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




                   Vibration sensor based on optical fibre
                         Conception and simulation
                                                Pierre Masure, Laurent-Yves Kalambayi




   Abstract—This project concerns the development of a vibration            the fiber at a given frequency. The purpose of our proposed
sensor based on the use of optical fibres, which will compensate             solution will be to recover the excitation spectrum in the
for the limitations of sensors based on classical technologies.             optical domain. In order to do this, we should try to have the
The approach considered concerns the realisation of a vibration
sensor based on the polarisation properties of optical fibres. The           best similarity between the excitation and the optical domain
polarisation state of the observed light at the output of a fibre            in an spectrum point of view: the recovery of the excitation
varies with the vibrations. In this context, it is proposed to design       spectrum in frequency and in amplitude at the optical output
a simple sensor using out of the shelf equipment. The model has             of the sytem should be obtained. The model of the Fiber
been implemented on Matlab. The practical testing of the model              Stretcher used is the PZ2-PM-1.5-FC/APC-E operating at a
has also been included to this project. We can conclude that
the vibration sensor created provides the results we expected               wavelength of 1550 nm with a fiber stretch of 3.8µm/V, the
theorically. We obtain similar results for the model implemented            fiber length is 40 meters and the fiber wind is a 2-Layer wind.
on Matlab compared to the real device. These results prove that             The fiber used in the stretcher is a polarisation maintaining
the sensor is operational.                                                  fiber which inhibits the polarisation mode coupling present in
  Index Terms—optical, fibre, sensor, vibration.                             normal optical fibers.

                         I. I NTRODUCTION                                                        III. T HE SENSOR MODEL

T     HIS project concerns the development of a vibration
      sensor based on the use of optical fibres, which will
compensate for the limitations of sensors based on classical
                                                                              To recover the excitation spectrum, we must create a system
                                                                            which can provide us a way to recover this spectrum in the
                                                                            optical domain.
technologies. Sensors based on optical fibres are insensitive
to electromagnetic perturbations, usable in harsh environ-
ment (flammable environment, high temperatures, corrosion
risks,...) and are appropriate for distributed measurements. The
approach considered concerns the realisation of a vibration
sensor based on the polarisation properties of optical fibres.
The polarisation state of the observed light at the output of a
fibre varies with the vibrations. In this context, it is proposed to
design a simple sensor using out of the shelf equipment. After
designing the system, the project will consist in simulating
his efficiency on Matlab. Beforehand, it will be necessary to
implement a fibre model taking into account the vibration
effects on the polarisation properties. An additionnal part,
which is the practical testing of the model, is included to this
project.                                                                    Fig. 1.   Sensor model


                    II. T HE FIBER STRECTHER                                   On Fig.1 we show the measurement tool we created to
   The OPTIPHASE PZ2 High-efficiency Fiber Stretcher is a                    perform the given task, the vibration measurement. To perform
fiber wound piezo-electric element for use in a wide range                   the task, we need:
of optical interferometric measurement and sensing system                      • a laser
applications. Typical uses include open loop demodulation,                     • a polarizer
sensor simulation, white-light scanning interferometers and                    • the piezo-electric fiber stretcher
large angle modulation of interferometric phase. PZ2 Fiber                     • an analyzer
stretcher are available with SMF-28e+ or PM [PANDA] fiber                       • an oscilloscope+FFT
types. We will use the fiber stretcher as a device to simulate               We suppose that a laser launches any polarisation state at
the vibration: the piezo-electric element which will stretch                the input of the fiber. Therefore, we use a linear polarizer
                                                                            in order to have a defined state of polarisation at the input of
  M. Masure and Kalambayi are with the Department of Electrical Engineer-
ing, UMons, Belgium                                                         the piezo-electric fiber stretcher. The light travels through the
  Manuscript received December 14, 2009; revised December 19, 2009.         fiber 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 the       of the stretcher and q is the azimuth of the fastest linear
generator. The elongation has for effect to modifies the light      polarization.
polarisation state. After the piezo-electric fiber stretcher, we                          z = zi + kxexc                    (5)
put a polariser which is used to analyze the light coming from
                                                                      where z is the full length of the fiber in the fiber stretcher, zi
the stretcher. This particular polariser is denoted by the term
                                                                   the initial length of the fiber (40m), kxexc the elongation of
’analyzer’.
                                                                   the fiber due to the mechanical excitation, k the coefficient
   The physical idea behind this construction is that the power
                                                                   describing the linear stretch of the fiber with the voltage
of the light will have an initial value if there is no stretch.
                                                                   (3.8µm/V), xexc the excitation signal of the piezo-electric
   When we induce a stretch to the fiber, the light polarisation
                                                                   element in Volt.
state of the light will be modifiedin time. As a consequence,
                                                                      We make a simplification and suppose that q = 0 (x and y
the power transmitted by the analyzer will vary in time.
                                                                   axes aligned with the eigenmodes of the PMF-fiber):
Physically, we feel that we could recover some spectrum
information in the optical domain. Let us put some math-
                                                                                                                    
