Multiple Reflections in Acoustic Surface Wave Reflective Arrays

IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SW-22, NO. 4, JULY 1975 251

Multiple Reflections in Acoustic Surface Wave

Reflective Arrays OBERDAN W. OTTO

Abstract-The effects of multiple reflections in two-dimensional acoustic surface wave reflective arrays are examined via a computational model which accounts for all multiple reflections. A weak reflection unit cell scattering matrix is developed in detail for scattering on an anisotropic surface. Several reflecting geometries are solved for the weak reflection case providing a comparative basis against which the distortions of frequency responses and spatial beam profiles induced by multiple reflections are compared. By comparing computed frequency responses and beam profiles to experiment, it is found that single reflector scattering coefficients can be deduced.

I. INTRODUCTIOK

T WO-DIRIEKSIONAL acoustic surface wave reflective arrays have been under investigation for their applica-

tions in surface wave filters [l, 23. In the interest of device efficiency i t is desirable to make the reflective arrays as highly reflective as possible. It, is thus important to know at what point multiple reflections become important and what the multiple reflection effects are in terms of distortion of frequency responses and beam profiles. These questions are addressed in this paper via a computational algorithm which takes into account all multiple reflections in a reflective array.

Weak coupling models [ 2 4 ] which assume no multiple reflections are useful for predicting the magnitude of a reflection and frequency response when the total array reflectivity is small. The calculation is based on the assumption that the transmitted wave is unaffected by the number of grooves traversed. Any departure from this assumption requires a complete solution to the problem, taking into account all multiple reflections.

11. RIULTIPLE REFLECTION MODEL

To model multiple reflect,ions in two dimensions, the reflective array is superimposed on a parallelogram grid, where the grid lines are parallel to the incident or reflected power flow directions (Fig. la) . Each "unit cell" in the grid may contain one or more or a noninteger number of reflectors. If the number of reflectors per unit cell differs from unity, then the total reflectivity of each cell must be much smaller than unity, so that' the unit cells are adequately described by a weak reflection model. Assuming that a wave incident upon a unit cell scatters into a single mode, the unit cell may be described by a 4 X 4 scattering

Manuscript received October 29, 1974; revised February 17, 1975. The author is with Hughes Research Laboratories, Malibu, Calif.

90265.

matrix which couples four modes (Fig. l b ) :

From the form of the S-matrix it is clear that the modes factor into two uncoupled set,s which are denoted by solid and broken arrows in Fig. lb. Since the two sets of modes are independent one can, without loss of generality, reduce to a 2 x 2 represent,ation of the S-matrix (subscripts have been relabeled) :

where the variables are defined as in Fig. IC. It should be emphasized that this is a reduction of a 4-port scattering matrix and is not a standard 2-port scattering matrix. Although this form of the S-matrix is the same as for collinear scattering C53, two-dimensional scattering is funda- mentally different from collinear, since the reflected wave does not propagat'e in the same path as the incident wave. The collinear case, as is shown in the Appendix, can be solved in a simple analytical closed form. Although in the collinear case the inputs and outputs of a unit cell are coupled by reflections in all the neighboring unit cells, the outputs of a given two-dimensional unit cell are coupled to the inputs only by the scattering element in that unit cell. Thus the number of multiple reflect'ions in a two- dimensional array is finite and equal to the number of reflective stripes along the narrow dimension of the array.

The solut,ion to the two-dimensional scattering problem for an array of N X M unit cells may be sct up formally as

where a and a' are M-vectors, b and b' are A--vectors, and S is an ( N + M ) X ( N + M ) matrix. The elements of S are polynominals in the elements of the unit cell S-matrix. For large arrays, the evaluation of (3) is a computational nightmare. Assuming that the elements of S are known, they must be stored in an ( N + M ) x ( N + M ) complex array. Then the matrix product is evaluated requir-

252 IEEE TRAN3ACTIONB ON 3ONICS AND UJA'RASONICS, JULY 1975

UNIT CELL GRID LINES

! <

OUTLINEOF REFLECTIVE ARRAY POWER FLOW

REFLECTED

Io1

i' E B,* - - a-T';\i-..

*-- A 3

40' 02 Az 2

l b ) ( C )

Fig. 1. Decomposition of reflective array into unit cells. (a) Super- position of array onto arallelogram grid. (b) Unit cell viewed aa a &port. (c) Uncouple$tw+mode representation of unit cell.

ing 4( N + M)2 complex multiplications and additions. It would take on the order of 4NM operations to set up S. The need for a streamlined computational technique is readily apparent, from the point of view of both storage requirements and speed.

