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Radiation from coupled open-ended waveguides Driessen, Peter F. 1981-12-31

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RADIATION FROM COUPLED OPEN-ENDED  WAVEGUIDES  by  Peter F. D r i e s s e n B.Sc., U n i v e r s i t y of B r i t i s h  Columbia, 1976  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in the Department of Electrical  We accept t h i s to  THE  Engineering  t h e s i s as conforming  the r e q u i r e d standard  UNIVERSITY OF BRITISH COLUMBIA February 1981  ©  Peter Frank D r i e s s e n , 1981  In  presenting  requirements of  B r i t i s h  i t  freely  agree for  this f o r  an  a v a i l a b l e  that  I  understood  that  f i n a n c i a l  copying  her or  s h a l l  The 2075  U n i v e r s i t y Wesbrook  Vancouver, V6T  Date  DE-6  (2/79)  of  Electrical of  Canada  1W5  29 A p r i 1 1981  the  L i b r a r y  copying  p u b l i c a t i o n  the  be  allowed  Engineering Columbia  of  the  s h a l l  and study.  by  of  University  I of  this  It  this  without  make  further  head  representatives.  not  B r i t i s h  Place  at  granted  permission.  Department  f u l f i l m e n t  the  extensive  may be  by h i s or  gain  that  reference  f o r  purposes  or  degree  agree  f o r  permission  scholarly  i n p a r t i a l  advanced  Columbia,  department  for  thesis  thesis  o f my i s  thesis my  written  i i  ABSTRACT  Ray-optical between  open-ended  radiation plate  methods  methods larger  arrays,  of  with  require  finite  only  extensive  to  used  to  parallel-plate  patterns  waveguides  are  take  calculate  waveguides, arrays  the  central  ray  tracing,  into  the  account  of  as  coupling  well  coupled  guide  as  p a r a l l e l -  driven.  These  particularly  the  many  the  for  the  possible  ray  paths. The guides  coupling in  coefficients  isolation  previously  derived  groundplanes  etc.,  of  the  The  parallel-plate  the  five  aperture  staggered plane  array  agreed  well  presence a  general  coefficients  an  patterns  H-plane  to  of  the  and  three  with  those  other  lack  of  guides,  sensitivity  details  arrays  plane,  as  with  the  patterns  could  depth,  and  optical  methods  number may  antennas  of  for  outer the be  a  be  with  horn  to  Radiation  with  well  the  thus  compared  array.  element  with  of  were  sectoral  waveguide  well  variety  waveguide  the  two  of  the  structure.  using  and  both  remarkably  indicating  calculated  patterns  three  in  coupling  surrounding  agreed  between  a l l  as  patterns  of  not  outer useful  variety  of  by  in  the  both  edges  the  the  aperture A '  wide  width,  array.  development  applications.  in  element  patterns.  the  the  of  three  varying  guides in  a  in  experimental  obtained  simulate  waveguide  that  edges  experimental  Rayof  iii  TABLE OF CONTENTS  ABSTRACT  i i  LIST OF TABLES  v  LIST OF FIGURES  vi  ACKNOWLEDGEMENTS  x  1  1  I n t r o d u c t i o n And M o t i v a t i o n 1.1 1.2 1.3 1.4 1.5  2  3  5  6  1 3 8 10 12  Method Of A n a l y s i s  19  2.1 2.2 2.3 2.4  19 21 24 28  General D e s c r i p t i o n C a n o n i c a l Problems L i m i t a t i o n s Of The Method Summary  Coupling 3.1 3.2 3.3 3.4 3.5 3.6  4  Introduction L i t e r a t u r e Review Approach To The Problem Experimental Arrangement Summary  Between Two Adjacent  Waveguides  Formulation Single D i f f r a c t i o n Multiple Diffraction C a l c u l a t i o n Of The Coupling Numerical R e s u l t s Summary  Coupling  Between Separated  Coefficient  Guides  31 31 34 34 38 42 46 58  4.1 C a l c u l a t i o n Of The C o u p l i n g C o e f f i c i e n t 4.2 A n a l y t i c a l And Numerical R e s u l t s 4.3 Summary  58 64 68  R a d i a t i o n P a t t e r n Of A S i n g l e Guide  81  5.1 5.2 5.3 5.4 5.5  81 81 84 89 90  Introduction C a n o n i c a l Problems C a l c u l a t i o n Of R a d i a t i o n P a t t e r n Numerical And Experimental R e s u l t s Summary  R a d i a t i o n P a t t e r n Of Three Element Waveguide Array 6.1 I n t r o d u c t i o n 6.2 R a y - o p t i c a l Formulation 6.3 R a d i a t i o n P a t t e r n With Guides Of I n f i n i t e Depth  . 96 96 97 98  iv  6.4 6.5 6.6 7  8  9  10  R a d i a t i o n With Outer Guides Shorted Numerical And Experimental R e s u l t s Summary  R a d i a t i o n P a t t e r n Of F i v e Element  104 107 109  Waveguide Array ..  120  7.1 7.2  R a y - o p t i c a l Formulation R a d i a t i o n P a t t e r n With Outer Guides Of I n f i n i t e Depth 7.3 R a d i a t i o n With Outer Guides Shorted 7.4 Numerical And Experimental R e s u l t s 7.5 Summary  120  C o u p l i n g Between Staggered Guides  148  8.1 8.2 8.3 8.4  148 150 154 155  C a n o n i c a l Problem C a l c u l a t i o n Of The Coupling C o e f f i c i e n t Numerical R e s u l t s Summary  121 125 129 132  R a d i a t i o n From A Staggered P a r a l l e l P l a t e Waveguide  163  9.1 9.2 9.3 9.4  163 164 171 172  Formulation C a l c u l a t i o n Of R a d i a t i o n P a t t e r n Numerical And Experimental R e s u l t s Summary  R a d i a t i o n From Array  Multi-element  Staggered  Waveguide  184  10.1 I n t r o d u c t i o n 184 10.2 R a d i a t i o n With Outer Guides Of I n f i n i t e Depth . 185 10.3 R a d i a t i o n With Outer Guides Shorted 188 10.4 Numerical And Experimental R e s u l t s 192 10.5 Summary 194 11  General C o n c l u s i o n s And D i s c u s s i o n 11.1 11.2 11.3  Discussion Suggestions For F u r t h e r Work C o n c l u s i o n s In B r i e f  205 205 211 213  REFERENCES  215  APPENDICES  220  A B C D  S c a t t e r e d F i e l d s Along A Shadow Boundary 220 Higher Order D i f f r a c t i o n Terms For Coupling Between Adjacent P a r a l l e l P l a t e Waveguides i n The TEM Mode 222 C o n t i n u i t y Of F i e l d s Across Shadow Boundaries . 227 D e s c r i p t i o n of Antenna P a t t e r n Range 229  V  LIST  OF  TABLES  table  I  II  III  page  coupling  c o e f f i c i e n t  p a r a l l e l  plate  coupling  c o e f f i c i e n t  p a r a l l e l  plate  radiation  between  adjacent  waveguides between  waveguides  pattern  parameters  •  57  separated 80 147  vi LIST  OF  FIGURES  figure 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 2.1 2.2 2.3 2.4 2.5 3.1 3.2 3.3 3.4 3.5 3.6  page sector-shaped r a d i a t i o n p a t t e r n near optimum and t y p i c a l r a d i a t i o n p a t t e r n s for a feed antenna p a r a l l e l p l a t e waveguide a r r a y with d e s i r e d aperture f i e l d l i n e feed f o r a p a r a b o l i c c y l i n d e r r e f l e c t o r a r r a y of c o n c e n t r i c c i r c u l a r waveguides staggered a r r a y of p a r a l l e l p l a t e waveguides c r o s s - s e c t i o n of c o n c e n t r i c c i r c u l a r waveguide feed p a r a l l e l p l a t e waveguide two separated p a r a l l e l p l a t e waveguides staggered p a r a l l e l p l a t e waveguide f l a n g e d p a r a l l e l p l a t e waveguides two separated f l a n g e d p a r a l l e l p l a t e waveguides f i n i t e a r r a y of p a r a l l e l p l a t e waveguides embedded i n a groundplane i n f i n i t e a r r a y of p a r a l l e l p l a t e waveguides f i n i t e a r r a y of p a r a l l e l p l a t e waveguides embedded i n a simulated groundplane f i n i t e a r r a y of p a r a l l e l p l a t e waveguides with t h i c k w a l l s embedded i n a groundplane f i n i t e a r r a y of p a r a l l e l p l a t e waveguides in i s o l a t i o n N - f u r c a t e d waveguide d i f f r a c t e d rays two adjacent p a r a l l e l p l a t e waveguides two adjacent staggered p a r a l l e l p l a t e waveguides H-plane s e c t o r a l horn  13  i n c i d e n t and r e f l e c t e d shadow boundaries i n c i d e n t , r e f l e c t e d and d i f f r a c t e d f i e l d s l i n e source i n c i d e n t on a h a l f - p l a n e l i n e source i n c i d e n t on two p a r a l l e l half-planes l i n e source i n c i d e n t on two staggered half-planes  29 29 30  ray paths of mode f i e l d s i n two adjacent p a r a l l e l p l a t e waveguides ray paths from the d r i v e n guide to the p a r a s i t i c guide two s u c c e s s i v e rays i n a ray path from the d r i v e n guide t o a guide a p e r t u r e two s u c c e s s i v e rays i n a ray path, both rays in a guide a p e r t u r e two s u c c e s s i v e rays i n a ray path from the guide aperture to the p a r a s i t i c guide TEM-TEM c o u p l i n g between adjacent waveguides (a=d)  13 13 14 14 14 14 15 15 15 15 16 16 16 16 17 17 17 18 17 17 18  30 30 48 48 49 49 49 50  vii 3.7  TEM-TEM  coupling  between  adjacent  waveguides  coupling  between  adjacent  waveguides  (a=d/2) 3.8  TEM-TEM  52  (a=2d) 3.9  TE ,-TE|  coupling  between  adjacent  waveguides  (a=d) coupling coupling coupling coupling  c o e f f i c c o e f f i c at low at low  i e n t (a=d, N=0) i e n t (a=d, N=l) frequencies' frequencies  3.10 3.11 3.12 3.13  TEM-TMr, TEi-TE n TEM-TEM TEi-TEi  4.1  r a y p a t h s o f mode f i e l d s i n two s e p a r a t e d p a r a l l e l plate waveguides ray paths from the d r i v e n guide to the p a r a s i t i c guide TEM-TEM c o u p l i n g b e t w e e n s e p a r a t e d w a v e g u i d e s (a = b = 0 . 3 3 8 7 0 TEM-TEM c o u p l i n g b e t w e e n s e p a r a t e d w a v e g u i d e s (a=b=0.761X) TEi-TE / c o u p l ng b e t w e e n s e p a r a t e d w a v e g u i d e s (a=b=0.761X) TEM-TEM c o u p l n g b e t w e e n s e p a r a t e d w a v e g u i d e s (d=a=b) TEM-TEM c o u p l i n g b e t w e e n s e p a r a t e d w a v e g u i d e s (a=2d, b=d) TEM-TEM c o u p l : n g b e t w e e n s e p a r a t e d w a v e g u i d e s (a=d, b=2d) TEM-TEM c o u p l i n g b e t w e e n s e p a r a t e d w a v e g u : d e s  4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12  (a=d/2,b=d)  TE,-TE  | coupling wave  staggered plane  5.6  6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8  ray  paths  ray  path  normally p a r a l l e l  wave  staggered 5.1 5.2 5.3 5.4 5.5  between  separated  waveguides  (a=d=b) plane  51  normally p a r a l l e l in  (see  ray path (see continuity of ray-optical a a p a r a l l e l pl experimental a p a r a l l e l pl  a  incident  on  two n o n -  on  three  plates incident  non-  plates  p a r a l l e l  plate  waveguide  text) text) f i e l d s across nd exact radia ate waveguide and exact r a d i ate waveguide  shadow b o u n d a r y tion patterns for (a = 0.45"X) ation patterns for (a=0.45M  three element array of p a r a l l e l plate waveguides with outer guides shorted three element array of p a r a l l e l plate waveguides with outer guides of i n f i n i t e depth ray paths from c e n t r a l g u i d e s to edge 1 ray paths from c e n t r a l g u i d e s t o edge 3 ray paths from outer g u i d e s t o edge 1 ray paths from outer g u i d e to edge 3 radiation pattern of three element array with outer guides of i n f i n i t e depth (a=d=0.407X) radiation pattern of three element array  53 54 55 56 56  70 70 71 72 73 74 75 76 77 78 79 79 91 91 91 92 93 95  111 111 112 112 112 112 113  viii  a=d=0.450A s=0.856A A=0.131*0 r a d i a t i o n p a t t e r n of t h r e e element a r r a y a=d=0.339X s=0.645* A=0. 371 £. -171° r a d i a t i o n p a t t e r n of t h r e e element a r r a y a=d=0.356X s=0.677A A=0.308 ^-134° r a d i a t i o n p a t t e r n of t h r e e element a r r a y a=d=0.373X s=0.709A A=0.247 L +104° r a d i a t i o n p a t t e r n of t h r e e element a r r a y a=d=0.389X s=0.459X A=0.199^+38° r a d i a t i o n p a t t e r n of t h r e e element a r r a y a=d=0.44lA s = 0.597A A=0. 223 +166* U  6.9 6.10 6.11. 6.12 6.13 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12  f i v e element a r r a y of p a r a l l e l p l a t e waveguides w i t h o u t e r g u i d e s s h o r t e d f i v e element a r r a y of p a r a l l e l p l a t e waveguides w i t h o u t e r g u i d e s of i n f i n i t e depth r a y p a t h (see t e x t ) r a y p a t h s (see t e x t ) r a y p a t h s (see t e x t ) r a y p a t h s (see t e x t ) r a y p a t h s (see t e x t ) r a d i a t i o n p a t t e r n of f i v e element a r r a y w i t h o u t e r g u i d e s of i n f i n i t e depth d=a=b=0.450X r a d i a t i o n p a t t e r n of f i v e element a r r a y w i t h o u t e r g u i d e s of i n f i n i t e depth d=a=b=0.339X r a d i a t i o n p a t t e r n of f i v e element a r r a y w i t h o u t e r g u i d e s of i n f i n i t e depth d=a=b=0.450X comparison w i t h [38] r a d i a t i o n p a t t e r n of f i v e element a r r a y w i t h o u t e r g u i d e s of i n f i n i t e depth d f a = b=0.450X comparison w i t h [29] r a d i a t i o n p a t t e r n of f i v e element a r r a y d=a=b=0.441^ s,=0.838X s =0.597X A , = 0.137^ -12° A =0.074 ^-12° r a d i a t i o n p a t t e r n of f i v e element a r r a y d=a=b=0.407X s,=0.774* s = 0 . 5 5 l X z  7.13 7.14  2  A,=0.171 ^-55°  2  A =0.076 ^-57° 2  r a d i a t i o n p a t t e r n of f i v e element a r r a y d=a=b=0. 373 >s s,=0.709A s =0.505> A, =0.247 ^-103° A =0.096^ -98° r a d i a t i o n p a t t e r n of f i v e element a r r a y d=a=b=0.356X s,=0.677X S!=0.487X 2  7.15 7.16  2  A,=0.308 C -134° A =0.126^ -123° 2  a p e r t u r e f i e l d and r a d i a t i o n p a t t e r n of a f i v e element a r r a y A,=-0.215 A =-0.090 d=a = b=0. 450X S,=0.610X Si=0.356X a p e r t u r e f i e l d and r a d i a t i o n p a t t e r n of a f i v e element a r r a y A,=+0.131 A =+0.073 d=a = b=0. 450X s,=0. 8577\ s = 0.610X a p e r t u r e f i e l d and r a d i a t i o n p a t t e r n of a f i v e element a r r a y A,=-0.131 A =+0.045 d=a = b=0. 450A S,=0 . 857X S = 0 .356X 2  7.17  2  2  7.18  2  2  8.1 8.2  two a d j a c e n t s t a g g e r e d p a r a l l e l p l a t e waveguides l i n e source and p l a n e wave i n c i d e n t on a  114 115 116 117 118 119 134 134 135 135 135 136 136 137 138 139 139 140  141 142  '143 144 145 146 156  ix  8.3 8.4  half-plane ray paths for ray path (see  8.5  TEM-TEM  9.1 9.2 9.3  staggered p a r a l l e l plate waveguide ray paths for radiation pattern calculation shadow b o u n d a r i e s for a staggered p a r a l l e l plate waveguide  174 174  9.4 9.5  c o n t i n u i t y of f i e l d s a c r o s s shadow b o u n d a r i e s staggered p a r a l l e l plate waveguide radiation pattern of staggered waveguide ^=45° radiation pattern of staggered w a v e g u i d e ^=45° single and double d i f f r a c t i o n only radiation pattern of staggered waveguide for various stagger angles  175 176  9.6 9.7 9.8  10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8  coupling  10.10 10.11 10.12 10.13 10.14  156 157 157  c o e f f i c i e n t  between  staggered  array of staggered p a r a l l e l pl with outer guides shorted array of staggered p a r a l l e l pl with outer guides of i n f i n i t e r a y p a t h f o r c a l c u l a t i n g S, ray paths for calculating S ray paths for calculating S3 shadow b o u n d a r i e s f o r an a r r a y p a r a l l e l plate waveguides ray paths (see text) radiation pattern of staggered with outer guides of i n f i n i t e  ate  waveguides  V =45°  of  waveguide depth  array  '  waveguide  array  radiation  pattern  of  staggered  waveguide  array  radiation  pattern  of  staggered  waveguide  array  radiation  pattern  of  staggered  waveguide  array  radiation  pattern  of  staggered  waveguide  array  radiation  pattern  of  staqgered  waveguide  arrays  d=a=0.407* d = a = 0.373?\ d=a=0.339> d=a=0.450A  s=1.083>  s=1.000X s = 0.916A s=0.833A s=1.107A  ^=45° y=45°  1^ = 4 5 °  ^=45°  183  195 196 196 196  198  staggered  d=a=0.441>  182  197 197  of  Y = 45°  177  staggered  pattern  s = 1.166A  175  195 ate waveguides depth  radiation  d = a = 0.4747v  158  waveguides  3  d=a=0.450A  10.9  coupling text)  199 200 201 202 203 204  X  ACKNOWLEDGEMENTS  The  author  perceived thesis  this  and  work  is  thankful  to  investigation,  his  many  valuable  Dr. for  Edward his  comments  V.  J u l l ,  who  f i r s t  of  this  supervision and  suggestions  as  the  progressed. Thanks  the  sectoral  of  the  also  horn  antenna,  National  E l e c t r i c a l range,  are  to  to  Dave  to  Dr.  Research  Engineering  and  due  for  John  Fletcher J.Y.  Council  of  permission Hazell  Wong  to  for  for and  constructing W.  Canada use  Lavrench  Division  their  a s s i s t i n g  of  antenna with  the  measurements. The support Council  author of of  the Canada.  g r a t e f u l l y National  acknowledges  Sciences  and  the  Engineering  f i n a n c i a l Research  1 Chapter INTRODUCTION  AND MOTIVATION  INTRODUCTION  1.1  The  c a l c u l a t i o n  coupled  open-ended  of  whose  solution  arrays  with  waveguide  have  been  phased  used  in  arrays  amplitude  and  p a r a s i t i c dimensions.  phase  alter  radiation  the  ignored.  The  modified  by  p a r a s i t i c possible phased  array  ,  to  between  them.  For a  complexity  One as  with  a  power  the  many  of  each  and  a  each fixed  new  width  in  the  fixed the  is  parabolic  and  antenna the  which  array  depth"  w i l l  can of  than  by  the  range  of and  separately is  were be the  guide.  p a r a s i t i c  coupled  the  coupling  less  s h i f t e r s a  the  driven  adequate,  element  including  f i e l d s  and  somewhat  arrays  arrays,  the  central  of  antenna  a l t e r i n g  by  parasitic  a  basic  element,  of  phase  a p p l i c a t i o n  by  if  is  array  feeding  for  in  purposes  d i v i d e r s  antenna  changed  in  a  antenna  that  energy  element  design  parasitic  a  is  of  is  number,  p a r a s i t i c  possible feed  from  adjustment  the  excite  pattern  the  driven  be  and  w i l l  is  a p p l i c a t i o n s ,  pattern  phased  because  the  can  fields  waveguides  Waveguide  of  surrounding  of  in  f i e l d s  pattern  changing  range  associated  the  radiation  r e l a t i v e  added  of  elements  guides  available  pattern  both  between  assist  elements.  whose  coupling  electromagnetic  variety  whose  In  may  a  arrays  the  p a r a l l e l - p l a t e  problem  is  1  The  for  a  elements coupling adjustment thus  the  with  the  avoided.  waveguide  r e f l e c t o r .  For  array this  2  case,  a  sector-shaped  desirable, level  radiation  i . e . a p a t t e r n which p r o v i d e s  over  a  given  angular  ( F i g . 1 . 1 ) . A sector-shaped f o r space a t t e n u a t i o n ) reflector  surface  range  pattern  because  it  noise  from  ratio,  the  the  and  figure  Thus  of  merit  power  i s zero elsewhere (except  the  but does not  entire  spill  maximum d i r e c t i v i t y  i n t e r f e r e n c e pickup  ground.  usually  constant  illuminates  almost u n i f o r m l y  noise and  a  is  i s n e a r l y optimum  the s i d e s ( F i g . 1 . 2 ) . T h i s p r o v i d e s avoiding  pattern  the  i n c l u d i n g the  gain/noise  over while  thermal  temperature  for a s a t e l l i t e earth s t a t i o n  antenna, i s maximized. A p a r a l l e l - p l a t e waveguide a r r a y with a small number of elements ( F i g . 1 . 3 ) p o t e n t i a l l y s a t i s f i e s these for end  two-dimensional of the a r r a y may  f i e l d s . The  aperture  f i e l d at the open  be made to approximate a t r u n c a t e d  x)/x  curve by a d j u s t i n g the a r r a y parameters  and  number  of  parasitic  d r i v e n g u i d e ) . The transform  of  The  directly  cylinder  reflectors a  (Fig.1.5).  good p a t t e r n s [25].  by  depth  the c e n t r a l the  Fourier  f i e l d , w i l l then approximate  this  parallel-plate  to the design  guide  radiation pattern guides  surrounding  (sin  the  shape.  parameters of  as  (width,  r a d i a t i o n p a t t e r n , which i s  applied  used  guides  the aperture  required sector  requirements  array  of l i n e feeds  with Such  achieving  the  an  of  array  same  adjustment  concentric  of  also  be  sector-shaped  an antenna has produced  empirical  be  for parabolic  ( F i g . 1 . 4 ) . These parameters may in  may  the  circular reasonably parameters  3 Good  patterns  have  a l s o been obtained using an a r r a y  where the edges are not a l l i n the same plane, but staggered slightly  [13] ( F i g . 1 . 6 ) . Measurements of  various  feed antennas [15] has shown that there i s l i t t l e in  the  resulting  c i r c u l a r guides [15]  pattern  of  an  of  difference  a r r a y of staggered  (Fig.1.6) and a f o u r - r i n g c o r r u g a t e d s u r f a c e  (Fig.1.7) when both  amount  between  types  stagger  arrays  are  the  same  size.  The  i s an a d d i t i o n a l parameter which can be  a d j u s t e d to optimize the p a t t e r n . The design of waveguide feeds with has  been  array  of  parallel-plate of  optimum  f o r two dimensional  analysis  elements  mainly e m p i r i c a l to date. Thus the a n a l y s i s of an  development least  parasitic  could  be  waveguides feeds  for  may  assist  in  the  r e f l e c t o r antennas, at  s t r u c t u r e s . The  results  of  this  a p p l i e d not only to feed d e s i g n , but to  any coupled waveguide antenna.  1.2  LITERATURE REVIEW Many  different  waveguide  structures  types  of  open-ended  parallel-plate  have been analyzed using a v a r i e t y of  methods. The s t r u c t u r e s i n c l u d e i n f i n i t e a r r a y s of p a r a l l e l p l a t e waveguides, f i n i t e a r r a y s i n i s o l a t i o n or embedded a  groundplane  in  or other s u r f a c e ( F i g s . 1.8-1.17), and c l o s e d  r e g i o n problems i n c l u d i n g b i , t r i  and  N-furcated  waveguides  (Fig.1.18). The  fields  always be found  in  coupled  open-ended  r i g o r o u s l y except  for  waveguides cannot  certain  geometries,  4  because t h e r e s u l t i n g boundary v a l u e problem does not have a known c l o s e d form s o l u t i o n . Hence f o r many cases approximate methods have been- developed When  t o o b t a i n the f i e l d s .  r i g o r o u s s o l u t i o n s can be found, the methods used  i n c l u d e both the Wiener-Hopf techniques  [ 3 7 ] . These  technique  methods  can  or  infinite  matrix  mode  matching  a l s o be used  approximate s o l u t i o n s f o r those cases integral  and  where  equations  the  to f i n d  resulting  cannot  be s o l v e d  e x a c t l y . Approximate s o l u t i o n s a r e a l s o obtained u s i n g r a y optical  techniques  diffraction theoretic  based  on  [26], m o d i f i e d  techniques  the  geometrical  residue  calculus  theory  and  of  function  [38], and numerical techniques based on  moment methods [ 2 1 ] . The  choice  of method t o be a p p l i e d to a given problem  depends on the p a r t i c u l a r generally  limited  methods  are  suitable  dimensions  in  relatively  geometries,  for  methods  small s t r u c t u r e s .  (Fig.1.8) has been  The  the  and  Heins  be  used  literature  parallel used  plate  with  any  as  a  test  of  parallel-  in  isolation  below.  waveguide geometry  and r a y - o p t i c a l methods, p a r t i c u l a r l y  refinements of r a y - o p t i c a l t e c h n i q u e s . for  whose  p r i n c i p l e , but are g e n e r a l l y s u i t a b l e only f o r  single  numerical  are  ray-optical  structures  can  p l a t e waveguide s t r u c t u r e s i s reviewed A  methods  are on the order of a wavelength  or g r e a t e r , and numerical geometry  Rigorous  t o very s p e c i f i c  most  characteristic  geometry.  The  f o r exact, for several  exact  solution  r e f l e c t i o n c o e f f i c i e n t was given by Weinstein [45] [22] using  the  Wiener-Hopf  technique.  The  5  reflection and  c o e f f i c i e n t was found using moment methods by Wu  Chow [48] and was  [20].  Montgomery  and  residue c a l c u l u s . reflection  improved Chang  Rudduck  coefficient  by  Gardiol  Haldemann  [38] found i t using  and  Tsai  using  a  modified  [40] c a l c u l a t e d the  r a y - o p t i c a l procedure but  depending a l s o on a r e c i p r o c i t y argument. Keller  and  Yee,  Felsen  [50] found the r e f l e c t i o n c o e f f i c i e n t by t r a c i n g rays  directly  and  using  a ray-to-mode c o n v e r s i o n  improved t h e i r  s o l u t i o n f o r low  frequencies  [49].  [8]  out  Bowman  [9] p o i n t e d  that  formula. in  the exact Boersma  They  [18] and  the r a y - o p t i c a l  s o l u t i o n of [50] d i d not agree with the asymptotic  form  of  s o l u t i o n . The r a y - o p t i c a l s o l u t i o n was improved by [3]  Ahluwalia  [4] using  et  improvement asymptotic a  and  the  al.[l],  uniform  Boersma's  asymptotic  results  theory of  showed  much  i n the mode, t r a n s i t i o n regions and recovered the form of the exact  modified  s o l u t i o n . Lee [30] [31] d e r i v e d  diffraction  coefficient  which  includes  i n t e r a c t i o n s between the two h a l f - p l a n e edges a u t o m a t i c a l l y , so  that  these  interactions  c a l c u l a t e d . T h i s method reflection  theory  recovered  coefficient.  waveguides (Fig.1.9) by  waveguide  was  and  radiation  Tsai  methods  to  be  explicitly  s o l u t i o n f o r the  edge  [17] and  diffraction a l s o using  and Chang [ 3 8 ] .  pattern  given by Weinstein  the Wiener-Hopf technique. ray-optical  the exact  was c a l c u l a t e d using  Dybdal,Rudduck  exact  not  The c o u p l i n g between two separated  r e s i d u e c a l c u l u s by Montgomery The  need  a  parallel  plate  [45] and Heins [22] using  Yee and derive  of  a  Felsen  [19]  [49] used  r a d i a t i o n p a t t e r n which  6 agreed  well  plane.  Rudduck  aperture [30]  with  plane  the  exact  and  Wu  by  recovered  using  the  d i f f r a c t i o n  gain  calculated  The  f i e l d s  plate  waveguide  from  outside  [32]  using  modified for  a  a  the  line  spectral pattern by  a  surrounded  in  reflection an  waveguide  with by  Keller  using  his of [37]  [50]  modified the  using  Weiner-Hopf  by  a  and  this  of  f i e l d s  and  a  p a r a l l e l -  incident Lee  and  theory  it  Boersma the  the  f i e l d s  p a r a l l e l  plates  plane  wave  Mittra problem.  wedge  on  and  found  incident  [39] The  use  a  radiation  d i f f r a c t i o n  embedded  from  angle and  a  and  was  have  and  and  techniques,  by  The  by  a  flanged  (Fig.1.11b)  and  calculated  been  embedded  [40]  methods  also  waveguide  flange  Tsai  groundplane  methods.  (Fig.1.11a) of  a  ray-optical  single  c o e f f i c i e n t . was  in  waveguides  numerical  ray-optical  waveguide  of  staggered  using  groundplane  d i f f r a c t i o n  flanged  for  other  Rudduck  Lee  [42].  c o e f f i c i e n t  calculated  two  waveguides  arbitrary  the  method  aperture  also  considers  guide  by  his  normally  They  Rahmat-Samii  Rudduck  in  c o e f f i c i e n t .  asymptotic  on  also  rigorous,  i n f i n i t e  is  c o e f f i c i e n t .  plate  The  aperture  pattern  using  calculated  uniform  [5]  the  boundary  wave  were  approach  and  using  shadow  plane  staggered  Ryan  analyzed The  a  guide.  P a r a l l e l and/or  the  the  [24].  incident  Boersma  of  d i f f r a c t i o n  J u l l  the  domain  slope  solution  guide  source  staggered  found  when  d i f f r a c t i o n  (Fig.1.10). on  by  in  improved  c o e f f i c i e n t .  along  both  a  except  [41]  exact  modified were  solution  Yee, Lee  Felsen [30]  radiation  by Yee  Mittra and  was and  using pattern and  Lee  Felsen  [49]  7 using  ray-optical  d i f f r a c t i o n  coefficient  methods.  The  embedded  in  by  by  Lee  an  only  Harrington  the  and  with guide  the  Chang and  a l l  with  a l l  [36]  found  thick  walls  excited  both  only  Mittra  were  a l .  in  a  outer  a  array  in  for  the  [10]  pattern  groundplane  guides  in  excited  . Luzwick  radiation  and also  embedded guide  and and  Weiner-Hopf  excited.  embedded  and  residue  f i n i t e et  been  excited  one  a  Burnside  the  by  array  for  the  i n f i n i t e has  patterns  with  guides  and  the  and  technique  guides  f i n i t e  [38]  an  modified  with  by  adjacent  and  calculated  (Fig.1.15)  guides  calculated  between  were  a  such  was  using  for  moment  waveguides  Radiation  [29]  using  three  Weiner-Hopf  using  modified  and for  a  with  reactively  (Fig.1.16).  The ( b i , t r i  coupling and  calculated technique residue  array  excited  and  [35]  central  two  a  [48]  (Fig.1.13)  [38]  patterns  Lee  but  array  loaded  Chang  (Fig.1.17)  array  Chow  plate  using  techniques.  Montgomery  f i n i t e  [29]  groundplane  i s o l a t i o n  f i n i t e  p a r a l l e l  guide  by  simulated  The a  i n f i n i t e  one  calculated  same  both  Radiation  mode-matching  by  [47].  of  and  with  ,  in  Montgomery  for  between  and  coupling  Lee  [37]  Wu  using  Wu  by  calculus.  by  [30]  (Fig.1.12)  array  calculated  and  Lee  groundplane  guides  (Fig.1.14)  ,by  coupling a  numerically separated  methods  between  N-furcated by  and  Mittra by  calculus.  waveguides  waveguide) and  Montgomery  Lee and  [37] Chang  in  a  closed  (Fig.1.18)  has  using  the  [38]  using  region been  Weiner-Hopf modified  8  1.3  APPROACH  It array may  appears  in  the  waveguide  these  previous is  methods  structures  results  r e l a t i v e l y  and  require  large  size  Ray-optical  for  d i f f r a c t i o n  have  as  for  wave  d i f f r a c t i o n  plane  plane. 1.6  The  of  appear  several  to  be  Ray-optical  the  of  order  that  the the  f i e l d s  geometrical geometrical is  optics  rays  d i f f r a c t i o n asymptotic  travel  can  in  in  of  strike  the  of  a  the  basis  edges  exact  is  of  and  more.  (GTD)  for  for  of  solution h a l f and  r a y - o p t i c a l a n a l y s i s . solve  for  d i f f r a c t i n g  edges  the  structure methods  c a l l e d is  are  conductors  solution  so  to  introduced  the  theory  an  are assume  rays.  