Kentaro Nakamura

Marie Nakazawa

Sadayuki Ueha

Takahiro Aoyagi

Tokyo Tech

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Numerical analysis for ultrasonic beam of variable line focus
transducer
Takahiro Aoyagi‡, Marie Nakazawa, Kentaro Nakamura and Sadayuki Ueha
(Tokyo Tech.)
1 Introduction
Line-focus-beam ultrasonic transducer has been used
for material characterization system[1][2]. Such sys-
tem has solid acoustic lens that has an arc of a true
circle. On the other hand, submicron film of polyurea
was synthesized by vapor deposition process[3]. This
technique enables evaporatation of polyurea piezo-
electric film onto a polyimide film. Bending the
film to curved shape, we obtained variable line focus
transducer[4]. The bent film transducer has an si-
nusoidal cross section[5], however, radiated ultrasonic
beam was not studied. In this report, radiated ultra-
sonic beam from the variable line focus transducer is
numerically analyzed by FDTD (Finite difference time
domain method)[6] and by the wave theory for liquid-
solid interfaces[7].
2 Problem description
Fig.1 shows the geometry for the numerical calcula-
tion. The problem is treated as two-dimensional, that
has infinite length for the depth. The 5-mm-length
film transducer is bent between fixtures at the base.
The cross section of the film results a sinusoidal curve.
Polyurea piezoelectric film is deposited on the center
3mm area of the polyimide film. In this report, this
area is treated as the transmitting and receiving re-
gion. An aluminum plate is fixed at the distance of
0.8mm from the baseline of the sinusoidal curve. The
space between the transducer and the aluminum plate
is filled with water. When the baseline length of sinu-
soidal curve is changed, the film transducer is bent and
the height of the curve is changed. This shape change
of the sinusoidal curve causes the change of the focal
length of the film transducer. The sinusoidal shape of
the film transducer is calculated by the elliptic inte-
gral of second kind[5]. The calculation is carried out
at the frequency of 38.1MHz, which is the experimental
resonance frequency of the polyurea transducer.
3 Calculation method
Fig.2 shows the geometry for the FDTD calculation.
In this report, the FDTD method for elastic body is
applied. The space discrete step ∆d is 1.5µm, and the
time discrete step ∆t is 116.8 ps. These values are de-
termined in consideration of the acoustic phase velocity
in water(1500m/s) and the ultrasonic frequency. Since
FDTD analysis is restricted to finite calculation area,
aoyagi@cradle.titech.ac.jp
w(base width)
a(height)
0.8mm
Liquid(water)
Elastic body(aluminum)
Polyurea film(3mm)
Fixture(movable)
Polyimide film(5mm)
Figure 1: Problem geometry.
the absorbing boundary condition for the propagating
wave[8] is applied to the outer boundary of the water.
An elastic vibration is given as the normal component
of the particle velocity on the transducer surface. The
50 cycles of 38.1MHz sinusoidal waves are inputted.
Total received intensity is obtained by numerically in-
tegrating the normal component of particle velocity on
the lens surface. FFT is applied to the total received
signal on the lens, and the intensity at 38.1MHz is
extracted. In addition to the FDTD calculation, we
executed the numerical calculation by liquid-solid in-
terface reflection theory according to ref.[9].
7.5mm
w(base width)
a wave source(3mm)
6mm
0.8mm
1.5mm
Liquid(water)
Absorbing boundaryA
b
s
o
rb
in
g
 b
o
u
n
d
a
ry
A
b
s
o
rb
in
g
 b
o
u
n
d
a
ry
Elastic body(aluminum)
Figure 2: FDTD geometry.
4 Calculation Results
Fig.3 shows the relative intensity vs. lens width calcu-
lated by FDTD. The result has a peak value at the lens
