Reflective arrays of rectangular thin metal 'dots' spaced in a regular or near regular grid pattern, such as have been used in surface acoustic wave resonators and in-line dot RACs on LiNbO//3 , are analyzed. The half-wavelength spacing of dots required leads to capacitative coupling between adjacent dots. The charge distribution induced on the dots by the passage of the SAW is presented including the coupling. The 180 degree SAW reflectivity of the dots and the SAW velocity perturbation through the dot array are given for a range of dot sizes. These are shown to be in good agreement with experimental results. The work establishes that the method of analysis can form the basis of the design of 180 degree reflective weighted dot arrays, allowing them to be implemented through varying dot density and/or by varying dot dimensions

Huang, F

Paige, EGS

Selviah, DR

English

UCL Discovery

THE 180  DEGREE REFLECTIVITY AND  VELOCITY PERTURBATION OF THIN METAL DOTS Abstract F.  HUANG,  E.  G.  S.  PAIGE AND D.  R.  SELVIAH Department of Engineering Science University of Oxford England Reflective  arrays  of  rectangular thin metal 'dots'  spaced  in  a regular or near regular grid pattern,  such  as  have  been  used  in  surface acoustic  wave  resonators [l]  and in-line  dot  RACs [2]  on LiNbO , are  analysed.  The  half wavelength spacing  of  Jots  required  leads to capacitative coupling  between  adjacent  dots.  The  charge distribution induced on the dots by the passage of the  SAW is presented including the coupling.  The 1800  SAW  reflectivity  of  the  dots and the SAW velocity  perturbation  through  the dot array are given  for  a  range of dot sizes.  These are shown to  be  in good  agreement with experimental results 111. The  work  establishes  that  the  method  of analysis  can form  the basis  of the design of  1800 reflective weighted dot arrays allowing them to be implemented  through varying dot  density and/or  by varying dot dimensions. 1.  Introduction Electronic  devices based  on the reflection of surface  acoustic waves  (SAWS)  by reflective array structures  are of  importance in  signal processing applications.  This  paper  concentrates on those devices  in  which  the SAWs are reflected  through 1800.  Such devices  include SAW resonators  111 and In-Line  Reflective Array compressors (ILRACs)  123. A  variety  of  SAW  reflectors have  been used within  the  reflecting  arrays including grooves, [3],  thin  metal strips [4],  thick metal dots [5] and  more  recently thin metal dots  [6]. The  thin metal  dots  have a number  of practical advantages over other techniques: (i)  They  are very simple to fabricate  unlike grooves;  only  a  single  lithographic step being required  to  fabricate  both  transducers  and reflective thin metal dot arrays. (ii)  Very  tight tolerances on metal thickness are not required since the reflection mechanism is based  on  the  electrical  shorting effect unlike thick  metal  dots  where  the reflectivity depends predominantly  on  the  mass  loading  and  hence thickness.  Thin metal dots simply need  to be thin enough  to avoid mass  loading (thickness <  0.005  A) 0090-5607/85/0000-0011  $1.00 0  1985 IEEE and  yet  thick  enough  to  provide  a conducting region. (iii) Thin  metal  dots arrays can give lower loss.  In  metal strip arrays modes  are set up  in which  charge  flows along the  length of the strip leading  to  ohmic  loss.  In metal dot arrays this is  avoided  resulting in higher Q's in resonators [l].  Dot  arrays have about 1/5th of  the loss of strip arrays.  In addition multistrip coupler type behaviour  can be  avoided. Although  impressive results have been obtained for  both  resonators and  ILRACs employing SAW 1800 reflective  thin  metal  dot arrays, a theoretical basis  that  enables  weighted  arrays  to  be accurately  designed  has  not been available. For even  if  the  dot  density in a  reflecting row is used  as  the  basis  for designing in reflectivity weighting  [5],  effects of mutual coupling and of velocity  perturbation  are  overlooked.  It is the aim  of  the  paper  to present a theoretical model and  resulting design curves for 1800 reflectivity and  velocity perturbation to enable such weighted arrays  to  be  designed.  In section 2,  a model  is described  for  the  reflection  of  SAWs from thin metal  dot  arrays  taking  full  account  of  the electrical  interactions  between dots.  Section 3 coinpares  the theoretical predictions of  the model with  experimental  results  obtained  from  SAW resonators  on lithium niobate.  