Acoustic lens for use in acoustic microscope

An acoustic lens for use in an acoustic microscope including a solid state medium for propagating an acoustic wave having a wavelength .lambda. and having opposed end surfaces, an electric-acoustic transducer applied on one end surface of the solid state medium and having a radius a, and a spherical lens portion formed in the other end surface of the solid state medium and having an aperture of a radius w. The length l and the aperture radius w are normalized by the transducer radius a such that Z=l.lambda./a.sup.2 and W=w/a. Values of Z and W are selected from such a region in a first quadrant of the Z-W coordinate system that desired power and/or phase are obtained. The region neighboring the point Z=1 and W=1 is excluded.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
This invention relates to an acoustic lens for use in an acoustic 
microscope comprising an ultrasonic wave propagating solid state medium 
having opposite end surfaces, an electric-acoustic piezoelectric 
transducer applied on one end surface of said solid state medium, and a 
lens portion formed in the other end surface of said solid state medium. 
2. Description of the Prior Art 
Measurements utilizing acoustic energy have been applied in various 
applications such as sonar, defect detector and fish finder technologies. 
In medical applications, ultrasonic diagnosing apparatus has been widely 
used. Recently there has been developed an acoustic microscope which 
utilizes the transmissibility of an ultrasonic wave through a specimen as 
well as the modulation of the ultrasonic wave due to the elastic 
characteristics of the specimen. With the aid of such an acoustic 
microscope it is possible to observe an image of the elastic specimen at a 
high resolution. The frequency of the ultrasonic wave used in the acoustic 
microscope is usually set to several hundred megahertz, but recently an 
acoustic microscope using ultrasonic waves of very high frequency, e.g., 
up to the order of gigahertz has been developed. For instance, when water 
is used as the liquid medium between the acoustic lens and the specimen, 
it is possible to obtain a high resolution of about 1 .mu.m by using an 
ultrasonic wave of 1 GHz. Such a resolution is comparable with that of 
usual optical microscopes. If liquid helium or liquid nitrogen is inserted 
between the acoustic lens and the specimen, there is a possibility that a 
higher resolution than 1 .mu.m could be attained. 
FIG. 1 is a schematic view showing a typical known acoustic microscope. An 
acoustic lens 1 comprises an ultrasonic wave propagating solid state 
medium 2 made of material such as sapphire and fused quartz having a high 
acoustic propagation velocity, an electric-acoustic piezoelectric 
transducer 3 applied on one end surface of the solid state medium 2, and a 
lens portion 4 formed in the other end surface of the solid state medium 
2. A high frequency pulse generated by a high frequency pulse generator 5 
is supplied to the transducer 3 via a circulator 6, and the transducer 3 
produces a plane ultrasonic wave. The ultrasonic wave propagates within 
the solid state medium 2 and is converged into a spherical wave by the 
spherical lens portion 4. Between the acoustic lens 1 and a specimen 9 is 
placed an acoustic wave propagating liquid medium 10 such as water, and 
the converted spherical wave is projected onto the specimen 9 as a 
microscopic spot via the liquid medium 10. In the acoustic microscope of 
the reflection type, the ultrasonic wave reflected by the specimen 9 is 
collected by the lens portion 4, and then is made incident upon the 
transducer 3 which converts the received ultrasonic wave into an electric 
signal. The electric signal is then supplied to a signal processing 
circuit 7 via the circulator 6 and the signal processing circuit produces 
a video signal. The video signal is then supplied to a monitor 8 to 
display an ultrasonic image of the specimen 9. When the acoustic lens 1 
and specimen 9 are moved two-dimensionally relative to each other to 
effect the mechanical scan, a two-dimensional image of the specimen due to 
the elasticity can be displayed. 
In the reflection type acoustic microscope, when the acoustic beam is 
focused onto a surface of the specimen, it is possible to obtain the 
acoustic image having a construct in accordance with the difference in the 
reflection factor for the acoustic wave of the specimen surface. When the 
specimen is brought closer to the acoustic lens, the incident angle of the 
spherical acoustic wave emanating from the acoustic lens and impinging 
upon the specimen changes continuously from 0.degree. to an angle formed 
between the outermost beam and a principal axis of the acoustic wave. Then 
the acoustic wave reflected by the specimen is modulated by various 
components in the specimen in different manners, and the reflected 
acoustic wave has a phase variation specific to the composition of the 
specimen. Therefore, by effecting the X-Y scan, it is possible to obtain 
an image having a contrast in accordance with the acoustic property of 
substances in the specimen. Further, when the acoustic lens is moved in 
the direction Z normal to the surface of the specimen to effect a linear 
scan and an output signal from the acoustic lens is plotted versus the 
distance in the direction Z, it is possible to attain a so-called V(Z) 
curve which is specific to the specimen. The above mentioned three 
functions of the acoustic microscope are very important. For instance, 
from the acoustic image of the specimen surface, it is possible to detect 
defects in the specimen surface. When the specimen surface is placed 
closer to the acoustic lens than the focal point, crystal construction and 
crystal boundary can be detected from the acoustic image. Moreover, from 
the V(Z) curve, one can specify or identify or more components in the 
specimen. 
Various studies have been done for the acoustic lens for use in the 
acoustic microscope, and various acoustic lenses and analyses thereof have 
been disclosed in the following references. 
