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Physics Procedia 00 (2009) 000–000 Physics Procedia 3 (2010) 585–591 www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
International Congress on Ultrasonics, Universidad de Santiago de Chile, January 2009
Bessel-like response in transducer based on homogeneously poled piezoelectric disks: Modeling and experimental analysis H. Calása,*, J. A. Eirasb, D. Contib, L. Castellanosa, A. Ramosa, and E. Morenoc b
a Instituto de Acústica-CSIC, Serrano 144, 28006 Madrid, España UFSCar, Rod. Washington Luis, km 235, 13.565-905 San Carlos- SP Brasil c LEIA, Leonardo Da Vinci, 11, 01510, Alava, España
Elsevier use only: Received date here; revised date here; accepted date here
Abstract The study of ultrasonic Bessel transducers is motivated by their ability to produce limited diffraction ultrasonic beams, providing improved lateral resolution and large penetration depth in the related ultrasonic field. There are three principal methods to construct a Bessel transducer. On the other hand, it is well known that radial modes in homogeneously poled piezoelectric disks (HPPD) are governed by a typical Bessel differential equation. In this paper, the possibility to generate Bessel radiation using a thickness HPPD is proposed and experimentally evaluated, by analyzing the field effects of the disk vibrations when it is driven in its radial modes PACS PACS: 4335
Keywords: Bessel transducer, radial modes, limited diffraction beam, piezoelectric disk
1. Introduction The achieving of a good lateral resolution in pulsed acoustic beams, maintained along a large field depth, is a quite important problem in many medical and industrial ultrasonic imaging applications. To solve it, in the necessary broadband conditions, by using the rather simple conventional ultrasonic devices is practically impossible. For this motive, complex array systems for dynamic electronic focusing and special transducers with limited diffraction have been studied and proposed by many researchers in papers and patents. The Bessel beam solution, for the scalar wave equation in isotropic/homogeneous media, is a particular case of limited diffraction waves. It was found in the optics by J. Durnin 0 in 1987 and was extended to acoustic by D. K. Hsu 0 for Bessel transducers, in 1989. Three principal methods were proposed to construct Bessel transducers: to apply a Bessel polarization over a piezoelectric disc implemented by D. K. Hsu 0, to construct an annular array
* Corresponding author. Tel.: +34-91-5618806; fax: +34-91-4117641. E-mail address:
[email protected]
doi:10.1016/j.phpro.2010.01.075
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using a composite 1-3 implemented by J. Lu 0, and to construct an annular array from a piezoelectric disc of discrete Bessel polarization presented by J. A. Eiras 0. On the other hand, it is well known that the radial modes in homogeneously poled piezoelectric discs (HPPD), related to one-dimensional models, appears as governed by a typical Bessel differential equation. The possibility to exploit the Bessel behaviour of radial modes in a homogeneous piezoelectric disk as modal transducers was presented by H. Calás 0 in a recent report. In this work, the viability of using the radial modes in HPPD to generate a Bessel beam is investigated. For analyzing the radial modes, we use the classical model Meitzler-O’Bryan-Tiersten 0, because this model has an implicit relation with thickness modes. In addition, acoustic field simulations, using the Rayleigh-Sommerfeld formulation, are applied for evaluating the ultrasonic fields indirectly generated by the radial modes deformations. This Bessel-like indirect radiation was experimentally confirmed by exciting piezoelectric disc transducers in their radial resonance frequencies. Experimental and simulated results, for radial vibration modes and acoustic radiations are presented and discussed.