The computational algorithm proposed here is illus- trated by Fig. 2. It may be represented mathematically by

The required storage is an M-vector for a and an N-vector for b, a factor of ( N + M ) smaller than for the S-matrix approach. The computation proceeds from upper left to lower right, the outputs of a unit cell become the inputs to the cells directly to the right and directly below. The number of operations required is 4NM, at least a factor of 5 faster than for the &matrix approach. This computational algorithm thus achieves a considerable saving in digital storage requirements and speed.

111. WEAK REFLECTION CASE

Before proceeding to multiple reflection computations it is worthwhile to examine the weak reflecting case. The solution to the weak reflecting case will provide a point of reference against which to compare the multiple reflection computations. In addition, as a by-product, the scattering parameters for unit cells having a number of reflecting stripes other than unity will be obtained. Only periodic arrays will be considered. Nonperiodic arrays may be analyzed by breaking them down into nearly periodic sections.

To begin the analysis the unit cell is taken as a "primitive cell" in which there is exactly one reflector as shown in Fig. 3. The incident and scattered wave vectors are respectively k , and k,. These wave vectors are determined from the effective inverse velocity surface in the periodic region. The determination of this- velocity surface is the subject of another paper and will not be discussed here [S].

Fig. 2. Mathematical algorithm for treating multiple reflections.,

Fig. 3. Primitive unit cell. Ri and R, are incident and scattered

flow directions. d is the unit vector along the edge of the reflective wavevectore. ui and U, lie along the incident and scattered power

stripes. 6 is the normal spacing between reflectors.

The edges of the unit cell ui and U, are parallel to the incident and scattered power flow directions, respectively. The magnitudes of ui and U. are determined by

where d is the center-to-center separation of the reflector stripes and 2 is the unit vector along d. The incident and scattered wave vectors are related by Snell's Law:

where is the unit vector parallel to the reflector stripes. The propagation phase delay from the center of one unit cell to the centers of adjacent cells along the i and S directions are

#c = ki*u; #, = k,*u,. (7)

Thus the primitive unit cell S-matrix is

ir exp [-i(#i + $ * ) D ] (1 - rZ)l/zexp C-ig.1

(1 - exp [-i#i] ir exp [-i(#i + $. ) /2 ] 1 (8)

S = [

OTTO: REFLECTIVE ARRAYS

where ir is the single stripe scattering coefficient. Wave amplitudes are normalized to the square root of a reference power per primitive unit cell. The above form of the S-matrix, which is unitary, satisfies the total power conservation requirement. Omitting the propagation phase factors exp [ -i($, + $#) 121, the phase requirement on S11 and S22 imposed by the power conservation requirement is

S11 = - s 2 2 * (9)

which has an infinite number of solutions. The appropriate solution is determined by the physical situation. For a perfectly symmetric reflector, the single stripe scattering coefficients for S11 and SZ2 are both equal to ir . For a perfectly asymmetric reflector, a step say, the single stripe scattering coefficients for Sll and SZ2 are equal to f r and T r , respectively. Computationally and experi- mentally the symmetry of the reflector makes no differ- ence; only an absolute phase measurement can make it apparent.

The weak reflection assumption is equivalent to assuming that the waves transmitted through the reflective array are not noticeably reduced in amplitude by the reflectors in their paths. It will be shown later that, if n is the number of reflectors along the length and m is the number of reflectors along the width of a filled-in (mathematically rectangular) array, the weak reflection assumption can be quantified as

nmrz << 1. (10)

In the weak reflecting case, one can easily construct the matrix of (3) from the primitive matrix in (6) . One must be careful, however, to refer the input and output wave amplitudes to constant phase fronts of the k; and k, waves. For cells which are on different reflecting stripes, the phase slope of the waves relative to ui and U, must be taken into account. For the i and S waves the phase taper per unit cell is

e, = k;*u,

0, = ka .Ui . (11)

Invoking the weak reflection assumption the elements of S are found to be

( S d q P = irgp exp [-i($,q + $.(m - p ) - esq + 4 p ) l

( S d qq = exp C-i$ml

(&l) p p = exp C-itLinl

( & 2 ) p q = irqp exp C-i($dn - q) + #*p + &q - eip)l.

(12) The matrix S can be applied to the computation of the

frequency response for various reflective array device configurations. Here four types of propagation paths are considered :

1) T-path is transmission through a single grating along ki;

253

L p a t h is reflection from a single grating from ki to k,; U-path is reflection from two mirror image gratings where k, is normal to the mirror plane. U, must there- fore be parallel to k,; 2-path is reflection from two identical gratings which are spatially displaced along U*.

