UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Diffraction of antenna radiation patterns by local obstacles Cheung, Hong D. 1999

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_1999-463311.pdf [ 5.06MB ]
Metadata
JSON: 831-1.0065036.json
JSON-LD: 831-1.0065036-ld.json
RDF/XML (Pretty): 831-1.0065036-rdf.xml
RDF/JSON: 831-1.0065036-rdf.json
Turtle: 831-1.0065036-turtle.txt
N-Triples: 831-1.0065036-rdf-ntriples.txt
Original Record: 831-1.0065036-source.json
Full Text
831-1.0065036-fulltext.txt
Citation
831-1.0065036.ris

Full Text

DIFFRACTION OF ANTENNA RADIATION PATTERNS BY LOCAL ~ OBSTACLES by HONG D. CHEUNG 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 MARCH 1999 © HONG D. CHEUNG, 1999 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 ZLECTtfZCAC k*J> QoMPUTZR £ACZHE£*zU&) The University of British Columbia Vancouver, Canada Date JULY 2S , DE-6 (2/88) Supervisor: Dr. Edward V. Jul! ABSTRACT Radiowave blockage by buildings is a problem common to mobile radio and cellular telephone systems in high-density urban areas. Plane wave incidence is an approximation which simplifies analysis of the scattering problem, but this is not an accurate assumptions if the obstacle is near the antenna. If the transmitting antenna has an omnidirectional pattern and the obstacle is large in wavelengths the scattered field can be calculated by the techniques of high frequency diffraction theory such as the geometrical theory of diffraction. If the antenna pattern is directive and the obstacle is local, the omnidirectional source solution can be converted to a beam source solution by the complex source point (CSP) method. Antenna patterns with sidelobes can be synthesized from arrays of CSP beams, each with appropriate amplitude, phase and direction. Their individual scattering from apertures and buildings may be calculated and the result summed for the total field. The complex source point method is used here to produce the basis elements of an array of linearly and directionally equispaced two-dimensional CSP beams and compared with Gaussian beam results obtained by others earlier. For efficiency a limited number of significant beams and beam directions is required. Approximately twice as many beams as the aperture width in wavelengths, with all beam directions normal to the aperture, is found to be sufficient here for simple uniform and cosinusoidal distributions in apertures of moderate size at ranges outside the evanescent field zone of the aperture. Now the exact solution for the far field of a line source, or here a beam source in the presence of a i i conducting half plane, is used as our basis element, to give the solution for antenna pattern diffraction by a local half plane. In this work a beam source in the presence of a conducting half plane, is used as our basis element, to give the solution for antenna pattern diffraction by a local half plane. Antenna pattern diffraction by an aperture near a wide slit is presented as simply a superposition of the solutions for two coplanar half planes with separated parallel edges. Antenna pattern distortion by various other local obstacles, such as conducting wedges and rectangular cylinders, can be obtained similarly. Numerical results, including both extended apertures and a single beam source, represented by arrays of complex source point beams and by a single CSP beam are given. From scattering patterns the latter are shown to accurately represent in some cases aperture distributions such as cosine-squared and with less accuracy uniform aperture distributions. The exact series solution and the moment method solution are used to verify the accuracy of the solution for conducting wedges and a rectangular cylinder of moderate size. Two dimensional scattering by conducting circular cylinders of small to moderate size in wavelengths, in the presence of local directive sources, is also rigorously analyzed. Calculated scattering cross sections show that for ^-polarized local sources the cross section first decreases and then increases with increasing directivity whereas for H-polarized sources the reverse occurs. i i i TABLE OF CONTENTS ABSTRACT , • ( i i ) ACKNOWLEDGEMENT (till) TABLE OF CONTENTS (Lv) LIST OF FIGURES ( v i i ) 1. MOTIVATION AND LITERATURE REVIEW 1.1 I n t r o d u c t i o n (1) 1.2 Complex Source Point (CSP) Method (4) 1.3 Gabor's Expansion (6) 1.4 Overview of The Thesis (7) 2. SYNTHESIS OF ANTENNA PATTERNS 2.1 A n a l y t i c a l Expressions (9) 2.2 Numerical Results (11) 2.2.1 Far F i e l d C a l c u l a t i o n s (11) 2.2.2 Near F i e l d C a l c u l a t i o n s (13) 3. APERTURE DIFFRACTION BY A CONDUCTING HALF^PLANE 3.1 A n a l y t i c a l Expressions (23) 3.2 Numerical Results (25) 3.2.1 ^ - p o l a r i z a t i o n Far F i e l d C a l c u l a t i o n (25) 3.2.2 H - p o l a r i z a t i o n Far F i e l d C a l c u l a t i o n (28) 4. APERTURE DIFFRACTION BY A WIDE SLIT 4.1 A n a l y t i c a l Expressions (42) 4.2 Numerical Results (43) 4.2.1 £-polarization Far F i e l d C a l c u l a t i o n (43) i v 4.2.2. H - p o l a r i z a t i o n Far F i e l d C a l c u l a t i o n (45) 5. APERTURE DIFFRACTION BY A CONDUCTING WEDGE 5.1 A n a l y t i c a l Expressions (55) 5.2 Numerical Results ; (57) 5.2.1 £-polarization Far F i e l d C a l c u l a t i o n (57) 5.2.2 H - p o l a r i z a t i o n Far F i e l d C a l c u l a t i o n (59) 6. APERTURE DIFFRACTION BY A CONDUCTING SQUARE CYLINDER 6.1 A n a l y t i c a l Expressions (72) 6.2 Numerical Results (76) 6.2.1 £-polarization Far F i e l d C a l c u l a t i o n (76) 6.2.2 H - p o l a r i z a t i o n Far F i e l d C a l c u l a t i o n (80) 7. APERTURE DIFFRACTION BY A CIRCULAR CYLINDER 7.1 A n a l y t i c a l Expressions (101) 7.2 Numerical Results (105) 7.2.1 S c a t t e r i n g Patterns (105) 7.2.2 S c a t t e r i n g Cross Section • (109) 7.2.3 Current Density (112) 8. CONCLUSIONS AND RECOMMENDATIONS 8.1 Conclusions • (128) 8.2 Future P o s s i b i l i t i e s (131) REFERENCES (133) TABLE 1. Aperture C o e f f i c i e n t s Am For Various Aperture D i s t r i b u t i o n (138) APPENDIX A. UNIFORM THEORY OF DIFFRACTION FOR A WEDGE-(139) v APPENDIX B. MOMENT METHOD LIST OF FIGURES Figure Title p a g e 2.1a Geometry of multiple complex line sources and real far observation point, in polar coordinates. - 15 2.1b Geometry of synthesis of an array of complex line sources for an arbitrary aperture distribution. 15 2.2 The far field patterns for a cosine-squared aperture distribution. 16 2.3 The far field patterns for a uniform aperture distribution. 17 2.4 The far field patterns for a cosine aperture distribution. 18 2.5 The far field patterns for the compound aperture distribution. 18 2.6 Near field patterns for a cosine-squared aperture distribution. — 19 2.7 Near field patterns for a uniform aperture distribution. 