The  extension  d i f f r a c t i o n  rays  of  Fig.1.3  their  These  l i n e s  numerical  exact  and  of of  hand.  wavelengths.  of  for  at  conducting  arrays edges  these  because  the  used  straight  accounts  that  of  perfectly  presence  or  successfully  geometrical  be  d i f f r a c t e d the  terms  dimensions  c o e f f i c i e n t form  the  d i f f r a c t i o n  which  that  in  here  techniques.  problem  matrices  choice  wavelength  theory  postulated  incident  natural  characteristic a  array  half-plane  f i e l d s  of  large  by  the  geometry,  waveguide  methods  electromagnetic provided  a  expected  canonical  p a r a l l e l - p l a t e  consist  methods  their  ray-optical  for  waveguide  considered  be  this  using  be  analyzed  useful  the  methods  to  plate  been  may  very  of  1.6)  have  be  solution  p a r a l l e l  using  it  w i l l  exact  would  and  analyzed  methods,  no  small  (Fig.1.3  e f f e c t i v e l y  Similar  There  PROBLEM  that  i s o l a t i o n  be  using  TO THE  [26].  produced  It when  (Fig.1.19). based  d i f f r a c t i o n  of  on  A the  of  a  9 plane  wave  by  D i f f r a c t i o n is  is  a  a  local  determined  other to  parts  edge  l i k e  becomes  shadow. [28]  not  of  analysis  than  a  A  ray-optical  corresponding  to  d i f f i c u l t y ,  1.3  a  for  on  appears  a  straight  that  the  in  asymptotic  the  but  d i f f r a c t i o n  between  have  may  waves  been  with  l i m i t ,  light  and  advocated  some  extended  to  f i e l d  of  or  be  [1]  sacrifice  to  allow  incident are  is  in  on  f i e l d s  the  required  in  the  turn  open  f i e l d s  in  the  h a l f -  in  the  d i f f r a c t e d  central  driven  guide.  The  tracing  Incident  that  the end.  many  or  by  The  of  may  edge.  ray  paths by  guides these  be  traced source  and  f i e l d s  surrounding f i e l d s  The  arrays  the  waveguide  be The  d i f f r a c t e d .  d i f f r a c t e d  parasitic  magnitude  an  paths  waveguide  excited  central  ray  f i e l d s  from  reflected  Fields  outer  by  p a r a l l e l - p l a t e  require  down  excite  begin  d i f f r a c t e d  again  f i e l d s .  travel  may  f i e l d s .  be  1.6  the  the  been  extensions  surface,  and  at  have  the  analysis  represent  d i f f r a c t e d  dependent  f i e l d  which  boundaries  analysis  f i e l d s  transmitter  not  edge.  to  from  F i g s .  v a l i d  d i f f r a c t e d  half-plane  of  is  theories  These  another  ray-optical  edge  coefficient  rays.  GTD  plane  [6].  resulting  is  d i f f r a c t e d  of  the  methods  when  reflected  at  and  [7],  a p p l i c a t i o n .  general [50]  The  uniformly  this  Ray-optical  plane  in  uniform  overcome  alone,  source  half-plane  d i f f r a c t i o n  d i f f r a c t i n g  line  i n f i n i t e  s i m p l i c i t y  more  the  Various  to  edge  flaw  is  the  structure.  a  major  c o e f f i c i e n t and  the  conducting  effect;  the  from  looks One  by  of  emanate  perfectly  may  are w i l l the be  10 calculated.  These  be  at  reflected  end.  These  the  f i e l d s  pattern  the  The 1-  w i l l  shorted  d i f f r a c t e d  d i f f r a c t e d is  f i e l d s  coupling  end,  and  f i e l d s  the  process  of  a l l  these  thus  divides  between  two  down  the  outer  d i f f r a c t e d  w i l l  and  sum  analysis  travel  in  turn  continues.  d i f f r a c t e d naturally  adjacent  at  guides,  the  open  excite  The  more  radiation  f i e l d s .  into  several  p a r a l l e l - p l a t e  parts:  waveguides  (Fig.1.20) 2-  coupling  3-  r e f l e c t i o n  4-  radiation  5-  between  two  from  the  pattern  radiation  separated open  of  end  single  pattern  of  guides  of  a  (Fig.1.9)  single  guide  guide  (Fig.1.8)  (Fig.1.8)  multi-element  waveguide  array  (Fig.1.3) 6-  coupling  waveguides 7-  between  two  adjacent  staggered  end  single  p a r a l l e l - p l a t e  (Fig.1.20)  r e f l e c t i o n  from  open  of  staggered  guide  (Fig.1.10) 8-  radiation  9-  radiation  array  of  pattern  single of  staggered  guide  multi-element  (Fig,1.10)  staggered  waveguide  ( F i g . 1 . 6 ) .  Parts  1,2,3  6,7,  and  been  pattern  and 8  are  considered  4  are  needed  needed in  the  solutions  presented  improvements  to  these  to  to  solve  solve  9.  l i t e r a t u r e ,  5  and  Parts  s i m i l a r l y  2,3,4,7  however,  here  (parts  earlier  results.  2,  and  some 8)  parts 8  have  of  the  constitute  11  EXPERIMENTAL  1.4  It  would  comparing  be  numerical  c o e f f i c i e n t s  d i f f i c u l t  array  desirable  to  approximate  of  dimension  1.6)  of  the  waveguide  i s  sectoral  flared  a  is  large  i n one  a  two-dimensional  aperture.  Thus  the  H-plane  sectoral  horn  approximates  the  p a r a l l e l - p l a t e  outer The  angle  horn  of WR-90  and  of  in  same  used  3 0 ° with  with  performed  the  for an  waveguide.  various at  Appendix D.  TE  to  i f  in  Figs.1.3  If  a  of  an  and  H-plane  the  center  pattern a  the  rectangular  in  the  to  [17].  measured  radiation that  is  two-dimensional  form  in  and  structures  mode  0  Appendages  by  possible  a  page  1  obtained-  guide of  the  of  an  p a r a l l e l - p l a t e  the  horn  simulate  guides.  the  measurements  aperture The outer  depths.  X-band  E-plane  width.  of  T E Mmode  antenna  of  is  may be  dimension  by  coupling  enough.  (Fig.1.22),  waveguide  of  but  the  analysis  those  patterns  to  the  dimensional  waveguides  (perpendicular  horn  approximates  two  radiation  guides  to  accurately,  p a r a l l e l - p l a t e  third  confirm  Measurement of  perform  to  results  measurement.  r e f l e c t i o n  However,  very  the  experimental  very  ARRANGEMENT  of  guides  Radiation  on the  1.016  outdoor  had  a  x  cm,  50  were  pattern antenna  total  flare  a n d was f e d  1.016  cm  wide  measurements  were  range  described  12 1 . 5  SUMMARY  The of  calculation  p a r a l l e l  designing and  plate  appears  element  arrays  to to  into  compared H-plane  to  When by  experiment  of  the  analysis the  because  of  expected  to  waveguide  well  parts  however  suited  to  here.  The  which  The  are  the  the  numerical  ray-optical  three  considered  made  various  and  analysis  theoretical  measurements  the  complete depth,  p a r a l l e l  optimize A  wide  the be  is  width,  of  to  applications.  structures,  to  in  five  divides  in  turn  in  patterns  w i l l  be  with  a  wide  angle  horn.  elements  adjusted  analyze  been  arrays  useful  used  experimental  the  p o t e n t i a l l y  have  chapters.  sectoral  small  Exact,  considered  several  following  is  of  elements.  be  be  patterns  waveguide  waveguide  method  the  with  methods  plate  naturally  radiation  waveguides  antennas  ray-optical  p a r a l l e l  of  useful  antennas.  the  results  number  and  amount  plate  pattern  of  patterns  parameters for  waveguide  the  variety  many  and  the  of  of  array  for may  a v a i l a b l e .  design  v e r i f i e d stagger may  be  particular be  The  several  obtained  results  are  types  of  13  near optimum pattern typical pattern  F i g . 1.1 sector-shaped radiation pattern  F i g . 1.2 near optimum and t y p i c a l r a d i a t i o n f o r a feed antenna  F i g . 1.3 p a r a l l e l p l a t e waveguide a r r a y with d e s i r e d aperture f i e l d  patterns  14  F i g 1.4 l i n e feed f o r a p a r a b o l i c cylinder reflector  F i g . 1.5 array of concentric waveguides  circular  F i g . 1.6 staggered array of p l a t e waveguides  parallel  Fig. 1.7 c r o s s - s e c t i o n of concentric circular waveguide feed  parallel  Fig. plate  1.8 waveguide  F i g . 1.10 staggered p a r a l l e l plate waveguide  Fig. 1.9 two s e p a r a t e d p a r a l l e l p l a t e waveguides  Fig. l . H flanged p a r a l l e l p l a t e waveguides  16  I 1  F i g . 1.12 two separated f l a n g e d p a r a l l e l p l a t e waveguides F i g . 1.13 f i n i t e array of p a r a l l e l p l a t e waveguides embedded i n a groundplane  F i g . 1.15 f i n i t e array of p a r a l l e l p l a t e waveguides embedded i n a s i m u l a t e d groundplane  F i g . 1.14 i n f i n i t e a r r a y o^ p a r a l l e l p l a t e waveguides  17  Fig. 1.18 N-furcated waveguide  Fig. 1.16 f i n i t e array of p a r a l l e l p l a t e waveguides w i t h t h i c k w a l l s embedded i n a g r o u n d p l a n e  Fig. 1.20 two a d j a c e n t p a r a l l e l p l a t e waveguides  Fig. 1.17 f i n i t e array of p a r a l l e l p l a t e waveguides i n i s o l a t i o n  Fig. 1.21 two a d j a c e n t staggered p a r a l l e l p l a t e waveguides  F i g . 1.19 d i f f r a c t e d rays  19 Chapter METHOD  2.1  GENERAL  In to  chapter  calculate  be  the  analysis  obtained the  case  others of  analysis mode  in  plane  waves  the  the  at  d i f f r a c t e d  analysis  be  w i l l  and  some  are  based  parts  of  considered  compared  related  which  from  again  d i f f r a c t e d ,  the  In  adequate.  from  with  in  those  geometries,  and  be  to  and  edges,  in  practice These  same  the  compared  of  single guide the  is  in the  thus  only line  of  from  the  from  a  the few  source  into  open  l i n e  are  of the  source  in  f i e l d s  process  two  end  additional  turn line are  continues  d i f f r a c t i o n s f i e l d s  for the  edge a  f i e l d s  [50]  F i r s t  an  additional  p r i n c i p l e  in  decomposed  exciting  These  analysis  guide.  edges  source  the  procedure  emanate  line  edges.  part  kr>>l  to  These  other  f i e l d s  i n d e f i n i t e l y .  at  driven  appear  edge.  any  a  distances  f i e l d s the  for  d i f f r a c t At  of  essentially  incident  source  usually  or  the  w i l l  be  patterns  reflection  waveguide.  centred  1  of  of  results.  the  d i f f r a c t e d  Chapter w i l l  methods  each  used  patterns  general,  ray-optical  similar  following  of  in  procedure  radiation  in  chapters  results  ray-optical by  which  later  for  or  described  radiation  experimental  begins  be  mentioned  numerical by  The  In  analysis  c o e f f i c i e n t s  upon  discussed.  The  ray-optical  w i l l  problems  w i l l  turn.  the  coupling  antennas  canonical  ANALYSIS  DESCRIPTION  this  waveguide  OF  2  also  are excite  20 modes  i n  space.  the  outer  The  relative  complex  to  the  c o e f f i c i e n t f i e l d s the  at  can  be  coupling  ray  of  of  into  between  path  of  guides  start  traced  i s  each  or  in a  to  the  a l l  To  radiation  f i e l d s to  when  a  a  find  the  in  one  pattern,  the  and end up r a d i a t i n g  the  coupling  produces  start  into  ray  strikes  an  of  possible  ray  The number  only  source  these  path.  would  the  possible  but  line  corresponding  ray  guide  the  coupling  point  with  path  find  the  modes  array.  group  ray  To  a l l  distant  associated  i n d i v i d u a l l y .  i n f i n i t e ,  s i g n i f i c a n t l y  a  into  waveguide to  T h e sum o f at  radiate  these  waveguide  the  paths  also  corresponds  rays,  in another.  would  are  paths  rays  groups,  The alternate  edge  the  and  of  observed  sequence  and end  space.  as  pattern  divided  mode  guides.  t o t a l i t y  p a r t i c u l a r  guide  incident  edges  radiation  guides  amplitudes  between  the  The  parasitic  f i r s t  few  c o e f f i c i e n t s  contribute or  radiation  patterns. A l l  ray  mentioned  paths  above  contribution (Figs. and  3.2,  10.7).  rays  which  f i r s t . sequence ray rays  or  the i n  f i e l d ray  be  be  or  these  In  found.  can  7.3-7.7,  8.3,  by  this  way t h e  From  this  the  be  9.2,  the  w i l l  any  found  of  ray  from  f i e l d total  problems  and  their  calculated 10.3-10.5,  computations,  paths  be  of  w i l l  combinations  ray  then  each  e x p l i c i t l y  these  various  for  radiation  represented  path  i t .  traced  6.3-6.6,  f a c i l i t a t e  occur  preceding can  5.1,  by  The  considered  coupling  4.2,  represented  be  w i l l  to  To  to  the  f i e l d s  two  successive  be  calculated  in a  p a r t i c u l a r  .  the  f i e l d  represented f i e l d  of by  solution,  the a l l the  21 coupling This of  approach  two  the  coefficients gives  successive  following  solutions each  of  Chapter  2.2  rays  then  to  a  a  patterns  series  of  sequence,  as  they  used  be  derived.  d i f f r a c t i o n  problems  which  are  to  are  required.  assemble  waveguide  can  The  the  problems  l i s t e d  solved  in  canonical  solutions  to  at  of  the  end  1.  PROBLEMS  Ray-optical  methods  solution  for  have  plane  conducting  half-plane.  Sommerfeld  [43]  and  radiation  in  be  coupled  CANONICAL  exact  rise  sections  w i l l the  and  and  their  wave  This  was  as  d i f f r a c t i o n  problem  solved  canonical  was  more  by  basis a  o r i g i n a l l y  simply  by  the  perfectly solved  Clemmow  by [14]  others. TM  Consider  the  H  half-plane x  _ .  =  u.  x k  on is  = 2TI/A  it the  at free  suppressed,  where  presence  of  total the  a  —  f i e l d  - i k r COS(6-GQ)  arbitrary  space  equations  The  and  a  is  at  a l l  half-plane H  angle  6  wavenumber.  time  w = 2r,f f i e l d  ^  = e  an  subsequent  wave.  z>0  1  ^  incident  y=0,  is  factor the  In e  angular  points given  Q  ~  1 L ) t  (Fig.2.1), this is  (x,y,z)  in  where  and  a l l  implied  frequency space  ^  and  of  the  in  the  by  *  *  t  - u  t  - G(r,e ,e), 0  (2.2)  22  /•,/ « «\ e G ( r , 6 ,6) =  where  r - l k r c o s ( 9 - S ) r A~T~~ x 0\ i — — {e °'F[-/2kr c o s ( — ^ ) ] /n n  ±  i x e  00  F(a)  = J" a  = /T7 e  F(-a)  and  )  F  [  _ ^  c  o  s  ^ 0  (  )  ]  }  >  (  dx,  2  >  3  )  +  "  /  (2.A)  (2.5)  - (a),  4  F  signs  c a n be evaluated  the Fresnel  0  2 X  the top(bottom)  result of  1  c o s ( 6 - e  -ikr  g  refer  to  f o r kr>>l  TM(TE)  using  f i e l d s .  the asymptotic  This form  integral . 2 F(a)  to  „ i § ^ - ,  1,  a »  obtain  u  where  i s  u 8  f  e  -  i  k  = u  t  g  + D(6 ,6)E(r)  the geometrical r  c o s  e- i k r  ( - o) e  e  ±  ,  0  e  ~  i  k  r  optics  (2.7)  f i e l d  cos(e+e ) 0  f  cos(6-6 )  A  This of  6  asymptotic  waves,  e <  ^_e  o  n  i s the d i f f r a c t e d f i e l d +in/« e-e e+8 ( s e c ( - ^ ) ± s e c ( ~ ) }  the incident  plane  <  Q  (2.8)  0  E(r)- e ^ ' / ^ i  and  0  f o r Ti+e <6<2Ti  D(e , e ) E ( r ) D(9 ,6) = -  r  , f,o r H-6Q<6<H+6Q  u n  _0 and  o  given by  7  qiven  by (2.9)  .  < ' 2  result f i e l d and a  shows and a  that  the total  reflected  d i f f r a c t e d  f i e l d  f i e l d  f i e l d  which  which  i s a  1 0  i s made u p are  both  c y l i n d r i c a l  '  23  wave  ( F i g . 2 . 2 ) .  coefficient  0  giving  emanating  E(r) edge  thus  c a n be  D(e ,e) the  from  the  appears  interpreted  as  magnitude  of  half-plane  edge.  l i k e  a  l i n e  the  a  d i f f r a c t i o n  c y l i n d r i c a l The  source  of  wave  d i f f r a c t i n g  the  d i f f r a c t e d  f i e l d . Unfortunately uniformly  v a l i d ,  boundaries  asymptotic  and  becomes  between  the  (Fig.2.1).  e = fr ± d i f f r a c t e d used  the  to  f i e l d  predict  as  the  l i t  expression i n f i n i t e  and  However, a  line  along  shadowed  (2.7)  does  source  d i f f r a c t e d  (2.7)  at  f i e l d  the  not  shadow  regions  at  characterize  the  away  is  edge  from  the  a n d c a n be the  shadow  boundary. The  expression  introducing i n f i n i t y to  a  transition  at  e = n ± e  rewriting  edge.  Then  which  looks  for  To  (2.2)  the  predict  line  that  at  This  r=a  source  uniformly  precisely i s  observation  f i e l d  the  again  the  when  f i e l d s has  u =EO) f i e l d  C  f o r  v  v  a a  been i n  t  total  [49].  Q  be made  i n  point  cancels  fact at  v a l i d  by the  equivalent r=a  from  the  is  f i e l d  and i s  when  a  uniformly  v a l i d  .  d i f f r a c t e d  It  a  can  function  f o r an  total  l i k e  e  a l l  (2.7)  f i e l d s  second line  shown  the  time  source [7]  requires i s  that  presence  k(r +r)>>l  i s  0  of  an  d i f f r a c t e d f o r an a  by a  isotropic  by  /Q r  r  +  R  f i e l d  expression  half-plane  given  /k(r +r+R) 0  d i f f r a c t e d  2  i s f o r  half-plane. line  source  (Fig.2.3)  the  24  ikS  kr r  F[-2  /k(r +r+S)  cos(-y4]}  p  n  If  r>>r  0  (2.12)  and  t  G  U %  Note  that  by  (2.12) d i f f r a c t e d at  least  incident the  (2.13)  i s  before  by an  the  predicted  by  of  the  a  d i f f r a c t e d edge,  calculated Consider planes  a i n  of  wave  where  solution  (2.2)  a  l i n e  doubly This [24]  edge).  l i n e  If  f(6)E(r) i s  this  source  and edge  to  gives l i e s  on  i s  ,  where  the  other  f i e l d has  not on  f i e l d  f($)E(r)  approximation  second  d i f f r a c t e d  edge  d i f f r a c t e d  [50]  the  that  source  multiply  been  source  ( i f  f i r s t  the  predict  already  edge  l i n e  to  can been  be used  acceptable the  shadow  f i r s t .  O F T H E METHOD  another the  using l i n e  l i k e  from  d i f f r a c t e d by  has  the  the  used  which  isotropic  others  LIMITATIONS  When  be  d i f f r a c t i n g  (2.12).  except  boundary  f i r s t  looks  then  using  results,  plane  can  d i r e c t i o n edge,  successfully  the  A f i e l d  boundary  d i f f r a c t i n g  2.3  simply  on another  shadow  (2.13)  6 ) E ( r )  E(r)  f i e l d s .  approximated $  i s  and  once  r  becomes  0  (2.13)  m u l t i p l i e d  ( 'V  (2.12)  f i e l d edge  resultant this  source  F i g . 2 . 4 ,  near  where  edge  shadow cannot  l i n e  incident the  an  the  f i e l d  isotropic f i e l d  from  source on  i s  boundary be  again of  accurately  approximation.  two p a r a l l e l  observation  the  point  and  h a l f h a l f -  25 plane is  edges  that  and  are  the  is  c o l l i n e a r .  f i e l d  incident  i . e .  a l l  on  not  which  the  of  is  second  the  The  essence  d i f f r a c t e d  edge  is  not  a  of  the  d i f f i c u l t y  from  the  f i r s t  "ray  f i e l d "  [33],  form  ikf(r)  near at  the  the  shadow  f i r s t  The  ray  only  v a l i d the  source  by  edge  when  and  a  half-plane  the  mentioned  on  a  on  the  l i e is  cannot  accuracy  both  be  10  ray  so  f i e l d s  are  does  appear  to  d i f f r a c t e d  f i e l d  is  very  Thus  coupling  6)  and  (parts  the the  doubly  parts  l i n e  either. be of  are Thus  anisotropic  may  d i f f r a c t e d then  and  the  accurately a l l  potentially be  serious along  l i t t l e  this  [39]).  f i e l d .  accurate  the  focus  extensions  ray an  of  some  f i e l d ,  the  This  expected the  and  method  using  to  analysis  inaccurate.  the  unless  guides  radiation y i e l d  from  f i e l d source order  However, the  this t r i p l y  boundary,  the  f i n a l  1  single  acceptable  it  result.  (parts a  ray  d i f f r a c t e d  higher  shadow to  any  line  and  because,  in  d i f f r a c t e d the  t r i p l y  adjacent  should  f i e l d  doubly  t r i p l y  numerically  between  reflection  3,4,7,8)  of  of  i t s e l f  contributes for  not  a  boundary  that  d i f f r a c t e d not  is  with  boundary  uniform  d i f f r a c t i o n  boundary,  calculated  approximation,  f i e l d  is  parabola  shadow  shadow  and  shadow a  their  a  .  singly  not  and  [6]  the  page  the  here  for  within  along  incident  along  compromise  If  used  the  (i.e.  axis  solution  d i f f i c u l t y  f i e l d  boundary  methods  using  path  (2.14)  m  (ik)  edge  and guide  results.  26 However, 2)  the  for  the  t r i p l y  shape  of  coupling  and  the  between  higher  order  coupling  contribution  patterns  the  of  guide  to  the  multi-element  width  to  of  the  interactions  along  cumulative  because  the  if  incident  are  repeatedly  calculated  as  near  the  the  shadow  the  make  a  radiation  arrays  aperture  the  and  The  non-staggered  (part  affect  curve  result.  expected  errors  inaccurate  guides  f i e l d s  is  many  be  separated  d i f f r a c t e d  versus  s i g n i f i c a n t  two  (part  plane  5)  because  boundary.  d i f f r a c t e d  The  f i e l d s  f i e l d s  are  are  ray  f i e l d s . In  certain  expanding term  the  (non-ray)  f i e l d  applying  However, f i e l d s  in  the  two in  a  method  there  the  are  an  i n f i n i t e  half-plane turn  f i e l d .  an  d i f f r a c t e d  times  central f i e l d s .  on  a  summation  of  the  two  applying for  the  edge a l l  given  f i e l d  is  u  a  on  then  or  each  a  which  is  produces  two  d i f f r a c t e d  derived the  compared  f i e l d  half-plane  (n+1)  an  f i e l d s .  three  a  (n+1)  times  a 4  .  of much  Appendix  with  neatly  one  u  the  d i f f r a c t e d  d i f f r a c t e d  term-by-term  be  d i f f r a c t e d  f i e l d s  f i e l d  and  basis.  only  requires in  each  (2.14),  longer  times  the  by  h a l f - p l a n e s ,  times  in  with  term-by-term  no  produces  overcome  form  more  n  be  series  the  on  [50]  n  edge  a  of  sum  f i e l d  h a l f - p l a n e s , UAT to  wave  of  may  Taylor  can  case,  type  a  plane  contrast,  represent  incident  only  By  half-plane To  of  d i f f r a c t e d  d i f f r a c t e d n  in  three  aperture  by  d i f f i c u l t y  c y l i n d r i c a l  the  when  represented  when  this  representing  then  In  cases  B. (  is  basis. Ansatz  times case,  from  the  d i f f r a c t e d a l l  orders  more  complex  When  there  are  determined The and  by  expression leads  to  a  27 set  of  recurrence  process case  may  be  greatly  because  Ignoring  there  the  fact  mechanically  versus  as  B  exact  which  never  solution,  u-^.  are  cases  with  to  solved.  f i e l d s  high  [12]  the  of  also  in  and  coupling  random  known  the  and  orders  and  apparently  of  considered.  ray  very  This  half-plane  be  reflection  approaches  regardless  to  not to  an  be three  Ciarkowski  both  curves  the  f i e l d s  theory  by  must  in  f i e l d s  the  in  width  many  the  done  leads  guide  structure  are  that  was  which  complicated  applying  d i f f r a c t i o n Appendix  relations  shape  number  fine of  of  the  terms  considered. A  similar  d i f f i c u l t y  staggered.  Consider  staggered  p a r a l l e l  u.£ of  in the  these  The y i e l d there  and  plane  edges  be  i t .  is  (8.1)  plates  in  are  on  two  d i f f r a c t e d the  on  the  is  not  f i e l d  neighborhood incident v a l i d  or  and  the  expected  to  used. [50]  near  a  is  terms  edges,  away  of from  there  not  boundary  contribute  accuracy  if  thus  shadow  half-plane  boundary  the  incident  The  f i e l d  edge  [24]  acceptable  shadow on  The  f i e l d  ray  Hence  be  of  more  result.  to the  or  a the  boundary.  d i f f r a c t i o n  numerical  near  because  may  if  (Fig.2.5).  not  results  three  order  expected  is  approximation  are  source  plates  approach  accurate  higher the  edge,  shadow  term-by-term  l i n e  figures  second  reflected  a  occurs  this  and  along  which  t r i p l e  or  s i g n i f i c a n t l y approximation  the  shadow  are  only  to is  boundaries two  half-  28 2.4  SUMMARY  The waveguide The  ray-optical  analysis  of  any  parallel  plate  s t r u c t u r e r e q u i r e s that many ray paths be t r a c e d .  f i e l d s represented by these rays are c a l c u l a t e d  from the  c a n o n i c a l problems of a plane wave or a l i n e source  incident  on  a  half-plane.  calculations  This  method  shadow  boundaries.  l i n e source d i f f r a c t i o n succeeding  inaccurate  for  field  on or near shadow boundaries, which l i m i t s i t s  a p p l i c a t i o n to s i t u a t i o n s where along  is  chapters  v a r i o u s combinations  is  not  required  The s o l u t i o n s f o r plane wave and  by a  to  accuracy  find  half-plane  will  be  used  in  the f i e l d s represented by the  of two s u c c e s s i v e rays i n a  ray  path,  and thus the c o u p l i n g c o e f f i c i e n t s and r a d i a t i o n p a t t e r n s .  Fig. 2.1 i n c i d e n t and r e f l e c t e d  Fig. incident,  shadow b o u n d a r i e s  2.2  reflected  and  diffracted  fields  F i g . 2.3 l i n e s o u r c e i n c i d e n t on a half-plane  5  I  u  2  / u  2  6 F i g . 2.A l i n e s o u r c e i n c i d e n t on two p a r a l l e l h a l f - p l a n e s  F i g . 2.5 l i n e s o u r c e i n c i d e n t on two s t a g g e r e d h a l f - p l a n e s  31  Chapter 3 COUPLING BETWEEN TWO ADJACENT WAVEGUIDES  3.1 FORMULATION The  f i r s t problem t o be s o l v e d i s t h a t of t h e c o u p l i n g  between  two  adjacent  p a r a l l e l p l a t e waveguides  The r a y p a t h s from t h e lower parasitic  guide  driven  guide  (Fig.3.1).  t o t h e upper  a r e d e t e r m i n e d by i n s p e c t i o n and shown i n  Fig.3.2 . Consider waveguides  two  adjacent  consisting  of  semi-infinite the t h r e e  parallel  plate  p e r f e c t l y conducting  p a r a l l e l h a l f p l a n e s i n z>0 a t y=-d, y=0, y=a ( F i g . 3 . 1 ) .  We  w i s h t o determine t h e f i e l d c o u p l e d from t h e d r i v e n guide (d<y<0;  z>0)  incident  \  into  the p a r a s i t i c  guide  (0<y<a; z>0) . An  field  = e"  l  i k  N  Z  c  (3.1)  ( ^ ) sin d o  s  x  i n t h e d r i v e n guide w i l l e x c i t e f i e l d s of t h e form H  C  X E  c  _  =  ,  n^O ^ n  6  +11^2 COS-nTTV. sin a (  (3.2)  }  x  i n t h e p a r a s i t i c g u i d e . The c o u p l i n g  coefficient  the  of t h e n t h mode i n t h e  relative  amplitude  and  phase  p a r a s i t i c guide t o t h e N t h mode i n t h e i n c i d e n t  is  guide  at  32  z=0.  The propagation c o n s t a n t s ^  - /k -(,N/d) 2  kn - / ^ - ( r r n / a )  k N  »  are given by  k n  < '  2  3  ^ >  2  Here we use the r a y - o p t i c a l method of Yee, F e l s e n and [50] to f i n d  A  3 )  Keller  .  Following  [50]  the i n c i d e n t  field  (3.1) i s decomposed  i n t o two plane waves H  I  x  £  .  = .  =1_  u  { e  -i(k z-Nrry/d) N  +  x e  -i(k z+NTry/d) N  }  (  3  r  )  x  traveling  i n the d i r e c t i o n s  TI ± e  relative  to  h a l f plane guide boundaries, where sin  *  {  ,1 TM f i e l d s i TE f i e l d s  7 >  '  U  = q .  term  direction  11  in  ~ N 6  (3.5) i s a plane wave t r a v e l i n g  i n the  which has the value  ±k  u  -  =  (3 9)  at the edge y=0, direction  ;  (3-8)  2  first  the (3.6)  6„ = N-n/kd , N  =  T  The  >  z=0 The other term, a  n + Q  plane  wave  in  the  has the value  N  (-D T  (3.10)  N  U  i  =  ~~2^~  at the edge y=-d, by the i n c i d e n t The  u. d  These two plane waves are r e p r e s e n t e d  rays i n the lower guide of F i g . 3.1.  fields  guide appear  z=0.  excited  to o r i g i n a t e  f(6)E(r)  by d i f f r a c t i o n i n t o the p a r a s i t i c from l i n e sources of the  form (3.11)  33  located  a t t h e e d g e s z = 0 ; y=0  distance the  from  diffracted  excites  field.  in  In  (3.11)  r  i s the radiation  Radiation  the p a r a s i t i c  in  guide  in  the  z>0;  i s  pattern  direction  0<y<a.  due t o a l i n e  the  The  of  0 - 9r\ fields  source  (3.11)  one o f i t s edges a r e [ 5 0 ] j—  c c F  = u  d  x  c  =  I  ,  n=0  ,1 n " 2  y  /2k a I n ifn = 0 ifn i 0 '  {  coupling  of  (3.12)  employed refers  i  x u  under  a line  incident i s  immediately  a  linesource  chapter,  the following  x  of the f i g u r e  notations are  the  superscript  showing  the  x  particular  consideration. wave  i n the driven  source u  line u  the  n  letter  field  m  yields  A^ .  this  the  i s a plane is  (3.2)  t o r e p r e s e n t t h e f i e l d s , where to  path  with  coefficient  Throughout  ray  (3.13)  = 0 or a .  Q  Comparison  u  a.  f(6)  t h e n t h w a v e g u i d e mode  excited at  t h e edge and  or  .  m-l  field  (ie  ±  source (ie u  x  m  guide.  arising  from  i s a singly field  arising  i s an m  times  diffraction diffracted from  of field).  diffraction  multiply ^  -  1  the  of a  diffracted  field). A ^ ^ i s  the contribution  diffracted  j times.  t o A^  n  of (3.2) a r i s i n g  from  fields  34  3.2  SINGLE  The  DIFFRACTION  coupling  calculated by  a  half  f i e l d an  from  in  angle  from  the  plane. the  solution  One o f  driven  Q  guide  edge,  is  single  incident  f i e l d  wave  waves  comprising  the  on  edge  i s  of  the  a  y=0,z=0  At distances  given  e  sin  we  is  n  given  kr>>l  (3.14)  v  D ( 6 ,e)  at  by  uf = D(2n-e ,e ) E ( r ) , 1 2q N n where  is  plane  F i g . 3 . 2 a ) .  d i f f r a c t e d  d i f f r a c t i o n  d i f f r a c t i o n  two p l a n e  (see  the  due to  for  the  = 2n -  Q  this  contribution  by  (2.9).  Substituting  = nti/ka  (  3  >  1  5  )  find  "  The  /^K , ,  1  a l  "  2  F  i  ^  E  radiation  comparison  pattern  with  contribution d i f f r a c t i o n  to only  MULTIPLE  The multiple  r  f(s)  (3.11). the  >  -  of  (3.16)  is  this  into  Putting  coupling  coefficient  of  obtained  by  (3.12),  the  (3.2)  for  single  is  /k+Tk._  /-.N  (1) _ n / ^Sln " 4k a n  3.3  <  W  /k+ik  N •  k  n  (3.17)  +k  N  n  DIFFRACTION  contributions d i f f r a c t i o n  to are  the  coupling  obtained  coefficient  separately  for  each  for ray  35 path of  i n F i g . 3 . 2 . two  then  successive  pieced  f i e l d s  i n each  3.4  In  the  the  and  u  l  C  N  i s ,  =  i  u  l  N  C  and the w i l l  Q  = e  e = j  from  T  (  sequence  the  be  plane  at 6  direction  edge  i n a  d i f f r a c t i o n w i l l  coupling total  be  problems  solved,  contributed  coupling.  considered  and  by the  The pairs  are  of  shown  in  3.5. a  edge  individual  find  which  F i g . 