IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-31, NO. 2 , MARCH 1984
105
Ray-Optical Evaluation of V(z)in the Reflection Acoustic Microscope Abstract-When viewing materials having Rayleigh velocity greater
than the velocity in the coupling fluid, the output voltage of the reflection acoustic microscope varies with defocus distance as a result of
coupling to the Rayleigh waves at the object surface. In this analysis,
the fields reflected by the surface are separated into a geometrically
reflected part, and one due t o the excitation and reradiation of the
Rayleigh wave. Using a ray-optical approach, simple formulas are
derived for the contribution to the output voltage from the geometrically reflected and Rayleigh fields. The formulas give good overall
agreement with measurements. Because the results are in analytic form,
it is easily seen how the various geometric and acoustic parameters of
the lens and object affect the output voltage.
0.8
U
'U
-27r
0.4 I. INTRODUCTION
W
ORKING in the reflection mode, the acoustic microscope with water couplant has been used to image the
surfaces of solids, for example, in the study of integrated circuits [ 11 -[4] , and in the investigation of grain structure in
metals [SI, [ 6 ] . For acoustic waves in water, the reflection
coefficient at the surface of most metals and crystaline materials is greater than 0.8. Because the reflection coefficient
is so high, its variation from point to point on the surface is
small and is not sufficient t o produce the contrast observed in
the images of integrated circuits and metals grains. It has been
recognized that the contrast in these images is due t o the sensitivity of the microscope's output voltage t o height variations
151 ,171 [91.
In studying the contrast mechanism, it was found that as the
microscope is moved towards an object with a smooth surface,
the output voltage exhibits a series of minima and maxima,
which was dubbed the acoustic material signature (AMS) [7].
The displacement of the microscope between minima was
found to be related t o the Rayleigh wave velocity of the object
[ 101 , [ 1I ] . While these observations were originally made on
spherical lenses, they have also been observed with cylindrical
lenses, whose focus is a line rather than a point [12], [13].
Two equivalent explanations of the AMS were given in terms
of Rayleigh critical angle phenomena [14] , [ 151 . From a rayoptical viewpoint, the AMS is due t o interference between the
fields of near-axial rays reflected from the surface, and the
fields along ray paths that include the excitation and reradiaManuscript received January 21, 1984. This work was supported in
part by a Visiting Fellowship from the Royal Society, London, and in
part by the Institute of Imaging Sciences of the Polytechnic Institute of
New York, and was carried out while the author was on sabbatic leave
at the Department of Electrical and Electronic Engineering, University
College, London.
The author is with the Department of Electrical Engineering and
Computer Science, Polytechnic Institute of New York, Brooklyn, NY
11201.
0.0
SIN 8
Fig. 1 . Magnitude and phase of reflection coefficient R (0) for plane
waves incident from water onto YIG-after Quate et al. [ 5 ] .
tion of the leaky Rayleigh wave [ 141 . Alternatively, Fourier
optics has been used t o explain the AMS in terms of the rapid
variation of the phase of the reflection coefficient R ( 0 ) for
angles of incidence 8 in the vicinity of the Rayleigh critical
angle OR = sin-' ( V w / V R ) ,where V , and VR are the wave
velocities for water and for the Rayleigh wave [ 151 . The
connection between the rapid variation of the phase of R ( 0 )
near
and the excitation of the Rayleigh wave is discussed
in [16], [17].
In this paper, we employ ray optics t o develop simple analytic expressions giving the variation V ( z )of the output voltage with displacement z of the object surface from the focal
plane. The previous ray-optical study of the AMS [I41 predicted the spacing Az of the minima of V ( z )based on changes
in phase along the different ray paths. The study did not compute the absolute phase or the amplitudes, so that it was not
possible t o predict the shape of the curve V ( z ) ,including the
depth of the minima. Fourier optics has been used to accurately predict the V ( z )dependence [8] , [ 181, [ 191 . However, because the Fourier transforms are carried out numerically, one cannot easily see from this approach how the
various geometric and acoustic parameters influence V ( z ) .
11. SEPARATION
OF GEOMETRICAND LEAKY
WAVE EFFECTS
The reflection coefficient for waves incident on a substrate
having Rayleigh velocity greater than about twice that of the
couplant is similar t o that shown in Fig. 1 for yttrium iron
garnet (YIG) and water. The magnitude and phase of R ( 0 )
have some variation in the vicinity of the longitudinal and
shear critical angles and 8,. For 8 >Os, R(8) = 1 for materials with no acoustic damping, but the phase of R ( 8 ) undergoes a rapid change of 2n as 0 increases past OR.
0018-9537/84/0300-0105$01.00 0 1984 IEEE
IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-31, NO. 2, MARCH 1984
106
.
"
0
Qp
-
"
,.
"
-
U
I
\
@
0
solid. then to first order
we have
n
Re k X
k,
%
CY
is the sum ad t al. Since 0 x k R .
kR t i ( f f d t 011).
(3)
It has been found that the effect of dissipation in the solid on
the zero ko of R(k,) is to increase its imaginary part by nearly
where the transverse wavenumber k , = k , sin 0 . In (l), p w
and p , are the mass densities of the couplant (water) and the
substrate, and kl = w/Vl,k, = w/V, and k , = w / V w ,where
w is the radian frequency of the harmonic time dependence
exp (- iwt), V l ,and V, are the longitudinal and shear wave
velocities of the substrate, and V , is the acoustic velocity of
the couplant.
It is seen from (1) that R(k,) has branch point singularities
at k k l , kk,, and kk,. In addition it has off axis poles and
zeros, two of which +k, and ?ko lie close to the real k , axis
in the complex k, plane. The location of the singularities is
shown in Fig. 2. Other poles and zeros lie far from the real
k , axis and do not significantly influence the field reflected
from the surface.