2. Bessel beam solution The isotropic/homogenous scalar wave equation for the velocity potential I (r , M , z, t ) , in cylindrical coordinates, and assuming circular symmetry around the propagation axis of the acoustic perturbation through the radiated medium, is given by:
ª 1 w § w · w2 1 w2 º « ¨ r ¸ 2 2 2 » I(r, z, t ) 0. ¬ r wr © wr ¹ wz c wt ¼
(1)
Where, r , z, t and c are: the radial coordinate, the axial coordinates, the time and the sound velocity, respectively. For a given value of wavenumber, k Z / c , the equation (1) has Bessel beam solutions, of the form:
I(r, z, t) J0 (D r)eiE z eiZt D2 E
2
k2
(2)
For obtaining a Bessel beam in the irradiated medium, a Bessel-like excitation is necessary over the transducer emitting surface. But when such a transducer is constructed with this purpose, in the practice, the excitation function in the radiating aperture must be quantized with annular approaches, or well, alternatively, a difficult polarization process must be implemented. 2.1. Acoustic field calculation The Rayleigh-Sommerfeld integral 0 is used to simulate the acoustic field generated by a finite aperture transducer. This integral can be written in the following form:
G I (r , t )
§ 1 ¨ c ( ) v t ³ 1 ¨ 2S ©
G v2 (rT )G (t t c R ) · c ds ¸ dt c T ³³S ¸ R T ¹
(3)
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Where v1 and v 2 are the piston velocity waveform and spatial distribution of velocity, respectively,
G distance between the transducer differential element dsT and the point described by the vector r .
R
is the
2.2. Theoretical model for radial modes The vibration modes in homogeneously poled piezoelectric discs (HPPD) have been intensively studied in the past. For a qualitative analysis of radial modes, we used the Meitzler–O’Bryan–Tiersten (MOT) model 0, which considers the coupling between thickness and radial vibrations in a disk of radius a and thickness 2b, assuming that a >>2b. Free stress is also considered, and the angular dependence is neglected (/ș = 0, uș= 0). For this model, the field of displacement (ur), the stress in the radial direction (Trr) and the electrical impedance Z, can be written in the following form 0: §Z AJ 1 ¨ ¨v © p
ur
§Z §Z A ¨ rJ 0 ¨ ¨ vp ¨ vp © ©
Trr
Z
· §Z r ¸ (V p 1) J1 ¨ ¸ ¨ vp ¹ ©
p 2 § Z ·ª ° 2( e31 ) J1 ¨ a ¸ « 1V ® p p ¨ ¸ S a 2H 33 Z ° c11p H 33 © v p ¹ «¬ ¯ i 2b
p
· r ¸ e iZ t ¸ ¹
(4)
·· ep r ¸ ¸ 31 V ¸ ¸ 2b ¹¹
§ Z ·º J1 J 0 ¨ a ¸» ¨ v p ¸» © ¹¼
1
(5)
½ ° 1¾ ° ¿
1
(6)
Where,
ª 1 V p J §Z ° A «® 1¨ ¨v «° a © p ¯ ¬
· Z §Z r ¸ p J0 ¨ ¸ v ¨v ¹ © p
1 · ½° e31p a V º » r ¸¾ p ¸ » 2 c b 11 ¹ ¿° ¼
Vp
vp
2
p c12
(7)
(8)
p c11
p c11
(9)
U
Also, using this MOT model, it is possible to obtain an expression for the surface velocity (vz) 0. b
vz
w ³0 wt S zz dz
ª cE §Z iZ b « 13E AJ 0 ¨ ¨ vp c © ¬« 33
· e V º r ¸ 33E ¸ c 2b »» ¹ 33 ¼
(10)
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p
E
p
p
where: cij and eij are elastic and piezoelectric constants, respectively; cij , eij , H ij are planar elastic, planar piezoelectric and planar dielectric constants, respectively (for more details, to see 0, eq. 10); U is the mass density and V is the voltage.
