Detection of Pseudo-singularities by Wavelet

Progress In Electromagnetics Research Symposium, Beijing, China, March 23–27, 2009

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Detection of Pseudo-singularities by Wavelet Technique for Extracting Leaky and Bulk Waves in Piezoelectric Material D. Benatia1 , T. Fortaki1 , and M. Benslama2 1

D´epartement d’Electronique, Facult´e des Sciences de l’Ing´enieur Universit´e de Batna, Algeria 2 D´epartement d’Electronique, Facult´e des Sciences de l’Ing´enieur Universit´e de Constantine, Algeria

Abstract— In this paper, we propose a new numerical method for leaky and bulk wave detection of an acoustic microwave signal during the propagation of acoustic microwaves in a piezoelectric substrate. Moreover, we know that the Fourier transform presents a global spectral study of signal, this is not interesting if we want to study a signal locally and know its features in a more precise manner. By the use of wavelet transform, we can reduce this drawback. The originality of the wavelet transform consists of the local analysis of signal singularities (or signal pseudosingularities) where abrupt events appear and hence access to hidden information by using the scale of this transform as up scaling parameters. These pseudo-singularities (correspond to abrupt variations) inform us of presence of leaky and bulk waves in piezoelectric materials. Furthermore, this transform proved its efficiency in many applications, such as signal processing and the analysis of waves in microstrip structures. Hence, it can play an important role in the modeling of pseudo-singularities in acoustoelectronic. 1. INTRODUCTION

The investigation of bibliography in micro-acoustic area permits us to point the state of the art. Two major works can be mentioned, the first one is Greeb’ paper [1], which explore the interaction using effective permittivity concept, another work of Lakin [2] which elaborate a perturbation theory to explain the interaction phenomena. Following the work of Greeb, Milson [3] in 1977 elaborates a relation based of the charge density, and develops a formalism based on the Fourier Transform. In spite of it, this method doesn’t permit to distinguish easily the different modes of propagation by a numerical methods based on the inverse Fourier transform. These works were taken by Junjhunwalla [4] that granted a particular attention to the SSBW (Surface Skimming Bulk Waves). Yashiro and Goto [5] introduce the method of the stationary phase of Lightill [6] to calculate the pseudo-singularities that inform us on the presence of the pseudo- waves, particularly the leaky and bulk waves. In our case, we propose another approach for the modelling of the acoustic microwaves with a complementary vision to the literature mentioned above. In this approach we interested especially in the detection of pseudo-singularities by the use of a wavelet transform as detection tool [7, 8] in order to mark the mode of a leaky waves and the mode of a bulk waves [9–12]. 2. PHENOMENOLOGICAL TENSORIAL PIEZOELECTRIC EQUATIONS

The signal to be treated will be applied to the electrodes of the transducer that generate the compression and dilatation, so a piezoelectric wave is generated and propagated in the X direction (Figure. 1). We consider the space coordinates: X1 = X, X2 = Y , X3 = Z. The mechanical state of the medium is defined by two magnitudes of tensorial type, the stress Tij and the mechanical deformation (Strain) Sij (i, j = 1, 2, 3). The electric state of the medium is defined by two vectors, the electric field Ek and the electric induction Di . The stress tensor and the electric induction are given by: Tij = Cijkl · Skl − ekij · Ek Di = eikl · Skl + εik · Ek

(1) (2)

with i, j, k, l = 1, 2, 3. Where εik : permittivity tensor (F/m), ejkl : piezoelectric tensor (c/m), Cijkl : elastic tensor (N/m2 )

PIERS Proceedings, Beijing, China, March 23–27, 2009

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Y Interdigital transducer Leaky waves (Propagation out of the crystal)

Input of signal X Output of signal Surface elastic waves

Z

Bulk waves (into the crystal) Piezoelectric substrate (LiNbO3)

Figure 1: LiNbO3 crystal excited by transducer.

