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High frequency beam diffraction by apertures and reflectors 1987

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H I G H F R E Q U E N C Y B E A M D I F F R A C T I O N B Y A P E R T U R E S A N D R E F L E C T O R S by GIBREEL A. SUEDAN M. SC., CALIFORNIA INSTITUTE OF TECHNOLOGY A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA JUNE 1987 © GIBREEL A. SUEDAN, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of £LBcT/2.fCH L ^ &tNEEgjN£ The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Seft- WK1 DE-6(3/81) - ii - ABSTRACT Most solutions for electromagnetic wave diffraction by obstacles and apertures assume plane wave incidence or omnidirectional local sources. Solutions to diffraction problems for local directive sources are needed. The complex source point representation of directive beams together with uniform solutions to high frequency diffraction problems is a powerful combination for this. Here the method is applied to beam diffraction by planar structures with edges, such as the half-plane, slit, strip, wedge and circular aperture. Previously used restrictions to very narrow beams and paraxial regions, are removed here and the range of validity increased. Also it is shown that the complex source point method can give a better approximation to broad antenna beams than the Gaussian function. The solution derived for the half-plane problem is uniform, accurate and valid for all beam orientations. This solution can be used as a reference solution for other uniform or asymptotic solutions and is used to solve for the wide slit and complementary strip problems. Uniform solutions for omidirectional sources are developed and extended analytically to become solutions for directive beams. The uniform theory of diffraction is used to obtain uniform solutions where there are no simple exact solutions, such as for the wedge and circular aperture. Otherwise rigorously correct solutions at high frequencies for singly diffracted far fields are used, such as for the half-plane, slit and strip. The geometrical theory of diffraction and equivalent line currents are used to include interaction between edges. Extensive numerical results including the limiting cases; e.g. plane wave incidence, line and point sources are given. These solutions are compared with previous solutions, wherever possible and good agreement is evident Beam (Effraction by a wedge with its edge on the beam axis is analysed. This solution completes a previous asymptotic solution which is infinite on the shadow boundaries and inaccurate in the transition regions. Finally, the diffraction by a circular aperture illuminated by normally incident acoustic beam, is derived and the singularity along the axial caustic is removed using Bessel functions and a closed form expression for multiple diffraction is derived. - iv - ACKNOWLEDGMENTS I wish to express my heart felt gratitude to Dr. E. V. Ml for both his academic and personal assistance, advice and encouragement throughout the course of this investigation. I am also indebted to him for taking the time to read and evaluate my thesis through his invaluable suggestions. Full recognition and thanks to the Libyan people for their generous financial support throughout my years of study. The computing support provided by the Natural Sciences and Engineering Research Council (NSERC) of Canada, is gratefully acknowledged. I wish to thank the general office staff of the Department of Electrical Engineering at the University of British Columbia for their kind help and cooperation. Special thanks and deep appreciation to my family who waited patiently during my long course of study. - v - TABLE OF CONTENTS ABSTRACT ii ACKNOWLEDGEMENTS iv TABLE OF CONTENTS . v LIST OF FIGURES vii 1 INTRODUCTION AND LITERATURE REVIEW 1 1.1 Introduction 1 1.2 General Assumptions 5 1.3 Literature Review 5 1.3.1 Beam Representation by Current Distributions 6 1.3.2 Spectral Theory of Diffraction 6 1.3.3 Kirchhoff-Fresnel Method 7 1.3.4 Boundary Diffraction Wave Theory 8 1.3.5 Uniform Asymptotic Theory of Diffraction 8 1.3.6 Inhomogeneous (Evanescent) Wave Tracking 10 1.3.7 Complex Ray Tracing 10 1.3.8 Complex Source Point 11 1.4 Overview of the Thesis 15 2 COMPLEX SOURCE POINT METHOD 17 2.1 Beam Evolution from Complex Line Source 17 2.2 Half Power Beam Width 19 2.3 Comparison with Gaussian and Typical Aperture Beam patterns 20 2.4 Multiple Complex Line Sources 22 3 BEAM DIFFRACTION BY A CONDUCTING HALF-PLANE 27 3.1 Uniform Solution for Field Radiation Pattern 27 3.2 Shadow and Reflection Boundaries 28 3.3 Numerical Results for the Half-Plane 31 4 BEAM DIFFRACTION BY A WIDE SLIT AND COMPLEMENTARY STRIP . . 38 4.1 Beam Diffraction by a Wide Slit 38 4.1.1 Far Field Calculation 38 4.1.2 Multiple Diffraction Calculation 40 4.1.3 Numerical Results for the Slit 43 4.2 Beam Diffraction by a Wide Conducting Strip 48 4.2.1 Far Field Calculation 48 4.2.2 Numerical Results for the Strip 50 - vi - 5 BEAM DIFFRACTION BY A CONDUCTING WEDGE 55 5.1 Real Line Source Solution 55 5.2 Uniform Solution for Beam Source 58 5.3 Numerical Results for the Wedge 59 6 BEAM DIFFRACTION BY A CIRCULAR APERTURE 67 6.1 Uniform Point-Source Solution 67 6.1.1 Single Diffraction Solution 68 6.1.2 Multiple Diffraction Solution 70 6.2 Uniform Beam Solution 73 6.2.1 Far Field Calculation 73 6.2.2 Shadow and Reflection Boundary Calculations 74 6.3 Numerical Results 74 7 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 80 7.1 Summary 80 7.2 Conclusions 81 7.3 ' Recommendations for Future Work 84 REFERENCES 86 APPENDIX 91 - vii - LIST OF FIGURES Figure Title Page 1.1a Branch cut and branch points of the CSP 14 Lib The paraxial region of a Gaussian beam 14 2.1a Geometry of a complex line source and real far field point 24 2.1b Geometry of multiple complex line sources and real far field point 24 2.2 Comparison of normalized patterns of a typical aperture, Gaussian beam and complex source point 25 2.3 Normalized patterns of multiple complex line sources 26 3.1 Geometry of a complex line source diffraction by a half-plane 34 3.2 Comparison of a uniform and asymptotic solutions of beam diffraction by a half-plane ( The edge on the beam axis ) 35 3.3 Comparison of a uniform and asymptotic solutions of beam diffraction by a half-plane ( The edge off the beam axis ) 36 3.4 Comparison of uniform solutions of a plane wave and limiting beam diffraction by a half-plane 37 4.1 Geometry of a complex line source diffraction by a slit 45 4.2 Normalized total field pattern of a beam diffraction by a slit 46 4.3 Comparison of patterns of a plane wave and limiting beam diffraction by a slit 47 4.4 Geometry of a complex line source diffraction by a strip 52 4.5 Normalized total field patterns of a beam diffraction by a strip ( Normal incidence ) 53 4.6 Normalized total field patterns of a beam diffraction by a strip ( Non-normal incidence ) 54 - viii - Figure Title Page 5.1 Geometry of a complex line source diffraction by a wedge 62 5.2 Normalized total field patterns of a beam diffraction by a rectangular wedge ( # o <0 c r ) 63 5.3 Normalized total field patterns of a beam diffraction by a rectangular wedge ( ^0>"^cr ) 64 5.4 Normalized total field patterns of a beam diffraction by a rectangular wedge ( ^ 0 = ^ c r ) 65 5.5 Normalized total field patterns of a beam diffraction by wedges of different angles ( 0 o ># c r ) 66 6.1 Geometry of a complex point source diffraction by a circular aperture 77 6.2 Normalized total field patterns of a beam diffraction by a circular aperture ( Normal incidence ) 78 6.3 Comparison of patterns of a single and mutiple diffraction of a beam by a circular aperture ( Normal incidence ) 79 E.1 Geometry of a complex line source diffraction by a parabolic reflector 103 E.2 Normalized Diffracted field component of a beam diffraction by a parabolic reflector 104 - 1 - CHAPTER I INTRODUCTION AND LITERATURE REVIEW 1.1 Introduction Electromagnetic wave d i f f r a c t i o n by conducting r e f l e c t o r s and by apertures i n conducting screens has been stud i e d e x t e n s i v e l y f o r many years. S o l u t i o n s f o r plane wave in c i d e n c e , or a d i s t a n t source, and i s o t r o p i c l o c a l sources i n r e a l space, have been obtained f o r the h a l f - p l a n e , f o r the wedge, f o r the s l i t and complementary d i s c (see Bowman et a l . , 1969). Of the two ca t e g o r i e s of s o l u t i o n s , low frequency and high frequency or asymptotic s o l u t i o n s , t h i s t h e s i s i s concerned w i t h the l a t t e r . Uniform asymptotic s o l u t i o n s f o r the h a l f screen and wedge have been obtained f o r o m n i d i r e c t i o n a l l o c a l sources (e.g. Boersma and Lee, 1977; Kouyoumjian and Pathak, 1974). Uniform asymptotic s o l u t i o n s are u s e f u l i n the d i f f r a c t i o n s o l u t i o n s f o r d i r e c t i v e beams considered i n t h i s t h e s i s . D i f f r a c t i o n by simple shapes when i l l u m i n a t e d by d i r e c t i v e l o c a l sources such as Gaussian beams and antenna beams, using d i f f e r e n t techniques, r e c e n t l y , has been e x t e n s i v e l y s t u d i e d . One of these techniques i s the beam f i e l d r e p r e s e n t a t i o n by a current d i s t r i b u t i o n . Anderson (1978) a p p l i e d t h i s technique to solve f o r antenna beam d i f f r a c t i o n by a conducting h a l f - p l a n e . The d i f f i c u l t y of o b t a i n i n g the current d i s t r i b u t i o n , which represents the e f f e c t of the antenna beam e x a c t l y or approximately, i s one of the disadvantages of t h i s approach. Another i s the d i f f i c u l t y of s o l v i n g the r e s u l t i n g boundary value - 2 - problem. C e r t a i n l y the accuracy depends on the f i e l d r e p r e s e n t a t i o n and on the approximations made to solve the i n t e g r a l i n v o l v e d . However, t h i s approach gives continuous f i e l d s at the shadow boundaries. The K i r c h h o f f method i s another technique used to solve f o r Gaussian beam d i f f r a c t i o n by h a l f - s c r e e n (Pearson et a l . , 1969). Because of double i n t e g r a t i o n introduced i n t h i s method, asymptotic s o l u t i o n s are complicated and numerical s o l u t i o n s very c o s t l y when la r g e s c a t t e r e r s are assumed. Another shortcoming i s poor accuracy i n the region o f f the beam a x i s . However, i t p r e d i c t s no s i n g u l a r i t y on the c a u s t i c a x i s . The Boundary D i f f r a c t i o n Wave Theory (BDWT) proposed by Miyamoto and Wolf (1962), overcomes the problem of a double i n t e g r a t i o n i n the K i r c h h o f f method, but makes the integrand (the vector p o t e n t i a l ) more complicated. Consequently the i n t e g r a t i o n becomes very d i f f i c u l t i n a d d i t i o n to the d i f f i c u l t y i n o b t a i n i n g the vector p o t e n t i a l i t s e l f . This approach has the same accuracy as the K i r c h h o f f method or l e s s when the vector p o t e n t i a l i s approximate. Using the BDWT, O t i s and L i t (1975) gave the s o l u t i o n to 2-dimensional Gaussian beam d i f f r a c t i o n by a h a l f - s c r e e n and the 3-dimensional case was given by Takenaka and Fukumitsu (1982). The s i n g l e d i f f r a c t i o n by a c i r c u l a r aperture when i l l u m i n a t e d by a normally i n c i d e n t Gaussian beam was obtained by O t i s et a l . (1977) and co r r e c t e d by Takenaka et a l . (1980). A l s o the same problem was solved by Belanger and Couture (1983), using the BDWT wi t h the Gaussian beam represented by a complex source p o i n t . - 3 - The Inhomogeneous (evanescent) Wave Tracking (IWT) proposed by Choudhary and Felsen (1973) and r e f i n e d by E i n z i g e r and Raz (1980), i s another approach used to solve the problem of d i r e c t i v e f i e l d s . The main advantage of t h i s method i s that i t gives a p h y s i c a l e x p l a n a t i o n of the propagation and s c a t t e r i n g mechanism. Because of the d i f f i c u l t y of o b t a i n i n g the phase paths, i t has been r a r e l y used. Choudhary and Felsen (1974) a p p l i e d the IWT method to the problem of Gaussian beam r e l f e c t i o n by a conducting c i r c u l a r c y l i n d e r . R e f l e c t i o n by a p a r a b o l i c r e f e l c t o r was given by Hasselmann and Fe l s e n (1982). Also F e l s e n (1976) studi e d the propagation of Gaussian beams i n f r e e space using the same method. The Complex Ray Tracing (CRT) method was invented to overcome the d i f f i c u l t y of determining the phase paths i n the IWT method by t r a c i n g the d i r e c t i v e f i e l d s i n the complex space. This technique was a p p l i e d by t r a c i n g the Gaussian beam i n f r e e space by K e l l e r and S t r e i f e r (1971), Deschamps (1971, 1972) and Wil l i a m s (1973). Ghione et a l . (1984) used the CRT method to study the r a d i a t i o n from l a r g e apertures w i t h tapered i l l u m i n a t i o n s . S c a t t e r i n g of evanescent plane waves by a •«-..' conducting c i r c u l a r c y l i n d e r was given by Wang and Deschamps (1974). A l s o Chione et a l . (1984) a p p l i e d the same technique to a r e f l e c t o r antenna i l l u m i n a t e d by a beam f i e l d . The CRT method i s an o p t i c a l (asymptotic) s o l u t i o n v a l i d only f o r high frequencies. The re p r e s e n t a t i o n of d i r e c t i v e beams w i t h complex source points along with using e x i s t i n g (exact or approximate) s o l u t i o n s f o r r e a l sources, which i s c a l l e d the Complex Source Point (CSP) method, - 4 - can give (exact or asymptotic) s o l u t i o n s to many ca n o n i c a l and l e s s simple problems i n v o l v i n g d i r e c t i v e sources, w i t h no e x t r a e f f o r t , provided a n a l y t i c a l c o n t i n u a t i o n i n t o complex space i s p o s s i b l e . The only d i f f i c u l t y w i t h the CRT and CSP methods, e s p e c i a l l y f o r non-planar s u r f a c e s , i s to f i n d an a - p r i o r i s e l e c t i o n r u l e to d i s t i n g u i s h the re l e v a n t from spurious ray c o n t r i b u t i o n s . Now t h i s can be done by studying the saddle p o i n t s and steepest descent paths (Ghione et a l . , 1984). Otherwise these techniques are easy to apply, accurate, and need no i n t e g r a l e v a l u a t i o n i n asymptotic s o l u t i o n s . Furthermore these techniques are uniform on the shadow boundaries, except f o r asymptotic s o l u t i o n s when the beam a x i s passes through the d i f f r a c t i n g edge. The CSP method uses e x i s t i n g s o l u t i o n s , so i t needs l e s s e f f o r t and i t can be used f o r exact s o l u t i o n s . Because of the above, the CSP method i s adopted everywhere i n t h i s t h e s i s . The asymptotic s o l u t i o n s f o r the Gaussian beam d i f f r a c t i o n by a conducting wedge (F e l s e n , 1976) and by a h a l f - s c r e e n (Green et a l . , 1979), are i n v a l i d when the beam a x i s passes through the d i f f r a c t i n g edge or when broad beams are assumed. They are in a c c u r a t e i n the t r a n s i t i o n regions and s i n g u l a r on the shadow boundaries. One of the goals here i s to o b t a i n a uniform s o l u t i o n f o r the wedge us i n g the Uniform Theory of D i f f r a c t i o n and the CSP r e p r e s e n t a t i o n , and f o r the h a l f - s c r e e n , based on a simple s o l u t i o n exact i n the f a r f i e l d l i m i t , u sing the CSP method. A more simple convenient formula f o r the shadow boundary l o c a t i o n s a l s o w i l l be d e r i v e d . s o l u t i o n t o 2-dimensional antenna beam d i f f r a c t i o n by a s l i t , i n c l u d i n g double - 5 - d i f f r a c t i o n , and complementary conducting s t r i p w i l l be given. This problem has not been studied before. F i n a l l y beam d i f f r a c t i o n by c i r c u l a r aperture for normal incidence, including i n t e r a c t i o n between the edges, i s analyzed using the UTD and CSP representation. For a l l the above examples, numerical r e s u l t s include the l i m i t i n g cases of plane wave incidence or i s o t r o p i c sources. 1.2 General Assumptions Through a l l the subsequent a n a l y s i s , the following are assumed: a) The time dependence i s harmonic (exp[ju>t]) and i s suppressed. b) The medium i s homogeneous, i s o t r o p i c , nondispersive and non-dissipative. c) The frequencies are very high and observation points are i n the far f i e l d of the scatterer ( k r » l ) . d) Perfect conductors (screen, half-screen, etc.) e) Scalar f i e l d s (U) .are assumed. f) Soft boundary conditions are assumed. 1.3 Literature Review Scattering by simple r e f l e c t o r s and apertures as a half-plane, wedge, c i r c u l a r aperture, parabolic and paraboloidal antennas, and c i r c u l a r cylinders when illum i n a t e d by d i r e c t i v e sources, which are approximately Gaussian beams i n the para x i a l region, have been studied by many researchers using the d i f f e r e n t techniques summarized below. As - 6 - only the complex source point method of secton 1.3.8 i s used i n t h i s t h e s i s , the reader may choose to omit s e c t i o n s 1.3.1-1.3.7. 1.3.1 Beam Representation by Current D i s t r i b u t i o n s By t h i s method, the i n c i d e n t beam i s represented by a non-uniform current sheet d i s t r i b u t i o n . Then the boundary value problem of current element i n presence of the s c a t t e r e r i s solved by i n t e g r a t i n g the obtained s o l u t i o n over the whole current sheet. Then the i n t e g r a l i s evaluated n u m e r i c a l l y or a s y m p t o t i c a l l y . The s o l u t i o n w i t h t h i s approach i s continuous at the shadow boundaries. This approach i s d i f f e r e n t from p h y s i c a l o p t i c s and s p e c t r a l theory of d i f f r a c t i o n . Anderson (1978) used t h i s technique to solve antenna beam d i f f r a c t i o n by a conducting h a l f - s c r e e n . 