3 . 3  half-plane  to  ray path  rays  F i g s . 3 . 3 ,  the  rays  together  successive  in  F i r s t  u  wave  or  or  e  = 2TT -  0  6 = ~  at  is  incident  .  The d i f f r a c t e d  a  point  r=a  on  a  f i e l d  away  from  (2.11)  a  >  )  (3.18)  where (  a  )  V  =  ) / E ( a )  a  -in/4  C (a)  =  N  1  —  (3.19)  , . /' 2 , 2 " N *  + {  ,  e  k  r  k  F  [  / |  (  ^  _  AZ^  )]  V TT  + ie  1  Y  k  k  N  3  F [ / | (v^k+k^ + /k+k^") ]}  (3.20)  and  T  For  l  1  k a » l  U  l  =  r-' 1  i f |e-6o|<~/2 otherwise  (3.18)  U  i  T  l  C  N  E  s i m p l i f i e s  (  r  )  (3.21)  -  to  >  (3-22)  where +i-/4 C  /k  /k-tTk  ~ N  In  *  Fig.3.4  .  (3.23)  k  /2T  an  isotropic  line  source  u.  w  _ , - i= (r)  36  located that  a t S, a d i s t a n c e r 0  same.  o r in fx  e = TT/X  The  Consider  cos(-  2  t h e two c a s e s  cos(—f)  ik(r-r ) 0  from  r>r„  field  i nthe  (2.12).  and r < r . 0  If  R=r-d,  e  -i,/4  = 0  r>r  0  S=r+d,  e  to find  ik(r-r ) Q  f i  0  -ITI/4  ik(r -r)  r /T.  ) ATQ  K— » kr x  Q  i  i s s u b t r a c t e d out, both  = + \ E(r+r ) 2 0 n  This f i e l d original field  t  (3.24)  Q  0  F[/2kT] + 1 E(r+r ) ' 2 " 0' l  J  and kr>>l, and the f i e l d  kr >>i  source  _ + i E(r+r ) .  , R = d - r , S=r+d, a n d  ik(r -r)  m  _ F[/2kr ] Q  u = "m  2  V  from  by  ,  l  n  .  (3.25)  the incident  line  E(r ) E(r) .  ^  U  source  diffracted  1  ( 3 . 2 4 ) a n d ( 3 . 2 5 ) become (3.26)  n  h a s two c o m p o n e n t s : t h e f i e l d line  originate of  total  (2.12):  =  r<r  u  i s found  in  = 1 e  If  The  e+e  e  u  .  i s i n c i d e n t on  r e s u l t s f o r F i g s . 3 . 4 a a n d 3.4c a r e f o u n d t o be t h e  substitute  If  o r ITT/X  edge a t e> ^ T / I  directions  from a g u i d e w a l l  0  reduced  i n amplitude  t h e edge.  These  a t d i f f e r e n t p o i n t s i nspace,  image o f t h e o r i g i n a l  o f t h e image o f t h e by h a l f , p l u s a  two l i n e  sources  one a t t h e l o c a t i o n  l i n e s o u r c e a t d i s t a n c e r„ f r o m t h e  edge, a n d t h e o t h e r a t t h e edge. The  r e s u l t s f o r F i g s . 3 . 4 b a n d 3.4d a l s o a r e f o u n d  t h e same. C o n s i d e r t h e t w o c a s e s substitute cos(-/) =  in o , c o s ( - / ) = -1  r>r  0  (2.12): to find  and r < r . 0  R=r+d,  t o be  If  r>r  0  S=r-d,  37  ,  1  U  m  2  E(r+r  =  0  a " "  7  4  i k ( r _ r  7=  e  } + T  r<r  u M  If  2  )  n  0  /T7  /krT^  and kr>>l both  Q  = i E(r+r ) + T 2  .  (3.28)  (3.27) and (3.28) become (3.29)  2/2T  0  0  This f i e l d field  (3.27)  E(r.)E(r).  n  m  .  F[/2k7]  T  +  0  kr >>i u  ik(r -r)  -iTt/4  4E(r+r  =  nfik^l  R=r+d, S=d-r, and  ,  Q  ,  /kr  /TT  If  0)  has two  reduced  components:  i n amplitude  the o r i g i n a l  by h a l f , p l u s a f i e l d  by the edge. These two l i n e sources o r i g i n a t e points  in  space,  one a t the l o c a t i o n  source a t d i s t a n c e r  line  diffracted  at  different  of the o r i g i n a l  from the edge-, and the other  c  source  line  at the  edge. In u  m-l  is  =  £  Fig.3.5a  ( )  incident  This f i e l d  on  that  an  isotropic  a  0  line  source  from the  edge  edge at an angle e = TT/2 or 3TT/2 Q  i sdiffracted  0 =e  u  3.5b  l o c a t e d a t S, a t a d i s t a n c e r  r  angle  and  or 2r, - e  n  i n t o a waveguide of width a . From  a t an  (3.18), the d i f f r a c t e d  field  i s , f o r kr>>l m U  m  =  where  l n 0 (r>>  T  C  c  ( r n  ( r  o  (3.30)  ) E  )  i  9  s  i v e n  b  Y  (3.20).  For  k r > > l (3.30) 0  s i m p l i f i e s to U  m  =  T  ln ( V C  E  E  (  r  )  •  (3.31)  38  3.4  CALCULATION  The  f i e l d s  successive  the  waveguides.  and  The  For  each  The  of  In  is  l  for  w i l l  of  now b e  adjacent  these  possible  located  (m-1)  two  used  to  p a r a l l e l - p l a t e  calculations,  combinations  guide  of  shown  rays  in  up  and one c o m b i n a t i o n  F i g . 3.2b  ( a ) E ( r )  at  the  d i f f r a c t e d  at  excited  by  and  f i e l d s  at  be  the  to with  the  ray  i s  from  the  modes  in  calculated  (3.2).  are  edge  d i f f r a c t e d  of  last  represented  incident the  f i e l d s  calculated  The amplitude  the  the  Fig.3.2, can  (3.12)  d i f f r a c t e d  laf N  in  ray  ray.  (3.11),  y=a,z=0,  C  rays  mth  d i f f r a c t e d  at  =  of  the  parasitic  and c.  U  two  d i f f r a c t i o n ,  sequence  doubly  observed  by  f i e l d  ,  y=z=0.  From  f i e l d  is  a  Figures  given  by  (3.18),  as  line  source  <' ) 3  d i f f r a c t i n g the  edge  edge.  y=a,  z=0,  This  f i e l d  and from  is  (3.30)  32  again gives  a  source  U  located  2  =  laf C ' ( a ) C ( a ) E ( r )  (3.33)  n  at  f i e l d z=0.  a l l  u ^ ^ o f the  (3.9),  paths  combinations  above  between  t r i p l e  comparison  line  ray  with  upper  3.2b  by v a r i o u s  d i f f r a c t i o n .  associated  the  COEFFICIENT  presented  coupling  including  f i e l d s  as  represent  quadruple  by  represented  rays  calculate  F i g . 3 . 2 ,  OF T H E COUPLING  , From  the  d i f f r a c t i n g  edge.  given  by  (3.10),  (3.18),  as  observed  i s at  In  F i g . 3.2c  d i f f r a c t e d y=z=0,  the  at  the the  incident edge  y=-d,  d i f f r a c t e d  f i e l d  39 is  a  line C 1  source  =  c  , N  :  2q  located  at  the  diffracted  at  (  d  )  E  (  r  )  (3.34)  d i f f r a c t i n g the  edge  edge.  y=z=0,  This  and from  f i e l d  (3.30)  is  gives  again a  line  source  2  U  located the  Z  =  ^ - ^(d)C (d)E(r)  i  at  the  d i f f r a c t i n g  coupling  by t r e a t i n g  using  (3.11),  N  n  (  2  = ^  )  The 3.2d,  e  (3.10), observed  U  i  d  n  [ ( - D  4  n  C ^ ) C  diffracted resultant  d  £  2  2d,  is  at  z=0,  diffracted  and  Thus  N  fields  .  n  the  the  are  the  (3.36)  represented  incident  f i e l d  y=z=0.  From  edge  diffracted  f i e l d  i s  a  by ,  Figures given  E  by  (3.18), line  as  source  3  d i f f r a c t i n g  the  N< > (  made  is  < - >  edge  edge.  y=-d,  This  z=0,  and  f i e l d from  is  37  again  (3.26)  the  is  d + r  >  +  A q T=J  C  N  (  d  )  E  (  d  )  E  (  r  )  •  ° '  3  8  )  V 2 TT  4  u  sources  r  the  d  line  N( > ( >  f i e l d  C  at  y=-d,z=0,  at  diffraction  + (-D C^(d)C (d)]  ( a )  F i g . 3.2d  diffracted  C  as  to  a  In  f ?  n  contribution  double  (3.35)  and (3.2).  and f.  =  ' £  7  and  diffracted  at  The t o t a l from  t r i p l y  located  4  (3.33)  (3.12)  y 2/2k  is  edge.  coefficient  found  A  (3.35)  n  up of the at  two l i n e other  the  edge  at  source y=~d,  y=z=0,  f i e l d s , z=0.  and from  one c e n t e r e d  at  This  f i e l d  is  (3.30)  gives  two  y=-  again line  40 sources +in/A  3 " fqj C * ( d ) C ( 2 d ) E ( r ) + ^  U  S —  n  located  at  the d i f f r a c t i n g  f i e l d  i s  (3.32).  This  (3.26)  the  given f i e l d  resultant  '<d)C (d)E(d)E(r)  (3.9)  again f i e l d  (3.39)  n  edge.  by  is  C  Fig.3.2e  In  and the  diffracted  the  f i e l d at  incident  i s  y=a,  z=0  given  by  and  from  i s  +iir/A  "2  = " i is  y=2a,  C^(a)E(a+r) -  -  made z=0,  the  d i f f r a c t e d  3  In  at  4i  =  f i e l d  2  other y=z=0  at  =  source  y=a,  and from  (3.40)  fields,  z=0.  one centered  This  (3.30)  the  f i e l d  is  resultant  at  again  f i e l d  i s  +iTi/4  + ^  n  ^  u  y=z=0  u  two l i n e  C'(a)C (2a)E(r)  Fig.3.2f  at  of  .  C^(a)E(a)E(r)  -1  e  U  up  j-  the  i s  incident-  given  by  and from  " V "N C  -  I  1  C^(a)C (a)E(a)E(r) n  f i e l d  (3.34).  (3.29)  ( d ) E ( d + r )  -  i s This  the  given  f i e l d  resultant  by  is  .  (3.41)  (3.10)  again  f i e l d  and the  diffracted  is  Z r ~ ^ 7 = — C^(d)E(d)E(r) .  (3.42)  /27T U  i s  2  made  up of  two l i n e  source  d,  z=0,  the  other  at  y=z=0.  This  at  y=a,  z=0  and from  (3.30)  the  u  f 3  The  =  n  (3.43)  N  contribution  d i f f r a c t i o n as  f i e l d  +W4  line  is  +  to  found  sources  the  one c e n t e r e d is  again  resultant  <  C'(d)C (d+a)E(r) n  total  t r i p l e  Thus  TC  fields,  C^(d)C  coupling  by t r e a t i n g  and using  f i e l d  y=-  diffracted is  (3.43)  (a)E(d)E(r).  coefficient (3.39),  (3.11),  at  (3.12)  A  N n  (3.41) and  from and  (3.2).  41 +iTr/4  r V  ^  =  3  - N  [  C  n  ( d ) C  " S  ( 2 d )  n  ( a ) C  ( 2 a )  (-D  +  N + n  C'(d)C (d a)] n  +  n  + g ^ - [CjJ(d)C (d)E(d) + C^(a)C (a)E(a) + (-1) n n  N + n  n  CjJ(d)C (a)E(d) ] . n  (3.44) A  quadruply  d i f f r a c t e d  f i e l d  i s  represented  i n  g Fig.3.2g. f i e l d s y=0,  Here  in  (3.38)  z=0,  source  u  so  f i e l d s  i s  2  are  given  d i f f r a c t e d  that  from  centered  by  (3.38).  independently  (3.29),  at  Both  line  by  consists  of  ( - 2 d , 0 ) , ( 0 , 0 ) , ( - d , 0 )  source  the  edge  four  line  and  (0,0)  g respectively.  which  Oik'^^ ) 1  the  Note  asymptotic  solution.  If  consistency Keller  that  i s  of  terms  of  d i f f r a c t i o n  not  theory  expected  used  contribute From by  only  number  of  here. very  (3.26) half  order  It  s i g n i f i c a n t l y  to  turns  arising  the  be  order  the  that  i s  3  in  d i f f r a c t i o n i n  terms  added  be  and  &  then  N n  to  the  also  the  order  within  u  term  included  Thus  meaningful  i n  second  plane  and higher  out  f i e l d s  (2.9)  (3.31).  quadruple be  as  half  to  higher  in  0(k' ) lfl  d i f f r a c t i o n  the  ray-optical  these  terms  numerically. the  d i f f r a c t e d  d i f f r a c t i o n  the  l  and (3.29)  the  order  0(ic' ^)are  and  l i t t l e  grows  but only  same  these  coefficient  to  when  terms  d i f f r a c t i o n order,  due to  of  of  that  (3.23) , (3.26) , (3.29)  are  the  expansion  requires  contributions  one  from  f i e l d  along  amplitudes  the  shadow  quadruple  and  rapidly  for  each  0{u" )  and  O  be  expected  /x  may value  of  the  are  boundary. higher  successively  coupling  terms to  reduced  from  The order  higher higher  contribute  c o e f f i c i e n t .  42  Contributions  Oik'' ),  calculated  12  ( =l,  N=0,  T  3.5  to  n=0),  from  for  are  given  NUMERICAL  RESULTS  Numerical  values  were  calculated  and  TE polarizations.  double,  and t r i p l e  The mode  in  guide, a=d,  is  shown  i n  coupling  from  the  both  of  d i f f r a c t i o n  the When  width  a  and d  to  A^  for  n  both TM  due to  single,  included. ,the  coupling  mode  in  and 3.8  Values  F i g . 3 . 9 .  of  and  from  the for  of  a TEM  parasitic  guide  widths  (TE, - T E/  ft,,  Figs.3.10  and 3.11  modes  into  guides  are  terms  reveals  the  the  a  of  shape  the  show higher  the  T E Mc a s e  driven  guide  various  the  with  the  cutoff  the  double near  The  to  widths,  same  average  minima  modes.  correction  the  the  Adding  structure  some  near of  provides  c o e f f i c i e n t . fine  provide  ,especially  in  term  coupling  widths  basic  parasitic  d i f f r a c t i o n  the  cutoff  structure  driven  terms  d i f f r a c t i o n  alter  3.7  respectively. in  0 0  TEM  F i g . 3 . 6 ,  coefficient  TM a n d T E f u n d a m e n t a l  single  behaviour  the  a  A  coupling  modes. When  width  to  plotted the  of  TEM-TEM  of  B.  widths  were  d i f f r a c t i o n  of  coupling  guide  and phase  a n d a=2d  case  Contributions  guide  are  the  various  driven  order  in Appendix  for  the  a=d/2  special  d i f f r a c t i o n  amplitude  coupling)  order  for  the  higher  t r i p l e  this but  fine do  not  curve. parasitic  (a=d/2)  guide  (Fig.3.7)  is the  half  the  minima  near  43 the  mode  transition  parasitic (a=2d) for  guide  is  (Fig.3.8)  the  both  correction  compared  the  Exact not  case  the  to  r e f l e c t i o n  single  [50]  with  the  as  small  as  0.1  wavelengths presented to  [50]  been  here and  may  [18]  TEM  frequencies  be  results  the  In  a  more  result  the  c o e f f i c i e n t  A  N  as  exact  ray-optical  to same  for  accuracy  method  of  widths  and  values  a  [18]  guide  coupling  The  have  for  correction  result  TEM-TEM  are  a  c o e f f i c i e n t frequency  for  expected  TE/  mode  obtained  widths  than  less This a  0.6  of  A ^  comparable  calculation  similar  also, Higher  the  When  to  compared in  to  the  be  consistent  the  terms  TEM case  was  asymptotic deviate of  with  deviate  comparison  has  ,/2  for  the  the  low  asymptotic  non-asymptotic and  3.13.  The  s i g n i f i c a n t l y  at  and  the  T E ,  of  [18]  with made  0.7"X the for  in  results the  approximation  even  0(1<."  using  F i g s . 3 . 1 2  in  results  order  c o e f f i c i e n t s  calculated  (3.18)  begin  0.3X  appears  c o e f f i c i e n t s .  are  using  results  coupling  cutoff  (3.22)  asymptotic  used  provides  However,  coupling.  because  and  near  approximation  where  with  T E /- T E /  widths  used. The  case.  low  wavelengths  for  guide  ( n = l , 2 , 3 . . . ) .  d i f f r a c t i o n  comparison.  agreement  the  driven  t r a n s i t i o n  terms  coupling  the  good  the  When  a=d.  for  show  of  d=n/2  double  expressions guide  depth.  mode  d i f f r a c t i o n  the  for  the  or  the  for  in  width  at  a=n  where  results  available  at  t r i p l e  substantial to  reduced  appear  guide  adding  are  double  minima  parasitic  cases  widths  guide  r e f l e c t i o n (3.31)  is  more.  ) and  0(k~')  were  calculated  44 but  they  also  found  were  been  noted  open-ended  higher  order they  exact  solution  do  (  depth  of  closer  to  at  the  TM modes  these  minima  mode  behaviour  similar  taken  r e f l e c t i o n surmised  give  a It  cusps  out  widths.  does  have  any  terms  solution,  even  with  for  the  comparison, w i l l used  here  is  no  however  contain  by  in  a  and  error,  expansion  a  cusps many  cusps. unable  the  of  to  is It  for  TE  and  the  single  f i r s t  derivative  account.  are  found  terms  (see  reasonable  provide  an  a l .  [50]  for  that  for  has  at  assume  the  accurate  the sharp  the  [50]  the  not  might  width.  mode however  numbers  S i m i l a r l y  is to  It  eventually  [45]  Appendix  case  appears  This  large  in  of  3.9).  solution very  moves  orders  solution  waveguide  into  null  would  exact  The  and  transition  when  transition  waveguide.  terms  known  even  the  et  width  modes).  higher  Yee  mode  guide  mode  single  coupling it  a  ray-optical  taken  the  as  by  more  cusps,  are  case,  of  the  of  The  coupling  solution  be  (Figs.3.6  found  at  discontinuous  d i f f r a c t i o n  has  quadruple  versus  increases  account  s t i l l  that  /2  width  that  exactly  transition not  into  below  (2n-l)"X  (nulls)  to  coefficient  with  the to  coupling  s l i g h t l y and  taking  null  turns  r e f l e c t i o n  l i k e l y  It  scattering  asymptotic  of  c o e f f i c i e n t  that  sharp  are  transition  are  be  of  waveguide  the  curves  widths  d i f f r a c t i o n  the  plate  with  for  is  contributions.  calculation  f i e l d s  agree  a l l  the  the  small  [9].  that  nX  only  p a r a l l e l  not  minima  widths  in  d i f f r a c t e d  since  have  give  that  single  Note  to  in  of the  ray-optical  B).  An  exact  available that  it  ray-optical solution  for also  method  near  the  45 mode  transition  acceptable  case  consideration  asymptotic  of  the  c o e f f i c i e n t  of  0(k~ ).  essentially  f u l l  expression  to  be  F i g .  2]  [4,  No guides  a l l  terms  applying  the  of  occurs  the  the  Each of  rapid  growth  be  better find.  that  the  of  not  slowly provide  in  U A T may Only  the this  series  guide  and  in  number  the  Since found  i s ,  near  the  a  of  general with  B),  a UAT  accuracy  would  however, be  with  even  Appendix  It  the  is  (though  numerical  to  cutoff  the  (see  terms  values  included. was  for  to  term  edges  improvement.  near  ray-optical  solution  exact  the  reflection  this  guides  method  widths  by  In  p a r a l l e l - p l a t e  required  0(^~9  d i f f i c u l t  problem.  the  series  are  a  A N / Vproviding  other in  with  is  only  expected  marginal,  fundamental  at to  if  any,  mode  [3,  Fig.2] calculations  i s o l a t i o n  interesting here  to  substantial  improvement  this  between  have  very  this  for  accelerated).  that  to  open-ended  with  is  t r a n s i t i o n  provide  but  eqns.(8.7),(8.8)]  for  ray-optical  for  to  obtained  [4,  there  for  simpler  mode  widths  guides  interactions  appear  that  terms  interaction  adjacent  solution  to  50  be  [1]  single  agreement  may  represents  the  to  transition  convergence  the  a  However  l  up  two  expected,  given  expression  similar  convergent;  mode  from  eqns.(35),(36)]  )and  / z  are  been  (UAT)  f i n a l  is  [3,  Oa~'  cusps  has  theory  r e f l e c t i o n  waveguide  method  where  elsewhere.  Some uniform  widths  to  those  has  compare for  of  the  been the  adjacent  coupling found;  coupling guides  in  between however  adjacent it  is  c o e f f i c i e n t s  obtained  the  of  presence  other  46 guides.  Montgomery  between  adjacent  Fig.1.18  using  the  coupling  using  guides  function  similar  results  s i m i l a r .  i s  guides  i n the  array  the  other  guides  ray-optical asymptotic  form  be  each  of  (3.29)  The  coupling  geometry  Lee  [29]  of  F i g .  structure  i n  found 1.15  structures  consideration  are  that  a  expected.  case,  l i s t e d  c o e f f i c i e n t  i n Table  i n  the  to  which  considered.  It  i s  derived  well  gives  the  region  insensitive  quite  the  techniques.  under  c o e f f i c i e n t  coupling  with  the  Fresnel  here  these  best  are  of  not  i n the other  of  adjacent  unexpected absence  results.  agreement  integral  and (3.31)  presence  pair  thus  , are  I  when  scattering  of The the  functions  used.  SUMMARY  Coupling been  between  calculated  driven  guide  to  contributions parasitic number number results rays  i s  agrees  (3.23),(3.26),  3.6  might  quite  coupling  closed  the  that  The coupling  guides  that  f o r  to  for  other  the  techniques.  comparison  The  calculated  theoretic  Weiner-Hopf  reasonable  [38]  for  coefficients  s u f f i c i e n t l y  very  and Chang  of of  by f i r s t the  to  guide  adjacent  from  are  (single  i n each  obtained and  f i n a l  sequences sequence even  double  when  of  ray of  plate  sequences  guide  amplitude  the  different rays  tracing  parasitic the  p a r a l l e l  and the i n  rays  rays  then  from  adding  excited sequence.  grows  rapidly  are  However, such  has the the  i n the  each  three  d i f f r a c t i o n )  of  mode  increases. only  waveguides  The  as  the  useful  sequences  considered.  of An  47 exact  solution  comparison,  however,  agree  remarkably  when  the  two This  s e n s i t i v i t y  of  surrounding  results waveguide  are  this the  well  coupling those  are  the  not  and in  obtained  by by  a  c o e f f i c i e n t s gives the  available  c o e f f i c i e n t s  indicates  coupling  structure,  is  surrounded  agreement  applicable  array.  geometry  with  guides  halfplanes.  the  for  obtained other  other general to  confidence  context  of  the  here  methods guides  or  lack  of  nature  of  that the  for  these complete  48  F i g . 3.2 ray paths from the d r i v e n to the p a r a s i t i c guide  guide  F i g . 3.3 two s u c c e s s i v e r a y s i n a r a y p a t h from the d r i v e n guide to a guide a p e r t u r e  U  m  Um-1  |U  m  U m-1  F i g . 3.4 two s u c c e s s i v e r a y s i n a r a y p a t h , b o t h rays i n a guide aperture  F i g . 3.5 two s u c c e s s i v e r a y s i n a r a y p a t h f r o m the guide aperture to the p a r a s i t i c guide  -45  0.5  0  1.0  d/A  1.5  2.0  180 -i  Fig.  3.6  TEM-TEM c o u p l i n g between  ....  adjacent  waveguides  single diffraction s i n g l e and d o u b l e d i f f r a c t i o n s i n g l e and d o u b l e and t r i p l e diffractio  0 i  180  -j  135  •  -|—•—,—,——i  0  0.5  * * *—•—i—•—•  1.0  •—•— —•—• 1  1.5  • •  2.0  d/A 3.7  TEM-TEM c o u p l i n g between single single single  adjacent  waveguides  diffraction and d o u b l e d i f f r a c t i o n and d o u b l e and t r i p l e d i f f r a c t i o n  (a-d/2  180 -i  &  135  o o  90  rO  45  0.5  Fig.  3.8  1.0  TEM-TEM c o u p l i n g between single single single  1.5  2.0  waveguides  (a=2d)  d/A adjacent  diffraction and d o u b l e d i f f r a c t i o n and double and t r i p l e d i f f r a c t i o n  53  180  1  single diffraction s i n g l e and d o u b l e d i f f r a c t i o n s i n g l e and d o u b l e and t r i p l e d i f f r a c t i o n  10  i  TABLE  I  Coupling Coefficients A (d,a) Between A d j a c e n t P a r a l l e l P l a t e Waveguides a = d = 0.45 X m  9 excited +# c o u p l e d into  Montgomery 20  log|A| (dB)  Lee  [38] Lk (degrees)  20  log|A| (dB)  [29] LA (degrees)  1+2 & 2+1  -15.32  +105.2  -15.76  +102.3  3+2 & 3+4  -15.57  +107.8  -15.78  +106;8  -15.33  +105.3  -15.78  +106.8  2+3  this theory using the asymptotic form (3.22) 20 l o g | A | LA (degrees) (dB)  -15.74  +102.3  t h i s theory u s i n g the non-asymptotic. form (3.17) 20  logJA] (dB)  -15.80  Lk (degrees)  +97.7  58 Chapter COUPLING  4.1  CALCULATION  The w i l l  guides.  found F i r s t  quadruple  from  between i n  The f i e l d s f i e l d s  expressions  derived  represented  found  by  [38]  and a using  Consider waveguides planes widths  of  the  ( F i g . 4 . 1 ) .  (3.9)  These B  D  at  plane  - %rr-&  N  and  the  into  edge  waves 0o=£/v  y=~d, are  z>0  z=0  respectively  the  f i e l d s  and hence  [17]  to  the those  using  edge  Montgomery  and  d  into  conducting  and f i e l d the  waves,  and by  p a r a l l e l - p l a t e  and y=a,  incident  incident  the  compared  and  y=0  the  the  using  way  a l .  b,  two p l a n e  y=-d, are  3  calculated  calculus.  determine  As i n Chapter  by  et  be  inspection  i s  found  adjacent including  i t  perfectly  four  -d-b<y<-d,  decomposed  ray  of  0<y<a, i s  w i l l  for  by  this are  (Fig.4.1)  and  s e m i - i n f i n i t e  guides to  to  adjacent  guide  guide  In  residue  transmitting z>0.  2.  argument  y=-d-b,  We w i s h  up  preceding  Dybdal  modified  to  by each  ray  reciprocity  three  that  i n Fig.4.2  by  at  similar  The results  three  z>0  the  rays  consisting  i n  guides  represented  c o e f f i c i e n t .  d i f f r a c t i o n  separated  determined  i n Chapter  a l l  GUIDES  COEFFICIENT  ray paths are  of  previously  Chang  manner  a l l possible  the  coupling  a  two  d i f f r a c t i o n  ( F i g . 4 . 2 ) .  SEPARATED  OF T H E COUPLING  coupling  be  BETWEEN  4 .  a  so  that  coupled  from  parasitic mode  i n the  with  value  (3.10)  at  0* i s  the  respectively  given  the  guide driven given  y=-d-b,  on the h a l f - p l a n e  where  h a l f -  z=0.  edges  by  at  (3.6).  59 The  doubly  F i g . 4 . 2 a .  The  diffracted y=z=0  diffracted  at  incident  the  edge  the  diffracted  u, 1  =  3  located  fields  represented  f i e l d  ,  given  z=0.  From  (3.18)  y=-d,  f i e l d  are  is  a  line  by as  (3.9)  the  diffracted  at  at  source  (4-1)  d i f f r a c t i n g the  i s  observed  ^C-(d)E(r) 2q N  at  by  edge  edge.  y=z=0  This  and from  f i e l d  (3.30)  i s  gives  again a  line  source  The  coupling  treating (3.2).  coefficient  (4.2)  as  a  from  line  double  source  d i f f r a c t i o n  and using  i s  (3.11),  found  (3.12)  by and  Thus  T~ +iTT/4 A.  =  Nn  e  e V'TT — _ rr , 2/2 k a  (4.3)  C ' ( d ) C (d) . N  n  n  The  t r i p l y  F i g s . 4 . 2 b by  (3.9)  (4.1). from  and c. and the  This (3.29)  2  b  i s  In  the  Fig.4.2b  singly is  again  resultant  made  f i e l d  diffracted  f i e l d  at  represented f i e l d u^  b  the  u  i s  by given  i s  given  by  edge  y=z=0  and  i s  +iTT/4  7Z C ' ( d ) E ( d + r ) 4q  are  incident  diffracted  1 =  2 u  f i e l d s  f i e l d the  b S  diffracted  +  N  up of  -  4q two l i n e  the  other  at  y=z=0.  This  z=0  and from  (3.30)  the  C'(d)E(d)E(r) .  1  ^  sources, f i e l d  i s  resultant  (4.4)  N one c e n t r e d  at  again  diffracted  f i e l d  i s  y=-d, at  z=0, y=a,  60 U  3  b  —T icT C ' ( d ) C ( d a ) E ( r )  =  n  In From  Fig.4.2c  (3.18),  is  a  line  =  1  located  c  u  2  i s  z=0, at  y=z=0  c U  3  N  b  +  ±  T  /  4  -  C" (d) C (a)E (d)E (r) .  (4.5)  n  f i e l d u  y=-d,  z=0,  i s  given  the  by  (3.10).  diffracted  f i e l d  (4.6)  d i f f r a c t i n g edge  edge.  This  z=0  and  y= d, -  )  c  E  (  b  +  r  at  (-1) e ~4q~ ^ 7 = ~  "  )  two l i n e  y=-d,  and from  f i e l d from  i s  again  (3.29)  the  i s  u 2  up of  other  -(-1) ~^a7  "  e  N  (  made  the  at  N C  c  —  '(b)E(r) N  the  f i e l d  -(-1)  2  C  the at  =  1  source  at  resultant  —  incident  observed  2q  diffracted  U  the  Z±Z±L  c u  as  -  +  z=0.  (3.30)  the  +  i  V  4  C'(b)E(b)E(r) .  sources, This  f i e l d  f _ ^ n  +  -  is  resultant  N  C'(b)C (b d)E(r)  one c e n t r e d again  f i e l d  (4.7)  at  y=-d-b,  diffracted  i s  +i*/4 ~Z— C'(b)C (d)E(b)E(r) . >'2TT N  T  n  (4.8) The  total  found  by  (3.11),  contribution  treating (3.12) r  A Yin  (  3  )  =  —  given  e,  by  from  t r i p l e  d i f f r a c t i o n  i s  r  and (4.8)  [-(-l)  . rz , 4/2 k a n  [-(-DV  N  quadruply  Figs.4.2d,  A Nn  as  line  sources  and  using  Thus  +iTT/4  8k a n  is  (4.5)  and (3.2).  £ i T  The  to  f  C  ' ( d ) C (d+a) N n  (d)C n  (a)E(d)  diffracted  and g.  (4.6)  n  and ^  In d 2  -  ( - l ) C ' ( b ) C (b+d)] N n N  -  ( - l ) C ' ( b ) C (d) E (b) ] (4.9) N n N  fields  Fig.4.2d  u  ±  i s  are  represented  given  by  (3.10), d  i s  given  by  (4.7).  in  The f i e l d *  2  •  i s  61 again  d i f f r a c t e d  f i e l d d  u  y=z=0  and  from  (3.29)  the  resultant  i s -(-1)  =  3  at  (_i) U  N  C^(b)E(b d r) +  -  +  t  -  (-1) -(JZ-  1  N  + e  e  W  4  -__c'(b)E(b d)E(r) +  e  N  +  i  T  '  /  4  C^(b)E(b)E(d+r)  .  (4.10)  -3/2 Note  that  the  reasons  of  mentioned  sources This  term  at  (3.30) , , = 4  the  +  -i-ii8q ^  u  z=0;  is  3  )  in Chapter  y=-d-b,  f i e l d  0(k  3.  y=z=0;  again  resultant  has  f i e l d  "  been is  d  3  made  and y=~d,  d i f f r a c t e d  up of  z=0  at  .  u  /2TT  consists  4  of  three  The  contribution  is  found  (3.11),  (4) ^  by  8q  to \  sources  from  n  treating  (3.12) j-  [ {  _i,  this  E  i T ( - l )  and  from  (4.11)  quadruply line  at  y=a,  z=0.  d i f f r a c t e d  f i e l d  sources  and  using  Thus  N + n  c.(b)C  <bfd a>] +  n  N  +  n  [c;(b)E(b d)C (a)  ^ I S k a  +  z=0  .  as  8 ^ k a  n  line  C'(b)E(b+d)C (a)E(r) N n  /r—  a l l centred  (4.11)  and (3.2). +iTt/4  fnJL!  =  line  three  +iir/4  N  +  C ' ( b ) E ( b ) C (d+a)E(r) N n  /r  the  respectively.  y=a,  N +1TT/4 2  for  i s  ^ C ' ( b ' ) C (b+d+a)E(r) 8q N n  (-D  u  deleted  +  n  +  C " (b)E(b) C  r  ( d a ) ]. +  (  4  >  1  2  )  n In  Fig.4.2e  (4.1). and  This  from e U  u  e 2  2  i s  is  u  f i e l d  (3.26)  ^  the  given i s  again  (3.9)  f i e l d  1 / 4 £ - = T /2TT +  and  d i f f r a c t e d  resultant  x =  by  i  l  T  ^ at  is the  given edge  by  y=z=0,  i s  4  P  ^  made  C  N  (  d  )  up of  E  (  d  +  r  )  -  two l i n e  sources,  C£(d)E(d)E(r) .  one  at  y=d,  (4.13)  z=0,  the  62  other  at  y=z=0.  This  field u  e 2  i s again d i f f r a c t e d  edge y=-d, z=0, and from (3.26) the r e s u l t a n t U  3  1 iq"  =  e C  N  (  d  )  E  (  2  d  +  r  ~  )  +  i  7  T  /  at the  field is  4  c' ( d ) E ( 2 d ) E ( r )  ? - — —  +171/4  -  T  C' (d)E(d)E(d+r) . N  rz—  /2TT  i s made up of three l i n e sources  U;J  z=0,  at  and y=-2d, z=0 r e s p e c t i v e l y . Again  (4.14)  y=-3d,  z = 0,  y=7d, ~3/2  the term of 0(k  )  e  has been d e l e t e d . T h i s f i e l d edge  y=z=0,  and  from  =  6  4  3  (3.30),  s e p a r a t e l y , the r e s u l t a n t u  i s again d i f f r a c t e d  u  ~ C (d)C ( 3 d ) E ( r ) Bq N n  treating  at  each l i n e  the  source  field is + iTT/4 -r C'(d)E(2d)C (d)E(r) 8q ^ N n  -  1  +i7i/4  -  — 8q ^  In F i g . 4 . 2 f the i n c i d e n t diffracted  at  =  f  •1  resultant f U  field  2  by  ( 3 . 1 8 ) , as observed  i s a line  source (4.16)  L  the  (4.15)  (3.9) i s  - — c' ( b ) E ( r ) 2q N  edge  This  y=-d-b,  field  is  again  z=0, and from (3.26) the  field is 1 " 4q"  =  given  the edge y=-d, z=0. From  at  .  /2TT  l o c a t e d at the d i f f r a c t i n g edge. diffracted  C'(d)E(d)C (2d)E(r) N n  field  at y=-d-b, z=0, the d i f f r a c t e d u  rr-  +i  T C  N  (  b  )  E  (  b  +  r  )  +  ^  T T  /4 C^(b)E(b)E(r)  6  ,  (4.17)  > 2TT .  u  f 2  i s made up of two l i n e sources, one at y=-2b-d, z=0,  other  at  y=-d-b, z=0. T h i s f i e l d  the  i s again d i f f r a c t e d at  the edge y=-d, z=0, and from (3.29) the r e s u l t a n t  field is  f U  3  -1 i ^  -  "  / i ^ — = —  +  ^  +  T  C  N  (  b  )  E  (  2  b  +  ^  i  7  1  4  6  3  e  C'(b)E(2b)E(r)  +iiT/4  3  u  i s made u p o f t h r e e  z=0,  a n d y=-d-b,  diffracted line  u. 4  line  sources  (4.18)  a t y=-2b-d,  z = 0, y = - d , u  z=0 r e s p e c t i v e l y . T h i s f i e l d  a t t h e edge y=z=0, a n d from  source  =  p - C ^ ( b ) E ( b ) E ( b + r ) .  L  (3.30),  f 3  i s again  t r e a t i n g each  separately, the resultant f i e l d i s +iTt/4  1 — C ' ( b ) C (2b+d)E(r) 8q N n  -  8q  C ' ( b ) E ( 2 b ) C (d)E(r) N n  +  • — 8q  1  + iTT/4 C' ( b ) E ( b ) C N  ^  (b+d)E(r)  .  (4.19) g  In F i g . 4 . 2 g  the incident f i e l d  i s given by (3.9),^  g  is given b y (4.1), and  q  2  is again diffracted a t t h e edge y=a, the r e s u l t a n t f i e l d i s +iTt/4 u  9 3  _1 8q  =  c  . ( )E(d+a+r) N d  -  field u "  i s given by (4.4). This  u  8q  2  z=0,a n d from  (3.26)  C ' ( d ) E ( d + a ) E (r) N  ^  +in/4  u  9 3  +  - i !  