Analysis of Generation and Detection of Surface and Bulk...

147 IEEE TRANSACTIONS ON SONICS A N D ULTRASONICS, VOL. SU-24,NO. 3, MAY l977

Analysis of Generation and Detectionof Surface and Bulk Acoustic Wavesby

lnterdigital Transducers ROBERT F. MILSOM, N. H. C. REILLY, AND MARTIN REDWOOD

Ahrroct-A method of analysis which uses a combination of analyti- coupling hasso far only been solved for the specific cases of an cal and numerical techniques has been developed to obtain an accurate infinite uniform array on a semi-infinite substrate[ l 11 and a solution to the coupled electromagnetic and acoustic fields set up by anlong uniform array on a rectangular substrate [131. This latterinterdigital transducer on the surface of a piezoelectric substrate. Full account is taken of the coupling t o bulk modes as well as surface modes, and the solution for the chargeon the electrodes includes both electrostatic charge and piezoelectrically regenerated charge. Programs have been written for interdigital arrays with uniform aperture but varying electrode width and pitch and arbitrary electrical connections. The theory is also valid for arbitrary crystal orientations. Generation and detection may be analyzed separately with informationbeing pro-vided on the partition of power into the various acoustic modes and the external load impedance, and the bulk wave radiation patterns are also computed. The program may also be used to find the insertion loss of a pair of transducers. Results are presented for the Bleustein- Gulyaev orientation of PZT-4 ceramic and the YZ and 41” rotated YX orientations of lithium niobate.

A I . INTRODUCTION

CONSIDERABLE number of methods of theoretical analysis of interdigital transducers have been reported,

including the use of electrical equivalent circuits [ l]-[3] as well as field analysis [4] -[131. However, none of these methods is exact, and although many show good agreement with experiment, under some conditions their applicability is necessanly restricted by the approximations which are made.

Two disadvantages of existing theories stand out, namely, the failure t o satisfy the very complex electrical boundary conditions while simultaneously taking the piezoelectric cou-pling fully into accountin the most general case, and secondly, the omission of bulk modes which are undesirable in surface wave devices but which nevertheless occur and cause consider-able problems in wide-band applications. The “self-consistent” theory presented by Emtage [5] goes some way towards over-coming the former difficulty but is, in fact, only self-consistent for the fundamental spatial component of the solu-tion which tends to limit the method to uniform narrow-band transducers. This method is, however, accurate for high-coupling materials, whereas many other methods only apply for weak piezoelectric coupling. The problem of bulk wave

Manuscript received February 24, 1976; revised June 10, 1976. This work was supported by the U.K. Science Research Council and thePro-curement Executive, Ministry of Defence, sponsored by DCVD.

R. F. Milsom is with the Allen Clark Research Centre, the Plessey Company Limited, Caswell, Towcester, Northants., NN12 8EQ, England.

N. H. C. Rellly and M. Redwood are with the Department of Electri-cal and Electronic Engineering, Queen Mary College, University of London, London, E l 4NS, England.

method is not applicable to a substrate whose back surface has been treated to absorb bulk wave radiation.

It seems clear that an exact solution to the mostgeneral field problem is impossible using analytical methods alone, and we have therefore developed a method of analysis which uses a combination of analytical and numerical techniques to obtain the solutions to both generation and detection by in-terdigital arrays. At present, application is limited by the available computer size to transducers with comparatively short arrays, but it is just such devices which suffer most from bulk wave problems.

11. THEINTERDIGITALTRANSDUCERS An interdigital transducer typically consists of an array of

parallel electrodes on the surface of a piezoelectric solid. AI-ternate electrodes are interconnected, and piezoelectrically generated surface acoustic waves are launched in the direction normal to the electrodes when an electrical signal is applied to the input. The generation is strongest at that frequency for which one wavelength is equal to one period of the array. Detection is achieved by means of a second interdigital array spaced at a short distance from the first.

Two separate problems will be considered. Firstly, the sur-face between input and output transducers will be assumed to be electrically free as in Fig. I (a), and secondly, this region will be assumed to be coated with a metal Wm as in Fig. l(b). In this latter instance the gaps between the electrodes rather than the electrodes themselves are considered as forming the array. In addition to launching surface waves, an interdigital transducer also couples to bulk waves which radiate into the substrate. At any particular frequency, the principle lobe of radiation is at that angle for which Brag-type reinforcement occurs, although in general there is some radiation at all angles. Because bulk wave velocities are generally higher than that of the surface wave, strong radiation into the substrate only occurs above the fundamental frequency of the array, but in wide-band devices bulk wave loss can be quite considerable at frequencies within the passband. Apart from providing a means of power loss, bulk waves create further problems if detected at the output transducer. Reflections from the bot-tom face and ends of the solid can usually be removed by scat-tering or absorption, but bulk waves traveling at a shallow angle to the surface directly to the output transducer cannot

148 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, MAY 1977

INPUT TRANSDUCER OUTPUTTRANSDUCER

Free Surface

P l e z o e l e c t r l c s o i l d

Fig. 1 . Interdigitaltransducersforgeneration and detection of surface waves. (a)Intermediateregionelectricallyfree. (b) Intermediateregion metallized.

be so easily dealt with. A third bulk wave problem is the coupling between surface and bulk modes which distorts the surface wave response.

It is generally assumed that the piezoelectric generalization of the theorem of reciprocity [141 may be invoked to solve the problem of detection once the solution to the generation problem is known [8 ] , or vice versa [6]. This is quite valid if only the surface mode is included, since each array has only two acoustic ports and the theorem is simple to apply. How-ever, when bulk modes are included there is a continuous distribution of acoustic power in the sagital plane, and applica-tion of this theorem is virtually impossible. Detection of bulk waves is a different physical process from that of generation and is a function of the separation of the two transducers in addition to the properties of the detecting array. As a result, in the theory presented here, generation and detection are solved independently from first principles.

A number of different types of surface and bulk waves couple to the electric field produced by an interdigital array. These depend on the crystallographic symmetry of the sub-strate and the orientationof the surface plane and fingers of the array. The type of surface wave most commonly em-ployed is the Rayleigh wave [15 ] which has elliptical particle motion with components normal to the surface and parallel to the surface in the direction of propagation. Bleustein-Culyaev [161, [171 waves are also used, and these have a somewhat simpler particle motion which is parallel to the sur-face and normal to the direction of propagation. An impor- tant feature of this type of wave is the great difference in penetration depth under free and metallized surfacesof high permittivity materials. This is the reason for considering the two distinct setsof electrical boundary conditions shown in Fig. 1. More general surface waves having components of

particle motion in all three directions can also exist, typical examples being the leaky surface waves described by Yaman-ouchi and Shibayama [181. The particle motion associated with the surface wave also gives an indication of which bulk modes couple to the applied electric field. For example, a Rayleigh wave transducer will couple to both longitudinal and vertically polarized shear modes, whereas a Bleustein-Gulyaev wave transducer will couple only to the horizontally polarized shear mode. The leaky wave is accompanied by all three bulk modes. The relative strength of bulk and surface wave coupling will be considered in detail later.

111. EQUATIONSOF MOTION AND BOUNDARYCONDITIONS

Acoustic waves in a typical material are some five orders of magnitude slower than electromagnetic waves, so the piezo-electrically coupled electric field is assumed to be quasi-static. Maxwell's equations therefore reduce t o

D.. = pI , I S

and

where D, E , #J and ps are electric flux density, field, electro-static potential and free charge density, and the comma de-notes differentiation in the usual tensor notation. Equations (1) and ( 2 ) must be satisfied both for the piezoelectric solid which occupies the half-space(x2 <0) and for the free space (x2 >0). The solid is assumed to be a perfect insulator so that p s is zero. In the free space region, (1) reduces to Laplace's equation

@,ii = 0.

