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Modifications to horn antennas which reduce beamwidth and back radiation Jazi, Ahmad Safaai 1974

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MODIFICATIONS  TO  HORN  BEAMWIDTH  AND  ANTENNAS BACK  WHICH  REDUCE  RADIATION  by  AHMAD B . A . S c ,  A  Arya-Mehr  THESIS THE  S&FAAI-J&ZI  U n i v e r s i t y  SUBMITTED  IN  i n  OF  FOR  THE  APPLIED  the  T e c h n o l o g y ,  PARTIAL  REQUIREMENTS MASTER  of  FULFILMENT DEGREE  19  OF  OF  SCIENCE  Department of  E l e c t r i c a l  We  accept  r e q u i r e d  THE  t h i s  t h e s i s  E n g i n e e r i n g  a s  conforming  to  s t a n d a r d  UNIVERSITY  OF  June  BRITISH  197U  COLUMBIA  the  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives.  It is understood that copying or publication  of this thesis for financial gain shall not be allowed without my written permission.  Department of The University of British Columbia Vancouver 8, Canada  ABSTRACT  Modifications  to horn  antennas through  parabolic  cylinder  reflectors  introduced  to narrow  the  the  back  geometrical  Kirchhoff patterns  method of  the  reflectors  In  choked  the  are  the  in  the  i n the  reduction axis  gain  on in  the  from  region Very  horn  method  i s  of  where  the  E-plane to  parabolic geometrical  agreement i n the  the  applied  c h o k e s and  of  reduce  and  calculation  level  are  length.  concepts  patterns  lobe  gain  the  close  general  h o r n s by  i s  obtained  front  region.  back  radiation for  structure for  chokes r e s u l t s  back r a d i a t i o n , and  the  the  other  the  beamwidth  i f the  Considerably  the  Kirchhoff  approximate  of  beamwidth  reflectors,  and  the  and  horns  with  predicted.  Modification reduction  increase  in  measured  d i r e c t i o n s the  h o r n s and  reflectors  and  latter  theory  reflected  fails.  computed  rear  The  forward  theory  the  the  horns.  fields  in  diffraction  employed  flanges  i n v e s t i g a t e d . The  increasing  diffraction  are  modified  determine  between  beamwidth,  r a d i a t i o n without  The  are  choke  focal  lower  back  but  gain.  hand,  only  length  of  result  considerable  slight  improvement  in  by a  parabolic significant  substantial increase the  reflectors  radiation i s also  ii  a  Modification  will  and  a  in  achieved.  i s  i n the  on-  optimal.  TABLE  OF  CONTENTS  ABSTRACT TABLE  i i  OF CONTENTS  i i i  LIST  OF TABLES  i v  LIST  OF I L L U S T R A T I O N S  V  ACKNOWLEDGMENT 1.  INTRODUCTION  2.  RADIATION  v i i 1  MECHANISMS  AND E-PLANE  ANTENNAS . 2. 1 D i f f r a c t i o n by a Wedge 2.2 R a d i a t i o n Mechanisms 2.3 F o r m u l a t i o n o f R a d i a t i o n 3 .  AN ANALYSIS FOR E-PLANE ANTENNAS 3.1 I n t r o d u c t i o n 3.2 A N o n i s o t r o p i c L i n e 3.3 3.1*  4.  PATTERNS  OF  HORN 6 6 7 9  P a t t e r n  PATTERNS  O F CHOKED  HORN 16 16  Source  i n  a  P a r a l l e l - p l a t e  K a v s g u i d s ..........................................  R e f l e c t i o n D i f f r a c t e d  from Chcke F i e l d s  A NEW M O D I F I C A T I O N  TO HORN  I n t e r i o r  ANTENNAS  THROUGH  PARABOLIC CYLINDER FLANGES 4.1 I n t r o d u c t i o n 4.2 R e f l e c t i o n from P a r a b o l i c F l a n g e s 4.3 D i f f r a c t e d F i e l d s and R a d i a t i o n P a t t e r n 4.4 O p t i m i z a t i o n o f F o c a l Length 5.  NUMERICAL AND EXPERIMENTAL 5.1 Comparison o f Measured 5.2 Comparison o f P a t t e r n s 5.3  6.  1o 18 22  26 26 27 32 33  RESULTS and Computed P a t t e r n s Before and A f t e r  M o d i f i c a t i o n E r r o r s  . . . . . . . .  36 39 45 53  CONCLUSIONS  . . .  51  REFERENCES  55  APPENDIX  57  i i i  LIST  OF  TABLES  Table  page  I  Dimensions  II  O p t i m a l  III  Dimensions  IV  Measured i n  of  F o c a l  the  of  and  T e s t  Antennas  Lengths Chokes  foe i n  Computed  O n - a x i s  G a i n  iv  36  T e s t  the  37  E - p l a n e  V a l u e s and  Horns  of  39  Beamwidths,  B a c k - t o - F r o n t  I n c r e a s e s  R a t i o s  ....  45  LIST  OF  ILLUSTRATIONS  F i g u r e  Page  2.1  C o o r d i n a t e s  f o r  a  L i n e  S o u r c e  2.2  C o o r d i n a t e s  f o r a  Horn  i n  2.3  R a d i a t i o n  2.4  Geometry  3.1  A  3.2  Geometry  o f  a  Choked  4.1  Geometry  o f  a  Horn  Line  i n 5.1  a  Hedge  t h e E - P l a n e  •  . . . . . . . . . . . . .  Mechanisms o f  Images  Source  i n  i n  a  o f  a  P a r a b o l i c  7 8 9  t h e Lower  K a i l  P a r a l l e l - P l a t e Horn  Waveguide  Antenna  M o d i f i e d  by  12  i n  . . . . . . .  t h e F - P l a n e  P a r a b o l i c  ..  18 19  F l a n g e s  t h e E - P l a n e  Diagram and  Near  27 P y r a m i d a l  Horn  Alone  a n d with  Choke  Flanges  37  5.2  Antenna  Range  5.3  E - P l a n e  P a t t e r n s  o f  Horn  A  w i t h  Chokes  40  5.4  E - P l a n e  P a t t e r n s  o f  Horn  B  with  Chokes  41  5.5  E - P l a n e P a t t e r n s R e f l e c t o r s  o f  Horn  A  w i t h  P a r a b o l i c  E - P l a n e P a t t e r n s R e f l e c t o r s  o f  5.6  a n d Test  Horns  38  43 Horn  B w i t h  P a r a b o l i c 44  5.7  A Comparison o f Measured E - P l a n e B e f o r e and A f t e r M o d i f i c a t i o n s  5.8  A Comparison o f Computed E - P l a n e P a t t e r n s o f Hern A B e f o r e and A f t e r M o d i f i c a t i o n s . . . . . . . . . . . . . . . . . . . .  47  P a t t e r n s o f Horn A w i t h at S e v e r a l Wavelengths  49  5.9  5.10  Beamwidths  5.11  A Comparison o f H-Plane p a r a b o l i c R e f l e c t o r s  A.1  C o o r d i n a t e s  o f  Horn  f o r a  A  P a r a b o l i c  w i t h  L i n e  v  P a t t e r n s  Hern  A  46  R e f l e c t o r s  P a r a b o l i c  P a t t e r n s  o f  R e f l e c t o r s  o f  Horn  A  . . . .  49  w i t h 50  Source  Fear  a  Half  P l a n e  . . .  57  A.2  E r r o r  i n  I  (Y)  f o r  r<  r  60  A.3  E r r o r  i n  I  (y)  f o r  r<  r  61  v i  AHKNOWLf-DfiMftNT  I  am  thankful  Interest, writing  advice this  The  financial  the  form  Research  of  to  Dr. and  R.  V.  Jull,  guidance  my  supervisor,  during  the  for  r e s e a r c h work  his and  thesis. support UBC  Council  from  the  Utnverslty  graduate  fellowship  of  during  Canada  acknowledged.  vii  the  of  British  and  from  months  of  Columbia  the May  In  National  and June  Is  1  INTEODUCTIOH  1.  R e c t a n g u l a r communications C o n s i d e r a b l e  horn  antennas  and  r a d i o  a t t e n t i o n  has  p a t t e r n s  of  horns  f o r  beamwidth  and  h i g h e r  g a i n .  waveguide  s u p p o r t i n g  d i s t r i b u t i o n  i s  s i n u s o i d a l l y t h a t  magnetic  f i e l d ,  the  c a l l e d  H - p l a n e  the  H - p l a n e  edges  a r e  t h e  r a y s  s o u r c e  o b l i q u e l y  any  r e s u l t s  an  i n  p r i m a r i l y  w i t h  s h o r t  d i f f r a c t e d  by  the  as  edges  back to  the  can  E - p l a n e  antennas  horn  edges  t h o s e  of  the  E - p l a n e to  p y r a m i d a l p a t t e r n  of  a r e  and  at  f i e l d s  H - p l a n e  p a t t e r n ,  The  l a s t  by  and  a r i s e s  p r o d u c i n g p a t t e r n .  antennas h i g h e r  by  edges,  H - p l a n e  horn  the  e x c i t e d  edges  the  i s  e l e c t r i c  d i f f r a c t e d  E - p l a n e  p a t t e r n .  i t  i l l u m i n a t e d  d i f f r a c t e d  the  H - p l a n e  r a d i a t i o n  edges  f i e l d  i n c i d e n t  i n c i d e n t  f i r s t  the  of  a r e  a  words,  i l l u m i n a t i o n  E - p l a n e  c o n t r i b u t e t o  whereas  the  the  by  e l e c t r i c  t o  r a y s  the  fed  p a r a l l e l  Hence,  upon  antenna  o t h e r  the  l o b e s  narrower  i n  to  than  r a d i a t i o n  T h i s ,  p a r a l l e l  f e e d s .  l o b e s ,  E - p l a n e ,  edges,  the  microwave  the  the  E-plane  waves.  which  improved  horn  p r i n c i p a l  by  s t r o n g e r  horn  i n primary  back  mode  Indeed,  w h i l e  of  and  1q  edges  m o d i f i c a t i o n  t h e  TE  as  improve  H-plane.  i n c i d e n t  f i e l d s  Consequently  I n  are  to  p y r a m i d a l  u s u a l l y  s i g n i f i c a n t l y  d i f f r a c t e d  i n  o p t i c s  edges  a  used  and  s i d e  dominant  edges.  j u n c t i o n ,  a c t u a l  c o n t r i b u t e from  i s  g e o m e t r i c a l E - p l a n e  In  the  the  paid  lower  a p e r t u r e  than  horn-waveguide t h e  i n  w i d e l y  astronomy been  d e s i g n a t e d  s t r o n g e r  f i e l d ,  uniform  tapered  i m p l i e s  much  the  a r e  g a i n  which d e a l s  p a t t e r n .  a  s i g n i f i c a n t  i n t o  the  amount  l a t e r a l  and  of r e a r  energy  i s  d i r e c t i o n s  2  which  r e s u l t s  back  l o b e s  and  s u b s t a n t i a l  have  R o b e r t s  q u a r t e r  i n  [1]  l o n g  i n  s u b s t a n t i a l l y u t i l i z e d  choke  the  d i r e c t i o n s ,  horn  which  i s  reduced.  Lawrie  which  c o n d u c t i n g  the  E - p l a n e  and  i d e n t i c a l The  over  and  and  the  of  f l a n g e s  on  In  i f  a  the  the  c h i e f  r e s u l t i n g  horns.  a c c o u n t s  f o r  et  p a t t e r n s f o c u s i n g ,  of  i n  horn  the  a  horn  a p e r t u r e  broadening,  n e a r l y  a p e r t u r e . reduced  ot  f a l l  s t u d i e d  i n The  are  H - p l a n e  not  i s  s u r f a c e s .  square  and  [ 3 ]  r e a r  antennas  advantage  do  a l  are  e x t e r i o r  s i g n i f i c a n t l y  E  can  d i s t r i b u t i o n s ,  has  w i t h  horns  Koshy  r a d i a t i o n  horn  i d e n t i c a l  c o r r u g a t e d  r a d i a t i o n  c o r r u g a t e d  but  a  the  edges;  t h e  a r e  u t i l i z e d  i s  t h i s  r a d i a t i o n i n t o  the  t h e  e f f e c t  antennas. and  p a r a b o l i c t o  r e f l e c t  d i r e c t i o n .  horn  of  T h e i r  t i l t i n g  waves  the  T h i s  the  r a y s  Consequently  r a y s  w i t h i n  the  t h r o u g h  m o d i f i c a t i o n  the  f i e l d s  w i t h  r e f l e c t o r s a l l  antennas  improving  Indeed,  c y l i n d r i c a l  to  i n t r o d u c e d .  beamwidth,  r a d i a t i o n .  t h e r e f o r e ,  forward  m o d i f i c a t i o n  r e f l e c t o r s  back  edges  be  new  narrowing  the  E - p l a n e  the  of  edges  the  c o r r u g a t e d  l o b e s ,  t h e s i s  f o r  t h e  i n t o  p a t t e r n s  are  t h a t  c h o k e s  on  p y r a m i d a l  be  c y l i n d e r  r e d u c i n g  can  w a l l s  LaGrone  beam.  p a r a b o l i c  edges  used  a l s o  the  t h i s  accounts  [2]  may  the  f a c t ,  c u r r e n t  p a t t e r n  f l a n g e d  r a d i a t i o n  these  beam  i s  i n v e s t i g a t i o n  In  of  showed  a p e r t u r e  d i s t r i b u t i o n  by  P e t e r s  a t t e n t i o n .  