i s made u p o f t h r e e  line  z=0,  a n d y=2a,  diffracted line  C (d)E(d)E(a+r)  sources  (4.20)  a t y=2a+d,  z=0 r e s p e c t i v e l y .  This field  a t t h e edge y=z=0, a n d from  source  .  (3.30),  z=0, u  9 3  i s  y=a,  again  t r e a t i n g each  s e p a r a t e l y , t h er e s u l t a n t f i e l d i s +1TT/4  u  9  4  = -  — C ' (d)C (d+2a)E(r) 8q N n  +  -J-8q  -  T ~ —  C'(d)E(d+a)C (a)E(r) N n  +irr/4 C ' ( d ) E ( d ) C (2a)E(r) ,  (4.21)  V 2 TT  The  contributions  to  \ . n  from  these  quadruply  64 d i f f r a c t e d (4.21)  f i e l d s  as  line  i s  found  sources  by t r e a t i n g  and using  (4.15),  (3.11),  (4.19)  (3.12)  and  and  (3.2).  Thus f— +i7T/4  .. (4) \n  V *  6  r 8/2  =  [  C  k a n  N  (  d  )  C  n  (  3  d  -  )  -  iftv lbK  a  a  n  A  The  sum  4  4.2  -  )  of  coefficient including  guide of  A  )  C ( b ) E ( b ) C (b+d) N n  -  ( d ) E (d) C (2a) ]  1  N  to  (4)  from  quadruple  (4.9),  (4.12)  separated  those  are  coupling  up  calculated  et  by c o n s i d e r i n g  the  argument line  of  (4.9),  Dybdal  source'incident  the  the  to  and  RESULTS  coefficients  obtained  reciprocity  i s  waveguides  (4.3),  A  i s  d i f f r a c t i o n .  AND NUMERICAL  with  d i f f r a c t i o n  and (4.22)  The expressions  were  to  n  (4)  between  n  \  (4.22)  n  and compared  a  guide  d  +  and separations  as  the  +  C A ( b ) E ( 2 b ) C (d) N n  coupling  compared  line  b  -  quadruple  widths  results  2  C ' ( d ) E ( d ) C (2d) N n  (4.3),  others.  were  (  -  *  ANALYTICAL  The  n  C  C'(d)E(2d)C (d) N n  contribution  (  )  t"  n  total  b  C ' ( d ) C (d+2a)] N n  + C^(d)E(d+a)C (a) The  (  i T  z +  N  C  on  the  receiving  was u s e d  source.  to  with  for the  (4.12) a l .  and  [17],  transmitting parasitic  obtain  the  various results (4.22) Their guide guide.  response  of  65  It  turns  identical paths TEM  out  that  results  in  to  Figs.4.2a,  b,  A  (2)  V  +  Q  and  in  V  ~  the  T E ,  -  0  c  derived  and  d  are  gives  here,  if  considered,  v i r t u a l l y  only  the  ray  i . e .  in  the  shown  the  ray  were the  The  <  and  4.4,  widths  of  0.338"X  ranging  from  of  and  0.4X the  and  to  interaction  terms  interaction  terms  in  compared  to  TEM  have  24  those  function  of  the  results  [38]  and  Lee  TE i c o u p l i n g  are  If  TEM  and  interaction  decreases is by  Dybdal  [29]. is  shown  guide  between  guides  the  monotonically  Dybdal  show more  in  for  between  increased.  becomes  of  Fig.4.5  separation  o0  separation  coupling  in  A  guides as  Addition an  the  of  the  o s c i l l a t i o n  pronounced  as  more  added. of  Fig.4.3  this in  obtained at  for  0.761X  the  with  for  which  cusps  a  A o o  neglected  g. c o e f f i c i e n t s  Chang  guides  guides  coupling  as  and  and  between  the  of  limitations  i l l u s t r a t e d  for  2.0\.  A /2,  interaction  F i g s . 4 . 2 e , f ,  coupling  between  the  compared  and  separation  case  neglect  amplitude  period  4  •  )  calculated  Fig.4.3  The  3  Montgomery  neglected,  <- >  00  values  guides,  a l . [ 1 7 ] ,  with  n  paths  numerical  An  between  »  expressions  The and  00  •  12  in  +  < 3 )  case  V '  Dybdal's  is  those  method  case  7* nn I 00  et  Dybdal's  which  here.  guide  ray-optical  method  Montgomery's  Montgomery's  separations  of  are  results  results n X/2,  in  are the  (n=l,2,...),  66 but  the  r a y - o p t i c a l  results  approximate  agreement.  o s c i l l a t i o n  to  the  cusps  were  not  even  with  large  number  a  While  the  the  q u a l i t a t i v e  Higher  curve  obtained of  addition  of  agreement  separation.  These  higher  d i f f r a c t i o n  the  that  optical dB. the  i s  a  the  applied  and  Boersma  two  a  line  on  (Fig.2.5). f i e l d f i e l d .  i s  here  along  the  These  find  doubly f i e l d  incident both  of  though  are  these  shadow  and  guide  accurately  one-quarter  i n  be  one-half .will or  a l .  be  (3.29) of  the  [33].  Lee  boundary  of  wave  normally  (Fig.4.11)  p a r a l l e l doubly of  2  singly  i n view  plates  -  the  (3.26)  plane  the  0  f i e l d  shadow  staggered  Boersma  the ray-  expected  w i l l  and Lee et  a  theory.  boundary.  that  on t h e  a the  of  may be  incorrect  two c a s e s  than  results,  by about  d i f f r a c t e d  p a r a l l e l  two  gives  accurate,  boundary  two c a s e s :  on  Lee  theory,  r a y - o p t i c a l  coupling  [32]  f i e l d s  for  greater  add an  a l l quadruple and  (equations  are  two n o n - s t a g g e r e d  In  this  shadow  results  the  terms  calculated  predicts  and Boersma  plates  somewhat Even  f i e l d s  incident  [32]  source  be  results  the  used  Lee  p a r a l l e l  incident  with  are  the  the  of  terms  and cannot  and the  only  However,  values  along  twice).  analysis  some  which  f i e l d  one-quarter  for  error  the  cusps.  Montgomery's  measure of  terms  r a y - o p t i c a l  the  f i e l d  incident  this  underestimates  method  d i f f r a c t e d  the  interaction  less  boundary  calculation The  the  of  show  terms.  Montgomery's  theory  This  with  interaction  shadow  Assuming  i s  here  d i f f r a c t i o n  with  with  quantitative  along  order  i n phase  agreement  order  obtained  the  and  plates  d i f f r a c t e d incident  do not consider  the  67 specific  case  p a r a l l e l  plates,  Fig.(4))  indicate  the  f i e l d s  indicate  which  to  the  in  for  guides  from  0.4"X  those  with 1-  some  the  to  important of  t r a n s i t i o n  widths.  the  minima  deeper  are  when  neglected  these  ii  .  and  Both  hand  i s  results  occur  with  not  at  the of  also  d=n"X  results  the  /2 of  method  t r i v i a l ,  case  ([32],  underestimates  Their  w i l l  half-plane  also  calculated  same  of  these  calculated  of  width  the  of  a n d may be  (Fig.4.12)  for  the  and this  same  order  of  exactly  i f  minima  results  those  here  The a p p l i c a t i o n  at  higher  up to  However,  These  results  on two  which  case  width  The results width  that  a l l  i s  varied  are  similar  (Fig.3.6-3.9),  but  differences:  shape  account.  with  the  two g u i d e s  to  much  their  boundary.  consistent  [38].  incident  used  2 . OA ( F i g s . 4 . 6-4 .10 ) .  addition  the  i s  i s  of  difference  2-  analysis  source  chapters.  are  for  4.2d),  coupling  three  coupling  three  to  the  (line  shadow  problem  later  The  the  enhanced  and Chang  here or  the  which  intractable arises  that  near  ( n = l , 2 , . . . ) ,  [32]  Fig.2.3  that  Montgomery  arises  for  five  at  the  quite  the  case et  shallow  a l .  array  was  Lee's  array  embedded  i n i n  and  more  the  mode  widths taken  the  and into  guides  i s  broad.  a=d=b=0.45/\  are  compared  [38]  [29]  i n  and Lee  five  between  Montgomery's  i s  between  makes  at  transition  d i f f r a c t i o n  considered  guides.  terms  especially  mode  times  are  coupling  was  curve  interaction  Montgomery authors  the  d i f f r a c t i o n  adjacent  each free a  pair  space  waveguides of  separated  (Fig.1.17)  simulated  Table  and  groundplane  68 (Fig.1.15) . As  in  the  c o e f f i c i e n t s guides  is  absence in  the  of  other  in  a  of  surrounding  other  well  of  separated  pair  derived  quite  coupling  here  with  in  these  the  results  guides.  of in  the  which  the  the  the  coupling  presence  groundplane  coupling  between of  two  other  indicates  coefficients  a  to  separated  guides  and  general  lack  the  nature  of  structure.  SUMMARY The  coupling  waveguides  has  adjacent shape  of  the  to  adjacent  To  of  separated  by  the  obtain  an  coupling  to  same  versus  terms  compared  coupling  those  between  up  p a r a l l e l method  plate  used  for  approximately  correct  guide  it  to  only  the  of :  c o e f f i c i e n t s  calculated  interactions  by  guides shows  at  width least  double  boundary  and  method is  obtained  Dybdal  is  only  Montgomery  ray-optical  shadow  calculated  include  as  two  was  quadruple  d i f f r a c t i o n  for  guides.  The  results  been  curve  d i f f r a c t i o n ,  with  between  waveguides.  necessary  the  of  to  guides  coupling  agrees  simulated  s e n s i t i v i t y  4.3  The  guides  i s o l a t i o n , in  adjacent  insensitive  s i m i l a r i t y  embedded  the  for  considered.  This  of  are  presence  guides  case  et  neglected.  Chang  [38].  there  apparent.  agree  al.[17]  approximate  when  here  if  very  interaction  Including agreement  Thus  are  However,  four the  the  well  those  with  inaccuracy  edges results  along  the of a  obtained  69 here other  are  similar guides  c o e f f i c i e n t s  those  or  a  between  calculations considering  to  separated  details  guides  in  groundplane.  involving the  for  the of  the  guides  Thus may  array other  the  of  guides  be  presence the used  F i g . 1 . 3 in  the  of  coupling in  the  without array.  70  F i g . 4.2 ray path from the d r i v e n t o the p a r a s i t i c guide  guide  71  -15 i  -20 ^  -25  -30 J  -35  0  0.5  2.0  1 .5  1.0  d/A  180 i  90  -90  -180  4.3  TEM-TEM c o u p l i n g between waveguides For  Figs. ,  4.3  -  separated  338X).  4.9  as  applicable  (2)  A  A  OO  (2)  + A  +A  CO  (  2  )  +  A  (  3  - o - Montgomery  +A  A  A  (a=b=0.  (3) ,(4) + A  CO  +A  *  Dybdal et  )  +  A  (  4  )  +  A  (  and Chang al.  [17]  5  )  [38]  (measured)  72  Fig.  4.4  TEM-TEM c o u p l i n g between waveguides See F i g .  (a=b=0.  4.3  for  761X).  key.  separated  25  30  35  •40  -45  -^•h  0  1.0  0.5  2.0  1.5 d/A  180  i  /  V / /  90  •7  / // / / / /  -90  • /  // -180  0.5  0  1.0  1.5 d/A  Fig.  4.5  T F ^ - T F ^ c o u p l i n g between waveguides See F i g . 4 . 3  (a=b=0. for  761X).  key.  separated  2.0  74  Fig.  4.6  TEM-TEM c o u p l i n g between See F i g .  4.3 f o r  key.  separated waveguides  (d=a=b).  76  Fig.  4.9  TEM-TEM c o u p l i n g between waveguides See  (a=d/2, b=d) .  F i g . 4.3 f o r  key.  separated  78  §  - 3 0  cn o o  -60  CM  -90  1.5  1.0  0.5  2.0 dA  i  180  if)  90  <b ^_  cn  7  fl  X3 0  ^  //  < no -90  4 /  /  /  - /  4-  -180  1.5  1.0  0.5  2.0 d/A  F i g . 4.10  TE -TE  c o u p l i n g between  1  waveguides A  ....  ll  (a=d=b).  (2)  A n i  2  ' ^ ^ ) ^ ^ )  separated  l 2 u  Fig. 4.11 p l a n e wave n o r m a l l y i n c i d e n t on two n o n - s t a g g e r e d p a r a l l e l plates  i 3 u  Fig. 4.12 l i n e source normally i n c i d e n t on t h r e e n o n - s t a g g e r e d p a r a l l e l plates  TABLE I I Coupling Coefficients A^b.d.a) Between S e p a r a t e d P a r a l l e l P l a t e Waveguides a = d = b = 0.45 X  t excited -nit coupled Into  Montgomery [38] 20 log|A| (dB)  LA  (degrees)  Dybdal [17]  Lee [29] 20 log|A| (dB)  LA  (degrees)  -20.69  -74.8  -21.27  -77.7  1-3  -20.69  -74.9  -21.27  -77.7  2+4  -20.83  -72.4  -21.24  -73.8  3+1 & 3-»5  20 log|A| (dB)  -21.34  LA  (degrees)  -74.6  this theory using the asymptotic form (3.17) 20 log|A| LA (dB) (degrees)  -22.26  -75.8  this theorjr using the non-asymptot: c form (3.22) 20 log|Al LA (de reeo) (dB) E  -22.64  -84.6  81 Chapter RADIATION  5.1  plate  radiation  employed  pattern  was  Because  many  it  is  The  to  f i e l d s  are  guide  f i e l d s  A  SINGLE  to  in  can  particular sum  this of  TEM  case.  from  of  the  edges  CANONICAL  by  GUIDE  of  w i l l  the  into  the  rays  mode  radiation  represented  from  w i l l  be  up  of  d i f f r a c t e d the  mode  ( F i g . 5 . 1 ) . in  in  a  ray  each  Several in  a l l  find array,  made  These  is  ray  in The  path  ray  are path  different  the  pattern by  [49],  to  analysis  paths  space.  Felsen  are  to  radiation  waveguide  The  ray  similar  required  consideration  waveguide.  PROBLEMS  be  edges.  ray  f i n a l  from  and  waveguide  waveguide  p a r a l l e l -  This  Yee  convenience.  The  f i e l d s the  by  successive  total  methods  analysis.  tracing  radiated  open-ended  multi-element  under by  The  ray-optical  from  the  traced  edge.  by  the  for  turn.  be  single  analysis  here  edge  f i e l d s  a  obtained  radiated  the  of  coupling  of  represented  represents  5.2  the  calculated  calculated  both  the  d i f f r a c t e d  f i e l d s  found  pattern  repeated  restricted  paths  for  parts  radiation  f i e l d s  is  previously  the  to  pattern  waveguide  those  the  OF  INTRODUCTION  The  the  PATTERN  5  guide made  paths  up  ray to  a of  leading  82 The  f i e l d s  combinations Fig.5.1  of  have  of  by  the  In  some  any ray  rays  in a  sequence  and  3.  were  the  be  paths  the  found  i t . not  various  ray  Thus  can  preceding 5.3)  the  in  in Chapter  immediately  (Fig.5.2  of  of  f i e l d s from  the  Other  needed  included  and are  here.  Fig.5.2a  waveguide f i e l d  by  successive  calculated  ray  combinations presented  two  been  represented fields  represented-  the  u  =  plane  wave  y=z = 0  edge  in  a  at  direction  e o  is  6  u  =0.  is  = 1/2 From  given  incident  (2.7),  the  on  a  diffracted  by  +iV4 u  D(0,6)E(r)  =  -  sec(^)E(r),  u  1  In  F i g 5.2b  y=a,z =0  a  at  direction  e  6  plane 0  =  .  2 l T  is  wave  u  From  given  (5.1)  2  is  incident  the  diffracted  = 1/2  (2.7),  on  the  f i e l d  edge  in  the  by  + i7T/4 u  u.  =  x  D(27!,6)E(r)  =  +u  £  sec(f-)E(r)  /2T In  F i g  located edge the  a  at  an  total)  5.3a  an  distance  r  angle  Q  f i e l d  in  from  0  = 3TT/2 .  Q  U  isotropic  the  m 1  =  +  where  a  line  guide  From  G(  V  wall  (2.12),  direction  .  (  is  u  e  ,6)  =  -iir/4  )  on  given  that (not  by (5.3)  Q  Q  Ftvkr  = E(r)  diffracted  e)E(r),  +ikr sine (+e  2  m  incident  the is  e  source  f o r o < e < TT/2  G(r  5  2  0  (cos -  Q  -  s i n -) ]  83 -ikr  +e  sine  fl  °  fl  F [ « £ r ~ (cos j + s i n | ) ] }  (5.4a)  }  f o r TT/2 < G < 3TI/2  -iu/4  G(r  ,6)  =  +ikr  i— vT 'T  0  sinS  Q  e  F[> kT~ ( - c o s - + s i n - ) ] u ^ ^  {-e  /  -ikr  +e  sine  „  °  F [ A 7 ^ (cos |  fi  + s i n j ) ]}  (5.4b)  and f o r 3TT/2 < 6 < 2TT  -i7i/4 G(r  . 6) O  +ikr  =  r  sin6  {-e  F[*£r~  (-cos  0  —  -ikr  sine  -e  ^ g - + s i n -) ] Z  Z  p  F f v ^ (-cos j  U  fl  - s i n | ) ]}  ,  (5.4c)  f o r k r >> 1 0  G(r 0  In  Fig  f i e l d  e  an  5.3b  distance angle  .e) ^  r  0  from  o  = v/2  in  the  .  e  +  i 7 T  /  4  /—  „,  > cos(e/2)  E ( r 0)  isotropic a  guide  From  direction  c o s '6- -  line  wall  (2.12), &  is  is the  (5.5)  source incident diffracted  given  by  = E(r) on  that (not  located edge the  at  a an  total)  84 -G(r  m+1  5.3  ,6)E(r) .  (5.6)  CALCULATION  OF RADIATION  The  radiation  plate  total  waveguide  semi-infinite perfectly  width  y=+d/2.  the  space  pattern  d  is  parallel-plate  conducting  and  As  of  We w i s h  outside  before,  the  p  (  0  of  d )  calculated waveguide  p a r a l l e l  to  the  PATTERN  here.  the  of  i n z>0  f i e l d s  p a r a l l e l -  Consider  consisting  half-planes  determine  a  the  at  a two  y=-d/2  radiated  into  guide.  incident  f i e l d  -ikz  i  (5.7)  in  the  the  t o p and bottom  value  guide  is  decomposed edges  into  at  9 = a  two p l a n e 0  or  2~r  waves  incident  respectively,  on  whose  is  1 U  at  both  In  edge  singly  F i g . 5 . l a y=+d/2,  u.  In the  (5.8)  edges.  The e.  2  ~  i  edge  3  =  diffracted the  2=0.  ^  incident From  (5.2)  D(2TT , 6 ) E ( r )  F i g . 5 . l b y=-d/2,  f i e l d s  the z=0.  are  f i e l d the  shown  (5.8)  is  diffracted  i n F i g 5.1a diffracted f i e l d  by t h e  is  .  (5.9)  incident From  and  f i e l d  (3.18),  as  (5.8)  is  observed  diffracted at  by  y=+d/2,  85 2=0  the  d i f f r a c t e d  u  =  b  1  located  f i e l d  u  i s  a  line  source  - - C (d)E(r) 2 0  at  the  (5.10)  d i f f r a c t i n g  edge,  where  from  (3.19)  and  (3.20) 2F(^d)  e ^ = _ _ _ 4  .( )  c  d  7  _  _  r  .  (  5  >  n  )  /T7  If  k d » l ,  +in/4 C  (d) ^ -  •  This and  line  source  from  (5.3)  h =  2  In  n  c  (  d  the  again a  d i f f r a c t e d  at  the  edge  y=+d/2,  2=0  f i e l d  )  ~  G(d,6)E(r).  Fig.5.1c  edge  2=0  i s  gives _  u °  the  (5.12)  /r7  0  the  y=+d/2,  incident 2=0.  d i f f r a c t e d  (5.13)  f i e l d  f i e l d  From ^  c a  (5.8)  (3.18), s  line  a  as  i s  d i f f r a c t e d  observed  at  by  y=+d/2,  source  -c; (d) U  °  l  located  =  at  resultant  i s  other edge  the  _  edge  C  ( 0  d  y=-d/2,  )  E(d+r)  made  up of  two l i n e  at  y=-d/2,  y=+d/2,  edge.  This 2=0  f i e l d  and from  i s  again  (3.26)  the  i s  —4  2  c  at  =  (5.14)  d i f f r a c t i n g  f i e l d  c  u  ^  E(r)  the  d i f f r a c t e d  u  ~  2=0  2=0.  +  +  i  V  4  C'(d)E(d)E(r)  sources,  This  and from  e  f i e l d  (5.3)  one at i s  the  again  .  y=-3d/2,  (5.15)  2=0,  d i f f r a c t e d  resultant  f i e l d  i s  at  the the  -  c u  =  3  C  (  0  d  )  +  —1  G(2d,6)E(r)  V 4  i  +  8  C'(d)E(d)G(d,6)E(r)  which l o o k s l i k e two l i n e sources a t t h e d i f f r a c t i n g In F i g 5.Id t h e i n c i d e n t f i e l d u u  the f i e l d  i s given  diffracted  by  a t t h e edge  (5.10).  (5.16)  edge.  i s g i v e n by (5.8) and This  y=+d/2,  6  z=0  field  i s again  and from (3.26) t h e  resultant f i e l d i s d  u  u  " —  C  =  2  (  d  e  )  0  E(d+r)  +  +  i  l  T  /  4  —  C' (d)E(d)E(r)  .  (5.17)  i s made up of two l i n e s o u r c e s , one a t y=+3d/2, z=0, t h e  2  other  a t y=+d/2, z=0. T h i s f i e l d  i s again d i f f r a c t e d  at the  edge y=-d/2, z=0 and from (3.26) t h e r e s u l t a n t f i e l d i s u  d  " 0 — C  =  (  d  e  )  E(2d+r)  +  +  L  v  /  4  C' (d)E(2d)E(r)  '' +iTT/4 +  u  C' (d)E(d)E(d+r)  i s made up of t h r e e l i n e sources a t y=-5d/2,  d 3  y=-3d/2, z=0 r e s p e c t i v e l y . T h i s f i e l d  .  (5.18)  y=-d/2 and  i s again d i f f r a c t e d a t  the edge y=+d/2, z=0 and from (5.3) t h e r e s u l t a n t f i e l d i s d  - Q — C  =  u 4  (  d  )  + e  G(3d,6)E(r)  W 4  +  C ' ( d ) E ( 2 d ) G ( d , 6 ) E (r)  +iTT/4 +  •  C'(d)E(d)G(2d,6)E(r)  8/2TT  which l o o k s l i k e t h r e e l i n e sources a t the d i f f r a c t i n g The  total fields  z=0 up t o quadruple (5.9),  (5.13),  p  o diffracted  diffraction  are  edge  edge.  from t h e edge y=+d/2, given  by  t h e sum of  (5.16) and ( 5 . 1 9 ) . These f i e l d s appear l i k e  l i n e sources a t t h e edge. The f i e l d the  (5.19)  U  y=-d/2, z=0 and r a d i a t e d  diffracted  from  i n t o space i s c a l c u l a t e d  87 s i m i l a r l y given  using  the  ray  =  The  and  i s  (5.20)  at  the  pattern  two edges  -ikd  — — P  =  the  second  terms  are lengths  the  shadowed  +ikd  — +  from  given  edges  .  (5.21)  which  to  line  by  P" e 0  factors  quadrant  i s  apparent  — sin6  CTT/2 < e < TT) ,  the  by these  waveguide  .„  array  f i r s t  the  sin6  quadrant  the  formed  of  P' e 0  0  In  F i g . 5 . l e , f , g , h ,  -Pi . 0,  radiation  sources  path  i n  by  P" 0  in  paths  a  where  the  adjust  distant  for  exponential the  different  observation  (o < e < TT/2) ,  one  of  the  point. edges  i s  and  -ikd . . —z s i n e P  The  =  o  pattern  P. e  is  .  symmetrical  P (d,2n-6)  =  n  so  e = n  that (5.23) d i f f r a c t i o n  V  The  radiation  be  about  p(d,6) 0 * pattern  0  may  (5.22)  for  single  written (1) p  e  =  0  Note  that  given  by  at  +  i3Ti/4 — ~ sec(-) /2TT e  e = TT  (1)  P  o  k  d  s i n (— s i n e ) E ( r )  the  value  of  P  q  .  ( 1 )  (5.24)  i s  f i n i t e  and i s  +13TT/4  (d,e=,>  = ? ~ -  k  d E  (  r  )f  /2TT so  that  width.  the  maximum  F o r o < e < TT/2  radiation  i s  proportional  to  the  guide  88 • (i) p  At  i =  0  =  e  -  +  l  /„  V  2 ~ ~  ~  =  r  there  7r/2  -ikd —  «  4  e S  E  2  C  is  )  . „ s  i  n  6  •  E  a  (  d i s c o n t i n u i t y  in  the  5  '  2  6  )  single  e d i f f r a c t i o n e < TT/2 . order  radiation  This  it  can  the  the  shown S  X  n  u  e  u-i  U  2  The  2  )  Z i M  b TT+  discontinuity  because  -ikd — -  n f l  i  double  d  , d  2  e  n  d is c on tin u ity  at  6 =  TT/2  is  in given  d i f f r a c t i o n  term  s i n 9  which k  The  . s i n  . . . s  i  „[49] n  .  D  The  the  pattern  is  addition  of  discontinuity  at  6 =  about  half  t r i p l e  d i f f r a c t i o n  in  the  because  n/2  double  s i m i l a r l y  of term  d i f f r a c t i o n it  can  be  (Fig.5.4b)  E  +  U  ( 2  d isco ntinuity  pattern  at  about  half  in I S  e = TT/2 of  J  )  E  =  the  2  by i  successively  higher  d i s c o n t i n u i t y  at  applies  disco ntinuit y  the  (  2  (  2  at  be  e  [49]  —-T" ^ .  reduced. 0 =  *  E  +ikd  d i f f r a c t i o n  may  )  d i f f r a c t i o n  <3 IL~)  u  ~ sine  order  e = TT/2  U  t r i p l e  given  ^ (j) e  f  u  to  double  sine e i  higher  of  pattern  that  (  this  for  taking  addition  radiation  eliminates  shown  eliminates  shadowed  by  (Fig.5.4a)  e  radiation  The  that  z  ,TT(j )  2  account.  remaining  +  reduced  d i f f r a c t i o n  —T~ 2  be  used.  _  2  can  is  u  of (5.12) i s  +ikd  f  because  form  the  (H )  2  b  asymptotic  by  u  ^  ± i M sine 2  double  1  into  term  been  e,ir-»-  if  discontinuity  d i f f r a c t i o n  d i f f r a c t i o n  pattern,  3V2  .  Thus  which by  -  2  8  )  is  taking  terms, Similar  5  radiation  .  sin6  (  the  reasoning  89  5.4  NUMERICAL  The  AND EXPERIMENTAL  RESULTS  radiation  pattern  a  single  waveguide  calculated  by r a y - o p t i c a l  methods  the  exact  known  pattern[Weinstein,  plane  pattern  of  These  patterns  are  and  5.6.  The  The consist gain  forward  the  into the  direction  i s  never  i s  less  shadow  also  much  (5.28)  reduced higher  to  exact  when  1  dB and i s  only  an  a r b i t r a r i l y  pattern  for for  Each  as  the  the  width  of  constant  and the then  i s  more  form  of  The discontinuity continuity  f o r the asymptotic small  value  successive  0.5  d i f f r a c t i o n  pattern  the  pattern  for guide  waveguide  in  exact dB i n  The discontinuity  asymptotic  by approximately  experimental  Fig.5.5  be e x p e c t e d ,  less  quadruple  the  since  plate  the r a y - o p t i c a l  ( F i g . 5.5b).  terms.  1.4).  plotted  essentially  ( F i g . 5.5a).  true  discontinuity The  then  are  order  in  E -  0'<90°).  1 dB when  smaller  but are  As might  is  The ray - o p t i c a l  throughout  with  calculated  decreases  The phase  between  more  boundary  (section  widths  p a r a l l e l  lobe.  (-90°<  error  then  a  beamwidth  d i r e c t i o n  account.  used  and  increased.  maximum  of  single  and the  forward  6 = n/2  is  i s  compared  e - TT)  a  the  pattern  are  of  guide  i s  horn  patterns  patterns  the  sectoral  measure,  p a r a l l e l - p l a t e  and the measured  guide  radiation only  1969]  for various  radian  C ©' =  increases  The  shown  i n  e' < 1 8 0 ° .  <  the H-plane  radiation  o < e < 2r -180°  of  i s  taken  accurate  at  (5.12)  i s  at  6 = TT/2  relations form,  (5.27)  a n d may b e  by considering  interaction  at  s t i l l  reduces  the  dB  the  one-half. is  widths  also from  within 0.339^  1 to  0.441^  of  (f*10-  90 13 B  GHz)  and  >150°  over  where  because  of  the  the  entire  patterns  blockage  by  angular  could  not  the  antenna  pattern  of  range,  be  except  measured  for  accurately  mounting  arrangement  ( F i g s . 5 . 6 a , b , c , d ) .  5.5  SUMMARY  The  radiation  p a r a l l e l  plate  compared  to  sectoral  be  to  in  the  the  The results sectoral  and  that  gives The  experimental  and  experimental  more  complex  chapters.  an  good  less  taking  the  higher  that  of  a  to  may be  be  that  considered  can  form  of  of  the  the  with in  terms.  experimental  exact,  both  extended  and  and  accuracy  between  suggests  account  terms  d i f f r a c t i o n  exact  the  shadow  when  p a r a l l e l - p l a t e  of  agreement  dB  and  H-plane  the  asymptotic  pattern  estimate  methods  1  into  order  the  an  along  then  taken  methods  with  expected  E-plane  patterns  structures  is  non-staggered  r a y - o p t i c a l  obtained  between  the  single  are  adding  simulates  and  approximation.  by  agreement  horn  well,  plane  further  confirms  by  d i s c o n t i n u i t y  d i f f r a c t i o n  and/or  good  found  patterns  aperture  s t i l l  expression  quite  The  quadruple  reduced  is  experimental  horn.  boundary up  waveguide  a  H-plane waveguide of  this  r a y - o p t i c a l r a y - o p t i c a l  confidence the  to  following  \  ray paths  \  t  F i g . 5.1 in a parallel  e  tr  ray  Fig. path  5.2 (see  5.3 (see  tit.  p l a t e waveguide  e Fig. ray path  \_  text)  text)  92  F i g . 5.4 c o n t i n u i t y of f i e l d s a c r o s s shadow boundary see eqns. (5.27) and (5.28).  CL  I  -20  a>  angle (degrees) 5.5a  R a y - o p t i c a l and e x a c t r a d i a t i o n p a t t e r n s f o r a p a r a l l e l p l a t e waveguide using  (5.11)  exact ray-optical  (a=0.  45A).  94  270 i  180  90  -90 -180 Fig.  5.5b  -90 0 angle (degrees) R a y - o p t i c a l and e x a c t for  a parallel  using  plate  asymptotic exact ray-optical  form  180  90  radiation waveguide (5.12)  patterns (a=0.  45X)  F i g . 5.6  E x a c t r a d i a t i o n p a t t e r n s f o r a p a r a l l e l p l a t e waveguide compared w i t h measured E-plane p a t t e r n s o f H-plane s e c t o r a l h o r n .  96  Chapter 6 RADIATION PATTERN OF THREE ELEMENT WAVEGUIDE ARRAY  6.1  INTRODUCTION In  this  chapter  the  radiation  is  found  of a m u l t i -  element  waveguide  array  similar  to  used f o r a s i n g l e waveguide, and w i l l be  those  (Fig.6.1)  pattern  by  methods  compared t o experimental r e s u l t s using the H-plane  sectoral  horn. Calculation plate  of  waveguide  pattern depth  the r a d i a t i o n p a t t e r n of the p a r a l l e l -  array  begins  by  first  c a l c u l a t i n g the  f o r the case when a l l three guides are of i n f i n i t e (Fig.6.2).  parallel-plate  A l l edges  of  the  resulting  of  waveguides are e x c i t e d by the c e n t r a l d r i v e n  guide. The f i e l d s d i f f r a c t e d by these edges to y i e l d  array  are c a l c u l a t e d  the r a d i a t i o n p a t t e r n .  If the p a r a s i t i c guides are of f i n i t e depth, the f i e l d s coupled  into  them  also  c o n t r i b u t e to the t o t a l r a d i a t i o n  p a t t e r n . For t h i s case, the c o u p l i n g c o e f f i c i e n t c e n t r a l d r i v e n guide and the p a r a s i t i c  between the  guide on e i t h e r  must be c a l c u l a t e d . The coupled f i e l d s are r e f l e c t e d open  and  waveguide  radiated  between  c l o s e d ends of the p a r a s i t i c guide and e v e n t u a l l y  r a d i a t e d from the open end. The t o t a l the  side  array  i s calculated  radiation by  pattern  of  adding the f i e l d s  from the edges e x c i t e d by the c e n t r a l d r i v e n  guide  and the f i e l d s r a d i a t e d from the outer guides. In the TE \ case the c o u p l i n g between adjacent guides i s  97  very  weak,  parasitic the  and  guide  pattern.  the  TEM  6.2  not  for i t s radiation For t h i s  Consider waveguides  y=d/2,  and  three  y=d/2+a  (Fig.6.2).  i n the central  edges a t  significantly  the analysis i s much  the  affect  i s restricted to  stronger.  four in As  perfectly  z>0  at  waves  0 = o o r 2T?  y=-d/2,  5 the incident  -d/2<y<d/2,  two p l a n e  conducting  y=-d/2-a,  i n Chapter  guide  parallel-plate  z>0,  given  incident  respectively.  TEM by  on t h e t o p These  plane  l  2  (6.1)  at  t h e edge y=d/2,  by  the incident  The  z=0,  rays  These plane  waves  be d i f f r a c t e d a g a i n  are  into  Fig.6.1  diffracted  of  Fig.6.2.  by  the  waveguide  may  be r a d i a t e d  by a n o t h e r  waveguide edge. Thus a l l  edges  are  excited  guide  and  contribute  Sequences of rays  radiated  guide  represented  fields  pattern.  are  and a r e  diffracted  driven  they  z=0,  i n the central  resulting  waveguide  and y=-d/2,  central  as  into  have the v a l u e  V "  four  driven  i s decomposed i n t o  edges.  to  coupled  semi-infinite  of  half-planes  waves b o t h  may  adjacent  thin  bottom  i s  FORMULATION  consisting  infinitely  (5.7)  reason  case where t h e c o u p l i n g  RAY-OPTICAL  mode  power  enough  diffracted  called  by  the  fields  to  the  ray paths  trace  from v a r i o u s  edges and  or  i n the  radiation the  fields  eventually  space. s h o w s some o f  the  ray  paths  which  must  be  98  c o n s i d e r e d t o c a l c u l a t e the r a d i a t i o n p a t t e r n of the d r i v e n guide 6.4  central  i n the absence of the o u t e r g u i d e s . F i g . 6 . 3  and  shows some of the ray paths which must be c o n s i d e r e d , t o  c a l c u l a t e the edges  additional  excited  by  diffraction  the  central  from  driven  the guide  waveguide when  the  waveguides of i n f i n i t e depth are added.  6.3  RADIATION PATTERN WITH GUIDES OF INFINITE DEPTH The  t o t a l r a d i a t i o n p a t t e r n P • of the  parallel  field  fields  from  scattered  n  p  i s made up of the  a l l the  sum  of  edges. P ( d ) r e p r e s e n t s o  represents  the the and  the t o t a l f i e l d s c a t t e r e d from  the n t h edge (as l a b e l l e d i n F i g . 6 . 