The strain is bound to the relative displacements of the particles of the material environment is defined by: µ ¶ ∂Uj 1 ∂Ui Sij = + (3) 2 ∂Xj ∂Xi where Ui represents the elastic displacement of the particle (i = 1, 2, 3). Note that in the quasi-static approximation, we can define an electric field of components: Ei = −

∂U4 ∂Xi

(4)

where U4 is the electric potential (with i = 1, 2, 3) In the quasi-static approximation, the Maxwell’s equation amount to the Poisson’s equation: ~ = ∂Di = 0 div · D ∂Xi

(5)

The movement of the particles under the action of stress (constraints), is described by the following: ∇T =

∂Tij ∂ 2 Uj =ρ· ∂Xi ∂t2

where ρ is the mass density of medium. Replacing (3) and (4) in (1) and (2), we obtain: µ ¶ 1 ∂Uk ∂Ul ∂U4 Tij = Cijkl · + + ekij · 2 ∂Xl ∂Xk ∂Xk µ ¶ 1 ∂Uk ∂Ul ∂U4 + − εik · Di = eikl · 2 ∂Xl ∂Xk ∂Xk

(6)

(7) (8)

Replacing (7) and (8) in (5) and (6), we obtain the piezoelectric tensorial equations: ∂ 2 uj ∂ 2 uk ∂ 2 U4 + elij = ρ 2 ∂Xi ∂Xl ∂Xk ∂Xi ∂t 2 2 ∂ uk ∂ U4 eikl − εik = 0 ∂Xi ∂Xl ∂Xk ∂Xi

Cijkl

(9) (10)

3. THE FORM OF SOLUTION

Consider the following form of the surface wave (partial wave): Ui = ui exp (jk · αi Y ) exp −j [ω · t − k(1 + jγ)X]

(11)

where ui (i = 1, 2, 3) are the displacement amplitudes, ui (i = 4) is the amplitude of the electric potential, k is the constant of propagation, the αi are the penetration coefficients of the wave inside

Progress In Electromagnetics Research Symposium, Beijing, China, March 23–27, 2009

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the piezoelectric substrate (Figure 1), γ is the coefficient of longitudinal attenuation and ω is the angular pulsation. Equations (9) and (10) can be written in a matrix form as: [A][U ] = [0]

(12)

with [U ] = [u1 , u2 , u3 , u4 ]T , [A] is a matrix (4 × 4). The determinant of the matrix [A] must be zero to ensure a non trivial solution, it can be written as: 8 X βi · αi = 0 (13) i=0

where Bi depends on the piezoelectric material features (Cijkl , εik , elij ) and of the acoustic velocity VS . The determinant of the matrix [A] must be zero, we have eight complex roots (i = 1 . . . 8): αi = ai + jbi

(14)

where ai : is the real part and bi : is the imaginary part, with am = am+1 and bm = −bm+1 where m = 1, 3, 5, 7. In the surface mode (or Rayleigh wave) the αi (i = 1 . . . 8) are conjugated by pairs and only the complex roots with negative imaginary part are taken into consideration (for convergence reasons). Let us first neglect the longitudinal attenuation (γ = 0) and insert (14) in (11) to obtain: Ui = ui exp −(bi k · Y ) exp −j [ω · t − k (X + ai Y )]

(15)

If we go inside the crystal (Y tends to −∞), the wave Ui tends to zero. This corresponds to surface acoustic waves (S.A.W) (Figure 1). In the opposite case (Y tends to +∞), Ui tends +∞ (without physical signification). 4. LEAKY AND BULK WAVES

The variation of the acoustic velocity VS allows us to obtain bi = 0 (imaginary part) and (15) becomes: Ui = ui · exp −j [ω · t − k · (X + ai Y )] (16) The wave nature bulk waves (B.W) and leaky waves (L.W) depends on the sign of the real part of αi (ai ). If ai is negative, we have bulk waves (Propagation inside the crystal (Figures 1 and 2)). If ai is positive, we have leaky waves (radiation out of the crystal (Figures 1 and 2)). 0.6

