A
NEW
DIRECTIONAL ACOUSTIC LENS:
V-GROOVE LENS
A.
Bozkurt, G. Yaraliozlu,
A.
Atalar and
H.
Koymen
Bilkent University, Ankara, Turkey
06533
ABSTRACT
A
new directional acoustic lens
is
introduced. The geometry
is very similar to the line-focus lens except the lens cavity,
which is shaped as
a
groove with flat-bottom V cross section.
The slanted planar edges
of
the groove are inclined in order
to generate waves incident on the object surface at
a
critical
angle. Hence, the edges of the groove act like two wedge
transducers facing each other. The cross section of the lens
is
the same as that of the Lamb Wave Lens. Therefore, it en-
joys the same sensitivity to surface wave excitations. On the
other hand, since the cross section remains the same along
one of the lateral directions, it has directional properties very
similar to that
of
the Line Focus Beam Lens. The waves
normally incident on the object surface generated from the
flat-bottom, interfere with those at the critical angle, giving
rise to
a
V(Z)
effect. Calculated responses
of
the lens are
presented for silicon
(001)
surface as
a
function
of
crystal
orientation. The calculated curves are compared with mea-
surement results. The leaky wave velocities are extracted
from the measurement results using the conventional
FFT
algorithm.
A
new model based algorithm is proposed for
extracting the velocity information from
V(
2)
data.
Introduction
A
directional lens such
as
line focus beam (LFB) lens
[I]
produce
a
V(2)
pattern from which surface wave velocity
in
a
particular direction can be deduced with high accuracy.
The accuracy depends on the number of oscillations in the
pattern. For high velocity materials the period is large, and
thus there will be
a
limited number of oscillations, resulting
in
a
low measurement accuracy. Moreover, the signal level
in
V(
2)
gets smaller as the defocus distance is increased.
The conventional spherical lens was originally designed for
imaging purposes. After it
is
discovered that
V(2)
has unique
material characterization properties, this lens
is
used mostly
for characterization despite its deficiency in exciting leaky
waves. Lamb wave lens combines the
V(2)
effect with its
higher efficiency of leaky wave excitation. It maintains
a
large signal level at an extended defocus range. However, it
does not have a direction sensitivity which
is
required for the
characterization of anisotropic materials.
In this paper, we introduce
a
new directional lens, V-groove
lens, with
a
higher sensitivity as compared to other
directional lenses. It inherits the high efficiency of the Lamb
wave lens, while providing the high directionality
of
the LFB
lens. First, the geometry of the V-groove lens is described
and design considerations are discussed. Then, the response
of the lens
as
a
function
of
defocus distance is investigated.
Theoretical simulations are performed for an anisotropic crys-
tal to deduce the accuracy of such
a
lens. Experimental
results are presented and compared with the theory. More-
over,
a
refinement in the procedure of extracting velocity
information from
V(2)
data is presented. The new proce-
dure involves
a
model based analysis approach rather than
a
simple FFT analysis on the data.
Figure
1:
Geometry of the V-groove lens
V-Groove
Lens
V-groove lens differs from the line-focus-beam lens in the way
the refracting element
of
the lens
is
fabricated.
As
opposed
to the cylindrical cavity of the LFB lens, V-groove lens has
a
V
shaped groove with flattened bottom
as
shown in Fig.
1.
Essentially, the relationship between the LFB lens and the
conventional lens is the same as the relationship between V-
groove lens and the Lamb wave lens
[2].
In other words, leaky
waves are selectively excited on the surface of the object
when V-groove lens is used, while, in case of LFB lens, all of
the modes are excited due
to
its wide angular spectrum.
The transducer
is
designed
so
as to insonify a substantial por-
tion of the groove with minimal waste of power elsewhere.
The flat bottom part does not cause any refraction, and thus
a
part of the incident beam insonifies the object surface at
normal incidence. Symmetrical sides
of
the groove causes
a
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1993
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1993
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583
refraction, and hence two symmetrical beams insonify the ob-
ject surface at the same incidence angle. The interference
of
the refracted beams. which encounter leaky wave modes
on
the object surface, with the reference beam resulting from
the specular reflection of the normally incident beam pro-
duces the
V(2).