2 ) in p  three  s c a t t e r e d from the open end of the c e n t r a l g u i d e , (n=l,2,3,4)  p  of  p l a t e waveguides w i t h o n l y the c e n t e r guide d r i v e n  i s c a l c u l a t e d . The  field  array  not  already  included  . o  The  t o t a l radiation pattern P  i s g i v e n by the sum rear  f o r the a r r a y of  Fig.6.2  of the f i e l d s from a l l the edges. In the  h a l f s p a c e a l l but the outermost edges are shadowed, so  t h a t f o r a n g l e s o < e < TT/2 o n l y the edge at is v i s i b l e , P  t  and -  P e  ,d+2a. . r, - i k ( — — ) sine  -lk T  -  P  3  e  ,d+2a, . „ (—— )sin6  z=0  ( 6 2  3  For a n g l e s ~/2 < e < -n a l l edges a r e v i s i b l e  P  y=(d+2a)/2,  and  )  99  -ik — sine + P  e  l  +ik — sine +  P  2  e +ik ( d+2a•)sin6  +  The  pattern The  paths  P  4  2  e  (6.3)  i s s y m m e t r i c a l about 6 = T T . P  field  q  in Fig.5.1.  was  calculated  The  field  p a t h s i n F i g . 6 . 3 . The paths are and  ray  image a b o u t t h e  solution  successive  using  is calculated  p 3  using  paths used to c a l c u l a t e  z-axis  for  of  rays i n the  ray  the  those  shown  ray  the  ray  the p  ray  anc 2  in  path  and  thus  the  ^  P  4  Figs.6.3  (labelled  contribute  t o any  to  represent  the  letter  under  of  the the  . The  by  two  6.4  can  now  with  a  ray  (n=l,2,3,4) from  p  there are  a,b,c,etc. in PN  represented  fields associated  contributions  waveguide edges. Note t h a t paths  fields  p a t h s o f F i g s . 6 . 3 and  used t o c a l c u l a t e a l l the  u.  j _ is calculated  the  6.4. The  be  field  i n F i g . 6 . 4 . The  the  p  in Chapter 5 using  n  several  Figs.  6.3  different and  following notations  f i e l d s , where t h e f i g u r e showing  the  the  6.4)  are  superscript  ray which  employed  x refers  particular  ray  to  path  consideration. = an  incident  field  i n the  a  i n a ray  central driven  guide  1  u  X  nm has  =  field  been d i f f r a c t e d m  times.  path c o n t r i b u t i n g  to  p n  which  100 The  f i e l d  To  in  by  a  as  line  located  calculated  contributions  i s  In  at  p  ,  3  Fig.6.4a  d i f f r a c t e d  observed  to  i n Chapter  5.  consider  the  at  the  edge  y=d/2+a,  z=0  the  the  incident  ray  f i e l d  y=d/2,  u  z=0.  d i f f r a c t e d  ,  i  From  f i e l d  u *  source  31  U  been  F i g . 6 . 4 .  (6.1),  (3.18), is  has  q  calculate  diagrams given  P  ~~2  =  at  ( r )  the  d i f f r a c t e d  at  resultant  ( 6  d i f f r a c t i n g the  f i e l d  edge  edge.  This  y=d/2+a,  z=0,  f i e l d  -  i s  and from  4 )  again  (5.3)  the  i s  C ! (a) =  from  the In  A —  F i g . 6 . 4 b  y=+d/2,  the  edge  edge  z=0  the  d i f f r a c t e d  C  =  y=d/2,  z=0  f i e l d  3  2  i s  d/2,  =  made  z=0,  d i f f r a c t e d f i e l d  i s  up of  the at  y=-d/2,  u  z=0.  f i e l d  ,  given  From u  i s  by  (3.18) a  (6.1), as  line  i s  observed  source  E(r)  line and  (6.6)  source  i s  from  again  (3.29)  the  d i f f r a c t e d  at  resultant  the  doubly  i s  - C ' (d) _2 E(d+r)  b  u  - ~  This  u  f i e l d  ( d )  edge.  d i f f r a c t e d  incident  the  " n  u  the  at  b  at  (6.5)  edge.  d i f f r a c t e d at  G(a,6)E(r)  -  two l i n e  other  at  „  /  4  C' (d)E(d)E(r) .  source  y=d/2,  y=(d+2a)/2,  i l T  z=0.  z=0,  f i e l d s ,  one c e n t r e d  The f i e l d  and from  (6.7)  (6.7)  (5.3)  the  i s  at  y=-  again  resultant  101  -C-(d) u  33  In the  4  Fig.6.4c  f i e l d  u ^  d i f f r a c t e d resultant  the i s  at  y=+d/2+2a,  -  +  up  of  3  3  f i e l d =  z=0  by  (6.1)'and  f i e l d  and from  is  again  (3.26)  C'(a)E(a)E(r) .  two  0  line  source  at  z=0  the  (6.9)  fields  respectively.  y=+d/2,  z=0  and  centred This  from  at  f i e l d  (3.26)  is the  is  C'(a) u  given  This  ^  and y=+d/2+a,  d i f f r a c t e d  resultant  (6.4).  is  i  (6.8)  i.T,/A E  z=0  u  y=+d/2+a,  4  made  again  edge  -A_ (a r)  =  by  (a)  32  f i e l d  C ' (d) E (d) G ( a , 6 ) E (r) , 0  is  C u  given  iir/4  -  incident  the  f i e l d  c  is  G(d+a,6)E(r)  in/4  — g — E(2a+r)  -  C!(a)E(2a)E(r)  irr/4 ~  Ci (a)E(a)E(a+r) .  ~T  8/27 is 2a,  made  y=+d/2  again the  up of  three  and y=+d/2-a,  d i f f r a c t e d  resultant  u  c  line  at  f i e l d  r> f a i V!i  .  G ( 3 a (  34  source  z=0  the  fields  edge  y=+d/2+a,  e  represents up  to  three  edge. the  total  and including  z=0  f i e l d  and from  is (5.3)  S_c'(a)E(2a)G(a,6)E(r) /2TT  f.  in/4  8/2TT"  d i f f r a c t i n g  This  y=+d/2-  in/4  .  ) E ( r )  8  as  at  is  8  appears  centred  respectively.  _  which  (6.10)  U  The  line sum  f i e l d  P3  quadruple  sources of  C' (a)E(a)G(2a,6)E(r)  a l l  (6.5), from  (6.11)  °  the  centred (6.8),  edge  d i f f r a c t i o n .  at  and  y=+d/2+a,  the (6.11) z=0  102  To  calculate  diagrams  in F i g . 6 . 3 .  given  by  (6.4)  and  the  contributions  edge  (6.1), is  In  the  given  y=+d/2,  z = 0,  Fig.6.3a  singly by  In  and from  f i e l d  is  f i e l d  is  (3.26)  b u  the given  again  the c  1 3  ( d )  the  i s  the  f i e l d u  ; Q i s  "  given  d i f f r a c t e d  resultant  u  and at  i  is  ±  is  the  given  given  edge  by  is  (6.12)  (6.1),  (6.7).  y=+d/2+a,  z=0  by at  f i e l d  by  ray is  i  Cj(a)E(a)G(a,6)E(r) .  f i e l d  " o = — g ~ E(d+a+r)  f i e l d  f i e l d  (5.3)  f i e l d  (6.6)  diffracted  resultant  incident  This  +  incident by  the  consider  1TT/4  - ~ ~ G(2a,e)E(r)  Fig.6.3b  ^  d i f f r a c t e d  (6.9).  -CA(a) =  to  the This  and  from  is W  A  + -  C'(d)E(d+a)E(r)  ir/4  ^ — "UC ' ( d ) E ( d ) E ( a + r ) .  -  u  k  consists  y=+d/2+a again  b.  three  line  and y=+d/2+2a,  d i f f r a c t e d  resultant  u  of  f i e l d  C'(d)  =- V -  at  (6.13)  8/2T  the  source  f i e l d s  z=0  respectively.  edge  y=+d/2,  at  y=+3d/2+2a,  This and from  z=0  f i e l d  is  (5.3)  the  is  G(d+2a,9)E(r)  -  l*/4  C'(d)E(d+a)  G(a,6)E(r)  in/4 +  In  Fig.6.3c  d i f f r a c t e d at  the  source  at  edge  the the  -  incident edge  C'(d)E(d)G(2a,8)E(r).  f i e l d  y=-d/2,  y=-d/2-a,  z=0  z=0.  the  given From  f i e l d  u  by  (3.18), is  (6.14)  (6.1) as  given  i s  observed by a  line  = located  -S^—  at  the  d i f f r a c t e d  at  resultant  f i e l d  u  =  1 2  y=-d/2-a, at  the  edge  edge.  of  source  a  )  two l i n e  z=0  edge  f i e l d  i s  z=0 and from  again  (3.26)  the  is  -  (  This  y=-d/2-a,  0 —7^-E(a+r)  consists  l 2  (6-15)  d i f f r a c t i n g the  C  u  E(r)  e " -S  /  4  Cj(a)E(a)E(r) .  f i e l d s  respectively.  y=-d/2,  z=0  This  and from  at  f i e l d  (3.29)  (6.16)  y=-d/2-2a, i s the  again  z=0  and  d i f f r a c t e d  resultant  f i e l d  is  u.,  C'(a) - ^ 5 — E(2a+r)  =  in/4 C'(a)E(2a)E(r)  +  in/4 -  C'(a)E(a)E(a+r) . 8V/2T7  (6.17)  U  c u  consists  1 3  and  of  three  y=-d/2-a,  d i f f r a c t e d  at  line  z=0  the  source  f i e l d s  respectively.  edge  resultant f i e l d is C(a) <; = - V G(2a+d,6)E(r) u  y=+d/2,  +  y=-d/2-2a,  This  z=0  in/4 ~~~~~  at  f i e l d  and  y=-d/2  is  from  again  (5.3)  the  C'(a)E(2a)G(d,9)E(r)  i n /4  _  ^  c:(a)E(a)G(a+d,6)E(r) .  8v^ The ?2  sum from  of the  (6.12), edge  (6.14)  y=d/2,  z=0  (6.18)  0  and (6.18)  represents  the  up to  including  quadruple  and  f i e l d  d i f f r a c t i o n . Similar the  edge  calculations  y=-d/2,  z=0  a n d P4  for from  the  fields  y=-d/2-a,  ?  2  scattered  show  that  from  104 P  2  P,  (6.19)  =  -P  =  - P  4  3  The to  •  (6.20)  results  give  the  for  total  ?  parasitic  guides  6.4  RADIATION  WITH  the  array  f i n i t e  of  the  array  The  depth  of  the  guides  radiation  and  thus  coupling  driven width  guide a  travel end. is  the  has  the  reflected The  w i l l  alter  F i g . 6 . 1 , that  and change  the  d  then  back  guide travel  and  part  A  of  aperture  plane  z=0  relative  guide  the  is  sum  of  the  a  A  Q 0  the in  (6.3)  of  Fig.6.2  equivalent  guides  these  are  guides  w i l l  Adjusting  the  phase  relative  between  adjacent  towards  = oo A  elZKS  +  A  of  their  parasitic The  3.  open  central guide  coupled  reflected the  the  at  the  end  R  e  =  f i e l d s  +  of  f i e l d s shorted  ,where  part  radiated.  to  in  the  those  geometric  oo oo ^  be the  outer  from  guides  the  at  central  the driven  series  A„„e A  of  pattern.  (d,a)  are  to  now  reradiated.  Chapter  and  is  value  into  and  array  is  outer  radiation  and  calculated  which  the  coupled  c o e f f i c i e n t  outer  f i e l d s  SHORTED  of  (6.2)  the  GUIDES  end  width  been  down  The  of  far  of  depth.  f i e l d s  from  pattern  in  i n f i n i t e  except  reflected  The  of  OUTER  Fig.6.2  depth.  substituted  radiation  with  Consider  are  n  W """ e  +  1 -  ••• , . 7 7 ^ =  00  (6.21)  where R  0 Q  p a r a l l e l  is  the plate  reflection waveguide  coefficient [45]  [50],  at  the  and  S  open is  end  the  of  depth  the of  105 the  outer  guide.  The  t o t a l  calculated f i n i t e the  f i e l d s and  by  depth  array  r a d i a t i o n adding  of  of  the  f i e l d s  from  (Figs.6.5  guides  have  di  outer  been and  guides  to  from  edges  array  the  the  excited  6.6).  d i f f r a c t e d 6.6  the  additional  d i r e c t l y  and  of  r a d i a t i o n  The  other  guide  F i g s . 6 . 5  the  F i g . 6 . 2 .  radiated  pattern  Since at  represent  caused  f i e l d s  radiated  f i e l d s  f i e l d s  the  by  the from of  ( F i g . 5 . 1 ) ,  i n  f i e l d s  once,  i s  consist  guides  f i e l d s  a l l  least  F i g . 6 . 1  f i e l d  outer by  of  a i n  ray  including  shorted the  outer  paths  i n  quadruple  f f r a c t i o n . Radiation  simply plate  the  edges  calculated In  due  l t  P (a)i n  to  the  the  f i e l d  ray  from  shown  i n  and (5.22)  by A of  r a d i a t i o n  (6.21).  F i g . 5 . 1  of  a  Scattering  from  the  paths  in F i g s . 6 . 5  the  edge  outer  y=d/2,  z=0  the  f i e l d  from  guides and  i s  6.6.  i s  (6-22)  from  the  same  edge  i s  in/A  G(2d,e)E(r) -  =  i s  p a r a l l e l -  G(d,6)E(r).  =  C'(d)  3  guides  (5.21)  Q  by c o n s i d e r i n g  F i g . 6 . 5 b i  u  outer  m u l t i p l i e d  Fig.6.5a  u 1-2  the  pattern  waveguide  other  In  from  C^(d)E(d)G(d,6)E(r) .  (6.23)  A /2T7 In  U  F i g . 6 . 5 c  1'  In  c  "  C  = 3  0 — (  F i g . 6 . 6  the  a  f i e l d  )  = -iL 3  3  4  the  same  edge  i s  e - — — C'(a)E(a)G(d,6)E(r) . A/2TT i 7 ! / 4  G(d+a,6)E(r)  the  f i e l d  from  the  edge  y=(d+2a)/2,  (6.24)  z=0  i s  ITT/4  Cl(d) u  from  G(d+a,6)E(r)  + Av^7  Cl(d)E(d)G(a,6)E(r) . 0  (6.25)  106 The give  f i e l d s  the  f i e l d  radiation f i e l d of  from  the  z=0  P-  outer  the  f i e l d s P^  d/2-a,  (6.23) from  from  P^  the  (6.22),  show  the  edge  guides.  edge  from  and (6.24)  edge  y=d/2,  Similarly  y=d/2+a,  the  must  z=0.  y=-d/2,  be z=0  (6.25)  Similar z=0,  added  to  excited  by  gives  the  calculations  and  p  ^  from  y=-  that  = _<  (6.26)  P  2  1  p. _ _ .  (6-27)  P  The  ray  about  the The  t  o  t  a  for  the  the  at  P^ a n d ?l  in Fig.6.5  by t h e  f i n i t e  sum  of  This  radiation  obtain  P  and  q  the  are  and  depth P^ must  complete  the  image  6.6. the  outer  multiplied  by the  be  of  added  radiation  to  P  pattern  F i g . 6 . 1 .  quadrant  y=d/2+a,  radiated  to  of  f i r s t  A .  calculate shown  caused  array  array  the  to  those  and (6.3)  the  In that  of  coefficient  (6.2) l  z-axis  is  coupling  p  used  radiation  guides  in  paths  z=0  f i e l d s ,  are  so  for  o < 6 < TT/2  shadowed  a  \ \  and do not  edges  except  contribute  to  that  -ik(——)smo P  total  = P  k  + API e  3  • i /d+a, .. -ik(—TT-J s i n o  + AP e  .  2  Q  In  the  second  quadrant  for  TT/2  < e < T T  (6.26)  107  ., .d+2a. . -ik(——)sm6 = P + API e t 3 c  P  total  -ik(~-)sine + AP ( a ) e u - i k ^ sin6  + A?^ e +ik y sine + AP^ e +ik(^-)sin6 + AP (a) e Q  ., ,d+2a, . +ik(—2~)sine + AP! e 4  (6.29)  0  This P  pattern  i s  about  that  6 = TI S O  , (2TT-6) = P^ _ , ( 6 ) .  total  6.5  total  NUMERICAL  The p a r a l l e l is  symmetrical  radiation plate  compared  sectoral  AND E X P E R I M E N T A L  to  pattern  waveguides  with  the  the  outer  radiation  E-plane  The  0—  P  are ,  t  narrower  beamwidth  The  addition  of  of  central  depending outer  on  guides The  their varies  pattern  the  the of  the  added  of  then  a  by  to  A  and  the  three-element  H-plane  the  outer f o r  (Fig.6.2), (6.3),  has  a  guide.  in  in  of  methods  plotted  depth  altered  of  an '  simulate  (6.2)  guide  Adjustment of  of  are  single  guides  array  r a y - o p t i c a l  i n f i n i t e  driven  phase  by  patterns  given  outer  depth.  element  patterns  IT) .  guides  pattern  the  three  calculated  somewhat  pattern  a  appendages  guides (Fig. 1.2). -180° < 6 ' < 1 8 0 ° (0 ' = When  of  measured  horn  RESULTS  the  radiation  d i f f e r e n t depth  of  ways the  (6.21). array  with  outer  108  guides s h o r t e d shows a pronounced  r e d u c t i o n of the beamwidth  compared t o that of a s i n g l e guide when the all  three  guides  Adjustment  is  in  phase  (arg  A  radiation =  0)  from  (Fig.6.8).  of the frequency such that t h i s phase  difference  i s near 180° produces a p a t t e r n with a n u l l on the beam a x i s and  two  main  lobes  to  either  p a t t e r n shapes with v a r y i n g A from  a  Fourier  another value essentially  of  relatively radiation  (Figs.6.9,6.13)  expected  t r a n s f o r m of the a p e r t u r e f i e l d s  [ 2 7 ] . For  this  phase over  in  the  pattern  amplitude  the  phase  of  the  pattern  i s s u b s t a n t i a l l y reduced compared  variety  of  is  the forward d i r e c t i o n and the back to  that  s i n g l e guide. Thus t h i s p a r a l l e l - p l a t e waveguide can a  was  an angular s e c t o r of a b o u t ± 6 0 °  a l l cases  constant  are what would  These  be  constant  ( F i g . 6 . 1 0 ) . In  side  patterns  by  adjustment  of  a  produce  of the depth of the  outer guides or the frequency. The  calculated  accurate  at  patterns  about 6 = 90  are  expected  to  be  least  o f f the beam a x i s near the shadow  boundary i n the a p e r t u r e plane f o r the reasons mentioned Chapter  2.  outermost at  At  angles  slightly  less  then  90°.  (6.28),  90° a l l edges c o n t r i b u t e  consequently a d i s c o n t i n u i t y 6=  g r e a t e r then 90° only the  edge c o n t r i b u t e s t o the r a d i a t i o n  angles  in  T h i s d i s c o n t i n u i t y may  the  pattern  reduce  the  to be s u f f i c i e n t ,  discontinuity  whereas  (6.29), and  is  found  at  be reduced by i n c l u d i n g more  ray paths along the shadow boundary. Terms up d i f f r a c t i o n appear  in  to  quadruple  s i n c e more terms do not  s i g n i f i c a n t l y , and the number of  terms grows very r a p i d l y f o r higher o r d e r s  of  diffraction.  109 The  discontinuity  depending  on  frequency. discussed To  the  The in  using  method  Lee  near  the  problem  of  planes  is  a  between  the  outer  problems  on  the  shadow  d e t a i l  in  Chapter  s i g n i f i c a n t ray-optical  and  shadow  varies  of  a  accuracy  90°  6=  depth  more  effect  of  at  Boersma  boundary.  line  source  needed,  be  where  solution  the  half-plane  edges  are  a l l  solution,  which  would  be  Figs.6.3b,  6.3c,  6.4b,  6.5c  on  the  boundaries  were  two  find  pattern  the  that  the  f i e l d s  non-staggered  Such  a  evaluate  and  6.6,  is  were  generally  the  d i f f r a c t i o n  observation  c o l l i n e a r . to  the  require  to  for  source,  used  +4dB  and  in  may  applied  incident  and  2.  techniques  A  guides  improvement  [32]  -4dB  h a l f -  point  and  ray-optical the  f i e l d s  not  in  presently  a v a i l a b l e . The the  experimental  calculated  8 = - 1 5 0 ° . are could  caused not  antenna  6.6  Any  be  results  pattern by  site  mounting  up  to  asymmetry  measured  ,  9=±60  and  within  within  and  the  small  r e f l e c t i o n s .  The  patterns  accurately  because  of  1  dB  of  3dB  up  to  o s c i l l a t i o n s near  blockage  ©=180° by  the  arrangement.  SUMMARY  Radiation aperiodic been  patterns  array  analyzed  tractable reasonable  from of by  however  a  f i n i t e ,  rather  than  open-ended  p a r a l l e l  plate  ray-optical  methods.  Only  for  accuracy  the rapidly  numbers  of  increases  an  i n f i n i t e  waveguides small  arrays  ray  required  with  the  number  has are for of  110 d i f f r a c t i n g analyzed  with  The off  the  edges. t r i p l e  beam  axis  in  the  be  reduced  by  be  eliminated number  The  of  expected  of when  aperture  the  in  increased  or  dimensions  of  and  of  has  edges  d i s c o n t i n u i t y  This  into  at  90  edges  are  discontinuity  account,  and  been  included.  d i f f r a c t i n g  d,a  so  large  s  ,  but  even  can  cannot with  a  to  the  width.  of  a  reasonably  antenna  and  waveguide  A/3  are  or  f i e l d s  the  less,  in  the  with  those  horn  with  p a r a l l e l - p l a t e well.  The  be  waveguide  outer  depending  frequency. may  agree  sectoral  beamwidth  array  to  the  generally  simulate  guides  for  Inaccuracies  reduced  H-plane  the  v a l i d  boundary.  patterns  that  only  are  shadow  decreased the  is  calculation the  T E M mode  waveguide  a p p l i c a t i o n .  the  paths  analysis  the  appears  the  ray  dimensions  along  ones,  appendages  of  four  a  d i r e c t i o n .  values  experimental  calculated  this  a l l  the  array  rays.  in  plane  The  of  more  by  contain  most  backward  moderate  p a r t i c u l a r l y  array  because  ray-optical  dimensions  three-element  results  taking for  a  d i f f r a c t i o n  calculated  shadowed  larger  Here  Thus  adjusted  on  the to  guides the  pattern  suit  the  —>  z  Fig. 6.1 three element a r r a y of p a r a l l e l p l a t e waveguides w i t h outer guides shorted  d/2 d/2  Fig. 6.2 three element a r r a y of p a r a l l e l p l a t e waveguides w i t h outer guides of i n f i n i t e depth  112  \  it  NJ1 t.  ii F i g . 6.3 ray paths from c e n t r a l g u i d e s t o edge 1  \  \  F i g . 6.4 ray paths from c e n t r a l t o edge 3  guide  F i g . 6.6 r a y path from o u t e r g u i d e t o edge 3  \  n  ray  F i g . 6.5 p a t h s f r o m o u t e r g u i d e s t o edge  1  113  -I-—•  -90  -180  •  1  -90  •  •  ' —  0  •  •  90  •  180  angle (degrees) Fig.  6.7  R a d i a t i o n p a t t e r n o f three element a r r a y w i t h o u t e r g u i d e s o f i n f i n i t e d e p t h (a=d=0. 401X).  270 i  angle (degrees) Fig.  6.8  R a d i a t i o n p a t t e r n o f three element a r r a y a=d=0. 450X s=0. 8 5 6 X A=0.131 L 0 ° .  115  270  i  Fig.  6.9  R a d i a t i o n p a t t e r n o f three element a r r a y a=d=0. 339X s=0. 645X A=0. 371 L - 1 7 l ! using using  (5.12) (5.11)  116  _90 -I -180  Fig.  6.10  .  . -90  •  •  • 0  angle (degrees)  — 90  — l 180  R a d i a t i o n p a t t e r n o f three element a r r a y a=d=0. 356X s=0. 677X A=0. 308 L -134°.  117  Fig  6.11  R a d i a t i o n p a t t e r n o f three element a r r a y a=d=0. 373X s=0. 709X A=0. 247 L +104  .  118  -90  H -180  —  • -90  •  •  • 0  -  ' 90  ' 180  angle (degrees) F i g . 6.12  R a d i a t i o n p a t t e r n o f t h r e e element a r r a y a=d=0. 389X s=0. 459X A = 0 . 199 L +38°.  119  -90 -I -180  •  . -90  •  •  •—  0  •  • 90  •  • 180  angle (degrees) Fig.  6.13  R a d i a t i o n p a t t e r n of three element a r r a y a=d=0. 441X s=0. 597X A=0. 223 L + 1 6 6 ° .  120 Chapter RADIATION  7.1  RAY-OPTICAL  The  results The  the  additional formation  into  of  account.  guides and  array  for  of  w i l l  [29]  be  [38]  of  element  array  outer  guides  many  more  five  WAVEGUIDE  ray  at  ARRAY  incident  mode  in  by  is  the  by  be  central  be  taken  for  outer  other  depth  methods  respectively.  perfectly  two  As  p a r a l l e l  plate  conducting  y=-d/2-a,  guide  into  6.  allow  (Fig.7.1)  s e m i - i n f i n i t e  six  array  must  by  the  i n Chapter  calculated  found  plate  extending  element  which  ( F i g . 7 . 2 ) .  decomposed  p a r a l l e l  and f i n i t e  results  y=-d/2-2a,  y = d / 2 + 2a  five  w i l l  adjacent  and  found  paths  results  of  element  considered  the  (Fig.7.2)  with  consisting z>0  in  patterns  d=+d/2+a  (5.7),  five  three  compared  in  a  may be  and experimental  waveguides  ELEMENT  (Fig.7.1,7.2)  i n f i n i t e  Consider  planes  pattern  The  both  OF F I V E  FORMULATION  radiation  waveguide  the  PATTERN  7  y=-d/2,  in  y=+d/2,  Chapter  -d/2<y<+d/2, plane  h a l f -  z>0  waves  6  the  ,  given  each  with  amplitude  (7.1)  at  the  edges  waves  are  guide  of  represented F i g . 7 . 2 .  considered Chapter  y=-d/2,  6.  to  z=0  by  the  Many  calculate  and y=+d/2,  the  The a d d i t i o n a l  incident  of  the  ray  paths  rays  ray  radiation  z=0.  These i n  paths  pattern which  two  the  central  which are  occur  plane  must  shown in  the  be in five  121 element in  7.2  array  Figs.7.3-7  Fig.7.2  so  f i r s t  given  the  three  element  GUIDES  pattern  by  of  the  sum  a l l  0<e<ii/2  and  OUTER  radiation  quadrant  for  in  PATTERN WITH  total is  that  not  array  are  shown  .7.  RADIATION  The  the  but  the  ?  OF  INFINITE  for  t  fields  but  the  outermost  only  the  edge  the  from  a l l  edges  are  y=d/2+2a,  z=0  DEPTH  array  of  edges.  In  shadowed is  v i s i b l e  are  v i s i b l e  . , ,d+4a. . -ik(—)sxn9  Q  P  In  the  t  =  P  5  d  second  quadrant  (  Tt/2<e<n  )  a l l  edges  and  _  i k (  d±ia  ) s i n e  - l k (——) sine + P  3  d  d  - i k — sine + P +  d  o  p  d  + P„ e 2  +ik - sine  . . ,d+2a.  . .  + i k (——) sine + p  e  4  + +  P  6  e  i  k  {  ^ 2  )  s  i  n  e  (7.3)  122 where  ?  i s  n  symmetrical The be  a  about  created  by  presence  a d d i t i o n a l  To the  i n  the  not  in Chapter  6  additional  modified  up to  must  ray  guides  now paths  ( F i g . 7 . 3 ) .  because  quadruple be  i s  t  new o u t e r  be  pattern  = P <e) .  t  new a n d m u s t  the of  d i f f r a c t e d  observed  of  The  there  are  d i f f r a c t i o n .  calculated  The  using  the  F i g . 7 . 4 .  diagram  i s  #n.  P (2n-e)  account  paths  are  c  edge  c a l c u l a t e d  4  into  need  2  ray  calculate  ray  (7.1)  and P  P  and P  paths  P  the  no  ray  and  3  from  that  o  take  f i e l d s  c  P  s  to  The  P  radiated  e = n  f i e l d s  modified  f i e l d s  f i e l d  at  additional F i g . 7 . 3 .  at  the  y=d/2+a,  c o n t r i b u t i o n  The incident  edge  z=0  y=+d/2,  the  to  P  f i e l d  u  z=0.  d i f f r a c t e d  From  3  consider given  (3.18)  f i e l d  i s  a  by as l i n e  source  A _0  a u  3  c  -5—  =  1  located  at  the  d i f f r a c t e d f i e l d  at  u  3  2  i s  z=0,  E ( r )  d i f f r a c t i n g y=d/2+a,  edge.  z=0  C'(a)  V!i (a r)  32  the  E  4  made  d i f f r a c t e d f i e l d  in A\ (7.4)  ,.  This  and from  f i e l d  (3.29)  i s  the  again  resultant  i s  a  u  ( a )  up of other at  +  e  S — 4/2T7  +  two l i n e at  (7-5)  i 7 , / 4  C'(a)E(a)E(r)  sources,  y=+d/2+a,  y=+d/2+2a,  z=0  one c e n t r e d  z=0.  and from  at  This  f i e l d  (3.26)  the  y=+d/2, i s  again  resultant  i s C'(a) a u  33  =  - 0 8  E(a+b+r)  -  pi  7 1  /  4  ^Z  C' ( a ) E ( a + b ) E ( r )  Bl  in/4 +  C  8/2^  °  (a)E(a)E  (b+r)  .  (7.6)  123  This  f i e l d  (5.3)  the  i s  again  resultant  diffracted  f i e l d  at  y=+d/2+a,  z=0  and  from  i s  -C'(a) =  a  u  °  3 4  G(a+2b,6)E(r)  +  -  C'(a)E(a+b)G(b,6)E(r)  8v^  8  °  iit/4 -  C (a)E(a)G(2b,6)E(r) ,  S 8>2TT  (7.7)  0  /  a u  3  must  4  be  added  to P  (7.3)  which  up  and including  to  To  (7.1),  the  In a  u  f i e l d  represents  calculate  F i g . 7 . 4 .  i s  i s  again  0 — C  5  =  3  (  a  the  the  by  6  from  (7.4) at  and  u  diffracted at  e  +  the  h  " n c  u * at  diffracted f i e l d  b  3  in  y=d/2+a,z=0  the  2  is  ray diagrams i  u  i s  given z=0  by  and  given  of by  (7.5).  This  from  (5.3)  i  V  4  C'(a)E(a)G(b,6)E(r) .  incident  f i e l d  y=-d/2,z=0.  diffracted  (7.8)  °  f i e l d  u  given  From  (3.18)  is  line  a  by  (7.1)  as  is  observed  source  { d )  the  y=+d/2,  =  -2  (7.9)  b  d i f f r a c t i n g  at  z=0  edge.  and  ^  S  m  E(d+r)  from  -  4  52 52  edge  P  This  f i e l d  (3.29)  u  the  5  i s  1  again  resultant  is u  u  f i e l d  = -VB(r)  5  located  the  the edge  y=+d/2,z=0  the  is  G(a+b,6)E(r)  at  5  y=+d/2+a,  )  Fig.7.4b  the  f i e l d a  8/27 In  give  consider  incident  diffracted  to  d i f f r a c t i o n .  f i e l d  the  given  f i e l d  f i e l d  quadruple  Fig.7.4a  resultant  u  i n Chapter  3  a  ^  e  U  P  ° ^  t  w  0  (7.10)  C'(d)E(d)E(r) . .  pr~  line  0  sources,  one c e n t r e d  at  y=-  124  d/2 z=0,  the  r  d i f f r a c t e d f i e l d  other at  at  y=+d/2,  y=+d/2+a,  z=0.  z=0  This  and from  f i e l d  (3.29)  i s  the  again  resultant  i s  5  -C'(d) -°-  =  b  u  3  in/4 E  (d+a+r)  -  C'(d)E(d+a)E(r) °  8>^"r7  8  iTT/4 -  C (d)E(d)E(a+r) 8/2TT  i s  u.^  3  made  y=+d/2  z=0  line  This  and from  -C'(d) -2  =  b  three  r e s p e c t i v e l y .  y=+d/2+2a,  u  up of  G  sources f i e l d  (5.3)  (d+a+b,6)E(r)  i  c  -  7  at  i s  the  y=-d/2, again  resultant T  /  y=+d/2+a  and  d i f f r a c t e d f i e l d  at  i s  4  C'(d)E(d+a)G(b,0)E(r)  8^  8  5 4  (7.11)  1  0  iir/4 _  £  C (d)E (d)G(a+b,0)E(r)  °  8^7 The  sum  the  edge  of  (7.8)  and (7.12)  y=d/2+2a,z=0  up  represents  to  (7.12)  and  the  f i e l d  P  5  including  from  quadruple  d i f f r a c t i o n . Similar  c a l c u l a t i o n s  contributions  to  show  f o r  P  6  and  the  additional  that (7.13)  P  4  P  6 The  ^ 3  =  =  _  ray paths  F i g s . 7 . 3 The  to with  give  5 . used  contributions in  to  to  calculate  Pif a r e  the  image  ?(,  and  about  the  the  additional  z-axis  of  those  and  (7.3)  and 7.4.  results the  (7.14)  P  for  total  p a r a s i t i c  P  n  are  radiation  guides  of  substituted pattern  i n f i n i t e  of  depth.  i n  the  (7.2) array  of  Fig.7.2  125  7.3  RADIATION  WITH  Consider Fig.7.2 depth.  the array  except  that  The f i e l d s  reflected  width  separated  into  end  and  between  \  2  adjacent  guides  of  now o f w i l l  of  to  f i n i t e now  be  The TEM  guides  of  b and a  i n Chapters  3 and 4  are  t h e ' f i e l d s  relative  to  width  coefficients  and &  plane  are  reradiated.  calculated  coupling  equivalent  guides  between  (d,a)  i s  guides  these  d were  the aperture  the  used  to  i n the  f i e l d s  outer  from  the  guide.  value  &  +d/2<y<+d/2+a  Q 0  which  outer  (d,a,b)  values  driven  The  0 0  These  the  at  central  A  SHORTED  Fig.7.1  shorted  by d i s t a n c e  determine guides  the  and B  respectively.  of the  coefficient  d and a  GUIDES  coupled  from  coupling  OUTER  of  1  i s ,  the  from  f i e l d s  (6.21)  i n  the  given  f i r s t  outer  guide  by  i2ks A  A  l  the  R  (  0  guide  outer  the outer  6  OklT  a  *  )  of  guide.  second  into  0  '  a )  =  1  where  0Q  ( d  outer  driven guide the  " oo R  s  t  e  width  (7  reflection a,  coefficient  a n d s^^  i s  the  the value  guide  +d/2+a<y<+d/2+a+b,  guide  and the  must  second  be taken outer  +  - > 15  6  To determine  B (d,a,b) 0 Q  n  ( a )  1  f i r s t into  guide  i s  A ^ A (a,b)  the  depth  A  2  of  open  of  the  account.  guide  f i e l d s  The  into  end of  the  the coupling  outer  given  at  f i r s t i n the  from the  energy  both  second coupled  by  (7.16)  126 The  value  of  a  of  the  geometric  f i e l d  A  a  2  t  t  n  aperture  e  plane  i s  t h e sum  series  iks A  2  f oo  =  B  ( d  ' ' a  b )  +  1-R  where  s  i s  2  The  the  t o t a l  c a l c u l a t e d f i n i t e the  array  f i e l d s and  f i e l d s  guide  i s  p a r a l l e l  plate  In given  In  5"2  F i g . 7 . 5 b  given  =  the  P  b  =  f i e l d s  5  1 7 )  to  array  the  the  from p  by  i n  f i e l d s  radiated  f i e l d s  the  ray  by the  paths  from  i n  the  A  a  guides  (5.21)  by t h e from of  ( F i g . 5 . 1 ) ,  i n  outer  i s  consist  guides  f i e l d s  from  F i g . 7 . 1  caused  m u l t i p l i e d  r a d i a t i o n  of  f i e l d  outer  the  0  guide.  shorted shown  and (5.22) x  o  r  A  of  i n a  Scattering  2  outer  guides  i s  F i g . 7 . 5 - 7 . 7 .  edge  y=+d/2+a+b,z =0  (b) G(b,8)E(r> . 5  „  (7.18)  from  a  the  edge  y=+d/2+a+b,z=0  i s  by —  5"3  The  outer the  excited  f i e l d  f i e l d u  -C'(a) u  '  by  cU  of  from  pattern  by c o n s i d e r i n g the  ( 7  a d d i t i o n a l  Radiation  due to  F i g . 7 . 5 a  e  second  The  edges  the  ]  guides  waveguide  edges  c a l c u l a t e d  is  other  )  r a d i a t i o n  d i r e c t l y  simply  other  the  outer  (Fig.7.5-7.7)  F i g . 5 . 1  from  from  b  e  the  F i g . 7 . 2 .  radiated  a  pattern  adding the  (  of  r a d i a t i o n  of  of  0 0 ' ., , i k s 2 A  (b)  0 0  depth  by  depth  l  A  "  radiation  4  u  5  iV4 G(a+b,6)E(r)  -  .  „  4  and u  a  from from  -  the  the  5  „  are  b  edge  f i r s t  outer  i  / ^  C'  (a)E  (a) G  0  added  to  (b, e ) E  give  y=+d/2+a+b,z=0 guide.  In  ( r ) .  the excited  F i g . 7 . 6 a  the  (7.19)  f i e l d by f i e l d  127 u  a 3"3  from a  the  - C (b) _0  =  Fig.7.6b  given  u  is  a  n  C  =  3"3  a  )  the  second outer  (  —  radiation  (2b,6)E(r)  f i e l d  +  -  given  by  C ' ( b ) E (b) G ( b , 6) E ( r ) .  ^  3"3  (7.20)  0  from  b  u  from  In  p" 31  the  outer  edge  given  G(2a,6)E(r)  f i e l d  guide  same  the  edge  y=+d/2+a,z=0  is  from  I  T  L  /  4  C ' (a) E (a) G ( a , 6) E (r) .  the  edge  f i r s t  outer  guide  and would  was m i s s i n g .  