General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright
owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
Users may download and print one copy of any publication from the public portal for the purpose of private study or research.
You may not further distribute the material or use it for any profit-making activity or commercial gain
You may freely distribute the URL identifying the publication in the public portal
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Downloaded from orbit.dtu.dk on: Aug 09, 2022
Calculation of pressure fields from arbitrarily shaped, apodized, and excited
ultrasound transducers
Jensen, Jørgen Arendt; Svendsen, Niels Bruun
Published in:
I E E E Transactions on Ultrasonics, Ferroelectrics and Frequency Control
Link to article, DOI:
10.1109/58.139123
Publication date:
1992
Document Version
Publisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):
Jensen, J. A., & Svendsen, N. B. (1992). Calculation of pressure fields from arbitrarily shaped, apodized, and
excited ultrasound transducers. I E E E Transactions on Ultrasonics, Ferroelectrics and Frequency Control,
39(2), 262-267. https://doi.org/10.1109/58.139123
262
IEEE TRANSACTIONS ON ULTRASONICS. FERROELECTRICS. AND FREQUENCY CONTROL. VOL.
39.
NO.
2,
MARCH
1992
Calculation
of
Pressure Fields from Arbitrarily
Shaped, Apodized, and Excited
Ultrasound Transducers
J~rgen
Arendt
Jensen
and
Niels
Bruun
Svendsen
Abstract-
A
method for the simulation
of
pulsed pressure
fields from arbitrarily shaped, apodized and excited ultrasound
transducers
is
suggested. It relies on the Tupholme-Stepanishen
method for calculating pulsed pressure fields, and can also handle
the continuous wave and pulse-echo case. The field is calculated
by dividing the surface into small rectangles and then summing
their response.
A
fast calculation is obtained by using the far-
field approximation. Examples
of
the accuracy of the approach
and actual calculation times are given.
T
I.
INTRODUCTION
HE
MOST
IMPORTANT component
in
acquiring high
quality images for medical ultrasound scanners is the
probing transducer. Ultimately it determines the quality of
the data acquired, and
thus
the quality of the images and
parameters displayed. Considerable effort has therefore been
spend on designing transducers and characterizing the field
emitted and received
[l],
[2].
Several methods for calculating the pressure field have been
developed for assisting
in
the design and characterization
of
various transducer geometries. Most of the methods can be
traced
to
the fundamental solutions
of
Rayleigh, King, and
Schoch, of which a review can be found in
[3].
The most powerful approach seems to be the method
developed by Tupholme and Stepanishen
[4]-[6],
which gives
an exact solution for a transducer modeled as a planar piston
vibrating uniformly
in
an infinite rigid, planar baffle. Analytic
expressions for several transducer types have been found
[5],
[7],
but closed form solutions can not be found for all types.
Especially the introduction
of
exotic geometries or apodization
of the transducer surface leads
to
analytically unsolvable
integrals.
In this paper we will develop
a
simulation approach based
on
the Tupholme-Stepanishen approach, which can simulate
transducers with any apodization of the transducer surface and
with any excitation
of
the transducer.
Manuscript received June
12,
1991; revised October
21,
1991;
accepted
October
22,
1991. This work was supported
in
part by the Danish Technical
Research Council, with grant 164218, in part by Briiel and
Kjzr
A/S,
in part
by Novo’s Foundation, in part by H.C. 0rsted’s Foundation, and in part by
Trane’s Foundation.
The paper proceeds along the following lines. Section
I1
for-
mulates the problem and details the underlying theory. Section
111
gives various implementation details and Section IV lists
a number of examples for different transducer geometries and
apodization functions.
It
will be shown that the method is fast
and gives accurate answers.
11.
THEORY
The purpose of this paper is
to
devise a fast and accu-
rate method for calculating the pulsed pressure field emitted
from an arbitrarily shaped, apodized, and excited ultrasound
transducer.
It
is assumed that the transducer
is
mounted in an infinite,
rigid baffle. Enforcing appropriate boundary conditions, the
emitted field can be found by solving the wave equation for
the velocity potential
4)
[4],
[5]:
from which the pressure is calculated as:
where
p0
is the mean density of the media,
CO
is the propaga-
tion velocity. and
p1
is the over pressure.
The coordinate system shown in Fig.