1.3.2 S p e c t r a l Theory of D i f f r a c t i o n The basic concepts of the s p e c t r a l theory of d i f f r a c t i o n (STD) proposed by M i t t r a et a l . (1976), were i l l u s t r a t e d by the f a m i l i a r conducting h a l f - p l a n e i l l u m i n a t e d by plane wave. The p r i n c i p a l c o n t r i b u t i o n of STD i s the i n t r o d u c t i o n of the s p e c t r a l d i f f r a c t i o n c o e f f i c i e n t which i s defined as the F o u r i e r transform of the current induced on the s c a t t e r e r . This c o e f f i c i e n t i s ass o c i a t e d w i t h the i n t e g r a l r e p r e s e n t a t i o n of the s c a t t e r e d and t o t a l f i e l d s . Although the s p e c t r a l d i f f r a c t i o n c o e f f i c i e n t tends to i n f i n i t y at the shadow boundaries, the f i e l d s obtained by the STD are f i n i t e . The s c a t t e r e d f i e l d can be constructed by convolving, i n the space domain, the induced - 7 - current and the r a d i a t e d f i e l d of an elementary point or l i n e source current (Green's f u n c t i o n ) . The t o t a l f i e l d i s the sum of the s c a t t e r e d f i e l d and, whenever a p p l i c a b l e , the i n c i d e n t f i e l d . When the i n t e g r a l s i n v o l v e d i n the s c a t t e r e d and t o t a l f i e l d s are a s y m p t o t i c a l l y evaluated using the saddle point technique, the le a d i n g term y i e l d s K e l l e r ' s GTD f i e l d . A r b i t r a r y i n c i d e n t f i e l d s , a l s o can be assumed using the STD technique by applying the s u p e r p o s i t i o n p r i n c i p a l . The spectrum of the i n c i d e n t a r b i t r a r y f i e l d i s m u l t i p l i e d by the s p e c t r a l d i f f r a c t i o n c o e f f i c i e n t of the plane wave then i n t e g r a t i n g over the e n t i r e spectrum to give a double i n t e g r a l r e p r e s e n t a t i o n of the s c a t t e r e d and t o t a l f i e l d s . This i n t e g r a l may be evaluated a s y m p t o t i c a l l y or nu m e r i c a l l y . Rahmat-Sammii and M i t t r a (1977) give d e t a i l e d c a l c u l a t i o n s and a p p l i c a t i o n s . 1.3.3 Kirchhoff-Fresnel Method In t h i s method the t o t a l f i e l d behind an aperture i n a conducting plane i s given i n terms of the double i n t e g r a l of the i n c i d e n t f i e l d and i t s d e r i v a t i v e i n the plane of the aperture. This i n t e g r a t i o n i s taken over the aperture (Born and Wolf, 1974, p. 375-386). Then the i n t e g r a l i s evaluated numerically or a s y m p t o t i c a l l y to y i e l d the t o t a l f i e l d . Pearson et a l . (1969) a p p l i e d t h i s technique to the d i f f r a c t i o n of a fundamental-mode Gaussian beam (Kogelnik, 1965) by a s e m i - i n f i n i t e conducting screen. An asymptotic s o l u c t i o n i n the F r e s n e l l i m i t was derived. - 8 - 1.3.4 Boundary D i f f r a c t i o n Wave Theory Through the use of Stoke's theorem and an a s s o c i a t e p o t e n t i a l v e c t o r , Miyamoto and Wolf (1962) showed t h a t , i n general, the K i r c h h o f f surface i n t e g r a l , mentioned i n the previous s e c i t o n , can be s p l i t i n t o two separate l i n e i n t e g r a l s . One represents a wave o r i g i n a t i n g from the boundary of the d i f f r a c t i n g aperture c a l l e d the boundary d i f f r a c t i o n wave (BDW) and the other represents the geometrical wave o r i g i n a t i n g from the source. The l a t t e r i s zero i f the observation p o i n t s l i e i n the shadow region. The t o t a l f i e l d i s given by the sum of the BDW and geometrical wave f i e l d s . The BDW method has the same l i m i t a t i o n s and approximations as the K i r c h h o f f - F r e s n e l method. A p p l i c a t i o n of the technique to a Gaussian beam w i t h c y l i n d r i c a l symmetry (Siegman, 1971, Ch. 8) normally i n c i d e n t on a c i r c u l a r aperture i n a conducting plane i s given by O t i s (1974) under the p a r a x i a l f a r f i e l d approximations. 1.3.5 Uniform Asymptotic Theory of D i f f r a c t i o n A uniform asymptotic theory of d i f f r a c t i o n (UAT) which provides the c o r r e c t asymptotic s o l u t i o n f o r an a r b i t r a r y i n c i d e n t f i e l d on a h a l f - p l a n e has been developed by A l h u w a l i a et a l . (1968) and Lewis and Boersma (1969), c o r r e c t s defects of the geometricl theory of d i f f r a c t i o n (GTD); such as s i n g u l a r i t i e s at the shadow boundaries and at the d i f f r a c t i n g edge. I t a l s o provides higher order terms i n the d i f f r a c t e d f i e l d expansion. - 9 - Boersma and Lee (1977) a p p l i e d UAT to the problem of c y l i n d r i c a l wave from a l i n e source p a r a l l e l to the edge of a conducting h a l f - p l a n e . In t h e i r approach a l l f i e l d s are expanded a s y m p t o t i c a l l y i n i n v e r s e powers of the wave number which i s assumed l a r g e . The c o e f f i c i e n t s of expansion are derived by s u b s t i t u t i n g i n the reduced wave equation. The post u l a t e d t o t a l f i e l d i s a uniform asymptotic expansion based on the exact s o l u t i o n of a plane wave i n c i d e n t on a h a l f - p l a n e , so the UAT reduces to the e x a c t s o l u t i o n of p l a n e wave d i f f r a c t i o n by a h a l f - p l a n e . E x c luding the c a u s t i c p o ints at the source and i t s image, the UAT s o l u t i o n f o r the t o t a l f i e l d i s f i n i t e and c o n t i n u o u s at a l l observation p o i n t s . Away from the shadow boundaries, the l e a d i n g term of the UAT s o l u t i o n reduces to the GTD s o l u t i o n . Since the UAT s o l u t i o n remains f i n i t e at the d i f f r a c t i n g edge, i t can a l s o be used f o r n e a r - f i e l d c a l c u l a t i o n s . U n l i k e the GTD where the d i f f r a c t i o n c o e f f i c i e n t i s taken from the Sommerfield's h a l f - p l a n e s o l u t i o n , the d i f f r a c t i o n c o e f f i c i e n t of UAT i s derived by e n f o r c i n g the edge c o n d i t i o n . The UAT has been extended to electromagnetic d i f f r a c t i o n by a curved wedge by Lee and Deschamps (1976) but there i t i s approximate. The main disadvantage of the UAT i s i t s complexity i n determining higher order terms, when very d i r e c t i v e i n c i d e n t f i e l d s are assumed and when i n t e r a c t i o n between edges are s i g n i f i c a n t . In such cases the uniform geometrical theory of d i f f r a c t i o n by Kouyouimjian and Pathak (1974), which gives l e s s accurate r e s u l t s , may be used i n s t e a d . - 10 - 1.3.6 Inhomogeneous (Evanescent) Wave Tracking Here inhomogeneous waves, such as Gaussian beams, can be tracked from the source or the i n i t i a l surface to the observation point v i a the s c a t t e r e r , t o t a l l y i n r e a l space. By s o l v i n g the d i f f e r e n t i a l equations f o r the r e a l and imaginary parts of the phase and amplitude f u n c t i o n s , which r e s u l t from s a t i s f y i n g the reduced wave equation, the t o t a l f i e l d can be completely determined. N e g l e c t i n g the wave length squared term i n the above d i f f e r e n t i a l equations enables one to c a l c u l a t e the phase independently from the amplitude. The s o l u t i o n obtained i n t h i s way i s approximate, but the accuracy increases w i t h decreasing the wave len g t h . For d e t a i l s see Choudhary and Felsen (1973), Felsen (1976) and E i n z i g e r and Raz (1980). The inhomogeneous wave t r a c k i n g method i s a p p l i e d to Gaussian beam r e f l e c t i o n by conducting c i r c u l a r c y l i n d e r as given by Choudhary and Felsen (1974) without i n c l u d i n g d i f f r a c t i o n from edges. 1.3.7 Complex Ray Tracing Since i t has been noted that a Gaussian beam can be represented i n terms of a bundle of complex rays, by Deschamps (1971) and K e l l e r and S t r e i f e r (1971), complex ray t r a c i n g (CRT) was introduced and complex geometrical o p t i c s has been developed. In the CRT method, the phase, amplitude and space coordinates are allowed to take complex values, as i n the IWT method. The mathematical basis of t h i s method i s the process of a n a l y t i c c o n t i n u a t i o n . The t r a c i n g of the f i e l d from the complex source to the r e a l observation point v i a the s c a t t e r e r (complex i n general) i s i n complex space. The study of a Gaussian beam, simulated - 11 - by a complex l i n e or point source, and propagation i n f r e e space from an assigned i n i t i a l f i e l d d i s t r i b u t i o n have been e x t e n s i v e l y d e a l t w i t h (Ghione, Montrosset and Orta, 1984). The e i k o n a l and transport equations used i n the IWT method are a p p l i c a b l e here. Without s e p a r a t i n g the phase and amplitude f u n c t i o n s i n t o r e a l and imaginary p a r t s , the d i f f e r e n t i a l equations can be solved by the method of c h a r a c t e r i s t i c s to o b t a i n the phase and amplitude of the t o t a l f i e l d . 1.3.8 Complex Source Point In the complex ray t r a c i n g method, asymptotic s o l u t i o n s are obtained f o r high frequency ( l a r g e wave numbers) and f a r f i e l d s . But the Complex Source P o i n t (CSP) method can be used to o b t a i n exact as w e l l as asymptotic s o l u t i o n s f o r low or high frequencies and near or f a r f i e l d s , as long as s o l u t i o n s to corresponding r e a l sources e x i s t and can be a n a l y t i c a l l y continued i n t o complex space. On as s i g n i n g complex values to the source coordinate l o c a t i o n s of an o s c i l l a t i n g i s o t r o p i c p o i n t or l i n e source, one may generate a h i g h l y c o l l i m a t e d f i e l d that behaves i n the v i c i n i t y of i t s maximum (beam a x i s ) l i k e a 3-dimensional (point source) or 2-dimensional ( l i n e source) Gaussian beam (Deschamps 1971, Jones 1979; Couture and Belanger 1981, A l b e r t s e n et a l . , 1983, and F e l s e n 1976, 1984). This i m p l i e s that the CSP s u b s t i t u t i o n converts point or l i n e source Green's f u n c t i o n s f o r propagation and d i f f r a c t i o n i n various environments i n t o f i e l d s o l u t i o n s f o r i n c i d e n t Gaussian beams. Thus without f u r t h e r e f f o r t , the whole rigorous and asymptotic s o l u t i o n s y i e l d the f i e l d response f o r beam - 1 2 - e x c i t a t i o n , p r o v i d e d t h a t t h e r e c a n b e a n a n a l y t i c c o n t i n u a t i o n o f t h e s o l u t i o n s f r o m r e a l t o c o m p l e x s p a c e . L e t u s t a k e , a s a n e x a m p l e , t h e e v o l u t i o n o f a 2 - d i m e n s i o n a l G a u s s i a n b e a m u s i n g t h e C S P . T h e f i e l d o f a n i s o t r o p i c p o i n t s o u r c e i n f r e e s p a c e , i s g i v e n b y G r e e n ' s f u n c t i o n G ( R ) w h i c h i s a s o l u t i o n t o t h e w a v e e q u a t i o n - j k R G ( R ) = 1 , ( 1 . 1 ) R w h e r e R i s t h e d i s t a n c e b e t w e e n t h e s o u r c e a n d o b s e r v a t i o n p o i n t , w h i c h c a n b e r e a l o r c o m p l e x . F o r a w a v e p r o p a g a t i n g i n t h e z - d i r e c i o n , l e t t h e s o u r c e b e l o c a t e d a t ( 0 , 0 , - j b ) w h e r e b i s a r e a l p o s i t i v e n u m b e r . T h e n R = [ p 2 + ( z + j b ) 2 ] 1 / 2 ; p2 = x 2 + y 2 , R e a l ( R ) X ) ( 1 . 2 ) F r o m ( 1 . 2 ) , R i s a m u l t i v a l u e d f u n c t i o n a n d v a n i s h e s a t t h e b r a n c h l i n e z=0, p=b. T o m a k e R s i n g l e v a l u e d a n d G ( R ) a n a l y t i c , a b r a n c h c u t ( s u r f a c e ) a t z=0 a n d p<b s h o u l d b e i n t r o d u c e d ( s e e F i g . 1 . 1 a ) . I n t h e p a r a x i a l r e g i o n (p « z "•" b ) a n d f o r z>0, R c a n b e s i m p l i f i e d t o R - j [ b ~ _ b - P 2 ] + z [ l + 91 ] ( 1 . 3 ) 2 ( z 2 + b 2 ) 2 ( z 2 + b 2 ) I n s e r t i n g ( 1 . 3 ) i n ( 1 . 1 ) g i v e s k b e -CP/W) 2 " J B ( P ' Z > G(p,z) * 1 . e Q P / w ; . e ( 1 . 4 ) / z 2 + b 2 w h e r e w i s t h e e - 1 h a l f b e a m w i d t h a n d B(p,z) i s t h e p h a s e w = [ 2 ( z 2 + b 2 ) / k b ] 1 / 2 ( 1 . 5 ) B(p,z) = 1 [ k b + (p/w) 2] + t a n - 1 ( b / z ) ( 1 . 6 ) b - 1 3 - T h e p r o p a g a t i n g w a v e , d e f i n e d b y ( 1 . 4 ) , i s s u b j e c t t o a n e x p o n e n t i a l d e c a y p e r p e n d i c u l a r t o t h e z - a x i s p r o p o r t i o n a l t o p . T h u s a G a u s s i a n b e a m i s f o r m e d i n t h e p a r a x i a l r e g i o n . F o r b > > z , t h e w a v e p r o p a g a t e s p a r a l l e l t o t h e z - a x i s w i t h d i s t o r t i o n o f t h e w a v e f r o n t ; a n d f o r b<<z t h e p h a s e p a t h s ( l o c u s o f e - 1 p o i n t s ) a r e h y p e r b o l o i d s g i v e n b y ( p / w Q ) 2 - ( z / b ) 2 = 1 , ( 1 . 7 ) w h e r e w i s t h e e _ 1 h a l f - b e a m w i d t h a t t h e b e a m w a i s t ( z = 0 ) . o x ' W Q = ( 2 b / k ) 1 / 2 i s o f t e n c a l l e d t h e s p o t s i z e a t t h e b e a m w a i s t ( s e e F i g . 1 . 1 b ) . T h i s d e r i v a t i o n i s a l s o v a l i d f o r t w o d i m e n s i o n a l f i e l d s . T h e i m p l i c a t i o n s o f t h i s a r e t h a t f i e l d s o l u t i o n s f o r t w o o r t h r e e - d i m e n s i o n a l G r e e n ' s f u n c t i o n s c a n b e c o n t i n u e d a n a l y t i c a l l y i n t o c o m p l e x s p a c e t o p r o v i d e t h e s o l u t i o n s f o r d i r e c t i v e b e a m s . T h i s c a n b e a p p l i e d t o b o t h n u m e r i c a l a n d a n a l y t i c a l s o l u t i o n s , o r t o e i t h e r l o w o r h i g h f r e q u e n c y d i f f r a c t i o n s o l u t i o n s . A t l o w f r e q u e n c i e s b e a m d i f f r a c t i o n c a n a l s o b e s o l v e d n u m e r i c a l l y . A t h i g h f r e q u e n c i e s , n u m e r i c a l m e t h o d s g e n e r a l l y a r e i n e f f i c i e n t o r f a i l a n d t h e g e o m e t r i c a l t h e o r y o f d i f f r a c t i o n t o g e t h e r w i t h t h e c o m p l e x s o u r c e p o i n t m e t h o d p r o v i d e s t h e m o s t c o n v e n i e n t s o l u t i o n f o r b e a m d i f f r a c t i o n s . F o r h i g h e r m o d e s o f G a u s s i a n b e a m s , s e e S h i n a n d F e l s e n ( 1 9 7 6 ) , H a s h i m o t o ( 1 9 8 5 ) a n d L u k a n d Y u ( 1 9 8 5 ) . R e p r e s e n t a t i o n o f m o r e c o m p l i c a t e d b e a m s h a s b e e n s t u d i e d b y M a n t i c a e t a l . ( 1 9 8 6 ) , a n d w a v e s o l u t i o n s u n d e r c o m p l e x s p a c e - t i m e s h i f t s h a s b e e n , l a t e l y , p r o p o s e d b y E i n z i g e r a n d R a z ( 1 9 8 7 ) . - 1 4 - P A z F i g . 1 . 1 a B r a n c h c u t a n d b r a n c h p o i n t s o f t h e C S P Hyperboloid z F i g . 1 . 1 b T h e p a r a x i a l r e g i o n o f a G a u s s i a n b e a m - 15 - 1.4 Overview of the Thesis An i n t r o d u c t i o n i s given i n Secti o n (1.1) and some of e x i s t i n g work i n the l i t e r a t u r e on the r e p r e s e n t a t i o n and d i f f r a c t i o n of d i r e c t i v e source f i e l d s by simple shapes i s summarized i n Sectio n (1.3). In Chapter I I , beam e v o l u t i o n from a Complex Source Point (CSP) i s given i n p o l a r coordinates. This i s more convenient than r e p r e s e n t a t i o n i n c a r t e s i a n coordinates and d i r e c t l y r e l a t e s the beam parameters ( o r i e n t a t i o n and d i r e c t i v i t y ) to the complex coordinates of the source. The beam f i e l d generated by the CSP, a Gaussian beam and a t y p i c a l antenna aperture are compared and i l l u s t r a t e d . D e r i v a t i o n of more complicated beams; e.g. a beam w i t h s i d e l o b e s , i s a l s o achieved. Chapter I I I i s devoted to o b t a i n i n g a simple s o l u t i o n , uniform everywhere and f o r a l l beam o r i e n t a t i o n s , f o r antenna beam d i f f r a c t i o n by a h a l f - s c r e e n , based on the exact f a r f i e l d s o l u t i o n of l i n e source d i f f r a c t i o n by a h a l f - s c r e e n . A comparison of t h i s s o l u t i o n w i t h the asymptotic s o l u t i o n given by Green et a l . (1979) i s i l l u s t r a t e d . Also a simpler formula f o r shadow boundary l o c a t i o n i s derived. R e s u l t s obtained here are used i n Chapter IV of the problem of beam d i f f r a c t i o n by a s l i t In a conducting plane and by a complementary s t r i p . In Chapter V, beam d i f f r a c t i o n by a conducting wedge, when the beam a x i s passes through the edge, i s derived using the UTD. Shadow boundaries are obtained and numerical r e s u l t s f o r d i f f e r e n t angles of incidence and wedge angles are given. In the above examples, only 2-dimensional beams and s t r a i g h t edges are assumed. In Chapter VI, a 3-dimensional beam d i f f r a c t e d by a - 16 - c i r c u l a r aperture i n a p e r f e c t l y conducting plane, i n c l u d i n g m u l t i p l e d i f f r a c t i o n . Normal i n c i d e n c e , i . e . the beam a x i s c o i n c i d e n t w i t h the aperture a x i s , i s assumed. Numerical r e s u l t s f o r d i f f e r e n t beam w a i s t s , i n c l u d i n g the plane wave as a l i m i t i n g case, are given. The l a t t e r i s compared w i t h K e l l e r ' s s o l u t i o n (1957). A summary, conclusions and recommendations f o r fu t u r e work are given i n Chapter V I I . Appendix A contains an e v a l u a t i o n of F r e s n e l i n t e g r a l s w i t h complex arguments i n terms of e r r o r f u n c t i o n s and some important p r o p e r t i e s of F r e s n e l i n t e g r a l s are given. I n Appendix B, the r e a l and i m a g i n a r y p a r t s of r , the complex s dis t a n c e from the source to the edge, i n terms of the r e a l d i s tance of the source and the i n c i d e n t beam parameters are deri v e d . For comparison reasons the asymptotic s o l u t i o n of Gaussian beam d i f f r a c t i o n by conducting h a l f - s c r e e n , and the shadow boundary p o s i t i o n s given by Green et a l . (1979) are summarized i n Appendix C. Appendix D, shows the s i n g u l a r i t y c a n c e l l a t i o n i n the wedge d i f f r a c t i o n c o e f f i c i e n t at the shadow boundaries, when i l l u m i n a t e d by a r e a l l i n e source or a beam source i t s a x i s passing through the edge. Appendix E contains the a n a l y s i s of beam d i f f r a c t i o n by a conducting p a r a b o l i c r e f l e c t o r . In Appendix F, the d e r i v a t i o n of arctangent of a complex number, i n the proper quadrant Is given i n terms of a complex angle i n the f i r s t quadrant, that can be determined by the UBC computer f u n c t i o n s . Appendix G i s a l i s t of computer programs f o r the problems discussed i n Chapters I I - V I and i n Appendix E. - 17 - CHAPTER II COMPLEX SOURCE POINT METHOD By a s s i g n i n g complex values to the source coordinate l o c a t i o n s of a time harmonic i s o t r o p i c p o i n t or l i n e source i n a homogeneous unbounded medium, one may generate a c o l l i m a t e d f i e l d that behaves l i k e a 3-dimensional (point source) or 2-dimensional ( l i n e source) d i r e c t i v e beam. This i m p l i e s that the complex source point s u b s t i t u t i o n converts p o i n t or l i n e source Green's f u n c t i o n s (wave equation s o l u t i o n s ) f o r propagation and d i f f r a c t i o n i n various environments i n t o f i e l d s o l u t i o n s f o r i n c i d e n t d i r e c t i v e beams. Thus, without f u r t h e r e f f o r t , the whole rig o r o u s and asymptotic d i f f r a c t i o n s o l u t i o n s y i e l d the f i e l d response f o r beam e x c i t a t i o n , provided there can be an a n a l y t i c c o n t i n u a t i o n of the s o l u t i o n s from r e a l space to complex space. 