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High frequency beam diffraction by apertures and reflectors Suedan, Gibreel A. 1987

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HIGH  FREQUENCY  B E A M DIFFRACTION  BY  APERTURES  REFLECTORS  by GIBREEL A. M. SC., CALIFORNIA  INSTITUTE  A THESIS SUBMITTED THE  SUEDAN OF TECHNOLOGY  IN PARTIAL FULFILMENT OF  REQUIREMENTS  FOR THE DEGREE OF  DOCTOR OF  PHILOSOPHY  in THE  F A C U L T Y OF GRADUATE STUDIES  DEPARTMENT OF ELECTRICAL ENGINEERING  We  accept this thesis as conforming to the required standard  THE  UNIVERSITY  OF BRITISH COLUMBIA  JUNE  1987  © GIBREEL A. SUEDAN,  1987  AND  In  presenting  degree  this  at the  of  department publication  partial  fulfilment  British Columbia,  for reference and study.  this or of  in  University of  freely available copying  thesis  thesis by  this  for scholarly  his  or  thesis  her  of  the  I agree  I further agree  purposes  may  representatives.  It  £LBcT/2.fCH  L  ^  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  DE-6(3/81)  Seft-  is  for  an  WK1  &tNEEgjN£  advanced  that the Library shall make it that permission  for extensive  granted  head  by the  understood  that  for financial gain shall not be allowed without  permission.  Department of  be  requirements  of  my  copying  or  my written  -  ii -  ABSTRACT  Most solutions  for electromagnetic  wave  diffraction by obstacles and apertures  assume plane wave incidence or omnidirectional local sources. problems  for  local  directive  sources  are  representation of directive beams together  needed.  Solutions to diffraction  The  complex  source  point  with uniform solutions to high frequency  diffraction problems is a powerful combination for this.  Here the method is applied  to beam diffraction by planar structures with edges, such as the half-plane, slit, strip, wedge and circular aperture. Previously  used  restrictions  to  very narrow beams  removed here and the range of validity increased.  and paraxial regions, are  Also it is shown that the complex  source point method can give a better approximation to broad antenna beams than the Gaussian function. The solution derived for the half-plane problem is uniform, accurate and valid for all beam orientations. uniform  or  asymptotic  This solution can be used as a reference solution for other solutions  and  is  used  to  solve  for  the  wide  slit and  developed  and  extended  complementary strip problems. Uniform  solutions  for  omidirectional  sources  analytically to become solutions for directive beams.  are  The uniform theory of diffraction  is used to obtain uniform solutions where there are no simple exact solutions, such as for the wedge and circular aperture.  Otherwise rigorously correct solutions at high  frequencies for singly diffracted far fields are used, such as for the half-plane, slit and strip.  The geometrical theory of diffraction and equivalent line currents are used  to include interaction between edges.  Extensive numerical results including the limiting cases; e.g. plane wave incidence, line  and point  sources  are  given.  These  solutions  are  compared  with  previous  solutions, wherever possible and good agreement is evident Beam (Effraction by a wedge with its edge on the beam axis is analysed. solution  completes a previous asymptotic solution which is infinite  boundaries and inaccurate in the  transition regions.  Finally, the  on the  This shadow  diffraction by a  circular aperture illuminated by normally incident acoustic beam, is derived and the singularity along the axial caustic is removed using Bessel functions and a closed form expression for multiple diffraction is derived.  -  iv -  ACKNOWLEDGMENTS  I wish to express my heart felt gratitude to Dr. E. V. Ml for both his academic and personal assistance, advice and encouragement throughout the course o this investigation. I am also indebted to him for taking the time to read and evaluate my thesis through his invaluable suggestions.  Full recognition and thanks to the Libyan people for their generous financial support throughout my years of study. The computing support provided by the Natural Sciences and Engineering Research Council (NSERC) of Canada, is gratefully acknowledged. I wish to thank the general office staff of the Department of Electrical Engineering at the University of British Columbia for their kind help and cooperation. Special thanks and deep appreciation to my family who waited patiently during my long course of study.  -  v -  TABLE OF CONTENTS ABSTRACT  ii  ACKNOWLEDGEMENTS  iv  TABLE OF CONTENTS  .  LIST OF FIGURES 1 INTRODUCTION AND LITERATURE REVIEW 1.1 1.2 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 1.3.6 1.3.7 1.3.8 1.4  Introduction General Assumptions Literature Review Beam Representation by Current Distributions Spectral Theory of Diffraction Kirchhoff-Fresnel Method Boundary Diffraction Wave Theory Uniform Asymptotic Theory of Diffraction Inhomogeneous (Evanescent) Wave Tracking Complex Ray Tracing Complex Source Point Overview of the Thesis  2 COMPLEX SOURCE POINT METHOD 2.1 2.2 2.3 2.4  Beam Evolution from Complex Line Source Half Power Beam Width Comparison with Gaussian and Typical Aperture Beam patterns Multiple Complex Line Sources  3 BEAM DIFFRACTION BY A CONDUCTING HALF-PLANE 3.1 3.2 3.3  Uniform Solution for Field Radiation Pattern Shadow and Reflection Boundaries Numerical Results for the Half-Plane  v vii 1 1 5 5 6 6 7 8 8 10 10 11 15 17 17 19 20 22 27 27 28 31  4 BEAM DIFFRACTION BY A WIDE SLIT AND COMPLEMENTARY STRIP . . 38 4.1 4.1.1 4.1.2 4.1.3 4.2 4.2.1 4.2.2  Beam Diffraction by a Wide Slit Far Field Calculation Multiple Diffraction Calculation Numerical Results for the Slit Beam Diffraction by a Wide Conducting Strip Far Field Calculation Numerical Results for the Strip  38 38 40 43 48 48 50  -  vi -  5 BEAM DIFFRACTION BY A CONDUCTING WEDGE 5.1 5.2 5.3  55  Real Line Source Solution Uniform Solution for Beam Source Numerical Results for the Wedge  55 58 59  6 BEAM DIFFRACTION BY A CIRCULAR APERTURE  67  6.1 6.1.1 6.1.2 6.2 6.2.1 6.2.2 6.3  Uniform Point-Source Solution Single Diffraction Solution Multiple Diffraction Solution Uniform Beam Solution Far Field Calculation Shadow and Reflection Boundary Calculations Numerical Results  7 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 7.1 Summary 7.2 Conclusions 7.3 ' Recommendations for Future Work  67 68 70 73 73 74 74 80 80 81 84  REFERENCES  86  APPENDIX  91  -  vii -  LIST OF FIGURES Figure  Title  Page  1.1a  Branch cut and branch points of the CSP  14  Lib  The paraxial region of a Gaussian beam  14  2.1a  Geometry of a complex line source and real far field point  24  2.1b  Geometry of multiple complex line sources and real far field point  24  2.2  Comparison of  normalized patterns  of  a typical aperture,  Gaussian beam and complex source point  25  2.3  Normalized patterns of multiple complex line sources  26  3.1  Geometry of a complex line source diffraction by a half-plane Comparison of a uniform and asymptotic solutions of beam diffraction by a half-plane ( The edge on the beam axis )  35  Comparison of a uniform and asymptotic solutions of beam diffraction by a half-plane ( The edge off the beam axis )  36  3.2 3.3 3.4  Comparison of  uniform  solutions  of  a  plane  34  wave and  limiting beam diffraction by a half-plane  37  4.1  Geometry of a complex line source diffraction by a slit  45  4.2  Normalized total field pattern of a beam diffraction by a slit Comparison of patterns of a plane wave and limiting beam  46  diffraction by a slit  47  4.4  Geometry of a complex line source diffraction by a strip  52  4.5  Normalized total field patterns of a beam diffraction by a strip ( Normal incidence ) Normalized total field patterns of a beam diffraction by a strip ( Non-normal incidence )  4.3  4.6  53 54  -  Figure  viii -  Title  Page  5.1  Geometry of a complex line source diffraction by a wedge  62  5.2  Normalized total field patterns of a beam diffraction by a rectangular wedge ( # < 0 )  63  Normalized total field patterns of a beam diffraction by a rectangular wedge ( ^ >"^ )  64  Normalized total field patterns of a beam diffraction by a rectangular wedge ( ^ = ^ )  65  Normalized total field patterns of a beam wedges of different angles ( 0 > # )  66  o  5.3  cr  0  5.4  0  5.5  cr  c r  o  6.1 6.2 6.3 E.1 E.2  diffraction by  c r  Geometry of a complex point source diffraction by a circular aperture  77  Normalized total field patterns of a beam diffraction by a circular aperture ( Normal incidence )  78  Comparison of patterns of a single and mutiple diffraction of a beam by a circular aperture ( Normal incidence )  79  Geometry of a parabolic reflector  complex  line  source  diffraction  by a  Normalized Diffracted field component of a beam diffraction by a parabolic reflector  103 104  - 1 -  CHAPTER I INTRODUCTION AND LITERATURE REVIEW  1.1  Introduction Electromagnetic  apertures years.  wave d i f f r a c t i o n  i n conducting  s c r e e n s has been s t u d i e d e x t e n s i v e l y  S o l u t i o n s f o r p l a n e wave i n c i d e n c e , o r a d i s t a n t  isotropic  local  half-plane,  sources  high  i n real  space,  have  f o r t h e wedge, f o r t h e s l i t  Bowman e t a l . , 1969). and  by c o n d u c t i n g r e f l e c t o r s  frequency  with the l a t t e r .  been  and by  f o r many  s o u r c e , and  obtained  and complementary  for  the  disc (see  Of t h e two c a t e g o r i e s o f s o l u t i o n s , low frequency or asymptotic  solutions,  this  thesis  i s concerned  U n i f o r m a s y m p t o t i c s o l u t i o n s f o r t h e h a l f s c r e e n and  wedge have been o b t a i n e d f o r o m n i d i r e c t i o n a l l o c a l s o u r c e s ( e . g . Boersma and  L e e , 1977;  solutions  Kouyoumjian  are useful  considered i n t h i s Diffraction sources  such  techniques, techniques Anderson  i n the d i f f r a c t i o n  by s i m p l e  Uniform  solutions  shapes when i l l u m i n a t e d  recently,  beams  has been  i s t h e beam f i e l d applied  and antenna extensively  for directive  beams  by d i r e c t i v e  local  distribution,  this  which  beams, u s i n g studied.  i s the d i f f i c u l t y  different  One of these  technique  t o solve  f o r antenna  represents the e f f e c t  of s o l v i n g  beam  The d i f f i c u l t y o f o b t a i n i n g t h e o f t h e antenna  e x a c t l y o r a p p r o x i m a t e l y , i s one of t h e d i s a d v a n t a g e s o f t h i s Another  asymptotic  r e p r e s e n t a t i o n by a c u r r e n t d i s t r i b u t i o n .  d i f f r a c t i o n by a c o n d u c t i n g h a l f - p l a n e . current  1974).  thesis.  as G a u s s i a n  (1978)  and Pathak,  the r e s u l t i n g  beam  approach.  boundary  value  - 2 -  problem. on  C e r t a i n l y t h e a c c u r a c y depends on t h e f i e l d r e p r e s e n t a t i o n and  t h e a p p r o x i m a t i o n s made t o s o l v e t h e i n t e g r a l  involved.  However,  t h i s approach g i v e s c o n t i n u o u s f i e l d s a t the shadow b o u n d a r i e s . The  Kirchhoff  Gaussian Because  beam  method  i s another  diffraction  o f double  technique  by h a l f - s c r e e n  integration  used  (Pearson  introduced i n this  to solve f o r e t a l . , 1969).  method,  asymptotic  s o l u t i o n s a r e c o m p l i c a t e d and n u m e r i c a l s o l u t i o n s v e r y c o s t l y when l a r g e scatterers  a r e assumed.  Another  r e g i o n o f f t h e beam a x i s .  shortcoming  i s poor  accuracy  i n the  However, i t p r e d i c t s no s i n g u l a r i t y on t h e  caustic axis. The  Boundary D i f f r a c t i o n  and  Wolf  (1962),  the  Kirchhoff  overcomes  method,  more c o m p l i c a t e d .  Wave Theory t h e problem  (BDWT) proposed o f a double  by Miyamoto  integration i n  but makes t h e i n t e g r a n d ( t h e v e c t o r p o t e n t i a l )  Consequently  i n a d d i t i o n t o the d i f f i c u l t y  t h e i n t e g r a t i o n becomes v e r y  difficult  i n obtaining the vector p o t e n t i a l  itself.  T h i s approach has t h e same a c c u r a c y as t h e K i r c h h o f f method o r l e s s when the  vector potential  i s approximate.  Using  t h e BDWT, O t i s  and L i t  (1975) gave t h e s o l u t i o n t o 2 - d i m e n s i o n a l G a u s s i a n beam d i f f r a c t i o n by a half-screen Fukumitsu  and t h e 3 - d i m e n s i o n a l  (1982).  The s i n g l e  case  diffraction  was g i v e n  by Takenaka and  by a c i r c u l a r  a p e r t u r e when  i l l u m i n a t e d by a n o r m a l l y i n c i d e n t G a u s s i a n beam was o b t a i n e d by O t i s e t al.  (1977) and c o r r e c t e d  by Takenaka  e t a l . (1980).  problem was s o l v e d by B e l a n g e r and Couture  A l s o t h e same  ( 1 9 8 3 ) , u s i n g t h e BDWT w i t h  the G a u s s i a n beam r e p r e s e n t e d by a complex source p o i n t .  - 3 -  The  Inhomogeneous  (evanescent)  Wave  Tracking  (IWT) proposed  by  Choudhary and F e l s e n (1973) and r e f i n e d by E i n z i g e r and Raz (1980), i s another  approach used  to solve  the problem o f d i r e c t i v e  fields.  The  main advantage of t h i s method i s t h a t i t g i v e s a p h y s i c a l e x p l a n a t i o n o f the p r o p a g a t i o n and s c a t t e r i n g mechanism. obtaining Felsen  the phase  paths,  Because of the d i f f i c u l t y o f  i t has been  (1974) a p p l i e d the IWT method  rarely  used.  t o t h e problem o f G a u s s i a n  r e l f e c t i o n by a c o n d u c t i n g c i r c u l a r c y l i n d e r .  the propagation  of Gaussian  beam  R e f l e c t i o n by a p a r a b o l i c  r e f e l c t o r was g i v e n by Hasselmann and F e l s e n (1982). studied  Choudhary and  A l s o F e l s e n (1976)  beams i n f r e e space u s i n g t h e same  method. The  Complex Ray T r a c i n g (CRT) method was i n v e n t e d  difficulty  of d e t e r m i n i n g  the d i r e c t i v e f i e l d s by  tracing  (1971),  the phase p a t h s i n t h e IWT method by t r a c i n g  i n the complex space.  the Gaussian  Deschamps (1971,  beam  i n free  •«-..'  conducting Also  Chione  illuminations.  circular  T h i s technique was a p p l i e d  space  1972) and W i l l i a m s  (1984) used the CRT method t o study w i t h tapered  t o overcome t h e  by K e l l e r (1973).  and  Streifer  Ghione  et  al.  t h e r a d i a t i o n from l a r g e a p e r t u r e s  S c a t t e r i n g o f evanescent plane waves by a  c y l i n d e r was  given  e t a l . (1984) a p p l i e d  by Wang and Deschamps  t h e same technique  to a  (1974).  reflector  antenna i l l u m i n a t e d by a beam f i e l d . The high  CRT method i s an o p t i c a l  frequencies.  (asymptotic)  s o l u t i o n v a l i d only f o r  The r e p r e s e n t a t i o n o f d i r e c t i v e  beams w i t h  complex  source p o i n t s a l o n g w i t h u s i n g e x i s t i n g ( e x a c t or approximate) s o l u t i o n s f o r r e a l s o u r c e s , which i s c a l l e d the Complex Source P o i n t (CSP) method,  - 4 -  can  give  simple  (exact  or a s y m p t o t i c )  problems  involving  solutions  directive  t o many  sources,  c a n o n i c a l and  with  no  extra  less  effort,  p r o v i d e d a n a l y t i c a l c o n t i n u a t i o n i n t o complex space i s p o s s i b l e . The  only d i f f i c u l t y  non-planar  surfaces,  w i t h t h e CRT and CSP methods, e s p e c i a l l y f o r  is  to  find  an  a-priori  selection  d i s t i n g u i s h t h e r e l e v a n t from s p u r i o u s r a y c o n t r i b u t i o n s .  rule  Now t h i s can  be done by s t u d y i n g t h e s a d d l e p o i n t s and s t e e p e s t descent paths et  a l . , 1984).  and  Otherwise  techniques  are uniform  on  t h e shadow  boundaries,  a s y m p t o t i c s o l u t i o n s when t h e beam a x i s passes edge. and  The CSP method uses  i t can be used  f o r exact s o l u t i o n s .  asymptotic  c o n d u c t i n g wedge 1979), edge  are i n v a l i d o r when  solutions  (Felsen,  except f o r  the d i f f r a c t i n g  Because of t h e above, the CSP  f o r the G a u s s i a n  beam d i f f r a c t i o n  1976) and by a h a l f - s c r e e n (Green  beams  effort  thesis.  when t h e beam a x i s  broad  through  Furthermore  e x i s t i n g s o l u t i o n s , so i t needs l e s s  method i s adopted everywhere i n t h i s The  (Ghione  t h e s e t e c h n i q u e s a r e easy t o a p p l y , a c c u r a t e ,  need no i n t e g r a l e v a l u a t i o n i n a s y m p t o t i c s o l u t i o n s .  these  to  a r e assumed.  passes  through  They  by a  et a l . ,  the d i f f r a c t i n g  are inaccurate i n the  t r a n s i t i o n r e g i o n s and s i n g u l a r on t h e shadow b o u n d a r i e s . One of the g o a l s here i s t o o b t a i n a u n i f o r m s o l u t i o n f o r t h e wedge u s i n g t h e U n i f o r m Theory o f D i f f r a c t i o n and t h e CSP r e p r e s e n t a t i o n , and for  t h e h a l f - s c r e e n , based  limit, shadow  on a s i m p l e s o l u t i o n e x a c t i n t h e f a r f i e l d  u s i n g t h e CSP method. boundary  2-dimensional  locations  antenna  beam  A more s i m p l e c o n v e n i e n t f o r m u l a f o r t h e also  will  diffraction  be by  derived. a  slit,  solution  including  to  double  - 5 -  diffraction,  and complementary  conducting  problem  not been  before.  has  circular  aperture  f o r normal  the  edges,  i s analyzed  the  above  examples,  plane wave i n c i d e n c e  1.2  studied  using  strip  will  Finally  incidence,  be g i v e n .  beam  including  diffraction  results  or i s o t r o p i c  include  by  i n t e r a c t i o n between  the UTD and CSP r e p r e s e n t a t i o n .  numerical  This  the l i m i t i n g  For a l l cases of  sources.  General Assumptions Through a l l the subsequent a n a l y s i s , the f o l l o w i n g a r e assumed: a)  The time dependence i s harmonic (exp[ju>t]) and i s suppressed.  b)  The  medium  is  homogeneous,  isotropic,  nondispersive  and  non-dissipative. c)  The f r e q u e n c i e s the f a r f i e l d  are very  high  of the s c a t t e r e r  and o b s e r v a t i o n  are i n  (kr»l).  d)  P e r f e c t conductors  e)  Scalar f i e l d s  f)  S o f t boundary c o n d i t i o n s a r e assumed.  1.3  points  (screen, h a l f - s c r e e n , e t c . )  (U) .are assumed.  L i t e r a t u r e Review Scattering  wedge, circular  circular  by  simple  reflectors  aperture,  parabolic  cylinders  approximately  when  Gaussian  by many r e s e a r c h e r s  illuminated  and a p e r t u r e s and by  as  paraboloidal  directive  half-plane,  antennas,  sources,  beams i n the p a r a x i a l r e g i o n ,  u s i n g the d i f f e r e n t techniques  a  and  which are  have been  studied  summarized below.  As  - 6 -  only  the complex source  p o i n t method o f s e c t o n  1.3.8 i s used i n t h i s  t h e s i s , the reader may choose t o omit s e c t i o n s 1.3.1-1.3.7.  1.3.1  Beam R e p r e s e n t a t i o n  by C u r r e n t D i s t r i b u t i o n s  By t h i s method, the i n c i d e n t beam i s r e p r e s e n t e d c u r r e n t sheet d i s t r i b u t i o n . element  i n presence  obtained  Then t h e boundary v a l u e problem of c u r r e n t  of the s c a t t e r e r  numerically  approach  i s continuous  different  from  or  a t the shadow  physical  a conducting  sheet.  asymptotically.  optics  by  i n t e g r a t i n g the  Then t h e i n t e g r a l i s  The  solution  boundaries.  with  this  This  approach i s  of  diffraction.  and s p e c t r a l theory  Anderson (1978) used t h i s t e c h n i q u e  1.3.2  i s solved  s o l u t i o n over t h e whole c u r r e n t  evaluated  by a non-uniform  t o s o l v e antenna beam d i f f r a c t i o n by  half-screen.  