                                                                                           1 0         0          0
ematics behind this to prove that our system works. The                                 0 1           0          0 
formalism used to analyze this system is the Stokes formalism.                   Ms =  0 0 cos(δ) cos(δ)
                                                                                                                                  (6)
The Stokes formalism describes the polarisation state of the                               0 0 −sin(δ) cos(δ)
light through a 4-dimensional real vector. Let us analyze our
vibration tool mathematically:                                                                sint = Ms s                         (7)
The polarisation state launched by the laser is random.
                                                                     At the output of the fiber stretcher, we obtain:
   We use a polariser to modify that random polarisation state
to obtain a well-defined polarisation state:                                                                           T
                                                                                 sint = 1     0   cos(δ) −sin(δ)                  (8)
                                                     T
             sin = 1    cos(2φ) sin(2φ)          0           (1)     The analyzer can be described by an 4x4 Mueler matrix:
   with φ corresponding to the polarisation angle with respect                                                        
to the x-Axis. Let us assume that φ=45 ˚ :                                      1          cos(2θ)       sin(2θ)     0
                                                                         1 cos(2θ)       cos(2θ)2   sin(2θ)cos(2θ) 0
                                            T                      Ma =                                               
                    sin = 1    0    1   0                    (2)         2 sin(2θ) sin(2θ)cos(2θ)      sin(2θ)2     0
                                                                                0              0            0        0
  We describe the piezo-electric    fiber stretcher with the fol-                                                      (9)
lowing 4x4 Mueller matrix:                                          If we consider that θ=45 ˚ :
                                                                                                    
                    m11 m12          m13        m14                                            1 0 1 0
                  m21 m22           m23        m24                                      1  0 0 0 0
           Ms =                                            (3)                   Ma =                           (10)
                  m31 m32           m33        m34                                      2  1 0 1 0
                    m41 m42          m43        m44                                            0 0 0 0

                                                                                            sout = Ma sint                       (11)
                     m11 = 1, m12 = 0, m13 = 0
           m14 = 0, m21 = 0, m31 = 0, m41 = 0                                        1                                     T
                                                                            sout =     1 + cos(δ)     0   1 + cos(δ) 0           (12)
                              δ        δ                                             2
                    m22 = cos2 + sin2 cos4q
                              2        2
                                       δ                                     1                                          T
                           m23 = sin2 sin4q                          sout =     1 + cos(∆βz) 0 1 + cos(∆βz) 0               (13)
                                       2                                     2
                           m24 = −sinδsin2q                          The global power at the output is given by the first element
                                       δ                           of sout :
                           m32 = sin2 sin4q                                                  1
                                       2                                            sout0 = (1 + cos(∆βz))                  (14)
                              δ        δ                                                     2
                    m33 = cos2 − sin2 cos4q
                              2        2                             ∆β is constant with the stretch because we are working with
                             m34 = sinδcos2q                       a PMF-fiber. This has been proven in the reference [1][2].
                                   m42 = sinδsin2q                                     1
                                                                                 sout0 = (1 + cos(∆β(zi + kxexc ))               (15)
                               m43 = −sinδcos2q                                        2
                                        m44 = cosδ                   Let us suppose that the excitation is sinusoidal:

