RADIATION FROM COUPLED OPEN-ENDED WAVEGUIDES by Peter F. D r i e s s e n B.Sc., U n i v e r s i t y of B r i t i s h Columbia, 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Electrical We accept t h i s to THE Engineering t h e s i s as conforming the r e q u i r e d standard UNIVERSITY OF BRITISH COLUMBIA February 1981 © Peter Frank D r i e s s e n , 1981 In presenting requirements of B r i t i s h i t freely agree for this f o r an a v a i l a b l e that I understood that f i n a n c i a l copying her or s h a l l The 2075 U n i v e r s i t y Wesbrook Vancouver, V6T Date DE-6 (2/79) of Electrical of Canada 1W5 29 A p r i 1 1981 the L i b r a r y copying p u b l i c a t i o n the be allowed Engineering Columbia of the s h a l l and study. by of University I of this It this without make further head representatives. not B r i t i s h Place at granted permission. Department f u l f i l m e n t the extensive may be by h i s or gain that reference f o r purposes or degree agree f o r permission scholarly i n p a r t i a l advanced Columbia, department for thesis thesis o f my i s thesis my written i i ABSTRACT Ray-optical between open-ended radiation plate methods methods larger arrays, of with require finite only extensive to used to parallel-plate patterns waveguides are take calculate waveguides, arrays the central ray tracing, into the account of as coupling well coupled guide as p a r a l l e l - driven. These particularly the many the for the possible ray paths. The guides coupling in coefficients isolation previously derived groundplanes etc., of the The parallel-plate the five aperture staggered plane array agreed well presence a general coefficients an patterns H-plane to of the and three with those other lack of guides, sensitivity details arrays plane, as with the patterns could depth, and optical methods number may antennas of for outer the be a be with horn to Radiation with well the thus compared array. element with of were sectoral waveguide well variety waveguide the two of the structure. using and both remarkably indicating calculated patterns three in coupling surrounding agreed between a l l as patterns of not outer useful variety of by in the both edges the the aperture A ' wide width, array. development applications. in element patterns. the the of three varying guides in a in experimental obtained simulate waveguide that edges experimental Rayof iii TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES v LIST OF FIGURES vi ACKNOWLEDGEMENTS x 1 1 I n t r o d u c t i o n And M o t i v a t i o n 1.1 1.2 1.3 1.4 1.5 2 3 5 6 1 3 8 10 12 Method Of A n a l y s i s 19 2.1 2.2 2.3 2.4 19 21 24 28 General D e s c r i p t i o n C a n o n i c a l Problems L i m i t a t i o n s Of The Method Summary Coupling 3.1 3.2 3.3 3.4 3.5 3.6 4 Introduction L i t e r a t u r e Review Approach To The Problem Experimental Arrangement Summary Between Two Adjacent Waveguides Formulation Single D i f f r a c t i o n Multiple Diffraction C a l c u l a t i o n Of The Coupling Numerical R e s u l t s Summary Coupling Between Separated Coefficient Guides 31 31 34 34 38 42 46 58 4.1 C a l c u l a t i o n Of The C o u p l i n g C o e f f i c i e n t 4.2 A n a l y t i c a l And Numerical R e s u l t s 4.3 Summary 58 64 68 R a d i a t i o n P a t t e r n Of A S i n g l e Guide 81 5.1 5.2 5.3 5.4 5.5 81 81 84 89 90 Introduction C a n o n i c a l Problems C a l c u l a t i o n Of R a d i a t i o n P a t t e r n Numerical And Experimental R e s u l t s Summary R a d i a t i o n P a t t e r n Of Three Element Waveguide Array 6.1 I n t r o d u c t i o n 6.2 R a y - o p t i c a l Formulation 6.3 R a d i a t i o n P a t t e r n With Guides Of I n f i n i t e Depth . 96 96 97 98 iv 6.4 6.5 6.6 7 8 9 10 R a d i a t i o n With Outer Guides Shorted Numerical And Experimental R e s u l t s Summary R a d i a t i o n P a t t e r n Of F i v e Element 104 107 109 Waveguide Array .. 120 7.1 7.2 R a y - o p t i c a l Formulation R a d i a t i o n P a t t e r n With Outer Guides Of I n f i n i t e Depth 7.3 R a d i a t i o n With Outer Guides Shorted 7.4 Numerical And Experimental R e s u l t s 7.5 Summary 120 C o u p l i n g Between Staggered Guides 148 8.1 8.2 8.3 8.4 148 150 154 155 C a n o n i c a l Problem C a l c u l a t i o n Of The Coupling C o e f f i c i e n t Numerical R e s u l t s Summary 121 125 129 132 R a d i a t i o n From A Staggered P a r a l l e l P l a t e Waveguide 163 9.1 9.2 9.3 9.4 163 164 171 172 Formulation C a l c u l a t i o n Of R a d i a t i o n P a t t e r n Numerical And Experimental R e s u l t s Summary R a d i a t i o n From Array Multi-element Staggered Waveguide 184 10.1 I n t r o d u c t i o n 184 10.2 R a d i a t i o n With Outer Guides Of I n f i n i t e Depth . 185 10.3 R a d i a t i o n With Outer Guides Shorted 188 10.4 Numerical And Experimental R e s u l t s 192 10.5 Summary 194 11 General C o n c l u s i o n s And D i s c u s s i o n 11.1 11.2 11.3 Discussion Suggestions For F u r t h e r Work C o n c l u s i o n s In B r i e f 205 205 211 213 REFERENCES 215 APPENDICES 220 A B C D S c a t t e r e d F i e l d s Along A Shadow Boundary 220 Higher Order D i f f r a c t i o n Terms For Coupling Between Adjacent P a r a l l e l P l a t e Waveguides i n The TEM Mode 222 C o n t i n u i t y Of F i e l d s Across Shadow Boundaries . 227 D e s c r i p t i o n of Antenna P a t t e r n Range 229 V LIST OF TABLES table I II III page coupling c o e f f i c i e n t p a r a l l e l plate coupling c o e f f i c i e n t p a r a l l e l plate radiation between adjacent waveguides between waveguides pattern parameters • 57 separated 80 147 vi LIST OF FIGURES figure 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 2.1 2.2 2.3 2.4 2.5 3.1 3.2 3.3 3.4 3.5 3.6 page sector-shaped r a d i a t i o n p a t t e r n near optimum and t y p i c a l r a d i a t i o n p a t t e r n s for a feed antenna p a r a l l e l p l a t e waveguide a r r a y with d e s i r e d aperture f i e l d l i n e feed f o r a p a r a b o l i c c y l i n d e r r e f l e c t o r a r r a y of c o n c e n t r i c c i r c u l a r waveguides staggered a r r a y of p a r a l l e l p l a t e waveguides c r o s s - s e c t i o n of c o n c e n t r i c c i r c u l a r waveguide feed p a r a l l e l p l a t e waveguide two separated p a r a l l e l p l a t e waveguides staggered p a r a l l e l p l a t e waveguide f l a n g e d p a r a l l e l p l a t e waveguides two separated f l a n g e d p a r a l l e l p l a t e waveguides f i n i t e a r r a y of p a r a l l e l p l a t e waveguides embedded i n a groundplane i n f i n i t e a r r a y of p a r a l l e l p l a t e waveguides f i n i t e a r r a y of p a r a l l e l p l a t e waveguides embedded i n a simulated groundplane f i n i t e a r r a y of p a r a l l e l p l a t e waveguides with t h i c k w a l l s embedded i n a groundplane f i n i t e a r r a y of p a r a l l e l p l a t e waveguides in i s o l a t i o n N - f u r c a t e d waveguide d i f f r a c t e d rays two adjacent p a r a l l e l p l a t e waveguides two adjacent staggered p a r a l l e l p l a t e waveguides H-plane s e c t o r a l horn 13 i n c i d e n t and r e f l e c t e d shadow boundaries i n c i d e n t , r e f l e c t e d and d i f f r a c t e d f i e l d s l i n e source i n c i d e n t on a h a l f - p l a n e l i n e source i n c i d e n t on two p a r a l l e l half-planes l i n e source i n c i d e n t on two staggered half-planes 29 29 30 ray paths of mode f i e l d s i n two adjacent p a r a l l e l p l a t e waveguides ray paths from the d r i v e n guide to the p a r a s i t i c guide two s u c c e s s i v e rays i n a ray path from the d r i v e n guide t o a guide a p e r t u r e two s u c c e s s i v e rays i n a ray path, both rays in a guide a p e r t u r e two s u c c e s s i v e rays i n a ray path from the guide aperture to the p a r a s i t i c guide TEM-TEM c o u p l i n g between adjacent waveguides (a=d) 13 13 14 14 14 14 15 15 15 15 16 16 16 16 17 17 17 18 17 17 18 30 30 48 48 49 49 49 50 vii 3.7 TEM-TEM coupling between adjacent waveguides coupling between adjacent waveguides (a=d/2) 3.8 TEM-TEM 52 (a=2d) 3.9 TE ,-TE| coupling between adjacent waveguides (a=d) coupling coupling coupling coupling c o e f f i c c o e f f i c at low at low i e n t (a=d, N=0) i e n t (a=d, N=l) frequencies' frequencies 3.10 3.11 3.12 3.13 TEM-TMr, TEi-TE n TEM-TEM TEi-TEi 4.1 r a y p a t h s o f mode f i e l d s i n two s e p a r a t e d p a r a l l e l plate waveguides ray paths from the d r i v e n guide to the p a r a s i t i c guide TEM-TEM c o u p l i n g b e t w e e n s e p a r a t e d w a v e g u i d e s (a = b = 0 . 3 3 8 7 0 TEM-TEM c o u p l i n g b e t w e e n s e p a r a t e d w a v e g u i d e s (a=b=0.761X) TEi-TE / c o u p l ng b e t w e e n s e p a r a t e d w a v e g u i d e s (a=b=0.761X) TEM-TEM c o u p l n g b e t w e e n s e p a r a t e d w a v e g u i d e s (d=a=b) TEM-TEM c o u p l i n g b e t w e e n s e p a r a t e d w a v e g u i d e s (a=2d, b=d) TEM-TEM c o u p l : n g b e t w e e n s e p a r a t e d w a v e g u i d e s (a=d, b=2d) TEM-TEM c o u p l i n g b e t w e e n s e p a r a t e d w a v e g u : d e s 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 (a=d/2,b=d) TE,-TE | coupling wave staggered plane 5.6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 ray paths ray path normally p a r a l l e l wave staggered 5.1 5.2 5.3 5.4 5.5 between separated waveguides (a=d=b) plane 51 normally p a r a l l e l in (see ray path (see continuity of ray-optical a a p a r a l l e l pl experimental a p a r a l l e l pl a incident on two n o n - on three plates incident non- plates p a r a l l e l plate waveguide text) text) f i e l d s across nd exact radia ate waveguide and exact r a d i ate waveguide shadow b o u n d a r y tion patterns for (a = 0.45"X) ation patterns for (a=0.45M three element array of p a r a l l e l plate waveguides with outer guides shorted three element array of p a r a l l e l plate waveguides with outer guides of i n f i n i t e depth ray paths from c e n t r a l g u i d e s to edge 1 ray paths from c e n t r a l g u i d e s t o edge 3 ray paths from outer g u i d e s t o edge 1 ray paths from outer g u i d e to edge 3 radiation pattern of three element array with outer guides of i n f i n i t e depth (a=d=0.407X) radiation pattern of three element array 53 54 55 56 56 70 70 71 72 73 74 75 76 77 78 79 79 91 91 91 92 93 95 111 111 112 112 112 112 113 viii a=d=0.450A s=0.856A A=0.131*0 r a d i a t i o n p a t t e r n of t h r e e element a r r a y a=d=0.339X s=0.645* A=0. 371 £. -171° r a d i a t i o n p a t t e r n of t h r e e element a r r a y a=d=0.356X s=0.677A A=0.308 ^-134° r a d i a t i o n p a t t e r n of t h r e e element a r r a y a=d=0.373X s=0.709A A=0.247 L +104° r a d i a t i o n p a t t e r n of t h r e e element a r r a y a=d=0.389X s=0.459X A=0.199^+38° r a d i a t i o n p a t t e r n of t h r e e element a r r a y a=d=0.44lA s = 0.597A A=0. 223 +166* U 6.9 6.10 6.11. 6.12 6.13 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 f i v e element a r r a y of p a r a l l e l p l a t e waveguides w i t h o u t e r g u i d e s s h o r t e d f i v e element a r r a y of p a r a l l e l p l a t e waveguides w i t h o u t e r g u i d e s of i n f i n i t e depth r a y p a t h (see t e x t ) r a y p a t h s (see t e x t ) r a y p a t h s (see t e x t ) r a y p a t h s (see t e x t ) r a y p a t h s (see t e x t ) r a d i a t i o n p a t t e r n of f i v e element a r r a y w i t h o u t e r g u i d e s of i n f i n i t e depth d=a=b=0.450X r a d i a t i o n p a t t e r n of f i v e element a r r a y w i t h o u t e r g u i d e s of i n f i n i t e depth d=a=b=0.339X r a d i a t i o n p a t t e r n of f i v e element a r r a y w i t h o u t e r g u i d e s of i n f i n i t e depth d=a=b=0.450X comparison w i t h [38] r a d i a t i o n p a t t e r n of f i v e element a r r a y w i t h o u t e r g u i d e s of i n f i n i t e depth d f a = b=0.450X comparison w i t h [29] r a d i a t i o n p a t t e r n of f i v e element a r r a y d=a=b=0.441^ s,=0.838X s =0.597X A , = 0.137^ -12° A =0.074 ^-12° r a d i a t i o n p a t t e r n of f i v e element a r r a y d=a=b=0.407X s,=0.774* s = 0 . 5 5 l X z 7.13 7.14 2 A,=0.171 ^-55° 2 A =0.076 ^-57° 2 r a d i a t i o n p a t t e r n of f i v e element a r r a y d=a=b=0. 373 >s s,=0.709A s =0.505> A, =0.247 ^-103° A =0.096^ -98° r a d i a t i o n p a t t e r n of f i v e element a r r a y d=a=b=0.356X s,=0.677X S!=0.487X 2 7.15 7.16 2 A,=0.308 C -134° A =0.126^ -123° 2 a p e r t u r e f i e l d and r a d i a t i o n p a t t e r n of a f i v e element a r r a y A,=-0.215 A =-0.090 d=a = b=0. 450X S,=0.610X Si=0.356X a p e r t u r e f i e l d and r a d i a t i o n p a t t e r n of a f i v e element a r r a y A,=+0.131 A =+0.073 d=a = b=0. 450X s,=0. 8577\ s = 0.610X a p e r t u r e f i e l d and r a d i a t i o n p a t t e r n of a f i v e element a r r a y A,=-0.131 A =+0.045 d=a = b=0. 450A S,=0 . 857X S = 0 .356X 2 7.17 2 2 7.18 2 2 8.1 8.2 two a d j a c e n t s t a g g e r e d p a r a l l e l p l a t e waveguides l i n e source and p l a n e wave i n c i d e n t on a 114 115 116 117 118 119 134 134 135 135 135 136 136 137 138 139 139 140 141 142 '143 144 145 146 156 ix 8.3 8.4 half-plane ray paths for ray path (see 8.5 TEM-TEM 9.1 9.2 9.3 staggered p a r a l l e l plate waveguide ray paths for radiation pattern calculation shadow b o u n d a r i e s for a staggered p a r a l l e l plate waveguide 174 174 9.4 9.5 c o n t i n u i t y of f i e l d s a c r o s s shadow b o u n d a r i e s staggered p a r a l l e l plate waveguide radiation pattern of staggered waveguide ^=45° radiation pattern of staggered w a v e g u i d e ^=45° single and double d i f f r a c t i o n only radiation pattern of staggered waveguide for various stagger angles 175 176 9.6 9.7 9.8 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 coupling 10.10 10.11 10.12 10.13 10.14 156 157 157 c o e f f i c i e n t between staggered array of staggered p a r a l l e l pl with outer guides shorted array of staggered p a r a l l e l pl with outer guides of i n f i n i t e r a y p a t h f o r c a l c u l a t i n g S, ray paths for calculating S ray paths for calculating S3 shadow b o u n d a r i e s f o r an a r r a y p a r a l l e l plate waveguides ray paths (see text) radiation pattern of staggered with outer guides of i n f i n i t e ate waveguides V =45° of waveguide depth array ' waveguide array radiation pattern of staggered waveguide array radiation pattern of staggered waveguide array radiation pattern of staggered waveguide array radiation pattern of staggered waveguide array radiation pattern of staqgered waveguide arrays d=a=0.407* d = a = 0.373?\ d=a=0.339> d=a=0.450A s=1.083> s=1.000X s = 0.916A s=0.833A s=1.107A ^=45° y=45° 1^ = 4 5 ° ^=45° 183 195 196 196 196 198 staggered d=a=0.441> 182 197 197 of Y = 45° 177 staggered pattern s = 1.166A 175 195 ate waveguides depth radiation d = a = 0.4747v 158 waveguides 3 d=a=0.450A 10.9 coupling text) 199 200 201 202 203 204 X ACKNOWLEDGEMENTS The author perceived thesis this and work is thankful to investigation, his many valuable Dr. for Edward his comments V. J u l l , who f i r s t of this supervision and suggestions as the progressed. Thanks the sectoral of the also horn antenna, National E l e c t r i c a l range, are to to Dave to Dr. Research Engineering and due for John Fletcher J.Y. Council of permission Hazell Wong to for for and constructing W. Canada use Lavrench Division their a s s i s t i n g of antenna with the measurements. The support Council author of of the Canada. g r a t e f u l l y National acknowledges Sciences and the Engineering f i n a n c i a l Research 1 Chapter INTRODUCTION AND MOTIVATION INTRODUCTION 1.1 The c a l c u l a t i o n coupled open-ended of whose solution arrays with waveguide have been phased used in arrays amplitude and p a r a s i t i c dimensions. phase alter radiation the ignored. The modified by p a r a s i t i c possible phased array , to between them. For a complexity One as with a power the many of each and a each fixed new width in the fixed the is parabolic and antenna the which array depth" w i l l can of than by the range of and separately is were be the guide. p a r a s i t i c coupled the coupling less s h i f t e r s a the driven adequate, element including f i e l d s and somewhat arrays arrays, the central of antenna a l t e r i n g by parasitic a basic element, of phase a p p l i c a t i o n by if is array feeding for in purposes d i v i d e r s antenna changed in a antenna that energy element design parasitic a is of is number, p a r a s i t i c possible feed from adjustment the excite pattern the driven be and w i l l is a p p l i c a t i o n s , pattern phased because the can fields waveguides Waveguide of surrounding of in f i e l d s pattern changing range associated the radiation r e l a t i v e added of elements guides available pattern both between assist elements. whose coupling electromagnetic variety whose In may a arrays the p a r a l l e l - p l a t e problem is 1 The for a elements coupling adjustment thus the with the avoided. waveguide r e f l e c t o r . For array this 2 case, a sector-shaped desirable, level radiation i . e . a p a t t e r n which p r o v i d e s over a given angular ( F i g . 1 . 1 ) . A sector-shaped f o r space a t t e n u a t i o n ) reflector surface range pattern because it noise from ratio, the the and figure Thus of merit power i s zero elsewhere (except the but does not entire spill maximum d i r e c t i v i t y i n t e r f e r e n c e pickup ground. usually constant illuminates almost u n i f o r m l y noise and a is i s n e a r l y optimum the s i d e s ( F i g . 1 . 2 ) . T h i s p r o v i d e s avoiding pattern the i n c l u d i n g the gain/noise over while thermal temperature for a s a t e l l i t e earth s t a t i o n antenna, i s maximized. A p a r a l l e l - p l a t e waveguide a r r a y with a small number of elements ( F i g . 1 . 3 ) p o t e n t i a l l y s a t i s f i e s these for end two-dimensional of the a r r a y may f i e l d s . The aperture f i e l d at the open be made to approximate a t r u n c a t e d x)/x curve by a d j u s t i n g the a r r a y parameters and number of parasitic d r i v e n g u i d e ) . The transform of The directly cylinder reflectors a (Fig.1.5). good p a t t e r n s [25]. by depth the c e n t r a l the Fourier f i e l d , w i l l then approximate this parallel-plate to the design guide radiation pattern guides surrounding (sin the shape. parameters of as (width, r a d i a t i o n p a t t e r n , which i s applied used guides the aperture required sector requirements array of l i n e feeds with Such achieving the an of array same adjustment concentric of also be sector-shaped an antenna has produced empirical be for parabolic ( F i g . 1 . 4 ) . These parameters may in may the circular reasonably parameters 3 Good patterns have a l s o been obtained using an a r r a y where the edges are not a l l i n the same plane, but staggered slightly [13] ( F i g . 1 . 6 ) . Measurements of various feed antennas [15] has shown that there i s l i t t l e in the resulting c i r c u l a r guides [15] pattern of an of difference a r r a y of staggered (Fig.1.6) and a f o u r - r i n g c o r r u g a t e d s u r f a c e (Fig.1.7) when both amount between types stagger arrays are the same size. The i s an a d d i t i o n a l parameter which can be a d j u s t e d to optimize the p a t t e r n . The design of waveguide feeds with has been array of parallel-plate of optimum f o r two dimensional analysis elements mainly e m p i r i c a l to date. Thus the a n a l y s i s of an development least parasitic could be waveguides feeds for may assist in the r e f l e c t o r antennas, at s t r u c t u r e s . The results of this a p p l i e d not only to feed d e s i g n , but to any coupled waveguide antenna. 1.2 LITERATURE REVIEW Many different waveguide structures types of open-ended parallel-plate have been analyzed using a v a r i e t y of methods. The s t r u c t u r e s i n c l u d e i n f i n i t e a r r a y s of p a r a l l e l p l a t e waveguides, f i n i t e a r r a y s i n i s o l a t i o n or embedded a groundplane in or other s u r f a c e ( F i g s . 1.8-1.17), and c l o s e d r e g i o n problems i n c l u d i n g b i , t r i and N-furcated waveguides (Fig.1.18). The fields always be found in coupled open-ended r i g o r o u s l y except for waveguides cannot certain geometries, 4 because t h e r e s u l t i n g boundary v a l u e problem does not have a known c l o s e d form s o l u t i o n . Hence f o r many cases approximate methods have been- developed When t o o b t a i n the f i e l d s . r i g o r o u s s o l u t i o n s can be found, the methods used i n c l u d e both the Wiener-Hopf techniques [ 3 7 ] . These technique methods can or infinite matrix mode matching a l s o be used approximate s o l u t i o n s f o r those cases integral and where equations the to f i n d resulting cannot be s o l v e d e x a c t l y . Approximate s o l u t i o n s a r e a l s o obtained u s i n g r a y optical techniques diffraction theoretic based on [26], m o d i f i e d techniques the geometrical residue calculus theory and of function [38], and numerical techniques based on moment methods [ 2 1 ] . The choice of method t o be a p p l i e d to a given problem depends on the p a r t i c u l a r generally limited methods are suitable dimensions in relatively geometries, for methods small s t r u c t u r e s . (Fig.1.8) has been The the and Heins be used literature parallel used plate with any as a test of parallel- in isolation below. waveguide geometry and r a y - o p t i c a l methods, p a r t i c u l a r l y refinements of r a y - o p t i c a l t e c h n i q u e s . for whose p r i n c i p l e , but are g e n e r a l l y s u i t a b l e only f o r single numerical are ray-optical structures can p l a t e waveguide s t r u c t u r e s i s reviewed A methods are on the order of a wavelength or g r e a t e r , and numerical geometry Rigorous t o very s p e c i f i c most characteristic geometry. The f o r exact, for several exact solution r e f l e c t i o n c o e f f i c i e n t was given by Weinstein [45] [22] using the Wiener-Hopf technique. The 5 reflection and c o e f f i c i e n t was found using moment methods by Wu Chow [48] and was [20]. Montgomery and residue c a l c u l u s . reflection improved Chang Rudduck coefficient by Gardiol Haldemann [38] found i t using and Tsai using a modified [40] c a l c u l a t e d the r a y - o p t i c a l procedure but depending a l s o on a r e c i p r o c i t y argument. Keller and Yee, Felsen [50] found the r e f l e c t i o n c o e f f i c i e n t by t r a c i n g rays directly and using a ray-to-mode c o n v e r s i o n improved t h e i r s o l u t i o n f o r low frequencies [49]. [8] out Bowman [9] p o i n t e d that formula. in the exact Boersma They [18] and the r a y - o p t i c a l s o l u t i o n of [50] d i d not agree with the asymptotic form of s o l u t i o n . The r a y - o p t i c a l s o l u t i o n was improved by [3] Ahluwalia [4] using et improvement asymptotic a and the al.[l], uniform Boersma's asymptotic results theory of showed much i n the mode, t r a n s i t i o n regions and recovered the form of the exact modified s o l u t i o n . Lee [30] [31] d e r i v e d diffraction coefficient which includes i n t e r a c t i o n s between the two h a l f - p l a n e edges a u t o m a t i c a l l y , so that these interactions c a l c u l a t e d . T h i s method reflection theory recovered coefficient. waveguides (Fig.1.9) by waveguide was and radiation Tsai methods to be explicitly s o l u t i o n f o r the edge [17] and diffraction a l s o using and Chang [ 3 8 ] . pattern given by Weinstein the Wiener-Hopf technique. ray-optical the exact was c a l c u l a t e d using Dybdal,Rudduck exact not The c o u p l i n g between two separated r e s i d u e c a l c u l u s by Montgomery The need a parallel plate [45] and Heins [22] using Yee and derive of a Felsen [19] [49] used r a d i a t i o n p a t t e r n which 6 agreed well plane. Rudduck aperture [30] with plane the exact and Wu by recovered using the d i f f r a c t i o n gain calculated The f i e l d s plate waveguide from outside [32] using modified for a a the line spectral pattern by a surrounded in reflection an waveguide with by Keller using his of [37] [50] modified the using Weiner-Hopf by a and this of f i e l d s and a p a r a l l e l - incident Lee and theory it Boersma the the f i e l d s p a r a l l e l plates plane wave Mittra problem. wedge on and found incident [39] The use a radiation d i f f r a c t i o n embedded from angle and a and was have and and techniques, by The by a flanged (Fig.1.11b) and calculated been embedded [40] methods also waveguide flange Tsai groundplane methods. (Fig.1.11a) of a ray-optical single c o e f f i c i e n t . was in waveguides numerical ray-optical waveguide of staggered using groundplane d i f f r a c t i o n flanged for other Rudduck Lee [42]. c o e f f i c i e n t calculated two waveguides arbitrary the method aperture also considers guide by his normally They Rahmat-Samii Rudduck in c o e f f i c i e n t . asymptotic on also rigorous, i n f i n i t e is c o e f f i c i e n t . plate The aperture pattern using calculated uniform [5] the boundary wave were approach and using shadow plane staggered Ryan analyzed The a guide. P a r a l l e l and/or the the [24]. incident Boersma of d i f f r a c t i o n J u l l the domain slope solution guide source staggered found when d i f f r a c t i o n (Fig.1.10). on by in improved c o e f f i c i e n t . along both a except [41] exact modified were solution Yee, Lee Felsen [30] radiation by Yee Mittra and was and using pattern and Lee Felsen [49] 7 using ray-optical d i f f r a c t i o n coefficient methods. The embedded in by by Lee an only Harrington the and with guide the Chang and a l l with a l l [36] found thick walls excited both only Mittra were a l . in a outer a array in for the [10] pattern groundplane guides in excited . Luzwick radiation and also embedded guide and and Weiner-Hopf excited. embedded and residue f i n i t e et been excited one a Burnside the by array for the i n f i n i t e has patterns with guides and the and technique guides f i n i t e [38] an modified with by adjacent and calculated (Fig.1.15) guides calculated between were a such was using for moment waveguides Radiation [29] using three Weiner-Hopf using modified and for a with reactively (Fig.1.16). The ( b i , t r i coupling and calculated technique residue array excited and [35] central two a [48] (Fig.1.13) [38] patterns Lee but array loaded Chang (Fig.1.17) array Chow plate using techniques. Montgomery f i n i t e [29] groundplane i s o l a t i o n f i n i t e p a r a l l e l guide by simulated The a i n f i n i t e one calculated same both Radiation mode-matching by [47]. of and with , in Montgomery for between and coupling Lee [37] Wu using Wu by calculus. by [30] (Fig.1.12) array calculated and Lee groundplane guides (Fig.1.14) ,by coupling a numerically separated methods between N-furcated by and Mittra by calculus. waveguides waveguide) and Montgomery Lee and [37] Chang in a closed (Fig.1.18) has using the [38] using region been Weiner-Hopf modified 8 1.3 APPROACH It array may appears in the waveguide these previous is methods structures results r e l a t i v e l y and require large size Ray-optical for d i f f r a c t i o n have as for wave d i f f r a c t i o n plane plane. 