A THEORETICAL INVESTIGATIOK OF THE FOCUSSING OF THE SCT

M. E. BARNARD, M. J. LANCASTER AND E. G. S. PAIGE

X p s r t m e n t of Engineering Science University of Oxford

England

Abstract

The Slanted Chirped Transducer ( S C T ) has been shown t o focus a SAW t o a frequency dependent p o i n t upon the substrate surface [l], and forms the basis of a spectrum a n a l y s e r [ Z ] . Although a number of experimental and numerical r e s u l t s have been obtained for t h e behaviour of the focussing SCT, only very simple a n a l y t i c a l theories have been developed t o expla in the focussing behaviour .

In t h i s paper an a n a l y t i c a l model of the focuss ing a c t i o n of a straight l i n e a r l y chirped a r r a y of isotropic radiators upon an isotropic s u r f a c e is presented . The a n a l y t i c a l r e s u l t s are shown t o be c o n s i s t e n t w i t h t h e numerical ly p r e d i c t e d r a d i a t i o n f ie ld .

The effect of the aniso t ropy of the substrate (Y-cut LiNbO3) upon the isotropic focussing a c t i o n can then be seen by in t roducing the an iso t ropy s lowly i n t o t h e numerical predictions. This method a l l o w s the formation of the "tail" reported previous ly t o be observed and explained i n terms of the substrate aniso t ropy .

1. In t roduct ion

The s l a n t e d Chirped Transducer (SCT) is an i n t e r d i g i t a l t ransducer i n w h i c h t he spacing between f i n g e r s varies monotonically ( c h i r p e d ) and i n which a l l f i n g e r s are i n c l i n e d a t an a n g l e t o the bus-bars w h i c h is o t h e r than 900. An example is shown i n F igure l a . I n t h i s paper we shall restrict cons idera t ion t o a l i n e a r chirp. I f t h e f i n g e r s are short (about t e n a c o u s t i c wavelengths or less), it h a s been shown tha t the s u r f a c e a c o u s t i c waves (SAW) launched from t he SCT are focused, t he p o s i t i o n of the focus being frequency dependent [l]. This p r o p e r t y has been e x p l o i t e d i n the invent ion and development of a SAW spectrum a n a l y s e r [2 ,3 ] .

I n t h i s paper we report a theoretical i n v e s t i g a t i o n of the focuss ing behaviour . The focussing a c t i o n is not straightforward even on an isotropic substrate; on an h i g h l y anisotropic s u b s t r a t e material such as lithium niobate it is complicated. Considerat ion of real materials is confined t o lithium niobate because very a c c u r a t e va lues of the v e l o c i t y are a v a i l a b l e [4 ] and the high electrolaechanical coupl ing makes it a very strong contender as a device substrate. Furthermore, very s t r i k i n g behaviour h a s been

32 - 1985 ULTRASONICS SYMPOSIUM

observed i n which the focussing region becomes g r e a t l y extend& i n t o a "tail" 111.

I n the next s e c t i o n w e cons ider the isotropic case. Analy t ica l express ions f o r the focussing behaviour are compared w i t h p r e d i c t i o n s of a simple model 111 and w i t h numerical r e s u l t s . Then, i n s e c t i o n 3, aniso t ropy is introduced by a progress ive change f r o m a d i r e c t i o n a l l y independent v e l o c i t y t o the v e l o c i t y v a r i a t i o n which occurs on Y-cut lithium niobate . In t h i s l i m i t a comparison w i t h experiments on lithium niobate shows good agreement. Resul t s are d iscussed and the phys ica l o r i g i n of the "tail" is described i n s e c t i o n 4.

2 . Isotropic Behaviour

An explana t ion of the focuss ing a c t i o n of the SCT t o g e t h e r w i t h t h e d e r i v a t i o n of an expression for focal length has a l r e a d y been presented for t h e isotropic case [l]. W e sha l l refer t o t h i s focal length later.