                                                                                        xexc = Asin(2πfexc t)                    (16)
                          δ = ∆βz                            (4)
   where ∆β is the fiber birefringence and δ is the phase                         1
retardance between the two eigenmodes of the PMF-fiber                  sout0 =     (1 + cos (∆β (zi + kAsin(2πfexc t))))         (17)
                                                                                 2
REPORT OF THE MULTIMEDIA PROJECT 2009, UMONS MA2, DECEMBER 2009                                                                       3



  We see in this equation that the temporal evolution of the         Finally, for the equation (with x cosinusoidal) cos(p1 + x),
output power depends on fexc . The information related to the        we have the following spectral components (frequency:
mechanical excitation is somehow comprised in equation (18).       amplitude)
Let us analyze equation (19) in detail to see if the optical                                        1 1
                                                                                    DC : cos(p1 ) − p2 cos(p1 )              (26)
spectrum is a good image of the excitation spectrum.                                                2 22
  To perform this, we will use the Taylor expansion in the
                                                                                                  1        p3     1
next section.                                                                 fexc : −sin(p1 )p2 + sin(p1 ) 2 (1 + )              (27)
                                                                                                  6         2     2
           IV. N ON - LINEARITIES IN THE SENSOR                                                   1        p2
                                                                                         2fexc : − cos(p1 ) 2                     (28)
  Let us develop equation (18):                                                                   2         2
                                                   
                                                                                            1         p3
 sout0
          1
         = 1 + cos ∆βzi + ∆βkA sin2πfexc t (18)
                                                                                           sin(p1 ) 2
                                                                                          3fexc :                                 (29)
          2                                                                                 6          4
                           p1       p2                               When performing simplifications, you have:
  Let us consider the Taylor expansion of this function:                                                      p2
                                                                                                               2
                                                                                        DC : cos(p1 )(1 −        )                (30)
  cos(p1 + x) = cos(p1 ) − sin(p1 )x                                                                          4
                 1              1                                                                                   p2
               − cos(p1 )x2 + sin(p1 )x3                                            fexc : −sin(p1 )p2 (−1 +         2
                                                                                                                       )          (31)
                 2              6                                                                                   8
                                    1
                                + cos(p1 )x4 + . . .       (19)
                                   24                                                                 p2
                                                                                                       2
                                                                                          2fexc : −      cos(p1 )                 (32)
  when considering that                                                                               4
                     x = p2 cos(2πfexc t)                  (20)                                   p3
                                                                                                   2
                                                                                           3fexc :   sin(p1 )                     (33)
                          p1 = ∆βzi                        (21)                                   8
                                                                      Let us analyze these results for a sinusoidal excitation at a
                         p2 = ∆βkA                         (22)    frequency fexc :
   p2 contains the information about the amplitude of the             • The system is non-linear: the optical spectrum is not the