20 2.8 Near field patterns for a uniform aperture distribution, (comparison between complex sources solution and Fresnel transform solution for La - 2.5X) 21 2.9 Near field patterns for a uniform aperture distribution, (comparison between complex sources solution and Fresnel transform solution for L0 - 5X) 22 3.1 Geometry of an array of complex sources for beam diffraction by a conducting half-plane. — 30 3.2 Normalized E-polarized far field diffraction patterns of an array of complex line sources above a perfectly conducting half-plane, (kb = 0) -— 31 3.3 Normalized E-polarized far field diffraction patterns of an array of complex line sources above a perfectly conducting half-plane, (kb = 4) 32 3.4 A comparison of normalized E-polarized far field diffraction patterns of a cosine-squared aperture distribution (L0 = 2.5X) and a single complex source point above a perfectly conducting half-plane. 33 3.5 A comparison of normalized E-polarized far field diffraction patterns of a cosine-squared aperture distribution (L0 = 5X) and a single complex source point above a perfectly conducting half-plane. 34 VII'; 3.6 A comparison of normalized £-polarized far field diffraction patterns o f a uniform aperture distribution (L0 = 2.5X) and a single complex source point above a perfectly conducting half-plane. 3 5 3.7 A comparison of normalized £-polarized far field diffraction patterns of a uniform aperture distribution (L0 = 5X) and a single complex source point above a perfectly conducting half-plane. 36 3.8 A comparison of normalized tf-polarized far field diffraction patterns of a cosine-squared aperture distribution (L„ = 2.5X) and a single complex source point above a perfectly conducting half-plane. 37 3.9 A comparison of normalized //-polarized far field diffraction patterns o f a cosine-squared aperture distribution (L0 = 5X) and a single complex source point above a perfectly conducting half-plane. 38 3.10 A comparison of normalized //-polarized far field diffraction patterns of a uniform aperture distribution (L0 = 2.5X) and a single complex source point above a perfectly conducting half-plane. 39 3.11 A comparison of normalized H-polarized far field diffraction patterns of a uniform aperture distribution (L0 = 5\) and a single complex source point above a perfectly conducting half-plane. 40 3.12 Convergence of the exact series solution and the uniform solution for a cosine-squared aperture diffraction by a conducting half-plane. (£-polarization) 41 4.1 Geometry of an array of complex source beams diffraction by a slit. 47 4.2 A Comparison of normalized £-polarized far field diffraction patterns of a slit excited by a cosine-squared distribution and a single complex line source at y 0 behind the center of the slit. (y0 = LJ2) 48 4.3 A Comparison of normalized £-polarized far field diffraction patterns of a slit excited by a cosine-squared distribution and a single complex line source at y0 behind the center of the slit. (y0 = 2Lo) 49 4.4 A Comparison of normalized £-polarized far field diffraction patterns of a slit excited by a cosine-squared distribution and a single complex line source at y0 behind the center of the slit. (y0 = 10Lo) 50 4.5 A Comparison of normalized £-polarized far field diffraction patterns of a slit excited by a uniform distribution and a single complex line source at y0 behind the center of the slit. (y«, = U2) 51 viii 4.6 A Comparison of normalized E-polarized far field diffraction patterns of a slit excited by a uniform distribution and a single complex line source at y„ behind the center of the slit. (y0 = 1L0) 52 4.7 A Comparison of normalized //-polarized far field diffraction patterns of a slit excited by a cosine-squared distribution and a single complex line source at behind the center of the slit. 53 4.8 A Comparison of normalized //-polarized far field diffraction patterns of a slit excited by a uniform distribution and a single complex line source at behind the center of the slit. 54 5.1 Geometry of an array of complex source beams for diffraction by a conducting wedge. 62 5.2 A comparison of normalized E-polarized far field of a right angled wedge excited by a cosine squared aperture distribution and a single complex source beam. (L* = 2.5X). 63 5.3 A comparison of normalized E-polarized far field of a right angled wedge excited by a uniform aperture distribution and a single complex source beam. (La = 2.5X). 64 5.4 A comparison of normalized E-polarized far field of a right angled wedge excited by a cosine squared aperture distribution and a single complex source beam. ( La - 5X). 65 5.5 A comparison of normalized E-polarized far field of a right angled wedge excited by a uniform aperture distribution and a single complex source beam. (La = 5X). 66 5.6 Convergence of the exact series solution and the UTD solution for E-polarized far field of a uniform aperture distribution above a perfectly conducting wedge. 67 5.7 A comparison of normalized //-polarized far field of a right angled wedge excited by a cosine squared aperture distribution and a single complex source beam at kr0 = 16 from the edge. 68 5.8 A comparison of normalized //-polarized far field of a right angled wedge excited by a cosine-squared aperture distribution and a single complex source beam, at kr0 = 32 from the edge. 6 9 5.9 A comparison of normalized //-polarized far field of a right angled wedge excited by a uniform aperture distribution and a single complex source beam at kr0 = 16 from the edge. 7 0 ix 6.1 6.2 5.10 A comparison of normalized //-polarized far field of a right angled wedge excited by a uniform aperture distribution and a single complex source beam at kr0 = 32 from the edge. 7 \ Geometry of an array of complex source beams diffraction by a conducting square cylinder. 83 Coordinate systems of the diffraction problem. 84 6.3 E-polarized far field of a square cylinder for a single beam source. (kr0 - 16, a = 2X, 60 = 371/4). 85 6.4 E-polarized far field of a square cylinder for a single beam source. (kra = 16, a , = 5X,60 = 3nJ4). 86 6.5 E-polarized far field of a square cylinder for a single beam source. (kr0 = 32, a = 2X, 60 = 3;t/4). 87 6.6 E-polarized far field of a square cylinder for a single beam source. (kr0 = 32, a = 2X, So - nil). 88 6.7 E-polarized far field of a square cylinder for a single beam source. (kr0 = 16, a = 2X,60 = rt2). 89 6.8 E-polarized far field of a square cylinder for a uniform aperture distribution. 90 (6O = 3K/4) 6.9 //-polarized far field of a square cylinder for a single beam source. {kr0 = 16, a = 2X, 6O = 3TC/4). 91 6.10 //-polarized far field of a square cylinder for a single beam source. (kr0 = 16, a = 5X, 60 = 3itJ4). 92 6.11 //-polarized far field of a square cylinder for a single beam source. {kr0 = 32, a = 2X, 60 = 37t/4). 93 6.12 Far zone diffracted field by a square cylinder for a single beam source. (H-polarization, kr0 = 16, a = 2X, 60 - nil). 94 6.13 //-polarized far field of a square cylinder for a single beam source. (kr„ = 32, a = 2X, 60 = TC/2). 