ray-optical evaluation of v(z) in the reflection acoustic...

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 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 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 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.

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 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 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 While these observations were originally made on spherical lenses, they have also been observed with cylindrical lenses, whose focus is a line rather than a point [12] , [13].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 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 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 numerically, 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 materials with no acoustic damping, but the phase of R(8) under-goes a rapid change of 2n as 0 increases past OR.

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106 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-31, NO. 2, MARCH 1984

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

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 pressure 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 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 ( k , (1 1)COS e ) p i ( x , z i ) -

The field incident in the vicinity of the point xi

108 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-31, NO. 2,MARCH 1984

- f < z

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 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 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 illumination 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 transducer. 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 order 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


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

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 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 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 coefficient 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 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

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 factor 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 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 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).


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

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 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 voltage 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 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, 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 measurements, except in the range - 1


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 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 / / 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 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 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 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 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 geometrically 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

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 )

116 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL.SU-31,NO. 2,MARCH 1984

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 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 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 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 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, ultrasonics, and radio propagation. His research has dealt with various as- pects 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.

ray-optical evaluation of v(z) in the reflection acoustic...

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