1
is used in the
calculation. The particle velocity normal to the transducer
surface is denoted by
,[)(F2
+
$3.
t).
The solution to the
homogeneous wave equation using Greens function is
[S]:
where
S
denotes
the
transducer surface.
g
is the time-dependent
Green’s function and
is
University
of
Denmark, 2800 Lyngby, Denmark, and he
is
currently visiting
where
I
‘1
-
‘2
-
‘3
I
is
the
distance
from
the
’lUface
J.
A.
Jensen
is
with the Electronics Institute, Building
349,
Technical
the Department
of
Biomedical Engineering, Duke University, Durham, NC
to the point where the field is calculated The integral is a
27706.
Building
356,
DK-2800 Lyngby, Denmark.
N. B. Svendsen is with the Danish Acoustical Institute, Akademivej,
statement of Huygens’ principle, that for a planar vibrating
surface each point on the source generates
a
spherical wave,
IEEE
Log
Number
9105552.
and the resulting field
is
found by integrating these waves at
0885-3010i92$03.00
Q
1992 IEEE
Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on November 13, 2009 at 09:53 from IEEE Xplore. Restrictions apply.
JENSEN
AND
SVENDSEN:
CALCULATION
OF
PRESSURE
FIELDS
FROM ULTRASOUND
TRANSDUCERS
263
/
Transducer
Fig.
1.
Coordinate system
for
calculating the incident
field
the point of interest under the assumption
of
radiation into an
isotropic, homogeneous, nondissipative medium.
If
a slightly curved transducer is used, an additional term
is introduced as shown in Morse and Feshbach [S]. This term
is called the second order diffraction term in Penttinen and
Luukkala [7].
It
can be shown to vanish for a planar transducer,
and as long as the transducer is only slightly curved and large
compared to the wavelength
of
the ultrasound, the resulting
expression is a good approximation to the pressure field [7].
If
it
is assumed that surface vibration accounting for the
excitation function and electromechanical impulse response
can be split into a spatial component
a(T;
+Tfj)
and a temporal
component
U,
(f2
)
then:
where
a(,?)
is denoted the spatial source velocity distribution
[
101. This implies that the vibration amplitude at a certain point
on
the surface does not depend
on
time,
so
the amplitude of
vibration is not influenced by the shape of the excitation.
The function
hcL(F1.Fz>t-tz)
=
n(Fz+F3)g(T;.t
I
F'2+F3,t2)dzF3
(6)
L
is called the apodized spatial impulse response and it relates
the transducer geometry to the acoustical field. By this function
we can write
$(F1,?5>t)
=
lJE(t)
*ha(r;:F2,tt)
(7)
t
where
~~(t)
is the piston velocity waveform, and the velocity
potential is written as a convolution in time between this and
the apodized spatial impulse response.
If
the particle velocity is assumed
to
be uniform over the
surface
of
the transducer,
(5)
can be reduced to [6]:
dj(71.,72.t)
=
ltwe(t2)Lg(F1,t
I
F2+?3,tz)d2F3
dt2
(8)
where the last integral equals the traditional spatial impulse
response.
Note that
11,
depends on the difference between
71
and
?2,
thus it is spatially varying. To emphasize this
h,
is written
ha(?l,
f2>
t).
The sound pressure for the incident field then is
or
Note here the separation between the excitation and the
transducer geometry.
The
7le(t)
includes the electromechanical
impulse response
of
the transducer
[2].
Explicit solutions for a number
of
transducer geometries
have been found. Analytical expressions for the circular, flat
transducer can be found
in
[5],
and for the circular, concave
geometry in [7], [9].
It must be emphasized that only two approximations are
used here. The first is the assumption of a large and slightly
curved transducer, and the second assumption is that
of
sepa-
rability between excitation and transducer geometry. Trans-
ducers can be constructed in which this is a very good
approximation,
so
that the pressure field calculated by this
method is in good agreement with the measured field.
The geometric features of the transducer are contained in the
apodized spatial impulse response
h,(T;,
Fz,
t)
and from this
the field for any excitation function, including the continuous
wave case, can be calculated. Further, it has been shown that
the pulse echo field received by the emitting transducer can
be calculated by [ll]:
where
vPe
is the pulse-echo electromechanical impulse re-
sponse including the excitation function.