95 6.14 A comparison between UTD solution and Moment method for //-polarized far field of a square cylinder for a single beam source. (kr0 = 16, a = 2X, 60 = 3rc/4). 96 6.15 A comparison between UTD solution and Moment method for //-polarized far field of a square cylinder for a single beam source. (kr0 = 32, a = 2K, 60 = 3TI/4). 9 7 6.16 A comparison between UTD solution and Moment method for //-polarized far field of a square cylinder for a single beam source. (kr0 = 16, a = 2K, 90 = TC/2). 98 6.17 A comparison between UTD solution and Moment method for //-polarized far field of a square cylinder for a single beam source. (kr0 = 32, a = 2K, 60 = TT/2). 99 6.18 //-polarized far field of a square cylinder for a uniform aperture distribution. (60 = 3TT/4) 100 7.1 Geometry of an array of complex source beams diffraction by a conducting circular cylinder. 114 7.2 Normalized E-polarized total far field diffraction patterns of a circular cylinder, (ka = 2.5rc, kxa = 5TT). 115 7.3 Normalized E-polarized total far field diffraction patterns of a circular cylinder, (ka = 2.5TT, kx0 = 25re). 116 7.4 Normalized E-polarized total far field diffraction patterns of a circular cylinder for a cosine-squared aperture distribution. 117 7.5 Normalized E-polarized total far field diffraction patterns of a circular cylinder for a uniform aperture distribution. 118 7.6 Normalized //-polarized total far field diffraction patterns of a circular cylinder, (ka = 2.5K, kx0 = 5n). 119 7.7 Normalized //-polarized total far field diffraction patterns of a circular cylinder, (ka = 2.5n, kxa = 25*). 120 7.8 Normalized //-polarized total far field diffraction patterns of a circular cylinder for a cosine-squared aperture distribution. 121 7.9 Normalized E-polarized total far field diffraction patterns of a circular cylinder for a uniform aperture distribution. 122 7.10 Normalized total scattering cross sections for a conducting cylinder illuminated by an E-polarized line source. 123 7.11 Normalized total scattering cross sections for a conducting cylinder illuminated by an //-polarized line source. — 124 xi 7.12 Normalized total scattering cross sections for a conducting cylinder with radius ka = 3.5. - 125 7.13 Normalized total scattering cross sections for a conducting cylinder with the source at a distance kx0 - 16. — 126 7.14 Current density on a conducting cylinder of radius ka = 6. 127 xii ACKNOWLEDGMENTS The author wishes to express his deepest gratitude to Dr. E. V . M l of the Department of Electrical and Computer Engineering for all his guidance and encouragement, academic and personal assistance throughout the graduate program leading to this dissertation. Xiii CHAPTER 1 MOTIVATION AND LITERATURE REVIEW 1.1 Introductions In a variety of applications of high frequency communications, the information carrying signals encounter complicated environmental conditions along the propagation path through interaction with obstacles. For example, radio blockage by local obstacles such as fences, building parapets and buildings occurs for many paraboloidal reflector antennas in high density urban areas. This thesis presents procedures for systematically predicting the effects of certain simple structures in the near field of antenna apertures. The analytical and numerical techniques developed are in two dimensions only. However these techniques may be applied directly to the prediction of scattering in the principle planes of radiating rectangular apertures such as pyramidal horns or linearly polarized rectangular arrays of horns or dipoles, where the aperture fields are separable in the aperture coordinates. Plane wave incidence is an approximation which simplifies analysis of a scattering problem, but is not an accurate assumption if the obstacle is near the antenna. If the transmitting antenna is omnidirectional and the obstacle is large in wavelengths the scattered field can be calculated by the techniques of high frequency diffraction theory such as geometry theory of diffraction (GTD) and uniform theory of diffraction (UTD) [1]. If the antenna beam is directive, the omnidirectional source solution can be converted to a beam source solution by the complex source point method [2], [3]. Uniform asymptotic solutions for simple obstacles such as a half plane, a slit and a wedge have been obtained for single omnidirectional and directive local sources, [4], [5]. Antenna patterns with sidelobes can be synthesized from arrays of such 1 complex sources with appropriate amplitude and phase [6], [7]. Here their scattering patterns from apertures and cylindrical scatterers are predicted. This thesis begins by synthesizing antenna patterns from arrays of CSP sources in two dimensions. Their radiation patterns are examined in the near and far field. The beam scattering patterns of nearby obstacles are then calculated. The obstacles initially are canonical shapes, such as cylinders, half-planes, wedges, and simple combinations thereof follow. Decomposition of aperture field into basis elements is one approach to solve the difficulties of tracking the radiated field from large aperture through a complicated environment. Radiation from large apertures is usually expressed as a superposition of either a continuous angular spectrum of unidirectional plane waves or a discrete sum of omnidirectional point sources. It has been shown that a superposition of sources of variable directivity, Gaussian beams, is efficient for synthesising the radiation fields of extended sources. [6], [7]. A two dimensional array of Gaussian beams, equispaced linearly and directionally, with each beam direction having a particular beamwidth and phase and with both real and imaginary beam directions included, can represent the propagating and evanescent fields of any aperture distribution to any desired accuracy. For efficiency it is usually necessary to limit the number of beams, beamwidths and directions to those which contribute significantly to the total field in a particular application. However, when Gaussian beams are used to simulate non-Gaussian beam type data, there arise ambiguities due to choice of beam width, spacing and direction, which are the user's disposal [6], [7], It is well known that assigning a complex value to a source coordinate converts an omnidirectional point or line source into a beam that is paraxially Gaussian [2], [3], [8], [9]. The asymptotic solution for this complex source Gaussian beam diffraction by a conducting wedge 2 and by a half-plane are available [4], [10]. Since the source coordinate appears only as a parameter in the field equations for the Green's function, the complex source point (CSP) substitution converts a point or line source wave solution into field solutions for propagation and diffraction in various environments for incident Gaussian beams. However, it should be noted that the complete complex ray field furnishes a rigorous