In the absence of acoustic dissipation in the solid, the square
bracket in (1) vanishes at the Rayleigh wavenumber kR =
w / V R . For k, real and I k, I > k,, the term in the square
bracket is real, while the second term in the denominator is
small and imaginary. Thus, the denominator of R(k,) will
vanish at complex points k, = ? k , close to k k R , where k , =
0 + i f f . It is found that f f / k R<< 1 and that (0 - kR)/kR <<
a/kR [ 1 6 ] , Because the second term in the numerator in (1)
is of opposite sign t o that in the denominator, R(k,) will have
a zero at the points k, = + k o ,where ko is the conjugate of k,.
Because the poles and zeros are so close t o k k R , the variation of R(k,) for I k , near kR will be approximately given by
I
This expression has unit magnitude and undergoes a phase
shift of nearly -2n as Ik , I increases past k R , i.e., as 10 I increases past O R . Thus, the presence of the pole-zero pair account for the major features of the reflection coefficient near
the Rayleigh critical angle.
In the absence of dissipation in the solid, a will be nonzero
because of the leakage of energy into the water. Dissipation in
the solid will cause an additional attenuation of the Rayleigh
wave. If ad is the attenuation constant of a Rayleigh wave
propagating on a free surface, and aIis the attenuation due to
B. Extraction of the Leaky Wave Poles
As discusse'd above, the behavior of R(k,) for k , near +kR
is contained in the factor (2). Thus, we define Ro(k,) such
that R(k,) is the product of Ro(k,) and the factor in (2), i.e.,
In the absence of dissipation, Ro(k,) will have the same magnitude as R(k,). The phase of Ro(k,) will be nearly the same
as that of R(k,) for I k, I < k,, but will be nearly constant for
k , < 1 k, I < k,, except for a sharp increase by n as Ik, I approaches k,. In the presence of damping, the phase and magnitude of Ro(k,) will be nearly constant for I k , I in the
vicinity of kR since the variation of R(k,) in that region is
accounted for by the pole-zero factor.
The term (k; - k:)/(ki - k f , )is small compared t o unity
except for I k, I near kR , where Ro(k,) = 1. We can, therefore, approximate R(k,) with little error as
The above form of R(k,) separates it into a part containing
the Rayleigh pole singularity and a portion Ro(k,) that contains the variations due to the branch points.
C. Separation of the Reflected Field
Consider a field incident on the substrate, as shown in Fig. 3.
The field is assumed t o have no variation alongy, and to be
described by a family of rays that have a line focus at x =
z = 0. Provided that zi is larger than about one wavelength
A, = 2n/k,, the pressure along the surface of such an incident
field is given by
(7)
where G ( 0 ) is a slow varying amplitude function. A twodimensional field is treated here for simplicity, since the rayoptical expressions derived for the excitation of the leaky
BERTONI: RAY-OPTICAL EVALUATION OF V ( Z )
,
107
TRANSDUCER
REFLECTED R A Y S
\
\---
L E A K Y RAYLEIGH
Y
r
T
1
PHASE
&
D
II
I
I
'
Fig. 3. Ray structure of reflected fields produced by a convergent field
incident on surface of a solid.
wave can then be applied t o the microscope case using the
procedures of ray optics.
Let Pi@,, z i ) be the Fourier transform of the incident pressure field p i ( x , z i ) given in (7). The reflected pressure field
p r ( x , z i ) along the surface can then be found by taking the inverse transform of the product of P i ( k x ,z i ) and R ( k x ) . Using
the approximation (6) for R(k,), we can separate the pressure
of the reflected field into a geometrical part p ~ ( xz ,i ) and a
leaky wave part p L ( x , z i ) . Thus,
Pr(xi zi) = P G ( X , Z i )
+ P L ( X ,Zi), (8)
where
and
The branch point singularities of R o ( k x )in (9) give rise t o
lateral ray contributions t o p ~ ( xz i, ) [23] , [24] . Ignoring
these contributions, the steepest descent evaluation of (9)
yields the ordinary ray-optical representation of p c ( x , zi), in
which the reflected ray field is found as the product of the incident ray field and the local reflection coefficient [23], [24].
Thus, the total reflected field contains a geometrical optics
term
~ ~ ( x , z i ) = R o (COS
k , e ) p i ( x , z i ) -
(1 1)
The field incident in the vicinity of the point xi < 0, at
which the ray incident at the angle OR intersects the surface,
will excite the leaky Rayleigh wave propagating to the right.
In the Appendix, it is shown that to the right of the excitation
region about x i , the field p i (x, z i ) along the interface associated with the leaky wave propagating t o the right is given by
The leaky wave fields at a finite distance above the surface in
Fig. 3 can be found by tracking along the parallel family of
I
SURFACE
214
4z
Fig. 4. Geometrically reflected rays in the microscope for defocus distance z < 0.
leaky wave rays. Ignoring diffraction effects in the vicinity of
the ray that leaves the surface from the point xi, the field amplitude is constant along the parallel family of rays and the
phase increases linearly with distance.
In the acoustic microscope, the leaky Rayleigh waves excited
in the substrate have cylindrical symmetry, converging from
the point of excitation x i to the z axis (x = 0), and then diverging as they pass the z axis. The field variation associated with
this cylindrical focusing is given by the additional factor
d m . As the field passes the focus it acquires an additional phase given by the factor exp (- in/2), corresponding t o
a line focus [25].
111. VOLTAGEDUE TO GEOMETRICALLY
REFLECTED FIELDS
The transducer and lens geometry is shown in Fig. 4. The
transducer is assumed to be circular with radius R T , and to be
a distance D from the spherical lens surface. The lens has
radius R and aperture radius a , which is assumed to be less
than R T . The focal length f of the lens is given by f = n R /
(n - l), where n is the ratio of the longitudinal wave velocity
in the lens rod t o the wave velocity in water.