1.6 The of appear several to be Ray-optical the of order that the the f i e l d s geometrical geometrical is optics rays d i f f r a c t i o n asymptotic travel can in in of strike the of a the basis edges exact is of and more. (GTD) for for of solution h a l f and r a y - o p t i c a l a n a l y s i s . solve for d i f f r a c t i n g edges the structure methods c a l l e d is are conductors solution so to introduced the theory an are assume rays. The extension d i f f r a c t i o n rays of Fig.1.3 their These l i n e s numerical exact and of of hand. wavelengths. of for at conducting arrays edges these because the used straight accounts that of perfectly presence or successfully geometrical be d i f f r a c t e d the terms dimensions c o e f f i c i e n t form the d i f f r a c t i o n which that in here techniques. problem matrices choice wavelength theory postulated incident natural characteristic a array half-plane f i e l d s of large by the geometry, waveguide methods electromagnetic provided a expected canonical p a r a l l e l - p l a t e consist methods their ray-optical for waveguide considered be this using be analyzed useful the methods to plate been may very of 1.6) have be solution p a r a l l e l using it w i l l exact would and analyzed methods, no small (Fig.1.3 e f f e c t i v e l y Similar There PROBLEM that i s o l a t i o n be using TO THE [26]. produced It when (Fig.1.19). based d i f f r a c t i o n of on A the of a 9 plane wave by D i f f r a c t i o n is is a a local determined other to parts edge l i k e becomes shadow. [28] not of analysis than a A ray-optical corresponding to d i f f i c u l t y , 1.3 a for on appears a straight that the in asymptotic the but d i f f r a c t i o n between have may waves been with l i m i t , light and advocated some extended to f i e l d of or be [1] sacrifice to allow incident are is in on f i e l d s the required in the turn open f i e l d s in the h a l f - in the d i f f r a c t e d central driven guide. The tracing Incident that the end. many or by The of may edge. ray paths by guides these be traced source and f i e l d s surrounding f i e l d s The arrays the waveguide be The d i f f r a c t e d . d i f f r a c t e d parasitic magnitude an paths waveguide excited central ray f i e l d s from reflected Fields outer by p a r a l l e l - p l a t e require down excite begin d i f f r a c t e d again f i e l d s . travel may f i e l d s . be 1.6 the the been extensions surface, and at have the analysis represent d i f f r a c t e d dependent f i e l d which boundaries analysis f i e l d s transmitter not edge. to from F i g s . v a l i d d i f f r a c t e d half-plane of is theories These another ray-optical edge coefficient rays. GTD plane [6]. resulting is d i f f r a c t e d of the methods when reflected at and [7], a p p l i c a t i o n . general [50] The uniformly this Ray-optical plane in uniform overcome alone, source half-plane d i f f r a c t i o n d i f f r a c t i n g line i n f i n i t e s i m p l i c i t y more the Various to edge flaw is the structure. a major c o e f f i c i e n t and the conducting effect; the from looks One by of emanate perfectly may are w i l l the be 10 calculated. These be at reflected end. These the f i e l d s pattern the The 1- w i l l shorted d i f f r a c t e d d i f f r a c t e d is f i e l d s coupling end, and f i e l d s the process of a l l these thus divides between two down the outer d i f f r a c t e d w i l l and sum analysis travel in turn continues. d i f f r a c t e d naturally adjacent at guides, the open excite The more radiation f i e l d s . into several p a r a l l e l - p l a t e parts: waveguides (Fig.1.20) 2- coupling 3- r e f l e c t i o n 4- radiation 5- between two from the pattern radiation separated open of end single pattern of guides of a (Fig.1.9) single guide guide (Fig.1.8) (Fig.1.8) multi-element waveguide array (Fig.1.3) 6- coupling waveguides 7- between two adjacent staggered end single p a r a l l e l - p l a t e (Fig.1.20) r e f l e c t i o n from open of staggered guide (Fig.1.10) 8- radiation 9- radiation array of pattern single of staggered guide multi-element (Fig,1.10) staggered waveguide ( F i g . 1 . 6 ) . Parts 1,2,3 6,7, and been pattern and 8 are considered 4 are needed needed in the solutions presented improvements to these to to solve solve 9. l i t e r a t u r e , 5 and Parts s i m i l a r l y 2,3,4,7 however, here (parts earlier results. 2, and some 8) parts 8 have of the constitute 11 EXPERIMENTAL 1.4 It would comparing be numerical c o e f f i c i e n t s d i f f i c u l t array desirable to approximate of dimension 1.6) of the waveguide i s sectoral flared a is large i n one a two-dimensional aperture. Thus the H-plane sectoral horn approximates the p a r a l l e l - p l a t e outer The angle horn of WR-90 and of in same used 3 0 ° with with performed the for an waveguide. various at Appendix D. TE to i f in Figs.1.3 If a of an and H-plane the center pattern a the rectangular in the to [17]. measured radiation that is two-dimensional form in and structures mode 0 Appendages by possible a page 1 obtained- guide of the of an p a r a l l e l - p l a t e the horn simulate guides. the measurements aperture The outer depths. X-band E-plane width. of T E Mmode antenna of is may be dimension by coupling enough. (Fig.1.22), waveguide of but the analysis those patterns to the dimensional waveguides (perpendicular horn approximates two radiation guides to accurately, p a r a l l e l - p l a t e third confirm Measurement of perform to results measurement. r e f l e c t i o n However, very the experimental very ARRANGEMENT of guides Radiation on the 1.016 outdoor had a x cm, 50 were pattern antenna total flare a n d was f e d 1.016 cm wide measurements were range described 12 1 . 5 SUMMARY The of calculation p a r a l l e l designing and plate appears element arrays to to into compared H-plane to When by experiment of the analysis the because of expected to waveguide well parts however suited to here. The which The are the the numerical ray-optical three considered made various and analysis theoretical measurements the complete depth, p a r a l l e l optimize A wide the be is width, of to applications. structures, to in five divides in turn in patterns w i l l be with a wide angle horn. elements adjusted analyze been arrays useful used experimental the p o t e n t i a l l y have chapters. sectoral small Exact, considered several following is of elements. be be patterns waveguide waveguide method the with methods plate naturally radiation waveguides antennas ray-optical p a r a l l e l of useful antennas. the results number and amount plate pattern of patterns parameters for waveguide the variety many and the of of array for may a v a i l a b l e . design v e r i f i e d stagger may be particular be The several obtained results are types of 13 near optimum pattern typical pattern F i g . 1.1 sector-shaped radiation pattern F i g . 1.2 near optimum and t y p i c a l r a d i a t i o n f o r a feed antenna F i g . 1.3 p a r a l l e l p l a t e waveguide a r r a y with d e s i r e d aperture f i e l d patterns 14 F i g 1.4 l i n e feed f o r a p a r a b o l i c cylinder reflector F i g . 1.5 array of concentric waveguides circular F i g . 1.6 staggered array of p l a t e waveguides parallel Fig. 1.7 c r o s s - s e c t i o n of concentric circular waveguide feed parallel Fig. plate 1.8 waveguide F i g . 1.10 staggered p a r a l l e l plate waveguide Fig. 1.9 two s e p a r a t e d p a r a l l e l p l a t e waveguides Fig. l . H flanged p a r a l l e l p l a t e waveguides 16 I 1 F i g . 1.12 two separated f l a n g e d p a r a l l e l p l a t e waveguides F i g . 1.13 f i n i t e array of p a r a l l e l p l a t e waveguides embedded i n a groundplane F i g . 1.15 f i n i t e array of p a r a l l e l p l a t e waveguides embedded i n a s i m u l a t e d groundplane F i g . 1.14 i n f i n i t e a r r a y o^ p a r a l l e l p l a t e waveguides 17 Fig. 1.18 N-furcated waveguide Fig. 1.16 f i n i t e array of p a r a l l e l p l a t e waveguides w i t h t h i c k w a l l s embedded i n a g r o u n d p l a n e Fig. 1.20 two a d j a c e n t p a r a l l e l p l a t e waveguides Fig. 1.17 f i n i t e array of p a r a l l e l p l a t e waveguides i n i s o l a t i o n Fig. 1.21 two a d j a c e n t staggered p a r a l l e l p l a t e waveguides F i g . 1.19 d i f f r a c t e d rays 19 Chapter METHOD 2.1 GENERAL In to chapter calculate be the analysis obtained the case others of analysis mode in plane waves the the at d i f f r a c t e d analysis be w i l l and some are based parts of considered compared related which from again d i f f r a c t e d , the In adequate. from with in those geometries, and be to and edges, in practice These same the compared of single guide the is in the thus only line of from the from a the few source into open l i n e are of the source in f i e l d s process two end additional turn line are continues d i f f r a c t i o n s f i e l d s for the edge a f i e l d s [50] F i r s t an additional p r i n c i p l e in decomposed exciting These analysis guide. edges source the procedure emanate line edges. part kr>>l to These other f i e l d s i n d e f i n i t e l y . at driven appear edge. any a distances f i e l d s the for d i f f r a c t At of essentially incident source usually or the w i l l be patterns reflection waveguide. centred 1 of of results. the d i f f r a c t e d Chapter w i l l methods each used patterns general, ray-optical similar following of in procedure radiation in chapters results ray-optical by which later for or described radiation experimental begins be mentioned numerical by The In analysis c o e f f i c i e n t s upon discussed. The ray-optical w i l l problems w i l l turn. the coupling antennas canonical ANALYSIS DESCRIPTION this waveguide OF 2 also are excite 20 modes i n space. the outer The relative complex to the c o e f f i c i e n t f i e l d s the at can be coupling ray of of into between path of guides start traced i s each or in a to the a l l To radiation f i e l d s to when a a find the in one pattern, the and end up r a d i a t i n g the coupling produces start into ray strikes an of possible ray The number only source these path. would the possible but line corresponding ray guide the coupling point with path find the modes array. group ray To a l l distant associated i n d i v i d u a l l y . i n f i n i t e , s i g n i f i c a n t l y a into waveguide to T h e sum o f at radiate these waveguide the paths also corresponds rays, in another. would are paths rays groups, The alternate edge the and of observed sequence and end space. as pattern divided mode guides. t o t a l i t y p a r t i c u l a r guide incident edges radiation guides amplitudes between the The parasitic f i r s t few c o e f f i c i e n t s contribute or radiation patterns. A l l ray mentioned paths above contribution (Figs. and 3.2, 10.7). rays which f i r s t . sequence ray rays or the i n f i e l d ray be be or these In found. can 7.3-7.7, 8.3, by this way t h e From this the be 9.2, the w i l l any found of ray from f i e l d total problems and their calculated 10.3-10.5, computations, paths be of w i l l combinations ray then each e x p l i c i t l y these various for radiation represented path i t . traced 6.3-6.6, f a c i l i t a t e occur preceding can 5.1, by The considered coupling 4.2, represented be w i l l to To to the f i e l d s two successive be calculated in a p a r t i c u l a r . the f i e l d represented f i e l d of by solution, the a l l the 21 coupling This of approach two the coefficients gives successive following solutions each of Chapter 2.2 rays then to a a patterns series of sequence, as they used be derived. d i f f r a c t i o n problems which are to are required. assemble waveguide can The the problems l i s t e d solved in canonical solutions to at of the end 1. PROBLEMS Ray-optical methods solution for have plane conducting half-plane. Sommerfeld [43] and radiation in be coupled CANONICAL exact rise sections w i l l the and and their wave This was as d i f f r a c t i o n problem solved canonical was more by basis a o r i g i n a l l y simply by the perfectly solved Clemmow by [14] others. TM Consider the H half-plane x _ . = u. x k on is = 2TI/A it the at free suppressed, where presence of total the a — f i e l d - i k r COS(6-GQ) arbitrary space equations The and a is at a l l half-plane H angle 6 wavenumber. time w = 2r,f f i e l d ^ = e an subsequent wave. z>0 1 ^ incident y=0, is factor the In e angular points given Q ~ 1 L ) t (Fig.2.1), this is (x,y,z) in where and a l l implied frequency space ^ and of the in the by * * t - u t - G(r,e ,e), 0 (2.2) 22 /•,/ « «\ e G ( r , 6 ,6) = where r - l k r c o s ( 9 - S ) r A~T~~ x 0\ i — — {e °'F[-/2kr c o s ( — ^ ) ] /n n ± i x e 00 F(a) = J" a = /T7 e F(-a) and ) F [ _ ^ c o s ^ 0 ( ) ] } > ( dx, 2 > 3 ) + " / (2.A) (2.5) - (a), 4 F signs c a n be evaluated the Fresnel 0 2 X the top(bottom) result of 1 c o s ( 6 - e -ikr g refer to f o r kr>>l TM(TE) using f i e l d s . the asymptotic This form integral . 2 F(a) to „ i § ^ - , 1, a » obtain u where i s u 8 f e - i k = u t g + D(6 ,6)E(r) the geometrical r c o s e- i k r ( - o) e e ± , 0 e ~ i k r optics (2.7) f i e l d cos(e+e ) 0 f cos(6-6 ) A This of 6 asymptotic waves, e < ^_e o n i s the d i f f r a c t e d f i e l d +in/« e-e e+8 ( s e c ( - ^ ) ± s e c ( ~ ) } the incident plane < Q (2.8) 0 E(r)- e ^ ' / ^ i and 0 f o r Ti+e <6<2Ti D(e , e ) E ( r ) D(9 ,6) = - r , f,o r H-6Q<6<H+6Q u n _0 and o given by 7 qiven by (2.9) . < ' 2 result f i e l d and a shows and a that the total reflected d i f f r a c t e d f i e l d f i e l d f i e l d which which i s a 1 0 i s made u p are both c y l i n d r i c a l ' 23 wave ( F i g . 2 . 2 ) . coefficient 0 giving emanating E(r) edge thus c a n be D(e ,e) the from the appears interpreted as magnitude of half-plane edge. l i k e a l i n e the a d i f f r a c t i o n c y l i n d r i c a l The source of wave d i f f r a c t i n g the d i f f r a c t e d f i e l d . Unfortunately uniformly v a l i d , boundaries asymptotic and becomes between the (Fig.2.1). e = fr ± d i f f r a c t e d used the to f i e l d predict as the l i t expression i n f i n i t e and However, a line along shadowed (2.7) does source d i f f r a c t e d (2.7) at f i e l d the not shadow regions at characterize the away is edge from the a n d c a n be the shadow boundary. The expression introducing i n f i n i t y to a transition at e = n ± e rewriting edge. Then which looks for To (2.2) the predict line that at This r=a source uniformly precisely i s observation f i e l d the again the when f i e l d s has u =EO) f i e l d C f o r v v a a been i n t total [49]. Q be made i n point cancels fact at v a l i d by the equivalent r=a from the is f i e l d and i s when a uniformly v a l i d . d i f f r a c t e d It a can function f o r an total l i k e e a l l (2.7) f i e l d s second line shown the time source [7] requires i s that presence k(r +r)>>l i s 0 of an d i f f r a c t e d f o r an a by a isotropic by /Q r r + R f i e l d expression half-plane given /k(r +r+R) 0 d i f f r a c t e d 2 i s f o r half-plane. line source (Fig.2.3) the 24 ikS kr r F[-2 /k(r +r+S) cos(-y4]} p n If r>>r 0 (2.12) and t G U % Note that by (2.12) d i f f r a c t e d at least incident the (2.13) i s before by an the predicted by of the a d i f f r a c t e d edge, calculated Consider planes a i n of wave where solution (2.2) a l i n e doubly This [24] edge). l i n e If f(6)E(r) i s this source and edge to gives l i e s on i s , where the other f i e l d has not on f i e l d f($)E(r) approximation second d i f f r a c t e d edge d i f f r a c t e d [50] the that source multiply been source ( i f f i r s t the predict already edge l i n e to can been be used acceptable the shadow f i r s t . O F T H E METHOD another the using l i n e l i k e from d i f f r a c t e d by has the the used which isotropic others LIMITATIONS When be d i f f r a c t i n g (2.12). except boundary f i r s t looks then using results, plane can d i r e c t i o n edge, successfully the A f i e l d boundary d i f f r a c t i n g 2.3 simply on another shadow (2.13) 6 ) E ( r ) E(r) f i e l d s . approximated $ i s and once r becomes 0 (2.13) m u l t i p l i e d ( 'V (2.12) f i e l d edge resultant this source F i g . 2 . 4 , near where edge shadow cannot l i n e incident the an the f i e l d isotropic f i e l d from source on i s boundary be again of accurately approximation. two p a r a l l e l observation the point and h a l f h a l f - 25 plane is edges that and are the is c o l l i n e a r . f i e l d incident i . e . a l l on not which the of is second the The essence d i f f r a c t e d edge is not a of the d i f f i c u l t y from the f i r s t "ray f i e l d " [33], form ikf(r) near at the the shadow f i r s t The ray only v a l i d the source by edge when and a half-plane the mentioned on a on the l i e is cannot accuracy both be 10 ray so f i e l d s are does appear to d i f f r a c t e d f i e l d is very Thus coupling 6) and (parts the the doubly parts l i n e either. be of are Thus anisotropic may d i f f r a c t e d then and the accurately a l l potentially be serious along l i t t l e this [39]). f i e l d . accurate the focus extensions ray an of some f i e l d , the This expected the and method using to analysis inaccurate. the unless guides radiation y i e l d from f i e l d source order However, the this t r i p l y boundary, the f i n a l 1 single acceptable it result. (parts a ray d i f f r a c t e d higher shadow to any line and because, in d i f f r a c t e d the t r i p l y adjacent should f i e l d doubly t r i p l y numerically between reflection 3,4,7,8) of of i t s e l f contributes for not a boundary that d i f f r a c t e d not is with boundary uniform d i f f r a c t i o n boundary, calculated approximation, f i e l d is parabola shadow shadow and shadow a their a . singly not and [6] the page the here for within along incident along compromise If used the (i.e. axis solution d i f f i c u l t y f i e l d boundary methods using path (2.14) m (ik) edge and guide results. 26 However, 2) the for the t r i p l y shape of coupling and the between higher order coupling contribution patterns the of guide to the multi-element width to of the interactions along cumulative because the if incident are repeatedly calculated as near the the shadow the make a radiation arrays aperture the and The non-staggered (part affect curve result. expected errors inaccurate guides f i e l d s is many be separated d i f f r a c t e d versus s i g n i f i c a n t two (part plane 5) because boundary. d i f f r a c t e d The f i e l d s f i e l d s are are ray f i e l d s . In certain expanding term the (non-ray) f i e l d applying However, f i e l d s in the two in a method there the are an i n f i n i t e half-plane turn f i e l d . an d i f f r a c t e d times central f i e l d s . on a summation of the two applying for the edge a l l given f i e l d is u a on then or each a which is produces two d i f f r a c t e d derived the compared f i e l d half-plane (n+1) an f i e l d s . three a (n+1) times a 4 . of much Appendix with neatly one u the d i f f r a c t e d d i f f r a c t e d term-by-term be d i f f r a c t e d f i e l d s f i e l d and basis. only requires in each (2.14), longer times the by h a l f - p l a n e s , times in with term-by-term no produces overcome form more n be series the on [50] n edge a of sum f i e l d h a l f - p l a n e s , UAT to wave of may Taylor can case, type a plane contrast, represent incident only By half-plane To of d i f f r a c t e d d i f f r a c t e d n in three aperture by d i f f i c u l t y c y l i n d r i c a l the when represented when this representing then In cases B. ( is basis. Ansatz times case, from the d i f f r a c t e d a l l orders more complex When there are determined The and by expression leads to a 27 set of recurrence process case may be greatly because Ignoring there the fact mechanically versus as B exact which never solution, u-^. are cases with to solved. f i e l d s high [12] the of also in and coupling random known the and orders and apparently of considered. ray very This half-plane be reflection approaches regardless to not to an be three Ciarkowski both curves the f i e l d s theory by must in f i e l d s the in width many the done leads guide structure are that was which complicated applying d i f f r a c t i o n Appendix relations shape number fine of of the terms considered. A similar d i f f i c u l t y staggered. Consider staggered p a r a l l e l u.£ of in the these The y i e l d there and plane edges be i t . is (8.1) plates in are on two d i f f r a c t e d the on the is not f i e l d neighborhood incident v a l i d or and the expected to used. [50] near a is terms edges, away of from there not boundary contribute accuracy if thus shadow half-plane boundary the incident The f i e l d edge [24] acceptable shadow on The f i e l d ray Hence be of more result. to the or a the boundary. d i f f r a c t i o n numerical near because may if (Fig.2.5). not results three order expected is approximation are source plates approach accurate higher the edge, shadow term-by-term l i n e figures second reflected a occurs this and along which t r i p l e or s i g n i f i c a n t l y approximation the shadow are only to is boundaries two half- 28 2.4 SUMMARY The waveguide The ray-optical analysis of any parallel plate s t r u c t u r e r e q u i r e s that many ray paths be t r a c e d . f i e l d s represented by these rays are c a l c u l a t e d from the c a n o n i c a l problems of a plane wave or a l i n e source incident on a half-plane. calculations This method shadow boundaries. l i n e source d i f f r a c t i o n succeeding inaccurate for field on or near shadow boundaries, which l i m i t s i t s a p p l i c a t i o n to s i t u a t i o n s where along is chapters v a r i o u s combinations is not required The s o l u t i o n s f o r plane wave and by a to accuracy find half-plane will be used in the f i e l d s represented by the of two s u c c e s s i v e rays i n a ray path, and thus the c o u p l i n g c o e f f i c i e n t s and r a d i a t i o n p a t t e r n s . Fig. 2.1 i n c i d e n t and r e f l e c t e d Fig. incident, shadow b o u n d a r i e s 2.2 reflected and diffracted fields F i g . 2.3 l i n e s o u r c e i n c i d e n t on a half-plane 5 I u 2 / u 2 6 F i g . 2.A l i n e s o u r c e i n c i d e n t on two p a r a l l e l h a l f - p l a n e s F i g . 2.5 l i n e s o u r c e i n c i d e n t on two s t a g g e r e d h a l f - p l a n e s 31 Chapter 3 COUPLING BETWEEN TWO ADJACENT WAVEGUIDES 3.1 FORMULATION The f i r s t problem t o be s o l v e d i s t h a t of t h e c o u p l i n g between two adjacent p a r a l l e l p l a t e waveguides The r a y p a t h s from t h e lower parasitic guide driven guide (Fig.3.1). t o t h e upper a r e d e t e r m i n e d by i n s p e c t i o n and shown i n Fig.3.2 . Consider waveguides two adjacent consisting of semi-infinite the t h r e e parallel plate p e r f e c t l y conducting p a r a l l e l h a l f p l a n e s i n z>0 a t y=-d, y=0, y=a ( F i g . 3 . 1 ) . We w i s h t o determine t h e f i e l d c o u p l e d from t h e d r i v e n guide (d<y<0; z>0) incident \ into the p a r a s i t i c guide (0<y<a; z>0) . An field = e" l i k N Z c (3.1) ( ^ ) sin d o s x i n t h e d r i v e n guide w i l l e x c i t e f i e l d s of t h e form H C X E c _ = , n^O ^ n 6 +11^2 COS-nTTV. sin a ( (3.2) } x i n t h e p a r a s i t i c g u i d e . The c o u p l i n g coefficient the of t h e n t h mode i n t h e relative amplitude and phase p a r a s i t i c guide t o t h e N t h mode i n t h e i n c i d e n t is guide at 32 z=0. The propagation c o n s t a n t s ^ - /k -(,N/d) 2 kn - / ^ - ( r r n / a ) k N » are given by k n < ' 2 3 ^ > 2 Here we use the r a y - o p t i c a l method of Yee, F e l s e n and [50] to f i n d A 3 ) Keller . Following [50] the i n c i d e n t field (3.1) i s decomposed i n t o two plane waves H I x £ . = . =1_ u { e -i(k z-Nrry/d) N + x e -i(k z+NTry/d) N } ( 3 r ) x traveling i n the d i r e c t i o n s TI ± e relative to h a l f plane guide boundaries, where sin * { ,1 TM f i e l d s i TE f i e l d s 7 > ' U = q . term direction 11 in ~ N 6 (3.5) i s a plane wave t r a v e l i n g i n the which has the value ±k u - = (3 9) at the edge y=0, direction ; (3-8) 2 first the (3.6) 6„ = N-n/kd , N = T The > z=0 The other term, a n + Q plane wave in the has the value N (-D T (3.10) N U i = ~~2^~ at the edge y=-d, by the i n c i d e n t The u. d These two plane waves are r e p r e s e n t e d rays i n the lower guide of F i g . 3.1. fields guide appear z=0. excited to o r i g i n a t e f(6)E(r) by d i f f r a c t i o n i n t o the p a r a s i t i c from l i n e sources of the form (3.11) 33 located a t t h e e d g e s z = 0 ; y=0 distance the from diffracted excites field. in In (3.11) r i s the radiation Radiation the p a r a s i t i c in guide in the z>0; i s pattern direction 0<y<a. due t o a l i n e the The of 0 - 9r\ fields source (3.11) one o f i t s edges a r e [ 5 0 ] j— c c F = u d x c = I , n=0 ,1 n " 2 y /2k a I n ifn = 0 ifn i 0 ' { coupling of (3.12) employed refers i x u under a line incident i s immediately a linesource chapter, the following x of the f i g u r e notations are the superscript showing the x particular consideration. wave i n the driven source u line u the n letter field m yields A^ . this the i s a plane is (3.2) t o r e p r e s e n t t h e f i e l d s , where to path with coefficient Throughout ray (3.13) = 0 or a . Q Comparison u a. f(6) t h e n t h w a v e g u i d e mode excited at t h e edge and or . m-l field (ie ± source (ie u x m guide. arising from i s a singly field arising i s an m times diffraction diffracted from of field). diffraction multiply ^ - 1 the of a diffracted field). A ^ ^ i s the contribution diffracted j times. t o A^ n of (3.2) a r i s i n g from fields 34 3.2 SINGLE The DIFFRACTION coupling calculated by a half f i e l d an from in angle from the plane. the solution One o f driven Q guide edge, is single incident f i e l d wave waves comprising the on edge i s of the a y=0,z=0 At distances given e sin we is n given kr>>l (3.14) v D ( 6 ,e) at by uf = D(2n-e ,e ) E ( r ) , 1 2q N n where is plane F i g . 3 . 2 a ) . d i f f r a c t e d d i f f r a c t i o n d i f f r a c t i o n two p l a n e (see the due to for the = 2n - Q this contribution by (2.9). Substituting = nti/ka ( 3 > 1 5 ) find " The /^K , , 1 a l " 2 F i ^ E radiation comparison pattern with contribution d i f f r a c t i o n to only MULTIPLE The multiple r f(s) (3.11). the > - of (3.16) is this into Putting coupling coefficient of obtained by (3.12), the (3.2) for single is /k+Tk._ /-.N (1) _ n / ^Sln " 4k a n 3.3 < W /k+ik N • k n (3.17) +k N n DIFFRACTION contributions d i f f r a c t i o n to are the coupling obtained coefficient separately for each for ray 35 path of i n F i g . 