1207

TRANSDUCER CHARACTERISTICS FOR ULTRASONIC STEREOHOLOGRAPHY* MARTIN FEDER School of Engineering Princeton University Princeton, N.J.

BRUCE D. SOLLISH, Ph.D Department of Electronics Weizmann Institute of Science Rehovot, Israel

U LTRASONIC imaging, the process whereby the sound transmission and reflection properties of a structure are displayed on a cathode-ray tube, had its first biomedical application in the mid- 1940s when F. A. Firestone used his "supersonic reflectoscope" to provide images of human bones.' In this first application of the pulse-echo technique to ultrasound, Firestone applied short electrical pulses to a quartz crystal to generate bursts of sound waves at a frequency of 5 MHz. Any boundary of two substances having different acoustic impedances will reflect a portion of the sound back to the source; after emitting the ultrasonic wave train, the crystal became an ultrasound detector. Small voltages generated by the reflected sound were amplified and displayed as amplitude of signal versus time on a cathode-ray oscilloscope, a method of display that since has been called the A-scan.2 J. J. Wild and S. M. Reid first applied the B-scan system, ultrasonic tomography, to biological tissues, viewing an excised piece of beef kidney cortex and subsequently observing and detecting in vivo a tumor (myoblastoma) in a human thigh.3 Again, the pulse-echo technique was used (recording reflected ultrasonic echoes as opposed to recording sound picked up on the other side of the structure by another transducer), but the method of display was different. In the B-scan mode of display the transducer is scanned across the target area and the returning echoes are *Presented in part at a Scientific Session held by the Section on Biomedical Engineering of the New York Academy of Medicine April 9, 1974. This research was conducted at the Weizmann Institute of Science, Rehovot, Israel, and was supported in part by the Israel Cancer Association, Tel Aviv, Israel.

Vol. 52, No. 10, December 1976

M. FEDER AND B. D. SOLLISH

1208

D.

1208

LITHIUM SULFATE CRYSTAL

SOLLISH

S

_

E

EPOXY

00W

Y AXIS

ACOUSTIC LENS X AXIS

Fig. 1. Cylindrically focused ultrasonic transducer.

b)

Fig. 2. Geometry of the transducer: a) The geometry of the focus where am is the flare angle and F is the focal length. b) The minimum distance S, and maximum distance S2 of the observation point Q from the cylindrical radiator.

displayed on the oscilloscope in an x-versus-y configuration. The position of the echo on the face of the cathode-ray tube face is related to the spatial orientation of the echo-producing interface, and the brightness is proportional to the amplitude of the returned signal, thus generating a twodimensional display. These techniques have been and are being utilized in the diagnosis of abdominal injuries, the detection of cysts and tumors, obstetrical and gynecological diagnoses (especially observation of fetal growth and development), as well as in fields such as cardiology, ophthalmology, and encephalography.4 In 1972 ultrasonic stereoholography, a method for three-dimensional visualization, was described. In this method a sequence of B-scans taken of the target from different perspectives is used to construct a multipleexposure hologram. This reconstructs the original three-dimensional object Bull. N. Y. Acad. Med.

ULTRASONIC STEREOHOLOGRAPHY

1209

8

ionCZ

Ii

I

I

P4it,

_t

-G.iO

-O.u

-0.35

a,.-

tI.Gt q..L ?

fl.

.is

-

0o .0

NORMALIZED DISTANCE FROM FOCUS ON X AXIS Fig. 3. Normalized pressure versus normalized distance from focus along the x axis for the sinusoidal steady-state, plotted by computer.

when illuminated by coherent light.5 The development of ultrasonic stereoholography will markedly enhance the range and value of ultrasound in its application to medicine and biology. Since the frequency of the mechanical vibrations that are useful in medical diagnosis range from approximately 1 to 15 MHz., the most common method of ultrasound generation is the excitation of a piezoelectric transducer by a high-frequency electrical generator. This transducer is connected to the structure under study through a coupling medium (water or oil) which will allow uniform contact. Sound produced by the action of the electrical field on the crystal will be transmitted accurately to the target organ, and the reflected echoes, likewise, will deform the transducer Vol. 52, No. 10, December 1976

1210

1210

M. FEDER AND M. FEDER AND B. D. SOLLISH B.

D.