In the discussion which follows all gratings are filled-in uniform gratings, since this is the simplest geometry to analyze.

A . T-Path In transmission, power is reflected out of the main beam

causing a resonant dip in the frequency response. The T-path power reflection coefficient in weak reflection is

" 1 m

-1 m p-l I rT l 2 = c - I C (Sll),, l2

where @ = t,h - Bi = $; - e, = ( k , - ki ) .d . The frequency a t which @ = 2un, where n is an integer, is the resonant frequency of the grating. Since CP does not depend on U, and U,, the resonance is independent of power walkoff.

B. &Path For reflection from a single grating, the average am-

plitude of the reflected wave along a k, phase front is of interest because this is what would be observed if a transducer were placed along that phasefront. The amplitude reflection coefficient for this configuration is

= imr exp ( -iy/2) sin (n@/2) sin (m@/2) n sin ( @ / 2 ) m sin ( @ / 2 ) (14)

where y = n$i + m$#. The bandwidth is determined by the longer of the two grating dimensions. In general, the output transducer will not line up with the phasefront of the reflected wave. If the phase taper at the transducer per unit cell is 6, then in the term sin (nCP/2) /sin (CP/2), the @ is replaced by @ + 6. This causes a shift in the resonance and introduces asymmetry to the bandshape.

C. U-Path In U-path, reflection from two mirror image gratings

is considered. The amplitude reflection coefficient for the average amplitude of the second reflected wave is

= -nmr2 exp (-iy)

254

Typically, gratings in this configuration are longer than they are wide so that the bandwidth is determined by grating length.

In general, the mirror plane between t’he two gratings may not be in a cryst.alline symmetry plane. Furthermore, the wavevector ( k , ) of t,he wave scattered from the first grating may not be normal to the mirror plane. Thus the resonant condit’ion for the second grating will be different from the first. Letting

@’ = (R,’ - k;’j .d’ (16)

where k( = R , and the primes denote quantities applicable to the second grating, then in one of the two sin ( w 1 @ / 2 ) / sin ( @ / 2 ) factors the @ is replaced by a’. In addition, the output transducer may not lie on a k,’ phasefront so that CP’ is replaced by @’ + 6, where 6 is defined the same as for L-path.

D. Z-Path For 2-path, reflection from two identical gratings is

considered. The amplitude reflection coefficient average amplitude of the second reflected wave is

l m n m

for the

(17)

The bandwidth is determined by the grating width. The Z-path does not share any generalizing complicat,ions such as occur for the L- and U-paths. The wave k, is scat,tered by the second grating back into ki so that, the resonant conditions for the two gratings are the same. Also assuming that the input and output transducers are parallel, the output transducer is parallel to t’he output wavefront’.

IV. SCALING THE S-MATRIX Computation time for the multiple reflection algorithm

proposed here is directly proportional to the ratio of array area to unit cell area. It is thus economically important to know how to scale the unit cell to include a large number of reflectors. In all cases the condition on the allowable scale factor is that the unit cell be adequately characterized by a weak reflection S-matrix. Referring to (12)- (15) it is possible to construct the analog to the matrix in (8) for a unit cell which is n primitive cells long by m primitive cells wide. The appropriate elements to that matrix are

Sllnm = imr sin (m@/2) m sin (@/2) exp ( - i Y P )

S22nm = inr sin (n@/2) ?L sin (@/2)

exp ( - i y / 2 )

= (1 - I SllnmS22nm 1 ) exp ( -im$,)

s21nm = ( 1 - 1 SllnmS22nm 1 ) l I 2 exp ( -in$;). (18)

IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, JULY 1975

The square root factor for S1znrn and SZlnm has been chosen arbitrarily so that t,he scaled S is unitary. Power conservation, however, is satisfied only if n = m ; otherwise, S l l n m and S d m do not satisfy (9). One must be certain when using the scaled unit cell Snm matrix t’hat the weak reflection condition (10) is satisfied. Typically, one can take nmr2 = 0.01 as an upper limit. If the total number of primitive cells is 31X, then the condition on the total number of scaled unit cells is N M 2 100 Xmr2, for T 5 0.1. For r > 0.1, the primitive cell S matrix should be used.