MILSOM e t al. : SURFACE BULK ACOUSTIC WAVES AND INTERDIGITAL TRANSDUCERS AND 149

The elastic field quantities must satisfy Newton's stress equa-tions of motion which, in the absence of internal body forces, become

T . . . - PG. = 0Y . 1 l (4)

where T ,U and p are stress, elastic displacement, and density, and the dot denotes differentiation with respect to time. Strain S is defined by

sij = 4 ( U i , j + U j , J (5)

The above relationships are coupled through the piezoelectric equations of state

T..=C!? SI ] tJkl k l -e k i j E k (6)

D, = eiklSkf f €fk E , (7 )

where c E ,e and aretheelastic,piezoelectric, andpermit-tivity tensors of the solid. Substituting ( 2 ) and (5) into (6) and (7) and hence into (1 ) and (4), the homogeneous second-order differential equations of the system are obtained:

-pc'l + c t k , u k , l i+ e k i j @ , k ;=o eikluk,4 , k ili - G k = 0. (8)

In obtaining a solution it will be assumed that the electrodes are massless and perfectly conducting and of sufficient length in the x1-direction to make all differentials with respect to x1 negligible. In addition, all quantities will be assumed to vary as exp (jut)and therefore the time dependence will be sup- pressed through the analysis.

These approximations give rise to the homogeneous mechan-ical boundary condition at the interface,

N2Ti, = 0, at x 2 = 0, (9)

where N, is the unit vector. The electrical boundary condi-tions may be summarized as follows.

a) Free charge density at the surface u(x3) is equal to the dis-

L

continuity in the normal componentof flux which must be zero on the free surfaces. Therefore

D2(O+) - D2(O-) = U (metallizedsurfaceat x 2 = 0) (10)

D2(O+)- D2(0-) = 0 (freesurfaceat x 2 = 0). (11)

b) Tangential electric field E3 and hence potential4, must be continuous atx 2 = 0. (It has already been assumed that E , = 0.)

c) Potential @ must be constant (apart from time variation) over all electrodes that are connected.

d) The relationship between potential difference across each set of electrodes and total current per set (= integral of jwu) must satisfy Kirchhoff's laws for the external circuit of each transducer.

Iv. GREEN'S FUNCTIONSFOR FREEAND METALLIZED SURFACES

The first stage of the analysis is to obtain the responseof a line source of charge on an otherwise electrically free surface. The method used is an extension of that described originally for an isotropic solid by Lamb [191 and subsequently applied to surface wave problems in a limited way by Tseng [7] . The most convenient approach is to first derive an effective surface permittivity function similar to that defined by Greebeet al. [ 2 0 ]. This function relates the Fourier transforms of charge density and potential at the surfacex2 = 0 and embodies an exact solution to ( 8 ) and (9). Once i t has been found, the complete system of equations reduces to the one-dimensional problem of satisfying the electrical boundary conditions.

Fourier transforms and inverse transforms will now be defined, respectively, by

;i; ( k ,x2) = 2; 1;$ (x3 ,x2) exp dx3

$ (x3 , x21=J-5( k , x 2 )exp (-jkx3) dk (12 ) - m

where $ is any of the field variables in ( 6 ) ,(7), and (8). As-suming for the present that the x2-dependence for x 2 <0 is of the form exp (&x,) where cy is a dimensionless decay coefficient, the transformed equations(8) become

= O

2 L 2

where U(= o / k )is the component of phase velocity in the x3-direction corresponding to the wave number k and where the material constants c E ,e , and have been reduced to the stan-dard matrix notation [ 2 1 ] . For a nontrivial solution, the determinant of the left-hand matrix, whichis an eighth order polynomial in a, is zero. Each of the eight roots represents an elementary mode of the half-space, but for the given boundary conditions only four of these modes apply. For positive k (or U) those roots which are real or complex must


Fig. 2 . Effective permittivity for YZ LiNb03. __ Real part. ---- Imaginary part.

have positive real parts to ensure decay into the substrate, while imaginary roots which represent nondecaying bulk modes must also be positive for propagation away from the surface. For negative k (or U) the remaining four roots are ap-propriate. The four elementary solutions to (13) correspond-ing to the selected roots a, are,therefore,exp (ankx2) (i,n = 1,2, 3 , 4 ) where each solution is normallized with re-spect to $(n), which has been rewritten as i$) for convenience. A more general solution is given by a linear combination of the elementarysolutions.Thus

- 4 ui= C Ani$") exp (ankxZ) , i = 1 , 2 , 3 , 4 (14)

n = l

where the ratios An/A4 (n = 1 , 2 , 3 ) may now be found by substituting (14) into (9) to give a solution satisfying the traction-free boundary condition.

The potential in the free-space region is given by

45=C A , exp (- Iklx2), x2 2 0 (1 5)n = l

which clearly satisfies Laplace's equation and the condition of continuity of potential atx2 = 0. The transform of the free charge density a at the surface x 2 = 0 is given by the discon-tinuity in D2 which is found by substituting the solution into (21, (54, and (7).

The effective permittivity e,, which for a particular crystal orientation is a function only of the horizontal component of phase velocity U , is defined by

-U

f, = (16)Ikl $lx,=ol

This differs slightly from Greebe's definition [20] in that con-tributions from both the internal and externalfield are now included in the same function. In addition, the function em-

braces all values of u including those for which bulk waves form part of the solution. This function is introduced be-cause, for a given substrate orientation, it effectively stores all the relevant mechanical information in a scalar electrical quan-tity and is also readily computed. In the foregoing analysis it will be found more convenientto regard E , as a function of slowness S(= l / u ) rather than velocity U. Fig. 2 shows a typical form of the function, in this case for Y-cut 2-propagating lithium niobate. Positive values of S only are shown because the function is even. The poles at *S, and zeros at bocor-responding to zero$ and 5are, respectively, the metallized and free surface Rayleigh wave slownesses as found by Camp- bell et al. [ 2 2 ] using essentially the same analysis. The discon-tinuities in de,/ds at * sI and Ls2 are the cutoff slownesses of bulk-longitudinal and vertically polarized bulk-shear waves (slow shear waves). A discontinuity arises when one decay co-efficient an changes from purely imaginary below cutoff t o either real or complex above. The type of bulk wave associated with each imaginary an is determined by the direction of the corresponding particle displacement vectur up). For YZ lithium niobate, the third typeof bulk wave (namely, hori-zontally polarized or fast shear) decouples from the electric field and therefore does not enter into E,. Finally, i t is worth noting that E , is purely real above the shear wave cutoff slowness s 2 .

Applying the convolution theorem [ 2 3 ] to (16) now gives a relationship between the spatial distributions of surface voltage and charge. Thus

$J(x~)=JsG(x3 - x;) u(x;) dx; -s

where the Green's function G ( x 3 )is given by

MILSOM et al.: SURFACE BULK ACOUSTIC WAVES AND INTERDIGITAL TRANSDUCERS AND 1 5 1

Although G is clearly a function of frequency this dependence may be removed by making the substitutions

x = 0 x 3

and

s = k f w (1 9)

giving the frequency independent Green’s function

1 I--~ ( x j r(s) exp ( - j sX)ds=

2n _ _ where

From (17) it is apparent that G is the surface voltage resulting from a delta functionor line source of charge parallel to the x 1 axis. This voltage response is crucial to the method of analysis and must be evaluated explicitly before the solution for an interdigital array is attempted. This is achieved by in-tegrating the poles analytically and the remaining function numerically. The numerical part of the function is then ap-proximated using an appropriate curve-fitting technique. De- tails of the integration are given in Appendix A.