S p e c t r a l Theory o f D i f f r a c t i o n The  basic  proposed  by M i t t r a  conducting  coefficient  of STD which  of t h e s p e c t r a l t h e o r y  e t a l . (1976),  half-plane  contribution  induced  concepts  illuminated  were by  of d i f f r a c t i o n  illustrated plane  by t h e f a m i l i a r  wave.  The  i s the i n t r o d u c t i o n of the s p e c t r a l  i s defined  on t h e s c a t t e r e r .  as the F o u r i e r This  coefficient  transform  diffraction  boundaries,  the f i e l d s  coefficient obtained  tends  to  principal diffraction  of the c u r r e n t  i s associated  i n t e g r a l r e p r e s e n t a t i o n of the s c a t t e r e d and t o t a l f i e l d s . spectral  (STD)  infinity  by the STD a r e f i n i t e .  w i t h the  A l t h o u g h the  at  the  shadow  The s c a t t e r e d  f i e l d can be c o n s t r u c t e d by c o n v o l v i n g , i n t h e space domain, t h e induced  - 7 -  current  and the r a d i a t e d  current  (Green's f u n c t i o n ) .  field  field  of an elementary p o i n t  using  When t h e i n t e g r a l s  i n t h e s c a t t e r e d and t o t a l f i e l d s a r e a s y m p t o t i c a l l y  the s a d d l e p o i n t  source  The t o t a l f i e l d i s the sum of the s c a t t e r e d  and, whenever a p p l i c a b l e , the i n c i d e n t f i e l d .  involved  or l i n e  technique,  evaluated  the l e a d i n g term y i e l d s K e l l e r ' s GTD  field. Arbitrary  incident  t e c h n i q u e by a p p l y i n g incident  arbitrary  fields,  also  can be  assumed  the s u p e r p o s i t i o n p r i n c i p a l . field  i s multiplied  by  using  t h e STD  The spectrum of t h e  the s p e c t r a l  diffraction  c o e f f i c i e n t of the p l a n e wave then i n t e g r a t i n g over t h e e n t i r e spectrum to  give  fields.  a double This  Rahmat-Sammii  i n t e g r a l representation  i n t e g r a l may be e v a l u a t e d and  Mittra  (1977)  of the s c a t t e r e d  asymptotically  give  detailed  and  total  or n u m e r i c a l l y .  calculations  and  applications.  1.3.3  Kirchhoff-Fresnel Method I n t h i s method t h e t o t a l  field  behind an a p e r t u r e  i n a conducting  plane i s g i v e n i n terms of the double i n t e g r a l of t h e i n c i d e n t f i e l d and its  d e r i v a t i v e i n the p l a n e of the a p e r t u r e .  over the a p e r t u r e i s evaluated  This  i n t e g r a t i o n i s taken  (Born and W o l f , 1974, p. 375-386).  numerically  Then t h e i n t e g r a l  or a s y m p t o t i c a l l y t o y i e l d the t o t a l  field.  P e a r s o n e t a l . (1969) a p p l i e d t h i s t e c h n i q u e t o t h e d i f f r a c t i o n of a  fundamental-mode  conducting derived.  screen.  Gaussian  beam  An a s y m p t o t i c  (Kogelnik,  1965) by a s e m i - i n f i n i t e  s o l u c t i o n i n the F r e s n e l  limit  was  - 8 -  1.3.4  Boundary D i f f r a c t i o n Wave Theory Through  the  use  of  Stoke's  theorem  and  an  associate  potential  v e c t o r , Miyamoto and Wolf (1962) showed t h a t , i n g e n e r a l , the K i r c h h o f f s u r f a c e i n t e g r a l , mentioned i n the p r e v i o u s s e c i t o n , can be two s e p a r a t e l i n e i n t e g r a l s .  One  boundary  aperture  wave  of  (BDW)  the and  diffracting the  from the s o u r c e .  other  The  the shadow r e g i o n .  The  approximations  The  called the  the  boundary  diffraction  g e o m e t r i c a l wave  originating  i s z e r o i f the o b s e r v a t i o n p o i n t s l i e i n  total field  g e o m e t r i c a l wave f i e l d s .  BDW  i s g i v e n by the sum method has  of the BDW  and  the same l i m i t a t i o n s  and  as the K i r c h h o f f - F r e s n e l method.  Application  of  the  technique  t o a Gaussian  beam w i t h  cylindrical  symmetry (Siegman, 1971, Ch. 8) n o r m a l l y i n c i d e n t on a c i r c u l a r in  a conducting  field  into  r e p r e s e n t s a wave o r i g i n a t i n g from the  represents  latter  split  plane  i s g i v e n by  aperture  O t i s (1974) under the p a r a x i a l f a r  approximations.  1.3.5  U n i f o r m A s y m p t o t i c Theory of D i f f r a c t i o n A u n i f o r m a s y m p t o t i c t h e o r y of d i f f r a c t i o n (UAT) w h i c h p r o v i d e s  correct  asymptotic  h a l f - p l a n e has  solution  been developed  for  an  arbitrary  incident  field  on  by A l h u w a l i a et a l . (1968) and Lewis  the a and  Boersma (1969), c o r r e c t s d e f e c t s of the g e o m e t r i c l t h e o r y of d i f f r a c t i o n (GTD);  such  as  d i f f r a c t i n g edge. field  expansion.  singularities  at  the  shadow  boundaries  and  at  the  I t a l s o p r o v i d e s h i g h e r o r d e r terms i n the d i f f r a c t e d  - 9 -  Boersma and  Lee  (1977) a p p l i e d  UAT  t o the problem  of  cylindrical  wave from a l i n e source p a r a l l e l t o the edge of a c o n d u c t i n g h a l f - p l a n e . In  their  approach  a l l fields  are  powers of the wave number which  expanded  asymptotically i n inverse  i s assumed l a r g e .  The  c o e f f i c i e n t s of  e x p a n s i o n are d e r i v e d by s u b s t i t u t i n g i n the reduced wave e q u a t i o n .  The  postulated  on  the  the  UAT  exact  total  solution  reduces  to  field of  the  i s a uniform asymptotic expansion  a p l a n e wave i n c i d e n t exact  solution  of  on  based  a half-plane,  plane  wave  so  diffraction  by  a  half-plane. E x c l u d i n g the c a u s t i c p o i n t s at the source and solution  for  the  total  observation points. of  the UAT  remains  field  is  and  continuous  UAT  at a l l  Away from the shadow b o u n d a r i e s , the l e a d i n g term  s o l u t i o n reduces to the GTD  finite  finite  i t s image, the  at  the  diffracting  near-field  calculations.  coefficient  i s taken  diffraction  coefficient  edge,  Unlike  from of  the  solution.  the  i t can GTD  is  also  where  Sommerfield's  UAT  S i n c e the UAT be  the  solution used  for  diffraction  half-plane solution,  derived  by  enforcing  the  the edge  condition. The  UAT  has  been  curved wedge by Lee  extended  and  to  electromagnetic d i f f r a c t i o n  Deschamps (1976) but  The main d i s a d v a n t a g e of the UAT  a  approximate.  i s i t s complexity i n determining higher  o r d e r terms, when v e r y d i r e c t i v e  incident  interaction  between edges are s i g n i f i c a n t .  geometrical  theory  of  there i t i s  by  diffraction  by  which g i v e s l e s s a c c u r a t e r e s u l t s , may  fields  I n such  Kouyouimjian be used  are assumed and cases and  instead.  when  the  uniform  Pathak  (1974),  - 10 -  1.3.6  Inhomogeneous (Evanescent) Wave Tracking Here  inhomogeneous waves,  from the source or the i n i t i a l  such as G a u s s i a n beams, can be  s u r f a c e t o the o b s e r v a t i o n p o i n t v i a the  s c a t t e r e r , t o t a l l y i n r e a l space. for  the r e a l and  tracked  By s o l v i n g the d i f f e r e n t i a l e q u a t i o n s  i m a g i n a r y p a r t s of the phase and a m p l i t u d e f u n c t i o n s ,  w h i c h r e s u l t from s a t i s f y i n g the reduced wave e q u a t i o n , the t o t a l can be c o m p l e t e l y determined. in  field  N e g l e c t i n g the wave l e n g t h squared  the above d i f f e r e n t i a l e q u a t i o n s enables one  i n d e p e n d e n t l y from the a m p l i t u d e .  The  term  t o c a l c u l a t e the phase  s o l u t i o n o b t a i n e d i n t h i s way i s  a p p r o x i m a t e , but the a c c u r a c y i n c r e a s e s w i t h d e c r e a s i n g the wave l e n g t h . For  d e t a i l s see Choudhary and F e l s e n (1973), F e l s e n (1976) and  and Raz The  Einziger  (1980). inhomogeneous wave t r a c k i n g method i s a p p l i e d t o G a u s s i a n beam  reflection  by  conducting c i r c u l a r  cylinder  as  g i v e n by  Choudhary  and  F e l s e n (1974) w i t h o u t i n c l u d i n g d i f f r a c t i o n from edges.  1.3.7  Complex Ray Tracing S i n c e i t has been noted t h a t a G a u s s i a n beam can be r e p r e s e n t e d i n  terms  of a bundle  Streifer  of complex r a y s , by Deschamps (1971) and K e l l e r  (1971), complex r a y t r a c i n g  geometrical optics a m p l i t u d e and  analytic  source  to  been d e v e l o p e d .  was  i n t r o d u c e d and  I n the CRT  method, the  complex phase,  space c o o r d i n a t e s are a l l o w e d t o t a k e complex v a l u e s , as  i n the IWT method. of  has  (CRT)  and  The m a t h e m a t i c a l b a s i s of t h i s method i s the p r o c e s s  continuation. the  real  The  tracing  observation point  g e n e r a l ) i s i n complex space.  The  of  the f i e l d  v i a the  from the  scatterer  complex  (complex  in  study of a G a u s s i a n beam, s i m u l a t e d  - 11 -  by a complex l i n e or p o i n t s o u r c e , and p r o p a g a t i o n i n f r e e space from an assigned  initial  field  distribution  have been e x t e n s i v e l y  dealt  with  (Ghione, Montrosset and O r t a , 1984). The  eikonal  and  a p p l i c a b l e here.  transport  Without  e q u a t i o n s used  i n the  IWT  method  s e p a r a t i n g the phase and a m p l i t u d e  are  functions  i n t o r e a l and i m a g i n a r y p a r t s , the d i f f e r e n t i a l e q u a t i o n s can be s o l v e d by  the method of c h a r a c t e r i s t i c s  the t o t a l  1.3.8  to o b t a i n the phase and a m p l i t u d e of  field.  Complex Source Point In  the  complex  ray  tracing  obtained f o r high frequency the Complex Source  method,  asymptotic  ( l a r g e wave numbers) and  P o i n t (CSP) method can be used  solutions  are  far fields.  But  to o b t a i n e x a c t  as  w e l l as a s y m p t o t i c s o l u t i o n s f o r low or h i g h f r e q u e n c i e s and near o r f a r f i e l d s , as l o n g as s o l u t i o n s t o c o r r e s p o n d i n g r e a l sources e x i s t and  can  be a n a l y t i c a l l y c o n t i n u e d i n t o complex space. On  a s s i g n i n g complex v a l u e s to the source c o o r d i n a t e l o c a t i o n s  an o s c i l l a t i n g i s o t r o p i c p o i n t or l i n e s o u r c e , one may  of  generate a h i g h l y  c o l l i m a t e d f i e l d t h a t behaves i n the v i c i n i t y of i t s maximum (beam a x i s ) like  a  Gaussian  3-dimensional  beam (Deschamps 1971,  A l b e r t s e n et a l . , CSP  (point  substitution  1983,  source)  or  Jones  1979;  and F e l s e n 1976,  c o n v e r t s p o i n t or  line  2-dimensional Couture  1984).  and  (line  Belanger  incident  rigorous  and  Gaussian  beams.  asymptotic  source Green's f u n c t i o n s  Thus w i t h o u t  solutions  yield  1981,  T h i s i m p l i e s t h a t the  p r o p a g a t i o n and d i f f r a c t i o n i n v a r i o u s environments i n t o f i e l d for  source)  the  further field  effort, response  for  solutions the  whole  f o r beam  -  e x c i t a t i o n , solutions  from  Let  us  Gaussian free  provided real  beam  there  t o complex  take,  space,  wave  that  as  using  an  -  c a n be  an  a n a l y t i c  continuation  of t h e  space.  example,  t h e CSP.  i s given  12  The  by Green's  the  e v o l u t i o n  f i e l d  of  f u n c t i o n  an  of  a  i s o t r o p i c  G(R) which  2-dimensional  point  i s a  source  s o l u t i o n  i n  t o t h e  equation -jkR  G(R)  1  =  ,  (1.1)  R where  R  i s the distance  can  be  real  the  source  or be  between  complex. located  t h e source  F o r a  wave  (0,0,-jb)  a t  and observation  propagating where  b  p o i n t ,  which  i n t h e z - d i r e c i o n ,l e t  i s a  r e a l  p o s i t i v e  number.  Then R From  [p +(z+jb) ] 2  =  (1.2),  z=0,  p=b.  (surface) In  2  R  i s a  To  2  multivalued  make  a t z=0  ; p  1 / 2  R  t h e p a r a x i a l  (p  + y  2  ,  2  valued  should  region  x  f u n c t i o n  single  a n d p<b  =  be  « z  Real(R)X)  and vanishes  and  G(R)  introduced  "•" b  )  and  (1.2)  a t  t h e branch  a n a l y t i c ,  ( s e eF i g .  f o r z>0,  R  a  branch  l i n e c u t  1.1a).  c a n be  s i m p l i f i e d  to R  j[b ~ _  -  -P  b  2  ] +  1  9  z [ l +  2(z +b ) 2  I n s e r t i n g  w w  kb e  1  *  / z  i s t h e e =  2  +  b  2  (1.3)  )  (1.3)i n (1.1) gives  G(p,z) where  2 ( z  2  ]  [ 2 ( z  B(p,z)  =  +  2  1 b  2  -  b  +  -CP/W) Q  2  P  /  w  2  ;  e  half  1  b  .  "J ( ' > . e B  P  Z  (1.4)  2  ) / k b ]  [kb +  beam 1  /  width  and  B(p,z)  i s the phase (1.5)  2  (p/w) ] 2  +  t a n  -  1  ( b / z )  (1.6)  -  The  propagating  decay  perpendicular  beam  i s  formed  p a r a l l e l the ( p / w  t o  phase  Q  wave,  )  2  w h e r e  i n  the  ( z / b )  w  i s  t o  the  the  (locus =  2  by  w i t h of  e  -  (1.4),  z-axis  p a r a x i a l  z-axis  paths  -  defined  13  region.  p o i n t s )  1  subject  p r o p o r t i o n a l F o r  d i s t o r t i o n -  i s  of  t o  t o  p  .  b>>z,  an  exponential  Thus  a  t h e wave  t h e wave  front;  a r e hyperboloids  Gaussian propagates  and  given  f o r b<<z  by  1,  t h e  e  (1.7) _  h a l f - b e a m  1  w i d t h  a t  t h e  beam  w a i s t  ( z = 0 ) .  o W  Q  = ( 2 b / k )  F i g .  '  x  1  /  i s o f t e n  2  c a l l e d  t h e  spot  s i z e  a t  t h e  beam  w a i s t  ( s e e  1.1b). This  d e r i v a t i o n  i m p l i c a t i o n s  o f  three-dimensional complex  space  a p p l i e d  t o  high  theory provides For  higher  E i n z i g e r  be  modes  beams  under and  Raz  dimensional  f i e l d s .  s o l u t i o n s  f o r  f i e l d can  a n a l y t i c a l  be  continued  a r e  and  Luk  has  been  complex (1987).  w i t h  s o l u t i o n  Gaussian and  Yu  s t u d i e d  space-time  the  or  beams,  see  (1985). by  Mantica  s h i f t s  A t f a i l  complex  f o r beam  has  or  low  n u m e r i c a l l y .  i n e f f i c i e n t  together  of  At  The  two  o r  a n a l y t i c a l l y beams.  s o l u t i o n s ,  s o l u t i o n s .  solved  convenient  (1985)  complicated  and  g e n e r a l l y  t h e most  t h a t  two  the s o l u t i o n s f o r d i r e c t i v e  d i f f r a c t i o n  Hashimoto  s o l u t i o n s  a l s o  f o r  f u n c t i o n s  d i f f r a c t i o n  can  v a l i d  a r e  numerical  methods  of  t h i s Green's  frequency  numerical  a l s o  t o provide  both  d i f f r a c t i o n  i s  t o  This  into  can  e i t h e r  low or  frequencies high and  beam  frequencies,  the  source  be  geometrical  point  method  d i f f r a c t i o n s . Shin  and  Felsen  (1976),  Representation  of  more  e t  and  wave  been,  a l . (1986), l a t e l y ,  proposed  by  -14-  P  A  z  F i g .  1.1a  Branch  c u ta n d b r a n c h  points  o f t h e CSP  Hyperboloid  z  F i g .  1.1b  The p a r a x i a l  region  o f a Gaussian  beam  - 15 -  1.4  Overview of the Thesis An i n t r o d u c t i o n i s g i v e n i n S e c t i o n (1.1) and some o f e x i s t i n g work  in  the l i t e r a t u r e  source  on t h e r e p r e s e n t a t i o n  and d i f f r a c t i o n  of d i r e c t i v e  f i e l d s by s i m p l e shapes i s summarized i n S e c t i o n ( 1 . 3 ) .  I n Chapter I I , beam e v o l u t i o n from a Complex Source P o i n t (CSP) i s given i n polar coordinates. in  cartesian  coordinates  T h i s i s more c o n v e n i e n t and  directly  relates  than r e p r e s e n t a t i o n t h e beam  parameters  ( o r i e n t a t i o n and d i r e c t i v i t y ) t o t h e complex c o o r d i n a t e s o f t h e s o u r c e . The  beam f i e l d  antenna  generated  aperture  complicated  by t h e CSP, a G a u s s i a n beam and a  a r e compared  and i l l u s t r a t e d .  Derivation  beams; e.g. a beam w i t h s i d e l o b e s , i s a l s o  Chapter  I I I i s devoted  to obtaining  a simple  typical o f more  achieved. solution,  uniform  everywhere and f o r a l l beam o r i e n t a t i o n s , f o r antenna beam d i f f r a c t i o n by  a h a l f - s c r e e n , based on t h e exact  diffraction asymptotic simpler  by a h a l f - s c r e e n .  far field  s o l u t i o n of l i n e  source  A comparison o f t h i s s o l u t i o n w i t h t h e  s o l u t i o n g i v e n by Green e t a l . (1979) i s i l l u s t r a t e d .  formula  f o r shadow  boundary  l o c a t i o n i s derived.  Also a Results  o b t a i n e d h e r e a r e used i n Chapter IV of t h e problem o f beam d i f f r a c t i o n by a s l i t  I n a conducting  p l a n e and by a complementary s t r i p .  I n Chapter V, beam d i f f r a c t i o n by a c o n d u c t i n g wedge, when t h e beam axis  passes  boundaries  through  t h e edge,  are obtained  i s derived  and n u m e r i c a l  using  results  t h e UTD.  for different  Shadow angles of  i n c i d e n c e and wedge a n g l e s a r e g i v e n . I n t h e above examples, o n l y 2 - d i m e n s i o n a l are  assumed.  I n Chapter  V I , a 3-dimensional  beams and s t r a i g h t edges beam  diffracted  by a  - 16 -  circular  aperture  diffraction. aperture  Normal i n c i d e n c e ,  axis,  waists,  i n a p e r f e c t l y conducting  i s assumed.  plane,  including multiple  i . e . t h e beam a x i s c o i n c i d e n t w i t h t h e Numerical  results  i n c l u d i n g t h e p l a n e wave as a l i m i t i n g  for different  beam  case, are given.  The  l a t t e r i s compared w i t h K e l l e r ' s s o l u t i o n (1957). A  summary, c o n c l u s i o n s  and recommendations  f o r future  work a r e  g i v e n i n Chapter V I I . Appendix A c o n t a i n s an e v a l u a t i o n o f F r e s n e l i n t e g r a l s w i t h complex arguments i n terms o f e r r o r f u n c t i o n s  and some i m p o r t a n t p r o p e r t i e s o f  Fresnel i n t e g r a l s are given. I n A p p e n d i x B, t h e r e a l and i m a g i n a r y p a r t s distance  o f r , t h e complex s  from t h e s o u r c e t o t h e edge, i n terms o f t h e r e a l d i s t a n c e of  the source and t h e i n c i d e n t beam parameters a r e d e r i v e d . For  comparison  reasons  the asymptotic  s o l u t i o n of Gaussian  beam  d i f f r a c t i o n by c o n d u c t i n g h a l f - s c r e e n , and t h e shadow boundary p o s i t i o n s g i v e n by Green e t a l . (1979) a r e summarized i n Appendix C. Appendix  D,  shows  the  singularity cancellation  i n t h e wedge  d i f f r a c t i o n c o e f f i c i e n t a t t h e shadow b o u n d a r i e s , when i l l u m i n a t e d by a r e a l l i n e s o u r c e o r a beam s o u r c e i t s a x i s p a s s i n g Appendix  E  contains  conducting p a r a b o l i c  the  analysis  of  through the edge.  beam  diffraction  by  a  reflector.  I n Appendix F, t h e d e r i v a t i o n o f a r c t a n g e n t o f a complex number, i n the  p r o p e r quadrant I s g i v e n  i n terms o f a complex a n g l e i n t h e f i r s t  q u a d r a n t , t h a t can be determined by t h e UBC computer f u n c t i o n s . Appendix discussed  G  is a  list  of  computer  programs  i n C h a p t e r s I I - V I and i n Appendix E.  f o r t h e problems  - 17 -  CHAPTER II COMPLEX SOURCE POINT METHOD  By a s s i g n i n g complex v a l u e s  t o t h e source c o o r d i n a t e  l o c a t i o n s of a  time harmonic i s o t r o p i c p o i n t o r l i n e s o u r c e i n a homogeneous unbounded medium,  one may  3-dimensional beam.  (point  or l i n e  propagation  source  Green's f u n c t i o n s  i n c i d e n t d i r e c t i v e beams.  beam e x c i t a t i o n ,  that (line  behaves source)  like  a  directive  (wave e q u a t i o n  solutions)  for  Thus, w i t h o u t  further effort,  d i f f r a c t i o n s o l u t i o n s y i e l d the f i e l d  provided  t h e whole response  t h e r e c a n be an a n a l y t i c c o n t i n u a t i o n o f  r e a l space t o complex space.  