Good agreement is obtained.  Section  4  presents  sets  of  design curves for both  sparse and more  dense arrays.  The paper  concludes  with a discussion and  sununary  in section 5. 2.  Theoretical Model  including Interactions The  model  is  an  adaptation  of  that successfully  employed  for  900  reflections  from thin metal dots arrays [6].  Two cases of  interest are considered in this section: (i)  SAWS incident normally on one free face of an isolated rectangular metal dot ( ii) SAWS incident on a dot array (see Figure 1)  where  the dots are assumed  to lie on  reflecting rows  normal  to  the  propagation  direction  and columns  parallel to the propagation direction and the  dot  faces  are  aligned  with  the  rows and colunins.  The  row separation is chosen to be A  /2 so  that  1800  reflections  add  in  phase  as in typical  resonator  and  ILRAC  structures.  The - 0. 1985 ULTRASONICS SYMPOSIUM -  11 column  spacing  is  also chosen to  be  A  /2 making what has been termed,  a  'waffle  iron'  array  0  [l]. The  model  and  calculation can  conveniently be divided into two parts: (i)  Calculation of the charge distribution on the dot. (ii)  calculation of the scattered SAW  field. 2.1  charge distribution calculation On  a  piezoelectric  substrate  a  SAW  is accompanied  by  a  potential  wave  with the same velocity  and  wavelength.  As  the  SAW  with its associated potential passes  through a metallic dot the  charge  on  the dot  flows so as  to maintain a constant  potential  over  the  dot.  The  charge distribution  on  the dot is found  by  treating the problem  as purely electrostatic.  That is:- (i)  Assume the dot has a potential Vm  at some instant. (ii)  Find  the difference between the potential V  on a surface with a dot present and that which would  have prevailed on a  free surface at the same instant m AV  =  V  -  V  COS(wt - kx)  (1)  mo where  w  and k are the angular frequency and  wave vector  respecLively  and V  is the peak  potential of  the wave. 0 (iii) Attribute  the  potential difference to redistribution  of  charge  on  the  dot.  This difference  potential  must  therefore be  generated by  the repositioned charge on  the dot. (iv)  Subdivide the dot  into rectangular cells, the  jth  cell  containing  a  charge q..  Then the difference potential AV. on the ith cell is due to the  charge in all the ;ells  of  the dot  itself and the  charge in all the cells of  neighbouring dots, so  taking  account  of  electrostatic  coupling between  dots.  This  can  be  expressed  in matrix formalism: AV1  =  M.  .q. 13 3 where  M..  comes  directly from the potential given by a chaise. (v)  Solve  this  equation  for  q.  by  matrix inversion  and  find  V  with  the  ald of charge conservation  (total  charge on the dot is zero) to yield  the  charge  distribution on the dot the dot potential V m 5 and m' Since the dot is a  thin lamina and it is known that  charge  tends  to concentrate near  edges and pints, it may  be  expected that charge would  build up  along  the  dot  edges and particularly at the corners.  Thus the calculated charge distribution in  Figures  2 and  3 for an isolated dot,  one face being  normal to the incident SAW  beam,  is much  as expected.  The  dot  has  a  length parallel to the incident  SAW  of  a =  0.5A0  and  a width normal  to the  incident  SAW  of  b =  0.25A0  where  A  is the incident SAW  wavelength.  Figures 2  and  3 show the charge  distribution  at two instants when  the SAW potential  wave  lies  symnetrically under the dot and  antisymnetrically.  For similar dots within a 'waffle  iron'  array the charge is enhanced at the dot  ends  in the symmetric  case  and  suppressed  in the antisymmetric case due to coupling. 0 2.2  Scattered SAW  field calculation By  extending  the one dimensional analysis of Milsom  et a1 171 and  Morgan  [8] to two dimensions an  oscillatory point charge generates a  scattered SAW  field given by  the point SAW  Green's  function, G.  When  the  radiating  source  is,  however,  a charge distribution the scattered  SAW  potential is given by V(K)  =  E(_k).cr(_k)  (3) where  these  quantities  are  the two dimensional Fourier transforms of  the equivalent quantities as a  function  of distance.  Therefore,  at any point in  the SAW  radiation field,  the SAW  amplitude and phase  are  found  by  summing  the contributions from the  point  SAW  Green's  functions due to the point charges  within  each  rectangular cell of  the dot taking account of  relevant phase delays. The  forward scattered wave  is 900  out of phase relative  to  the  incident  wave.  