the  Consequently  E - p l a n e  Furthermore,  c a t e g o r y  on  main  back  m o d i f i c a t i o n beams.  caused  H - p l a n e  r a d i a t i o n  s i d e  c u r r e n t  s u p p r e s s i o n  i n v e s t i g a t i o n  r a d i a t i o n .  antenna.  The  c o n s i d e r a b l e  f l a n g e s  back  s u p p r e s s  of  r a d i a t i o n .  e x p e r i m e n t a l  reduce  to  s u r f a c e s  r e c e i v e d  an  wavelength  back  g a i n  and  d i f f r a c t e d emanating  f o c i r e a r  r a d i a t e d  by from  a t  these  d i r e c t i o n s  power  i n  the  3  forward d i r e c t i o n increases, r e s u l t i n g i n narrower beamwidth and higher gain. Furthermore, back radiation depends e n t i r e l y on the f i e l d s d i f f r a c t e d by the r e f l e c t o r edges, these least  doubly  diffracted  and  fields  are  weak. Hence, s u b s t a n t i a l l y  at  lower  back radiation i s a l s o a n t i c i p a t e d . The analysis of E-plane patterns of flanged based  upon  can  directions,  however,  this  method  fails  to predict t h e radiation pattern, because the aperture  f i e l d d i s t r i b u t i o n i s assumed to be that of the incident and  be  the Kirchhoff method i n only t h e forward region. In  the l a t e r a l and rear entirely  horns  zero  outside  the  aperture,  and  this  is  not  fields a  assumption i n these regions. The geometrical d i f f r a c t i o n concepts,  on  the  other  hand,  have  recently  been  valid theory  used  to  successfully compute t h e uadiation patterns of aperture antennas i n a l l regions. It has also  been  used  in  other  antenna  and  waveguide problems. I n i t i a l l y , K e l l e r [ 4 ] introduced h i s 'geometrical theory of diffraction*  and  later  used i t to study the d i f f r a c t i o n by an  aperture [ 5 ] . This theory , sometimes strictly  valid  for  the  abbreviated  for  diffraction*  the  asymptotic  analyse the E-plane diffraction  of  is  'the  geometrical  to compute the gain and the radiation  pattern of a corner r e f l e c t o r antenna. In 1965, used  GTD,  d i f f r a c t i o n of incident waves of very  short wavelengths. In 1963, Ohba [ 6 ] employed method  as  solution patterns  for of  Russo et a l [7]  l i n e source d i f f r a c t i o n to  horns.  They  refer  to  this  l i n e source as 'edge d i f f r a c t i o n theory*. Using  •edge d i f f r a c t i o n theory*, i n  1966,  Yu e t a l [ 8 ]  presented  a  comprehensive i n t o  a c c o u n t  i n t e r i o r . a  a n a l y s i s  Yee  The  d i f f r a c t i o n  e t  c a l c u l a t e d  a l  d e r i v e d  by  of  The  the  of  d i f f i c u l t y  d i f f r a c t i o n  t h e  of  o t h e r  t h e  a  l i n e  hand,  the  which  s o u r c e  near  f r e e  of  problem.  p a r a b o l i c  f l a n g e d the  c a l c u l a t i o n s . K i r c h h o f f o n - a x i s i n  our  the  method  an  t h e on  at  of  d i f f r a c t i o n r e f l e c t i o n  been  r e c e n t l y  c o n c e p t s  of  the  f u r t h e r  f i e l d s  a t  h a l f but  the  t o  a  p a r a b o l i c  (wedge),  on of  remains  s i m i l a r f o r  of  o n - a x i s and  the  c a l c u l a t i o n  r e f l e c t o r s  a  problem.  t h e o r y  the  f a r  p a t t e r n s  r e l i a b l e  f o r  t h e  s t i l l  d i f f r a c t i o n  The  d i f f i c u l t y  r a d i a t i o n  i s  e n c o u n t e r s  p l a n e the  has  c a u s t i c s .  s o l u t i o n  encounter  edge  d i f f r a c t i o n  c a u s t i c s  method  and  the  and  the  a l t e r n a t i v e  chokes  c o e f f i c i e n t  have  of  c o n d u c t i n g  we  K i r c h h o f f  of  of  r e s t s a  •  horn  .  1  a n a l y z i n g  horns  a s  of  t a k i n g  from  t h e  horn  b o u n d a r i e s  f i e l d s  In  and  s i n g u l a r i t i e s ,  Consequently  f i e l d s  v i l l  be  of used  a n a l y s i s .  Host been  of  use  horns,  g e o m e t r i c a l  t h e o r y  c a l c u l a t i o n  t h e o r y  the  made  shadow  r e f l e c t i o n  horns  d i f f r a c t i o n  the  c h a l l e n g i n g  F o r t u n a t e l y  who  of  r e f l e c t i o n  s e c t o r a l  g e o m e t r i c a l  i s  e v a l u a t i o n  [11]  of  K e l l e r ' s  t h e  u s i n g  E - p l a n e  p a t t e r n s and  p y r a m i d a l  long  theory  on  f i e l d  of  [ 1 0 ] ,  s i n g u l a r i t i e s  edge  waveguide  a  J u l l  g e o m e t r i c a l  [ 9 ]  g a i n  c o e f f i c i e n t  E - p l a n e  m u l t i p l e  p a r a l l e l - p l a t e  t h e o r y .  f o r  of  s t u d i e d  a n a l y z e p r i n c i p a l  the  the  e x i s t i n g  m o d i f i c a t i o n s  e x p e r i m e n t a l l y . E - p l a n e  p l a n e  t w o - d i m e n s i o n a l  p a t t e r n s  r a d i a t i o n one  In  i f  t h e  t h i s of  t h e s i s  f l a n g e d  p a t t e r n s antenna  t o  has  c a n  horn  antennas  a t t e m p t s horns. be  The  are  made  t o  problem  of  reduced  s e p a r a b l e  have  a p e r t u r e  to  a f i e l d  5  d i s t r i b u t i o n s . t h e  Rhodes  E-plane  p l a n e  p a t t e r n s  and f e d  e s s e n t i a l l y assuming  by  t h a t  In  a r e  by  t h e  horns  on  c h a p t e r  4 .  a r e  f a r  t h e  i s  s t u d i e d .  i n  i n  t h a t  mode,  10  i n  t h e i s  a r e  H - p l a n e , c o n s t a n t .  d i s t r i b u t i o n s  t h e  problem  d i f f r a c t i o n  r a d i a t i o n  r a d i a t i o n The  f o r  o f  a  t h e  p a t t e r n o f  p a t t e r n  o f  5 .  o f  a  p e r f e c t l y f o r  a  h o r n  p a t t e r n  e f f e c t  computed c h a p t e r  by  mechanisms  o p t i m i z a t i o n and  a n g l e  horn  f i e l d  r a d i a t i o n  The  p r e s e n t e d  t h e  f i e l d  3  The measured  angle  and r a d i a t i o n  t h e  f l a r e  t h a t  t w o - d i m e n s i o n a l .  t h e s i s  t h e E-plane  r e f l e c t o r s i n  a s  demonstrated  t h e TE  ; t h e r e f o r e ,  t h i s  f l a n g e s  o f  a p e r t u r e  regarded  c h a p t e r  f i x e d  f l a r e  l e n g t h  r e v i e w e d  Then  w i t h  s u p p o r t i n g  s e p a r a b l e  o f  i s  choke  v e r i f i e d .  m o d i f i e d  be  2  In  i n v e s t i g a t e d a l s o  o f  s l a n t  c a n b e  wedge  f o r m u l a t e d .  c y l i n d e r  t o  c h a p t e r  d i s c u s s e d .  m o d i f i e d  horn  t h e a p p r o x i m a t e  p a t t e r n  c o n d u c t i n g  a  waveguide  t h e  a r e known  E - p l a n e  a  h a s e x p e r i m e n t a l l y  f o r  independent  F u r t h e r m o r e , horn  [ 1 2 ]  f o c a l  r a d i a t i o n  o f  a  i s horn  p a r a b o l i c a  horn  i s  l e n g t h  i s  p a t t e r n s  of  6  2.  RADIATION  I n by  a  MECHANISMS  t h i s  c h a p t e r  p e r f e c t l y  E - p l a n e review  t h e b a s i s  n  As  i s  e s t a b l i s h  an  and o f  t h e l a t e r  known,  Sommerfeld f o r  wave  n o r m a l l y  i n c i d e n t  h a l f  a s y m p t o t i c  f o r m u l a  t h e d i f f r a c t i o n  zone  w i t h  d i f f r a c t i o n case,  when  e x a c t  *  d  b y  a  h a l f  ( r , 6 )  I n  e  x  p  (  "  j  i  -  y  k  2  r  r  Q  )  H f o r  where  E  a n  [ 7 ]  s o u r c e o f  a r e r e v i e w e d .  o f  plane  i s  t o  p o l a r i z a t i o n ,  # 9  [v(r ,6 0  a  t h e T h i s  a  n  +  be  t h e r e i n  zero.  There a  s i m p l e  w r i t t e n  o  t o  source  0 ,n) + v(r ,0 o  f i r s t  l i n e  o f  t o  a  plane  edge  [ 1 1 ]  P a u l i ' s  l i n e  o f  t h e  plane  r e c i p r o c i t y  t o  t h e  P a u l i  of  wave  o b t a i n b y  a  a s a  o  i n f a r  wedge.  T h e  p a r t i c u l a r  source  0 ,n)J,  a  t h e  form a t  by  an  a  h a l f  f o r  f a r  r,  due  9  a s  -  an  by  i s , however,  f a r f i e l d  a  d e r i v e d  f o r m u l a t i o n  c l o s e d  t h e d i f f r a c t e d c  0  a  upon  of  c o n t a i n e d  tends  was  d i f f r a c t i o n  used  i s o t r o p i c  reduces  a t  horn  L a t e r ,  t h e d i f f r a c t i o n  (kr) .  a l  F i g . 2 . 1 .  source  -  e t  angle  f o r  which  [ 1 1 ] .  l i n e  o f  t h e wedge  [ 1 5 ] ,  f i e l d s a  Russo  plane.  t h e p r i n c i p a l  s o l u t i o n  plane  t o  f o r  d i f f r a c t i o n  l i n e  t h e f o r m u l a t i o n  t h e  c o n d u c t i n g  c o n j u n c t i o n  ANTENNAS.  i s o t r o p i c  [ 1 3 ]  s o l u t i o n  1 9 6 5 ,  HORN  c h a p t e r s .  p e r f e c t l y  I n  a  e x a c t  e l e c t r o m a g n e t i c  Hedge.  an  OF  §...flg32ga.  u  w e l l  p a t t e r n  f o r  o f  wedge  r a d i a t i o n  S ^ f £E§£lii9 ..... ftK.  2aJ  PATTERNS  t h e d i f f r a c t i o n  c o n d u c t i n g  f a r f i e l d forms  AND E-PLANE  (2-1)  7  _a v(r ,a,n)  -  Q  exp [ . i ( k r c o s a + 0  IT  21 cos 21 s i n n  4)]  nfcos  (f)  cos —) n  n  (2-2)  1/2, * F t ( k r _ ( l + cosa))  w i t h  F(T) = / ™ e x p ( - j y ) d y  f i e l d  o f  2  t h e l i n e .  kr»i  m  with  f r e e - s p a c e  propagation  r e p r e s e n t s c l o s e  F i g .  g 2 &  f e d  F r e s n e l  i s o l a t i o n  a n d( 2 - 2 ) ,  r  t o t h e d i f f r a c t i n g c o n s t a n t  t h ewedge order  and x  angle  terms  i n t e g r a l . i s ^  ,  = exp(-j  1  6  edge,  T h e  kr)/(kr)  a r e k= ^ —  p o l a r  i st h e  i s t h e wavelength,  e q u a l  t o (2-n)Tr  n e g l i g i b l e  f o r kr  Q  f a r  n  i s  a n dR l a r g e  Q  and n  t o2.^  2 . 1 . C o o r d i n a t e s wedge,  R a d i a t i o n  & an  higher  i n  ( 2 - 1 )  r e s p e c t  b y p u t t i n g  t h e complex  source  c o o r d i n a t e s  obtained  ] + RQ}  magnetic  a p p r o p r i a t e b y  r e f l e c t o r  a  f o r a  l i n e  s o u r c e  near  a  c o n d u c t i n g  Mechanisms  l i n e model  s o u r c e  o f a c o r n e r  f o ra t w o - d i m e n s i o n a l  p a r a l l e l - p l a t e i s formed  a t t h e apex  waveguide  E-plane  s u p p o r t i n g  b y t w o c o n d u c t i n g  h a l f  r e f l e c t o r i s horn  antenna  t h e TEM mode. T h e  p l a n e s  i n t e r s e c t i n g  1 / 2  8  a t  an  a n g l e  S,  a  uniform  ~  e  E  <8<e  .  E  e x c i t e s  29  edges  horn  r a y s  are  from  A The  an  The  i l l u m i n a t i o n  edge  B  edges  r e s u l t s  d i f f r a c t e d d i f f r a c t e d A  and  B  d i f f r a c t e d  lower  wedges.  e s t a b l i s h from t h e  (2-1) weaker  P a r t  t h e  of  r a y s  from  f i e l d s .  