A
good match between median direction
of
the refracted beam and the critical angle for the object
improves the measurement accuracy.
Simulations and Experiments
Simulating the performance
of
a
lens involves propagation
of
acoustic waves between the transducer and the refracting
element. The wavefront is then propagated through the re-
fracting element using ray theory. The wave front is then re-
flected from object surface upon propagation in liquid. This
analysis
is
similar to the one developed
for
Lamb wave lens
[a],
except for the circular symmetry. While the circular
symmetry
of
the Lamb lens allows the use of fast Hankel
transform for propagation purposes, propagation problem in
V-groove lens requires the more costly two dimensional FFT.
Also, the calculation
of
the reflection coefficient at
a
liquid
anisotropic solid interface is considerably more complicated
131.
0.00
-5.00
m^
E
h
2
-10.00
-15.00
-6.000
-4.000
-2.000
0.000
2.000
z
(mm)
Figure
2:
Calculated and measured
V(2)
values along
[loo]
direction on
(001)
surface
of
Si
at
f=25
klHz
A V-groove lens
is
designed to measure surface acoustic wave
velocity on silicon. Lens operates at
25
hlHz.
The silicon
sample
is
a
529
microns thick silicon wafer.
The Rayleigh
wavelength
is
approximately
200
microns, and hence the
wafer thickness
is
large enough
for
measurement purposes.
Water temperature
was
stable within
0.2
degrees. The align-
ment of the lens is achieved easily by maximizing the signal
from the flat part of the lens, since the maximum signal
is
reached when the object surface is perfectly parallel to the
flat part
of
V-groove.
A
simulation
is
done for this lens
for the
(001)
surface
of
silicon
as
the reflector. The result-
ing
V(2)
curve
is
depicted in Fig.
2
together with the mea-
surements along
[loo]
direction. The
V(2)
simulations are
repeated along varying directions for the same surface
of
sil-
icon.
Leaky wave velocity extraction
The conventional procedure
[4]
adapted for extracting SAW
velocity from measured
V(2)
data can be summarized
as
follows:
10
0
50
Y
Y
z
h
'c
00
2
-5
0
Model
"
~-
-10.0
Z
(mm.)
Figure
3:
Experimentaldata points
V2(Z)-V:e,(Z)
for
(001)
surface
of
Si
(F=25
MHz)
and least squared fitted curve
0
Measure
V(Z)
for the object
0
Obtain
a
L!.e,(Z)
using an object with no leaky wave
generation at the V-groove excitation range.
0
Find
L"(2)
-
K:,(Z)
0
Filter out any unwanted interference frequencies
0
Pad data with zeros
584
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1993
ULTRASONICS SYMPOSIUM
0
Use
a
proper window function
0
Apply
FFT
to find the period of oscillation
0
Determine velocity from period
I
This method yields its best results
if
there is only one leaky
wave mode. It is difficult to get accurate results particularly
when there are two modes with close velocities. Unfortu-
nately, many anisotropic materials support pseudo surface
waves along particular directions
[5]
with a velocity very close
to the
SAW
velocity. FFT algorithm gives biased results in
such
a
case.
To
alleviate this problem, an alternative proce-
dure
is
proposed.
A
model based algorithm
[6]
is adopted,
which suits better to the physical nature of the
V(2):
5600.0
Actual
o------..
..
o
FFT
extracted
'.
5400.0
h
5200
0
\
E
v
.-
0
0
0
Q)
50000
-
t
-
0
Q)
5200.0
2
E
v
-
oeo
w
-
0
.-
0
0
0
-
>"
5000.0
4800
0
t
L
0
\
0
4600
0
00
10
0
200
300
400
I
Direction (deg)
I
Figure
4:
Calculated and measured velocity (using
FFT
method) values on
(001)
surface of Si
as
a
function
of
di-
rection
0
Measure
V(Z)
for the object
0
Obtain
a
V,,j(Z)
as above
0
Fit the model parameters
to
Vre,(Z)
in the least mean
square sense, using Nelder-Meade simplex search.
0
Find
V'(2)
-
v,",,(Z)
0
Fit the model parameters using the same algorithm to
the squared difference. Find the period of oscillation.
