficients of silica (p₁₁–p₁₂=−0.14 [23

A. Bertholds and R. Dandliker, “Determination of the individual strain-optic coefficients in single mode optical fibre,” J. Lightwave Technol. 6, 17–20 (1988) [CrossRef] .

, 24

A. Galtarossa, D. Grosso, and L. Palmieri, “Accurate characterization of twist-induced optical activity in single-mode fibers by means of polarization-sensitive reflectometry,” IEEE Photon. Technol. Lett. 21, 1713–1715 (2009) [CrossRef] .

]), ν_P is the silica Poisson’s ratio (0.17), b is the outer radius of the fiber (125 μm), E is the silica Young’s modulus (70 GPa), f is the applied force per unit of length (N/m) and L corresponds to the crushed length, 3×10⁻³ m. This equation shows that the birefringence varies with the applied force and thus, to the rhythm of the vibration. This birefringence then induces a SOP modification. The signals reflected from the 3 FBGs are guided back to the circulator through the linear polarizer (now used as an analyzer) which transforms the SOP variation induced by the MTs into a power variation. All these contributions then reach one tunable optical filter which is adjusted such that it only selects one of the three reflected signals. The filtered signal is finally converted in the electrical domain via a photodiode and displayed on an oscilloscope. It clearly appears that this sensor measures the vibrations in a quasi-distributed manner as the optical filter selects only one of the three selected signals. Moreover, at the input of the optical filter, the power of the three reflected signals varies to the rhythm of the vibrations due to the presence of the analyzer. This ability is depicted in Figs. 3(a) – 3(c).

Fig. 3 FFTs of the three reflected signals (FBG₁ (a); FBG₂ (b) and FBG₃ (c)).

We experimentally applied a 160 Hz sine vibration on MT₁, a 110 Hz sine vibration on MT₂ and a 210 Hz sine vibration on MT₃. This figure gives the frequency spectrum of the three converted signals. From this figure, we can see that the signal reflected by FBG₁ has a contribution at 160 Hz, as this frequency is applied on MT₁ [Fig. 3(a)]. The signal reflected by FBG₂ has a contribution at 110 Hz, as this frequency is applied on MT₂ [Fig. 3(b)] and the signal reflected by FBG₃ has a contribution at 210 Hz, as we have applied this frequency on MT₃ [Fig. 3(c)]. However we also observe that the signal reflected by FBG₂ has also a contribution at 160 Hz while this frequency is only applied on S₁. Similarly the signal reflected by FBG₃ has contributions at 160 and 110 Hz while these frequencies are applied on S₁ and S₂ respectively. These artefacts are the consequence of the SOP modification. Indeed when the SOP is modified at a position z of the fiber, it is also modified between z and the fiber end. This means that if we had effectively applied a 160 Hz vibration on S₂ (or S₃), this vibration could not have been distinguished from an artefact resulting from a previous sensor. Note that this problem has also been observed in another polarimetric vibration sensor [18

Z. Zhang and X. Bao, “Distributed optical fiber vibration sensor based on spectrum analysis of polarization-OTDR system,” Opt. Express 16, 10240–10247 (2008) [CrossRef] [PubMed] .

]. To distinguish artefacts from really applied vibrations, as shown in Fig. 4, we now use a second FBG situated before each MT (except for MT₁ as the frequency spectrum of the vibration applied on S₂ is univocally deduced from the measurement of FBG₂) and Fig. 1 becomes Fig. 4. As it can be observed, except for S₁, two FBGs now surround each MT. Their spatial and wavelength separations are 30 cm and 10 nm, respectively. The Bragg wavelengths are respectively λ₁₂ = 1571.5 nm, λ₂₁ = 1560.8 nm, λ₂₂ = 1550.6 nm, λ₃₁ = 1540.3 nm and λ₃₂ = 1530.9 nm.

Fig. 4 Experimental setup (plain lines: optical links ; dashed lines: electrical links, FUT: fiber under test ; MT: mechanical transducer ; S_i: sensor i ; FBG: fiber Bragg grating ; OSC: oscilloscope).

The distinction between artefacts and really applied vibrations, at a specific sensor S_i (i=2,3), is realized by comparing the phase shift between the electrical signals corresponding to FBG_i1 and FBG_i2. We now use two tunable optical filters to select the signals reflected by the two FBGs surrounding the mechanical transducer MT_i. If there is no vibration between these FBGs and if a frequency is applied on one of the previous sensors, the phase shift between these two signals only corresponds to the time delay induced between the two FBGs. As this delay is very weak (the spatial separation is 30 cm, the round-trip time delay is equal to 3 ns, which is much smaller than the period of the vibration signals), the two reflected optical signals can be considered to have identical phases. On the other hand if there is a vibration the SOP is modified and the phases are no longer similar. The relation between the phase of the electrical signal and the SOP can be understood with the following example, depicted in Fig. 5 and presenting simulation results in two different conditions. Figure 5(a) shows the evolutions (in the electrical domain) of the signals reflected by FBG₂₁ and by FBG₂₂, when a 120 Hz vibrations is applied on MT₁ only. Because of this vibration the signal reflected by FBG₂₁ varies to the rhythm of this vibration, as well as the signal reflected by FBG₂₂. As there is no vibration on MT₂, the SOP does not change between these two FBGs and the evolution of the power transmitted through the polarizer does not change either, which explains why the signals are in phase. Figure 5(b) shows the evolutions (in the electrical domain) of the signals reflected by these FBGs when 120 Hz vibrations are applied both on MT₁ and on MT₂. As in the previous situation [Fig. 5(a)], the signal reflected by FBG₂₁ varies to the rhythm of the vibration applied on MT₁. The signal reflected by FBG₂₂ also varies to the rhythm of this vibration applied on MT₁. However contrary to the first situation, the SOP (at λ₂₂) is now also modified by the vibration applied on MT₂ (which has the same frequency). The transmitted power through the polarizer then varies in a sinusoidal way but at a given instant the value of this transmitted power is different because the SOP has been modified by the vibration applied on MT₂ and is then different. Over a vibration period this leads to a signal varying at the same frequency but phase shifted from the signal reflected by FBG₂₁.