Beam Evolution from a Complex Line Source Fig.  origin  2.1a shows a 2 - d i m e n s i o n a l l i n e of coordinates.  represent any  field  and d i f f r a c t i o n i n v a r i o u s environments i n t o f i e l d s o l u t i o n s  the s o l u t i o n s from  2.1  collimated  source) o r 2-dimensional  r i g o r o u s and a s y m p t o t i c for  a  T h i s i m p l i e s t h a t t h e complex s o u r c e p o i n t s u b s t i t u t i o n c o n v e r t s  point  for  generate  U  1  The f i e l d s a r e u n i f o r m  an o m n i d i r e c t i o n a l c y l i n d r i c a l wave.  i n t h e z d i r e c t i o n and The f i e l d  intensity at  p o i n t r , Q w h i c h i s a s o l u t i o n o f wave e q u a t i o n may be  observation  w r i t t e n as  s o u r c e a t r , 8 from t h e o o  -jn/4  = J%TI e  <P H  ( 2 )  °  (kR) »  kR  ;  kR » 1  ,  (2.1)  m  where R i s the d i s t a n c e o f t h e o b s e r v a t i o n p o i n t from t h e s o u r c e .  - 18 -  R = [r + r o 2  - 2rr o  2  L  In the f a r f i e l d  cos(0 - 0 ) 1 o  (r>>r ), R = r - r Q  (2.2) '  1 / 2  Q  c o s ( Q - 0 ) a p p l i e s i n the phase  term and R~r i n the a m p l i t u d e term of (2.1) g i v i n g -jk[r-r U = e  °  cos(0-0 )] °  0 < 0  ;  /k"r  < it  o  v  (2.3) '  By m a k i n g t h e s o u r c e c o o r d i n a t e s ( r , 0 ) c o m p l e x ( r , 0 ) the o o s s omnidirectional  wave  becomes  a  directive  beam  uniform  in  the  z  direction.  F  = 7  - SS  (2.4)  where "r , r" and "6" are t h e complex source p o s i t i o n , r e a l source p o s i t i o n and  beam parameter v e c t o r s g i v e n i n p o l a r c o o r d i n a t e s as r  = ( r , 0 ), o o o r* = ( r , 0 ) and "b" = ( b , 8 ) , where b d e f i n e s the sharpness of the beam s s s r  and (3 d e f i n e s i t s o r i e n t a t i o n .  A l l a n g l e s are measured from the x - a x i s .  r and r a r e m e a s u r e d from t h e o r i g i n w h i l e b i s measured from t h e r e a l s o p o i n t source as shown i n F i g . 2.1. r 0  g  s  = [r = cos  2  + 2 r ( - j b ) cos(p-0 ) + ( - j b ) ] 2  Q  -1  [  r  cos0  —  2  - jb cosP r  -jk[r-r U  g  s  (2.5) (2.6)  Replacing r ^ , 0  by r , 0  q  g  g  i n (2.3) g i v e s  cos(0-0 )] g  =  ;  cos(0 T 0 ) = r s s  and from ( 2 . 4 ) ,  g  ]  /kF r  ; Re(r ) > 0  s  where b > 0 and 0 <^ 8 << 2u. 1  1 / 2  Q  r »  |  I  r  (2.7)  S  cos0 s  cos0 + r —  sin0 sin0 s s  (2.8a)  - 19 -  r  s  r  s  cosO sinQ  s  = r cos0 - i b cosB o o  s  = r o  sinO  U s i n g these i n (2.8a) r  s  »  J  cos(0 + 0 ) = r s o  o  (2.8b)  - ib sinB J  yields  c o s ( 0 + 0 ) - j b c o s ( 0 + B) o  (2.8c)  S u b s t i t u t e (2.8c) i n (2.7) t o g e t -jk[r-r e  0  U  cos(0-0 ) ] kb cos(0-B) ° e  =  1  (2.9) /kr  By comparing  e q u a t i o n (2.9) w i t h (2.3) we f i n d t h a t (2.9) r e p r e s e n t s an  omnidirectional pattern  cylindrical  exp[kbcos(0-B)]  wave  with  (first  i t s maximum  term)  modulated  by a  i n the d i r e c t i o n  beam  0=8 and  minimum i n t h e d i r e c t i o n 0=8+n:.  2.2 Half Power Beam Width To  calculate  the half  power beam w i d t h  (HPBW), we n o r m a l i z e t h e  f i e l d of (2.9) t o i t s peak v a l u e . -kb[l = e  U (r,0)  - cos(0-B)] (2.10)  U (r,B) HPBW A t 0-8 = the n o r m a l i z e d f i e l d a m p l i t u d e of (2.10) e q u a l s 1//2. A  ....  -kb[l  ,HPBW..  - cos(  a,  Thus t h e h a l f HPBW  2  )]  1  (2.11)  72  power beam w i d t h i s r e l a t e d  ln/2\ , , . An/2 = 2 c o s - ^ l - £2^) ) ; ; kb kb >:> kb 2  t o t h e beam parameter  kb by  (2.12)  - 20 -  (2.12)  shows  that  as kb  increases  t h e beam  width  decreases.  If  kb < ^ An2 t h e beam does not decay t o t h e h a l f power p o i n t . S p e c i a l Case: r 8=0  o  and 0  g  or 0  a r e complex u n l e s s b=0, c o r r e s p o n d i n g t o a r e a l source o r  g  + ix.  o  To show t h e l a s t  case  substitute  f o r 8 = 0 o r 0 +n i n o o  (2.5) and (2.6) t o get r  = r  s  o  +jb  ; '  J  r  B =  0  o'  , 0  o  (2.13a)  +TI  and 0  s  =  (2.13b)  0  o  Therefore 0  becomes r e a l whenever t h e beam a x i s l i e s a l o n g r . s  2.3  o  Comparison w i t h G a u s s i a n and T y p i c a l A p e r t u r e Beam P a t t e r n s S i n c e near t h e beam a x i s , 0 - 8 i s s m a l l , we can w r i t e cos ( 9 - ) - l - <  Q  P  2  V  (2.14)  2  U s i n g (2.14) i n (2.10) g i v e s -kb(0-8)  lT(r,0)  2  T"  = e  (2.15)  U (r,8) x  Showing, as i s w e l l known (Green that  a complex source  paraxial region.  e t a l . (1979) and Hasselman ( 1 9 8 0 ) ) ,  p o i n t p r o v i d e s a beam which  i s Gaussian  i n the  I t i s i m p o r t a n t t o a p p r e c i a t e t h a t t h i s complex source  p o i n t r e p r e s e n t a t i o n o f a beam i s not l i m i t e d t o t h e p a r a x i a l r e g i o n . Fig.  2.2a shows t h e f a r f i e l d  radiation  p a t t e r n o f a source a t  k r = 8 f o r s e v e r a l v a l u e s o f t h e kb c o r r e s p o n d i n g Q  w i d t h s r a n g i n g from 68.5°(kb=2) t o 10.4°(kb=85).  t o h a l f power beam  - 21 -  I n F i g . 2.2b t h e broken  curve  i s a t y p i c a l a p e r t u r e antenna beam  p a t t e r n , t h a t o f an i n p h a s e c o s i n u s o i d a l d i s t r i b u t i o n i n an a p e r t u r e o f w i d t h 2a.  I t s normalized pattern I s  Ar.G)  _ cos[ka sin(9-B)]  U (r,B)  1 -  i  [ i ^ . sin(9-B)]  2  it  F o r ka=4, i t s h a l f - p o w e r 2.2b  is a  complex  beam w i d t h i s 55.7°.  source  point  pattern  The s o l i d c u r v e i n F i g .  of  t h e same  beam  width  (HPBW=55.7° o r kb=3) w i t h kb=  '  (2.17)  [l-co.(™)] 2 The  dashed  curve  i n F i g . 2.2b i s a G a u s s i a n  beam w i t h  t h e same beam  width.  U-V.G)  All  = e  A n ^ l i ^ ]  2  HPBW  (  t h r e e c u r v e s o v e r l a p i n t h e p a r a x i a l r e g i o n (9-8 s m a l l ) .  well  o f f t h e beam a x i s ,  t h e r e i s some d i f f e r e n c e ,  specially  2.i8)  At angles f o r broad  beams (kb s m a l l ) , between (2.10) and (2.18) b u t here t h e complex s o u r c e point  pattern given  (2.16).  by (2.10) I s a s l i g h t l y  b e t t e r approximation t o  The complex s o u r c e p o i n t r e p r e s e n t a t i o n appears  approximation  t o be a v a l i d  t o an antenna main beam p a t t e r n over t h e f o r w a r d  range (|0-B| < i t / 2 ) .  angular  Of c o u r s e i t cannot r e p r e s e n t p a t t e r n s i d e l o b e s .  - 22 -  2.4  Multiple Complex Line Sources More  c o m p l i c a t e d beams such  as beams w i t h s i d e l o b e s a l s o  d e r i v e d by u s i n g the complex source p o i n t method. one  source  at different r^, 0  locations  q  complex  locations  a n d t h e beam parameters  c a n be  By p u t t i n g more than  and changing  the r e a l  b , 8 we get a v a r i e t y o f beam  shapes. Let  us have M sources  located  a t M complex p o s i t i o n s .  The mth  i s located at r , 0 and i t s c o r r e s p o n d i n g beam parameters a r e om* om ° as shown i n F i g . 2.1b. Then the f a r f i e l d due t o the mth complex  source  r  b ,8 nr m  r  source U^" i s g i v e n by (2.9) and r e w r i t t e n here as m 4  U  1  - J ^ o m - -1-  COB  < - om>l 9  9  k b  m  ^s(0-?J  6  ;  r  /k7  m  » om' m r  b  Then the r e s u l t a n t f a r f i e l d , due t o the M w e i g h t e d s o u r c e s ,  (2.19) is  M  =2 . % m mm  U  U  m=l  ~  j k r  e  /kr  M j k r cos(0-0 ) kb c o s ( 0 - 6 ) v n om om m m I Q„ • e • e m=l  ,„ (2.20)  N  o  n  N  where 0^ are the w e i g h t i n g f a c t o r s . In  Fig.  different  2.3 t h e f i e l d  beam  parameters  due t o 3 l i n e b,8  while  sources  the real  i s derived  locations  for  are kept  c o n s t a n t , 0 .=0, 0 =TC/2, 0 =tr, k r . =kr = 1 , k r =0. The w e i g h t i n g ol o2 ' o3 o l o3 o2 0  f a c t o r 0,2=1 or  while  l e s s than 1 ) .  and Q  3  o  are v a r i a b l e s ( p o s i t i v e o r n e g a t i v e , g r e a t e r  Because s o u r c e s 1 and 3 are symmetric w i t h r e s p e c t t o  s o u r c e 2, the f i e l d s shown i n F i g . 2.3 are symmetric.  Asymmetric f i e l d s  - 23 -  a l s o can be d e r i v e d from asymmetric s o u r c e s . line  source d i f f r a c t e d  looks  by a s l i t  A beam w h i c h resembles a  and a beam w i t h f i r s t  l i k e a p l a n e wave d i f f r a c t e d  by a s l i t  sidelobes which  a r e shown i n F i g . 2.3.  More s i d e l o b e s can be d e r i v e d i f more complex sources a r e i n c l u d e d . In t h i s very is  chapter  we s t u d i e d  the l i n e source.  The p o i n t s o u r c e i s  much t h e same w i t h term —L_ i s r e p l a c e d by — — and two dimensions  r e p l a c e d by t h r e e d i m e n s i o n s . A  general  description  of  multiple  complex  r e p r e s e n t a t i o n of beams has been g i v e n by Hashimoto  source  (1985).  point  Fig.  2.1a  Geometry  of  a  complex  of  m u l t i p l e  l i n e  source  and  real  f a r  f i e l d  point  Fig.  2.1b  Geometry f i e l d  point  complex  l i n e  sources  and  real  f a r  t y p i c a l a p e r t u r e antenna (2.16, ka=4) Complex Source P o i n t (2.10, kb=3) G a u s s i a n (2.18, HPBW=55.7°)  ( 3 = 1 8 0 ° , Q =1.0, b!=b =b , Qi=Q3,Bl=B2-6 and 6 -3 +5) 2  2  2  3  3  2  - 27 -  CHAPTER III BEAM DIFFRACTION BY A CONDUCTING HALF-PLANE  The  total far field  beam i s d e r i v e d  diffraction  g i v e n by Born & Wolf  (1978) and Clemmow (1950) f o r a r e a l l i n e s o l u t i o n  and compared w i t h t h e  3.1  solution  far field  h a l f - s c r e e n of a  solution  asymptotic  from t h e exact  by a c o n d u c t i n g  g i v e n by Green e t a l (1979) (see Appendix C ) .  Uniform S o l u t i o n F a r F i e l d Radiation Suppose t h e o m n i d i r e c t i o n a l  the  edge o f an i n f i n i t e l y  thin  x>0 a s shown i n F i g . 3.1. r,Q  Pattern  source g i v e n  perfectly  by (2.1) i s p a r a l l e l t o  conducting  half-plane  i n y=0,  I f k ( r + r ) » l , t h e t o t a l f i e l d a t any p o i n t o  f a r from t h e edge ( r » r ) i s g i v e n e x a c t l y as Q  -j(kr-Ti/4) {  e  j k r cos(9-0 ) e ° °  0-0 cos( ° ) ]  F[-/2kr ~  U(r,0) = jkr  -e  cos(0+0 )  °  °  2 0+0  F[-/2k7 c o s ( 2.)]} (3.1) ° 2  where F[w]  -  /  e  dx  (3.2)  w i s the Fresnel i n t e g r a l  ( s e e Appendix A ) .  By making t h e c o o r d i n a t e s in and  ( > ® ) °^ r  0  0  t  n  e  source,  complex ( r , 0 ) as g  (2.5) and ( 2 . 6 ) , t h e o m n i d i r e c t i o n a l s o u r c e becomes a d i r e c t i v e the s o l u t i o n  i n (3.1)  g  beam  i s s t i l l v a l i d with r , 0 replacing r , 0 . s s o o  - 28 -  The for  total  far field  of a beam d i f f r a c t e d  by c o n d u c t i n g h a l f - s c r e e n  normal and non-normal i n c i d e n c e i s g i v e n as 0-9  -j(kr-nM)  jkr  e  s  cos(9-9 )  ^ —  s  i e  s  L  „  U(r,9) =  (3.3) /itkr  jkr  -e  cos(9+9 )  S  0+0  F[-/2kT cos( — ) ] }  S  2  s  where  F[w]  provide  i s the complex  values  boundaries  finite  Fresnel  integral.  and continuous  of the source  across  and h a l f - p l a n e .  g i v e n by Green e t a l (1979) the f i e l d when the beam  axis h i t s  Efficient  computer  Fresnel integrals  Fresnel  the shadow and  integrals reflection  In the asymptotic  solution  i s s i n g u l a r a l o n g the boundaries  the d i f f r a c t i n g  neighbourhood of the shadow  The  edge and i s i n a c c u r a t e i n the  boundaries.  subroutines  are a v a i l a b l e  for calculating  the  i n terms of e r r o r f u n c t i o n s w i t h the complex arguments  (see Appendix A ) .  3.2  Shadow and Reflection Boundaries F a r from  the h a l f - s c r e e n edge simple e x p r e s s i o n s f o r the shadow and  r e f l e c t i o n boundaries et  a l (1979).  only  the r e a l  imaginary 9  s  These e x p r e s s i o n s part  of r  g  to those of Green  are s i m p l e r and more a c c u r a t e because  need be c a l c u l a t e d , whereas b e f o r e r e a l p a r t ,  p a r t and a b s o l u t e v a l u e of r  were u s e d .  boundaries relative  a r e t o be d e r i v e d here analogous  r  s  and 0 s  i n general  and r e a l and imaginary p a r t s of s a r e t h e s o u r c e complex c o o r d i n a t e s . These  a r e not  straight  lines.  They  depend  on the  p o s i t i o n of the h a l f - s c r e e n edge w i t h r e s p e c t t o the beam a x i s  - 29 -  and the s o u r c e . and  The shadow and r e f l e c t i o n boundaries a r e s t r a i g h t  c o i n c i d e w i t h those  of  the  lines  source when i t s c o o r d i n a t e s a r e  real,  o n l y when the beam a x i s passes through the edge of the h a l f - s c r e e n . U s i n g the Green e t a l . (1979) d e f i n i t i o n  of shadow and  reflection  b o u n d a r i e s , we have Real(w^  / 4  ) = 0  (3.4)  where w i s the F r e s n e l i n t e g r a l argument of (3.3) g i v e n as 9+0  w.  = -/2kT  1)  cos(  (3.5)  2  r  L  where the s u b s c r i p t s i , r r e f e r  t o i n c i d e n c e and r e f l e c t i o n .  (3.4) i s  satisfied i f JTt/4  Imag[(we  ) ] = 0  (3.6)  2  and JTt/4  Real[(we Letting  r  s  ) ] _< 0  (3.7)  2  = R - j l as i n Appendix B and u s i n g the i d e n t i t i e s g i v e n by  (2.8c), i t i s easy t o show t h a t jn/4 (we Y' =~ j k [ R + r V.wc > 2  Q  COS(0+O )] Q  + k[I+b c o s ( 0 + B ) ]  (3.8)  Hence jit/4 Imag[(we ) ] = k[R+r 2  cos(0+0 ) ]  (3.9)  and jit/4 Real[(we  ) ] = k[I+b c o s ( 0 + 8 ) ]  (3.10)  2  S u b s t i t u t i n g (3.9) i n (3.6) and s o l v i n g for 0 r e f l e c t i o n b o u n d a r i e s , i . e . 0 = 0 . and 0 , gives ' si sr ' °  at  the  shadow  and  - 30 -  "I _ + cos(_) o R  9  = ± %  s i  sr  s i n c e R=Real(r ) and r s o  (3.11)  a r e r e a l and p o s i t i v e we can w r i t e r  -1 G  si sr  - °o  =  +  [ %  -  C O S  (  R  F~  ) ]  ( 3  °  '  1 2 )  or the shadow boundary p o s i t i o n i s R + [-re + cos ( _ ) ] o — r o _ 1  0 = 0 si  1  /  J  ; '  T  < B 2. 0 o  + n  (3.13)  + n  (3.14)  v  /  and the r e f l e c t i o n boundary p o s i t i o n i s "I 0  g r  = -e  Q  R  + [ii T cos (1-)] o  ;  B < 0 >  q  From (3.13) and (3.14) we can see t h e symmetry of shadow and r e f l e c t i o n boundaries w i t h respect to the h a l f - s c r e e n .  This property i s v a l i d f o r  o m n i d i r e c t i o n a l and d i r e c t i v e s o u r c e s as w e l l . gives 0 + 0 = 2n si sr To  satisfy  , for a l l  (3.7),  A d d i n g (3.13) and (3.14)  B  substitute  (3.15) for R  from  (3.9) and  (3.6) i n (B.6)  of  Appendix B y i e l d i n g I = -b  cos(P-0 ) — cos(0+0 ) — o  (3.16)  E x p a n d i n g cos(B-0o) i n terms of c o s i n e s and s i n e s  of (0+P)  and  (0+0o)  and s u b s t i t u t i n g i n (3.10) we get JTI/4  Real[(we  ) ] = -kb s i n ( 0 + B ) . tan(0+€> 2  o  )  From e q u a t i o n (3.13) and Appendix B we can show  (3.17a)  - 31 -  IT < 0  s1  - 0  Hence t a n ( 0  < —  i f  2  o  0 < 0  sl  - 6 < it  (3.17b)  -0 )>0 and s i n ( 0 -8)>0 o s i  si  A l s o we can show 1 < B - Q 2 s i  < it  o  - * < 0 ,-B < 0 2 s i  i f  (3.17c)  Hence, t a n ( 0 - 0 ) < 0 and s i n ( 0 -B) < 0. ' si o s i Therefore  (3.7) i s s a t i s f i e d  f o r t h e shadow boundary.  (3.14) and Appendix B we can s a t i s f y  Similarly  from  (3.7).  As  a c h e c k on t h e a b o v e f o r m u l a s f o r 0 . and 0 , l e t us d i s c u s s si sr the f o l l o w i n g s p e c i a l c a s e s . i)  R e a l l i n e s o u r c e i . e . b=0 from (3.10)  R=r , and o'  0 , = it + 0 si — o sr ii)  v  (3.18) '  Beam a x i s passes through t h e s c r e e n edge 8 = 0  +  o  it  from ( 3 . 1 3 ) , r = r + j b and ' s o v  0  which  ,  si sr  =  J  TC  + 0 ,  —  o'  i s t h e same  as  the r e a l  line  source  shadow  and  reflection  boundaries.  3.3  Numerical Results for the Half-Plane The  total  solid  field  curves  i n F i g . 3.2 and F i g . 3.3 r e p r e s e n t t h e u n i f o r m  calculated  from  ( 3 . 3 ) w i t h k r =16 a n d 0 =n/2 w h i l e t h e o o  - 32 -  dashed  curves  Appendix  represent  (C.5).  the asymptotic  F i g . 3.2.  calculated  from  The development of a beam from an  Q  o m n i d i r e c t i o n a l l i n e source (kb=0) in  field  The two d i m e n s i o n a l beam i s n o r m a l l y i n c i d e n t upon t h e  p l a n e and a t a d i s t a n c e kr =16.  half  total  F o r t h e case  t o a d i r e c t i v e beam (kb=12) i s shown  of kb=0 the p a t t e r n  oscillations  i n the  i l l u m i n a t e d r e g i o n (-rc/2<<K0) a r e f a m i l i a r , showing i n t e r f e r e n c e between ,  the d i r e c t wave from t h e source and a d i f f r a c t e d wave emanating  I n t h e shadow r e g i o n (0<<KV2) t h e r e i s o n l y a d i f f r a c t e d  edge.  which decreases w i t h <> t to. become z e r o on t h e c o n d u c t o r . the above o s c i l l a t i o n s occurs  because  a r e suppressed  the incident  r e g i o n as d i r e c t i v i t y  field  i s suppressed  how  This  i n the i l l u m i n a t e d  increases.  i n a c c u r a t e the asymptotic  boundary.  field,  As kb i n c r e a s e s  i n the i l l u m i n a t e d region.  F i g . 3.2, where t h e beam a x i s passes  In see  from t h e  solution  through the edge, we can becomes near  When kb i n c r e a s e s t h e r e i s l i t t l e improvement.  boundary the a s y m p t o t i c f i e l d  t h e shadow  On t h e shadow  i s s i n g u l a r f o r a l l v a l u e s of kb i n F i g .  3.2. In  F i g . 3.3  asympototic boundary  where the beam a x i s i s o f f the edge by an a n g l e 6, t h e  solution  especially  s o l u t i o n improved  i s finite  f o r s m a l l kb or 6 .  angle  and on the shadow  I n F i g . 3.3  the a s y m p t o t i c  g r e a t l y when t h e o f f edge a n g l e i n c r e a s e d from 15° t o  45° f o r a f i x e d kb=12. edge  but i n a c c u r a t e near  i s fixed  I n t h e lower graphs to  6=30°,  the  of F i g . 3.3  asymptotic  c o n s i d e r a b l y when kb i n c r e a s e d from kb=4 t o kb=16.  where t h e o f f  solution  improved  From the above we  can conclude t h a t the a s y m p t o t i c s o l u t i o n i s a good a p p r o x i m a t i o n t o t h e u n i f o r m s o l u t i o n whenever t h e beam a x i s i s w e l l o f f the d i f f r a c t i n g edge  - 33 -  and  the  beam  sufficiently Fig. distant  3.4  shows source  diffracted  the f a r f i e l d s  by a d a s h e d  line  field  source a l s o does.  and  kb a r e  plane  curve  d i f f r a c t i o n by a  and by a narrow beam  F o r b>>r  Q  t h e wave u n i f o r m l y and i n a s i m i l a r  So i n F i g . 3.4b we can see the  components are s i m i l a r . region  I n F i g . 3.4a the t o t a l  fields  (-n/2<<K0), because f a r from the  the d i r e c t i n c i d e n t wave of the narrow beam i s almost z e r o a few  degrees total  curve.  6  d i f f r a c t i o n by the edge.  edge and i t s neighbourhood  are d i f f e r e n t i n the i l l u m i n a t e d source  i . e . when  f o r half  r e p r e s e n t e d by a s o l i d  the d i f f r a c t i n g  the d i s t a n t  directive,  f o r then t h e r e i s l i t t l e  represented  illuminates way  sufficiently  large,  line  source  is  o f f the beam a x i s , field.  uniform,  The  direct  consequently  diffracted  wave  so the d i f f r a c t e d f i e l d wave  of the d i s t a n t  interference  occurs  between  and appears  i s e s s e n t i a l l y the  line  source  the d i r e c t  i s almost  wave  and the  as o s c i l l a t i o n s i n the i l l u m i n a t e d  region. To Fresnel  obtain  numerical  integral  values  subroutine  r e l a t i o n between the F r e s n e l  from  that  (3.3) i t i s necessary  can  handle  complex  arguments.  i n t e g r a l and the e r r o r f u n c t i o n  Appendix A.  