mechanical excitation. We will assume that non-linearities are          same as the excitation spectrum.
non-negligble for the first three orders of the Taylor expansion.          – fexc at the excitation creates fexc , 2fexc , 3fexc ,... in
   Let us modify this equation to analyze the contribution of                the optical domain.
the higher orders on the lower orders.                                    – The amplitudes aren’t directly recoverable because
   The second order term gives:                                              we want to have access to p2 which has the infor-
                                                                             mation about the amplitude of the mechanical exci-
                                1
  x2 = p2 cos2 (2πfexc t) = p2 (1 + cos(2π(2fexc )t)) (23)
         2                    2
                                                                             tation. There is no term directly proportional to p2 .
                                2                                            We need to perform some mathematical operations
  The third order term gives:                                                to recover the good amplitudes.
                                                                      • The linear term provides us fexc .
  x3 = p3 cos3 (2πfexc t) =
        2                                                             • The linear term doesn’t provides us directly the good
                   1
                p3 (1 + cos(2π(2fexc )t))cos(2πfexc t) (24)
                 2
                                                                        amplitude.
                   2                                                  • All the coefficients depends on p1 .
  The Simpson product gives:                                          • By playing on the p1 parameter, we can suppress or
                                                                        maximize the amplitude of the fundamental and the
  cos(2π(2fexc )t)cos(2πfexc t) =                                       higher orders.
                  1                                                   • To recover the p2 amplitude at the first order, we need
                    (cos(2π(3fexc )t) + cos(2πfexc t)) (25)
                  2                                                     to solve the equation with provides us more than one
                                                                        solution.
                                                                          – 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 of
                                                                        the amplitude at 2fexc to obtain an equation which can
                                                                        provide us p2 .
                                                                      • To perform this, we should maximize cos(p1 ): the mea-
                                                                                                  p2
                                                                        sure at 2fexc gives us − 42
REPORT OF THE MULTIMEDIA PROJECT 2009, UMONS MA2, DECEMBER 2009                                                                                       4



  •   The maximization/minimization of cos(p1 ) is the same
      as 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.

                               2π         2πfopt
            p1 = ∆βzi =            ∆nzi =        ∆nzi                (34)
                              λopt          c
where fopt is the frequency of the laser,λopt the wavelength
of the laser,∆n is the difference in refractive indexes between
the x and y axes of the PMF fiber.
   Now, we know that the coefficients varies with cos(p1 ) or
sin(p1 ). Changing fopt linearly will modify the coefficients
periodically. This result is very important if we want to
influence the performance of the vibration sensor: we can act
on the linearity of the system by modifying the wavelength of
the source.

                       V. M ATLAB RESULTS
   We implemented the vibration sensor model on Matlab                       Fig. 3.   y(f)(m): excitation elongation, sout(f)(dB): power in the optical
to simulate its working in order to validate the theory we                   domain; suppression of H2
developped during this project. We launched a sinusoidal
excitation into the system at 120Hz. We validated the theory
first for a random frequency of the laser in order to see                                        VI. E XPERIMENTAL RESULTS
the generation of harmonics until the third order (the other
                                                                                We experimentally implemented the vibration sensor to test
are neglegible). We see that there is a generation of these
                                                                             the model in order to validate the theory we developped during
frequencies fexc , 2fexc , 3fexc ,... This result is provided in
                                                                             this project. We launched a sinusoidal excitation thanks to the
Fig. 2. On the other hand, we tuned the frequency (fopt ) of the
                                                                             generator into the system at 100Hz. We valitade the theory
                                                                             first for a random frequency of the laser in order to see
                                                                             the generation of harmonics until the third order (the other
                                                                             are neglegible). We see that there is a generation of these
                                                                             frequencies fexc , 2fexc , 3fexc ,... This result is provided in
                                                                             Fig. 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 harmonic
                                                                             at 2fexc increases. This result is provided in Fig. 5. However,
                                                                             the system we experimentally implemented isn’t completely
Fig. 2. y(f) (m): excitation elongation, sout(f)(dB): power in the optical   the same as the model tested on Matlab because the parameter
domain; generation of H1 H2 H3                                               θ isn’t equal to 45 ˚ for the analyzer. Therefore, we couldn’t
                                                                             have an exact replica of the amplitudes we simulated with
laser in order to maximize the amplitude of the fundamental                  Matlab. These results prove that the spectrum depends on the
frequency fexc . The first harmonic at 2fexc is then suppressed.              wavelength of the source and that by modifying its wavelength
This result is provided in Fig. 3.                                           we can suppress or maximize the fundamental and the higher
REPORT OF THE MULTIMEDIA PROJECT 2009, UMONS MA2, DECEMBER 2009               5



orders.




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



                         VII. P ERSPECTIVES
  We don’t have the exact experimental replica of the model
implemented on Matlab. Therefore, we could modify our
model implemented on Matlab and generalize it with a variable
θ angle. When we will be able to have the exact replica of the
model implemented on Matlab on the experimental way, we
will have the possibility to compare the amplitudes obtained
experimentally with the simulated results.