Fig. 5 Evolution of the signals reflected by FBG₂₁ and FBG₂₂ when: (a) a 120 Hz vibration is applied on S₁; (b) 120 Hz vibrations are applied on MT₁ and MT₂.

However, during its propagation between FBG_i1 and FBG_i2 (in the example, i=2), the light SOP can be modified via other effects than the vibration:

It can be modified when propagating along the optical fiber due to the presence of intrinsic birefringence Δn_int and polarization mode coupling, as shown in [21
M. Wuilpart, P. Mégret, M. Blondel, A. J. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett. 13, 836–838 (2001) [CrossRef] .
].
It can also be modified when being transmitted and reflected by the FBGs [25
S. Bette, C. Caucheteur, M. Wuilpart, and P. Mégret, “Spectral characterization of differential group delay in uniform fiber Bragg gratings,” Opt. Express 13, 9954–9960 (2005) [CrossRef] [PubMed] .
].
It can be modified by inadvertently twisting the fiber [22
S. C. Rashleigh, “Origins and control of polarization effects in single-mode fibers,” J. Lightwave Technol. 1, 312–331 (1983) [CrossRef] .
].
The phase retardance δ_int due to the intrinsic birefringence Δn_int and in turn the SOP evolution depend on the wavelength: δ_int = 2πΔn_intL/λ, where Δn_int is the intrinsic birefringence (refractive index difference) and L is the length of the birefringent section. Concretely, this means that if the separation between the wavelengths is too large, the SOP evolutions of these two wavelengths along the optical fiber (not only between the two FBGs) will be different. Note that the phase retardance due to the applied vibration (Eq. (2)) also depends on the wavelength but as the crushed length is small (3 mm), this dependance can be assumed to be weak.

These undesired SOP variations will cause false detections in case of identical frequencies and have to be reduced as much as possible. Therefore:

to avoid SOP modifications during the propagation, we use a Low-Birefringence fiber whose intrinsic birefringence Δn_int is equal to 4×10⁻⁹, i.e. 25 times smaller than in a conventional singlemode fiber. In addition, we use a short distance between the FBGs (30 cm), this means that the SOP does not practically change between two adjacent FBGs.
to avoid modifications during the transmissions and reflections, the FBGs have been inscribed so that their birefringence is as weak as possible. Their birefringence is here equal to Δn_FBG= 5×10⁻⁶.
great care is taken in order to avoid twist (note that by properly designing each sensor, this effect can be avoided).
the wavelength separation between two adjacent FBGs (10 nm) is such that the SOP evolutions during their propagation along the fiber are not significantly different for two adjacent wavelengths.

These wavelength, birefringence and distance parameters ensure that the depolarisation process is weak. This has been experimentally verified by measuring the degree of polarization (DOP) of the light reflected by FBG₃₂ (most remote FBG). This measured DOP is equal to 0.9, which shows that the depolarisation is weak.

3. Simulations

In order to assess the experimental results and the analysis method described in section 4, the developed sensor has been simulated by using the Stokes formalism for modelling the different parts of the system. In this formalism, the SOPs are modelled by four-dimensional vectors and the optical devices are represented by 4×4 matrices. The propagation and modification of the different SOPs along the optical fiber are obtained by matrix products. Figure 6 shows that there are three different parts in the set-up that have to be modelled: the fiber sections (dashed lines), the mechanical transducers (black dots) and the fiber Bragg gratings (vertical plain lines) FBG_ij (i=1,2,3; j=1,2).

Fig. 6 Simulation of the proposed vibration sensor.

The simulation consists in calculating, for the different Bragg wavelengths λ_ij, the Stokes vectors at different places along the fiber in order to eventually obtain the optical power transmitted through the analyzer P_out,_λ_ij, when vibrations are applied on the different MTs. This can be summarized as follows:

The input SOP S_in,λij is a linear polarization state described by a Stokes vector with an azimuth φ such that, if P_in is the input power [26
R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
]:

$S_{in, λ i j} = (\begin{matrix} 1 \\ cos 2 φ \\ sin 2 φ \\ 0 \end{matrix}) P_{in}$
(3)
The first component of a Stokes vector corresponds to the optical power. This input SOP is fixed by the input polarizer. The Mueller matrix M_Pol,θ of a linear polarizer oriented at θ radians with respect to the reference horizontal x axis is given by:

$M_{Pol, θ} = \frac{1}{2} (\begin{matrix} 1 & cos 2 θ & sin 2 θ & 0 \\ cos 2 θ & {cos}^{2} 2 θ & sin 2 θ cos 2 θ & 0 \\ sin 2 θ & sin 2 θ cos θ & {sin}^{2} 2 θ & 0 \\ 0 & 0 & 0 & 0 \end{matrix})$
(4)
The different fiber sections M_fiber between the different optical devices (represented as dashed lines in Fig. 6) are modelled as linear retarders (we suppose there is no twist), such that [26
R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
]:

$M_{Fiber} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & {cos}^{2} \frac{δ_{int}}{2} + {sin}^{2} \frac{δ_{int}}{2} cos 4 q & {sin}^{2} \frac{δ_{int}}{2} sin 4 q & - sin δ_{int} sin 2 q \\ 0 & {sin}^{2} \frac{δ_{int}}{2} sin 4 q & {cos}^{2} \frac{δ_{int}}{2} - {sin}^{2} \frac{δ_{int}}{2} cos 4 q & sin δ_{int} cos 2 q \\ 0 & sin δ_{int} sin 2 q & - sin δ_{int} cos 2 q & cos δ_{int} \end{matrix})$
(5)
In this matrix, q is the azimuth of the fastest eigenmode and is randomly chosen as the exact orientation of the fiber compared to a reference x axis is unknown, the phase retardance due to the intrinsic birefringence δ_int is equal to δ_int = 2πΔn_intL_s/λ with L_s, the length of the fiber section, λ, the wavelength in vacuum and Δn_int the fiber intrinsic birefringence (equal to 4×10⁻⁹ as a low-birefringence fiber is used).
The mechanical transducers (black dots) have for effect, as already mentioned, to crush the optical fiber and as we suppose it induces no twist, the MTs can then be modelled as linear retarders (Eq. (5)) [26
R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
]:

$M_{Sensor} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & {cos}^{2} \frac{δ_{i}}{2} + {sin}^{2} \frac{δ_{i}}{2} cos 4 q & {sin}^{2} \frac{δ_{i}}{2} sin 4 q & - sin δ_{i} sin 2 q \\ 0 & {sin}^{2} \frac{δ_{i}}{2} sin 4 q & {cos}^{2} \frac{δ_{i}}{2} - {sin}^{2} \frac{δ_{i}}{2} cos 4 q & sin δ_{i} cos 2 q \\ 0 & sin δ_{i} sin 2 q & - sin δ_{i} cos 2 q & cos δ_{i} \end{matrix})$
(6)
Where δ_i is the phase retardance induced by the vibration applied on the sensor i and has the expression given by Eq. (2), in which the applied force per unit of length f_i depends on the applied acceleration level a_i (i=1,2,3) by the following equation [20
N. Linze, P. Tihon, O. Verlinden, P. Mégret, and M. Wuilpart, “Linearity considerations in polarization-based vibration sensors,” Appl. Opt. 51, 6997–7004 (2012) [CrossRef] [PubMed] .
]:

$f_{i} = \frac{m}{L} a_{i}$
(7)
L is the crushed length (3 mm) and m is the total mass fixed to the vibrating beam and includes the screw, the rubber edge, and the part of the beam that crushes the fiber. This total mass is around 4×10⁻³ kg. In this equation we also neglect the influence of intrinsic birefringence δ_int (at 1 m/s², the intrinsic birefringence is 10 times weaker than the induced birefringence δ_i).
The matrices representing the reflection and transmission mechanisms of the FBGs are given in [25
S. Bette, C. Caucheteur, M. Wuilpart, and P. Mégret, “Spectral characterization of differential group delay in uniform fiber Bragg gratings,” Opt. Express 13, 9954–9960 (2005) [CrossRef] [PubMed] .
]. The FBGs parameters are as follows: their birefringence Δn_FBG is 5×10⁻⁶, their refractive index modulation is 5×10⁻⁴ and their length is 3 mm. The five FBGs have the Bragg wavelengths given in the previous section.
During their propagation along the optical fiber, the different wavelengths λ_ij are reflected by the different FBGs FBG_ij (i=1,2,3 ; j=1,2). These wavelengths then go backward to the analyzer. The matrix corresponding to the propagation backwards of a specific wavelength $\overset{\leftarrow}{M_{λ i j}}$ can be deduced from the forward matrices $\vec{M_{λ i j}}$ by the following relation [27
M. Wuilpart, “Rayleigh Scattering in Optical Fibers and Applications to Distributed Measurements,” in Advanced Fiber Optics: Concepts and Technology , L. Thévenaz, eds. (EPFL Press, 2011), pp. 207–262.
]:

$\overset{\leftarrow}{M_{λ i j}} = M_{S} {\vec{M_{λ i j}}}^{T} M_{S}$
(8)
In this equation M_S is such that [27
M. Wuilpart, “Rayleigh Scattering in Optical Fibers and Applications to Distributed Measurements,” in Advanced Fiber Optics: Concepts and Technology , L. Thévenaz, eds. (EPFL Press, 2011), pp. 207–262.
]:

$M_{S} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix})$
(9)
The forward matrix $\vec{M_{λ i j}}$ is equal to the product of the Mueller matrices of the devices that interact with λ_ij during the forward propagation.
Knowing all these matrices, it is possible to find the SOP at the analyzer input S_rt,_λ_ij as:

$S_{r t, λ i j} = \overset{\leftarrow}{M_{λ i j}} \vec{M_{λ i j}} S_{in, λ i j} = M_{S} {\vec{M_{λ i j}}}^{T} M_{S} \vec{M_{λ i j}} S_{in, λ i j}$
(10)
Once S_rt,_λ_ij has been calculated, it is possible to calculate the optical power transmitted through the analyzer P_out,_λ_ij as S_out,_λ_ij(1), i.e. the first component of the SOP transmitted through the analyzer S_out,_λ_ij, which is given by:

$S_{out, λ i j} = M_{Pol, θ} S_{r t, λ i j}$
(11)

Note that this multi-step process gives the total power P_out at a specific time t, corresponding to specific values of a_i(t) (i=1,2,3). In order to recover the output power evolution, this process has to be repeated so as to simulate several periods of the applied vibrations a_i(t). Moreover, as the system is based on wavelength multiplexing, this process has to be repeated for the five Bragg wavelengths λ_ij, taking into account their respective paths.

This process can be illustrated for λ₁₂. The transmitted optical power P_out,_λ₁₂ corresponds to the first component of the Stokes vector S_out,_λ₁₂, which is given by:

S_{out, λ 12} = M_{Pol, θ} S_{r t, λ 12} = M_{Pol, θ} M_{S} {\vec{M_{λ 12}}}^{T} M_{S} {\vec{M_{λ 12}}}^{T} S_{in, λ 12}

(12)

In this equation

\vec{M_{λ 12}}

is equal to:

\vec{M_{λ 12}} = M_{F B G 12, r} M_{fiber 2} M_{Sensor 1} M_{fiber 1}

(13)

In this equation M_fiber1, M_fiber2 and M_fiber3 are the Mueller matrices corresponding to the fiber sections between the polarizer and MT₁ and between MT₁ and FBG₁₂, respectively. M_Sensor1 is the Mueller matrix corresponding to the first mechanical transducer (MT₁). M_FBG12,r represents the effect of the reflection by FBG₁₂. At the end of the whole process the simulation program gives the time evolution of the output power P_out,λij(t) reflected by the different Bragg wavelengths λ_ij, when different vibrations a_i(t) are applied on the mechanical transducers MT_i. It also calculates the FFT of these different signals and compares in phase the signals coming from FBG_i1 and FBG_i2 (i=1,2,3). The situation depicted in Fig. 7 has been simulated. A 300 Hz sine vibration is applied on MT₁, 175 Hz and 240 Hz sine vibrations are applied on MT₂ and 175 Hz and 300 Hz sine vibrations are applied on MT₃. The acceleration amplitudes are 30 m/s². The magnitude spectra obtained are given in Figs. 8(a) – 8(e).

Fig. 7 Multi-point sensing example: 300 Hz on MT₁, 175 Hz and 240 Hz on MT₂, 175 Hz and 300 Hz on MT₃. The acceleration amplitudes are 30 m/s².