SOLLISH~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

ID

o1 0

R

*

rn

_z 0.02

0.00

_

0.04

0.G6

0.38

I

0-10

0.22

0, 4

0IC 3I16

0.,0

0,22

A 0,i 'A

6

id28

NORMALIZED DISTANCE FROM FOCUS ON Z AXIS

Fig. 4. Normalized pressure versus normalized distance from focus along z axis for sinusoidal

steady-state, plotted by computer.

'*

//

I, \

\

\\

I

On V]

/I

/

\c

N

z

,:

I

.:iz

40; aiAd of

,r

I 5X.X:

?2

.-' 2J6 2.18 TIME AT FOCUS(microseconds)

. 1 .w 'sH

:

I

.,

0.@

C.24

5.X

.

.2

Fig. 5. Normalized pressure versus time (in microseconds) for focus in the quasi-steady-state, plotted by computer. Horizontal line denotes zero for normalized pressure, while points A, B, and C correspond to times selected for Figures 9, 10, and 11, respectively.

1211

ULTRASONIC STEREOHOLOGRAPHY

PULSE DURATION-50 NANOSECONDS

P14 N

0.00

G.02

0.09

0.06

0.08

0,10

0.12

0 14

0.16

0.18

0.20

0.22

0.24

0.26

0.28

TIME AT FOCUS(microseconds)

Fig. 6. Normalized pressure versus time for focus in the quasi-steady-state-50 nsec. simulated pulse duration, plotted by computer.

-3

9.E

PULSE DURATION-100 NANOSECONDS

O ,I, .4i

l\

jS

/i e

En

,E

I

I

p4

Ii

i i.. C .I

z

z

RI IT

r.02

-,

0.4ot G .0a

0.10

0.12

0.14

0.16

0.18

.20

0.22

0 24

u.26

0.28

TIME AT FOCUS(microseconds)

Fig. 7. Normalized pressure versus time for focus in the quasi-steady state plotted by computer-100 nsec. simulated pulse duration, plotted by computer.

1212

4.

M. FEDER AND B. D. SOLLISH

1212

M.

FEDER

AND

B.

D.

SOLLISH~~~~~~~~~~~~~~~~~~~~~~~~

C)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i

H

0.500

.

0.00

0.12

0*

0.20

.

0.24

03

0.2

0.b

0.40

0.

0.4 0. 0. 2 .%

TIME(microseconds) 1.48mm FROM FOCUS ON X AXIS

Fig. 8. Normalized time-varying pressure against time in microseconds for an observation point 1.48 mm. along the focal plane, plotted by computer.

and produce electrical signals that can be interpreted intelligently once they are shown on the cathode-ray tube. Ultrasonic steroholography, unlike other ultrasonic techniques in which spherical and spherically focused transducers are used, requires the use of a cylindrically focused ultrasonic transducer for generating projection-type B-scans. Since ultrasonic systems for medical diagnosis depend on measurement of the intensity of transmitted or reflected energy from biological tissue, it is extremely important to know the exact field distribution pattern of the vibrational radiation produced by the transducer and, hence, its relation to the geometry of the object under investigation.6 The equations describing the sound field in the focal region of a cylindrically-focused transducer have been derived under two conditions: 1) the sinusoidal steady state, in which the sound pressure generated by the transducer at any point in the focal region is a sinusoidal function of time and 2) the transient case, in which, in general, the sound pressure at any point in the focal region is a more complex function of time. The transient analysis, thus, describes the sound field before the sinusoidal steady state is reached. In order to illustrate more clearly the differences between the transient and sinusoidal steady state fields in the focal region, this study presents a computer-generated series of graphs showing the transient and sinusoidal Bull. N. Y. Acad. Med.