When the elements in (16) are applied to the multiple reflection model unit cell ( 4 j , the phase slope of the incident and scattered waves with respect to the unit cell edges must be taken into account as was done in the con- struction of the weak reflection matrix (12). In addition, when determining the average amplitude across a phasefront for the L, U , and 2 paths one must apply the scaling weighting fact,ors:

L-path sin (n@/2) n sin ( @ / 2 )

sin ( ? L @ )

n sin (@) U-path

2-path 1. (19)

In general, it may be desirable to allo\v t8he scaling factors n and m to be noninteger, as would be the case for the computation of a nonperiodic array. It is possible to make analytical ext’ensions or approximations to the sin ( n z j ,/sin( x) type terms; however, it is equally important to account for the shifting of the center of mass of the reflect’ors from unit cell to unit cell so that the co- herent nature of the scattering is maintained.

. V. COIMI’UTATIONAL RESCLTS

To illustrate the multiple reflection effects that occur within reflective arrays we consider the simple case of a periodic reflective array 10 reflectors long by 10 reflectors nide. It is assumed t’hat, the power flow vectors and wave normals are collinear. The angle through which the incident. wave is scattered is assumed to be 90’. In the computation, the unit cell grid is chosen so that t,here is exactly one reflector per unit cell. The S-matrix is thus

ir ( 1 - 7-2) 1’2

( l - i r S = [ l exp Ci2~flf01 (20)

where f is the operating frequency and .fo is the frequency for synchronous reflection.

Figure 4 shows the frequency response for the four propagation paths for the case iV = IP = 10, with single stripe reflectivities of r = 0.01, 0.0316, 0.1, 0.316. The curves for r = 0.01 and 0.0316 correspond quite well to the analytical expressions in (13)-(17). For the higher reflectivities the sharp nulls disappear and the sidelobes merge. For T = 0.316 in the L- and 2-paths, t’here is a dip in the center band response, but no such dip occurs for

OTIO: REFLECTIVE ARRAYS

0

g IO

Ej 20 m

z 3 0

LL W m

40

50

60 1 3

l - PATH

-

0.5 I O 15 f / f , I C )

I

0 5 10 15 f / f o l B l

f / f , ID)

Fig. 4. Insertion loss versus frequency for T-, L, U-, and 2-paths. NMrZ = O.Ol,O.l,l,lO, for 1,2,3,4, respectively.

I BEAM PROFILES

Fig. 5. Spatial output beam profiles a t synchronism = f ~ ) for

1,2,3,4, respectiveg. T-, L, U-, and 2- aths. N = M = 10; NMrZ = O.Ol,O.l,l,lO, for

001 01 10 100 001 0-1 ~- I IO 1 0 0

R = N M I ~ f A l

Fig. 6. Insertion loss versu8 reflectivity at synchronism = l o ) . N = M = 10. (a) T- and Lpaths. (b) U- and 2-paths.

255

the U-path. The reason for the centerband dips in the L- and 2-paths is severe wavefront dist,ortion caused by the multiple reflections.

Figure 5 shows the spatial beam profiles a t synchronism. The large dips in the case r = 0.316 indicate actual amplitude nulls in the profiles. The lobes on either side of a null are 180" out, of phase. Thus when the profile is aver- aged by the output transducer, cancellation between profile lobes on either side of a null can cause a substantial reduction in the transduced power. As the reflectivity is increased, more profile nulls appear. For the U-path, however, the second reflective array tends to undo the distortion introduced by the first. The output beam is thus relatively free of distortion, and there is no dip at center band.

The effects of beam distortion at synchronism on the insertion loss of the four paths versus reflectivity is shown in Fig. 6. The T- and 2-paths have null responses a t various reflectivities; whereas the L-path only undergoes oscillations. The insertion loss of the U-path versus T is quite well behaved. I t can be approximated except for some minor oscillations by the analytical expression

Y = ( X j 2 ) + ( ( X / 2 ) 2 + 17.5)lI2 (21 )

where X = -20 log,, ( N A f r 2 ) . This analytical approxi- mation for U-path multiple reflections has been success- fully applied to the design of efficient' dispersive reflective arrays [4]. Although the exact locations of oscillations or nulls depend on S , AI, and r separately, the overall features of the curves in Fig. 6 are adequately characterized by the parameter R = N M r 2 and are relatively insensitive to the array aspect ratio AV/&! and the single stripe reflectivity except for r very close to unity.