3 . 2 . two then successive pieced f i e l d s i n each 3.4 In the the and u l C N i s , = i u l N C and the w i l l Q = e e = j from T ( sequence the be plane at 6 direction edge i n a d i f f r a c t i o n w i l l coupling total be problems solved, contributed coupling. considered and by the The pairs are of shown in 3.5. a edge individual find which F i g . 3 . 3 half-plane to ray path rays F i g s . 3 . 3 , the rays together successive in F i r s t u wave or or e = 2TT - 0 6 = ~ at is incident . The d i f f r a c t e d a point r=a on a f i e l d away from (2.11) a > ) (3.18) where ( a ) V = ) / E ( a ) a -in/4 C (a) = N 1 — (3.19) , . /' 2 , 2 " N * + { , e k r k F [ / | ( ^ _ AZ^ )] V TT + ie 1 Y k k N 3 F [ / | (v^k+k^ + /k+k^") ]} (3.20) and T For l 1 k a » l U l = r-' 1 i f |e-6o|<~/2 otherwise (3.18) U i T l C N E s i m p l i f i e s ( r ) (3.21) - to > (3-22) where +i-/4 C /k /k-tTk ~ N In * Fig.3.4 . (3.23) k /2T an isotropic line source u. w _ , - i= (r) 36 located that a t S, a d i s t a n c e r 0 same. o r in fx e = TT/X The Consider cos(- 2 t h e two c a s e s cos(—f) ik(r-r ) 0 from r>r„ field i nthe (2.12). and r < r . 0 If R=r-d, e -i,/4 = 0 r>r 0 S=r+d, e to find ik(r-r ) Q f i 0 -ITI/4 ik(r -r) r /T. ) ATQ K— » kr x Q i i s s u b t r a c t e d out, both = + \ E(r+r ) 2 0 n This f i e l d original field t (3.24) Q 0 F[/2kT] + 1 E(r+r ) ' 2 " 0' l J and kr>>l, and the f i e l d kr >>i source _ + i E(r+r ) . , R = d - r , S=r+d, a n d ik(r -r) m _ F[/2kr ] Q u = "m 2 V from by , l n . (3.25) the incident line E(r ) E(r) . ^ U source diffracted 1 ( 3 . 2 4 ) a n d ( 3 . 2 5 ) become (3.26) n h a s two c o m p o n e n t s : t h e f i e l d line originate of total (2.12): = r<r u i s found in = 1 e If The e+e e u . i s i n c i d e n t on r e s u l t s f o r F i g s . 3 . 4 a a n d 3.4c a r e f o u n d t o be t h e substitute If o r ITT/X edge a t e> ^ T / I directions from a g u i d e w a l l 0 reduced i n amplitude t h e edge. These a t d i f f e r e n t p o i n t s i nspace, image o f t h e o r i g i n a l o f t h e image o f t h e by h a l f , p l u s a two l i n e sources one a t t h e l o c a t i o n l i n e s o u r c e a t d i s t a n c e r„ f r o m t h e edge, a n d t h e o t h e r a t t h e edge. The r e s u l t s f o r F i g s . 3 . 4 b a n d 3.4d a l s o a r e f o u n d t h e same. C o n s i d e r t h e t w o c a s e s substitute cos(-/) = in o , c o s ( - / ) = -1 r>r 0 (2.12): to find and r < r . 0 R=r+d, t o be If r>r 0 S=r-d, 37 , 1 U m 2 E(r+r = 0 a " " 7 4 i k ( r _ r 7= e } + T r<r u M If 2 ) n 0 /T7 /krT^ and kr>>l both Q = i E(r+r ) + T 2 . (3.28) (3.27) and (3.28) become (3.29) 2/2T 0 0 This f i e l d field (3.27) E(r.)E(r). n m . F[/2k7] T + 0 kr >>i u ik(r -r) -iTt/4 4E(r+r = nfik^l R=r+d, S=d-r, and , Q , /kr /TT If 0) has two reduced components: i n amplitude the o r i g i n a l by h a l f , p l u s a f i e l d by the edge. These two l i n e sources o r i g i n a t e points in space, one a t the l o c a t i o n source a t d i s t a n c e r line diffracted at different of the o r i g i n a l from the edge-, and the other c source line at the edge. In u m-l is = £ Fig.3.5a ( ) incident This f i e l d on that an isotropic a 0 line source from the edge edge at an angle e = TT/2 or 3TT/2 Q i sdiffracted 0 =e u 3.5b l o c a t e d a t S, a t a d i s t a n c e r r angle and or 2r, - e n i n t o a waveguide of width a . From a t an (3.18), the d i f f r a c t e d field i s , f o r kr>>l m U m = where l n 0 (r>> T C c ( r n ( r o (3.30) ) E ) i 9 s i v e n b Y (3.20). For k r > > l (3.30) 0 s i m p l i f i e s to U m = T ln ( V C E E ( r ) • (3.31) 38 3.4 CALCULATION The f i e l d s successive the waveguides. and The For each The of In is l for w i l l of now b e adjacent these possible located (m-1) two used to p a r a l l e l - p l a t e calculations, combinations guide of shown rays in up and one c o m b i n a t i o n F i g . 3.2b ( a ) E ( r ) at the d i f f r a c t e d at excited by and f i e l d s at be the to with the ray i s from the modes in calculated (3.2). are edge d i f f r a c t e d of last represented incident the f i e l d s calculated The amplitude the the Fig.3.2, can (3.12) d i f f r a c t e d laf N in ray ray. (3.11), y=a,z=0, C rays mth d i f f r a c t e d at = of the parasitic and c. U two d i f f r a c t i o n , sequence doubly observed by f i e l d , y=z=0. From f i e l d is a Figures given by (3.18), as line source <' ) 3 d i f f r a c t i n g the edge edge. y=a, z=0, This f i e l d and from is (3.30) 32 again gives a source U located 2 = laf C ' ( a ) C ( a ) E ( r ) (3.33) n at f i e l d z=0. a l l u ^ ^ o f the (3.9), paths combinations above between t r i p l e comparison line ray with upper 3.2b by v a r i o u s d i f f r a c t i o n . associated the COEFFICIENT presented coupling including f i e l d s as represent quadruple by represented rays calculate F i g . 3 . 2 , OF T H E COUPLING , From the d i f f r a c t i n g edge. given by (3.10), (3.18), as observed i s at In F i g . 3.2c d i f f r a c t e d y=z=0, the at the the incident edge y=-d, d i f f r a c t e d f i e l d 39 is a line C 1 source = c , N : 2q located at the diffracted at ( d ) E ( r ) (3.34) d i f f r a c t i n g the edge edge. y=z=0, This and from f i e l d (3.30) is gives again a line source 2 U located the Z = ^ - ^(d)C (d)E(r) i at the d i f f r a c t i n g coupling by t r e a t i n g using (3.11), N n ( 2 = ^ ) The 3.2d, e (3.10), observed U i d n [ ( - D 4 n C ^ ) C diffracted resultant d £ 2 2d, is at z=0, diffracted and Thus N fields . n the the are the (3.36) represented incident f i e l d y=z=0. From edge diffracted f i e l d i s a by , Figures given E by (3.18), line as source 3 d i f f r a c t i n g the N< > ( made is < - > edge edge. y=-d, This z=0, and f i e l d from is 37 again (3.26) the is d + r > + A q T=J C N ( d ) E ( d ) E ( r ) • ° ' 3 8 ) V 2 TT 4 u sources r the d line N( > ( > f i e l d C at y=-d,z=0, at diffraction + (-D C^(d)C (d)] ( a ) F i g . 3.2d diffracted C as to a In f ? n contribution double (3.35) and (3.2). and f. = ' £ 7 and diffracted at The t o t a l from t r i p l y located 4 (3.33) (3.12) y 2/2k is edge. coefficient found A (3.35) n up of the at two l i n e other the edge at source y=~d, y=z=0, f i e l d s , z=0. and from one c e n t e r e d at This f i e l d is (3.30) gives two y=- again line 40 sources +in/A 3 " fqj C * ( d ) C ( 2 d ) E ( r ) + ^ U S — n located at the d i f f r a c t i n g f i e l d i s (3.32). This (3.26) the given f i e l d resultant '<d)C (d)E(d)E(r) (3.9) again f i e l d (3.39) n edge. by is C Fig.3.2e In and the diffracted the f i e l d at incident i s y=a, z=0 given by and from i s +iir/A "2 = " i is y=2a, C^(a)E(a+r) - - made z=0, the d i f f r a c t e d 3 In at 4i = f i e l d 2 other y=z=0 at = source y=a, and from (3.40) fields, z=0. one centered This (3.30) the f i e l d is resultant at again f i e l d i s +iTi/4 + ^ n ^ u y=z=0 u two l i n e C'(a)C (2a)E(r) Fig.3.2f at of . C^(a)E(a)E(r) -1 e U up j- the i s incident- given by and from " V "N C - I 1 C^(a)C (a)E(a)E(r) n f i e l d (3.34). (3.29) ( d ) E ( d + r ) - i s This the given f i e l d resultant by is . (3.41) (3.10) again f i e l d and the diffracted is Z r ~ ^ 7 = — C^(d)E(d)E(r) . (3.42) /27T U i s 2 made up of two l i n e source d, z=0, the other at y=z=0. This at y=a, z=0 and from (3.30) the u f 3 The = n (3.43) N contribution d i f f r a c t i o n as f i e l d +W4 line is + to found sources the one c e n t e r e d is again resultant < C'(d)C (d+a)E(r) n total t r i p l e Thus TC fields, C^(d)C coupling by t r e a t i n g and using f i e l d y=- diffracted is (3.43) (a)E(d)E(r). coefficient (3.39), (3.11), at (3.12) A N n (3.41) and from and (3.2). 41 +iTr/4 r V ^ = 3 - N [ C n ( d ) C " S ( 2 d ) n ( a ) C ( 2 a ) (-D + N + n C'(d)C (d a)] n + n + g ^ - [CjJ(d)C (d)E(d) + C^(a)C (a)E(a) + (-1) n n N + n n CjJ(d)C (a)E(d) ] . n (3.44) A quadruply d i f f r a c t e d f i e l d i s represented i n g Fig.3.2g. f i e l d s y=0, Here in (3.38) z=0, source u so f i e l d s i s 2 are given d i f f r a c t e d that from centered by (3.38). independently (3.29), at Both line by consists of ( - 2 d , 0 ) , ( 0 , 0 ) , ( - d , 0 ) source the edge four line and (0,0) g respectively. which Oik'^^ ) 1 the Note asymptotic solution. If consistency Keller that i s of terms of d i f f r a c t i o n not theory expected used contribute From by only number of here. very (3.26) half order It s i g n i f i c a n t l y to turns arising the be order the that i s 3 in d i f f r a c t i o n i n terms added be and & then N n to the also the order within u term included Thus meaningful i n second plane and higher out f i e l d s (2.9) (3.31). quadruple be as half to higher in 0(k' ) lfl d i f f r a c t i o n the ray-optical these terms numerically. the d i f f r a c t e d d i f f r a c t i o n the l and (3.29) the order 0(ic' ^)are and l i t t l e grows but only same these coefficient to when terms d i f f r a c t i o n order, due to of of that (3.23) , (3.26) , (3.29) are the expansion requires contributions one from f i e l d along amplitudes the shadow quadruple and rapidly for each 0{u" ) and O be expected /x may value of the are boundary. higher successively coupling terms to reduced from The order higher higher contribute c o e f f i c i e n t . 42 Contributions Oik'' ), calculated 12 ( =l, N=0, T 3.5 to n=0), from for are given NUMERICAL RESULTS Numerical values were calculated and TE polarizations. double, and t r i p l e The mode in guide, a=d, is shown i n coupling from the both of d i f f r a c t i o n the When width a and d to A^ for n both TM due to single, included. ,the coupling mode in and 3.8 Values F i g . 3 . 9 . of and from the for of a TEM parasitic guide widths (TE, - T E/ ft,, Figs.3.10 and 3.11 modes into guides are terms reveals the the a of shape the show higher the T E Mc a s e driven guide various the with the cutoff the double near The to widths, same average minima modes. correction the the Adding structure some near of provides c o e f f i c i e n t . fine provide ,especially in term coupling widths basic parasitic d i f f r a c t i o n the cutoff structure driven terms d i f f r a c t i o n alter 3.7 respectively. in 0 0 TEM F i g . 3 . 6 , coefficient TM a n d T E f u n d a m e n t a l single behaviour the a A coupling modes. When width to plotted the of TEM-TEM of B. widths were d i f f r a c t i o n of coupling guide and phase a n d a=2d case Contributions guide are the various driven order in Appendix for the a=d/2 special d i f f r a c t i o n amplitude coupling) order for the higher t r i p l e this but fine do not curve. parasitic (a=d/2) guide (Fig.3.7) is the half the minima near 43 the mode transition parasitic (a=2d) for guide is (Fig.3.8) the both correction compared the Exact not case the to r e f l e c t i o n single [50] with the as small as 0.1 wavelengths presented to [50] been here and may [18] TEM frequencies be results the In a more result the c o e f f i c i e n t A N as exact ray-optical to same for accuracy method of widths and values a [18] guide coupling The have for correction result TEM-TEM are a c o e f f i c i e n t frequency for expected TE/ mode obtained widths than less This a 0.6 of A ^ comparable calculation similar also, Higher the When to compared in to the be consistent the terms TEM case was asymptotic deviate of with deviate comparison has ,/2 for the the low asymptotic non-asymptotic and 3.13. The s i g n i f i c a n t l y at and the T E , of [18] with made 0.7"X the for in results the approximation even 0(1<." using F i g s . 3 . 1 2 in results order c o e f f i c i e n t s calculated (3.18) begin 0.3X appears c o e f f i c i e n t s . are using results coupling cutoff (3.22) asymptotic used provides However, coupling. because and near approximation where with T E /- T E / widths used. The case. low wavelengths for guide ( n = l , 2 , 3 . . . ) . d i f f r a c t i o n comparison. agreement the driven t r a n s i t i o n terms coupling the good the When a=d. for show of d=n/2 double expressions guide depth. mode d i f f r a c t i o n the for the or the for in width at a=n where results available at t r i p l e substantial to reduced appear guide adding are double minima parasitic cases widths guide r e f l e c t i o n (3.31) is more. ) and 0(k~') were calculated 44 but they also found were been noted open-ended higher order they exact solution do ( depth of closer to at the TM modes these minima mode behaviour similar taken r e f l e c t i o n surmised give a It cusps out widths. does have any terms solution, even with for the comparison, w i l l used here is no however contain by in a and error, expansion a cusps many cusps. unable the of to is It for TE and the single f i r s t derivative account. are found terms (see reasonable provide an a l . [50] for that for has at assume the accurate the sharp the [50] the not might width. mode however numbers S i m i l a r l y is to It eventually [45] Appendix case appears This large in of 3.9). solution very moves orders solution waveguide into null would exact The and transition when transition waveguide. terms known even the et width modes). higher Yee mode guide mode single coupling it a ray-optical taken the as by more cusps, are case, of the of The coupling solution be (Figs.3.6 found at discontinuous d i f f r a c t i o n has quadruple versus increases account s t i l l that /2 width that exactly transition not into below (2n-l)"X (nulls) to coefficient with the to coupling s l i g h t l y and taking null turns r e f l e c t i o n l i k e l y It scattering asymptotic of c o e f f i c i e n t that sharp are transition are be of waveguide the curves widths d i f f r a c t i o n the plate with for is contributions. calculation f i e l d s agree a l l the the small [9]. that nX only p a r a l l e l not minima widths in d i f f r a c t e d since have give that single Note to in of the ray-optical B). An exact available that it ray-optical solution for also method near the 45 mode transition acceptable case consideration asymptotic of the c o e f f i c i e n t of 0(k~ ). essentially f u l l expression to be F i g . 2] [4, No guides a l l terms applying the of occurs the the Each of rapid growth be better find. that the of not slowly provide in U A T may Only the this series guide and in number the Since found i s , near the a of general with B), a UAT accuracy would however, be with even Appendix It the is (though numerical to cutoff the (see terms values included. was for to term edges improvement. near ray-optical solution exact the reflection this guides method widths by In p a r a l l e l - p l a t e required 0(^~9 d i f f i c u l t problem. the series are a A N / Vproviding other in with is only expected marginal, fundamental at to if any, mode [3, Fig.2] calculations i s o l a t i o n interesting here to substantial improvement this between have very this for accelerated). that to open-ended with is t r a n s i t i o n provide but eqns.(8.7),(8.8)] for ray-optical for to obtained [4, there for simpler mode widths guides interactions appear that terms interaction adjacent solution to 50 be [1] single agreement may represents the to transition convergence the a However l up two expected, given expression similar convergent; mode from eqns.(35),(36)] )and / z are been (UAT) f i n a l is [3, Oa~' cusps has theory r e f l e c t i o n waveguide method where elsewhere. Some uniform widths to those has compare for of the been the adjacent coupling found; coupling guides in between however adjacent it is c o e f f i c i e n t s obtained the of presence other 46 guides. Montgomery between adjacent Fig.1.18 using the coupling using guides function similar results s i m i l a r . i s guides i n the array the other guides ray-optical asymptotic form be each of (3.29) The coupling geometry Lee [29] of F i g . structure i n found 1.15 structures consideration are that a expected. case, l i s t e d c o e f f i c i e n t i n Table i n the to which considered. It i s derived well gives the region insensitive quite the techniques. under c o e f f i c i e n t coupling with the Fresnel here these best are of not i n the other of adjacent unexpected absence results. agreement integral and (3.31) presence pair thus , are I when scattering of The the functions used. SUMMARY Coupling been between calculated driven guide to contributions parasitic number number results rays i s agrees (3.23),(3.26), 3.6 might quite coupling closed the that The coupling guides that f o r to for other the techniques. comparison The calculated theoretic Weiner-Hopf reasonable [38] for coefficients s u f f i c i e n t l y very and Chang of of by f i r s t the to guide adjacent from are (single i n each obtained and f i n a l sequences sequence even double when of ray of plate sequences guide amplitude the different rays tracing parasitic the p a r a l l e l and the i n rays rays then from adding excited sequence. grows rapidly are However, such has the the i n the each three d i f f r a c t i o n ) of mode increases. only waveguides The as the useful sequences considered. of An 47 exact solution comparison, however, agree remarkably when the two This s e n s i t i v i t y of surrounding results waveguide are this the well coupling those are the not and in obtained by by a c o e f f i c i e n t s gives the available c o e f f i c i e n t s indicates coupling structure, is surrounded agreement applicable array. geometry with guides halfplanes. the for obtained other other general to confidence context of the here methods guides or lack of nature of that the for these complete 48 F i g . 3.2 ray paths from the d r i v e n to the p a r a s i t i c guide guide F i g . 3.3 two s u c c e s s i v e r a y s i n a r a y p a t h from the d r i v e n guide to a guide a p e r t u r e U m Um-1 |U m U m-1 F i g . 3.4 two s u c c e s s i v e r a y s i n a r a y p a t h , b o t h rays i n a guide aperture F i g . 3.5 two s u c c e s s i v e r a y s i n a r a y p a t h f r o m the guide aperture to the p a r a s i t i c guide -45 0.5 0 1.0 d/A 1.5 2.0 180 -i Fig. 3.6 TEM-TEM c o u p l i n g between .... adjacent waveguides single diffraction s i n g l e and d o u b l e d i f f r a c t i o n s i n g l e and d o u b l e and t r i p l e diffractio 0 i 180 -j 135 • -|—•—,—,——i 0 0.5 * * *—•—i—•—• 1.0 •—•— —•—• 1 1.5 • • 2.0 d/A 3.7 TEM-TEM c o u p l i n g between single single single adjacent waveguides diffraction and d o u b l e d i f f r a c t i o n and d o u b l e and t r i p l e d i f f r a c t i o n (a-d/2 180 -i & 135 o o 90 rO 45 0.5 Fig. 3.8 1.0 TEM-TEM c o u p l i n g between single single single 1.5 2.0 waveguides (a=2d) d/A adjacent diffraction and d o u b l e d i f f r a c t i o n and double and t r i p l e d i f f r a c t i o n 53 180 1 single diffraction s i n g l e and d o u b l e d i f f r a c t i o n s i n g l e and d o u b l e and t r i p l e d i f f r a c t i o n 10 i TABLE I Coupling Coefficients A (d,a) Between A d j a c e n t P a r a l l e l P l a t e Waveguides a = d = 0.45 X m 9 excited +# c o u p l e d into Montgomery 20 log|A| (dB) Lee [38] Lk (degrees) 20 log|A| (dB) [29] LA (degrees) 1+2 & 2+1 -15.32 +105.2 -15.76 +102.3 3+2 & 3+4 -15.57 +107.8 -15.78 +106;8 -15.33 +105.3 -15.78 +106.8 2+3 this theory using the asymptotic form (3.22) 20 l o g | A | LA (degrees) (dB) -15.74 +102.3 t h i s theory u s i n g the non-asymptotic. form (3.17) 20 logJA] (dB) -15.80 Lk (degrees) +97.7 58 Chapter COUPLING 4.1 CALCULATION The w i l l guides. found F i r s t quadruple from between i n The f i e l d s f i e l d s expressions derived represented found by [38] and a using Consider waveguides planes widths of the ( F i g . 4 . 1 ) . (3.9) These B D at plane - %rr-& N and the into edge waves 0o=£/v y=~d, are z>0 z=0 respectively the f i e l d s and hence [17] to the those using edge Montgomery and d into conducting and f i e l d the waves, and by p a r a l l e l - p l a t e and y=a, incident incident the compared and y=0 the the using way a l . b, two p l a n e y=-d, are 3 calculated calculus. determine As i n Chapter by et be inspection i s found adjacent including i t perfectly four -d-b<y<-d, decomposed ray of 0<y<a, i s w i l l for by this are (Fig.4.1) and s e m i - i n f i n i t e guides to to adjacent guide guide In residue transmitting z>0. 2. argument y=-d-b, We w i s h up preceding Dybdal modified to by each ray reciprocity three that i n Fig.4.2 by at similar The results three z>0 the rays consisting i n guides represented c o e f f i c i e n t . d i f f r a c t i o n separated determined i n Chapter a l l GUIDES COEFFICIENT ray paths are of previously Chang manner a l l possible the coupling a two d i f f r a c t i o n ( F i g . 4 . 2 ) . SEPARATED OF T H E COUPLING coupling be BETWEEN 4 . a so that coupled from parasitic mode i n the with value (3.10) at 0* i s the respectively given the guide driven given y=-d-b, on the h a l f - p l a n e where h a l f - z=0. edges by at (3.6). 59 The doubly F i g . 4 . 2 a . The diffracted y=z=0 diffracted at incident the edge the diffracted u, 1 = 3 located fields represented f i e l d , given z=0. From (3.18) y=-d, f i e l d are is a line by as (3.9) the diffracted at at source (4-1) d i f f r a c t i n g the i s observed ^C-(d)E(r) 2q N at by edge edge. y=z=0 This and from f i e l d (3.30) i s gives again a line source The coupling treating (3.2). coefficient (4.2) as a from line double source d i f f r a c t i o n and using i s (3.11), found (3.12) by and Thus T~ +iTT/4 A. = Nn e e V'TT — _ rr , 2/2 k a (4.3) C ' ( d ) C (d) . N n n The t r i p l y F i g s . 4 . 2 b by (3.9) (4.1). from and c. and the This (3.29) 2 b i s In the Fig.4.2b singly is again resultant made f i e l d diffracted f i e l d at represented f i e l d u^ b the u i s by given i s given by edge y=z=0 and i s +iTT/4 7Z C ' ( d ) E ( d + r ) 4q are incident diffracted 1 = 2 u f i e l d s f i e l d the b S diffracted + N up of - 4q two l i n e the other at y=z=0. This z=0 and from (3.30) the C'(d)E(d)E(r) . 1 ^ sources, f i e l d i s resultant (4.4) N one c e n t r e d at again diffracted f i e l d i s y=-d, at z=0, y=a, 60 U 3 b —T icT C ' ( d ) C ( d a ) E ( r ) = n In From Fig.4.2c (3.18), is a line = 1 located c u 2 i s z=0, at y=z=0 c U 3 N b + ± T / 4 - C" (d) C (a)E (d)E (r) . (4.5) n f i e l d u y=-d, z=0, i s given the by (3.10). diffracted f i e l d (4.6) d i f f r a c t i n g edge edge. This z=0 and y= d, - ) c E ( b + r at (-1) e ~4q~ ^ 7 = ~ " ) two l i n e y=-d, and from f i e l d from i s again (3.29) the i s u 2 up of other -(-1) ~^a7 " e N ( made the at N C c — '(b)E(r) N the f i e l d -(-1) 2 C the at = 1 source at resultant — incident observed 2q diffracted U the Z±Z±L c u as - + z=0. (3.30) the + i V 4 C'(b)E(b)E(r) . sources, This f i e l d f _ ^ n + - is resultant N C'(b)C (b d)E(r) one c e n t r e d again f i e l d (4.7) at y=-d-b, diffracted i s +i*/4 ~Z— C'(b)C (d)E(b)E(r) . >'2TT N T n (4.8) The total found by (3.11), contribution treating (3.12) r A Yin ( 3 ) = — given e, by from t r i p l e d i f f r a c t i o n i s r and (4.8) [-(-l) . rz , 4/2 k a n [-(-DV N quadruply Figs.4.2d, A Nn as line sources and using Thus +iTT/4 8k a n is (4.5) and (3.2). £ i T The to f C ' ( d ) C (d+a) N n (d)C n (a)E(d) diffracted and g. (4.6) n and ^ In d 2 - ( - l ) C ' ( b ) C (b+d)] N n N - ( - l ) C ' ( b ) C (d) E (b) ] (4.9) N n N fields Fig.4.2d u ± i s are represented given by (3.10), d i s given by (4.7). in The f i e l d * 2 • i s 61 again d i f f r a c t e d f i e l d d u y=z=0 and from (3.29) the resultant i s -(-1) = 3 at (_i) U N C^(b)E(b d r) + - + t - (-1) -(JZ- 1 N + e e W 4 -__c'(b)E(b d)E(r) + e N + i T ' / 4 C^(b)E(b)E(d+r) . (4.10) -3/2 Note that the reasons of mentioned sources This term at (3.30) , , = 4 the + -i-ii8q ^ u z=0; is 3 ) in Chapter y=-d-b, f i e l d 0(k 3. y=z=0; again resultant has f i e l d " been is d 3 made and y=~d, d i f f r a c t e d up of z=0 at . u /2TT consists 4 of three The contribution is found (3.11), (4) ^ by 8q to \ sources from n treating (3.12) j- [ { _i, this E i T ( - l ) and from (4.11) quadruply line at y=a, z=0. d i f f r a c t e d f i e l d sources and using Thus N + n c.(b)C <bfd a>] + n N + n [c;(b)E(b d)C (a) ^ I S k a + z=0 . as 8 ^ k a n line C'(b)E(b+d)C (a)E(r) N n /r— a l l centred (4.11) and (3.2). +iTt/4 fnJL! = line three +iir/4 N + C ' ( b ) E ( b ) C (d+a)E(r) N n /r the respectively. y=a, N +1TT/4 2 for i s ^ C ' ( b ' ) C (b+d+a)E(r) 8q N n (-D u deleted + n + C " (b)E(b) C r ( d a ) ]. + ( 4 > 1 2 ) n In Fig.4.2e (4.1). and This from e U u e 2 2 i s is u f i e l d (3.26) ^ the given i s again (3.9) f i e l d 1 / 4 £ - = T /2TT + and d i f f r a c t e d resultant x = by i l T ^ at is the given edge by y=z=0, i s 4 P ^ made C N ( d ) up of E ( d + r ) - two l i n e sources, C£(d)E(d)E(r) . one at y=d, (4.13) z=0, the 62 other at y=z=0. This field u e 2 i s again d i f f r a c t e d edge y=-d, z=0, and from (3.26) the r e s u l t a n t U 3 1 iq" = e C N ( d ) E ( 2 d + r ~ ) + i 7 T / at the field is 4 c' ( d ) E ( 2 d ) E ( r ) ? - — — +171/4 - T C' (d)E(d)E(d+r) . N rz— /2TT i s made up of three l i n e sources U;J z=0, at and y=-2d, z=0 r e s p e c t i v e l y . Again (4.14) y=-3d, z = 0, y=7d, ~3/2 the term of 0(k ) e has been d e l e t e d . T h i s f i e l d edge y=z=0, and from = 6 4 3 (3.30), s e p a r a t e l y , the r e s u l t a n t u i s again d i f f r a c t e d u ~ C (d)C ( 3 d ) E ( r ) Bq N n treating at each l i n e the source field is + iTT/4 -r C'(d)E(2d)C (d)E(r) 8q ^ N n - 1 +i7i/4 - — 8q ^ In F i g . 4 . 2 f the i n c i d e n t diffracted at = f •1 resultant f U field 2 by ( 3 . 1 8 ) , as observed i s a line source (4.16) L the (4.15) (3.9) i s - — c' ( b ) E ( r ) 2q N edge This y=-d-b, field is again z=0, and from (3.26) the field is 1 " 4q" = given the edge y=-d, z=0. From at . /2TT l o c a t e d at the d i f f r a c t i n g edge. diffracted C'(d)E(d)C (2d)E(r) N n field at y=-d-b, z=0, the d i f f r a c t e d u rr- +i T C N ( b ) E ( b + r ) + ^ T T /4 C^(b)E(b)E(r) 6 , (4.17) > 2TT . u f 2 i s made up of two l i n e sources, one at y=-2b-d, z=0, other at y=-d-b, z=0. T h i s f i e l d the i s again d i f f r a c t e d at the edge y=-d, z=0, and from (3.29) the r e s u l t a n t field is f U 3 -1 i ^ - " / i ^ — = — + ^ + T C N ( b ) E ( 2 b + ^ i 7 1 4 6 3 e C'(b)E(2b)E(r) +iiT/4 3 u i s made u p o f t h r e e z=0, a n d y=-d-b, diffracted line u. 4 line sources (4.18) a t y=-2b-d, z = 0, y = - d , u z=0 r e s p e c t i v e l y . T h i s f i e l d a t t h e edge y=z=0, a n d from source = p - C ^ ( b ) E ( b ) E ( b + r ) . L (3.30), f 3 i s again t r e a t i n g each separately, the resultant f i e l d i s +iTt/4 1 — C ' ( b ) C (2b+d)E(r) 8q N n - 8q C ' ( b ) E ( 2 b ) C (d)E(r) N n + • — 8q 1 + iTT/4 C' ( b ) E ( b ) C N ^ (b+d)E(r) . (4.19) g In F i g . 4 . 