(1) "ACOUSTIC MICROSCOPY BY MECHANICAL SCANNING", by R. A. Lemons, May 
1975, Microwave Laboratory, W. W. Hansen Laboratories of Physics, Stanford 
University Stanford, Calif. 
(2) "CHARACTERISTIC MATERIAL SIGNATURES BY ACOUSTIC MICROSCOPE" by R. D. 
Weglein and R. G. Wilson in "ELECTRONICS LETTERS", Vol. 14, No. 12, June 
6, 1978. 
(3) "An Angular-spectrum approach to contrast in reflection acoustic 
microscopy" by Abdallah Atalar in "JOURNAL OF THE APPLIED PHYSICS", Vol. 
49. No. 10, pp 5130-5139, October 1978. 
(4) "MODULATION TRANSFER FUNCTION FOR THE ACOUSTIC MICROSCOPE" by Abdallah 
Atalar in "ELECTRONICS LETTERS", Vol. 15, No. 11, May 24, 1979. 
(5) "RAY INTERPRETATION OF THE MATERIAL SIGNATURE IN THE ACOUSTIC 
MICROSCOPE" by W. Parmon and H. L. Berton in "ELECTRONICS LETTERS", Vol. 
15, No. 21, Oct. 11, 1979. 
(6) Japanese Patent Application Laid-Open Publication (Kokai) No. 
58-44,343. 
(7) Japanese Patent Application Laid-Open Publication No. 60-149,963, 
Japanese Patent Publication No. 59-50,937 and Japanese Utility Model 
Application Laid-Open Publication No. 57-120,250. 
In reference (1), there is disclosed an acoustic lens as shown in FIG. 2 of 
the present application. The acoustic lens comprises a sapphire rod 
(Al.sub.2 O.sub.3) 11, an Au electrode 12 applied on one end surface of 
the rod, a piezoelectric film 13 (ZnO) applied on the Au electrode 12, and 
an Al electrode 14 applied on the ZnO film 13. In the other end surface of 
the rod 11 there is formed a spherical lens portion 15. The dimension of 
the electric-acoustic transducer is defined by the dimension of the 
uppermost Al electrode 14. As the acoustic lens for 1 GHz, the following 
parameters have been proposed: 
l=2.00 mm 
r=0.135 mm 
.theta..sub.max =50.degree. 
D=0.207 mm 
d=0.156 mm 
wherein l is the length of the acoustic wave propagating solid state medium 
11, r is the radius of curvature of the spherical lens portion 15, .theta. 
is the aperture angle, D is the aperture diameter and d is the focal 
distance. This known acoustic lens has the F/number, defined by d/D, of 
0.75. In this acoustic lens, the acoustic energy impinging upon portions 
outside the aperture of the lens portion 15 becomes useless and might 
interfere with the acoustic energy passing through the lens portion 15. 
Therefore, when designing the acoustic lens, the dimension of the 
transducer, i.e. the diameter of the Al electrode 14, has to be adjusted 
such that the above-mentioned disturbing acoustic energy is minimized. 
Further, in order to protect the acoustic lens from damage or breakdown, 
the above dimension must be determined such that the acoustic energy is 
spread as widely as possible. In order to satisfy such requirements, it 
has been recommended that the diameter of the Al electrode 14 be made 
substantially equal to the aperture diameter D of the lens portion 15 and 
the length l of the medium 11 be selected such that the lens aperture is 
situated just in a Fresnel focal point or slightly longer than that. Here, 
the Fresnel focal distance l.sub.0 is given by l.sub.0 =.rho..sub.0.sup.2 
/.lambda., where .rho..sub.0 is the radius of the Al electrode 14 and 
.lambda. is the wavelength of the acoustic wave to be used. In this case, 
the diameter of the acoustic wave becomes substantially equal to the 
diameter of the transducer at the Fresnel focal distance. As stated above, 
in the known acoustic lens, the diameter of the transducer is made 
substantially equal to the aperture of the spherical lens portion 15 and 
the length of the medium 11 is made substantially equal to the Fresnel 
focal distance, so that uniform intensity distribution of acoustic energy 
can be obtained at the lens portion 15. This is the basic design principle 
of the known acoustic lens. This principle has been equally applied to 
known acoustic lenses described in references (2) to (5) and (7). 
In reference (6) there is disclosed an acoustic lens in which the length of 
the ultrasonic wave propagating rod is set to the inverse of an odd 
number, particularly one third (1/3) of the Fresnel focal distance and the 
aperture diameter of the lens portion is set also to the inverse of an odd 
number, particularly one third (1/3) of the diameter of the transducer. 
This known acoustic lens has been developed in order to solve the 
following problem. In order to reduce the dumping of the acoustic energy 
in the water inserted between the lens and specimen, it is advantageous to 
shorten the working distance. Then, the radius of the lens portion and the 
aperture diameter have to be reduced, so that the radius of the transducer 
becomes shorter accordingly. However, an acoustic lens having such a small 
transducer and lens portion cannot be practically manufactured or can be 
manufactured only with difficulty. In the acoustic lens shown in the 
reference (6), the above-mentioned problem is solved by increasing the 
dimension of the transducer. However, it should be noted that in this 
known acoustic lens, the previously mentioned principle that the amplitude 
of the acoustic energy becomes uniform at the lens portion has been 
equally applied. 