asymptotic solution of the field equations whereas the paraxial Gaussian beam is only an approximate solution. Hence this CSP method provides an inherently more rigorous approach to the synthesis of antenna patterns. Moreover if the point or line source Green's function is known exactly or asymptotically in a given environment, its analytical continuation from real space to complex space also provides an exact or asymptotically correct solution in the same environment [3]. This implies the behaviour of direct, reflected and diffracted beams can be found from the behaviour of real ray fields on replacement of real source positions by complex source positions in the ray phase and amplitude function, and in the reflection and diffraction coefficients. This approach of using rigorous solutions for point or line diffraction by canonical structures is taken here. Decomposition of a signal into basis elements can be done by utilizing the complex spectrogram. The complex spectrogram of a signal is defined as the Fourier transform of the product of the signal and a shifted complex conjugated window function. It is completely determined by the values on the points of a certain wavenumber-space lattice. This lattice was originally suggested by Gabor [11], [12] for the spectral decomposition of signals. In the antenna context the Gabor series may be viewed as an extended spectral representation comprising a discrete, two-dimensional superposition of elementary beam waves that is characterized by a finite width and a linear phase shift between neighbouring beams. In the paraxial regime of the beam, an aperture distribution described by a Gabor series may be considered as equivalent to a 3 field contribution by a discrete collection of point sources located in a complex grid [7]. The complex source point representation of directive beams with complex positions that can be used to obtain exact and asymptotic solutions for low or high frequency and far or near fields, as long as the corresponding real source solution exists and analytic continuation into complex space is possible. This technique is uniform on the shadow boundaries, provided the asymptotic solution is. This thesis begins by obtaining the near and far field antenna patterns for arbitrary aperture field distributions by using complex source point simulations. Uniform solutions of the half-plane based on a simple solution exact in the far field limit and of the wedge using the UTD and a CSP solution for a single directive beam were obtained by Suedan and Jull [4], [5]. Here this technique is extended to calculate the solution of two-dimensional antenna beams represented by an array of complex sources, and diffracted by a conducting half-plane, a slit, a wedge, and by rectangular and circular cylinders. In this thesis the following established analytical methods are uniquely combined in obtaining solutions for the two-dimensional near field scattering of these obstacles in the proximity of antenna apertures. 1.2 Complex Source Point (CSP) Method By assigning complex values to the source coordinate locations of an isotropic point or line source, one may generate a highly collimated field in the paraxial region like a three-dimensional (point source) or two-dimensional (line source) Gaussian beam [2], [3], [8], [9], [13], [14]. The field of a two-dimensional line source at r0, d0 from the origin of coordinates at any observation point r, 9 may be written as 4 -jkR kR»l, (1.2.1) where R is the distance of the observation point to the source. By making the source coordinates complex with fs=r0— jb , the omnidirectional wave becomes a directive beam uniform along in the z-axis. Here fs = {rs,9s), f0 =(r0,d0), and b ={b,0) are the complex source position, real source position, and beam parameter vectors, respectively, given in polar coordinates with all angles measured from jc-axis. The beam sharpness and its orientation are defined by b and (3. Then we get [16] rs=^-2jbr0cos(80-p)-b2, 0. =cos _ 1 | Re(r f)>0 ' r0 cosd0 - jb cos /? A (1.2.2) R,=Jr2+r?-2rr,cos(e-0s) In the far field limit (r » r0), eq. (1.2.1) can be written as 1 V ^ T jtfccos(e-^ ) 4kr (1.2.3) which represents an omnidirectional cylindrical wave modulated by a beam pattern g** 0 0^ 9 -^ with its maximum in the direction 6 = 0 and minimum in the direction 9 = n+pl. The half power beam width (HPBW) is related to the beam parameter kb by HPBW = 2 cos' In 2 V 2kb) kb>-In 2 (1.2.4) 1.3 Gabor's Expansion Decomposition of aperture fields into basis elements has been one approach to cope with 5 the difficulties for the tracking of fields generated in large aperture through a complicated environment. Two options may be used. One is a spectral representation that relies on plane wave superposition. An alternative is a superposition of local point source contributions. Such a formulation is the Gabor representation that may be written formally as [6], [7], [12], and [15], *,(*.<>)= S X A m ,„w(x- mL)e-j2'"tt'L (1.3.1) m-—« n--°° Here Ez(x,Q) is the aperture electric field distribution in y = 0 plane, w(x) = is a finite energy window function which is a Gaussian function in the Gabor series representation, and L is the spacing between Gaussian beams along ;e-axis. Although this was proposed by Gabor in 1946, use of this nonorthogonal representation has been limited by the difficulties in computing the series coefficients A ^ . These difficulties were partly removed by Bastiaans [12] and Janssen [16]. The coefficients A m , n may be obtained by convolving the desired aperture distribution with a biorthogonal function y(x) as described in [15, eqs. (35-7)]. Thus K» = ]Ez(x,0)f(x-mL)e-i2n)a'Ldx (1.3.2) where y(x) the is defined by the relation £ W(x)y(x - mDe-^dx = SJn, 8m = jj ^ °Q (1.3.3 ) The behaviour of A m , n can be known through the analytic and asymptotic properties of Fourier integral. Under some symmetry conditions, the spectral range m, n > 0 contains the entire information [15], such as 6 K-n=K,n if Im[£ z(x,0)7 (x -mL)]=0 A-m,n = <•, i f Et(-x,OY(rx + mL) = [Et(x,Oy(x-mL)]' ( 1.3.4) A-m* = - < * if Et (- x,OY (-x + mL) = -[Ez (x,0)f (x - mL)]* Then the field in y > 0 can be written as Ez(*>y)= X \K,nBmAx>y) (i-3-5) where B^ix.y) are the elementary beam fields of the source functions. These are Gaussian functions in a Gabor series representation [7, eq. (64)] and the paraxially Gaussian CSP beams of eq. ( 1.2.3 ) in this thesis. For an obstacle near the radiating aperture we use expressions for Bm,n(x,y) appropriate to the situation. For example, for E-polarized incidence on a half plane eq. ( 3.1.3 ), for a slit eq. (4.1.3 ), for a wedge eq. (5.1.5 ), for a rectangular cylinder eq. ( 6.1.8 ), and for a circular cylinder eq. (7.1.16 ). 