                         VIII. C ONCLUSION
   This project concerned the development of a vibration sen-
sor based on the use of optical fibres. The approach considered
concerned the realisation of a vibration sensor based on the
polarisation properties of optical fibres. The polarisation state
of the observed light at the output of a fibre varies with the
vibrations. After designing the system, the project consisted
in simulating his efficiency on Matlab. The practical testing
of the model has also been included to this project.
   We can conclude that the vibration sensor created provides
the results we expected theorically. We obtain coherent results
between the model implemented and the real device. These
results prove that the sensor is operational.

                         ACKNOWLEDGMENT
   The authors would like to thank the professor Marc Wuilpart
for his teaching method.

                             R EFERENCES
[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, The
    Netherlands,
[2] N. Ashby, D. A. Howe, J. Taylor, A. Hati, C. Nelson [National Institute
    of Standards and Technology],Optical Fiber Vibration and Acceleration
    Model, pp. 1 to 5.

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Vibration sensor simulation and design

  • 1. REPORT OF THE MULTIMEDIA PROJECT 2009, UMONS MA2, DECEMBER 2009 1 Vibration sensor based on optical fibre Conception and simulation Pierre Masure, Laurent-Yves Kalambayi Abstract—This project concerns the development of a vibration the fiber at a given frequency. The purpose of our proposed sensor based on the use of optical fibres, which will compensate solution will be to recover the excitation spectrum in the for the limitations of sensors based on classical technologies. optical domain. In order to do this, we should try to have the The approach considered concerns the realisation of a vibration sensor based on the polarisation properties of optical fibres. The best similarity between the excitation and the optical domain polarisation state of the observed light at the output of a fibre in an spectrum point of view: the recovery of the excitation varies with the vibrations. In this context, it is proposed to design spectrum in frequency and in amplitude at the optical output a simple sensor using out of the shelf equipment. The model has of the sytem should be obtained. The model of the Fiber been implemented on Matlab. The practical testing of the model Stretcher used is the PZ2-PM-1.5-FC/APC-E operating at a has also been included to this project. We can conclude that the vibration sensor created provides the results we expected wavelength of 1550 nm with a fiber stretch of 3.8µm/V, the theorically. We obtain similar results for the model implemented fiber length is 40 meters and the fiber wind is a 2-Layer wind. on Matlab compared to the real device. These results prove that The fiber used in the stretcher is a polarisation maintaining the sensor is operational. fiber which inhibits the polarisation mode coupling present in Index Terms—optical, fibre, sensor, vibration. normal optical fibers. I. I NTRODUCTION III. T HE SENSOR MODEL T HIS project concerns the development of a vibration sensor based on the use of optical fibres, which will compensate for the limitations of sensors based on classical To recover the excitation spectrum, we must create a system which can provide us a way to recover this spectrum in the optical domain. technologies. Sensors based on optical fibres are insensitive to electromagnetic perturbations, usable in harsh environ- ment (flammable environment, high temperatures, corrosion risks,...) and are appropriate for distributed measurements. The approach considered concerns the realisation of a vibration sensor based on the polarisation properties of optical fibres. The polarisation state of the observed light at the output of a fibre varies with the vibrations. In this context, it is proposed to design a simple sensor using out of the shelf equipment. After designing the system, the project will consist in simulating