2.1 Beam Evolution from a Complex Line Source F i g . 2.1a shows a 2 - d i m e n s i o n a l l i n e s o u r c e a t r , 8 from the o o o r i g i n of coordinates. The f i e l d s are uniform i n the z d i r e c t i o n and represent an o m n i d i r e c t i o n a l c y l i n d r i c a l wave. The f i e l d i n t e n s i t y at any observation p o i n t r , Q which i s a s o l u t i o n of wave equation may be w r i t t e n as -jn/4 <PkR U 1 = J%TI e H ( 2 ) ( k R ) » ; kR » 1 , (2.1) ° m where R i s the distance of the observation point from the source. - 18 - R = [ r 2 + r 2 - 2 r r cos(0 - 0 ) 1 1 / 2 (2.2) L o o o ' I n the f a r f i e l d ( r > > r Q ) , R = r - r Q cos (Q - 0 ) a p p l i e s i n the phase term and R~r i n the amplitude term of (2.1) g i v i n g - j k [ r - r cos(0-0 )] U = e ° ° ; 0 < 0 < it (2.3) o v ' /k"r By making the source c o o r d i n a t e s ( r , 0 ) complex (r , 0 ) the o o s s o m n i d i r e c t i o n a l wave becomes a d i r e c t i v e beam uniform i n the z d i r e c t i o n . F = 7 - SS (2.4) where "r , r" and "6" are the complex source p o s i t i o n , r e a l source p o s i t i o n and beam parameter vectors given i n p o l a r coordinates as r = ( r , 0 ), r o o o r* = ( r , 0 ) and "b" = (b, 8), where b defines the sharpness of the beam s s s and (3 defines i t s o r i e n t a t i o n . A l l angles are measured from the x - a x i s . r and r are measured from the o r i g i n w h i l e b i s measured from the r e a l s o po i n t source as shown i n F i g . 2.1. r g = [ r 2 + 2 r Q ( - j b ) cos(p-0 Q) + ( - j b ) 2 ] 1 / 2 ; R e ( r g ) > 0 (2.5) -1 r cos0 - jb c o sP 0 = cos [ — 2 ] (2.6) s r s where b > 0 and 0 <̂  8 << 2u. Replacing r ^ , 0 q by r g , 0 g i n (2.3) gives - j k [ r - r g c o s ( 0 - 0 g ) ] U 1 = ; r » | r I (2.7) /kF S r cos(0 T 0 ) = r cos0 cos0 + r s i n 0 s i n 0 (2.8a) s s s s — s s and from (2.4), - 19 - r cosO = r cos0 - ib cosB s s o o J » (2.8b) r sinQ = r sinO - ib sinB s s o o J Using these i n (2.8a) y i e l d s r cos(0 + 0 ) = r cos(0 + 0 ) - jb cos(0 + B) (2.8c) s s o o Su b s t i t u t e (2.8c) i n (2.7) to get - j k [ r - r cos(0-0 )] kb cos(0-B) e 0 ° e U 1 = (2.9) /kr By comparing equation (2.9) w i t h (2.3) we f i n d that (2.9) represents an o m n i d i r e c t i o n a l c y l i n d r i c a l wave ( f i r s t term) modulated by a beam pa t t e r n exp[kbcos(0-B)] w i t h i t s maximum i n the d i r e c t i o n 0=8 and minimum i n the d i r e c t i o n 0=8+n:. 2.2 Half Power Beam Width To c a l c u l a t e the h a l f power beam width (HPBW), we normalize the f i e l d of (2.9) to i t s peak value. U ( r , 0 ) At 0-8 = U A(r,B) HPBW - k b [ l - cos(0-B)] = e (2.10) the normalized f i e l d amplitude of (2.10) equals 1//2. . . . . ,HPBW.. - k b [ l - cos( )] a, 2 1 72 (2.11) Thus the h a l f power beam width i s r e l a t e d to the beam parameter kb by ln/2\ , , . An/2 ) ; kb :> kb 2 (2.12) HPBW = 2 c o s - ^ l - £2^) > - 20 - (2.12) shows that as kb increases the beam width decreases. I f kb < ̂  An2 the beam does not decay to the h a l f power p o i n t . S p e c i a l Case: r g and 0 g are complex unless b=0, corresponding to a r e a l source or 8 = 0 or 0 + ix. To show the l a s t case s u b s t i t u t e f o r 8=0 or 0 +n i n o o o o (2.5) and (2.6) to get and r = r + j b ; B = 0 , 0 + T I s o J ' r o' o 0 = 0 s o (2.13a) (2.13b) Therefore 0 becomes r e a l whenever the beam a x i s l i e s along r . s o 2.3 Comparison w i t h Gaussian and T y p i c a l Aperture Beam Patterns Since near the beam a x i s , 0 - 8 i s s m a l l , we can w r i t e cos ( 9 - P ) - l - < Q - V2 2 (2.14) Using (2.14) i n (2.10) gives - k b ( 0 - 8 ) 2 l T ( r , 0 ) T" = e (2.15) U x ( r , 8 ) Showing, as i s w e l l known (Green et a l . (1979) and Hasselman (1980)), that a complex source point provides a beam which i s Gaussian i n the p a r a x i a l r e g i o n . I t i s important to appreciate that t h i s complex source point r e p r e s e n t a t i o n of a beam i s not l i m i t e d to the p a r a x i a l region. F i g . 2.2a shows the f a r f i e l d r a d i a t i o n p a t t e r n of a source at k r Q = 8 f o r s e v e r a l v a l u e s of the kb c o r r e s p o n d i n g to h a l f power beam widths ranging from 68.5°(kb=2) to 10.4°(kb=85). - 21 - In F i g . 2.2b the broken curve i s a t y p i c a l aperture antenna beam p a t t e r n , that of an inphase c o s i n u s o i d a l d i s t r i b u t i o n i n an aperture of width 2a. I t s normalized p a t t e r n Is Ar.G) _ cos[ka s i n ( 9 - B ) ] U i ( r , B ) 1 - [ i ^ . s i n ( 9 - B ) ] 2 it For ka=4, i t s half-power beam width i s 55.7°. The s o l i d curve i n F i g . 2.2b i s a complex source point p a t t e r n of the same beam width (HPBW=55.7° or kb=3) w i t h k b = ' (2.17) [ l - c o . ( ™ ) ] 2 The dashed curve i n F i g . 2.2b i s a Gaussian beam w i t h the same beam width. U-V.G) - A n ^ l i ^ ] 2 = e HPBW ( 2 . i 8 ) A l l three curves overlap i n the p a r a x i a l r e gion (9-8 s m a l l ) . At angles w e l l o f f the beam a x i s , there i s some d i f f e r e n c e , s p e c i a l l y f o r broad beams (kb s m a l l ) , between (2.10) and (2.18) but here the complex source point p a t t e r n given by (2.10) Is a s l i g h t l y b e t t e r approximation to (2.16). The complex source point r e p r e s e n t a t i o n appears to be a v a l i d approximation to an antenna main beam p a t t e r n over the forward angular range (|0-B| < i t / 2 ) . Of course i t cannot represent p a t t e r n s i d e l o b e s . - 22 - 2.4 Multiple Complex Line Sources More complicated beams such as beams w i t h sidelobes a l s o can be derived by using the complex source point method. By p u t t i n g more than one source at d i f f e r e n t complex l o c a t i o n s and changing the r e a l l o c a t i o n s r ^ , 0 q and the beam parameters b , 8 we get a v a r i e t y of beam shapes. Let us have M sources l o c a t e d at M complex p o s i t i o n s . The mth source i s located at r , 0 and i t s corresponding beam parameters are om* om r ° r b , 8 as shown i n F i g . 2.1b. Then the f a r f i e l d due to the mth complex nr m source Û " i s given by (2.9) and r e w r i t t e n here as 4  m - J ^ o m C O B< 9- 9om>l k bm ^s(0-?J U 1 - -1- 6 ; r » rom'bm (2.19) m /k7 Then the r e s u l t a n t f a r f i e l d , due to the M weighted sources, i s M U =2 % Um . m m m=l ~ j k r M j k r cos ( 0 - 0 ) kb cos ( 0 - 6 ) e v n om om m N m ,„ o n N I Q„ • e • e (2.20) /k r m=l where 0^ are the weighting f a c t o r s . In F i g . 2.3 the f i e l d due to 3 l i n e sources i s derived f o r d i f f e r e n t beam parameters b , 8 w h i l e the r e a l l o c a t i o n s are kept c o n s t a n t , 0 .=0, 0 = T C / 2 , 0 =tr, k r . =kr 0=1, kr o=0. The weighting o l o2 ' o3 o l o3 o2 f a c t o r 0,2=1 while and Q3 are v a r i a b l e s ( p o s i t i v e or negative, greater or l e s s than 1). Because sources 1 and 3 are symmetric w i t h respect to source 2, the f i e l d s shown i n F i g . 2.3 are symmetric. Asymmetric f i e l d s - 23 - a l s o can be derived from asymmetric sources. A beam which resembles a l i n e source d i f f r a c t e d by a s l i t and a beam w i t h f i r s t s idelobes which looks l i k e a plane wave d i f f r a c t e d by a s l i t are shown i n F i g . 2.3. More sidelobes can be derived i f more complex sources are in c l u d e d . In t h i s chapter we studi e d the l i n e source. The point source i s v e r y much the same w i t h term —L_ i s replaced by — — and two dimensions i s replaced by three dimensions. A g e n e r a l d e s c r i p t i o n of m u l t i p l e complex s o u r c e p o i n t r e p r e s e n t a t i o n of beams has been given by Hashimoto (1985). F i g . 2.1a G e o m e t r y o f a c o m p l e x l i n e s o u r c e a n d r e a l f a r f i e l d p o i n t F i g . 2.1b G e o m e t r y o f m u l t i p l e c o m p l e x l i n e s o u r c e s a n d r e a l f a r f i e l d p o i n t t y p i c a l aperture antenna (2.16, ka=4) Complex Source P o i n t (2.10, kb=3) Gaussian (2.18, HPBW=55.7°) ( 3 2 = 1 8 0 ° , Q2=1.0, b!=b2=b3, Qi=Q3,Bl=B2-6 and 63-32+5) - 27 - CHAPTER III BEAM DIFFRACTION BY A CONDUCTING HALF-PLANE The t o t a l f a r f i e l d d i f f r a c t i o n by a conducting h a l f - s c r e e n of a beam i s derived from the exact f a r f i e l d s o l u t i o n given by Born & Wolf (1978) and Clemmow (1950) f o r a r e a l l i n e s o l u t i o n and compared w i t h the asymptotic s o l u t i o n given by Green et a l (1979) (see Appendix C). 3.1 Uniform S o l u t i o n Far F i e l d R a d i a t i o n P a t t e r n Suppose the o m n i d i r e c t i o n a l source given by (2.1) i s p a r a l l e l to the edge of an i n f i n i t e l y t h i n p e r f e c t l y conducting h a l f - p l a n e i n y=0, x>0 as shown i n F i g . 3.1. I f k(r+r o)»l, the t o t a l f i e l d at any point r , Q f a r from the edge (r»r Q) i s given e x a c t l y as - j ( k r - T i / 4 ) j k r cos(9-0 ) 0-0 e { e ° ° F[-/2kr cos( ° ) ] U(r,0) = ~ 2 j k r cos(0+0 ) 0+0 -e ° ° F[-/2k7 cos( 2.)]} (3.1) ° 2 where F[w] - / e dx (3.2) w i s the F r e s n e l i n t e g r a l (see Appendix A). By making the coordinates ( r 0>® 0) °^ t n e source, complex ( r g , 0 g ) as i n (2.5) and (2.6), the o m n i d i r e c t i o n a l source becomes a d i r e c t i v e beam and the s o l u t i o n i n (3.1) i s s t i l l v a l i d w i t h r , 0 r e p l a c i n g r , 0 . s s o o - 28 - The t o t a l f a r f i e l d of a beam d i f f r a c t e d by conducting half-screen for normal and non-normal incidence i s given as 0-9 -j(kr-nM) j k r s cos(9-9 s) ^ — e i e L s „ U(r,9) = (3.3) /itkr jkr cos(9+9 ) 0+0 -e S S F[-/2kT cos( — ) ] } s 2 where F[w] i s the complex Fresnel i n t e g r a l . The Fresnel i n t e g r a l s provide values f i n i t e and continuous across the shadow and r e f l e c t i o n boundaries of the source and half-plane. In the asymptotic s o l u t i o n given by Green et a l (1979) the f i e l d i s singular along the boundaries when the beam axis h i t s the d i f f r a c t i n g edge and i s inaccurate i n the neighbourhood of the shadow boundaries. E f f i c i e n t computer subroutines are a v a i l a b l e for c a l c u l a t i n g the Fresnel i n t e g r a l s i n terms of error functions with the complex arguments (see Appendix A). 3.2 Shadow and Reflection Boundaries Far from the half-screen edge simple expressions for the shadow and r e f l e c t i o n boundaries are to be derived here analogous to those of Green et a l (1979). These expressions are simpler and more accurate because only the r e a l part of r g need be calculated, whereas before r e a l part, imaginary part and absolute value of r and r e a l and imaginary parts of s 9 were used. r and 0 are the source complex coordinates. These s s s boundaries i n general are not st r a i g h t l i n e s . They depend on the r e l a t i v e p o s i t i o n of the half-screen edge with respect to the beam axis - 29 - and the source. The shadow and r e f l e c t i o n boundaries are s t r a i g h t l i n e s and c o i n c i d e w i t h those of the source when i t s coordinates are r e a l , only when the beam a x i s passes through the edge of the h a l f - s c r e e n . Using the Green et a l . (1979) d e f i n i t i o n of shadow and r e f l e c t i o n boundaries, we have R e a l ( w ^ / 4 ) = 0 (3.4) where w i s the F r e s n e l i n t e g r a l argument of (3.3) given as 9 + 0 w. = -/2kT cos( 1) (3.5) 2 r L where the s u b s c r i p t s i , r r e f e r to incidence and r e f l e c t i o n . (3.4) i s s a t i s f i e d i f JTt/4 Imag[(we ) 2 ] = 0 (3.6) and JTt/4 Real[(we ) 2 ] _< 0 (3.7) L e t t i n g r = R - j l as i n Appendix B and using the i d e n t i t i e s given by s (2.8c), i t i s easy to show that jn/4 V.wc >' 2 ~ Hence ( e Y = j k [ R + r Q C O S ( 0 + O Q ) ] + k[I+b c o s ( 0 + B ) ] (3.8) jit/4 Imag[(we ) 2 ] = k[R+r c o s ( 0 + 0 ) ] (3.9) and jit/4 Real[(we ) 2 ] = k[I+b c o s ( 0 + 8 ) ] (3.10) S u b s t i t u t i n g (3.9) i n (3.6) and s o l v i n g f o r 0 at the shadow and r e f l e c t i o n boundaries, i . e . 0 = 0 . and 0 , gives ' s i s r ' ° - 30 - " I _ R 9 s i = ± % + c o s ( _ ) (3.11) s r o sin c e R=Real(r ) and r are r e a l and p o s i t i v e we can w r i t e s o r -1 R G s i = - °o + [ % - C O S ( F ~ ) ] ( 3 ' 1 2 ) s r ° or the shadow boundary p o s i t i o n i s _ 1 R < 0 = 0 + [-re + cos ( _ ) ] ; B 2. 0 + n (3.13) s i o 1 — r / J ' T o v / o and the r e f l e c t i o n boundary p o s i t i o n i s " I R 0 g r = -eQ + [ii T cos (1-)] ; B < 0 q + n (3.14) o > From (3.13) and (3.14) we can see the symmetry of shadow and r e f l e c t i o n boundaries w i t h respect to the h a l f - s c r e e n . This property i s v a l i d f o r o m n i d i r e c t i o n a l and d i r e c t i v e sources as w e l l . Adding (3.13) and (3.14) gives 0 + 0 = 2n , f o r a l l B (3.15) s i s r To s a t i s f y (3.7), s u b s t i t u t e f o r R from (3.9) and (3.6) i n (B.6) of Appendix B y i e l d i n g cos ( P-0 ) I = -b — (3.16) cos(0+0 ) — o Expanding cos(B-0o) i n terms of cosines and sines of (0+P) and (0+0o) and s u b s t i t u t i n g i n (3.10) we get JTI/4 Real[(we ) 2 ] = -kb sin(0+B). tan(0+€>o ) (3.17a) From equation (3.13) and Appendix B we can show - 31 - IT < 0 - 0 < — i f 0 < 0 - 6 < it (3.17b) s 1 o 2 s l Hence tan(0 -0 )>0 and s i n ( 0 -8)>0 s i o s i Also we can show 1 < B - Q < it i f - * < 0 ,-B < 0 (3.17c) 2 s i o 2 s i Hence, tan(0 - 0 ) < 0 and s i n ( 0 -B) < 0. ' s i o s i Therefore (3.7) i s s a t i s f i e d f o r the shadow boundary. S i m i l a r l y from (3.14) and Appendix B we can s a t i s f y (3.7). As a check on the above f o r m u l a s f o r 0 . and 0 , l e t us discus s s i s r the f o l l o w i n g s p e c i a l cases. i ) Real l i n e source i . e . b=0 from (3.10) R=r , and o' 0 , = it + 0 (3.18) s i — o v ' sr i i ) Beam a x i s passes through the screen edge 8 = 0 + it o from (3.13), r = r + jb and v ' s o J 0 , = TC + 0 , s i — o' sr which i s the same as the r e a l l i n e source shadow and r e f l e c t i o n boundaries. 3.3 Numerical Results for the Half-Plane The s o l i d curves i n F i g . 3.2 and F i g . 3.3 represent the uniform t o t a l f i e l d c a l c u l a t e d from (3.3) w i t h k r =16 and 0 =n/2 w h i l e the o o - 32 - dashed curves represent the asymptotic t o t a l f i e l d c a l c u l a t e d from Appendix (C.5). The two dimensional beam i s normally i n c i d e n t upon the h a l f p l a n e and at a distance krQ=16. The development of a beam from an o m n i d i r e c t i o n a l l i n e source (kb=0) to a d i r e c t i v e beam (kb=12) i s shown i n F i g . 3.2. For the case of kb=0 the pa t t e r n o s c i l l a t i o n s i n the i l l u m i n a t e d region (-,rc/2<<K0) are f a m i l i a r , showing i n t e r f e r e n c e between the d i r e c t wave from the source and a d i f f r a c t e d wave emanating from the edge. In the shadow region (0<<KV2) there i s only a d i f f r a c t e d f i e l d , which decreases w i t h <t> to. become zero on the conductor. As kb increases the above o s c i l l a t i o n s are suppressed i n the i l l u m i n a t e d r e g i o n . This occurs because the i n c i d e n t f i e l d i s suppressed i n the i l l u m i n a t e d r e gion as d i r e c t i v i t y i n c r e a s e s . In F i g . 3.2, where the beam a x i s passes through the edge, we can see how inaccurate the asymptotic s o l u t i o n becomes near the shadow boundary. When kb increases there i s l i t t l e improvement. On the shadow boundary the asymptotic f i e l d i s s i n g u l a r f o r a l l values of kb i n F i g . 3.2. In F i g . 3.3 where the beam a x i s i s o f f the edge by an angle 6, the asympototic s o l u t i o n i s f i n i t e but inaccurate near and on the shadow boundary e s p e c i a l l y f o r small kb or 6. In F i g . 3.3 the asymptotic s o l u t i o n improved g r e a t l y when the o f f edge angle increased from 15° to 45° f o r a f i x e d kb=12. In the lower graphs of F i g . 3.3 where the o f f edge angle i s f i x e d to 6=30° , the asymptotic s o l u t i o n improved considerably when kb increased from kb=4 to kb=16. From the above we can conclude that the asymptotic s o l u t i o n i s a good approximation to the uniform s o l u t i o n whenever the beam a x i s i s w e l l o f f the d i f f r a c t i n g edge - 33 - and the beam i s s u f f i c i e n t l y d i r e c t i v e , i . e . when 6 and kb are s u f f i c i e n t l y large, for then there i s l i t t l e d i f f r a c t i o n by the edge. F i g . 3.4 shows the far f i e l d s for half plane d i f f r a c t i o n by a distant l i n e source represented by a s o l i d curve and by a narrow beam source r e p r e s e n t e d by a dashed curve. For b>>r Q the wave uniformly illuminates the d i f f r a c t i n g edge and i t s neighbourhood and i n a s i m i l a r way the distant l i n e source also does. So i n F i g . 3.4b we can see the d i f f r a c t e d f i e l d components are s i m i l a r . In F i g . 