A . Ray Structure for z <0
The transducer radiates a family of rays, which in the absence of diffraction are parallel to the z axis. These rays are
refracted at the lens surface and focused on the point z = 0
along the axis. If the object surface is located above the focus
( z < 0), as shown in Fig. 4, the rays reflected geometrically by
the object coverage back on the z axis. If z is in the range
- f/2 < z < 0, the rays cross the axis at a distance 2 Iz I above
the focus before they reach the lens. After refraction at the
lens surface, the ray in the lens rod appear t o diverge from a
virtual focus located a distance I U I below the lens surface. For
IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-31, NO. 2, MARCH 1984
108
- f < z < - f / 2 , the rays reflected from the object surface are
refracted by the lens before crossing the z axis. In this case,
the refracted rays cross the z axis at a focus located a distance
U above the lens surface. For all values of z , the value of U is
found from the imaging formula for a spherical surface t o be
U
=f(f+ 2z)/(2nz),
T
r T = 2(D- U ) / ( - u ) = a [ l - 2 z ( n D - f ) / f 2 ] .
-LIMITING
For z < z m i n ,the entire transducer is illuminated by the
returning field, while for z > z m i n only a limited area of radius
rT is illuminated. As suggested in Fig. 4, when 1 z I is not near
zero, the reflected rays illuminating the transducer are produced by incident rays that originate from a small area at the
center of the transducer.
<0
The transducer is assumed to excite fields that reach the lens
surface with unit amplitude stress Tzz. The wave is transmitted through the lens surface with transmission coefficient
T I . For an uncoated sapphire lens, and water couplant, the
variation of T I across a lens of 50" half angle is about 2 dB
[ 2 6 ] . If a quater wave matching layer is deposited on the lens
surface, the variation T 1 is still approximately 2 dB for a 50"
half angle lens, but the variation increases rapidly for half
angles above 50" [ 2 6 ] . In this analysis, we treat T I as a constant. The error introduced by this approximation decreases
rapidly as the microscope is moved towards the object, since
the reflected rays illuminating the transducer cross the interface in a small region near the center of the lens. For the same
reason, the reflection coefficient at the object surface is taken
to be the value at normal incidence R,(O) for all rays, and the
transmission coefficient T2 from water back into the lens is
taken to be a constant.
The amplitude of the pressure field in the water is affected
by the convergence, and subsequent divergence of the rays.
To conserve power, the field amplitude along a ray bundle
must vary inversely as the square root of the cross-sectional
area of the bundle [25]. Thus, the field returning t o the lens
surface differs from the field transmitted into the water by the
RAY
f
\OBJECT
SURFACE
(14)
The condition rT = RT is obtained at the object displacement
B. Ray Fields for z
--
(13)
which gives the virtual focus (U < 0) for - f / 2 < z < 0 , and a
real focus (U > 0 ) for -f < z < - f / 2 .
The rays returning to the transducer plane illuminate a circular spot of radius r T , which is defined by the limiting ray
refracted at the edge of the lens, as shown in Fig. 4. For I z I
small and RT > a , the illumination spot will be smaller than
the transducer. In this case, integration of the field over the
transducer to obtain the output voltage will be limited t o the
radius rT of the illuminating spot. For rT > R T , the integration must be taken over the entire transducer. To find rT for
Iz I small, we approximate 2 in Fig. 4 by a(f+ 2 z ) / f , recalling
that z < 0. Then for U < 0,
T
PHASE FRONT
22
Fig. 5. Geometrically reflected rays in the microscope for defocus distance z > 0.
factor f/lf+
22 I. The field returning t o the transducer from
the lens surface is further reduced by the factor I u/(D - u ) I
due t o ray spreading in the lens rod. An additional reduction
in the field amplitude due t o attenuation in the water is given
by the factor exp [-2a, ( f + z ) ] , where a, is the attenuation
constant in water.
Due to the spherical curvature of the phase fronts of the
field illuminating the transducer, the phase of the field arriving at the transducer is equal t o the sum of the phase [ 2 k ,
(D/n t f+ z ) ] of the central ray, and a curvature correction,
which for D >>RT is given by k,p2/[2n(D - U ) ] . Note that
k,/n is the wavenumber of longitudinal waves in the lens rod.
When the rays reflected from the object surface pass through
the focus along the axis, the fields experience an additional
phase change of -n [ 2 5 ] .
When taken together, the various amplitude and phase factors described above give the following expression for the field
at the transducer:
f
ARo(0)exp [(ik, - a,) 2 z ] If+zl
.~
ID-
UI
where
A = - T , T 2 exp [ i 2 k W ( D / n + f ) -2 a , f ] .
C. Ray Fields f o r z > 0
(17)
The ray structure for the case z > 0 is shown in Fig. 5. The
focal point for the rays returning t o the transducer now lies
above the lens surface a distance U given by (13). For z small,
U will be greater than D so that the focal point lies above the
transducer, and the ray field is convergent at the transducer,
rather than divergent as shown in Fig. 5. The condition U = D
BERTONI: RAY-OPTICAL EVALUATION OF V ( z )
109
is obtained for z = zo , where from (1 3 )
20
=
f2 2 n ( D - f/n)'
F = n ( D - f / n )h,/R$.
(18)
For this condition, the focal spot lies o n the transducer, and
the integration of the ray-optical field over the transducer will
vanish. Accounting for diffraction, the focal spot will have
finite size so that the integration, and hence the output voltage, will have a nonzero minimum.
As z increases above zo , U decreases below D and the rays
reaching the transducer will be divergent, as shown in Fig. 5.
Further increase in z causes an increase in the radius IT of the
illuminated spot o n the transducer plane, until at some value
z = zma, the entire transducer is illuminated. For D >>a, it
is seen from Fig. 5 that rT can be approximated as
rT = U
ID - U I/u.