2 g the incident f i e l d i s given by (3.9),^ g is given b y (4.1), and q 2 is again diffracted a t t h e edge y=a, the r e s u l t a n t f i e l d i s +iTt/4 u 9 3 _1 8q = c . ( )E(d+a+r) N d - field u " i s given by (4.4). This u 8q 2 z=0,a n d from (3.26) C ' ( d ) E ( d + a ) E (r) N ^ +in/4 u 9 3 + - i ! i s made u p o f t h r e e line z=0, a n d y=2a, diffracted line C (d)E(d)E(a+r) sources (4.20) a t y=2a+d, z=0 r e s p e c t i v e l y . This field a t t h e edge y=z=0, a n d from source . (3.30), z=0, u 9 3 i s y=a, again t r e a t i n g each s e p a r a t e l y , t h er e s u l t a n t f i e l d i s +1TT/4 u 9 4 = - — C ' (d)C (d+2a)E(r) 8q N n + -J-8q - T ~ — C'(d)E(d+a)C (a)E(r) N n +irr/4 C ' ( d ) E ( d ) C (2a)E(r) , (4.21) V 2 TT The contributions to \ . n from these quadruply 64 d i f f r a c t e d (4.21) f i e l d s as line i s found sources by t r e a t i n g and using (4.15), (3.11), (4.19) (3.12) and and (3.2). Thus f— +i7T/4 .. (4) \n V * 6 r 8/2 = [ C k a n N ( d ) C n ( 3 d - ) - iftv lbK a a n A The sum 4 4.2 - ) of coefficient including guide of A ) C ( b ) E ( b ) C (b+d) N n - ( d ) E (d) C (2a) ] 1 N to (4) from quadruple (4.9), (4.12) separated those are coupling up calculated et by c o n s i d e r i n g the argument line of (4.9), Dybdal source'incident the the to and RESULTS coefficients obtained reciprocity i s waveguides (4.3), A i s d i f f r a c t i o n . AND NUMERICAL with d i f f r a c t i o n and (4.22) The expressions were to n (4) between n \ (4.22) n and compared a guide d + and separations as the + C A ( b ) E ( 2 b ) C (d) N n coupling compared line b - quadruple widths results 2 C ' ( d ) E ( d ) C (2d) N n (4.3), others. were ( - * ANALYTICAL The n C C'(d)E(2d)C (d) N n contribution ( ) t" n total b C ' ( d ) C (d+2a)] N n + C^(d)E(d+a)C (a) The ( i T z + N C on the receiving was u s e d source. to with for the (4.12) a l . and [17], transmitting parasitic obtain the various results (4.22) Their guide guide. response of 65 It turns identical paths TEM out that results in to Figs.4.2a, b, A (2) V + Q and in V ~ the T E , - 0 c derived and d are gives here, if considered, v i r t u a l l y only the ray i . e . in the shown the ray were the The < and 4.4, widths of 0.338"X ranging from of and 0.4X the and to interaction terms interaction terms in compared to TEM have 24 those function of the results [38] and Lee TE i c o u p l i n g are If TEM and interaction decreases is by Dybdal [29]. is shown guide between guides the monotonically Dybdal show more in for between increased. becomes of Fig.4.5 separation o0 separation coupling in A guides as Addition an the of the o s c i l l a t i o n pronounced as more added. of Fig.4.3 this in obtained at for 0.761X the with for which cusps a A o o neglected g. c o e f f i c i e n t s Chang guides guides coupling as and and between the of limitations i l l u s t r a t e d for 2.0\. A /2, interaction F i g s . 4 . 2 e , f , coupling between the compared and separation case neglect amplitude period 4 • ) calculated Fig.4.3 The 3 Montgomery neglected, <- > 00 values guides, a l . [ 1 7 ] , with n paths numerical An between » expressions The and 00 • 12 in + < 3 ) case V ' Dybdal's is those method case 7* nn I 00 et Dybdal's which here. guide ray-optical method Montgomery's Montgomery's separations of are results results n X/2, in are the (n=l,2,...), 66 but the r a y - o p t i c a l results approximate agreement. o s c i l l a t i o n to the cusps were not even with large number a While the the q u a l i t a t i v e Higher curve obtained of addition of agreement separation. These higher d i f f r a c t i o n the that optical dB. the i s a the applied and Boersma two a line on (Fig.2.5). f i e l d f i e l d . i s here along the These find doubly f i e l d incident both of though are these shadow and guide accurately one-quarter i n be one-half .will or a l . be (3.29) of the [33]. Lee boundary of wave normally (Fig.4.11) p a r a l l e l doubly of 2 singly i n view plates - the (3.26) plane the 0 f i e l d shadow staggered Boersma the ray- expected w i l l and Lee et a theory. boundary. that on t h e a the of may be incorrect two c a s e s than results, by about d i f f r a c t e d p a r a l l e l two gives accurate, boundary two c a s e s : on Lee theory, r a y - o p t i c a l coupling [32] f i e l d s for greater add an a l l quadruple and (equations are two n o n - s t a g g e r e d In this shadow results the terms calculated predicts and Boersma plates somewhat Even f i e l d s incident [32] source be results the used Lee p a r a l l e l incident with are the the of terms and cannot and the only However, values along twice). analysis some which f i e l d one-quarter for error the cusps. Montgomery's measure of terms r a y - o p t i c a l the f i e l d incident this underestimates method d i f f r a c t e d the interaction less boundary calculation The the of show terms. Montgomery's theory This with interaction shadow Assuming i s here d i f f r a c t i o n with with quantitative along order i n phase agreement order obtained the and plates d i f f r a c t e d incident do not consider the 67 specific case p a r a l l e l plates, Fig.(4)) indicate the f i e l d s indicate which to the in for guides from 0.4"X those with 1- some the to important of t r a n s i t i o n widths. the minima deeper are when neglected these ii . and Both hand i s results occur with not at the of also d=n"X results the /2 of method t r i v i a l , case ([32], underestimates Their w i l l half-plane also calculated same of these calculated of width the of a n d may be (Fig.4.12) for the and this same order of exactly i f minima results those here The a p p l i c a t i o n at higher up to However, These results on two which case width The results width that a l l i s varied are similar (Fig.3.6-3.9), but differences: shape account. with the two g u i d e s to much their boundary. consistent [38]. incident used 2 . OA ( F i g s . 4 . 6-4 .10 ) . addition the i s i s of difference 2- analysis source chapters. are for 4.2d), coupling three coupling three to the (line shadow problem later The the enhanced and Chang here or the which intractable arises that near ( n = l , 2 , . . . ) , [32] Fig.2.3 that Montgomery arises for five at the quite the case et shallow a l . array was Lee's array embedded i n i n and more the mode widths taken the and into guides i s broad. a=d=b=0.45/\ are compared [38] [29] i n and Lee five between Montgomery's i s between makes at transition d i f f r a c t i o n considered guides. terms especially mode times are coupling was curve interaction Montgomery authors the d i f f r a c t i o n adjacent each free a pair space waveguides of separated (Fig.1.17) simulated Table and groundplane 68 (Fig.1.15) . As in the c o e f f i c i e n t s guides is absence in the of other in a of surrounding other well of separated pair derived quite coupling here with in these the results guides. of in the which the the the coupling presence groundplane coupling between of two other indicates coefficients a to separated guides and general lack the nature of structure. SUMMARY The coupling waveguides has adjacent shape of the to adjacent To of separated by the obtain an coupling to same versus terms compared coupling those between up p a r a l l e l method plate used for approximately correct guide it to only the of : c o e f f i c i e n t s calculated interactions by guides shows at width least double boundary and method is obtained Dybdal is only Montgomery ray-optical shadow calculated include as two was quadruple d i f f r a c t i o n for guides. The results been curve d i f f r a c t i o n , with between waveguides. necessary the of to guides coupling agrees simulated s e n s i t i v i t y 4.3 The guides i s o l a t i o n , in adjacent insensitive s i m i l a r i t y embedded the for considered. This of are presence guides case et neglected. Chang [38]. there apparent. agree al.[17] approximate when here if very interaction Including agreement Thus are However, four the the well those with inaccuracy edges results along the of a obtained 69 here other are similar guides c o e f f i c i e n t s those or a between calculations considering to separated details guides in groundplane. involving the for the of the guides Thus may array other the of guides be presence the used F i g . 1 . 3 in the of coupling in the without array. 70 F i g . 4.2 ray path from the d r i v e n t o the p a r a s i t i c guide guide 71 -15 i -20 ^ -25 -30 J -35 0 0.5 2.0 1 .5 1.0 d/A 180 i 90 -90 -180 4.3 TEM-TEM c o u p l i n g between waveguides For Figs. , 4.3 - separated 338X). 4.9 as applicable (2) A A OO (2) + A +A CO ( 2 ) + A ( 3 - o - Montgomery +A A A (a=b=0. (3) ,(4) + A CO +A * Dybdal et ) + A ( 4 ) + A ( and Chang al. [17] 5 ) [38] (measured) 72 Fig. 4.4 TEM-TEM c o u p l i n g between waveguides See F i g . (a=b=0. 4.3 for 761X). key. separated 25 30 35 •40 -45 -^•h 0 1.0 0.5 2.0 1.5 d/A 180 i / V / / 90 •7 / // / / / / -90 • / // -180 0.5 0 1.0 1.5 d/A Fig. 4.5 T F ^ - T F ^ c o u p l i n g between waveguides See F i g . 4 . 3 (a=b=0. for 761X). key. separated 2.0 74 Fig. 4.6 TEM-TEM c o u p l i n g between See F i g . 4.3 f o r key. separated waveguides (d=a=b). 76 Fig. 4.9 TEM-TEM c o u p l i n g between waveguides See (a=d/2, b=d) . F i g . 4.3 f o r key. separated 78 § - 3 0 cn o o -60 CM -90 1.5 1.0 0.5 2.0 dA i 180 if) 90 <b ^_ cn 7 fl X3 0 ^ // < no -90 4 / / / - / 4- -180 1.5 1.0 0.5 2.0 d/A F i g . 4.10 TE -TE c o u p l i n g between 1 waveguides A .... ll (a=d=b). (2) A n i 2 ' ^ ^ ) ^ ^ ) separated l 2 u Fig. 4.11 p l a n e wave n o r m a l l y i n c i d e n t on two n o n - s t a g g e r e d p a r a l l e l plates i 3 u Fig. 4.12 l i n e source normally i n c i d e n t on t h r e e n o n - s t a g g e r e d p a r a l l e l plates TABLE I I Coupling Coefficients A^b.d.a) Between S e p a r a t e d P a r a l l e l P l a t e Waveguides a = d = b = 0.45 X t excited -nit coupled Into Montgomery [38] 20 log|A| (dB) LA (degrees) Dybdal [17] Lee [29] 20 log|A| (dB) LA (degrees) -20.69 -74.8 -21.27 -77.7 1-3 -20.69 -74.9 -21.27 -77.7 2+4 -20.83 -72.4 -21.24 -73.8 3+1 & 3-»5 20 log|A| (dB) -21.34 LA (degrees) -74.6 this theory using the asymptotic form (3.17) 20 log|A| LA (dB) (degrees) -22.26 -75.8 this theorjr using the non-asymptot: c form (3.22) 20 log|Al LA (de reeo) (dB) E -22.64 -84.6 81 Chapter RADIATION 5.1 plate radiation employed pattern was Because many it is The to f i e l d s are guide f i e l d s A SINGLE to in can particular sum this of TEM case. from of the edges CANONICAL by GUIDE of w i l l the into the rays mode radiation represented from w i l l be up of d i f f r a c t e d the mode ( F i g . 5 . 1 ) . in in a ray each Several in a l l find array, made These is ray in The path ray are path different the pattern by [49], to analysis paths space. Felsen are to radiation waveguide The ray similar required consideration waveguide. PROBLEMS be edges. ray f i n a l from and waveguide waveguide p a r a l l e l - This Yee convenience. The f i e l d s the by successive total methods analysis. tracing radiated open-ended multi-element under by The ray-optical from the traced edge. by the for turn. be single analysis here edge f i e l d s a obtained radiated the of coupling of represented represents 5.2 the calculated calculated both the d i f f r a c t e d f i e l d s found pattern repeated restricted paths for parts radiation f i e l d s is previously the to pattern waveguide those the OF INTRODUCTION The the PATTERN 5 guide made paths up ray to a of leading 82 The f i e l d s combinations Fig.5.1 of have of by the In some any ray rays in a sequence and 3. were the be paths the found i t . not various ray Thus can preceding 5.3) the in in Chapter immediately (Fig.5.2 of of f i e l d s from the Other needed included and are here. Fig.5.2a waveguide f i e l d by successive calculated ray combinations presented two been represented fields represented- the u = plane wave y=z = 0 edge in a at direction e o is 6 u =0. is = 1/2 From given incident (2.7), the on a diffracted by +iV4 u D(0,6)E(r) = - sec(^)E(r), u 1 In F i g 5.2b y=a,z =0 a at direction e 6 plane 0 = . 2 l T is wave u From given (5.1) 2 is incident the diffracted = 1/2 (2.7), on the f i e l d edge in the by + i7T/4 u u. = x D(27!,6)E(r) = +u £ sec(f-)E(r) /2T In F i g located edge the a at an total) 5.3a an distance r angle Q f i e l d in from 0 = 3TT/2 . Q U isotropic the m 1 = + where a line guide From G( V wall (2.12), direction . ( is u e ,6) = -iir/4 ) on given that (not by (5.3) Q Q Ftvkr = E(r) diffracted e)E(r), +ikr sine (+e 2 m incident the is e source f o r o < e < TT/2 G(r 5 2 0 (cos - Q - s i n -) ] 83 -ikr +e sine fl ° fl F [ « £ r ~ (cos j + s i n | ) ] } (5.4a) } f o r TT/2 < G < 3TI/2 -iu/4 G(r ,6) = +ikr i— vT 'T 0 sinS Q e F[> kT~ ( - c o s - + s i n - ) ] u ^ ^ {-e / -ikr +e sine „ ° F [ A 7 ^ (cos | fi + s i n j ) ]} (5.4b) and f o r 3TT/2 < 6 < 2TT -i7i/4 G(r . 6) O +ikr = r sin6 {-e F[*£r~ (-cos 0 — -ikr sine -e ^ g - + s i n -) ] Z Z p F f v ^ (-cos j U fl - s i n | ) ]} , (5.4c) f o r k r >> 1 0 G(r 0 In Fig f i e l d e an 5.3b distance angle .e) ^ r 0 from o = v/2 in the . e + i 7 T / 4 /— „, > cos(e/2) E ( r 0) isotropic a guide From direction c o s '6- - line wall (2.12), & is is the (5.5) source incident diffracted given by = E(r) on that (not located edge the at a an total) 84 -G(r m+1 5.3 ,6)E(r) . (5.6) CALCULATION OF RADIATION The radiation plate total waveguide semi-infinite perfectly width y=+d/2. the space pattern d is parallel-plate conducting and As of We w i s h outside before, the p ( 0 of d ) calculated waveguide p a r a l l e l to the PATTERN here. the of i n z>0 f i e l d s p a r a l l e l - Consider consisting half-planes determine a the at a two y=-d/2 radiated into guide. incident f i e l d -ikz i (5.7) in the the t o p and bottom value guide is decomposed edges into at 9 = a two p l a n e 0 or 2~r waves incident respectively, on whose is 1 U at both In edge singly F i g . 5 . l a y=+d/2, u. In the (5.8) edges. The e. 2 ~ i edge 3 = diffracted the 2=0. ^ incident From (5.2) D(2TT , 6 ) E ( r ) F i g . 5 . l b y=-d/2, f i e l d s the z=0. are f i e l d the shown (5.8) is diffracted i n F i g 5.1a diffracted f i e l d by t h e is . (5.9) incident From and f i e l d (3.18), as (5.8) is observed diffracted at by y=+d/2, 85 2=0 the d i f f r a c t e d u = b 1 located f i e l d u i s a line source - - C (d)E(r) 2 0 at the (5.10) d i f f r a c t i n g edge, where from (3.19) and (3.20) 2F(^d) e ^ = _ _ _ 4 .( ) c d 7 _ _ r . ( 5 > n ) /T7 If k d » l , +in/4 C (d) ^ - • This and line source from (5.3) h = 2 In n c ( d the again a d i f f r a c t e d at the edge y=+d/2, 2=0 f i e l d ) ~ G(d,6)E(r). Fig.5.1c edge 2=0 i s gives _ u ° the (5.12) /r7 0 the y=+d/2, incident 2=0. d i f f r a c t e d (5.13) f i e l d f i e l d From ^ c a (5.8) (3.18), s line a as i s d i f f r a c t e d observed at by y=+d/2, source -c; (d) U ° l located = at resultant i s other edge the _ edge C ( 0 d y=-d/2, ) E(d+r) made up of two l i n e at y=-d/2, y=+d/2, edge. This 2=0 f i e l d and from i s again (3.26) the i s —4 2 c at = (5.14) d i f f r a c t i n g f i e l d c u ^ E(r) the d i f f r a c t e d u ~ 2=0 2=0. + + i V 4 C'(d)E(d)E(r) sources, This and from e f i e l d (5.3) one at i s the again . y=-3d/2, (5.15) 2=0, d i f f r a c t e d resultant f i e l d i s at the the - c u = 3 C ( 0 d ) + —1 G(2d,6)E(r) V 4 i + 8 C'(d)E(d)G(d,6)E(r) which l o o k s l i k e two l i n e sources a t t h e d i f f r a c t i n g In F i g 5.Id t h e i n c i d e n t f i e l d u u the f i e l d i s given diffracted by a t t h e edge (5.10). (5.16) edge. i s g i v e n by (5.8) and This y=+d/2, 6 z=0 field i s again and from (3.26) t h e resultant f i e l d i s d u u " — C = 2 ( d e ) 0 E(d+r) + + i l T / 4 — C' (d)E(d)E(r) . (5.17) i s made up of two l i n e s o u r c e s , one a t y=+3d/2, z=0, t h e 2 other a t y=+d/2, z=0. T h i s f i e l d i s again d i f f r a c t e d at the edge y=-d/2, z=0 and from (3.26) t h e r e s u l t a n t f i e l d i s u d " 0 — C = ( d e ) E(2d+r) + + L v / 4 C' (d)E(2d)E(r) '' +iTT/4 + u C' (d)E(d)E(d+r) i s made up of t h r e e l i n e sources a t y=-5d/2, d 3 y=-3d/2, z=0 r e s p e c t i v e l y . T h i s f i e l d . (5.18) y=-d/2 and i s again d i f f r a c t e d a t the edge y=+d/2, z=0 and from (5.3) t h e r e s u l t a n t f i e l d i s d - Q — C = u 4 ( d ) + e G(3d,6)E(r) W 4 + C ' ( d ) E ( 2 d ) G ( d , 6 ) E (r) +iTT/4 + • C'(d)E(d)G(2d,6)E(r) 8/2TT which l o o k s l i k e t h r e e l i n e sources a t the d i f f r a c t i n g The total fields z=0 up t o quadruple (5.9), (5.13), p o diffracted diffraction are edge edge. from t h e edge y=+d/2, given by t h e sum of (5.16) and ( 5 . 1 9 ) . These f i e l d s appear l i k e l i n e sources a t t h e edge. The f i e l d the (5.19) U y=-d/2, z=0 and r a d i a t e d diffracted from i n t o space i s c a l c u l a t e d 87 s i m i l a r l y given using the ray = The and i s (5.20) at the pattern two edges -ikd — — P = the second terms are lengths the shadowed +ikd — + from given edges . (5.21) which to line by P" e 0 factors quadrant i s apparent — sin6 CTT/2 < e < TT) , the by these waveguide .„ array f i r s t the sin6 quadrant the formed of P' e 0 0 In F i g . 5 . l e , f , g , h , -Pi . 0, radiation sources path i n by P" 0 in paths a where the adjust distant for exponential the different observation (o < e < TT/2) , one of the point. edges i s and -ikd . . —z s i n e P The = o pattern P. e is . symmetrical P (d,2n-6) = n so e = n that (5.23) d i f f r a c t i o n V The radiation be about p(d,6) 0 * pattern 0 may (5.22) for single written (1) p e = 0 Note that given by at + i3Ti/4 — ~ sec(-) /2TT e e = TT (1) P o k d s i n (— s i n e ) E ( r ) the value of P q . ( 1 ) (5.24) i s f i n i t e and i s +13TT/4 (d,e=,> = ? ~ - k d E ( r )f /2TT so that width. the maximum F o r o < e < TT/2 radiation i s proportional to the guide 88 • (i) p At i = 0 = e - + l /„ V 2 ~ ~ ~ = r there 7r/2 -ikd — « 4 e S E 2 C is ) . „ s i n 6 • E a ( d i s c o n t i n u i t y in the 5 ' 2 6 ) single e d i f f r a c t i o n e < TT/2 . order radiation This it can the the shown S X n u e u-i U 2 The 2 ) Z i M b TT+ discontinuity because -ikd — - n f l i double d , d 2 e n d is c on tin u ity at 6 = TT/2 is in given d i f f r a c t i o n term s i n 9 which k The . s i n . . . s i „[49] n . D The the pattern is addition of discontinuity at 6 = about half t r i p l e d i f f r a c t i o n in the because n/2 double s i m i l a r l y of term d i f f r a c t i o n it can be (Fig.5.4b) E + U ( 2 d isco ntinuity pattern at about half in I S e = TT/2 of J ) E = the 2 by i successively higher d i s c o n t i n u i t y at applies disco ntinuit y the ( 2 ( 2 at be e [49] —-T" ^ . reduced. 0 = * E +ikd d i f f r a c t i o n may ) d i f f r a c t i o n <3 IL~) u ~ sine order e = TT/2 U t r i p l e given ^ (j) e f u to double sine e i higher of pattern that ( this for taking addition radiation eliminates shown eliminates shadowed by (Fig.5.4a) e radiation The that z ,TT(j ) 2 account. remaining + reduced d i f f r a c t i o n —T~ 2 be used. _ 2 can is u of (5.12) i s +ikd f because form the (H ) 2 b asymptotic by u ^ ± i M sine 2 double 1 into term been e,ir-»- if discontinuity d i f f r a c t i o n d i f f r a c t i o n pattern, 3V2 . Thus which by - 2 8 ) is taking terms, Similar 5 radiation . sin6 ( the reasoning 89 5.4 NUMERICAL The AND EXPERIMENTAL RESULTS radiation pattern a single waveguide calculated by r a y - o p t i c a l methods the exact known pattern[Weinstein, plane pattern of These patterns are and 5.6. The The consist gain forward the into the direction i s never i s less shadow also much (5.28) reduced higher to exact when 1 dB and i s only an a r b i t r a r i l y pattern for for Each as the the width of constant and the then i s more form of The discontinuity continuity f o r the asymptotic small value successive 0.5 d i f f r a c t i o n pattern the pattern for guide waveguide in exact dB i n The discontinuity asymptotic by approximately experimental Fig.5.5 be e x p e c t e d , less quadruple the since plate the r a y - o p t i c a l ( F i g . 5.5b). terms. 1.4). plotted essentially ( F i g . 5.5a). true discontinuity The then are order in E - 0'<90°). 1 dB when smaller but are As might is The ray - o p t i c a l throughout with calculated decreases The phase between more boundary (section widths p a r a l l e l lobe. (-90°< error then a beamwidth d i r e c t i o n account. used and increased. maximum of single and the forward 6 = n/2 is i s compared e - TT) a the pattern are of guide i s horn patterns patterns the sectoral measure, p a r a l l e l - p l a t e and the measured guide radiation only 1969] for various radian C ©' = increases The shown i n e' < 1 8 0 ° . < the H-plane radiation o < e < 2r -180° of i s taken accurate at (5.12) i s at 6 = TT/2 relations form, (5.27) a n d may b e by considering interaction at s t i l l reduces the dB the one-half. is widths also from within 0.339^ 1 to 0.441^ of (f*10- 90 13 B GHz) and >150° over where because of the the entire patterns blockage by angular could not the antenna pattern of range, be except measured for accurately mounting arrangement ( F i g s . 5 . 6 a , b , c , d ) . 5.5 SUMMARY The radiation p a r a l l e l plate compared to sectoral be to in the the The results sectoral and that gives The experimental and experimental more complex chapters. an good less taking the higher that of a to may be be that considered can form of of the the with in terms. experimental exact, both extended and and accuracy between suggests account terms d i f f r a c t i o n exact the shadow when p a r a l l e l - p l a t e of agreement dB and H-plane the asymptotic pattern estimate methods 1 into order the an along then taken methods with expected E-plane patterns structures is non-staggered r a y - o p t i c a l obtained between the single are adding simulates and approximation. by agreement horn well, plane further confirms by d i s c o n t i n u i t y d i f f r a c t i o n and/or good found patterns aperture s t i l l expression quite The quadruple reduced is experimental horn. boundary up waveguide a H-plane waveguide of this r a y - o p t i c a l r a y - o p t i c a l confidence the to following \ ray paths \ t F i g . 5.1 in a parallel e tr ray Fig. path 5.2 (see 5.3 (see tit. p l a t e waveguide e Fig. ray path \_ text) text) 92 F i g . 5.4 c o n t i n u i t y of f i e l d s a c r o s s shadow boundary see eqns. (5.27) and (5.28). CL I -20 a> angle (degrees) 5.5a R a y - o p t i c a l and e x a c t r a d i a t i o n p a t t e r n s f o r a p a r a l l e l p l a t e waveguide using (5.11) exact ray-optical (a=0. 45A). 94 270 i 180 90 -90 -180 Fig. 5.5b -90 0 angle (degrees) R a y - o p t i c a l and e x a c t for a parallel using plate asymptotic exact ray-optical form 180 90 radiation waveguide (5.12) patterns (a=0. 45X) F i g . 5.6 E x a c t r a d i a t i o n p a t t e r n s f o r a p a r a l l e l p l a t e waveguide compared w i t h measured E-plane p a t t e r n s o f H-plane s e c t o r a l h o r n . 96 Chapter 6 RADIATION PATTERN OF THREE ELEMENT WAVEGUIDE ARRAY 6.1 INTRODUCTION In this chapter the radiation is found of a m u l t i - element waveguide array similar to used f o r a s i n g l e waveguide, and w i l l be those (Fig.6.1) pattern by methods compared t o experimental r e s u l t s using the H-plane sectoral horn. Calculation plate of waveguide pattern depth the r a d i a t i o n p a t t e r n of the p a r a l l e l - array begins by first c a l c u l a t i n g the f o r the case when a l l three guides are of i n f i n i t e (Fig.6.2). parallel-plate A l l edges of the resulting of waveguides are e x c i t e d by the c e n t r a l d r i v e n guide. The f i e l d s d i f f r a c t e d by these edges to y i e l d array are c a l c u l a t e d the r a d i a t i o n p a t t e r n . If the p a r a s i t i c guides are of f i n i t e depth, the f i e l d s coupled into them also c o n t r i b u t e to the t o t a l r a d i a t i o n p a t t e r n . For t h i s case, the c o u p l i n g c o e f f i c i e n t c e n t r a l d r i v e n guide and the p a r a s i t i c between the guide on e i t h e r must be c a l c u l a t e d . The coupled f i e l d s are r e f l e c t e d open and waveguide radiated between c l o s e d ends of the p a r a s i t i c guide and e v e n t u a l l y r a d i a t e d from the open end. The t o t a l the side array i s calculated radiation by pattern of adding the f i e l d s from the edges e x c i t e d by the c e n t r a l d r i v e n guide and the f i e l d s r a d i a t e d from the outer guides. In the TE \ case the c o u p l i n g between adjacent guides i s 97 very weak, parasitic the and guide pattern. the TEM 6.2 not for i t s radiation For t h i s Consider waveguides y=d/2, and three y=d/2+a (Fig.6.2). i n the central edges a t significantly the analysis i s much the affect i s restricted to stronger. four in As perfectly z>0 at waves 0 = o o r 2T? y=-d/2, 5 the incident -d/2<y<d/2, two p l a n e conducting y=-d/2-a, i n Chapter guide parallel-plate z>0, given incident respectively. TEM by on t h e t o p These plane l 2 (6.1) at t h e edge y=d/2, by the incident The z=0, rays These plane waves be d i f f r a c t e d a g a i n are into Fig.6.1 diffracted of Fig.6.2. by the waveguide may be r a d i a t e d by a n o t h e r waveguide edge. Thus a l l edges are excited guide and contribute Sequences of rays radiated guide represented fields pattern. are and a r e diffracted driven they z=0, i n the central resulting waveguide and y=-d/2, central as into have the v a l u e V " four driven i s decomposed i n t o edges. to coupled semi-infinite of half-planes waves b o t h may adjacent thin bottom i s FORMULATION consisting infinitely (5.7) reason case where t h e c o u p l i n g RAY-OPTICAL mode power enough diffracted called by the fields to the ray paths trace from v a r i o u s edges and or i n the radiation the fields eventually space. s h o w s some o f the ray paths which must be 98 c o n s i d e r e d t o c a l c u l a t e the r a d i a t i o n p a t t e r n of the d r i v e n guide 6.4 central i n the absence of the o u t e r g u i d e s . F i g . 6 . 3 and shows some of the ray paths which must be c o n s i d e r e d , t o c a l c u l a t e the edges additional excited by diffraction the central from driven the guide waveguide when the waveguides of i n f i n i t e depth are added. 6.3 RADIATION PATTERN WITH GUIDES OF INFINITE DEPTH The t o t a l r a d i a t i o n p a t t e r n P • of the parallel field fields from scattered n p i s made up of the a l l the sum of edges. P ( d ) r e p r e s e n t s o represents the the and the t o t a l f i e l d s c a t t e r e d from the n t h edge (as l a b e l l e d i n F i g . 6 . 2 ) in p three s c a t t e r e d from the open end of the c e n t r a l g u i d e , (n=l,2,3,4) p of p l a t e waveguides w i t h o n l y the c e n t e r guide d r i v e n i s c a l c u l a t e d . The field array not already included . o The t o t a l radiation pattern P i s g i v e n by the sum rear f o r the a r r a y of Fig.6.2 of the f i e l d s from a l l the edges. In the h a l f s p a c e a l l but the outermost edges are shadowed, so t h a t f o r a n g l e s o < e < TT/2 o n l y the edge at is v i s i b l e , P t and - P e ,d+2a. . r, - i k ( — — ) sine -lk T - P 3 e ,d+2a, . „ (—— )sin6 z=0 ( 6 2 3 For a n g l e s ~/2 < e < -n a l l edges a r e v i s i b l e P y=(d+2a)/2, and ) 99 -ik — sine + P e l +ik — sine + P 2 e +ik ( d+2a•)sin6 + The pattern The paths P 4 2 e (6.3) i s s y m m e t r i c a l about 6 = T T . P field q in Fig.5.1. was calculated The field p a t h s i n F i g . 