h i g h e r  (2-2)  t h a t of  d i f f r a c t e d  g e o m e t r i c a l r i s e  t o  o p t i c s  2 . 3 ( a ) . s o u r c e  A  edge  A  s i n g l y  by  h i t  the  B  and  shows  h i g h e r  the  d i f f r a c t e d  from  t h e  edges  from  may  now  be  other  one.  f i e l d  L i k e w i s e ,  p r o d u c i n g r a y s  t h o s e  waves I t  order  of  both  f o u r  more  of  the  c o n t i n u e  may  the  a n o t h e r  from  g i v e  o n l y  d i f f r a c t i o n s .  as  d i f f r a c t e d  the  W to  d i f f r a c t e d  o r d e r  d i f f r a c t e d  d i f f r a c t e d S  I t  E - p l a n e  s i n g l y  t h e  r a y s  edge  edge  wave.  emanating s i n g l y  a t  r e g i o n  d e s i g n a t e d  d i f f r a c t i o n .  2.3(b)  o r d e r  a  i n  the  f i r s t  i l l u m i n a t i n g  wedges  o r d e r  and  source  o p t i c s  the  Each  antenna  o r d e r  F i g .  second  i n  waves  s t r i k e  i n t e n s i t y  the  The  r a d i a t e d  c y l i n d r i c a l  edge  second  primary  f i e l d s ,  horn  of  f i e l d .  and  and  a  i n  The  t h i r d  l i n e  the  d i f f r a c t e d  F i g .  f o r  f u r t h e r  doubly  the  g e o m e t r i c a l  induced  From  i s  g i v i n g  are  i n  C o o r d i n a t e s  doubly  wave  B,  The  2 . 2 .  c a l l e d  f i e l d s ,  2 . 2 .  s i n g l y  F i g .  r e s u l t i n g  i l l u s t r a t e d as  i n  i s  and  edges.  c o n s i d e r e d  F i g .  wave  d i f f r a c t e d  t h e  shown  c y l i n d r i c a l  T h i s  d i f f r a c t i o n . s i n g l y  a s  E  be  to seen  d i f f r a c t i o n ,  f i e l d .  A  and  B  e n t e r  the  9  horn  and  F i g .  2 . 3 .  r a y s  may  images  are  r e f l e c t e d  R a d i form (b) the (c) f i r s  a t i o n S and Doubly i n c i d R e f l e c t t image  be  t r e a t e d  i s  the  determined  by  the  take  horn  When  p l a c e  o b t a i n e d  the by  d i f f r a c t e d  w a l l s .  2 . 3 ( c ) .  ms, (a) G e o m e t r i c d i f f r a c t e d waves e d waves from A, n g l y d i f f r a c t e d from horn i n t e lower w a l l .  method the  when  F i g .  of  images  f l a r e  some  of  the  of  r e f l e c t e d  a l o p t i c s wave from A and B. S and H due to wave from B. r i o r f o r the  [ 8 ] .  a n g l e  The  The the  number  horn.  r e f l e c t e d  of  F u r t h e r  r a y s  s t r i k e  edges.  the  c o m p l e t e d ,  i t s  mechanis s i n g l y d i f f r a c t e n t s i e d wave i n the  by  d i f f r a c t i o n s back  by  p r o c e s s f a r  f i e l d  of  r a d i a t i o n  s u p e r i m p o s i n g  and  d i f f r a c t i o n  r e f l e c t e d  the  f i e l d s  p a t t e r n  and of  the  g e o m e t r i c a l as  r e f l e c t i o n horn  o p t i c s  d e s c r i b e d  i n  can  and  the  i s be  v a r i o u s f o l l o w i n g  s e c t i o n .  2. 3_Formulation  In  t h i s  d i f f r a c t e d  o_f_Radiatipn  f o r m u l a t i o n  f i e l d s .  The  P a t t e r n  (2-2)  i s  employed  c y l i n d r i c a l  wave  t o  d e t e r m i n e  the  p r o p a g a t i o n  f a r  f a c t o r  1/2 exp(-j  kr)/(kr)  angular the  Z  t  o  dependence  a x i s ,  F i g .  2. 2.  t  n  ^  e  a  r  f i e l d  i s  i s  of  i n t e r e s t .  ,  i s  f u r t h e r  o m i t t e d ,  The  used  because  symmetry t o  o n l y  p r o p e r t y  s i m p l i f y  the  the about  problem  10  by  c o n s i d e r i n g  f i r s t  only  s u b s c r i p t s  o c c u r s ,  w h i l e  o r i g i n  o f  t h e  r e f e r t h e  i n c i d e n t  upper  t o  t h e  second  rays.  problem  Yu  which  i s  [ 8 ] ,  u n r e a l i s t i c  primary  n o r m a l i z e d  H  Cdgcs  A z e r o  the  horn,  H  A S  where  1  )  1  9  )  n=2.  a r a  <«  a n g l e .  d i f f r a c t i o n o f  t h e order  o f  f o r m u l a t e d  by  o m i t t i n g  some  B y  •  u n i f o r m  o p t i c s  wave  c y l i n d r i c a l  from wave,  •  E  (2-3)  '  6  by  E x c l u d i n g  d i f f r a c t e d  w r i t t e n  ^ E ' ^ E *  =  1  t h e  waves  f i e l d s  from  wave  from  d i f f r a c t e d t h e s e  edges  C  i n t o i n t h e  a s  )  1  t h e c y l i n d r i c a l  »  0 < 8 < ir  ,  (2-4)  symmetry,  <  ' ( - e"' )  0 < 8 <  »  N  ~  „ H  The  by  A l l t h e  t h e p o i n t s  i n d i c a t e  here  a  i l l u m i n a t e d  t h e s i n g l y  "AS  ( 1 ,  t o  t h e g e o m e t r i c a l  i s  0 < 8 < 6  H A. Cy — HBS  r e f e r  o r i g i n a l l y  m o d i f i e d  a t S  (  c a n be  (  which  .  u n i t y  i n c i d e n t  zone  was  F i g . 2 . 2 .  s o u r c e  and D  w i t h  f a r  t o  t o  (6) -  s  a t  0 < 8 < n  terms.  R e f e r r i n g t h e  p o i n t s  T h e s u p e r s c r i p t s  T h i s  a l  r e g i o n  s u b s c r i p t s  d i f f r a c t i o n . e t  h a l f  AS  d i f f r a c t e d  (  2  i  r  -  6  )  waves  >  1  r  - E 6  <  9  r e f l e c t e d  *  * »  by  t h e  (2-5)  horn  w a l l s  may  be  11  v i e w e d F i g .  as  c y l i n d r i c a l  2.4.  shows  I n s o f a r  as  c o n c e r n e d , i n t o  the t h e  lower  t o  t h e  i  being  the  S i m i l a r l y , t h e  H^  l a s t  The  =  l o w e r  i s  not  HJ  lower  and an  w a l l  t h e  the  l a s t  image  i n  i n  upper  i n  by  the  images t h e  lower  The  t h e  w a l l  t h e  p a t t e r n .  o b t a i n e d  1,2  edge lower  h a l f  upper  be  w a l l waves  s u b s t i t u t i n g  9  w a l l .  r e g i o n  s h o u l d  image  images.  i n  i s  t a k e n  may  a l s o  from  t h e  (2-4)  f  ,  -  (i+i)  e  E  <  e < f - i e  by  ;  E  h,  l a r g e s t  (2-6)  i n t e g e r  image  wave  YQ~  -  •  from  the  ,  0  upper  w a l l  c o r r e s p o n d i n g  to  i s  (-2h9  1}  s u b s c r i p t s  t h e  from  E  t h e  -  waves  <-2ie -e)  image  (9)  1}  of  from  g i v i n g  HJ^CB) = H J ^  h  emanating  p a t t e r n  r a d i a t i o n  a r e  ,  E  r a d i a t i o n  The  w a l l  -2i6 -6  geometry  image  a c c o u n t .  c o n t r i b u t e  the  waves  L  the  and  U  upper  i n t e g e r s h o u l d  +9)  E  i n d i c a t e w a l l s  the  be  <  9 <  (h+l)9  t h a t  the  r e s p e c t i v e l y .  v a l i d  m o d i f i e d  r e g i o n  f o r  E  -  f  image When t h e  •  (2-7)  waves t h e  l a s t  a r e  r a t i o image  from TT/29  i n  e  the  to  (2-8) 0  <  6 < We  IT now  i l l u m i n a t e d  (2h+l)9  E  examine by  t h e  • the  second  s i n g l y  o r d e r  d i f f r a c t e d  d i f f r a c t i o n s . f i e l d  of  an  Edge  A  i n t e n s i t y  i s of  12  F i g .  2.4.  Geometry  H^g^(~) f r o m v a r y i n g i t  i s  by  a  a  edge  B.  f u n c t i o n s  i n  reasonable  uniform  second  of  i n  S i n c e  the  the  d i f f r a c t e d  = H f  the  wave  from  B.  f i e l d  from  edge  n)  8,  A  lower  i s  w a l l ,  f i e l d s  H  f u r t h e r  ( 2 )  g i v e n  i l l u m i n a t e d  g i v i n g  r i s e  to  of  <  s e t  any  Dnder  8 <  the  waves  A  s l o w l y  p a r t i c u l a r  t h i s  of  edge  A  assumption,  28  +  E  8,  n)]  (6) - H.^  (2-9)  from  the  second  images  o r d e r  i n  Summing  t h e  the  d i f f r a c t e d  ( f " i6 ) [vUl. f - « E  + 8, n ) ]  E  + > 8  E  ,  n )  0  +  <  V  (  U  '  2 (2-10)  6 < ir  where  = l±-l  the  by  - (i+2) 8  11  i s  ,  •  IT  a n g l e ,  i s  -  rays  of  are  i l l u m i n a t i o n  + v ( b , f  by  a  w a l l .  d i f f r a c t e d  o Edge  lower  t h a t  [v(b,f +  (f)  }  B  the  neighbourhood  assumption  c y l i n d r i c a l  order  U ™ W  images  cos8  E  + l  second  Q  cos  order  i8  E  ,  i  d i f f r a c t e d  = l,2,....h-l  f i e l d s  from  .  edge  A  g i v e s  13  H  A  (  2  The  )  (9)  = H ^  second  image  c a n  a p p r o p r i a t e  angles The  of  u  -  Moreover, because  e  be  t c  they  a n g l e s  of  e t  a l  I t f i e l d s  of  c a n  i l l u m i n a t i n g  be  be  I t  i s  r a y  i s a  employed.  In  doing  edge  d e r i v e d  s o , a s  (2-6)  by  r e m a i n d e r s  p a i r s  t h e s e  used  i s  a r e  t r e a t e d  b e l i e v e  p r e f e r a b l e  t o  omit  of  do  p a t t e r n . v a l i d  n o t  u n l i k e l y  c l o s e  t c  wedges  t h i s  t h e s e  not  not  a r e  these  He  .  wedges  wedges  (2-2)  E*s  by  r e s p e c t i v e l y .  (2-2) i n  of  o r d e r e  (2-7)  r a d i a t i o n  ).  1 ?  second  and  i n t e r i o r  f i e l d  f i e l d s  t h e  s u b s t i t u t i n g  two  and  f a r  i s  f i e l d s  be  as  i f  not  a  i n  t h e  p a t t e r n . now  how A t  t h e each  c a l c u l a t e d .  l i n e  ,  t o  however,  determined.  of  i s  t h e  and  by  (2-5)  be  (2-11)  B  forms  t h e  + ^E. t o Tf  [ 8 ] ,  e v i d e n t  d i f f r a c t i o n  A  =i  r a d i a t i o n  s h o u l d  i n  due  t h e s e n  from  e x t e r i o r  t o  < ir  (2-11)  o f 29^, a n d 2 ( i r - 9  assumption.  c a l c u l a t i o n  o f  < 9  f i e l d s  f i e l d s  t h e r e f o r e Yu  were  horn  arguments  n e g l i g i b l e .  0  j u n c t i o n  t h e  e v a l u a t i o n  ,  those  s i g n i f i c a n t l y  2  ( 9 )  from  l i k e  d i f f r a c t e d  t h e  )  o b t a i n e d  on  E  2  d i f f r a c t e d  waveguide-horn  c o n t r i b u t e  v a l i d  H ^  + | ^  (6)  arguments  doubly  enough  )  order  waves  The  2  s o u r c e t h e  t o t a l  h i g h e r s t a g e  Then by  a f a r  o r d e r  t h e  the wedge  d i f f r a c t e d  i n t e n s i t y s o l u t i o n  (or  h a l f  d i f f r a c t e d  of to  plane)  f i e l d  t h e t h e i s from  14  V  6 )  where  1 ) ( 8 )  ^  +  H i n d i c a t e s  i n t o  a c c o u n t ,  AB  H  AS  H  =  H ^  m  m  )  )  (  6  )  t h e order  U  +  o f  ^  (  9  )  *  1  ( 2  t h e d i f f r a c t e d  f i e l d s  t o  "  1 2 )  be  taken  and  =  (6)  ( 6 )  [  = c j  1  " " ^  v  (  b  1  »  [v(£  l f  +  *>  |  + i 6  n  )  +  E  V  +  (  b  *  I F "  2  9  E  +  e, n) + v(£ , ±  0  ,  n  ^  )  ]  '  (2"13a)  (i+2)6  -  E  +  6, n)]  (2-13b)  w i t h  ^  =  ^  It  s h o u l d  By  symmetry,  (  "  "  f  be n o t i c e d t h e  i  E  9  )  '  t h a t  t o t a l  (  f o r  m=1  d i f f r a c t e d  ,  H'(e)  f i e l d  «  (e)  from  edge  2  _  1  A  b  )  . B  i s  d e r i v e d  as  H (-8)  ,  0 < 9 < |  H (2Tf-6)  ,  ir-8  A  H (6) B  =  i A  S i m i l a r l y ,  ,  t h e  t o t a l  E  < 9 <i ir .  d i f f r a c t e d  f i e l d s  (2-15)  r e f l e c t e d  b y  t h e  l o w e r  ,  15  w a l l  H  due  the  i  (9) - H ( - 2 i 9  L i  and  t o  A  t h a t  of  t h e  t h  image  -9)  E  i s  , \ - (i+l)8  l a s t  image  i n  E  the  < 0 < •£ - i 0  upper  w a l l  (2-16)  ,  E  i s  (2-17)  The  b o u n d a r i e s  f o l l o w  of  t h e  l a s t  image  t h e  r e g i o n s .  