As explained above, upon designing the acoustic lens it is preliminarily 
noted that the simplest or uniform distribution of the acoustic energy can 
be attained at the lens portion and that the acoustical field at other 
portions has been neglected. Particularly, the known acoustic lenses have 
been designed without taking into account the phase of the acoustical 
field. Therefore, it is practically impossible to design various acoustic 
lenses which can be advantageously used in various applications and 
satisfy various requirements. In practice, almost all acoustic lenses have 
been manufactured in such a manner that the aperture diameter of the lens 
portion is made substantially equal to the diameter of the transducer and 
the length of the ultrasonic wave propagating solid state medium is made 
substantially equal to the Fresnel focal distance. That is to say, the 
known acoustic lenses have been manufactured by determining various 
parameters such as frequency, aperture diameter and aperture angle in 
accordance with the above mentioned design principle and the lenses thus 
manufactured were set to actual acoustic microscopes to check whether or 
not the required conditions would be satisfied. In general, the known 
acoustic lenses manufactured in the manner explained above were not 
satisfactory. Then new acoustic lenses had to be manufactured again by 
changing one or more parameters. In this manner, the known acoustic lenses 
were manufactured by a trial and error method. It is apparent that such a 
process is quite cumbersome and requires a very long time, and sometimes 
desired acoustic lenses could not be obtained. Particularly, in the 
acoustic lens for obtaining the V(Z) curve the phase of the acoustical 
field is very important, and not only does the acoustic wave have to be 
in-phase at the spherical lens portion, but also the amplitude of the 
acoustic energy has to be sufficiently large at the spherical lens 
portion. However, it is practically difficult to obtain an acoustic lens 
satisfying such conditions. This is mainly due to the fact that, according 
to the known design principle, the lens aperture has to be small for 
making the acoustic wave in-phase at the lens aperture, and therefore the 
amplitude or power of the acoustic wave becomes weak. However, no study 
has been done for finding the maximum permissible phase difference. 
SUMMARY OF THE INVENTION 
The present invention has for its object to provide a novel and useful 
acoustic lens which can satisfy various requirements for various 
applications, by statically analyzing the amplitude and phase properties 
of acoustic energy in the propagation path from the transducer to the 
specimen and from the specimen to the transducer. 
It is another object of the invention to provide an acoustic lens which can 
attain a contrast due to variations in the amplitude and phase of the 
acoustic wave reflected by the specimen surface by normalizing the 
dimension of the transducer and the dimension of the lens aperture and the 
transducer can receive the acoustic wave modulated by the specimen with 
effective power and/or phase. 
According to the invention, an acoustic lens for use in an acoustic 
microscope comprises an ultrasonic wave propagating solid state medium 
having opposed end surfaces, an electric-acoustic piezoelectric transducer 
provided on one end surface of the solid state medium, and a lens portion 
formed in the other end surface of the solid state medium; wherein the 
radius of the transducer is defined as a, the length of the solid state 
medium measured in an ultrasonic wave propagating direction from the 
transducer to the lens portion is l, the aperture radius of the lens 
portion is w, the wavelength of the ultrasonic wave is .lambda., 
Z(=l.lambda./a.sup.2) and W(=w/a), being set to such a region in a first 
quadrant of Z-W coordinate system, excluding the region near the point 
(1,1), that an acoustical field having desired power and/or phase is 
obtained in the solid state medium. The first quadrant of a Z-W coordinate 
system is defined as the region of a graph of Z vs. W where both Z and W 
are positive values. 
The inventors have confirmed that the point (Z,W) can be advantageously set 
in a region other than a region defined by the W axis, a line expressed by 
W=Z, and a line represented by W=-5Z+3. Furthermore, the known region near 
the point Z=1, W=1 is outside the scope of the invention. By selecting 
points (Z,W) within such a preferable region, it is possible to attain 
acoustic lenses having particularly large power. 
Further, the inventors have found that the point (Z,W) is advantageously 
set within such a region in the first quadrant of the Z-W coordinate 
system that the phase difference is limited within 50.degree.. Such an 
acoustic lens is particularly suitable for obtaining the V(Z) curve. 
According to the known principle for designing the acoustic lens, the lens 
portion has to be arranged at a strictly defined position without taking 
into account the phase of the acoustic wave. According to the invention, 
the acoustic lens is designed by taking into account the phase and 
amplitude of the acoustic wave impinging upon the transducer. 
Particularly, in the acoustic lens for obtaining the V(Z) curve, the phase 
is much more important than the amplitude.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
Before explaining the present invention, the acoustical field distribution 
will be first explained. In order to derive an acoustical field u(x) of 
the acoustic energy emitted from an electric-acoustic piezoelectric 
transducer and propagating in an acoustic wave propagating solid state 
medium, an acoustical field due to a flat piston-shaped sound source 
having a circular cross section will be considered. It should be noted 
that the Lommel approximation for diffraction of light is also applied to 
the acoustical field. FIG. 3 is a schematic view showing a principal 
construction of the acoustic lens. In FIG. 3, a represents the radius of 
an electric-acoustic piezoelectric transducer 22 applied on one end 
surface of an acoustic wave propagating solid state medium 21, l 
represents the distance from the transducer 22 to the end of the solid 
state medium, measured along a central axis, o, x represents the distance 
from the central axis o to the edge of the propagating medium 21 in a 
direction perpendicular to the axis, and .lambda. represent the is a 
wavelength of the acoustic wave. Two normalized amounts, X and Z, are 
defined as follows: X=x/a and Z=.lambda.l/a.sup.2. The a sound pressure P 
can be expressed as follows: 
EQU P=.rho..multidot.c.multidot..omega..sub.0 .multidot.e.sup.i(.omega.t-kz) 
.multidot.(u.sub.1 +iu.sub.2) (1) 
wherein 
##EQU1## 
In the above equation, .rho. is the density of the liquid medium between 
the acoustic lens and the specimen, C is the velocity in the liquid medium 
and k=2.pi./.lambda.. 