VI. EXPERIMENT

General agreement is found between the predictions of the model and measuremcnt,s on many reflective array gratings. Gratings consisting of shorting aluminum stripes on Y-cut LiNbOs were employed in the study, since the single stripe reflection coefficient is relatively large. Device fabrication was particularly simple since the transducers and gratings were produced in the same 1it.hography step. The best correlation between experimental and computed frequency responses and beam profiles for scattering between the z- and x-directed Rayleigh waves was obtained for a single stripe reflectivity of

r = 0.027 (22 )

for stripe-to-period ratios near 0.5. Figures 7 and S illustrate a typical comparison between

experiment and theory for a 2-path configuration. The A1 reflectors with a stripe-to-period ratio of 0.4 are oriented a t 46.80" to the z-axis. The velocity anisotropy for the 40y0 metallized periodic region is taken as (uz /u t )e f feo t ive =

1.0804 based on experiments performed by the author. When the reflection is close to 90" the computation is ex- tremely insensitive to the actual velocity anisotropy. The

256 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, JULY 1975

I 1 I 1 1 1 170 172 I 74 l76

FREOUENCY. MHz

Fig. 7. Comparison of experimental and theoretical frequency responses in Z-path for periodic rectangular metallic reflective arrays, N = 110, M = 53, T = 0.0257. Scattering is from +Z t o + X to +Z on Y-cut LiNbOs by A1 stripea oriented at 46.8" to the z-axis.

l I

BE&M PROFILE

Fig. 8. Comparison of experimental and theoretical output beam profiles a t synchronism (172 MHz) for the same device &S in Fig. 6.

curves assuming an exact 90" reflection are indistinguish- able from Figs. 7 and 8. The actual gratings are 110 reflectors long by 53 reflectors wide. The computation is performed for arrays of dimensions 112 X 56 for scaling convenience, with the scaled unit cell eight primitive cells square. The error introduced by the scaling is less than 0.5 dB. The stripe reflectivity in the computation which gives the best agreement with experiment is r = 0.0248. For t'he actual 110 X 53, 40% metallized arrays this translates to r = 0.0257 assuming a constant W M r 2 parameter.

Although not indicated by the computational data in Section V, neither non-90' reflectors nor multiple reflection distortions nor the combination of the two produce asymmetric frequency responses. The asymmetry of the frequency response in Fig. 7 is due to rolloff in the input/ output transducer bandshape. The center band dip is clearly shown by Fig. S to be due to partial cancellation between profile lobes on either side of an amplitude null.

The disparity between the frequency sidelobes is appar- ently due to the neglect of diffraction in the modeling.

Diffraction has been neglected for a very good reason. The multiple reflections model proposed here is basically incompatible wit,h a rigorous accounting for diffraction. The incident and scattered waves can be decomposed into a superposition of plane waves. To begin with, the multi- plicity of plane waves introduces an unavoidable am- biguity to the definition of t>he unit cell. Furthermore, since the multiple reflections distort the beam profiles, the plane wave decomposition changes mith position in the reflective array. Thus the superposition of plane waves principle is inapplicable inside the array. B y computing the diffraction of the wave up to and away from the reflective arrays, however, better agrcenlent bet,ween experiment and theory is likely to be obtained.

VII. CONC,LUSIOKS The multiple reflection model which has been presented

here is applicable to a wide variety o f reflective array configurations and surface anisotropies. For the condition where the unit cells are weakly reflecting, the model can be scaled to include more t'han a single reflector per unit cell. h noninteger number of reflectors per unit cell is also possible, so that the model can be applied to nonperiodic arrays and arrays of complicat,ed shapes.

The comput,at>ions perfornzed here for a 10 X 10 array illustrate the general features o f multiple reflection effects. The parameter R = S M r 2 has been shoxn to roughly characterize the nlultiple reflection behavior for all four configurations considered. The transition from the weak reflection to the 'multiple reflection region, as indicated by Fig. 6, occurs around t,he point N M r 2 = 1. In the nlultiple reflection region, a center band dip occurs for the L- and 2-paths due to severe distortion of the beam profile and amplitude averaging by the output transducer. Similarly, the transmission loss (T-path) through a single grating can be considerably greater than would be ex- pected from the actual pouw that is reflected by the grating. For the li'-pat,h, however, the beam profile distortions introduced by the first grating arc, for the most part, removed by the second grating.

Finally, it has been shown that the single st,ripe reflection coefficient r can be deduced by matching the shapes of the computed and experimental frequency responses.

