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REPORT OF THE MULTIMEDIA PROJECT 2009, UMONS MA2, DECEMBER 2009 1

Vibration sensor based on optical fibreConception and simulation

Pierre Masure, Laurent-Yves Kalambayi

Abstract—This project concerns the development of a vibrationsensor based on the use of optical fibres, which will compensatefor the limitations of sensors based on classical technologies.The approach considered concerns the realisation of a vibrationsensor based on the polarisation properties of optical fibres. Thepolarisation state of the observed light at the output of a fibrevaries with the vibrations. In this context, it is proposed to designa simple sensor using out of the shelf equipment. The model hasbeen implemented on Matlab. The practical testing of the modelhas also been included to this project. We can conclude thatthe vibration sensor created provides the results we expectedtheorically. We obtain similar results for the model implementedon Matlab compared to the real device. These results prove thatthe sensor is operational.

Index Terms—optical, fibre, sensor, vibration.

I. INTRODUCTION

THIS project concerns the development of a vibrationsensor based on the use of optical fibres, which will

compensate for the limitations of sensors based on classicaltechnologies. Sensors based on optical fibres are insensitiveto electromagnetic perturbations, usable in harsh environ-ment (flammable environment, high temperatures, corrosionrisks,...) and are appropriate for distributed measurements. Theapproach considered concerns the realisation of a vibrationsensor based on the polarisation properties of optical fibres.The polarisation state of the observed light at the output of afibre varies with the vibrations. In this context, it is proposed todesign a simple sensor using out of the shelf equipment. Afterdesigning the system, the project will consist in simulatinghis efficiency on Matlab. Beforehand, it will be necessary toimplement a fibre model taking into account the vibrationeffects on the polarisation properties. An additionnal part,which is the practical testing of the model, is included to thisproject.

II. THE FIBER STRECTHER

The OPTIPHASE PZ2 High-efficiency Fiber Stretcher is afiber wound piezo-electric element for use in a wide rangeof optical interferometric measurement and sensing systemapplications. Typical uses include open loop demodulation,sensor simulation, white-light scanning interferometers andlarge angle modulation of interferometric phase. PZ2 Fiberstretcher are available with SMF-28e+ or PM [PANDA] fibertypes. We will use the fiber stretcher as a device to simulatethe vibration: the piezo-electric element which will stretch

M. Masure and Kalambayi are with the Department of Electrical Engineer-ing, UMons, Belgium

Manuscript received December 14, 2009; revised December 19, 2009.

the fiber at a given frequency. The purpose of our proposedsolution will be to recover the excitation spectrum in theoptical domain. In order to do this, we should try to have thebest similarity between the excitation and the optical domainin an spectrum point of view: the recovery of the excitationspectrum in frequency and in amplitude at the optical outputof the sytem should be obtained. The model of the FiberStretcher used is the PZ2-PM-1.5-FC/APC-E operating at awavelength of 1550 nm with a fiber stretch of 3.8µm/V, thefiber length is 40 meters and the fiber wind is a 2-Layer wind.The fiber used in the stretcher is a polarisation maintainingfiber which inhibits the polarisation mode coupling present innormal optical fibers.

III. THE SENSOR MODEL

To recover the excitation spectrum, we must create a systemwhich can provide us a way to recover this spectrum in theoptical domain.

Fig. 1. Sensor model

On Fig.1 we show the measurement tool we created toperform the given task, the vibration measurement. To performthe task, we need:

• a laser• a polarizer• the piezo-electric fiber stretcher• an analyzer• an oscilloscope+FFT

We suppose that a laser launches any polarisation state atthe input of the fiber. Therefore, we use a linear polarizerin order to have a defined state of polarisation at the input ofthe piezo-electric fiber stretcher. The light travels through thefiber winded up the piezo-electric element which stretches the


fiber with an elongation driven by the signal provided by thegenerator. The elongation has for effect to modifies the lightpolarisation state. After the piezo-electric fiber stretcher, weput a polariser which is used to analyze the light coming fromthe stretcher. This particular polariser is denoted by the term’analyzer’.