According to the invention, in the acoustical field generated by the 
electric-acoustic transducer having a radius of a, a lens aperture w is 
arranged at a distance Z from the transducer and then the influence of the 
lens aperture upon the acoustic field is calculated, while the 
normalization of W=w/a is taken place. 
By using the parameters W and Z thus normalized, the known acoustic lenses 
will be first analyzed. The first reference (1) mentions the W=1 and Z=1 
or Z&gt;1 (but near 1). The other references also describe the same principle 
in design that W is set to 1 and Z is set to 1 or slightly larger than 1. 
The inventors have found that points other than Z=1, W=1 can yield 
acoustic lenses having unexpected properties. 
The above equation (1) was calculated to derive the amplitude and phase of 
the acoustic wave. Amplitude and phase are represented three-dimensionally 
in FIGS. 4A and 4B, respectively. Where 2 is less than 1 the amplitude and 
phase fluctuate largely and at Z-1 the maximum sound pressure is obtained. 
In order to show the condition of the sound pressure in greater detail, 
FIGS. 5A and 5B illustrate the amplitude and phase properties, 
respectively at X=0.2, 0.4, 0.8, 1.0, 1.2 and 1.4. Further, according to 
the invention, the phase is the important property, so that the phase 
variations at Z=1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5 are also 
shown in FIGS. 6A.about.6L. In these graphs the phase at X=0 is normalized 
into 0.degree.. From the graphs shown in FIGS. 4, 5 and 6, it can be 
understood that the acoustic wave becomes in-phase to a greater extent in 
accordance with the increase of Z, but the amplitude becomes gradually 
smaller. 
In order to derive the power of the acoustic wave immediately after the 
lens aperture, a value (u) of a summation of all sound pressure within the 
aperture radius w at a position separated from the transducer by a 
distance Z is first calculated and then a value of 20 log (u) is 
calculated. FIG. 7 illustrates a relationship between the intensity, i.e., 
power of the acoustic wave, and the distance Z at W=0.2, 0.4, 0.6, 0.8, 
1.0, 1.2 and 1.4, while the normalization of W=w/a is effected. In FIG. 7, 
the vertical axis denotes the power, i.e., the intensity of sound, and the 
power becomes larger in accordance with the increase in W. But when W is 
increased, the phase difference becomes larger. In the case of deriving 
the V(Z) curve, the phase of the acoustic field becomes important. For the 
acoustic lens, the acoustic wave is in-phase and has a large power at the 
aperture of the lens portion. In order to investigate this further in 
detail, the relationship between W and Z as well as the relationship 
between the power and Z were derived at various phase differences. FIGS. 
8A and 8B illustrate the relationship between W and Z and the power and Z 
at a phase difference at 5.degree.. At first, a value of Z(Z=1.25) which 
gives the maximum power was derived from the graph shown in FIG. 8B, and 
then a value of W(W=0.39) corresponding to the thus derived Z was found 
from the graph illustrated in FIG. 8A. In this manner, the values of W and 
Z giving the maximum power can be derived. The following Table 1 shows 
various values of W and Z for phase differences of 10.degree., 15.degree., 
20.degree., 25.degree., 30.degree., 40' and 60.degree.. 
TABLE 1 
______________________________________ 
maximum 
phase power 
difference Z W (dB) 
______________________________________ 
5.degree. 1.25 0.39 26.7 
10.degree. 1.25 0.69 29.6 
15.degree. 1.25 0.93 31 
20.degree. 1.5 1.11 31.3 
25.degree. 1.5 1.14 31.4 
30.degree. 1.5 1.18 31.5 
40.degree. 1 1.2 32 
60.degree. 1 1.3 32.3 
______________________________________ 
In the above Table 1, the maximum power is represented by 20 log (u), so 
that the power of the acoustical field becomes larger in accordance with 
the increase of the maximum power. For instance, the power at the phase 
difference of 10.degree. is larger than that at the phase difference of 
5.degree. by 2.9 dB (=29.6-26.7). However, the inventors have further 
confirmed that calculated values and characteristics of acoustic lenses 
calculated by W.noteq.1, i.e., a.noteq.w do not correspond to those of 
actual acoustic lenses. 
The inventors have further investigated and found a process of 
approximating theoretically calculated acoustic lens to actual lenses for 
wide variations other than W=1 and Z=1 on the basis of the calculation 
method disclosed in the above mentioned reference (3). It should be noted 
that the reference (3) merely teaches a method of estimating acoustic 
lenses manufactured in accordance with the known design principle of W=1 
and Z=1 or slightly larger than 1, and does not teach a general guideline 
for designing acousting lenses. By using the newly developed approximating 
method, the inventors have explored the possibility of practical acoustic 
lenses having values of Z and W which vary over a wide region other than 
region near the point (Z,W)=(1,1). 