0

Piezoelectric cristal LiNbO3 Y-X Cut

Piezoelectric crystal LiNbO3 Y-X Cut -0.1

0.4 Singularities

L.W

0.2

-0.2

0

-0.3

-0.2

-0.4

-0.4

-0.5

B.W -0.6

2000

Singularities 2500

-0.6 2000

Nul Imaginary part

Acoustic velocity (m/s)

3000 3500 4000 Acoustic velocity (m/s)

(a)

(b)

3000

3500

4000

4500

5000

2500

Figure 2: Penetration coefficient α4 . (a) Real part. (b) Imaginary part.

4500

5000

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5. GENERAL FORM OF THE ACOUSTIC WAVE SOLUTION

The general form of Ui (i = 1, 2, 3, 4) is expressed by: Ui (k, Y ) =

4 X

(n)

Cn · Ai

· exp−j[ωt−k·αn ·Y ]

(17)

n=1 (n)

where Ai are the components of eigen vectors associated to coefficients αn [3] Cn are the constants determined by the traction-free boundary conditions [3]. (n) the product Cn · Ai depends on Cijkl , εik , elij and of the acoustic velocity VS . For i = 4, we have a potential electric given by: U4 (k, Y ) =

4 X

(n)

Cn · A4 · exp−j[ωt−k·αn ·Y ]

(18)

n=1

6. WAVELET TRANSFORM OF SIGNAL

The signal chosen for analysis by wavelet transform is given by Equation (18). It represents an electric potential “U4 ” coupled with an elastic wave of components “U1 , U2 , U3 ”. This wave with the frequency “f ” propagates along the X direction and guided on the free surface of the piezoelectric (Figure 1). The space-scale type wavelet transform of U4 (in the neighbourhood of the surface: Y ≈ 0) is given by this convolution product [8]: µ ¶ 1 X TU4 (Y ≈ 0, X, a) = U4 (Y ≈ 0, X) ⊗ √ Ψ∗ (19) a a where U4 (Y ≈ 0, X) is a signal in the neighbourhood of the material surface (Y ≈ 0). 2 Ψ∗ (X) is the complex conjugate of the wavelet (Mexican-hat): Ψ(X) = d2 /dX 2 (e−X /2 ) The frequency-scale type wavelet transform of U4 becomes a simple product: √ TU4 (Y ≈ 0, f, a) = aU4 (Y ≈ 0, f ) · Ψ∗ (a · f )

(20)

where U4 (Y ≈ 0, f ) is the Fourier transform of a signal in the neighbourhood of the material surface 2 2 Ψ∗ (f ) is the Fourier transform of Ψ∗ (X): Ψ∗ (f ) = (2 · π)1/2 · e−(4·π ·f /2) · (2 · π · f )2 . ∗ Replacing the expression of Ψ (a · f ) in Equation (20), the wavelet transform of U4 in this case becomes: √ 4·π 2 a2 ·f 2 TU4 (Y ≈ 0, f, a) = |U4 (k, 0)| · 2 · a · π · e− 2 · (2πf · a)2 (21) | {z } Ψ∗ (a·f )

With |U4 (k, 0)| =

4 P n=1

(n)

Cn · A4 , k = 2 · π · f /VS

7. RESULTS AND DISCUSSION

The detection of leaky and bulk waves appears at the level of the penetration coefficients when the acoustic velocity Vs change its value. This change results in an annulation of the imaginary parts of the penetration coefficients. We note that when the imaginary part becomes null, it appears at the level of signal wavelet transform an abrupt variations called pseudo-singularities. These pseudo-singularities are not always observable. The use of the scale of this transform permits to visualise them. Once these pseudo-singularities are detected, the sign of the real part can inform us about the leaky wave (ai > 0) and the bulk wave (ai < 0). The analysis of the signal by wavelet transform (Equation (21)) clearly shows these pseudosingularities at the level of the contour of three-dimensional figure (the above view). This figure englobes the Wavelet Transform of “U4 ”, the Acoustic Velocity “VS ” and the Scale “a” with the frequency “f ” as parameter (Figures 3, 4 and 5). At the frequency f = 1 GHZ, it is impossible to detect the pseudo-singularities for a scale superior to 10−8 , In this case we can’t detect the pseudo-singularities (Figure 3). For a good detection, it is necessary to reduce the scale from 10−8 to 10−10 (Figures 4 and 5), these pseudo-singularities appear more and more clearly.