6 . 3 . The paths are and ray image a b o u t t h e solution successive using is calculated p 3 using paths used to c a l c u l a t e z-axis for of rays i n the ray the those shown ray the ray the p ray anc 2 in path and thus the ^ P 4 Figs.6.3 (labelled contribute t o any to represent the letter under of the the . The by two 6.4 can now with a ray (n=l,2,3,4) from p there are a,b,c,etc. in PN represented fields associated contributions waveguide edges. Note t h a t paths fields p a t h s o f F i g s . 6 . 3 and used t o c a l c u l a t e a l l the u. j _ is calculated the 6.4. The be field i n F i g . 6 . 4 . The the p in Chapter 5 using n several Figs. 6.3 different and following notations f i e l d s , where t h e f i g u r e showing the the 6.4) are superscript ray which employed x refers particular ray to path consideration. = an incident field i n the a i n a ray central driven guide 1 u X nm has = field been d i f f r a c t e d m times. path c o n t r i b u t i n g to p n which 100 The f i e l d To in by a as line located calculated contributions i s In at p , 3 Fig.6.4a d i f f r a c t e d observed to i n Chapter 5. consider the at the edge y=d/2+a, z=0 the the incident ray f i e l d y=d/2, u z=0. d i f f r a c t e d , i From f i e l d u * source 31 U been F i g . 6 . 4 . (6.1), (3.18), is has q calculate diagrams given P ~~2 = at ( r ) the d i f f r a c t e d at resultant ( 6 d i f f r a c t i n g the f i e l d edge edge. This y=d/2+a, z=0, f i e l d - i s and from 4 ) again (5.3) the i s C ! (a) = from the In A — F i g . 6 . 4 b y=+d/2, the edge edge z=0 the d i f f r a c t e d C = y=d/2, z=0 f i e l d 3 2 i s d/2, = made z=0, d i f f r a c t e d f i e l d i s up of the at y=-d/2, u z=0. f i e l d , given From u i s by (3.18) a (6.1), as line i s observed source E(r) line and (6.6) source i s from again (3.29) the d i f f r a c t e d at resultant the doubly i s - C ' (d) _2 E(d+r) b u - ~ This u f i e l d ( d ) edge. d i f f r a c t e d incident the " n u the at b at (6.5) edge. d i f f r a c t e d at G(a,6)E(r) - two l i n e other at „ / 4 C' (d)E(d)E(r) . source y=d/2, y=(d+2a)/2, i l T z=0. z=0, f i e l d s , one c e n t r e d The f i e l d and from (6.7) (6.7) (5.3) the i s at y=- again resultant 101 -C-(d) u 33 In the 4 Fig.6.4c f i e l d u ^ d i f f r a c t e d resultant the i s at y=+d/2+2a, - + up of 3 3 f i e l d = z=0 by (6.1)'and f i e l d and from is again (3.26) C'(a)E(a)E(r) . two 0 line source at z=0 the (6.9) fields respectively. y=+d/2, z=0 and centred This from at f i e l d (3.26) is the is C'(a) u given This ^ and y=+d/2+a, d i f f r a c t e d resultant (6.4). is i (6.8) i.T,/A E z=0 u y=+d/2+a, 4 made again edge -A_ (a r) = by (a) 32 f i e l d C ' (d) E (d) G ( a , 6 ) E (r) , 0 is C u given iir/4 - incident the f i e l d c is G(d+a,6)E(r) in/4 — g — E(2a+r) - C!(a)E(2a)E(r) irr/4 ~ Ci (a)E(a)E(a+r) . ~T 8/27 is 2a, made y=+d/2 again the up of three and y=+d/2-a, d i f f r a c t e d resultant u c line at f i e l d r> f a i V!i . G ( 3 a ( 34 source z=0 the fields edge y=+d/2+a, e represents up to three edge. the total and including z=0 f i e l d and from is (5.3) S_c'(a)E(2a)G(a,6)E(r) /2TT f. in/4 8/2TT" d i f f r a c t i n g This y=+d/2- in/4 . ) E ( r ) 8 as at is 8 appears centred respectively. _ which (6.10) U The line sum f i e l d P3 quadruple sources of C' (a)E(a)G(2a,6)E(r) a l l (6.5), from (6.11) ° the centred (6.8), edge d i f f r a c t i o n . at and y=+d/2+a, the (6.11) z=0 102 To calculate diagrams in F i g . 6 . 3 . given by (6.4) and the contributions edge (6.1), is In the given y=+d/2, z = 0, Fig.6.3a singly by In and from f i e l d is f i e l d is (3.26) b u the given again the c 1 3 ( d ) the i s the f i e l d u ; Q i s " given d i f f r a c t e d resultant u and at i is ± is the given given edge by is (6.12) (6.1), (6.7). y=+d/2+a, z=0 by at f i e l d by ray is i Cj(a)E(a)G(a,6)E(r) . f i e l d " o = — g ~ E(d+a+r) f i e l d f i e l d (5.3) f i e l d (6.6) diffracted resultant incident This + incident by the consider 1TT/4 - ~ ~ G(2a,e)E(r) Fig.6.3b ^ d i f f r a c t e d (6.9). -CA(a) = to the This and from is W A + - C'(d)E(d+a)E(r) ir/4 ^ — "UC ' ( d ) E ( d ) E ( a + r ) . - u k consists y=+d/2+a again b. three line and y=+d/2+2a, d i f f r a c t e d resultant u of f i e l d C'(d) =- V - at (6.13) 8/2T the source f i e l d s z=0 respectively. edge y=+d/2, at y=+3d/2+2a, This and from z=0 f i e l d is (5.3) the is G(d+2a,9)E(r) - l*/4 C'(d)E(d+a) G(a,6)E(r) in/4 + In Fig.6.3c d i f f r a c t e d at the source at edge the the - incident edge C'(d)E(d)G(2a,8)E(r). f i e l d y=-d/2, y=-d/2-a, z=0 z=0. the given From f i e l d u by (3.18), is (6.14) (6.1) as given i s observed by a line = located -S^— at the d i f f r a c t e d at resultant f i e l d u = 1 2 y=-d/2-a, at the edge edge. of source a ) two l i n e z=0 edge f i e l d i s z=0 and from again (3.26) the is - ( This y=-d/2-a, 0 —7^-E(a+r) consists l 2 (6-15) d i f f r a c t i n g the C u E(r) e " -S / 4 Cj(a)E(a)E(r) . f i e l d s respectively. y=-d/2, z=0 This and from at f i e l d (3.29) (6.16) y=-d/2-2a, i s the again z=0 and d i f f r a c t e d resultant f i e l d is u., C'(a) - ^ 5 — E(2a+r) = in/4 C'(a)E(2a)E(r) + in/4 - C'(a)E(a)E(a+r) . 8V/2T7 (6.17) U c u consists 1 3 and of three y=-d/2-a, d i f f r a c t e d at line z=0 the source f i e l d s respectively. edge resultant f i e l d is C(a) <; = - V G(2a+d,6)E(r) u y=+d/2, + y=-d/2-2a, This z=0 in/4 ~~~~~ at f i e l d and y=-d/2 is from again (5.3) the C'(a)E(2a)G(d,9)E(r) i n /4 _ ^ c:(a)E(a)G(a+d,6)E(r) . 8v^ The ?2 sum from of the (6.12), edge (6.14) y=d/2, z=0 (6.18) 0 and (6.18) represents the up to including quadruple and f i e l d d i f f r a c t i o n . Similar the edge calculations y=-d/2, z=0 a n d P4 for from the fields y=-d/2-a, ? 2 scattered show that from 104 P 2 P, (6.19) = -P = - P 4 3 The to • (6.20) results give the for total ? parasitic guides 6.4 RADIATION WITH the array f i n i t e of the array The depth of the guides radiation and thus coupling driven width guide a travel end. is the has the reflected The w i l l alter F i g . 6 . 1 , that and change the d then back guide travel and part A of aperture plane z=0 relative guide the is sum of the a A Q 0 the in (6.3) of Fig.6.2 equivalent guides these are guides w i l l Adjusting the phase relative between adjacent towards = oo A elZKS + A of their parasitic The 3. open central guide coupled reflected the the at the end R e = f i e l d s + of f i e l d s shorted ,where part radiated. to in the those geometric oo oo ^ be the outer from guides the at central the driven series A„„e A of pattern. (d,a) are to now reradiated. Chapter and is value into and array is outer radiation and calculated which the coupled c o e f f i c i e n t outer f i e l d s SHORTED of (6.2) the GUIDES end width been down The of far of depth. f i e l d s from pattern in i n f i n i t e except reflected The of OUTER Fig.6.2 depth. substituted radiation with Consider are n W """ e + 1 - ••• , . 7 7 ^ = 00 (6.21) where R 0 Q p a r a l l e l is the plate reflection waveguide coefficient [45] [50], at the and S open is end the of depth the of 105 the outer guide. The t o t a l calculated f i n i t e the f i e l d s and by depth array r a d i a t i o n adding of of the f i e l d s from (Figs.6.5 guides have di outer been and guides to from edges array the the excited 6.6). d i f f r a c t e d 6.6 the additional d i r e c t l y and of r a d i a t i o n The other guide F i g s . 6 . 5 the F i g . 6 . 2 . radiated pattern Since at represent caused f i e l d s radiated f i e l d s f i e l d s the by the from of ( F i g . 5 . 1 ) , i n f i e l d s once, i s consist guides f i e l d s a l l least F i g . 6 . 1 f i e l d outer by of a i n ray including shorted the outer paths i n quadruple f f r a c t i o n . Radiation simply plate the edges calculated In due l t P (a)i n to the the f i e l d ray from shown i n and (5.22) by A of r a d i a t i o n (6.21). F i g . 5 . 1 of a Scattering from the paths in F i g s . 6 . 5 the edge outer y=d/2, z=0 the f i e l d from guides and i s 6.6. i s (6-22) from the same edge i s in/A G(2d,e)E(r) - = i s p a r a l l e l - G(d,6)E(r). = C'(d) 3 guides (5.21) Q by c o n s i d e r i n g F i g . 6 . 5 b i u outer m u l t i p l i e d Fig.6.5a u 1-2 the pattern waveguide other In from C^(d)E(d)G(d,6)E(r) . (6.23) A /2T7 In U F i g . 6 . 5 c 1' In c " C = 3 0 — ( F i g . 6 . 6 the a f i e l d ) = -iL 3 3 4 the same edge i s e - — — C'(a)E(a)G(d,6)E(r) . A/2TT i 7 ! / 4 G(d+a,6)E(r) the f i e l d from the edge y=(d+2a)/2, (6.24) z=0 i s ITT/4 Cl(d) u from G(d+a,6)E(r) + Av^7 Cl(d)E(d)G(a,6)E(r) . 0 (6.25) 106 The give f i e l d s the f i e l d radiation f i e l d of from the z=0 P- outer the f i e l d s P^ d/2-a, (6.23) from from P^ the (6.22), show the edge guides. edge from and (6.24) edge y=d/2, Similarly y=d/2+a, the must z=0. y=-d/2, be z=0 (6.25) Similar z=0, added to excited by gives the calculations and p ^ from y=- that = _< (6.26) P 2 1 p. _ _ . (6-27) P The ray about the The t o t a for the the at P^ a n d ?l in Fig.6.5 by t h e f i n i t e sum of This radiation obtain P and q the are and depth P^ must complete the image 6.6. the outer multiplied by the be of added radiation to P pattern F i g . 6 . 1 . quadrant y=d/2+a, radiated to of f i r s t A . calculate shown caused array array the to those and (6.3) the In that of coefficient (6.2) l z-axis is coupling p used radiation guides in paths z=0 f i e l d s , are so for o < 6 < TT/2 shadowed a \ \ and do not edges except contribute to that -ik(——)smo P total = P k + API e 3 • i /d+a, .. -ik(—TT-J s i n o + AP e . 2 Q In the second quadrant for TT/2 < e < T T (6.26) 107 ., .d+2a. . -ik(——)sm6 = P + API e t 3 c P total -ik(~-)sine + AP ( a ) e u - i k ^ sin6 + A?^ e +ik y sine + AP^ e +ik(^-)sin6 + AP (a) e Q ., ,d+2a, . +ik(—2~)sine + AP! e 4 (6.29) 0 This P pattern i s about that 6 = TI S O , (2TT-6) = P^ _ , ( 6 ) . total 6.5 total NUMERICAL The p a r a l l e l is symmetrical radiation plate compared sectoral AND E X P E R I M E N T A L to pattern waveguides with the the outer radiation E-plane The 0— P are , t narrower beamwidth The addition of of central depending outer on guides The their varies pattern the the of the added of then a by to A and the three-element H-plane the outer f o r (Fig.6.2), (6.3), has a guide. in in of methods plotted depth altered of an ' simulate (6.2) guide Adjustment of of are single guides array r a y - o p t i c a l i n f i n i t e driven phase by patterns given outer depth. element patterns IT) . guides pattern the three calculated somewhat pattern a appendages guides (Fig. 1.2). -180° < 6 ' < 1 8 0 ° (0 ' = When of measured horn RESULTS the radiation d i f f e r e n t depth of ways the (6.21). array with outer 108 guides s h o r t e d shows a pronounced r e d u c t i o n of the beamwidth compared t o that of a s i n g l e guide when the all three guides Adjustment is in phase (arg A radiation = 0) from (Fig.6.8). of the frequency such that t h i s phase difference i s near 180° produces a p a t t e r n with a n u l l on the beam a x i s and two main lobes to either p a t t e r n shapes with v a r y i n g A from a Fourier another value essentially of relatively radiation (Figs.6.9,6.13) expected t r a n s f o r m of the a p e r t u r e f i e l d s [ 2 7 ] . For this phase over in the pattern amplitude the phase of the pattern i s s u b s t a n t i a l l y reduced compared variety of is the forward d i r e c t i o n and the back to that s i n g l e guide. Thus t h i s p a r a l l e l - p l a t e waveguide can a was an angular s e c t o r of a b o u t ± 6 0 ° a l l cases constant are what would These be constant ( F i g . 6 . 1 0 ) . In side patterns by adjustment of a produce of the depth of the outer guides or the frequency. The calculated accurate at patterns about 6 = 90 are expected to be least o f f the beam a x i s near the shadow boundary i n the a p e r t u r e plane f o r the reasons mentioned Chapter 2. outermost at At angles slightly less then 90°. (6.28), 90° a l l edges c o n t r i b u t e consequently a d i s c o n t i n u i t y 6= g r e a t e r then 90° only the edge c o n t r i b u t e s t o the r a d i a t i o n angles in T h i s d i s c o n t i n u i t y may the pattern reduce the to be s u f f i c i e n t , discontinuity whereas (6.29), and is found at be reduced by i n c l u d i n g more ray paths along the shadow boundary. Terms up d i f f r a c t i o n appear in to quadruple s i n c e more terms do not s i g n i f i c a n t l y , and the number of terms grows very r a p i d l y f o r higher o r d e r s of diffraction. 109 The discontinuity depending on frequency. discussed To the The in using method Lee near the problem of planes is a between the outer problems on the shadow d e t a i l in Chapter s i g n i f i c a n t ray-optical and shadow varies of a accuracy 90° 6= depth more effect of at Boersma boundary. line source needed, be where solution the half-plane edges are a l l solution, which would be Figs.6.3b, 6.3c, 6.4b, 6.5c on the boundaries were two find pattern the that the f i e l d s non-staggered Such a evaluate and 6.6, is were generally the d i f f r a c t i o n observation c o l l i n e a r . to the require to for source, used +4dB and in may applied incident and 2. techniques A guides improvement [32] -4dB h a l f - point and ray-optical the f i e l d s not in presently a v a i l a b l e . The the experimental calculated 8 = - 1 5 0 ° . are could caused not antenna 6.6 Any be results pattern by site mounting up to asymmetry measured , 9=±60 and within within and the small r e f l e c t i o n s . The patterns accurately because of 1 dB of 3dB up to o s c i l l a t i o n s near blockage ©=180° by the arrangement. SUMMARY Radiation aperiodic been patterns array analyzed tractable reasonable from of by however a f i n i t e , rather than open-ended p a r a l l e l plate ray-optical methods. Only for accuracy the rapidly numbers of increases an i n f i n i t e waveguides small arrays ray required with the number has are for of 110 d i f f r a c t i n g analyzed with The off the edges. t r i p l e beam axis in the be reduced by be eliminated number The of expected of when aperture the in increased or dimensions of and of has edges d i s c o n t i n u i t y This into at 90 edges are discontinuity account, and been included. d i f f r a c t i n g d,a so large s , but even can cannot with a to the width. of a reasonably antenna and waveguide A/3 are or f i e l d s the less, in the with those horn with p a r a l l e l - p l a t e well. The be waveguide outer depending frequency. may agree sectoral beamwidth array to the generally simulate guides for Inaccuracies reduced H-plane the v a l i d boundary. patterns that only are shadow decreased the is calculation the T E M mode waveguide a p p l i c a t i o n . the paths analysis the appears the ray dimensions along ones, appendages of four a d i r e c t i o n . values experimental calculated this a l l the array rays. in plane The of more by contain most backward moderate p a r t i c u l a r l y array because ray-optical dimensions three-element results taking for a d i f f r a c t i o n calculated shadowed larger Here Thus adjusted on the to guides the pattern suit the —> z Fig. 6.1 three element a r r a y of p a r a l l e l p l a t e waveguides w i t h outer guides shorted d/2 d/2 Fig. 6.2 three element a r r a y of p a r a l l e l p l a t e waveguides w i t h outer guides of i n f i n i t e depth 112 \ it NJ1 t. ii F i g . 6.3 ray paths from c e n t r a l g u i d e s t o edge 1 \ \ F i g . 6.4 ray paths from c e n t r a l t o edge 3 guide F i g . 6.6 r a y path from o u t e r g u i d e t o edge 3 \ n ray F i g . 6.5 p a t h s f r o m o u t e r g u i d e s t o edge 1 113 -I-—• -90 -180 • 1 -90 • • ' — 0 • • 90 • 180 angle (degrees) Fig. 6.7 R a d i a t i o n p a t t e r n o f three element a r r a y w i t h o u t e r g u i d e s o f i n f i n i t e d e p t h (a=d=0. 401X). 270 i angle (degrees) Fig. 6.8 R a d i a t i o n p a t t e r n o f three element a r r a y a=d=0. 450X s=0. 8 5 6 X A=0.131 L 0 ° . 115 270 i Fig. 6.9 R a d i a t i o n p a t t e r n o f three element a r r a y a=d=0. 339X s=0. 645X A=0. 371 L - 1 7 l ! using using (5.12) (5.11) 116 _90 -I -180 Fig. 6.10 . . -90 • • • 0 angle (degrees) — 90 — l 180 R a d i a t i o n p a t t e r n o f three element a r r a y a=d=0. 356X s=0. 677X A=0. 308 L -134°. 117 Fig 6.11 R a d i a t i o n p a t t e r n o f three element a r r a y a=d=0. 373X s=0. 709X A=0. 247 L +104 . 118 -90 H -180 — • -90 • • • 0 - ' 90 ' 180 angle (degrees) F i g . 6.12 R a d i a t i o n p a t t e r n o f t h r e e element a r r a y a=d=0. 389X s=0. 459X A = 0 . 199 L +38°. 119 -90 -I -180 • . -90 • • •— 0 • • 90 • • 180 angle (degrees) Fig. 6.13 R a d i a t i o n p a t t e r n of three element a r r a y a=d=0. 441X s=0. 597X A=0. 223 L + 1 6 6 ° . 120 Chapter RADIATION 7.1 RAY-OPTICAL The results The the additional formation into of account. guides and array for of w i l l [29] be [38] of element array outer guides many more five WAVEGUIDE ray at ARRAY incident mode in by is the by be central be taken for outer other depth methods respectively. perfectly two As p a r a l l e l plate conducting y=-d/2-a, guide into 6. allow (Fig.7.1) s e m i - i n f i n i t e six array must by the i n Chapter calculated found plate extending element which ( F i g . 7 . 2 ) . decomposed p a r a l l e l and f i n i t e results y=-d/2-2a, y = d / 2 + 2a five w i l l adjacent and found paths results of element considered the (Fig.7.2) with consisting z>0 in patterns d=+d/2+a (5.7), five three compared in a may be and experimental waveguides ELEMENT (Fig.7.1,7.2) i n f i n i t e Consider planes pattern The both OF F I V E FORMULATION radiation waveguide the PATTERN 7 y=-d/2, in y=+d/2, Chapter -d/2<y<+d/2, plane h a l f - z>0 waves 6 the , given each with amplitude (7.1) at the edges waves are guide of represented F i g . 7 . 2 . considered Chapter y=-d/2, 6. to z=0 by the Many calculate and y=+d/2, the The a d d i t i o n a l incident of the ray paths rays ray radiation z=0. These i n paths pattern which two the central which are occur plane must shown in the be in five 121 element in 7.2 array Figs.7.3-7 Fig.7.2 so f i r s t given the three element GUIDES pattern by of the sum a l l 0<e<ii/2 and OUTER radiation quadrant for in PATTERN WITH total is that not array are shown .7. RADIATION The the but the ? OF INFINITE for t fields but the outermost only the edge the from a l l edges are y=d/2+2a, z=0 DEPTH array of edges. In shadowed is v i s i b l e are v i s i b l e . , ,d+4a. . -ik(—)sxn9 Q P In the t = P 5 d second quadrant ( Tt/2<e<n ) a l l edges and _ i k ( d±ia ) s i n e - l k (——) sine + P 3 d d - i k — sine + P + d o p d + P„ e 2 +ik - sine . . ,d+2a. . . + i k (——) sine + p e 4 + + P 6 e i k { ^ 2 ) s i n e (7.3) 122 where ? i s n symmetrical The be a about created by presence a d d i t i o n a l To the i n the not in Chapter 6 additional modified up to must ray guides now paths ( F i g . 7 . 3 ) . because quadruple be i s t new o u t e r be pattern = P <e) . t new a n d m u s t the of d i f f r a c t e d observed of The there are d i f f r a c t i o n . calculated The using the F i g . 7 . 4 . diagram i s #n. P (2n-e) account paths are c edge c a l c u l a t e d 4 into need 2 ray calculate ray (7.1) and P P and P paths P the no ray and 3 from that o take f i e l d s c P s to The P radiated e = n f i e l d s modified f i e l d s f i e l d at additional F i g . 7 . 3 . at the y=d/2+a, c o n t r i b u t i o n The incident edge z=0 y=+d/2, the to P f i e l d u z=0. d i f f r a c t e d From 3 consider given (3.18) f i e l d i s a by as l i n e source A _0 a u 3 c -5— = 1 located at the d i f f r a c t e d f i e l d at u 3 2 i s z=0, E ( r ) d i f f r a c t i n g y=d/2+a, edge. z=0 C'(a) V!i (a r) 32 the E 4 made d i f f r a c t e d f i e l d in A\ (7.4) ,. This and from f i e l d (3.29) i s the again resultant i s a u ( a ) up of other at + e S — 4/2T7 + two l i n e at (7-5) i 7 , / 4 C'(a)E(a)E(r) sources, y=+d/2+a, y=+d/2+2a, z=0 one c e n t r e d z=0. and from at This f i e l d (3.26) the y=+d/2, i s again resultant i s C'(a) a u 33 = - 0 8 E(a+b+r) - pi 7 1 / 4 ^Z C' ( a ) E ( a + b ) E ( r ) Bl in/4 + C 8/2^ ° (a)E(a)E (b+r) . (7.6) 123 This f i e l d (5.3) the i s again resultant diffracted f i e l d at y=+d/2+a, z=0 and from i s -C'(a) = a u ° 3 4 G(a+2b,6)E(r) + - C'(a)E(a+b)G(b,6)E(r) 8v^ 8 ° iit/4 - C (a)E(a)G(2b,6)E(r) , S 8>2TT (7.7) 0 / a u 3 must 4 be added to P (7.3) which up and including to To (7.1), the In a u f i e l d represents calculate F i g . 7 . 4 . i s i s again 0 — C 5 = 3 ( a the the by 6 from (7.4) at and u diffracted at e + the h " n c u * at diffracted f i e l d b 3 in y=d/2+a,z=0 the 2 is ray diagrams i u i s given z=0 by and given of by (7.5). This from (5.3) i V 4 C'(a)E(a)G(b,6)E(r) . incident f i e l d y=-d/2,z=0. diffracted (7.8) ° f i e l d u given From (3.18) is line a by (7.1) as is observed source { d ) the y=+d/2, = -2 (7.9) b d i f f r a c t i n g at z=0 edge. and ^ S m E(d+r) from - 4 52 52 edge P This f i e l d (3.29) u the 5 i s 1 again resultant is u u f i e l d = -VB(r) 5 located the the edge y=+d/2,z=0 the is G(a+b,6)E(r) at 5 y=+d/2+a, ) Fig.7.4b the f i e l d a 8/27 In give consider incident diffracted to d i f f r a c t i o n . f i e l d the given f i e l d f i e l d quadruple Fig.7.4a resultant u i n Chapter 3 a ^ e U P ° ^ t w 0 (7.10) C'(d)E(d)E(r) . . pr~ line 0 sources, one c e n t r e d at y=- 124 d/2 z=0, the r d i f f r a c t e d f i e l d other at at y=+d/2, y=+d/2+a, z=0. z=0 This and from f i e l d (3.29) i s the again resultant i s 5 -C'(d) -°- = b u 3 in/4 E (d+a+r) - C'(d)E(d+a)E(r) ° 8>^"r7 8 iTT/4 - C (d)E(d)E(a+r) 8/2TT i s u.^ 3 made y=+d/2 z=0 line This and from -C'(d) -2 = b three r e s p e c t i v e l y . y=+d/2+2a, u up of G sources f i e l d (5.3) (d+a+b,6)E(r) i c - 7 at i s the y=-d/2, again resultant T / y=+d/2+a and d i f f r a c t e d f i e l d at i s 4 C'(d)E(d+a)G(b,0)E(r) 8^ 8 5 4 (7.11) 1 0 iir/4 _ £ C (d)E (d)G(a+b,0)E(r) ° 8^7 The sum the edge of (7.8) and (7.12) y=d/2+2a,z=0 up represents to (7.12) and the f i e l d P 5 including from quadruple d i f f r a c t i o n . Similar c a l c u l a t i o n s contributions to show f o r P 6 and the additional that (7.13) P 4 P 6 The ^ 3 = = _ ray paths F i g s . 7 . 3 The to with give 5 . used contributions in to to calculate Pif a r e the image ?(, and about the the additional z-axis of those and (7.3) and 7.4. results the (7.14) P for total p a r a s i t i c P n are radiation guides of substituted pattern i n f i n i t e of depth. i n the (7.2) array of Fig.7.2 125 7.3 RADIATION WITH Consider Fig.7.2 depth. the array except that The f i e l d s reflected width separated into end and between \ 2 adjacent guides of now o f w i l l of to f i n i t e now be The TEM guides of b and a i n Chapters 3 and 4 are t h e ' f i e l d s relative to width coefficients and & plane are reradiated. calculated coupling equivalent guides between (d,a) i s guides these d were the aperture the used to i n the f i e l d s outer from the guide. value & +d/2<y<+d/2+a Q 0 which outer (d,a,b) values driven The 0 0 These the at central A SHORTED Fig.7.1 shorted by d i s t a n c e determine guides the and B respectively. of the coefficient d and a GUIDES coupled from coupling OUTER of 1 i s , the from f i e l d s (6.21) i n the given f i r s t outer guide by i2ks A A l the R ( 0 guide outer the outer 6 OklT a * ) of guide. second into 0 ' a ) = 1 where 0Q ( d outer driven guide the " oo R s t e width (7 reflection a, coefficient a n d s^^ i s the the value guide +d/2+a<y<+d/2+a+b, guide and the must second be taken outer + - > 15 6 To determine B (d,a,b) 0 Q n ( a ) 1 f i r s t into guide i s A ^ A (a,b) the depth A 2 of open of the account. guide f i e l d s The into end of the the coupling outer given at f i r s t i n the from the energy both second coupled by (7.16) 126 The value of a of the geometric f i e l d A a 2 t t n aperture e plane i s t h e sum series iks A 2 f oo = B ( d ' ' a b ) + 1-R where s i s 2 The the t o t a l c a l c u l a t e d f i n i t e the array f i e l d s and f i e l d s guide i s p a r a l l e l plate In given In 5"2 F i g . 7 . 5 b given = the P b = f i e l d s 5 1 7 ) to array the the from p by i n f i e l d s radiated f i e l d s the ray by the paths from i n the A a guides (5.21) by t h e from of ( F i g . 5 . 1 ) , i n outer i s consist guides f i e l d s from F i g . 7 . 1 caused m u l t i p l i e d r a d i a t i o n of f i e l d outer the 0 guide. shorted shown and (5.22) x o r A of i n a Scattering 2 outer guides i s F i g . 7 . 5 - 7 . 7 . edge y=+d/2+a+b,z =0 (b) G(b,8)E(r> . 5 „ (7.18) from a the edge y=+d/2+a+b,z=0 i s by — 5"3 The outer the excited f i e l d f i e l d u -C'(a) u ' by cU of from pattern by c o n s i d e r i n g the ( 7 a d d i t i o n a l Radiation due to F i g . 7 . 5 a e second The edges the ] guides waveguide edges c a l c u l a t e d is other ) r a d i a t i o n d i r e c t l y simply other the outer (Fig.7.5-7.7) F i g . 5 . 1 from from b e the F i g . 7 . 2 . radiated a pattern adding the ( of r a d i a t i o n of of 0 0 ' ., , i k s 2 A (b) 0 0 depth by depth l A " radiation 4 u 5 iV4 G(a+b,6)E(r) - . „ 4 and u a from from - the the 5 „ are b edge f i r s t outer i / ^ C' (a)E (a) G 0 added to (b, e ) E give y=+d/2+a+b,z=0 guide. In ( r ) . the excited F i g . 7 . 6 a the (7.19) f i e l d by f i e l d 127 u a 3"3 from a the - C (b) _0 = Fig.7.6b given u is a n C = 3"3 a ) the second outer ( — radiation (2b,6)E(r) f i e l d + - given by C ' ( b ) E (b) G ( b , 6) E ( r ) . ^ 3"3 (7.20) 0 from b u from In p" 31 the outer edge given G(2a,6)E(r) f i e l d guide same the edge y=+d/2+a,z=0 is from I T L / 4 C ' (a) E (a) G ( a , 6) E (r) . the edge f i r s t outer guide and would was m i s s i n g . u , 3 ( is b the f i e l d i n the not by r a d i a t i o n Fig.7.7a y=+d/2+a,z=0 (7.21) excited by 2 guide, excited P - be the from ^ ^ v 2 f r present f i e l d the o presence m t i f h e the p ^ second edge of the second from outer the guide. y=+d/2,z=0 is by U In i s by b u the y=+d/2+a,z =0 XT,/A G 4 3"3 In f-.dge -C'(a) l"2 2~~ ' G(a = Fig.7.7b the f i e l d u • 6)E(r) 1 , ! ^ from ' (7 the edge y=+d/2,z=0 is 22) given by b u C = n ( b ) - — — G(a+b,6)E(r) + B i T / 4 C ' (b) E (b) G (a ,6) E ( r ) . (7.23) 4v57 In Fig.7.7c the f i e l d u from 1 1 1 3 the edge y=+d/2,z=0 i s given by C'(a) = c u G ( a + d , 6 ) E (r) -2 1 , 1 3 C ' (a) E (a) G ( d , 6) E ( r ) . + 4v 7 4 / 2 The fields (7.22), f i e l d from only from the second one outer (7.23) the outer guide. and (7.24) edge guide are y=+d/2,z=0 which (7.24) ° would added excited not be to by give the radiation present with 128 Similar calculations y=-d/2,z=0, P and from 6 P" 2 P = 41 the P^ edge from the from 2 f i e l d s the y=-d/2-a-b P^ edge show for the edge y=-d/2-a,z=0 and that (7.25) -p" 1 - 3'l = P" 42 P = ( ' 7 2 6 ) (7.27) -P" 42 (7.28) The ray are the paths used image F i g s . 