3.4a the t o t a l f i e l d s are d i f f e r e n t i n the illuminated region (-n/2<<K0), because f a r from the source the d i r e c t incident wave of the narrow beam i s almost zero a few degrees o ff the beam axis, so the d i f f r a c t e d f i e l d i s e s s e n t i a l l y the t o t a l f i e l d . The d i r e c t wave of the distant l i n e source i s almost uniform, consequently interference between the d i r e c t wave and the d i f f r a c t e d wave occurs and appears as o s c i l l a t i o n s i n the illuminated region. To obtain numerical values from (3.3) i t i s necessary to have a Fresnel i n t e g r a l subroutine that can handle complex arguments. A r e l a t i o n between the Fresnel i n t e g r a l and the error function i s given i n Appendix A. Subroutines for the error function with complex argument are a v a i l a b l e i n the UBC computing center General L i b r a r y . Also tables of the error function with complex arguments by Gautschi (1964) and tables of the modified Fresnel i n t e g r a l with complex arguments by Clemmow and Munford (1952) agree with our subroutine whenever comparison i s possible. -34- Line Source Beam Axis F i g . 3.1 a) Geometry of a complex l ine source diffract ion by a half-plane b) Beam orientation with respect to the edge.   FTg. 3.4 Uniform s o l u t i o n (eq. 3.3) comparison of a d i s t a n t source ( s o l i d ) and a l i m i t i n g beam of l a r g e kb kb=o , k r =85 , 6 =90° kb=85 k r =8 , 8 =90° , B=270 o o - 38 - CHAPTER IV BEAM DIFFRACTION BY A WIDE SLIT AND COMPLEMENTARY STRIP 4.1 Beam Diffraction by a Wide Slit With the r e s u l t s f o r beam d i f f r a c t i o n by a h a l f - p l a n e we can solve the problem of beam d i f f r a c t i o n by a s l i t i n a conducting plane and i t s complement, a conducting s t r i p . The s l i t between two coplanar h a l f - p l a n e s w i t h p a r a l l e l edges, or i t s complement, the s t r i p , are a t r a d i t i o n a l t e s t of the o r i e s i n v o l v i n g m u l t i p l e d i f f r a c t i o n by edges. 4.1.1 Far Field Calculation F i g . 4.1 shows a l i n e source p a r a l l e l to a s l i t i n y=0, |x|<_ a. With an i n c i d e n t f i e l d given by (2.1) the t o t a l f a r f i e l d of the ha l f - p l a n e on the r i g h t side i n i s o l a t i o n U i ( r l f Q ^ ) i s given by (3.3) w i t h r l f 0 i r e p l a c i n g r , 0 and r 0 1 r e p l a c i n g r , 0 . S J. S i . s s S i m i l a r l y the t o t a l f a r f i e l d of the l e f t h a l f - p l a n e i n i s o l a t i o n U 2 ( r 2 , 0 2 ) i s g i v e n by (3.3) w i t h r 2 , 0 2 r e p l a c i n g r , 0 and r ^ . © ^ r e p l a c i n g r , 0 . A l l the coordinates are shown i n F i g . 4.1. 8 S These expressions f o r the f i e l d s of the two ha l f - p l a n e s c o n t a i n both i n c i d e n t and d i f f r a c t e d f i e l d s behind the s l i t . Consequently the t o t a l n o n - i n t e r a c t i o n f a r f i e l d s f o r the s l i t are t h e i r sum l e s s an i n c i d e n t f i e l d U 1. u s = Ui(ri,©i) + U 2 ( r 2 , 0 2 ) - U 1 (4.1) In the f a r f i e l d of the s l i t ( r » a) r^ = r - a cos0 1 , 0 1 = 0 r 2 - r - a cos©2 , 0 2 = n - 0 ; 0 < 0 < it (4 .2a) = 3n - 9 ; n < 0 < 2 i x - 39 - These f a r f i e l d s ubstitutions for r j , r 2 are used i n the exponential terms of U 1 and U 2 while r^ - r 2 - r i s used i n the amplitude terms. From F i g . 4.1 we can write the following geometrical r e l a t i o n s for r s l , r „, 0 , 0 9 , measured from the two edges i n terms of r , 0 . r = [ r 2 + a 2 T 2ar cos© ] 1 / 2 ; Real(r ) > 0 , (4.2b) S i s s s si. 2 2 a r sin0 s, - * _ s i n " 1 ( * 1) , (4.2c) 2 4 where r and 0 are measured from the centre of the s l i t and given by s s (2.5) and (2.6) r e s p e c t i v e l y . We may use the Fresnel i n t e g r a l i d e n t i t y . _jn/4 F[-w] = /if e - F[w] (4.3) to include the extra incident f i e l d - j k [ r i _ r g l c o s ( 0 1 _ 0 s l ) ] - j k [ r _ r s cos(©_0 s)] U 1 = — = -1- (4.4) /kr /kT i n a Fresnel i n t e g r a l te rm. Then the non_interaction f a r f i e l d s of the s l i t are - 40 - -j(kr-n/4) u t s jka cos9, j k r , cos(9,_9 ,) e 1 {" e S l 1 3 1 F [ - w n ] jkr^cosCGi+e^) F [ w r l ] } e j k a cos9 2 j k r „ c o s ( 9 2 _ 9 ) + e { e S l S l F [ w i 2 ] j k r c o s ( 9 2 + 9 s 2 ) - e F t w r 2 ^ ̂  ^ ^ - e where . 0 X T 9 w±1 = - ^ Z k r a l cos( —) (4.6) r l 2 and s i m i l a r l y f o r w^ w i t h s u b s c r i p t 2 r e p l a c i n g 1 i n (4.6). r2 This i s an accurate s o l u t i o n f o r s l i t s s u f f i c i e n t l y wide that i n t e r a c t i o n between edges i s n e g l i g i b l e . In order to i n d i c a t e the accuracy i t i s u s e f u l to i n c l u d e a l s o i n t e r a c t i o n between the s l i t edges. E a r l i e r r e s u l t s f o r plane wave incidence using the geometrical theory of d i f f r a c t i o n show that s i n g l e and double d i f f r a c t i o n provide accurate r e s u l t s f o r s l i t widths ka > 2 ( K e l l e r (1957), F i g . 9). 4.1.2 M u l t i p l e D i f f r a c t i o n C a l c u l a t i o n To include higher order i n t e r a c t i o n s between the edges the f i e l d s i n g l y d i f f r a c t e d from each edge i n the d i r e c t i o n of the opposite edge i s replaced by the f i e l d of a l i n e source of equal amplitude l o c a t e d at the edge from which the s i n g l y d i f f r a c t e d f i e l d o r i g i n a t e s . For example - 41 - the doubly diffracted f i e ld from the left edge is produced by the singly diffracted f ie ld from the right edge in the 0^=71 direction and vice-versa for the right edge. This can be repeated inf in i te ly many times. Adding a l l contributions from the two edges gives the multiply diffracted fields of the s l i t . The singly diffracted component of the f i e ld given by (3.3) can be written as U d = U* D(0 r O, r) , (4.7) where i s the incident f i e ld calculated at the diffracting edge given by - j k r J s i e U = , Ikr I » 1 (4.8) e AT 5 s and D(0 , r , 0, r) is the dif fract ion coefficient of the edge s s D(6.. V e. r , . ^ f ' 4 , .fl . fl , 1^ ( 4. 9 ) where ŵ  and w f are given by (4.6). Following the same procedure given by J u l l (1981, p.91) in calculating multiple di f fract ion for plane wave incidence on a s l i t , the multiply diffracted f i e ld excluding single di f fract ion, can be written as U e x D l ( Do D l + D 2 > + Ue2 D 2 ( D 0 D 2 + D l ) U d = I (4.10) (1 - D 2) o - 42 - I f the l i n e source coordinates are on the s l i t a x i s ( y - a x i s ) , i . e . , the s o u r c e i s symmetric w i t h r e s p e c t to both edges of the s l i t . Then r .. = r „ and 9 = 9 - consequently U i = and D* = D 2. Then (4.10) can be s i m p l i f i e d to U ^ D J (Df + D 2) ua = — I t (4.11) ( i - »„> where U^, i s given by (4.8) w i t h r , replaces r and e l s l s D Q = D(T C, 2a, it, 2a) j(2ka+it/4) = - /47TT . e . F[/4ka] (4.12) D l " D ( 9 s l ' r s l ' *» 2 a ) j ( - 2 k a + it/4) - j k r s l c o s 9 g l j k r g l =-/2kr n/ixka e sl Df = D(it, 2a, 9 ^ ) F[/2kr" sin(M)] (4.13) s i 2 j ( 2 k a + n/4) - j k a c o s 9 1 9 j k r =-/8ka/Ti . e . e . F[/4ka s i n ( — ) ] . - — (4.14) 2 Jkr While D | i s given by (4.14) w i t h r 2 , 9 2 r e p l a c i n g r ^ 9^ For a wide range of s l i t widths (2a), higher order i n t e r a c t i o n f i e l d s of the s l i t give l i t t l e or no improvement i n accuracy over the f i r s t order i n t e r a c t i o n . In p r a c t i c e i t i s s u f f i c i e n t to i n c l u d e the f i r s t order i n t e r a c t i o n only. That i s p a r t l y because the above higher - 43 - order i n t e r a c t i o n c a l c u l a t i o n s are approximate, f o r the edge d i f f r a c t e d f i e l d s are not o m n i d i r e c t i o n a l as assumed. I t i s a l s o because higher order i n t e r a c t i o n i s weak except f o r narrow s l i t s , f o r which the whole process diverges and a d i f f e r e n t method Is r e q u i r e d . Now the t o t a l f i e l d i n c l u d i n g m u l t i p l e d i f f r a c t i o n i s given by U t = + U d , (4.15) m s m where i s g i v e n by (4.5) and U d i s g i v e n by (4.10). The t o t a l f a r f i e l d given by (4.15) i s continuous and f r e e from shadow boundary s i n g u l a r i t i e s because the F r e s n e l i n t e g r a l s are r e t a i n e d . 4.1.3 Numerical Results for the Slit The d i f f r a c t i o n patterns of F i g . 4.2 are c a l c u l a t e d f o r a l i n e source p a r a l l e l to and at a height k r Q = 8 above a s l i t of width 2ka=16. The s o l i d curves are the n o n - i n t e r a c t i o n d i f f r a c t i o n f i e l d s c a l c u l a t e d from (4.5) and the dashed curves c a l c u l a t e d from (4.15) i n c l u d e higher order i n t e r a c t i o n between the edges of the s l i t . C l e a r l y i n t e r a c t i o n f i e l d s are of minor importance f o r t h i s width of s l i t . For an o m n i d i r e c t i o n a l source (kb=0) the i n c i d e n t f i e l d has a s u b s t a n t i a l symmetric phase v a r i a t i o n across the aperture r e s u l t i n g i n a broad main beam w i t h high shoulders. As the source becomes d i r e c t i v e , beam d e f i n i t i o n improves. For moderate source d i r e c t i v i t y (e.g., kb=8) the aperture i l l u m i n a t i o n i s e s s e n t i a l l y Gaussian and so i s the p a t t e r n . For a very d i r e c t i v e source (kb=85) the aperture i l l u m i n a t i o n i s e s s e n t i a l l y plane wave and the d i f f r a c t i o n p a t t e r n i s very l i k e that f o r plane wave i l l u m i n a t i o n of a s l i t . F i g . 4.3 compares r e s u l t s from - 44 - K e l l e r ' s geometrical theory of d i f f r a c t i o n (1957, F i g . 7) and beam d i f f r a c t i o n f o r kb=85. The s i n g l y d i f f r a c t e d f i e l d patterns are almost i d e n t i c a l , but the i n t e r a c t i o n f i e l d s d i f f e r . K e l l e r ' s m u l t i p l e d i f f r a c t i o n f i e l d i s a summation of a l l f i e l d s r e s u l t i n g from the f i r s t term In an asymptotic expansion of the F r e s n e l i n t e g r a l ; higher order F r e s n e l i n t e g r a l asymptotic expansion terms are omitted. I t i s a l s o s i n g u l a r at shadow boundaries of the d i f f r a c t e d f i e l d , as i s evident h e re at <t>=-90°. These are l i m i t a t i o n s of the g e o m e t r i c a l theory of d i f f r a c t i o n .  -46- - 9 0 -45 0 45 * - 90 -45 0 45 90° . 4.2 Normalized t o t a l f i e l d p a t t e r n of a beam d i f f r a c t i o n by a s l i t , s i n g l e ( s o l i d ) and double (dashed) d i f f r a c t i o n s (ka=8=kr , 9 =90° , 13=270°) 0 .8 - 0 .6 - 0.4- 0 .2 - 0.0 (a) L i m i t i n g beam incidence (kr =8, kb=85) o (b) Plane wave incidence ( K e l l e r , 1957) F i g . 4.3 Comparison of a plane wave and a l i m i t i n g beam d i f f r a c t i o n by a s l i t (ka=8) ( s o l i d ) s i n g l e d i f f r a c t i o n and (dashed) m u l t i p l e d i f f r a c t i o n - 48 - 4.2 Beam D i f f r a c t i o n by a Wide Conducting S t r i p As another a p p l i c a t i o n of l i n e source d i f f r a c t i o n by a h a l f - p l a n e i s the l i n e source d i f f r a c t i o n by a conducting s t r i p . F i g . 4.4 shows a l i n e source above and p a r a l l e l to a conducting s t r i p i n the y=o plane and |x|<a. 4.2.1 Far F i e l d C a l c u l a t i o n I n the f a r f i e l d of the s t r i p (r»r Q,a) we use the approximation given by (4.2). The s i n g l y d i f f r a c t e d t o t a l f i e l d of a l i n e source above a s t r i p can be c a l c u l a t e d from the t o t a l f i e l d of a l i n e source over a h a l f - p l a n e f o r each of the edges. This t o t a l f i e l d U C may be w r i t t e n s u ! = UlCri.Oj.) + U 2 ( r 2 , e 2 ) - [ U i ( r , 0 ) + U r ( r , 0 ) ] ; O<0<n = U ^ r ^ © ^ + U 2 ( r 2 , 0 2 ) ; n<0<2n (4.16) where i s t o t a l f i e l d of the l i n e source over a h a l f _ p l a n e at y=0 and x>-a and U 2 i s the t o t a l f i e l d of the same l i n e source over a h a l f - p l a n e at y=0 and x<a. U 1 ( r 1 , 0 1 ) and U 2(r 2,© 2) are given by (3.3) w i t h ri,®i and r 2 , 0 2 r e p l a c i n g r,0, r e s p e c t i v e l y . While U* and U r are the i n c i d e n t and r e f l e c t e d f i e l d s , r e s p e c t i v e l y . U* i s given by (4.4) and - J k [ r i _ r cos(0 1+© s l)] - j k [ r - r g cos(0+0 g)] e e U r = - = - (4.17) / k T / k r " Using (4.3), the s i n g l y d i f f r a c t e d t o t a l f i e l d given by (4.16) can be s i m p l i f i e d to - 49 - For O<0<u - j ( k r - n 7 4 ) j k a c o s 0 1 j k r 1cos(e 1 _ 0 ) {e (- e 8 1 8 1 F[-w 1 TJ = , s /ixkr" j k r s l c o s ( 0 1 + 0 s l ) j k a cos0 2 j k r cos( 0 2 - 0 7 ) + e (e S / S Z F [ w i 2 ] j k r cos(0 2+0 ) - e F t w r 2 ] ^ » ( 4* 1 8 a> and f o r it<0<2"n; -j(kr--n:/4) j k a cos0 1 j k r . cos(Oi_0 ..) t _ e {e (e 8 1 F [ w ± 1 ] U s = ^ k 7 J k r s l - s ( 0 1 + 0 s l ) " e F [ w r l ] J j k a cos©2 j k r „ cos(0 2-0 9 ) + e (e S / S l F[w. 2] j k r cos(0 2+0 ) - e S Z F [ w r 2 ] ) } (4.18b) Where ŵ  and w^are the F r e s n e l i n t e g r a l arguments given by (4.6). I f i n t e r a c t i o n f i e l d s of the s t r i p edges are c a l c u l a t e d i n a s i m i l a r way as f o r the i n t e r a c t i o n f i e l d s of the s l i t i t i s found that they vanish on the conducting s t r i p . Consequently a new d i f f r a c t i o n c o e f f i c i e n t i s re q u i r e d (e.g. Karp and K e l l e r , 1961) based on the normal d e r i v a t i v e of the d i f f r a c t e d f i e l d s i n the d i r e c t i o n of the edge opposite. These i n t e r a c t i o n f i e l d s are much weaker than f o r the s l i t and so are omitted here. - 50 - 4.2.2 Numerical Results for the Strip The f i e l d p a t t e r n shown i n F i g s . 4.5 and 4.6 are c a l c u l a t e d from (4.18) a f t e r n o r m a l i z a t i o n to the f i e l d on the peak of the p a t t e r n . F i g . 4.5 i l l u s t r a t e s the development of a beam s o l u t i o n from an o m n i d i r e c t i o n a l l i n e s o u rce (kb=0) at kr Q=8, Q^=n/2 above a s t r i p of width 2ka=16, to a beam source (kb=12) perpendicular to the s t r i p . The t o t a l f a r f i e l d i s maximum i n the i l l u m i n a t e d region at G=it/2 where the r e f l e c t e d f i e l d i s combined w i t h the d i f f r a c t e d f i e l d s from both edges. The d i f f r a c t e d f i e l d s from the two edges at the f i e l d point 9=n/2 or 3%/2 add i n phase. In the shadow region at 0=3TC/2 the t o t a l f i e l d i s a r e l a t i v e maximum. As kb increases from 0 to 12 the i n c i d e n t beam becomes narrower and the edges are l e s s i l l u m i n a t e d . So the t o t a l f i e l d behind the s t r i p , which i s mainly the d i f f r a c t e d f i e l d s from the edges, decreases. The t o t a l f i e l d i n the i l l u m i n a t e d region becomes more d i r e c t i v e w i t h fewer sidelobes because there i s l i t t l e i n t e r a c t i o n between the r e f l e c t e d f i e l d and the d i f f r a c t e d f i e l d s . Now i f the i n c i d e n t beam of kb=8 i s o f f the perpendicular to the s t r i p by an angle 6, as shown i n F i g . 4.6, the t o t a l f i e l d p a t t e r n i s t i l t e d . As 6 increases the t o t a l f i e l d p a t t e r n Is t i l t e d more to the same si d e of the i n c i d e n t beam and i t becomes l a r g e r behind the s t r i p as shown. The d i f f r a c t e d f i e l d c o n t r i b u t i o n from the r i g h t edge i s s m a l l and the main c o n t r i b u t i o n i s from the l e f t edge and the i n c i d e n t f i e l d . The d i f f r a c t e d f i e l d from the l e f t edge i s a maximum when the i n c i d e n t beam a x i s i s d i r e c t e d at the l e f t edge as shown i n F i g . 4.6 f o r the case of 6=45°. - 51 - In the lower r i g h t graph of F i g . 4.6, the s t r i p width i s increased to 2ka=160 as a l i m i t of a plane screen. The r e s u l t i n g t o t a l f i e l d p a t t e r n i s a t i l t e d sharp beam making an angle 135° w i t h the s t r i p , and i s simply the r e f l e c t e d f i e l d .   -54 - - 55 - CHAPTER V BEAM DIFFRACTION BY A CONDUCTING WEDGE A p e r f e c t l y conducting wedge of e x t e r i o r angle n t i s i l l u m i n a t e d by a l i n e source at r Q , 0 q p a r a l l e l to the edge, as shown i n F i g . 5.1. For t h i s c o n f i g u r a t i o n , l e t us have the f o l l o w i n g l i m i t a t i o n s : OO ; (5.1a) ° 2 0<9<n7t ; (5.1b) 1.5<n_<2 or 1 > a > 0 , (5.1c) 2 W where a i s the i n t e r i o r wedge angle. S i n c e 9 = nTt/2 i s the l i n e of w symmetry of the above c o n f i g u r a t i o n , the s o l u t i o n f o r 0<QQ<rm/2 w i t h 9 measured from the upper s u r f a c e , i s the same as that f o r nit/2<9 o<mt when 0 i s measured from the lower wedge surface. Therefore 9 = nTt/2 i s sym c a l l e d angle of symmetry. 5.1 Real Line Source Solution E x t e r i o r to the wedge, the t o t a l f a r f i e l d Ufc i s given as U* = U 1. S ( 9 s i - 9 ) + U* S ( 9 g r l - 9 ) + U 2 S ( 9 _ 9 g r 2 ) + U d (5.2) Here S ( x ) i s the u n i t step f u n c t i o n , 9 9 and 9 are the shadow v ' f » s i ' s r l sr2 boundary, r e f l e c t i o n boundary f o r the upper surface and r e f l e c t i o n boundary f o r the lower s u r f a c e , r e s p e c t i v e l y . A l l the boundary angles are measured from the upper surface of the wedge and are given by 9 = it + 9 ; (5.3a) - 56 - °srl " % ~ e o 5 < 5' 3 b> 9 s r 2 = ( 2 n ~ 1 ) 7 t " 0 o > ( 5- 3 c> i d. T r U/, U , U^, and U 2 are the i n c i d e n t , d i f f r a c t e d and r e f l e c t e d f i e l d s from the upper and lower s u r f a c e s , r e s p e c t i v e l y . —jkR^" U 1 = /ix7T H ^ ^ k R 1 ) = — ; kR 1 » 1 (5.4a) 4 d T j k r cos(0 - e ) - j k r e 0 ° 1 ; r » r (5.4b) /kT / O N j ^ r cos(6+0 ) - j k r UJ - - H<2>(kR*) - -e ° 0 e _ /kr (5.5) The d i f f r a c t e d f i e l d i s j k r ^ cos[2nit - ( O + 0 J ] - j k r e (5.6) /kr - j k r U d = U* . D ( 0 Q , r o , 0 ) JL— , (5.7) /kr where u*e i s the i n c i d e n t f i e l d at the edge of the wedge given by = /^72 H< 2 ) ( k r Q ) (5.8a) - j k r Q * — — ; k r » 1 (5.8b) /kT ° o and D ( Q 0 > r Q , 0) i s the u n i f o r m d i f f r a c t i o n c o e f f i c i e n t g i v e n by Kouyoumjian and Pathak (1974), w i t h some m o d i f i c a t i o n s , as - 57 - -JTc/4 D(G r 0) - { [cotdi ) G(w ) - cot(T 2 ) G(w2)] ° 2n/2T i + [cot(T 3) G(w3) - cotCL\) G(W L ()]}, (5.9) Here ( jw2 jic/4 G(w) = 2j e w F[w] ; Real(we ) > 0 (5.10a) j w 2 JTE/4 = - 2 j e w F[-w] ; Real(we ) < 0 (5.10b) F[w] i s the F r e s n e l i n t e g r a l given by (3.2), and 2TcnM* - (0 + 0 ) w. , = - / 2 k T cos[ 1 —] (5.11a) 2TC n M* - (0 + 0 ) w3 4 - -» / i 2kr 0 cos[ - —] (5.11b) TC - (0 T 0 ) T. . = ° _ (5.12a) i , z 2n TC + (0 T 0 ) T- , = — (5.12b) J ' * 2n T T In (5 .11) Hi and M 2 are integers which most nearly sa t i s fy the equations. 