In the above analysis, the charge on the electrodes has been regarded as the source of waves, while the waves themselves are represented by their associated voltage distribution. This is the most convenient representation when the surface region outside the transducer is unmetallized as in Fig. l(a), since the sources occupy a finite region. However, when the external region is metallized as in Fig. l(b), it is desirable to change to a different representation where the tangential component of electric field E3(=-&5/ax3) in the gaps between electrodes is regarded as the source and the waves are represented by the total charge at a given point defined by

which will clearly be constant in the gaps. From FourieI transform theory [ 2 3 ] ,

0 =jZ/k

and

i? =jk7 ( 2 3 )

where the subscript has been dropped fromE 3 and where the overbar denotes the Fourier transform as before. Substituting ( 2 3 ) into (16),

which again on application of the convolution theorum gives

where the metallized Green’s function G , expressed as a func-tion of X (= o x 3 )is given by

1 I--

and

The same methodof integration is used to evaluate ( 2 6 ) and (20),the principle difference between the two integrands be. ing that the surface wave poles are shifted from the free to the metallized slownesses.

V. NUMERICALSOLUTIONFOR SURFACECHARGEAND

FIELDDISTRIBUTION

A , Generation I t has already been shown that the effective permittivity,

and hence the Green’s function which is derived from it, con-tains an exact solution to the mechanical part of the problem. It remains therefore to find the charge distribution on the elec-trodes which on substitution into(17) gives a voltage equal t o the known applied voltage on each electrode.

The chosen method of finding this charge distribution em-ploys a numerical integration procedure which is a modified form of Simpson’s rule [ 2 4 ]. As shown in Fig. 3 , each of the M electrodes is first divided into R equal intervals (whereR is even) giving a matrix of pointsx, (n = 1 , 2 , . . N ) ,whereS ,

N =M(R + 1) and where each interval on the electrode con-taining the nth point is of width h,. The values of potential and charge density at these points are designated by 4, and U,

(n = 1, 2 , . . N ) . To illustrate the method it is assumedS ,

initially that there are no singularities in the integralin (17). Under these circumstances, Simpson’s rule gives a simple set of relationships

N

j = 1

where

Ai, = 4 P,hi C ( X ~- x i ) (29)

and Piis equal to4, 1, or 2 depending, respectively, on whether xi is an even point on the electrode, on the edge of the electrode, or an odd point other than at the edge. If al- .

ternate electrodes of a symmetrical array are connected to-gether and a voltage V, is applied, then & is equal to+V,/2, where xi lies on an even electrode and - Vz/2 on an odd elec-trode. Values of the coefficientsAi, are computed rapidly from the previously derived approximation to G(x) (see Ap- pendix A) and the complex linear equations (28) solved for the unknown values of charge density U, .

This simplified approach cannot in fact be used directly since the singularities in the integration have been ignored. These must now be taken into account. Purely electrostatic analysis [9] shows that the value of a(x) approaches infinity at the electrode edges and it is reasonable to assume that this


lh”T l B # U 8 h l 1 , , , , . 1 ----- X ” - - -X I 1 2 1 3 =L XS 1 6 x7 ra ~~~~~~~~

Fig. 3. Subdivision of interdigital array for numerical integration procedure.

remains true when the charge distribution is perturbed by piezoelectric coupling to the acoustic waves.

In the normal application of Simpson’s rule the charge dis-tribution over a double interval (e.g. from xi to x j + J is approximated to a parabola

u’(x) = axz t bx t c (30) with the coefficientsa, b , c chosen so that u‘(x) is equal to U .I ’ o . + ~and u j + z at x j , x j + , andx i+ , . However, to take ac-I count of infinite charge density at electrode edges, the form of the function over a double interval adjacent to the edge xi will be approximated by

a’ (x ) = ax2 t bx t uj In Ix - xi I (3 1)

where II and b are again related to u j , u j+ and ai+, such that theapproximation is exact a t and x i + * . Theunknowns ai in equations (28) are now the coefficients of the logarith-mic functions if xi is at an electrode edge, and again equal to the actual values of charge density if xi is not an edge point.

As shown in Appendix A, the Green’s function G(x) ap-proaches infinity atx = 0, and this introduces a further singu-larity into the integral in (17). An additional modification to the numerical integration procedure along the same lines is therefore needed to allow for this. The effect of these modifi- cations is t o greatly complicate the form of the matrix ele- ments A j j in (28). However, once these have been obtained for all possible cases relating to the relative positions of x; and xi , they are valid for any array structure and any Green’s function of the same basic form.

Denoting the revised coefficients in the modified integra-tion procedure byA b , the correct solution to theelectrical boundary value problem is given by the linear equations

N @i= c A .!.

11 ai (32)j =l

rather than by (28). If the array is asymmetrical it cannot be assumed that the two sets of electrode voltages are equal to +VI/2. In this more general case the two voltages are denoted by V;= (VI/2 + A V) and Vi = (- V1/2t A V), and the extra unknown A V i s found from the condition of continuity of charge (i.e. the total charge on all electrodes must be zero) which provides the necessary extra equation to be solved simultaneously with (32). A similar technique to that de-scribed here has been used to find a numerical solution for the purely electrostatic charge (21.

The solution to the metallized surface problem shownin Fig. l(b) is directly analogous to the free surface problem, and the same numerical integration algorithm is used to find values of tangential field E, as was previously used to find values of ai. The sampling points xi are now chosen to be in the M gaps between electrodes. Thus (32) now becomes

where Qiand Eiare the values of total charge (as defined by (25)) and tangential field at the pointxi and the matrix ele- ments A:, are the same as before but with values of the Green’s function G, ( x ) replacing G(x) . However, (33) is not the com-plete solution since, althoughQ is constant over a gap, its value at each gap is unknown. Consequently there are M extra unknowns requiring an extra M equations. These are provided by integrating the tangential field over each of the M gaps and putting the integral equal to the known potential difference. Thus

m(R + 1)c B j E j ~ ( - l ) ~ m = 1 , 2 ; . . , MV I ,j = (m - 1)(R + 1) + 1

(34)

where the modified Simpson’s Rule coefficients Bj are ob-tained from the approximations of (30) and (3 1 ) in the appro-priate regions. Equations (33) and (34) are now solved simul-taneously for Ei and Q;.

In the free surface problem, the total charge (per unit finger length in the x1-direction) on the mth electrode is given by

(35)

and for the metallized surface problem,

q m = Q ~ ( R + I ) + I- Q ~ ( R + I ) . (36)

The current in a transducerof aperture W is therefore

IT = i w w C q m (37)

where the summation is taken over those electrodes which have a positive voltage applied to them. The input admittance of the transducer is then

B. Detection Consider the problem in Fig. l(a) where the output trans-

ducer is separated from the input transducer by a region of free surface. Once the solution to the input transducer is known, (17) may be used to find the potential at an arbitrary point on the surface x 2 = 0. In particular, at each sampling point xi of the output transducer, the integral in (17) gives a potentialdue solely to charges on theinputtransducer. This potential is the sum of electrostatic, surface wave, and bulk wave terms, all of which are included in the Green’s function G as described in Appendix A. The potential due to the charge on the output transducer itself is given by (32) as before. The total potential at each xi is therefore given by

N

(39) N

Qi = C A ! .E . where uj relates to the charge at the pointxj on the output‘ I I (33)

j = 1 transducer, N is the total number of sampling points on the

153 MILSOM et al. :SURFACE AND BULK ACOUSTIC WAVES AND INTERDIGITAL TRANSDUCERS

(QUADRATURE 1

Fig. 4. Charge distribution for four electrode-pair input transducer on YZ LiNb03 at centerfrequency. (a)Approximate Engan solution. (b) Exact solution-components in phase and quadrature with applied voltage.