his efficiency on Matlab. Beforehand, it will be necessary to implement a fibre model taking into account the vibration effects on the polarisation properties. An additionnal part, which is the practical testing of the model, is included to this project. Fig. 1. Sensor model II. T HE FIBER STRECTHER On Fig.1 we show the measurement tool we created to The OPTIPHASE PZ2 High-efficiency Fiber Stretcher is a perform the given task, the vibration measurement. To perform fiber wound piezo-electric element for use in a wide range the task, we need: of optical interferometric measurement and sensing system • a laser applications. Typical uses include open loop demodulation, • a polarizer sensor simulation, white-light scanning interferometers and • the piezo-electric fiber stretcher large angle modulation of interferometric phase. PZ2 Fiber • an analyzer stretcher are available with SMF-28e+ or PM [PANDA] fiber • an oscilloscope+FFT types. We will use the fiber stretcher as a device to simulate We suppose that a laser launches any polarisation state at the vibration: the piezo-electric element which will stretch the input of the fiber. Therefore, we use a linear polarizer in order to have a defined state of polarisation at the input of M. Masure and Kalambayi are with the Department of Electrical Engineer- ing, UMons, Belgium the piezo-electric fiber stretcher. The light travels through the Manuscript received December 14, 2009; revised December 19, 2009. fiber winded up the piezo-electric element which stretches the
  • 2. REPORT OF THE MULTIMEDIA PROJECT 2009, UMONS MA2, DECEMBER 2009 2 fiber with an elongation driven by the signal provided by the of the stretcher and q is the azimuth of the fastest linear generator. The elongation has for effect to modifies the light polarization. polarisation state. After the piezo-electric fiber stretcher, we z = zi + kxexc (5) put a polariser which is used to analyze the light coming from where z is the full length of the fiber in the fiber stretcher, zi the stretcher. This particular polariser is denoted by the term the initial length of the fiber (40m), kxexc the elongation of ’analyzer’. the fiber due to the mechanical excitation, k the coefficient The physical idea behind this construction is that the power describing the linear stretch of the fiber with the voltage of the light will have an initial value if there is no stretch. (3.8µm/V), xexc the excitation signal of the piezo-electric When we induce a stretch to the fiber, the light polarisation element in Volt. state of the light will be modifiedin time. As a consequence, We make a simplification and suppose that q = 0 (x and y the power transmitted by the analyzer will vary in time. axes aligned with the eigenmodes of the PMF-fiber): Physically, we feel that we could recover some spectrum information in the optical domain. Let us put some math-   1 0 0 0 ematics behind this to prove that our system works. The 0 1 0 0  formalism used to analyze this system is the Stokes formalism. Ms =  0 0 cos(δ) cos(δ)  (6) The Stokes formalism describes the polarisation state of the 0 0 −sin(δ) cos(δ) light through a 4-dimensional real vector. Let us analyze our vibration tool mathematically: sint = Ms s (7) The polarisation state launched by the laser is random. At the output of the fiber stretcher, we obtain: We use a polariser to modify that random polarisation state to obtain a well-defined polarisation state: T sint = 1 0 cos(δ) −sin(δ) (8) T sin = 1 cos(2φ) sin(2φ) 0 (1) The analyzer can be described by an 4x4 Mueler matrix: with φ corresponding to the polarisation angle with respect   to the x-Axis. Let us assume that φ=45 ˚ : 1 cos(2θ) sin(2θ) 0 1 cos(2θ) cos(2θ)2 sin(2θ)cos(2θ) 0 T Ma =   sin = 1 0 1 0 (2) 2 sin(2θ) sin(2θ)cos(2θ) sin(2θ)2 0 0 0 0 0 We describe the piezo-electric fiber stretcher with the fol- (9) lowing 4x4 Mueller matrix: If we consider that θ=45 ˚ :     m11 m12 m13 m14 1 0 1 0 m21 m22 m23 m24 1  0 0 0 0 Ms =   (3) Ma =   (10) m31 m32 m33 m34 2  1 0 1 0 m41 m42 m43 m44 0 0 0 0 sout = Ma sint (11) m11 = 1, m12 = 0, m13 = 0 m14 = 0, m21 = 0, m31 = 0, m41 = 0 1 T sout = 1 + cos(δ) 0 1 + cos(δ) 0 (12) δ δ 2 m22 = cos2 + sin2 cos4q 2 2 δ 1 T m23 = sin2 sin4q sout = 1 + cos(∆βz) 0 1 + cos(∆βz) 0 (13) 2 2 m24 = −sinδsin2q The global power at the output is given by the first element δ of sout : m32 = sin2 sin4q 1 2 sout0 = (1 + cos(∆βz)) (14) δ δ 2 m33 = cos2 − sin2 cos4q 2 2 ∆β is constant with the stretch because we are working with m34 = sinδcos2q a PMF-fiber. This has been proven in the reference [1][2]. m42 = sinδsin2q 1 sout0 = (1 + cos(∆β(zi + kxexc )) (15) m43 = −sinδcos2q 2 m44 = cosδ Let us suppose that the excitation is sinusoidal: xexc = Asin(2πfexc t) (16) δ = ∆βz (4) where ∆β is the fiber birefringence and δ is the phase 1 retardance between the two eigenmodes of the PMF-fiber sout0 = (1 + cos (∆β (zi + kAsin(2πfexc t)))) (17) 2
  • 3. REPORT OF THE MULTIMEDIA PROJECT 2009, UMONS MA2, DECEMBER 2009 3 We see in this equation that the temporal evolution of the Finally, for the equation (with x cosinusoidal) cos(p1 + x), output power depends on fexc . The information related to the we have the following spectral components (frequency: mechanical excitation is somehow comprised in equation (18). amplitude) Let us analyze equation (19) in detail to see if the optical 1 1 DC : cos(p1 ) − p2 cos(p1 ) (26) spectrum is a good image of the excitation spectrum. 2 22 To perform this, we will use the Taylor expansion in the 1 p3 1 next section. fexc : −sin(p1 )p2 + sin(p1 ) 2 (1 + ) (27) 6 2 2 IV. N ON - LINEARITIES IN THE SENSOR 1 p2 2fexc : − cos(p1 ) 2 (28) Let us develop equation (18): 2 2    1 p3 sout0 1 = 1 + cos ∆βzi + ∆βkA sin2πfexc t (18)   sin(p1 ) 2 3fexc : (29) 2 6 4 p1 p2 When performing simplifications, you have: Let us consider the Taylor expansion of this function: p2 2 DC : cos(p1 )(1 − ) (30) cos(p1 + x) = cos(p1 ) − sin(p1 )x 4 1 1 p2 − cos(p1 )x2 + sin(p1 )x3 fexc : −sin(p1 )p2 (−1 + 2 ) (31) 2 6 8 1 + cos(p1 )x4 + . . . (19) 24 p2 2 2fexc : − cos(p1 ) (32) when considering that 4 x = p2 cos(2πfexc t) (20) p3 2 3fexc : sin(p1 ) (33) p1 = ∆βzi (21) 8 Let us analyze these results for a sinusoidal excitation at a p2 = ∆βkA (22) frequency fexc : p2 contains the information about the amplitude of the • The system is non-linear: the optical spectrum is not the mechanical excitation. We will assume that non-linearities are same as the excitation spectrum. non-negligble for the first three orders of the Taylor expansion. – fexc at the excitation creates fexc , 2fexc , 3fexc ,... in Let us modify this equation to analyze the contribution of the optical domain. the higher orders on the lower orders. – The amplitudes aren’t directly recoverable because The second order term gives: we want to have access to p2 which has the infor- mation about the amplitude of the mechanical exci- 1 x2 = p2 cos2 (2πfexc t) = p2 (1 + cos(2π(2fexc )t)) (23) 2 2 tation. There is no term directly proportional to p2 . 2 We need to perform some mathematical operations The third order term gives: to recover the good amplitudes. • The linear term provides us fexc . x3 = p3 cos3 (2πfexc t) = 2 • The linear term doesn’t provides us directly the good 1 p3 (1 + cos(2π(2fexc )t))cos(2πfexc t) (24) 2 amplitude. 2 • All the coefficients depends on p1 . The Simpson product gives: • By playing on the p1 parameter, we can suppress or maximize the amplitude of the fundamental and the cos(2π(2fexc )t)cos(2πfexc t) = higher orders. 1 • To recover the p2 amplitude at the first order, we need (cos(2π(3fexc )t) + cos(2πfexc t)) (25) 2 to solve the equation with provides us more than one solution. – 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 of the amplitude at 2fexc to obtain an equation which can provide us p2 . • To perform this, we should maximize cos(p1 ): the mea- p2 sure at 2fexc gives us − 42