2TC n - (0 T 9 ) = -TC (5.13a) 2TC n - (0 T 0 ) = TC (5.13b) Here we have two cases, depending on whether the reflected f i e ld from the lower wedge surface ex is ts or does not, i . e . 0 > 0 , where the & ' o < cr ' c r i t i c a l angle for illumination of the lower wedge surface is 0 =(n-l)Tt. cr v ' - 58 - Now the total f i e ld u f c can be written as, for O<0 <0 ; (5.14) o cr u C = u d + u 1 + u£ , ° < 0 < 0 s r l d , i A = u + u * srl<0<0 . (5.15) s i d , 0 X0<mt = u s i = 0 , tin <0<2it and for 0 <0 <0 (5.16) cr— o sym ; u C - u d + u 1 + ui , 0 < 9 < Q s r l " u < 1 + u l > 0 s r l < 9 < Q s r 2 ( 5 ' 1 7 ) , = u d + u 1 + uT2 , © s r 2 < 0 <nn = 0 , nix<0<2ix 5.2 Uniform Solution for a Beam Source To get the two-dimensional beam solution from the omnidrectional l ine source solution, we replace r , 0 by r , 0 in a l l above equations o o s s except (5.1), (5.3), (5.13), (5.14) and (5.16). In (5.1), 0 q i s left as i t i s and in (5.13), 0 is replaced by R e a l ( 0 ) . (5.14) and (5.16) are o s replaced, respectively, by 0 . > mi ' (5.18) sr2 < - 59 - F i n a l l y (5.3) i s given by (3.13) and (3.14) and r e w r i t t e n here as 0 = 0 + [TC T c o s - 1 ( R / r ) , B +TC (5.19) s i o o s o v / Q s r l = _ 0 o + [ 7 t - c o s " 1 ( R / r o ) ] ' P ^ ( 0 o + T t ) » ( 5 , 2 0 ) 9 s r 2 = ~ ( n 7 t " 9 o ) + [ % T C O S ~ l ( R / r o ) ] ' P T ( 0 o + T l ) » ( 5 , 2 1 ) where r and 0 are given by (2.5) and (2.6) r e s p e c t i v e l y . s s I f the edge of the wedge l i e s on the beam a x i s , r and 0 a r e s s s i m p l i f i e d to r = r + jb and 0 = 0 , (5.22) s o J s o' v ' and one of the cotangent f u n c t i o n s i n (5.9) i s s i n g u l a r on the shadow or r e f l e c t i o n boundaries, but when m u l t i p l i e d by the corresponding G(w) f u n c t i o n , i t becomes f i n i t e (see Appendix D). Hence the d i f f r a c t i o n c o e f f i c i e n t given by (5.9) i s always f i n i t e , u n l i k e the asymptotic s o l u t i o n s . When the beam a x i s does not pass through the wedge edge, a l l the cotangent f u n c t i o n s are f i n i t e everywhere. 5*3 Numerical R e s u l t s f o r the Wedge In a l l the f o l l o w i n g f i g u r e s the source Is p a r a l l e l to the edge and at a d i s t a n c e k r Q = 16. A l s o the edge l i e s on the beam a x i s ; i . e . 6 = 0 +TC. o In F i g s . 5.2, 5.3 and 5.4 the wedge a n g l e a = ( 2 - n ) i t i s k e p t w c o n s t a n t a t a = 90° f o r which n=1.5. T h e r e f o r e the c r i t i c a l angle w 0 =90° and the angle of symmetry 0 = 135°. cr o J J S V M F i g . 5.2 i l l u s t r a t e s how the normalized t o t a l f i e l d v a r i e s when the i n c i d e n t f i e l d on a r i g h t angled conducting wedge changes from - 60 - o m n i d i r e c t i o n a l (kb=0) to a d i r e c t i v e beam (kb=12). As kb increases two beams appear, one a l o n g the r e f l e c t i o n boundary at © g r^ = 120°, and a n o t h e r a l o n g t h e shadow boundary at 0 g^ = 240°. I n the r e g i o n i l l u m i n a t e d by the i n c i d e n t and r e f l e c t e d f i e l d s c o n s i d e r a b l e c o n s t r u c t i v e and d e s t r u c t i v e i n t e r f e r e n c e between i n c i d e n t and r e f l e c t e d f i e l d s i s observed when kb=0. As kb increases t h i s I n t e r f e r e n c e decreases because i n c i d e n t and r e f l e c t e d f i e l d s become d i r e c t i v e . The d i f f r a c t e d f i e l d does not change s i g n i f i c a n t l y because the edges l i e on the beam a x i s . Since the i n c i d e n t angle 0 = 60° i s l e s s than 0 =90°, o c r ' there i s no r e f l e c t e d f i e l d from the lower wedge surface. In the shadow reg i o n (240 o<9<270°) there i s only a d i f f r a c t e d f i e l d , which vanishes on the lower wedge surface. F i g . 5.3 i s s i m i l a r to F i g . 5.2 except the angle of incidence 0 =120° i s greater than 0 so both faces of the wedge are i l l u m i n a t e d , o c r In t h i s case the r e f l e c t e d f i e l d s from both wedge surfaces c o n t r i b u t e to the t o t a l f i e l d . I n t e r f e r e n c e between 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 occurs i n O<0<0 . = 60° and 27O°<0<0 _ = 240°, a l s o s r l sr2 ' between i n c i d e n t and d i f f r a c t e d f i e l d s i n the region 6O°<0<24OO. Here a l l the region e x t e r i o r to the wedge i s i l l u m i n a t e d by i n c i d e n t , d i f f r a c t e d and f o r 0<6O° and 0>24O° r e f l e c t e d f i e l d s , so there i s no shadow regions when kb i s s m a l l . When kb i s s u f f i c i e n t l y l a r g e , shadow regions may e x i s t . I n F i g . 5.4 the angle of incidence i s chosen to equal the angle of symmetry; I.e. 0 q = © s v m = 135°. The arrangement was used as a p a r t i a l v e r i f i c a t i o n of the v a l i d i t y of our equations and computer programs i n t h i s a n a l y s i s . The symmetry of the f i e l d about 0=135° i s c l e a r . When - 61 - kb i s s u f f i c i e n t l y l a r g e , say kb=12, two d i r e c t i v e beams appear at G=9 = 4 5 ° and 0 0 = 225°. These are the r e f l e c t e d f i e l d s from the s r l s r z upper and lower wedge surfaces r e s p e c t i v e l y . But when kb=0 or i s s m a l l , say kb=2, the In t e r f e r e n c e between r e f l e c t e d , i n c i d e n t and d i f f r a c t e d f i e l d s are more s i g n i f i c a n t i n the regions O<0<45° and 225°<0<27O O. Most of the t o t a l f a r f i e l d i n the region 45°<0<225° when kb i s s m a l l , but not zero, i s due to the d i f f r a c t e d f i e l d . A l l of i t i s d i f f r a c t e d f i e l d when kb i s s u f f i c i e n t l y l a r g e , because the r e f l e c t e d f i e l d s from both wedge surfaces do not c o n t r i b u t e to the t o t a l f i e l d i n t h i s r e g i o n . F i g . 5.5 shows how the t o t a l f i e l d s f o r an o m n i d i r e c t i o n a l source (kb=0) and a beam source (kb=4) change w i t h the i n t e r i o r wedge angle a . w F o r 0 =120°, t h e wedge a n g l e i s changed from a =90°(n=1.5) t o a o w h a l f - p l a n e 0^=0°(n=2), comparing the case of h a l f - p l a n e s o l u t i o n to the beam d i f f r a c t i o n by h a l f plane s o l u t i o n given i n chapter 3, gives another check on the v a l i d i t y and accuracy of our a n a l y s i s and computer programs. From F i g . 5.5 we can n o t i c e that the t o t a l f i e l d i n the region c l o s e r to the upper wedge surface; i . e . 0<n7t/2, i s not s i g n i f i c a n t l y a f f e c t e d w i t h the change of wedge angle because i n t h i s r e g i o n the i n c i d e n t and r e f l e c t e d f i e l d s do not change w i t h the wedge angle. In the region c l o s e r to the lower surface; i . e . {Ill < 0 < nu), 2 the t o t a l f i e l d i s n o t i c e a b l y changed w i t h the change of the wedge angle, because the r e f l e c t e d f i e l d changes s i g n i f i c a n t l y w i t h the change of the wedge angle.      - 67 - CHAPTER VI BEAM DIFFRACTION BY A CIRCULAR APERTURE (NORMAL INCIDENCE) F i g . 6.1 shows a c ircular aperture of radius a in a conducting plane (xy-plane) and centred at the or ig in . For normal incidence, the point source l ies on the aperture axis (z-axis) . Because of symmetry, without loss of generality the problem can be treated as 2-dimensional. 6.1 Uniform Point-Source Solution For a point source located at a distance z from the origin and r * o & o from the aperture edges making an angle 0 q with the aperture plane, as shown in F i g . 6 . i , we have the following relations: ) r = / z 2 + a 2 (6.1) o o QQ = TC _ c o s _ 1 ( a / r o ) (6.2) For an observation point in the far f i e l d at r ,Q from the origin , or at r^, from one edge and r 2 , Q 2 ^ r o m t n e opposite edge, we have the following approximations: r n . - r T a cos G ; r » a , (6.3) I»^ R - r - z sinQ ; r » z . (6.4) o o Where R is the distance from the source to the observation point. © 1 = 0 (6.5a) Q 2 = TC _ Q ; 0<e<it ; (6.5b) - 68 - 9 2 = 3n - 0 ; Tt<0<2n ; 0 = 1.5n + (J) ; -1.5n<_<Kn/2. (6.5c) 6.1.1 Single Diffraction The incident and r e f l e c t e d f i e l d s at a distant f i e l d point (r,0) due to an i s o t r o p t i c point source at (0, z ) and a c i r c u l a r aperture i n i r a co n d u c t i n g plane at z=0 are u and u , r e s p e c t i v e l y , are given by < - j k R jkz sin0 " J k r i e „ o e • r ^ u = « e — t — — (6.6) kR K r and -jkz sin0 - j k r U r = " e 4 — (6.7) kr The resultant d i f f r a c t e d component of the f i e l d by a curved edge i s given by K e l l e r (1957) as u d » u* . D ( 0 q , r Q , 0) . /p/r(r+p) e" j k r , (6.8) where u* i s the incident f i e l d at the d i f f r a c t i n g edge - j k r o and D(0 Q, r » 0) i s the d i f f r a c t i o n c o e f f i c i e n t given by Kouyoumjian and Pathak (1974) as kr + j i c/4 ' o J D(0 r 0) = J — - e [G(w ) - G(w )] (6.10) o o N n i r G(w) i s given by - 69 - j w 2 G(w) = - e F [±w] (6.11) The (+) s i g n a p p l i e s i n the shadow region, and (-) s i g n i n the i l l u m i n a t e d r e g i o n . F[w] i s the F r e s n e l i n t e g r a l given by (3.2) w i t h complex arguments w. given by i » ' w = -/2kT cos(_°.) (6.12) i , r o 2 /p/r'(r'+p) i s the c u r v a t u r e f a c t o r w i t h r ' as the distance from the d i f f r a c t i n g edge to the observation p o i n t ; i . e . r ^ , r 2 i n our case. For normal i n c i d e n c e , K e l l e r ( 1 9 5 7 ) showed p = a/cos© (6.13) and t h i s holds a l s o f o r a point source on the c i r c u l a r aperture a x i s . S u b s t i t u t i n g f o r p and r ' i n the curvature f a c t o r and w i t h the f a r f i e l d approximations i n (6.3) one can show that /p/r^r^f-p) = I /a/cos©! (6.14) 2 2 r 2 Where r i s measured from the o r i g i n and ©i, 2 given by (6.5). The s i n g l y d i f f r a c t e d component of the f a r f i e l d U d i s the sum of the d i f f r a c t e d f i e l d component by one edge and i t s opposite which are g i v e n by (6.8). iTc/4 ^ k a c o s &~'"-^) -j(kacos©-Tt/4) - j k r T i d , T T i r D i e + Do e i e ,,, 1 C v U - ka e U I — i £ J (6.15) s ° A 7\ k r vka cos9 where D l » 2 E D(© 0> r 0 » Q l » 2 > (6.16) are given by (6.10). - 70 - On the a x i a l c a u s t i c ; i . e . 9 = — or — , (6.15) i s s i n g u l a r , so near 2 2 t h i s a x i s the f i e l d i s in a c c u r a t e . The s i n g l y d i f f r a c t e d f a r f i e l d i n (6.15) can be r e w r i t t e n as U d - ka U 1 [ (D X + D 2) cos(ka cos9 -TC/4) /ka" cos9 s i n ( k a cos9 -TC/4) + j ( D l-D 2) # 1 ] ± - (6*17) /ka cos9 k r As ka cos9 •* 0, i . e . , 9 •> TC/2 or 3TC/2, we use the asymptotic expansion of the B e s s e l f u n c t i o n s cos(ka cos9 -TC/4) „ /—r- T o N n o \ J: = /TC/2 J (ka cos9) (6.18a) •ka cos0 o s i n ( k a COS9-TC/4) _ r-pr . n. _ . — - /TC/2 J ^ ( k a cos9) (6.18b) /ka cos9 Here J and J i are B e s s e l f u n c t i o n s of the f i r s t k i n d , of order zero and o 1 one, r e s p e c t i v e l y . S u b s t i t u t i n g (6.18) i n (6.17) gives jtt/4 , I JTC/4 U g « J— e ka U 1 [ ( 0 ^ 2 ) J Q ( k a c o s 9 ) - j k r + j ( D 1 - D 2 ) . J 1 ( ( k a c o s 9 ) ] 1_ (6*19) k r 6.1.2 Multiple Diffraction Solution The d i f f r a c t e d f i e l d component at (r,9) due to a point source at ( r Q , 9 Q ) from a curved edge i s given by (6.8). To s i m p l i f y the a n a l y s i s we r e w r i t e (6.8) as f o l l o w s : U d = U 1 D(9 , r , 9', r') (6.20) o o* o' ' where r ' , 9' are observation point p o l a r coordinates measured from the edge and - 71 - - j k r ' D(0 r 0', r') = D(0 r 0') / k a 6 ; 0 > U. ( 6 . 2 1 ) ° ° ° ° j/+cos0 / k r . k r ' < 2 Here D(® Q> r Q , ©') i s g i v e n by (6.10) and r , 0 are measured from the centre of the curved edge. F o l l o w i n g the same procedure given by J u l l (1981), the m u l t i p l y d i f f r a c t e d f i e l d , e xcluding s i n g l e d i f f r a c t e d f i e l d , can be w r i t t e n as ) ,d 1 „i U = U D [(D + D D ™ + D D 1 0 + ) m o o o z u o i U + ( D Q + D Q D 1 0 + D 2 D 2 0 ) ] , (6.22) where D Q , D ^ , D 1 0 and D 2 Q are given as f o l l o w i n g : , j ( 2 k a - .n/4) D Q = D(TC, 2a, TC, 2a) = /4/TC e F|/4kaJ (6.23) D 1 = D ( 0 , r , TC, 2a) o o' _j(2ka+n/4) j k r ( l - c o s 0 ) = /2r /ita e e ° ° F[/2kr~ s i n ( 0 / 2 ) ] , (6.24) and D^g = D (T t , 2a, 0 L, r^) j2ka Tj(kacos0 - T t/4) - j k r D 1 0 = T T t ( 2 k a / n ) 3 / 2 e e F[/4ka~ sin(0,/2) ] f _ 20 . 2 k r /ka cos0 (6.25) Now (6.22) can be summed, provided DQ*1» g i v i n g „d . B i D l ( D 1 0 + D 2 0) " ° <!-».) Again near the a x i a l c a u s t i c ; i . e . 0 = TC/2 or 3TC/2, the f i e l d must be modified as f o r s i n g l e d i f f r a c t i o n . F o l l o w i n g the procedures given before to get the m u l t i p l y d i f f r a c t e d component, r e w r i t e D^g, D 2 0 i n terms of D 1 0, D 2 0 as - 72 - T j ( k a cosO-Tt/4) - j k r Din = + D' - 6 , (6-27) 20 28 /ka cosG k r where j2ka D' = n ( 2 k a / n ) 3 / 2 e F[/4ka sin ( e 1/2) ]. (6.28) 28 2 Then D 1 0 + D20 c a n ^ e s i m p l i f i e d to , - j ( k a cos9-Tt/4) , j ( k a C O S O - T C/4) D10 e ~ D20 e - j k r D10+ D20 = ~[ = 1 (6-29) /ka cosO k r r , ' ' .cos(ka C O S G - T C/4) . ' 1 . s i n ( k a C O S © - T C/4)- I e ^ k r = -L(D10~D20^ / -J(D10+D20^ -== J /ka cos© /kacos© o r > N (o•Ju) S u b s t i t u t i n g from (6.18) i n (6.30) gives 1 t ,) : - j ( D i 0 + D 2 0 ) J x ( k a cos©)] e j k r D10 + D20 = [(n10_D20) J n ( k a c o s©) (6.31) S u b s t i t u t e f o r (6.31) i n (6.26) to get the m u l t i p l y d i f f r a c t e d f i e l d component near the a x i a l c a u s t i c as . D* , , , , - j k r IT = S^TT— [ ( D 1 0 - D 2 0 ) J (ka cos©) - j ( D 1 0 + D 2 0 ) J , (ka cos©)] 1 m (D -1) ° 1 k r o (6.32) The t o t a l f i e l d , i n c l u d i n g s i n g l e d i f f r a c t i o n only, i s U C = U d + U 1 S(0 -0) + U r S(0 -0) (6.33a) s s i s r and i n c l u d i n g m u l t i p l e d i f f r a c t i o n i s U* = U d + U d + U 1 S(0 g i-©) + U r S ( 9 g r - 0 ) , (6.33b) - 73 - where S(x) i s the p o s i t i v e u n i t step f u n c t i o n . U , U , U and U are s m the i n c i d e n t , r e f l e c t e d , s i n g l y d i f f r a c t e d and m u l t i p l y d i f f r a c t e d f a r f i e l d s . 6.2 Uniform Beam Solution For normal incidence the beam a x i s i s perpendicular to the aperture plane and c o i n c i d e s w i t h the aperture a x i s . 6.2.1 Far Field Calculation To change from an i s o t r o p i c point source s o l u t i o n to a d i r e c t i v e beam s o l u t i o n , complex values appropriate to the beam width and beam d i r e c t i o n are g i v e n t o the so u r c e c o o r d i n a t e s . Then z , r and 0 ° 0 ' o o become complex and are c a l l e d z , r and 0 , r e s p e c t i v e l y . s s s z g = Z q - jb sin8 ; b>0, 0<8<2TC (6.34) r = Vz1 + a 2 ; R e a l ( r ) > 0 (6.35) s s s ~~ 0 = TC - c o s " 1 (a/r ) (6.36) s s' Where Z q , b and 8 a r e r e a l v a l u e s , d e f i n i n g the source l o c a t i o n , beam w i d t h and beam d i r e c t i o n , r e s p e c t i v e l y . By r e p l a c i n g Z q, r ^ and 0 q by z , r and 0 , r e s p e c t i v e l y i n (6.1 to 6.33) we get the beam s o l u t i o n s s s f o r a l l above cases, s i n g l e and m u l t i p l e d i f f r a c t i o n , near or f a r from the a x i a l c a u s t i c . - 74 - 6.2.2 Shadow and R e f l e c t i o n Boundary C a l c u l a t i o n s Since the diffract ion phenomena is loca l , we assume the shadow and reflection boundaries for a curved edge are the same as for a straight edge at the point of di f fract ion. So the results given in (3.13) and (3.14) are val id here. The shadow and illuminated regions mentioned in (6.11) and Appendix D are given as, for shadow region: Real(w ) - Imag(w ) > 0 (6.37) and for illuminated region: Real(w ) - Imag(w ) < 0 (6.38) where w are the Fresnel integral arguments given as I » r 6 T 8 v . - - / 2 k F cos( !-) (6.39) i ,r s 2 Also the shadow and illuminated regions for incident or reflected f ields can be defined, respectively, by 0 ^ 0 or 0 (6.40) < s i sr ' where 0 and 0 are given by (3.13) and (3.14), respectively, s j- s r 6.3 Numerical R e s u l t s A point source on the c ircular aperture axis is at a distance kz = 3it from the centre of the aperture which is of a radius ka=3iu In o the following figures, the horizontal axis, <t> in degrees, is the angle measured from the aperture axis as shown in F i g . 6.1, and the vert ica l - 75 - a x i s i s the normalized f a r t o t a l f i e l d p a t t e r n , i n c l u d i n g s i n g l e d i f f r a c t i o n only or i n c l u d i n g s i n g l e and m u l t i p l e d i f f r a c t i o n . In F i g . 6.2 the dashed curves represent the non-modified s o l u t i o n c a l c u l a t e d from (6.25), (6.26) and (6.33b) which i s v a l i d f a r from the a x i a l c a u s t i c . The s o l i d curves represent the modified s o l u t i o n near the a x i a l c a u s t i c , c a l c u l a t e d from (6.32) and (6.33b). This f i g u r e shows how the t o t a l f a r f i e l d i n c l u d i n g m u l t i p l e d i f f r a c t i o n , i s modified near the a x i a l c a u s t i c ( z - a x i s ) f o r two d i f f e r e n t cases. One i s the l i m i t i n g case of a point source (kb=0) and the other i s a d i r e c t i v e beam (kb=8). F i g . 6.3 i l l u s t r a t e s the development of the beam s o l u t i o n from the point source (kb=0) to a d i r e c t i v e beam (kb=16), and an e s s e n t i a l l y p l a n e wave (kb=85). When kb Is s m a l l compared to k z Q i n t e r a c t i o n between the i n c i d e n t and d i f f r a c t e d f i e l d s i n the i l l u m i n a t e d r e gion i s evident because the aperture edge i s s t r o n g l y i l l u m i n a t e d , consequently the d i f f r a c t e d f i e l d i s s i g n i f i c a n t . As kb i n c r e a s e s , e.g. kb=16, the i n c i d e n t beam becomes narrower and the edge i s weakly i l l u m i n a t e d so the d i f f r a c t e d f i e l d i s i n s i g n i f i c a n t and the i n t e r a c t i o n decreases. In the shadow region the i n t e r a c t i o n occurs between the d i f f r a c t e d f i e l d s from the two opposite d i f f r a c t e d p o ints on the aperture edge. This i n t e r a c t i o n i s s i g n i f i c a n t when kb small or when b » z Q , eg. kb=85 where the f i e l d i n c i d e n t on the aperture becomes uniform, l i k e a plane wave, the edge i s s t r o n g l y i l l u m i n a t e d again. In t h i s f i g u r e the dashed curves represent the s i n g l e d i f f r a c t i o n s o l u t i o n given by (6.33a), and - 76 - the sol id curves include for multiple diffraction solution given by (6.33b). For this choice of aperture radius (ka=3Tc), the singly and multiply diffracted fields are very much the same except at the conductor, i . e . <()=90o, the multiply diffracted f ie ld vanishes on the conductor, satisfying the boundary condition of a perfect conductor while the singly diffracted f i e ld does not. F i e l d P o i n t F i g . 