1.4 Overview of The Thesis An introduction and motivation of the thesis has been given in section (1.1) and some literature related to this project reviewed in sections (1.2) to (1.3) and the appendices. In chapter 2, various antenna aperture distributions are synthesized from arrays of complex sources modelled in two-dimensional polar coordinates. Gabor's expansion is used to determine the amplitude of each individual complex source. As this has been done previously with Gaussian beams [6], [7], a comparison between complex source and Gaussian beam results is made. Near and far field antenna patterns for arbitrary field distributions obtained by using arrays of complex source beams are compared with results from Gaussian beams and with truncated Fresnel transform and Fourier transform results from Kirchhoff s method. 7 Chapter 3 is devoted to a uniform solution for scattering by a half-plane in the presence of local aperture distributions. It is based on a simple uniform solution exact in the far field limit. A comparison of this solution with the exact series solution verifies the accuracy and the validity of the computer programs. In Chapter 4, results obtained from Chapter 3 are for beam diffraction by a wide slit consisting of two coplanar half planes with separated parallel edges. The total diffracted field will be the field scattered from the two half planes in isolation plus their interaction scattered fields. In Chapter 5, the UTD is combined with the CSP method for aperture diffraction by a local right angled conducting wedge. Then such arrays of complex sources are used to calculate the scattering patterns of local moderate or large rectangular cylinders in Chapter 6. The total scattered field is the superposition of the solution for the far field of arrays of complex sources in the presence of four right angled wedges and first order interaction between the edges is included when significant. A moment method solution is used to verify the accuracy of the solution for cylinders of moderate size. Two dimensional beam diffraction by circular cylinder problems are considered in Chapter 7. The field intensity at any observation point is calculated from the exact series solutions in terms of cylindrical waves. Sometimes it may suffice to determine only certain average characteristics of a scattering obstacle for many practical purposes. The total scattering cross section is one such important quantity and is obtained for different cylinder sizes, beam directivities and source distances. A conclusion, discussion and future possibilities are given in the last chapter. 8 CHAPTER 2 SYNTHESIS OF ANTENNA PATTERNS 2.1 Analytical Expressions The complex source point method [2], [3] converts an omnidirectional source diffraction solution into a beam solution by an appropriate choice of complex source coordinates. Rays emanating from a complex source point are known to generate a real beam with no radiation pattern sidelobes. More realistic antenna radiation patterns having sidelobes can be handled by radiation from more than one source. It has already been shown that antenna patterns with sidelobes can be synthesized from arrays of Gaussian beams [6], [7] and this representation applies to both near and far radiation patterns of the apertures. Others have given a direct complex source representation of plane aperture radiation and proposed simulation procedures to match a CSP beam to an aperture radiation pattern [21], [22]. Such arrays of complex sources located at complex locations (r„ ft) are controlled by real source coordinates (x = mL), beam parameters bn =[Lcos(/?n - 3 ; r / 2 ) ] 2 IX, which define beam directivity with X the wavelength and /?„ = 3n I 2 + sin"1 (nX I L), is the direction of the elementary beam axis measured from the positive x axis [7] (see Figs. 2.1a, b). The intermediate and far fields arise from beams with real direction angles (/3, real), and the beams with complex beam direction (p\ complex) contribute in the very near field and in the aperture plane. The accuracy and efficiency of the computational sequence is controlled by the beam width b„, the location of the beam waist identified through the index m, and the direction of the beam maximum identified through index n. The geometry of multiple 9 complex line source is shown in Fig. 2.1a Thus, the total field can be written as m N ^ = 2 X A ^ N zfmjij (2.1.1) Here, E\ 'z[ m,n ] is given by eq. (1.2.3 ) with R replacing Rs and (2.1.2) , - ! f ro[mJ cos 60[m, -jb„ cos/3„ \lm*l = C 0 S It has been pointed out that a reasonable and possibly the best choice for accurate far field radiation patterns is to select the separation between two source points L = (N + Yi)X, (N is the largest integer n for real pX) [7]. Here to simplify our analysis we choose all spacings L = A/2 and let iv* = 0 so that all beam directions are real and orthogonal to the line of array (Fig. 2.1b). Therefore we omit the index n to simplify the notation in the following discussion. Then eq. ( 2.1.1) becomes and the subscripts [m,0] become [m] in eq. ( 2.1.2 ). The weighting factors Am in ( 2.1.3 ) are calculated from eq. (1.3.2) for Gaussian window functions w(x) with the appropriate aperture distribution Ez(x,0) and n = 0 substituted. Table 1 gives the results for several simple distributions and one compound distribution. We also examine here the effect of using CSP beams instead of Gaussian beams and the effect of the aperture distribution on the required of number beams. M Ez = 2An>K (2.1.3) m=-M 10 2.2 Numerical Results 2.2.1 Far Field Calculation Simulations of the radiation patterns of in-phase cosine-squared and uniform distributions in an aperture of width LQ = 2.5X 5X, and 9X, are shown in Figs. 2.2 and 2.3, respectively. The solid curves in Figs. 2.2 and 2.3 are the uniform total far field calculated from the complex sources with different weighting factors located along aperture plane. The results are compared with the reference patterns calculated from the truncated Fourier transform of a cosine-squared distribution in an aperture of width L0, with zero field assumed in the aperture plane outside the aperture [eg. 23, pp. 13, 21] Et(r,t)= C 0 S ^ ~ * M ) fc" c o s ' ( « / L > * * ' * cos (pe •j(kr-it!4) £ S m | kL0 sin^ 7T (2.2.1) 4Tx and of a uniform field distribution kL0 sin0 K1-1 k L 0 sin0^2 4r~X jkx sin $ dx sin kL0 sin^^ 2 , kLa sin (j) (2.2.2) In Fig. 2.2a the radiation far field for an in-phase cosine-squared aperture distribution in an aperture of width L0 = 2.5X is calculated. The ratio LJX takes the value 0.5, in accordance with the guidelines proposed above. Then Ba = 3TC/2, kb0 = n/2 and each CSP half power beamwidth is 77.6° according to eq. ( 1.2.4 ). With only 5 (M = 2) beam sources the beam 11 series computations agree with the Fourier transform well over the main beam. Note that the truncated Fourier transform represents an antenna radiation pattern most accurately on the beam axis and progressively less accurately off it. In Fig. 2.2b the radiation far field for an in-phase cosine-squared aperture (L0 = 5X) distribution is calculated. 