  • 4. REPORT OF THE MULTIMEDIA PROJECT 2009, UMONS MA2, DECEMBER 2009 4 • The maximization/minimization of cos(p1 ) is the same as 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. 2π 2πfopt p1 = ∆βzi = ∆nzi = ∆nzi (34) λopt c where fopt is the frequency of the laser,λopt the wavelength of the laser,∆n is the difference in refractive indexes between the x and y axes of the PMF fiber. Now, we know that the coefficients varies with cos(p1 ) or sin(p1 ). Changing fopt linearly will modify the coefficients periodically. This result is very important if we want to influence the performance of the vibration sensor: we can act on the linearity of the system by modifying the wavelength of the source. V. M ATLAB RESULTS We implemented the vibration sensor model on Matlab Fig. 3. y(f)(m): excitation elongation, sout(f)(dB): power in the optical to simulate its working in order to validate the theory we domain; suppression of H2 developped during this project. We launched a sinusoidal excitation into the system at 120Hz. We validated the theory first for a random frequency of the laser in order to see VI. E XPERIMENTAL RESULTS the generation of harmonics until the third order (the other We experimentally implemented the vibration sensor to test are neglegible). We see that there is a generation of these the model in order to validate the theory we developped during frequencies fexc , 2fexc , 3fexc ,... This result is provided in this project. We launched a sinusoidal excitation thanks to the Fig. 2. On the other hand, we tuned the frequency (fopt ) of the generator into the system at 100Hz. We valitade the theory first for a random frequency of the laser in order to see the generation of harmonics until the third order (the other are neglegible). We see that there is a generation of these frequencies fexc , 2fexc , 3fexc ,... This result is provided in Fig. 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 harmonic at 2fexc increases. This result is provided in Fig. 5. However, the system we experimentally implemented isn’t completely Fig. 2. y(f) (m): excitation elongation, sout(f)(dB): power in the optical the same as the model tested on Matlab because the parameter domain; generation of H1 H2 H3 θ isn’t equal to 45 ˚ for the analyzer. Therefore, we couldn’t have an exact replica of the amplitudes we simulated with laser in order to maximize the amplitude of the fundamental Matlab. These results prove that the spectrum depends on the frequency fexc . The first harmonic at 2fexc is then suppressed. wavelength of the source and that by modifying its wavelength This result is provided in Fig. 3. we can suppress or maximize the fundamental and the higher
  • 5. REPORT OF THE MULTIMEDIA PROJECT 2009, UMONS MA2, DECEMBER 2009 5 orders. Fig. 5. Power in the optical domain; maximization of H2 VII. P ERSPECTIVES We don’t have the exact experimental replica of the model implemented on Matlab. Therefore, we could modify our model implemented on Matlab and generalize it with a variable θ angle. When we will be able to have the exact replica of the model implemented on Matlab on the experimental way, we will have the possibility to compare the amplitudes obtained experimentally with the simulated results. VIII. C ONCLUSION This project concerned the development of a vibration sen- sor based on the use of optical fibres. The approach considered concerned the realisation of a vibration sensor based on the polarisation properties of optical fibres. The polarisation state of the observed light at the output of a fibre varies with the vibrations. After designing the system, the project consisted in simulating his efficiency on Matlab. The practical testing of the model has also been included to this project. We can conclude that the vibration sensor created provides the results we expected theorically. We obtain coherent results between the model implemented and the real device. These results prove that the sensor is operational. ACKNOWLEDGMENT The authors would like to thank the professor Marc Wuilpart for his teaching method. R EFERENCES [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, The Netherlands, [2] N. Ashby, D. A. Howe, J. Taylor, A. Hati, C. Nelson [National Institute of Standards and Technology],Optical Fiber Vibration and Acceleration Model, pp. 1 to 5.