1213

ULTRASONIC STEREOHOLOGRAPHY

21.3703 MICROSECONDS AFTER PULSE APPLICATION (n

CZ I

Un

UM CD

n

U,

m C£:

0. U,

U,

z

Ce

.Z

LI

N

U' -T,

06

0 . 00 ),

0,3"4

., .,,i

-"-,

Ili

I

;93

1.141

C.

za

2.03 Re

3X06

DISTANCE FROM FOCUS ON X AXIS (mm) Fig. 9. Normalized time-varying pressure against time in microseconds for the entire focal plane, plotted by computer-time = 21.3703 Asec. after pulse application.

steady-state behavior of a model cylindrically focused transducer. An experiment in visualizing the sinusoidal steady-state is described also. MODEL TRANSDUCER

The design of the cylindrically focused transducer shown in Figure 1 is such that a concave acoustic lens causes the in-phase ultrasonic wavefronts to converge inward at the lens-water interface along the entire length dimension (y axis), coming to a focus at some distance from the face of the transducer along the axis of propagation (z axis). The transducer used as a model was a lithium-sulfate cylindrically Vol. 52, No. 10, December

1976

M. FEDER AND B. D. SOLLISH

1214

AND

B.

D.

SOLLISH

U,

21.4708 MICROSECONDS AFTER PULSE APPLICATION In p.

CybS.

I

£11

r%.

rz~ cm 03.

N 1 U,

Ns

z

U,l CY

N.

K.3.06

.Zc-

I

J.38

G

.76

--- I_-

Ir

i .is

i 9S3

i .9i

2-9

-.63

3;

'f

DISTANCE FROM FOCUS ON X AXIS (mm)

Fig. 10. Normalized time-varying pressure against time in microseconds for the entire focal plane, plotted by computer-time = 21.4708 Asec. after pulse application.

focused ultrasonic transducer designated model no. 57A8686 by the manufacturer, Automation Industries of Colorado. The focal length (F) was 32 mm. in water (see Figure 2), the diameter (D) of the curved face was 8.0 mm., the flare angle (am) is one-eighth radian, and at an operating frequency of 10 MHz. the acoustic wavelength (A) was 0.15 mm.7 Steady-state behavior. Initial efforts were devoted to examining the output in the sinusoidal steady-state, where excitation of the transducer to output ultrasound at its natural frequency is accomplished by means of a continuously applied high frequency sine wave. The pressures generated are given by the equations in Appendix A. For purposes of computer plotting, normalized equations were used. The calculation of pressure along the focal plane (x axis) normalized with Bull. N. Y. Acad. Med.

1215

ULTRASONIC STEREOHOLOGRAPHY

21.5708 MICROSECONDS AFTER PULSE APPLICATION

N Z r.

I-

.06

0.00

0.38

0.76

.

15

1.53

I'_______

1.91

2.029

2.b8

306

DISTANCE FROM FOCUS ON X AXISn(mm) Fig. I1. Normalized time-varying pressure against time in microseconds for entire focal plane, plotted by computer-time = 21.5700 pusec. after pulse application. Notice flattening compared to Figure 9.

respect to the pressure at the surface (Appendix B, equation 1) was performed by iterating x from -1/10 focal length (-3.2 mm.) to +1/10 focal length (3.2 mm.) in steps of 1/10 wavelength (0.015 mm.). This pressure was plotted against distance normalized with respect to the diameter (x/D) as shown in Figure 3. The sound pressure is a maximum at the focus, with the main lobe extending to x/D and additional side lobes of width x/D. Steady-state pressure along the axis of propagation similarly was normalized with respect to pressure at the surface (Appendix B, equation 2). Iterated along the z axis in the same way as the iteration for the focal Vol. 52, No. 10, December 1976

1216

1216

M. FEDER AND B. D. SOLLISH SOLLISH~~~~~~~~~~~~~~~~~~~~~~~~

M. FEDER AND B. D.

holographic plate

Fig: 12. Block diagram of qualitative analysis of sound field. Destructive interference between object beam and reference beam was caused by ultrasonic vibrations generated by the cylindrically focused transducer. The optical image of water in the tank was recorded on a holographic plate.