7 . 5 , 7 . 6 depth of and p A to of . ^ the 7.7. outer guides This obtain radiation the must complete p z-axis The multiplied i s the f i r s t y=+d/2+a+b radiated are f i e l d p total = P the by the be added P 4]/ 4 of array to ? and P 2 those sum o f t pattern by the ( ) , 0 a t (7.2) o t a i in f i n i t e p ( 0 c o e f f i c i e n t of p p P^ shown caused appropriate radiation quadrant shadowed so o < e < TT/2 a l l and ) or A and of b (7'.3) the array do P t l + + the second = P t A P (b) 2 quadrant + except contribute that to the . sme Q A, P" e 1 5 _ p total not edges that -ik F ' radiation ,d+2a+2b, -ik ( ) In 2 F i g . 7 . 1 . In at calculate about and the to A P" e 15 i k ( Q for e ,d+2a+b, . . — )sin6 2 TT/2 < e < TT ±t2|±2b ) s i n e . (7 ( / 29) '^> 129 . , ,d+2a+b. . . -ik ( —) s i n e + A P (b) 2 e 0 -ik l y s i n e tV3 + V 3 1 + V32 + ] e - i k f ^ J s i n e + A . P (a) 1 0 e + [ A ^ + A P^]e + 1 [A, P i 1 2 + A P"]e 2 2 + AjP -ik d — sine +ik — sine 2 J + i k (~^-) (a) sine e ., ,d+2a, . . + i k (——) s m S + [A P X 4 A + ^ + A 2 +ik ( where + A.P"e about measured appendages e . , ,d+2a+2b, . „ +ik ( ) sine given e = TT S in with patterns simulate the of other of (7.30) Chapter that O patterns compared to • AND E X P E R I M E N T A L E-plane 2 ) sine 1 6 radiation are 4 (b) are NUMERICAL The array A P P^ symmetrical 7.4 + P ' ]e an outer p t otai The 6. ( 2 l 1 " e ) = P pattern totai ( 6 ) is - RESULTS a five element theoretical H-plane guides waveguide results sectoral and horn (e'-0-Tr). the with 130 When consists the outer of a essentially degrees The patterns residue to a forward d i r e c t i o n Taking higher eliminate In except for the the simulated range of 4 5 ° in the agree remarkably be angles, zero at control guides the the because are 90-120 a axis all. in small in the is in but the plane. does compared element direction not of is to that array Over the presence groundplane patterns aperture [29]. agreement found boundary. five the that The reduces pattern the to angles the shadow an is angular two patterns the simulated not so requires the good f i e l d for to . are of phase of the aperture f i n i t e depth that in the pattern may be adjusted (Figs.7.12-7.18). possible for the five compared to the three which f i e l d s is Fig.7.10. terms to application parameters 90° the despite the pattern about beam at guides relative radiation dB groundplane expected, of compared in e = forward well, ©=±90 When a that a larger [38] ray-optical in As are d i f f r a c t i o n case The There the here on the boundary. near embedded groundplane. off of discrepancy Fig.7.11 calculated 90° shadow fraction depth range frequency. methods order this angular calculated calculus within an e'= the i n f i n i t e ( F i g . 7 . 8 , 7 . 9 ) . the at along of lobe on (<ldB) plane are over depending aperture agree single constant discontinuity by guides may element be is f i e l d in driven More element it control array, adjusted. the guide. to array possible suit of parasitic Hence a the In pattern there general the particular considered because to here are the is as more five 131 element then may patterns the be axis, three adjusted whereas single null patterns and have is the the three the beamwidth patterns. produce two axis. more radiation The in less then is patterns can produce of the slopes element about phase constant steeper nulls patterns The and five adjacent element beam generally back smaller element to at a the beam only five forward the the a element direction three element patterns. As in accurate The at 90° three off discontinuity ray is the paths noted into in which case. shadow d e t a i l edges in expected a to discussed The in be the an patterns s t i l l 1 dB is up calculation made Additional s i g n i f i c a n t 2. depending then for to then the for improvement the array i n f i n i t e depth discussed larger these more number d i f f i c u l t i e s may the three element in of be array chapter. patterns for agree quite 0=60° Thus of boundary. discontinuity on the least taking d i f f i c u l t i e s were array, here by The -3dB Because generally comparable. pattern array ( no d i f f r a c t i o n . are shadow however and not the reduced preceding accuracy along patterns is element experimental the patterns boundary greater case, axis larger Chapter (Figs.7.12-7.15) within is five in the +3dB parameters, more beam .quadruple between The the account; beyond varies element appears with as and the the five the good 3dB accuracy to be element calculated as up the to of degraded three array ones element e'=150°), the to but ray-optical s l i g h t l y when the larger. calculations are performed for arrays for 132 which in the guide the phase p a r a s i t i c with the in be top. of and a l l When phase two the aperture lobes Note to that the are aperture or fields exactly f i e l d s shown patterns not case a . would out of and the in Figs.7.16- are tabulated the When the the array A that on the because might across the whose the are out beam axis f i e l d is small the is a good size aperture approximation of radiation a pattern shape. phase of the dimensions further each as aperture considering crude and beamwidth guide the pattern sector amplitude f l a t parasitic null (Fig.1.3), a very deep a 3dB (Fig.7.17), not 7.18) are curve require narrowest s u r p r i s i n g independently. patterns is shape, approximate by phase (Fig. is the in in is sector x)/x the beam 6? = ± 3 8 a t h i s determined adjusted at to (sin expected are there This for truncated 7.5 phase has f i e l d s i n sign array. phased the aperture these pattern f i e l d s (Fig.7.16) f i e l d s are The of make in patterns However, approximation is guide. the aperture expected. the exactly parameters that alternating of guides to i l l . Note when chosen radiation Relevant Table was driven corresponding 7.18). depth element aperture and cannot improvement be f i e l d excited be in as the in a array of array. SUMMARY The p a r a l l e l radiation plate from waveguides a five element has been analyzed f i n i t e the same ray- 133 optical 6. methods The radiation however, more because ray paths When the results that for more taken into guides the of in more d i f f r a c t i n g edges of i n f i n i t e result method size array is are ray-optical this element Chapter complex and hence account. ray-optical calculus the three calculation are outer residue the pattern be between confidence for there to the agreement by used is and quite method array. that the calculated good, can The depth which give gives acceptable pattern also agrees the quite for the a forward When as general and for of complexity of can element by in of to three accuracy the a wider a of of are patterns the structure adjustment to type element of A Wiener—Hopf simulated guides experimental the a by s e n s i t i v i t y the outer produce approximation array. lack direction degradation array calculated embedded the calculated well that array indicates the with well the sector groundplane, of the of do array, of agree shaped quite a method The patterns parameters, depth indicating increases. five then for the as slight as the element the three a good a small including pattern in structure. f i n i t e not which pattern surrounding ray-optical variety technique, such 134 Fig. 7.1 f i v e element array of p a r a l l e l p l a t e waveguides with outer guides shorted AV r 1 -ri b i Fig. 7.2 f i v e element a r r a y o f p a r a l l e l p l a t e waveguides with outer guides of i n f i n i t e depth d/2 d/2 —*— i b t a Fig. ray path 7.3 b F i g . 7.A ray paths(see (see t e x t ) \ t F i g . 7.5 ray paths(see t text) text) 11 b Fig. ray paths 7.6 (see text) \ V Fig. ray paths 7.7 (see text) 137 angle (degrees) Fig. 7.8 R a d i a t i o n p a t t e r n of f i v e element a r r a y with outer guides of i n f i n i t e depth d=a=b=0. 450X. 138 angle (degrees) angle (degrees) ig. 7.9 R a d i a t i o n p a t t e r n o f f i v e element a r r a y w i t outer guides of i n f i n i t e depth d=a=b=0. 3 139 0 CD 1 10 i Q. £ -20 JTJ 0. -30 -90 Fig. 7.11 CD Si_ -45 0 angle (degrees) 45 90 R a d i a t i o n p a t t e r n o f f i v e element a r r a y w i t h outer guides o f i n f i n i t e d e p t h , d=a=b=0. 4 5 0 A , c o m p a r i s o n w i t h [ 2 9 ] . -10 H O CL I> OI -20 H -30 -90 Fig. 7.10 -45 0 angle (degrees) 45 90 R a d i a t i o n p a t t e r n o f f i v e element a r r a y w i t h outer guides o f i n f i n i t e d e p t h , d=a=b=0.450X,comparison w i t h [ 3 8 ] . this theory comparison 140 angle (degrees) Fig. 7.12 R a d i a t i o n p a t t e r n o f f i v e element a r r a y d=a=b=0.441X s,=0.838X s =0.597X. A=0.137 L -12° A=0.074 L - 1 2 ° . 141 CD 32. -10 § CL -20 H > -30 -180 •90 180 0 angle (degrees) 180 -i 90 in cn a> ro _c CL -90 -180 -180 -90 0 90 angle (degrees) Fig. 7.13 R a d i a t i o n p a t t e r n o f f i v e element a r r a y d=a=b=0.407X s =0.774X s_=0.551X ft-0.171 L -55° A=0.076 L - 5 7 ° . t 2 180 142 -180 -I -180 • • 1 • -90 • ' • • 1 0 ^ 90 angle (degrees) F i g . 7.14 R a d i a t i o n p a t t e r n o f f i v e element a r r a y d=a=b=0.373X S!=0.709X s =0.505X A=0.247 L -103° A=0.096 L - 9 8 ° . i z 2 ' 180 - 3 0 - 1 8 0 - 9 0 0 180 90 angle (degrees) 180 -i 90 -90 - 1 8 0 - 1 8 0 -90 180 90 0 angle (degrees) Fig. 7 . 1 5 Radiation pattern of five d=a=b=0.356X A=0.308 i L - 1 3 4 ° S;L=0.677A A=0.126 Z element array s =0.487X 2 L - 1 2 3 ° . 144 Fig. 7.16 Aperture f i e l d and r a d i a t i o n p a t t e r n o f a f i v e element a r r a y A=-0.215 A=-0.090 d=a=b=0.450X =0.610X s =0.356X. S l 2 145 Fig. 7.17 A p e r t u r e f i e l d and r a d i a t i o n p a t t e r n o f a f i v e element a r r a y A=+0.131 A=+0.073 d=a=b=0.450X S]_=0.857X s =0.610X. 2 146 Fig. 7.18 A p e r t u r e f i e l d and r a d i a t i o n p a t t e r n of a f i v e element a r r a y A=-0.131 A=+0.045 d=a=b=0.450X s =0.857X s =0.356X. 1 2 TABLE I I I R a d i a t i o n P a t t e r n Parameters Figure 7.8a A A l 0 A 2 0 null(dB) Beamwidth (de:grees) -lOdB -3dB -ldB back radiation(dB) 0 39 54 76 -25 7.13a +0.131 +0.073 0 12 21 64 -28 7.13b -0.215 -0.090 -9 50 59 78 -26 7.13c -0.131 +0.045 0 31 43 67 -32 148 Chapter COUPLING The coupling waveguides that for First reflection a l l are which establish to the 8.1 each i t . e Q r . The in to d i f f r a c t i o n ray used and from problems paths here. similar w i l l The The the of be is of the successive solved analysis t r i p l e f i e l d f i e l d two to guides. including inspection. calculated these manner non-staggered up by a f i r s t to restricted PROBLEM an from 0 The isotropic an edge d i f f r a c t e d p l i n e is source incident (not the on total) = E(r)located that f i e l d edge at observed at an at 6±8 such u at of paths is in staggered case. Fig.8.2a distance ray adjacent found coupling ray notation CANONICAL angle and GUIDES for be determined occur the TEM In a by preceding rays w i l l possible represented STAGGERED coefficient (Fig.8.1) d i f f r a c t i o n ray BETWEEN 8 the that . m+1 = c o s ( __o } Q G.(r ,e ,6)E(r) n d 0 d i f f r a c t i n g 0 edge, where is given by (8.1) 149 -in/4 G,(r.6 d 0 .6) = 0 f~ /iT _ i k cos(9-6 ) r n -e cos(0+6) ° ° ^ 6+6 F [ / 2 k r ^ c o s (——)]) , (8.2 a > 0 F(a) i s g i v e n by (2.4) a < 0 and E ( r ) ) (8.2a) = -F(-a) F(a) (—)] u -ikr F'(ci) 6-9 F [ v ^ k r ~ cos {-e i s g i v e n by (2.10) 150 In each at the case the diffracted d i f f r a c t i n g f i e l d is a line on a Fig.8.2b half-plane d i f f r a c t e d may, U a = G d ( a be ' at e 0 ' wave angle which (2.2), l plane edge f i e l d from centred edge. -ikr In source is u 6 = e . Q ) is 0 An uniformly written cos(6-6 expression v a l i d for a l l for the angles 6 as ' e ) E ( r ) incident (8.3) where G (a,6 G and a d is point. ( a 'V the If 6 ) distance where 8.2 o D(e , 6 ) „ CALCULATION waveguides half-planes of determine z>0) into incident 8 4 from the edge to the observation D(6 .8i, ( given by adjacent consisting of y=a,z>l, y=0,z>0 the the of the f i e l d s f i e l d U i = guide guide _ i v 1 three ) from and l=a the (0<y<a, p a r a l l e l - p l a t e perfectly and y=-d,z=0, coupled e 5 COEFFICIENT s e m i - i n f i n i t e the top staggered - (2.9). OF T H E COUPLING two stagger 8 0 i s Consider angle <-' ka>>l, G (a,e ,6) A ,9) —iuo = conducting where tan y driven z>0). f > o . We w i s h guide As i s the to (-d<y<0, before the z (8.6) 151 in the guide i s decomposed the half-plane 2TT respectively. edges into y=-d,z=0 Both waves the at have waves incident angles 6 = o on and value 1 = (8.7) 2 i at a n d y=z=0 plane u two plane edges. The singly incident f i e l d y=z=0. diffracted given u From (2.7) f i e l d by (8.7) i s i s shown i n Fig.8.3a. The diffracted the diffracted f i e l d i s at given the edge by a line source u j D(2TT,0)E (r) = a located at the d i f f r a c t i n g coupling f o r as source a line single Q The i s (3.11), contribution found (3.12) to by treating and (3.2). the (8.8) Thus in/4 « ( 1 ) edge. d i f f r a c t i o n and using . . jz A (8.8) 0(2,,0), (8.9) 2> 2 k a / which i s The c. identical doubly In Fig.8.3b d i f f r a c t e d edge y=a,z=l u located the at = b 1 at distance diffracted source to (3.17) diffracted f i e l d s the incident the edge 2 f a i at shown u From f i e l d given (8.3), i s i n Figs.8.5b and a as line by (8.7), observed t h e two the edge i s at the source -*)E(r) the d i f f r a c t i n g between are f i e l d y=z=0. the diffracted iG'(c.27T f i n t h e TEM case.. < 8 edge, where edges. y=a,z=l This a n d from c = = 2 /2 /£ + /£ + a f i e l d i s i s (8.1) g i v e s 1 0 ) again a line 152 u Fig.8.3c in = b j G^<C,2TT,J - v ) G ( c , ^ -Y,2n)E(r) . L d 2 shows a ray path which i s i d e n t i c a l t o Fig.3.2c (8.7), f o r a non-staggered contribution to diffraction X = the c o u p l i n g i 9 s that shown i s given by by (3.35). i v e n coefficient A g from The double i s given by e 1 1 T / * 4 ( 2/1 k a S C ) 2 _ y ) 2 d 31 G d - , , 2 , ) E - I K £ 2 + c^(d)C (d)] . (8.12) \jf = O , (8.12) i s e q u i v a l e n t t o (3.36) i n the TEM case. If The t r i p l y d i f f r a c t e d e. Fig.8.3d shown and i n Fig.3.2d u i s again d i f f r a c t e d source 3 j G ^ ( c ( 2 » ^ - ? ) G a ( c ( that guide. Here ^ i s (3.38) f o r the TEM case. In e i s given by (8.10) and ^ 2 2-n r e p l a c e d by 2JL _y by (8.11) with the f i n a l a line to i s given by ( 3 . 3 7 ) , i s given by e This f i e l d = i s identical non-staggered i s given by (8.7), u given U for a which i s given by (3.39) s p e c i a l i z e d 3 Fig.8.3e is f i e l d s are shown i n Figs.8.5d and shows a ray path given by (8.7), at guide. Here ^ i s given by (3.34) and ^ (2) (8.11) at y=z=0 and from (8.1) g i v e s ^ - T ^ - * ) G d ( c i - t O ) E ( r ) (8.13) the edge. In F i g . 8 . 3 f the i n c i d e n t diffracted at = u given by (8.7) i s the edge y=-d,z=0. From (8.3) as observed a t y=z=0 the d i f f r a c t e d uf 1 field field ;GMd,0,^)E(r) 2 d 2 i s a line source (8.14) 153 located a t the d i f f r a c t i n g edge. This field d i f f r a c t e d a t y=z=0 and from (8.1) g i v e s a l i n e u o = 2 located ^ G ' ( d 2 d ( 0 ^ ) G . ( d , ^ 2 d 2 2 , 3 a t the d i f f r a c t i n g = k G 2 A d (8.15) edge. (d,0,J)G. ( a , - V ) G . ( c , ^ J 2 d g i v e n by (8.10) and by 3TT/2 looks l i k e a l i n e u| 2 the i n c i d e n t f i e l d u In F i g . 8 . 3 g replaced d 2 2 = source -¥)E(r) This field d i f f r a c t e d a t y=z=0 and from (8.1) g i v e s a l i n e u i s again i s again source -?,2Ti)E(r) . (8.16) i s g i v e n by ( 8 . 7 ) , is i s g i v e n by (8.11) w i t h t h e f i n a l 2TT . This f i e l d i s reflected a t y=0,z = l and source i G ^ ( c , 2 T T ^ - H ' ) G d ( c , ^ - ^ ^ ) E ( r ) (8.17) x at t h e image p o i n t y=-a,z=l, where t h e n o t a t i o n a field i n the f i g u r e l a b e l l e d x t h a t has been d i f f r a c t e d m times and a l s o r e f l e c t e d . This f i e l d y=a,z=l and from (8.1) g i v e s a l i n e 3 at = 2 G ' (c ,2TT d t h e edge. coefficient (3.39), 2 - T ) G (c ,~r d u— denotes i s again d i f f r a c t e d a t source ,~r) G ( 2 a ,^r,2 n) E ( r ) 2 2 The t o t a l (8.18) 2. d contribution from t r i p l e d i f f r a c t i o n t o the coupling i s found by ( 8 . 1 3 ) , (8.16) and (8.18) a s l i n e sources and u s i n g ( 3 . 1 1 ) , (3.12) and ( 3 . 2 ) . Thus r treating 1^/4 , + G^(c,2TT,i - ) G ( c , ^ T d - y , ^ L -Y)G (c d 1 -w, ) 0 154 + G total d i f f r a c t i o n (8.19). Note staggered If Y along is no longer a It but arise s ^ o of does as a angles less of i function guides, of by (8.9), for the coupled when for the of which i s pronounced terms as v (3.12) a n ray 5> n w h e r e & =0, n considered. d i f f r a c t i o n increased. A (a,a) a and d and for As fine is = ~n~/2- Y, coefficient widths a B here were coupling i n Fig.8.4. i s to guide angle ( i . e . modes reveals not the modal the coupling coefficient. d i f f r a c t i o n apply conversion field y the guide single (non- v = o i n t h e TEM c a s e waveguide (8.12) and i n the staggered the ray-to-mode angle including (3.44). f i e l d arises the up to and g given s a r e shown v A (d,a) the restrictions not occur other (8.19) not . v a l i d the stagger i f behaviour double 3 ) , by t h e sum of i s because values staggered given V i k d ^)E(d)] boundary Numerical becomes d the coupled RESULTS calculated the i f A NUMERICAL average ( d i f f i c u l t y complement would i s °> = valid This various 0 because shadow Fig.8.4). 8.3 C (8.19) that is the that guides) Note f i e l d . t d coupling coefficient t r i p l e (8.1). -«C (cf-T,f)G (2a^2,)e- ( d , 2 , l 8x7 + The d for the non- provides the Addition structure of which 155 Exact r e s u l t s available f o r comparison. and without i n accuracy coefficients are not However, on comparing curves i t appears as the stagger there i s with some increases. SUMMARY The c o u p l i n g parallel the coefficient p l a t e w a v e g u i d e s was The n u m e r i c a l from coupling triple diffraction, deterioration 8.4 f o r the r e s u l t s show a non-staggered increased. These calculations involving between found adjacent by r a y - o p t i c a l moderate and will methods. gradual r e s u l t s as the angle calculations staggered be change of s t a g g e r i s useful an a r r a y o f s t a g g e r e d w a v e g u i d e s . for Fig. 8.1 two a d j a c e n t s t a g g e r e d p a r a l l e l p l a t e waveguides Fig. 8.2 l i n e s o u r c e and p l a n e w a v e i n c i d e n t on a h a l f p l a n e 157 u ray paths ray F i g . 8.3 for coupling coefficient F i g . 8.4 path (see t e x t ) 158 0 i 180 i F i g . 8.5a TEM-TEM c o u p l i n g between s t a g g e r e d waveguides <y=cr s i n g l e d i f f r a c t i o n s i n g l e and d o u b l e s i n g l e and double d i f f r a c t i o n and t r i p l e d i f f r a c t i o n -45 1.5 1.0 0.5 2.0 d/A 180 i 8.5b TEM-TEM c o u p l i n g b e t w e e n s t a g g e r e d waveguides 4*= 15°. single diffraction s i n g l e and d o u b l e d i f f r a c t i o n s i n g l e and d o u b l e and t r i p l e d i f f r a c t i o n 160 -45 0.5 1.0 1.5 2.0 d/A Fig. 8.5c TEM-TEM c o u p l i n g b e t w e e n staggered waveguides Y=30°. single single single diffraction and d o u b l e d i f f r a c t i o n and d o u b l e and t r i p l e d i f f r a c t i o n -45 I —> 0.5 0 1.0 1.5 2.0 d/A 180 -i 1 3 5 •{ Fig. 8.5d TEM-TEM c o u p l i n g between staggered waveguides Y=45°. single single single diffraction and double d i f f r a c t i o n and d o u b l e and t r i p l e d i f f r a c t i o n 162 0 1 Fig. 8.5e TEM-TEM c o u p l i n g b e t w e e n staggered waveguides 4<=60°. single single single diffraction and double d i f f r a c t i o n , and d o u b l e and t r i p l e d i f f a c t i o n 163 Chapter RADIATION FROM A STAGGERED PARALLEL-PLATE WAVEGUIDE FORMULATION 9.1 The radiation waveguide pattern (Fig.9.1) in methods similar to 5). pattern was This [42] using equivalent double up including and staggered guide d i f f r a c t i o n pattern a l l i s the ray shadowed the path made w i l l pattern be w i l l width a taking found a for of and l=a tan radiated into the space before decomposed the into here one. in f i e l d s a l l in Fig.9.2 shown a Note that given order total that then , f i e l d s angular the of radiation represented some the by are directions. by each total ray radiation angles. p a r a l l e l - p l a t e where . paths represented two p e r f e c t l y 'b/ ray for some Rudduck The f i e l d s account and terms The the (Chapter including are paths and y=+a/2,z=l, stagger As into the Ryan only. sum o f s e m i - i n f i n i t e and by ray-optical guide d i f f r a c t i o n . contribute section consisting y=-a/2,z>0 the by theory ray calculated, be Consider previously more found p a r a l l e l - p l a t e non-staggered t r i p l e up o f do not following a non-staggered paths, and for i s considered h a s many a staggered d i f f r a c t i o n f i e l d s then a d i f f r a c t i o n the to of T E M mode found edge to the those representing In 9 outside the incident f i e l d two p l a n e waves conducting Y We w i s h waveguide * > 0 to s t n of half-planes e determine angle the of f i e l d s guide. (8.6) incident in the on the guide i s half-plane 164 edges. 6 =0 has o top T h e wave edge value incident on the (8.7) that y=+a/2,z=l at at bottom edge. has 6 =2T 0 edge y=-a/2,z=0 T h e wave incident at on the value -ikJ, at that 9.2 edge. CALCULATION The b. In singly F i g . 9 . 2 a at f i e l d line a the (8.7) at the (8.7) D f i e l d y=+a/2,z=l. shown u From i n Figs.9.2a given by (2.7) the and (9.1) i s diffracted T , , e ) E ( r ) 2 ( Fig.9.2b i s at a the the edge line incident f i e l d y=-a/2,z=0. From In f i e l d i s a line 2 ) by the source Fig.9.2c the - (2.7) { at 9 given u D d i f f r a c t e d d i f f r a c t e d are i (0,6)E(r) edge. i s { In f i e l d = b u edge d i f f r a c t e d d i f f r a c t e d f i e l d s source edge. i s PATTERN incident the = A U at d i f f r a c t e d the d i f f r a c t e d i s OF RADIATION the edge incident f i e l d y=+a/2,z=l. u From ± 9 - given (2.7) 3 ) by the source -ik£ u = 1 — D(2Ti,2Tr-e)E(r) located at from lower f i e l d the u-r the i s d i f f r a c t i n g guide a line at (9.4) edge. y=-a/2,z=l-a source (9.4) This tan f located f i e l d . at i s The the reflected reflected image point 165 y=-3a/2,z=l. is The radiation formed located by the at the three two pattern line from sources guide edges single (9.2), and d i f f r a c t i o n (9.3) the and image (9.4) point respectively. The In doubly Fig.9.2d This f i e l d gives a u = d 2 at is i is line located f i e l d u = -ik£ ^-z the y=+a/2,z=l, at e 2 located (9.1) = at and \ y=-a/2,z=0 located at shown u is d in Figs.9.2d-h. found y=+a/2,z=l (8.3). from and from (8.1) (9.1) the looks point e- at diffracted f i e l d a line at is , « d i f f r a c t i n g found from and from G (8.1) . (c 2v 21 f i e l d a This incident the line edge source as is line at observed at located at again diffracted source „( 2 d 3t ,^,6)E(r) » /> A fi (9.7) F r d edge. In (8.3). gives F i g . 9 . 2 f This a -*>G ( c , J d i f f r a c t i n g reflected (9.6) f i e l d a is and, source gives G ' ( 2 d , 2 T i , 3^ -1),G. 2 This y=-a/2,z=l (8.1) - ^ u , . , . , is the the (9.6) edge. y=-3a/2,z=l. -iki. the is Fig.9.2e diffracted like and from y In (9.5) GU2d,2nAE(r) a 2 S L ^ l f at edge. d i f f r a c t i n g u| y=+a/2,z=l u by half-plane image and diffracted (8.1) 2 lower (8.7) d i f f r a c t i n g From 1 at by are j G M : , 0 , T - f l G J c , 7 - ? , e ) E ( r ) 2 d 2 d 2 given u* the again the y=+a/2,z=l. the given fields source u located diffracted edge. f i e l d line u is ± is by diffracted at source -Y,6)E(r) In given Fig.9.2g (9.8) VL is given by 166 (8.7) and ^ y=+a/2,z=l i and from u^ = the at the lower f i e l d h a l f - p l a n e like a h _ £ = located at the like image point those above, u i line = ^ u 3 . f i e l d i s tan i d i f f r a c t e d . 0 given s reflected The located (9.9) ^ is at by at r e f l e c t e d at the image and (9.1) y=+a/2,z=l and u^ from (9.10) at edge. This f i e l d y=-a/2,z=l-a line source tan is reflected The e (9.10) at reflected located at the d i f f r a c t e d in the by radiated Figs.9.2i-p. expressions inspection. In f i e l d s are represented Using methods the f o r Fig.9.2i the similar to f i e l d s can be radiated f i e l d i s source G ; located line a diagrams determined a Fig.9.2h (9.9) y=-3a/2,z=l. t r i p l y ray This source at source y=-a/2,z=l-a d i f f r a c t i n g u- the d i f f r a c t e d d f i e l d by line is 3TT 3TT U\-elr\ G' (2d,2TT,^-)G (2d,—,2T,-6)E(r) h a l f - p l a n e The a f i e l d source lower looks at f i e l d line _ I K 5 u This edge. line In This gives (8.1) the a y=-3a/2,z=l. (9.6). by gives (8.1) d i f f r a c t i n g u2 l o o k s point , ~ G' (c,0,~ -v)G (c,~r - f , 2 7 1 - 6 ) E ( r ) 2 d 2 d I 2 located found from ( 8 . 3 ) . s ( c ' at 2 i ' T , , , G d ( y=+a/2,z=l. c ' r In , , ' r - ) G Fig.9.2j d ( c the ^ ' w l E ( r l radiated ( 9 f i e l d - i s u ) a source Ic (c.0.|-«G <c,2l. a.„ located i a at y=-a/2,z=0. T In G a ( e .l-,..,E(r, Fig.9.2k the radiated f i e l d is a 167 line source k e located line „, - at = ^ line at -u, T r - 6 ) E ( r ) (9.13) - » ) G . ( c , ^ -V.2iT-e)E(r) Fig.9.21 In the d 2 radiated f i e l d is a - ^ e ) E ( r ) ( c , f Fig.9.2m ( c ^ - ! ^ ) G a y=+a/2,z=l. J ^ ( c . O ^ - m - at the (9.14) radiated f i e l d is a In a ( 2 a , ^ 8 ) E ( r ) Fig.9.2n the (9.15) radiated f i e l d is a Figs Fig.9.2o The from u y=-3a/2,z=l _ " the e " i k is and these is n d ( 2 a , ^ M ) E ( r ) is by (9.1) and which d i f f r a c t e d ^ d (2a,2Tr, is paths given resultant (8.1) ray f i e l d £ 2 ( c ^ - f , f ) G 9.1m reflected (3.26) d (9.16) y=-3a/2,z=l. contribution o 2 i - .3JL _1 source In U v ( 2 a , ^ - Y ) G d y=-a/2,z=l. j G ' ( c , 0 i - T ) G = located In _ source line 3 1 In G - ( 2 a , 2 T r , ^ ) G at located u . y=-3a/2,z=l. — located 3 „,._ - ¥ > G Ac , J source u* U 3_l G' (c , 2 TT at f i e l d not calculated u° appears the valid edge is like and separately. given a the line y=+a/2,z=l by (9.6). source and at from is 3TT 1 i —) [— E ( 2 a + r ) * — / 4 — E ( 2 a ) E ( r ) ] . (9.17) 2V^TT The at f i e l d u° y=+5a/2,z=l, which is y=-3a/2,z=l is the made is made other up of again up of at two two line sources, y=+a/2,z=l. line d i f f r a c t e d sources at one centered The r e f l e c t e d at y=-7a/2,z=l y=+a/2,z=l and from f i e l d and (8.1) 168 the ° u resultant _ e = 3 _ 4 G f i e l d i s ' ( 2 a , 2 T T ^ - ) G (4a,^|,6)E(r) d 2 d 2 f -ikJ. iTr/4 eIn tan ^v^,^, i s , e ^ ; 2 F i g . 9 . 