## 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.

© 2013 OSA

## 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 [1–3, 6]. 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 [1–3] or on the movement of one or two flexural beams [4, 5]. 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 [7–12]. 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], a Michelson [9] or a Fabry-Perot configuration [10]. The vibration can also be detected with reflectometry techniques measuring the variation of the optical phase [11, 12]. 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 [13–15]. Some other systems are based on optical power modulation by micro-bending [16]. Finally some systems are based on the state of polarization (SOP) of light [17–20], whose one approach is Polarization Optical Time Domain Reflectometry (POTDR) [21].

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], 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, 18]. 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, 18]. With these sensors, accelerations as low as 1 m/s^{2} 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] 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).

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 (

*λ*

_{1}=1571.5 nm,

*λ*

_{2}=1550.6 nm and

*λ*

_{3}= 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.

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]:

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

In Eqs. 1 and 2 *k* = 2*π*/*λ*, *λ* is the light wavelength in vacuum, *n*_{0} is the effective refractive index (about 1.45), *p*_{11} and *p*_{12} are the elasto-optic coefficients of silica (*p*_{11}–*p*_{12}=−0.14 [23, 24]), *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).

We experimentally applied a 160 Hz sine vibration on MT_{1}, a 110 Hz sine vibration on MT_{2} and a 210 Hz sine vibration on MT_{3}. This figure gives the frequency spectrum of the three converted signals. From this figure, we can see that the signal reflected by FBG_{1} has a contribution at 160 Hz, as this frequency is applied on MT_{1} [Fig. 3(a)]. The signal reflected by FBG_{2} has a contribution at 110 Hz, as this frequency is applied on MT_{2} [Fig. 3(b)] and the signal reflected by FBG_{3} has a contribution at 210 Hz, as we have applied this frequency on MT_{3} [Fig. 3(c)]. However we also observe that the signal reflected by FBG_{2} has also a contribution at 160 Hz while this frequency is only applied on S_{1}. Similarly the signal reflected by FBG_{3} has contributions at 160 and 110 Hz while these frequencies are applied on S_{1} and S_{2} 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 S_{2} (or S_{3}), 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]. To distinguish artefacts from really applied vibrations, as shown in Fig. 4, we now use a second FBG situated before each MT (except for MT_{1} as the frequency spectrum of the vibration applied on S_{2} is univocally deduced from the measurement of FBG_{2}) and Fig. 1 becomes Fig. 4. As it can be observed, except for S_{1}, two FBGs now surround each MT. Their spatial and wavelength separations are 30 cm and 10 nm, respectively. The Bragg wavelengths are respectively *λ*_{12} = 1571.5 nm, *λ*_{21} = 1560.8 nm, *λ*_{22} = 1550.6 nm, *λ*_{31} = 1540.3 nm and *λ*_{32} = 1530.9 nm.

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

*. 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 FBG*

_{i}_{21}and by FBG

_{22}, when a 120 Hz vibrations is applied on MT

_{1}only. Because of this vibration the signal reflected by FBG

_{21}varies to the rhythm of this vibration, as well as the signal reflected by FBG

_{22}. As there is no vibration on MT

_{2}, 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 MT

_{1}and on MT

_{2}. As in the previous situation [Fig. 5(a)], the signal reflected by FBG

_{21}varies to the rhythm of the vibration applied on MT

_{1}. The signal reflected by FBG

_{22}also varies to the rhythm of this vibration applied on MT

_{1}. However contrary to the first situation, the SOP (at

*λ*

_{22}) is now also modified by the vibration applied on MT

_{2}(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 MT

_{2}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 FBG

_{21}.

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
and polarization mode coupling, as shown in [21]._{int} - It can also be modified when being transmitted and reflected by the FBGs [25].
- It can be modified by inadvertently twisting the fiber [22].
- The phase retardance
*δ*due to the intrinsic birefringence Δn_{int}and in turn the SOP evolution depend on the wavelength:_{int}*δ*= 2_{int}*π*Δ*n*, where Δ_{int}L/λ*n*is the intrinsic birefringence (refractive index difference) and_{int}*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
is equal to 4×10_{int}^{−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 Δ
*n*= 5×10_{FBG}^{−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 FBG_{32} (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).