Subroutines  are a v a i l a b l e  i n the UBC computing c e n t e r G e n e r a l L i b r a r y .  of  function  the e r r o r  tables  of  the  is  possible.  with  modified  Clemmow and Munford  f o r the e r r o r  complex  Fresnel  function  arguments  integral  with  with  t o have a  i s given i n  complex  complex  argument  Also  by G a u t s c h i  tables  (1964) and  arguments  (1952) agree with our s u b r o u t i n e whenever  A  by  comparison  -34-  Line Source  Beam Axis  Fig.  3.1  a) Geometry of a complex l i n e source d i f f r a c t i o n by a half-plane b) Beam orientation with respect to the edge.  FTg. 3.4  Uniform s o l u t i o n (eq. 3.3) comparison o f a d i s t a n t s o u r c e ( s o l i d ) and a l i m i t i n g beam of l a r g e kb kb=o kb=85  ,  k r =85  ,  6 =90°  k r =8 o  ,  8 =90° o  ,  B=270  - 38 -  CHAPTER IV BEAM DIFFRACTION BY A WIDE SLIT AND COMPLEMENTARY STRIP  4.1 Beam Diffraction by a Wide S l i t W i t h t h e r e s u l t s f o r beam d i f f r a c t i o n by a h a l f - p l a n e we can s o l v e the problem o f beam d i f f r a c t i o n by a s l i t complement,  a  conducting  half-planes  with  parallel  strip.  i n a c o n d u c t i n g p l a n e and i t s  The  slit  between  two  coplanar  edges, o r i t s complement, t h e s t r i p , a r e a  t r a d i t i o n a l t e s t of t h e o r i e s i n v o l v i n g m u l t i p l e d i f f r a c t i o n by edges.  4.1.1  Far Field Calculation Fig.  With  4.1 shows a l i n e  an i n c i d e n t  half-plane with r  l  f  field  source p a r a l l e l  given  on t h e r i g h t  by (2.1) t h e t o t a l  side i n i s o l a t i o n  0 i r e p l a c i n g r , 0 and r  0  S J.  Similarly 2  These  r  by (3.3)  s  s  2  ,0  2  i n isolation  r e p l a c i n g r , 0 and r ^ .  ©^  a r e shown i n F i g . 4.1.  S  expressions  f o r the f i e l d s  b o t h i n c i d e n t and d i f f r a c t e d total  l f  of the  replacing r , 0 .  1  r e p l a c i n g r , 0 . A l l the coordinates 8  f a rfield  of the l e f t half-plane  by ( 3 . 3 ) w i t h  2  i n y=0, |x|<_ a.  U i ( r Q ^ ) i s given  S i.  the t o t a l f a r f i e l d  U ( r , 0 2 ) i s given  to a s l i t  non-interaction  of t h e two h a l f - p l a n e s  f i e l d s behind t h e s l i t .  far fields  f o r the s l i t  contain  Consequently t h e  are t h e i r  sum l e s s an  incident f i e l d U . 1  u  s  =  Ui(ri,©i) + U ( r , 0 ) - U 2  2  I n the f a r f i e l d of the s l i t r^ = r - a cos0 r  2  1  - r - a cos©  2  ,  0  ,  0  (4.1)  1  2  (r »  a)  = 0  1  2  = n - 0 ; = 3n - 9  0 < 0 < ;  it  n<0<2ix  (4 .2a)  - 39  These  far  terms  of  field U  and  1  From F i g . 4.1 r „,  r  substitutions U  we  while  2  r^  for - r  -  rj, r  are  - r i s used  2  used  i n the  i n the  can w r i t e the f o l l o w i n g g e o m e t r i c a l  0  , 0 ,  measured from the two  S i  = [r + a s  9  2  2  T 2ar  2  cos© ] s  s  1  /  s, - * _ s i n "  1  r ( *  sin0 1)  amplitude  terms.  relations for r  sl,  edges i n terms of r , 0 .  ;  2  2  a  exponential  Real(r ) > 0 , si. 2  (4.2b)  ,  (4.2c)  4  2  where  r  (2.5)  and  s  and  0  are  s  (2.6)  measured  from the c e n t r e of the s l i t  respectively.  We  may  use  the  and  Fresnel  given  by  integral  identity. _jn/4 F[-w]  = /if e  - F[w]  (4.3)  to i n c l u d e the e x t r a i n c i d e n t -jk[  U  1  r i  _ r  g  l  cos(0 _0 )] 1  = — /kr  in  a F r e s n e l i n t e g r a l te rm.  slit  are  field -jk[r_r  s l  =  -1-  s  cos(©_0 )] s  (4.4)  /kT  Then the n o n _ i n t e r a c t i o n  f a r f i e l d s of  the  - 40 -  -j(kr-n/4)  u  j k a cos9, j k r , c o s ( 9 , _ 9 ,) e {" e F[-w ]  t  1  S l  1  3 1  n  s  jkr^cosCGi+e^)  F[w ]} r l  - ee jka cos9 + e  j k r „ cos(9 _9 { e  2  )  2  S l  F[w ]  S l  jkr  cos(9  2 +  i 2  9  s 2  )  - e  F  t r2^ ^  ^  w  ^  where . w  = -^Zkr  ±1  0 a l  cos(  This  i s an  interaction  edges. theory  (4.6)  2  s i m i l a r l y f o r w^ r2  accuracy  9 —)  rl and  T  X  with subscript  accurate  between  edges  i t i s useful  solution  2 replacing for s l i t s  i s negligible.  to include  also  1 i n (4.6). s u f f i c i e n t l y wide  I n order interaction  to i n d i c a t e between  the  that the slit  E a r l i e r r e s u l t s f o r p l a n e wave i n c i d e n c e u s i n g the g e o m e t r i c a l of d i f f r a c t i o n show t h a t  single  and double d i f f r a c t i o n p r o v i d e  a c c u r a t e r e s u l t s f o r s l i t w i d t h s ka > 2 ( K e l l e r (1957), F i g . 9).  4.1.2  Multiple Diffraction Calculation To  singly  include  higher  order i n t e r a c t i o n s  between the edges the  field  d i f f r a c t e d from each edge i n t h e d i r e c t i o n of the o p p o s i t e edge  i s r e p l a c e d by the f i e l d of a l i n e source of e q u a l a m p l i t u d e l o c a t e d a t the edge from w h i c h the s i n g l y d i f f r a c t e d f i e l d o r i g i n a t e s .  For example  - 41 -  the  doubly  singly  diffracted  diffracted  vice-versa times.  for  field  field  the  from the  from the  right  right  edge.  edge  is  produced  by  the  edge i n the 0^=71 d i r e c t i o n and  This can be repeated  Adding a l l contributions  diffracted f i e l d s of the  left  infinitely  many  from the two edges gives the multiply  slit.  The singly d i f f r a c t e d component of the f i e l d given by (3.3)  can be  written as U  = U*  d  where  is  D(0  r  O, r)  ,  (4.7)  the incident f i e l d calculated at the d i f f r a c t i n g edge given  by -jkr J  U  i  s  e  =  e  , Ikr I »  ATs  1  5  (4.8)  and D(0 , r , 0, r) is the d i f f r a c t i o n coefficient of the edge s s D(6..  e. r , . ^ f '  V  where w^ and w  , .fl  4  , 1^ .  . fl  (4  9)  are given by (4.6).  f  Following  p.91)  in  c a l c u l a t i n g multiple d i f f r a c t i o n for plane wave incidence on a s l i t ,  the  multiply  the  same  diffracted  procedure  field  given  excluding  by  Jull  (1981,  single d i f f r a c t i o n ,  can be written  as U  U  d  =  e  D x  l  ( D  o l D  + D  2>  +  I  U  e2  D  2(  D  D 0  2  + D  l) (4.10)  (1 - D ) o 2  - 42 -  If  the l i n e source  source r  c o o r d i n a t e s a r e on the s l i t  i s symmetric w i t h  .. = r „ and 9 = 9 -  respect  to both  consequently  U  a x i s ( y - a x i s ) , i . e . , the  edges of the s l i t . and D* = D .  i =  2  Then  Then (4.10)  can be s i m p l i f i e d t o (Df +  U^DJ  u  a  =  D) 2  —It ( i - »„>  (4.11)  where U^, i s g i v e n by (4.8) w i t h r , r e p l a c e s r and el sl s D  2a, i t , 2a)  D(TC,  =  Q  j(2ka+it/4)  = - /47TT . e D  l  "  D  (  9  sl'  r  . F[/4ka]  s l ' *»  sl  =-/2kr / i x k a n  (4.12)  2 a )  j ( - 2 k a + it/4) e  -jkr  s l  cos9  F[/2kr" si Df = D(it, 2a,  jkr  g l  g l  sin(M)]  (4.13)  2  9 ^ )  j ( 2 k a + n/4) - j k a c o s 9 9 =-/8ka/Ti . e . e . F [ / 4 k a s i n ( — ) ] . - — (4.14) 2 Jkr j k r  1  W h i l e D | i s g i v e n by (4.14) w i t h r , 9 2  For fields  a wide  range of s l i t  of the s l i t  give  little  first  order  interaction.  first  order  i n t e r a c t i o n only.  2  widths  replacing r ^ (2a), higher  or no improvement  9^ order  i n accuracy  In practice i t i s s u f f i c i e n t That i s p a r t l y  interaction over the  t o i n c l u d e the  because t h e above  higher  - 43 -  order i n t e r a c t i o n fields  are not  calculations  are approximate,  o m n i d i r e c t i o n a l as  o r d e r i n t e r a c t i o n i s weak except  assumed.  f o r the edge d i f f r a c t e d  I t i s a l s o because h i g h e r  f o r narrow s l i t s ,  f o r which the whole  process d i v e r g e s and a d i f f e r e n t method Is r e q u i r e d . Now U  the t o t a l f i e l d i n c l u d i n g m u l t i p l e d i f f r a c t i o n =  t  m  s  where field  + U  d  m  ,  (4.15)  i s g i v e n by given  by  i s g i v e n by  (4.5)  (4.15)  is  and  U  d  i s g i v e n by ( 4 . 1 0 ) .  continuous  and  free  from  The t o t a l f a r  shadow  boundary  s i n g u l a r i t i e s because the F r e s n e l i n t e g r a l s are r e t a i n e d .  4.1.3  Numerical Results for the S l i t The  d i f f r a c t i o n patterns  of  F i g . 4.2  source p a r a l l e l t o and a t a h e i g h t k r The  s o l i d curves  from (4.5) order  and  are  an  = 8 above a s l i t  calculated  between the edges of  f i e l d s are of minor importance For  calculated  for a  the s l i t .  (kb=0)  calculated  from (4.15) i n c l u d e h i g h e r  f o r t h i s w i d t h of  o m n i d i r e c t i o n a l source  line  of w i d t h 2ka=16.  the n o n - i n t e r a c t i o n d i f f r a c t i o n f i e l d s  the dashed c u r v e s  interaction  Q  are  the  Clearly  interaction  slit. incident f i e l d  has  a  s u b s t a n t i a l symmetric phase v a r i a t i o n a c r o s s the a p e r t u r e r e s u l t i n g i n a broad  main beam w i t h h i g h s h o u l d e r s .  beam d e f i n i t i o n improves.  As  the source becomes d i r e c t i v e ,  For moderate source d i r e c t i v i t y ( e . g . , kb=8)  the a p e r t u r e i l l u m i n a t i o n i s e s s e n t i a l l y G a u s s i a n and so i s the p a t t e r n . For  a  very  directive  source  (kb=85)  the  aperture  illumination  is  e s s e n t i a l l y plane wave and the d i f f r a c t i o n p a t t e r n i s very l i k e t h a t f o r plane  wave  illumination  of  a  slit.  F i g . 4.3  compares  results  from  - 44  Keller's  geometrical  d i f f r a c t i o n f o r kb=85. identical,  but  diffraction field term  the  singular  integral  of  The s i n g l y interaction  expansion  asymptotic  diffraction.  (1957,  F i g . 7) and beam  d i f f r a c t e d f i e l d p a t t e r n s a r e almost fields  differ.  Keller's  terms a r e o m i t t e d .  of t h e d i f f r a c t e d  These are l i m i t a t i o n s  field,  multiple  from t h e f i r s t  of t h e F r e s n e l i n t e g r a l ;  expansion  a t shadow boundaries  h e r e a t <t>=-90°.  diffraction  i s a summation of a l l f i e l d s r e s u l t i n g  I n an a s y m p t o t i c  Fresnel  theory  -  higher  order  I t i s also as i s e v i d e n t  o f t h e g e o m e t r i c a l t h e o r y of  -46-  -90 . 4.2  -45  0  45 *  -90  -45  0  45  90°  N o r m a l i z e d t o t a l f i e l d p a t t e r n o f a beam d i f f r a c t i o n by a s l i t , s i n g l e ( s o l i d ) and double (dashed) d i f f r a c t i o n s (ka=8=kr  , 9 =90° , 13=270°)  0.8-  0.6-  0.4-  0.2-  0.0  (b) P l a n e wave i n c i d e n c e ( K e l l e r , 1957) (a) L i m i t i n g beam i n c i d e n c e ( k r =8, kb=85) o F i g . 4.3 Comparison o f a plane wave and a l i m i t i n g beam d i f f r a c t i o n by a s l i t (ka=8) ( s o l i d ) s i n g l e d i f f r a c t i o n and (dashed) m u l t i p l e d i f f r a c t i o n  - 48 -  4.2  Beam D i f f r a c t i o n As  by a Wide C o n d u c t i n g  another a p p l i c a t i o n of l i n e  i s the l i n e source d i f f r a c t i o n line and  Strip  source d i f f r a c t i o n  by a h a l f - p l a n e  by a c o n d u c t i n g s t r i p .  source above and p a r a l l e l  F i g . 4.4 shows a  to a conducting s t r i p  i n t h e y=o p l a n e  |x|<a.  4.2.1  Far Field In  Calculation  the f a r f i e l d  of the s t r i p  ( r » r , a ) we use t h e a p p r o x i m a t i o n Q  g i v e n by ( 4 . 2 ) . The can  singly  be  diffracted  calculated  from  total  field  the t o t a l  h a l f - p l a n e f o r each o f t h e edges.  of a l i n e  field  of a  s o u r c e above a s t r i p line  This t o t a l f i e l d U  source  over  a  may be w r i t t e n  C  s = U l C r i . O j . ) + U ( r , e ) - [ U ( r , 0 ) + U ( r , 0 ) ] ; O<0<n  !  u  i  = U^r^©^ where x>-a at  2  2  + U (r ,0 ) 2  2  ; n<0<2n  2  (4.16)  i s t o t a l f i e l d o f t h e l i n e s o u r c e over a h a l f _ p l a n e a t y=0 and and U  i s t h e t o t a l f i e l d o f t h e same l i n e s o u r c e over a h a l f - p l a n e  2  y=0 and x<a.  and r , 0 2  2  U (r ,0 ) 1  1  1  and U ( r , © ) a r e g i v e n by (3.3) w i t h 2  -Jk[ e U  r  r i  _r  1  2  W h i l e U* and U  -jk[r-r e  sl  g  = -  to  the s i n g l y  are the i n c i d e n t  cos(0+0 )] g  (4.17) /kr"  /kT  (4.3),  r  ri,®i  U* i s g i v e n by (4.4) and  cos(0 +© )]  = -  simplified  2  replacing r,0, respectively.  and r e f l e c t e d f i e l d s , r e s p e c t i v e l y .  Using  r  2  diffracted  total  field  g i v e n by (4.16) can be  - 49 -  For O<0<u -j(kr-n74) TJ  =  s  j k a cos0 {e  1  j k r cos(e _0 (- e  )  1  1  8  1  8  1  F[-w  1  , /ixkr"  jkr jka cos0 j k r + e (e 2  S  s l  cos(0 0 ) 1 +  cos(0 -0 ) 2  7  /  jkr  s l  F[w ]  S Z  i 2  cos(0 +0  )  2  -e  F  t ]^ » ( * > w  4  18a  r2  and f o r it<0<2"n; -j(kr--n:/4) j k a c o s 0 j k r . c o s ( O i _ 0 ..) e {e (e F[w ] 1  t _ U  s  =  8 1  ± 1  ^ k 7  J sl " e k r  -s(0  1 +  0  s l  ) F[w ]J r l  j k a cos© j k r „ c o s ( 0 - 0 ) + e (e F[w. ] 2  2  S  /  9  S l  2  jkr - e  cos(0 +0 2  ) F[w ])}  S Z  r 2  (4.18b)  Where w^ and w^are t h e F r e s n e l i n t e g r a l arguments g i v e n by ( 4 . 6 ) . If  interaction  fields  of t h e s t r i p  s i m i l a r way as f o r t h e i n t e r a c t i o n they  edges  are calculated  f i e l d s of the s l i t  v a n i s h on t h e c o n d u c t i n g s t r i p .  Consequently  in a  i t i s found t h a t a new d i f f r a c t i o n  c o e f f i c i e n t i s r e q u i r e d ( e . g . Karp and K e l l e r , 1961) based on t h e normal derivative  of t h e d i f f r a c t e d  opposite.  These i n t e r a c t i o n  and so a r e o m i t t e d h e r e .  fields  i n the d i r e c t i o n  of t h e edge  f i e l d s a r e much weaker than f o r t h e s l i t  - 50 -  4.2.2  Numerical Results for the Strip The  field  pattern  shown i n F i g s .  (4.18) a f t e r n o r m a l i z a t i o n Fig.  4.5  line  and  4.6  are c a l c u l a t e d  t o the f i e l d on the peak of the  illustrates  omnidirectional  4.5  the development  of  from  pattern.  a beam s o l u t i o n  from  s o u r c e ( k b = 0 ) a t k r = 8 , Q^=n/2 above a s t r i p of Q  w i d t h 2ka=16, t o a beam s o u r c e (kb=12) p e r p e n d i c u l a r t o the s t r i p . total far field  i s maximum i n the i l l u m i n a t e d r e g i o n  reflected field  i s combined w i t h  The  diffracted fields  3%/2  add i n phase.  relative  maximum.  from  t h e two  As  kb  increases  directive  with  total fewer  field  i n the  sidelobes  from  0  to  12  illuminated  because  point  9=n/2 o r  a t 0=3TC/2 the t o t a l f i e l d i s a the  incident  beam  So t h e t o t a l  field  b e h i n d the s t r i p , w h i c h i s m a i n l y the d i f f r a c t e d f i e l d s The  from b o t h edges.  edges a t t h e f i e l d  I n t h e shadow r e g i o n  The  a t G=it/2 where t h e  the d i f f r a c t e d f i e l d s  becomes narrower and the edges a r e l e s s i l l u m i n a t e d .  decreases.  an  there  from t h e edges,  region  becomes  is little  more  interaction  between the r e f l e c t e d f i e l d and the d i f f r a c t e d f i e l d s . Now strip  i f the i n c i d e n t  beam o f kb=8 i s o f f the p e r p e n d i c u l a r t o t h e  by an a n g l e 6, as shown i n F i g . 4.6,  tilted.  As  6 increases  the t o t a l  field  the t o t a l f i e l d  pattern  Is t i l t e d  pattern  is  more t o t h e  same s i d e of the i n c i d e n t beam and i t becomes l a r g e r b e h i n d the s t r i p as shown. The  diffracted field  contribution  from the r i g h t edge i s s m a l l  the main c o n t r i b u t i o n i s from t h e l e f t edge and the i n c i d e n t f i e l d . diffracted field  from the l e f t  axis  a t the l e f t  i s directed  6=45°.  and The  edge i s a maximum when t h e i n c i d e n t beam edge as shown i n F i g . 4.6  f o r t h e case of  - 51 -  In the lower r i g h t graph of F i g . 4.6, to  2ka=160 as  a limit  pattern i s a t i l t e d  of a p l a n e  screen.  the s t r i p w i d t h i s i n c r e a s e d The  resulting  sharp beam making an a n g l e 135°  i s s i m p l y the r e f l e c t e d  field.  total  field  w i t h the s t r i p ,  and  -54-  - 55 -  CHAPTER V BEAM DIFFRACTION BY A CONDUCTING WEDGE  A p e r f e c t l y c o n d u c t i n g wedge of e x t e r i o r a n g l e n t i s i l l u m i n a t e d by a l i n e source a t r , 0 Q  p a r a l l e l t o t h e edge, as shown i n F i g . 5.1. F o r  q  t h i s c o n f i g u r a t i o n , l e t us have t h e f o l l o w i n g OO  limitations:  ;  (5.1a)  0<9<n7t  ;  (5.1b)  1.5<n_<2  or 1 2  °  2  > a  >0  ,  (5.1c)  W  where a i s t h e i n t e r i o r wedge a n g l e . w symmetry  Since  9 = nTt/2 i s t h e l i n e o f  o f t h e a b o v e c o n f i g u r a t i o n , t h e s o l u t i o n f o r 0<Q <rm/2 w i t h 9 Q  measured from t h e upper s u r f a c e , i s t h e same as t h a t f o r nit/2<9 <mt when o  0 i s measured  f r o m t h e l o w e r wedge  surface.  Therefore 9 = nTt/2 i s sym  c a l l e d a n g l e of symmetry.  5.1  Real Line Source Solution E x t e r i o r t o t h e wedge, t h e t o t a l f a r f i e l d U U* = U . S ( 9 - 9 ) + U* S ( 9 1  s i  Here  g r l  -9) + U  2  S(9_9  fc  g r 2  i s g i v e n as ) + U  d  (5.2)  S ( x ) i s t h e u n i t step f u n c t i o n , 9 9 and 9 a r e t h e shadow ' f » s i ' s r l sr2 v  boundary,  reflection  boundary  f o r t h e upper  boundary f o r t h e l o w e r s u r f a c e , r e s p e c t i v e l y .  surface  and r e f l e c t i o n  A l l t h e boundary  angles  a r e measured from the upper s u r f a c e o f the wedge and a r e g i v e n by 9  = it + 9  ;  (5.3a)  - 56 °srl " 9  sr2  ~ o  %  e  (  =  2 n  < ' >  5  ~ ) 1  5  " o >  7 t  3b  ( - >  0  5  i d. T r U/, U , U^, and U a r e t h e i n c i d e n t , 2  d i f f r a c t e d and r e f l e c t e d  3 c  fields  from the upper and lower s u r f a c e s , r e s p e c t i v e l y .  U  —jkR^" = /ix7T H ^ ^ k R ) = — ; 4dT  1  kR  1  jkr e  cos(0-e ) °  0  -jkr 1  ;  »  1  r »  1  (5.4a)  r  (5.4b)  /kT  /ON j ^ r cos(6+0 ) - j k r H< >(kR*) - -e ° e_  UJ - -  2  0  (5.5)  /kr  jkr^  cos[2nit -  (O+0J]- j k r  e  (5.6)  /kr The d i f f r a c t e d f i e l d i s -jkr U  d  = U* . D ( 0 , r , 0 ) JL— /kr Q  o  ,  (5.7)  where u* i s t h e i n c i d e n t f i e l d a t t h e edge of t h e wedge g i v e n by e  = /^72 H < -jkr * — — /kT  and  D  (  Q 0  >  r  Q  2 )  (kr )  (5.8a)  Q  Q  ; kr » 1  o  (5.8b)  °  , 0) i s the uniform d i f f r a c t i o n  coefficient  Kouyoumjian and Pathak ( 1 9 7 4 ) , w i t h some m o d i f i c a t i o n s , as  g i v e n by  - 57 -  -JTc/4 D(G  r  0) -  °  { [ c o t d i ) G(w ) - c o t ( T ) G(w )] 2  2n/2T  2  i  + [cot(T ) G(w ) - cotCL\) G( 3  3  Here  WL(  )]},  (5.9)  (  jw = 2j e w F[w]  jic/4 Real(we ) > 0  2  G(w)  jw  = -2j e  ;  (5.10a)  JTE/4  2  w F[-w]  ;  Real(we  ) < 0  (5.10b)  F[w] i s the F r e s n e l i n t e g r a l g i v e n by (3.2), and  w. , = - / 2 k T cos[  2TcnM* - (0 + 0 ) 1 —]  (5.11a)  2TC n M* - (0 + 0 )  w  - -» 2kr /i  3  4  TC -  T. . i  ,  =  z  TC +  T- , '*  =  J  0  cos[  (5.11b)  (5.12a)  (0 T 0 ) — 2n  (5.12b)  T  Hi  (5.11)  —]  (0 T 0 ) °_ 2n  T In  -  and M  2  are  integers  which most n e a r l y s a t i s f y  the  equations. - (0 T 9 ) = -TC  2TC n  - (0 T 0 ) = TC  2TC n  Here we have two cases, the  lower wedge s u r f a c e &  critical 0  cr  angle  =(n-l)Tt. v  '  (5.13a)  for  (5.13b)  depending on whether the reflected f i e l d from e x i s t s or does not, i . e . '  illumination  of  the  lower  0  > 0 , where the o < cr'  wedge  surface  is  - 58 -  Now the t o t a l f i e l d u  for  can be written as,  fc  O<0 <0 ; o cr  u  = u  C  (5.14)  + u  d  d , =  u  +  1  + u£  ,  "  u < 1 +  , = u  d  u  = 0  1  0 X0<mt si  tin <0<2it (5.16)  + ui  ,  0< < 9  > srl 0  1  + u  (5.15)  si  ;  l  + u  s r l  A  = 0  + u  0  srl<0<0 . ,  d  0  *  d = u  - u  C  °< <  i  u  and for 0 <0 <0 cr— o sym  u  ,  T 2  < 9 < Q  Q s r l  sr2  , ©  s r 2  ,  nix<0<2ix  (  5  '  1  7  )  < 0 <nn  5.2 Uniform Solution for a Beam Source To  get  the two-dimensional  beam solution  from the omnidrectional  l i n e source s o l u t i o n , we replace r , 0 by r , 0 i n a l l above equations o o s s except (5.1), (5.3), (5.13), (5.14) and (5.16). In (5.1), 0 i s l e f t as q  it  i s and i n (5.13), 0 i s replaced by R e a l ( 0 ) . o s replaced, respectively, by 0 . > mi sr2 <  '  (5.14) and (5.16) are  (5.18)  - 59 -  F i n a l l y (5.3) i s g i v e n by (3.13) and (3.14) and r e w r i t t e n here as 0 = 0 si  + [TC T c o s ( R / r ) o  ,  - 1  o  Q  srl  =  9  sr2  =  _ 0  o  ~  +  -  [ 7 t  ( n 7 t  c  " o 9  o  s  ) +  "  1  [  %  (  R  T  /  r  o  C O S  )  ]  ~  l  B o  s  +TC  ' o  ( R / r  v  ^  P  '  ) ]  (5.19)  P  ( 0  T  o  ( 0  + T t )  o  + T l )  »  (  »  (  5  5  ,  ,  2  2  0  1  /  )  )  and 0  where r  a r e g i v e n by (2.5) and (2.6) r e s p e c t i v e l y . s t h e edge o f t h e wedge l i e s on t h e beam a x i s , r  s If  and 0 s  are s  simplified to r  s  = r  o  and 0 = 0 , s o'  + jb J  v  and one o f t h e cotangent f u n c t i o n s reflection function,  boundaries,  coefficient solutions.  