6.1 Geometry of a complex point source d i f f r a c t i o n by a c i r c u l a r aperture F i g . 6.2 Normalized t o t a l f i e l d p attern of a point source (kb=0) and moderate beam (kb=8) ( s o l i d ) modified and (dashed) non-modified on the c a u s t i c a x i s (Normal incidence) F i g . 6.3 Single (solid) and multiple (dashed) total patterns of a beam diffract ion by a circular aperture (Normal incidence) - 80 - CHAPTER VII SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 7.1 Summary The complex source point method was used to represent a d i r e c t i v e beam which i s Gaussian i n the p a r a x i a l r e g i o n . Uniform s o l u t i o n s f o r o m n i d i r e c t i o n a l sources were developed and extended a n a l y t i c a l l y to become s o l u t i o n s f o r d i r e c t i v e beams. The geometrical theory of d i f f r a c t i o n and e q u i v a l e n t l i n e c u r r e n t s were used t o i n c l u d e i n t e r a c t i o n between the edges of the s l i t and c i r c u l a r aperture. Numerical r e s u l t s i n c l u d i n g the l i m i t i n g cases; e.g. plane wave incidence (kb •*• c o) and l i n e or point sources (kb = 0 ) , were given f o r every case s t u d i e d . A l s o comparisons w i t h e x i s t i n g s o l u t i o n s were made wherever p o s s i b l e . I n Chapter I I , a d i r e c t i v e beam was derived i n polar coordinates and compared w i t h a Gaussian beam and a t y p i c a l antenna beam. An expression f o r the half-power beam width was derived , and a simple d i s c u s s i o n of the use of m u l t i p l e complex source points to derive more complicated beams was given. The s o l u t i o n of beam d i f f r a c t i o n by a h a l f - s c r e e n , derived i n Chapter I I I from a simple s o l u t i o n exact i n the f a r f i e l d l i m i t , was used to solve the problem of beam d i f f r a c t i o n by wide s l i t and complementary s t r i p . A l s o a convenient, simple formula was derived f o r the l o c a t i o n of the shadow boundaries of a s t r a i g h t edge. - 81 - Beam diffract ion by a wedge with its edge on the beam axis was analysed using the uniform theory of di f fract ion. This uniform solution completes the asymptotic solution, for the same problem, mentioned by Felsen (1976), whose solution is inf in i te on the shadow boundaries and inaccurate in the transition regions. Also the shadow boundaries are given here for any beam orientation. F ina l ly , the diffraction by c ircular aperture when illuminated by normally incident beam, was derived using the uniform theory of dif fract ion and along the axial caustic, Bessel functions were used to remove the s ingularity. Multiple diffractions were considered and a closed form expression was derived. 7*2 Conclusions The beam derived in Chapter II using the complex source point method, can represent a typical antenna beam better than the Gaussian function especially for wide antenna beams (small kb). When the imaginary part (b) of the complex source position vector (F =r"Q-jF) is very large compared to the real part ( r Q ) , i . e . b » r Q , the beam tends to a plane wave. Other authors assume very narrow beams and solve diffraction problems in the paraxial region. This kind of assumption makes the contribution of the diffracted f ie ld negligible, unless the beam axis passes through the diffracting edge. In our analysis this assumption was removed and the range of va l id i ty was increased to cover the whole region of interest. - 82 - The synthesis of more complicated beams such as one with sidelobes or a nearly square beam, was given in section (2.3) by using multiple complex source points. But the width and sidelobe level of the resultant beam are yet to be calculated and related to the complex coordinates. The study of simulating any beam in terms of Gaussian beams and Complex Source Points, given by Mantica et a l (1986), was not rigorous and some assumptions were made to simplify the analysis. Our solution of beam diffract ion by half-screen is accurate, uniform everywhere and val id for a l l beam orientations and widths. This solution can be used as a reference solution for other uniform or asymptotic solutions which are inaccurate in the transition regions and inf in i te on the shadow boundaries. The l imiting case kb = 0 of our solution to the strip when illuminated by omnidirectional source, is in very good agreement with solutions of l ine source diffract ion by a s t r ip , given by Vankoughnett and Wong (1981) and by Shafai and Elmoazzen (1972). Tsai et a l . (1972) have shown, by comparison with numerical results, that the geometrical theory of diffract ion yields satisfactory results for reflector widths as small as 0.2X (wave length) when double diffraction is included. For our choice of str ip width 2.5X., single diffract ion is sufficient. Since the contribution to the diffracted f i e ld for directive sources is always less than or equal (for the edge on the beam axis) to that of an omnidirectional source, the accuracy for directive beams is at least as high as that for omnidirectional sources. - 83 - In a l l cases s t u d i e d here the i n c i d e n t f i e l d was normal to the d i f f r a c t i n g edge. S o l u t i o n s can be extended to i n c l u d e oblique incidence on a s t r a i g h t edge or wedge. The ordinary UTD was used i n s o l v i n g the problems of the wedge and c i r c u l a r aperture. For b e t t e r accuracy one may use the UTD augmented by slope d i f f r a c t i o n (Kouyoumjian et a l . , 1981) or the improved v e r s i o n (Buyukdhura and Kouyoumjian, 1985), i n s t e a d . The d i f f r a c t i o n of a beam by p a r a b o l i c c y l i n d e r r e f l e c t o r w i t h an edge was a l s o considered (see appendix E) before we were aware that Ghione et a l . , (1984) had published t h e i r s o l u t i o n to t h i s problem. However, the d i f f r a c t e d f i e l d and r e f l e c t e d f i e l d , w i t h some approximations, without using the computer search technique i s given i n Appendix E. The problem was not pursued f u r t h e r , although as they suggest f u r t h e r i n v e s t i g a t i o n i s needed to c l a r i f y and s i m p l i f y the method. The uniform theory of d i f f r a c t i o n was used to o b t a i n uniform s o l u t i o n s where there were no simple exact s o l u t i o n s , such as f o r the wedge and c i r c u l a r aperture. Otherwise r i g o r o u s l y c o r r e c t s o l u t i o n s at high frequencies f o r f a r s i n g l y d i f f r a c t e d f i e l d s were used, such as f o r the h a l f - s c r e e n , s l i t and s t r i p . A l l the s o l u t i o n s obtained f o r the above cases are uniform, f o r F r e s n e l i n t e g r a l s provide a smooth t r a n s i t i o n through shadow and r e f l e c t i o n boundary regions. For s i m p l i c i t y s c a l a r ( a c o u s t i c ) f i e l d s were assumed through out t h i s t h e s i s . The r e s u l t s a p p l y d i r e c t l y to t w o - d i m e n s i o n a l electromagnetic f i e l d s i n the case of the h a l f - p l a n e , s l i t or s t r i p and wedge. For the c i r c u l a r aperture extension to vector electromagnetic f i e l d s can be made by c o n s i d e r i n g the s c a l a r f i e l d as one component of - 84 - the vector f i e l d or as a s c a l a r p o t e n t i a l from which vector f i e l d s are deriv e d . 7.3 Recommendations for Future Work So f a r we have d e a l t w i t h problems that assume p e r f e c t conductors, simple beams and normal i n c i d e n c e , to g e n e r a l i z e the i n c i d e n t beam and the r e f l e c t o r s boundary c o n d i t i o n s the f o l l o w i n g may be considered: i ) To make the a n a l y s i s by the complex ray t r a c i n g method more complete, e s p e c i a l l y f o r non-planar s u r f a c e s , a general a - p r i o r i c r i t e r i o n ; i . e . , one which does not re q u i r e the study of steepest d e s c e n t s paths used by Ghione et a l . (198 4 ) , i s needed f o r two-dimensional d i f f r a c t i o n . i i ) D i f f r a c t i o n by simple shapes when i l l u m i n a t e d by more complicated beams w i t h s i d e l o b e s . When these beams are represented by m u l t i p l e complex source p o i n t s , s o l u t i o n s may e a s i l y be obtained u s i n g the s u p e r p o s i t i o n p r i n c i p l e . i i i ) The problem of beam d i f f r a c t i o n by s t r a i g h t wedge where the edge does not l i e on the beam a x i s , using the UTD to assess the asymptotic s o l u t i o n by Felsen (1976), i s yet to be done. A l s o d i f f r a c t i o n by a conducting curved wedge has not been studied yet. The s o l u t i o n f o r r e a l source d i f f r a c t i o n by a curved wedge by Lee and Deschamps (1976) or Deschamps (1985) may be used. i v ) A l l e x i s t i n g s o l u t i o n s f o r beam d i f f r a c t i o n by a c i r c u l a r a p e rture, assume symmetrical i n c i d e n c e ; i . e . the beam a x i s c o i n c i d e s w i t h the aperture a x i s . The more general non-symmetrical incidence case w i t h the beam a x i s s h i f t e d from the aperture a x i s by some distance or at some angle, apparently has not been reported y e t . - 85 - v) To cover a wide range of problems, using the CSP method, a rigorous simulation of an arbitrary beam in terms of complex source points needs to be derived. What exists in the l i terature now is based on assumptions and experience; i . e . t r i a l and error. vi) Solutions to beam diffraction by simple shapes such as half-plane, str ip and wedge, under impedance boundary conditions may be obtained using the corresponding solutions for omnidirectional sources by Bucci and Franceschetti (1976), and Tiberio et a l . (1982 and 1985), respectively. v i i ) Extensions of this method to three-dimensional dif fract ion by curved surfaces with edges need to be addressed. - 86 - REFERENCES ALBERTSEN, N., NIELSEN, Per. and PONTOPPIDAN, K., "New concepts i n m u l t i - r e f l e c t o r antenna a n a l y s i s , f i n a l r e p o r t " , TICRA A/S Engineering Consultants, (Copenhagen, Denmark, Sept., 1983), pp. 28-40. ANDERSON, I . (1978): "The d i f f r a c t i o n of an antenna beam by a nearby conducting h a l f - p l a n e " , I n t . Conf. Ant. Prop. ICAP, UK., pp. 244-246. ARNAUD, J . 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V o l . 12, No. 4, pp. 872-876. - 91 - APPENDIX A THE FRESNEL INTEGRAL WITH A COMPLEX ARGUMENT A.I E v a l u a t i o n of F r e s n e l I n t e g r a l s from E r r o r Functions For a r e a l or complex argument w, F r e s n e l I n t e g r a l F[w] i s defined as oo - J T 2 F[w] = / e dx (A.1) w By changing v a r i a b l e s ^ - j i t / 4 -JTC/4 T = y e , dx = e dy (A.2) F[w] - i j l l / 4 / i y dy (A.3) JTC/4 we The complementary e r r o r f u n c t i o n i s defined as 2 , -Y 2 e r f c ( v ) = -± J e y dy = 1 - e r f ( v ) (A.4) v where v can be r e a l or complex and e r f ( v ) i s the e r r o r f u n c t i o n , o v 2 e r f ( v ) = _£ / i y dy (A.5) / i t Q j i t / 4 From (A.3) and (A.4) w i t h v = w e we can w r i t e -JTC/4 j i t / 4 F[w] =/iT/2 e erfc(we ) (A.6) Subroutines f o r complementary e r r o r f u n c t i o n s w i t h complex arguments are a v a i l a b l e i n UBC Computing Centre L i b r a r y . A.2 Some P r o p e r t i e s of F r e s n e l I n t e g r a l s : i ) Symmetry r e l a t i o n F[w] + F[-w] = ZiTsJ1 1/^ (A.7) - 92 - i i ) Special values -J*M -I F [ - » ] = /TT e , F[0] = ± F [ - » ] and F[<=°] = 0 (A.8) 2 i i i ) Asymptotic expansion. F[w] ~ S(-w) + F[w] ; |w| -»• » , (A.9) where S(x) is the unit step function, and . - jw 2 0 0 F[w] = I i - ! I r(n + l /2)(-jw)" n (A.10) 2w/ix n=0 where T(x) is the gamma function. T(n + 1/2) = /TT (l/2)(3/2) (n - 1/2) (A.11) - 93 - APPENDIX B CALCULATION OF REAL AND IMAGINARY PARTS OF r B.l Analysis Using (2.5) which i s r e w r i t t e n here as r = [ r 2 + 2r (-.1b) cos(B-0 ) + ( - j b ) 2 ] 1 / 2 ; Re(r )>0 ( B . l ) S O O \j ' o Let us w r i t e r as s r = R - j I ; R > 0 (B.2) Where R and I are r e a l . By squaring ( B . l ) and (B.2) and equating the r e a l parts and the imaginary p a r t s , r e s p e c t i v e l y , we get R 2 - I 2 = r 2 - b 2 (B.3) o and R.I = r b cos(B-0 ) (B.4) o o So l v i n g (B.3) and (B.4) f o r R and I gives r2 _ b2 R = [(-2 ) + I / ( r 2 - b 2 ) 2 + [2r b cos(B-0 ) ] 2 ] 1 / 2 ; (B.5) 2 and r b I = — cos(B - 0 ) (B.6) R 0 From (B.6) we n o t i c e t h a t : I > 0 i f |8 - 0 | < TC/2. (B.7) < o > - 94 - (B.9) B.2 Special Cases: i) b=0 gives R=rQ and 1=0 ; (B.8) ii) b=rQ gives R=rQ V | cos (B-e )|, and I = +R ; IB-0 I > n/2 ' o1 s ii i ) B = 0 or 0 + it gives R = r ' o o ° o and I = ^ <B- 1 0> iv) |B - 0 q | « u/2 gives R = Vr2 - b 2 and I = 0 if r > b ; o o ' R = 0 and I = 0 if r = b ; ( B . l l ) o R - 0 and I = ̂  b 2-r 2 if r < b o o APPENDIX C GAUSSIAN BEAM DIFFRACTION BY HALF-SCREEN (ASYMPTOTIC SOLUTION) The asymptotic s o l u t i o n given by Green et a l . (1979) f o r a Gaussian beam d i f f r a c t i o n by a h a l f - s c r e e n i s summarized here with some changes i n coordinates and n o t a t i o n f o r comparison with the s o l u t i o n given i n Chapter 3. By r e p l a c i n g 9 ' , 9 , p', p and E by (- - 9 ) , ( — - 9 ) , r , 2 S 2 S r and U, r e s p e c t i v e l y , we can w r i t e the i n c i d e n t U 1, r e f l e c t e d U r, d i f f r a c t e d U dand t o t a l Ufc f a r f i e l d s as j k r cos (9-G ) - j k r U 1 - e S S e /kr ( C . l ) j k r cos (9+9 ) - j k r U r = - e S S e (C.2) /kr - j ( kr s-37 t/4) Q _ Q 0+9 - j k r [sec( ') (1-A +) - S e c ( - 9 - i ) (1-A~)] e 2/2rtkr A " s vkr where (C.3) + _. , O T 0 A = — 1 - sec ( — r — ) ; r » |r | (C.4) 4kr 1 s U 1 = U d + U 1 . S ( 9 s i - 0) + U r. S ( 0 g r - 0) (C.5) where S(x) i s the u n i t step f u n c t i o n , and 0 . and 0 are the shadow and s i s r - 96 - r e f l e c t i o n boundaries measured from the i l l u m i n a t e d side of the h a l f - s c r e e n . -lr ^Mlm(9 s)] . R e ( r s ) ^ ^ 0 = it + Re(0 ) + tan [ Z Z ] S | r s j + I m ( r g ) . cosh[Im(©s)] 0 = 2Tt - 0 , (C7) s r s i ' where Re(x) and Im(x) are the r e a l and imaginary p a r t s . The accuracy of + the above s o l u t i o n depends on how small i s | A | compared to 1. - 97 - APPENDIX D THE SINGULARITY CANCELLATION IN THE WEDGE DIFFRACTION COEFFICIENT One of the cotangent f u n c t i o n s of the d i f f r a c t i o n c o e f f i c i e n t given by (5.9) i s s i n g u l a r on the shadow or r e f l e c t i o n boundaries, at the same time the corresponding G(w) f u n c t i o n i s zero. So the term cot(t).G(w) i s f i n i t e everywhere. Therefore the s i n g u l a r i t y i s c a n c e l l e d and the d i f f r a c t i o n c o e f f i c i e n t i s f i n i t e everywhere. Near the shadow boundary or r e f l e c t i o n boundary from the upper wedge sur f a c e , we can w r i t e from (5.3a,b), 0 T 0 = TC + e (D.l) o where e -*• -^0. e>0 and e<0 d e f i n e the shadow and i l l u m i n a t e d r e g i o n s , r e s p e c t i v e l y . S u b s t i t u t i n g (D.l) i n (5.13a) gives = 0 (D.2) and ( D . l ) , (D.2) i n (5.11a) and (5.12a) gives cot(T. ,) = c o t ( l i ) = — (D.3) L f j L 2n e w, = - /2kT c o s ( 1 t + E ) = /zkr~ . I (D.4) 1,2 ° 2 ° 2 S u b s t i t u t e (D.4) i n (D.10) to get kr e2 J ( - ° — ) G( v. ?) = 2j e 1 . ( i /2k7 1) P[± /2k7 1] , eCO (D.5) From (D.3) and (D.5) we get as e + - 0 -JTC/4 c o t ( T 1 2).G(w 1 2) = T n /2rckr o e (D.6) - 98 - Where (-) f o r shadow region and (+) f o r i l l u m i n a t e d r e g i o n . S i m i l a r l y , near the r e f l e c t i o n boundary due to the lower wedge sur f a c e , we can w r i t e from (5.3c) 0 + 0 = (2n - 1) TC - E (D.7) o A g a i n £ •*• ̂ 0 . e>0 and e<0 define the shadow and i l l u m i n a t e d regions of the r e f l e c t e d f i e l d , r e s p e c t i v e l y . I n s e r t i n g (D.7) i n (5.13b) gives M+ = 1 (D.8) and s u b s t i t u t i n g (D.7) and (D.8) i n (5.11b), (5.12b) and (5.10) gives c o t ( T H ) = cot(Tt _ 4r-) = "5- (D.9) and _ jTt/4 cot(T l +) G(w H) = + n/2rckF e (D.10) o where (-) and (+) r e f e r to the shadow and i l l u m i n a t e d regions of the r e f l e c t e d f i e l d , r e s p e c t i v e l y . Notice that the t h i r d term of the d i f f r a c t i o n c o e f f i c i e n t i s f i n i t e everywhere, because of the r e s t r i c t i o n on 0 , (O<0Q<^2-) . T h e r e f o r e , a l l s i n g u l a r i t i e s of the d i f f r a c t i o n c o e f f i c i e n t are c a n c e l l e d and the d i f f r a c t i o n c o e f f i c i e n t i s f i n i t e everywhere. - 99 - APPENDIX E BEAM DIFFRACTION BY A PARABOLIC REFLECTOR The c a l c u l a t i o n of the r e f l e c t e d f i e l d from a conducting p a r a b o l i c c y l i n d e r when i l l u m i n a t e d by a Gaussian beam i s given by Hasselmann & Fe l s e n (1982). In t h e i r a n a l y s i s , they assumed a very sharp Gaussian beam and an i n f i n i t e p a r a b o l i c r e f l e c t o r . so they d i d not include the d i f f r a c t e d f i e l d from the edge of a f i n i t e r e f l e c t o r . They used the method of computer search f o r the complex r e f l e c t i o n p o i n t s . To i n c l u d e the d i f f r a c t e d f i e l d from the edges, h a l f - p l a n e tangent to the r e f l e c t o r at i t s edges are used i n s t e a d . A l s o i n the f a r f i e l d , w i t h some approximation, the r e f l e c t e d f i e l d can be c a l c u l a t e d without computer research f o r the r e f l e c t i o n p o i n t s . F i g . E . l , shows a l i n e source p a r a l l e l to the r e f l e c t o r a x i s at a complex p o i n t S ( r g , 0 g ) . A ( r , 0 ) i s a f a r f i e l d p o i n t , p ( r p » Q p ) i s a t y p i c a l p o i n t on the r e f l e c t o r , E ( r g , O e ) i s the edge of the r e f l e c t o r and 0(0,0) i s i t s focus. The equation of the