13 (M = 6) beam sources provide a good agreement with the truncated Fourier transform result over the main beam and first sidelobe. Fig. 2.2c shows that the radiation far field for an in-phase cosine-squared aperture (L0 = 9X) distribution with 19 (M = 9) beam sources included provides a good agreement with the truncated Fourier transform result over the main beam and first three sidelobes. In each case complex source point beams provide less accuracy at wide angles off the beam axis. In Figs. 2.3a, b, c and d respectively, the far field of a uniform distribution in the aperture of width L0= 2.5X, 5X, 9X and 1IX is shown. With 5 (M = 2) beam sources for L0 = 2.5X, 17 (M = 8) beam sources for L0 = 5X, 23 (Af = 11) beam sources for L0 = 9X and 27 (A/ = 13) beam sources for L0 = \\X, the beam series solutions show similar agreement with the reference result over the same range of angle <j> as the cosine-squared aperture distribution. Comparing the two cases, it is apparent that more beam sources need to be included in the beam series computation for an uniform distribution than for a cosine- squared distribution with the same aperture size. Generally the aperture width and the amplitude profile of the aperture field determine the number of beam sources needed. For smaller apertures and for aperture distributions tapered toward the edge of the profile, fewer line sources are needed to for the same accuracy. Diffraction by local obstacles is of course simpler to calculate if fewer sources represent the radiating aperture. The computations for the radiation far field for an in-phase cosine aperture distribution 12 Ez (x,6) = cos(m I L0 )rect(x I L0) and a compound aperture distribution .1 + 0.9 cos{nx / La ))recr(x / L0) in apertures of width LQ = 5X, and 9\ are also shown in Figs. 2.4 and 2.5 respectively. The numerical results show negligible differences between beam array solutions and reference patterns over the main beam and, for the larger apertures, the first sidelobes of the pattern. As the aperture size increases more beams are used and both Gaussian and CSP beams accurately produce the reference pattern over increasing numbers of sidelobes. At angles far off the beam axis, however, larger discrepancies occur in both Gaussian beam and CSP solutions. The CSP beam yield values above the reference solution and Gaussian beam solutions as <p -» 90° because they are only paraxially Gaussian and do not approach zero far off the beam axis. Consequently for small apertures with wide-angle sidelobes, as in Fig. 2.2a, the first sidelobe level is inaccurately predicted by this CSP beam solution. Elementary beams which vanish as (j)—> 90° can accurately produce the reference pattern at large angles <fi, but convenient solutions for diffraction of these beams, or even by purely Gaussian beams, by local canonical structures, are not available. By using narrower CSP beams with the same amplitudes it may be possible to improve the accuracy of the pattern representation of wide angles, but this has not been investigated. 2.2.2 Near Field Calculations Although the beam series computation does not agree with the reference solution at large observation angles, the latter is inaccurate there and the relative advantage of the CSP beam series representation is evident when diffraction by a local object is to be calculated. 13 The computation sequence associated with the beam series can be carried out not only in the far field but also arbitrarily close to the aperture in the Fresnel region. From Figs. 2.2 to 2.5 the radiation patterns are at distance from the aperture sufficiently large that all rays paths from the aperture plane to the field point are essentially parallel. Now this assumption is removed and the radiation field in the Fresnel zone of the aperture is examined. In Figs. 2.6 and 2.7 the radiation near fields at ranges r = L20 13/1, L] 12A, L] / X, 2L] / X, and <», for a cosine-squared distribution and an uniform distribution in an aperture of width L0 = 2.5 A and 5A, are shown. The effects of broadening of the main beam, raising the sidelobe levels and filling in of pattern nulls due to finite range are evident. Figs. 2.8 and 2.9 show the comparison between CSP method and truncated Fresnel transform [23, p. 32] E 2 ( r ,0 ) = ^ = 0 « W>)« * e'^dx ( 2.2.3 ) for a uniform aperture distribution in the aperture size L0 = 2.5A and 5A. respectively. With the same number of complex beam sources as the radiation far field calculation, it provides a good agreement with the approximate truncated Fresnel transform result over the main beam and first sidelobes. The Fresnel transform result is based on including only first order near field effects through a quadratic phase term [23] and like the truncated Fourier transform result is progressively less accurate off the beam axis. Previous numerical results for similarly small apertures but with wider beam spacing and more beams [7, Figs. 15, 16] show Gaussian beam solutions also fail to accurately predict the reference pattern at large angles <p off the beam axis. •I 14 Fig. 2.1a Geometry of multiple complex line sources and a real far observation point in polar coordinates. Fig. 2.1b Geometry for synthesis of an array of complex line sources for an arbitrary aperture distribution. 15 (gp) J3M0J 3A)IVPX 16 17 i M u •o II 1 ^  as H o c o 'is •c •a £ o U fN tin as CO II JO in o co c o P 1 3 t o o co O o co (gp) JSMOJ dxvmpg NO 00 (L) II 9H A S3 U ;a '35 o o ca s-«a co s H fN da to c o o co <u o o co a 4) —-1 I • O U as ll o co fe 18 Neco2 l.xls -60 -I— 1 —'— I— 1 —«JH— 1 — 1 — I 0 30 60 90 <f> (degrees) -70 A 1 ' H - 1 —I 1 1 1 0 30 60 90 <f> (degrees) Fig. 2.6 Near field patterns for a cosine-squared aperture distribution. L = Xll. (a) M = 2, L 0 - 2.5X, (b) M = 6, L 0 = 5X. 19 Neco2_l .xls-neco2_l </> (degrees) <f> (degrees) Fig. 2.7 Near field patterns for a uniform aperture distribution. L = XI2. (a) M - 2, L 0 = 2.5X, (b )M = i , L 0 = 5X. (gp) 'jZMOJ BAtfVpU 21 0\ 22 CHAPTER 3 APERTURE DIFFRACTION BY A CONDUCTING HALF-PLANE 3.1 Analytical Expressions The exact total far field solution for plane wave diffraction by a conducting half-plane was obtained by Sommerfeld [24] by his method of many-valued wave functions. A more direct approach to the exact solution by solving integral equations was used by Clemmow [25]. The exact total far field for a uniform line source parallel to the edge of a conducting half plane can be obtained by integration of the corresponding plane wave solution over all incident angles [26], [27]. The far field result can be deduced more simply by reciprocity from the exact solution for plane wave diffraction by a half plane [23, p. 83]. This has been used with the complex source point method in solving the single beam diffraction by a half plane [4], [5]. Now, we extend this with the Gabor series to solve half plane diffraction by an aperture