given  by  ( s e e Appendix  (5.9) i s always  by t h e c o r r e s p o n d i n g D).  G(w)  Hence t h e d i f f r a c t i o n  finite,  unlike  the asymptotic  When t h e beam a x i s does not pass through the wedge edge, a l l  the cotangent f u n c t i o n s  5*3  i n (5.9) i s s i n g u l a r on t h e shadow o r  but when m u l t i p l i e d  i t becomes f i n i t e  (5.22) '  are f i n i t e  everywhere.  N u m e r i c a l R e s u l t s f o r t h e Wedge I n a l l t h e f o l l o w i n g f i g u r e s the s o u r c e I s p a r a l l e l t o t h e edge and  at  a distance k r  6 = 0  Q  = 16.  A l s o t h e edge l i e s  on t h e beam a x i s ; i . e .  +TC.  o In F i g s .  c o n s t a n t a t a = 90° f o r w h i c h w  n=1.5.  0 c r =90° and t h e a n go l e of symmetry 0 J J Fig. incident  = (2-n)it i s kept w T h e r e f o r e the c r i t i c a l angle  5.2, 5.3 and 5.4 t h e wedge a n g l e  S  V  M  a  = 135°.  5.2 i l l u s t r a t e s how t h e n o r m a l i z e d t o t a l f i e l d v a r i e s when t h e field  on  a  right  angled  conducting  wedge  changes  from  - 60 -  o m n i d i r e c t i o n a l (kb=0) t o a d i r e c t i v e beam (kb=12). As kb i n c r e a s e s  two  beams a p p e a r , one  and  another  along  illuminated  along  the  by  the  shadow  the  reflection  boundary at © ^  boundary at 0 ^  = 240°.  g  incident  and  = 120°,  gr  reflected  In the  fields  considerable  c o n s t r u c t i v e and d e s t r u c t i v e i n t e r f e r e n c e between i n c i d e n t and fields  is  observed  when  kb=0.  d e c r e a s e s because i n c i d e n t and  As  kb  increases  region  this  reflected  Interference  r e f l e c t e d f i e l d s become d i r e c t i v e .  The  d i f f r a c t e d f i e l d does not change s i g n i f i c a n t l y because the edges l i e on the  = 60°  S i n c e the i n c i d e n t a n g l e 0  beam a x i s .  i s l e s s than 0  o t h e r e i s no r e f l e c t e d f i e l d f r o m the lower wedge s u r f a c e .  =90°, cr '  I n the shadow  r e g i o n (240 <9<270°) t h e r e i s o n l y a d i f f r a c t e d f i e l d , w h i c h v a n i s h e s o  on  the lower wedge s u r f a c e . 0  o  Fig. 5.3 i s s i m i l a r =120° i s g r e a t e r than 0  t o F i g . 5.2 except the angle of i n c i d e n c e so b o t h f a c e s of the wedge a r e i l l u m i n a t e d , cr  I n t h i s case the r e f l e c t e d f i e l d s from b o t h wedge s u r f a c e s c o n t r i b u t e t o the  total  diffracted  field.  Interference  f i e l d s o c c u r s i n O<0<0 . = 60° srl  between i n c i d e n t and all  the  region  diffracted  and  diffracted  exterior  to  f o r 0<6O° and  Incident,  and 27O°<0<0  reflected  _ = 240°, a l s o sr2 '  i n the r e g i o n 6O°<0<24O . O  wedge  is  i l l u m i n a t e d by  0>24O° r e f l e c t e d  fields,  and  Here  incident,  so t h e r e  is  no  When kb i s s u f f i c i e n t l y l a r g e , shadow  exist.  I n F i g . 5.4  the a n g l e  s y m m e t r y ; I.e. 0  q  = ©  = s v m  of i n c i d e n c e i s chosen t o e q u a l the a n g l e 135°.  v e r i f i c a t i o n of the v a l i d i t y this  fields  the  shadow r e g i o n s when kb i s s m a l l . r e g i o n s may  between  analysis.  The  The  of our  arrangement was equations  symmetry of the f i e l d  and  of  used as a p a r t i a l  computer programs i n  about 0=135° i s c l e a r .  When  - 61 -  kb G=9  i s sufficiently srl  =45°  large,  say kb=12, two d i r e c t i v e beams  and 0 = 225°. srz 0  These a r e t h e r e f l e c t e d f i e l d s from t h e  upper and lower wedge s u r f a c e s r e s p e c t i v e l y . say  kb=2, the I n t e r f e r e n c e  fields  a r e more  but  not z e r o ,  far field  i n the regions  wedge  O<0<45° and 225°<0<27O . O  i n t h e r e g i o n 45°<0<225° when kb i s s m a l l ,  i s due t o t h e d i f f r a c t e d f i e l d .  f i e l d when kb i s s u f f i c i e n t l y both  But when kb=0 o r i s s m a l l ,  between r e f l e c t e d , i n c i d e n t and d i f f r a c t e d  significant  Most of the t o t a l  surfaces  do  appear a t  A l l of i t i s d i f f r a c t e d  l a r g e , because t h e r e f l e c t e d f i e l d s  not c o n t r i b u t e  to the t o t a l  field  from  i n this  region. Fig.  5.5 shows how t h e t o t a l f i e l d s  f o r an o m n i d i r e c t i o n a l  source  (kb=0) and a beam s o u r c e (kb=4) change w i t h t h e i n t e r i o r wedge a n g l e a . w F o r 0 = 1 2 0 ° , t h e w e d g e a n g l e i s c h a n g e d f r o m a =90°(n=1.5) t o a o w half-plane beam  0^=0°(n=2), comparing t h e case o f h a l f - p l a n e s o l u t i o n t o t h e  diffraction  by  half  plane  solution  given  i n chapter  3,  gives  a n o t h e r check on t h e v a l i d i t y and a c c u r a c y o f our a n a l y s i s and computer programs. region  From F i g .  closer  significantly region angle. the  the  upper  affected with  can n o t i c e wedge  that  the t o t a l  surface;  i . e . 0<n7t/2,  do not change w i t h  I n t h e r e g i o n c l o s e r t o t h e lower s u r f a c e ; field  field  i n the i s not  t h e change o f wedge a n g l e because i n t h i s  t h e i n c i d e n t and r e f l e c t e d f i e l d s  total  angle,  to  5.5 we  i s noticeably  changed w i t h  t h e wedge  i . e . {Ill < 0 < n u ) , 2  t h e change  o f t h e wedge  because t h e r e f l e c t e d f i e l d changes s i g n i f i c a n t l y w i t h t h e change  of t h e wedge a n g l e .  - 67 -  CHAPTER VI BEAM DIFFRACTION BY A CIRCULAR APERTURE (NORMAL INCIDENCE)  Fig.  6.1  shows a c i r c u l a r  plane (xy-plane)  aperture  of  and centred at the o r i g i n .  point source l i e s on the aperture axis  radius a i n a conducting For normal incidence,  (z-axis).  the  Because of symmetry,  without loss of generality the problem can be treated as 2-dimensional.  6.1  Uniform Point-Source Solution For a point source located at a distance z from the o r i g i n and r * o o &  from the aperture edges making an angle 0  q  with the aperture plane, as  shown i n F i g . 6 . i , we have the following relations: ) r  o  Q  = /z  2  o  + a  (6.1)  2  = TC _ c o s ( a / r )  (6.2)  _ 1  Q  o  For an observation point i n the far f i e l d at r , Q from the o r i g i n , or at r^,  from one edge and r , 2  Q  2  ^  r  o  m  t  n  e  opposite  edge, we have  the  following approximations: r  . - r T a cos G ; r » a , I»^ R - r - z sinQ ; r»z . o o n  (6.3) (6.4)  Where R is the distance from the source to the observation point. ©  1  = 0  Q  2  = TC _ Q  (6.5a) ;  0<e<it  ;  (6.5b)  - 68 -  9  = 3n - 0 ;  2  0  (J)  = 1.5n +  6.1.1  ;  -1.5n<_<Kn/2.  ;  (6.5c)  Single Diffraction The  due  Tt<0<2n  incident  and r e f l e c t e d  fields  t o an i s o t r o p t i c p o i n t source  at a distant  conducting  u  < i  =  e  j  plane k  R  „  a t z=0 a r e u  jkz sin0 " J o e  k  r and u , r e s p e c t i v e l y ,  kR  K  a r e g i v e n by  r  •  — t — —  « e  point (r,0)  a t ( 0 , z ) and a c i r c u l a r a p e r t u r e i n i  a  field  r  ^  (6.6)  r  and -jkz U  r  =  "  sin0  -jkr  4 —  e  (6.7)  kr The  resultant  diffracted  component  of the f i e l d  by a curved  edge i s  g i v e n by K e l l e r (1957) as u  » u* . D ( 0 , r , 0) .  d  q  Q  where u * i s t h e i n c i d e n t  field  /p/r(r+p) e "  j k r  ,  (6.8)  a t t h e d i f f r a c t i n g edge  -jkr  o and  D ( 0 , r » 0) i s the d i f f r a c t i o n c o e f f i c i e n t g i v e n by Kouyoumjian and Q  Pathak (1974) as kr  D(0 G(w)  o  r o  0) =  i s g i v e n by  +jic/4  ' o J — - e N n  J  [G(w ) - G(w ) ] i r  (6.10)  - 69 -  jw  2  G(w) = - e The  F [±w]  (+) s i g n  illuminated  (6.11)  applies  region.  i n t h e shadow  region,  F[w] i s t h e F r e s n e l  complex arguments w.  and  integral  (-)  sign  i n the  g i v e n by (3.2) w i t h  g i v e n by  i»'  = -/2kT cos(_°.)  w  i,r  (6.12)  2  o  /p/r'(r'+p) i s t h e c u r v a t u r e  f a c t o r w i t h r ' as t h e d i s t a n c e from t h e  d i f f r a c t i n g edge t o t h e o b s e r v a t i o n p o i n t ; i . e . r ^ , r For normal i n c i d e n c e , K e l l e r ( 1 9 5 7 )  2  i n our c a s e .  showed  p = a/cos© and  this  holds  (6.13)  also f o r a point  s o u r c e on t h e c i r c u l a r a p e r t u r e  axis.  S u b s t i t u t i n g f o r p and r ' i n t h e c u r v a t u r e f a c t o r and w i t h t h e f a r f i e l d a p p r o x i m a t i o n s i n (6.3) one c a n show t h a t /p/r^r^f-p) = I  2 2  /a/cos©!  (6.14)  2  r  Where r i s measured from t h e o r i g i n and © i , g i v e n by ( 6 . 5 ) . 2  The s i n g l y the  diffracted  given  d i f f r a c t e d component of t h e f a r f i e l d U field  component  d  i s t h e sum o f  by one edge and i t s o p p o s i t e w h i c h a r e  by ( 6 . 8 ) . iTc/4  Tid , U - ka e  ^ i r ie U I—i  k  D  T  T  °  s  a  cos&  ~'"-^)  -j(kacos©-Tt/4) - j k r + Do e ie £ J A 7\ kr vka c o s 9  ,,, v (6.15) 1 C  where D  l»2  E  D  (© > 0  r 0  »  Q  are g i v e n by (6.10).  l»2>  (6.16)  - 70 -  On t h e a x i a l c a u s t i c ; i . e . 9 = — o r — , 2 2 this axis  the f i e l d  i s inaccurate.  (6.15) i s s i n g u l a r , so near  The s i n g l y d i f f r a c t e d f a r f i e l d i n  (6.15) can be r e w r i t t e n as U  - ka  d  U  1  [ ( D + D ) c o s ( k a cos9 -TC/4) /ka" cos9 X  2  s i n ( k a cos9 -TC/4) + j ( D l  -D ) 2  1 ] ±-  #  /ka cos9  17)  kr  9 •> TC/2 o r 3TC/2, we use t h e a s y m p t o t i c e x p a n s i o n  As k a c o s 9 •* 0, i . e . , of  *  (6  the B e s s e l f u n c t i o n s  c o s ( k a c o s 9 -TC/4) „ /—ro \ J: = /TC/2 J ( k a cos9) (6.18a) •ka cos0 o s i n ( k a COS9-TC/4) _ r-pr . . _ . — - /TC/2 J ^ ( k a c o s 9 ) (6.18b) /ka cos9 Here J and J i a r e B e s s e l f u n c t i o n s o f t h e f i r s t k i n d , o f o r d e r z e r o and o T  N  n o  n  1  one, r e s p e c t i v e l y . ,  U  I  «  g  J—  S u b s t i t u t i n g (6.18) i n (6.17) g i v e s  jtt/4 JTC/4 e  ka U  1  [(0^2)  J (kacos9) Q  -jkr +j(D -D ).J ((kacos9)] 1  2  1  1_ kr  6.1.2  *  19)  Multiple Diffraction Solution The  diffracted  field  component a t ( r , 9 ) due t o a p o i n t s o u r c e a t  ( r , 9 ) f r o m a c u r v e d edge i s g i v e n by ( 6 . 8 ) . Q  (6  Q  To s i m p l i f y t h e a n a l y s i s  we r e w r i t e (6.8) as f o l l o w s : U  d  = U  1  o  D(9 , r , 9', r ' ) o* o' '  where r ' , 9' a r e o b s e r v a t i o n p o i n t p o l a r c o o r d i n a t e s measured edge and  (6.20) from t h e  - 71 -  -jkr' D(0  r ° °  Here  0', r ' ) = D(0 r ° °  D(® >  r  Q  Q  , ©')  i  (1981), field,  k  given  s  c e n t r e of t h e c u r v e d  0') / j/+cos0  edge.  the m u l t i p l y  a  diffracted  field,  excluding  single  diffracted  D  1 0  +  )  i U  ,  (6.22)  , j ( 2 k a - .n/4) = D(TC, 2a, TC, 2a) = /4/TC e F|/4kaJ  (6.23)  +(D  D  (6.21)  )  z u  Q  Q  ; 0 > U. < 2  F o l l o w i n g t h e same p r o c e d u r e g i v e n by J u l l  ,d 1 „i U = U D [(D + D D ™ + D m o o o o  D  /kr.kr'  by ( 6 . 1 0 ) and r , 0 a r e measured from t h e  can be w r i t t e n as  where D  6  ,D^,D  1  0  +  Q  D  Q  D  1  0  +D  2  D  2  )]  0  and D Q a r e g i v e n as f o l l o w i n g : 2  D(0 , r ,  TC, 2a) o' _j(2ka+n/4) j k r ( l - c o s 0 ) = / 2 r /ita e e ° ° F[/2kr~ s i n ( 0 / 2 ) ] , =  1  o  (6.24)  and  D^g  =  D  = T Tt(2ka/n)  1 0  20  2a, 0 , r ^ )  D(Tt,  L  j2ka Tj(kacos0-Tt/4) -jkr e e F[/4ka~ s i n ( 0 , / 2 ) ] f _ . 2 kr /ka cos0 (6.25)  3 / 2  Now (6.22) can be summed, p r o v i d e d D *1» Q  „d .  i  B  " Again  l (D  °  1 0  +  D ) 2 0  <!-».)  near the a x i a l c a u s t i c ;  modified before  D  as f o r s i n g l e  10  D  2 0  as  i . e . 0 = TC/2 o r 3TC/2, t h e f i e l d  diffraction.  t o get the m u l t i p l y  terms o f D ,  giving  Following  must be  the procedures  d i f f r a c t e d component, r e w r i t e  D^g,  D  given 2 0  in  - 72 -  T j ( k a cosO-Tt/4) - j k r Din = + D' 20  -  ,  6  28  /ka cosG  (6-27)  kr  where j2ka D'  = n(2ka/n)  F [ / 4 k a s i n ( e / 2 ) ].  e  3 / 2  28 Then D  + 20 D  1 0  c  a  ^  n  simplified to  e  ,  10  D  D  10 20 = ~[ +D  r,  (6.28)  1  2  '  D  ,  ~ 20 D  =  ' .cos(ka  = -L( 10~ 20^ D  - j ( k a cos9-Tt/4) e  COSG-TC/4)  /  /ka cos©  /ka cosO . D ' +D  1  j(ka  COSO-TC/4)  -jkr  e  1  (- ) 6  kr . s i n ( k a COS©-TC/4)-I e ^  -J( 10 20^  -==  /kacos©  29  k r  J  o r > N  (o•Ju)  S u b s t i t u t i n g from (6.18) i n (6.30) g i v e s t  1  10  D  +  20  D  [( 10 20) ,) ( :  =  n  _D  J  k a  n  cos  ©)  - j ( D i + D ) J ( k a cos©)] e 0  2 0  j k r  x  (6.31) Substitute  f o r (6.31) i n (6.26) t o get t h e m u l t i p l y  diffracted  field  component near the a x i a l c a u s t i c as .  D*  IT = S^TT— m  (D -1) o  ,  ,  ,  ,  -jkr  [ ( D - D ) J ( k a cos©) - j ( D + D ) J , (ka cos©)] 1 ° 1 0  2 0  1 0  2 0  1  k  r  (6.32)  The t o t a l f i e l d , i n c l u d i n g s i n g l e d i f f r a c t i o n o n l y , i s U  C  = U  d  s  + U  1  S ( 0 -0) + U s i  r  S(0 -0) sr  (6.33a)  and i n c l u d i n g m u l t i p l e d i f f r a c t i o n i s U* = U  d  + U  d  + U  1  S(0 -©) + U gi  r  S(9 -0), g r  (6.33b)  - 73 -  where S ( x ) i s t h e p o s i t i v e u n i t s t e p f u n c t i o n .  U , U , U s  the  incident,  reflected,  singly  d i f f r a c t e d and m u l t i p l y  and U a r e m  diffracted far  fields.  6.2  Uniform Beam Solution For normal i n c i d e n c e t h e beam a x i s i s p e r p e n d i c u l a r t o t h e a p e r t u r e  p l a n e and c o i n c i d e s  6.2.1  with  the a p e r t u r e a x i s .  Far Field Calculation To  change  beam s o l u t i o n ,  from an i s o t r o p i c p o i n t complex v a l u e s  source s o l u t i o n  appropriate  t o t h e beam w i d t h  d i r e c t i o n are given to the source coordinates. °  to a d i r e c t i v e  Then z , r and 0 0' o o  become complex and a r e c a l l e d z , r and 0 , r e s p e c t i v e l y . s s s z = Z - j b sin8 ; b>0, 0<8<2TC g  r 0  q  = Vz  1  s s  s  + a  ; Real(r  2  = TC - c o s "  1  and beam  ) > 0 s ~~  (6.34) (6.35)  (a/r) s'  (6.36)  Where Z , b and 8 a r e r e a l v a l u e s , d e f i n i n g t h e source l o c a t i o n , beam q  w i d t h and beam d i r e c t i o n , r e s p e c t i v e l y . z , r and 0 , r e s p e c t i v e l y s s s  Z , r ^ and 0 q  q  by  i n (6.1 t o 6.33) we get t h e beam s o l u t i o n  f o r a l l above c a s e s , s i n g l e and m u l t i p l e the a x i a l c a u s t i c .  By r e p l a c i n g  diffraction,  near o r f a r from  - 74 -  6.2.2  Shadow and R e f l e c t i o n Boundary C a l c u l a t i o n s Since the d i f f r a c t i o n phenomena i s  reflection edge at  l o c a l , we assume the shadow and  boundaries for a curved edge are the same as for a straight  the point  of d i f f r a c t i o n .  So the results  given i n (3.13) and  (3.14) are v a l i d here. The shadow and illuminated regions mentioned i n (6.11) and Appendix D are given as, for shadow region: Real(w  ) - Imag(w  ) > 0  (6.37)  ) < 0  (6.38)  and for illuminated region: Real(w where w I»  r  ) - Imag(w  are the Fresnel i n t e g r a l arguments given as 6 T 8  v . i ,r  - / 2 k F cos( s  !-)  (6.39)  2  Also the shadow and illuminated regions for incident or reflected can be defined, respectively,  by  0^0 or 0 < si sr where 0  6.3  s j-  and 0  o  (6.40) '  are given by (3.13) and (3.14),  respectively,  Numerical Results A point  kz  sr  fields  source  on  the  c i r c u l a r aperture  axis  is  at  a  distance  = 3it from the centre of the aperture which is of a radius ka=3iu  the following figures,  the horizontal a x i s ,  <> t i n degrees,  measured from the aperture axis as shown i n F i g . 6.1,  In  i s the angle  and the v e r t i c a l  - 75 -  axis  i s the  normalized  far total  field  pattern,  including  single  d i f f r a c t i o n o n l y o r i n c l u d i n g s i n g l e and m u l t i p l e d i f f r a c t i o n . In  F i g . 6.2  calculated axial the  t h e dashed c u r v e s  The s o l i d  axial caustic, how  the  curves  calculated  total  represent  from  far field  m o d i f i e d near t h e a x i a l c a u s t i c is  t h e non-modified  solution  from ( 6 . 2 5 ) , (6.26) and (6.33b) which i s v a l i d f a r from t h e  caustic.  shows  represent  t h e l i m i t i n g case  the modified  (6.32) and  including (z-axis)  of a p o i n t  source  solution  (6.33b).  multiple  This  near figure  diffraction, is  f o r two d i f f e r e n t c a s e s . (kb=0) and  the other  One is a  d i r e c t i v e beam (kb=8). Fig. point  6.3 i l l u s t r a t e s t h e development of t h e beam s o l u t i o n from t h e  source  plane  (kb=0) t o a d i r e c t i v e  wave ( k b = 8 5 ) .  between t h e i n c i d e n t  beam  (kb=16),  and an e s s e n t i a l l y  When kb I s s m a l l c o m p a r e d t o k z  interaction  Q  and d i f f r a c t e d f i e l d s i n t h e i l l u m i n a t e d  e v i d e n t because t h e a p e r t u r e edge i s s t r o n g l y the d i f f r a c t e d f i e l d  i s significant.  illuminated,  region i s  consequently  As kb i n c r e a s e s , e.g. kb=16, t h e  i n c i d e n t beam becomes narrower and t h e edge i s weakly i l l u m i n a t e d so t h e d i f f r a c t e d f i e l d i s i n s i g n i f i c a n t and t h e i n t e r a c t i o n d e c r e a s e s .  In the  shadow r e g i o n t h e i n t e r a c t i o n o c c u r s between t h e d i f f r a c t e d f i e l d s  from  the  This  two  opposite  diffracted  points  on  the  aperture  edge.  i n t e r a c t i o n i s s i g n i f i c a n t when kb s m a l l o r when b » z , eg. kb=85 where Q  the f i e l d  incident  the  i s strongly  edge  curves  represent  on t h e a p e r t u r e becomes u n i f o r m , illuminated  again.  In this  l i k e a plane wave, figure  t h e dashed  t h e s i n g l e d i f f r a c t i o n s o l u t i o n g i v e n by ( 6 . 3 3 a ) , and  - 76 -  the  solid  curves  include  for  multiple  diffraction  solution  given  by  (6.33b). For this choice of aperture radius (ka=3Tc), the singly and multiply diffracted <()=90 , o  the  satisfying  fields  are very much the same except at the conductor,  multiply the  diffracted  boundary  condition  singly diffracted f i e l d does not.  field of  vanishes a  perfect  on  the  conductor  i.e.  conductor, while  the  Field Point Fig.  6.1  Geometry of a complex p o i n t s o u r c e d i f f r a c t i o n by a c i r c u l a r aperture  F i g . 