APPENDIX COLLIKEAR lfU1,TIPLE REFLECTIONS

The case for collinear scattering from a periodic array can be solved in closed form [ S ] . This is accomplished by transforming from an S-mat,rix unit cell representation to a T-matrix (transmission) representation and c,ascading the 7'-matrices. Starting with the S-matrix representation for a single cell as

IEEE TRANSACTIONS ON SONICS AND rLTRASONICS, VOL. SW-22, NO. 4, JULY 1975 257

we transform to a T-matrix representation

If the S-matrix is unitary it can be expressed in the form

i tanh (x) l/cosh (x)

l/cosh (x) i tanh ( 5 )

S(1) =

The corresponding 7’-matrix is

[ i

cosh (x) i sinh (5)

-1: sinh (x) cosh (x) 1 . T(1j =

The T-matrix for 1V unit cell periods is

cosh (Xs) i sinh (1Nx)

T(N) = T ( l j N =

-2 sinh (Xz) cosh ( N Z )

Thus the S-matrix for A T uniform periods becomes

i tanh (11-2) l/cosh (XZ)

ljcosh (‘Vs) i tanh ( S x ) S ( N ) =

If the single cell reflection coefficient is Sll( 1) = ir, then the reflection coefficient for X cells is & , ( N ) = i tanh ( W tanh-’ r ) . The transcendental functional form of the solution relates to the fact, that in the collinear case there are an infinite number of multiple reflect’ions.

ACKNOWLEDGMENT The author wishes to thank R. Dimon for his able tech-

nical assistance in the preparation of the test devices.

REFERENCES [l] R. C. Williamson and H. I . Smith, “The Use of Surface-Elastic-

Wave Reflection Gratings in Large Time-Bandwidth Pulse- Compression Filters,” IEEE Trans. Microwave Theory Tech.

121 R. D. Weglein and 0. W. Otto, “Characteristics of Periodic Acoustic Surface Wave Grating Filters,” Electron. Lett. 10, 68-

[3] T. A. Martin, “The IMCON Pulse Compression Filter and its 69, 1974.

Applications,” IEEE Trans. Microwave Theory Tech. MTT-21,

[4] H. M. Gerard, 0. W . Otto, and R. D. Weglein, “Development

Filter,” Proceedings 1974 IEEE I7ltrasonics Symposium, pp. of a Broadband Reflective Array 10,OOO:l Pulse compression

[5] E. K. Sittig and G. A. Coquin, “Filters and Dispersive Delay Lines Using Repetitively Mismatched Ultrasonic Transmission

IS] 0. W. Otto, “Phase Matching Condition for Scattering From Lines,” IEEE Trans. Sonics Ultrason. SLr-15, 111-119, 1968.

Acoustic Surface Reflective Arrays,” Appl. Phys. Lett. 26, 215- 217, 1975.

MTT-21, 195-205, 1973.

186-194, 1973.

197-201, 1974.

Pump Requirements for Parametric Amplification and

Generation of Surface Waves on YZ LiNbO,

Absfract-Expressions are developed for estimating the threshold electric fields required for parametric amplification or generation of surface waves in piezoelectric materials utilizing the nonlinear characteristics of the material. When applied to Y Z LiNbOa, they yield values in excess of 20 kV/cm under the most optimistic, yet still realistic, conditions. These values are too Iarge for amplification or generation to be considered for practical applications. Several pump electrode geometries and circuit matching techniques were investigated to determine the best obtainable pump circuit per- formance. An interdigital electrode separated from the crystal by a small gap was found to be the most efficient pump electrode geometry and was capable of producing approximately ZkV/cm of field per watt’‘2 of input pump power.

ported in part by the National Science Foundation under Grant Manuscript received September 23, 1974. This work was sup-

GK32690 and the Graduate School of the Universitv of Minnesota, Minneapolis, Minn.

University of Minnesota, Minneapolis, Minn. 55455. The author is with the Department of Electrical Engineering,

S I. INTRODUCTION

EVERAL new surfacc wave devices including convolvers and correlators have been developed which

utilize the nonlinear interaction of electric fields and acoustic surface waves on piezoelectric materials [l]. Since t,his interaction can be described as a parametric process [l] (specifically, a mixing process for convolvers and correlators) the possibility of paramet,ric amplification and generation exists. These latter processes have pot,ential device applications not only for amplifiers and oscillators but also for new and unique devices such as frequency selective limiters [ 2 ] and spectrum clippers [ 3 ] .

Unlike simple mixing, however, these M t e r interactions do not, occur until a critical or threshold value of the electric field pump is reached. The feasibility of dcvices based on these processes depends on the magnitude of the threshold

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Multiple Reflections in Acoustic Surface Wave Reflective Arrays