f i e l d s  The  s u p e r i m p o s i n g phase  d e s c r i b e d  f a r a l l  r e f e r e n c e ,  zone t h e  say  (9) = H ( 9 )  r e l e v a n t edge  a r e  zero  o u t s i d e  w a l l  s h o u l d  t h e  d e f i n e d  the  =  e x  l o c a l  A  Y  f a c t o r s  P l ~ J ^ E c° ( - E>3 k  terms  +  B  phase  s  0  e  Y . _ - e x p [ - j k b s i n 6]  p a t t e r n and  i s  now  c o n s i d e r i n g  o b t a i n e d a  h  Y ^  (2-18)  B h  r e f e r r e d  t o  edge  A  a r e  »  »  - e x p [ - j k U s i n ( 1 0 + 6)] E  »  and  Y  Bh  "  e  x  P M  k  * h  sin  (h9  E  -6)]  .  by  common  A.  Bh  Y  lower  r a d i a t i o n  + H (e) ^  Where  above  + H (9) + H (9)  s  AS  t h e  (2-8).  A l l  Y  i n  (2-19)  16  3.  AN A N A L Y S I S  3^J_In  problem  f a r - f i e l d choke  t h i s  t o  r a d i a t i o n f l a n g e s .  problem,  g e o m e t r i c a l  method  o f  o n - a x i s  a  s a t i s f a c t o r y used  l i n e  OF  CHOKED  r e f l e c t e d f i e l d s  from  a  o f  HORN  ANTENNAS  d i f f r a c t e d  on  E - p l a n e  antenna  m o d i f i e d  e x p e r i m e n t a l l y  made  t o  s t u d i e d  t h e o r e t i c a l l y  t h e r a d i a t i o n  p a t t e r n .  The  method  f o r  t h e  D i f f i c u l t y ,  however,  a r i s e s  when  t h e  t h e chokes  on  a i s .  an o v e r a l l  a r e t o  t h e o t h e r  hand,  Consequently a n a l y s i s  f i r s t  i s  be  both  t h e  t o  methods  be a r c  c h o k e s .  t h e f i e l d  produced  Then  determined.  edges  determined.  a r e known  waveguide.  i n t e r i o r  t h e antenna  o f  r e v i e w  p a r a l l e l - p l a t e  a t  t h e  p r o m i s i n g  r e s u l t s ,  i s  horn  i s  a  from  we  c h a p t e r  [ 1 ]  was  chokes  horn.  t h e choke  t h i s  p y r a m i d a l  d i f f r a c t i o n  f o l l o w s i n  a  attempt  or. t h e bears i n  o f  i n  and Roberts  r e f l e c t e d  t h e o r y  what  s o u r c e  o f  choked  a l t e r n a t e l y  In  no  e f f e c t  f i e l d s  K i r c h h o f f  s t u d i e d  p a t t e r n  b u t t h e  treatment  be  LaGrone  i n v e s t i g a t e  The  PATTERNS  trod uct ion  The  by  FOR E - P L A N E  by  t h e  A t  a  f i e l d  t h e end t h e  and t h e t o t a l  f i e l d  a r e  c a l c u l a t e d .  3.2_A_NonisotroBic_Line„Source  Yee i n  a  e t  a l [ 9 ]  p a r a l l e l - p l a t e  f i e l d s  on  c o n t e n t  o u r s e l v e s  when  have  t h e  l i n e  s t u d i e d  waveguide.  t h e m u l t i p l y with  source  i n  c o n v e r t e d r a y s  r e s u l t s  produces  a  P a r a l l e l - P l a t e  t h e b e h a v i o u r  They  r e f l e c t e d  t h e i r  a  f o r  i n t o a  t r a n s v e r s e  o f t h e  modal  ffayeguide  a  l i n e sum  f o r m .  p a r t i c u l a r magnetic  s o u r c e o f  t h e  Here  we  case  ;  f i e l d .  The  17  f a r be  f i e l d  a t r,  expressed  t h e source  produce  i n f r e e  space c a n  u(9)  .  (3-1)  (kr)  $ eare  w h e r e r  f r e e - s p a c e shown  p o l a r  c o o r d i n a t e s  propagation  t h e source  L e t t h e s o u r c e  a c c o r d i n g  a n d  k  b e a t  i s  t h e  y = y , z=o Q  t o Y e ee t a l , t h e f i e l d  a t  i s  1 /o  w  exp(-j |) Z  (y,z) = ( j )  x  from  c o n s t a n t .  i n F i g . 3 . 1 . Then,  z i n t h e guide  H  would  a s  °  y»  t h a t  H (r,6)-  '  as  6  e  O^d)"  m  1  [u(e ) m  exp(jnnry /d) 0  m=o  + u(-6 )  exp(-jmTry /d)] e x p ( - ; j K j z|)  m  0  cos^Y/d) (3-2)  where  e  K  e  0  M  ' -  i s  a n de  K  1 ,  e  m  a r e c o n s t a n t s  -  = [k2- -fL)2]  sin9  When  ,  (  m  = —  ,  t h e l i n e  obtained  2 ;  1  cos9  s o u r c e  /  m =f  =  0 ,  (sgnz) ( ^ )  .  (3-3)  i s p u t o n e i t h e r  b y s e t t i n g  b y  ,  2  m  g i v e n  (_e )=0  u  m  ,  i n  c o n d u c t i n g (3-2),  [9].  w a l l ,  H  (y,z)  18  F i g .  3 . 1 . Coordinates f o r a magnetic p a r a l l e l - p l a t e waveguide.  3 ^ 3 _ R e f l g c t i p n ^  A  p a r a l l e l - p l a t e o f  a  source  i n  a  f£gjn_Choke_Interior  two-dimensional  geometry  l i n e  choke  waveguide choked  horn  C o n s i d e r i n g  t h e  upper  n o n i s o t r o p i c  l i n e  s o u r c e s  may  be  short antenna  choke,  viewed  a s  an  open-ended  c i r c u i t e d  a t  one  e n d .  i s  i l l u s t r a t e d  edges  which  A a n d A*  e x c i t e  may be modes  TM  i n  The  F i g . 3 . 2 .  regarded  i n  a s  t h e  choke.  t h e  choke  a r e  on t h e  om The  e x c i t e d  depth.  n o t i c i n g  c o n d u c t i n g w r i t t e n  H  x  r  modes  a r e r e f l e c t e d t h a t  w a l l s  and u s i n g  (y,z)  y,  «  z  aperture,H r e p r e s e n t  ( f )  1  /  t r a v e l l i n g  s o u r c e s  (3-2),  A  a n d A*  t h e r e f l e c t e d  TM w a v e s  2  e x p ( - j j ) Z CjnO^d)m=o  c a n  be  [a^^-6 )  1  m  + H ?(-ir + 6 ) e x p ( - j m T f ) ]  • ex I-jK (z + 2d )]  •  < o .  m  cos [ 2 J (y - | ) ]  denote c  A  and H 9  A  wave  ,  r e c t a n g u l a r #  t h e d i f f r a c t e d  c y l i n d r i c a l o m i t t e d .  l i n e  a f t e r  a s  A  where  both  back  t o  T h e s u p e r s c r i p t  z  f i e l d s  c  1  m  (3-4)  c o o r d i n a t e s be  propagation  P  from  determined from f a c t o r ,  i n d i c a t e s  edges  i n A  i , e, that  t h e c e n t r e t h e  a n d A* i n exp(-j  these  next  t h e  s t e p ,  which  k r ) / ( k r )  f i e l d s  o f  1  /  r  2  belong  t h e i s t o  19  a  choked  horn.  From  (3-3)  8  i s  m  o b t a i n e d  as  (3-5)  The  modes  f o r  which  k<  o r  d  e q u i v a l e n t l y  K  m  i s  y  z  F i g .  3 . 2 .  Geometry  of  a  choked  i m a g i n a r y ,  are  nonpropagating  f r e q u e n c y  i s  not  f i r s t a  evanescent  high  mode  i n  (3-4)  Z  t h e  a x i s .  problem  which  TEM  £16],  of  f a r  t h e  a  t o  o n l y a  the  i n  f o r w a r d  f i e l d  U s i n g  t h e  K i r c h h o f f  the  f a r  f i e l d  of  of  u s i n g t o  K i r c h h o f f  f o r  the  choke  as  s i n g l e  term  i n i n  i n  t h e  method  r e g i o n .  chokes  the  o n l y  f i r s t  boundary  s o l u t i o n  a  i f  of  the  wave.  d i s t r i b u t i o n  the  f r e q u e n c y  the  TEM  shadow  employ the  i d e a  E - p l a n e .  r a p i d l y  a p p l i c a b l e  c o n s i d e r  has  the  c u t - o f f  the  u s u a l l y  we  i n  v a n i s h i n g  a d d i t i o n ,  wave  f i e l d  a p e r t u r e  t o  c o r r e s p o n d s  we  antenna  modes  equal  i s  Thus,  t h e r e f o r e ,  c a l c u l a t i o n c h o k e s ,  .  r e f l e c t e d  d i r e c t i o n ;  In  element  o p e r a t i o n  The  t h e  mode.  impedance  s e r i e s  n e a r l y  horn  i s  f o r w a r d f o r  the  C o n s i d e r i n g  both  s y m m e t r i c a l a  about  t w o - d i m e n s i o n a l  i s o l a t i o n  i s  20  H  C 1  (6)  \r(2TT)  e x p ( - j j ) g(sin6)  (3-6)  ,  '  where  g(si:i8) =2/  •  1  H (y,o) ' ccs(k )dy  ,  r  i y  b  k  x  .  = k sir.9  (3-7)  x  2 From  (3-4)  and  H (y,o) = ( £ ) r  (3-5)  (kd)"  1 / 2  x  Upon  H  u s i n g  C 1  f o r  (3-7)  and  ( 6 ) = (kd s i n e )  - 1  m=0  r  [H (TT) + H ? ( - * ) ] e x p [ - j ( | + 2kd')]  1  (3-8)  C  A  a  (3-8)  i n  (3-6)  we  [H (TT) + H ? ( - i t ) ) C  A  a  o b t a i n  • {sin[k(|+  d)sin0]  (3-9)  - s i n ( ^ - s i n e ) }exp ( - j 2 k d ' ) .  A t 0-0  ,  H  F o r w e l l  ^ C  0  )  =  [  l a r g e  p e r f e c t l y  d i f f r a c t e d  TEH  H  ^ C  U  )  +  H  a n g l e s  approximated  r e f l e c t e d a  (  by  wave.  a r e  0  the  9  U  )  ,  e ^ ^ j a ^ . )  ]  t h e  s i n g l y  I n v o k i n g  c o n d u c t i n g f i e l d s  a  f a r  f i e l d  d i f f r a c t e d  the  wedge  [ 14]  d e r i v e d  a s  (  o f  f i e l d s  d i f f r a c t i o n and  using  t h e  of (3-8)  chokes due  a  p l a n e  ,  these  3  _  1  0  )  can t o  be t h e  wave  by  s i n g l y  21  H H  1A  <> 9  (  r x  o)exp(-jf)sin^  y >  (2it)  1  /  (3-lla)  n ( c o s — - cos  2  H (y,o)exp(-jf) r  x  (8TT) 2  sin  1 /  where  n = 2 -  - ^ a n d IT  c o r r e s p o n d i n g  H  H  > >  1B«  "  <  6 )  H  =  R  T a k i n g f i e l d  1A'  C  s  t  0 < 0 <  <"*>  A  a s  b y  *  9  a  common  t h e chokes  Y  1A«  (  o f  wedge  choke  A.  By symmetry,  6  )  '  Y  AA'  A A  a r e  *  (2~12b)  2  r e f e r e n c e ,  t h e  t o t a l  f a r  i s  o <e <e  ,  n  ,  t h e  (3-12a)  phase  ,  A 0  t  H  a n g l e  e  t h e lower  (6) + H ^ ( e ) Y  C  n  o <e<  1 < >  H (6) - 1  where  f o r  9  edge  H  *-  E  <- >  s c a t t e r e d  '  e  f i e l d s  1 A  (3-llb)  |  +  (6)Y  2  *  6  s  *  A B  + 1 ^ , ( 6 ) T^,  , 9, < 9 < f  (3-13)  22  Y  A0  Y  AA'  Y  =  e x p I - ; i  -  i  a r e  83  I  k  S  i  n  6  *  ]  exp[-j  kd sin8]  expt-j  k  t h e phase  (b+d)  ,  sine]  f a c t o r s  .  (3-14)  r e f e r r e d  t o  A  and  i s  y  g i v e n  b y  ( 2 - 1 9 ) .  AB Two being i s  p a t t e r n s  a  s m a l l  generated  r e g i o n 0  < 6  e  a n g l e , i n  d i f f r a c t i o n  a r e generated,  theory  t h e r e g i o n  t h e K i r c h h o f f  t h e r e g i o n  < 9 < 8^  2  from  one i n  e <6  r e s u l t .  g i v i n g  <TT , 8  2  r e s u l t .  < 8i ,  2  a  complete  A  from  T h e t w o p a t t e r n s  0 < 6 < Q^, second  t h e  then  p a t t e r n  p a t t e r n  g e o m e t r i c a l  o v e r l a p  i n  6-±  i n  t h e  t h e  r e g i o n  < TT.  3^4 J ) i f f r a c t e d _ F i e l d s  In  c h a p t e r  d i f f r a c t e d  by  i m p l i c i t l y  a  from  (2-4) ,  2  we c a l c u l a t e d  t h e upper  f u n c t i o n  (2-12)  a n d  a f t e r  m o d i f i c a t i o n  a n g l e .  F o r c o n v e n i e n c e  H (8)  b y  A  using  d i r e c t i o n H  A  Of)  .  -f By  h i t s  o f  n  ^  -§Jin  (r » »n)  where  0  •  a  horn.  T h i s  m o d i f i c a t i o n e„  ® (8) A  i s o  D  i s  H A  c a n be  h=2  o n e h a l f t  f i e l d  o f  s e e n  ,  w h i l e  t h e  f l a r e  o b t a i n e d  e  from  i t .  upper  edge  A». (2-1)  v  t h e t o t a l  unmodified  B e f o r e  we d e f i n e  a s b e i n g  a n  t h r o u g h  ,  t h e  u s i n g  o f  ( 2 - 1 3 ) .  n=2 -  n = 2 -  C o n s i d e r i n g  edge  H (8)  c h o k e ,  T h e f i e l d a n d  (2-2)  a  r a y  from  edge  o n  t h i s  i n t e n s i t y t h e f i e l d  i n  t h e  r a y  i s  k  d i f f r a c t e d  a t  A»  23  can  b e  H ?  w r i t t e n  (6) = H ( f ) [ v ( d , \  A  d i f f r a c t e d  second  order  f i e l d t h e a  the  shadow  A .  