6.2  N o r m a l i z e d t o t a l f i e l d p a t t e r n of a p o i n t s o u r c e (kb=0) and moderate beam (kb=8) ( s o l i d ) m o d i f i e d and (dashed) non-modified on the c a u s t i c a x i s (Normal i n c i d e n c e )  Fig.  6.3  Single (solid) and multiple (dashed) t o t a l patterns of a beam d i f f r a c t i o n by a c i r c u l a r aperture (Normal incidence)  - 80 -  CHAPTER VII SUMMARY, CONCLUSIONS AND RECOMMENDATIONS  7.1 Summary The  complex source  p o i n t method was used t o r e p r e s e n t  beam w h i c h i s G a u s s i a n i n t h e p a r a x i a l r e g i o n . omnidirectional become  solutions  diffraction  Numerical  developed  f o r directive  between results  incidence  were  and e q u i v a l e n t  interaction  every  sources  t h e edges including  (kb •*• ) and l i n e  analytically to  The g e o m e t r i c a l  currents  of t h e s l i t the l i m i t i n g  or p o i n t sources  co  case s t u d i e d .  line  Uniform s o l u t i o n s f o r  and extended  beams.  a directive  were  used  theory  to include  and c i r c u l a r cases;  of  aperture.  e.g. p l a n e  wave  (kb = 0 ) , were g i v e n f o r  A l s o comparisons w i t h e x i s t i n g s o l u t i o n s were made  wherever p o s s i b l e . I n Chapter I I , a d i r e c t i v e and  compared  expression  with  a Gaussian  f o r the half-power  beam was d e r i v e d beam  and a t y p i c a l  beam w i d t h  The Chapter used  antenna  was d e r i v e d ,  d i s c u s s i o n o f t h e use of m u l t i p l e complex source complicated  i n polar  coordinates beam.  and a  An  simple  p o i n t s t o d e r i v e more  beams was g i v e n .  s o l u t i o n o f beam I I I from a s i m p l e  t o solve  diffraction  s o l u t i o n exact  t h e problem  complementary s t r i p .  by a h a l f - s c r e e n , d e r i v e d i n  o f beam  Also a convenient,  i n the f a r f i e l d  diffraction  by wide  l i m i t , was slit  and  s i m p l e f o r m u l a was d e r i v e d f o r  the l o c a t i o n of the shadow b o u n d a r i e s of a s t r a i g h t edge.  - 81 -  Beam d i f f r a c t i o n by a wedge with  its  edge on the  analysed using the uniform theory of d i f f r a c t i o n . completes the  asymptotic  solution,  Felsen (1976), whose solution i s inaccurate i n the  for the  beam axis was  This uniform solution  same problem, mentioned by  i n f i n i t e on the shadow boundaries and  t r a n s i t i o n regions.  Also the shadow boundaries are  given here for any beam o r i e n t a t i o n . Finally, normally  the d i f f r a c t i o n by c i r c u l a r aperture when illuminated by  incident  beam,  was  derived  using  d i f f r a c t i o n and along the a x i a l c a u s t i c , remove  the  singularity.  Multiple  the  uniform  theory  of  Bessel functions were used  diffractions  were considered  to  and a  closed form expression was derived.  7*2  Conclusions The  method,  beam derived can represent  function  especially  in  Chapter II  using  the  complex  a t y p i c a l antenna beam better for  wide  antenna  beams  source  point  than the Gaussian  (small  kb).  When  the  imaginary part (b) of the complex source position vector (F =r" -jF) i s Q  very large compared to the r e a l part ( r ) , i . e . Q  b»r ,  the beam tends to  Q  a plane wave. Other problems  authors  in  contribution  the of  assume  very  narrow  paraxial region. the  and  This kind of  diffracted f i e l d  passes through the d i f f r a c t i n g edge.  beams  negligible,  solve  diffraction  assumption makes  the  unless the beam axis  In our analysis  this  assumption  was removed and the range of v a l i d i t y was increased to cover the whole region of  interest.  - 82 -  The synthesis of more complicated beams such as one with or a nearly square beam, was given i n section  (2.3)  complex  sidelobe  source  resultant  points.  beam are  coordinates.  But  yet  to  The study  be  of  the  width  and  calculated  sidelobes  by using multiple  and related  level to  of  the  simulating any beam i n terms  the  complex  of Gaussian  beams and Complex Source Points, given by Mantica et a l (1986), was not rigorous and some assumptions were made to simplify the analysis. Our  solution  of  beam  diffraction  by  half-screen  is  accurate,  uniform everywhere and v a l i d for a l l beam orientations and widths. solution  can  be  used  as  a reference  solution  for  other  This  uniform  or  asymptotic solutions which are inaccurate i n the t r a n s i t i o n regions and i n f i n i t e on the shadow boundaries. The  limiting  case  kb  =  0  of  our  illuminated by omnidirectional source, solutions  solution  is  to  the  strip  when  in very good agreement  with  of l i n e source d i f f r a c t i o n by a s t r i p , given by Vankoughnett  and Wong (1981) and by Shafai and Elmoazzen (1972). have shown,  by comparison with numerical r e s u l t s ,  theory of d i f f r a c t i o n yields  satisfactory  results  Tsai et a l . (1972) that the geometrical for r e f l e c t o r  widths  as small as 0.2X (wave length) when double d i f f r a c t i o n i s included. our choice of s t r i p width 2.5X., single d i f f r a c t i o n i s s u f f i c i e n t .  For Since  the contribution to the d i f f r a c t e d f i e l d for d i r e c t i v e sources is always less  than  or  equal  (for  omnidirectional source,  the  edge  on the  beam axis)  to  that  of  an  the accuracy for d i r e c t i v e beams is at least as  high as that for omnidirectional sources.  - 83 -  In  a l l cases  diffracting  s t u d i e d here  edge.  Solutions  the i n c i d e n t f i e l d can  be  extended  was to  normal t o t h e  include  oblique  i n c i d e n c e on a s t r a i g h t edge or wedge. The o r d i n a r y UTD was used i n s o l v i n g the problems of the wedge and c i r c u l a r aperture. slope  For b e t t e r accuracy  diffraction  (Kouyoumjian  one may use t h e UTD augmented by  e t a l . , 1981) o r the improved  version  (Buyukdhura and Kouyoumjian, 1985), i n s t e a d . The d i f f r a c t i o n of a beam by 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 w i t h an edge was Ghione  also  considered  ( s e e appendix E) b e f o r e  e t a l . , (1984) had p u b l i s h e d  However,  the  diffracted  approximations, Appendix suggest  E.  field  their  and  we were aware  solution  reflected  to t h i s field,  The  problem  was  further investigation  not pursued i s needed  problem.  with  w i t h o u t u s i n g t h e computer s e a r c h t e c h n i q u e  some  i s given i n  f u r t h e r , although  to c l a r i f y  that  as  and s i m p l i f y  they the  method. The  uniform  theory  of  diffraction  s o l u t i o n s where t h e r e were no s i m p l e wedge and c i r c u l a r a p e r t u r e .  was  exact  used  to obtain  uniform  s o l u t i o n s , such as f o r the  Otherwise r i g o r o u s l y c o r r e c t s o l u t i o n s at  h i g h f r e q u e n c i e s f o r f a r s i n g l y d i f f r a c t e d f i e l d s were used, such as f o r the  half-screen, s l i t  above  cases  are  uniform,  t r a n s i t i o n through For this  electromagnetic wedge. fields  A l l the s o l u t i o n s o b t a i n e d  for Fresnel  integrals  provide  f o r the a  smooth  shadow and r e f l e c t i o n boundary r e g i o n s .  simplicity  thesis.  and s t r i p .  scalar (acoustic) fields  The  results  apply  were assumed through  directly  to  two-dimensional  f i e l d s i n the case of the h a l f - p l a n e , s l i t  F o r the c i r c u l a r  aperture  extension  to vector  can be made by c o n s i d e r i n g t h e s c a l a r f i e l d  out  or s t r i p and  electromagnetic  as one component of  - 84 -  the v e c t o r f i e l d  o r as a s c a l a r p o t e n t i a l from which v e c t o r f i e l d s a r e  derived.  7.3  Recommendations for Future Work So f a r we have d e a l t w i t h problems t h a t assume p e r f e c t c o n d u c t o r s ,  s i m p l e beams and normal i n c i d e n c e , t o g e n e r a l i z e t h e i n c i d e n t beam and the r e f l e c t o r s boundary c o n d i t i o n s the f o l l o w i n g may be c o n s i d e r e d : i)  To make t h e a n a l y s i s by t h e complex r a y t r a c i n g method more  complete,  especially  criterion;  i . e . , one w h i c h  descents  paths  two-dimensional ii)  used  by  does  surfaces,  not r e q u i r e  Ghione  et  a  general  t h e study  a l . (1984),  a-priori  of  steepest  i s needed f o r  diffraction.  Diffraction  complicated multiple  f o r non-planar  by  simple  shapes  beams w i t h s i d e l o b e s .  complex source  when  illuminated  by  more  When these beams a r e r e p r e s e n t e d by  p o i n t s , s o l u t i o n s may e a s i l y be o b t a i n e d  using  the s u p e r p o s i t i o n p r i n c i p l e . i i i ) The problem o f beam d i f f r a c t i o n edge  does  not l i e on t h e beam  asymptotic  solution  diffraction The  solution  by  Felsen  by a c o n d u c t i n g for real  source  axis, (1976),  curved  by s t r a i g h t wedge where t h e  using  t h e UTD  i s y e t t o be  wedge has not been  diffraction  t o assess the done.  Also  studied yet.  by a curved wedge by Lee and  Deschamps (1976) o r Deschamps (1985) may be used. iv) aperture,  A l l existing  s o l u t i o n s f o r beam  assume s y m m e t r i c a l  w i t h the aperture a x i s .  diffraction  by a  circular  i n c i d e n c e ; i . e . the beam a x i s c o i n c i d e s  The more g e n e r a l non-symmetrical  i n c i d e n c e case  w i t h t h e beam a x i s s h i f t e d from the a p e r t u r e a x i s by some d i s t a n c e o r a t some a n g l e , a p p a r e n t l y has not been r e p o r t e d y e t .  - 85 -  v) rigorous  To cover a wide simulation  of  range  an arbitrary  points needs to be derived. on assumptions and experience; vi)  Solutions  half-plane,  to  of  beam  problems, beam i n  using the CSP method, a terms  of  complex  source  What exists i n the l i t e r a t u r e now is based i.e.  t r i a l and error.  diffraction  by  simple  shapes  such  as  s t r i p and wedge, under impedance boundary conditions may be  obtained using the corresponding solutions  for omnidirectional  by Bucci and Franceschetti (1976), and Tiberio et  al.  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SIEGMAN, A.E., "An i n t r o d u c t i o n t o l a s e r s and masers", ( M c G r a w - H i l l , New York, 1971), c h . V I I I .  - 90 -  TAKENAKA, T., KAKEYA, M. and FUKUMITSU, 0. (1980): "Asymptotic r e p r e s e n t a t i o n o f t h e boundary d i f f r a c t i o n wave f o r a G a u s s i a n beam i n c i d e n t on a c i r c u l a r a p e r t u r e " , J . Opt. S o c Am., V o l . 70, No. 11, pp. 1323-1328. TAKENAKA, T. and FUKUMITSU, 0. (1982): "Asymptotic r e p r e s e n t a t i o n of the b o u n d a r y - d i f f r a c t i o n wave f o r a t h r e e - d i m e n s i o n a l Gaussian beam i n c i d e n t upon a K i r c h h o f f h a l f - s c r e e n " , J . Opt. Soc. Am. V o l . 72, No. 3, pp. 331-336. TIBERI0, R., BESSI, F., MANARA, G. and PEL0SI, G. (1982): "Scattering by a s t r i p w i t h f a c e impedances a t edge on i n c i d e n c e " , Radio S c i . , Vol. 17, pp. 1199-1210. TIBERI0, R., PEL0SI, G. and MANARA, G. (1985): "a u n i f o r m GTD f o r m u l a t i o n f o r t h e d i f f r a c t i o n by a wedge w i t h impedance f a c e s " , IEEE T r a n s . A n t . Prop., V o l . AP-33, No. 8, pp. 867-873. TSAI, L . L . , WILTON, R.D., HARRISON, M.G. and WRIGHT, E.H. (1972): "A comparison o f g e o m e t r i c a l t h e o r y of d i f f r a c i t o n and i n t e g r a l e q u a t i o n f o r m u l a t i o n f o r a n a l y s i s of r e f l e c t o r antennas", IEEE Trans. A n t . Prop., V o l . AP-20, No. 6, pp. 705-712. WANG, W.Y.D. and DESCHAMPS, G.A. (1974): t r a c i n g t o s c a t t e r i n g problems", P r o c 1541-1551.  " A p p l i c a t i o n of complex r a y IEEE, V o l . 62, No. 11, pp.  WILLIAMS, C S . (1973), "Gaussian beam f o r m u l a s from d i f f r a c t i o n t h e o r y " , J . App. Opt. V o l . 12, No. 4, pp. 872-876.  - 91 -  APPENDIX A THE FRESNEL INTEGRAL WITH A COMPLEX ARGUMENT  A.I  E v a l u a t i o n o f F r e s n e l I n t e g r a l s from E r r o r F u n c t i o n s For  a real  o r complex argument  w, F r e s n e l  I n t e g r a l F[w] i s  d e f i n e d as oo  F[w] = / w  - J T  2  e  dx  (A.1)  By changing v a r i a b l e s  ^ -JTC/4  -jit/4 T  = y e  F[w] - i  ,  /  j l l / 4  dx = e  i  y  dy  (A.2)  dy  (A.3)  JTC/4  we The complementary e r r o r f u n c t i o n i s d e f i n e d as  ,  2 erfc(v) = -±  -Y  J v  e  2  dy = 1 - e r f ( v )  y  (A.4)  where v can be r e a l o r complex and e r f ( v ) i s t h e e r r o r o  erf(v) = _£ /it  v /  function,  2  i  y  dy  (A.5)  Q  jit/4  From (A.3) and (A.4) w i t h v = w e -JTC/4  F[w] =/iT/2 e Subroutines  for  jit/4  erfc(we  we can w r i t e  )  complementary  (A.6) error  functions  with  complex  arguments a r e a v a i l a b l e i n UBC Computing Centre L i b r a r y .  A.2  Some P r o p e r t i e s o f F r e s n e l i)  Integrals:  Symmetry r e l a t i o n  F[w] + F[-w] = ZiTsJ /^ 11  (A.7)  - 92 -  ii)  Special values  -J*M  F [ - » ] = /TT e  ,  -I  F[0] = ± F [ - » ] and F[<=°] = 0 2  (A.8)  i i i ) Asymptotic expansion. F[w] ~ S(-w) + F[w]  ;  |w| -»• »  ,  (A.9)  where S(x) is the unit step function, and . -jw  2  00  F[w] = I i - ! I r(n + l / 2 ) ( - j w ) " 2w/ix n=0 where T(x) is the gamma function. T(n + 1/2)  = /TT ( l / 2 ) ( 3 / 2 )  n  (n - 1/2)  (A.10)  (A.11)  - 93 -  APPENDIX B CALCULATION OF REAL AND IMAGINARY PARTS OF r  B.l  Analysis U s i n g (2.5) w h i c h i s r e w r i t t e n here as r  = [r  + 2 r (-.1b) cos(B-0 ) + ( - j b ) ] S O O \j ' L e t us w r i t e r as s 2  2  r = R - j I  ;  1 / 2  ; R e ( r )>0  (B.l)  o  R > 0  Where R and I a r e r e a l .  (B.2)  By s q u a r i n g  ( B . l ) and (B.2) and e q u a t i n g t h e  r e a l p a r t s and t h e i m a g i n a r y p a r t s , r e s p e c t i v e l y , we get R  2  - I  2  = r - b o 2  (B.3)  2  and R.I  = r b cos(B-0 ) o o  (B.4)  S o l v i n g (B.3) and (B.4) f o r R and I g i v e s r  2 _ 2  R = [(-2  b  )+ I / (r 2  2  - b ) + [ 2 r b cos(B-0 ) ] 2  2  2  ]  1  /  2 ;  (B.5)  and r b I =— cos(B - 0 ) R 0  From (B.6) we n o t i c e I > 0 <  (B.6)  that:  i f |8 - 0 | < TC/2. o >  (B.7)  - 94  -  B.2 Special Cases: i)  b=0  ii)  b=r  Q  gives  R=r and 1=0  ;  gives  R=r V | c o s ( B - e )|,  Q  (B.8)  Q  (B.9)  and I = +R  IB-0 I > n/2  ;  '  i i i ) B = 0 or 0 + it ' o o  o s 1  gives R = r ° o  and I = ^  iv)  < - > B  10  |B - 0 | « u/2 gives q  R = R =  R -  Vr - b o 2  0  0  2  and  I = 0  if  r >b ; o '  and  I = 0  if  r  and  I = ^ b -r i f o 2  2  = b  o r <b o  ;  (B.ll)  APPENDIX C GAUSSIAN BEAM DIFFRACTION BY HALF-SCREEN (ASYMPTOTIC SOLUTION)  The a s y m p t o t i c s o l u t i o n g i v e n by Green e t a l . (1979) f o r a G a u s s i a n beam d i f f r a c t i o n in  coordinates  by a h a l f - s c r e e n i s summarized here w i t h and n o t a t i o n  f o r comparison w i t h  By r e p l a c i n g 9 ' ,  C h a p t e r 3.  9,  p',  some changes  the s o l u t i o n g i v e n i n  p and E by (- - 9 ) , ( — - 9 ) , 2 2  r ,  S  r  and U, r e s p e c t i v e l y , we  d i f f r a c t e d U and t o t a l U d  U  1  fc  can w r i t e  the i n c i d e n t U , 1  reflected  S  U , r  f a r f i e l d s as  cos(9-G ) - j k r e  jkr - e  S  S  (C.l)  /kr  U  r  j k r cos(9+9 ) - j k r = - e e S  (C.2)  S  /kr  -j(kr -37 /4) s  Q  t  _  0+9  Q  [sec( 2/2rtkr  -  j  k  r  ') ( 1 - A ) - e c ( - - i ) ( 1 - A ~ ) ] e +  S  s  9  "  vkr A  where  (C.3) + _. A = —14kr  U  1  = U  d  ,OT0 sec ( — — ) 1  r  ; r »  + U . S ( 9 - 0) + U . S ( 0 1  |r |  (C.4)  - 0)  (C.5)  s  r  s i  g r  where S ( x ) i s the u n i t s t e p f u n c t i o n , and 0 . and 0 a r e the shadow and si sr  - 96 -  reflection  boundaries  measured  from  the  illuminated  side  of t h e  half-screen.  -lr  0  =  it  S  0  sr  =  ^Mlm(9 )] . Re(r ) s  + Re(0 ) + tan [  2Tt -  0  ^  s  Z  Z  ]  ^  | r j + I m ( r ) . cosh[Im(© )] s  g  s  (C7) '  ,  si  where Re(x) and Im(x) a r e t h e r e a l and i m a g i n a r y p a r t s .  The a c c u r a c y of  + the above s o l u t i o n depends on how s m a l l i s | A | compared t o 1.  - 97 -  APPENDIX D THE SINGULARITY CANCELLATION IN THE WEDGE DIFFRACTION COEFFICIENT  One of the cotangent f u n c t i o n s of the d i f f r a c t i o n c o e f f i c i e n t by (5.9) i s s i n g u l a r on t h e shadow o r r e f l e c t i o n b o u n d a r i e s , time the c o r r e s p o n d i n g is  finite  G(w) f u n c t i o n i s z e r o .  everywhere.  Therefore  So t h e term  the s i n g u l a r i t y  given  a t t h e same cot(t).G(w)  i s c a n c e l l e d and t h e  d i f f r a c t i o n c o e f f i c i e n t i s f i n i t e everywhere. Near  the shadow  boundary  or r e f l e c t i o n  wedge s u r f a c e , we can w r i t e from 0  T  0  =  o  TC  from  + e  (D.l) t h e shadow and i l l u m i n a t e d r e g i o n s ,  S u b s t i t u t i n g ( D . l ) i n (5.13a) g i v e s  = 0 and  the upper  (5.3a,b),  w h e r e e -*• -^0. e>0 and e<0 d e f i n e respectively.  boundary  (D.2)  ( D . l ) , (D.2) i n (5.11a) and (5.12a) g i v e s c o t ( T . ,) = c o t ( l i ) = — 2n e  (D.3)  L f j L  w, = - /2kT cos( 1,2 °  1 t +  E  2  ) = /zkr~  °  . I  (D.4)  2  S u b s t i t u t e (D.4) i n (D.10) t o get kr G(v. ) = ?  2j  e  2  J ( - ° — )  e  1  . ( i  /2k7  From (D.3) and (D.5) we get as e +  1) P[±  /2k7 1]  ,  eCO  (D.5)  -0 -JTC/4  cot(T  1  2  ).G(w  1  2  )  = T n /2rckr e o  (D.6)  - 98 -  Where (-) f o r shadow r e g i o n near  and (+) f o r i l l u m i n a t e d r e g i o n .  t h e r e f l e c t i o n boundary  write  due  t o t h e lower wedge  Similarly,  surface,  we can  from (5.3c) 0+0  o  = (2n - 1)  A g a i n £ •*• ^0  TC -  (D.7)  E  . e>0 and e<0 d e f i n e  t h e shadow and i l l u m i n a t e d r e g i o n s of  the r e f l e c t e d f i e l d , r e s p e c t i v e l y .  I n s e r t i n g (D.7) i n (5.13b) g i v e s  M+ = 1  (D.8)  and s u b s t i t u t i n g (D.7) and (D.8) i n ( 5 . 1 1 b ) , (5.12b) and (5.10) g i v e s  c o t ( T ) = cot(Tt _ 4r-) H  = "5-  (D.9)  and _  jTt/4  c o t ( T ) G(w ) = + n/2rckF e o l+  where  (-)  reflected  and  (+) r e f e r  field,  t o t h e shadow and i l l u m i n a t e d  respectively.  diffraction coefficient i s finite on 0 , (O<0 <^2-) . Q  coefficient everywhere.  (D.10)  H  Therefore,  are c a n c e l l e d  and  Notice  that  r e g i o n s of t h e  the t h i r d  term  of t h e  everywhere, because of t h e r e s t r i c t i o n a l l s i n g u l a r i t i e s of the d i f f r a c t i o n the d i f f r a c t i o n c o e f f i c i e n t  is  finite  - 99 -  APPENDIX E BEAM DIFFRACTION BY A PARABOLIC REFLECTOR  The  c a l c u l a t i o n of t h e r e f l e c t e d f i e l d  c y l i n d e r when i l l u m i n a t e d  by a G a u s s i a n  Felsen  analysis,  (1982).  In their  beam i s g i v e n by Hasselmann &  they assumed a v e r y sharp  beam and an i n f i n i t e p a r a b o l i c r e f l e c t o r . diffracted  field  from  from a c o n d u c t i n g p a r a b o l i c  Gaussian  so they d i d n o t i n c l u d e t h e  t h e edge o f a f i n i t e  reflector.  They used t h e  method o f computer s e a r c h f o r t h e complex r e f l e c t i o n p o i n t s . To to  include the d i f f r a c t e d f i e l d  from t h e edges, h a l f - p l a n e tangent  t h e r e f l e c t o r a t i t s edges a r e used i n s t e a d .  with  some a p p r o x i m a t i o n , t h e r e f l e c t e d  computer r e s e a r c h f o r the r e f l e c t i o n Fig.  E . l , shows a l i n e  field  Also i n the f a r f i e l d ,  c a n be c a l c u l a t e d  points.  source p a r a l l e l  to the r e f l e c t o r axis at a  complex p o i n t S ( r , 0 ) . A ( r , 0 ) i s a f a r f i e l d point, ( p  g  typical  without  g  r p  »  Q p  )  i s  a  p o i n t on t h e r e f l e c t o r , E ( r , O ) i s t h e edge of t h e r e f l e c t o r g  and 0(0,0) i s i t s f o c u s .  