width of 4.35mm. The ripples caused by leaky surface
-  219  -
P2-6 Proc. Symp. Ultrason. Electron., Vol. 27, (2006) pp. 219-22015-17 November, 2006
acoustic wave can be observed. Fig.4 shows the results
calculated being based on the liquid-elastic wave reflec-
tion theory. The leaky saw velocity is assumed to be
2480m/s, and the loss coefficient α is varied from 0.001
to 0.01. The positions of the local maximums and the
local minimums of the ripple have good agreement with
Fig.3. In Fig.4, the solid line shows the calculated re-
sult without leaky saw effects. Unlike true circular arc
lens case, the ripple caused by leaky saw is observed at
both side of the focal point(w = 4.37mm). The curva-
ture of the sinusoidal lens is not constant, therefore the
leaky saw excites around the focal point. Fig.5 shows
the case of α = 0.01, and the leaky saw velocity is var-
ied from 2840m/s to 5000m/s. Fig.4 and Fig.5 show
reasonable results like the true circle arc lens case as
in the former paper[9].
5 Conclusion
In this report, the characteristics of ultrasonic beam
radiated by the variable line focus transducer was nu-
merically calculated using FDTD and the liquid-elastic
wave reflection theory. The position of the local min-
imums and the local maximums of the ripple showed
good agreement in both results. The interference ripple
caused by leaky-saw was observed in lens width vs. re-
ceived intensity curve. As the conventional V(z) curve,
the ripple changed according to the loss parameter α
and the leaky saw velocity. Comparison of these results
with experiments are left for further study.
4.25 4.3 4.35 4.4 4.45 4.5 4.55 4.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Transducer base width(mm)
R
el
at
iv
e 
re
ce
iv
ed
 in
te
ns
ity
Figure 3: V(z) curve calculated by FDTD.
Acknowledgement
FDTD calculations in this report were performed on
TSUBAME supercomputer of Global Scientific Infor-
mation Center, Tokyo Tech. during test use period.
References
[1] J. Kushibiki and N. Chubachi: IEEE Trans. Sonics
Ultrason. SU-32 (1985) No. 2, 189.
 0
 0.2
 0.4
 0.6
 0.8
 1
 1.2
 4.25  4.3  4.35  4.4  4.45  4.5  4.55  4.6
R
el
at
iv
e 
re
ce
iv
ed
 in
te
ns
ity
Transducer base width(mm)
without SAW
α=0.001
α=0.003
α=0.005
α=0.010
α=0.030
α=0.050
Figure 4: V(z) curve calculated by wave
theory(vLSAW = 2840m/s).
 0
 0.2
 0.4
 0.6
 0.8
 1
 1.2
 1.4
 4.25  4.3  4.35  4.4  4.45  4.5  4.55  4.6
R
el
et
iv
e 
re
ce
iv
ed
 in
te
nc
ity
Transducer base width(mm)
without SAW
2840m/s
3000m/s
4000m/s
5000m/s
Figure 5: V(z) curve calculated by wave theory(α =
0.01).
[2] Y. Ohashi, M. Arakawa and J. Kushibiki: Jpn. J.
Appl. Phys. 45 (2006) No. 5B, 4505.
[3] T. Hattori, Y. Takahashi, M. Iijima and E. Fukada:
Jpn. J. Appl. Phys. 79 (1996) 1713.
[4] T. Kosugi, H. Nagatsuka, T. Ishii, K. Nakamura, S.
Ueha and Y. Hirao: Proc. Intl. Conf. on New Piezo-
electric Materials and High Performance Acoustic
Wave Devices (2002) p.179.
[5] M. Nakazawa, T. Kosugi, A. Maezawa, K. Naka-
mura and S. Ueha: Proc. Spring Meet. Acoust. Soc.
Jpn. (2005) p.973[in Japanese].
[6] M. Sato: FDTD hou niyoru danseisindou·hadou no
kaiseki nyuumon (Introduction to analysis for elas-
tic vibration/wave by FDTD) (Morikita shuppan,
Tokyo, 2003)[in Japanese].
[7] H. L. Bertoni and T. Tamir: Appl. Phys. 2 (1973)
157.
[8] C. J. Randall: Geophysics, 54 (1989) No. 9, 1141.
[9] J. Kushibiki, A. Ohkubo and N. Chubachi : Proc.
IEEE Ultranson. Symp. ,1982, p.623.
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Numerical analysis for ultrasonic beam of variable line focus transducer

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Numerical analysis for ultrasonic beam of variable line focus transducer

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