FIG. 9 is a schematic view for explaining a theoretical calculation process 
performed by the inventors. In this process, the acoustic fields at four 
planes H.sub.0 .about.H.sub.3 are considered. H.sub.0 is a plane of a 
transducer 31 having a radius a and H.sub.1 and H.sub.2 are back and front 
focal planes of the lens. H.sub.3 is a plane separated from H.sub.2 by a 
distance Z. The reflection of the acoustic wave is carried out at this 
plane H.sub.3. A lens portion 32 has an aperture radius of w, pupil 
function P.sub.1 for the acoustic wave impinging upon the specimen and a 
pupil function P.sub.2 for the acoustic wave reflected by the specimen. 
The planes H.sub.0 and H.sub.1 are separated from each other by a distance 
d. Then acoustical fields u.sub.1.sup.+, u.sub.2.sup.+, u.sub.3.sup.+, 
u.sub.1.sup.-, u.sub.2.sup.- and u.sub.3.sup.- of the incident acoustic 
wave and the reflected acoustic wave at these planes are calculated. 
u.sub.1.sup.+ is the acoustical field emitted by the transducer 31 and 
impinging upon the plane H.sub.1. Assuming that the acoustic lens is 
sufficiently thin, the acoustic lens can be considered to be a phase 
converting element which converts an incident plane wave into a spherical 
wave. Then, the acoustical field u.sub.2.sup.+ at the front focal point 
plane H.sub.2 can be expressed as follows: 
##EQU2## 
In this equation (2), k.sub.0 is equal to 2.pi./.lambda..sub.0 
(.lambda..sub.0 is the wavelength of the acoustic wave in the liquid 
medium), f is a focal distance, R.sub.e is the radius of curvature of the 
lens portion 32, and c is the ratio of the velocity of the acoustic wave 
in water to that in the solid state medium. Then the following relation is 
given: 
EQU f=R.sub.e .vertline.(1-c) 
The propagation of the acoustic energy from the plane H.sub.2 to the plane 
H.sub.3 can be simply calculated by using angular-spectrum. When the 
acoustical field u.sub.2 .sup.+ (x,y) is Fourier transformed, the 
following equation is obtained: 
EQU u.sub.2.sup.+ (k.sub.x,k.sub.y)=F[u.sub.2.sup.+ (x,y)] 
Then u.sub.3.sup.+ (k.sub.x,k.sub.y) can be expressed as follows. 
EQU u.sub.3.sup.+ (k.sub.x,k.sub.y)=u.sub.2.sup.+ (k.sub.x,k.sub.y) exp 
[ik.sub.z 'z] (3) 
Now, it is assumed that k.sub.z '=k.sub.z +.alpha..sub.z and k.sub.z 
=.sqroot.k.sub.0.sup.2 -k.sub.x.sup.2 -k.sub.y.sup.2, the equation (3) can 
be rewritten in the following manner: 
##EQU3## 
wherein .alpha. is an attenuation constant. 
Here, the following approximation can be applied: 
##EQU4## 
Then the equation (3) can be rewritten in the following manner: 
##EQU5## 
Therefore, the acoustical field u.sub.3.sup.- reflected by the specimen 
surface plane H.sub.3 can be expressed as follows: 
EQU u.sub.3.sup.- (k.sub.x,k.sub.y)=u.sub.3.sup.+ (k.sub.x,k.sub.y)R(k.sub.x 
/k.sub.0,k.sub.y /k.sub.0) (5) 
In this equation (5) R denotes the reflective function. Next, the 
acoustical field u.sub.2.sup.- impinging upon the plane H.sub.2 can be 
represented by the following equation (6): 
##EQU6## 
In order to derive the acoustical function u.sub.1.sup.- (x,y), 
u.sub.2.sup.- (k.sub.x,k.sub.y) is first inversely Fourier transformed to 
derive u.sub.2.sup.- (x,y). That is to say, u.sub.2.sup.- (x,y) may be 
derived by the following equation (7): 
EQU u.sub.2.sup.- (x,y)=F.sup.-1 [u.sub.2.sup.- (k.sub.x,k.sub.y)](7) 
u.sub.1.sup.- (x,y) at the plane H.sub.1 can be expressed by the following 
equation (8) similar to the equation (2): 
##EQU7## 
Further, u.sub.0.sup.- at the plane H.sub.0 can be given by the following 
equation (9): 
EQU u.sub.0.sup.- (k.sub.x,k.sub.y)=u.sub.1.sup.- (k.sub.x,k.sub.y) exp 
[ik.sub.z d] (9) 
The above equation (9) may be rewritten into the following equation (10) by 
using the convolution theorem: 
EQU u.sub.0.sup.- (x,y)=u.sub.1.sup.- (x,y) F.sup.-1 [exp (ik.sub.z d)](10) 
It should be noted that the voltage generated by the piezoelectric 
transducer is an integration of products of weight function S(x,y) of the 
piezoelectric transducer and u.sub.0.sup.- (x,y). Here, the weight 
function S(x,y) represents an acoustical field which is generated by the 
transducer when a unit voltage is applied to the transducer and can be 
expressed as follows: 
EQU S(x,y)=U.sub.0.sup.+ (x,y) 
Therefore, the output voltage V(Z) from the transducer can be expressed as 
follows: 
##EQU8## 
Now the above equation V(Z) can be rewritten as follows by effecting the 
replacement of R(x/f, y/f)=R(k.sub.1 /k.sub.0), u.sub.1.sup.+ 
(x,y)=u.sub.1.sup.+ (r), P(x,y)=P(r) and r=(x.sup.2 +y.sup.2).sup.1/2 : 
##EQU9## 
Further values of V(Z) are theoretically calculated for various values of W 
and Z by taking into account the pupil functions P.sub.1 and P.sub.2 
together with anti-reflection layer and spherical aberration of the lens 
portion. An example of a V(Z) curve thus calculated is shown in FIG. 10. 
