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105 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-31, NO.
2 , MARCH 1984
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
re- flection 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 to 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 geometri- cally 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.
I. INTRODUCTION ORKING in the reflection mode, the acoustic
micro- Wscope with water couplant has been used to image the
surfaces of solids, for example, in the study of integrated cir-
cuits [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 ma-
terials 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 to 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 sen-
sitivity of the microscope's output voltage to 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 to the
Rayleigh wave velocity of the object [101 , [1 I ] . 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observations were originally made on spherical lenses, they have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 ray- optical
viewpoint, the AMS is due to interference between the fields of
near-axial rays reflected from the surface, and the fields along
ray paths that include the excitation and reradia-
Manuscript 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.
0.8U
-27r'U 0.4 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 to develop simple ana- lytic
expressions giving the variation V(z )of the output volt- age with
displacement z of the object surface from the focal plane. The
previous ray-optical study of the AMS [I41 pre- dicted the spacing
Az of the minima of V(z)based on changes in phase along the
different ray paths. The study did not com- pute the absolute phase
or the amplitudes, so that it was not possible to predict the shape
of the curve V(z) ,including the depth of the minima. Fourier
optics has been used to accu- rately predict the V(z)dependence [8]
, [181, [191 . How-ever, because the Fourier transforms are carried
out numer- ically, one cannot easily see from this approach how the
various geometric and acoustic parameters influence V(z) .
11. SEPARATION AND LEAKYOF GEOMETRIC WAVE EFFECTS
The reflection coefficient for waves incident on a substrate
having Rayleigh velocity greater than about twice that of the
couplant is similar to 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 > O s , R(8)= 1 for ma- terials with no
acoustic damping, but the phase of R(8) under-goes a rapid change
of 2n as 0 increases past OR.
0018-9537/84/0300-0105$01.000 1984 IEEE
http:0018-9537/84/0300-0105$01.00
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106 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-31, NO.
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solid. then to first order CY is the sum ad t al . Since 0 x k R
. we have
@ " . 0 - ,. " " U - I \ 0 n Re k X Qp k , % 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), pw and
p , are the mass densities of the couplant (water) and the
substrate, and kl = w/Vl,k, = w/V, and k , = w/Vw,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
kk 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/VR. 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 kkR , where k , = 0+ i f f . It is found that f f /
kR
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107BERTONI: RAY-OPTICAL EVALUATION OF V ( Z )
, REFLECTED RAYS
LEAKY RAYLEIGH
Fig. 3. Ray structure of reflected fields produced by a
convergent field incident on surface of a solid.
wave can then be applied to the microscope case using the
procedures of ray optics.
Let Pi@, , z i ) be the Fourier transform of the incident pres-
sure field p i ( x ,z i )given in (7) . The reflected pressure
field pr (x ,z i ) along the surface can then be found by taking
the in- verse transform of the product of Pi(kx , z i ) and R(kx) .
Using the approximation (6) for R(k,), we can separate the pressure
of the reflected field into a geometrical part p ~ ( x ,z i ) and a
leaky wave part p L (x ,z i ) . Thus,
Pr(xi zi) = P G ( X , Z i ) + P L ( X , Z i ) , (8)
where
and
The branch point singularities of R o ( k x )in (9) give rise t
o lateral ray contributions to p ~ ( x ,z 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
in- cident ray field and the local reflection coefficient [23],
[24]. Thus, the total reflected field contains a geometrical optics
term
~ ~ ( x , z i ) = R o ( k , (1 1)COS e ) p i ( x , z i ) -
The field incident in the vicinity of the point xi
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108 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-31, NO.
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- f < z
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109 BERTONI: RAY-OPTICAL EVALUATION OF V ( z )
is obtained for z = zo ,where from (13 )
f2
20 = (18)2n(D - f/n)'
For this condition, the focal spot lies on 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 volt- age, 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 on 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 . (19) Equating TT and the transducer radius
RT, 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 to 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.
D.Output Voltage The output voltage VG produced by the
geometrically re-
flected ray fields falling on the transducer is found by
integrat- ing (16)over the transducer. For z z,,, the transducer is
fully illuminated and the upper limit of the integration over p
must be taken as RT. Otherwise, the illu- mination is limited to a
circle radius rT given by (14) or (19).