u  ,  3  (  is  b  the  f i e l d  i n the  not  by r a d i a t i o n  Fig.7.7a  y=+d/2+a,z=0  (7.21)  excited  by 2  guide,  excited  P  -  be  the  from  ^  ^  v  2  f  r  present  f i e l d the  o  presence  m  t  i f  h  e  the  p ^  second edge  of  the  second  from outer  the  guide.  y=+d/2,z=0  is  by U  In  i s  by  b  u  the  y=+d/2+a,z =0  XT,/A G  4  3"3  In  f-.dge  -C'(a)  l"2  2~~ ' G(a  =  Fig.7.7b  the  f i e l d  u  •  6)E(r)  1  , ! ^ from  '  (7  the  edge  y=+d/2,z=0  is  22)  given  by b  u  C  =  n  (  b  )  - — — G(a+b,6)E(r)  +  B  i  T  /  4  C ' (b) E (b) G (a ,6) E ( r ) .  (7.23)  4v57 In  Fig.7.7c  the  f i e l d u  from  1 1 1 3  the  edge  y=+d/2,z=0  i s  given  by C'(a) =  c u  G ( a + d , 6 ) E (r)  -2  1 , 1 3  C ' (a) E (a) G ( d , 6) E ( r ) .  +  4v 7  4  /  2  The  fields  (7.22),  f i e l d from only  from the  second  one outer  (7.23) the  outer  guide.  and (7.24)  edge guide  are  y=+d/2,z=0 which  (7.24)  °  would  added excited not  be  to by  give  the  radiation  present  with  128 Similar  calculations  y=-d/2,z=0, P  and  from  6 P" 2  P  =  41  the  P^  edge  from  the  from  2  f i e l d s  the  y=-d/2-a-b  P^  edge  show  for  the  edge  y=-d/2-a,z=0  and  that (7.25)  -p" 1  - 3'l  =  P" 42  P  =  (  '  7  2  6  )  (7.27)  -P" 42  (7.28)  The  ray  are  the  paths  used  image  F i g s . 7 . 5 , 7 . 6 depth  of  and  p  A to of  .  ^  the  7.7.  outer  guides  This obtain  radiation the  must  complete  p  z-axis  The  multiplied  i s  the  f i r s t  y=+d/2+a+b  radiated  are  f i e l d  p total  =  P  the  by  the  be  added  P  4]/ 4  of  array  to  ?  and  P  2  those  sum o f  t  pattern  by  the  ( )  ,  0  a  t  (7.2) o  t  a  i  in  f i n i t e p  (  0  c o e f f i c i e n t  of p  p  P^  shown  caused  appropriate  radiation  quadrant  shadowed  so  o < e < TT/2 a l l  and  )  or  A  and of  b  (7'.3)  the  array  do  P  t l  +  +  the  second  =  P  t  A P (b) 2  quadrant  +  except  contribute  that  to  the  . sme Q  A, P" e 1 5  _ p total  not  edges  that  -ik  F  '  radiation  ,d+2a+2b, -ik ( )  In  2  F i g . 7 . 1 . In  at  calculate  about  and  the  to  A P" e 15  i k (  Q  for  e  ,d+2a+b, . . — )sin6 2  TT/2 < e < TT  ±t2|±2b  ) s i n e  .  (7  ( /  29) '^>  129  . , ,d+2a+b. . . -ik ( —) s i n e +  A P (b) 2  e  0  -ik l y s i n e  tV3  +  V 3 1  +  V32  +  ] e  - i k f ^ J s i n e +  A . P (a) 1 0  e  +  [ A ^  +  A P^]e  +  1  [A, P i 1 2  +  A P"]e 2 2  +  AjP  -ik  d — sine  +ik  — sine  2  J  + i k (~^-) (a)  sine  e  ., ,d+2a, . . + i k (——) s m S  +  [A P X  4  A  +  ^  +  A 2  +ik (  where  +  A.P"e  about  measured  appendages  e . , ,d+2a+2b, . „ +ik ( ) sine  given e = TT  S  in  with  patterns  simulate  the  of other  of  (7.30)  Chapter  that  O  patterns  compared  to  •  AND E X P E R I M E N T A L  E-plane  2  ) sine  1 6  radiation  are  4  (b)  are  NUMERICAL  The array  A P  P^  symmetrical  7.4  +  P ' ]e  an  outer  p t  otai  The  6. ( 2 l 1  "  e )  =  P  pattern  totai  ( 6 )  is  -  RESULTS  a  five  element  theoretical H-plane guides  waveguide  results  sectoral  and  horn  (e'-0-Tr).  the with  130  When consists  the  outer  of  a  essentially degrees  The  patterns  residue to  a  forward  d i r e c t i o n  Taking  higher  eliminate In  except  for  the  the  simulated  range  of  4 5 °  in  the  agree  remarkably  be  angles,  zero  at  control guides  the the  because  are  90-120 a  axis  all. in  small in  the  is  in  but  the  plane.  does  compared  element  direction  not  of  is  to  that  array  Over  the  presence  groundplane  patterns  aperture  [29].  agreement  found  boundary.  five  the  that  The  reduces  pattern  the  to  angles  the  shadow  an  is  angular  two  patterns  the  simulated  not  so  requires  the  good f i e l d  for to  . are  of  phase  of  the  aperture  f i n i t e  depth  that  in  the  pattern  may  be  adjusted  (Figs.7.12-7.18).  possible  for  the  five  compared  to  the  three  which  f i e l d s  is  Fig.7.10.  terms  to  application  parameters  90°  the  despite  the  pattern  about  beam  at  guides  relative  radiation  dB  groundplane  expected,  of  compared  in  e =  forward  well,  ©=±90  When  a  that  a  larger  [38]  ray-optical  in  As  are  d i f f r a c t i o n  case  The  There  the  here  on  the  boundary.  near  embedded  groundplane.  off  of  discrepancy  Fig.7.11  calculated  90°  shadow  fraction  depth  range  frequency.  methods  order  this  angular  calculated  calculus  within  an  e'=  the  i n f i n i t e  ( F i g . 7 . 8 , 7 . 9 ) .  the  at  along  of  lobe  on  (<ldB)  plane  are  over  depending  aperture  agree  single  constant  discontinuity  by  guides  may  element be  is  f i e l d  in  driven  More  element  it  control  array,  adjusted.  the  guide.  to  array  possible  suit of  parasitic Hence  a  the  In  pattern  there  general  the  particular  considered because  to  here are the  is as more five  131 element then may  patterns  the be  axis,  three  adjusted whereas  single  null  patterns and  have  is  the  the  three  the  beamwidth  patterns.  produce  two  axis.  more  radiation  The  in  less  then  is  patterns  can  produce  of  the  slopes  element about  phase  constant  steeper  nulls  patterns The  and  five  adjacent  element  beam  generally  back  smaller  element to  at  a  the  beam  only  five  forward  the  the  a  element  direction  three  element  patterns. As  in  accurate The  at  90°  three off  discontinuity  ray is  the  paths noted  into  in  which  case.  shadow  d e t a i l  edges  in  expected  a to  discussed The  in  be  the  an  patterns s t i l l  1  dB  is  up  calculation made  Additional  s i g n i f i c a n t  2.  depending  then  for  to  then  the  for  improvement  the  array  i n f i n i t e  depth  discussed  larger  these  more  number  d i f f i c u l t i e s may  the  three  element  in of be  array  chapter.  patterns  for  agree quite  0=60°  Thus  of  boundary.  discontinuity on  the  least  taking  d i f f i c u l t i e s were  array,  here  by  The  -3dB  Because  generally  comparable.  pattern array  (  no  d i f f r a c t i o n .  are  shadow  however  and  not  the  reduced  preceding  accuracy  along  patterns  is  element  experimental  the  patterns  boundary  greater  case,  axis  larger  Chapter  (Figs.7.12-7.15) within  is  five  in  the  +3dB  parameters,  more  beam  .quadruple  between  The  the  account;  beyond  varies  element  appears  with as  and  the  the  five the  good 3dB  accuracy  to  be  element  calculated  as up  the to  of  degraded  three  array ones  element  e'=150°), the  to  but  ray-optical  s l i g h t l y  when  the  larger. calculations  are  performed  for  arrays  for  132 which in  the  guide  the  phase  p a r a s i t i c  with  the  in  be top. of and  a l l  When phase two  the  aperture  lobes  Note  to  that  the  are  aperture  or  fields  exactly  f i e l d s  shown  patterns  not  case  a .  would  out  of  and  the  in  Figs.7.16-  are  tabulated  the  When  the  the  array A  that  on  the  because  might  across  the  whose  the  are  out  beam  axis  f i e l d  is  small  the  is  a good size  aperture  approximation  of  radiation  a  pattern  shape.  phase  of  the  dimensions further  each  as  aperture  considering  crude  and  beamwidth  guide  the pattern  sector  amplitude  f l a t  parasitic null  (Fig.1.3), a  very  deep  a  3dB  (Fig.7.17),  not  7.18)  are  curve  require  narrowest  s u r p r i s i n g  independently.  patterns  is  shape,  approximate  by  phase  (Fig.  is  the  in  in  is  sector  x)/x  the  beam  6? = ± 3 8  a  t h i s  determined  adjusted  at  to  (sin  expected  are  there  This  for  truncated  7.5  phase  has  f i e l d s  i n sign  array.  phased  the  aperture  these  pattern  f i e l d s  (Fig.7.16)  f i e l d s  are  The  of  make  in  patterns  However,  approximation  is  guide.  the  aperture  expected.  the  exactly  parameters  that  alternating  of  guides  to  i l l .  Note when  chosen  radiation  Relevant  Table  was  driven  corresponding 7.18).  depth  element  aperture and  cannot  improvement be  f i e l d  excited  be  in as  the in  a  array  of  array.  SUMMARY  The p a r a l l e l  radiation plate  from  waveguides  a  five  element  has  been  analyzed  f i n i t e the  same  ray-  133 optical 6.  methods  The  radiation  however, more  because  ray  paths  When  the  results  that  for  more  taken  into  guides  the  of  in  more  d i f f r a c t i n g  edges  of  i n f i n i t e  result  method  size  array  is  are  ray-optical  this  element  Chapter complex  and  hence  account.  ray-optical  calculus the  three  calculation  are  outer  residue  the  pattern  be  between  confidence  for  there  to  the  agreement by  used  is  and  quite  method  array.  that  the  calculated  good,  can  The  depth  which  give  gives  acceptable  pattern  also  agrees  the quite for  the  a  forward When  as  general  and for of  complexity  of  can  element  by  in of  to  three  accuracy the a  wider  a  of  of are  patterns  the  structure  adjustment to  type  element  of  A  Wiener—Hopf  simulated  guides  experimental the  a  by  s e n s i t i v i t y  the  outer  produce  approximation array.  lack  direction  degradation  array  calculated  embedded  the  calculated well  that  array  indicates the  with  well  the  sector  groundplane,  of  the  of do  array,  of  agree  shaped  quite a  method The  patterns  parameters,  depth  indicating  increases.  five  then  for  the as  slight as  the  element  the  three  a  good  a  small  including  pattern  in  structure.  f i n i t e not  which  pattern  surrounding  ray-optical  variety  technique,  such  134  Fig. 7.1 f i v e element array of p a r a l l e l p l a t e waveguides with outer guides shorted  AV r  1 -ri  b i  Fig. 7.2 f i v e element a r r a y o f p a r a l l e l p l a t e waveguides with outer guides of i n f i n i t e depth  d/2 d/2 —*—  i b t  a Fig. ray path  7.3  b F i g . 7.A ray paths(see  (see t e x t )  \  t  F i g . 7.5 ray paths(see  t  text)  text)  11  b Fig. ray paths  7.6 (see  text)  \  V  Fig. ray paths  7.7 (see  text)  137  angle (degrees) Fig.  7.8  R a d i a t i o n p a t t e r n of f i v e element a r r a y with outer guides of i n f i n i t e depth d=a=b=0. 450X.  138  angle (degrees)  angle (degrees) ig.  7.9  R a d i a t i o n p a t t e r n o f f i v e element a r r a y w i t outer guides of i n f i n i t e depth d=a=b=0. 3  139  0  CD  1  10  i Q. £  -20  JTJ 0.  -30  -90 Fig.  7.11  CD  Si_  -45  0  angle (degrees)  45  90  R a d i a t i o n p a t t e r n o f f i v e element a r r a y w i t h outer guides o f i n f i n i t e d e p t h , d=a=b=0. 4 5 0 A , c o m p a r i s o n w i t h [ 2 9 ] .  -10 H  O  CL  I>  OI  -20 H  -30  -90 Fig.  7.10  -45  0  angle (degrees)  45  90  R a d i a t i o n p a t t e r n o f f i v e element a r r a y w i t h outer guides o f i n f i n i t e d e p t h , d=a=b=0.450X,comparison w i t h [ 3 8 ] . this theory comparison  140  angle (degrees) Fig.  7.12  R a d i a t i o n p a t t e r n o f f i v e element a r r a y d=a=b=0.441X s,=0.838X s =0.597X. A=0.137 L -12° A=0.074 L - 1 2 ° .  141  CD 32.  -10  § CL  -20 H  >  -30  -180  •90  180  0  angle (degrees) 180  -i  90 in  cn a>  ro _c CL  -90  -180 -180  -90  0  90  angle (degrees) Fig.  7.13  R a d i a t i o n p a t t e r n o f f i v e element a r r a y d=a=b=0.407X s =0.774X s_=0.551X ft-0.171 L -55° A=0.076 L - 5 7 ° . t 2  180  142  -180  -I -180  •  •  1  •  -90  •  '  •  •  1  0  ^  90  angle (degrees) F i g . 7.14  R a d i a t i o n p a t t e r n o f f i v e element a r r a y d=a=b=0.373X S!=0.709X s =0.505X A=0.247 L -103° A=0.096 L - 9 8 ° . i z 2  '  180  - 3 0 - 1 8 0  - 9 0  0  180  90  angle (degrees)  180  -i  90  -90  - 1 8 0 - 1 8 0  -90  180  90  0  angle (degrees) Fig.  7 . 1 5  Radiation pattern of five d=a=b=0.356X A=0.308 i  L  - 1 3 4 °  S;L=0.677A  A=0.126 Z  element  array  s =0.487X 2  L  - 1 2 3 ° .  144  Fig.  7.16  Aperture f i e l d and r a d i a t i o n p a t t e r n o f a f i v e element a r r a y A=-0.215 A=-0.090 d=a=b=0.450X =0.610X s =0.356X. S l  2  145  Fig.  7.17  A p e r t u r e f i e l d and r a d i a t i o n p a t t e r n o f a f i v e element a r r a y A=+0.131 A=+0.073 d=a=b=0.450X S]_=0.857X s =0.610X. 2  146  Fig.  7.18  A p e r t u r e f i e l d and r a d i a t i o n p a t t e r n of a f i v e element a r r a y A=-0.131 A=+0.045 d=a=b=0.450X s =0.857X s =0.356X. 1  2  TABLE I I I R a d i a t i o n P a t t e r n Parameters  Figure 7.8a  A A  l  0  A  2  0  null(dB)  Beamwidth (de:grees) -lOdB -3dB -ldB  back radiation(dB)  0  39  54  76  -25  7.13a  +0.131  +0.073  0  12  21  64  -28  7.13b  -0.215  -0.090  -9  50  59  78  -26  7.13c  -0.131  +0.045  0  31  43  67  -32  148 Chapter COUPLING  The  coupling  waveguides that  for  First  reflection  a l l  are  which  establish to  the  8.1  each i t .  e  Q  r  .  The in  to  d i f f r a c t i o n ray  used  and  from  problems  paths  here.  similar  w i l l  The  The  the of be  is  of  the  successive  solved  analysis  t r i p l e f i e l d  f i e l d two  to  guides.  including  inspection.  calculated  these  manner  non-staggered  up  by  a  f i r s t  to  restricted  PROBLEM  an  from  0  The  isotropic an  edge  d i f f r a c t e d  p  l i n e  is  source  incident  (not  the  on  total)  =  E(r)located  that f i e l d  edge  at  observed  at an at  6±8 such  u  at  of  paths  is  in  staggered  case.  Fig.8.2a  distance  ray  adjacent  found  coupling  ray  notation  CANONICAL  angle  and  GUIDES  for  be  determined  occur  the  TEM  In a  by  preceding  rays  w i l l  possible  represented  STAGGERED  coefficient  (Fig.8.1)  d i f f r a c t i o n  ray  BETWEEN  8  the  that  .  m+1  =  c o s (  __o  }  Q  G.(r ,e ,6)E(r) n  d  0  d i f f r a c t i n g  0  edge,  where  is  given  by  (8.1)  149  -in/4 G,(r.6  d  0  .6)  =  0  f~ /iT  _  i  k  cos(9-6 )  r n  -e  cos(0+6) °  °  ^ 6+6  F [ / 2 k r ^ c o s (——)]) , (8.2  a > 0  F(a)  i s g i v e n by  (2.4)  a  < 0  and E ( r )  )  (8.2a)  = -F(-a)  F(a)  (—)]  u -ikr  F'(ci)  6-9 F [ v ^ k r ~ cos  {-e  i s g i v e n by  (2.10)  150 In  each  at  the  case  the  diffracted  d i f f r a c t i n g  f i e l d  is  a  line  on  a  Fig.8.2b  half-plane  d i f f r a c t e d may,  U  a  =  G  d  ( a  be  '  at  e 0  '  wave  angle  which  (2.2),  l  plane  edge  f i e l d  from  centred  edge. -ikr  In  source  is  u 6  = e .  Q  ) is  0  An  uniformly  written  cos(6-6  expression  v a l i d  for  a l l  for  the  angles  6  as  '  e ) E ( r )  incident  (8.3)  where  G (a,6 G  and  a  d  is  point.  ( a  'V  the  If  6 )  distance  where  8.2  o  D(e , 6 )  „  CALCULATION  waveguides half-planes of  determine z>0)  into  incident  8 4  from  the  edge  to  the  observation  D(6 .8i,  (  given  by  adjacent  consisting  of  y=a,z>l,  y=0,z>0  the the  of  the  f i e l d s  f i e l d U i  =  guide  guide _ i v 1  three  )  from  and l=a the  (0<y<a,  p a r a l l e l - p l a t e  perfectly  and y=-d,z=0,  coupled  e  5  COEFFICIENT  s e m i - i n f i n i t e  the  top  staggered  -  (2.9).  OF T H E COUPLING  two  stagger  8  0  i s  Consider  angle  <-'  ka>>l,  G (a,e ,6) A  ,9)  —iuo  =  conducting  where tan y  driven z>0).  f > o .  We w i s h  guide As  i s  the to  (-d<y<0,  before  the  z  (8.6)  151 in  the guide  i s decomposed  the  half-plane  2TT  respectively.  edges  into  y=-d,z=0  Both  waves  the  at  have  waves  incident  angles  6 = o  on and  value  1  =  (8.7)  2  i at  a n d y=z=0  plane  u  two plane  edges.  The  singly  incident  f i e l d  y=z=0.  diffracted given  u  From  (2.7)  f i e l d  by (8.7)  i s  i s  shown  i n Fig.8.3a. The  diffracted  the diffracted  f i e l d  i s  at  given  the  edge  by a  line  source u  j D(2TT,0)E (r)  =  a  located  at  the d i f f r a c t i n g  coupling  f o r  as  source  a  line  single  Q  The i s  (3.11),  contribution  found  (3.12)  to  by treating and (3.2).  the (8.8)  Thus  in/4  «  ( 1 )  edge.  d i f f r a c t i o n  and using  . . jz A  (8.8)  0(2,,0),  (8.9)  2> 2 k a /  which  i s The  c.  identical doubly  In Fig.8.3b  d i f f r a c t e d edge  y=a,z=l  u  located the  at  =  b  1  at  distance  diffracted source  to  (3.17)  diffracted  f i e l d s  the incident the edge  2  f  a  i  at  shown  u  From  f i e l d  given  (8.3), i s  i n Figs.8.5b and  a  as  line  by  (8.7),  observed  t h e two  the edge  i s  at  the  source  -*)E(r)  the d i f f r a c t i n g between  are  f i e l d  y=z=0.  the diffracted  iG'(c.27T f  i n t h e TEM case..  < 8  edge,  where  edges.  y=a,z=l  This  a n d from  c  = =  2  /2  /£ + /£ + a  f i e l d  i s  i s  (8.1) g i v e s  1 0 )  again a  line  152  u  Fig.8.3c in  =  b  j G^<C,2TT,J - v ) G ( c , ^  -Y,2n)E(r) .  L  d  2  shows a ray path which i s i d e n t i c a l t o  Fig.3.2c  (8.7),  f o r a non-staggered  contribution  to  diffraction X  =  the c o u p l i n g  i  9  s  that  shown  i s given by  by (3.35).  i v e n  coefficient  A  g  from  The double  i s given by e  1  1  T  /  *  4 (  2/1 k a  S  C  )  2  _  y  )  2  d  31  G  d  - , ,  2  ,  )  E  -  I  K  £  2  +  c^(d)C (d)]  .  (8.12)  \jf = O , (8.12) i s e q u i v a l e n t t o (3.36) i n the TEM case.  If  The t r i p l y d i f f r a c t e d e. Fig.8.3d shown  and  i n Fig.3.2d  u  i s again d i f f r a c t e d  source  3  j G ^ ( c  (  2 » ^ - ? ) G  a  ( c  (  that  guide. Here ^ i s (3.38)  f o r the TEM case. In e  i s given by (8.10) and ^  2  2-n r e p l a c e d by 2JL _y  by (8.11) with the f i n a l  a line  to  i s given by ( 3 . 3 7 ) , i s given by  e  This f i e l d  =  i s identical  non-staggered  i s given by (8.7), u  given  U  for a  which  i s given by (3.39) s p e c i a l i z e d  3  Fig.8.3e is  f i e l d s are shown i n Figs.8.5d and  shows a ray path  given by (8.7),  at  guide. Here ^  i s given by (3.34) and ^  (2)  (8.11)  at y=z=0 and from (8.1) g i v e s  ^ - T ^ - * ) G  d  ( c i - t O ) E ( r )  (8.13)  the edge. In F i g . 8 . 3 f the i n c i d e n t  diffracted  at  =  u  given  by  (8.7) i s  the edge y=-d,z=0. From (8.3) as observed a t  y=z=0 the d i f f r a c t e d uf 1  field  field  ;GMd,0,^)E(r) 2 d 2  i s a line  source (8.14)  153  located  a t the d i f f r a c t i n g  edge.  This  field  d i f f r a c t e d a t y=z=0 and from (8.1) g i v e s a l i n e u  o  =  2  located  ^ G ' ( d 2  d  (  0 ^ ) G . ( d , ^ 2  d  2  2  , 3  a t the d i f f r a c t i n g  =  k  G  2  A  d  (8.15)  edge.  (d,0,J)G. ( a , - V ) G . ( c , ^ J 2  d  g i v e n by (8.10) and by  3TT/2  looks l i k e a l i n e u|  2  the i n c i d e n t f i e l d u  In F i g . 8 . 3 g  replaced  d  2 2  =  source  -¥)E(r)  This  field  d i f f r a c t e d a t y=z=0 and from (8.1) g i v e s a l i n e u  i s again  i s again source  -?,2Ti)E(r) .  (8.16)  i s g i v e n by ( 8 . 7 ) ,  is  i s g i v e n by (8.11) w i t h t h e f i n a l 2TT  . This f i e l d i s reflected  a t y=0,z = l and  source  i G ^ ( c , 2 T T ^ - H ' ) G  d  ( c , ^ - ^ ^ ) E ( r )  (8.17)  x  at t h e image p o i n t y=-a,z=l, where t h e n o t a t i o n a  field  i n the f i g u r e  l a b e l l e d x t h a t has been d i f f r a c t e d m  times and a l s o r e f l e c t e d .  This f i e l d  y=a,z=l and from (8.1) g i v e s a l i n e 3  at  =  2  G ' (c ,2TT d  t h e edge.  coefficient (3.39),  2  - T ) G (c ,~r d  u— denotes  i s again d i f f r a c t e d a t source  ,~r) G ( 2 a ,^r,2 n) E ( r )  2  2  The t o t a l  (8.18)  2.  d  contribution  from t r i p l e d i f f r a c t i o n  t o the coupling  i s found  by  ( 8 . 1 3 ) , (8.16) and (8.18) a s l i n e sources and u s i n g  ( 3 . 1 1 ) , (3.12) and ( 3 . 2 ) . Thus r  treating  1^/4 ,  +  G^(c,2TT,i - ) G ( c , ^ T  d  - y , ^ L -Y)G (c d  1 -w, ) 0  154  +  G  total  d i f f r a c t i o n  (8.19).  Note  staggered If  Y  along  is  no longer  a  It  but  arise  s  ^  o  of  does  as a  angles  less  of i  function  guides, of  by  (8.9),  for  the coupled when  for  the  of  which  i s  pronounced  terms as  v  (3.12) a  n  ray  5>  n  w h e r e & =0, n  considered.  d i f f r a c t i o n  increased.  A (a,a)  a and d and for  As  fine  is  = ~n~/2- Y,  coefficient  widths  a  B  here  were  coupling  i n Fig.8.4.  i s  to  guide  angle  ( i . e .  modes  reveals  not  the modal  the coupling coefficient.  d i f f r a c t i o n  apply  conversion  field  y  the guide  single  (non-  v = o  i n t h e TEM c a s e  waveguide  (8.12) and  i n the staggered  the ray-to-mode  angle  including  (3.44).  f i e l d  arises  the  up to and  g  given  s  a r e shown  v  A (d,a)  the restrictions  not occur other  (8.19)  not . v a l i d  the stagger  i f  behaviour  double  3 )  ,  by t h e sum of  i s  because  values  staggered  given  V  i k  d  ^)E(d)]  boundary  Numerical  becomes  d  the coupled  RESULTS  calculated  the  i f  A  NUMERICAL  average  (  d i f f i c u l t y  complement  would  i s  °>  =  valid  This  various  0  because  shadow  Fig.8.4).  8.3  C  (8.19)  that  is  the  that  guides)  Note  f i e l d .  t  d  coupling coefficient  t r i p l e  (8.1).  -«C (cf-T,f)G (2a^2,)e-  ( d , 2 , l  8x7  +  The  d  for  the  non-  provides  the  Addition structure  of  which  155  Exact r e s u l t s available  f o r comparison.  and without  i n accuracy  coefficients  are  not  However, on comparing curves i t appears  as the stagger  there i s  with some  increases.  SUMMARY The c o u p l i n g  parallel  the  coefficient  p l a t e w a v e g u i d e s was  The n u m e r i c a l from  coupling  triple diffraction,  deterioration  8.4  f o r the  r e s u l t s show a  non-staggered  increased.  These  calculations  involving  between found  adjacent  by r a y - o p t i c a l  moderate  and  will  methods.  gradual  r e s u l t s as the angle  calculations  staggered  be  change  of s t a g g e r i s useful  an a r r a y o f s t a g g e r e d w a v e g u i d e s .  for  Fig.  8.1  two a d j a c e n t s t a g g e r e d p a r a l l e l p l a t e waveguides  Fig. 8.2 l i n e s o u r c e and p l a n e w a v e i n c i d e n t on a h a l f p l a n e  157  u  ray paths  ray  F i g . 8.3 for coupling  coefficient  F i g . 8.4 path (see t e x t )  158  0  i  180  i  F i g .  8.5a  TEM-TEM  c o u p l i n g  between  s t a g g e r e d  waveguides  <y=cr s i n g l e  d i f f r a c t i o n  s i n g l e  and  d o u b l e  s i n g l e  and  double  d i f f r a c t i o n and  t r i p l e  d i f f r a c t i o n  -45  1.5  1.0  0.5  2.0  d/A 180  i  8.5b  TEM-TEM c o u p l i n g b e t w e e n s t a g g e r e d  waveguides  4*= 15°. single diffraction s i n g l e and d o u b l e d i f f r a c t i o n s i n g l e and d o u b l e and t r i p l e d i f f r a c t i o n  160  -45  0.5  1.0  1.5  2.0  d/A Fig.  8.5c  TEM-TEM c o u p l i n g b e t w e e n  staggered  waveguides  Y=30°. single single single  diffraction and d o u b l e d i f f r a c t i o n and d o u b l e and t r i p l e d i f f r a c t i o n  -45 I  —>  0.5  0  1.0  1.5  2.0  d/A  180  -i  1 3 5 •{  Fig.  8.5d  TEM-TEM c o u p l i n g between  staggered  waveguides  Y=45°. single single single  diffraction and double d i f f r a c t i o n and d o u b l e and t r i p l e d i f f r a c t i o n  162  0 1  Fig.  8.5e  TEM-TEM c o u p l i n g b e t w e e n  staggered waveguides  4<=60°. single single single  diffraction and double d i f f r a c t i o n , and d o u b l e and t r i p l e d i f f a c t i o n  163 Chapter RADIATION  FROM  A STAGGERED  PARALLEL-PLATE  WAVEGUIDE  FORMULATION  9.1  The  radiation  waveguide  pattern  (Fig.9.1)  in  methods  similar  to  5).  pattern  was  This  [42]  using  equivalent  double  up  including  and  staggered  guide  d i f f r a c t i o n pattern a l l  i s  the  ray  shadowed the  path  made  w i l l  pattern  be  w i l l  width  a  taking  found  a  for  of  and  l=a  tan  radiated  into  the  space  before  decomposed  the  into  here  one.  in  f i e l d s  a l l  in  Fig.9.2  shown  a  Note  that  given  order  total  that  then  ,  f i e l d s  angular  the  of  radiation  represented  some  the  by are  directions. by each  total  ray  radiation  angles. p a r a l l e l - p l a t e  where .  paths  represented  two p e r f e c t l y  'b/  ray  for  some  Rudduck  The  f i e l d s  account  and  terms  The  the  (Chapter  including  are  paths  and  y=+a/2,z=l,  stagger  As  into  the  Ryan  only.  sum o f  s e m i - i n f i n i t e  and  by  ray-optical  guide  d i f f r a c t i o n .  contribute  section  consisting  y=-a/2,z>0  the  by  theory  ray  calculated,  be  Consider  previously  more  found  p a r a l l e l - p l a t e  non-staggered  t r i p l e  up o f  do not  following  a  non-staggered  paths,  and  for  i s  considered  h a s many a  staggered  d i f f r a c t i o n  f i e l d s  then  a  d i f f r a c t i o n  the  to  of  T E M mode  found  edge to  the  those  representing  In  9  outside  the  incident  f i e l d  two p l a n e  waves  conducting  Y  We w i s h  waveguide  *  > 0  to  s  t  n  of  half-planes e  determine  angle the  of  f i e l d s  guide. (8.6)  incident  in  the  on the  guide  i s  half-plane  164 edges. 6 =0  has  o  top  T h e wave  edge  value  incident  on the  (8.7)  that  y=+a/2,z=l  at  at  bottom  edge. has  6 =2T 0  edge  y=-a/2,z=0  T h e wave  incident  at  on  the  value  -ikJ,  at  that  9.2  edge.  CALCULATION  The b.  In  singly  F i g . 9 . 2 a at  f i e l d  line  a  the  (8.7)  at  the  (8.7)  D  f i e l d  y=+a/2,z=l.  shown  u From  i n Figs.9.2a  given  by  (2.7)  the  and  (9.1)  i s  diffracted  T , , e ) E ( r )  2  (  Fig.9.2b  i s  at a  the  the  edge  line  incident  f i e l d  y=-a/2,z=0.  From  In  f i e l d  i s  a  line  2  )  by the  source  Fig.9.2c the  -  (2.7)  {  at  9  given  u  D  d i f f r a c t e d  d i f f r a c t e d  are  i (0,6)E(r)  edge.  i s  {  In  f i e l d  =  b u  edge  d i f f r a c t e d  d i f f r a c t e d  f i e l d s  source  edge.  i s  PATTERN  incident  the  =  A U  at  d i f f r a c t e d the  d i f f r a c t e d i s  OF RADIATION  the  edge  incident  f i e l d  y=+a/2,z=l.  u  From  ±  9  -  given  (2.7)  3  )  by the  source  -ik£ u  =  1  —  D(2Ti,2Tr-e)E(r)  located  at  from  lower  f i e l d  the  u-r  the  i s  d i f f r a c t i n g  guide a  line  at  (9.4)  edge.  y=-a/2,z=l-a  source  (9.4)  This tan f  located  f i e l d . at  i s  The the  reflected reflected  image  point  165 y=-3a/2,z=l. is  The radiation  formed  located  by the  at  the  three two  pattern  line  from  sources  guide  edges  single  (9.2), and  d i f f r a c t i o n  (9.3)  the  and  image  (9.4) point  respectively. The In  doubly  Fig.9.2d  This  f i e l d  gives  a  u  =  d  2  at  is  i  is  line  located f i e l d  u  =  -ik£ ^-z  the  y=+a/2,z=l,  at  e 2  located (9.1)  =  at  and \  y=-a/2,z=0  located  at  shown  u  is  d  in  Figs.9.2d-h.  found  y=+a/2,z=l  (8.3).  from  and from  (8.1)  (9.1) the  looks  point  e-  at  diffracted f i e l d  a  line  at  is  , «  d i f f r a c t i n g found from  and from  G  (8.1)  . (c 2v 21  f i e l d  a  This  incident the  line  edge source  as  is  line  at  observed  at  located  at  again  diffracted  source  „( 2 d 3t ,^,6)E(r)  » />  A  fi  (9.7)  F r  d  edge.  In  (8.3). gives  F i g . 9 . 2 f  This a  -*>G ( c , J  d i f f r a c t i n g  reflected  (9.6)  f i e l d a  is  and,  source  gives  G ' ( 2 d , 2 T i , 3^ -1),G. 2  This  y=-a/2,z=l  (8.1)  - ^ u , . , . ,  is  the  the  (9.6)  edge.  y=-3a/2,z=l.  -iki.  the  is  Fig.9.2e  diffracted  like  and from  y  In  (9.5)  GU2d,2nAE(r) a 2  S L ^ l  f  at  edge.  d i f f r a c t i n g  u|  y=+a/2,z=l  u  by  half-plane  image  and  diffracted  (8.1)  2  lower  (8.7)  d i f f r a c t i n g  From  1 at  by  are  j G M : , 0 , T - f l G J c , 7 - ? , e ) E ( r ) 2 d 2 d 2  given  u*  the  again  the  y=+a/2,z=l.  the  given  fields  source  u  located  diffracted  edge.  f i e l d  line  u is  ±  is  by  diffracted  at  source  -Y,6)E(r)  In  given  Fig.9.2g  (9.8)  VL  is  given  by  166  (8.7)  and ^  y=+a/2,z=l  i  and from  u^ =  the  at  the  lower  f i e l d  h a l f - p l a n e like  a  h  _  £  =  located  at  the  like  image  point  those  above,  u  i  line  = ^  u  3  .  f i e l d  i s  tan  i  d i f f r a c t e d  .  0  given  s  reflected The  located  (9.9)  ^  is  at  by  at  r e f l e c t e d  at  the  image  and  (9.1)  y=+a/2,z=l  and  u^ from  (9.10)  at  edge.  This  f i e l d  y=-a/2,z=l-a  line  source  tan  is  reflected The  e  (9.10)  at  reflected  located  at  the  d i f f r a c t e d in  the  by  radiated  Figs.9.2i-p.  expressions  inspection.  In  f i e l d s  are  represented  Using  methods  the  f o r  Fig.9.2i  the  similar  to  f i e l d s  can  be  radiated  f i e l d  i s  source  G  ;  located line  a  diagrams  determined a  Fig.9.2h  (9.9)  y=-3a/2,z=l.  t r i p l y ray  This  source  at  source  y=-a/2,z=l-a  d i f f r a c t i n g  u-  the  d i f f r a c t e d  d  f i e l d  by  line  is  3TT 3TT U\-elr\ G' (2d,2TT,^-)G (2d,—,2T,-6)E(r)  h a l f - p l a n e  The  a  f i e l d  source  lower  looks  at  f i e l d  line  _ I K 5  u  This  edge.  line  In  This  gives  (8.1)  the  a  y=-3a/2,z=l.  (9.6).  by  gives  (8.1)  d i f f r a c t i n g  u2 l o o k s  point  ,  ~ G' (c,0,~ -v)G (c,~r - f , 2 7 1 - 6 ) E ( r ) 2 d 2 d I  2  located  found from ( 8 . 3 ) .  s  (  c  '  at  2  i  ' T  ,  ,  ,  G  d  (  y=+a/2,z=l.  c  ' r  In  ,  ,  ' r -  )  G  Fig.9.2j  d  (  c  the  ^ '  w  l  E  (  r  l  radiated  ( 9  f i e l d  -  i s  u )  a  source  Ic (c.0.|-«G <c,2l. a.„  located  i  a  at  y=-a/2,z=0.  T  In  G a ( e  .l-,..,E(r,  Fig.9.2k  the  radiated  f i e l d  is  a  167  line  source  k  e  located line  „, -  at  =  ^  line  at  -u, T r - 6 ) E ( r )  (9.13)  - » ) G . ( c , ^ -V.2iT-e)E(r)  Fig.9.21  In  the  d  2  radiated  f i e l d  is  a  - ^ e ) E ( r )  ( c , f  Fig.9.2m  ( c ^ - ! ^ ) G  a  y=+a/2,z=l.  