The r a d i a t i o n f i e ld of the SCT w i l l be i n v e s t i g a t e d w i t h t he aid of a s i m p l i f i e d model. The SCT i s replaced by a continuum of p o i n t sources each of w h i c h is fed w i t h a phase sh i f t ed vers ion of the input s i g n a l . The phase s h i f t v a r i e s w i t h d i s t a n c e along the SCT, q, according t o the r e l a t i o n

rl, = aq2 + w (1)

where a and f3 are c o n s t a n t s . This is the basis of an accura te r e p r e s e n t a t i o n of the l i n e a r chirp SCT i n the l i m i t of it having very short f i n g e r s ( less than half a wavelength) and being operated a t fundamental, as d i s t i n c t f r o m harmonic, f requencies . Figure lb shows the source a r r a y located at p = 0 i n a p , q coord ina te system. W i t h the source a r r a y excited at frequency f , the c o n t r i b u t i o n of an e lementa l source at q t o the amplitude a t pl,ql is given by

2 1/2 and r = [ ~ f + ts - q,) I

0090-5607/85/0000-0032 $1.00 0 1985 IEEE

k is the wave v e c t o r a t frequency f and A. the s t r e n g t h of SAW e x c i t a t i o n due t o an elemental source. I n this equat ion and elsewhere it is implicit that the real part is t o be taken .

I f the p o i n t p and q is at the focus o f the a r r a y f o r the e x c i t a t i o n frequency f then waves a r r i v i n g at ( p ,q ) should be i n phase, i .e. @

should be independent of q and r . To exploit t h i s condi t ion i n the l o c a t i o n of the focus a va lue of q is def ined , %, . such that the d e r i v a t i o n of p w i t h respect t o q is zero. From equat ion 3 t h i s g i v e s the r e l a t i o n

1. 1

1 1

2 1 /2 where ro = [pf + (go - q l ) I .

By intxoducing the angle s) (see Figure lb) and not ing s inqo = (go - ql) / ro , t h e n equat ion 4 becomes

( 5 )

showing 40 is the primary source of r a d i a t i o n for any poin t pl,q which fa l l s on the s t r a i g h t l i n e def ined by go ana qO.through equat ion 5. F igure 2 sketches the d i s t r l b u t i o n of these l i n e s for a p o s i t i v e .

0 Z a g 0 + P = ks inq

Reverting t o the phase factor p, we may o b t a i n an expression f o r it i n the v i c i n i t y of a p o i n t 90 by expanding about that p o i n t . Thus

01Aq' +

[a - 2ra k cos's)

-t- cos q0 Aq3 + . . . ( 6 ) ksins) ' I 0

where Aq = 90 - ql. From equat ion 5 the first order term is zero. Phase v a r i a t i o n is f u r t h e r reduced when h igher order terms are zero. The second order term is z e r o when

r 0 = kcosz(q0)/2a ( 7 )

This i d e n t i f i e s t he p o i n t on a l i n e , determined by

go and qo f r o m equat ion 5, where i n t e n s i t y can be expected t o be h igh due t o c o n s t r u c t i v e i n t e r f e r e n c e . Such p o i n t s are indica ted i n Figure 2 where it can be seen t h e y d i s t r i b u t e themselves i n a crescent-shaped focal region. The th i rd order term does not go t o zero for acceptable parameters. However, i f it is minimized w i t h respect t o q a p o i n t i n the r a d i a t i o n f ie ld ( p

F' g ~ ) is found a t

for which

Since the phase v a r i a t i o n of r a d i a t i o n a r r i v i n g at t h i s p o i n t is a minimum it is i n t e r p r e t e d as the focus.

The va lue of r ( p &,>) can be i n t e r p r e t e d as a f o c a l length , F, e e n ing from the focus t o the p o i n t on the a r r a y of sources which makes the maximum c o n t r i b u t i o n t o the amplitude at the focus. It is r e a d i l y shown that equat ion 11 can be r e w r i t t e n as

where v is t h e s u r f a c e wave v e l o c i t y , f i s t h e frequency and B/T is the chirp rate referred t o the d i r e c t i o n of ro. w i t h t h a t given earlier (see equat ion 5, e = 450 of re ference [l]).

This r e s u l t is c o n s i s t e n t

The complete r a d i a t i o n f i e ld of t h e a r r a y can be obtained numerically. The continuum of sources is replaced by a l i n e of discrete sources each w i t h an i s o t r o p i c r a d i a t i o n field g iven by equat ion 2. The ampli tudes are summed a t each p o i n t i n the f i e l d . F igure 3 s h o w s a contour plot of the i n t e n s i t y i n the r a d i a t i o n f ie ld for a source d i s t r i b u t i o n conforming t o a bandwidth B of 25 MHz, t i m e d u r a t i o n T of 2 .5 @sec, c e n t r e frequency f of 60 MHz and e x c i t a t i o n frequency 98 60 MHz. and k = 1.98 x 10' m ). N o t e a new coord ina te system ( x , y ) has been introduced which is the o l d (p,q) rotated through 450 and cent red on the a r r a y of sources . I n the new system the focal axis is i n the x - d i r e c t i o n . Superimposed on the r a d i a t i o n p a t t e r n is a dashed curve showing the crescent - shaped focal region given by equat ions 5 and 7 . The focal p o i n t p r e d i c t e d by equat ion E is shown as a cross.