I (QUADRATURE)

Fig. 5. Charge distribution for split-finger pitch weighted transducer on YZ LiNb03 at center frequency.

output transducer, and Vi = V,' i f xi lies on one set of elec- surface wave incident on an isolated transducer. In this case trodes and V; on the other set. These two voltages (V,' and the known potentials at the sampling points xi are equal to VG) are unknown initially; therefore, two extra equationsare A. exp (Yosoxi),where A. is the voltage amplitude of the in-required. These are provided by the condition of continuity cident surface wave and the sign of the argument is determined of charge and by the external circuit whichgives an additional by whether this wave is incident from the left (negative) or the relationship between charge and voltage determined by h r c h - right (positive). hoff's laws. As with generation, the problemof detection with a metal-

The method used to solve detection of the field set up by lized surface is directly analogous to the free surface problem, the input transducer may also be applied to the detection of a with the additional unknowns again found using (34). It is


[ I N P H A S E )

( Q U A D R A T U R E )

(a)

( I N P H A S E )

( Q U A D R A T U R E )

(C) Fig. 6. Charge distribution for four electrode-pair output transducer on YZ L a b 0 3 at center frequency-components in

phase and quadrature with output voltage. (a) With open-circuit termination. (b) With short-circuit termination. (c) With matched termination.

155 MILSOM et al. : SURFACE AND BULK ACOUSTIC WAVES AND INTERDIGITAL TRANSDUCERS

Elec t r i ca l

Power

I-I SurfaceWave

-1 SurfaceWave 1

Power Power I I I I l

I

I

Electrical4 Power

Tranrmctted I Ref lected SurfaceWave I

1-Incident

ISurface wave Power l Surface Power IWave

I Power PIEZOELECTRIC SOLID l l

worth noting that the charge distribution and electric field on a detecting transducer are asymmetrical even when the array itself is symmetrical. There are therefore twice as many un-knowns ui(or Ej) in detection as in generation by a symmetrical array of the same length.

Fig. 4(b) illustrates the exact charge distribution on a four electrode-pair uniform transducer onYZ lithium niobate oper-ated at its center frequency. This is compared with a standard approximation [9] (Fig. 4(a) based on the solution for an in-finite array under dc conditions, which ignores end effects and takes no account of piezoelectric regeneration. In the exact solution the regenerated charge gives rise t o a quadrature com-ponent and also some distortionof the in-phase component. Unlike the theory presented here, the approximate theory is not applicable to a more general array geometry, an example of which is shown in Fig. 5 which illustrates the charge distri-bution on a pitch-weighted split-finger transducer. Fig. 6 shows the charge density on a4 electrode pair detecting array under open-circuit, short-circuit, and matched conditions, and this clearly indicates the variation of charge distribution with the impedance of the external circuit. Comparison of Fig. 4 ,

acoustic modes within the substrate. Such information is im-portant since only the direct surface wave component is useful. Bulk waves and reflected surface waves give rise to spurious signals and undesirable loss. It is therefore necessary to examine in detail the electromagnetic and acoustic fields in the piezoelectric solid.

Consider the transducer in Fig. 7(a) which will be assumed to be of uniform aperture W in the x1-direction, but having in general electrodes of varying width and pitch and arbitrary electrical connections. The time average electrical power flow-ing in at the surface x2 = 0 is given by

m

Pi,= ~ x j ) (40)W Re l_ i(x3)* dxj

where @ and i are, respectively, the voltage and current density at the surface. I f jwu is now substituted for i and the convolu-tion theorem applied to (40),

Pi,= -awW Re jr(s);(us)E(us)* ds1: (41)

(S), and (6) emphasises the need for ageneral numerical solu-tion to the charge density on an interdigital array.

VI. PARTITIONOF POWER A. Surface Waves

Equation (38) provides an expression for the electrical input admittance of an interdigital transducer. It does not, however, provide detailed information on the relative amplitudes of the

where the Fourier transforms are expressed as functions of slowness S using (19), and &(us)has been replaced by r(s)i?(us) using (16) and (21). The integral in (41) has es-sentially the same singularities as that in(20) (see Appendix A). The principal difference is that the integrand in (41) is finite at the origin since the total charge at the surface, whch is proportion to $O) , is zero. Fig. 2 illustrates that r(s) (=l/ Is1 e,) is a purely real function for numerical values of s

156 IEEE TRANSACTIONS ON SONICS AND MAY 1977ULTRASONICS,

greater than the cutoff slowness sN of the slowest bulk wave (where the types of bulk wave coupled to the electric field have been numbered 1 to N in order of increasing slowness). In addition, r(s) has poles at the surface wave slownesses f so where so > sN. Integrating these poles analytically as dis-cussed in Appendix A,

where

(43)

The first and second terms in (42), which result from the poles a t +so and -so,are, respectively, the amounts of input power converted to positive and negative travelling surface waves. The third term which must be integrated numerically is the total power converted to bulkwaves.

It will be found useful to express the power carried by a travelling surface wave as a function of its associated voltage amplitude at the surface. From (17), (19), and (A14), that part of the potential4 at the surface which is associated solely with the positive travelling surface wave is given by

4, = -jC, exp (-jwsox2) u(x;) exp ~osox; )dx; . (44)1: Therefore, from (12), 4, may be expressed in terms of the Fourier transform of the charge distribution

4, = - j 2nGsG(wso)exp ( - josox3) (45)

so that the voltage amplitude of the wave is

V, =(46)- j 2nG,o(wso).

Substituting for;(uso) from (46) into the first term of ( a ) , the power P, carried by the surface wave travelling in the positive direction is

P, = - a wWV, V;/Gs. (47)

Similarly the power carried by the surface wave travelling in the negative direction is

P, = -4WWV, V,*IG, (48)

where the amplitude of the negative wave is

V, = -j 2nG,i?(-(49)u s oj.

It is interesting to note that the electrostatic charge on the electrodes is in phase with the applied voltage and can therefore be expressed as a real function. If it is assumed that the piezoelectrically regenerated charge, which is a complex func-tion, is negligible by comparison, then U is real and its Fourier transform obeys the relationship [23]

-U( -k)= Z(k)* (50)

so that the forward and backward surface waves carry equal power. In general, this will not be true if the regenerated charge is significant compared to the electrostatic charge,un-

less the transducer itself is either symmetrical or anti-symmetrical.

Now consider the partitionof power in detection of surface waves by a single transducer as shown in Fig. 7(b). A surface wave of voltage amplitude A . is incident from the left. This gives rise to a charge distribution U which is determined by the method described in Section V, and this charge in turn regen- erates a backward wave of amplitude Vn (49) and a forward wave of amplitude V, (46). The regenerated backward wave is identifiable as the reflected surface wave, while the trans-mitted surface wave is the sum of incident and regenerated forward waves. The powers carried by the incident, reflected, and transmitted surface waves are then given, respectively, by

P2 = -4wWAoAo*/G,

P , = -4 wWV, Vz/G,

P, = - 4 wW(A0 .t VP)(A$+ V,*)/G,. (5 1)

The power delivered to the load is given by

PL = -:Re ( V i - V i ) I F (52)

where V: and V i are the voltages on the two sets of electrodes and the transducer current IT is given by (37) with the summa-tion taken over the positive electrodes. The negative sign in (52) arises because power is flowing out of the piezoelectric substrate. The voltage, current, and load impedance ZL also satisfy the condition

(V,' - V i )= ZLI, (53 )

which has been used as one of the set of linear equations in the solution for the charge on the transducer.