2 p , 2TT-e G'(2a,2irA -» — / - » — , d , r e f l e c t e d at The r e f l e c t e d , — E(2a)G ( 2 a , ^ , 6 ) E ( r ) . given by the lower f i e l d u| (9.18) 2 (9.8) with half-plane looks replaced e at l i k e a by y=-a/2,z=l-a line source -ik£ U3 = — G (2a,2Tt,^-)G (4a,^,2Ti-e)E(r) ~ d e~ - ikt TTT d at The t o t a l up to (9.5), as t r i p l e the (9.4), as The sources radiation f i e l d s , (9.16) which i s These f i e l d s to (9.3), appear Q the up sum o f s (9.(3), (9.18). The total f i e l d s upper by y=-a/2,z=0 the y=-3a/2,z=l, pattern | - < e < ^ + 4 , i s d 0 These the given and edge. edge from as given by and the line the (9.|9), image of edge. for = the by (9.10), at are (9.15) lower The t o t a l - i k — sin6 S„ at given (9.9), l i n e (9.11), the (9.(4). edge. d i f f r a c t e d Q sources are and ' d i f f r a c t i o n (9.7), line (9.19) 2 y=-3a/2,z=l. S d i f f r a c t i o n at -3 point from upper sources 4 E(2a)G_(2a,^-,6)E(r) d i f f r a c t e d of appear / f i e l d s (9.12) sources the image appear t r i p l e sum the (9.2), f i e l d s (9.8), T /2T 2 y=+a/2,z=l of 7 f located sum I Gl(2a,2Ti -J) — 4 edge d S' 0 e formed given -ikJicosd e by by these apparent line 169 +ik — s i n e S" e o + + +ik — " ' e S e 0 where the adjust for to distant a some of exponential the the be path the l i n e radiation considered 0 = S 0 6 = For ^ - other sources angular are shadowed are four into l i n e and shadow TT and S' e * < ranges, do boundaries five regions which For o < e < n/2 - v The o ~o b Y<e< -j-* £ d +ik — s i n e -ik£cosO e + e (9.22) 2 < 2, 9 ^-j - y which terms i s the there may into radiation this c l e a r l y not shadow boundaries are be reduced pattern The symmetrical d i s c o n t i n u i t i e s account. discontinuity e < j - ? { 9 2 3 ) e pattern At must 37T — + +ikfsine b not (9.21) d • - i k — sine 0 the which -ik£cos6 TI — " y < 6 < — s factors e TT For There array from In (Fig.9.3). - i k — sine S the point. pattern separately are sources pattern. (9-20) lengths observation to the - terms d i f f e r i n g apparent contribute dividing -ik£cos6 sine 2 at For e = ~ - y, i n the taking by the (9.2), because e = TT/2 of because double i t u ^, order i s shown pattern, d i f f r a c t i o n and (9.4) shadowed d i f f r a c t i o n u c a n be and d i f f r a c t i o n single (9.3) b ^ + y radiation higher example given addition discontinuity by 6 =T . about that 2 has a for eliminates (Fig.9.4a) + b 1 i k f s i n e 1 -,) ] e <+ [ 6 M „ a + 2 u if the u [ e - ^ - Y M d asymptotic discontinuity can c u x be e 7 0 -ik£cose e 2 form (9.24) is (8.5) by by adding 3d + + i k— sin6 it , 2 [6= ] e 1 + at 6 = TT/2caused eliminated -ik£cos6 [6-<J-¥> ] e d - i k - sine = " i k f s i n e + used. shadowing because -ik£cose e + u S i m i l a r l y e r 2 for e < n/2 (Fig.9.4b) TT o of the + [6= j ] e d - i k — sine 2 -ik£.cos9 e d - i k — sine [6= \ } e = Similarly the C ^ discontinuity -ik£cos9 e . ate = (9.25) +^ c a u s e d by shadowing f TT for 1 of 8 > — + y can be eliminated by adding " because 2 (Fig.9.4c) [6=(j+Y)~] 3d — +ik e sine e + +ik = [ 6 = ( j +^) ] e + Similarly for the — -y can be u 2 [e=(j+«n~] sin6 e sine . 2 discontinuity e > d +ik— -ik2.cosS ate (9.26) = -^-ycaused eliminated by by shadowing adding u of because 2 (Fig.9.4d) d — sine -ik uJ [6=(^ e = However not the 2 r because of + u* [6=(^- -*')'] e these eliminate the and d +ik — , <3 .„ +ik — sine -3TT + 2 [ e = ( — -H-) ] e addition regions e 2 o completely pattern, some f u -ik2cos9 added have . double (9.27) d i f f r a c t i o n discontinuities terms sin? 2 are in themselves discontinuities at terms the does radiation shadowed the in shadow 171 boundaries. However, d i f f r a c t i o n terms d i f f r a c t i o n terms, double the be the eliminated. double terms by to the those in effect reduce in double the the the of double the single adding the d i s c o n t i n u i t i e s terms d i f f r a c t i o n d i f f r a c t i o n are terms. can t r i p l e discontinuities terms be much In added is pattern calculated in in terms which less then reduce the can each remain those summary, to terms, in in higher the order discontinuities s for o the exactly staggered the same way as guide of . is s Q it that ~ (B) Q ray mirror paths image which about NUMERICAL The plate sectoral consist horn of about a (2TT-9) = S be considered z-axis of AND E X P E R I M E N T A L waveguide with must the radiation compared at is overall certain Again, radiation F i g . 9 . 5 9.3 the then in boundaries. The The less adding d i f f r a c t i o n shadow found that d i f f r a c t i o n d i f f r a c t i o n at much so d i s c o n t i n u i t i e s t r i p l e discontinuities pattern. S i m i l a r l y , the is d i f f r a c t i o n radiation the pattern calculated the for a single 9 = 4 0 ° measured stagger of off the those find shown s in are o the F i g . 9 . 2 . RESULTS a single by staggered ray-optical E-plane angle asymmetrical to patterns of ^ = 4 5 ° . lobe guide with axis p a r a l l e l - methods of the is H-plane The patterns maximum amplitude (Figs.9.6a-d). The 172 d i s c o n t i n u i t i e s which might be expected on the four shadow i boundaries very is small not but when used a l l 6=-135°, at there other a agreed except large very are to remarkably a changes is The experimental by Ryan here i f to £ = + 4 5 ° varies contrast essentially patterns within 1 dB and Rudduck only [42] are the ray paths using only single in Fig.9.7. pronounced the double The d i s c o n t i n u i t i e s d i f f r a c t i o n and is very single at pattern similar and the has to the 0° to pattern. patterns guide and shown d i s c o n t i n u i t i e s radiation fixed calculated has but gradually width from becomes for stagger are that more shown of a in angles F i g . 9 . 8 . non-staggered from The guide pattern as asymmetrical. SUMMARY The plate is theory (8.5) at in range. obtained pattern small increased 9.4 (|©|<90°) ray-optical If phase phase are d i f f r a c t i o n at The the patterns boundaries, The small. 3dB are considered. d i f f r a c t i o n t r i p l e used. about direction calculated d i f f r a c t i o n shadow is (Fig.9.3) where the those Radiation double of +45° angles. patterns s i m i l a r 9.1a-h with (8.5) remain forward this generally The form and discontinuity guide over at a the non-staggered constant 75° is in - 4 5 ° , asymptotic d i s c o n t i n u i t i e s considerably to the - 9 0 ° , radiation waveguide pattern is of a calculated single by staggered ray-optical p a r a l l e l methods. The 173 d i s c o n t i n u i t i e s are very double between expected small, even d i f f r a c t i o n this confidence geometries. when are theory that along both the terms four shadow up and considered. and may the be to The applied including good experimental to boundaries agreement results more only gives complex d/2 H F i g . 9.1 s t a g g e r e d p a r a l l e l p l a t e waveguide \ \ \ \ /Ti Tin m n F i g . 9.2 ray paths f o r r a d i a t i o n p a t t e r n calculation vUTi 175 II III ; H iv shadow b o u n d a r i e s r/ F i g . 9.3 f o r staggered p a r a l l e l plate + + F i g . 9.4 c o n t i n u i t y f i e l d s a c r o s s shadow s e e e q n s . (9.24) - ( 9 . 2 7 ) . boundaries waveguides 176 1 —j d/2 d/2 F i g . 9.5 staggered parallel plate waveguide -180 90 -90 0 90 angle (degrees) n angle (degrees) Fig. 9.6a R a d i a t i o n p a t t e r n o f s t a g g e r e d w a v e g u i d e ¥=45°. u s i n g (8.5) u s i n g (8.4) d-0.3 178 -270 -I -180 • • • • -90 . 0 • 90 • • 180 angle (degrees) Fig. 9.6b Radiation pattern of d=0.373X s t a g g e r e d w a v e g u i d e ¥=45 . 179 -30 -H—• -180 • —' -90 ' ' ' 0 angle (degrees) ' ~~L 90 180 1 90 i angle (degrees) Fig. 9.6c R a d i a t i o n p a t t e r n o f staggered waveguide ¥ = 4 5 d=0.407X . 180 181 angle 90 (degrees) i angle (degrees) F i g . 9.6e R a d i a t i o n p a t t e r n of staggered d=0.474X waveguide ¥=45 . 182 -270 -I -180 Fig. 9.7 • • 1 • • . -90 0 angle (degrees) • • . 90 • • . 180 R a d i a t i o n p a t t e r n o f staggered waveguide ¥=45° s i n g l e and double d i f f r a c t i o n o n l y . d=0.441A single diffraction s i n g l e and double d i f f r a c t i o n 183 CD 33. O CL > -180 •90 0 angle (degrees) 90 90 -90 H Cl -180 -270 -180 F i g . 9.8 90 -90 0 angle (degrees) R a d i a t i o n p a t t e r n o f staggered f o r v a r i o u s stagger a n g l e s . ... — ^=15' • ¥=30' waveguide 4/=45 ¥=60 c c 184 Chapter RADIATION FROM M U L T I - E L E M E N T STAGGERED WAVEGUIDE ARRAY INTRODUCTION 10.1 The radiation (Fig.10.1) array similar The difference main pattern with methods to those a l l edges staggered behind the aperture The stagger calculation the case f i r s t . are when The a l l for the two array arrays are the array in is the waveguide is found by of Chapter 6. that in aperture outer waveguide the radiation the plane, edges are plane. does change guides additional are multi-element waveguides s i g n i f i c a n t l y , however. shorted the used the whereas the a between array in of staggered non-staggered to 10 are of f i e l d s calculated before A s i n f i n i t e radiated later. depth when The the pattern pattern is the analysis for calculated outer is guides restricted TEM-case. Consider waveguides into two d/2,z=0 Fig.10.2. of four and y=d/2+a,z>l (Fig.10.2). plane and waves driven each y=+d/2,z=0. by the guide with These incident conducting y=-d/2,z>0, y=-d/2-a,z>l, at central p a r a l l e l - p l a t e perfectly half-planes the represented s e m i - i n f i n i t e thin y=+d/2,z>0 in adjacent consisting i n f i n i t e l y (8.6) three The -d/2<y<d/2 value two rays in is at (8.7) plane the incident f i e l d decomposed the edges waves central guide y=are of 185 10.2 RADIATION The total p a r a l l e l is WITH radiation plate The central total the Fig.10.2) scattered s-(n=3,4) r e p r e s e n t s then reflected (Fig.10.5). of the nth found . separately angles. because the S t For The for pattern e $ not the array center of nth in (d) open appear the d i f f e r e n t a l l edges and - i k ^ ^ s i n e edge central from and guide the image must angular images are calculated 1 edge be regions v i s i b l e for at o < e < n e = n . are shadowed and -ikJccosG e 2 nth the pattern except the and seven edges ^ (Fig.10.3,10.4), of about of 0 in radiation symmetrical p labelled emanate be the images. end (as to only of the of need driven represents from surface three sum their edge P of guide the and the scattered s- DEPTH (n=l,2,3,4) n outer each edges from the total is a l l S, - the included pattern o < e < j up d, the because The made a l l fields edge is f i e l d s from The (Fig.10.6), all the of t the from already s INFINITE only t width OF with scattered of not s from f i e l d guide f i e l d pattern f i e l d scattered represents GUIDES waveguides calculated. f i e l d s OUTER . dO.l) 3 j -w < e < ~ -ik(^ S t = S 3 e + S 1 e + P(J - i k (d) — + 2 a )sin6 -ik£cos6 smo (10.2) 186 For — < B < -ik( -^ p-)sin6 d -ik y + S e l 0 -ik( S- + d j l S + +* 2 < e < | -)sine -ik2,cos6 a e e +ik j sin6 P (d) + For -ik£cos6 2 — 2 sin6 (10.3) 6 T; •ik( ^|p-)sine -ik".cos9 d -ik + S y sine e P (d) + Q +ik + S 2 sine 6 + i k (—-~) s i n e + S e -ik2.cos6 e (10.4) 4 The are s f i e l d s P q calculated are 3 s- The paths about using calculated f i e l d s ray were the are calculated the ray using calculated in Chapter paths the using used to calculate z-axis of those in ray the s shown in The Fig.10.3. paths ray and 2 5. in fields The Fig.10.4. in s are the and 10.4. Figs.10.3 1 fields paths 4 s The Fig.10.5. image The / 187 solutions for are in found To Here In U In U To ( 2 33 " ' f c d In 33 2 7 ; G d ( d a the line In To a l l in ray these Chapter diagram ray 8 paths and in 9. Fig.10.3. source -4-)G Ac ,\ - Y , 6 ) E ( r ) o\ , calculate Fig.10.4a d ( c ' 2 - ^ 1 radiated s the (10.5) consider 3 radiated f i e l d the ray u 3 G d ( d ' f 2 " 1 , radiated E ( r • ) u ( is 3 3 a line is 2 V ) G d ( c u , 1 the 2 3 a line e E is a ( r diagram in line Fig.10.5. 2 G d ( c ' ^'f 2 Fig.10.5c | Similar the radiated -^GjCc.^jradiated f i e l d a line d ) calculations is a line the f i e l d s 8 In source d for - > (10.10) L 2 G ( d , 0 , | ) G ( d ^ , | -¥)G (c,^ -H,2,-6)E(r). d 1 0 source -y,^j)-G (2a,^ ,27i-e)E(r) . f i e l d ) (10.9) is d 6 source i G l ( c , 2 i r , f -V)G.(c,-2f -Y,2ir-9)E(r). 2 d z a z the ' source -'> > ( > • ray f i e l d is 3 0 (10.7) d f i e l d 1 source 1 consider s- ) -^. f)G (2a,^,e)E(r) . radiated ) 6 f i e l d C 0 the ) G . | -*'>V HT * -2 Fig.10.5b = -* the calculate = a the ( ' G 2 = = U ' c d Fig.10.4c 32 U . Fig.10.4. Fig.10.5a In is 2 in G 2 = 33 + ^ 2 that consider f i e l d X Fig.10.4b 33 to by source 2 = similar 1 = c line 32 s radiated diagram u manner represented ^ G ' ( C , 2 T T ^ - V ) G AC,^ -V,^r 2 a 2 a. 2. z = where a a f i e l d s calculate the u, 13 the s 2 scattered (10.11) from the 188 edge y=-d/2,z=0 S 2 = -S _ = -S. S. These ] and for radiation pattern RADIATION Consider of be f i n i t e of of radiation guide guide of coupled at end the alter of travel that z=0 sum o f to give that change the and which the the i s been down the the relative guides these relative g a r e now guides w i l l Adjusting phase to of the their between the central adjacent staggered outer calculated in Chapter 8. T h e outer then f i e l d s geometric total pattern. A (d,a) back to the equivalent outer into radiation the into SHORTED end and reradiated. reflected a substituted Fig.10.2. coupled end. The f i e l d s of now Fig.10.1, c o e f f i c i e n t has A' the is w i l l a value guide (10.4) GUIDES far d is plane and array of width part aperture are except the guides shorted ~ s The f i e l d s from of d OUTER array width ,where the Fig.10.2 f i e l d s The of coupling driven n (10.3) and thus The a n WITH depth. the s the r e f l e c t e d depth show (10.13) (10.2), array y=-d/2-a,z=l (10.12) (10.1), the from 4 t results 10.3 s guide and are travel towards and part i n those the from series i s outer the reflected the open radiated. guides central at the driven 189 whereR (a)is the g the S staggered i s the total calculated array from the radiation by depth f i e l d s of waveguide outer at the of width guide open a measured of end [40], and from the plane. The the c o e f f i c i e n t p a r a l l e l - p l a t e depth aperture f i n i t e reflection of adding of the other the outer Fig.10.2. radiated edges pattern of the array radiation guides The to the from excited by the caused f i e l d s radiated f i e l d s outer f i e l d s Fig.10.1 f i e l d additional d i r e c t l y of a by the from consist guides, i n is and of f i e l d s shorted guide (Fig.10.7). Radiation s (9.19-9.22) 0 multiplied due to by f i e l d U from A* 1'2 the the the of radiation considering In from guides a staggered of (10.14). from the ray edge outer Scattering guides i n y=+d/2,z=0 simply the p a r a l l e l - p l a t e outer paths i s Fig.10.7. can be waveguide from i s pattern other calculated by Fig.10.7a the In written J G (d,2ir,^)G (d,2j,6)E(r) . = d Fig.10.7b the (10.15) d f i e l d from the edges edge y=+d/2,z=0 can be written in/ 4 U 1'3 4 G'(d,0,^)G (2d,^,e) = G (d,0,^)E(d)G (d,^,e). - ^ d d d A /2 TT In Fig.10.7c " .$ V3 n In = the -ikJ. Z-z— f i e l d from the edge (10.16) y=+d/2,z=0 i s G l (V cL , 0u , - ^ +r V + 4 ' , ^ 0) Gv u( d , ^ , 6 ) . , ) l G ( c , , 4 (10.17) 3 " ^~T~ ° d ' ' 2 " ' ' d ^ ~ 2 ^ ' 2 ' d ' ~ 2 Fig.10.7d the J f i e l d from J the edge y=+d/2+a,z=l i s U 1'3 = | In Fig.10.7e = e 3'3 u The from t h e edge y=+d/2-a,z=l i s d f c d w (10.15), (10.16) a n d (10.17) a r e added t o g i v e from t h e edge y=d/2,z=0 e x c i t e d by guides. from calculations t h e image of the fields = S a n d (10.19) gives t h e p o i n t y=+d/2-a,z=l. that _s' 1 (10.20) K - -H 4 °- (1 3• The a d d i t i o n a l r a d i a t i o n caused by t h e f i n i t e depth outer Similar from t h e edge y=-d/2,z=0 a n d f r o m y = - d / 2 - a , z = l show c2 radiation S i m i l a r l y (10.18) g i v e s t h e f i e l d from t h e edge y=+d/2+a,z=l, 4 (10.19) H d 2 d t h e outer 3 (10.18) d j L. ^ _- , -. 22 7i .1--e6 )) i G ( d . 2 , . | ) G ( d I. 2n| . f - * .) G. 3( c field S d thefield field from ( '2^|)G (d,^-¥)G (c,^--¥,e). d d fields the S G guides i sthearray s u m o f s , S^, Q 21) ofthe and s £ m u l t i p l i e d b y t h e c o e f f i c i e n t A* o f ( 1 0 . 1 4 ) . T h i s r a d i a t i o n i s a d d e d t o S i n (10.1)-(10.4) pattern For t o obtain t h e complete radiation for thearray o f Fig.10.1. s t o t a l 0 < 6 < — 2 -ik( S , total + = t d j ^)sine -ik£cos6 A'S' e 3 - i k ( -^") s i n 6 e • d + A'S 2 0 (10.22) 191 IT For - -¥ Ti < 6 < -ikj—^sine s ^total = S + t A'S' e 3 e -ik + A'S Q l y s i n e e -ik — + For — < 6 < 2 2 -ik£cos6 A'S^ sine (10.23) e +¥ -ik ( total S t + A'S' ) sine d-f2 a e -ik£cos6 e 3 -ik(~.)sin6 + A'S + A'S^ 0 e -ik — -ik( + A'S- e + A'S' e + A S e 3 sin6 e d 2 a sine + i k ( ~ — ) sine 2 0 -ik£cos6 e +ik — 2 )sin6 (10.24) 192 For j +¥ <e< TI - i k ( "'" ) s i n S d S . , total = S t + -ik£cos6 2a A'S' e 3 e ., ,d+a, . „ - i k (——) sine + e A'S 0 ., -ik + A'S^ e + A'S' 2 . „ sine +ik — sine e 2 . , ,d+a. . „ +ik(——-)sine -_ A'S + d — e 0 +ik ( " d + NUMERICAL 10.4 The compared array with horn guides. Radiation i n f i n i t e depth y . s -ik£cos9 • e 4 a calculated by ray-optical E-plane appendages patterns are shown y in changes three patterns added to when the Fig.10.8 from (10.25) RESULTS of with As ) sin6 e pattern measured sectoral angles ' AND E X P E R I M E N T A L radiation waveguide A f 2 a 0° of outer 90° staggered methods an simulate for to element various the H-plane the guides shape is outer are of stagger of the 193 pattern changes staggered array expected. the The to are that general the outer generally staggered from to The boundaries then and of a of a three single element guide, increasing &= less then array as arrays 1 dB 6 as is \jf ones to arrangement effect depth gradually a shown from the has lobe single a with guide. the dB 3 non- might to be broaden element along is (Fig.10.9- the 2 dB in shadow 6 - ~> /2 T there the the smaller at because in non- ranging generally boundary edges patterns frequency then This six three were less the beamwidths the + y shadow are staggered non-staggered 7. 1 dB up patterns generally agree to and at because of 0=160° e' = 1 8 0 ° blockage with within could the 3 not dB be by the antenna 60° for a {e'-9-ir). increasing is which single of /2 patterns The mounting depth, and within accurately 1 I case: or depth expected 6= any four for on © = "n"/2 + y . experimental measured guide and f i n i t e have depending at to e'=-150°. The patterns along Chapters calculated of those non-staggered compared of are to TT/2 edges The to The * 90° at the two guides d i s c o n t i n u i t i e s in only of that of effect similar array. ± 2 0 ° 10.13). a from pattern. When up gradually in y from Fig.10.14. non-staggered null on the beamwidth beam 0° to The patterns pattern for axis, considerably to a fixed change that guide pattern with narrower then that 194 10.5 SUMMARY The plate radiation pattern waveguides was compared with sectoral horn with complex then the shadow the edges of are in multiple The are hand, the staggered case the calculation is plane, avoiding at the staggered array, shadowed staggered be in a the case, to to a is time as observed then not are give single a to generally be H-plane is more are more considered. because the the On not a l l d i f f i c u l t i e s shadow observed the only shadow or boundaries in because three one nonedge boundaries five which in may array. broader large s i g n i f i c a n t guide, at traversing for there simpler those compared especially to unexpected when an and boundary. non-staggered patterns adjusted compared shadow smaller This shadowed The a thus methods using because more along paths p a r a l l e l calculation and is be staggered obtained non-staggered ray of ray-optical The much case. by results d i s c o n t i n u i t i e s generally array appendages. same edges an calculated experimental boundaries other of ty" . then The reduction however. in for the non- parameters can beamwidth as 10.1 Fig. array of staggered p a r a l l e l p l a t e waveguides w i t h outer guides shorted T a d/2 d/2 4— 10.2 Fig. array of staggered p a r a l l e l p l a t e waveguides w i t h outer guides o f i n f i n i t e depth 196 \ ray F i g . 10.3 path f o r c a l c u l a t i n g \ \ ray v/ paths Fig. for y/n ray paths \ 10.4 calculating v/ F i g . 10.5 for calculating II Ill / I \ VII VI F i g . 10.6 hadow boundaries f o r an a r r a y o f taggered p a r a l l e l p l a t e waveguides \ d F i g . 10.7 ray paths (see t e x t ) £-20 -30 -180 -90 0 90 angle (degrees) 180 180 4 Si 33. 9 0 0 -90 4 -180 Fig. 10.8 -90 0 angle (degrees) -180 90 R a d i a t i o n p a t t e r n o f s t a g g e r e d waveguide a r r a y w i t h guides of i n f i n i t e depth d=a=0.450X ¥=45°. ¥=15 £ ¥=30* ¥=75 ¥=45 ¥=60 c outer Cu > > -20 -30 -180 -90 0 90 angle (degrees) 180 n -180 -180 -90 0 90 angle (degrees) Fiq. 10.9 R a d i a t i o n p a t t e r n o f s t a g g e r e d wave d=a=0.474X s=1.166X ¥=45 . 200 angle (degrees) 180 i angle (degrees) 10.10 R a d i a t i o n p a t t e r n o f staggered waveguide d=a=0.441X s=1.083X ¥=45°. array 201 angle (degrees) 180 i angle (degrees) 10 1 1 R a d i a t i o n p a t t e r n o f staggered waveguide d=a=0.407X s=1.000X ¥=45°. array 202 angle (degrees) -180 -180 -90 0 90 180 angle (degrees) Fig. 10.12 R a d i a t i o n p a t t e r n o f staggered waveguide d=a=0.373X s=0.916X ¥=45°. array 203 angle (degrees) 180 n -180 -180 -90 90 0 180 angle (degrees) Fig 10 13 Radiation pattern d=a=0.339X of s=0.833X staggered waveguide ¥=45°. array 204 0i CD -10 I Q. £ -20 4 03 0» -30 -180 -90 0 angle (degrees) 90 180 i 90 0) cn SI CL -90 1 -180 Fig. 10.14 Radiation stagger 90 -90 0 angle (degrees) -180 pattern angles ¥=15 ¥ c ¥=30' of staggered d=a=0.450X waveguide a r r a y s=1.107X. ¥=45 ¥=60' 180 for various 205 Chapter GENERAL 11 CONCLUSIONS AND D I S C U S S I O N DISCUSSION 11.1 Ray-optical d i f f r a c t i o n p a r a l l e l both have plate and patterns staggered, plane been based used were for geometrical the Coupling small horn the analyze separated calculated sectoral on to waveguides. adjacent Radiation methods f i e l d s and to both also in c o e f f i c i e n t s waveguides arrays, theory were simulate the coupled between calculated. staggered measured of and using non- an H - two-dimensional structures. Ray-optical largely various plate methods because simple of of anticipated that more of the and other that edges S p e c i f i c a l l y , the quite presence well of by complex, plate methods The obtained here results where can successful be of be various it was to agreement experimental a v a i l a b l e , if even extended good with for p a r a l l e l here, Thus might generally others and consist waveguides. analysis indicates the number of small. the coupling with other obtained considered results is this structures structures. extension for h a l f - p l a n e s more theoretical d i f f r a c t i n g results r a y - o p t i c a l r a y - o p t i c a l chosen involving p a r a l l e l complex this agreed The considerably combinations these good structures waveguides. though the were those guides c o e f f i c i e n t s calculated and/or a by obtained other simulated here methods in groundplane 206 [29] [38]. guides Other in isolation comparison. in using wedge guides obtained not which recently in good the more complex The guides interaction in the of this coupling versus similar [38] using residue modified agreement. coupling A to the observed, results may to that calculus, general coefficients was patterns i n f i n i t e by. o t h e r i s o l a t i o n or patterns for using methods embedded in arrays an obtained depth (Figs.6.1,7.1,10.1) calculated the obtained lack details which be applied here for of of gives d i r e c t l y to structures. calculated patterns those if for separated q u a l i t a t i v e l y these radiation of two inclusion o s c i l l a t i o n structure that The adjacent available with [17] two was in surrounding confidence agreed quantitative of not between techniques an between were ignored. yielded s e n s i t i v i t y the (Fig.4.1) was curve coupling coupling d i f f r a c t i o n interaction but for isolation separation for (Fig.3.1) Results guides between results agreed [29] for [38] a H-plane for these horn arrays with same those array guides agreement sectoral with in groundplane. outer good well the simulated with showed patterns (Fig.7.2) arrays with The shorted experimental in a l l cases. Other were not available for comparison. The these basic geometries boundaries as d i f f i c u l t y the are that the f i e l d order applying f i e l d s combinations incident Consequently, is in of and of each d i f f r a c t e d f i e l d s higher ray-optical of the order subsequent methods along same shadow order d i f f r a c t e d d i f f r a c t e d to of k f i e l d s . f i e l d 207 is not diminished. ray-optical methods number possible of Thus to which must half-plane edges are the rays containing be may only considered contrast, two calculations including more for complex very the to use calculate waveguides. The widths, has a this n=50 the with excellent only a improvement widths is exact over to as of more from waveguide ray the paths to r e f l e c t i o n d i f f r a c t i o n . many ray Thus By paths ray-optical require become terms intractable coupling the of a for curve coupling agreement only of are included. d i f f r a c t i o n r e f l e c t i o n simpler case at solution the versus does at ray-optical use not the to for the transition guide derivative. The (UAT) waveguide. mode obtained be obtaining plate even may adjacent in exact obtained is theory successful known it between p a r a l l e l of why asymptotic f i r s t the the reason discontinuous orders in the very with d i f f r a c t i o n few of uniform is c o e f f i c i e n t where part the UAT agreement times 2 which d i f f r a c t i o n the agreement cusp only B). the order plate require Appendix that structure times geometries of is accurately Excellent n is given the require extending rapidly into w i l l of geometries. d i f f i c u l t r e f l e c t i o n grows representing simple d i f f i c u l t y excellent a contribution see orders to p a r a l l e l w i l l guides n, high This edges terms on up A result structures considered d i f f r a c t . adjacent (depending complex introduced find from immediate paths be two to c o e f f i c i e n t more ray d i f f r a c t i o n which one if width However, terms of up UAT to with give s i g n i f i c a n t mode transition method used here, and 208 thus would coupling not be expected coefficient Another result complex structures at shadow the p a r a l l e l plane less of is 0.5 t r i p l e d i f f r a c t i o n reduced to ray an paths. this a dB, parameters, even are used accuracy of complexity of The agreement is into small value up 3 20 occur single the aperture paths and up to may be more non-staggered depending not up on to more shadow method guides the array quadruple possible taking the more considering paths by near to