The physical idea behind this construction is that the powerof the light will have an initial value if there is no stretch.

When we induce a stretch to the fiber, the light polarisationstate of the light will be modifiedin time. As a consequence,the power transmitted by the analyzer will vary in time.Physically, we feel that we could recover some spectruminformation in the optical domain. Let us put some math-ematics behind this to prove that our system works. Theformalism used to analyze this system is the Stokes formalism.The Stokes formalism describes the polarisation state of thelight through a 4-dimensional real vector. Let us analyze ourvibration tool mathematically:The polarisation state launched by the laser is random.

We use a polariser to modify that random polarisation stateto obtain a well-defined polarisation state:

sin =(1 cos(2φ) sin(2φ) 0

)T(1)

with φ corresponding to the polarisation angle with respectto the x-Axis. Let us assume that φ=45 ˚ :

sin =(1 0 1 0

)T(2)

We describe the piezo-electric fiber stretcher with the fol-lowing 4x4 Mueller matrix:

Ms =

m11 m12 m13 m14m21 m22 m23 m24m31 m32 m33 m34m41 m42 m43 m44

(3)

m11 = 1,m12 = 0,m13 = 0m14 = 0,m21 = 0,m31 = 0,m41 = 0

m22 = cos2δ

2+ sin2 δ

2cos4q

m23 = sin2 δ

2sin4q

m24 = −sinδsin2q

m32 = sin2 δ

2sin4q

m33 = cos2δ

2− sin2 δ

2cos4q

m34 = sinδcos2qm42 = sinδsin2q

m43 = −sinδcos2qm44 = cosδ

δ = ∆βz (4)

where ∆β is the fiber birefringence and δ is the phaseretardance between the two eigenmodes of the PMF-fiber

of the stretcher and q is the azimuth of the fastest linearpolarization.

z = zi + kxexc (5)

where z is the full length of the fiber in the fiber stretcher, zi

the initial length of the fiber (40m), kxexc the elongation ofthe fiber due to the mechanical excitation, k the coefficientdescribing the linear stretch of the fiber with the voltage(3.8µm/V), xexc the excitation signal of the piezo-electricelement in Volt.

We make a simplification and suppose that q = 0 (x and yaxes aligned with the eigenmodes of the PMF-fiber):

Ms =

1 0 0 00 1 0 00 0 cos(δ) cos(δ)0 0 −sin(δ) cos(δ)

(6)

sint = Mss (7)

At the output of the fiber stretcher, we obtain:

sint =(1 0 cos(δ) −sin(δ)

)T(8)

The analyzer can be described by an 4x4 Mueler matrix:

Ma =12

1 cos(2θ) sin(2θ) 0

cos(2θ) cos(2θ)2 sin(2θ)cos(2θ) 0sin(2θ) sin(2θ)cos(2θ) sin(2θ)2 0

0 0 0 0

(9)

If we consider that θ=45 ˚ :

Ma =12

1 0 1 00 0 0 01 0 1 00 0 0 0

(10)

sout = Masint (11)

sout =12(1 + cos(δ) 0 1 + cos(δ) 0

)T(12)

sout =12(1 + cos(∆βz) 0 1 + cos(∆βz) 0

)T(13)

The global power at the output is given by the first elementof sout :

sout0 =12

(1 + cos(∆βz)) (14)

∆β is constant with the stretch because we are working witha PMF-fiber. This has been proven in the reference [1][2].

sout0 =12

(1 + cos(∆β(zi + kxexc)) (15)

Let us suppose that the excitation is sinusoidal:

xexc = Asin(2πfexct) (16)

sout0 =12

(1 + cos (∆β (zi + kAsin(2πfexct)))) (17)


We see in this equation that the temporal evolution of theoutput power depends on fexc. The information related to themechanical excitation is somehow comprised in equation (18).Let us analyze equation (19) in detail to see if the opticalspectrum is a good image of the excitation spectrum.