B. Bulk Waves The total power generated by the interdigital transducer in

the form of bulk waves is given by the last term in (42). This bulk wave radiation will now be considered in more detail, and expressions for the radiation patterns of the different bulk waves will be derived.

In general, the time average electromagnetic power density is given by the Poynting vector [ 2 5 ]

[ P e ] ,= 4 Re [E* X H ] (54)

where E and H are the electric and magnetic fields. However, using the quasistatic approximation to Maxwell's equations and a standard identity of vector algebra [26] , this reduces to

[P,] = Re (b,&*). (55)

In addition, the time average mechanical power density is given by [6]

[Pm],= - 5 Re (Tj jG?) . (56)

The aggregate power density in the piezoelectric substrate is therefore

Pi = [Pe] t [ P m ] i= f Re (-i>i$I* - TijUT). (57)

Consider the power flowing acrossa surface of width W in the x1-direction surrounding the entire half-spacex2 f 0 in Fig. 7(a). The electrical power flowing in at the surface x2 = 0 is

- -

157 MILSOM et a l . : SURFACEAND BULKWAVESACOUSTIC AND INTERDIGITAL TRANSDUCERS

Since only those field variables associated with surface waves do not decayin the positive and negative xg-directions, the first two terms in(58) give the surface wave powers which have already beenderived in (48) and (47) , respectively. The third term gives the bulk wave power Pb. Therefore, substitut-ing (57) into this term,

pb -- - _ W Re (D2@*- Ti, =- - I dX3, (59)I: .

and applying the convolution theorem and replacingalar by iw ,

Pb = -nW Re jw (D2$*t T j 2UT),,, = - - l d k . (60)S: The solutions_for 3 a t d iii given by (14) and the corresponding solutions forD2 and T i 2 ,found by substituting3 and Uiinto the transforms of (2), (5),(6) ,and (7):are now substituted into (60) giving

Because this integral is evaluated at x2 = -m, many terms vanish over certain ranges. For example, if a, (n = 1 , 2 , or 3 ) is the decay coefficient associated with the nth bulk wave, then this is imaginary in the range -as, <k <ws,, but other-wise has a real part of the same sign a s k , so that those terms containing CY,, vanish for I kl >ws,. Similarly a 4 ,which rep-resents the essentially electrostatic part of the field, has areal part with the samesign as k , so that terms containinga4 vanish for all values of k . Also, if n # m but a,,and am are both imaginary, then the term containing exp (a, t a:) kx2 vanishes due to the infinitely rapid oscillations of the inte-grand a t x2 =-m. Equation ( 6 1 ) therefore reduces to

3

= c n = l

where the power in the nth bulkwave P,, is given by

J-WS,

It has been found convenient to define a"resistance density" function R, and "conductance density" function G, for each of the three bulk waves. Thus for n = 1, 2 , and 3,

R,($) = -wnRe { j (sy)(s)$(S)'"'* t Ti2(s)(")i$)(s)*)/

( ~ ( s ) W * ) } ( 6 4 )

and

where E is the tangential component of electricfield at the surface x2 = 0 and all the Fourier transforms offield variables have been expressed as functions of slownessS (=k/w).The power in the nth bulkwave is therefore given simply by

P,, = wW R,(s)$s)z(s)* dsl: or

As with the effective permittivity, the functionsR,(s) and G,(s) depend solely on the substrate orientation and there-fore need be evaluated only once for a given cut. The value of P, at any frequency therefore immediately follows from the computed charge or field distribution by numerically in-tegrating (66)or (67) . The resistance densities for slow shear and longitudinalwaves on YZ lithium niobate are shown in Fig. 8. (The fast shear wave is decoupled from the electric field, and therefore, itsresistance density is zero.)

Having obtained the power carried by the acoustic modes, it is nowpossible to obtain the contributions to the input con-ductance from each of these modes. This conductance Gin is given by

where Pi, is the power generated for a peak appliedvoltage V,. Therefore,

where PS is the total surface wave power given by the first two terms in (42) and P , . . .Pv are the powers in theN bulk waves coupled to the applied field. The input admittance of a transducer on YZ lithium niobate, for example, may therefore be represented by the equivalent circuit inFig. 9, where the real part Ginhas been separated into the contributions from Rayleigh waves C R ,longitudinal waves G,, and slow shear waves, G T ,where

The equivalence of the expression in (69)and the real part of (38) provide a useful check on the accuracy and self-consistency of the solution. Substantial disagreement between the two computedvalues is an indication that more sampling points per electrode are requiredin the solution for charge density. Normally it is found that 5 samples are sufficient for


R ,-3 x 1o'? n m' I

Fig. 9. Equivalent circuit for input admittance of transducer on YZ LiNb03. B = susceptance, G, = Rayleigh wave contribution to con-ductance, G L = longitudnd wave contribution to conductance, GT = slow shear wave contribution to conductance.

frequencies up to about21 times the fundamental and 7 samples for the third harmonic. This increase is a consequence of the more rapid spatial variation of charge at the higher fre-quency. The accuracy of the solution for detection can be as-sessed by checking that the sumof transmitted, reflected, mode converted, and load poweris equal to the incident power.

The distribution of bulk wave power with angle will not be considered. If the decay coefficient a, in (14) is imaginary representing a bulk mode travelling into the substrate, then the wave vector is at an angle 8, to the normal where

3 ? I 0 2 1 2 3 x l 0 '

Fig. 10. Slowness surface in the sagitd plane of YZ L a b 0 3 for longitudinal waves Sb") and slow shear waves Sb(*).

and where the sign is the same as that of the value of slowness S being considered. The actual slowness s t ) of the bulk mode measured parallel to the wave vector is given by

s t ) (8,) = S cosec 0, = s(1 + la,I2). (72)

Thus the computation for every value of slowness S provides a point on the slowness surface of each bulk modeso long as S is below the cutoff slowness of that particular mode. The slowness surfaces for longitudinal and slow shear waves in the lower half of the ~ 2 ~ 3 - p l a n e of lithium niobate are shownin Fig. 10. From (66) and (67) it follows that

dp,- (73)-- W WR,(s)E(s)O*(s)ds

or alternatively

5= oWG,(s)E(s)E*(s) (74)ds

but

159 MILSOM et al . : SURFACE AND BULK ACOUSTIC WAVES AND INTERDIGITAL TRANSDUCERS

l l '.l

I 'i

Fig. 11. Contributions to conductance for3-period BleusteinCulyaev wave transducer on metallized and free surfaces of PZT-4 ceramic. __ BleusteinCulyaev contribution on metallized surface. ---- Fast shear wave contribution on metallized surface. -.-.- Fast shear wave contribution on free surface. BleusteinCulyaev wave contribution on free sur-face is negligible. (Electrode spacing lmm, center frequencyfo is computed from free surface velocity, and conduc-tances are for unit transducer aperture.)

Therefore the distribution of power with angle is given by substituting either (73) or (74) into (75). It should be noted that power is expressed as a function of wave normal rather than Poynting vector angle, so the distribution is only approximate where there is strong beam-steering of the bulk wave. In the interests of improving the speed of computation, a series of polynomials was fitted to the resistance and conductance densities and also to the slowness surfaces using a least squares procedure [27]. In order to obtain a satisfactory fit, it was necessary to divide the functions in Fig. 8 into a number of separate ranges of values of S. Each slowness surface, however, was approximated by a single polynomial in On.