The quantity f / n is the focal length inside the lens rod. Thus
D - f/n in (23) represents the distance from the transducer to
the back focal plane of the lens. Since nh, is the wavelength
in the lens rod, R$/(nh,) is the Fresnel distance for the transducer. Thus F in ( 2 3 ) represents the ratio of the separation
between the transducer and back focal plane to the Fresnel
distance of the transducer. In typical lens designs, F is near
unity.
Making use of ( 1 S ) , (1 8), (20), and (23), and assuming
D >>f/n, as is frequently the case, we can express z o , zmin,
and z,,
as
zo = (f/RT)*h,/(2F),
Zmin
(19)
Equating TT and the transducer radius R T , and using (13),
gives z,,
as
For z > z,,
the transducer is fully illuminated.
Using the same arguments as given in the previous section,
the reflected ray field at the transducer is again found t o be
given by (16), except for an additional -n phase shift for
z > zo that the fields acquire as they pass through the second
focus at U.
(23)
=-zo [ ( R T / -~ )11
zmax x zo
1
[ ( R T / ~+)1 1 .
(24)
The value of RT is typically chosen to lie between a and 2a,
while f i s somewhat larger than a. Thus, for F near unity. it
is seen from ( 2 4 ) that zo , I zmin I and z,,
are all on the order of A,, which is usually much smaller than f. As a result,
2z/f << 1 for 0 < z < z,,, , and the middle expressions for
X in (22) holds approximately over the entire range z,,, <
z
<Zmax-
The variation of I VG(z)l is shown in Fig. 6 for F = 1 and
two values of RT/a. Note that for given values of a and F ,
doubling RT implies that D must be increased by a factor of
four. For I z I >> zo , sin X = 1 so that the variation in I V , ( z ) I
is due to the factor (l/z) in ( 2 1 ) ,and the attenuation in water
D. Output Voltage
exp ( - 2 q V z ) ,which has been neglected in drawing Fig. 6. The
zero of V,(Z) at z = zo results from the focusing of the reThe output voltage VG produced by the geometrically reflected ray fields falling on the transducer is found by integrat- flected rays onto the transducer by the lens. In measurements
made with various spherical and cylindrical lenses, a minimum
ing ( 1 6 ) over the transducer. For z < zmin and for z > z,,,
of I V G ( Z I)occurs just to the right of the peak. as suggested in
the transducer is fully illuminated and the upper limit of the
integration over p must be taken as RT. Otherwise, the illuFig. 6 [ 5 1 , P I , [121, ~ 7 1 .
The choice F = 1 places the lens at the point where the ramination is limited to a circle radius rT given by (14) or (19).
diated
fields change from near field to far field dependence.
We assume that the output voltage that would be produced
As
a
result,
diffraction effects will be significant in the propaby a uniform field of unit amplitude is V o . Carrying out the
gation
between
the lens and transducer. Diffraction, together
integration over the transducer, as discussed above and simpliwith
variation
in
T 1 , T , , and Ro(O)that have been neglected,
fying, leads t o the expression
are expected to have the effect of reducing the voltage for z
small. However, spherical divergence of the reflected rays for
IzI >>zo reduces the significance of these effects. The zero
at z = 2z0 for RT = a results from phase cancellation across
exp [(ik, - a,) 22 + i X ] .
( 2 1 ) the transducer. Unlike the one at z = z o . the existance of the
Here,
zero at z = 2 z 0 , and even its location, will be sensitive t o the
exact size of the transducer, variations of T 1 ,T,, and Ro(B),
and to diffraction effects.
When F < the term sin X in (21) has one or more zeros in
the range z <O. ThFse zeros are the result of phase cancellation across the transducer. This effect is shown in measurements made by I. Smith [28] with a 12.5 MHz fused quartz
lens whose aperature had been reduced to eliminate Rayleigh
wave excitation on Dural. The measured response for a Dural
object, shown in Fig. 7, is therefore due entirely to the geowhere
metrically reflected field. The deep minimum for z < 0 is con-
i,
IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-31, NO. 2, MARCH 1984
110
Fig. 6. Output voltage V G ( Z ) due to the geometrically reflected fields for a lens with F = 1. The solid curve is for RT = a .
If RT = 2 a , V G ( Z )is modified in the range -1 < z < 3 as shown by the dashed curve, but is unchanged outside this range.
Dural V[z] p a r a x i a l lens
30
r
-4
-3
-2
-I
O
I
z (mm)
Fig. 7. Output voltage for a 12.5 MHz, limited aperture lens with F =
0.36 showing a zero of sin X for z < 0.
sistent with the short length of the lens rod compared to the
Fresnel distance. The dimensions of the lens used to make the
measurement in Fig. 7 give the parametersf= 21.4 mm, F =
0.36, and zo = 0.532 mm from (18). The stop radius was not
recorded when the measurements were made. Reasonable
agreement was obtained for a = RT/3 and the resulting curve
is shown dashed in Fig. 7 .
SURFACE
Fig. 8. Leaky Rayleigh wave rays in the microscope for defocus distance z < 0.
IV. VOLTAGE
D UE TO THE LEAKYWAVE F I E L DFOR z < 0
The transducer and lens geometry is shown in Fig. 8, together with the rays relevant for computing the field at the
transducer resulting from the leaky wave excited on the object
surface.
A . Structure of the Rayleigh Rays
As it propagates along the surface, the leaky Rayleigh wave
in the plane of the drawing reradiates at the angle O R , as
shown in Fig. 8. One of these rays, called the principal ray,
appears to come from the focus, and is refracted at the lens so
BERTONI: RAY-OPTICAL EVALUATION OF V ( z )
111
as t o travel parallel t o the lens axis. Rays that are reradiated
on either side of the principal ray in the plane of the drawing
travel parallel t o it in the water. At the lens surface they are
refracted so as to focus inside the lens at distance ( f / n )
cos' OR from the lens surface. Since the lens has cylindrical
symmetry about the z axis, the focus of the entire leaky wave
family is in the form of a ring of radius p R =:sin
OR.