e  The e q u a t i o n of the p a r a b o l a of a f o c a l l e n g t h  F i s g i v e n by r  P  = 2F/(1 + cos0 ) P  (E.l)  and t h e s l o p e of t h e t a n g e n t i a l h a l f - p l a n e i s g i v e n as dy dx  2F  y e e 0 = (TC - 0 )/2 . t e  (E.2a)  y J  and  = tan0  J  (E.2b)  From the geometry o f F i g . E . l and t h e f a r f i e l d a p p r o x i m a t i o n s , we have r  I  = r - r s  cos(0 - 0 ) , s  (E.3a)  - 100  r r  = r + r  r  l  r  =  r  +  r  e  c  o  * r + r  2  cos(9  p  (  s  9  +  /  -  © ) >  +  (E.3b)  p  Q e  ^  '  (E.3c)  cos(0 - 0 ) .  g  (E.3d)  U s i n g t h e complex source p o i n t method, an i n c i d e n t f i e l d o f a beam which makes an a n g l e 8 w i t h the x - a x i s i s g i v e n as ~J = —  U  k r  i  _j r - e k  AT.l  r  _  r  C O S  ( 0 - 0 )]  kb c o s ( 0 - B) . e  ,  H e r e r , 0 a r e t h e r e a l c o o r d i n a t e s of the l i n e s o u r c e . o o  (E.4)  The  reflected  f i e l d a t A from p o i n t P I s -jk(r  sp  J  U  *  r  ,  6  /k(r where r  r  r Z  Here R  £  z  a r e shown i n F i e . E . l and sp r R cos©. =P_£ i . (E.6) ( 2 r - R cos0.) p c i i s t h e l o c a l r a d i u s of c u r v a t u r e and ©^ i s the a n g l e between the b  r S  p ) and t h e normal  a t t h e r e f l e c t i o n p o i n t P.  = 2F/cos (© 12) p  (E.7)  3  c  0  To  (E.5)  + r )  r  and r  i n c i d e n t ray ( R  + r ) r'  - - ( 0 12 - 0 ) p sp  i  calculate  ;  R e a l ( 0 . ) > 0. i  t h e complex r e f l e c t i o n  (E.8)  point ( > r  p  9  p^'  w  e  a  PP^  v  t  n  e  s t a t i o n a r y phase c o n d i t i o n d dx  (r_ P S  P  + r j = 0 r  ,  (E.9)  - 101 -  where r  s  r  tv s (  •  P  x  )2  %  +  2  .  2 1/2  s  ,  x2,l/2  = [ ( x - x ) ' + (y - y ) ] ' > + (y - y ) ] x  f  •  y> i  -  p  Substituting cosG  E1 0 a  *  p  (E.lOb)  (E.10) i n ( E . 9 ) , g i v e s  + sin©  r  <->  r  . c o t ( 0 /2) = - [ cosG cosG - sinG . Cot(0 /2)1, ( E . l l ) p sp sp r> sp sp p 1  where x - x cosG  =  r  y - y ,  P  r  x sp  r  =  r  r  - x  = -E  cosG  sinQ  1  ,  r  E.  ,  y  - y  r  sinG = _E sp  p  r  ,  (E.12b)  sp  dy = — E = 2F/y dx P  .  (E.12c)  p  A f t e r some m a n i p u l a t i o n s sin(0  1  r  sp  c o t ( 0 12)  (E.12a)  on ( E . l l ) , one g e t s  + Q /2) = - s i n ( G /2 p p  - Q  ) sp'  v  (E.13) '  or Q  r  = -(Q - Q ) p sp  (E.14) '  From F i g . E . l , ( E . l ) and t h e s i n e l a w , sin(0 v  Inserting  sp  2F + 0 ) cos(Q ) = _ s i n ( 0 s' p p s K  x  v  r  + 0 ) sp'  K  (E.14) i n (E.15) and a f t e r some m a n i p u l a t i o n s , one g e t s  (E.15) '  - 102 -  0  =-  p  i { 0 n  v  + 0  s  2  In the far f i e l d 0  + sin [ — sin(0 ) + sin(0 r r' s s _ 1  r  L  = 0 and 0  r  d  0/  r  J  )]}  v  (E.1'6)  i s derived.  p  The d i f f r a c t e d f i e l d U (U  +  v  d  at A is the sum of the d i f f r a c t e d f i e l d  and U ) of each edge d  U  = U  d  + U  [St(G-n+0  d  St(it-0E-0)]  ) +  (E.17)  [St(0+rc-0T) + s t ( n + 0 E + 0 ) ]  d  where St( ) is a unit step function,  -JCkr - * / 4 )  U =f d  c  {e  8  o  s  (  9  1  8  0  )  0 1  s  Jkr  8 l  co8(e e ) 1 +  0  F[-/2kr  1 +  0  S L  ^  cos(_——)]} SX  (E.18)  n  i s given by (E.18) with subscript 2 replacing 1.  d  0  0  2  i  9 l  - e and U  li)]  lC  /ZkF  1  0  F[V2kT 08(-i  1  =  (it-© ) + 0 t  = (ii-G ) - 0  2  ;  0 <  2TC  ,  (E.19a)  ;  0 < © < 2TC  ,  (E.19b)  Q  <  l  2  and r  = [r  sl  1  2  e  + r  + 2r r c o s ( 0  2  s  e s  r 0  =  0  sin"  +  1  where r  sin(0  [ _!  S  s  s  s  1 / 2  / J  , „  O N A  .  ,  (E.20a)  ,  (E.20b)  + 0 )  ! r  and 0  + 0 ) ]  ^ e  !_]  i  sl are given by (2.5)  and (2.6),  respectively,  Fig E.2 shows the normalized d i f f r a c t e d f i e l d component for a beam represented by the Complex Source Point method with i t s axis directed to the apex of included. field.  the In  reflector. the  region  The incident and reflected 0>12O°,  the  diffracted  field  fields is  are not  the  total  -103-  F i e l d Point A  e el"  /  D  api  Line Source  e  f  x  (-F.0)  X  Parabolic Reflector Feeding Beam  Fig. E . l  Geometry of a beam i d i f f r a c t i o n by a parabolic cylinder  F i g . E.2  N o r m a l i z e d d i f f r a c t e d f i e l d component of a beam d i f f r a c t i o n by a p a r a b o l i c (kF  = IOTT, 6  = 60 )  cylinder  - 105 -  APPENDIX F EVALUATION OF ARCTANGENT OF A COMPLEX NUMBER  To  calculate  t h e complex  angle  0 i n the proper  quadrant; i . e .  —n<Real(0) <it, which a complex p o s i t i o n v e c t o r makes w i t h the x - a x i s tan9 = L x  ,  (F.l)  or tan^+jO^  = R +jl t  (F.2)  t  where 0 and 0 a r e t h e r e a l and i m a g i n a r y p a r t s o f 0, and R and I a r e R I t t the r e a l and i m a g i n a r y p a r t s o f ( y / x ) . CO8(0+J9 ) = - 4 _  where  R  c  Expanding  = R  c  - j l  and - I a r e t h e r e a l c ( F . 3 ) and e q u a t i n g  i m a g i n a r y p a r t s of both s i d e s ,  A l s o we can w r i t e (F.3)  and i m a g i n a r y  the r e a l  parts  parts  o f both  o f ( _====—») • J 2, 2 vx +y s i d e s and t h e  gives  cos0 . cosh0 = R R 1 c  (F.4)  sin0„. s i n h 0 = 1 R I c  (F.5)  and T  A l s o by expanding  t h e l e f t s i d e o f ( F . 2 ) , one g e t s  tan0„ + j tanh© R I T  = R  1 - j tan©,,. t a n h 0 R I J  T  t  + jl  t  (F.6)  - 106 -  A f t e r some m a n i p u l a t i o n s on (F.6) we can w r i t e sec © 2  tanhG  .  _ = I (1 + t a n e . t a n h 0 )  (F.7)  2  2  R  ] ;  and sech^O  tan©  .  — (1 + t a n 0 _ . t a n h 0 _ ) 2  = R  (F.8)  2  R  J.  From (F.7) we can say 9 ^ 0 Now l e t 0  s  If  I  ^ 0  (F.9)  be g i v e n as °  - coiV  e  t  s  -  J|I |)  c  the a r c c o s i n e  (  F  A  O  )  C  c  o f a complex number o f form (F.10) i s a v a i l a b l e i n t h e UBC  Computer L i b r a r y . 0 = T 0 ; s  U s i n g ( F . 4 ) , ( F . 5 ) and (F.9) we get I J 0 , I > 0 T  c  (  F  >  n  )  and 0 = ± ©* s  * where 0  g  ;  I < 0 t >  ,  I < 0 c  (F.12)  i s t h e complex COTIjugate of 0 . g  -  107 -  APPENDIX G  LIST OF COMPUTER PROGRAMS FOR CSP ANALYSIS  All the programs used in the Complex Source Point analysis and listed below are written in the language of FORTRAN.  G.l  Comparison of CSP, Gaussian and Typical Antenna Beams  This program makes use of expressions  (2.10), (2.16) and (2.18) with ka=4,  kb = 3 and HPBW = 55.7° to compare the normalized far fields of CSP, Gaussian, and typical antenna beams. Q p  Also it uses (2.20) with different weighting factors (Q^, Q ,  and beam parameters (b^, b  different beams.  2  2>  b^ and fS y Q  /3 , /3 J) to calculate far fields of q2  q  -  108 -  C PROGRAM CALCULATES AND COMPARES THE FIELDS OF THE CSP, GAUSSIAN AND C TYPICAL ANTENNA BEAMS. THIS PROGRAM I S CALLED "CSP.FTNC". C C C  C  C  The Time H a r m o n i c F a c t o r " e x p ( - i w t ) " i s s u p p r e s s e d . The Common F a c t o r " e x p ( i k r ) / S q r t ( k r ) " i s s u p p e r e s s e d . ==================================================================== KK = 181 KM = (KK+1)/2 H =1.0 =============== PI = 3.1415926 DTR = P I / 1 8 0 . 0 C = CMPLX(0.0,1.0) A  = 4.0 = 3.0 I F ( B .LT. 0.25*ALOG(2.0) ) STOP ==================================================================== HPBW I S THE HALH-POWER BEAM WIDTH ==================================================================== HPBW= 2.0*ACOS( 1.0 - 0.5*ALOG(2.0)/B )  B  C C C C  C C  II C 22 III C 1  DO 111 K=1,KK Y = H*(K-KM) FI = Y*DTR CSP = ABS( E X P ( B * ( C O S ( F I ) - 1 . 0 ) ) ) GB = EXP( -ALOG(4.0)*(FI/HPBW)**2 ) U = A*SIN(FI) I F ( ABS(U) .EQ. P I / 2 . 0 ) GO TO 11 CD = COS(U)/( 1.0 - ( 2 . * U / P I ) * * 2 ) GO TO 22 CD = P I / 4 . 0 WRITE(6,1)  Y , CD ,CSP ,GB  CONTINUE ============================ FORMAT( F6.1 ,3(1X, E14.7) ) STOP END  -  109 -  C********************************************************************* C PROGRAM FOR DEVELOPING A BEAM FROM SINGLE OR MULTIPLE LINE SOURCE(S) C LOCATED AT COMPLEX P O I N T ( S ) . THIS PROGRAM CALLED "CSP.FTNM". C C THE TIME HARMONIC FACTOR " e x p ( - i w t ) " I S SUPPRESSED. C THE COMMON FACTOR " e x p ( i k r ) / S q r t ( k r ) " I S SUPPRESSED. c  =  =  C  =  =  =  =  =  =  =  =  =  =  =  =  =  COMPLEX*8 REAL *4 KK H C PI DTR  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  =  = 361 =1.0 = CMPLX(0.0,1.0) = 3.1415926 = PI/180.0  Q2 = B2 = BET2 = R02 = TH02 =  1.0 4.0 PI 0.0 PI/2. 0  Q1 =--1.0 B1 = B2 R01 = 1.0 DLTA= P I / 4 . 0 GAMA= P I / 2 . 0 Q3 = QJ B3 = B1 R03 = R01  -  GAMA TH01 = TH02 TH03 = TH02 + GAMA  -  DLTA BTA 1 = BTA 2 BTA3 = BTA2 + DLTA C  BIG  = 0.0  DO 111 K=1, KK Y(K)= H*(K-1) TH = Y(K)*DTR U1 = CEXP( Bl*COS(TH-BTA1) - C*R01*COS(TH- THOl) ) U2 = CEXP( B2*COS(TH-BTA2) - C*R02*COS(TH-•TH02) ) U3 = CEXP( B3*COS(TH-BTA3) - C*R03*COS(TH-•TH03) ) AUT(K)= CABS( Q1*U1 + Q2*U2 + Q3*U3 ) I F ( AUT(K) .GT. BIG ) BIG == AUT(K)  CC CC CC C 111  =  =  =  =  =  =  =  =  =  =  =  =  =  C „Ui ,U2 ,U3 ,CASIN ,CACOS ,CATAN ,ARKTAN 7 ( 3 6 1 ) ,AUT(361) ,AU1(361) ,AU2(361) ,AU3(361)  AU1(K)= CABS( Q1*U1 ) AU2(K)= CABS( Q2*U2 ) AU3(K)= CABS( Q3*U3 ) CONTINUE DO 222 K=1,KK  - no -  CC CC CC CC 222 C 1 C2  AUTN= AUT(K)/BIG WRITE(6,1) Y ( K ) , AUTN AUN1= A U 1 ( K ) / B I G AUN2= AU2(K)/B1G AUN3= A U 3 ( K ) / B I G WRITE(6,2) Y ( K ) ,AUN1 ,AUN2 ,AUN3 ,AUN CONTINUE  FORMAT( F6.1 , I X , E14.7 ) FORMAT( F6.1 ,4(1X, E14.7) ) STOP END C*********************************************************************  - Ill -  G.2  Beam Diffraction by a Half-Plane  This program uses expressions (3.3) and (C.5) to compare the total (asymptotic and uniform) far Fields of beam diffraction by a half-plane with kr =16, o  and different values of the beam parameters kb and 0=^ + 0 - 6  .  6 =90° o  -  112 -  C P r o g r a m f o r c a l c u l a t i n g A n t e n n a Beam D i f f r a c t i o n by a H a l f S c r e e n . C The Complex S o u r c e P o i n t S o l u t i o n compared w i t h A s y m p t o t i c s o l u t i o n C by G r e e n e t a l . ( l 9 7 9 ) . P r o g r a m c a l l e d "HP.FTN1". 0===================================================================== C The Time H a r m o n i c F a c t o r " e x p ( - i w t ) " i s s u p p r e s s e d . C The Common F a c t o r " e x p ( i . k r ) / S q r t ( k r ) " i s s u p p r e s s e d . C===================================================================== COMMON C , PI C COMPLEX*8 WI ,WR ,UI ,UR ,UTG ,UDF ,CACOS ,CFR COMPLEX*8 C ,RS ,THS ,DL1 ,DL2 ,SC1 ,SC2 ,QST REAL*4 Y ( 1 8 1 ) ,AUF(181) A U G ( l B l ) C KK = 181 KM = (KK+1)/2 H =1.0 C C = CMPLX(0. , 1 . ) PI = 3.1415926 DTR = P I / I 80. C RO = 1 6 . THO = P I / 2 . C DO 999 J=1,3 B = 2**(J+1) CC I F ( J .LE. 1 ) B = 0.0 CC I F ( J .GE. 6 ) B = 12.0 C CC DO 999 J=1,3 CC BETA= THO + PI - J * P I / 1 2 . BETA= THO + PI - P I / 6 . C I F ( BETA .EQ. (TH0+PI) ) GO TO 11 C RS = CSQRT( R0**2 + 2.0*R0*C*B*COS(BETA-TH0) - B**2 ) I F ( REAL(RS) .LE. 0.0 ) RS = -RS THS = CACOS( (R0*COS(TH0) + C*B*COS(BETA)) /RS ) C CC U REAL(THS) CC V AIMAG(THS) CC RSA = CABS(RS) CC RSR = REAL(RS) CC RSI = AIMAG(RS) CC THSI= PI + ( U + ATAN( SINH(V)*RSR/(RSA + RSI*COSH(V)) ) ) C ==================================================================== C "THSI" a n d "THSR" a r e t h e shadow b o u n d a r y a n g l e s f o r i n c i d e n t a n d C r e f l e c t e d f i e l d s r e s p e c t i v e l y ( m e a s u r e d from t h e h a l f - s c r e e n ) . C ==================================================================== THSI= PI + THO + ACOS( REAL(RS)/R0 ) I F ( BETA .GT. (TH0+PI) ) THSI = 2.*(PI+TH0) - THSI GO TO 22 C 11 RS = R0 - C*B THS = THO r  -  113 -  THSI= PI + THO 22 THSR= 2.0*PI - THSI C ==================================================================== C "TH" t h e o b s e r v a t i o n a n g l e measured from t h e h a l f - s c r e e n ( X - a x i s ). C " F l " i s t h e o b s e r v a t i o n a n g l e measured from beam a x i s i n a n t i c l o c k . C ==================================================================== QST = CEXP( C*(RS - 0.75*PI) ) / CSQRT(8.0*PI*RS) c  C  C C C C  AMX = 0 . 0 DO 111 K=1,KK Y(K)= HMK-KM) Fl = Y(K)*DTR TH = 1 . 5*PI + F l ==================================================================== The f a r f i e l d d i s t a n c e "R" i s n o t used i n c a l c u l a t i n g t h e p a t t e r n . "UI" a n d "UR" a r e t h e i n c i d e n t a n d r e f l e c t e d s f i e l d , r e s p e c t i v e l y . ==================================================================== UI = CEXP( -C*R0*COS(TH-TH0) + B*COS(TH-BETA) ) UR =-CEXP( -C*R0*COS(TH+TH0) + B*COS(TH+BETA) )  c  C C C  C  C CC C 33 ' C 111 C  222 C 999 1  WI WR  = -CSQRT(2.0*RS) * CCOS((TH-THS)/2.0) = -CSQRT(2.0*RS) * CCOS((TH+THS)/2.0)  UTG = ( UI*CFR(WI) + UR*CFR(WR) ) * C E X P ( - C * P I / 4 . ) / S Q R T ( P I ) AUG(K)= CABS(UTG) IF( IF(  (AIMAG(THS).EQ. 0.).AND.(REAL(TH-THS) .EQ. P I ) ) (AIMAG(THS).EQ. 0.).AND.(REAL(TH+THS) .EQ. P I ) )  SC1 SC2 DL1 DL2 UDF  = = = = =  GO TO 33 GO TO 33  1,0/CCOS(.5*(TH-THS)) 1.0/CCOS(.5*(TH+THS)) C*SC1**2/(4.*RS) C*SC2**2/(4.*RS) QST * ( SC1 * ( 1 . - DL1) - S C 2 * ( 1 . - DL2) )  I F ( TH .LE. THSI ) I F ( TH .LE. THSR ) AUF(K)= CABS(UDF)  UDF = UDF + UI UDF = UDF + UR  IF((AUG(K).GT.AMX).OR.(AUF(K).GT.AMX)) AMX=AMAX1(AUG(K),AUF(K)) I F ( AUG(K) .GT. AMX ) AMX= AUG(K) GO TO 111 AUF(K)= 100.0*AMX ================= CONTINUE ============== DO 222 K=1,KK UGN = AUG(K)/AMX UFN = AUF(K)/AMX I F ( UFN .GT. 1.1 ) UFN = 1.1 WRITE(6,1) Y ( K ) , UGN , UFN CONTINUE ======== CONTINUE FORMAT( F6.1 ,2(1X ,E14.7) ) STOP END  -  114 -  C === ======== = ===== ===== = ============ = = === = ======= = C PROGRAM FOR CALCULATING FRESNEL INTEGRAL "CFR" OF COMPLEX ARGUMENT. ==================================================================== COMPLEX FUNCTION CFR(X) c  c  COMMON C , PI COMPLEX*8 C , X COMPLEX*16 Z , ERFZ Z = X*CEXP(-C*PI/4.0) CALL CERF(Z,ERFZ) CFR = 0 . 5 * S Q R T ( P I ) * C E X P ( C * P I / 4 . 0 ) * ( 1 . 0 - E R F Z ) RETURN END Q********************************************************************* Q************************************************  * * * * * * * * * * * * * * * * * * * * *  -  G.3  115 -  Beam Diffraction by a Wide Slit  Using expressions (4.5), (4.10) and (4.15), the total diffracted far field including interaction  between the  edges is  calculated  different values of the beam parameter, kb.  for  kr = ka = 8, 0 = 9O°, 0 = 2 7 0 ° Q  o  and  -  116'-'  £********************************************************************* C PROGRAM FOR CALCULATING ANTENNA BEAM (SINGLE & MULTIPLE) DIFFRACTION C BY A S L I T USING HALF-PLANE SOLUTION.THE PROGRAM I S CALLED "SLIT.FTN" C " NON-SYMETRICAL INCIDENCE " C C TIME DEPENDCE e x p ( - i w t ) 6 COMMON FACTOR e x p ( i ( k r - P I / 4 ) ) / S q r t ( P I * k r ) . C =================================================================== COMMON C ,PI COMPLEX*8 WI1 ,WI2 ,WR1 ,WR2 ,CFR , FR ,CACOS ,CASIN COMPLEX*8 DO ,DI1 ,DI2 ,DF1 ,DF2 ,UI ,UDD ,UDM ,US COMPLEX*8 UE1 ,UE2 ,U1 ,U2 ,UI0 ,UI1 ,UI2 ,UR1 ,UR2 COMPLEX*8 C ,F0 R S ,RS1 ,RS2 ,THS ,THS1 ,THS2 CMPLX REAL Y ( 1 8 1 ) , AUS(181) , AUT(181) C KK = 181 KM = (KK+1)/2 H = 1 . C C = CMPLX(0.0,1.0) PI = 3.1415926 DTR = P I / 1 8 0 . FO = S Q R T ( P I ) * C E X P ( C * P I / 4 . 0 ) r  t  A =8. RO =8. THO = P I / 2 . B BETA1= TH01 + P I BETA2= 3*PI - BETA 1 RS = CSQRT( R0**2 + 2.*R0*C*B*COS(BETA1-TH0) - B**2 ) IF(- REAL (RS ) .LE. 0.0 ) RS = -RS THS = CACOS( (R0*COS(TH0) + C*B*COS(BETA1)) /RS) KU--^ + - A**2 ft—* - 2.*R0*A* COS(THO) ) R01 = SQRT(v R0**2 '( R0**2 + A**2 + 2.*R0*A* COS(THO) ) R02 = SQRT( ASIN(R0*SIN(TH0)/R01) TH01 = PI ASIN(R0*SIN(TH0)/R02) TH02 = PI  RSI IF( RS2 IF(  C CC CC CC CC CC CC  c  = CSQRT( RS**2 REAL(RS1) .LE. = CSQRT( RS**2 REAL(RS2) .LE.  + A**2 - 2.*RS*A*CCOS(THS) ) 0.0 ) RS1 = -RS1 + A**2 + 2.*RS*A*CCOS(THS) ) 0.0 ) RS2 = -RS2  THS1= P I - CASIN(RS*CSIN(THS)/RS1) THS2= P I - CASIN(RS*CSIN(THS)/RS2) THSI1= P I + TH01.+ A C O S ( R E A L ( R S 1 ) / R 0 l ) THSI2= P I + TH02 + ACOS(REAL(RS2)/R02) I F ( BETA1 .GT. TH01+PI ) THSI1 = 2.*(PI+TH01) I F ( BETA2 .GT. TH02+PI ) THSI2 = 2.*(PI+TH02) THSR1 « 2.*PI - THSI1 THSR2 = 2.*PI - THSI2 UE1 = CEXP(C*RS1) /CSQRT(RS1) UE2 = CEXP(C*RS2) /CSQRT(RS2)  (  THSI 1 THSI 2  -  117  -  I F { REAL( CSQRT(2.*RS1) * C S I N ( T H S l / 2 . ) ) .LT. 0. ) I F ( REAL( CSQRT(2.*RS1) *CSIN(THS1/2.) ) .LT. 0. )  C  DO  C  STOP STOP  = - S Q R T U . / P I ) * CEXP(-C*(2.*A+PI/4.) ) * FR( SQRT(4.*A) )  DI1 = - C S Q R T ( 2 . * R S 1 / ( P I * A ) ) * CFR( CSQRT(2.*RS1)*CSIN(THS1/2.) ) D l 1 = DI1 * CEXP( C*(2.*A + RS1*CCOS(THS1) - RS1 - P I / 4 . ) ) DI2 = - C S Q R T ( 2 . * R S 2 / ( P I * A ) ) * CFR( CSQRT(2.*RS2)*CSIN(THS2/2.) ) DI2 = DI2 * CEXP( C*(2.*A + RS2*CCOS(THS2) - RS2 - P I / 4 . ) )  C C ALL ANGLES ARE IN RADIANS EXCEPT Y ( K ) IN DEGREES. C BIG = 0.0 DO 111 K=1,KK Y ( K ) = H*(K-KM) FI = Y(K)*DTR TH1 = 1. 5*PI + F I C TH2 = 3*PI - TH1 I F ( THI .LT. P I ) TH2 = PI - TH1 C C THE DISTANCES RI ,R2 ARE NOT USED IN CALCULATING THE PATTERN. C R1 = R-A*COS(TH1) C R2 = R+A*COS(TH1) C RS*COS(TH1-THS) = R0*COS(TH1-THO) + C*B*COS(THI-BETA) C THE U I , S AND UR,S ARE EXPONENTIAL FUNCTIONS. C FOR SYMMETRICAL NORMAL INCIDENCE " UI0 = UI1 = U I 2 & UR1 = UR2 " C UIO = CEXP( -C*R0 *COS(TH1-TH0 ) + B*COS(TH1-BETA1) ) UI 1 = CEXP( -C*R01*COS(TH1-TH01) + B*COS(TH1-BETA1) ) UR1 = CEXP( -C*R01*COS(TH1+TH01) + B*COS(TH1+BETA1) ) UI2 = CEXP( -C*R02*COS(TH2-TH02) + B*COS(TH2-BETA2) ) UR2 = CEXP( -C*R02*COS(TH2+TH02) + B*COS(TH2+BETA2) ) WI1 WR1 WI2 WR2  = = = =  -CSQRT(2.0*RS1) -CSQRT(2.0*RS1) -CSQRT(2.0*RS2) -CSQRT(2.0*RS2)  * * * *  CCOS((TH1-THS1)/2.0) CCOS((TH1+THS1)/2.0) CCOS((TH2-THS2)/2.0) CCOS((TH2+THS2)/2.0)  C C THE CFR(W) I S A SUBROUTINE CALCULATES FRESNEL INTEGRALS WITH COMPLEX C ARGUMENTS. UI = UI0*F0 U1 = ( UI1*CFR(WI1) - UR1*CFR(WR1) ) * CEXP(-C*A*COS(TH1)) U2 = ( UI2*CFR(WI2) - UR2*CFR(WR2) ) * CEXP(-C*A*COS(TH2)) C C UI UD AND US ARE THE INCIDENT, DIFFRACTED AND TOTAL SINGLE DIFFRACC TION FAR F I E L D PATTERNS, RESPECTIVELY. f  C C C CC  US = U l + U2 - UI AUS(K)= CABS(US) DF1 = -SORT(8.*A/PI) *CEXP( C * ( A * C O S ( T H 1 ) - 2 . * A - P I / 4 . ) ) DF1 = DF1 * FR( S Q R T ( 4 . * A ) * S I N ( T H l / 2 . ) ) DF2 = -SQRT(8.*A/PI) *CEXP( C * ( A * C O S ( T H 2 ) - 2 . * A P I / 4 . ) ) DF2 = DF2 * FR( SQRT(4.*A)*SIN(TH2/2.) ) -  UDD = UE1*DI1*DF2 + UE2*DI2*DF1 UDM =( UE1*DI1*(D0*DF1+DF2) + UE2*DI2*(D0*DF2+DF1) ) / ( l . - D 0 * * 2 )  AUT(K)=  C  118 -  CABS(US+UDM)  I F ( ( A U T ( K ) . G E . B I G ) . O R . ( A D S ( R ) . G E . B I G ) ) BIG=AMAX1(AUT(K),AUS(K))  C 111 C  CONTINUE DO 222 K=1,KK USN= A U S ( K ) / B I G UTN= A U T ( K ) / B I G WRITE(6,1) Y ( K ) , USN , UTN CONTINUE  222 C 1  FORMAT( F6.1 ,2(1X ,E14.7) ) STOP END C = = = = = = = = = = = = = = == = === = = = = = = = == = = = = = C PROGRAM FOR CALCULATING FRESNEL INTEGRAL "CFR" OF COMPLEX C COMPLEX FUNCTION CFR(X) COMMON COMPLEX*8 COMPLEX*16  ARGUMENTS.  C , PI C ,X Z , ERFZ  C C THE CERF(W,ERF) I S A SUBROUTINE CALCULATES ERROR FUNCTIONS WITH C COMPLEX ARGUMENTS. C Z = X*CEXP(-C*PI/4.D) CALL CERF(Z,ERFZ) CFR = 0.5 * SQRT(PI) * C E X P ( C * P l / 4 . 0 ) * (1.0-ERFZ) RETURN END C ==================================================================== C PROGRAM FOR CALCULATING FRESNEL INTEGRAL "FR" OF REAL ARGUMENTS. C COMPLEX FUNCTION F R ( X ) C ========================= COMMON C , PI COMPLEX*8 C COMPLEX*16 Z , ERFZ C • Z = X*CEXP(-C*PI/4.0) CALL CERF(Z,ERFZ) FR = 0.5 * SQRT(PI) * C E X P ( C * P l / 4 . 0 ) * (1.0-ERFZ) RETURN END Q  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  *  Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  -  G.4  119 -  Beam Diffraction by a Strip  This diffraction  program calculates  the  total  far  field  of  a  normally  incident  beam  by a conducting strip, neglecting the interaction between the edges, using  expression (4.18) with kr =ka = 8, t 9 = 9 0 ° and different values of beam parameters kb o  and B=ir + 6 - 5 .  