This gives an overall  phase  shift after the wave  has traversed one  row of  dots.  A  similar phase shift occurs at each  row  resulting  in  an  effective  velocity reduction  in a waffle iron array.  The  fractional velocity  perturbation  is  given  in terms of the forward scattered  reflectivity r(  Oo  ) by 3.  Comparison of Model  Predictions with Experimental Waffle Iron Results (4) Matthaei and  Barman  [I]  have performed  several experiments on waffle iron configurations in which they  placed  rectangular dots on a  regular square lattice  of  periodicity  A0/2  (Figure  1) and investigated  the  velocity  perturbation and  180° reflectivity  as  they  varied  the dot  length,  a, parallel  to  the  propagation  direction  whilst maintaining  the  dot  width  normal  to  the propagation  direction  at  b  =  A  /4.  They  only considered frequencies resonant with the structure i.e. SAWS  of wavelength  A  =  A 0 0' The  waffle  iron  structure  has been treated using  the model  described in Section 2.  In order to take full account of  the electrostatic coupling between  dots  the  number  of  near  neighbour interactions  was increased until the inclusion of further  interactions  produced negligible change. In  the  worst  case  (nearly  touching dots) this required  20 near neighbours to be included. Following  Matthaei  and  Barman the  'Fractional 12 -  1985 ULTRASONICS SYMPOSIUM column metallisation', t, is defined to be t =  a/(a+g)  (5) Figure  4  shows  how  model  predictions  of  the variation  of  1800  reflectivity of a waffle iron dot  row  as the dot length,  a, is increased is in good  agreement  with the experiment.  Figure 5 also shows  similar  good  agreement between experiment and theory for the fractional velocity change. 4.  Design curves Figures  6  and  7  have been calculated for a waffle  iron array as described above.  The  curves show contours of  constant 180° reflectivity and  Oo forward  scattering as a  function of  dot length (a) and  dot  width  (b).  For  the  range  of  dot dimensions plotted,  interactions between dots were found  to be weak.  The  strongest interactions were found  between adjacent dots in the same column  so that,  compared  to  an isolated  dot,  the greatest difference  in reflectivity occurred at a  = 0.4h , b  =  0.4h.  For  this case the waffle iron had  a reflectivity  of  2.55% compared  with 2.25% for the isolated  dot  and velocity perturbation of  -1.43% compared  with  -1.62%.  Thus these curves can be used  as  universal  design  curves for dot  arrays with  a  A0/2  spacing parallel to the propagation direction  and  a spacing greater  than or equal to A0/2  along  the  rows  themselves  since  for the ranges  specified the interaction between adjacent dots  in  the  same  row  are  negligible.  (For normallization  purposes,  the  separation between dots  in  a  row has been taken as  lho.)  Figures 8 and 9  show similar curves for isolated dots over a much  wider range of  dimensions.  These  curves can therefore  be  used  to  design  sparse dot arrays where interact  ions are small. 5.  Discussion and Conclusions The  velocity  perturbation  and  180° reflectivity  of  thin rectangular metal  dots both when  isolated and  in a waffle iron array have been predicted,  and  predictions have been  found  to be in  very  good  agreement  with  the  experimental results  available.  This  provides  validation of the  theoretical  method.  Predictions  have then been extended to form a basis  for designing arrays with  chosen  properties  e.g.  reflectivity  as a function  of  dot  dimensions  for the purposes  Of weighting.  The  universal design curves presented are  valid  for  a  row  spacing  of  A  /2 which is suitable for resonators and  ILRACs. 0 During  fabrication  slight changes  in the dot dimensions  can occur as a result of  over or under etching.  When  this  occurs  both a  and b can be expected  to  decrease  or  increase  by  the same amount.  By  choosing the dot dimensions to lie on one of  the contours at the point where  the contour becomes  a  tangent  to  the  line  a  = b the dot reflectivity  will  be  most  insensitive  to fabrication  tolerances.  