distribution [49], as indicated in Fig. 3.1. The total far field diffraction pattern is represented by a superposition of the diffraction patterns of discrete elementary beams linearly shifted with respect to one another by distances L along the aperture plane. M EJr,9)= ^KBJr.d) (3.1.1) For the basis elements BJr,9) here, we use the solution for the total field at r, $ in cylindrical coordinates due to an electric line source at r0[mj, 9B[n, and parallel to the edge with beam parameters b0, j5a defined as followings: 23 _[i«»(4~(*+g.))f (3.1.2) thus Bm(r,B) = -,-j(kr-x/i) _eikr,i~i'°'{e+o,t~,) J mj cos\ ml COS] e+e. '[ml (3.1.3) Where (r, 9) and rJ(m], 6s[m, defined in eq. ( 2.1.2 ) are the observation position and complex source position in polar coordinates, and F(w) is the complex Fresnel integral F(w)=f~' e'iridt (3.1.4) For computation of elementary beam fields (eq. (3.1.3 )), it is necessary to have a Fresnel integral subroutine that can handle complex arguments. A useful relation between the complementary error function and the Fresnel integral is given by F(w)=e-j'"* \ e-1 du = ^ - e-iKl\rfc[weiK") (3.1.5) we" A computer subroutine for the complementary error function with complex arguments is available. For //-polarization, Hz replaces Ez in the above equation, the (-) sign in the non-exponent part of the reflected wave term in eq. ( 3.1.3 ) becomes (+) and therefore ,-;(b-*/4) V 2 ~ 4lkr>lm) C 0 { 9 2lm' (3.1.6) 24 3.2 Numerical Results 3.2.1 ^-Polarization Far Field Calculation First, we examine the total far field diffraction patterns of an array of complex line sources with uniform amplitude and beam directivity above a perfectly conducting half-plane. In Fig. 3.2, an array of omnidirectional line sources spaced L = rJ2 apart are placed at a distance kr0 = 8 from and normally incident (60 - nil) to the half-plane. The results from 3.2a to 3.2d are obtained from eq. ( 3.1.1 ) and ( 3.1.3 ) with M = 0, 1,2, and 3 corresponding to 1, 3, 5 and 7 line sources included. From the results, it is evident that the more sources are included, the more interference lobes appear in the illuminated region (-n/2 < <f> < 0) due to the interaction between direct incidence from the line sources and a diffracted wave from the edge. Also the lobe levels in the normalized far field diffraction patterns are smaller for larger array of line sources. In the shadow region, (0 < <p < nJ2) the total far field pattern decreases monotonically as <j> increases and vanishes on the conductor for a small array. An additional lobe appears in this region for a large array of beam sources due to penetration of the outer line source. In Fig. 3.3, an array of omnidirectional line sources are replaced by sources with beam directivity kb = 4 and normally incident to the half-plane. There are similar characteristics between the two results but with smaller lobe levels both in the illuminated and shadow region for a more directive beam The solid curves in Fig. 3.4 represent the total field calculated from eq. ( 3.1.1 ) for a distance above the half plane kr0 ranging from 2 to 32 between the conducting edge and the center of the aperture plane for a cosine-squared distribution with aperture size in 25 wavelengths is L0 = 1.5X. The half-plane is in the Fresnel zone of the aperture. The angle of incidence 90 of the central beam with respect to the edge is nil. A l l the line source beams are spaced L = Xll apart. Then BQ = 3nll, corresponding to normal incidence on the half plane. The beam directivity for the above choice of L and BQ is b0 = 0.25A. and each CSP half power beam width is 77.6° (kb0 = nil). The relative weighting factors Am of each line source are calculated according to eq. ( 1.3.2 ). A narrower diffracted beam is obtained by moving the aperture plane further from the half plane for then the beam illuminating the half plane is narrower. For kr0 = 32 the pattern has more oscillations in the illuminated region (-nil < <p < 0), due to interference between the direct wave from the beam sources and a diffracted wave from the edge. In the shadow region (0 < <p < nil), the total far field pattern decreases monotonically as <j> increases and vanishes on the conductor. The field in the shadow region decreases more rapidly for larger kr0 because there is more blockage by the half plane. Also shown for comparison are the results for incidence of a single line source beam with the same FfPBW (Jkfe = 11.5) as the cosine-squared aperture far field radiation pattern in isolation (dashed curves). Clearly a single beam is a good approximation in this case, as might be expected for a cosine-squared aperture distribution. The dotted curve in Fig. 3.4a is the far field radiation pattern of a cosine-squared aperture isolated without the conducting half-plane present. The uniform total far field for a cosine-squared distribution in an aperture of width L0 = 5X has also been calculated from eq. ( 3.1.1 ) with proper Am substitution and is shown in Fig. 3.5. Here again kr0, the distance between the conducting edge and the aperture center ranges from 2 to 32. The major difference between Figs. 3.4 and 3.5 are a sharper main beam and more lobes in the illuminated region for the larger aperture case. 26 In the shadow region, a small lobe appears when the aperture is close to the half-plane (kr0, = 2, and 8) because of the interference between the diffracted wave and the incidence wave from the sources located at left end of the array. Notice that the edge of the conducting half-plane located outside the HPBW angular range of the \ni > 1 sources for kr0, = 2 and \m\ > 3 sources for kra, = 8 cases. Also included are the diffraction patterns for single line source beam with the same HPBW (kb = 46.85) as the cosine-squared aperture far field radiation pattern. Again a single beam is a good approximation in this case, as expected The total far field for a uniform distribution in an aperture of width L0 = 2.5X has also been calculated and is shown in Fig. 3.6. Here again kr0, the distance between the conducting edge and the aperture center ranges from 2 to 32. The major difference between Figs. 3.4 and 3.6 are the higher lobe levels in the illuminated region for the uniform aperture distribution case due to larger interaction between the direct wave from different line sources and a diffracted wave from the edge. Also included are the results for single beam source incidence (kb = 22.84). The HPBW of the single beam is the same as that of the uniform aperture far field radiation pattern. Clearly a single beam is not a good approximation in this case, as expected, since a single complex line source beam cannot accurately represent the far field of a uniform aperture distribution. In Fig. 3.7, The total far field has been calculated for a uniform distribution in an aperture of width La - 5X at distance kr0 from 2 to 32. Comparing to the L0 = 2.5A. case, the fields show sharper main beam, more lobes in the illuminated region, and a significant lobe appears in the shadow region when kr0 is small. 