I t  6  )  =  i s  t h e  +  A*  0 , 2 ) ] •, 0 < 8 < *  h i t back  c a l c u l a t e edge  A  on  t h a t  The  i t s  second  (3-15)  t h i s  boundary  o f  f i e l d  a n d  n o t  i n t e r a c t i o n  a  i n t e r a c t i o n  t h e d i f f r a c t e d  image  o r d e r  .  f u r n i s h i n g  a p p r o x i m a t i o n  Furthermore,  o r  A  t h e shadow  [ 9 ]  t h e a s y m p t o t i c f i e l d .  edge  t h e r e s u l t i n g  i s  1  shown  s o u r c e  edge.  a c c o r d i n g l y  To  t h a t  o p t i c s  from  edge  h a s been  boundary  d i f f r a c t i n g  (  from  n o t i c e  g e o m e t r i c a l  AA'  2) + v ( d , ^ |  i n t e r a c t i o n .  from  emanate  6,  r a y s  we s h o u l d  r a y s  on  +  A  The  H  a s  o f  f i e l d  o n e - h a l f  appears from  f i e l d  t o t h e  i s now  o b t a i n e d .  l  C  I  v  j  (  d  '  -  f  "  9  '  n  )  +  v  (  d  '  "  9, n) + v ( 2 d , 2 | _  0  >  n  9  )  '  ] ,  n  )  ]  +  °2  I  v  (  2  0 < 6 < |  d  '  ,  (3-16)  w i t h  C  l  =  v  The n e g l e c t e d . b e t t e r  < '°' > d  t r i p l y  2  a n d  I n c l u s i o n  a p p r o x i m a t i o n .  s o l u t i o n  f o r  h i g h e r o f  t h e s e  Bowman  s c a t t e r i n g  order  i n t e r a c t i o n  f i e l d s  does  [ 1 7 ] by  , a n  who  f i e l d s  n o t g e n e r a l l y compared  open-ended  t h e  a r e  y i e l d  a  e x a c t  p a r a l l e l - p l a t e  24  waveguide  w i t h  g e o m e t r i c a l t h e  A  By  H  c o r r e s p o n d i n g  d i f f r a c t i o n  c o n t r i b u t i o n s  The  H  t h e  (6)  C  -  (  6  )  =  (8)  I t  should  e x c l u d e  H  S  C  (8)  H  + H ^ ,  ,  < 9 < f  (0)  A  the  A  C  (  '  9  0  ,  A  t h e  f i e l d s o p t i c s  ;  i . e ,  -  1.  that  f i e l d  (2-16)  0  f i e l d s  and  t h e  i s  i s  now  o b t a i n e d .  .  B  and  edge  .  equal  t h e  i n  horn  that  f o r  (3-15)  and  t h e  B'  a r e  the  from  t o  3  _  2  0  )  (3-20)  chokes.  horn  .  which  (3-18)  (  t o  E  i n  i n t e r a c t i o n s .  .  wedge  the  (3-19)  i n t o  < 8 < 8  i n  A  of  d i s c r e p a n c i e s  order  boundaries  r e f l e c t e d  (2-17)  h i g h e r  means  ,  8 < \  d i f f r a c t e d  ,  from  <  by  found  edge  by  < 8 < f  0  n o t i c e d  0  d i f f r a c t e d  *  H ?(-6)  be  (6)  f i e l d s  )  d i f f r a c t e d  The  w i t h o u t  (3-21)  horn H (8) A  i n t e r i o r i s  a r e  o b t a i n e d  s u b s t i t u t e d  ; i . e ,  A  . \ i  and  from  The  H (6)  t r i p l y  f i e l d  g e o m e t r i c a l chokes  t h e  d e r i v e d  method,  d i f f r a c t e d  H  -  theory  t o t a l  symmetry,  B°  from  r e s u l t  (8)  = H  (8)  = H  A  A  (-2i6 -8)  ,  (-2h8  .  E  E  +0)  (3-22a)  (3-22b)  by  The  boundaries  (3-22)  i n  a r e t h e same  a s  those  i n  (2-16)  and  (2-17) . C o n s i d e r i n g f i e l d  i s  o b t a i n e d  d e s c r i b e d  .  above  " t o t <> 9  =  <>  Y.  R |  A  S  ,  *  6  H  +  Y  by  with  S  H  +  Where  t h e upper  H  Y ^  b y (3-1H).  B'<  6 ) Y  ° C 0 )  ,  Y  h a l f  r e g i o n  0 < 6  s u p e r i m p o s i n g  a l l  t h e l o c a l  Y  AS  +  AB«  H  A  C  (  6  )  phase  +  \ fll  +  H  (  m  6  B  )  C  Y  (  e  A i  t h e t h e  f a c t o r s  )  A B  Y  +  H  +  u S  (  A '  9  )  Y  (  6  A B  '  )  '  Y  Y  terms  t o A .  A A '  B h  ,  • M  f a r  r e l e v a n t  r e f e r r e d  H  t o t a l  (3-23) a  n  d  Y  B  h  a r e g i v e n  b y  (2-19)  a n d  T  M  ,  a n d  26  A  4.  NEW  MODIFICATION  TO  HORN  ANTENNAS  CYLINDER  4,.  s o u r c e  f i e l d s t h e  used  t o  i n  a  a  edge;  K i r c h h o f f  of  a  from  the  t h e  c a l c u l a t i o n t h e  f i r s t  i n  f l a n g e s ,  a  o f  t h e  t h e t h e  s t e p s ,  s t u d i e d .  Then  O p t i m i z a t i o n  edge  t h e  f o l l o w i n g  p r e v i o u s  the  i s  back  f o c a l  l e n g t h  a  r a y s  f a r  c y l i n d e r  d i r e c t i o n s  d i f f r a c t e d from o f f a r  wave  d i f f r a c t e d  we  v e r i f i e d  t h e  p a t t e r n the f i e l d s  d i f f r a c t i o n antenna  i s t h e  f o r edges.  f l a n g e s  e q u a t i o n s  a t  the  a t  r e f l e c t e d  the  f i e l d  region  employ  p a r a b o l i c t h e  the  e x p e c t e d .  r a d i a t i o n  o f  wave  a t  a d d i t i o n ,  f a r  a t  p a r a b o l i c  In  a l s o  be i n t o  c y l i n d r i c a l  f l a n g e s  method  may  f o r w a r d  f i e l d  of  these  the  f i r s t i s  edge  o r i g i n a t i n g  f l a n g e s  plane  i n  r a d i a t i o n  i s  r a y s  a  beamwidth.  t o t a l  by  E - p l a n e  r e a r  i n t o  power  some  p r i n c i p a l  c y l i n d r i c a l  f o c u s  g e o m e t r i c a l  t h e  w i t h  t h e  c a l c u l a t i o n  u s i n g  t h e  c o n v e r t i n g  by  the caused  p a r a b o l i c  a  r e f l e c t i o n by  i n  i n  E - p l a n e  f i e l d s  c h a p t e r s of  t h e  p a r a b o l i c  f o r and  of  narrower  of  by  r a y s  f a c t ,  at  lower  a n a l y s i s  method  c y l i n d r i c a l  t h a t  f i e l d s  s t r u c t u r e  r a d i a t e d  f l a n g e  modified  wave  In  source  t h e r e f o r e ,  the  horn  l i n e  the  Moreover,  property  r e s u l t i n g  i l l u m i n a t i o n  I n  the  r e v e a l e d  i s  d i f f r a c t e d  Consequently  i n c r e a s e s  horn  a l l  d i r e c t i o n .  has by  E - p l a n e .  Therefore  r e f l e c t  a p e r t u r e .  t h e  FLANGES  have  r a d i a t i o n  the  edges.  r e f l e c t o r  antennas  c y l i n d r i c a l  forward  r a d i a t e d  horn  back  have  from  t h e  of  o f  d i f f r a c t i o n  I n  PARABOLIC  1 , I n t r o d u c t i o n  S t u d i e s  of  THROUGH  d e r i v e d  i s i n  d e t e r m i n e d . e n d .  27  4 . 2  R e f l e c t i o n  -  The  f r o m _ P a r a b o l i c .  geometry  of  c y l i n d e r  f l a n g e s  i s  f l a n g e ,  edge  may  producing  a  a p e r t u r e .  For  A  a  horn  shown be  i n  antenna  F i g .  regarded  n o n i s o t r o p i c s m a l l  F l a n g e s  4. 1. as  a  6  ,  wave  the  by  C o n s i d e r i n g primary  c y l i n d r i c a l  a n g l e s  m o d i f i e d  f a r  f e e d  t o  p a r a b o l i c the  a t  upper  the  f o c u s  i l l u m i n a t e  r e f l e c t e d  the  f i e l d  i s  y  A'  ^  A  —-1%  0  "  z  B 2f  F i g .  4 . 1 .  Geometry  of  a  horn  m o d i f i e d  by  p a r a b o l i c  c y l i n d e r  f l a n g e s  o b t a i n e d  by  F o u r i e r  t r a n s f o r m  r e f l e c t i o n  employing  the  the  of  f i e l d  K i r c h h o f f  t h e  on  a  method;  a p e r t u r e  ray  may  be  i . e .  f i e l d  d e t e r m i n i n g  d i s t r i b u t i o n .  w r i t t e n  the  Before  as  (kp)  p where  r a d i a t i o n and  more  t o  be  p a t t e r n  elsewhere  p a r a b o l i c no  ,  (<(>)  H  of  means  f l a n g e s .  s p r e a d i n g  determined  i n  Sec.  l i n e  s o u r c e  A.  The  f i e l d  i s  f o r  t h a t  the  A f t e r  t a k e s  r e f l e c t i o n  p l a c e  out  t o  the the  4 . 3 ,  r e p r e s e n t s  s u p e r s c r i p t a r a y s  horn are  p  i n  the (4-1)  m o d i f i e d  by  c o l l i m a t e d  and  a p e r t u r e  p l a n e .  The  28  r e f l e c t i o n  c o e f f i c i e n t  t r a n s v e r s e  magnetic  p e r f e c t l y  c o n d u c t i n g .  the  a p e r t u r e  plane  r a y  path  i n  plane t h e  ARY  t h i s  plane  can  F i g .  be  -  both  s y m m e t r i c a l  about  s o l u t i o n  a  V  (6)  =  on  from  the  to  the  l i n e  source  r e f l e c t o r  i s  the  t o t a l  the  egual  f i e l d  A  2f,  f o c u s  i s  with  r e f l e c t e d  f  to  a  change i n  the  ray  produces  assumed  phase  f o c u s  A  to p o i n t  a l o n g  the f o c a l  at  a be at the  a p e r t u r e l e n g t h  the  of  a p e r t u r e  <•>  P  (4-2)  f l a n g e s , the  z  the  a x i s  i n  t w o - d i m e n s i o n a l [16],  determined  ff^  *  ex  k  f o r  When  i s  components  i s  the  the  d i s t a n c e  4 . 1 .  '  C o n s i d e r i n g  f l a n g e s  The  H  t a n g e n t i a l  and  f o r  w r i t t e n .  (kp)  i n  +1,  a c c o u n t s  The  now  V  f i e l d  d i s t a n c e  r e f l e c t o r .  i s  M  a p e r t u r e F i g .  4 . 1 .  problem the  f i e l d Using  w i t h  f a r  d i s t r i b u t i o n the  e l e c t r i c  r e f l e c t e d  i s  K i r c h h o f f  f i e l d  f i e l d  having  from  the  as  1 (sine)  )  (4-3)  (2ir) w here  | + 2 f . q(sine)  -  2 J  H  x  r  (p,<|>)  cos  (k^y)  dy,  k  x  = k  sine  (4-4)  b 2 1/2 The been a r e  c y l i n d r i c a l omitted used  whereas In  i n r,  wave i n  the  p r o p a g a t i o n  ( 4 - 3 ) .  A l s o ,  c a l c u l a t i o n  e  denote  F i g .  4 . 1 .  p o l a r The  of  f a c t o r p,<ji the  are  of  a t  kr)/(kr)  p o l a r  a p e r t u r e  c o o r d i n a t e s equation  exp(-j  f i e l d  the  t h e  c o o r d i n a t e s  f a r  upper  has which  d i s t r i b u t i o n , f i e l d . p a r a b o l a  may  be  29  d e s c r i b e d  P-  He  2  a s  f  l-cos<}>  (4-5)  f u r t h e r  y  -  have  psin<j> + j  From  '  (4-6)  ( 4 - 5 ) a n d ( 4 - 6 ) we o b t a i n  P = (4-7)  y  =. 2f  coti.  .  b .  2^2 Using  (4-C)  i n  (4-7)  SJ(P.«  y i e l d s  (4-2)  - - S E E t J ^ L  s  i  n  i  H  P  (4-9)  (kp) From  ( 4 - 2 ) ,  H/O)  ( 4 - 3 ) ,  = 2(f)  1  /  ( 4 - 8 ) a n d ( 4 - 9 ) we o b t a i n  exp[-j(2kf - J)] / »  2  H  /(*)  cos[k(2f  2  * cot  The  i n t e g r a l  For f l a n g e s horn  i n  l a r g e i s  edges  |  + | )  (4-10)  a n g l e s  sine]  d<fr/sin |  .  (  i s n u m e r i c a l l y  0  approximated  ,  t h e  f a r  edges.  _  1  0  )  e v a l u a t e d .  f i e l d  b y t h ei n t e r a c t i o n  a n dt h er e f l e c t o r  4  I n o r d e r  r a d i a t e d f i e l d s  from  t h e  between t h e  t o c a l c u l a t e  t h e s e  30  f i e l d s  we  t a n g e n t  to  f u r t h e r i t  a t  ray  from  edge  &  of  i l l u m i n a t i o n  approximate  the i n  edge.  the i s  d i r e c t i o n H  P  i s  the  a n g l e  of  determined  H  A  ? 0 )  The  •  (—)  H  (i)  order  <>  -  6  H  r a y s  ?  C- f>  n The By  t r i p l y  and  symmetry,  edges  f  2  W(2f,  -  Using  (2-1)  edge and  h a l f - p l a n e  r e f l e c t o r ,  A*.  The  (2-2)  a  i n t e n s i t y  and  n o t i c i n g  i s  2.  ,  +  v(2f,  4  the  f i e l d  d i f f r a c t e d  at  9.  2)  A»  *  +  s u b s e q u e n t l y  f i e l d  g i v e n  f  n)  -  6,  +  V  6,  2)]  h i t  .  (  edge  A  4  _  u  (e)  A  ?  A  a  by  (2f.  ^  -  9,  n)]  ,  (4-12)  order  i n t e r a c t i o n  c o r r e s p o n d i n g  f i e l d s  f i e l d s  a r e  d i f f r a c t e d  n e g l e c t e d . by  the  l o w e r  (-6)  ,  (2TT-8)  ,  0  <  8 <  \  ={ H ?  )  p r o d u c i n g  are  H  A'  •  h i g h e r  the  +  from  i n t e r a c t i o n  A  h i t s  L  upper  a  2  i n c i d e n c e  [ v(2f,  d i f f r a c t e d  A£'  t h e  by  a s  = H /  second  r e f l e c t o r  C o n s i d e r i n g  A. t h a t  each  <  e  <  u'  .  (  4  _  1  3  )  31  V  >  6  (  =  The t h e  AA'  H  t o t a l  r a d i a t i o n  common  9  f i e l d  c o n t r i b u t e d  p a t t e r n  phase  c a n  r e f e r e n c e ,  V H  (4-14)  " >  (  A  ?  (  6  )  now b e  s a y edge  A0  ( e ) Y  A A '  Y  B  H  ?  (  9  >  *  p  ^A  A0  '  Y  AA'  r e p l a c e d  2  H  A «  A  N  (  0  )  A A «  Y  D  Y  r e g i o n  e  +  A B '  f i e l d  B?< > 9  H  a  r  5.  0  A B '  c o n s i d e r i n g  a  .  i>  e  +  H  A A '  C  9  )  +  H  ™B'B< > 6  AB  <e <i ,  2  e  o  b  Y  t  AB"  a  i  n  62<  #  2  r a d i a t i o n  t h eK i r c h h o f f  g e o m e t r i c a l 9 <ir  ft  b y  t o  AA' '  f a r  i n c l u d i n g t h e  o b t a i n e d  r e f l e c t o r s  " T *  e  d  f  9  *  (4-15)  *  ^ o m (3-14)  i n  which  d i s  b y 2 f .  The  9 <  Y  e  H (6)  Y  Z  0  +  b y t h e p a r a b o l i c  <  9  K  0  i  r e s u l t  d i f f r a c t i o n  p a t t e r n  i n t h e r e g i o n t h e o r y  . T h e t w o c o r r e s p o n d i n g  ^ r e s u l t i n g  i s  i n a c o m p l e t e  o b t a i n e d 0 £ 9 < 6  r e s u l t  p a t t e r n s p a t t e r n .  1  b y  f i r s t  a n d  i n t h e o v e r l a p  t h e n  r e g i o n i nt h e  32  U 3 _ D i f f r a c t e d _ F i e l d s _ a n d _ R a i  E x c l u d i n g r e f l e c t o r s ,  t h e  the  f i e l d s  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  i n t o  from  the  edges  horn A  and  and B  i n t o  a r e  t h e  w r i t t e n  as  H (6) = H (6) P  A  A  = H (-G)  H (6) p  A  b  where  5. A  f i e l d  H P  by  H  to  be  P  c  and  t h e  (3-21)  t o t a l  •  Where given  o < e < f-  determined  i n  a r e (3-22)  f i e l d  ( 4  _  1 6 )  (4-17)  S e c . 3 . 4 .  d i f f r a c t e d  and  f a r  ,  f i e l d s e q u a l  t o  The  g e o m e t r i c a l  r e f l e c t e d those  f o r  a  o p t i c s  from  horn  choked  horn  .  i n  t h e  upper  h a l f  r e g i o n  o  < 6 < ir  c a n  o b t a i n e d .  < > 9  ,  and  The now  0 < 9 <f  was  i n t e r i o r given  ,  " a ' w i t f  Y^ by  ,  ( 2 - 1 9 ) .  +  A<>  H  P  ,  9  +  Y^  h ' W *  and  +  Y  U  a r e  t h e  l o c a l  phase  f a c t o r s  33  P a r a b o l i c u t i l i z e d  f o r  r e f l e c t o r s  are  r e f l e c t o r s  t h e  such  c o n f o c a l ,  p a r a b o l i c  the  far  of  horn  be  f a r  sum  of  g e o m e t r i c a l  the  f i e l d s .  A  The  i s  antenna.  them  power.  These  a c h i e v e d .  i s o l a t i o n  be  and  A l l  appear have  to f o c a l  A c c o r d i n g l y  t h a t  of  i t s  phase.  of  o n - a x i s  of  may  t h e  horn  o p t i c s  i s  w e l l  f i e l d  g e o m e t r i c a l  approximated  and  o p t i c s  by  the  s i n g l y  f i e l d  r e f e r r e d  i s  exp(-j  From  f i e l d  l e n g t h s  horn  some  power  i n  i n  f o c a l  g i v e n  o n l y  o n - a x i s a  should  a  o n - a x i s  o n - a x i s  =  8  the  The  edge  H  f i e l d  of  but  maximum  f l a n g e s  d i f f r a c t e d t o  i n c r e a s e t h a t  o n - a x i s  d i f f e r e n t  m o d i f i c a t i o n  s i g n i f i c a n t l y l e n g t h s  with  k£  c o s 9  E  and  (2-4)  E  ) .  the  (2-5)  t o t a l  o n - a x i s  s i n g l y  d i f f r a c t e d  f i e l d  i s  H  s  The  l  n  S  2va ,  =  E  o n - a x i s  s e t t i n g  H  P  =  E  f i e l d  1  /  2  n)  e x  P  ,  n  r e f l e c t e d  (4-10)  i n  9=0  2 ( f )  w-e ,  H ( 2 k f  ,  -  2 -  from  must  have  the  ( 4  f l a n g e s  i s  „  2 0 )  o b t a i n e d  y i e l d i n g  -  £)]  / J 2  Re  . '  P(*)  H  4  A  sin J  d+  (4_2i)  by  34  | g + 8in8 H  In  H  order  + 2Nir  |HP  =  t o e v a l u a t e  d i f f r a c t e d  f i e l d  from  H (<J.) - v ( % , T r - e  t h e a s y m p t o t i c  form  /" exp(-ju )dy 2  a p p l y i n g  H (<i») A  (2-2),  P  H  1,  ....  H  p  by  t h e  (4-23)  o f  F r e s n e l  i n t e g r a l ,  i . e . ,  T » 1 ,  ,  t o  Q U )  E  s i n g l y  f o l l o w s :  reduces  p  (4-22)  .  2  = ex [-j(k£  P  & a s  exp(-JT )/(2JT)  =  T  and  edge  n)  E  N • 0, !  we m a y a p p r o x i m a t e  f  +  P  A  Using  |H  ,  (  4  _  2  4  )  where  (2irkA ) E  Using  H  P  (4-24)  = 2 ( f )  1  /  2  n(cos  i n  — -  (4-21)  ex [-j(2kf P  cos  £—  ,  o b t a i n  we  + WE)]  n  §  J £ • <  i n t e g r a l f o c a l  i n  length  Q(<J>)/sin (4-25) i s  i s  )  /*  X  I  2 1  i s p o s i t i v e .  . s i n| always Hence,  (4-25) p o s i t i v e , from  (4-22)  t h e r e f o r e t h e  t h e  o p t i m a l  35  N  where  2  3  2  1  =  1  H  s  + H  f o r  c o n v e n i e n c e ,  a c t  l e s s  not  4  ir  s i n e  with  e f f e c t i v e l y .  recommended  b o u n d a r i e s  l i e  too  i n  (4-26)  '  i s  n u m e r i c a l l y  the  l i m i t a t i o n  t h a t  s m a l l  The  a s y m p t o t i c  form  of  the  c l o s e  e v a l u a t e d ,  c a l c u l a t i o n t o  the  o n - a x i s  of  and  i s  chosen  r e f l e c t o r s  F r e s n e l 3  N  ,  d i r e c t i o n .  w i l l  i n t e g r a l  because  i s  shadow  36  5.  NUMERICAL  M o d i f i c a t i o n s p a r a b o l i c  and  t h i s  r e s u l t s  are  gain  compute  were these  these  r a d i a t i o n  of  a  were  i n i n  of  the|e[ t h e | e |  by  programmed.  o r d e r s  as  < 25°region  and  > i 5 <> r e g i o n  were  s t u d i e d  e x p e r i m e n t a l  are  ( 2 - 1 8 ) ,  h i g h  a s  (3-23) p a t t e r n s  t h r e e .  g e o m e t r i c a l  used  i n  on  v e r i f i e d .  Computed  the  i n  m o d i f i c a t i o n s  horn  g i v e n  and  the  The method  computation  p a t t e r n s .  c o n s t r u c t e d are  waveguide  w i t h  diagram  t h e  of  of  e g u a t i o n s f i e l d s  f l a n g e s  and  e f f e c t s  t h e s e  r e c t a n g u l a r  horns  back  n u m e r i c a l  p a t t e r n s  d i f f r a c t i o n  Two  t h e  choke  t h e o r e t i c a l l y  r a d i a t i o n  method  modified  and  were  c h a p t e r  and  RESULTS  through  the  d i f f r a c t e d  K i r c h h o f f of  In  compared  (4-18),  i n c l u d e  of  4.  bearawidth,  and  antennas  r e f l e c t o r s  3  To  horn  EXPERIMENTAL  c y l i n d e r  c h a p t e r s  the  to  AND  f o r g i v e n  p y r a m i d a l the i n  antennas,  horn  e x p e r i m e n t a l  s t u d i e s .  Table  The  horns  were  dimensions  1.016  and  F i g .  5 . 1 .  i n t e r i o r antennas  horn  a r e  I.  shown  i n  The  A  and  horn  B,  d i m e n s i o n s  of  fed 2.286 The  by  a  C  m.  T  n  e  o p e r a t i n g  Table I  f r e q u e n c y  Horn  l (cm)  A  9.6  11.1  9.0  7.8  B  6.9  10-5  7.6  6.0  was  l {cm)  E  9.0  H  GHZ  f o r  a(cm)  horn  b(cm)  A  and  8.5  GHZ  f o r  horn  B.  TO  37  F i g ,  5 . 1 .  d e s i g n  t h e p a r a b o l i c  o p t i m a l These  (a) P y r a m i d a l horn d i m e n s i o n s (b) C h o k e d horn dimensions i n t h e E-plane (c) P a r a b o l i c f l a n g e d horn d i m e n s i o n s i n t h e E-plane  f o c a l a r e  r e f l e c t o r s ,  l e n g t h s  g i v e n  a t  i n  we f i r s t  t h e o p e r a t i n g  T a b l e  I I .  c a l c u l a t e d  f r e g u e n c i e s  The  a  s e r i e s  u s i n g  r e f l e c t o r s  o f  (4-26).  then  were  Table H  Horn  f(cm)  A  2.84  4.51  6.18  7.84  9.51  B  1.36  3.13  4.90  6*65  8.42  C o n s t r u c t e d  w i t h  a  f o c a l  l e n g t h  4.90  cm f o r horn  B.  T h e chokes  were  than  1 / 2 wavelength  a n d depths  e q u a l  i n t e g e r . i n  T a b l e  and the  The I I I .  *B = 3 . 5 3 same  The horns horns  a s  d i m e n s i o n s  depths  c m .  p a r a b o l i c  T h e  a s t h e horns  f i e l d  r a d i a t i o n  r e c e i v i n g  a r e shown  i n  i n  F i g . 5. 2 .  t o  ( | i n  c o r r e s p o n d  t o +  t h e  T h e  f o r horn  have ;  t o  H=1  A a n d  w i d t h s H  t h e E-plane  r e f l e c t o r s  p a t t e r n s  antennas.  6 . 1 0 cm  d e s i g n e d  t h e chokes  T h e choke  widths  f a r  o f  o f  ,  l e s s  b e i n g a r e  an g i v e n  *A=3.33  a n d t h e c h o k e s  c  m  have  H - p l a n e .  were  measured  antenna  range  u s i n g a n d t h e  t h e t e s t  38  F i g . 5 . 2 . (a) antenna range (b),horn w i t h chokes parabolic c y l i n d r e flanges.  5  (c)horn w i t h  39  Table II  In  what  compared.  f o l l o w s  Then  m o d i f i c a t i o n s computed  i s  p a t t e r n s  5^1_Comparison  a.  Horn  A  a  w i t h  c h o k e s  most  of  r a d i a t i o n i n  the  and  h i g h e r  g e n e r a l l y might  of  a t  measured  p a t t e r n s  w i t h  p a t t e r n s and  are  without  i n v e s t i g a t i o n  the  f o r  the  end.  &.g^_Com2utgd_Patterng  i n  F i g .  range  the  the  w i t h  p a t t e r n  t h e s e  e x t e r i o r  c o r n e r s  f i e l d s  a r e  of  f i e l d s ,  the  of  f i e l d s a s  chokes.  s m a l l  computed  and  of as  an  a c c u r a c y .  d i s c u s s e d p a t t e r n .  To  i n t e n s i t y measured  by  t h e a  A  i s  of  i s  t h e  i n c l u d i n g  back  the  choke  s e c .  3.4,  t r i p l y edges.  does  d i f f r a c t e d  n e g l e c t e d . f o r  not  a p p r o x i m a t i o n a t  a p p r o x i m a t i o n  p a t t e r n s  not  D i s c o n t i n u i t i e s  b e t t e r  f i e l d s  f i r s t and  i n  but  r a d i a t i o n  index  between  over  d i r e c t i o n s  back  A  Moreover,  0  r e a r  horn  r e g i o n ,  e = +g o  e l i m i n a t e d  i n c l u d i n g  the  t h e  l e v e l  be  r a d i a t i o n by  i n  of  agreement  f r o n t  b o u n d a r i e s  s u f f i c i e n t  i n t e r a c t i o n  p a t t e r n s  c l o s e  the  r a t i o  may  i s  i n  p a t t e r n  g e n e r a l  measured  There  t r a n s i t i o n  of  o b t a i n e d  and  5.3.  b a c k - t o - f r o n t  order  of  and  e r r o r  computed  the  obtained  comparison  at  a n g u l a r  improve  be  of  the  g i v e n  shosn  The  of  computed  i s  computed  I n c l u s i o n  f i r s t  2.6  An  but  i s  1.5  2.5  p r e s e n t e d .  s t r u c t u r e  p r e d i c t e d .  B  d(cm)  chokes  i s  r e p r e s e n t e d ,  1-1  r  the  f i n e  A  of _Measured .  d i s c o n t i n u i t i e s t h e  d(cm)  comparison  comparison  w i t h  Horn  A horn  the t h e s e  s i m i l a r B  w i t h  Uj  -50  180  -150  -J20  -90  -60  -30  0  30  60  90  120  6 (DEGREES) F i g .  