This  feature, which  can lead  to an improvement  in reproducibility, is not available for simple strip arrays. This  new  form  of  weighting can be  combined with  other  weighting  techniques  such  as  dot density 191,  phase weighting,  withdrawal weighting and row  length weighting  [lo]. New  forms  of  array  structure  can  also be accomodated  such as those using reflectors having alternate  polarity  reflection  coefficients recently  described  [ll].  To  date  grooves and ridges  or shorted and open metal  strips have been considered.  Figure  8  shows that by  choosing dot dimensions  suitably,  dots with both  positive and negative reflection coefficients can be  realised. 6.  References 1.  Matthaei,  G. L.  and  Barman,  F.,  'A  study of  the  Q  and Modes  of  SAW  resonators using metal "waffle-iron"  and strip arrays',  IEEE  Transactions on  Sonics and Ultrasonics,  Vol.  SU-25,  No.  3,  pp. 138-146,  May  1978. 2.  Woods,  R.  C.,  'Reflective  array compressors  using  180°  reflecting  metal  dot arrays',  J. Appl.  Phys.,  Vol.  54,  No. 11, November 1983. 3.  Chapnan,  R.  E.,  Chapnan,  R.  K.,  Morgan, D.  P.  and  Paige,  E.  G.  S.,  'In-Line  Reflective Array  Devices',  1978 IEEE  Ultrasonics Symposium, pp  728-733. 4.  Riha,  G.,  Stocker,  H.  R..  Veith,  R.  and Bulst,  W.  E., 'RAC-filters  with position weighted metallic  strip  arrays',  1982  IEEE  Ultrasonics Symposium,  pp  83-87. 5.  Solie,  L.  P.,  'Surface  acoustic wave reflective  dot  array  (RDA),  Applied  Physics Letters,  Vol.  28,  NO.  8, Pp 420-422,  1976 6.  HUang,  F.  and  Paige.  E.  G.  S., #Reflection  of  surface  acoustic  waves  by thin metal  dots',  1982  IEEE  Ultrasonics  Symposium Proceedings,  pp 72-82. 7.  Milsom,  R.  F,  Reilly,  N.  H.  C.  and Redwood,  M.  'Analysis of  generation and detection of  surface  and  bulk  acoustic  waves  by interdigital  transducers ,  IEEE  Transactions on Sonics  and  Ultrasonics,  Vol SU-24,  NO  3,  pp  147- 166,  May  1977. 8.  Morgan,  D. P.,  'Quasistatic analysis of generalised  SAW  transducers  using  the  Green's Function  method',  IEEE  Transactions on Sonics and Ultrasonics,  Vol  SU-27,  pp  111-123,  No  3, May 1980. 9.  Tanski,  W.  J.; Block,  M.  and vulcan,  A., 'High  performance  SAW  resonator  filters  for satellite  use',  1980  IEEE  Ultrasonics Symposium Proceedings,  pp 140-152. io.  Edmonson,  P.  J.;  Campbell,  C.  K. and Smith,  p.  M.,  "~row-band  SAW  filters  using stepped-resonators  with  tapered  gratings',  1984 IEEE Ultrasonics Symposium,  pp  235-238. 1985 ULTRASONICS SYMPOSIUM -  13 Huang  et al. 1.1 ' 1.0 . 0.1 ' 7.6 . 0.4. 0.1 , 0.0. 11.  Yamanouchi,  K.  and  Furuyaahiki,  H., 'Low-loss  SAW  filter  using  internal reflection types  of  new  single-phase  unidirectional transducer',  1984  I-  Ultrasonics Symposium; pp 68-71. -I Figure  I.  Waffle  iron reflective array. Figure  2.  Charge  distribution on a  dot with a  - 0.5X potentia?  symmetrical with respect  to dot. and b  - 0.25X0 - SAW Figure 3.  Charge  distribution on  a  dot with a  =  0.5X potentia9 antisymmetrical with respect to dot. and  b  - 0.25Ao  - SAW 1.0, 180'  Reflectivity of  a  dot row  as a percentage I  .o 0.0  0.1  1 .o L Figure 4.  180"  reflectivity as a  function of metallization for waffle  iron array; theory (-),  experiment  (x x). Figure  5.  Velocity change as a'funciton of metallization for a  waffle  iron array; theory  (--),  experiment  (x x) 14 -  1985 ULTRASONICS SYMPOSIUM Huang et al. bt). "b.I  0.2  0.3  0.4  0.9  0.6  0.7  I ./A  ./A 0.9  I Figure 6.  180°  reflectivity (percentage) as a  Figure 8.  180'  reflectivity (percentage) as a function of dot dimensions for a  function of dot dimension for a waffle iron array with row  period of  single row of dots. of  Xal2. b/  A *I  I Figure 7.  Percentage velocity reduction as  a function of dot dimensions for a waffle iron array with row period of ,4012. a/). Figure 9.  Percentage velocity reduction as  a function of dot dimensions for a single row of dots. 1985 ULTRASONICS SYMPOSIUM -  15 

A study of the Q and Modes of SAW resonators using metal &quot;waffle-iron&quot; and strip arrays',

Analysis of generation and detection of surface and bulk acoustic waves by interdigital transducers ,

High performance SAW resonator filters for satellite use',

In-Line Reflective Array Devices',

Low-loss SAW filter using internal reflection types of new single-phase unidirectional transducer',

Quasistatic analysis of generalised SAW transducers using the Green's Function method',

180 DEGREE REFLECTIVITY AND VELOCITY PERTURBATION OF THIN METAL DOTS.

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