As seen in Fig. 8, those rays radiated t o the right of the principal ray pass through the ring focus and illuminate the portion
of the transducer to the left of the principal ray. Provided
that the lens aperture a is not close to p R and, assuming
D >> a, these rays will illuminate the entire portion of the
transducer to the left of the principal ray. Considering the
entire cylindrically symmetric ray family, it is seen from Fig. 8
that each point of the transducer is illuminated by two rays,
one from each side of the ring focus. One set of rays crosses
the axis, which is a line focus for the set.
B. Ray Fields
As in the case of the geometrically reflected rays, the lens is
assumed t o be illuminated by a family of rays parallel to the
axis and carrying a stress field T,, of unit amplitude. This
field is transmitted into the water with transmission coefficient TI. Because of the spherical convergence, the field of
the ray incident on the object surface at the angle OR will have
an amplitude increased by the factorf/(l zI sec e,). Attenuation in the water results in the additional factor exp [-a!,(f1 z 1 sec e,)] . The phase at the surface of the ray incident at
OR is equal to the phase k , (f+D/n) at the focus in the absence of the substrate, less the phase change k , Iz I sec O R , for
propagation from the surface plane to the focus. Using the
foregoing, the field at the surface of the ray incident at the
angle eR is
p ~ ) l / ' due t o cylindrical spreading of the rays (out of the
plane of the drawing in Fig. 8). Water attenuation introduces
the additional amplitude factor exp [-a!,(f- ( z I sec e,)].
Finally, transmission through the interface is accounted for
by the coefficient Tz .
The rays in the lens rod experience convergence and divergence in the plane of the drawing of Fig. 8, as well as in
the radial direction. The convergence t o the ring focus, and
subsequent divergence, in the plane of the drawing effects the
field reaching. the transducer by the amplitude factor [fcos'
OR /(no)]'I2, for D >>f/n. The focus introduces the phase
factor exp (-in/2) corresponding t o a line focus. The radial
spreading effects the field at the transducer by the amplitude
factor ( p R / p ) ' l z , where p is the radial distance at which the
ray intercepts the transducer plane. An additional phase factor exp (- in/2) must be included for those rays that cross the
lens axis.
The phase change due t o the path length along the principal
ray from the object surface to the transducer is given by
k,(f t D/n) less the phase kw I z I sec OR for propagation from
the focus t o the object surface. Because neighboring rays also
pass through the ring focus, they will have the same phase except for a correction due to phase front curvature. For rays
that do not cross the lens axis the correction is given by
k,(p - pR)'/(2nD). Rays that cross the axis have phase correction k w ( p + pR)'/(2nD).
The various factors described in the preceding three paragraphs can be combined into a single factor that gives the
variation of the field in going from the object surface t o the
transducer plane. This factor is given by
T2(IZltaneR
. %) 11'
nD
PR
P
. e-in/z exp [ik,(f+ D/n -
L.
*
. exp
fcos'eR
+ D / n - IzI sec e,)].
[ik,(f
(25)
The field given by ( 2 5 ) excites the leaky Rayleigh wave in
the vicinity of the radius I z I tan O R , which then propagates
cylindrically inward towards the z axis. After passing the focus
at the z axis, with the attendent phase change exp ( - i n / 2 )
appropriate to a line focus [ 2 S ] t, he leaky wave diverges.
When it again reaches the radius I z I tan O R , it launches the
principal ray of Fig. 8, whose initial field is given by (12), with
x - x i replaced by 2 Iz I tan O R , and multiplied by the factor
exp ( - h / 2 ) .
As discussed in the Appendix, expression (12) gives the initial field of the principal ray provided that z < z , where
z1 = -h,/(4 cos
(26)
sin' OR).
For I z I < I z 1, the initial field of the principal ray is modified
by the factor [ 1 erfc (s)] . For example, when z = 0,
then s = 0 for the principal ray and the factor is or 6 dB.
The field propagating along the principal ray back t o the
lens surface decreases in amplitude by the factor (lz I tan e,/
(3)
exP [-
+
e-in/z
( f - IZI sec
Iz I sec
e,)]
)I ( ~ X P[ikw ( P - PR)' /(2nD)I
exp [ikw ( P + PR l2 /(2nD)11.
(27)
In this expression, the first term in the bracket arises from the
rays that do not cross the lens axis, while the second term
comes from rays that do cross the axis.
The field at the transducer is found by multiplying (27) by
(12), with p i substituted from (2S), and finally by the factor
exp (-in/2). After some manipulation the field is found t o be
. exp [ 2 ( a ,
- a! sin 0,)
IzI sec O R ]
1
. exp [- i 2 k , I z I cos OR ] -
6
{exp [ik, ( p - p ~ ) ' / ( 2 n D > +
]
3
where A is defined in (17).
IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-31, NO. 2, MARCH 1984
112
I .4
T'"'
1.2
1.0
0.8
-T
R
0
- -200
-
- -404
Fig. 9. Variation of the integral K with1
.,'
1
1
1
1
1
1
,
1
1
1
1
,
1
1
1
1
-3
-2
-I
0
Fig. 10. Output voltage due to leaky Rayleigh wave fields on a fused quartz object.