So
the emitted pulsed
field, the received field, and the continuous wave case can
be derived from the apodized spatial impulse response as the
electromechanical impulse response usually can be determined
from a simple measurement.
Thus, the original problem is transformed into calculating
the apodized spatial impulse response. Closed form solutions
have been found for some cases as mentioned previously, but
not for all geometries and rarely when apodization is used.
A.
Simulation
Method
In analytic calculations the solution is found by evaluating
which part of a sphere with center at the field point that
intersects the transducer surface [9]. The area of the strip on the
radiator surface divided by the distance to the field point gives
the spatial impulse response at that time instance in the case
of uniform vibration. When apodization is used, the different
areas on the strip should be suitably weighted.
In this simulation method the problem is reversed. A spher-
ical wave is emitted from a point on the aperture and all
spherical waves are summed at the field point. weighted by
the inverse of the distance from the aperture point to the
Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on November 13, 2009 at 09:53 from IEEE Xplore. Restrictions apply.
264
IEEE TRANSACTIONS
ON
ULTRASONICS,
FERROELECTRICS.
AND
FREQUENCY CONTROL.
VOL.
39,
NO.
2,
MARCH
1992
field point. The apodized spatial impulse response is then
approximated by
where
r,
denotes the points on the transducer surface.
Just dividing the surface into points has the drawback
that quite a large number of points must be used as the
variance of
h,
at a time instance depends on the number of
spherical waves received in one time interval.
The
number
of
points can be drastically reduced by dividing the surface
into small rectangles and then summing the responses from
these rectangles. This is the approach used here.
A
similar
approach has also been studied by Ocheltree and Frizzel for
the continuous wave case [12]. Here we are studying the pulsed
field case in which the continuous wave case can be calculated
as a special case.
Dividing the transducer surface into squares introduces an
approximation to the true geometry, and the field will deviate
from the true one. The problem is reduced by using small
squares, where the distance to the field point is large compared
to
the size of the squares. Thus,
it
is appropriate
to
use a far-
field approximation, when calculating the contribution from
each individual element. The exact solution for the impulse
response from a rectangular piston is derived in
[6]
so
only
an intuitive explanation for the far-field solution
is
given here.
As
the impulse response at a point in front of a piston is
proportional to how large part of the piston that contributes
to
the response at a given time, the problem of deriving the
response is reduced to geometric considerations concerning
the distance between the field point and the different parts
of the transducer. From a point near the piston surface the
isodistance curves looks like shown in Fig. 2(a), but if the
distance increases the curves tends
to
straight lines, which is
shown in Fig 2(b). The first is the near-field situation and the
latter is the far-field situation.
To calculate the far-field response from the rectangle a
description of the piston and the location
of
the field point
as shown in Fig.
3
is needed. The piston is described by its
length and width and to fix the location of
the
field point, the
piston is placed in a coordinate system in the XY-plane with
the center at the origin. Then the location is defined by the field
point’s position vector, split up into a unit vector
(xe,
ye,
z,)
and a distance,
1.
In general the far-field spatial impulse response has the
shape of a trapezoid as shown in Fig.
4,
where
ti
is the time-
of-flight from the nearest corner of the piston
to
the field point.
Likewise
t2
and
t3
are the time-of-flight from the second and
third nearest corner, and
tl
is the time-of-flight from the corner
with
the
largest distance
to
the field point.
In
special cases
two
or more
of
the
t’s
are equal.
The trapezoid shape response can be calculated by con-
volving two rectangular pulses. The width of these pulses are
calculated by projecting the length and width
of
the piston
onto the line through the rectangles center and the field point.
(b)
Fig.
2.
Isodistance curves
on
the transducer surface.
(a)
Near-field.
(b)
Far-field.
Y
Fig.
3.
Description
of
the piston and its orientation.
Fig.
4.
Far-field response
Based on
the
mentioned description
of
the system we get
where
wy
and
W,
are the side lengths of the rectangle.
The arrival times are then calculated by
Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on November 13, 2009 at 09:53 from IEEE Xplore. Restrictions apply.
JENSEN
AND
SVENDSEN:
CALCULATION
OF
PRESSURE
FIELDS
FROM
ULTRASOUND
TRANSDUCERS
265
tJ
=
tl
+
at,
+
atz.