J ^ ( c . O ^ - m  -  at  the  (9.14)  radiated  f i e l d  is  a  In  a  ( 2 a , ^ 8 ) E ( r )  Fig.9.2n  the  (9.15)  radiated  f i e l d  is  a  Figs  Fig.9.2o  The  from u  y=-3a/2,z=l  _ "  the e "  i  k  is  and  these  is  n  d  ( 2 a , ^ M ) E ( r )  is  by  (9.1)  and  which  d i f f r a c t e d  ^ d (2a,2Tr,  is  paths  given  resultant  (8.1)  ray  f i e l d  £  2  ( c ^ - f , f ) G  9.1m  reflected  (3.26)  d  (9.16)  y=-3a/2,z=l.  contribution  o 2  i - .3JL  _1  source  In  U  v  ( 2 a , ^ - Y ) G  d  y=-a/2,z=l.  j G ' ( c , 0 i - T ) G  =  located  In  _  source  line  3  1  In  G - ( 2 a , 2 T r , ^ ) G  at  located  u  .  y=-3a/2,z=l.  —  located  3  „,._  - ¥ > G Ac , J  source  u*  U  3_l  G' (c , 2 TT  at  f i e l d  not  calculated u°  appears  the  valid  edge  is  like  and  separately.  given a  the  line  y=+a/2,z=l  by  (9.6).  source and  at from  is  3TT 1 i —) [— E ( 2 a + r )  * —  /  4  —  E ( 2 a ) E ( r ) ] .  (9.17)  2V^TT  The at  f i e l d  u°  y=+5a/2,z=l, which  is  y=-3a/2,z=l  is the  made is  made  other  up of  again  up of at  two  two  line  sources,  y=+a/2,z=l. line  d i f f r a c t e d  sources at  one  centered  The r e f l e c t e d at  y=-7a/2,z=l  y=+a/2,z=l  and from  f i e l d and (8.1)  168 the °  u  resultant _  e  =  3  _  4  G  f i e l d  i s  ' ( 2 a , 2 T T ^ - ) G (4a,^|,6)E(r) d 2 d 2 f  -ikJ.  iTr/4  eIn  tan  ^v^,^,  i s ,  e  ^  ;  2  F i g . 9 . 2 p  ,  2TT-e  G'(2a,2irA  -» — / - » — , d  ,  r e f l e c t e d  at  The r e f l e c t e d  ,  — E(2a)G ( 2 a , ^ , 6 ) E ( r ) . given  by  the  lower  f i e l d  u|  (9.18)  2  (9.8)  with  half-plane looks  replaced  e  at  l i k e  a  by  y=-a/2,z=l-a  line  source  -ik£  U3  =  — G (2a,2Tt,^-)G (4a,^,2Ti-e)E(r)  ~  d  e~ -  ikt TTT  d  at  The  t o t a l  up to (9.5), as  t r i p l e  the  (9.4), as  The  sources  radiation  f i e l d s  , (9.16)  which  i s  These f i e l d s to  (9.3),  appear  Q  the  up  sum o f  s  (9.(3),  (9.18).  The total  f i e l d s  upper  by  y=-a/2,z=0  the  y=-3a/2,z=l,  pattern  | - < e < ^ + 4  ,  i s  d  0  These  the  given  and  edge.  edge  from  as  given  by  and the  line the  (9.|9), image  of  edge.  for  =  the  by  (9.10), at  are  (9.15)  lower  The t o t a l  - i k — sin6  S„  at  given  (9.9),  l i n e  (9.11),  the  (9.(4).  edge.  d i f f r a c t e d  Q  sources  are  and  '  d i f f r a c t i o n  (9.7),  line  (9.19)  2  y=-3a/2,z=l. S  d i f f r a c t i o n  at  -3  point  from  upper  sources  4  E(2a)G_(2a,^-,6)E(r)  d i f f r a c t e d  of  appear  /  f i e l d s  (9.12)  sources  the  image  appear  t r i p l e  sum  the  (9.2),  f i e l d s  (9.8),  T  /2T  2  y=+a/2,z=l of  7  f  located  sum  I  Gl(2a,2Ti -J)  — 4  edge  d  S'  0  e  formed given  -ikJicosd e  by by  these  apparent  line  169  +ik — s i n e S" e o  +  +  +ik — " ' e  S  e  0  where  the  adjust  for  to  distant  a  some  of  exponential the  the  be  path  the  l i n e  radiation  considered  0  =  S  0  6  =  For  ^  -  other  sources  angular  are  shadowed  are  four  into  l i n e  and  shadow  TT  and  S' e * <  ranges, do  boundaries  five  regions  which  For  o < e < n/2 - v  The  o ~o b  Y<e< -j-* £  d +ik — s i n e  -ik£cosO e  +  e  (9.22)  2  < 2,  9  ^-j - y which terms  i s  the there may  into  radiation  this  c l e a r l y  not  shadow  boundaries  are be  reduced  pattern  The  symmetrical  d i s c o n t i n u i t i e s  account.  discontinuity e < j - ?  {  9  2  3  )  e  pattern At  must  37T  — +  +ikfsine b  not  (9.21)  d • - i k — sine 0  the  which  -ik£cos6  TI  — " y < 6 < —  s  factors  e  TT  For  There  array  from  In  (Fig.9.3).  - i k — sine S  the  point.  pattern  separately  are  sources  pattern.  (9-20)  lengths  observation  to  the  -  terms  d i f f e r i n g  apparent  contribute dividing  -ik£cos6  sine  2  at  For  e = ~ - y, i n the  taking  by  the  (9.2),  because  e = TT/2 of  because  double i t  u  ^,  order  i s  shown  pattern,  d i f f r a c t i o n  and  (9.4)  shadowed  d i f f r a c t i o n u  c a n be  and  d i f f r a c t i o n  single  (9.3) b  ^ + y  radiation  higher  example  given  addition  discontinuity  by  6 =T .  about  that  2  has  a  for  eliminates (Fig.9.4a)  +  b 1  i k f s i n e  1 -,) ] e <+  [  6  M  „  a  +  2  u  if  the  u  [ e - ^ - Y M  d  asymptotic  discontinuity can  c u x  be  e  7  0  -ik£cose e  2  form  (9.24) is  (8.5) by  by adding  3d + + i k— sin6 it , 2 [6= ] e  1  +  at 6 = TT/2caused  eliminated  -ik£cos6  [6-<J-¥> ] e  d  - i k - sine =  " i k f s i n e  +  used.  shadowing  because  -ik£cose e  +  u  S i m i l a r l y  e  r  2  for  e < n/2  (Fig.9.4b)  TT  o  of  the  +  [6= j ] e  d - i k — sine 2  -ik£.cos9 e  d - i k — sine [6= \ } e  = Similarly  the  C ^  discontinuity  -ik£cos9 e .  ate  =  (9.25)  +^ c a u s e d  by  shadowing f  TT  for  1  of  8 > — + y  can  be  eliminated  by  adding  "  because  2  (Fig.9.4c) [6=(j+Y)~]  3d —  +ik e  sine e  +  +ik =  [ 6 = ( j +^) ] e +  Similarly for  the  —  -y  can  be  u  2  [e=(j+«n~]  sin6  e  sine .  2  discontinuity  e >  d +ik—  -ik2.cosS  ate  (9.26)  = -^-ycaused  eliminated  by  by  shadowing  adding  u  of  because  2  (Fig.9.4d) d — sine  -ik  uJ [6=(^  e  = However not  the  2  r  because  of  + u* [6=(^- -*')'] e  these  eliminate the and  d +ik —  ,  <3 .„ +ik — sine -3TT + 2 [ e = ( — -H-) ] e  addition  regions  e  2  o  completely  pattern, some  f u  -ik2cos9  added have  .  double  (9.27) d i f f r a c t i o n  discontinuities terms  sin?  2  are  in  themselves  discontinuities  at  terms  the  does  radiation  shadowed the  in  shadow  171 boundaries.  However,  d i f f r a c t i o n  terms  d i f f r a c t i o n  terms,  double the  be the  eliminated.  double  terms  by  to  the  those  in  effect  reduce  in  double  the the  the  of  double  the  single  adding  the  d i s c o n t i n u i t i e s  terms  d i f f r a c t i o n  d i f f r a c t i o n  are  terms.  can  t r i p l e  discontinuities  terms  be  much In  added  is  pattern  calculated  in  in  terms  which  less  then  reduce  the  can  each  remain those  summary,  to  terms,  in  in  higher  the  order  discontinuities  s  for  o  the  exactly  staggered  the  same  way  as  guide  of  .  is  s  Q  it  that  ~ (B) Q  ray  mirror  paths image  which about  NUMERICAL  The plate  sectoral consist  horn of  about  a  (2TT-9)  =  S  be  considered  z-axis  of  AND E X P E R I M E N T A L  waveguide with  must the  radiation  compared  at  is  overall  certain  Again,  radiation  F i g . 9 . 5  9.3  the  then  in  boundaries.  The  The  less  adding  d i f f r a c t i o n  shadow  found  that  d i f f r a c t i o n  d i f f r a c t i o n at  much  so  d i s c o n t i n u i t i e s  t r i p l e  discontinuities  pattern.  S i m i l a r l y , the  is  d i f f r a c t i o n  radiation  the  pattern  calculated the  for  a  single 9 = 4 0 °  measured stagger  of  off  the  those  find  shown  s  in  are  o  the  F i g . 9 . 2 .  RESULTS  a  single  by  staggered  ray-optical  E-plane angle  asymmetrical  to  patterns  of  ^ = 4 5 ° .  lobe  guide  with  axis  p a r a l l e l -  methods of  the  is  H-plane  The  patterns  maximum  amplitude  (Figs.9.6a-d).  The  172 d i s c o n t i n u i t i e s  which  might  be  expected  on  the  four  shadow  i  boundaries very is  small  not  but  when  used  a l l  6=-135°,  at  there  other  a  agreed  except  large  very  are  to  remarkably  a  changes is  The  experimental  by  Ryan  here  i f  to  £ = + 4 5 ° varies contrast  essentially patterns  within  1  dB  and  Rudduck  only  [42]  are  the  ray  paths  using  only  single  in  Fig.9.7.  pronounced  the  double  The  d i s c o n t i n u i t i e s  d i f f r a c t i o n and  is  very  single at  pattern similar  and  the has  to  the  0°  to  pattern. patterns  guide  and  shown  d i s c o n t i n u i t i e s  radiation fixed  calculated  has  but  gradually  width  from  becomes  for  stagger  are  that more  shown  of  a  in  angles F i g . 9 . 8 .  non-staggered  from The guide  pattern as  asymmetrical.  SUMMARY  The plate  is  theory  (8.5)  at  in  range.  obtained  pattern  small  increased  9.4  (|©|<90°)  ray-optical  If  phase  phase  are  d i f f r a c t i o n  at  The  the  patterns  boundaries,  The  small.  3dB  are  considered.  d i f f r a c t i o n  t r i p l e  used.  about  direction  calculated  d i f f r a c t i o n  shadow  is  (Fig.9.3)  where  the  those  Radiation double  of  +45°  angles.  patterns  s i m i l a r  9.1a-h  with  (8.5)  remain  forward  this  generally  The  form  and  discontinuity  guide  over  at  a  the  non-staggered  constant  75°  is  in  - 4 5 ° ,  asymptotic  d i s c o n t i n u i t i e s  considerably to  the  - 9 0 ° ,  radiation  waveguide  pattern is  of  a  calculated  single by  staggered  ray-optical  p a r a l l e l  methods.  The  173 d i s c o n t i n u i t i e s are  very  double between  expected  small,  even  d i f f r a c t i o n this  confidence geometries.  when  are  theory  that  along  both  the terms  four  shadow  up  and  considered.  and may  the be  to The  applied  including  good  experimental to  boundaries  agreement  results more  only  gives complex  d/2 H  F i g . 9.1 s t a g g e r e d p a r a l l e l p l a t e waveguide  \  \  \ \  /Ti Tin m  n  F i g . 9.2 ray paths f o r r a d i a t i o n p a t t e r n  calculation  vUTi  175  II  III  ;  H  iv  shadow b o u n d a r i e s  r/  F i g . 9.3 f o r staggered p a r a l l e l  plate  +  +  F i g . 9.4 c o n t i n u i t y f i e l d s a c r o s s shadow s e e e q n s . (9.24) - ( 9 . 2 7 ) .  boundaries  waveguides  176  1 —j d/2 d/2  F i g . 9.5 staggered  parallel  plate  waveguide  -180  90  -90  0  90  angle (degrees)  n  angle (degrees) Fig.  9.6a  R a d i a t i o n p a t t e r n o f s t a g g e r e d w a v e g u i d e ¥=45°. u s i n g (8.5) u s i n g (8.4)  d-0.3  178  -270  -I -180  •  •  •  •  -90  . 0  • 90  •  • 180  angle (degrees) Fig.  9.6b  Radiation pattern of d=0.373X  s t a g g e r e d w a v e g u i d e ¥=45 .  179  -30 -H—• -180  •  —'  -90  ' ' '  0 angle (degrees)  ' ~~L 90 180 1  90 i  angle (degrees) Fig.  9.6c  R a d i a t i o n p a t t e r n o f staggered waveguide ¥ = 4 5 d=0.407X  .  180  181  angle  90  (degrees)  i  angle (degrees) F i g . 9.6e  R a d i a t i o n p a t t e r n of staggered d=0.474X  waveguide ¥=45 .  182  -270 -I -180 Fig.  9.7  •  •  1  •  •  .  -90 0 angle (degrees)  •  •  .  90  •  •  .  180  R a d i a t i o n p a t t e r n o f staggered waveguide ¥=45° s i n g l e and double d i f f r a c t i o n o n l y . d=0.441A  single diffraction s i n g l e and double d i f f r a c t i o n  183  CD  33.  O CL  >  -180  •90  0 angle (degrees)  90  90  -90 H Cl  -180  -270 -180  F i g . 9.8  90  -90 0 angle (degrees) R a d i a t i o n p a t t e r n o f staggered f o r v a r i o u s stagger a n g l e s . ... —  ^=15' • ¥=30'  waveguide  4/=45 ¥=60  c  c  184 Chapter RADIATION  FROM M U L T I - E L E M E N T  STAGGERED  WAVEGUIDE  ARRAY  INTRODUCTION  10.1  The  radiation  (Fig.10.1)  array  similar  The  difference  main  pattern  with  methods  to  those  a l l  edges  staggered  behind  the  aperture  The  stagger  calculation the  case  f i r s t . are  when The  a l l  for  the  two  array  arrays  are  the  array  in  is  the  waveguide  is  found  by  of  Chapter  6.  that  in  aperture  outer  waveguide  the  radiation  the  plane,  edges  are  plane. does  change  guides  additional are  multi-element waveguides  s i g n i f i c a n t l y , however.  shorted the  used the  whereas  the  a  between  array  in  of  staggered  non-staggered  to  10  are  of  f i e l d s  calculated  before  A s  i n f i n i t e  radiated  later.  depth  when  The  the  pattern  pattern  is  the  analysis  for  calculated  outer is  guides  restricted  TEM-case.  Consider waveguides  into  two  d/2,z=0  Fig.10.2.  of  four  and  y=d/2+a,z>l  (Fig.10.2).  plane and  waves  driven each  y=+d/2,z=0. by  the  guide with These  incident  conducting  y=-d/2,z>0,  y=-d/2-a,z>l,  at  central  p a r a l l e l - p l a t e  perfectly  half-planes  the  represented  s e m i - i n f i n i t e  thin  y=+d/2,z>0 in  adjacent  consisting  i n f i n i t e l y  (8.6)  three  The  -d/2<y<d/2  value two rays  in  is at  (8.7)  plane the  incident  f i e l d  decomposed  the  edges  waves  central  guide  y=are of  185  10.2  RADIATION  The  total  p a r a l l e l is  WITH  radiation  plate  The  central total  the  Fig.10.2)  scattered  s-(n=3,4) r e p r e s e n t s then  reflected  (Fig.10.5). of  the  nth  found  .  separately  angles.  because  the  S  t For  The for  pattern  e  $  not  the  array  center of  nth in  (d)  open  appear  the  d i f f e r e n t  a l l  edges  and  - i k ^ ^ s i n e  edge  central from  and guide  the  image  must angular  images  are  calculated  1  edge  be  regions  v i s i b l e  for  at  o < e < n  e = n . are  shadowed  and  -ikJccosG  e  2  nth  the  pattern  except  the  and  seven  edges  ^  (Fig.10.3,10.4),  of  about  of  0  in  radiation  symmetrical  p  labelled  emanate  be  the  images. end  (as  to  only  of  the  of  need  driven  represents  from  surface  three  sum  their  edge P  of  guide  the  and  the  scattered  s-  DEPTH  (n=l,2,3,4)  n  outer  each  edges  from  the  total  is  a l l  S,  -  the  included  pattern  o < e < j  up  d,  the  because The  made  a l l  fields  edge  is  f i e l d s  from  The  (Fig.10.6), all  the  of  t  the  from  already  s  INFINITE  only  t  width  OF  with  scattered  of  not  s  from  f i e l d  guide  f i e l d  pattern  f i e l d  scattered  represents  GUIDES  waveguides  calculated.  f i e l d s  OUTER  .  dO.l)  3  j -w < e < ~ -ik(^ S  t  =  S  3  e  +  S  1  e  +  P(J  - i k  (d)  —  + 2 a  )sin6  -ik£cos6  smo  (10.2)  186 For  —  <  B  <  -ik( -^ p-)sin6 d  -ik y +  S  e  l  0  -ik( S-  +  d j l  S  +  +*  2  < e <  | -)sine  -ik2,cos6  a  e  e  +ik  j  sin6  P (d)  +  For  -ik£cos6  2  — 2  sin6 (10.3)  6  T;  •ik( ^|p-)sine  -ik".cos9  d  -ik +  S  y  sine  e  P (d)  +  Q  +ik +  S  2  sine  6  + i k (—-~) s i n e +  S  e  -ik2.cos6 e  (10.4)  4 The are s  f i e l d s  P  q  calculated are  3  s-  The  paths  about  using  calculated  f i e l d s ray  were  the  are  calculated the  ray  using  calculated  in  Chapter  paths the  using  used  to  calculate  z-axis  of  those  in ray  the s  shown  in  The  Fig.10.3. paths  ray and  2  5.  in  fields The  Fig.10.4. in  s  are  the  and  10.4.  Figs.10.3  1  fields  paths  4  s  The  Fig.10.5. image The  /  187 solutions  for  are  in  found To  Here  In  U  In  U  To  (  2  33  " ' f  c  d  In 33  2 7 ;  G  d  (  d  a  the  line  In  To  a l l  in  ray  these  Chapter diagram  ray 8  paths  and  in  9.  Fig.10.3.  source  -4-)G Ac ,\ - Y , 6 ) E ( r ) o\ ,  calculate  Fig.10.4a  d  ( c  ' 2  - ^  1  radiated  s  the  (10.5)  consider  3  radiated  f i e l d  the  ray  u  3  G  d  (  d  ' f 2 " 1  ,  radiated  E  (  r  •  )  u  (  is  3 3  a  line  is  2  V  )  G  d  (  c  u  ,  1  the  2  3  a  line  e  E  is  a  (  r  diagram  in  line  Fig.10.5.  2  G  d  (  c  ' ^'f 2  Fig.10.5c |  Similar  the  radiated  -^GjCc.^jradiated  f i e l d  a  line  d  )  calculations  is  a  line  the  f i e l d s  8  In  source  d  for  - >  (10.10)  L  2  G ( d , 0 , | ) G ( d ^ , | -¥)G (c,^ -H,2,-6)E(r). d  1 0  source  -y,^j)-G (2a,^ ,27i-e)E(r) .  f i e l d  )  (10.9)  is  d  6  source  i G l ( c , 2 i r , f -V)G.(c,-2f -Y,2ir-9)E(r). 2 d z a z the  '  source  -'> > ( > •  ray  f i e l d  is  3  0  (10.7)  d  f i e l d  1  source  1  consider  s-  )  -^. f)G (2a,^,e)E(r) .  radiated  )  6  f i e l d  C  0  the  ) G  . | -*'>V HT  * -2  Fig.10.5b  =  -*  the  calculate  =  a  the  ( '  G  2  =  =  U  '  c  d  Fig.10.4c  32  U  .  Fig.10.4.  Fig.10.5a  In  is  2  in  G  2  =  33  + ^  2  that  consider  f i e l d  X  Fig.10.4b  33  to  by  source  2  =  similar  1  =  c  line  32  s  radiated  diagram  u  manner  represented  ^ G ' ( C , 2 T T ^ - V ) G AC,^ -V,^r 2 a 2 a. 2. z  =  where  a  a  f i e l d s  calculate  the  u, 13  the  s  2  scattered  (10.11)  from  the  188 edge  y=-d/2,z=0  S  2  =  -S _  =  -S.  S.  These  ]  and  for  radiation  pattern  RADIATION  Consider  of be  f i n i t e  of  of  radiation  guide  guide  of  coupled at end  the  alter  of  travel  that  z=0  sum o f  to  give  that  change the  and  which the  the  i s  been  down  the  the  relative  guides  these  relative  g  a r e now  guides  w i l l  Adjusting  phase  to  of  the their  between  the  central  adjacent  staggered  outer  calculated  in Chapter  8. T h e  outer then  f i e l d s  geometric  total  pattern.  A (d,a)  back  to  the  equivalent  outer  into  radiation  the  into  SHORTED  end and reradiated.  reflected  a  substituted  Fig.10.2.  coupled  end. The f i e l d s  of  now  Fig.10.1,  c o e f f i c i e n t  has  A'  the  is  w i l l  a  value  guide  (10.4)  GUIDES  far  d  is  plane  and array  of  width  part  aperture  are  except  the  guides  shorted  ~  s  The f i e l d s  from  of  d  OUTER  array  width  ,where  the  Fig.10.2  f i e l d s  The  of  coupling  driven  n  (10.3)  and thus  The  a  n  WITH  depth.  the  s  the  r e f l e c t e d  depth  show  (10.13)  (10.2),  array  y=-d/2-a,z=l  (10.12)  (10.1),  the  from  4  t  results  10.3  s  guide  and are  travel  towards  and part i n  those  the from  series  i s  outer the  reflected the  open  radiated. guides central  at  the  driven  189 whereR (a)is  the  g  the S  staggered i s  the  total  calculated  array  from  the  radiation  by  depth  f i e l d s  of  waveguide  outer  at  the  of  width  guide  open a  measured  of  end [40],  and  from  the  plane.  The  the  c o e f f i c i e n t  p a r a l l e l - p l a t e  depth  aperture  f i n i t e  reflection  of  adding  of  the  other  the  outer  Fig.10.2.  radiated edges  pattern  of  the  array  radiation  guides  The  to  the  from  excited  by  the  caused  f i e l d s  radiated  f i e l d s  outer  f i e l d s  Fig.10.1  f i e l d  additional  d i r e c t l y  of  a  by the from  consist  guides,  i n  is  and  of  f i e l d s  shorted  guide  (Fig.10.7). Radiation s  (9.19-9.22)  0  multiplied due  to  by  f i e l d  U  from  A*  1'2  the the  the  of  radiation  considering  In  from  guides  a  staggered  of  (10.14).  from  the  ray  edge  outer  Scattering guides  i n  y=+d/2,z=0  simply  the  p a r a l l e l - p l a t e  outer  paths  i s  Fig.10.7.  can  be  waveguide  from i s  pattern  other  calculated  by  Fig.10.7a  the  In  written  J G (d,2ir,^)G (d,2j,6)E(r) .  =  d  Fig.10.7b  the  (10.15)  d  f i e l d  from  the  edges  edge  y=+d/2,z=0  can  be  written  in/ 4 U  1'3  4 G'(d,0,^)G (2d,^,e)  =  G (d,0,^)E(d)G (d,^,e).  - ^  d  d  d  A /2 TT In  Fig.10.7c  " .$  V3 n  In  =  the  -ikJ. Z-z—  f i e l d  from  the  edge  (10.16)  y=+d/2,z=0  i s  G l (V cL , 0u , - ^ +r V + 4 ' , ^ 0) Gv u( d , ^ , 6 ) . , ) l G ( c , , 4  (10.17)  3  " ^~T~ ° d ' ' 2 " ' ' d ^ ~ 2 ^ ' 2 ' d ' ~ 2  Fig.10.7d  the  J  f i e l d  from  J  the  edge  y=+d/2+a,z=l  i s  U  1'3  =  |  In Fig.10.7e =  e  3'3  u  The  from t h e edge y=+d/2-a,z=l i s  d  f c  d  w  (10.15),  (10.16) a n d (10.17) a r e added t o g i v e  from t h e edge y=d/2,z=0 e x c i t e d by guides.  from  calculations  t h e image  of the fields  =  S  a n d (10.19)  gives t h e  p o i n t y=+d/2-a,z=l.  that  _s' 1  (10.20)  K - -H 4  °-  (1  3•  The a d d i t i o n a l r a d i a t i o n caused by t h e f i n i t e depth outer  Similar  from t h e edge y=-d/2,z=0 a n d  f r o m y = - d / 2 - a , z = l show c2  radiation  S i m i l a r l y (10.18) g i v e s t h e f i e l d  from t h e edge y=+d/2+a,z=l,  4  (10.19)  H  d  2 d  t h e outer  3  (10.18)  d  j L. ^ _- , -. 22 7i .1--e6 )) i G ( d . 2 , . | ) G ( d I. 2n| . f - * .) G. 3( c  field  S  d  thefield  field  from  ( '2^|)G (d,^-¥)G (c,^--¥,e). d  d  fields  the  S  G  guides i sthearray  s u m o f s , S^, Q  21)  ofthe  and s £ m u l t i p l i e d  b y t h e c o e f f i c i e n t A* o f ( 1 0 . 1 4 ) . T h i s r a d i a t i o n i s a d d e d t o S  i n (10.1)-(10.4)  pattern For  t o obtain  t h e complete  radiation  for thearray o f Fig.10.1.  s t o t a l  0 < 6 < — 2 -ik( S  , total  +  = t  d j  ^)sine  -ik£cos6  A'S' e 3 - i k ( -^") s i n 6 e • d  +  A'S  2  0  (10.22)  191 IT  For  -  -¥  Ti  < 6 <  -ikj—^sine  s  ^total  =  S  +  t  A'S' e 3  e  -ik +  A'S  Q  l y s i n e  e  -ik — +  For  — < 6 <  2  2  -ik£cos6  A'S^  sine  (10.23)  e  +¥  -ik ( total  S  t  +  A'S'  ) sine  d-f2 a  e  -ik£cos6 e  3  -ik(~.)sin6 +  A'S  +  A'S^  0  e  -ik —  -ik( +  A'S-  e  +  A'S'  e  +  A S  e  3  sin6  e  d  2 a  sine  + i k ( ~ — ) sine 2  0  -ik£cos6 e  +ik — 2  )sin6  (10.24)  192 For  j  +¥  <e<  TI  - i k ( "'" ) s i n S d  S  . , total  =  S t  +  -ik£cos6  2a  A'S' e 3  e  ., ,d+a, . „ - i k (——) sine  +  e  A'S 0  ., -ik +  A'S^ e  +  A'S' 2  . „ sine  +ik —  sine  e  2  . , ,d+a. . „ +ik(——-)sine  -_ A'S  +  d —  e  0  +ik ( " d  +  NUMERICAL  10.4  The  compared  array with horn  guides.  Radiation  i n f i n i t e  depth y  .  s  -ik£cos9 •  e  4  a  calculated  by  ray-optical  E-plane  appendages patterns  are  shown y  in  changes  three  patterns  added  to  when  the  Fig.10.8 from  (10.25)  RESULTS  of  with  As  ) sin6  e  pattern  measured  sectoral  angles  '  AND E X P E R I M E N T A L  radiation  waveguide  A  f 2 a  0°  of  outer  90°  staggered  methods an  simulate  for to  element  various the  H-plane  the  guides  shape  is  outer are  of  stagger of  the  193 pattern  changes  staggered  array  expected. the  The  to  are  that  general  the  outer  generally  staggered from  to  The  boundaries then and  of  a  of  a  three  single  element  guide,  increasing  &=  less  then  array  as  arrays  1  dB  6  as  is  \jf  ones  to  arrangement effect  depth  gradually  a  shown  from  the  has  lobe  single  a  with  guide.  the dB  3  non-  might  to  be  broaden  element  along  is  (Fig.10.9-  the  2  dB  in  shadow  6 - ~> /2 T  there  the  the  smaller  at  because in  non-  ranging  generally  boundary  edges  patterns  frequency  then  This  six  three  were  less  the  beamwidths  the  + y  shadow  are  staggered  non-staggered  7.  1  dB  up  patterns  generally  agree  to  and  at  because  of  0=160° e' = 1 8 0 °  blockage  with  within  could  the 3  not  dB be  by  the  antenna  60°  for  a  {e'-9-ir).  increasing  is  which  single  of  /2  patterns  The  mounting  depth,  and  within  accurately  1 I  case:  or  depth  expected  6=  any  four  for  on  © = "n"/2 + y .  experimental  measured  guide  and  f i n i t e  have  depending  at  to  e'=-150°.  The  patterns  along  Chapters  calculated  of  those  non-staggered  compared  of  are  to  TT/2  edges  The  to  The * 90°  at the  two  guides  d i s c o n t i n u i t i e s  in  only  of  that  of  effect  similar  array.  ± 2 0 °  10.13).  a  from  pattern. When  up  gradually  in  y  from  Fig.10.14.  non-staggered  null  on  the  beamwidth  beam  0°  to  The  patterns  pattern  for  axis,  considerably  to  a  fixed change  that  guide  pattern  with  narrower  then  that  194 10.5  SUMMARY  The plate  radiation  pattern  waveguides  was  compared  with  sectoral  horn  with  complex  then  the  shadow the edges of  are  in  multiple The  are  hand, the  staggered  case  the  calculation  is  plane,  avoiding  at  the  staggered  array,  shadowed  staggered be  in  a  the  case,  to  to a  is  time as  observed then  not  are  give single  a  to  generally  be  H-plane is  more  are  more  considered. because  the  the  On  not  a l l  d i f f i c u l t i e s  shadow  observed  the  only  shadow or  boundaries  in  because  three  one  nonedge  boundaries  five  which  in may  array. broader  large  s i g n i f i c a n t guide,  at  traversing  for  there  simpler  those  compared  especially  to  unexpected  when  an  and  boundary.  non-staggered  patterns  adjusted  compared  shadow  smaller  This  shadowed  The  a  thus  methods  using  because  more  along  paths  p a r a l l e l  calculation  and  is  be  staggered  obtained  non-staggered ray  of  ray-optical  The  much  case.  by  results  d i s c o n t i n u i t i e s  generally  array  appendages.  same  edges  an  calculated  experimental  boundaries  other  of  ty"  .  then The  reduction  however.  in  for  the  non-  parameters  can  beamwidth  as  10.1  Fig. array of staggered p a r a l l e l p l a t e waveguides w i t h outer guides shorted  T  a  d/2 d/2  4—  10.2  Fig. array of staggered p a r a l l e l p l a t e waveguides w i t h outer guides o f i n f i n i t e depth  196  \  ray  F i g . 10.3 path f o r c a l c u l a t i n g  \  \  ray  v/  paths  Fig. for  y/n  ray  paths  \  10.4 calculating  v/  F i g . 10.5 for calculating  II  Ill  /  I  \  VII  VI  F i g . 10.6 hadow boundaries f o r an a r r a y o f taggered p a r a l l e l p l a t e waveguides  \  d  F i g . 10.7 ray paths (see t e x t )  £-20  -30  -180  -90  0 90 angle (degrees)  180  180 4  Si  33.  9  0  0 -90 4 -180  Fig.  10.8  -90 0 angle (degrees)  -180  90  R a d i a t i o n p a t t e r n o f s t a g g e r e d waveguide a r r a y w i t h guides of i n f i n i t e depth d=a=0.450X ¥=45°. ¥=15  £  ¥=30*  ¥=75  ¥=45 ¥=60  c  outer  Cu >  > -20  -30 -180  -90  0  90  angle (degrees)  180 n  -180 -180  -90  0  90  angle (degrees) Fiq.  10.9  R a d i a t i o n p a t t e r n o f s t a g g e r e d wave d=a=0.474X s=1.166X ¥=45 .  200  angle (degrees)  180  i  angle (degrees) 10.10  R a d i a t i o n p a t t e r n o f staggered waveguide d=a=0.441X s=1.083X ¥=45°.  array  201  angle (degrees)  180  i  angle (degrees) 10 1 1  R a d i a t i o n p a t t e r n o f staggered waveguide d=a=0.407X s=1.000X ¥=45°.  array  202  angle (degrees)  -180 -180  -90  0  90  180  angle (degrees) Fig.  10.12  R a d i a t i o n p a t t e r n o f staggered waveguide d=a=0.373X s=0.916X ¥=45°.  array  203  angle  (degrees)  180 n  -180 -180  -90  90  0  180  angle (degrees) Fig  10 13  Radiation pattern d=a=0.339X  of  s=0.833X  staggered waveguide ¥=45°.  array  204  0i  CD  -10  I  Q.  £  -20 4  03 0»  -30 -180  -90  0 angle (degrees)  90  180 i  90  0)  cn  SI CL  -90 1  -180  Fig.  10.14  Radiation stagger  90  -90 0 angle (degrees)  -180  pattern  angles ¥=15  ¥ c  ¥=30'  of staggered d=a=0.450X  waveguide a r r a y s=1.107X. ¥=45 ¥=60'  180  for various  205 Chapter GENERAL  11  CONCLUSIONS  AND D I S C U S S I O N  DISCUSSION  11.1  Ray-optical d i f f r a c t i o n p a r a l l e l both  have  plate  and  patterns  staggered, plane  been  based  used  were  for  geometrical the  Coupling  small  horn  the  analyze  separated  calculated  sectoral  on  to  waveguides.  adjacent  Radiation  methods  f i e l d s  and to  both  also  in  c o e f f i c i e n t s  waveguides  arrays,  theory  were  simulate  the  coupled between  calculated.  staggered  measured  of  and  using  non-  an  H -  two-dimensional  structures. Ray-optical largely various plate  methods  because simple  of  of  anticipated  that  more  of  the  and  other  that  edges  S p e c i f i c a l l y ,  the  quite  presence  well of  by  complex,  plate  methods The  obtained  here  results  where  can  successful  be  of  be  various it  was to  agreement  experimental  a v a i l a b l e , if  even  extended  good  with  for  p a r a l l e l  here,  Thus  might  generally  others  and  consist  waveguides.  analysis  indicates  the  number  of  small.  the  coupling  with  other  obtained  considered  results  is  this  structures  structures.  extension  for  h a l f - p l a n e s  more  theoretical  d i f f r a c t i n g  results  r a y - o p t i c a l  r a y - o p t i c a l  chosen  involving  p a r a l l e l  complex  this  agreed  The  considerably  combinations  these  good  structures  waveguides.  though  the  were  those  guides  c o e f f i c i e n t s  calculated and/or  a  by  obtained other  simulated  here  methods  in  groundplane  206 [29]  [38].  guides  Other  in  isolation  comparison. in  using  wedge  guides  obtained not  which  recently in  good  the  more  complex The  guides  interaction  in  the  of  this  coupling  versus  similar  [38]  using  residue  modified  agreement.  coupling  A  to  the  observed,  results  may  to  that  calculus,  general  coefficients was  patterns  i n f i n i t e by. o t h e r  i s o l a t i o n  or  patterns  for  using  methods  embedded  in  arrays  an  obtained  depth  (Figs.6.1,7.1,10.1)  calculated  the  obtained  lack details  which  be  applied  here  for  of of  gives  d i r e c t l y  to  structures.  calculated  patterns  those  if  for  separated  q u a l i t a t i v e l y  these  radiation of  two  inclusion  o s c i l l a t i o n  structure  that  The  adjacent  available  with [17]  two  was  in  surrounding  confidence  agreed  quantitative  of  not  between  techniques  an  between  were  ignored.  yielded  s e n s i t i v i t y the  (Fig.4.1)  was  curve  coupling  coupling  d i f f r a c t i o n  interaction  but  for  isolation  separation  for  (Fig.3.1)  Results  guides  between  results  agreed  [29]  for  [38]  a  H-plane for  these  horn  arrays  with  same  those  array  guides  agreement  sectoral  with  in  groundplane.  outer  good  well  the  simulated  with  showed  patterns  (Fig.7.2)  arrays  with  The  shorted experimental  in  a l l  cases.  Other  were  not  available  for  comparison. The these  basic  geometries  boundaries as  d i f f i c u l t y  the  are  that  the  f i e l d order  applying  f i e l d s  combinations  incident  Consequently,  is  in  of  and of  each  d i f f r a c t e d  f i e l d s  higher  ray-optical  of  the  order  subsequent  methods  along same  shadow  order  d i f f r a c t e d d i f f r a c t e d  to  of  k  f i e l d s . f i e l d  207 is  not  diminished.  ray-optical  methods  number  possible  of  Thus to  which  must  half-plane  edges  are  the  rays  containing be  may  only  considered  contrast,  two  calculations including more  for  complex  very  the  to  use  calculate  waveguides.  