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

*, when vibrations are applied on the different MTs. This can be summarized as follows:*

_{out,}_{λ}_{ij}- The input SOP S
is a linear polarization state described by a Stokes vector with an azimuth_{in,λij}*φ*such that, if Pis the input power [26]:_{in}$${S}_{\mathit{in},\lambda ij}=\left(\begin{array}{c}1\\ \text{cos}2\phi \\ \text{sin}2\phi \\ 0\end{array}\right){P}_{\mathit{in}}$$The first component of a Stokes vector corresponds to the optical power. This input SOP is fixed by the input polarizer. The Mueller matrix Mof a linear polarizer oriented at_{Pol,θ}*θ*radians with respect to the reference horizontal*x*axis is given by:$${M}_{\mathit{Pol},\theta}=\frac{1}{2}\left(\begin{array}{cccc}1& \text{cos}2\theta & \text{sin}2\theta & 0\\ \text{cos}2\theta & {\text{cos}}^{2}2\theta & \text{sin}2\theta \text{cos}2\theta & 0\\ \text{sin}2\theta & \text{sin}2\theta \text{cos}\theta & {\text{sin}}^{2}2\theta & 0\\ 0& 0& 0& 0\end{array}\right)$$ - The different fiber sections M
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]:_{fiber}$${M}_{\mathit{Fiber}}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& {\text{cos}}^{2}\frac{{\delta}_{\mathit{int}}}{2}+{\text{sin}}^{2}\frac{{\delta}_{\mathit{int}}}{2}\text{cos}4q& {\text{sin}}^{2}\frac{{\delta}_{\mathit{int}}}{2}\text{sin}4q& -\text{sin}{\delta}_{\mathit{int}}\text{sin}2q\\ 0& {\text{sin}}^{2}\frac{{\delta}_{\mathit{int}}}{2}\text{sin}4q& {\text{cos}}^{2}\frac{{\delta}_{\mathit{int}}}{2}-{\text{sin}}^{2}\frac{{\delta}_{\mathit{int}}}{2}\text{cos}4q& \text{sin}{\delta}_{\mathit{int}}\text{cos}2q\\ 0& \text{sin}{\delta}_{\mathit{int}}\text{sin}2q& -\text{sin}{\delta}_{\mathit{int}}\text{cos}2q& \text{cos}{\delta}_{\mathit{int}}\end{array}\right)$$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*is equal to_{int}*δ*= 2_{int}*π*Δ*n*with_{int}L_{s}/λ*L*, the length of the fiber section,_{s}*λ*, the wavelength in vacuum and Δ*n*the fiber intrinsic birefringence (equal to 4×10_{int}^{−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]:$${M}_{\mathit{Sensor}}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& {\text{cos}}^{2}\frac{{\delta}_{i}}{2}+{\text{sin}}^{2}\frac{{\delta}_{i}}{2}\text{cos}4q& {\text{sin}}^{2}\frac{{\delta}_{i}}{2}\text{sin}4q& -\text{sin}{\delta}_{i}\text{sin}2q\\ 0& {\text{sin}}^{2}\frac{{\delta}_{i}}{2}\text{sin}4q& {\text{cos}}^{2}\frac{{\delta}_{i}}{2}-{\text{sin}}^{2}\frac{{\delta}_{i}}{2}\text{cos}4q& \text{sin}{\delta}_{i}\text{cos}2q\\ 0& \text{sin}{\delta}_{i}\text{sin}2q& -\text{sin}{\delta}_{i}\text{cos}2q& \text{cos}{\delta}_{i}\end{array}\right)$$Where
*δ*is the phase retardance induced by the vibration applied on the sensor_{i}*i*and has the expression given by Eq. (2), in which the applied force per unit of length*f*depends on the applied acceleration level a_{i}(i=1,2,3) by the following equation [20]:_{i}*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*(at 1 m/s_{int}^{2}, 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]. The FBGs parameters are as follows: their birefringence Δ
*n*is 5×10_{FBG}^{−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*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 $\overleftarrow{{M}_{\lambda ij}}$ can be deduced from the forward matrices $\overrightarrow{{M}_{\lambda ij}}$ by the following relation [27]: In this equation M_{ij}is such that [27]:_{S}$${M}_{S}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right)$$The forward matrix $\overrightarrow{{M}_{\lambda ij}}$ is equal to the product of the Mueller matrices of the devices that interact with*λ*during the forward propagation._{ij} - Knowing all these matrices, it is possible to find the SOP at the analyzer input S
as:_{rt,}_{λ}_{ij} - Once
*S*has been calculated, it is possible to calculate the optical power transmitted through the analyzer P_{rt,}_{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:_{out,}_{λ}_{ij}

Note that this multi-step process gives the total power P* _{out}* at a specific time

*t*, corresponding to specific values of a

*(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*

_{i}*λ*, taking into account their respective paths.

