# Projet Ma2

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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

AbstractThis 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 Termsoptical, fibre, sensor, vibration.

I. INTRODUCTION

THIS project concerns the development of a vibrationsensor based on the use of optical fibres, which willcompensate 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 3.8m/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

REPORT OF THE MULTIMEDIA PROJECT 2009, UMONS MA2, DECEMBER 2009 2

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 termanalyzer.

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(2) sin(2) 0

)T(1)

with corresponding to the polarisation angle with respectto the x-Axis. Let us assume that =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 = cos2

2+ sin2

2cos4q

m23 = sin2

2sin4q

m24 = sinsin2qm32 = sin2

2sin4q

m33 = cos2

2 sin2

2cos4q

m34 = sincos2qm42 = sinsin2q

m43 = sincos2qm44 = cos

= z (4)

where is the fiber birefringence and 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, zithe 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(3.8m/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() cos()0 0 sin() cos()

(6)sint = Mss (7)

At the output of the fiber stretcher, we obtain:

sint =(1 0 cos() sin())T (8)

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

Ma =12

1 cos(2) sin(2) 0

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

0 0 0 0

(9)

If we consider that =45 :

Ma =12

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

(10)sout = Masint (11)

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

)T(12)

sout =12(1 + cos(z) 0 1 + cos(z) 0

)T(13)

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

sout0 =12

(1 + cos(z)) (14)

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((zi + kxexc)) (15)

Let us suppose that the excitation is sinusoidal:

xexc = Asin(2pifexct) (16)

sout0 =12

(1 + cos ( (zi + kAsin(2pifexct)))) (17)

REPORT OF THE MULTIMEDIA PROJECT 2009, UMONS MA2, DECEMBER 2009 3

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 + coszi

p1

+ kA p2

sin2pifexct

(18)

Let us consider the Taylor expansion of this function:

cos(p1 + x) = cos(p1) sin(p1)x 1

2cos(p1)x2 +

16sin(p1)x3

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

when considering that

x = p2cos(2pifexct) (20)

p1 = zi (21)

p2 = kA (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 = p22cos2(2pifexct) = p22

12

(1 + cos(2pi(2fexc)t)) (23)

The third order term gives:

x3 = p32cos3(2pifexct) =

p3212

(1 + cos(2pi(2fexc)t))cos(2pifexct) (24)

The Simpson product gives:

cos(2pi(2fexc)t)cos(2pifexct) =12

(cos(2pi(3fexc)t) + cos(2pifexct)) (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)p322

(1 +12

) (27)

2fexc : 12cos(p1)p222

(28)

3fexc :16sin(p1)

p324

(29)

When performing simplifications, you have:

DC : cos(p1)(1 p22

4) (30)

fexc : sin(p1)p2(1 + p22

8) (31)

2fexc : p22

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