Progress In Electromagnetics Research Symposium, Beijing, China, March 23–27, 2009

Figure 3: The wavelet transform of the U4 Scale order (10−8 ).

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Figure 4: The wavelet transform of U4 Scale order (10−10 ).

Good detection of pseudo-singularities

Figure 5: The above view of Figure 4 (Good detection of pseudo-singularities).

8. CONCLUSION

In this work, we have developed a model for studying the behaviour of electroacoustic waves at the level of pseudo-singularities which appear for some velocities giving the leaky and bulk waves. This model that insures the detection of pseudo-singularities using the wavelet transform. This latter has the property of being locally maximal around the points where the signal is singular or pseudo-singular. In this paper, we were interested in the influence of the scale “a” of this transform. The decrease of the scale allowed us to detect the pseudo-singularities whatever the frequency so that we can obtain all the details about the propagation of acoustic waves in particular on the leaky and bulk waves. This information can be useful for many applications such us the antenna and the oscillator devices. REFERENCES

1. Greeb, C. A. J., et al., “Electrical coupling properties of acoustics and electric surface waves,” Physical Report, Vol. 1, No. 5, 235–268, 1971. 2. Lakin, K. M., “Perturbation theory for electromagnetic coupling to elastic surface waves on piezoelectric substrates,” J. Appl. Phys., Vol. 42, No. 3, 899–906, 1971. 3. Milson, R. F., N. H. C. Reilly, and M. Redwood, “Analysis of generation and detection of surface and bulk acoustic waves by interdigital transducers,” IEEE Trans. Sonics and Ultrason., Vol. 24, No. 3, 147–166, 1977. 4. Jhunjhunwalla, A. and J. F. Vetelino, “Spectrum of acoustic waves emanating from an IDT on a piezoelectric half space,” Proceedings of IEEE Ultrasonics Symposium, 945–952, 1979.

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5. Yashiro, K. and N. Goto, “Analysis of generation of acoustic waves on the surface of a semiifinite piezoelectric solid,” IEEE Trans. Sonics and Ultrason., Vol. 25, No. 3, 146–153, 1978. 6. Lightil, J., Waves in Fluid, University Press, Cambridge, 1978. 7. Mallat, S. and W. L. Wan, “Singularity detection and processing with wavelets,” IEEE Trans. on Information Theory, Vol. 38, 617–643, 1992. 8. Arneodo, A., F. Argoul, E. Bacry, J. Elezgaray, and J. F. Muzy, Ondelettes, Multifractales et Turbulences, Diderot (ed.), Art et Sciences, Paris, France, 1995. 9. Josse, F. and D. L. Lee, “Analysis of the excitation, interaction, and detection of bulk and surface waves on piezoelectric substrates,” IEEE Trans. Sonics and Ultrason., Vol. 29, 261–273, 1982. 10. Goodberlet, M. A. and D. L. Lee, “The excitation and detection of surface generated bulk waves,” IEEE Trans. Sonics and Ultrason., Vol. 31, 67–76. 1984. 11. Fusero, Y., S. Ballandras, J. Desbois, J. M. Hod´e, and P. Ventura, “SSBW to PSAW conversion in SAW devices using heavy mechanical loading,” IEEE Trans. on Ultrason., Ferroelectrics, and Frequency Control, Vol. 49, No. 6, 805–814, 2002. 12. Martin, F., “Propagation characteristics of harmonic surface skimming bulk waves on ST,” Electron Lett., Vol. 38, 941–942, August 2002.