0
Determine velocity from period
A
simple model is selected to reduce the computational com-
plexity
of
the search process. The model assumes uniform
insonification of the V-groove and ignores the diffraction in
the liquid. The output voltage
is
obtained as the absolute
value
of
a
sum
of
three complex terms. The first term
is
due
to the flat central portion
of
the V-groove lens and its phase
is assumed to change linearly with the defocus distance. The
second term arises from the specular reflection
of
obliquely
incident waves. The change in its phase depends on the incli-
nation angle of the V-groove as well as the defocus distance.
Variation in its amplitude is determined from geometrical
considerations. The last term exists only
if
a
leaky wave is
excited on the object surface. The phase
of
this term is as-
sumed to depend on the critical angle of the object material.
Since leaky waves are assumed to decay exponentially
[7],
the amplitude
of
the third term is exponentially dependent
on the defocus distance. Copper
is
used as the reference
material, since no leaky wave can be excited on its surface
by
this lens. Using the reference material data,
Vref(.Z),
the
unknown parameters in the first and second term are deter-
mined. The parameters of the third term -among them is
the leaky wave velocity
of
the object-
is
subsequently deter-
mined from the measured
V(2)
data on the object. Fig.
3
shows the measured squared difference data together with
the results
of
the simple model whose parameters are opti-
mized to best fit the data.
Si
(001)
5600.0
Actual
D------.-----Q
Model fitted
(1
st
order)
5400.0
4600
0
0.0
10
0
20
0
30.0
40
0
50.0
Direction (deg)
Figure
5:
Calculated and measured velocity (using model
fitting method) values
on
(001)
surface of Si as a function of
direction
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The velocity
is
extracted from the measured
V(2)
data for
(001)
surface
of
silicon along different directions using
FFT
method and the proposed method.
These experimental
re-
sults are given in Fig.
4
along with calculated leaky wave
velocities from elastic constants
[8].
The estimated velocity
values follows the variation predicted from tabulated elastic
constants within 1%. It
is
interesting to note that
for
angles
larger 25 degrees, presence of pseudo surface waves is indi-
cated by an emerging extra peak in the frequency spectrum.
However, the accuracy at points where the two modes are of
equal magnitude seems to be degraded.
The velocity estimates obtained from the same experimental
data using model fitting method
is
depicted in Fig.
5.
The
model assumes
a
single leaky wave excitation. It can be
observed that the agreement between the extracted values
and the calculations are significantly better, when there is no
pseudo surface wave excitation. Obviously, in the presence
of pseudo surface waves,
a
single model order can not match
the inherent complexity
of
the data. Hence, the estimated
velocity value falls somewhere between the two modes.
Conclusions
The proposed V-groove lens
is
proved to be a powerful tool
for measuring leaky velocities with
a
good directional sen-
sitivity. Directional properties
is
comparable to LFB lens
while its leaky wave excitation efficiency
is
as good
as
the
Lamb wave lens. Since its excitation angle is fixed, a given
V-groove lens can only be used for a limited range of veloci-
ties. Typically,
for
a
particular material
a
matching V-groove
lens must be used. Because of its high excitation efficiency,
the modulation index in the
V(2)
curve maintains
a
high
value over an extended defocus distance. This provides
a
high signal-to-noise ratio. Hence, a more accurate charac-
terization
is
possible.
The best
use
of
V(2)
data requires
a
processing procedure
which concur with nature of the physical problem. The tra-
ditional approach
is
to use
a
FFT based spectral algorithm
to find the velocities. This method yields particularly erro-
neous results
if
there are more than one leaky wave excita-
tion. The method proposed in this paper adopts
a
model
based approach to extract velocity. The model makes use
of
all
available physical and geometrical data with some un-
known parameters. The unknown parameters are found at
the end of a minimization procedure which fits the measured
data to the model prediction. The performance of the
V-
groove lens used along with the new extraction procedure
is
experimentally tested on silicon. Characterization
of
silicon
has inherent difficulties because of its relatively high SAW
velocity.
As
long
as
the surface wave velocity remains within the range
dictated by the fixed inclination angle
of
the V-groove lens, it
is possible to estimate the surface wave velocity as
a
function
of orientation with a very high accuracy.
Acknowledgment
This work is supported by Turkish Scientific and Technical
Research Council, TUBITAK.
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