This curve is calculated by using an acoustic lens having an acoustic wave 
propagating solid state medium made of fused quartz having a length of 
l=6.7 mm, a transducer having a diameter 2a=0.766 mm, a radius of 
curvature RA=0.5, and an aperture angle SI=60.degree.. The frequency of 
the acoustic wave is selected to be 200 MHz. 
Further, peak value V.sub.max of V(Z) for various values of W and Z and 
difference V.sub.max -V.sub.min between successive peak and valley are 
calculated and these values are shown in FIGS. 11 and 12, respectively. It 
has been confirmed that similar curves can be obtained when the aperture 
angle SI is varied from 45.degree. to 75.degree.. As can be understood 
from these graphs, superior acoustic lenses can be obtained in a wide 
region other than the region near W=1 and Z=1 to which the known acoustic 
lenses belong. Particularly in a region of W&lt;1 and Z&lt;1, it is possible to 
design acoustic lenses having large values of V.sub.max and V.sub.max 
-V.sub.min. The graphs further indicate that these are two semi-whirlpool 
areas about points of W=0, Z=1/5 and W=0, Z=1/3. In these areas, if W is 
changed slightly, the power, i.e. gain is changed largely. This means that 
in these regions desired characteristics could hardly be obtained owing to 
manufacturing error. Further, in these graphs regions denoted by broken 
lines are unstable regions and desired characteristics might not be 
obtained. The inventors have found that in a region of the graph of 
V.sub.max surrounded by a line W=Z, a line W=-5Z+3 and the W axis, 
acoustic lenses having good characteristics could not be obtained. 
Further, if Z and W are selected from a region surrounded by lines 
expressed by W=-1/9Z+1 and W=-4Z+10.5 and the Z axis, it is possible to 
obtain acoustic lenses having larger powers than those of the known 
acoustic lenses. Further in the acoustic lens disclosed in the reference 
(6), two points, Z=1/3, W=1/3 and Z=1/5, W=1/5 have been selected. 
Therefore, regions near these points should be considered to be out of the 
scope of the invention. 
In the graph of V.sub.max -V.sub.min, when the phase difference exceeds 
50.degree., V.sub.max -V.sub.min becomes too small and useful V(Z) curves 
could not be obtained. Therefore, it is preferable to select a phase 
difference smaller than 50.degree.. In order to design acoustic lenses 
having larger values of (V.sub.max -V.sub.min) then those of the known 
acoustic lenses, it is preferable to select points (Z,W) from a region 
surrounded by solid lines expressed by W=-6Z+3, W=-2/1.7Z+2 and W=1/2Z+0.2 
and the Z axis. Therefore, if points (Z,W) are selected from a region 
which is included in both the preferable regions in FIGS. 11 and 12, it is 
possible to obtain acoustic lenses which are advantageously used for 
attaining both the amplitude image and V(Z) curve. Such compatible lenses 
could never be proposed prior to the present invention. 
As explained above, according to the invention, values of W and Z are 
determined by taking into account the acoustic field. Next a process for 
practically manufacturing the acoustic lens according to the invention 
will be explained. 
FIG. 13 is a schematic view showing various parameters of the acoustic 
lens. 
a - - - radius of electric-acoustic piezoelectric transducer 22; 
l - - - whole length of acoustic wave propagating solid state medium 21; 
d - - - depth of a lens portion 23; 
RA - - - radius of curvature of lens portion; 
SI - - - aperture angle of lens portion 
w - - - radius of aperture: 
Further, a focal distance is denoted by f and the ratio of the velocity of 
the acoustic wave in the liquid medium so that in the solid state medium 
21 is represented by c. 
FIG. 14 is a flow chart showing the process of manufacturing the acoustic 
lens according to the invention. 
At first, the frequency of the acoustic wave to be used and values of W and 
Z are determined. 
Next, the radius of curvature RA of the lens portion is determined. In this 
case, the maximum value of RA is determined by loss in the liquid medium. 
For instance, the radius of curvature RA of the lens portion may be set to 
2 mm, 2.5 mm or 3 mm for the acoustic lens of 100 MHz, 0.5 mm, 0.75 mm, 
1.00 mm, 1.25 mm or 1.5 mm for 200 MHz, and 0.25 mm or 0.5 mm for 400 MHz. 
Then, the aperture angle SI is determined and further the radius of 
aperture w is calculated from RA and SI in accordance with the equation, 
w=RA.multidot.sin (SI). 
As explained above, since the normalization of W=w/a is effected, the 
radius a of the transducer is calculated from W and w (a=w/W). 