3. Simulations The piezoelectric material considered here is of PZT type; Table 1 shows the numerical values of the physical constants used for simulation. Table. 1 2
PZT
2
S12 (m /N) -5.74e-12
S11 (m /N) 1.64e-11
D31 (m/V) 1.71e-10
U (kg/m3 )
H33(F/m) 1700
7450
Figure 1.a shows the behaviour of its electrical impedance |Z| for a typical real uniformly-poled piezoelectric disk, with a radius a = 12.5 mm and a thickness 2b = 1 mm, remarking the resonance frequencies. Figure 1b shows its surface velocity in the z direction, under open-circuit electrical condition, for the radial modes 3, 4 and 5. It is interesting to note that, for these radial modes, vz behaviour is very close to a Bessel function of first kind and cero order. Based on this circumstance and on the above expression in (2), we have established the hypothesis of that a HPPD transducer could radiate ultrasonic beams in a similar way to a conventional Bessel transducer, under the condition that it is electrically excited in its radial modes frequency bands. G In figure 2, the simulation of the radiation field, using the expression (3), is shown; where, v 2 ( rT ) was calculated normalizing equation (10), and v1 (t c) is a tone burst with a carrier frequency of 453 KHz and formed by three cycles. It can be appreciated a typical Bessel radiation in both, xz (fig.2a) and xy (fig.2b) planes 0. -11
-7.114
5
10
x 10
f r5=575.4KHz
fr1=80.4KHz
-7.115
f r3=332.2KHz
4
10
f r4=453.4KHz
3
v z /w
|Z |
-7.116
-7.117
fr3=332.2KHz
10
f r5=575.4KHz
-7.118 2
10
-7.119
fr4=453.4KHz
fr2=208.6KHz
(b)
(a)
1
10
0
1
2
3
4 f(Hz)
5
6
7 5
x 10
-7.12
-0.01
-0.005
0 r(m)
0.005
0.01
Fig.1 Radial vibration modes as a function of the frequency for an uniformly poled disk: (a) Behaviour of the electrical impedance |Z|, and (b) radial distribution of the z velocity for modes 3 , 4 and 5 of a disk (a = 12.5 mm and 2b = 1 mm).
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However, in this point, an interesting question arises: is sufficient?, the energy given by the radial modes, to generate an appraisable ultrasonic radiation in the z direction. To answer this question, could be a complex task, using only simulation tools.
4. Experiments and measurements
4.1. Measurement procedure In order to assess the hypothesis presented in the previous section, a transducer with homogeneous polarization was constructed using a ceramic (PZT 53/47 + 1 wt % Nb) disk with thickness of 1 mm and a diameter of 25 mm. The so constructed transducer can be considered without backing. This transducer was excited with the frequency of the fourth radial mode (453 KHz). The driving pulse was generated with a Matec card TB-1000. The radiated signal was scanned by using a needle broadband piezoelectric hydrophone (0,6 mm in effective diameter), and the field RF signals were captured using a digital oscilloscope TSD 744 (Tektronix) of 2 GHz in sampling rate.
(a)
(b)
Fig.2 Simulation of the radiation field emitted in z-axis direction, from the radial vibration modes of a HPPD transducer. a) plane xz, and b) plane xy.
4.2. Experimental results In the figure 3a, a spatial distribution along x axis of the field RF signals measured by the broadband hydrophone, at 5 mm from the transducer radiation face, is shown. By considering a fixed time value in the waveform, we can analyze the field effects of the transducer surface vibrations.
15
0.12 0.1
10
0.1
0.08
(a)
0.06
0.08 0.02
0
0
-0.02
-5
-0.04
0.06 V(v)
x(mm)
(b)
0.04
5
0.04 0.02
-0.06
-10
-0.08
0
-0.1
-15 -15
-10
-5 t(s)
0
5 -6
x 10
-0.02 15
10
5
0 x(mm)
-5
-10
-15
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Fig.3 1-D spatial distributions measured in the x axis of the RF signals for the 4th e radial mode, experimentally measured by the hydrophone at z = 5 mm: a) scan in x direction (RF signal); b) profile of hydrophone voltage on time fixed..