parabola of a f o c a l length F i s given by r = 2F/(1 + cos0 ) ( E . l ) P P and the slope of the t a n g e n t i a l h a l f - p l a n e i s given as 2F dy dx y y J e J e = tan0 (E.2a) and 0 = (TC - 0 )/2 . (E.2b) t e From the geometry of F i g . E . l and the f a r f i e l d approximations, we have r = r - r cos(0 - 0 ) , (E.3a) I s s - 100 - r r = r + r p cos(9 + © p) > (E.3b) r l = r + r e c o s ( 9 + / Q e ^ ' (E.3c) r 2 * r + r g cos(0 - 0 ) . (E.3d) Using the complex source point method, an i n c i d e n t f i e l d of a beam which makes an angle 8 with the x-axis i s given as ~ J k r i _ j k r r _ r C O S ( 0 - 0 )] kb cos(0 - B) U = — - e . e , (E.4) AT. l Here r , 0 are the r e a l coordinates of the l i n e source. The r e f l e c t e d o o f i e l d at A from point P Is - j k ( r + r ) J sp r ' U r * 6 , (E.5) / k ( r r + r z ) where r and r are shown i n F i e . E . l and r sp b r R cos©. r =- P_£ i . (E.6) Z (2r - R cos0.) p c i Here R £ i s the l o c a l r adius of curvature and ©^ i s the angle between the i n c i d e n t ray ( r S p ) and the normal at the r e f l e c t i o n point P. R = 2F/cos3(© 12) (E.7) c p 0 - -(0 12 - 0 ) ; Real(0.) > 0. (E .8) i p sp i To c a l c u l a t e t h e complex r e f l e c t i o n p o i n t ( r p> 9 p ^ ' w e a P P ^ v t n e s t a t i o n a r y phase c o n d i t i o n d ( r _ + r j = 0 , (E .9) P dx  SP r - 101 - where r s P • t (vxs)2 + % - ys>2i1/2 • <E-10a> x2 . , x2,l/2 > + (y - y ) ] S u b s t i t u t i n g (E.10) i n (E.9), gives cosG . sp sp p r f = [(x - x p ) ' + (y - y p ) ] ' * (E.lOb) cosG + sin© . cot(0 /2) = -[ sG - sinG Cot(0 /2)1, ( E . l l ) r r p 1 sp sp r> where x - x y - y cosG r = P , s i n Q r = E. , (E.12a) r r r r x - x y - y cosG = -E 1 , sinG = _E 1 , (E.12b) sp r sp r sp sp dy cot(0 12) = — E = 2F/y . (E.12c) p dx p P A f t e r some manipulations on ( E . l l ) , one gets s i n ( 0 + Q /2) = - sin(G /2 - Q ) (E.13) r p p sp' v ' or Q = -(Q - Q ) (E.14) r p sp ' From F i g . E . l , ( E . l ) and the sine law, 2F s i n ( 0 + 0 ) cos(Q ) = _ s i n ( 0 + 0 ) (E.15) v sp s' K p x r v p sp' K ' s I n s e r t i n g (E.14) i n (E.15) and a f t e r some manipulations, one gets - 102 - 0 = - i { 0 + 0 + s i n _ 1 [ — s i n ( 0 ) + s i n ( 0 + 0 )]} (E.16) p n s r L r r ' v s r / J v ' v 2 s In the far f i e ld 0 = 0 and 0 is derived. r p The diffracted f i e ld U d at A is the sum of the diffracted f i e ld (U d and U d ) of each edge U d = U d [St(G-n+0 ) + S t ( i t - 0 E - 0 ) ] + U d [ S t ( 0 +rc - 0 T ) + s t ( n + 0 E + 0 ) ] (E.17) where St( ) is a unit step function, -JCkr - * / 4 ) c o s ( 9 0 ) 0 0 Ud=f { e 8 1 1 8 1 F[V2kT l C08(- i li)] 1 /ZkF s i 2 - e J k r 8 l c o 8 ( e 1 + e 9 l ) 0 1 + 0 S L ^ F[-/2kr cos(_——)]} (E.18) S X n and U d is given by (E.18) with subscript 2 replacing 1. 0 = (it-© t) + 0 ; 0 < Q l < 2TC , (E.19a) 0 2 = (ii-G ) - 0 ; 0 < © 2 < 2TC , (E.19b) and r = [ r 2 + r 2 + 2r r cos ( 0 + 0 ) ] 1 / 2 , „ O N A . s l 1 e s e s ^ e s / J , (E.20a) r s i n ( 0 + 0 ) 0 = 0 + s in" 1 [ _ ! ! !_] , (E.20b) S r i s l where r and 0 are given by (2.5) and (2.6), respectively, s s Fig E.2 shows the normalized diffracted f i e ld component for a beam represented by the Complex Source Point method with i ts axis directed to the apex of the reflector. The incident and reflected fields are not included. In the region 0 >12O°, the diffracted f ie ld is the total f i e l d . -103- Field Point A el" ef (-F.0) e / Dapi Line Source x X Parabolic Reflector Feeding Beam F i g . E . l Geometry of a beam idiffraction by a parabolic cylinder F i g . E.2 Normalized d i f f r a c t e d f i e l d component of a beam d i f f r a c t i o n by a p a r a b o l i c c y l i n d e r (kF = IOTT, 6 = 60 ) - 105 - APPENDIX F EVALUATION OF ARCTANGENT OF A COMPLEX NUMBER To c a l c u l a t e the complex angle 0 i n the proper quadrant; i . e . —n<Real(0) <it, which a complex p o s i t i o n vector makes with the x-axis tan9 = L , ( F . l ) x or t a n ^ + j O ^ = R t + j l t (F.2) where 0 and 0 are the r e a l and imaginary parts of 0, and R and I are R I t t the r e a l and imaginary parts of ( y / x ) . A l s o we can w r i t e CO8(0+J9 ) = - 4 _ = R c - j l (F.3) where R and - I a r e t h e r e a l and i m a g i n a r y p a r t s of ( _====—») • c c J 2, 2 vx +y Expanding (F.3) and equating the r e a l parts of both sides and the imaginary parts of both s i d e s , gives cos0 . cosh0 = R (F.4) R 1 c and sin0„. s i n h 0 T = 1 (F.5) R I c Also by expanding the l e f t s ide of (F.2), one gets tan0„ + j tanh©T R I 1 - j tan©,,. tanh0 T  J R I = R t + j l t (F.6) - 106 - A f t e r some manipulations on (F.6) we can w r i t e sec2© tanhG . _ = I (F.7) (1 + t a n 2 e R . t a n h 2 0 ] ; ) and sech^ O tan© . — = R (F.8) (1 + tan 20_.tanh 20_) R J. From (F.7) we can say 9 ^ 0 I f I t ^ 0 (F.9) Now l e t 0 be given as s ° e - c o i V - J | I C | ) ( F A O ) s c c the arccosine of a complex number of form (F.10) i s a v a i l a b l e i n the UBC Computer L i b r a r y . Using ( F . 4 ) , (F.5) and (F.9) we get 0 = T 0 s ; I T J 0 , I c > 0 ( F > n ) and 0 = ± ©* ; I < 0 , I < 0 (F.12) s t > c * where 0 g i s the complex COTIjugate of 0 g. - 107 - APPENDIX G LIST OF COMPUTER PROGRAMS FOR CSP ANALYSIS All the programs used in the Complex Source Point analysis and listed below are written in the language of FORTRAN. G.l Comparison of CSP, Gaussian and Typical Antenna Beams This program makes use of expressions (2.10), (2.16) and (2.18) with ka=4, kb = 3 and HPBW = 55.7° to compare the normalized far fields of CSP, Gaussian, and typical antenna beams. Also it uses (2.20) with different weighting factors (Q ,̂ Q 2 , Qp and beam parameters (b ,̂ b 2 > b̂  and fS Qy /3q 2, /3qJ) to calculate far fields of different beams. - 108 - C PROGRAM CALCULATES AND COMPARES THE FIELDS OF THE CSP, GAUSSIAN AND C TYPICAL ANTENNA BEAMS. THIS PROGRAM IS CALLED "CSP.FTNC". C The Time Harmonic F a c t o r " e x p ( - i w t ) " i s suppressed . C The Common F a c t o r " e x p ( i k r ) / S q r t ( k r ) " i s supperessed. C ==================================================================== KK = 181 KM = (KK+1)/2 H =1.0 C =============== PI = 3.1415926 DTR = PI/180.0 C = CMPLX(0.0,1.0) C A = 4.0 B = 3.0 IF ( B .LT. 0.25*ALOG(2.0) ) STOP C ==================================================================== C HPBW IS THE HALH-POWER BEAM WIDTH C ==================================================================== HPBW= 2.0*ACOS( 1.0 - 0.5*ALOG(2.0)/B ) C DO 111 K=1,KK Y = H*(K-KM) FI = Y*DTR C CSP = ABS( EXP(B*(COS(FI)-1.0)) ) GB = EXP( -ALOG(4.0)*(FI/HPBW)**2 ) C U = A*SIN(FI) IF ( ABS(U) .EQ. PI/2.0 ) GO TO 11 CD = COS(U)/( 1.0 - (2.*U/PI)**2 ) GO TO 22 II CD = PI/4.0 C 22 WRITE(6,1) Y , CD ,CSP ,GB III CONTINUE C ============================ 1 FORMAT( F6.1 ,3(1X, E14.7) ) STOP END - 109 - C********************************************************************* C PROGRAM FOR DEVELOPING A BEAM FROM SINGLE OR MULTIPLE LINE SOURCE(S) C LOCATED AT COMPLEX POINT(S). THIS PROGRAM CALLED "CSP.FTNM". C C THE TIME HARMONIC FACTOR " e x p ( - i w t ) " IS SUPPRESSED. C THE COMMON FACTOR " e x p ( i k r ) / S q r t ( k r ) " IS SUPPRESSED. c = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = COMPLEX*8 C „Ui ,U2 ,U3 ,CASIN ,CACOS ,CATAN ,ARKTAN REAL *4 7(361) ,AUT(361) ,AU1(361) ,AU2(361) ,AU3(361) C KK = 361 H =1.0 C = CMPLX(0.0,1.0) PI = 3.1415926 DTR = PI/180.0 Q2 = 1.0 B2 = 4.0 BET2 = PI R02 = 0.0 TH02 = P I / 2 . 0 Q1 =--1.0 B1 = B2 R01 = 1.0 DLTA= PI/4. 0 GAMA= PI/ 2 . 0 Q3 = QJ B3 = B1 R03 = R01 TH01 = TH02 - GAMA TH03 = TH02 + GAMA BTA 1 = BTA 2 - DLTA BTA3 = BTA2 + DLTA BIG = 0.0 C DO 111 K=1, KK Y(K)= H*(K-1) TH = Y(K)*DTR U1 = CEXP( Bl*COS(TH-BTA1) - U2 = CEXP( B2*COS(TH-BTA2) - U3 = CEXP( B3*COS(TH-BTA3) - C*R01*COS(TH- C*R02*COS(TH- C*R03*COS(TH- THOl) ) •TH02) ) •TH03) ) AUT(K)= CABS( Q1*U1 + Q2*U2 IF( AUT(K) .GT. BIG ) + Q3*U3 ) BIG = = AUT(K) CC AU1(K)= CABS( Q1*U1 ) CC AU2(K)= CABS( Q2*U2 ) CC AU3(K)= CABS( Q3*U3 ) C 111 CONTINUE DO 222 K=1,KK - no - AUTN= AUT(K)/BIG WRITE(6,1) Y(K) , AUTN CC AUN1= AU1(K)/BIG CC AUN2= AU2(K)/B1G CC AUN3= AU3(K)/BIG CC WRITE(6,2) Y(K) ,AUN1 ,AUN2 ,AUN3 ,AUN 222 CONTINUE C 1 FORMAT( F6.1 , IX, E14.7 ) C2 FORMAT( F6.1 ,4(1X, E14.7) ) STOP END C********************************************************************* - I l l - G.2 Beam Diffraction by a Half-Plane This program uses expressions (3.3) and (C.5) to compare the total (asymptotic and uniform) far Fields of beam diffraction by a half-plane with kro=16, 6 o = 9 0 ° and different values of the beam parameters kb and 0=^ + 0 - 6 . - 112 - C Program f o r c a l c u l a t i n g Antenna Beam D i f f r a c t i o n by a Ha l f Screen. C The Complex Source P o i n t S o l u t i o n compared w i t h Asymptotic s o l u t i o n C by Green et a l . ( l 9 7 9 ) . Program c a l l e d "HP.FTN1". 0===================================================================== C The Time Harmonic F a c t o r " e x p ( - i w t ) " i s suppressed . C The Common F a c t o r " e x p ( i . k r ) / S q r t ( k r ) " i s suppressed . C===================================================================== COMMON C , PI C COMPLEX*8 WI ,WR ,UI ,UR ,UTG ,UDF ,CACOS ,CFR COMPLEX*8 C ,RS ,THS ,DL1 ,DL2 ,SC1 ,SC2 ,QST REAL*4 Y(181) ,AUF(181) r A U G ( l B l ) C KK = 181 KM = (KK+1)/2 H =1.0 C C = CMPLX(0. ,1. ) PI = 3.1415926 DTR = PI / I 80. C RO = 1 6 . THO = PI / 2 . C DO 999 J=1,3 B = 2**(J+1) CC IF( J .LE. 1 ) B = 0.0 CC I F ( J .GE. 6 ) B = 12.0 C CC DO 999 J=1,3 CC BETA= THO + PI - J*PI/12. BETA= THO + PI -PI/6. C IF( BETA .EQ. (TH0+PI) ) GO TO 11 C RS = CSQRT( R0**2 + 2.0*R0*C*B*COS(BETA-TH0) - B**2 ) IF( REAL(RS) .LE. 0.0 ) RS = -RS THS = CACOS( (R0*COS(TH0) + C*B*COS(BETA)) /RS ) C CC U REAL(THS) CC V AIMAG(THS) CC RSA = CABS(RS) CC RSR = REAL(RS) CC RSI = AIMAG(RS) CC THSI= PI + ( U + ATAN( SINH(V)*RSR/(RSA + RSI*COSH(V)) ) ) C ==================================================================== C "THSI" and "THSR" ar e the shadow boundary an g l e s f o r i n c i d e n t and C r e f l e c t e d f i e l d s r e s p e c t i v e l y (measured from the h a l f - s c r e e n ) . C ==================================================================== THSI= PI + THO + ACOS( REAL(RS)/R0 ) IF( BETA .GT. (TH0+PI) ) THSI = 2.*(PI+TH0) - THSI GO TO 22 C 11 RS = R0 - C*B THS = THO - 113 - THSI= PI + THO 22 THSR= 2.0*PI - THSI C ==================================================================== C "TH" the o b s e r v a t i o n angle measured from the h a l f - s c r e e n ( X - a x i s ). C " F l " i s the o b s e r v a t i o n angle measured from beam a x i s i n a n t i c l o c k . C ==================================================================== QST = CEXP( C*(RS - 0.75*PI) ) / CSQRT(8.0*PI*RS) c AMX =0.0 DO 111 K=1,KK C Y(K)= HMK-KM) F l = Y(K)*DTR TH = 1 . 5*PI + F l C ==================================================================== C The f a r f i e l d d i s t a n c e "R" i s not used i n c a l c u l a t i n g the p a t t e r n . C "UI" and "UR" are the i n c i d e n t and r e f l e c t e d s f i e l d , r e s p e c t i v e l y . C ==================================================================== UI = CEXP( -C*R0*COS(TH-TH0) + B*COS(TH-BETA) ) UR =-CEXP( -C*R0*COS(TH+TH0) + B*COS(TH+BETA) ) c WI = -CSQRT(2.0*RS) * CCOS((TH-THS)/2.0) WR = -CSQRT(2.0*RS) * CCOS((TH+THS)/2.0) C UTG = ( UI*CFR(WI) + UR*CFR(WR) ) * CEXP(-C*PI/4.)/SQRT(PI) AUG(K)= CABS(UTG) C IF( (AIMAG(THS).EQ. 0.).AND.(REAL(TH-THS) .EQ. PI) ) GO TO 33 IF( (AIMAG(THS).EQ. 0.).AND.(REAL(TH+THS) .EQ. PI) ) GO TO 33 C SC1 = 1,0/CCOS(.5*(TH-THS)) SC2 = 1.0/CCOS(.5*(TH+THS)) DL1 = C*SC1**2/(4.*RS) DL2 = C*SC2**2/(4.*RS) UDF = QST * ( SC1 * ( 1 . - DL1) - SC2*(1.- DL2) ) C IF( TH .LE. THSI ) UDF = UDF + UI IF( TH .LE. THSR ) UDF = UDF + UR AUF(K)= CABS(UDF) C CC IF((AUG(K).GT.AMX).OR.(AUF(K).GT.AMX)) AMX=AMAX1(AUG(K),AUF(K)) IF( AUG(K) .GT. AMX ) AMX= AUG(K) GO TO 111 C 33 ' AUF(K)= 100.0*AMX C ================= 111 CONTINUE C ============== DO 222 K=1,KK UGN = AUG(K)/AMX UFN = AUF(K)/AMX IF( UFN .GT. 1.1 ) UFN = 1.1 WRITE(6,1) Y(K) , UGN , UFN 222 CONTINUE C ======== 999 CONTINUE 1 FORMAT( F6.1 ,2(1X ,E14.7) ) STOP END - 114 - C === ======== = ===== ===== = ============ = = === = ======= = C PROGRAM FOR CALCULATING FRESNEL INTEGRAL "CFR" OF COMPLEX ARGUMENT. c ==================================================================== COMPLEX FUNCTION CFR(X) c COMMON C , PI COMPLEX*8 C , X COMPLEX*16 Z , ERFZ Z = X*CEXP(-C*PI/4.0) CALL CERF(Z,ERFZ) CFR = 0.5*SQRT(PI)*CEXP(C*PI/4.0)*(1.0-ERFZ) RETURN END Q********************************************************************* Q************************************************ * * * * * * * * * * * * * * * * * * * * * - 115 - G.3 Beam Diffraction by a Wide Slit Using expressions (4.5), (4.10) and (4.15), the total diffracted far field including interaction between the edges is calculated for krQ = ka = 8, 0 o = 9O°, 0 = 2 7 0 ° and different values of the beam parameter, kb. - 116'-' £********************************************************************* C PROGRAM FOR CALCULATING ANTENNA BEAM (SINGLE & MULTIPLE) DIFFRACTION C BY A SLIT USING HALF-PLANE SOLUTION.THE PROGRAM IS CALLED "SLIT.FTN" C " NON-SYMETRICAL INCIDENCE " C C TIME DEPENDCE e x p ( - i w t ) 6 COMMON FACTOR e x p ( i ( k r - P I / 4 ) ) / S q r t ( P I * k r ) . C =================================================================== COMMON C ,PI COMPLEX*8 WI1 ,WI2 ,WR1 ,WR2 ,CFR , FR ,CACOS ,CASIN COMPLEX*8 DO ,DI1 ,DI2 ,DF1 ,DF2 ,UI ,UDD ,UDM ,US COMPLEX*8 UE1 ,UE2 ,U1 ,U2 ,UI0 ,UI1 ,UI2 ,UR1 ,UR2 COMPLEX*8 C ,F0 rRS ,RS1 ,RS2 ,THS ,THS1 ,THS2 tCMPLX REAL Y(181) , AUS(181) , AUT(181) C KK = 181 KM = (KK+1)/2 H = 1 . C C = CMPLX(0.0,1.0) PI = 3.1415926 DTR = PI/180. FO = SQRT(PI)*CEXP(C*PI/4.0) C CC CC CC CC CC CC c A =8. RO =8. THO = P I / 2 . B BETA1= TH01 BETA2= 3*PI RS = CSQRT( IF(- REAL (RS ) THS = CACOS( + PI - BETA 1 R0**2 + 2.*R0*C*B*COS(BETA1-TH0) - .LE. 0.0 ) RS = -RS (R0*COS(TH0) + C*B*COS(BETA1)) /RS B**2 ) ) R01 = R02 = TH01 = TH02 = SQRT( R0**2 + A**2 SQRT( R0**2 PI PI v K U - - ^ - ft—* - 2.*R0*A* COS(THO) ) '( + A**2 + 2.*R0*A* COS(THO) ) ASIN(R0*SIN(TH0)/R01) ASIN(R0*SIN(TH0)/R02) RSI = CSQRT( RS**2 + A**2 - 2.*RS*A*CCOS(THS) ) IF( REAL(RS1) .LE. 0.0 ) RS1 = -RS1 RS2 = CSQRT( RS**2 + A**2 + 2.*RS*A*CCOS(THS) ) IF( REAL(RS2) .LE. 0.0 ) RS2 = -RS2 THS1= PI - CASIN(RS*CSIN(THS)/RS1) THS2= PI - CASIN(RS*CSIN(THS)/RS2) THSI1= PI + TH01.+ ACOS(REAL(RS1)/R0l) THSI2= PI + TH02 + ACOS(REAL(RS2)/R02) IF( BETA1 .GT. TH01+PI ) THSI1 = 2.*(PI+TH01) IF( BETA2 .GT. TH02+PI ) THSI2 = 2.*(PI+TH02) THSR1 « 2.*PI - THSI1 THSR2 = 2.*PI - THSI2 THSI 1 THSI 2 UE1 = CEXP(C*RS1) UE2 = CEXP(C*RS2) /CSQRT(RS1) /CSQRT(RS2) ( - 117 - C C IF{ REAL( CSQRT(2.*RS1) *CSIN(THSl/2.) ) .LT. 0. ) STOP IF( REAL( CSQRT(2.*RS1) *CSIN(THS1/2.) ) .LT. 0. ) STOP DO = -SQRTU./PI) * CEXP(-C*(2.*A+PI/4.) ) * FR( SQRT(4.*A) ) DI1 = -CSQRT(2.*RS1/(PI*A))* CFR( CSQRT(2.*RS1)*CSIN(THS1/2.) ) Dl 1 = DI1 * CEXP( C*(2.*A + RS1*CCOS(THS1) - RS1 - PI/4.) ) DI2 = -CSQRT(2.*RS2/(PI*A))* CFR( CSQRT(2.*RS2)*CSIN(THS2/2.) ) DI2 = DI2 * CEXP( C*(2.*A + RS2*CCOS(THS2) - RS2 - PI/4.) ) C C ALL ANGLES ARE IN RADIANS EXCEPT Y(K) IN DEGREES. C BIG = 0.0 DO 111 K=1,KK Y(K)= H*(K-KM) FI = Y(K)*DTR TH1 = 1. 5*PI + FI C TH2 = 3*PI - TH1 IF( THI .LT. PI ) TH2 = PI - TH1 C C THE DISTANCES RI ,R2 ARE NOT USED IN CALCULATING THE PATTERN. C R1 = R-A*COS(TH1) C R2 = R+A*COS(TH1) C RS*COS(TH1-THS) = R0*COS(TH1-THO) + C*B*COS(THI-BETA) C THE UI,S AND UR,S ARE EXPONENTIAL FUNCTIONS. C FOR SYMMETRICAL NORMAL INCIDENCE " UI0 = UI1 = UI2 & UR1 = UR2 " C UIO = CEXP( -C*R0 *COS(TH1-TH0 ) + B*COS(TH1-BETA1) ) UI 1 = CEXP( -C*R01*COS(TH1-TH01) + B*COS(TH1-BETA1) ) UR1 = CEXP( -C*R01*COS(TH1+TH01) + B*COS(TH1+BETA1) ) UI2 = CEXP( -C*R02*COS(TH2-TH02) + B*COS(TH2-BETA2) ) UR2 = CEXP( -C*R02*COS(TH2+TH02) + B*COS(TH2+BETA2) ) WI1 = -CSQRT(2.0*RS1) * CCOS((TH1-THS1)/2.0) WR1 = -CSQRT(2.0*RS1) * CCOS((TH1+THS1)/2.0) WI2 = -CSQRT(2.0*RS2) * CCOS((TH2-THS2)/2.0) WR2 = -CSQRT(2.0*RS2) * CCOS((TH2+THS2)/2.0) C C THE CFR(W) I S A SUBROUTINE CALCULATES FRESNEL INTEGRALS WITH COMPLEX C ARGUMENTS. UI = UI0*F0 U1 = ( UI1*CFR(WI1) - UR1*CFR(WR1) ) * CEXP(-C*A*COS(TH1)) U2 = ( UI2*CFR(WI2) - UR2*CFR(WR2) ) * CEXP(-C*A*COS(TH2)) C C UI fUD AND US ARE THE INCIDENT, DIFFRACTED AND TOTAL SINGLE DIFFRAC- C TION FAR FIELD PATTERNS, RESPECTIVELY. C C US = Ul + U2 - UI AUS(K)= CABS(US) DF1 = -SORT(8.*A/PI) *CEXP( C*(A*COS(TH1)-2.*A-PI/4.)) DF1 = DF1 * FR( SQRT(4.*A)*SIN(THl/2.) ) DF2 = -SQRT(8.*A/PI) *CEXP( C*(A*COS(TH2)-2.*A -PI/4.)) DF2 = DF2 * FR( SQRT(4.*A)*SIN(TH2/2.) ) C CC UDD = UE1*DI1*DF2 + UE2*DI2*DF1 UDM =( UE1*DI1*(D0*DF1+DF2) + UE2*DI2*(D0*DF2+DF1) )/(l.-D0**2) - 118 - AUT(K)= CABS(US+UDM) C IF((AUT(K).GE.BIG).OR.(ADS(R).GE.BIG)) BIG=AMAX1(AUT(K),AUS(K)) C 111 CONTINUE C DO 222 K=1,KK USN= AUS(K)/BIG UTN= AUT(K)/BIG WRITE(6,1) Y(K) , USN , UTN 222 CONTINUE C 1 FORMAT( F6.1 ,2(1X ,E14.7) ) STOP END C = = = = = = = = = = = = = = == = === = = = = = = = == = = = = = C PROGRAM FOR CALCULATING FRESNEL INTEGRAL "CFR" OF COMPLEX ARGUMENTS. C COMPLEX FUNCTION CFR(X) COMMON C , PI COMPLEX*8 C , X COMPLEX*16 Z , ERFZ C C THE CERF(W,ERF) I S A SUBROUTINE CALCULATES ERROR FUNCTIONS WITH C COMPLEX ARGUMENTS. C Z = X*CEXP(-C*PI/4 .D) CALL CERF(Z,ERFZ) CFR = 0.5 * SQRT(PI) * CEXP(C*Pl/4.0) * (1.0-ERFZ) RETURN END C ==================================================================== C PROGRAM FOR CALCULATING FRESNEL INTEGRAL "FR" OF REAL ARGUMENTS. C COMPLEX FUNCTION FR(X) C ========================= COMMON C , PI COMPLEX*8 C COMPLEX*16 Z , ERFZ C • Z = X*CEXP(-C*PI/4.0) CALL CERF(Z,ERFZ) FR = 0.5 * SQRT(PI) * CEXP(C*Pl/4.0) * (1.0-ERFZ) RETURN END Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * - 119 - G.4 Beam Diffraction by a Strip This program calculates the total far field of a normally incident beam diffraction by a conducting strip, neglecting the interaction between the edges, using expression (4.18) with kro=ka = 8, t9 o =90° and different values of beam parameters kb and B=ir + 6 -5 . - 120 - Q********************************************************************* C PROGRAM FOR CALCULATING ANTENNA BEAM SINGLY DIFFRACTED BY A STRIP C USING HALF-PLANE SOLUTION. THE PROGRAM IS CALLED " STRP.FTN ". C = === = = = = = = = = = = = = = = = = = = = = = = = = = == = = = = = = = = = === = = = = = C TIME DEPENDCE " e x p ( - i w t ) " . C COMMON FACTOR " e x p ( i k r ) / S q r t ( k r ) " . C THE ANGLES (THS,THS1,THS2,TH1,TH2,Fl,BETA) ALL ARE IN RADIANS. C ==================================================================== COMMON C , PI COMPLEX*8 C ,F0 ,WI1 ,WR1 ,WI2 ,WR2 ,CASIN ,CACOS ,CFR COMPLEX*8 Ul ,U2 ,UI ,UR ,UD ,UT COMPLEX*8 RS ,RS1 ,RS2 ,THS ,THS1 ,THS2 C REAL*4 Y(361) ,AUT(361) ,AUD(36l) ,AUI(361) ,AUR(361) C Y0 = 00.0 H = 1 . 