VII. BLEUSTEIN-CULYAEVWAVE TRANSDUCERS

Application to the theory of transducers on the Bleustein- Gulyaev wave orientation of PZT-4 ceramic will now be considered. This type of wave may be useful in the design of sur-face wave devices because of the simple transverse particle motion and high coupling coefficient. The theory described here is necessary to provide an understandingof such devices since this cannot be drawn from direct analogy with Rayleigh wave devices due to the great different in penetration depth between the waves travelling under metallized and freesur-faces (about one wavelength and several hundred wavelengths, respectively) [161. The analysis of the preceding sections and also that of Greebe et d. [20] shows that apart from the Bleustein-Gulyaev mode, only horizontally polarizedshear waves can couple to the applied field. The relative contributions to the input conductance from the two possible modes will now be considered. Fig. 11 shows these contributions as functions of frequency for a three-period, 1 : 1 mark/space-ratio transducer with both metallized and free external sur-face regions. (Note that a three-period transducer has 5 elec-trodes and 6 gaps in the metallized case and 6 electrodes and

Fig. 12. Fast shear bulk wave angular power distribution for 3-period BleusteinCulyaev wave transducer on PZT-4 at center frequency. (a) Metallized surface. (b) Free surface.

5 gaps in the free surface case.) With the former electrical boundary condition, 95.1%of the input power is converted to surface waves at the fundamental frequency with more power converted t o shear waves at high frequencies. In the latter case there is negligible coupling to the deep penetrating surface wave, and almost 100%of the power is converted t o shear waves at all frequencies regardlessof the length of the trans-ducer. However, varying the number of electrodes of the transducer in the metallized case gives 76.1,90.5,97.9, and 98.4% surface wave coupling at the fundamental frequency for 1,2, 5 , and 10 periods, respectively.

The distribution of bulk-wave power with angle into the sub-strate at the frequencyof maximum input conductance is shown for the two electrical boundary conditions in Fig. 12. With a free surface, the principal lobe of radiation is almost


Fig. 13 . Open-circuit voltage transfer ratio of two 3-period BleusteinCulyaev transducers on PZT-4 separated by 10 periods. __ Freesurface between transducers. ---- Metallized surface between transducers. (Center frequencyfo is computed from metallized surface velocity.)

I v O ' g l ' '

0 61 /

l 0 L-

o 2 i

L 0

Fig. 14. Center frequency open-circuit voltage transfer ratio versus number of periods M of each BleusteinCulyaev transducer on PZT-4. __Metallized surface between transducers. Free surface be-tween transducers, separation = 20 periods. ---- Free surface be-tween transducers, separation = 50 periods.

parallel to the surface, but witha metallized surface this lobe is at about 20" to the surface. The total bulk-wave loss which is proportional to the area inside the graph is considerably less. (Note the different scales in the two diagrams.)

The ratio of open-circuit voltage detectedby a second three-period transducer to input transducer voltageis shown as a function of frequency for both free and metallized intermedi-ate regions in Fig. 13. The center-to-center separation of the transducers is ten periods. In the former instance, the signal is due almost entirelyto bulk waves. The comparative weakness of the signal is due to the leakage of bulk-wave energy into the substrate. The droop towards the high-frequency-side at the

passband is due to the transducer separation being greater in terms of number of bulk-wave wavelengths at higher fre-quencies. A center frequency voltage-transfer-ratio of 0.9 in the metallized case shows that a lowloss delay line is possible with a very small number of electrodes. Variation of center frequency signal with transducer length (and also transducer separation for the free surface condition) is shown in Fig. 14. The non-linearity of each curve is due to the strong interaction ofsourcesasshownby Mitchell er Saturationoccursfor a transducer length of about eight periods in all cases although the saturation occurs at a level which decreases with the separ-ation on a free surface. Examination of the bulk wave Green's function given by equation A.15 shows that for this orienta-tion the surface amplitude of the bulk wave is a Hankel func-tion which decays as the inverse square-root of distance. The maximum detected signal therefore decays in the same way with transducer separation. In the metallised case the detected signal is due only to the surface wave and is therefore indepen-dent of transducer separation apart from attenuation due t o electrical loss in the metal film and imperfections of the sub-strate.

VIII. TRANSDUCERSON LITHIUM NIOBATE Some results of applying the theory to a more anisotropic

piezoelectric will now be examined. Interdigital transducers on Y-cut 2-propagating lithium niobate couple to Rayleigh waves, horizontal shear and longitudinal waves. The propor-tions of electrical input power converted to the three modes for uniform 1 :1 mark-space-ratio arrays operated at their cen-ter frequencies are shown as functions of number of electrode pairs in Fig. 15. It is clear that the loss due to bulk waves, par-ticularly shear waves, can be very substantial for short arrays. The corresponding proportion of power convertedto Bleu-stein-Gulyaev waves on metallized PZT-4 is shown for com-parison, and it is apparent that this substrate is more favorable

161 MILSOM et al.: SURFACE AND BULK ACOUSTIC WAVES AND INTERDIGITAL TRANSDUCERS

1

Fig. 15. Proportions of input power converted to Rayleigh, slow shear, and longitudinal waves for 1: 1 mark space ratio transducers on YZ LiNb03 operated at their center frequency versus number ofelec-trode pairs M. __ Rayleigh wave power. -.-.- Slow shear wave power. ---- Longitudinal wave power. . . . . . Corresponding pro-portion of power converted to BleusteinCulyaev waves on metallized PZT-4.

than YZ lithium niobate for reducing bulk wave loss. Fig. 16 shows how the contributions to the input conductance (as defined by (70)) vary with frequency for a71 electrode pair transducer. The contribution from Rayleigh waves GR has the expected (sin x/x)’ form. The shear wave contribution is significant inside the Rayleigh passband, but increases further with frequency, reaching a peak at approximately twice the fundamental. Both longitudinal and shear waves contribute significantly above this frequency. The general behavior of the bulk waves can be inferred from (66)and from the form of the “resistance density” functions R , and R2 shown in Fig. 8. The Fourier transform of charge density o(k)has a peak at

i I

1 2 01

1.01 I I 201 l

. .

./ 0L,i-_L__0

L# 10 20 30 LO 50 50 ti0 70 80 90 1 0 0 120 MHz

Fig. 17. Theoretical and measured input conductance G and resistance R of a 7 1 electrode pair 1: 1 mark-space ratio transducer on YZ LiNb03 versus frequency. (Equivalent circuits appropriate for comparison over the two frequency ranges are shown including stray capacitanceC and inductance L . )

‘ m u ’ r

1 5r

01 - 1 L

20 LO 60 80 100 MHz

Fig. 16. Contributions to input conductance from Rayleigh waves ( G R ) ,slow shear waves ( G T ) ,and longitudinal waves (CL) for a 71 electrode pair 1 : 1 mark space ratio transducer on YZ LiNb03 versus frequency.

ko(= 2n/h),where h is the periodicity of the array, and there-fore it is clear that shear wave radiation will be greatest at the angular frequency o for which ko = a s I where s1 is the cutoff slowness for longitudinal waves and the peak of the shear wave resistance density R ? .