We assume that the output voltage that would be produced by a
uniform field of unit amplitude is V o . Carrying out the
integration over the transducer, as discussed above and simpli-
fying, leads to the expression
exp [(ik, - a,) 22 + i X ]. (21) Here,
where
F = n(D - f / n )h,/R$. (23) 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 trans- ducer. Thus F in (23) 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 ) , (18), (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 =-zo [ ( R T / ~ )- 11 1
zmax x zo [ ( R T / ~ ) (24)+ 1 1 . 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 (24) that zo , I zmin I and
z,,, are all on the or- der of A,, which is usually much smaller
than f. As a result, 2z/f zo reduces the significance of these
effects. The zero at z = 2z0 for RT = a results from phase
cancellation across the transducer. Unlike the one at z = z o . the
existance of the zero at z = 2 z 0 ,and even its location, will be
sensitive to the exact size of the transducer, variations of T 1
,T,, and Ro(B), and to diffraction effects.
When F
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110 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-31, NO.
2, MARCH 1984
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 , VG(Z)is modified in the range -1 < z < 3 as
shown by the dashed curve, but is unchanged outside this range.
Dura l V [ z ] p a r a x i a l lens 30
- 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/3and the resulting curve is shown dashed in Fig. 7 .
r SURFACE
Fig. 8. Leaky Rayleigh wave rays in the microscope for defocus
dis-tance z < 0.
IV. VOLTAGEDUE TO THE LEAKYWAVE F I E L DFOR z
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111 BERTONI: RAY-OPTICAL EVALUATION OF V ( z )
as to travel parallel to the lens axis. Rays that are reradiated
on either side of the principal ray in the plane of the drawing
travel parallel to 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 O R .
As seen in Fig. 8, those rays radiated to the right of the prin-
cipal 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 to 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 coeffi- cient TI.
Because of the spherical convergence, the field of the ray incident
on the object surface at the angle O R will have an amplitude
increased by the factorf/(l zIsec e,). Attenua-tion in the water
results in the additional factor exp [-a!,(f-1 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 ab- sence 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
L.
.exp [ik,(f + D / n- IzI sec e,)]. (25) The field given by
(25)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 ( - in /2)
appropriate to a line focus [ 2 S ] ,the 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 ini-tial
field of the principal ray provided that z >f/n. The focus
introduces the phase factor exp (-in/2) corresponding to 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 fac- tor exp (- in/2)must be included for those
rays that cross the lens axis.
The phase change due to 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 O R for propagation from the focus to
the object surface. Because neighboring rays also pass through the
ring focus, they will have the same phase ex- cept 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 cor- rection k w ( p + pR)'/(2nD).
The various factors described in the preceding three para-
graphs can be combined into a single factor that gives the
variation of the field in going from the object surface to the
transducer plane. This factor is given by
T2( IZl taneR fcos 'eR . %) 11' PR nD P
. e-in/z exp [ik,(f+ D/n - Iz I sec e,)] * exP [- ( f - IZI sec
)I ( ~ X P[ ikw( P - PR)' /(2nD)I + e-in/z 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 to be
.exp [2(a , - a! sin 0,) IzI sec O R ]
. exp [- i2k , Iz I cos OR ] -1 6
{exp [ik, ( p - p ~ ) ' / ( 2 n D > ]+
where A is defined in (17).
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112 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-31, NO.
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I.4 T'"' 1.2
1.0
0.8
-T R
0 -- -200 -- -404
Fig. 9. Variation of the integral K with '1,.
1 1 1 1 1 1 , 1 1 1 1 , 1 1 1 1
-7 -6 -5 -4 -3 -2 - I 0 Fig. 10. Output voltage due to leaky
Rayleigh wave fields on a fused quartz object.
C.Leaky Wave Voltage VL The output voltage due to the leaky wave
is found by inte-
grating the field (28) over the transducer. Again we let Vo be
and the voltage that would be produced by a uniform illumination of
unit amplitude. Changing the variable of integration, the voltage
can be written as The integral in (30) cannot be evaluate.. in
terms of simple
functions. We have computed K numerically as a function of 2 a l
h W ( f i f 3sin p R for various values of pT,i.e., various values
of F. The re- V, = VoAK [ (nh, D)3/4 sults of the calculations are
shown in Fig. 9 . It is seen that
. exp [-i2kwI z I cos e,] neither the magnitude or phase of K is
sensitive to p R . For z
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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 ob- tained from (29) due to
the factor [ l - ($) erfc (s)] . In Fig. 10 a straight line, drawn
dashed, has been used t o connect IVLI at the point z1 /zo and at
the point z = 0, in order to in- dicate the general behavior of IV,
1 in the range z1 zo. As a result, the volt- age is due entirely to
the leaky wave rays in this range of z.