Further, by using the equation Z=l.lambda./a.sup.2, the length l of the 
solid state medium is calculated in accordance with the following 
equation. 
EQU l=l'+f.multidot.c+d 
Next, it is judged that the acoustic wave reflected from the specimen is 
made incident upon the transducer without being affected by acoustic waves 
which have been multiple-reflected within the acoustic lens. That is to 
say, the acoustic wave reflected from the specimen has to be made incident 
upon the transducer for time intervals during which the multiply-reflected 
acoustic waves do not impinge upon the transducer. Conditions for 
effecting this judgment are determined by considering the minimum pulse 
repetition time defined by the resolution, timings at which the acoustic 
wave reflected from the specimen is made incident upon the transducer and 
timings at which the multiple reflection acoustic wves are made incident 
upon the transducer. This will be explained in detail hereinbelow. 
The theoretical resolution is given by 0.7 .lambda. when the convergence of 
beam, aberrations, etc. are ignored. Therefore, when a field of view 
having a width of 2 mm is to be displayed on a television monitor, a 
number of samplings N of 2000 .mu.m/0.7 .lambda. .mu.m is required. In 
general, the number of samplings N can be given by N=L.sub.s /0.7 
.lambda., wherein L.sub.s is the width of the field of view. Now, it is 
assumed that the transmitting pulse has a pulse period of T.sub.s, then 
EQU T.sub.s =(1/f.times.1/2).times.0.8(sec) 
can be obtained. At respective sides of the frame, there are overscan areas 
of 10%. Then the sampling time T.sub.1 is given as follows. 
EQU T.sub.1 =T.sub.s /N(sec) 
This time should be equal to a time T.sub.2 during which the acoustic wave 
reciprocates between the transducer and the specimen, so that the 
following equation is established. 
##EQU10## 
wherein V.sub.s is the velocity of the acoustic wave in the solid state 
medium, and V.sub.w is the velocity in the liquid medium situated between 
the acoustic lens and the specimen. From the above equations, the 
following equation (12) can be derived: 
##EQU11## 
In the above equation (12) the parameter C is a safety factor which is 
usually set to 2. The equation (12) starts from the condition that T.sub.1 
should be equal to T.sub.2. Here, T.sub.1 is the maximum permissible 
sampling time, so that the equation (12) gives the maximum lens length L, 
i.e. the axial length of the acoustic wave propagating solid state medium. 
A further condition is that the acoustic wave reflected from the specimen 
should not be coincident with the multiply -reflected acoustic waves 
within the acoustic lens. FIG. 15 illustrates a time relation between 
these acoustic waves. The lens length L should be determined such that the 
acoustic wave reflected from the specimen is situated between successive 
acoustic waves multiply-reflected by the acoustic lens. T.sub.1, T.sub.2 
and T.sub.3 are determined by the pulse period T.sub.s of the transmitted 
pulse and T.sub.s =T.sub.1 =T.sub.2 =T.sub.3. It is assumed that N waves 
are inserted in the transmitted signal, and then the following equation 
can be derived. 
##EQU12## 
In this equation, F is the frequency of the transmitted pulse. The 
inventors have confirmed from the analysis of the V(Z) curve that 
necessary marginal distances before and after the transmission are 40 
.lambda. and 20 .lambda., respectively, so that the following equation is 
established. 
EQU T.sub.4 =40.lambda./V.sub..omega. (14) 
Here, since .lambda.=V.sub.w /F, the above equation (14) can be rewritten 
into the following equation (15). 
EQU T.sub.4 =40/F (15) 
Similarly, the following equation (16) is derived. 
EQU T.sub.5 =20/F (16) 
From the above analysis the necessary conditions for obtaining acceptable 
lens length are expressed as follows: 
##EQU13## 
When the length l of the acoustic lens is judged to be inadequate, the 
aperture angle SI is redetermined as depicted in the flow chart shown in 
FIG. 14. When the lens length is judged to be correct, a first set of data 
values such as lens radius, aperture angle, lens depth, diameter of 
transducer and lens length is generated. Then, for the same values of W 
and Z, a next set of data values is determined in the same manner as that 
explained above. After a plurality sets of data values have been derived, 
one can select a suitable set of values. This last selection can be 
performed by taking into account the phase difference and power of the 
acoustic field. 
Finally, the diameter of the lens is determined by deriving the probability 
that the transducer receives acoustic waves reflected within the lens by 
means of ray-tracing acoustic waves emitted from all positions of the 
transducer. The diameter of lens A is deterined such that said probability 
is minimized. 
Now several examples of data values of the acoustic lenses designed in the 
manner explained above are shown in the following Table 2. 
TABLE 2 
__________________________________________________________________________ 
Phase difference 
small middle large 
aperture 
aperture 
aperture 
aperture 
aperture 
aperture 
Frequency 
small 
large small 
large small 
large 
__________________________________________________________________________ 
100 MHz 
.phi. 10.degree. 
.phi. 20.degree. 
SI 20.degree. 
SI 20.degree. 
RA 2 RA 2.5 
200 MHz 
.phi. 5.degree. 
.phi. 20.degree. 
.phi. 30.degree. 
SI 20.degree. 
SI 26.degree. 
SI 28.degree. 