plane, the normalized pressure was plotted against normalized distance (Appendix B, equation 3); Figure 4 shows a slow dropoff of pressure in the near focal region. Thus, the usual experimental results in relating pressure to distance in the focal region are reflected by the equations and accurately generated by the computer simulation. Transient behavior. The previous results described the pressure in the focal region of the cylindrically focused transducer when the transducer oscillation is of a continuous sinusoidal nature. In medical diagnosis and in ultrasonic stereoholography, pulsed operation of the transducer is required to interface successfully with the signal-processing equipment and produce a recognizable picture. The digital analysis of the transient, or pulsed operation, thus was undertaken. For analysis purposes it has been assumed that the transducer emits a sine wave of n half-cycles duration. The total duration of the pulse is then nT/2, where T is the period of the wave. The ideal transducer would have n = 1, 50 nsec. for this simulation, as minimum pulse-train width; the actual transducer we used had n = 2, or a pulse width of 100 nsec. As the number of half cycles increases, the pressure in the focal region approaches the value predicted on the basis of the steady-state analysis. The transient behavior of the cylindrically focused transducer has been analyzed in detail by B. D. Sollish; the computer simulation of the transient state was based on that analysis.9 Bull. N. Y. Acad. Med.

ULTRASONIC STEREOHOLOGRAPHY

1217

Fig. 13. Portion of beam progression (dark spot in center) reconstructed from an optical hologram by the method described in Figure 12, against background illumination from the water tank appearing as an irregular white shadow. The outline delineated the entire beam which was visible in the series of photographs which was taken.

For any given observation point Q in the focal region of the transducer there are several crucial time zones. For some time after the transducer is excited no change at all can be detected at the observation point. At a later time wavefronts from the part of the transducer closest to the point of observation will have reached that point, but not pressure from other sections of the transducer. At a still later time pressure from all parts of the transducer will reach the observation point. Finally, for an excitation period of finite length, the pressure will fall off to zero. Vol. 52, No. 10, December 1976

1218

M. FEDER AND B. D. SOLLISH APPENDIX A

Steady State Acoustic Pressures 8 1.

Pressure

amplitude at focus: P

=

2a pcU

(F/A)1/2

(1)

Pressureamplitude in the focal plane (along the x-axis): P

2a pcU

=

(F/A)2

sinc 2 X

(2)

Pressure amplitude on the axis of propagation in the focal region:

Pz

F 21/2 = 2 pcU ((.1)1/2 [C ( z sz

{;2)

2

+ S

(a

mY) (m

~~1/2 ;_2z ZA.3

AAI(3

APPENDIX B

COMPUTER PLOTING FORMULAE

Steady-State Acoustic Pressures 1.

Acoustic pressure along focal plane (x-axis) normalized with respect to acoustic pressure at surface of transducer

Px 2.

=

K sinc K 2 -x

(1)

Normalized steady-state acoustic pressure along the axis of propagation (z-axis) P --

3.

a~ [C (K6) + S (KS|)]1/ 2

Normalized distance along the z-axis (axis of propagation)

(3/

(2

For each observation point Q, with the projection of the transducer in the x-z plane there are defined distances S 1 and S2 in the plane containing Q, where SI is the distance from the observation point to the closest point on the face of the transducer and S2 the distance from the observation point to the furthest point on the transducer face (see Figure 2b). For time t < (S1/c), or less than the minimum propagation time from the transducer to the observation point, no disturbance is detected at the observation point. For (S 1/c) t < (S2/c), from when the first ultrasonic wavefront begins to reach the observation point to the time that the entire beam has had a chance to reach it, the transient state exists, in which the observation point is disturbed by only a part of the transducer face. After -

Bull. N. Y. Acad. Med.

ULTRASONIC STEREOHOLOGRAPHY

1219

Time-Varying Acoustic Prcssures9 Instantaneous pressure amplitudes at the focus, quasi-steady-state, normalized with respect to acoustic pressure at the surface of the transducer

po)

=

p0(t) =

F

4%(

1/2 ~~2n(t---F)2T( + Cc2C Tct-T ) 1/2 S [sin _ C

(4)

F -< nT valid for 0 < t' = -c t 2 Instantaneous pressure amplitude for a given point x in the focal plane

normalized with respect to surface pressure-transient region

Pxp5t) 1

Fc T

Pl$W

Slic

St4l

< t

cos

-

Sl2w >1

Ca[sin< T (t

+

valid for

~~~1/2