This experiment can answer in part the question formulated in the previous section 3. Figure 3b shows the field profile, by fixing the time at an instant with a maximum field value in x = 0. Both figures illustrate a Bessel-like behaviour of the transducer under radial driving. The fourth radial mode had been excited and four lobules are shown in figures 3b. Therefore, we can sustain that a homogeneous poled piezoelectric disk, properly driven, can radiate in a way very similar to a conventional Bessel transducer. However, a critical topic is the resulting depthRI ILHOG IRU WKLV VSHFLDO %HVVHOOLNH UDGLDWLRQ 7KH H[SUHVVLRQ IRU ILHOG depth LQ D %HVVHO WUDQVGXFHU LV ZHOO NQRZQ 2
DoF
§k· a ¨ ¸ 1 ©D ¹
(11)
In the case presented here, the depth of field is around 20 mm. This poor depth of field is a consequence of the proximity between the values of k and D . However, this limitation could be convenient in some applications, because, in the far field, a good beam lateral collimation (up to 300 mm) is maintained and lateral lobes are quite minimized. In figure 4, the mentioned behaviour is clearly shown.
campo acustico 3D 0 -2 -4 -6 -8 300
Amplitude (dB)
250 0
200
-20 150
-40 100 0
xEixo (mm)Y (mm)
50 -50 -100
-12 -14 -16 -18
100
50
-10
y(mm) Eixo X (mm)
-20 -22
0
Fig.4 Measurements of 2-D spatial distribution of the radiation field related to the fourth radial mode.
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5. Conclusions In this contribution, some secondary vibrations and ultrasonic radiations of a thickness HPPD, alternatively driven in its radial modes, were predicted, generated and experimentally confirmed. For simulating the radial vibration modes, the classical model of Meitzler-O’Bryan-Tiersten was applied, and, for calculating the radiation acoustic field involved, the Rayleigh-Sommerfeld integral was implemented for this particular radiating configuration. These Bessel-like radiations were experimentally obtained by exciting piezoelectric the disk transducers with tone bursts tuned in their resonant radial modes. It can be concluded that a homogeneously poled piezoelectric disk could work as a Bessel ultrasonic transducer if it is narrow-band excited in its radial modes.
Acknowledgements The authors would like to express their thanks to the projects SUCODIC (CYTED) and CITMA/CNPq for their founding. Science & Innovation Ministry of Spain (R&D Projects PN DPI2005-00124 & DPI2008-05213) also offered financial support for this work. Finally, our acknowledgement to Eng. Pedro T. Sanz, for his experimental assistance. The scientific stay of Dr. H. Calás in Instituto de Acústica is been supported by the JAE-CSIC Program.
References [1] J. Durnin; J. J. Miceli Jr., J. H. Eberly, “Diffraction-Free Beams”, Phys. Rev. Lett., Vol. 58 (15), p. 1499, 1987. [2] D. K. Hsu, F. J. Margetan, D. O. Thompson, “Bessel Beam Ultrasonic Transducer: Fabrication Method and Experimental Results”, Appl. Phys. Lett., Vol. 55 (20), p. 2066, 1989. [3] J. Lu and J.F. Greenleaf, ”Ultrasonic non-diffracting transducer for medical imaging,” IEEE Transactions on ultrasonic. Ferroelectrics, and frequency control. Vol 37 no 5, pp 438-447, 1990. [4] J.A. Eiras, E. Moreno, H. Calas, A. Aulet, C. A. Negreira, L. Leija, G. González, “Vibration modes in ultrasonic Bessel transducer”. IEEE Ultrasonic Symp., pp. 1314-1317, 2003. [5] H. Calás, E. Moreno, J. A. Eiras, A. Aulet, J. Figueredo and L. Leija, “Non-uniformly polarized piezoelectric modal transducer: Fabrication method and experimental results”, Smart Mater. Struct. 15. pp. 904-908, 2006. [6] H. Meitzler, H. M. O’Brian, H. F. Tiersten, “Definition and measurement of radial mode coupling factors in piezoelectric ceramic materials with large variation in Poisson’s ratio”, IEEE Trans. Sonics Ultrason., vol. SU-20, pp. 233–239, 1973. [7] G. R. Harris, “Review of transient theory for a baffled planar piston”, J. Acoust.Soc.Am. vol 70, pp.10-20, 1981.