-7
-6
-5
-4
C. Leaky Wave Voltage VL
The output voltage due to the leaky wave is found by integrating the field (28) over the transducer. Again we let Vo be
the voltage that would be produced by a uniform illumination
of unit amplitude. Changing the variable of integration, the
voltage can be written as V, = VoAK
[2 a l h W(n( f i f 3
sin
h, D)3/4
. exp [-i2kwI z I cos e,]
exp [2(a, where
(Y
sin 6,)
IzI sec e,],
(29)
and
The integral in (30) cannot be evaluate.. in terms of simple
functions. We have computed K numerically as a function of
p R for various values of pT,i.e., various values of F. The results of the calculations are shown in Fig. 9 . It is seen that
neither the magnitude or phase of K is sensitive to p R .
For z <zl , the only dependence of VL on z is through the
phase term exp (- ik, I zI cos 0,) and the exponential amplitude term. The dependence of I VLI on z is shown in Fig. 10
for a fused quartz object (e, = 26") and a lens with F = 1 and
a = f sin 45'. Curves are shown for two values of transducer
radius RT = a and R T = 2a. The difference in the slopes of
the two curves for I VLI is due to the fact that the absissa is
z/zo,
where for a fixed value of F it is seen from (24) that zo
varies with R T , Had the curves been plotted as a function of
z,the slopes would be the same.
113
BERTONI: RAY-OPTICAL EVALUATION OF V ( Z )
With the values of the lens parameters chosen, and for
OR = 26', Izl l/zo has value 0.31 for RT = a , and 1.24 for
RT = 2a. These values give the endpoints for the solid curves
in Fig. 10. At z = 0, I V, I is 6 dB lower than the value obtained from (29) due to the factor [ l - ($) erfc (s)] . In Fig.
10 a straight line, drawn dashed, has been used t o connect
IVL I at the point z1 /zo and at the point z = 0, in order to indicate the general behavior of I V, 1 in the range z1 < z < 0.
The exponential variation of I V, (z) I indicated in Fig. 10 has
been observed by Smith and Wickramasinghe [27]. Working
with a 12.5 MHz fused quartz lens ( R = 16 mm), they blocked
its central portion so that none of the geometrically reflected
rays reach the transducer for Iz I >> zo. As a result, the voltage is due entirely t o the leaky wave rays in this range of z.
The curve of I VG(Z)I in Fig. 10 is the same as shown in
Fig. 6 for z/zo < - 1, except for a 2 dB reduction corresponding t o R o ( 0 )of fused quartz. Deep minima in the AMs will
occur when I V, I and I VGI are nearly equal. For I z 1 >> zo
the value of I V, I will be greater than that of 1 VGI implying
that a method for selectively reducing I V, I is required t o
achieve a series of deep minima. Attenuation in water, which
has been neglected in drawing Fig. 10, is seen from (21) and
(29) t o slightly increase I V, I over I VGI, while acoustic damping in the object has the reverse effect.
In expression (29), the amplitude term in the brackets is
proportional t o the attenuation al of the Rayleigh wave resulting from reradiation into the water. The attenuation ad
of the Rayleigh wave due to acoustic damping in the object
does not appear in the amplitude term in the brackets, although it is in the exponential term. Dransfeld and Salzmann
[ 191 give a simple approximate expression for a1that leads to
the relation
(YlAw
I
-20
I
I
-16
I
I
-12
: I ,
I
-8
I
-4
I
1
0
I
I
4
I
I - ,
8
z
(pm)
Fig. 11. Total output voltage V ( z ) o f a 1.1 GHz lens with F = 1.1 for
a YIG object. Solid curves are calculated and dots are
experimental points taken from Quate et al. [ 5 ] .
was taken to be 0.204 dB/ym, and accounts for the fact that
1 V, I increases t o the left. The Rayleigh angle OR = 24.62'
was chosen for the calculation to fit the measured data. This
angle corresponds t o a Rayleigh velocity VR = 3600 m/s,
which is in the range of reported values for YIG. The theoretical curve gives reasonable agreement with the measurements, except in the range - 1 < z < 5 where diffraction effects in the lens rod are most important.
A. Acoustic Material Signature
= ( P w / P d (sin O R ) 2 ,
(32)
where ps and p w are the mass densities of the object and
water. Using this expression, the bracketed term in (29)
becomes
The series of maxima and minima for z < 0 constitutes the
AMs, and may be used to measure the Rayleigh velocity of the
object. The minima occur when the phase difference between
V, and V, is an odd multiple of n. The phase difference is
given by
2k,v(1
. (sin O R ) 5 1 2 (f / R T ) 3 J 2 ,
(33)
which allows simple estimation of the amplitude of the leaky
wave voltage relative to VG.
V. TOTALVOLTAGEA N D THE AMS
The total output voltage V ( z )of the transducer for z < 0 is
obtained by adding (21) and (29), taking into account their
relative phases. For z > 0, I V, I will decrease very rapidly to
zero, and hence the total voltage will be nearly equal to VG.
As an example, we have plotted I VG(z)I, I V,(z)I, and
I V(z) I in Fig. 11 for a 1.1 GHz saphire lens looking at YIG.
The dots in Fig. 11 represent measurements reported by
Quate, Atalar, and Wickramasinghe [ S I . The lens dimensions
R = 105 pm, a = 75 ym, and D = 1230 pm give the parameters
F = 1 . 1 , ~=0.82
~
y m , a n d f = 121 pm. Attenuation in water
- cose,)IzI+(n+&-X),
(34)
provided that F > so that sin X does not change the sign for
z < 0. When z is several times z o , X = n/(;!F)and the second
term in (34) is independent of z. In this case, the separation
between minima is given by
A,,,/(l - cos OR), which was
previously derived [ 141 . [ I S ] . For F < additional n phase
shift must be included in (34) as the zeros of sin X a r e passed.
The accuracy with which the Rayleigh velocity can be determined depends on the number of minima that can be measured and their narrowness. Narrowness of the minima depends on the peak-to-valley ratio, which in turn depends on
how close I VGI is to I V, 1. Thus, it is desirable to have I VGI
and I VL I as closely equal to possible over a large range z < 0.