Direct comparison between measured and predicted im-pedance is difficult for small low-admittance transducers due to the effects of stray capacitance and inductance. However, typical values of these stray components suggest that they wlll have minimal effect on the measured value of radiation resistance R in the series representation up to about 50 MHz and on the measured value of conductance G in the parallel representation above this frequency. Fig. 17 shows that there is good agreement for R and G in the two frequency


0 1 2 3 L . 5

Fig. 18. Angular bulk wave power distribution for 4 electrode-pair 1 : 1 markspace ratio transducer on YZ LiNb03. (a) Slow shear wave dis-tribution at center frequencyfo. (b) Longitudinal wave distribution at 2fo. (c) Slow shear wave distribution at 2fo.

ranges, respectively. The effective equivalent circuits includ-ing stray componentsC and L are also shown for the two sets of measurements.

The angular distributionsof bulk-wave power of a four electrode-pair 1: 1 mark-space ratio transducer are illustrated in Fig. 18, for both the fundamental frequencyfo and the frequency of maximum bulk wave loss 2f0. The radiation pat-

60,

LO

20

, . . . i

U---. II---~ ~ ~

0 20 LO -~

$0 80 100 MHz

Fig. 19. Insertion loss of two 4 electrode-pair 1 : 1 mark space ratio transducers on YZ L a b 0 3 terminated with 50 ohm resistances. ~ T h e o r y . . . . . . Experiment.

tern of each mode is asymmetrical due to the anisotropyof the substrate. This phenomenon has been observed experi- mentally by Schmidt [29]. However, the total field produced by the two modesat the surface is symmetrical, and no ad- vantage can be gained by positioning an output transdu,per on one side of the input transduceror the other. Were this the case, the theorem of reciprocity would be violated. At fo, shear wave power is concentrated near the surface and can therefore travel direct to an output transducer positioned near the input transducer. Similarly at 2f0, longitudinal wave power can travel direct to the output transducer, but shear waves which travel at a steep angle to the surface can only be detected after reflection from the bottom face of a finite solid. In practice, tlus mode can be absorbed or scattered, making the substrate in effect semi-infinite.

The insertion loss of two arbitrarily terminated transducers may be predicted using the detection theory described in Sec-tion V-B. Predicted and measured absolute insertion loss of two four electrode-pair1 : 1 mark-space ratio transducers each terminated with a 50-ohm resistance are shown in Fig. 19. The transducer separation was 37.6 periods. There is excellent agreement at all frequencies up to the third harmonic, includ-ing the detection of longitudinal waves in the region of twice the fundamental. The level of other spurious signals which are not identifiable in the frequency response may also be pre-dicted. Although multiply reflected signals are not specifi- cally included in the analysis, an estimate of the triple transit suppression may be inferred from the reflection coefficient of a single transducer. For four electrode-pairs the ratioof re-flected to incident power computed from(51) at the center frequency is 0.0144. The triple-transit signal, which undergoes two reflections, is therefore 36.8 dB down with respect to the direct signal. This compares with an average measured value of 37 dB. The shear wave signal which is generated just above the surface wave center frequency cannot be distinguished in the frequency response of Fig. 19,but its amplitude in relation to the maximum surface wave signal can be estimated from the

163 MILSOM e t al. : SURFACE AND BULK ACOUSTIC WAVES AND INTERDIGITAL TRANSDUCERS

1 V 20

lc 1-. I I J 0 20 LO 60 80 1CQ MHz

Fig. 20. Insertion loss of two split-finger I : 3 mark-space ratio 4 electrode-pair transducers on YZ LiNbO3 terminated with 50 ohm resistances. __Theory. . . . . . Experiment.

dB

L___-.L-..----l _L_-0 20 LO 60 80 100 MHz

Fig. 21. Theoretical insertion loss of two 4 electrode-pair 1:1 mark-space ratio transducers on 41" rotated YX LBbO3 terminated with 50 ohm resistances.

relevant far-field contributions to the Green's function in (A14) and (A15). At a distance corresponding to the actual transducer separation, the shear wave would be approximately 33 dB down on the surface wave. This compares with a mea-sured value of 35 dB obtained by time-resolving the two signals. Further comparison between predicted and measured insertion loss is illustrated in Fig. 20. In this instance both transducers had split-electrodes [30] with approximately 1:3 mark-space-ratio, and the terminations were again 50-ohm re-sistances. There is good general agreement, although the de-tail does show some discrepancies. Slight misorientation of the substrate and slight deviation from uniformity in the trans-ducer geometry may have accounted for this.

Finally, the theory was applied to the 4l0-rotated YXcutof lithium niobate. According to Yamanouchi et al. [ 181, an electrically free surface can support a surface wave of the Blew stein-Culyaev type which leaks an infinitesimal proportion of its energy into vertically polarized shear waves. The present theory shows that there is strong coupling to boththis leaky wave.and to a horizontally polarized shear wave. Also there is less than 0.5%difference in the velocities of these two waves, and they could only be resolved by a very narrow-band transducer. The far-field contributions to the Green's functions in (A14) and (A15) show that the twowaves are of equal amplitude at a distance of some 40 wavelengths from the source. In addition, thereis some coupling to a longitudinal

164

wave. The theoretical response of a transducer pair identical to that used on YZlithium niobate in Fig. 19 is shown in Fig. 21. As expected, the passband is grossly distorted by the pres-ence of the bulk shear wave, although the midband insertion loss of only 7 dB with 50-ohm terminations compared with22 dB on the YZ cut illustrates the very high effective coupling coefficient.

1X. CONCLUSION

A theory of interdigital surface wave transducers which is both more exact and more general than any previously reported has been developed. This theory includes the full ef-fect of coupling t o surface and bulk waves and interaction between these modes. The theory is applicable to arrays of electrodes of varying width and separation and arbitrary po-larity of connections, and is currently being extended to in-clude length-weighted transducers. The theory has been applied to problems with both free and metallized surface regions outside the transducers, and generation and detection have been solved independently. Some second-order effects such as the finite mass and conductivity of the electrodes and diffraction in the surface plane are as yet excluded. However, the most serious disadvantage of the theory is the large amount of computer storage and time required to set up and solve the linear equations(32)-(34).

Experimental evidence tends to confirm the accuracy of the theory, although it has not always been possible to obtain the degree of precision in the experiments that the exact theory requires. Further experiments are necessary to test the full generality of the method.

APPENDIX A EVALUATIONOF THE GREEN'S FUNCTION

The integral Green's function in (20) represents the electrical potential at the surface x2 = 0 resulting from a unit line source of charge parallel to the xI -axis. As described in Section IV, the integrand may be found exactly but only by using numeri-cal methods. It is therefore also necessary to use numerical analysis to evaluate the integral itself. However, Fig. 2 and (21) show that there are singularities at the origin and at the surface waves slownesses+so, and these have to be removed and treated analytically [24] . Equation (21) is therefore rewritten as

r(s) = rO(s) 'rds) 'rb(s ) (Al l

where

G, is given by (43), and ~ ~ ( 0 )is the value of the effective permittivity at S = 0. Equation (20) may now be expressed as

IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, MAY 1977

the sum of three Green's functions

1 -r,(s) exp ( - j sX) ds t -

The three terms on the right of (AS) are, respectively, the elec-trostatic, surface wave, and bulkwave contributions to the potential field.

Firstly, consider the electrostatic Green's function S

G o ( X )= 'J roexp ( - j sX)ds277 _ S

Strictly, this integral is infinite for all X . This apparently meaningless solution arises because an isolated line source of charge cannot reside in an electric field. It is therefore neces-sary initially to consider the potential due to equal and op-posite line sources at k X , . This is given by

c-

Co(X - X , ) - Go(X + X , ) = -' J ' .