_{ij}This process can be illustrated for *λ*_{12}. The transmitted optical power P_{out,}_{λ}_{12} corresponds to the first component of the Stokes vector S_{out,}_{λ}_{12}, which is given by:

In this equation M_{fiber1}, M_{fiber2} and M_{fiber3} are the Mueller matrices corresponding to the fiber sections between the polarizer and MT_{1} and between MT_{1} and FBG_{12}, respectively. M_{Sensor1} is the Mueller matrix corresponding to the first mechanical transducer (MT_{1}). M_{FBG12,r} represents the effect of the reflection by FBG_{12}. 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

*(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*

_{i}_{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 MT

_{1}, 175 Hz and 240 Hz sine vibrations are applied on MT

_{2}and 175 Hz and 300 Hz sine vibrations are applied on MT

_{3}. The acceleration amplitudes are 30 m/s

^{2}. The magnitude spectra obtained are given in Figs. 8(a) – 8(e).

The signal coming from FBG_{12}, depicted in Fig. 8(a), has a contribution at 300 Hz, as this frequency is applied on MT_{1}. As there is no new vibration between MT_{1} and MT_{2}, the signal reflected by FBG_{21} [Fig. 8(b)] has a spectrum similar to that of FBG_{12}, i.e. a contribution at 300 Hz. The signal reflected by FBG_{22} [Fig. 8(c)] has two additional frequency contributions as 175 Hz and 240 Hz sine vibrations are applied on MT_{2}. However, as a 300 Hz vibration is applied on MT_{1}, this frequency is also observed in the FBG_{22} frequency spectrum. The signal reflected by FBG_{31} [Fig. 8(d)] has a frequency spectrum similar to that of FBG_{22} as no mechanical transducer is situated in between. Finally, the signal from FBG_{32} [Fig. 8(e)] has contributions at 175 Hz and 300 Hz, as these vibrations are applied on MT_{3} but also at 240 Hz as this vibration is applied on MT_{2}.

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

*(t). Note that periodical disturbances below curves are the result of the use of a rectangular window for the FFT calculus.*

_{i}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 Δ*ϕ* 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.

For instance, for MT_{2}, 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 MT_{1}, it is not possible, from the magnitude spectra, to know if this frequency is really applied on MT_{2} or if it is an artefact. The phase difference between the signals coming from FBG_{21} and FBG_{22} 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 MT_{2}. For MT_{3}, 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 MT_{3} while the 240 Hz frequency is not present.

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 MT_{1}, while multi-sine vibrations are applied both on MT_{2} and MT_{3}: respectively 175 Hz and 240 Hz on MT_{2}, and 175 Hz and 300 Hz on MT_{3}. The Bragg wavelengths are respectively *λ*_{12} = 1571.5 nm, *λ*_{21} = 1560.8 nm, *λ*_{22} = 1550.6 nm, *λ*_{31} = 1540.3 nm and *λ*_{32} = 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].

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 FBG_{21} and FBG_{22} and at 240 Hz between the electrical signals coming from FBG_{31} and FBG_{32}, is not precisely equal to 0°, while there is no applied vibrations at these frequencies on MT_{2} and MT_{3}. 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 MT_{2} but no vibration on MT_{1} and MT_{3} [Fig. 11(a)]. Figure 11(b) shows the phase shift between the electrical signals coming from FBG_{31} and FBG_{32} for each of these tests. The acceleration level of the vibration applied on MT_{2} is equal to 30 m/s^{2}.

In theory, as there is no vibration between FBG_{31} and FBG_{32}, 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 MT_{2} and MT_{3} induces an absolute phase shift between the signals coming from FBG_{31} and FBG_{32} which is smaller than 5°, the vibration on MT_{3} 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 MT_{2} and MT_{3}, respectively [Fig. 12(a)]. For each frequency the phase of the vibration on MT_{2} is held constant (0°) while the phase of the vibration applied on MT_{3} is changed linearly between 0° to 360°. No vibration is applied on MT_{1}. The acceleration levels on MT_{2} and MT_{3} are both equal to 30 m/s^{2} 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 MT_{3}. 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.

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 MT_{2} and on MT_{3}, respectively [Fig. 13(a)]. The vibration applied on MT_{2} has an amplitude equal to 45 m/s^{2} while the amplitude of the vibration applied on *MT*_{3} is changed from 0.5 to 60 m/s^{2}. No vibration is applied on MT_{1}. Figure 13(b) shows the obtained evolution. It shows that under a certain amplitude equal to around 1 m/s^{2}, 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/s^{2}.

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 MT_{3} 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.

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/s^{2}. 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]. Using a 8 g mass would then allow to measure accelerations down to 0.5 m/s^{2}. 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]. 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/s^{2} 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*.

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