The  widths, has  a  this n=50  the  with  excellent  only  a  improvement widths  is  exact  over  to  as  of more from  waveguide  ray  the  paths  to  r e f l e c t i o n  d i f f r a c t i o n . many  ray  Thus  By  paths  ray-optical  require  become  terms  intractable  coupling  the  of  a  for  curve  coupling  agreement  only  of  are  included.  d i f f r a c t i o n  r e f l e c t i o n simpler  case  at  solution  the  versus  does at  ray-optical  use not  the  to  for  the  transition guide  derivative.  The  (UAT)  waveguide.  mode  obtained  be  obtaining  plate  even  may  adjacent  in  exact  obtained  is  theory  successful known  it  between  p a r a l l e l  of  why  asymptotic  f i r s t  the the  reason  discontinuous  orders in  the  very  with  d i f f r a c t i o n  few  of  uniform  is  c o e f f i c i e n t  where  part  the  UAT  agreement  times  2  which  d i f f r a c t i o n  the  agreement  cusp  only  B).  the  order  plate  require  Appendix  that  structure  times  geometries  of  is  accurately  Excellent  n  is  given  the  require  extending  rapidly  into  w i l l  of  geometries.  d i f f i c u l t  r e f l e c t i o n  grows  representing  simple  d i f f i c u l t y  excellent  a  contribution  see  orders  to  p a r a l l e l  w i l l  guides  n,  high  This  edges  terms  on  up  A  result  structures  considered  d i f f r a c t .  adjacent  (depending  complex  introduced  find  from  immediate  paths  be  two  to  c o e f f i c i e n t  more ray  d i f f r a c t i o n  which  one  if  width  However,  terms  of  up  UAT  to  with  give  s i g n i f i c a n t  mode  transition  method  used  here,  and  208 thus  would  coupling  not  be  expected  coefficient  Another  result  complex  structures  at  shadow  the  p a r a l l e l plane  less  of is  0.5  t r i p l e  d i f f r a c t i o n  reduced  to  ray  an  paths.  this  a  dB,  parameters,  even are  used  accuracy  of  complexity  of  The  agreement is  into  small  value  up  3  20  occur single  the  aperture  paths and  up  to  may  be  more  non-staggered  depending  not  up  on to  more  shadow method  guides  the  array  quadruple  possible  taking  the  more  considering  paths  by  near  to  reduce  terms  with  the  boundaries, decreases  the  as  the  increases.  from  by  the  the  shadow  array  ray-optical  equally  dB  was  in  ray  by of  ray  It  away  between about  to  6  to  a  account,  array  ray-optical  affected  only  taken  structure  accuracy  s l i g h t l y  which  the  the  which  For  when  further  the  larger.  even  be  Thus  methods  discontinuity  considered.  here.  improve  d i s c o n t i n u i t i e s  become  when  discontinuity  method  the  element  can  ray-optical  this  are  three  discontinuity  this  that  a r b i t r a r i l y  For  d i f f r a c t i o n  extending  waveguide  the  s i g n i f i c a n t l y  either.  boundaries  plate  is  to  good  size,  and  for  boundaries as  is  indicated  experimental  both  simple  only  and  by  the  results,  more  complex  arrays. The  basic  boundaries along  is  f i e l d .  that  shadow  represented then  reason  the  by  the  a  ray  when  in  used  is  the  f i e l d  this  this  method  boundaries  resultant  Thus  for  in  f i e l d  d i f f i c u l t y to  not  accurate.  aperture the is  in  calculate  plane  aperture turn  is  plane  at  the  shadow  the If  fields a  f i e l d  d i f f r a c t e d , is  d i f f r a c t e d  not in  a  ray the  209 aperture  plane,  accurate  the  value.  ray  This  applies  d i f f r a c t e d  f i e l d s  waveguides  calculations  For size  a  of  the  the  exact  are  the  are  ray  l i k e  look  Boersma  on  a  problem  to  an  higher  order  p a r a l l e l  plate  waveguide  the  by  the  comparison  because  if  y i e l d  measured  and  the  as  two  and  with  arrays, d i f f r a c t e d  incident  to  two  be  of  Chapters  in  the  somewhat.  However,  half-plane  for  the  doubly  incident inaccurate  non-staggered two  staggered  6  for  and  this  the  the  plates  f i e l d s  look  like  d i f f r a c t e d  f i e l d  f i e l d . for  case  Lee  the  plates.  which  double  case  Thus  of  and of  plates  a  it  line  arises  half-plane  might  accuracy  d i f f r a c t i o n  w i l l  singly  a  and may  source  repeately  7.  plane  better  the  p a r a l l e l  d i s c o n t i n u i t i e s  aperture for  that  boundary  the  non-staggered  available,  needed  f i e l d  on  predicts  shadow  inaccurate  solution  was  When  on  be  patterns  more  For  not  method. as  pattern  here  the  this  incident  arrays  If  been  open-ended  cumulative,  one-quarter  incident  source  incident  along  showed  expected  t r i p l e  the  calculated  incident  [32]  wave  the  and  ray-optical  small.  are  does  t r i p l y  inaccuracy,  in  used  f i e l d  w i l l  in  a l l  the  this  quite  method  the  be  is  errors  one-half  line  to  here  f i e l d s .  d i f f r a c t e d  a  by  guide  repeatedly  The  plane  used  non-staggered  discontinuity  result  however, f i e l d s  in  single  method  problem  a  in  d i f f r a c t i o n  the  have  radiation  been  solution  (Fig.4.12)  reduced to  would  the have  also.  the  complex  ray-optical p a r a l l e l  methods plate  used  here  waveguide  are  extended  structures,  to the  210 combination  of  inaccuracy  of  to  the  large  the  number  method  d i s c o n t i n u i t i e s  up  to  the  calculated  radiation  in  a  manner.  simple  from  the  well  shadow  with  methods  most  in  dB  the  in  the  can  plate  -10 a  beamwidth  five  element  only  s l i g h t l y  amplitude  was  was  Other  patterns These  beam  with  beam  in  applied  reflector. varied  two  could  angle  the  H-plane.  If  as the  dynamically,  quite  predicting  ray-optical  interest  methods  can,  the for  within  radiation  a  be  by  of  for  from  dB  with in  on  the  appendages  of the  for the  a  size  120°  a  angular array. axis. design  produce and  parabolic guides  loading  is  whose  the  pattern  parasitic  reactive  to to  E-plane  beam  a  with  pattern  over  of  with  obtained  A  sector-shaped  feed  variety  aperture  d i r e c t l y  the  a  non-staggered  n u l l s  applied  p a r a l l e l  pattern  was  the  element  with  The  depth e.g.  that  0.5  three  l i n e  arrays  s k i r t s  within  coverage  a  agree  for  wavelengths.  horns  wide  d i r e c t l y  away  the  the  in  eliminated  patterns  sector-shaped  steep  obtained  sectoral  case  that  suitable  considering  with  were  leads  greatest  small  good  and  constant  results  H-plane  134°  then  obtained  of  predict  p o t e n t i a l l y  array,  more  range  of  are  of  be  plane  useful  of  remarkably  dB  cannot  so  the  waveguides.  patterns  waveguides  and  boundaries  radiation  ray-optical  p a r a l l e l  A  the  Thus  coupled  applications.  which  regions  successfully  needed  non-staggered  aperture  be  l i m i t s ,  plate  the  results  these  radiation  in  paths  shadow  angular  a p p l i c a t i o n s .  The  the  However,  here  ray  pattern,  experimental  employed  patterns  along  ± 3  boundary  the  of  a  a  fan  narrow could  be  cylinder could  be  described  211 in  [35],  11.2  some  beam  steering  SUGGESTIONS  There  are  a  work  that  warrant  made  to  improve  f i e l d s  along  array  by  FOR  the  FURTHER  number  of  further the  the  questions  accuracy  boundaries  uniform  methods  the  assumption  used  ray  f i e l d s .  The  problem  a  f i e l d  p a r a l l e l  plates  may  obtained  solution but  be  for  the  a  plane  f i e l d  needed  here.  three  p a r a l l e l  three  and  number  ray  five  of  problem  The  to  rapidly.  when  more  is  required  Similarly, improved  number  of  on  points for  more  considered Ray-optical  to  obtained  the  coupling uniform  be  are  which  may  with  the  coefficient  of  the  [32],  plates shadow  are  incident  f i r s t .  required  The  far  using  the  in  [32],  boundary  is  incident for  on  both  However,  the  as  the  increases,  because  the  if  to  practical  better  accuracy  used  calculations  p a r t i c u l a r l y  near  cusps  are  expected,  must  be  considered  here. may  the  but  of  accuracy  be  methods  the  number  desired  cease  considered  be  f i e l d s  normally  source  the  may  Boersma  a l l  arrays.  obtain  methods,  where  and  boundary  methods  was  paths  also  this  non-staggered  argument  the  d i f f i c u l t ,  than  the  source  line  shadow  edges  of  p a r a l l e l  a  would  the  attempt  that  along  from  calculation  reciprocity two  two  ray  line  arise  the  Lee  here  non-staggered  much  regions  of  An  investigated  a  then  using  transition  be  by  solution  along  be  grows  a l l  element  becomes  paths  may  plates  edges  of  wave  at  which  of  avoiding  two  possible.  WORK  thus  on  be  investigation.  shadow  using  should  mode  again  for  be  the  better  212 accuracy  is  very  Other  large.  methods  radiation  pattern  separated  guides,  produced  the  and  more  is  here. of  This  the  coupling  of  the  is  d i f f r a c t e d  from  outer  in  the  5  the  central of  to  method  becomes there  c o e f f i c i e n t accuracy  using  extended  to  useful moment  in  this  methods  d i f f i c u l t i e s  [11] near  this  guides.  d i f f i c u l t  to  for  coupling moment  thickness  context.  shadow  [35]  p a r a l l e l in  a  help  to  boundaries  calculated may  be  this  r e f l e c t i o n with  which  may  may  radiation has  waveguide  overcome  good  approach  and  of  the  with  The  groundplane  if  arrays  method  this  plate  f i e l d  structures  [36]  A combination  may  the  calculated and  from  large  c o e f f i c i e n t s  method  loaded  [44]  [48],  f i e l d  However,  modes. was  to  for  outer  propagating  the  reradiated  shorted  use  pattern  coupling  been  and  method  f i n i t e  not  also  calculus,  moment  find  and  compared  have  a  reactively of  depth  is  place  resulting  the  patterns  waveguide  applied  walls  Radiation  single  Another  with  (-12dB)  a  patterns. to  which  0.25  the  because  used  in  radiation  dB),  [38]  approach  However,  large,  two  widths,  substituted the  and  method  transition  be  edge  many  from  ray-optical  outer  more  are  the  (-20  of  calculus  mode  array.  coupling between  the  0.1  residue case  residue  about  about  the  coupling  improve  be  i n f i n i t e  the  improve  may  to  guide.  modified  extended  than  not  is  at  element  only  guide  guides  using  be  cusps  one  to  For  c o e f f i c i e n t  may  c o e f f i c i e n t  used  modified  accurate  improvement  where  the  ray-optical  the  be  results.  expected  calculation  outer  may  been arrays  also  be  ray-optical  and  some  moment  of method  the is  213 used  near  from  i t .  the  The to  aperture  results  designing  ( F i g . 1 . 5 ) . patterns  obtained  an  c i r c u l a r  empirical  f i r s t  step  also  to  would  used  be  as  a  of  guide  waveguides  the  good  parameters.  open-ended  work  to  may  and  away  reasonably  required  analysis  method  circular  between  present  the  be  produce  adjustment  The  towards  ray-optical  concentric  c o e f f i c i e n t  pattern.  the  may  known  waveguides  radiation  of  are  TE,| -TEM coupling  and  here  array  These by  plane  The  coaxial  predict  be  the  considered  optimization  of  such  a an  array.  11.3  CONCLUSIONS  Ray-optical waveguide  and  except  are  BRIEF  methods  structures  Theoretical well  IN  near  with  shadow The  t r a n s i t i o n  regions  orders  d i f f r a c t i o n  the  accuracy  along of  a  is  shadow  greatest  transition  and  to  radiation  as  the is  regions  and  shadow  is  at  least  using  one-third  uniform  into  often  of  methods  even  a is  mode  when  many  half-plane  However, away  the  near  Consequently,  the  from  boundaries,  provided  quite  d i s c o n t i n u i t i e s  account. of  plate edges.  agree  accurate  increased.  most  small,  not  number  are  edges  where  boundaries,  taken  p a r a l l e l  patterns  shadow are  to  half-plane  is  interest  acceptably  six  method  boundary  are  by  up  applied  boundaries,  reduced  errors  results  been  experimental  observed.  of  have  and  regions  the  d i f f i c u l t  mode  here  spacing  wavelength.  edges  between  Improving because  the  of  the the  214  l a r g e number of edge i n t e r a c t i o n s which must be Larger  and  diffraction of  patterns  more than is  waveguide a r r a y  complex s m a l l e r and available with  a d j u s t i n g the width and  only  arrays  considered.  r e q u i r e higher orders of  simpler a r r a y s . 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M e n e n d e z , "GTD, r a y f i e l d , a n d comments on two p a p e r s " , IEEE Trans.Antennas P r o p a g a t . , vol.AP-26, pp.352-354, 1978.  [34] R.M.Lewis and of edge d i f f r a c t i o n " ,  1969.  J.Boersma, "Uniform asymptotic theory J.Math.Phys., vol.10, pp.2291-2305,  218  [35]  J.Luzwick  aperture  and  antenna  E l e c t r i c a l Syracuse,  and New  R.F.Harrington,  array",  Technical  Computer  York,  "A  Report  Engineering,  reactively No.  3,  Syracuse  loaded  Dept.  Of  University,  1976.  [36] J . L u z w i c k and R . F . H a r r i n g t o n , "A solution for a wide aperture reactively loaded antenna array", Technical Report No. 5, Dept. Of E l e c t r i c a l and Computer Engineering, Syracuse University, Syracuse, New Y o r k , 1977.  [37]  R.Mittra  and  Theory  of  Waves.  [38]  J.P.Montgomery  Guided  S.W.Lee, New  A n a l y t i c a l  York:  and  D.C.Chang,  boundary  value  problems  based  upon  calculus  and  function  theoretic  164,  Department  U . S .  of  Techniques  Macmillan,  a  the  "Electromagnetic  modification  techniques",  Commerce,  in  1971.  of  NBS  residue  Monograph  1979.  [39] Y.Rahmat-Samii and R . M i t t r a , "On the investigation of d i f f r a c t e d f i e l d s at the shadow boundaries of staggered p a r a l l e l plates", Radio Science, vol.12, pp.659-670, 1977.  [40] R.C.Rudduck and L . L . T s a i , "Aperture reflection c o e f f i c i e n t of TEM and T E mode p a r a l l e l - p l a t e waveguides", IEEE Trans.Antennas Propagat., vol.AP-16, pp.83-89, 1968. o  l  [41] R.C.Rudduck and D . C . F . W u , "Slope d i f f r a c t i o n analysis of TEM p a r a l l e l plate guide radiation patterns", IEEE Trans.Antennas Propagat., vol.AP-17, pp. 797-799, 1969.  [42]  C.E.Ryan,  analysis  of  waveguides",  Trans.Antennas  A.Sommerfeld,  moment  d i f f r a c t i o n " , pp.62-69,  "A  wedge  of  d i f f r a c t i o n  p a r a l l e l - p l a t e  Propagat  "Mathematische  Math.Ann.,  G.A.Thiele  combining  patterns  ,  vol.AP-16,  1968.  D i f f r a k t i o n " ,  [44]  And R.C.Rudduck,  radiation  IEEE  pp.490-491,  [43]  Jr.  the  1975.  and  with  Theorie  pp.317-374,  T.H.Newhouse,  methods IEEE  vol.47,  the  Trans.Antennas  "A  der  1896.  hybrid  technique  for  theory  of  geometrical Propagat.,  vol.AP-23,  219 [45] L.A.Weinstein, Factorization Method.  [46]  R.Wohlleben,  primary  feed  e f f i c i e n c y " ,  [47]  H.Mathes  large  in  "Numerical f i n i t e  and  opening  E l e c t . L e t t . ,  C.P.Wu,  waveguides 254,  for  The Theory of D i f f r a c t i o n and the Boulder, Colorado: Golem P r e s s , 1969.  O.Lochner,  angles  vol.8,  pp.474-476,  solution  arrays",  and  Radio  for  the  Science,  "Simple high  small  aperture  1972.  coupling  between  vol.14,  pp.245-  1969.  [48] S.L.Wu and Y.L.Chow, "An application of methods to waveguides scattering problems", Trans.Microw.Th.Tech., vol.MTT-20, pp.744-749, 1972.  [49]  H.Y.Yee  waveguide Inst.  and  L.B.Felsen,  d i s c o n t i n u i t i e s " ,  Brooklyn,  Farmingdale,  "Ray-optical Electrophys.  N . Y . ,  Rep.  techniques Dep.,  moment IEEE  for  Polytechnic  PIBEP-68-005,  June  1968.  [50] H.Y.Yee, L.B.Felsen and reflection from the open end J . A p p l . M a t h . , vol.16, pp.268-300,  J . B . K e l l e r , "Ray theory of of a waveguide", SIAM 1968.  220 APPENDIX A  SCATTERED F I E L D S ALONG A SHADOW BOUNDARY  The r e s u l t s relation  (3.26) a n d (3.29) may be c h e c k e d b y u s i n g a symmetry  for scattered  fields  u (2TT-9)  =  s  which i s exact by d e f i n i t i o n  field, is  s  for diffraction  A.l)  and then  the f i e l d  taking  =  [ 3 ] . The s c a t t e r e d fields,  +  the f i r s t  +  term,  where  u  s  2  field  if  (3.29)  6 o f f t h e shadow b o u n d a r y  ikr e  0  e  /kr_  ikr  (A.2)  /kr  ( 3 . 2 6 ) , a s i s shown  on a h a l f  plane  6 = y +6 i s g i v e n ,  ikR  TT  However,  representing the scattered  incident  observed at  '2 -iir/4 — e  (A.l).  incident  at the o b s e r v a t i o n p o i n t  i s g i v e n by  source E(r)  field  • ,* +17T/4  2/2TT  Q  a line  A . l ) , the t o t a l  e  2/k(r+r )  the remaining terms  (zr) s a t i s f i e s ' ( A . l ) , s 2 Consider  T  which i s the i n c i d e n t  o u t , then  but not the  field  (3.29) may be w r i t t e n  Q  Q  subtracted  at a small angle  ik(r+r )  /k(r+r )  (Fig.  and d i f f r a c t e d  6 -»• 0, t h e n  Q  u  screen  (3.26) a n d (3.29) a s w r i t t e n d o n o t s a t i s f y  ik(r+r )  If  by a p l a n e  includes reflected  d e r i v e d by f i n d i n g  (Fig.  (A.l)  -T u (6)  / k (r +r+R) Q  I/  k  r  0  from  r s  i  is  field below,  a t 6^=3^/2 (2.12) b y  . — «' n  r +r+R Q  ikS  +2 /  +x  /k(r +r+S) Q  cos — > V  r  +  S  (A.3)  221  i k R  /2~  + A  /2  -iir/4  I -  e  i k R  +2 /  j  ^  (r +r+R)  sin r  /k(r +r+R) Q  +r+R  0  Q  i k S  + T  where  S  •>  r  R , S , r , r  _  r  Q »  a  n  0  , 6  and  e  a r e  Q  d e f i n e d  +2 / / k ( r  Q  + r + S )  F i g . 2.3.  i n  cos — r  Q  l e t 6  +  r  +  (A.4)  S  -»• 0 ,  then  R  •*  r + r  Q  ,  d  i k ( r  +  r  Q  )  i k ( r + r  o  )  i  k  (  r  .  r  0  + T / k ( r + r  fc  Q  )  J  F[/2kr^]  2/k(r+r ) Q  (A.5)  I f  k r  Q  > > l  (A.5)  =  (A.2)  QED.  Fig. A . l  222  APPENDIX B  HIGHER ORDER D I F F R A C T I O N TERMS FOR COUPLING BETWEEN ADJACENT P A R A L L E L P L A T E WAVEGUIDES I N THE TEM MODE  A general  e x p r e s s i o n f o r TEM-TEM c o u p l i n g o f 0 ( k  cent waveguides  i s derived  (3.18),  (3.29)  (3.26),  calculate possible  f o r a l l orders  and (3.30)  the c o n t r i b u t i o n ray paths  systematically  of d i f f r a c t i o n .  First  adja-  the  equations  a r e s p e c i a l i z e d t o t h e TEM c a s e a n d u s e d t o  to the coupling f o r a generalized  from the d r i v e n  and t h e i r  ) between  guide  contributions  to the p a r a s i t i c  a r e added t o g i v e  ray path.  guide  are  All  traced  the coupling  coeffi-  cient .  From diffracted  (3.18)  =  L  as o b s e r v e d a t  j T  a point  edge,  2  T  other  edge  diffracted  ^ ^" i =  n  t  n  e  driven  guide  excites  a  travels  a distance  r^ away  from t h e  |e-e | = T T / 2  if  Q  (B.2)  along  a distance field  When t h e l i n e  plane  =  2  u  (B.l)  where  +i  field  u  C^r^ECr)  i n the aperture  -l  This  field  field  u  diffracting  the i n c i d e n t  2  r^  of 0(k~  =  ~  if  |e-e | 0  = 3TT/2  .  t h e shadow b o u n d a r y away 1 / 2  )  i n the aperture i s from  C^ (r )E(r +r) )  1  and i s a g a i n d i f f r a c t e d  1  (3.26)  plane.  or  The r e s u l t a n t  doubly  (3.29)  .  source u - i n the aperture m-1  at an-  (B.3)  plane  is diffracted  from  another  223  edge  the r e s u l t a n t  u  =  m  is  — C ' l r j E ^ + . - . + r ,+r) m 0 1 1 m-l  r  i s the distance  field  u may b e d i f f r a c t e d m the resultant  u  where and  r  m  m+1  x^ i s g i v e n  parasitic  from t h e l i n e into  field  field  guide  from t h e i n c i d e n t  u , m+1  from  contributes  (m+1)  m  All now t r a c e d  to A ^ of x  and t h e v a l u e s  2  Q  °  °  1  f o r the fields  edge  A o f t h e mode i n t h e 00  +...+r  )  (B.6)  1  o f any r a y path  and on the t o t a l  from t h e d r i v e n  of x ,  a simple pattern  x^ a n d  plane  path  a r e thus  length  z=0 a n d b a c k  back t o c e n t r e .  t o centre  The p a t h s  t o the p a r a s i t i c  are evaluated  starting  r  =  m+1 E n=l  where  r t  =S  f o r each r a y p a t h .  represented  1 represents  and 0 represents  length  guide are  and e n d i n g a t t h e c e n t r a l  w h i c h may b e u n i q u e l y  (0,1,00,01,10,11,000,001,etc.)  central  source t o the d i f f r a c t i n g  r  m  and x  p o s s i b l e ray paths  y=z=0 f o l l o w s  the  (B.5)  plane.  i n the aperture  y=-d,  line  2_J.  /2"ka  Each r a y path  number  a t 6=0 o r 2TT a n d f r o m  t o the amplitude  /£_e  on t h e s i g n s  the aperture  guide  The  diffraction  -1/2 The 0 ( k ) contributions  in  the p a r a s i t i c  edge.  is  00  only  source to the d i f f r a c t i n g  by (B.2)  (m+1)  dependent  (B.4)  T T C' (r ) C (r + . . . + r ) E ( r ) 2 3 0 1 0 1 m  =  i s the distance  The  ,  2  where  (3.30)  , m-l  field  by a  edge binary  a r a y g o i n g down t o  a r a y g o i n g u p t o y = a , z=0 a n d  o f any r a y path  starting  and ending at  e d g e y=z=0 may b e w r i t t e n  5 =  /  [2a(1 =1  "W  + 2d  w  (B  -  7)  224  where i"mkj  =  a binary  m  =  number  k  =  decimal value  j  =  digit  All fields  four  -d)  four  in  of in  the  ray  is  incident  binary  binary binary  field  starting  Table  Ray  is  into  c o u n t i n g from  least  first  diffracted  parasitic  left  quadruply  and e n d i n g a t  g i v e n by S .  the  a ray  path,  number, number  is  representing  number,  i n c l u d i n g at  length  diffracted  Groups of  Group #  path  number  the  to  in  right.  diffracted central  The r a y p a t h s  g r o u p s d e p e n d i n g on f r o m w h i c h edge  g r o u p s shown i n  TABLE B . l  digits  a binary  them a p a t h  which the  into  penultimate  in  p o s s i b l e ray paths  y = z = 0 for  (y = 0 o r  of  number  contain within  naturally  digit  the  edge  divide  driven  and f r o m w h i c h e d g e  guide  (y =  0 or  a).  Rath  Penultimate Ray D i f f r a c t e d a t y=  Ray a t y=  Length r  t  0  0  S  2  0  a  S +  3  -d  0  S + d  4  -d  a  S + d + a  T^T  ^  The q u a d r u p l e  coupling  coefficient  pression  is,  are  u s i n g the  The  Paths  1  eaph group.  the  B.l.  Incident Diffracted  The p r o d u c t  guide  s  determined  and h i g h e r  given  order  by terms  asymptotic  form  by i n s p e c t i o n o f  of  diffraction the  (5.12):  form  the  a  ray  paths  contributions  (B.6).  in  to  the  The c o m p l e t e  ex-  225  +  2  y  y _2  , m=l  m=l  terms  beyond s i x o r d e r s  Note is  given  there of  by 2  that  are only  two r a y p a t h s  the r e f l e c t i o n  In widths  2^  the expression  M  (B.8)  .  1,  orders  using  values  (3.31)  The c o n t r i b u t i o n  was n o t s i g n i f i c a n t ,  f o r each order  of diffraction  however.  of diffraction  f o r n odd and n ^ l .  and  n  By c o m p a r i s o n  i n the  calculation  i3Tr/4  are given  to  0 0  ikna  even  coupling coefficient diffraction  in F i g . B . l .  improved over the t r i p l e  are expected  equal  (3.23)  3 up t o and i n c l u d i n g t r i p l e  significantly  of  even  f o r the complete  of d i f f r a c t i o n  cusps which  )  C (na)  n  Chapter  ^  reduces  .  .  of  1  was n o t f o u n d .  f o r each order  n  Numerical  +  ,  ;  t h e s p e c i a l c a s e o f TEM-TEM c o u p l i n g between g u i d e s o f  (a=d)  ka »  m  i  + 1  coefficient.  i  For  ^  n  2  . .  2  of ray paths  and  (  i  C (S+d+a)  m  of d i f f r a c t i o n  t h e number  f o r n even  n / / 2  2  k=l  o f 0 ( k "S  _D  C (S+d)  m  k=l  2  these  (  L  mil  e x p r e s s i o n f o r terms  m  . , _(2m+l) k=l 2  L  2  A general  C (s+a)  -  and (B.10)  Note t h a t  diffraction  a t t h e mode t r a n s i t i o n  A ^ using the  higher  the s o l u t i o n  results  widths  for  results  of Chapter  i s not 3.  are not obtained.  The  226  -h A comparison of obtained  by Yee e t  al.  expressions for  [50]  and Boersma  the  0(k  )  reflection  shows t h a t  [3]  the  coefficient  UAT s o l u t i o n o f  [3]  3/2 may b e 2  n  It for  1  n  1/f2  was the  obtained in  from the  the  ray-optical  denominator  conjectured  that  of  the  (4_oo)  infinite  Performing  i3TT/4  ?  2/2T  (ka)  3 / 2  w h i c h may be  rewritten,  1(4-oo) A  x  00  In  the  case,  results  converge  slowly  pected,  for  work a t  all,  n £  50.  I  e  e  2 n  -jT-  this to  (ka)  .3/2  (B.10)  yields  ikna (B.ll)  3/2  in  the  test  „m 2  I  e  ik2ma  ^„ 3/2 m=2 2 2  substitution  show a c u s p  because a r a t i o  at  works the  m  very  well,  the  this  series  0  CQ  •25H  O O CM  -50-  —1—.—.—.—1—1—1  1.0 Fig. coupling  2.0  d/X  B.l  coefficient  and t h e  mode t r a n s i t i o n  coupling case,  shows t h a t  (B.12)  3/2  verge .  8  on  expression  even  i3iT/4  However,  <  an i m p r o v e d  substitution  for  (8.3) a n d (8.9)).  eqns.  yield  n  u s i n g n=2m  2/2TT  reflection  by s u b s t i t u t i n g  ([3],  might  _n/2 2  n=4  n  [50]  sum  this  I  e  ^  of  same s u b s t i t u t i o n  coupling coefficient. A„„ 00  the  solution  A  00  numerical  widths,  substitution (B.12)  as  ex-  does  does not  not  con-  227  APPENDIX C CONTINUITY OF FIELDS ACROSS SHADOW BOUNDARIES  The expression  (9.24), which shows how a d i s c o n t i n u i t y i n the single  d i f f r a c t i o n r a d i a t i o n pattern of a staggered p a r a l l e l plate waveguide i s eliminated by adding a double d i f f r a c t i o n term, i s derived here using  (8.2) and the IT  asymptotic  form (8.5).  I t w i l l be shown that (Fig. 9.3a) f o r 6 = — -¥ that as  6 -»• 0 +ik — sin8 u  ±  - i k — sin9  (9+6) e  +  u  2  (  8  +  6  -ik ^ sin8 =  Note  u^ (9-6)  2  -ik£cos9  2  = c  2  (Cl)  ; c o s ( ^ -V) 2  and B a r e d e f i n e d  -ik£cos6  6  e  e  f r o m F i g . 9.1 S , + d  ) 6  by r e f e r e n c e  = -; c  to the previous  sin(^- - ¥ ) =  A, A ' , A " , A"' c  2  equation.  +ik j sin{j -V+6) j  D ( 0 , j -¥+6)E(r)e  +  |  D ( 0 , j - 4 ' ) G ( c , ^ j - ¥ , j -¥+6)E(r)e  =  7 2  D(0  -ik  ~ -^)G (c,~A  a  2  2  2  2  2  A-G^c,^-  -¥-6)  .  j sin(~ -y-6) - i k H c o s ( | - -¥-6)  -4f-6)E(r)e  +ikdsin(^- -¥+6) +ik£cos(^ -¥+6) e +  - i k i l c o s (j -¥+6) e  -ik  A'e  j s i n {j -¥+6)  d  (C.2)  e  A"G (c,^- -¥,j -¥+6) d  (C3)  228  + i k d s i n ( j -V+6) + i k J l c o s ( j - ¥ + 6 ) A'e  ..  +  e  CHI  A  { -  -i =  /4  |  A"'  +  i  k  c  C  O  s  F[/2kc-sin(f)]  5  e  +  e  -ikccos6  p  [  /  2  i  r  ^  s  i  n  (  e"  +  . |  )  ]  i  +  k  C  e  °  C  -  i  k  S( 2  c o s <¥- f) ] }  *" > F  C  6  C  °  S  (  2  f  +  6  )  F[/2kc-cos(¥ f)]} +  (C.4)  let  A e  <5 -> 0 , t h e n A ' = A " = A ' " = A , a n d  +  i  k  C +  A  ^  i ( - e  -ITT/4  =  A"  A  +ikc A  The  e  expressions  (9.24)  ^  +  i  _  i  k  i  C  F[O]  F[O] +  A +ikc —e  -  (9.26)  i  k  _  k  +e  -  e"  +  , +  _ B  i  C  °  C  k  C  Q  S  ™ P [ ^ t e cos*]  2 V J (  F[/2kc  e  =  are derived  ,_  A +ikc —e  + B  in a precisely  |  cos¥j  >  (C.5)  _ „ » Q E D . i<~.t>) n  analogous  (  r  fi  manner.  229  APPENDIX  D  N a t i o n a l Research C o u n c i l of Canada D i v i s i o n of E l e c t r i c a l Engineering Antenna P a t t e r n Range  TRANSMITTER A s t a n d a r d 1 mW s i g n a l g e n e r a t o r output was f e d through a c o a x i a l c o u p l e r to a t r a v e l l i n g wave tube a m p l i f i e r w i t h a g a i n of 30 db. A sample from the c o u p l e r was used to monitor f r e quency, u s i n g a d i g i t a l c o u n t e r . A low-pass f i l t e r was i n s e r t e d a t the output o f the a m p l i f i e r as a s a f e g u a r d a g a i n s t any harmonics. The s i g n a l was then fed through l o w - l o s s s e m i - r i g i d coax up to the roof. The antenna used t o f l o o d the h o r n under t e s t was a s t a n d a r d X-band horn.  RECEIVER The o t h e r end of the l i n k b a s i c a l l y c o n s i s t s o f a l a r g e t u r n t a b l e on the r o o f , which i s r o t a t a b l e through 360 , and app r o p r i a t e r e c e i v i n g i n s t r u m e n t a t i o n . Antennas under t e s t are mounted on t h i s t u r n t a b l e , whose angle of r o t a t i o n i s l i n k e d by a synchro system to a l a b r e c o r d e r c h a r t d r i v e . A f t e r r e c e p t i o n by the horn under t e s t , the R.F. s i g n a l was c o n v e r t e d t o an I.F. o f 65 MHz by a waveguide c r y s t a l mixer a t t a c h e d to the horn f l a n g e . Frequency s e l e c t i v e t e e s w i t h i n the S c i e n t i f i c A t l a n t a S e r i e s 1600 Wideband r e c e i v e r , permit the use o f a s i n g l e RG214/U c o a x i a l c a b l e t o t r a n s f e r the L.O. s i g n a l from r e c e i v e r t o mixer, and, a f t e r harmonic mixing takes p l a c e , the I.F. s i g n a l from the mixer back down to the r e c e i v e r . D i r e c t l y below the t u r n t a b l e , above the l a b c e i l i n g , an R.F. r o t a t i n g j o i n t w i t h mercury c o n t a c t s , f a c i l i t a t e s r o t a t i o n . A f t e r the CW s i g n a l reaches the r e c e i v e r , and has passed through a s e r i e s of d i f f e r e n t c o n v e r s i o n s , 1 KHz m o d u l a t i o n i s added. T h i s m o d u l a t i o n i s then d e t e c t e d by a bolometer d e t e c t o r and the output fed to a S c i e n t i f i c A t l a n t a R e c t a n g u l a r - P o l a r P a t t e r n Recorder, S e r i e s 1580 for p l o t t i n g .  PATH LENGTH T r a n s m i s s i o n p a t h l e n g t h from the t u r n t a b l e c e n t r e o f r o t a t i o n to the a d j a c e n t v e r t i c a l 4"x4" support was 27 f e e t , 4 i n c h e s . A c t u a l a p e r t u r e t o a p e r t u r e s e p a r a t i o n was about 12 i n c h e s l e s s .  230  i  if* ^ UJ m  £3  .<5  CJ  V  ul  uj  1 o  o  o:  U)  L_.  I  

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