Fig. 8 Magnitude spectra of the signals coming from FBG₁₂ (a), FBG₂₁ (b), FBG₂₂ (c), FBG₃₁ (d) and FBG₃₂ (e) (simulation results).

The signal coming from FBG₁₂, depicted in Fig. 8(a), has a contribution at 300 Hz, as this frequency is applied on MT₁. As there is no new vibration between MT₁ and MT₂, the signal reflected by FBG₂₁ [Fig. 8(b)] has a spectrum similar to that of FBG₁₂, i.e. a contribution at 300 Hz. The signal reflected by FBG₂₂ [Fig. 8(c)] has two additional frequency contributions as 175 Hz and 240 Hz sine vibrations are applied on MT₂. However, as a 300 Hz vibration is applied on MT₁, this frequency is also observed in the FBG₂₂ frequency spectrum. The signal reflected by FBG₃₁ [Fig. 8(d)] has a frequency spectrum similar to that of FBG₂₂ as no mechanical transducer is situated in between. Finally, the signal from FBG₃₂ [Fig. 8(e)] has contributions at 175 Hz and 300 Hz, as these vibrations are applied on MT₃ but also at 240 Hz as this vibration is applied on MT₂.

From the magnitude spectra, we can observe that, due to the modification of the SOP at the different MTs, the frequencies of the applied vibrations cannot generally be univocally deduced (except if it is the first occurrence of these frequencies). They also show that other frequency contributions, which happen to result from intermodulation products between the applied frequencies, are present in the measured spectra. These additional frequencies are the result of the multi-sine aspect of the power variation. This multi-sine aspect is mathematically explained by the product between the different Mueller matrices representing the mechanical transducers, given in Eq. (6), in which the term δ_i varies to the rhythm of the vibration f_i(t). Note that periodical disturbances below curves are the result of the use of a rectangular window for the FFT calculus.

As proposed in the previous section, in order to distinguish really applied vibrations from artefacts, we calculate, for each frequency contained in the frequency spectrum, the phase shift between the signals coming from FBG_i1 and FBG_i2 (i=2,3). Table 1 gives the different phase shifts Δϕ for the three applied frequencies. Note that for some frequencies, the calculation of the phase shift is not needed (not relevant) as the vibration frequency is either not present or unequivocally determined from the magnitude spectra.

Table 1 Frequencies at the different MTs (Not relevant : the phase shift has not to be calculated; Δϕ: the phase shift has to be calculated in order to deduce if this frequency is really applied) (simulation results)

	175 Hz	240 Hz	300 Hz
MT₂ (FBG₂₁ – FBG₂₂)	Not relevant	Not relevant	Phase shift = 0.54°
MT₃ (FBG₃₁ – FBG₃₂)	Δϕ = 44.54°	Δϕ = 0.15°	Δϕ = 54.75°

For instance, for MT₂, the presence of the 175 Hz and 240 Hz contributions are determined from the magnitude spectrum given in Fig. 1(d), when comparing to Fig. 8(c) and calculating the phase shift is not relevant. On the contrary, as a 300 Hz vibration is also applied on MT₁, it is not possible, from the magnitude spectra, to know if this frequency is really applied on MT₂ or if it is an artefact. The phase difference between the signals coming from FBG₂₁ and FBG₂₂ at that frequency was calculated to be equal to 0.54°, which is practically in phase. The difference with the theoretical value of 0° comes from the fact that the SOP is slightly modified during its propagation by the different devices (intrinsic birefringence of fiber sections and FBGs). As it will be shown in the following section a threshold will be defined. We then deduce that no vibration at 300 Hz is applied on MT₂. For MT₃, we have to calculate the phase shift for the three frequencies of interest. Table 2 shows that the phase shift at 240 Hz is close to 0° while the phase shifts at 175 Hz and 300 Hz are totally different from 0° (around 44.54° and 54.72°). We can deduce that the 175 Hz and 300 Hz frequencies are applied on MT₃ while the 240 Hz frequency is not present.

Table 2 Frequencies at the different MTs (Not relevant : the phase shift has not to be calculated; Phase shift: the phase shift has to be calculated in order to deduce if this frequency is really applied)

	175 Hz	240 Hz	300 Hz
MT₂ (FBG₂₁ – FBG₂₂)	Not relevant	Not relevant	Δϕ = −2°
MT₃ (FBG₃₁ – FBG₃₂)	Δϕ = 100°	Δϕ = 1°	Δϕ = −120°

Figure 8 and Table 1 together show that combining the magnitude spectra with the phase shifts at the different frequencies contained in the spectra allows to know which vibration frequencies are applied on the different mechanical transducers. The following section is dedicated to the experimental validation of this method.

4. Experimental results

The experiments consisted in implementing the situation depicted in Fig. 7 and validating the proposed method. As a reminder, in this example, a 300 Hz sine vibration is applied on MT₁, while multi-sine vibrations are applied both on MT₂ and MT₃: respectively 175 Hz and 240 Hz on MT₂, and 175 Hz and 300 Hz on MT₃. The Bragg wavelengths are respectively λ₁₂ = 1571.5 nm, λ₂₁ = 1560.8 nm, λ₂₂ = 1550.6 nm, λ₃₁ = 1540.3 nm and λ₃₂ = 1530.9 nm. The FBGs are 30 cm apart from each other. The five reflected time signals and their magnitude spectra are given in Figs. 9 and 10, respectively. These two figures show the multi-sine aspect of the measured signals. From Fig. 10 we can observe that these five spectra have the same aspect as in the simulations [Fig. 8].

Fig. 9 Measured electrical signals corresponding to FBG₁₂ (a), FBG₂₁ (b), FBG₂₂ (c), FBG₃₁ (d) and FBG₃₂ (e).

Fig. 10 Magnitude spectra of the signals coming from FBG₁₂ (a), FBG₂₁ (b), FBG₂₂ (c), FBG₃₁ (d) and FBG₃₂ (e).