2W€,(O) _ _ IS1

. exp ( - j s X ) [exp ( jsX,) - exp (-jsXo)] ds (A71

which reduces to

since only the even part of the integrand contributes to the integral. Equation (A8) is a standard integration [31]. Thus

Go(X - X,)- Go(X+ X,)

When (19) is substituted into (A9), the frequency dependence vanishes, leaving

x01 -

where the sources are atx3 = *xo. Provided that for each line source of charge there is a cor-

responding equal and opposite parallel source at some other point on the surface, it is valid to write

This proviso is satisfied in the solution for the interdigital array by the condition of continuity at charge.

Equation (AI 1) does not, however, give the full solution for the electrostatic field, and this can be seen from the solution for W = 0. The potential due toa line source at zero fre-

MILSOM et al. :SURFACE BULK ACOUSTIC WAVES AND INTERDIGITAL TRANSDUCERSAND 165

quency Goo (x) is obtained from the effective permittivity at infinite and not zero slowness €,(m). This is a consequence of (19) which makes S infinite at zero frequency for all values of wavenumber k .

Thus

Since the limit ofG ( X )as W approaches zero must be equal to (A12) (plus a constant), an additional logarithmic function must arise from the numerical integration of the third term of (‘45).

Before considering this, the second term in (A5) will be evaluated. This represents the surface wave component of the field and is given by

In order to evaluate this integral i t is convenient to introduce an infinitesimal dissipation into the material which has the effect of displacing the first pole at so (positive travelling surface waves) into the negative half of the Argand diagram, and the second pole at -so (negative travelling surface waves) into the positive half. Taking the contour of integration firstly in the lower half-plane for X > 0 and secondly in the upper half-plane for X <0, the theorem of residues [23,ch. 7) gives

C,(X) = -jG, exp (-jsoIXI). (A14)

Finally, the third term in (AS) is a numerical integral which, on taking into account the even symmetry of I‘(s), reduces to

The function r b ( s ) was approximated over a suitable number of ranges of values of S using a least square curve-fitting rou-tine, and a Gaussian numerical integration formula [24] was used t o evaluate the integral. The integral was truncated at an appropriately large value of S. Division of the integral into ranges made i t possible to isolate the far-field contributions resulting from the cutoff discontinuities associated with each bulk wave, and it was found that these contributions could be approximated closely by functions of the form

where S, is the cutoff slowness of the nth bulk wave and the constants G,, p, , , and & were again obtained by curve fitting. However, in the near-field it was not possible to isolate the distinct contributions from each of t h e N bulk modes. The func-tion was therefore approximated by the sum of a logarithmic function and a single polynomial. Thus in the near field, let us say for X < X L ,

where the coefficient of the logarithmic function is chosen so that C ( X )approaches (A12) plus a constantas frequency tends to zero. The polynomial coefficients were found by ap-plying the least squares procedure to the difference function. In the far field, for X >X L ,

(A. 18)

These accurate approximations to the bulkwave Green’s function are similar to those used to generate Hankel functions [3 l ] . G b ( X ) for Y z lithium niobate is shown in Fig. 2 2 , which clearly demonstrates the shear and longitudinal waves

0-

.-, , -2

-l0 I 2 3 L 5 6 7 B 9 10 Raylelgh wavelengths

Fig. 22. Bulk wave Green’s function cb(x)for YZ LiNb03. __ Real part. ---- Imaginary part. CS = Rayleigh wave amplitude.

166

modulating one another in the far field and the logarithmic form of the real part of the function in the very near field.

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‘ 1 Tseng, C. C . , “Frequency response of an interdigital transducer for excitation of surface elastic waves,” IEEE Trans. ED-1.5, 1968, pp. 586-594. Joshi, S. G., and White,R . M,,“Excitation and detection of sur -face elastic waves in piezoelectric crystals,”J. Acousr. Soc. Amer., 4 6 , 1 9 6 9 , pp. 17-27.

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[ 101 Skeie, H . , “Electrical and mechanical loading of piezoelectric surface supporting surface waves,” J. Acoust. Soc. Amer., 48, 1970, pp. 1098-1104.

[ l11 Milsom, R. F., and Redwood, M., “Piezoelectric generation of surface waves by interdigital array ;variational method of analy-sis,”Proc. IEE, 118, 1971,pp. 831-840.

[ 121 Auld, B. A., and Kino,G. S., “Normal mode theory for acoustic waves and its application to the interdigital transducer,” IEEE Trans. ED-18, 1971, pp. 898-908.

[ 131 Wagers, R. S., “Analysis of acoustic bulk mode excitation by interdigital transducers,” Appl. Phys. Lerr.. 24, 1974,pp. 401-403.

[ 141 Lewis, J . A., “The effect of driving electrode shape of the electri-

IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, MAY 1 9 7 7

cal properties of piezoelectric crystals,”Bell System Tech. J . , 4 0 , 1 9 6 1 , ~ ~ .1259-1280.

[ 151 Lord Rayleigh, “On waves propagated along the plane surface of an elastic solid,” Proc. London Math. Soc.,1 7 , 1885, pp. 4-1 1.

[ 161 Bleustein, J . D.,“A new surface wave in piezoelectric materials,” Appl. Phys. Lett., 13, 1968,pp.412-413.

[ 171 Gulyaev, Y. V., “Electroacoustic surface waves in solids,”Sou. Phys. JETP,9 , 1 9 6 9 , pp. 63-65.

[ 181 Yamanouchi, K., and Shibayama, K . , “Propagation and amplifi-cation of Rayleigh waves and piezoelectric leaky surface waves in LiNb03,’;J. Appl. Phys. ,43 , 1972,pp. 856-862. Lamb. H., “On the propagation of tremors over the surface of an elastic solid,” Phil. Trans. Roy. Soc. London, A203, 1904, pp. 1-42. Greebe, G . A. A. J., Van Dalen, P. A., Swanenberg, T. J. B., and Wolter, J., “Electric coupling properties of acoustic and electric surface waves,” Phys. Reporrs (Sec. C, Phys. Leu.), 1 , 1971, 235-268. Nye, J . F . , Physical Propertiesof Crystals, Chaps. 7 and 8 , Ox-ford, 1957. Campbell, J. J . , and Jones, W. R . , “A method for estimating optimal crystal cuts and propagation directions for excitation of piezoelectric surface waves,” IEEE Trans. SU-1.5, 1968, pp. 209-217. Arfken, G.,MathematicalMethodsforPhysicists,Chap. 15, Academic Press, 1966. Noble, B.,Numerical Methods: 2 , Chap. 9 , Oliver and Boyd, 1964.-_.

1251 Bleaney, B. I . , and Bleaney, G., Electricit,v and Magnetism, Chap. 10,Oxford, 1957.

1261 Rutherford, D. E., VectorMethods, Chap. 1,Oliver and Boyd, 1939.

[27] Noble, B.,R‘umericaIMerhods: l , Chap. 5,Oliver and Boyd, 1964.

1281 Willis, W., Mitchell, R. F., and Redwood, M , , “Electrode inter-action in acoustic surface wave transducers,”Elect. Lett., 5 , 1969, pp. 456-457.

[ 2 9 ] Schmidt, R . V. , ‘The excitation of shear elastic waves by an in-terdigital transducer operated at its surface wave center fre-quency,” J. Appl. Phys., 43, 1972, pp. 2498-2501.

[30] Bristol, T. W.,Jones, W. R., Snow, P. B., and Smith, W . R., “Ap- plications of double electrodes in acoustic surface wave device design,”Proc. IEEE Ultrasonics Symposium, Boston,1972, pp, 343-345.

[ 3 1 ] Abromowitz, M . and Stegun, I . A., Handbook ofMathematica1 Functions, Dover, 1965.

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