0 KR =361 c C = CMPLX (0.0,1.0) PI = 3.1415926 DTR = PI/180.0 F0 = CEXP(-C*PI/4.0)/SQRT(PI) C R0 =8.0 A = 8.0 TH0 = PI/2.0 C B = 8.0 CC DO 999 1=1,6 CC B 2.0**(I-2) CC IPC (I .LE. 1).OR.(I .GE. 6) ) B = 2.4*(I-1) C DO 999 J=1,5 BETA= TH0 + PI - P I * ( J - 1 ) / 1 2 . I F ( J .GE. 5 ) GO TO 11 C GO TO 22 11 A = 10.*A BETA= TH0+PI-PI/4. C CC BET1= 3.0*PI - BETA CC • I F ( BETA .LT. PI ) BET1 = PI - BETA CC BET2= BETA C CC R01 = SQRT( R0**2 + A**2 - 2.0*R0*A*COS(TH0) ) CC R02 = SQRT( R0**2 + A**2 + 2.0*R0*A*COS(TH0) ) CC TH01= ASIN( R0*SIN(TH0) /R01 ) CC TH02= ASIN( R0*SIN(TH0) /R02 ) C 22 RS = CSQRT( R0**2 - B**2 + 2.0*C*B*R0*COS(BETA-THO) ) IF( REAL(RS) .LE. 0.0 ) RS = -RS THS = CACOS( (R0*COS(TH0) + C*B*COS(BETA)) /RS ) C RS1 = CSQRT( RS**2 + A**2 - 2.0*RS*A*CCOS(THS) ) RS2 = CSQRT( RS**2 + A**2 + 2.0*RS*A*CCOS(THS) ) IF( REAL(RS1) .LE. 0.0 ) RS1 = -RS1 - 121 - I F ( REAL(RS2) .LE. 0.0 ) RS2 = -RS2 THS1= CASIN( RS*CSIN(THS) /RS1 ) THS2= CASIN( RS*CSIN(THS) /RS2 ) C CC THI1= PI + TH01 + ARCOS(REAL(RSI)/R01) CC I F ( BET1 .GT. (TH01+PI) ) THI1 = 2.0*(PI+THO1) - THI1 CC THR1= 2.0*PI - THI1 C CC THI2 = PI + TH02 + ARCOS(REAL(RS2)/R02) CC I F ( BET2 .GT. (TH02+PI) ) THI2= 2.0*(PI+TH02) - THI2 CC THR2= 2.0*PI - THI2 C BIGT = 0.0 DO 111 K=1 , KK Y(K) = H*(K-1) TH = Y(K)*DTR TH2 = TH THI = PI - TH IF( THI .LT. 0. ) TH1 = TH1 + 2*PI C THE DISTANCE "R" IS NOT USED IN CALCULATING THE PATTERN. C R1 = R+A*COS(TH) C R2 = R-A*COS(TH) C R i = R-RS*CCOS(TH-THS) » C Rr = R-RS*CCOS(TH+THS) c ================================================== MI 1 = -CSQRT(2.0*RS1)*CCOS( (TH1-THS1)/2.0 ) WR1 = -CSQRT(2.0*RS1 )*CCOS( (TH1 +THS1 )'/2 . 0 ) WI2 = -CSQRT(2.0*RS2)*CCOS( (TH2-THS2)/2.0 ) WR2 = -CSQRT(2.0*RS2)*CCOS( (TH2+THS2)/2.0 ) C CC ARI1 = ABS( ATAN( AIMAG(WI1)/REAL(WI1) ) ) CC IF( ARI1 .GT. PI/4.0 ) STOP CC ARR1= ABS( ATAN( AIMAG(WR1)/REAL(WR1) ) ) CC I F ( ARR1 .GT. PI/4.0 ) STOP CC ARI2= ABS( ATAN( AIMAG(WI2)/REAL(WI2) ) ) CC I F ( ARI2 .GT. PI/4.0 ) STOP CC ARR2 = ABS( ATAN( AIMAG(WR2)/REAL(WR2) ) ) CC I F ( ARR2 .GT. PI/4.0 ) STOP C ==================================================================== C "UT" IS THE TOTAL SINGLE DIFFRACTION PATTERN BY STRIP. C "U1,E2" ARE THE TOTAL DIFF. PATTERN BY HALF PLANES FORMING THE STRIP C "UI & ER" ARE INCIDENT AND REFLECTED FIELD PATTERN RESPECTIVELY. C ==================================================================== UI =+CEXP( -C*RS*CCOS(TH-THS) ) UR =-CEXP( -C*RS*CCOS(TH+THS) ) Ul = UI*CFR(WI1) + UR*CFR(WR1) U2 = UI*CFR(WI2) + UR*CFR(WR2) UT = F0*(U1+U2) IF( TH .LT. PI ) UT = UT - (UI+UR) C CC UD = UT - UI CC I F ( (TH1 .LT. THR1).AND.(TH2 .LT. THR2) ) UD = UT - (UI+UR) CC I F ( (TH1 .GT. THI1).AND.(TH2 .GT. THI2) ) UD = UT C AUT(K)= CABS(UT) CC AUD(K)= CABS(UD) CC AUI(K)= CABS(UI) - 122 - CC AUR(K)= CABS(UR) IF( AUT(K) .GT. BIGT ) BIGT = AUT(K) 111 CONTINUE C ============== DO 222 K=1,KK UTN = AUT(K)/BIGT WRITE(6,1) Y(K) , UTN 222 CONTINUE C ========================= 999 CONTINUE C ========================= 1 FORMAT( F6.1 ,1X, E14.7 ) STOP END C PROGRAM FOR CALCULATING FRESNEL INTEGRAL "CFR" OF COMPLEX ARGUMENTS. C COMPLEX FUNCTION CFR(X) C ========================= COMMON C , PI COMPLEX*8 C , X COMPLEX*16 Z , ERFZ Z = X*CEXP(-C*Pl/4.0) CALL CERF(Z,ERFZ) CFR = 0.5*SQRT(PI)*CEXP(C*PI/4.0)*( 1 .0-ERFZ) RETURN END Q ************************************************** C********************************************************************* - 123 - G.5 Beam Diffraction by a Wedge The total diffracted field given by (5.15) or (5.17) is calculated using this program with kro=16, 0 = 6^+17 for different angles of incidence, 6Q and wedge angles, a . - 124 - C PROGRAM FOR CALCULATING BEAM DIFFRACTION BY A CONDUCTING WEDGE USING C THE UNIFORM THEORY OF DIFFRACTION (KOUYOUMJIAN & PATHAK 1974). THE C EDGE LIES ON THE BEAM AXIS. THIS PROGRAM CALLED " WEDG.FTN ". C ============================================== C The time harmonic f a c t o r " Exp(+jwt) " i s suppressed . C Common f a c t o r i s " E x p ( - j . k r ) / S q r t ( k r ) " . C =============== ================================================ COMPLEX*8 C ,RS ,WI1 ,WI2 ,WR1 ,WR2 ,GR ,CFR ,CASIN ,CACOS COMPLEX*8 UO ,UE ,UI ,UR1 ,UR2 ,UD1 ,UD2 ,UD ,UT REAL Y(361) ,AUT(361) ,N C KK = 361 H =1.0 N =1.50 ERR =0.01 C = CMPLX(0.0,1.0) PI = 3.1415926 DTR = PI/180.0 THSY= N*Pl/2.0 THCR= (N-1.0)*PI c R0 = 16.0 TH0 = ? BETA= TH0 + PI C ==================================== C ALW i s - t h e wedge i n t e r i o r a n g l e . CC ALW = (2.0 - N)*PI C c c IF( (TH0 .LT. 0.).OR.(TH0 .GT. THSY ) ) STOP CC I F ( (BETA .LE. PI).OR.(BETA .GT. 2.0*PI ) ) STOP DO 999 J = 1,4 ? B = 2.0*(J-1) ? C ================ RS = R0 + C*B THS = TH0 C ==================================================================: C "THSI" and "THSR" are the shadow boundary angles f o r i n c i d e n t and C r e f l e c t e d f i e l d s r e s p e c t i v e l y (measured from wedge upper s u r f a c e ) THSI = PI + TH0 THSR1= PI - TH0 THSR2= (2.0*N - 1.0)*PI - TH0 UE = CEXP(-C*RS)/CSQRT(RS) U0 = -CEXP(-C*Pl/4.0) / ( N * SQRT(8.0*PI) ) C C "TH" the o b s e r v a t i o n angle measured from the h a l f - s c r e e n ( X - a x i s ) C DO 111 K=1,KK C Y(K)= H*(K-1) TH = Y(K)*DTR Q ===================================================================: C The d i s t a n c e "R" i s not used i n c a l c u l a t i n g the p a t t e r n . - 125 - C " UI , UR1, UR2 " are the i n c i d e n t and r e f l e c t e d f i e l d s from upper C and lower Wedge s u r f a c e s , r e s p e c t i v e l y . C ================================== THI = TH - THS THR = TH + THS UI = CEXP( C*RS*COS(THI) •) UR1 =-CEXP( C*RS*COS(THR) ) UR2 =-CEXP( C*RS*COS(2.0*N*PI-THR) ) C IF( TH .GT. N*PI ) GO TO 88 C MR 1 = 0.0 MI 1 = 0.0 MR2 =0.0 MI 2 = 0.0 IF( (THI .GT. -THSY).AND.(THI .LT. -THCR) ) MI 1= -1.0 IF( (THI .GT. THCR).AND.(THI .LT. N*PI) ) MI 2= +1.0 IF( (THR .GT. THCR).AND.(THR .LT. 1.5*N*PI) ) MR2= +1.0 C WI1 = -CSQRT(2.0*RS) * COS( 0.5*(2*N*PI*MI1 - T H I ) ) WR1 = -CSQRT(2.0*RS) * COS( 0.5*(2*N*PI*MR1 - THR) ) WI2 = -CSQRT(2.0*RS) * COS( 0.5*(2*N*PI*MI2 - T H I ) ) WR2 = -CSQRT(2.0*RS) * COS( 0.5*(2*N*PI*MR2 - THR) ) C TI1 = 0.5*(PI-THI) / N TR1 = 0.5*(PI-THR) / N TI2 = 0.5*(PI+THI) / N TR2 = 0.5*(PI+THR) / N C IF ( ABS(TH-THSI ) .LE. ERR ) GO TO 11 IF ( ABS(TH-THSR1) .LE. ERR ) GO TO 22 C UD1 = GR(WI1)/TAN(TI1) - GR(WR1)/TAN(TR1) GO TO 33 C 11 UD1 = N * CSQRT(2.0*PI*RS) * CEXP(C*PI/4.0) - GR(WR1)/TAN(TR1) GO TO 33 C 22 UD1 = GR(WI1)/TAN(TI1) - N * CSQRT(2.0*PI*RS) * CEXP(C*PI/4.0) C 33 I F ( ABS(TH-THSR2) .LE. ERR ) GO TO 44 C UD2 = GR(WI2)/TAN(TI2) - GR(WR2)/TAN(TR2) GO TO 55 C 44 UD2 = GR(WI2)/TAN(TI2) + N * CSQRT(2.0*PI*RS) * CEXP(C*PI/4.0) C 55 UD = U0 * UE * (UD1+UD2) C IF( (THO .GT. (N-1.0)*PI).AND.(THO .LE. N*PI/2.0) ) GOTO 66 C UT = UD IF( TH .LE. THSR1 ) UT = UD + UI + UR1 IF( (TH .GT. THSR1).AND.(TH .LE. THSI) ) UT = UD + UI GO TO 99 C 66 UT = UD + UI IF( TH .LE. THSR1 ) UT = UT + UR1 IF ( TH .GT. THSR2 ) UT = UT + UR2 GO TO 99 - 126 - C 88 UT = 0.0 C 99 AUT(K)= CABS(UT) IF( AUT(K) .GT .BIG ) BIG = AUT(K) C ==================================== 111 CONTINUE C =============== DO 222 K=1,KK UTN = AUT(K)/BIG WRITE(6,1) Y(K) , UTN 222 CONTINUE C ========== 999 CONTINUE C ========================= 1 FORMAT( F6.1 ,IX, E14.7 ) STOP END C ==================================================================== C PROGR. FOR CALCULATING MODIFIED FRESNEL INTEGRAL OF COMPLEX ARGUMENT C COMPLEX FUNCTION GR(X) C ======================= COMPLEX*8 C , X , CFR C = CMPLX(0.0,1.0) PI = 3.1415926 C I F ( REAL(X*CEXP(C*PI/4.0)) .LT. 0.0 ) X = -X GR = 2.0 * C * X * CEXP(C*X*X) *CFR(X) RETURN END C ==================================================================== C PROGRAM FOR CALCULATING FRESNEL INTEGRAL "CFR" OF COMPLEX ARGUMENT. C ==================================================================== COMPLEX FUNCTION CFR(X) C ========================= COMPLEX*8 C , X COMPLEX*16 Z , ERFZ c C = CMPLX(0.0,1.0) PI = 3.1415926 C Z = X * CEXP(C*PI/4.0) CALL CERF(Z,ERFZ) CFR = 0.5 * SQRT(PI) * CEXP(-C*PI/4.0) * (1.0 - ERFZ) RETURN END Q******************************************************* * * * * * * * * * * * * * * Q********************************************************************* - 127 - G.6 Beam Diffraction by a Circular Aperture This program makes use of expressions (6.33a,b) to calculate the single and multiple diffraction total fields modified on the caustic axis with kz0=ka=3ir, 0 O =9O°, /3 =270° and different values of the beam parameter, kb. - 128 - C* C c c c c c c c= **************************************************** PROGRAM CALCULATES SINGLE AND MULTIPLE BEAM DIFFRACTION BY CIRCULAR APERTURE (NORMAL INCIDENCE), MODIFIED ON THE CAUSTIC AXIS, USING THE UNIFROM THEORY OF DIFFRACTION & COMPLEX SOURCE-POINT REPRESENTATION. THIS PROGRAM IS CALLED " CRCL.FTN2 ". THE TIME DEPENDENCE " e x p ( - i w t ) " AND A COMMON FACTOR " e x p ( i k r ) / k r n ARE SUPPRESSED. COMMON C ,PI ,THSI COMPLEX*8 C ,CASIN ,CACOS ,CATAN ,DC ,FR ,CFR COMPLEX*8 ZS ,RS1 ,THS1 ,WI ,B0S ,B1S ,B0M ,B1M COMPLEX*8 UE ,UI ,UR ,U1 ,U2 ,USD ,UMD ,US COMPLEX*8 DO ,D1 ,D2 ,DI ,DF1 ,DF2 REAL *4 Y(181) , AUS(181) , AUT(181) C C C C C C C C J J = 16 KK = 91 H =1.0 C = CMPLX(0.0,1.0) PI = 3.1415926 DTR = PI/180. A = 3*PI Z0 = A BETA = 1.5*PI • R01 = SQRT( Z0**2 + A**2 ) TH01 = PI - ATAN( ZO/A ) DO 999 1=1,6 B = 2.**(I-1) IF ( I .LE. 1 ) B IF ( I .GE. 6 ) B 0.0 85.0 ZS Z0 - C*B RSI = CSQRT( ZS**2 + A**2 ) IF ( REAL(RSI) .LE. 0.0 ) RS1 THSI = PI - CATAN( ZS/A ) -RSI THSI, THSR ARE THE SHADOW AND REFLECTION BOUNDARY ANGLES, THSI = PI + TH01 + ACOS( REAL(RS1)/R0l ) IF( BETA .GT. TH01+PI ) THSI = 2.MPI+TH01) -THSI THSR = 2.0*PI - THSI UE = CEXP(C*RS1) / RS1 DO = SQRTU./PI) * CEXP( C*(PI/4 - 2.*A) ) * FR( SQRT(4.*A) ) WI = CSQRT(2.0*RS1) *CSIN(THS1/2.0) IF( REAL( WI*CEXP(-C*PI/4.) ) .LT. 0.0 ) TOP Dl « CSQRT(2.*RS1/(PI*A)) *CEXP(C*(PI/4.+2.*A)) Dl = Dl * CEXP(-C*WI**2) *CFR(WI) 129 - Q K S B B C B S S B E B C S B S B B S S B B B S B B E S S S S S S E S BG «= 0.0 c DO 111 K=1,KK Y(K) = H*(K-1) FI « Y(K)*DTR C THI «= 1 .5*PI + FI TH2 = 1.5*PI - FI C WF1 «= SQRT(4.0*A) * SIN(THl/2.0) WF2 «= SORT(4.0*A) * SIN(TH2/2.0) DF1 = CEXP(-C*WF1**2) * FR(WF1) DF2 «= CEXP(-C*WF2**2) * FR(WF2) C D1 * DC(THS1,RS1,TH1) D2 « DC(THSI,RS1,TH2) C UI •= CEXP( C * ZS * COS(FI) ) UR * -CEXP(-C * ZS * COS(FI) ) C IF( K .GT. JJ ) GO TO 11 C BOS « (D2 + Dl ) * BESJ0(A*SIN(FI),1) B1S • (D2 - D1) * BESJ1(A*SIN(FI),1) USD « A * SQRT(PI/2.0) * CEXP(-C*PI / 4.0) * UE * (BOS + C*B1S) C BOM - (DF1 + DF2) * B E S J 0 ( A * S I N ( F I ) , I ) B1M - (DF1 - DF2) *B E S J 1 ( A * S I N ( F I ) , I ) UMD « 2. * (A**1.5) * UE * Dl * (B1M + C*B0M) / (1.0-D0) C GO TO 22 C II U1 CEXP( -C*(A*SIN(FI) - P I / 4 . ) ) U2 « 1.0/U1 USD « SQRT(A/SIN(FI)) *CEXP{-C*PI / 4.) *UE *(U1*D1 + U2*D2) C UMD - C *A *SQRT( 8.0 / ( P I * S I N ( F I ) ) ) UMD = UMD *UE *DI *( U1*DF1 + U2*DF2 ) / (1.0-D0) C US « USD 22 I F ( (THI .LT. THSI).AND.(TH2 .LT. THSI) ) US = USD + UI C AUS(K)= CABS(US) AUT(K)= CABS(US+UMD) IF( (AUS(K).GT.BG).OR.(AUT(K).GT.BG) ) BG=AMAX1(AUT(K),AUS(K)) Q r s c c c n c c = c = c t n = = = = n c = = = = = : = = = : = r = c E C = = = = = = = = = = = = = = = = = = : = = = = = = = I I I CONTINUE Q CESCBBeBBSSBeESEBSBB DO 222 K «= 1 , KK USN * AUS(K)/BG UTN «= AUT (K) /BG WRITE (6 ,1) Y(K) , USN , UTN 222 CONTINUE Q B C C C E C = C C 999 CONTINUE £ B K B B K B B B B E B B S B B B S B B B B B B B B S B B S S 1 FORMAT( F6.1 , 2(1X , E14 .7 ) ) STOP END - 130 - C = = = = = = = w = = = = = = = * = = = = = = = = = = = = = = = s = c: = = K = ti = = = = = z:c: = = = = = ^ C PROGRAM FOR CALCULATING DIFFRACTION COFFICIENT . C COMPLEX FUNCTION DC(THS,RS,TH) COMMON C ,PI ,THSI COMPLEX*8 C ,RS ,THS ,WI ,WR ,DI ,DR ,CFR WI = -CSQRT(2.0*RS) * CCOS( " (TH-THS)/2.0 ) WR = -CSQRT(2.0*RS) * CCOS( (TH+THS)/2.0 ) DI «= CFR(WI ) DR •= CFR(WR) IF( REAL(WI*CEXP(-C*PI/4.)) .LT. 0. ) DI = -CFR( -WI ) IF( TH .LT. THSI ) DI = -CFR(-WI ) DC = CSQRT(RS/PI) *CEXP(-C*PI/4.0) DC = DC *( DI*CEXP(-C*WI**2) - DR*CEXP(-C*WR**2) ) RETURN END C PROGRAM FOR CALCULATING FRESNEL INTEGRAL "CFR" OF COMPLEX ARGUMENTS. C s E B E E E S e E S S S B C C E E E E S E E e S B e e C E S S E B E E S C C e E e B C e z S S C E B e e C S E S E E S C Z S S E E S E S : COMPLEX FUNCTION CFR(X) C = = = = = = = = = = = = = = = = = = = = = = = COMMON C ,PI COMPLEX*B C , X COMPLEX*16 Z , ERFZ C Z «= X*CEXP(-C*PI/4.0) CALL CERF(Z,ERFZ) CFR = 0.5*SQRT(PI)*CEXP(C*Pl/4.0)*(1.0-ERFZ) RETURN END C C PROGRAM FOR CALCULATING FRESNEL INTEGRAL "FR" OF REAL ARGUMENTS. C E E E E E E E E B E B E E E E E E . E E E E E E E E E ^ E E Z E E B E E E E E E B E B B E E E E B E E E E E E B E E E E E E E E E E E E COMPLEX FUNCTION FR(X) C « " = " = = • = = " " === = = COMMON C ,PI ,THSI COMPLEX*8 C COMPLEX*16 Z , ERFZ C Z = X*CEXP(-C*PI/4.0) CALL CERF(Z,ERFZ) FR = 0.5*SQRT(PI)*CEXP(C*Pl/4.0)*(1.0-ERFZ) RETURN END Q ********************************************************* £********************************************************************* - 131 - G.7 Beam Diffraction by a Parabolic Reflector The following program uses (E17) to calculate the diffracted field component for a beam of parameters (kb=16, 0=180°), located at (kro=0, 0O=°°) incident on a conducting parabolic reflector (kF=107r, <? e=60°). Also, the program can be used to calculate the reflected and total far fields using (E5), (E4) and (E17), without using the computer search technique. - 132 - C********************************************************************* C PROGRAM CALCULATES ANTENNA BEAM DIFFRACTION BY A PARABOLIC REFLECTOR C NO COMPUTER SEARCH FOR REFLECTION POINTS. C THE PROGRAM IS CALLED "PRBL.FTN,NS". Q = c c = = = = = t = c c c c = : = = B C = = r: = = E = = = =:r:c:r:=; = c = = = = = = = = s: = = = — = = = = C C = = E S I = = = = = : = = = : = ; — = = 1 C TIME DEPENDCE " E x p ( - i w t ) ". C COMMON FACTOR " E x p ( i k r ) / S q r t ( k r ) ". COMPLEX*8 C ,UI rUR rUD1 ,UD2 ,UD ,UT ,CD ,CFR ,CASIN ,CACOS COMPLEX*8 THP ,THS ,THSP ,THS1 ,THS2 ,THI fTH1P ,THS1P ,WP COMPLEX*8 FO ,RI ,RR ,RS ,RP ,RSP ,RS1 ,RS2 ,RC ,RX ,XP ,YP COMPLEX*8 XS ,YS REAL* 4 2(361) ,AUI(361) ,AUR(361) ,AUD(361) ,AUT(36l) C KK = 181 H =1.0 Q ======= C = CMPLX(0.0,1.0) PI = 3.1415926 DTR « PI/180. FO = CEXP(-C*PI/4.)/SQRT(PI) C RO = 0.0 THO = 0.0 F = 10*PI THE = PI/3.0 BET = PI+TH0 B = 16.0 C ============================================================== C BW IS THE HALF-POWER BEAM WIDTH C ============================================================== CC BW «= 66.*DTR CC B 0.5*ALOG(2.0)/(1.0 - COS(BW/2.0)) C THT = (PI-THE)/2. RE = 2*F/(1.0+COS(THE)) R01 = SQRT( RE**2 + R0**2 + 2*RE*R0*COS(THE+TH0) ) TH01= THT + ASIN( R0*SIN(THE+TH0) /R01 ) C I F ( BET .EQ. THO+PI ) GO TO 11 RS = CSQRT( R0**2 - 2*C*R0*B*COS(BET-TH0) - B**2 ) IF( REAL(RS) .LE. 0.0 ) RS = -RS THS = CCOS( ( R0*COS(TH0) + C*B*COS(BET) ) /RS ) GO TO 22 C 11 RS = R0 - C*B THS * THO C 22 RS1 = CSQRT( RE**2 + RS**2 + 2*RE*RS*CCOS(THE+THS) ) IF ( REAL(RS1) .LE. 0.0 ) RSI = -RS1 THS1= THT + CASIN( RS*CSIN(THE+THS) /RS1 ) RS2 = RSI THS2= THS1 C ============================================================== C THBI, THBR ARE THE SHADOW AND REFLECTION BOUNDARY ANGLES C • THBI= TH01 + PI + ACOS(REAL(RS1)/R01) IF( BET .GT. (TH01+THT) ) THBI = 2*(PI+TH01) - THBI THBR= 2*PI - THBI - 133 - Q B B S S B S S S S E E S S S B S E BIG = 0.0 DO 111 K = 1,KK 2(K)= H*(K-1) TH «= Z(K)*DTR C TH 1 = (PI-THT) + TH IF( TH1 .GT. 2.0*PI ) TH1 = TH1 - 2.0*PI TH2 «= (PI-THT) - TH IF ( TH2 .LT. 0.0 ) TH2 = TH2 + 2.0*PI C UD1 «= FO * CEXP( C*RE*COS(TH+THE) ) * CD(RS1 ,THS1 ,TH1 ) IF( (TH .GT. PI/2.).AND.(TH .LT. (PI-THT)) ) GOTO 33 UD2 = FO * CEXP( C*RE*COS(TH-THE) ) * CD(RS2,THS2,TH2) GO TO 44 33 UD2 = 0.0 c 44 UD = UD1 + UD2 AUD(K)= CABS(UD) IF( AUD(K) .GT. BIG ) BIG = AUD(K) C TO CALCULATE THE DIFFRACTED FIELD COMPONENT ONLY GO TO 111 GO TO 111 C 110 I F ( TH .GT. (PI-THE) ) GO TO 55 THR « TH THP «=-.5 *( THS+THR + CASIN(4.*F*SIN(THR)/RS + CSIN(THS+THR)) ) C RP - 2.*F/(1.+CCOS(THP)) I F ( REAL(RP) .LE. 0.0 ) RP * -RP C RSP = CSQRT( RP**2 + RS**2 + 2.*RP*RS*CCOS(THP+THS) ) IF( REAL(RSP) .LE. 0.0 ) RSP = -RSP THSP= THR+THP C TH 1P «= (PI+THP)/2. + TH THS1P= (PI+THP)/2. - THSP WP « -CSQRT(2.*RSP) * CCOS((TH1P+THS1P)/2.) RWP = REAL( WP * CEXP(-C*PI / 4 . ) ) I F ( RWP .GT. 0.0 ) GO TO 55 C THI = THP/2.0 - THSP IF( REAL(THI) .LT. 0. ) THI =-THI RC = 2.0*F/CCOS(THP/2.0)**3 RX = -RC*CCOS(THI)*RSP/( 2.0*RSP-RC*CCOS(THI) ) C UR - CEXP( C M RSP + RP*CCOS(TH+THP) ) ) GO TO 66 55 UR « 0.0 66 AUR(K)= CABS(UR) CC I F ( AUR(K) .GT. BIG ) BIG «= AUR(K) C I F ( THI .GT. THBI ) GO TO 77 UI « CEXP( B*COS(TH-BET) - C*R0*COS(TH-TH0) ) GO TO 88 77 UI - 0.0 88 AUI(K)= CABS(UI) C UT * UI + UR + UD - 134 - AUT(K)= CABS(UT) CC IF( AUT(K) .GT. BIG ) BIG •= AUT{K) C . 111 CONTINUE Q c=c=ec=cec DO 222 K *= 1 , KK CC UIN = AUI(K)/BIG CC URN • AUR(K)/BIG UDN = AUD(K)/BIG CC UTN = AUT(K)/BIG 222 WRITE(6,1) Z(K) ,UDN c 1 FORMAT( F6.1 ,1X, E14.7 ) STOP END Ccesccccc=cEecccecsnctci=s:crce=c=ccccccc==c=c=s=c==i:c=s=cc=scc=====r= C PROGRAM CALCULATES THE DIFFRACTED FIELD COMPONENT BY A HALF SCREEN. COMPLEX FUNCTION CD(R0,TH0,TH) Q KSBSBSeBS&&&BSS&SSSSeBSS&SESSBS 'S COMPLEX*8 C ,TH0 ,R0 ,WI ,WR ,EI ,ER ,DI ,DR ,CFR C PI « 3.1415926 C * CMPLX(0.0,1.0) C EI • CEXP( -C*R0 *CCOS(TH-TH0) ) ER « CEXP( -C*R0 *CCOS(TH+TH0) ) WI «-CSQRT(2.0*RO) * CCOS((TH-THO)/2.0) WR —CSQRT(2.0*R0) * CCOS((TH+THO)/2.0) RWI « REAL( WI *CEXP(-C*Pl/4.) ) RWR - REAL( WR *CEXP(-C*PI/4.) ) C DI « CFR(WI) DR « CFR(WR) IF ( RWI .LT. 0.0 ) DI = -CFR(-WI) IF ( RWR .LT. 0.0 ) DR •= -CFR(-WR) CD « EI*DI - ER*DR RETURN END ^ t S X B B B B B B B B B B E S S B B B B S B B B S B B B B B B B S S B B S S e B B S S B B E C ' e B B B e S B B B S S B B B B S B B S B B S S C PROGRAM CALCULATES THE FRESNEL INTEGRAL "CFR" OF COMPLEX ARGUMENTS. Q C t S - B S B E S S S B S S B B S S B B B S S E E S S S E S S S S & e S S E S & S S S B S B S S S S & e S S S S B e s S S S E E E S S S E E COMPLEX FUNCTION CFR(X) £ B S S S B S S S S B B B B B B B S B S S E E B E S E COMPLEX*8 C , X COMPLEX*16 Z , ERFZ c PI « 3.1415926 C « CMPLX(0.0,1.0) C Z » X * CEXP( -C*Pl/4.0 ) CALL CERF(Z,ERFZ) CFR * 0.5 * SQRT(PI) * CEXP(C*PI/4.0) * (1.0 - ERFZ) RETURN END C********************************************************************* C*********************************************************************

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