(14)
The amplitude of the trapezoid is a function of piston area
and distance
to
the field point. The shape depends
on
At1
and
At,,
but the area of the trapezoid integrated over the whole
time interval is always equal
to
B.
Arbitrarily Shaped Piston
This quite simple result for the far-field response of a
rectangular piston can be used when designing a fast numerical
method for calculating the response of a piston of any shape.
As
for the point source representation the piston is split up
into small squares
(or
rectangles), but instead of representing
the small areas by point sources the far-field response from
the rectangles is used. This makes it possible
to
use larger and
therefore fewer squares
to
describe the piston. The resulting
response
is
calculated by summing up the responses from
all the squares. Apodizing is obtained by multiplying the
individual responses by an apodizing factor, that could be
a function of e.g. the radius
of
the transducer. In a similar
manner a time delay can be added to the response, giving a
different phase for different parts of the transducer, like in a
phased array.
C.
Far-Field Region
The size of the rectangles must be chosen
so
that. the field
point lies in the far-field region. This is given by [l]:
n
wL
I>>-
4x
where
1
is the distance
to
the field point,
W
the largest
dimension of the rectangle and
X
the wavelength, which equals
co/f,
where
f
is frequency.
Iff
is
the highest frequency in the
response simulated, the side length should obey the relation
(16)
W
<<
dw~.
(17)
Examples indicating how close the side length can be chosen
to
this limit and what accuracy then is obtained, are given in
Section IV.
111. OVERVIEW
OF
PROGRAM
The prime application of
this
program is
to
investigate fields
from transducers of shapes with unknown analytic solutions
and
to
study the influence
of
apodization and phasing
of
elements.
It
is very difficult
to
imagine the shape of these
fields, and therefore whether correct results are calculated.
In
order
to
solve this problem, the program has been divided
into two parts. The first calculates the position and orientation
of the small rectangles describing the transducer, and the
second performs the field calculation. The
two
parts are
independent thereby enabling the possibility of thoroughly
testing and calibrating the field calculation, which is transducer
independent. The only uncertainty is then the placement of
the rectangles. This part of the program can, however, be
interfaced
to
a CAD program, that can visualize the placement
Time
Is]
XIO-7
Fig.
5.
Simulated
(-)
and true
(-
-
-)
spatial
impulse
response
of
con-
cave transducer. The time
on
the y-axis
is
relative. Zero corresponds
to
t
=
;;.63pS.
of the rectangles. By this method accurate and reliable results
should be assured.
IV. EXAMPLES
In
this section several examples
of
use
of
the program
are shown. Responses are compared to analytic solutions
and guidelines for choosing the number of elements and the
resulting computing times are given.
The first example is for a concave, nonapodized transducer
with an aperture radius
of
8
mm and
a
focal distance of
150
mm.
An
analytic expression is found for this geometry [9] and
can thus be compared
to
simulated responses.
The spatial impulse response at a distance
of
120
mm from
the surface
is
shown
in
Fig.
5
from on the acoustical axis and
out in steps
of
1
mm. The transducer was divided into
3177
squares with a side length of
0.25
mm. Calculating the ten lines
of the spatial impulse response at a sampling frequency of l00
MHz took
1.5
S
on
an HP/Apollo 90001425t workstation', and
is shown as dashed lines in Fig.
5.
We see that the program
quite accurately tracks the theoretical spatial impulse response
off the acoustical axis. On the axis it is, however, more difficult
to
get the exact position and shape of the abrupt changes in
the spatial impulse response due
to
the employment
of
the
far-field approximation.
In the next example the concave transducer was apodized
with a Gaussian distribution function defined as
a(T)
=
e-a;(f)l
(18)
where
R
is the radius of the aperture and
7'
the distance from
the center.
ap
was chosen
to
be
2.
The spatial impulse response
is shown
in
Fig.
6.
The characteristic elimination
of
sharp
edges in the spatial impulse response is seen.
To
show that the pulse-echo response can be calculated
to good accuracy a single example is shown in Fig.
7.
The
measured and simulated responses were obtained at a distance
'This workstation has
roughly
the
calculation speed
of
a
40-MHz
486
PC.
Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on November 13, 2009 at 09:53 from IEEE Xplore. Restrictions apply.