RA 0.75 RA 1.25 RA 1.25 
.phi. 5.degree. 
SI 22.degree. 
RA 1.5 
420 MHz .phi. 10.degree. 
.phi. 20.degree. 
.phi. 40.degree. 
SI 60.degree. 
SI 56.degree. 
SI 62.degree. 
RA 0.25 RA 0.5 RA 0.5 
__________________________________________________________________________ 
.phi.: phase difference, 
SI: Aperture angle, 
RA: radius of curvature (mm) 
As explained above in detail, according to the invention, the acoustic lens 
having desired properties can be designed in an easy and accurate manner. 
The following Table 3 shows some embodiments of the acoustic lens 
according to the invention. In these embodiments, the frequency of the 
acoustic wave was selected to be 400 MHz and the radius of curvature RA is 
set to 0.5 mm. Further, since the aperture angle SI of the lens portion is 
usually set to 60.degree. for general specimens, the aperture angle was 
designed to be about 60.degree.. It should be noted that values of Z and W 
of examples Nos. 12 and 13 fall within known valves for acoustic lenses. 
TABLE 3 
______________________________________ 
radius of 
lens length l 
aperture transducer a 
No. (Z, W) (mm) angle (SI) 
(mm) 
______________________________________ 
1 0.8, 0.6 15.243 60.degree. 
0.722 
2 0.8, 0.9 6.957 60.degree. 
0.481 
3 0.8, 1.0 5.697 60.degree. 
0.433 
4 0.9, 0.4 29.796 50.degree. 
0.958 
5 0.9, 0.5 20.275 52.degree. 
0.788 
6 0.9, 0.6 17.107 60.degree. 
0.722 
7 0.9, 0.8 9.766 60.degree. 
0.541 
8 0.9, 1.0 6.369 60.degree. 
0.433 
9 0.9, 1.1 5.32 60.degree. 
0.394 
10 1.0, 0.6 18.971 60.degree. 
0.722 
11 1.0, 0.8 10.815 60.degree. 
0.541 
12 1.0, 1.0 7.04 60.degree. 
0.433 
13 1.0, 1.1 5.875 60.degree. 
0.394 
14 1.2, 0.6 22.7 60.degree. 
0.722 
15 1.2, 0.8 12.912 60.degree. 
0.541 
16 1.2, 1.0 8.382 60.degree. 
0.433 
17 1.2, 1.2 5.921 60.degree. 
0.361 
18 1.5, 0.8 16.058 60.degree. 
0.541 
19 1.5, 1.0 10.396 60.degree. 
0.433 
20 1.6, 0.5 41.503 58.degree. 
0.848 
21 2.0, 0.6 37.615 60.degree. 
0.722 
22 2.0, 1.0 13.751 60.degree. 
0.433 
23 2.0, 1.2 9.65 60.degree. 
0.361 
24 3.0, 0.6 49.093 54.degree. 
0.674 
25 3.0, 0.8 31.789 60.degree. 
0.541 
26 3.0, 1.0 20.463 60.degree. 
0.433 
27 3.0, 1.2 14.311 60.degree. 
0.361 
28 4.0, 0.8 42.276 60.degree. 
0.541 
29 4.0, 1.0 27.1744 60.degree. 
0.433 
30 4.0, 1.2 18.971 60.degree. 
0.361 
31 4.0, 1.4 14.025 60.degree. 
0.309 
______________________________________ 
Frequency: 400 MHz 
Radius of curvature RA: 0.5 mm 
FIG. 16 is a graph showing points (Z,W) of the embodiments Nos. 1 to 31 
depicted in the Table 3. In all the embodiments, it is possible to obtain 
large power V.sub.max and power difference V.sub.max -V.sub.min, so that 
they can be used as the power lens as well as V(Z) lens. Particularly, the 
group surrounded by broken circle A is preferable as the V(Z) lens and the 
group surrounded by broken circle B is preferable as the amplitude 
contrast lens. Therefore, the embodiments belonging to both groups A and B 
can be preferably used as both the V(Z) lens and the amplitude contrast 
lens. In FIG. 16 the region near the point (Z,W)=(1,1) belonging to the 
known acoustic lens is shown by broken line C. 
In the above embodiments, the frequency of the acoustic wave was 400 MHz. 
According to the invention it is possible to design various acoustic 
lenses to be used at any desired frequencies. For instance, an acoustic 
lens for a low frequency such as 50 MHz having the following data values 
is obtained. 
Z=0.8 
W=0.9 
radius of transducer a=4.811 mm 
lens length l=86.14 mm 
radius of curvature RA=5.0 mm 
aperture angle SI=60.degree. When such a low frequency acoustic lens is 
used, the acoustic wave can penetrate into a specimen to a depth of about 
3 mm, so that it can be advantageously used to detect defects in a bond in 
a semiconductor chip or internal defects of ceramic products. 
As explained above, according to the invention it is possible to obtain new 
acoustic lenses having various properties by designing on the bases of 
values of Z and W which are selected from the region outside the region 
near the point (Z,W)=(1,1) of the known acoustic lens. Therefore, optimum 
acoustic lenses for various applications can be easily and accurately 
selected. Further, it has been confirmed that the acoustic lens for 
obtaining the V(Z) curve may have a phase difference of up to 50.degree., 
and thus a V(Z) acoustic lens having a higher power can be obtained.