To perform this, we will use the Taylor expansion in thenext section.

IV. NON-LINEARITIES IN THE SENSOR

Let us develop equation (18):

sout0 =12

1 + cos

∆βzi︸︷︷︸p1

+ ∆βkA︸︷︷︸p2

sin2πfexct

(18)

Let us consider the Taylor expansion of this function:

cos(p1 + x) = cos(p1)− sin(p1)x

− 12cos(p1)x2 +

16sin(p1)x3

+124cos(p1)x4 + . . . (19)

when considering that

x = p2cos(2πfexct) (20)

p1 = ∆βzi (21)

p2 = ∆βkA (22)

p2 contains the information about the amplitude of themechanical excitation. We will assume that non-linearities arenon-negligble for the first three orders of the Taylor expansion.

Let us modify this equation to analyze the contribution ofthe higher orders on the lower orders.

The second order term gives:

x2 = p22cos

2(2πfexct) = p22

12

(1 + cos(2π(2fexc)t)) (23)

The third order term gives:

x3 = p32cos

3(2πfexct) =

p32

12

(1 + cos(2π(2fexc)t))cos(2πfexct) (24)

The Simpson product gives:

cos(2π(2fexc)t)cos(2πfexct) =12

(cos(2π(3fexc)t) + cos(2πfexct)) (25)

Finally, for the equation (with x cosinusoidal) cos(p1 + x),we have the following spectral components (frequency:

amplitude)

DC : cos(p1)− 12p22

12cos(p1) (26)

fexc : −sin(p1)p2 +16sin(p1)

p32

2(1 +

12

) (27)

2fexc : −12cos(p1)

p22

2(28)

3fexc :16sin(p1)

p32

4(29)

When performing simplifications, you have:

DC : cos(p1)(1− p22

4) (30)

fexc : −sin(p1)p2(−1 +p22

8) (31)

2fexc : −p22

4cos(p1) (32)

3fexc :p32

8sin(p1) (33)

Let us analyze these results for a sinusoidal excitation at afrequency fexc:

• The system is non-linear: the optical spectrum is not thesame as the excitation spectrum.

– fexc at the excitation creates fexc, 2fexc, 3fexc,... inthe optical domain.

– The amplitudes aren’t directly recoverable becausewe want to have access to p2 which has the infor-mation about the amplitude of the mechanical exci-tation. There is no term directly proportional to p2.We need to perform some mathematical operationsto recover the good amplitudes.

• The linear term provides us fexc.• The linear term doesn’t provides us directly the good

amplitude.• All the coefficients depends on p1.• By playing on the p1 parameter, we can suppress or

maximize the amplitude of the fundamental and thehigher orders.

• To recover the p2 amplitude at the first order, we needto solve the equation with provides us more than onesolution.

– The p2 amplitude isn’t easily recoverable at fexc.• If we consider the second order, we only have one

physical solution to the equation to find the p2 amplitude.– Why? A negative amplitude isn’t physical.

• To conclude: we should consider the measurement ofthe amplitude at 2fexc to obtain an equation which canprovide us p2.

• To perform this, we should maximize cos(p1): the mea-sure at 2fexc gives us −p2

24


• The maximization/minimization of cos(p1) is the sameas maximizing/minimizing the DC power.

– This can be pratical for an experimental use.• p1 is linked to the the wavelength of the source.

– We can tune p1 (and the amplitude of the harmonics)by changing the wavelength of the source.

p1 = ∆βzi =2πλopt

∆nzi =2πfopt

c∆nzi (34)

where fopt is the frequency of the laser,λopt the wavelengthof the laser,∆n is the difference in refractive indexes betweenthe x and y axes of the PMF fiber.