o  -  120 -  Q********************************************** C PROGRAM FOR CALCULATING ANTENNA BEAM SINGLY DIFFRACTED BY A STRIP C USING HALF-PLANE SOLUTION. THE PROGRAM I S CALLED " STRP.FTN ". C = === = = = = = = = = = = = = = = = = = = = = = = = = = == = = = = = = = = = === = = = = = C TIME DEPENDCE " exp(-iwt) " . C COMMON FACTOR " exp(ikr)/Sqrt(kr) " . C THE ANGLES (THS,THS1,THS2,TH1,TH2,Fl,BETA) A L L ARE I N RADIANS. C ==================================================================== COMMON C , PI COMPLEX*8 C ,F0 ,WI1 ,WR1 ,WI2 ,WR2 ,CASIN ,CACOS ,CFR COMPLEX*8 U l ,U2 ,UI ,UR ,UD ,UT COMPLEX*8 RS ,RS1 ,RS2 ,THS ,THS1 ,THS2 C REAL*4 Y ( 3 6 1 ) ,AUT(361) ,AUD(36l) ,AUI(361) ,AUR(361) C Y0 = 00.0 H =1. 0 KR = 3 6 1 c  C  C CC CC CC C  C 11 C CC CC • CC C CC CC CC CC C 22 C  C PI DTR F0  = CMPLX (0.0,1.0) = 3.1415926 = PI/180.0 = CEXP(-C*PI/4.0)/SQRT(PI)  R0 = 8 . 0 A = 8.0 TH0 = P I / 2 . 0 B  = 8.0 DO 999 1=1,6 B 2.0**(I-2) IPC ( I .LE. 1).OR.(I  .GE. 6) )  B = 2.4*(I-1)  DO 999 J=1,5 BETA= TH0 + P I - P I * ( J - 1 ) / 1 2 . I F ( J .GE. 5 ) GO TO 11 GO TO 22 A = 10.*A BETA= TH0+PI-PI/4. BET1= 3.0*PI - BETA I F ( BETA .LT. P I ) BET2= BETA R01 = SQRT( R02 = SQRT( TH01= ASIN( TH02= ASIN(  BET1 = PI - BETA  R0**2 + A**2 - 2.0*R0*A*COS(TH0) ) R0**2 + A**2 + 2.0*R0*A*COS(TH0) ) R0*SIN(TH0) /R01 ) R0*SIN(TH0) /R02 )  RS = CSQRT( R0**2 - B**2 + 2.0*C*B*R0*COS(BETA-THO) ) I F ( REAL(RS) .LE. 0.0 ) RS = -RS THS = CACOS( (R0*COS(TH0) + C*B*COS(BETA)) /RS ) RS1 = CSQRT( RS**2 + A**2 - 2.0*RS*A*CCOS(THS) ) RS2 = CSQRT( RS**2 + A**2 + 2.0*RS*A*CCOS(THS) ) I F ( REAL(RS1) .LE. 0.0 ) RS1 = -RS1  -  C CC CC CC C CC CC CC C  121 -  I F ( REAL(RS2) .LE. 0.0 ) THS1= CASIN( RS*CSIN(THS) /RS1 ) THS2= CASIN( RS*CSIN(THS) /RS2 )  RS2 = -RS2  THI1= P I + TH01 + A R C O S ( R E A L ( R S I ) / R 0 1 ) I F ( BET1 .GT. (TH01+PI) ) THI1 = 2.0*(PI+THO1) - THI1 THR1= 2.0*PI - THI1 THI2 = PI + TH02 + ARCOS(REAL(RS2)/R02) I F ( BET2 .GT. (TH02+PI) ) THI2= 2.0*(PI+TH02) - THI2 THR2= 2.0*PI - THI2 BIGT = 0.0 DO 111 K=1 , KK Y ( K ) = H*(K-1) TH = Y(K)*DTR TH2 = TH THI = P I - TH I F ( THI .LT. 0. ) TH1 = TH1 + 2*PI  C THE DISTANCE "R" I S NOT USED I N CALCULATING THE PATTERN. C R1 = R+A*COS(TH) C R2 = R-A*COS(TH) C R i = R-RS*CCOS(TH-THS) » C R r = R-RS*CCOS(TH+THS) c  C CC CC CC CC CC CC CC CC C C C C C  C CC CC CC C CC CC  ==================================================  MI 1 WR1 WI2 WR2  = = = =  -CSQRT(2.0*RS1)*CCOS( -CSQRT(2.0*RS1 )*CCOS( -CSQRT(2.0*RS2)*CCOS( -CSQRT(2.0*RS2)*CCOS(  (TH1-THS1)/2.0 ) (TH1 +THS1 )'/2 . 0 ) (TH2-THS2)/2.0 ) (TH2+THS2)/2.0 )  ARI1 = ABS( ATAN( AIMAG(WI1)/REAL(WI1) ) ) I F ( ARI1 .GT. P I / 4 . 0 ) STOP ARR1= ABS( ATAN( AIMAG(WR1)/REAL(WR1) ) ) I F ( ARR1 .GT. P I / 4 . 0 ) STOP ARI2= ABS( ATAN( AIMAG(WI2)/REAL(WI2) ) ) I F ( ARI2 .GT. P I / 4 . 0 ) STOP ARR2 = ABS( ATAN( AIMAG(WR2)/REAL(WR2) )) I F ( ARR2 .GT. P I / 4 . 0 ) STOP ==================================================================== "UT" I S THE TOTAL SINGLE DIFFRACTION PATTERN BY STRIP. "U1,E2" ARE THE TOTAL D I F F . PATTERN BY HALF PLANES FORMING THE STRIP "UI & ER" ARE INCIDENT AND REFLECTED F I E L D PATTERN RESPECTIVELY. ==================================================================== UI =+CEXP( -C*RS*CCOS(TH-THS) ) UR =-CEXP( -C*RS*CCOS(TH+THS) ) Ul = UI*CFR(WI1) + UR*CFR(WR1) U2 = UI*CFR(WI2) + UR*CFR(WR2) UT = F0*(U1+U2) I F ( TH .LT. P I )  UT = UT - (UI+UR)  UD = UT - UI I F ( (TH1 .LT. THR1).AND.(TH2 I F ( (TH1 .GT. THI1).AND.(TH2 AUT(K)= CABS(UT) AUD(K)= CABS(UD) AUI(K)= CABS(UI)  .LT. THR2) ) .GT. T H I 2 ) )  UD = UT - (UI+UR) UD = UT  CC 111 C  222 C 999 C 1  AUR(K)= CABS(UR) I F ( AUT(K) .GT. BIGT )  122 -  BIGT = AUT(K)  CONTINUE ============== DO 222 K=1,KK UTN = AUT(K)/BIGT WRITE(6,1) Y ( K ) , UTN CONTINUE ========================= CONTINUE ========================= FORMAT( F6.1 ,1X, E14.7 ) STOP END  C PROGRAM FOR CALCULATING FRESNEL INTEGRAL "CFR" OF COMPLEX ARGUMENTS. C COMPLEX FUNCTION CFR(X) C ========================= COMMON C , PI COMPLEX*8 C ,X COMPLEX*16 Z , ERFZ Z = X*CEXP(-C*Pl/4.0) CALL CERF(Z,ERFZ) CFR = 0 . 5 * S Q R T ( P I ) * C E X P ( C * P I / 4 . 0 ) * ( 1 .0-ERFZ) RETURN END Q ************************************************** C*********************************************************************  -  G.5  123 -  Beam Diffraction by a Wedge  The  total  diffracted field given  program with kr =16, o  angles, a .  0 = 6^+17  by (5.15) or (5.17) is  calculated using this  for different angles of incidence,  6  Q  and wedge  -  C C C C C C C  C  124 -  PROGRAM FOR CALCULATING BEAM DIFFRACTION BY A CONDUCTING WEDGE USING THE UNIFORM THEORY OF DIFFRACTION (KOUYOUMJIAN & PATHAK 1974). THE EDGE L I E S ON THE BEAM A X I S . THIS PROGRAM CALLED " WEDG.FTN ". ============================================== The t i m e h a r m o n i c f a c t o r " E x p ( + j w t ) " i s s u p p r e s s e d . Common f a c t o r i s " E x p ( - j . k r ) / S q r t ( k r ) " . =============== ================================================ COMPLEX*8 C ,RS ,WI1 ,WI2 ,WR1 ,WR2 ,GR ,CFR ,CASIN ,CACOS COMPLEX*8 UO ,UE ,UI ,UR1 ,UR2 ,UD1 ,UD2 ,UD ,UT REAL Y ( 3 6 1 ) ,AUT(361) ,N KK = 361 H =1.0 N =1.50 ERR = 0 . 0 1 C = PI = DTR = THSY= THCR=  CMPLX(0.0,1.0) 3.1415926 PI/180.0 N*Pl/2.0 (N-1.0)*PI  c  R0 = 16.0 TH0 = ? BETA= TH0 + P I C ==================================== C ALW i s - t h e wedge i n t e r i o r a n g l e . CC ALW = (2.0 - N ) * P I C I F ( (TH0 .LT. 0.).OR.(TH0 .GT. THSY ) ) CC I F ( (BETA .LE. PI).OR.(BETA .GT. 2.0*PI ) ) c  c C  C C C  STOP STOP  DO 999 J = 1,4 ? B = 2.0*(J-1) ? ================ RS = R0 + C*B THS = TH0 ==================================================================: "THSI" a n d "THSR" a r e t h e shadow b o u n d a r y a n g l e s f o r i n c i d e n t a n d r e f l e c t e d f i e l d s r e s p e c t i v e l y (measured from wedge u p p e r s u r f a c e ) THSI = PI + TH0 THSR1= PI - TH0 THSR2= (2.0*N - 1.0)*PI - TH0 UE = CEXP(-C*RS)/CSQRT(RS) U0 = - C E X P ( - C * P l / 4 . 0 ) / ( N * SQRT(8.0*PI) )  C C "TH" t h e o b s e r v a t i o n a n g l e m e a s u r e d f r o m t h e h a l f - s c r e e n ( X - a x i s ) C DO 111 K=1,KK C Y(K)= H*(K-1) TH = Y(K)*DTR Q =========================================================== C The d i s t a n c e "R" i s n o t used i n c a l c u l a t i n g t h e p a t t e r n .  -  125 -  C " UI , UR1, UR2 " a r e t h e i n c i d e n t a n d r e f l e c t e d f i e l d s from upper C a n d l o w e r Wedge s u r f a c e s , r e s p e c t i v e l y . C ================================== THI = TH - THS THR = TH + THS UI = CEXP( C*RS*COS(THI) •) UR1 =-CEXP( C*RS*COS(THR) ) UR2 =-CEXP( C*RS*COS(2.0*N*PI-THR) ) C I F ( TH .GT. N*PI ) GO TO 88 C MR 1 = 0.0 MI 1 = 0.0 MR2 = 0 . 0 MI 2 = 0.0 I F ( (THI .GT. -THSY).AND.(THI .LT. -THCR) ) MI 1= -1.0 I F ( (THI .GT. THCR).AND.(THI .LT. N*PI) ) MI 2= +1.0 I F ( (THR .GT. THCR).AND.(THR .LT. 1.5*N*PI) ) MR2= +1.0 C WI1 = -CSQRT(2.0*RS) * COS( 0.5*(2*N*PI*MI1 - T H I ) ) WR1 = -CSQRT(2.0*RS) * COS( 0.5*(2*N*PI*MR1 - THR) ) WI2 = -CSQRT(2.0*RS) * COS( 0.5*(2*N*PI*MI2 - T H I ) ) WR2 = -CSQRT(2.0*RS) * COS( 0.5*(2*N*PI*MR2 - THR) ) C TI1 = 0 . 5 * ( P I - T H I ) / N TR1 = 0.5*(PI-THR) / N T I 2 = 0.5*(PI+THI) / N TR2 = 0.5*(PI+THR) / N C I F ( ABS(TH-THSI ) .LE. ERR ) GO TO 11 I F ( ABS(TH-THSR1) .LE. ERR ) GO TO 22 C UD1 = GR(WI1)/TAN(TI1) - GR(WR1)/TAN(TR1) GO TO 33 C 11 UD1 = N * CSQRT(2.0*PI*RS) * C E X P ( C * P I / 4 . 0 ) - GR(WR1)/TAN(TR1) GO TO 33 C 22 UD1 = GR(WI1)/TAN(TI1) - N * CSQRT(2.0*PI*RS) * C E X P ( C * P I / 4 . 0 ) C 33 I F ( ABS(TH-THSR2) .LE. ERR ) GO TO 44 C UD2 = GR(WI2)/TAN(TI2) - GR(WR2)/TAN(TR2) GO TO 55 C 44 UD2 = GR(WI2)/TAN(TI2) + N * CSQRT(2.0*PI*RS) * C E X P ( C * P I / 4 . 0 ) C 55 UD = U0 * UE * (UD1+UD2) C I F ( (THO .GT. (N-1.0)*PI).AND.(THO .LE. N * P I / 2 . 0 ) ) G O T O 66 C UT = UD IF( TH .LE. THSR1 ) UT = UD + UI + UR1 I F ( (TH .GT. THSR1).AND.(TH .LE. THSI) ) UT = UD + UI GO TO 99 C 66 UT = UD + UI I F ( TH .LE. THSR1 ) UT = UT + UR1 I F ( TH .GT. THSR2 ) UT = UT + UR2 GO TO 99  -  C 88 C 99  UT  126 -  = 0.0  AUT(K)= CABS(UT) I F ( AUT(K) .GT .BIG ) BIG = AUT(K) C ==================================== 111 CONTINUE C =============== DO 222 K=1,KK UTN = A U T ( K ) / B I G WRITE(6,1) Y ( K ) , UTN 222 CONTINUE C ========== 999 CONTINUE C ========================= 1 FORMAT( F6.1 , I X , E14.7 ) STOP END C ==================================================================== C PROGR. FOR CALCULATING MODIFIED FRESNEL INTEGRAL OF COMPLEX ARGUMENT C COMPLEX FUNCTION GR(X) C ======================= COMPLEX*8 C , X , CFR C = CMPLX(0.0,1.0) PI = 3.1415926 C I F ( R E A L ( X * C E X P ( C * P I / 4 . 0 ) ) .LT. 0.0 ) X = -X GR = 2.0 * C * X * CEXP(C*X*X) *CFR(X) RETURN END C ==================================================================== C PROGRAM FOR CALCULATING FRESNEL INTEGRAL "CFR" OF COMPLEX ARGUMENT. C ==================================================================== COMPLEX FUNCTION CFR(X) C ========================= COMPLEX*8 C, X COMPLEX*16 Z , ERFZ c  C  C PI  = CMPLX(0.0,1.0) = 3.1415926  Z = X * CEXP(C*PI/4.0) CALL CERF(Z,ERFZ) CFR = 0.5 * SQRT(PI) * C E X P ( - C * P I / 4 . 0 ) * (1.0 - ERFZ) RETURN END  Q*******************************************************  * * * * * * * * * * * * * *  Q*********************************************************************  -  G.6  127 -  Beam Diffraction by a Circular Aperture  This program makes use  of expressions (6.33a,b) to calculate the single and  multiple diffraction total fields modified on the caustic axis with kz =ka=3ir, 0 =9O°, 0  /3 =270° and different values of the beam parameter, kb.  O  -  128 -  C* **************************************************** C PROGRAM CALCULATES SINGLE AND MULTIPLE BEAM DIFFRACTION BY CIRCULAR c APERTURE (NORMAL INCIDENCE), MODIFIED ON THE CAUSTIC A X I S , USING THE c UNIFROM THEORY OF DIFFRACTION & COMPLEX SOURCE-POINT REPRESENTATION. c THIS PROGRAM I S CALLED " CRCL.FTN2 ".  c c THE c ARE c=  TIME DEPENDENCE " e x p ( - i w t ) " AND A COMMON FACTOR SUPPRESSED.  "exp(ikr)/kr  n  COMMON  C ,PI ,THSI  COMPLEX*8 COMPLEX*8 COMPLEX*8 COMPLEX*8 REAL *4  C ,CASIN ,CACOS ,CATAN ,DC ,FR ,CFR ZS ,RS1 ,THS1 ,WI ,B0S ,B1S ,B0M ,B1M UE ,UI ,UR ,U1 ,U2 ,USD ,UMD ,US DO ,D1 ,D2 ,DI ,DF1 ,DF2 Y(181) , AUS(181) , AUT(181)  J J = 16 KK = 91 H =1.0 C PI DTR  = CMPLX(0.0,1.0) = 3.1415926 = PI/180.  A Z0 BETA  = 3*PI = A = 1.5*PI •  R01 = SQRT( Z0**2 + A**2 ) TH01 = PI - ATAN( ZO/A )  C C  C C C  C C C  DO 999 1=1,6 B = 2.**(I-1) I F ( I .LE. 1 ) B I F ( I .GE. 6 ) B ZS  0.0 85.0  Z0 - C*B  RSI = CSQRT( ZS**2 + A**2 ) I F ( R E A L ( R S I ) .LE. 0.0 ) RS1 THSI = PI - CATAN( ZS/A )  -RSI  T H S I , THSR ARE THE SHADOW AND REFLECTION BOUNDARY ANGLES, THSI = PI + TH01 + ACOS( R E A L ( R S 1 ) / R 0 l ) I F ( BETA .GT. TH01+PI ) THSI = 2.MPI+TH01) THSR = 2.0*PI - THSI UE DO  -THSI  = CEXP(C*RS1) / RS1 = S Q R T U . / P I ) * CEXP( C * ( P I / 4 - 2.*A) ) * FR( SQRT(4.*A) )  WI = CSQRT(2.0*RS1) *CSIN(THS1/2.0) I F ( REAL( WI*CEXP(-C*PI/4.) ) .LT. 0.0 ) TOP Dl « CSQRT(2.*RS1/(PI*A)) *CEXP(C*(PI/4.+2.*A)) Dl = D l * CEXP(-C*WI**2) *CFR(WI)  129 -  Q  K S B B C B S S B E B C S B S B B S S B B B S B B E S S S S S S E S  BG «= 0.0 c  C C  C C C C  C  C C II C C 22 C  Q  III Q  222 Q  DO 111 K=1,KK Y ( K ) = H*(K-1) FI « Y(K)*DTR THI TH2  «= 1 .5*PI + F I = 1.5*PI - F I  WF1 WF2 DF1 DF2  «= SQRT(4.0*A) * S I N ( T H l / 2 . 0 ) «= SORT(4.0*A) * SIN(TH2/2.0) = CEXP(-C*WF1**2) * FR(WF1) «= CEXP(-C*WF2**2) * FR(WF2)  D1 D2  * DC(THS1,RS1,TH1) « DC(THSI,RS1,TH2)  UI UR  • = CEXP( C * ZS * C O S ( F I ) ) * -CEXP(-C * ZS * C O S ( F I ) )  I F ( K .GT. J J )  GO TO 11  BOS B1S USD  « (D2 + D l ) * B E S J 0 ( A * S I N ( F I ) , 1 ) • (D2 - D1) * B E S J 1 ( A * S I N ( F I ) , 1 ) « A * SQRT(PI/2.0) * C E X P ( - C * P I / 4 . 0 ) * UE * (BOS + C*B1S)  BOM B1M UMD  - (DF1 + DF2) * B E S J 0 ( A * S I N ( F I ) , I ) - (DF1 - DF2) * B E S J 1 ( A * S I N ( F I ) , I ) « 2. * (A**1.5) * UE * D l * (B1M + C*B0M) / (1.0-D0)  GO TO 22 U1 U2 USD  CEXP( - C * ( A * S I N ( F I ) - P I / 4 . ) ) « 1.0/U1 « S Q R T ( A / S I N ( F I ) ) * C E X P { - C * P I / 4 . ) *UE *(U1*D1  UMD UMD  - C *A *SQRT( 8.0 / ( P I * S I N ( F I ) ) ) = UMD *UE *DI * ( U1*DF1 + U2*DF2 ) / (1.0-D0)  US « USD I F ( (THI .LT. THSI).AND.(TH2  .LT. THSI) ) US = USD + UI  AUS(K)= CABS(US) AUT(K)= CABS(US+UMD) I F ( (AUS(K).GT.BG).OR.(AUT(K).GT.BG)  ) BG=AMAX1(AUT(K),AUS(K))  rscccncc=c=ctn====nc=====:===:=r=cEC==================:=======  CONTINUE CESCBBeBBSSBeESEBSBB  DO 222 K «= 1 , KK USN * AUS(K)/BG UTN «= AUT (K) /BG W R I T E ( 6 , 1 ) Y ( K ) , USN , UTN CONTINUE BCCCEC=CC  999  CONTINUE  1  FORMAT( F 6 . 1 , 2(1X , E 1 4 . 7 ) ) STOP END  £  + U2*D2)  B K B B K B B B B E B B S B B B S B B B B B B B B S B B S S  C C  = == = == =w  130 -  === === =*= = = = == === == === =  s  =  c:  = = K=  ti  = = = = = z:c: = = = = =  ^  PROGRAM FOR CALCULATING DIFFRACTION COFFICIENT . COMPLEX COMMON COMPLEX*8 WI WR DI DR  FUNCTION  DC(THS,RS,TH)  C ,PI ,THSI C ,RS ,THS ,WI ,WR ,DI ,DR ,CFR  = -CSQRT(2.0*RS) * CCOS( " (TH-THS)/2.0 ) = -CSQRT(2.0*RS) * CCOS( (TH+THS)/2.0 ) «= CFR(WI ) •= CFR(WR)  -WI ) I F ( REAL(WI*CEXP(-C*PI/4.)) .LT. 0. ) DI = -CFR( I F ( TH .LT. THSI ) DI = -CFR(-WI ) DC = CSQRT(RS/PI) * C E X P ( - C * P I / 4 . 0 ) DC = DC * ( DI*CEXP(-C*WI**2) - DR*CEXP(-C*WR**2) ) RETURN END  C C PROGRAM FOR CALCULATING FRESNEL INTEGRAL "CFR" OF COMPLEX ARGUMENTS. C s E B E E E S e E S S S B C C E E E E S E E e S B e e C E S S E B E E S C C e E e B C e z S S C E B e e C S E S E E S C Z S S E E S E S :  C  C  C C  COMPLEX FUNCTION CFR(X) == ========= ============ COMMON C ,PI COMPLEX*B C ,X COMPLEX*16 Z , ERFZ Z «= X * C E X P ( - C * P I / 4 . 0 ) CALL CERF(Z,ERFZ) CFR = 0 . 5 * S Q R T ( P I ) * C E X P ( C * P l / 4 . 0 ) * ( 1 . 0 - E R F Z ) RETURN END PROGRAM FOR CALCULATING FRESNEL INTEGRAL "FR" OF REAL ARGUMENTS.  CEEEEEEEEBEBEEEEEE.EEEEEEEEE^EEZEEBEEEEEEBEBBEEEEBEEEEEEBEEEEEEEEEEEE  C  C  COMPLEX FUNCTION F R ( X ) «"="==•=="" === = = COMMON C ,PI ,THSI COMPLEX*8 C COMPLEX*16 Z , ERFZ  Z = X*CEXP(-C*PI/4.0) CALL CERF(Z,ERFZ) FR = 0.5*SQRT(PI)*CEXP(C*Pl/4.0)*(1.0-ERFZ) RETURN END Q ********************************************************* £*********************************************************************  -  G.7  131 -  Beam Diffraction by a Parabolic Reflector  The following program uses (E17) to calculate the diffracted field component for a beam of parameters (kb=16, 0=180°), located at (kr =0, o  conducting parabolic reflector (kF=107r, <? =60°). e  0 =°°) incident on a O  Also, the program can be used to  calculate the reflected and total far fields using (E5), (E4) and (E17), without using the computer search technique.  -  132 -  C********************************************************************* C PROGRAM CALCULATES ANTENNA BEAM DIFFRACTION BY A PARABOLIC REFLECTOR C NO COMPUTER SEARCH FOR REFLECTION POINTS. C THE PROGRAM I S CALLED "PRBL.FTN,NS". Q  = c c = = = = = t = c c c c = : = = B C = = r: = = E = = = =:r:c:r:=; = c = = = = = = = = s: = = = — = =  C C  C  TIME DEPENDCE " E x p ( - i w t ) ". COMMON FACTOR " E x p ( i k r ) / S q r t ( k r ) COMPLEX*8 COMPLEX*8 COMPLEX*8 COMPLEX*8 REAL* 4  Q  C  C PI DTR FO  = = « =  = = C C = = ESI= = = = =: = = =:=;—  =  ".  C ,UI UR UD1 ,UD2 ,UD ,UT ,CD ,CFR ,CASIN ,CACOS THP ,THS ,THSP ,THS1 ,THS2 ,THI TH1P ,THS1P ,WP FO ,RI ,RR ,RS ,RP ,RSP ,RS1 ,RS2 ,RC ,RX ,XP ,YP XS ,YS 2 ( 3 6 1 ) ,AUI(361) ,AUR(361) ,AUD(361) , A U T ( 3 6 l ) r  r  f  KK = 181 H =1.0 ======= CMPLX(0.0,1.0) 3.1415926 PI/180. CEXP(-C*PI/4.)/SQRT(PI)  RO = 0.0 THO = 0.0 F = 10*PI THE = P I / 3 . 0 BET = PI+TH0 B = 16.0 C ============================================================== C BW I S THE HALF-POWER BEAM WIDTH C ============================================================== CC BW «= 66.*DTR CC B 0.5*ALOG(2.0)/(1.0 - COS(BW/2.0)) C THT = ( P I - T H E ) / 2 . RE = 2*F/(1.0+COS(THE)) R01 = SQRT( RE**2 + R0**2 + 2*RE*R0*COS(THE+TH0) ) TH01= THT + ASIN( R0*SIN(THE+TH0) /R01 ) C I F ( BET .EQ. THO+PI ) GO TO 11 RS = CSQRT( R0**2 - 2*C*R0*B*COS(BET-TH0) - B**2 ) I F ( REAL(RS) .LE. 0.0 ) RS = -RS THS = CCOS( ( R0*COS(TH0) + C*B*COS(BET) ) /RS ) GO TO 22 C 11 RS = R0 - C*B THS * THO C 22 RS1 = CSQRT( RE**2 + RS**2 + 2*RE*RS*CCOS(THE+THS) ) I F ( REAL(RS1) .LE. 0.0 ) RSI = -RS1 THS1= THT + CASIN( RS*CSIN(THE+THS) /RS1 ) RS2 = RSI THS2= THS1 C ============================================================== C T H B I , THBR ARE THE SHADOW AND REFLECTION BOUNDARY ANGLES C • THBI= TH01 + P I + ACOS(REAL(RS1)/R01) I F ( BET .GT. (TH01+THT) ) THBI = 2*(PI+TH01) - THBI THBR= 2*PI - THBI  = 1  - 133 Q  C  C  33  B B S S B S S S S E E S S S B S E  BIG = 0.0 DO 111 K = 1,KK 2(K)= H*(K-1) TH «= Z(K)*DTR TH 1 IF( TH2 IF(  = (PI-THT) + TH TH1 .GT. 2.0*PI ) «= (PI-THT) - TH TH2 .LT. 0.0 )  TH1 = TH1 - 2.0*PI TH2 = TH2 + 2.0*PI  UD1 «= FO * CEXP( C*RE*COS(TH+THE) ) * CD(RS1 ,THS1 ,TH1 ) I F ( (TH .GT. PI/2.).AND.(TH .LT. ( P I - T H T ) ) ) G O T O 33 UD2 = FO * CEXP( C*RE*COS(TH-THE) ) * CD(RS2,THS2,TH2) GO TO 44 UD2 = 0.0  c  44  UD = UD1 + UD2 AUD(K)= CABS(UD) I F ( AUD(K) .GT. B I G )  C  TO CALCULATE  BIG = AUD(K)  THE DIFFRACTED F I E L D COMPONENT ONLY GO TO 111  GO TO 111 C 110 C C  C  C  C  I F ( TH .GT. (PI-THE) ) GO TO 55 THR « TH THP «=-.5 * ( THS+THR + CASIN(4.*F*SIN(THR)/RS + CSIN(THS+THR)) ) RP - 2.*F/(1.+CCOS(THP)) I F ( REAL(RP) .LE. 0.0 ) RP  RSP = CSQRT( RP**2 + RS**2 + 2.*RP*RS*CCOS(THP+THS) ) I F ( REAL(RSP) .LE. 0.0 ) RSP = -RSP THSP= THR+THP TH 1P «= ( P I + T H P ) / 2 . + TH THS1P= ( P I + T H P ) / 2 . - THSP WP « -CSQRT(2.*RSP) * CCOS((TH1P+THS1P)/2.) RWP = REAL( WP * C E X P ( - C * P I / 4 . ) ) I F ( RWP .GT. 0.0 ) GO TO 55 THI IF( RC RX  = THP/2.0 - THSP R E A L ( T H I ) .LT. 0. ) THI =-THI = 2.0*F/CCOS(THP/2.0)**3 = -RC*CCOS(THI)*RSP/( 2.0*RSP-RC*CCOS(THI)  UR 55 66 CC C  77 88 C  * -RP  - CEXP( C M RSP + RP*CCOS(TH+THP) ) GO TO 66 UR « 0.0 AUR(K)= CABS(UR) I F ( AUR(K) .GT. B I G ) BIG «= AUR(K)  )  I F ( THI .GT. THBI ) GO TO 77 UI « CEXP( B*COS(TH-BET) - C*R0*COS(TH-TH0) ) GO TO 88 UI - 0.0 AUI(K)= CABS(UI) UT  * U I + UR + UD  )  -  CC C 111 Q CC CC CC 222  134  -  AUT(K)= CABS(UT) I F ( AUT(K) .GT. BIG ) BIG •= AUT{K) . CONTINUE c=c=ec=cec DO 222 K *= 1 , KK UIN = A U I ( K ) / B I G URN • AUR(K)/BIG UDN = AUD(K)/BIG UTN = AUT(K)/BIG WRITE(6,1) Z(K) ,UDN  c  1  FORMAT( F6.1 ,1X, E14.7 ) STOP END  Ccesccccc=cEecccecsnctci=s:crce=c=ccccccc==c=c=s=c==i:c=s=cc=scc=====r=  C PROGRAM CALCULATES THE DIFFRACTED F I E L D COMPONENT BY A HALF SCREEN. COMPLEX Q  FUNCTION  CD(R0,TH0,TH)  KSBSBSeBS&&&BSS&SSSSeBSS&SESSBS'S  COMPLEX*8  C ,TH0  ,R0  ,WI  ,WR ,EI ,ER ,DI ,DR  ,CFR  C PI C  « 3.1415926 * CMPLX(0.0,1.0)  EI ER WI WR RWI RWR  • CEXP( -C*R0 *CCOS(TH-TH0) ) « CEXP( -C*R0 *CCOS(TH+TH0) ) « - C S Q R T ( 2 . 0 * R O ) * CCOS((TH-THO)/2.0) — C S Q R T ( 2 . 0 * R 0 ) * CCOS((TH+THO)/2.0) « REAL( WI * C E X P ( - C * P l / 4 . ) ) - REAL( WR *CEXP(-C*PI/4.) )  C  C DI « CFR(WI) DR « CFR(WR) I F ( RWI .LT. 0.0 ) I F ( RWR .LT. 0.0 ) CD « EI*DI - ER*DR RETURN END  DI = -CFR(-WI) DR •= -CFR(-WR)  ^tSXBBBBBBBBBBESSBBBBSBBBSBBBBBBBSSBBSSeBBSSBBEC'eBBBeSBBBSSBBBBSBBSBBSS  C PROGRAM CALCULATES THE FRESNEL INTEGRAL "CFR" OF COMPLEX ARGUMENTS. Q  C t S - B S B E S S S B S S B B S S B B B S S E E S S S E S S S S & e S S E S & S S S B S B S S S S & e S S S S B e s S S S E E E S S S E E  COMPLEX £  FUNCTION  CFR(X)  B S S S B S S S S B B B B B B B S B S S E E B E S E  COMPLEX*8 COMPLEX*16  C, X Z , ERFZ  c  PI C  « 3.1415926 « CMPLX(0.0,1.0)  C Z » X * CEXP( - C * P l / 4 . 0 ) CALL CERF(Z,ERFZ) CFR * 0.5 * SQRT(PI) * CEXP(C*PI/4.0) * (1.0 - ERFZ) RETURN END C********************************************************************* C*********************************************************************  

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