To know if these frequencies are really applied, we need to calculate, at these three frequencies, the phase shift between the electrical signals corresponding to the signals reflected by FBG_i1 and FBG_i2 (i=2,3). These phase shifts are given in Table 2. As in the simulations, it is not relevant to calculate the phase shift for all frequencies as we can deduce univocally the presence of some of them. As we can see, when the frequencies are really applied, the phase shift is different from 0° while this phase shift is close to 0° when the frequency is not applied (as it will be shown in section 5, due to the presence of noise, of intrinsic birefringence and experimental uncertainties on the phase measurement, a phase shift threshold equal to 5° will be defined). The difference between the simulated and the experimental phase shifts comes from the discrepancy between some parameters (principally, from the vibration phases and the input SOP). Note that the detection of the intermodulation frequencies is observed as explained in the simulations.

As in the simulations, Fig. 10 and Table 2 together show that the frequency components can be recovered univocally at the different mechanical transducers. As previously mentioned, the localization is done via the identification of the FBGs (position of the optical filters). This allows multi-point vibration sensing.

5. Discussion on the experimental results

In Table 2, we observed that the phase shifts at 300 Hz between the electrical signals coming from FBG₂₁ and FBG₂₂ and at 240 Hz between the electrical signals coming from FBG₃₁ and FBG₃₂, is not precisely equal to 0°, while there is no applied vibrations at these frequencies on MT₂ and MT₃. This means that a decision threshold must be implemented. To find this threshold, we conducted a large number of experiments whose principle is depicted in Fig. 11. In these experiments, we applied a sine vibration with a frequency f (here, 300 Hz) on MT₂ but no vibration on MT₁ and MT₃ [Fig. 11(a)]. Figure 11(b) shows the phase shift between the electrical signals coming from FBG₃₁ and FBG₃₂ for each of these tests. The acceleration level of the vibration applied on MT₂ is equal to 30 m/s².

Fig. 11 (a): System configuration (vibration on MT₂, no vibrations on MT₁ and MT₃); (b): Evolution of the measured phase shift vs. phase of the vibration applied on MT₂.

In theory, as there is no vibration between FBG₃₁ and FBG₃₂, the phase shift should be equal to 0°. From this figure, we can see that the phase shift can be as high as 4.5°. This discrepancy with the ideal value comes from the experimental uncertainties on the phase measurement and the presence of noise and intrinsic birefringence. We then state that a vibration is effectively applied on one MT if the absolute phase shift between the two signals is higher than 5°. This also means that if a vibration applied both on MT₂ and MT₃ induces an absolute phase shift between the signals coming from FBG₃₁ and FBG₃₂ which is smaller than 5°, the vibration on MT₃ is not considered as a real vibration but as an artefact (indetermination case). A solution is proposed hereunder in order to avoid these indetermination cases.

The phase shift between the electrical signals from two adjacent FBGs is related to the value of the phase delay δ induced by the applied vibration and given by Eq. (2). This equation shows that the time evolution of the phase delay depends on the applied force and thus depends on its amplitude, frequency and phase. Moreover, as explained, the phase shift corresponds to an SOP modification and thus also depends on the input SOP S_in,λij and the polarizer transmission axis. In the following of this section the effect of each of these four parameters is successively investigated.

The influence of the vibration phase and of the vibration frequency has been investigated as follows: two vibrations with a frequency f, which is successively modified from 100 Hz to 1000 Hz by 50 Hz steps, are applied on MT₂ and MT₃, respectively [Fig. 12(a)]. For each frequency the phase of the vibration on MT₂ is held constant (0°) while the phase of the vibration applied on MT₃ is changed linearly between 0° to 360°. No vibration is applied on MT₁. The acceleration levels on MT₂ and MT₃ are both equal to 30 m/s² and are held constant. Figure 12(b) shows that the vibration phase has an influence on the phase shift value. As it can be observed, for vibration phases around 180°, the measured phase shift is smaller than 5°, although a vibration is really applied on MT₃. Consequently, we cannot deduce, for these vibration phases, the presence of the vibration. It also shows that the vibration frequency has a weak influence on the phase shift value.

Fig. 12 (a): System configuration (vibrations on MT₂ and MT₃, no vibration on MT₁) ; (b) : Evolution of the measured phase shift vs. phase of the vibration on MT₃ for four vibration frequencies.

The influence of the vibration amplitude on the electrical phase shift has also been investigated. In this set of experiments, vibrations with a frequency f (here, 300 Hz) are applied on MT₂ and on MT₃, respectively [Fig. 13(a)]. The vibration applied on MT₂ has an amplitude equal to 45 m/s² while the amplitude of the vibration applied on MT₃ is changed from 0.5 to 60 m/s². No vibration is applied on MT₁. Figure 13(b) shows the obtained evolution. It shows that under a certain amplitude equal to around 1 m/s², the effect of the vibration is so weak that the phase shift becomes lower than 5° (dashed line). Several sets of experiments lead to a minimal peak acceleration detectable of 1 m/s².

Fig. 13 (a): System configuration (vibration on MT₂ and MT₃, no vibration on MT₁) ; (b) : Evolution of the measured phase shift vs. amplitude of the vibration on MT₃ (dashed line : 5° phase shift).

From these figures, we can observe that depending on the vibration amplitude, frequency and phase, indetermination cases can happen and applied vibrations could be considered as artefacts. A possible solution to avoid these indetermination cases is to change the orientation of the polarizer transmission axis. Figure 14 shows the situation depicted in Fig. 12 but with two different polarizer angles, here separated by 22.5°. Several sets of experiments were conducted at different transmission angles, from 0 to 180°, per 5° steps. It was observed that the effect of the angle modification was periodic with a period of 90° and a possible solution was to use two SOPs SOP_in,1 and SOP_in,2 separated by 22.5°. This situation is depicted in Fig. 14(b). We can therefore observe that rotating the transmission axis of the polarizer can solve the indetermination cases. As we can see, when launching SOP_in,1, for some vibration phases around 180°, the measured phase shift is lower than 5° while, with SOP_in,2, the measured phase shift is greater than 5° at this vibration phase. This different behaviour is due to the dependency of the electrical phase shift on the input SOP, as already mentioned. On the other hand we observe that, when no vibration is applied on MT₃ as discussed in Fig. 11, changing the input SOP has no influence and does not lead to measured phase shifts greater than 5°. This situation is shown in Fig. 15.