Now, we know that the coefficients varies with cos(p1) orsin(p1). Changing fopt linearly will modify the coefficientsperiodically. This result is very important if we want toinfluence the performance of the vibration sensor: we can acton the linearity of the system by modifying the wavelength ofthe source.

V. MATLAB RESULTS

We implemented the vibration sensor model on Matlabto simulate its working in order to validate the theory wedevelopped during this project. We launched a sinusoidalexcitation into the system at 120Hz. We validated the theoryfirst for a random frequency of the laser in order to seethe generation of harmonics until the third order (the otherare neglegible). We see that there is a generation of thesefrequencies fexc, 2fexc, 3fexc,... This result is provided inFig. 2. On the other hand, we tuned the frequency (fopt) of the

Fig. 2. y(f) (m): excitation elongation, sout(f)(dB): power in the opticaldomain; generation of H1 H2 H3

laser in order to maximize the amplitude of the fundamentalfrequency fexc. The first harmonic at 2fexc is then suppressed.This result is provided in Fig. 3.

Fig. 3. y(f)(m): excitation elongation, sout(f)(dB): power in the opticaldomain; suppression of H2

VI. EXPERIMENTAL RESULTS

We experimentally implemented the vibration sensor to testthe model in order to validate the theory we developped duringthis project. We launched a sinusoidal excitation thanks to thegenerator into the system at 100Hz. We valitade the theoryfirst for a random frequency of the laser in order to seethe generation of harmonics until the third order (the otherare neglegible). We see that there is a generation of thesefrequencies fexc, 2fexc, 3fexc,... This result is provided inFig. 4. On the other hand, we tuned the frequency (fopt) of the

Fig. 4. Power in the optical domain; generation of H1 H2 H3

laser in order to increase the amplitude of the DC component.The amplitude of fundamental decreases. The first harmonicat 2fexc increases. This result is provided in Fig. 5. However,the system we experimentally implemented isn’t completelythe same as the model tested on Matlab because the parameterθ isn’t equal to 45 ˚ for the analyzer. Therefore, we couldn’thave an exact replica of the amplitudes we simulated withMatlab. These results prove that the spectrum depends on thewavelength of the source and that by modifying its wavelengthwe can suppress or maximize the fundamental and the higher


orders.

Fig. 5. Power in the optical domain; maximization of H2

VII. PERSPECTIVES

We don’t have the exact experimental replica of the modelimplemented on Matlab. Therefore, we could modify ourmodel implemented on Matlab and generalize it with a variableθ angle. When we will be able to have the exact replica of themodel implemented on Matlab on the experimental way, wewill have the possibility to compare the amplitudes obtainedexperimentally with the simulated results.

VIII. CONCLUSION

This project concerned the development of a vibration sen-sor based on the use of optical fibres. The approach consideredconcerned the realisation of a vibration sensor based on thepolarisation properties of optical fibres. The polarisation stateof the observed light at the output of a fibre varies with thevibrations. After designing the system, the project consistedin simulating his efficiency on Matlab. The practical testingof the model has also been included to this project.

We can conclude that the vibration sensor created providesthe results we expected theorically. We obtain coherent resultsbetween the model implemented and the real device. Theseresults prove that the sensor is operational.

ACKNOWLEDGMENT

The authors would like to thank the professor Marc Wuilpartfor his teaching method.

REFERENCES

[1] C. Crunelle, M. wuilpart, P. Mgret,Sensitivity of Polarization Main-taining Fibres to Temperature and Strain for Sensing Applications, pp.205 to 208, in Proc. IEEE/LEOS Benelux Chapter 2006, Eindhoven, TheNetherlands,

[2] N. Ashby, D. A. Howe, J. Taylor, A. Hati, C. Nelson [National Instituteof Standards and Technology],Optical Fiber Vibration and AccelerationModel, pp. 1 to 5.