5 . 3 ,  E-plane  p a t t e r n s of measured,  horn  A  w i t h chokes computed-  at  *=3.33  cm  I  1*7  chokes horn  i s  A  shown  remain  i n  t r u e ,  p r e d i c t e d  p a t t e r n  because  i t  has  b.  with  Horn  A w i t h most t h e  the  of  t h i s  o f  one  i s  The  and  F i g .  computed  the  were  r a d i a t i o n .  produced  the  mount  system  p r e d i c t e d  and w i t h  i n t e r a c t i o n s i m i l a r with  r e a s o n a b l e  t h e  f i e l d s  6 =  +  QQ  between  comparison  r e f l e c t o r s  measured  approximate  a t  of i s  p a t t e r n s  t h a t  0  f o r  i n  the  f o r  horn  A,  o f  horn  A  t h i s  c o n s t r u c t i o n  of  the  agreement  over  however,  d e v i a t e s  from  l a r g e l y  because  the  lobe  horn horn.  i n i s  f o r  p l a n e s  i n  b a c k - t o - f r o n t There t r i p l y edges  measured  F i g .  5 . 6 .  m a i n l y  due  t o  the  can  be  s m a l l  h i g h e r  o r d e r  were  n e g l e c t e d . of  asymmetry t h e  r e a r  a l s o  p a t t e r n s The  the  d e v i a t i o n .  r a t i o  a r e and  i n  r e f l e c t i o n s  t h i s  s t r u c t u r e  r e f l e c t o r and  h a l f  u n d e s i r e d  account  because  the  good  by  A l s o ,  a c c u r a c y .  shown  of  a  > 120°,  the  computed  p a t t e r n  shows  p a r t l y  p a r t i c u l a r l y  d i s c o n t i n u i t i e s  measured  approximated  back  d i r e c t i o n s  drawn  e r r o r  with  p a t t e r n ,  region|e|  of  t h e  more  compared  5 . 5 ,  c a l c u l a t i o n  N e v e r t h e l e s s ,  c o n c l u s i o n s  r e f l e c t o r s  r e f l e c t o r s  by  same  g e n e r a l l y  horn  computed  i n  The  d i m e n s i o n s .  r e f l e c t o r s .  range.  measured  p a r a b o l i c  t h e r e  p a r a b o l i c  p a r a b o l i c  5 . 4 .  but  s m a l l e r  comparison  of  F i g .  horn  A B  i n  the  asymmetry  i n  d (DEGREES)  F i g .  5.5.  E-plane •at  A  p a t t e r n s  =3.33  cm.  of  horn  A  with  measured.  p a r a b o l i c  r e f l e c t o r s  computed  «t5  5^2_Com£arison  The  o f _ P a t t e r n s _ B e f o r e  measured  p a r a b o l i c  r e f l e c t o r s  compares  t h e  computed  v a l u e s  b a c k - t o - f r o n t m o d i f i c a t i o n back  but  r a t i o s  they  amount  a c h i e v e d  dimension  depends  was n o tg i v e n * .  Therefore,  5 . 8 . a n d  g a i n a n d  e v i d e n t  r e d u c t i o n  t h a t i n  t h e  a n d t h e o n - a x i s  [1] r e a c h e d  t h e same  o f choked  i n t h eb a c k  mainly  a n d w i t h  Measured  I V . I t i s  i n v e s t i g a t i o n  r e d u c t i o n  .  F i g .  i n t h eo n - a x i s  a n dRoberts  e x p e r i m e n t a l  r e d u c t i o n  F i g . 5 . 7 .  i n t h ebeamwidth  LaGrone  a g r e a t e r  chokes  i n a s u b s t a n t i a l  b u t improvement  i n t h e i r  o f t h i s  i n Table  r e s u l t s  w i t h  p a t t e r n s .  i n c r e a s e s  a r e g i v e n  b y chokes  i n  computed  o f beamwidths,  M o d i f i c a t i o n s  A a l o n e ,  a r e compared  i s i n s i g n i f i c a n t .  c o n c l u s i o n s  o f horn  c o r r e s p o n d i n g  r a d i a t i o n ,  g a i n  p a t t e r n s  a n dA f t e r  h o r n s ,  r a d i a t i o n . T h e  o n t h echoke  n o e x p l a n a t i o n  w i d t h .  T h i s  c a n b e  g i v e n  4  f o r  t h i s  d i f f e r e n c e . Tabte IV  ^\Measured Computed Horn A Horn A with  Beamwidth (Degrees) 23.0 23.0^\^ ^>«s.  21.0 ^ ^ V .  Horn A with  11.0  Horn B HornB with chokes Horn B with reflectors  9.5  29»0  N .  12.1  -29.4 -29.2  0.6 0.4  ^\-34.5  1.9 2.4  -  ^\-24.7 -20.1  27.0  27.8 ^v.^^  —  -19.5  -35.6  ^ ^ 3 0 - 0  On-axis gain increase (dB)  ^\-23.2  Z0.5  chokes  reflectors  Back-to-front ratio WB)  ^^-30.0 -29.6  15.0  ^ • ^ ^  oo  ^\-39.0 -34.9  0.5  1.8 2 , 1  >v  I n r e p l y t o a l e t t e r r e q u e s t i n g t h es i z e o f t h e choke, they wrote: " T h i s dimension i s n o t c r i t i c a l . I t depends e n t i r e l y o n t h e amount o f power being r a d i a t e d . T h i s power u s u a l l y . i s much t o o s m a l l t o cause a v o l t a g e a r c o v e r " . 1  -780  I  Li  -150  -120  i  i  -90  i  -60  I  i  0  -30  i  30  i  60  90  i  i  120  150  180  Q(DEGREES)  F i g .  5.7.  A  c o m p a r i s o n  b e f o r e •  and • horn horn  of  measured a f t e r  E-plane  m o d i f i c a t i o n s  a l o n e , w i t h  p a t t e r n s  p a r a b o l i c  horn  a t with  of  horn  A=3.33  A  cm.  chokes,  r e f l e c t o r s  ON  Q(DEGREES)  F i g .  5 . 8 .  A  c o m p a r i s o n  b e f o r e  and horn horn  of  computed  a f t e r  E-plane  p a t t e r n s  n o d i f i c a t i o n s  alone, w i t h p a r a b o l i c  horn r e f l e c t o r s .  a t w i t h  of  horn  A=3.33  A  cm.  chokes,  A  comparison  p a r a b o l i c  a n d back  o n - a x i s  g a i n  however,  depends  but  Compares w i t h  f o r which  t h e  GHZ  because  out  o f  o f  s i g n i f i c a n t  A  c o n s i d e r a b l e  o f  (4-10)  t h e f o c a l  beam  w i t h  t h e  beamwidth  i n c r e a s e  i n  t h e  i n t h e  beam  from  t h e  o f  f r e q u e n c y  f i e l d s , only  a t 5 . 9 .  f o r horn  a t 9 . 0 T h i s  GHZ.  A A t  s p l i t t i n g  i s n e a r l y  t h e horn  i s  t h e  F i g .  f r e q u e n c i e s  t w o l o b e s .  f i e l d  f r e q u e n c y  l a r g e r  o p t i m a l .  t h e r e f l e c t o r s  o n - a x i s with  a n d t h e  i s o b t a i n e d  i n t o  beamwidth,  i s a c h i e v e d  i s  a t s e v e r a l  i s s p l i t  f i e l d  t h e  t h e r e f l e c t e d  g a i n  l e n g t h  w i t h o u t i n  t h a t o f  a n d  r e d u c t i o n  t h e r e f l e c t o r s  t h e o n - a x i s  beams  w i t h  Improvement  from  i n  t h e o n - a x i s  phase  v a r i a t i o n  a  T h e narrowest  t h e main  t h e horn  t h e i n t e n s i t y  r a d i a t i o n  r e f l e c t o r s .  10.0 i s  i s e v i d e n t  i n c r e a s e  o f  a c h i e v e d .  t h e g r e a t e r  maximum  f r e q u e n c i e s  a l s o  on t h e s i z e  I t  l e n g t h ,  shows  r a d i a t i o n ,  w a s  o p e r a t i o n .  f o c a l  t h e p a t t e r n  r e f l e c t o r s  beamwidth  of  o f  180°  alone.  T h e  i l l u s t r a t e d  i n  F i g . 5 . 1 0 .  The e f f e c t  o f  m o d i f i e d l o n g e r  l e n g t h .  H - p l a n e  s l i g h t  I n  mayb e viewed  o t h e r  words,  appendages o f  w i t h  equal a  s l a n t  a  beamwidth  A  with  examine  p a t t e r n .  A before a  horn  we  horn  horn  ofi3.02°has o f  beamwidth  p a r a b o l i c  g a i n  Next  i s  t h e horn  unmodified  beamwidth  horn  o f  by  s t a n d a r d angle  r e d u c t i o n  A comparison  r e d u c t i o n  12.5  r e f l e c t o r s  t h e e f f e c t  a n d a f t e r  i s comparable  l e n g t h  o f  o f  o f  a t  3  2  9 .  . o  o f  with  a  horn  t h a t  i n s t a n c e , cm a n d a  G H Z ,  o f  a  t h e f l a r e  w h i l e  t h e  i s 11°.  p a r a b o l i c  measured  m o d i f i c a t i o n  F o r  o f 0  s h o r t e n i n g  t h e l e n g t h  beamwidth.  o f  a s a  H - p l a n e  i s shown  beamwidth  appendages  p a t t e r n s  i n F i g . 5 . 1 1 .  a n d s i d e l o b e  o nt h e o f There  l e v e l s , b u t  9 (DECREES)  F i g .  5 . 9 .  P a t t e r n s of r e f l e c t o r s  horn  FREQUENCY (GHZ)  A  with  p a r a b o l i c  F i g .  5.10.  Beamwidth  of  horn  A  w i t h  p a r a b o l i c  r e f l e c t o r s .  10  05  c o n s i d e r a b l e t h a t  the  r e d u c t i o n  E-plane  e s p e c i a l l y  i n  the  i n  the  f i e l d s r e a r  b a c k - t o - f r o n t  of  a  horn  r a t i o .  a f f e c t  the  T h i s  shows  H - p l a n e  p a t t e r n  d i r e c t i o n s .  5 . 3 _ E r r o r s  The l i n e  s o l u t i o n  s o u r c e  by  a  d e s c r i b e d  i n  s e c .  s o l u t i o n  i n  the  T h i s  s o l u t i o n  r a d i a t i o n c y l i n d e r i s  a  as In  a  s i n g l y  on  the  wedge  d i f f r a c t i n g  due  by  edge  t h e and  of  w i d e l y  E - p l a n e was  i s  f o r  The  amount  between  the  of  angle  i n t e r a c t i o n  c a l c u l a t i o n  can  or  of the  r e s u l t s  a  c o r n e r  s o l u t i o n some  e r r o r  source  i n  o b j e c t  c a l c u l a t e d  i s  the  of  p a r a b o l i c  i n  be  t h e r e  o b s e r v a t i o n .  f i e l d s  antennas.  approximate and  d i s t a n c e  c a n o n i c a l  horn  f i e l d s  horns  r e q u i r e d  f i e l d s .  of  (wedge)  d i f f r a c t i n g  d i f f r a c t e d  the  magnetic  a  chokes  the  p y r a m i d a l  i s  as  the by  when  however,  wedge  i n  a  plane  used  m o d i f i e d e x a c t  of  h a l f  p a t t e r n s  used  s i n g l y  horns, a  c o n d u c t i n g  horns  (2-2)  d i f f r a c t i o n  e r r o r depends  and  U s i n g  the  (2-2)  f u r t h e r  i n  e r r o r s  t o : D i f f r a c t e d  n o n i s o t r o p i c  f i e l d s  i n  assumption  then  t h a t  t h e the  r e s u l t i n g  i n  c y l i n d r i c a l  i s o t r o p i c  i n  of  d i f f r a c t e d  c a l c u l a t i o n  of  (2-2)  model  angle,  zone  been  T h e r e f o r e ,  d i f f r a c t i o n  t h e  has  by  m o d i f i e d  the  f a r  a n a l y s i s  given  p l a n e .  i n  1.  2.1  r e f l e c t o r s .  r e r l e c t o r  f o r  the  p e r f e c t l y  p a t t e r n s  h a l f  e x a c t l y .  f o r  higher  the  d i f f r a c t e d e r r o r .  i n  o r d e r  e r r o r s of  f i e l d s  However,  t w o - d i m e n s i o n a l  waves,'  c a l c u l a t i o n  r e s u l t s the  a  whereas of i n  t h e  and t h i s  they  f i e l d s .  the  more  c o n s e q u e n t l y e r r o r  are  i n t e r a c t i o n  t h e s e  d i f f r a c t i o n  problem  i s  not  a r e assumed  f i e l d s . ' T h i s I t  i s  e v i d e n t  n o n u n i f o r m i t y  the very  g r e a t e r  the  l a r g e  i n  a n g u l a r 2.  r e g i o n s  not  D i f f r a c t i o n  i n t e r a c t i o n t h e  ray  few  w a v e l e n g t h s .  i f  the  the  In  between horn  A,  waveguide  e r r o r width  to  (2-2)  w i t h  e l e c t r i c a l  used  b o u n d a r i e s . i n  v a l i d  f o r  i n t e r a c t i n g  s l a n t  i n  c a l c u l a t i o n  f a r  f i e l d ,  i s  u s u a l l y  Appendix  r a t i o  i n  i n t e r a c t i o n  s u f f i c i e n t  dimensions  the edges  l e n g t h  the  the  of  a  the  In  w h i l e only  a  r e v e a l s  t h a t  horn  i s  o n l y  can  be  f i e l d s  a c c u r a c y . horn,  a  of  g e n e r a l ,  more  the  a c c u r a t e  p a t t e r n .  