(his c o r r e s p y d s t o a = 1 . 2 9 x 10' m

I t is i n t e r e s t i n g t o see that the main side- lobe s t r u c t u r e lies on the p o s i t i v e y side of the c r e s c e n t focal region, a f e a t u r e which can be r e a d i l y understood f r o m a comparison of F igures 2 and 3.

To model the SCT mre e x a c t l y it is necessary t o in t roduce non-zero f i n g e r over lap . This is done by rep lac ing overlapping f i n g e r s by i s o t r o p i c a l l y r a d i a t i n g p o i n t sources placed along each over lap and seperated by h/2. Figure 4 shows contour plots for the same parameters as above, b u t w i t h o v e r l a p of 4h and f i n g e r i n c l i n a t i o n of 450. W i t h t h i s choice of f i n g e r i n c l i n a t i o n , the focussing proper ty has been re inforced . The f i n i t e overlap reduces sidelobes. This is because each region of o v e r l a p now c o n s i s t s of eight isotropic sources r a d i a t i n g i n phase and c r e a t i n g a r a d i a t i o n p a t t e r n w h i c h is no longer i s o t r o p i c . The r a d i a t i o n p a t t e r n favours r a d i a t i o n normal t o the f i n g e r and so, i n effect, creates a weighting func t ion w h i c h helps t o supress sidelobes.

3. Anisotropic Behaviour

The aniso t ropy of the electro-mechanical coupl ing cons tan t , K, and the an iso t ropy of SAW v e l o c i t y have t o be taken i n t o cons idera t ion when p r e d i c t i n g the behaviour of the SCT on a

1985 ULTRASONICS SYMPOSIUM - 33

c r y s t a l l i n e m a t e r i a l . An approach t o their i n c l u s i o n b y r e p r e s e n t i n g them by simple r e l a t i o n s which e n a b l e a n a l y t i c a l e x p r e s s i o n t o be found canno t be used because of the w i d e a n g l e spread of s i g n i f i c a n t wave vectors and because i n a m a t e r i a l such as l i t h i u m n i o b a t e simple r e l a t i o n s for the velocity a n i s o t r o p y are known to f a i l . For these r e a s o n s , p r e d i c t i o n s based on the numer i ca l method o n l y have been o b t a i n e d .

The a n i s o t r o p i c numer i ca l method of de te rmin ing the r a d i a t i o n f i e l d is modif ied by t h e i n c l u s i o n of a factor K ( x ) and the rep lacemen t of k (see e q u a t i o n 2) by kg(x) where

9( x 1 = voces I uJ( x 1 I/Vp( x ) ( 3 3 )

H e r e x is the a n g l e between t h e d i r e c t i o n of a s t ra ight l i n e between s o u r c e and f i e l d p o i n t ( d i r e c t i o n of Poyn t ing vector, 5 ) and a crystallographic r e f e r e n c e axis, @(x) is the angle between P and k, v ( x ) is the phase velocity associated w i t h iroyntqng vector i n c l i n e d a t x and v is 2nf /k . Data for the a n g u l a r v a r i a t i o n of K has been t a k e n from S lobodn ik e t a1 [SI and v e l o c i t y data f r o m r e f e r e n c e 4.