Fig. 14 (a): System configuration (vibration on MT₂ and MT₃, no vibration on MT₁) ; (b): Evolution of the measured phase shift vs. phase of the vibration on MT₃ for two different input SOPs separated by 45°.

Fig. 15 (a): System configuration (vibration on MT₂ and MT₃, no vibration on MT₁) ; (b): Evolution of the measured phase shift vs. phase of the vibration on MT₃ for two different input SOPs.

The complete analysis method is thus as follows: the signals reflected by the different pairs of FBGs are first measured, using two different input SOPs separated by 22.5°. The spectra of the electrically converted signals are then calculated. In a first step, with the magnitude spectrum of each of these signals, we deduce the frequencies which occur for the first time and can thus be univocally deduced (for the other frequencies, it is not possible to distinguish artefacts from really applied vibrations). In a second step, we calculate, for each pair of adjacent FBGs and for the two input SOPs, the phase shift at these frequencies. If one or two of these phase shifts are greater than 5°, this frequency is really applied. If the phase shifts are both lower than 5°, this frequency is an artefact (this is only possible as long as the acceleration level is high enough to generate such phase shifts, see Fig. 13). Note that there exist some fast polarization controllers which are commercially available and allow to rapidly (at a 4 kHz rate) change the light input SOP and to fix it at some particular values.

6. Performances

6.1. Number of sensors

The number of sensors depends on how many FBGs can be used. In our experiment, considering that the system bandwidth is limited to 40 nm (by the circulator) and that the FBGs have a 10 dB bandwidth (to avoid as much crosstalk as possible) equal to 3.5 nm, the number of FBGs is equal to 12. Adopting the disposition given in Fig. 4, the number of measurement points is equal to 6. This number of measurement points can be increased by using a broadband coupler instead of a circulator in order to have a broader bandwidth. An alternative configuration can also be used and consists in replacing each FBG_i,1 (i>1) by the measurement of FBG_i−1,2 (i>1), which allows to double the number of sensors to 12. However, in that case, to ensure the proper functioning of the phase shift method, the distance between every FBG (sensor) has to be as small as possible. On the contrary, using the configuration presented in the paper (two FBGs for one sensor), the distance between the sensors is not restrictive. The choice of one configuration or the other depends distance between the sensors required by the application.

6.2. Spatial separation between the FBGs

The method based on the measurement of the phase shift assumes that the surrounding FBGs are as close as possible. The spatial separation is here equal to 30 cm but could still be decreased. During the inscription process, the fiber was shifted by 30 cm from one FBG to another for convenience considerations, such as the minimal physical distance between the vibration sources (shakers).

6.3. Transducer characteristics

The dimensions of the mechanical transducer used are given in Fig. 2. The weight of the whole transducer is around 80 grams, which is higher than classical piezoelectric accelerometers which can weigh as low as several grams. However contrary to these classical accelerometers our transducers allow multiplexing and are then interesting for wiring reasons.

6.4. Minimal acceleration, maximal frequency and frequency resolution

As mentioned in the previous section, in the current configuration, the sensor can measure accelerations down to 1 m/s². This value is limited by the effect of the mechanical transducer. Measuring lower accelerations is still possible by increasing the mass applied on the fiber (here, 4 g) as the sensitivity of the transduction is directly proportional to the applied mass [20

N. Linze, P. Tihon, O. Verlinden, P. Mégret, and M. Wuilpart, “Linearity considerations in polarization-based vibration sensors,” Appl. Opt. 51, 6997–7004 (2012) [CrossRef] [PubMed] .

]. Using a 8 g mass would then allow to measure accelerations down to 0.5 m/s². The maximal measurable frequency is limited by the resonance of the mechanical transducer, which happens at around 1 kHz. On the other side the proposed signal processing assumes that the shape of the mechanical signal does not change between the two measurements, i.e. during around 100 ms. The frequency resolution can go down to 1 Hz and is mainly limited by the temporal window considered.

7. Conclusion

In this paper, we presented a polarimetric quasi-distributed vibration sensor which allows to localize and to measure the frequency content of the different vibrations applied along the fiber. This ability is made possible by combining mechanical transducers, which transform the mechanical perturbation in an SOP modification, and pairs of FBGs with different Bragg wavelengths, which surround each mechanical transducer and reflect light. Using direct voltage measurements followed by a quite simple processing method based on the calculation of the phase shift spectrum between two FBGs, frequencies up to 800 Hz have been detected. The presented sensor could measure frequencies up to 1 kHz, which corresponds to the resonance frequency of the transducer [20

N. Linze, P. Tihon, O. Verlinden, P. Mégret, and M. Wuilpart, “Linearity considerations in polarization-based vibration sensors,” Appl. Opt. 51, 6997–7004 (2012) [CrossRef] [PubMed] .

]. In the present configuration the system can be composed of up to 6 sensors (this number could be extended by modifying the configuration) and accelerations as low as 1 m/s² could be measured (lower accelerations could be measured by slightly modifying the transducer).

Acknowledgments

The authors would like to thank the financial support of the F.R.S.-FNRS (F.R.I.A). This research was supported by the Interuniversity Attraction Poles program of the Belgian Science Policy Office, under grant IAP P7-35 photonics@be. This project is within the frame of Actions de Recherche Concertées financées par le Ministère de la Communauté française – Direction générale de l’Enseignement non obligatoire et de la Recherche scientifique.