a d d i t i o n may  to  these  have  e s p e c i a l l y the  shadow  i n v e s t i g a t i o n  by  system  p a t t e r n  two  u n i t y ,  p r e d i c t e d  mount  An  the  than  t h e  the  s t r i c t l y  l e s s  c a l c u l a t e d l a r g e r  are  between  a p e r t u r e  s l i g h t l y  to  c o e f f i c i e n t s  f i e l d s  path  c l o s e  a i n  p r e d i c t e d  T a b l e f e e d .  IV,  i s  e r r o r s ,  n o t i c e a b l e the and  r e a r  the  e f f e c t  due  on  d i r e c t i o n s .  measured  l a r g e l y  waveguide  the The  b a c k - t o - I t o u t to  the  mount  feed  and  the  r a d i a t i o n d i f f e r e n c e r a t i o s  system  and  of the  53  6.  The  g e o m e t r i c a l  K i r c h h o f f  method  r a d i a t i o n and  by  p a t t e r n s  p a r a b o l i c  measured t h e  o v e r a l l  i n  o f  the  back  beamwidth  The  the  choked  s t r u c t u r e  by  the  lobe  chokes  of  i n t r o d u c e d  improve  gain  and  As  i n  p a t t e r n s a  was  very  o n - a x i s  i n  the  g a i n .  reduce  of  c l o s e  observed  r e d u c t i o n  the  showed  a  a  f i e l d f l a n g e s  computed  and  agreement  in  r e g i o n .  In  It  the  was  r e a r  observed  s u b s t a n t i a l  s l i g h t  the  f a r  choke  of  p r e d i c t e d . i n  o n l y  by  good  f r o n t  with  E - p l a n e  Comparison  was  r e s u l t e d  the  m o d i f i e d  t h e  l e v e l  i n  t h i s  horns  r e d u c t i o n  improvement  a  S u b s t a n t i a l l y c h i e f  advantage  w i t h  m o d i f i c a t i o n  most  by  can  narrow  i n  the  t h i s  chokes  of  t h e  r e s u l t  i n  a  the  and  r e f l e c t o r s  a n g u l a r  range.  s i g n i f i c a n t  i n c r e a s e  r a d i a t i o n  m o d i f i c a t i o n  i s  beamwidth,  Computed  p a r a b o l i c  back  c y l i n d e r  the  by  c o n s i d e r a b l e  lower of  p a r a b o l i c  r a d i a t i o n .  m o d i f i e d over  and  t o  back  m o d i f i c a t i o n  beamwidth  The  through o r d e r  the  agreement  a c h i e v e d . t h e  i n  horns  was  were  horns  horns  but  r e f l e c t o r s  measured  a n a l y z i n g  along  g a i n .  m o d i f i c a t i o n  t h e  d i f f r a c t i o n  r e f l e c t o r s .  r a d i a t i o n ,  and  i n  p y r a m i d a l  g e n e r a l  m o d i f i c a t i o n  of  employed of  lobe  the  method  c y l i n d e r  p a t t e r n s  d i r e c t i o n s t h a t  was  CONCLUSIONS  i n  i n was  the a l s o  comparison  r e s u l t i n g  narrower  beamwidth.  Consequently  a  been  a c h i e v e d .  antenna  has  a  may  f i n d  narrower  The  s h o r t  beamwidth  u s e f u l  horn i s  w i t h  l i g h t  than  a p p l i c a t i o n  s u b s t a n t i a l  d i r e c t i v i t y  has  c o n s t r u c t .  It  and  s i m p l e  to  o r d i n a r y  horns.  T h i s  i n  many  compact  s i t u a t i o n s  horn where  54  c o n v e n t i o n a l c o n v e n t i o n a l  r e f l e c t o r s horns  are  cannot  be  i n c o n v e n i e n t l y  e a s i l y l o n g .  used  and  where  55  REFERENCES  I. ,  H. LaGrone  A.  a n d G.  F.  R o b e r t s ,  a r e c t a n g u l a r horn antenna h i g h impedance choke P r o p a g a t . , v o l . A P - 1 4 , p p .  " M i n o r  t h r o u g h f l a n g e , " 102-104,  lobe  s u p p r e s s i o n  i n  t h e u t i l i z a t i o n o f a IEEE Trans. Antennas J a n . 1 9 6 6 .  2.  R. E. L a w r i e a n d L . P e t e r s , J r , " M o d i f i c a t i o n s antennas f o r l o w s i d e l o b e l e v e l s , " IEEE T r a n s . P r o p a g a t . , v o l . A P - 1 4 , p p . 605-610, Sept. 1 9 6 6 .  3.  V. K. K o s h y , K. G. N a i r , and G. P . S r i v a s t a v a , " A n a l y s i s o f r a d i a t i o n from a f l a n g e d a p e r t u r e a n t e n n a , " IEEE T r a n s . Antennas P r o p a g a t . , v o l . A P - 1 8 , p p . 4 0 7 - 4 1 1 , May 1 9 7 0 .  4.  J . B. K e l l e r , " T h e g e o m e t r i c a l J . O p t . S o c .Amer., v o l . 5 2 , p p .  5.  J . B. Phys.  K e l l e r , v o l . 28  ,  " D i f f r a c t i o n pp. 426-444,  theory o f d i f f r a c t i o n , " 116-130, 1 9 6 2 .  by an a p e r t u r e , " A p r .1 9 5 7 .  6.  Y . Ohba, "On t h e r a d i a t i o n p a t t e r n f i n i t e i n w i d t h , " IEEE Trans. v o l . A P - 1 1 , p p . 127-132, March 1 9 6 3 .  7.  P . M. R u s s f o r comput T r a n s . A March 1 9 6 5  8.  J . S . Y u , R. a n a l y s i s f o r t h e o r y , " IE pp. 138-149,  9.  H.  Y .  Y e e , L .  E.  62-68,  V.  C . Rudduck, a n d 1 . P e t e r s , J r . , "Comprehensive E - p l a n e o f horn antennas by edge d i f f r a c t i o n EE T r a n s . Antennas P r o p a g a t . , v o l . AP-14, March 1 9 6 6 . B .  F e l s e n ,  a n d J . B.  K e l l e r ,  " R a y  " E r r o r s  i n  IEEE T r a n s . pp. 25-31, J a n . 1973. Rhodes,  D.R.  IRE,  A .  W.  v o l .  " A n  p a t t e r n s  36,  p p .  Sommerfeld,  1954, 14.  theory  SIAM  J .  o f Appl.  J a n . 1972.  J u l l ,  r a d i a t i o n  13.  r e f l e c t o r P r o p a g a t . ,  o, R. c . R u d d u c k , a n d L . P e t e r s , J r . ,"A method ing E - p l a n e p a t t e r n o f horn a n t e n n a s , " IEEE ntennas P r o p a g a t . , v o l . A P - 1 3 , p p . 219-224, .  t h e  h o r n s , "  12.  A p p l .  E. V. J u l l , " R e f l e c t i o n from t h e a p e r t u r e o f a l o n g E - p l a n e s e c t o r a l h o r n , " IEEE Trans. Antennas P r o p a g a t . , v o l . A P - 2 0 ,  pp. II.  J .  o f a c o r n e r Antennas  r e f l e c t i o n a t t h e open e n d o f a w a v e g u i d e , " H a t h . , v o l . 1 6 , pp. 268-300, 1 9 6 8 . 10.  o f horn Antennas  p p .  P a u l i ,  p r e d i c t e d  Antennas  e x p e r i m e n t a l o f  O p t i c s ,  o f  p y r a m i d a l v o l .  i n v e s t i g a t i o n  e l e c t r o m a g n e t i c  1101-1105,  g a i n  P r o p a g a t . ,  horn  o f  a n t e n n a s , "  A P - 2 1 ,  t h e P r o c .  Sept. 1 9 4 8 .  New  Y o r k :  Academic  P r e s s , I n c . ,  245-265. "On  a s y m p t o t i c  s e r i e s  f o r  f u n c t i o n s  i n t h e  56  t h e o r y of d i f f r a c t i o n pp. 9 2 4 - 9 3 1 , Dec. 1938.  15.  H. Born and  Y o r k :  E. Pergamon  of  l i g h t , "  Wolf, P r i n c i p l e s P r e s s , 1965, c h .  of 11. Theory  Phys.  B e v . ,  o p t i c s ,  and  3rd  D e s i g n .  v o l .  54,  ed.  New  16.  S. S i l v e r , Microwave Antenna M c G r a w - H i l l , 1949, ch.6  Hew  York:  17.  J. J. Bowman, "Comparison of ray theory w i t h e x a c t t h e o r y f o r s c a t t e r i n g by an open w a v e g u i d e , " IEEE T r a n s . Antennas P r o p a g a t . , v o l . A P - 1 8 , pp. 131-132, J a n . 1970.  APPENDIX  AN  ERROR  INVESTIGATION OF  The near  a  g i v e n  V  A LINE  s o l u t i o n  p e r f e c t l y i n  -  d x  e x a c t  OF APPROXIHATE SOURCE  f o r t h e  c o n d u c t i n g  [ 1 6 , c h . 1 1 ]  BY A  e  O  o  )  ,  -  FOR  DIFFRACTION  HALF-PLANE  d i f f r a c t i o n  h a l f - p l a n e ,  a s o u t l i n e d  1(6 + 0 ) + 1(6 -  SOLUTIONS  o f  shown  a  l i n e  i n F i g .  s o u r c e A . 1 . , i s  below  for " polarization  ,  w i t h  «PU(J- «>] r  K v ) - ± 4>  1/2  and ¥x  d  >  2  [k(R!-R)]  4  + for c o s  -exp(-,1 u )  0  d  y  ( u + 2kR) ' 2  ,  ( A - l )  Ri " r + r . Q  denotes  f r e e - s p a c e  F i g .  t h e  d i f f r a c t e d  p r o p a g a t i o n  A . 1 C o o r d i n a t e s  f i e l d  c o n s t a n t .  f o r l i n e  a t I n  s o u r c e  p o i n t  P  a n d  t h e subsequent  T  near  a  k  i s  t h e  a n a l y s i s we  h a l f - p l a n e  58  c o n s i d e r  only  f o r o <  Although the  (A-1) i s e x a c t  a n a l y s i s  o f  however,  u s e f u l  I f  ,  kR >>i 1  Y  u  n o n e x p o n e n t i a l  E-plane  < ^  i t c a nn o t b e c o n v e n i e n t l y  p a t t e r n s  a p p r o x i m a t i o n s c a n  b e  term  o f  [kCRl+R)] / 1  antennas.  a s d e s c r i b e d  r e p l a c e d  b y  t h e i n t e g r a n d  C^exptKf-kR)], 1  o f horn  i t s o f  used  i n  There a r e ,  below. lower  l i m i t  ( A - 1 ) t o  i n  t h e  y i e l d  / exp(-jy2  2  ) d u  .  [k(Rl+R)]l/2  I f  f u r t h e r  the  r»r ,R Q  1  n o n e x p o n e n t i a l  e x p o n e n t i a l  a n d R  i n  terms  a n d  t e r e s ,  ( A - 2 ) may b e r e p l a c e d b y  R-^ = r + r  Q  b y  ,R = r - r  Q  i n  cosy  i n t h e  y i e l d i n g  1 (Y) - - exp(-1 k r ) 1/2 (kr)  «P[J.(f+kr cosY)3 " ^~T/2 ~ (|) o  *  /  exp(-jy^)dy  [kr (l+cosY))  1 > r 2  0  (A-3) (A-3)  i s i d e n t i c a l  Although s i t u a t i o n s For by  w i t h  ( A - 3 ) i s s u b j e c t  i t i s used  i n s t a n c e , employing  ( 2 - 2 ) i f  doubly  ( A - 3 ) ,  than  t h e  s l a n t  cases  a r e c o n s i d e r e d  without  l e n g t h . a s  i ni t .  t o t h e c o n s t r a i n t having  d i f f r a c t e d  while  n=2 i s u s e d  s a t i s f i e d  f i e l d s  t h e a p e r t u r e T o i n v e s t i g a t e f o l l o w s :  i n a w i d t h  r » r  t h i s  horn  ,  Q  some  c o n s t r a i n t .  a r e c a l c u l a t e d  i s u s u a l l y  t h e e r r o r s  i n  i n  s m a l l e r  i ( y )  t w o  59  a»  r  >r  which  t  Q  a m p l i t u d e  e r r o r s  r e s p e c t i v e l y F i g .  A . 2 .  e r r o r s l e s s  b.  I t  r< r  Q  7  t h e c o n s t r a i n t  d e f i n e d  a r e p l o t t e d i s  a r e w e l l  than  f a v o u r s  observed  below  f o r  t h a t  6.5°and  1.5  .  Q  T h e phase  [ l _  a n d  s e v e r a l  f o r d B .  v a l u e s  -201og  o f r  1 0  l'|"l  a n d r  a n d Y < 120°  r >o.5\ Q  The  a n d t h e  maximum  o  ,  t h e  e r r o r s  a r e  and 3. dB.  ,  t h e  c o r r e s p o n d i n g  a r e shown  e r r o r s  I t  r e s u l t .  l e s s  l l _ -  v e r s u s  c h a r a c t e r i s t i c s  a r e  a s  r » r  than  6t  i s  i n  seen  a n d 1.  dB.  phase  F i g . A.3.  t h a t  f o r r > 0  a n d  A s #  5 \  C o n s e q u e n t l y  i s  and y i f  a m p l i t u d e  r  >  e r r o r  e x p e c t e d K  l a r g e r  g o these  e r r o r s  0  0.5X  j _  sn  o  t  «  r  o  o and  i s  n o t c l o s e  c l o s e  t o  shadow  t o  180  b o u n d a r i e s ,  ; i . e ,t h e r e g i o n t h e r e s u l t i n g  o f  e r r o r s  c o n c e r n a r e  i s  n o t  n e g l i g i b l e .  60  45  90 V (degrees)  45  90  WO  (5  A.2  135  WO  Y (degrees)  J35  WO  45  Y (degrees)  F i g .  90  90 Y(degrees)  E r r o r  i n  I (y)  f o r  r<r  135  WO  F i g .  A . 3 .  E r r o r  i n  f o r  r<r  o  

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