In Fig. 11, crossing of the curves for I V, I and I VG I near a
point where the phase difference (34) is 3n results in a very
deep minimum. However, since I V, I becomes greater than
I Vc I to the left of this point, the next minimum is less deep.
To study the dept of the minima, consider the ratio of 1 V , I
(i)
3.
IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-31, NO. 2, MARCH 1984
114
to I VL I in the range I z I >> zo. In this range X = 71/(2F) so
that using (21), (241, (29) and taking F x X,D/R$
*
exp (212 I [a tan
-
a , (sec OR
-
I)]}.
(35)
The dependence of this ratio on Iz I is given by the last factor.
For the typical case where a tan OR is large compared to
a, (sec OR - 1). the ratio is large for I z I small and decreases
to a minimum at
lz1=(3)/[ataneR-a,(seceR-
I)].
(36)
If the minimum (36) is less than the distance from the edge of
the lens t o the focal plane, then I z I may be increased beyond
(36), in which case the ratio (35) will again increase.
Substituting the value of Iz I at the minimum into (35), and
neglecting a , (sec OR - 1) yields the minimum values of the
ratio as
asymuth angle in the transducer plane. The resulting phase
cancellation reduces the magnitude of the average.
Another possible effect of anisotropy is typified by the
(100) cut of Si. For this cut the Rayleigh wave exists only
for limited ranges of asymuth angle. Outside this range it
merges with a bulk shear wave [31]. Out of a total of 360”
of asymuth angle, the Rayleigh wave exists only over about
224”. This fact effects I VL I by the factor 224/360, or 4 dB.
In Fig. 12, we have plotted 1 VGIz, I VL1’ , normalized to
I VoA1 2 , for a 370 MHz saphire lens looking at Si. The calculations include the 4 dB reduction in I VLI discussed above. It .
was assumed that R = 380 pm, RT = a = R sin 55” and F = 0.8
in order t o compare with measurements made by Weglein [ 111
on Si using a lens of similar dimensions. The peak-to-valley
ratio compares well with Weglein’s measurements made o n
(100) Si. On the (1 11) cut of Si, on which Weglein also made
measurements, the Rayleigh wave exists for all angles, but has
a large variation of velocity with direction. Judging from the
peak-to-valley ratio found for the (1 11) cut, beam steering and
phase cancellation effects are more significant for this cut than
the effect cited above is for the (100) cut.
VI. CONCLUSION
(37)
The last factor in (37) depends entirely on material parameters. Since K is only weakly dependent on v R , the middle
factor depends primarily on the lens properties.
For the lens used to make the measurements shown in Fig.
11, and taking IK I = 1, the middle factor in (37) is 1.14.
Since YIG has low acoustic damping a e a[,and the last factor
has value 0.65. Hence, the minimum value of I VGI// VL I is
0.48 which corresponds to a peak-to-valley ratio in the AMS
of 9 dB. The AMS may be improved somewhat by the choice
of F and RT.
B. Effects of Damping and Anisotropy
Materials with intrinsic acoustic damping will have a >al.
For materials with small damping, having a > al makes the
ratio (37) closer t o unity and thus improves the peak-to-valley
ratio. However, materials with very large damping (a >> ar)
will have the ratio (37) greater than unity. In this case, the
peak-to-valley ratio improves as a decreases. This effect has
been seen on steels with attenuation up t o 6 dB per wavelength [30] .
Anisotropy in the substrate has several effects that tend t o
reduce I VL I. Beam steering of the Rayleigh wave in the surface will prevent surface waves from focusing on the lens axis.
As a result, some of the returning rays in the lens rod will not
travel in the plane containing the lens axis. Because of this
additional tilt, these rays will give a lower contribution t o the
output voltage, or may even miss the transducer. Even without beam steering, the variation of O R requires that the factor
exp (-i2k,u Iz I cos 6,) in (29) be replaced by its average over
Using a ray-optical approach, simple formulas have been derived that give the output voltage V G ( z )due to the fields geometrically reflected at the object surface, and the voltage
V L ( z )due to the excitation and reradiation of the leaky wave.
Contrary t o previous assertions [ 5 ] , ray optics does not lead
t o a simple sin ( k z ) / z variation for I VGI. Instead, I VGI shows
the marked asymmetry observed for actual lenses. In particular, for Iz I >> zo and Fsomewhat greater than
I V , 1 varies
as l / l z I. The expression for VL in the range z < 0 indicates
that its magnitude decreases as a simple exponential, except
for z near zero.
The total voltage V ( z )predicted by the ray-optical formulas
is in good agreement with measurements in the AMS region.
Near focus. the formulas are not as accurate because effects
such as diffraction in the lens rod have been neglected. Nevertheless, they do give a qualitative description of near focus
behavior.
The formulas make it relatively easy t o see how the various
geometric and acoustic parameters of the lens and object influence V(z). For example, it was shown how the parameters
determine the peak-to-valley ratio in the AMS. To improve
this ratio significantly, it is necessary t o reduce VL as compared to VG by some modification of the microscope. One
possible approach, suggested by the effects of anistropy, is t o
break the cylindrical symmetry of the Rayleigh wave fields
about the lens axis. This could be accomplished by notching
the transducer on one side down to a radius slightly less
than p R .