References and links

1.	X. Guo, Z. Yin, and N. Song, “Measuring vibration by using fiber Bragg grating and demodulating it by blazed grating,” Chinese Opt. Lett. 2, 393–395 (2004).
2.	A. Fender, W. N. Macpherson, R. R. J. Maier, J. S. Barton, D. S. George, R. I. Howden, G. W Smith, B. J. S. Jones, S. Mcculloch, X. Chen, R. Suo, L. Zhang, and I. Bennion, “Two-axis temperature-insensitive accelerometer based on multicore fiber Bragg gratings,” IEEE Sens. J. 8, 1292–1298 (2008) [CrossRef] .
3.	Y. Weng, X. Qiao, T. Guo, M. Hu, Z. Feng, R. Wang, and J. Zhang, “A robust and compact fiber Bragg grating vibration sensor for seismic measurement,” IEEE Sens. J. 12, 800–804 (2011) [CrossRef] .
4.	Y. Zhu, P. Shum, C. Lu, B. M. Lacquet, P. L. Swart, and S. J. Spammer, “Temperature-insensitive fiber Bragg grating accelerometer,” IEEE Photon. Technol. Lett. 15, 1437–1439 (2003) [CrossRef] .
5.	M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, “Flexural beam-based fiber Bragg grating accelerometers,” IEEE Photon. Technol. Lett. 10, 1605–1607 (1998) [CrossRef] .
6.	T. Guo, L. Shao, H-Y. Tam, P. A. Krug, and J. Albert, “Tilted fiber grating accelerometer incorporating an abrupt biconical taper for cladding to core recoupling,” Opt. Express 17, 20651–20660 (2009) [CrossRef] [PubMed] .
7.	K. Wada, H. Narui, D. Yakamoto, T. Matsuyama, and H. Horinaka, “Balanced polarization maintaining fiber Sagnac interferometer vibration sensor,” Opt. Express 19, 21467–21474 (2011) [CrossRef] [PubMed] .
8.	X. Fang, “Fiber-optic distributed sensing by two-loop Sagnac interferometer,” Opt. Lett. 21, 444–446 (1996) [CrossRef] [PubMed] .
9.	X. Hong, J. Wu, C. Zuo, F. Liu, H. Guo, and K. Xu, “Dual Michelson interferometers for distributed vibration detection,” Appl. Opt. 50, 4333–4338 (2011) [CrossRef] [PubMed] .
10.	R. M. Manuel, M. G. Shlyagin, and S. V. Miridonov, “Location of a time-varying disturbance using an array of identical fiber-optic interferometers interrogated by CW DFB laser,” Opt. Express 16, 20666–20675 (2008) [CrossRef] [PubMed] .
11.	J. C. Juarez and H. F. Taylor, “Field test of a distributed fiber-optic intrusion sensor system for long perimeters,” Appl. Opt. 46, 1968–1971 (2007) [CrossRef] [PubMed] .
12.	Y. Lu, T. Zhu, L. Chen, and X. Bao, “Distributed vibration sensor based on coherent detection of phase-OTDR,” J. Lightwave Technol. 28, 3243–3249 (2010).
13.	K. Hotate and S. S. L. Ong, “Distributed dynamic strain measurement using a correlation-based Brillouin sensing system,” IEEE Photon. Technol. Lett. 15, 272–274 (2003) [CrossRef] .
14.	P. Chaube, B. G. Colpitts, D. Jagannathan, and A. W. Brown, “Distributed fiber-optic sensor for dynamic strain measurement,” IEEE Sens. J. 8, 1067–1072 (2008) [CrossRef] .
15.	R. Bernini, A. Minardo, and L. Zini, “Dynamic strain measurement in optical fibers by stimulated Brillouin scattering,” Opt. Lett. 34, 2613–2615 (2009) [CrossRef] [PubMed] .
16.	F. Qin, H. Li, W. Fan, and Q. Sheng, “Experimental study on vibration frequency response of micro-bend optic-fiber sensor,” Chinese Opt. Lett. 7, 556–559 (2009) [CrossRef] .
17.	Z. Zhang and X. Bao, “Continuous and damped vibraton detection based on fiber diversity detection sensor by Rayleigh backscattering,” J. Lightwave Technol. 26, 832–838 (2008) [CrossRef] .
18.	Z. Zhang and X. Bao, “Distributed optical fiber vibration sensor based on spectrum analysis of polarization-OTDR system,” Opt. Express 16, 10240–10247 (2008) [CrossRef] [PubMed] .
19.	G. Rajan, M. Ramakrishnan, Y. Semenova, A. Domanski, A. Boczkowska, T. Wolinski, and G. Farrell, “Analysis of vibration measurements in a composite material using an embedded PM-PCF polarimetric sensor and an FBG sensor,” IEEE Sens. J. 12, 1365–1371 (2012) [CrossRef] .
20.	N. Linze, P. Tihon, O. Verlinden, P. Mégret, and M. Wuilpart, “Linearity considerations in polarization-based vibration sensors,” Appl. Opt. 51, 6997–7004 (2012) [CrossRef] [PubMed] .
21.	M. Wuilpart, P. Mégret, M. Blondel, A. J. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett. 13, 836–838 (2001) [CrossRef] .
22.	S. C. Rashleigh, “Origins and control of polarization effects in single-mode fibers,” J. Lightwave Technol. 1, 312–331 (1983) [CrossRef] .
23.	A. Bertholds and R. Dandliker, “Determination of the individual strain-optic coefficients in single mode optical fibre,” J. Lightwave Technol. 6, 17–20 (1988) [CrossRef] .
24.	A. Galtarossa, D. Grosso, and L. Palmieri, “Accurate characterization of twist-induced optical activity in single-mode fibers by means of polarization-sensitive reflectometry,” IEEE Photon. Technol. Lett. 21, 1713–1715 (2009) [CrossRef] .
25.	S. Bette, C. Caucheteur, M. Wuilpart, and P. Mégret, “Spectral characterization of differential group delay in uniform fiber Bragg gratings,” Opt. Express 13, 9954–9960 (2005) [CrossRef] [PubMed] .
26.	R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
27.	M. Wuilpart, “Rayleigh Scattering in Optical Fibers and Applications to Distributed Measurements,” in Advanced Fiber Optics: Concepts and Technology , L. Thévenaz, eds. (EPFL Press, 2011), pp. 207–262.