The curve of IVG(Z)I in Fig. 10 is the same as shown in Fig. 6
for z/zo >zo the value of I V, Iwill be greater than that of 1
VGI implying that a method for selectively reducing I V, I is
required to 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 IVGI,while acoustic damp-
ing 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 re-
sulting 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, al- though it is in the
exponential term. Dransfeld and Salzmann [191 give a simple
approximate expression for a1that leads to the relation
(YlAw = ( 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
.(sin O R ) 5 1 2 (f /RT)3J2 ,
(33) which allows simple estimation of the amplitude of the
leaky wave voltage relative to VG.
V. TOTAL VOLTAGE AND THE AMS The total output voltage V ( z )of
the transducer for z 0, IV, Iwill 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 IV(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
I I I I I : I , I I I 1 I I I I - , -20 -16 -12 -8 - 4 0 4 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 to the left. The Rayleigh angle OR = 24.62' was chosen
for the calculation to fit the measured data. This angle
corresponds to a Rayleigh velocity VR = 3600 m/s, which is in the
range of reported values for YIG. The theo- retical curve gives
reasonable agreement with the measure- ments, except in the range -
1
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114 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-31, NO.
2, MARCH 1984
to I VLI 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 [atan - 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
l z 1 = ( 3 ) / [ a t a n e R - a , ( s e c e R - I ) ] . (36)
If the minimum (36) is less than the distance from the edge of the
lens to 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
(37)
The last factor in (37) depends entirely on material parame-
ters. 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 / / VLI 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 ofDamping 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 to 6 dB per wave- length [30]
.
Anisotropy in the substrate has several effects that tend t o
reduce I VL I. Beam steering of the Rayleigh wave in the sur- face
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 to the output
voltage, or may even miss the transducer. Even with- out 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
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, IVL1’ ,normalized to I VoA1
2 , for a 370 MHz saphire lens looking at Si. The calcu- lations
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 to
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 Using a ray-optical approach, simple formulas
have been de-
rived that give the output voltage VG(z)due to the fields geo-
metrically reflected at the object surface, and the voltage VL(z
)due to the excitation and reradiation of the leaky wave. Contrary
t o previous assertions [5 ] , ray optics does not lead to a simple
sin ( k z ) / z variation for I VGI. Instead, I VGI shows the
marked asymmetry observed for actual lenses. In partic- ular, for
Iz I >>zo and Fsomewhat greater than 4,I V, 1 varies as l / l
z I. The expression for VL in the range z
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115 BERTONI: RAY-OPTICAL EVALUATION OF V ( Z )
lo-'
IO'
-3 IO
-4 In1"
-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 ] .
obtains
dk, dx'. ( A l )1 G(0) ea,'
p t ( x , Z i ) = - 2 q e i k p 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' >a, it is seen from ( 3 ) and (4) that
k2 - k ii P --- 2a, .
2kP
Thus, using expression (7) for the incident field in the
first
integral in (A2) gives the following expression for the field
of
leaky wave propagating in the + x direction:
exp {- i [ k , . \ l ( ~ ' ) ~+ (zi)' + k ~ x ' l }dx , , kh''
[(x')' + (zJ2 ]
The integrand of (A4) has a stationary phase point [25]
when
d 7[ - k , d(X')'-+ ( Z j ) 2 - k R x ' ] = 0.dx
The solution of this equation for x' is
x' = z i tan O R = xi . (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. Expand- ing 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
ik ( x - x i )p t ( x , z i ) = - 2 q e p Pi(xi, Z i )
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116 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL.SU-31,NO.
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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 [ I zi Ihw/(cos ] ‘
I 2 . (A10) If I zfl>> A,, then erfc (s) may be neglected a
short distance to the right of the point x i .
In evaluating the leaky wave contribution to the output volt-
age 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 xi . Using this value of x in (AlO), to-gether
with (A6), it is seen that erfc (s) can be neglected at - x i
provided that
Izi I >hw/[4 cos (sin OR)* 3 , (‘411) 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.
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Henry L. Bertoni (M’67-SM’79) was born in Chicago, IL, on
November 15, 1938. He re- ceived the B.S. degree in electrical
engineering from Northwestern University, Evanston, IL, in 1960,
and the M.S. degree in electrical engi- neering, in 1962, and the
Ph.D. degree in elec- trophysics, in 1967, both from the
Polytechnic Institute of Brooklyn (now of New York), Brooklyn,
NY.
In 1966 he joined the faculty of the Poly- technic 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 1981- 82. Since 1967 he has served as a consultant in
the areas of radar, ultra- sonics, and radio propagation. His
research has dealt with various as- pects of wave propagation and
scattering. These include the geo- metrical 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.