4,
APPENDI x
EVALUATION
O F LEAKYWAVE FIELDS
In order to evaluate the integral (lo), we substitute for
Pj(k,, zi)its Fourier representation in terms of pi(x, zi). With
this substitution, and changing order of integration, one
115
BERTONI: RAY-OPTICAL EVALUATION O F V ( Z )
lo-'
IO'
-3
IO
-4
I
n
1"
-2 00
-160
-120
- 80
- 40
0
40 Fig. 12. Computed output voltage of a 370 MHz, F = 0.8 lens for a (100) cut silicon object. Lens is similar to that of Weglein [ 11 ] . Since kR >>a, it is seen from ( 3 ) and (4) that
obtains
-
i k2 P- k i - - 2a,. 2kP Thus, using expression (7) for the incident field in the first integral in ( A 2 ) gives the following expression for the field of (Al)
leaky wave propagating in the + x direction: 1
dk, dx'.
p t ( x , Z i ) = - 2 q eikpx
G ( 0 ) ea,' exp {- i [ k , . \ l ( ~ ' ) ~+ (zi)' + k ~ x ' l d} x , ,
kh'' [(x')'
The integration over k , may be evaluated by deforming the
path of integration, which initially lies along the real k , axis,
into the upper half plane for x - x' > 0 , or into the lower half
plane for x - x' < 0 . In the first case the pole at kx = k, is intercepted, while in the second case the pole at k , = - k , is
intercepted. Taking the residues at the pole gives
+ (zJ2 ]
The integrand of (A4) has a stationary phase point [ 2 5 ] when d
dx 7
[-k,
d(X')'-+ ( Z j ) 2
- k R x ' ] = 0.
The solution of this equation for x' is
x' = z i tan O R = xi.
pL(x, z i ) = i 2kP i k p (x - x ' )
dx' The first integral in ( A 2 ) represents the field of the leaky wave
propagating in the +x direction, which is strongly excited by
the incident field to the left of the z axis in Fig. 3. The leaky
wave propagating in the - x direction is represented by the second integral in ( A 2 ) .
(A61
where x i is the point at which the incident ray making an angle
OR with the z axis intersects the surface of the solid. Expanding the phase to second order in ( x ' - x i ) about the stationary
point, and evaluating all amplitude terms at the stationary
point, the integral in ( A 4 ) may be approximately by the
method of stationary phase [25] as
p t ( x ,zi)= - 2 q e
ik ( x - x i )
p
Pi(xi, Z i )
IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-31,NO. 2, MARCH 1984
116
Making a change of variable in the integration, the integral can
be expressed in terms of the complementary error function
[ 3 2 ] . Thus,
s = (x
- Xi)
PW
(cos eR )3
21zj1
For ( x - x i ) large and positive, s has large magnitude and lies
in the first quadrant so that lerfc (s) I << 1. In this limit, (As)
reduces t o (1 2). Numerically, I erfc (s) I < 0.1 for I s I > fi
which occurs when
x
- xi > [I zi I hw/(cos
] ‘I2. (A10)
If I zfl>> A,, then erfc (s) may be neglected a short distance
t o the right of the point x i .
In evaluating the leaky wave contribution t o the output voltage we are primarily concerned with the field radiated about
the point - x i , which is located symmetrically about the z axis
from the launch point x i . Using this value of x in (AlO), together with (A6), it is seen that erfc (s) can be neglected at
- x i provided that
I zi I > hw/[4 cos
(sin
OR)*
3,
(‘41 1)
which is on the order of A,,
ACKNOWLEDGMENT
The author would like to thank Kumar Wickramsinghe and
Ian Smith for supplying him with experimental measurements
of V ( z ) and the corresponding lens dimensions.
REFERENCES
[ 11 R. A. Lemons and C. F. Quate, “Integrated circuits as viewed with an acoustic microscope,”Appl. Phys. Lett., vol. 25, pp. 251-253,1974. [ 21 V. Jipson and C. F. Quate, “Acoustic microscopy at optical wavelengths,”Appl. Phys. Lett., vol. 32, pp. 789-791, 1978. [3] R. L. Hollis and R. Hammer, “Imaging techniques for acoustic microscopy of microelectronic circuits,” IBM Watson Research Center, Rep. RC 8534 (#36855), Sept. 1980. [4] A. J. Miller, “Application of acoustic microscopy in the semi- conductor industry,” in Acoustic Imaging, vol. 12, E. A. Ash and C. R. Hill, Eds. London: Plenum, 1982, pp. 67-78. [SI C. F. Quate, A. Atalar, and H. K. Wickramasinghe, “Acoustic microscopy with mechanical scanning-A review,” Proc. IEEE. vol. 67, pp. 1092-1113,1979. [6] G.A.D. Briggs, C. Ilett, and M. G. Somekh, “Acoustic micros- copy for materials studies,” in Acoustic Imaging, vol. 12, E. A. Ash and C. R. Hill, Eds. London: Plenum, 1982, pp. 89-100. [7] R. S. Weglein and R. G . Wilson, “Characteristic materials signatures by acoustic microscopy,” Elec. Lett., vol. 14, pp. 352354,1978.
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Henry L. Bertoni (M’67-SM’79) was born in
Chicago, IL, on November 15, 1938. He received the B.S. degree in electrical engineering
from Northwestern University, Evanston, IL,
in 1960, and the M.S. degree in electrical engineering, in 1962, and the Ph.D. degree in electrophysics, in 1967, both from the Polytechnic
Institute of Brooklyn (now of New York),
Brooklyn, NY.
In 1966 he joined the faculty of the Polytechnic Institute of New York, and now holds
the rank of Professor in the Department of Electrical Engineering and
Computer Science. He was elected Speaker of the Faculty for 198182. Since 1967 he has served as a consultant in the areas of radar, ultrasonics, and radio propagation. His research has dealt with various aspects of wave propagation and scattering. These include the geometrical theory of diffraction, electromagnetic, and optical waves
at multilayered and periodic structures, magnetoelastic waves, and
ultrasonics as applied to both signal processing and nondestructive
evaluation.
Dr. Bertoni is a member of the Acoustical Society of America, the
International Scientific Radio Union, and Sigma Xi.