OCIS Codes
(060.2370) Fiber optics and optical communications : Fiber optics sensors
(280.4788) Remote sensing and sensors : Optical sensing and sensors

ToC Category:
Sensors

History
Original Manuscript: December 7, 2012
Revised Manuscript: February 15, 2013
Manuscript Accepted: February 15, 2013
Published: February 28, 2013

Citation
Nicolas Linze, Pierre Tihon, Olivier Verlinden, Patrice Mégret, and Marc Wuilpart, "Development of a multi-point polarization-based vibration sensor," Opt. Express 21, 5606-5624 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-5606

Sort: Author | Year | Journal | Reset

References

X. Guo, Z. Yin, and N. Song, “Measuring vibration by using fiber Bragg grating and demodulating it by blazed grating,” Chinese Opt. Lett.2, 393–395 (2004).
A. Fender, W. N. Macpherson, R. R. J. Maier, J. S. Barton, D. S. George, R. I. Howden, G. W Smith, B. J. S. Jones, S. Mcculloch, X. Chen, R. Suo, L. Zhang, and I. Bennion, “Two-axis temperature-insensitive accelerometer based on multicore fiber Bragg gratings,” IEEE Sens. J.8, 1292–1298 (2008). [CrossRef]
Y. Weng, X. Qiao, T. Guo, M. Hu, Z. Feng, R. Wang, and J. Zhang, “A robust and compact fiber Bragg grating vibration sensor for seismic measurement,” IEEE Sens. J.12, 800–804 (2011). [CrossRef]
Y. Zhu, P. Shum, C. Lu, B. M. Lacquet, P. L. Swart, and S. J. Spammer, “Temperature-insensitive fiber Bragg grating accelerometer,” IEEE Photon. Technol. Lett.15, 1437–1439 (2003). [CrossRef]
M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, “Flexural beam-based fiber Bragg grating accelerometers,” IEEE Photon. Technol. Lett.10, 1605–1607 (1998). [CrossRef]
T. Guo, L. Shao, H-Y. Tam, P. A. Krug, and J. Albert, “Tilted fiber grating accelerometer incorporating an abrupt biconical taper for cladding to core recoupling,” Opt. Express17, 20651–20660 (2009). [CrossRef] [PubMed]
K. Wada, H. Narui, D. Yakamoto, T. Matsuyama, and H. Horinaka, “Balanced polarization maintaining fiber Sagnac interferometer vibration sensor,” Opt. Express19, 21467–21474 (2011). [CrossRef] [PubMed]
X. Fang, “Fiber-optic distributed sensing by two-loop Sagnac interferometer,” Opt. Lett.21, 444–446 (1996). [CrossRef] [PubMed]
X. Hong, J. Wu, C. Zuo, F. Liu, H. Guo, and K. Xu, “Dual Michelson interferometers for distributed vibration detection,” Appl. Opt.50, 4333–4338 (2011). [CrossRef] [PubMed]
R. M. Manuel, M. G. Shlyagin, and S. V. Miridonov, “Location of a time-varying disturbance using an array of identical fiber-optic interferometers interrogated by CW DFB laser,” Opt. Express16, 20666–20675 (2008). [CrossRef] [PubMed]
J. C. Juarez and H. F. Taylor, “Field test of a distributed fiber-optic intrusion sensor system for long perimeters,” Appl. Opt.46, 1968–1971 (2007). [CrossRef] [PubMed]
Y. Lu, T. Zhu, L. Chen, and X. Bao, “Distributed vibration sensor based on coherent detection of phase-OTDR,” J. Lightwave Technol.28, 3243–3249 (2010).
K. Hotate and S. S. L. Ong, “Distributed dynamic strain measurement using a correlation-based Brillouin sensing system,” IEEE Photon. Technol. Lett.15, 272–274 (2003). [CrossRef]
P. Chaube, B. G. Colpitts, D. Jagannathan, and A. W. Brown, “Distributed fiber-optic sensor for dynamic strain measurement,” IEEE Sens. J.8, 1067–1072 (2008). [CrossRef]
R. Bernini, A. Minardo, and L. Zini, “Dynamic strain measurement in optical fibers by stimulated Brillouin scattering,” Opt. Lett.34, 2613–2615 (2009). [CrossRef] [PubMed]
F. Qin, H. Li, W. Fan, and Q. Sheng, “Experimental study on vibration frequency response of micro-bend optic-fiber sensor,” Chinese Opt. Lett.7, 556–559 (2009). [CrossRef]
Z. Zhang and X. Bao, “Continuous and damped vibraton detection based on fiber diversity detection sensor by Rayleigh backscattering,” J. Lightwave Technol.26, 832–838 (2008). [CrossRef]
Z. Zhang and X. Bao, “Distributed optical fiber vibration sensor based on spectrum analysis of polarization-OTDR system,” Opt. Express16, 10240–10247 (2008). [CrossRef] [PubMed]
G. Rajan, M. Ramakrishnan, Y. Semenova, A. Domanski, A. Boczkowska, T. Wolinski, and G. Farrell, “Analysis of vibration measurements in a composite material using an embedded PM-PCF polarimetric sensor and an FBG sensor,” IEEE Sens. J.12, 1365–1371 (2012). [CrossRef]
N. Linze, P. Tihon, O. Verlinden, P. Mégret, and M. Wuilpart, “Linearity considerations in polarization-based vibration sensors,” Appl. Opt.51, 6997–7004 (2012). [CrossRef] [PubMed]
M. Wuilpart, P. Mégret, M. Blondel, A. J. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett.13, 836–838 (2001). [CrossRef]
S. C. Rashleigh, “Origins and control of polarization effects in single-mode fibers,” J. Lightwave Technol.1, 312–331 (1983). [CrossRef]
A. Bertholds and R. Dandliker, “Determination of the individual strain-optic coefficients in single mode optical fibre,” J. Lightwave Technol.6, 17–20 (1988). [CrossRef]
A. Galtarossa, D. Grosso, and L. Palmieri, “Accurate characterization of twist-induced optical activity in single-mode fibers by means of polarization-sensitive reflectometry,” IEEE Photon. Technol. Lett.21, 1713–1715 (2009). [CrossRef]
S. Bette, C. Caucheteur, M. Wuilpart, and P. Mégret, “Spectral characterization of differential group delay in uniform fiber Bragg gratings,” Opt. Express13, 9954–9960 (2005). [CrossRef] [PubMed]
R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
M. Wuilpart, “Rayleigh Scattering in Optical Fibers and Applications to Distributed Measurements,” in Advanced Fiber Optics: Concepts and Technology, L. Thévenaz, eds. (EPFL Press, 2011), pp. 207–262.

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Click here to see a list of articles that cite this paper

Figures


Fig. 1	Fig. 2	Fig. 3


Fig. 4	Fig. 5	Fig. 6


Fig. 7	Fig. 8	Fig. 9


Fig. 10	Fig. 11	Fig. 12


Fig. 13	Fig. 14	Fig. 15