0

The f o c u s s i n g behav iour p r e d i c t e d and observed for l i t h i u m n i o b a t e u s i n g a v e r y similar model are i n f a i r l y good agreement and show dramatic d i f f e r e n c e s t o the isotropic behav iour [l]. Por t h i s r e a s o n we p r e s e n t i n F i g u r e 5 a sequence of r e s u l t s o b t a i n e d n u m e r i c a l l y i n which the anisotropy has been p r o g r e s s i v e l y changed from the isotropic l i m i t t o that of l i t h i u m n i o b a t e b y v a r i n g the factor E i n the f o l l o w i n g r e l a t i o n

9 ( x ) = Egl (X) + (1 - E ) g O ( X ) ( )

where g is the v a l u e of g ( x ) g i v e n b y e q u a t i o n 16 for n i o b a t e and g ( x ) is the v a l u e of an isotropic substrate i .e . 'uni ty . K ( x ) has been held c o n s t a n t ; its anisotropy p l a y s a minor role i n de t e rmin ing the r a d i a t i o n p a t t e r n . The SCT parameters are the same as those for P i g u r e 3 . The a r r a y of s o u r c e s are i n c l i n e d at an angle of 450 t o the z-axis of y-cut l i t h i u m n i o b a t e . P i n g e r overlap has been ignored .

lihium

Three f e a t u r e s stand o u t from the r e s u l t s : (1) The focal l e n g t h i n c r e a s e s w i t h anisotropy i n SO far as the f o c u s moves away from the SCl'; ( 2 ) the focal r e g i o n becomes p r o g r e s s i v e l y e l o n g a t e d i n t o the "tail" e x t e n d i n g over many millimetres; ( 3 ) as a n i s o t r o p y i n c r e a s e s the r a d i a t i o n pattern tends t o look more s y m n e t r i c a l .

I n P i g u r e 6 , the predicted and observed r a d i a t i o n f i e ld is shown for a t r a n s d u c e r w i t h the same parameters as for F i g u r e s 3 and 5 except the figure overlap is l h . The array is i n c l i n e d a t 450 t o the z-axis w i t h the normal to the f i n g e r s parallel to the Z - a x i s . G o o d agreement is o b t a i n e d .

s e e n t h a t d i f f e r e n t p o i n t s (%) on the a r r a y m a k e their main c o n t r i b u t i o n to the r a d i a t i o n f i e l d a l o n g straight l i n e s i n c l i n e d a t an a n g l e (q )

0 which depends on qo.. This s c a n of a n g l e w i t h q for the chirped a r r a y is the e q u i v a l e n t of s c a n o? a n g l e w i t h f r equency for a n unchirped a r r a y (a = 0 ) .

The p o s i t i o n of the focal p o i n t g i v e n by e q u a t i o n 8 is a l i n e a r f u n c t i o n of f r equency th rough k. The f requency dependence of the focus is the basis of the u s e of the SCT i n a spectrum analyser [ 2 ] .

For a m a t e r i a l w i t h the a n i s o t r o p y of l i t h i u m n i o b a t e , effects of a n i s o t r o p y canno t be modelled s imply by, for example, i n t r o d u c i n g the parabolic approximation for the slowness s u r f a c e [ 6 ] . If it is, a focal l e n g t h is predicted w h i c h is a n order of magnitude greater t h a n observed. On the other hand, by i n c l u d i n g the d e t a i l s of the e x p e r i m e n t a l l y de t e rmined s lowness s u r f a c e good agreement is found. A primary effect of a n i s o t r o p y is that the v a l u e of qo(%) w i l l change f r o m isotropic v a l u e s as a consequence of k and P b e i n g no l o n g e r parallel. The slowness s u r i a c e of Y-cut l i t h i u m n i o b a t e is, of c o u r s e , symmetrical a b o u t the Z - d i r e c t i o n . The P remains n e a r l y parallel t o the Z - a x i s as the angle of i n c l i n a t i o n of k t o the Z - a x i s i n c r e a s e s t o 1901. Above 1901 the -angle between k and P r educes ; t h e y become c o l i n e a r when k is i i ic l ined-at 230 t o the Z - a x i s . The effect is tha t for qo i n the r ange 360 t o 540 (for the isotropic case), l i n e s of c o n s t r u c t i v e phase g i v e n by e q u a t i o n 5 become s t r o n g l y deviated towards 450. O u t s i d e the range the d e v i a t i o n r educes u n t i l when q = 220 and 7 8 0 t he d e v i a t i o n is zero. For the example g i v e n , i n F i g u r e 5, most of the isotropic q v a l u e s f a l l i n the r a n g e 360 to 540 and there s t r o n g d e v i a t i o n towards 450 is t h e p r i m a r y c a u s e of the i n c r e a s e i n focal l e n g t h . The c o n t r i b u t i o n of qo < 360 is n e g l i g i b l e b u t that for q > 540 is of importance; it c o n t r i b u t e s t o p r e s e r v i n g t h e end of the c r e s c e n t - r e g i o n close t o the SCT. The n e t r e s u l t is t h a t c r e s c e n t - shape remains d i s c e r n a b l e close t o the ScT, t he c e n t r a l r e g i o n of the c r e s c e n t undergoes a major d i sp lacemen t and c o n t r i b u t e s t o the t a i l , t he c re scen t - shape o r i g i n a l l y due t o qo a p p r e c i a b l y less t h a n 450 is lost b y d e v i a t i o n i n t o the t a i l r e g i o n .