reduce terms with the boundaries, decreases the as the increases. from by the the shadow array ray-optical equally dB was in ray by of ray It away between about to 6 to a account, array ray-optical affected only taken structure accuracy s l i g h t l y which the the which For when further the larger. even be Thus methods discontinuity considered. here. improve d i s c o n t i n u i t i e s become when discontinuity method the element can ray-optical this are three discontinuity this that a r b i t r a r i l y For d i f f r a c t i o n extending waveguide the s i g n i f i c a n t l y either. boundaries plate is to good size, and for boundaries as is indicated experimental both simple only and by the results, more complex arrays. The basic boundaries along is f i e l d . that shadow represented then reason the by the a ray when in used is the f i e l d this this method boundaries resultant Thus for in f i e l d d i f f i c u l t y to not accurate. aperture the is in calculate plane aperture turn is plane at the shadow the If fields a f i e l d d i f f r a c t e d , is d i f f r a c t e d not in a ray the 209 aperture plane, accurate the value. ray This applies d i f f r a c t e d f i e l d s waveguides calculations For size a of the the exact are the are ray l i k e look Boersma on a problem to an higher order p a r a l l e l plate waveguide the by the comparison because if y i e l d measured and the as two and with arrays, d i f f r a c t e d incident to two be of Chapters in the somewhat. However, half-plane for the doubly incident inaccurate non-staggered two staggered 6 for and this the the plates f i e l d s look like d i f f r a c t e d f i e l d f i e l d . for case Lee the plates. which double case Thus of and of plates a it line arises half-plane might accuracy d i f f r a c t i o n w i l l singly a and may source repeately 7. plane better the p a r a l l e l d i s c o n t i n u i t i e s aperture for that boundary the non-staggered available, needed f i e l d on predicts shadow inaccurate solution was When on be patterns more For not method. as pattern here the this incident arrays If been open-ended cumulative, one-quarter incident source incident along showed expected t r i p l e the calculated incident [32] wave the and ray-optical small. are does t r i p l y inaccuracy, in used f i e l d w i l l in a l l the this quite method the be is errors one-half line to here f i e l d s . d i f f r a c t e d a by guide repeatedly The plane used non-staggered discontinuity result however, f i e l d s in single method problem a in d i f f r a c t i o n the have radiation been solution (Fig.4.12) reduced to would the have also. the complex ray-optical p a r a l l e l methods plate used here waveguide are extended structures, to the 210 combination of inaccuracy of to the large the number method d i s c o n t i n u i t i e s up to the calculated radiation in a manner. simple from the well shadow with methods most in dB the in the can plate -10 a beamwidth five element only s l i g h t l y amplitude was was Other patterns These beam with beam in applied reflector. varied two could angle the H-plane. If as the dynamically, quite predicting ray-optical interest methods can, the for within radiation a be by of for from dB with in on the appendages of the for the a size 120° a angular array. axis. design produce and parabolic guides loading is whose the pattern parasitic reactive to to E-plane beam a with pattern over of with obtained A sector-shaped feed variety aperture d i r e c t l y the a non-staggered n u l l s applied p a r a l l e l pattern was the element with The depth e.g. that 0.5 three l i n e arrays s k i r t s within coverage a agree for wavelengths. horns wide d i r e c t l y away the the in eliminated patterns sector-shaped steep obtained sectoral case that suitable considering with were leads greatest small good and constant results H-plane 134° then obtained of predict p o t e n t i a l l y array, more range of are of be plane useful of remarkably dB cannot so the waveguides. patterns waveguides and boundaries radiation ray-optical p a r a l l e l A the Thus coupled applications. which regions successfully needed non-staggered aperture be l i m i t s , plate the results these radiation in paths shadow angular a p p l i c a t i o n s . The the However, here ray pattern, experimental employed patterns along ± 3 boundary the of a a fan narrow could be cylinder could be described 211 in [35], 11.2 some beam steering SUGGESTIONS There are a work that warrant made to improve f i e l d s along array by FOR the FURTHER number of further the the questions accuracy boundaries uniform methods the assumption used ray f i e l d s . The problem a f i e l d p a r a l l e l plates may obtained solution but be for the a plane f i e l d needed here. three p a r a l l e l three and number ray five of problem The to rapidly. when more is required Similarly, improved number of on points for more considered Ray-optical to obtained the coupling uniform be are which may with the coefficient of the [32], plates shadow are incident f i r s t . required The far using the in [32], boundary is incident for on both However, the as the increases, because the if to practical better accuracy used calculations p a r t i c u l a r l y near cusps are expected, must be considered here. may the but of accuracy be methods the number desired cease considered be f i e l d s normally source the may Boersma a l l arrays. obtain methods, where and boundary methods was paths also this non-staggered argument the d i f f i c u l t , than the source line shadow edges of p a r a l l e l a would the attempt that along from calculation reciprocity two two ray line arise the Lee here non-staggered much regions of An investigated a then using transition be by solution along be grows a l l element becomes paths may plates edges of wave at which of avoiding two possible. WORK thus on be investigation. shadow using should mode again for be the better 212 accuracy is very Other large. methods radiation pattern separated guides, produced the and more is here. of This the coupling of the is d i f f r a c t e d from outer in the 5 the central of to method becomes there c o e f f i c i e n t accuracy using extended to useful moment in this methods d i f f i c u l t i e s [11] near this guides. d i f f i c u l t to for coupling moment thickness context. shadow [35] p a r a l l e l in a help to boundaries calculated may be this r e f l e c t i o n with which may may radiation has waveguide overcome good approach and of the with The groundplane if arrays method this plate f i e l d structures [36] A combination may the calculated and from large c o e f f i c i e n t s method loaded [44] [48], f i e l d However, modes. was to for outer propagating the reradiated shorted use pattern coupling been and method f i n i t e not also calculus, moment find and compared have a reactively of depth is place resulting the patterns waveguide applied walls Radiation single Another with (-12dB) a patterns. to which 0.25 the because used in radiation dB), [38] approach However, large, two widths, substituted the and method transition be edge many from ray-optical outer more are the (-20 of calculus mode array. coupling between the 0.1 residue case residue about about the coupling improve be i n f i n i t e the improve may to guide. modified extended than not is at element only guide guides using be cusps one to For c o e f f i c i e n t may c o e f f i c i e n t used modified accurate improvement where the ray-optical the be results. expected calculation outer may been arrays also be ray-optical and some moment of method the is 213 used near from i t . the The to aperture results designing ( F i g . 1 . 5 ) . patterns obtained an c i r c u l a r empirical f i r s t step also to would used be as a of guide waveguides the good parameters. open-ended work to may and away reasonably required analysis method circular between present the be produce adjustment The towards ray-optical concentric c o e f f i c i e n t pattern. the may known waveguides radiation of are TE,| -TEM coupling and here array These by plane The coaxial predict be the considered optimization of such a an array. 11.3 CONCLUSIONS Ray-optical waveguide and except are BRIEF methods structures Theoretical well IN near with shadow The t r a n s i t i o n regions orders d i f f r a c t i o n the accuracy along of a is shadow greatest transition and to radiation as the is regions and shadow is at least using one-third uniform into often of methods even a is mode when many half-plane However, away the near Consequently, the from boundaries, provided quite d i s c o n t i n u i t i e s account. of plate edges. agree accurate increased. most small, not number are edges where boundaries, taken p a r a l l e l patterns shadow are to half-plane is interest acceptably six method boundary are by up applied boundaries, reduced errors results been experimental observed. of have and regions the d i f f i c u l t mode here spacing wavelength. edges between Improving because the of the the 214 l a r g e number of edge i n t e r a c t i o n s which must be Larger and diffraction of patterns more than is waveguide a r r a y complex s m a l l e r and available with a d j u s t i n g the width and only arrays considered. r e q u i r e higher orders of simpler a r r a y s . 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M e n e n d e z , "GTD, r a y f i e l d , a n d comments on two p a p e r s " , IEEE Trans.Antennas P r o p a g a t . , vol.AP-26, pp.352-354, 1978. [34] R.M.Lewis and of edge d i f f r a c t i o n " , 1969. J.Boersma, "Uniform asymptotic theory J.Math.Phys., vol.10, pp.2291-2305, 218 [35] J.Luzwick aperture and antenna E l e c t r i c a l Syracuse, and New R.F.Harrington, array", Technical Computer York, "A Report Engineering, reactively No. 3, Syracuse loaded Dept. Of University, 1976. [36] J . L u z w i c k and R . F . H a r r i n g t o n , "A solution for a wide aperture reactively loaded antenna array", Technical Report No. 5, Dept. Of E l e c t r i c a l and Computer Engineering, Syracuse University, Syracuse, New Y o r k , 1977. [37] R.Mittra and Theory of Waves. [38] J.P.Montgomery Guided S.W.Lee, New A n a l y t i c a l York: and D.C.Chang, boundary value problems based upon calculus and function theoretic 164, Department U . S . of Techniques Macmillan, a the "Electromagnetic modification techniques", Commerce, in 1971. of NBS residue Monograph 1979. [39] Y.Rahmat-Samii and R . M i t t r a , "On the investigation of d i f f r a c t e d f i e l d s at the shadow boundaries of staggered p a r a l l e l plates", Radio Science, vol.12, pp.659-670, 1977. [40] R.C.Rudduck and L . L . T s a i , "Aperture reflection c o e f f i c i e n t of TEM and T E mode p a r a l l e l - p l a t e waveguides", IEEE Trans.Antennas Propagat., vol.AP-16, pp.83-89, 1968. o l [41] R.C.Rudduck and D . C . F . W u , "Slope d i f f r a c t i o n analysis of TEM p a r a l l e l plate guide radiation patterns", IEEE Trans.Antennas Propagat., vol.AP-17, pp. 797-799, 1969. [42] C.E.Ryan, analysis of waveguides", Trans.Antennas A.Sommerfeld, moment d i f f r a c t i o n " , pp.62-69, "A wedge of d i f f r a c t i o n p a r a l l e l - p l a t e Propagat "Mathematische Math.Ann., G.A.Thiele combining patterns , vol.AP-16, 1968. D i f f r a k t i o n " , [44] And R.C.Rudduck, radiation IEEE pp.490-491, [43] Jr. the 1975. and with Theorie pp.317-374, T.H.Newhouse, methods IEEE vol.47, the Trans.Antennas "A der 1896. hybrid technique for theory of geometrical Propagat., vol.AP-23, 219 [45] L.A.Weinstein, Factorization Method. [46] R.Wohlleben, primary feed e f f i c i e n c y " , [47] H.Mathes large in "Numerical f i n i t e and opening E l e c t . L e t t . , C.P.Wu, waveguides 254, for The Theory of D i f f r a c t i o n and the Boulder, Colorado: Golem P r e s s , 1969. O.Lochner, angles vol.8, pp.474-476, solution arrays", and Radio for the Science, "Simple high small aperture 1972. coupling between vol.14, pp.245- 1969. [48] S.L.Wu and Y.L.Chow, "An application of methods to waveguides scattering problems", Trans.Microw.Th.Tech., vol.MTT-20, pp.744-749, 1972. [49] H.Y.Yee waveguide Inst. and L.B.Felsen, d i s c o n t i n u i t i e s " , Brooklyn, Farmingdale, "Ray-optical Electrophys. N . Y . , Rep. techniques Dep., moment IEEE for Polytechnic PIBEP-68-005, June 1968. [50] H.Y.Yee, L.B.Felsen and reflection from the open end J . A p p l . M a t h . , vol.16, pp.268-300, J . B . K e l l e r , "Ray theory of of a waveguide", SIAM 1968. 220 APPENDIX A SCATTERED F I E L D S ALONG A SHADOW BOUNDARY The r e s u l t s relation (3.26) a n d (3.29) may be c h e c k e d b y u s i n g a symmetry for scattered fields u (2TT-9) = s which i s exact by d e f i n i t i o n field, is s for diffraction A.l) and then the f i e l d taking = [ 3 ] . The s c a t t e r e d fields, + the f i r s t + term, where u s 2 field if (3.29) 6 o f f t h e shadow b o u n d a r y ikr e 0 e /kr_ ikr (A.2) /kr ( 3 . 2 6 ) , a s i s shown on a h a l f plane 6 = y +6 i s g i v e n , ikR TT However, representing the scattered incident observed at '2 -iir/4 — e (A.l). incident at the o b s e r v a t i o n p o i n t i s g i v e n by source E(r) field • ,* +17T/4 2/2TT Q a line A . l ) , the t o t a l e 2/k(r+r ) the remaining terms (zr) s a t i s f i e s ' ( A . l ) , s 2 Consider T which i s the i n c i d e n t o u t , then but not the field (3.29) may be w r i t t e n Q Q subtracted at a small angle ik(r+r ) /k(r+r ) (Fig. and d i f f r a c t e d 6 -»• 0, t h e n Q u screen (3.26) a n d (3.29) a s w r i t t e n d o n o t s a t i s f y ik(r+r ) If by a p l a n e includes reflected d e r i v e d by f i n d i n g (Fig. (A.l) -T u (6) / k (r +r+R) Q I/ k r 0 from r s i is field below, a t 6^=3^/2 (2.12) b y . — «' n r +r+R Q ikS +2 / +x /k(r +r+S) Q cos — > V r + S (A.3) 221 i k R /2~ + A /2 -iir/4 I - e i k R +2 / j ^ (r +r+R) sin r /k(r +r+R) Q +r+R 0 Q i k S + T where S •> r R , S , r , r _ r Q » a n 0 , 6 and e a r e Q d e f i n e d +2 / / k ( r Q + r + S ) F i g . 2.3. i n cos — r Q l e t 6 + r + (A.4) S -»• 0 , then R •* r + r Q , d i k ( r + r Q ) i k ( r + r o ) i k ( r . r 0 + T / k ( r + r fc Q ) J F[/2kr^] 2/k(r+r ) Q (A.5) I f k r Q > > l (A.5) = (A.2) QED. Fig. A . l 222 APPENDIX B HIGHER ORDER D I F F R A C T I O N TERMS FOR COUPLING BETWEEN ADJACENT P A R A L L E L P L A T E WAVEGUIDES I N THE TEM MODE A general e x p r e s s i o n f o r TEM-TEM c o u p l i n g o f 0 ( k cent waveguides i s derived (3.18), (3.29) (3.26), calculate possible f o r a l l orders and (3.30) the c o n t r i b u t i o n ray paths systematically of d i f f r a c t i o n . First adja- the equations a r e s p e c i a l i z e d t o t h e TEM c a s e a n d u s e d t o to the coupling f o r a generalized from the d r i v e n and t h e i r ) between guide contributions to the p a r a s i t i c a r e added t o g i v e ray path. guide are All traced the coupling coeffi- cient . From diffracted (3.18) = L as o b s e r v e d a t j T a point edge, 2 T other edge diffracted ^ ^" i = n t n e driven guide excites a travels a distance r^ away from t h e |e-e | = T T / 2 if Q (B.2) along a distance field When t h e l i n e plane = 2 u (B.l) where +i field u C^r^ECr) i n the aperture -l This field field u diffracting the i n c i d e n t 2 r^ of 0(k~ = ~ if |e-e | 0 = 3TT/2 . t h e shadow b o u n d a r y away 1 / 2 ) i n the aperture i s from C^ (r )E(r +r) ) 1 and i s a g a i n d i f f r a c t e d 1 (3.26) plane. or The r e s u l t a n t doubly (3.29) . source u - i n the aperture m-1 at an- (B.3) plane is diffracted from another 223 edge the r e s u l t a n t u = m is — C ' l r j E ^ + . - . + r ,+r) m 0 1 1 m-l r i s the distance field u may b e d i f f r a c t e d m the resultant u where and r m m+1 x^ i s g i v e n parasitic from t h e l i n e into field field guide from t h e i n c i d e n t u , m+1 from contributes (m+1) m All now t r a c e d to A ^ of x and t h e v a l u e s 2 Q ° ° 1 f o r the fields edge A o f t h e mode i n t h e 00 +...+r ) (B.6) 1 o f any r a y path and on the t o t a l from t h e d r i v e n of x , a simple pattern x^ a n d plane path a r e thus length z=0 a n d b a c k back t o c e n t r e . t o centre The p a t h s t o the p a r a s i t i c are evaluated starting r = m+1 E n=l where r t =S f o r each r a y p a t h . represented 1 represents and 0 represents length guide are and e n d i n g a t t h e c e n t r a l w h i c h may b e u n i q u e l y (0,1,00,01,10,11,000,001,etc.) central source t o the d i f f r a c t i n g r m and x p o s s i b l e ray paths y=z=0 f o l l o w s the (B.5) plane. i n the aperture y=-d, line 2_J. /2"ka Each r a y path number a t 6=0 o r 2TT a n d f r o m t o the amplitude /£_e on t h e s i g n s the aperture guide The diffraction -1/2 The 0 ( k ) contributions in the p a r a s i t i c edge. is 00 only source to the d i f f r a c t i n g by (B.2) (m+1) dependent (B.4) T T C' (r ) C (r + . . . + r ) E ( r ) 2 3 0 1 0 1 m = i s the distance The , 2 where (3.30) , m-l field by a edge binary a r a y g o i n g down t o a r a y g o i n g u p t o y = a , z=0 a n d o f any r a y path starting and ending at e d g e y=z=0 may b e w r i t t e n 5 = / [2a(1 =1 "W + 2d w (B - 7) 224 where i"mkj = a binary m = number k = decimal value j = digit All fields four -d) four in of in the ray is incident binary binary binary field starting Table Ray is into c o u n t i n g from least first diffracted parasitic left quadruply and e n d i n g a t g i v e n by S . the a ray path, number, number is representing number, i n c l u d i n g at length diffracted Groups of Group # path number the to in right. diffracted central The r a y p a t h s g r o u p s d e p e n d i n g on f r o m w h i c h edge g r o u p s shown i n TABLE B . l digits a binary them a p a t h which the into penultimate in p o s s i b l e ray paths y = z = 0 for (y = 0 o r of number contain within naturally digit the edge divide driven and f r o m w h i c h e d g e guide (y = 0 or a). Rath Penultimate Ray D i f f r a c t e d a t y= Ray a t y= Length r t 0 0 S 2 0 a S + 3 -d 0 S + d 4 -d a S + d + a T^T ^ The q u a d r u p l e coupling coefficient pression is, are u s i n g the The Paths 1 eaph group. the B.l. Incident Diffracted The p r o d u c t guide s determined and h i g h e r given order by terms asymptotic form by i n s p e c t i o n o f of diffraction the (5.12): form the a ray paths contributions (B.6). in to the The c o m p l e t e ex- 225 + 2 y y _2 , m=l m=l terms beyond s i x o r d e r s Note is given there of by 2 that are only two r a y p a t h s the r e f l e c t i o n In widths 2^ the expression M (B.8) . 1, orders using values (3.31) The c o n t r i b u t i o n was n o t s i g n i f i c a n t , f o r each order of diffraction however. of diffraction f o r n odd and n ^ l . and n By c o m p a r i s o n i n the calculation i3Tr/4 are given to 0 0 ikna even coupling coefficient diffraction in F i g . B . l . improved over the t r i p l e are expected equal (3.23) 3 up t o and i n c l u d i n g t r i p l e significantly of even f o r the complete of d i f f r a c t i o n cusps which ) C (na) n Chapter ^ reduces . . of 1 was n o t f o u n d . f o r each order n Numerical + , ; t h e s p e c i a l c a s e o f TEM-TEM c o u p l i n g between g u i d e s o f (a=d) ka » m i + 1 coefficient. i For ^ n 2 . . 2 of ray paths and ( i C (S+d+a) m of d i f f r a c t i o n t h e number f o r n even n / / 2 2 k=l o f 0 ( k "S _D C (S+d) m k=l 2 these ( L mil e x p r e s s i o n f o r terms m . , _(2m+l) k=l 2 L 2 A general C (s+a) - and (B.10) Note t h a t diffraction a t t h e mode t r a n s i t i o n A ^ using the higher the s o l u t i o n results widths for results of Chapter i s not 3. are not obtained. The 226 -h A comparison of obtained by Yee e t al. expressions for [50] and Boersma the 0(k ) reflection shows t h a t [3] the coefficient UAT s o l u t i o n o f [3] 3/2 may b e 2 n It for 1 n 1/f2 was the obtained in from the the ray-optical denominator conjectured that of the (4_oo) infinite Performing i3TT/4 ? 2/2T (ka) 3 / 2 w h i c h may be rewritten, 1(4-oo) A x 00 In the case, results converge slowly pected, for work a t all, n £ 50. I e e 2 n -jT- this to (ka) .3/2 (B.10) yields ikna (B.ll) 3/2 in the test „m 2 I e ik2ma ^„ 3/2 m=2 2 2 substitution show a c u s p because a r a t i o at works the m very well, the this series 0 CQ •25H O O CM -50- —1—.—.—.—1—1—1 1.0 Fig. coupling 2.0 d/X B.l coefficient and t h e mode t r a n s i t i o n coupling case, shows t h a t (B.12) 3/2 verge . 8 on expression even i3iT/4 However, < an i m p r o v e d substitution for (8.3) a n d (8.9)). eqns. yield n u s i n g n=2m 2/2TT reflection by s u b s t i t u t i n g ([3], might _n/2 2 n=4 n [50] sum this I e ^ of same s u b s t i t u t i o n coupling coefficient. A„„ 00 the solution A 00 numerical widths, substitution (B.12) as ex- does does not not con- 227 APPENDIX C CONTINUITY OF FIELDS ACROSS SHADOW BOUNDARIES The expression (9.24), which shows how a d i s c o n t i n u i t y i n the single d i f f r a c t i o n r a d i a t i o n pattern of a staggered p a r a l l e l plate waveguide i s eliminated by adding a double d i f f r a c t i o n term, i s derived here using (8.2) and the IT asymptotic form (8.5). I t w i l l be shown that (Fig. 9.3a) f o r 6 = — -¥ that as 6 -»• 0 +ik — sin8 u ± - i k — sin9 (9+6) e + u 2 ( 8 + 6 -ik ^ sin8 = Note u^ (9-6) 2 -ik£cos9 2 = c 2 (Cl) ; c o s ( ^ -V) 2 and B a r e d e f i n e d -ik£cos6 6 e e f r o m F i g . 9.1 S , + d ) 6 by r e f e r e n c e = -; c to the previous sin(^- - ¥ ) = A, A ' , A " , A"' c 2 equation. +ik j sin{j -V+6) j D ( 0 , j -¥+6)E(r)e + | D ( 0 , j - 4 ' ) G ( c , ^ j - ¥ , j -¥+6)E(r)e = 7 2 D(0 -ik ~ -^)G (c,~A a 2 2 2 2 2 A-G^c,^- -¥-6) . j sin(~ -y-6) - i k H c o s ( | - -¥-6) -4f-6)E(r)e +ikdsin(^- -¥+6) +ik£cos(^ -¥+6) e + - i k i l c o s (j -¥+6) e -ik A'e j s i n {j -¥+6) d (C.2) e A"G (c,^- -¥,j -¥+6) d (C3) 228 + i k d s i n ( j -V+6) + i k J l c o s ( j - ¥ + 6 ) A'e .. + e CHI A { - -i = /4 | A"' + i k c C O s F[/2kc-sin(f)] 5 e + e -ikccos6 p [ / 2 i r ^ s i n ( e" + . | ) ] i + k C e ° C - i k S( 2 c o s <¥- f) ] } *" > F C 6 C ° S ( 2 f + 6 ) F[/2kc-cos(¥ f)]} + (C.4) let A e <5 -> 0 , t h e n A ' = A " = A ' " = A , a n d + i k C + A ^ i ( - e -ITT/4 = A" A +ikc A The e expressions (9.24) ^ + i _ i k i C F[O] F[O] + A +ikc —e - (9.26) i k _ k +e - e" + , + _ B i C ° C k C Q S ™ P [ ^ t e cos*] 2 V J ( F[/2kc e = are derived ,_ A +ikc —e + B in a precisely | cos¥j > (C.5) _ „ » Q E D . i<~.t>) n analogous ( r fi manner. 229 APPENDIX D N a t i o n a l Research C o u n c i l of Canada D i v i s i o n of E l e c t r i c a l Engineering Antenna P a t t e r n Range TRANSMITTER A s t a n d a r d 1 mW s i g n a l g e n e r a t o r output was f e d through a c o a x i a l c o u p l e r to a t r a v e l l i n g wave tube a m p l i f i e r w i t h a g a i n of 30 db. A sample from the c o u p l e r was used to monitor f r e quency, u s i n g a d i g i t a l c o u n t e r . A low-pass f i l t e r was i n s e r t e d a t the output o f the a m p l i f i e r as a s a f e g u a r d a g a i n s t any harmonics. The s i g n a l was then fed through l o w - l o s s s e m i - r i g i d coax up to the roof. The antenna used t o f l o o d the h o r n under t e s t was a s t a n d a r d X-band horn. RECEIVER The o t h e r end of the l i n k b a s i c a l l y c o n s i s t s o f a l a r g e t u r n t a b l e on the r o o f , which i s r o t a t a b l e through 360 , and app r o p r i a t e r e c e i v i n g i n s t r u m e n t a t i o n . Antennas under t e s t are mounted on t h i s t u r n t a b l e , whose angle of r o t a t i o n i s l i n k e d by a synchro system to a l a b r e c o r d e r c h a r t d r i v e . A f t e r r e c e p t i o n by the horn under t e s t , the R.F. s i g n a l was c o n v e r t e d t o an I.F. o f 65 MHz by a waveguide c r y s t a l mixer a t t a c h e d to the horn f l a n g e . Frequency s e l e c t i v e t e e s w i t h i n the S c i e n t i f i c A t l a n t a S e r i e s 1600 Wideband r e c e i v e r , permit the use o f a s i n g l e RG214/U c o a x i a l c a b l e t o t r a n s f e r the L.O. s i g n a l from r e c e i v e r t o mixer, and, a f t e r harmonic mixing takes p l a c e , the I.F. s i g n a l from the mixer back down to the r e c e i v e r . D i r e c t l y below the t u r n t a b l e , above the l a b c e i l i n g , an R.F. r o t a t i n g j o i n t w i t h mercury c o n t a c t s , f a c i l i t a t e s r o t a t i o n . A f t e r the CW s i g n a l reaches the r e c e i v e r , and has passed through a s e r i e s of d i f f e r e n t c o n v e r s i o n s , 1 KHz m o d u l a t i o n i s added. T h i s m o d u l a t i o n i s then d e t e c t e d by a bolometer d e t e c t o r and the output fed to a S c i e n t i f i c A t l a n t a R e c t a n g u l a r - P o l a r P a t t e r n Recorder, S e r i e s 1580 for p l o t t i n g . PATH LENGTH T r a n s m i s s i o n p a t h l e n g t h from the t u r n t a b l e c e n t r e o f r o t a t i o n to the a d j a c e n t v e r t i c a l 4"x4" support was 27 f e e t , 4 i n c h e s . A c t u a l a p e r t u r e t o a p e r t u r e s e p a r a t i o n was about 12 i n c h e s l e s s . 230 i if* ^ UJ m £3 .<5 CJ V ul uj 1 o o o: U) L_. I
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Radiation from coupled open-ended waveguides Driessen, Peter F. 1981
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Title | Radiation from coupled open-ended waveguides |
Creator |
Driessen, Peter F. |
Date | 1981 |
Date Issued | 2010-03-26T23:41:53Z |
Description | Ray-optical methods are used to calculate the coupling between open-ended parallel-plate waveguides, as well as the radiation patterns of finite arrays of coupled parallel-plate waveguides with only the central guide driven. These methods require extensive ray tracing, particularly for the larger arrays, to take into account the many possible ray paths. The coupling coefficients between both two and three guides in isolation agreed remarkably well with those previously derived in the presence of other guides, groundplanes etc., indicating a general lack of sensitivity of the coupling coefficients to the details of the surrounding structure. The calculated patterns were compared with experimental patterns using an H-plane sectoral horn to simulate the parallel-plate waveguide array. Radiation patterns of both three and five element arrays with all waveguide edges in the aperture plane, as well as that of a three element staggered array with the outer edges not in the aperture plane agreed well with the experimental patterns. A wide variety of patterns could be obtained by varying the width, depth, and number of the outer guides in the array. Ray-optical methods may thus be useful in the development of waveguide antennas for a variety of applications. |
Subject |
Wave guides |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2010-03-26 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0065489 |
URI | http://hdl.handle.net/2429/22785 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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