The r e s u l t s i n t h i s paper have been p r e s e n t e d for chirped s o u r c e s i n c l i n e d at 450 t o the Z - a x i s of Y-cut lithium n i o b a t e . This w a s t o explore the case where the isotropic focal axis w a s parallel t o a symnetry d i r e c t i o n . Pocussing a c t i o n o c c u r s for other o r i e n t a t i o n s of the array and t h i s has a l r e a d y been exploited i n device d e s i g n [2,3] where it has been shown that "off-axis" f o c u s s i n g c a n assist i n sidelobe r e d u c t i o n . For a l l cases, sidelobes may be s i g n i f i c a n t l y s u p r e s s e d by the e x t e n s i o n of f i n g e r overlap t o abou t t e n wave- l e n g t h s .

4. Discuss ion

It is t o be expected, even i n the isotropic case, that the r a d i a t i o n f ie ld w i l l lack symaetry because of the assyrrmetry of t h e s o u r c e s . We have

34 - 1985 ULTRASONICS SYMPOSIUM

References

1. Barnard, M. E., B l w h , P. D . , and Paige, E. G . s . , "The Focussing Slan ted Chirped Transducer", Proc. IEEE U l t r a s o n i c s Sym. 1983, pp. 205-208.

Paige, E. G . S., "A Novel Spectrum Analyser Based on t h e S lan ted Chirped Transducer", Proc. IEEE Ul t rasonics Symp. 1984, pp. 103-107.

3. Barnard, M. E., Lancaster , M. J., and Paige, E. G. S . , "A SAW Spectrum Analyser Based on t h e S l a n t e d Chirped Transducer", Proc. Conference on Technology o f H i g h Speed S i g n a l Processing, Trondheim, 1985.

4. A m e r i , S . , Ash, E. A . , H t o o , U., Murry, D., and Wickramasinghe, P r i v a t e Comnunication.

5. Slobodnilc, A. J., Conway, E. D., and Delmonico, R. T. "Microwave Acous t i c s HandbO0)c". V o l . lA. Air Force Cambridge Research Centre 1973.

6 . H a t h e w s , H . "Surface Wave F i l t e r s " , Wiley,

2. Barnard, M. E., Lancaster , M. J., and

1977 p. 48

Figure l(a) The Slanted Chirped Transducer

I s

(b) Coordinate systems for phased array. is plotted superimposed on the ,J)r

array.

1985 ULTRASONICS SYMPOSIUM - 35

Y

CX

2mm , I

F igu re 2. S t a t i o n a r y phase l i n e s of phased a r r a y . Crosses i n d i c a t e r e g i o n s of h igh in t en - s i t y . on .each l i n e .

- 3 to -6 db - 1 to - 3 db O to -1 db

1 I I 1 I I 24mm 34mm x 441x11 54rnlfl

(b )

F igu re 6 , I n t e n s i t y c o n t o u r s determined ( a ) numer i ca l ly and (b) e x p e r i m e n t a l l y . (See t e x t f o r d e t a i l s ) .

F igu re 3 . I n t e n s i t y con tour s f o r SCT f o c a l r e g i o n ; f o r ve ry s h o r t f i n g e r s . d e t a i l s ) .

(See t e x t fo r

2ml X

I I -2mn Om] , , ,

I 1 5 k 2 5 k 5nrm

F i g u r e 4 . I n t e n s i t y con tour s for f i n g e r o v e r l a p of 4 A .

36 - 1985 ULTRASONICS SYMPOSIUM

Y

0 to -1 .5 db

-1.5 t o -4.5 db

Figure 5 . Intensity contours showing e f f e c t of increasing anisotropy. Reading from top to bottom E = 0, 0 . 2 , 0 . 6 and 1 .

(-4.5 db U

1985 ULTRASONICS SYMPOSIUM - 37