3.2.2 //-Polarization Far Field Calculation 27 The //-polarization uniform total far field for a cosine-squared distribution Hz (*,0) = cos 2 (nx I L0 )rect(x I L0) and uniform distribution Hz (x,0) = rect(x I L0) in an aperture of width L0 = 2.5X, and 5X has also been calculated from eq. ( 3.1.1 ) with proper weighting factors Am and basis elements for //-polarization (eq. ( 3.1.6)) substitution and is shown in Figs. 3.8 to 3.11. It also can be done by using Babinet's principle from the results of E-polarization. Here again kr0, the distance between the conducting edge and the aperture center ranges from 2 to 32. There are apparent differences between E and //-polarization results. In the illuminated region, the interference between the direct incident from the beam sources and a diffracted wave from the edge decreases with field point moving away from the main lobe because of that the diffraction field vanishes at the direction of <$> - -nil. In the shadow region, the total far field pattern decreases as <p increases but does not vanishes on the conductor. However, the field in the shadow region also decreases more rapidly for larger kr0 because there is more blockages by the half plane. In order to verify the computer programs, the exact series solution for the total field due to a set of complex line sources diffracted by a conducting half plane was programmed; i.e., Ez = X Am^J,eip^Jp/q(krs[ml)sin m=-M y Kf p=l ^ i y \ 1 / Jk(z) is the Bessel function of the first kind of order k, and here q = 2 for half-plane. The Am are the amplitude coefficients for the distribution calculated form eq. ( 1.3.2 ) It was found that twenty terms in eq. ( 3.2.1 ) provides results essentially identical with those calculated from eq. (3.1.1 ) for a cosine-squared distribution at kr0 = 2 in Fig. 3.12. This 28 PVs(ml sin pff \ n (3.2.1) is an alternative formulation and capable of equal accuracy but it is much less efficient than using eq.( 3.1.3 ) for the basis function, particularly for larger apertures and larger ranges from the half plane. 29 30 HI 5.xls-hl <f> (degrees) <j> (degrees) Fig. 3.2 Normalized far field diffraction patterns of an array of complex line sources at a distance r 0 above a perfectly conducting half-plane, (kr 0 = 8, kb =0,L = r 012) 31 HI 5.xls-h2 <f> (degrees) <f> (degrees) Fig. 3.3 Normalized far field diffraction patterns of an array of complex line sources at a distance r 0 above a perfectly conducting half-plane, (kr 0 = 8, kb =4,L =r0/2) 32 HalfunC2.xls-HalfCo2 <j> (degrees) <f> (degrees) Fig. 3.4 A comparison of normalized far field diffraction patterns of a cosine-squared aperture distribution and a single complex source point (kb = 11.5) at a distance r0 from the edge of a perfectly conducting half plane. (M= 2, L0 = 2.5k, L = k/2, 90 = re/2). Cosine-squared, kb = 11.5, Aperture alone without half-plane present. 33 HalfunC2.xls~HalfCo2_l $ (degrees) <f> (degrees) Fig. 3.5 A comparison of normalized far field diffraction patterns of a cosine-squared aperture distribution and a single complex source point (kb - 46.85) at a distance r0 from the edge of a perfectly conducting half-plane. (M = 6, L0 = 5k, L = A/2, 6Q = nil). Cosine-squared, __kb- 46.85., Aperture alone without half-plane present. 34 HalfunC2.xls~HalflJn l l l l l (b)*r0 =8 -90 1 T -60 -30 0 30 60 90 (f> (degrees) (c)*r„ = 16 -90 1 T -60 -30 0 30 60 90 </> (degrees) (d)kra =32 -90 -60 -30 0 30 <f> (degrees) 60 90 -90 -60 -30 0 30 60 90 <j> (degrees) Fig. 3.6 A comparison of normalized far field diffraction patterns of a uniform aperture distribution and a single complex source point (kb = 22.84) at a distance r0 from the edge of a perfectly conducting half-plane. (M - 2, L0 = 2.5X, L = 7J2, 0o = 7i/2). Uniform, kb - 22.84., Aperture alone without half-plane present. 35 HalfunC2.xls-HalfUn2 $ (degrees) <f> (degrees) Fig. 3.7 A comparison of normalized far field diffraction patterns of a uniform aperture distribution and a single complex source point (kb = 89.08) at a distance r0 from the edge of a perfectly conducting half-plane. (M= 8, L0 = 5X,L = Xll, 90 = nil). Uniform, kb = 89.08., Aperture alone without half-plane present. 36 HalfunC2_H.xls-co2_l 0 (degrees) ^ (degrees) Fig. 3.8 A comparison of normalized far field diffraction patterns of a cosine-squared aperture distribution and a single complex source point (kb .= 11.5) at a distance r0 from the edge of a perfectly conducting half plane. (M= 2, L0 = 2.5X, L = X/2, 90 = TC/2). Cosine-squared, kb = 11.5. 37 HalfunC2_H.xls-co2_2 ^ (degrees) ^(degrees) Fig. 3.9 A comparison of normalized far field diffraction patterns of a cosine-squared aperture distribution and a single complex source point (kb = 46.85) at a distance r0 from the edge of a perfectly conducting half-plane. (M =6,L0 = 1.5X, L = Xll, 90 = nil). Cosine-squared, kb = 46.85. 38 HalfiinC2 H.xls-un 1 1 T ( C ) £ T - 0 = 8 -90 -60 -30 0 30 60 90 <f> ( d e g r e e s ) (c)*r0 = l6 0 r I 1 I 1 I 1 -90 -60 -30 0 30 60 90 <f> ( d e g r e e s ) (c)Ar0=32 •90 -60 -30 0 30 <f> ( d e g r e e s ) 60 90 -90 -60 -30 0 30 60 90 <f> ( d e g r e e s ) Fig. 3.10 A comparison of normalized far field diffraction patterns of a uniform aperture distribution and a single complex source point (kb = 22.84) at a distance r0 from the edge of a perfectly conducting half-plane. (M= 2, L0 = 2.5X, L = X/2, 0o = n/2). Uniform, kb = 22.84. 39 HalfunC2_H.xls-un_l 0 (degrees) 0 (degrees) Fig. 3.11 A comparison of normalized far field diffraction patterns of a uniform aperture distribution and a single complex source point (kb = 89.08) at a distance r0 from the edge of a perfectly conducting half-plane. (M= 8, L0 = 5X,L = Xll, 90 = Till). Uniform, to = 89.08. 40 CHAPTER 4 APERTURE DIFFRACTION BY A WIDE SLIT 4.1 Analytical Expressions A slit between two coplanar half planes with parallel edges has been a traditional test of diffraction theories. By using the results for beam diffraction by a half plane, we can solve the problem of beam diffraction by a wide slit in a conducting plane. Fig. 4.1 shows an aperture field parallel to and at a height y0 above the center of a slit in y - 0, jjcj < L0 / 2. The total non-interaction far field of the slit is the sum of the total far field of the half plane on the right side and on the left side less an incident field. Beam diffraction by a wide slit, with first order interaction between the slit edges excluded, is obtained from an exact far field uniform solution for an isotropic line source parallel to the edge of a half-plane by making the source position complex [5]. The total far field diffraction pattern of the half plane on the right side of the slit is given by replacing rx, 9X with r, 9 and rsX[mj, 8sX[m] with rs[m], 9s[m] in the elementary beam field Bm{r,9), eq. ( 3.1.3 ) for ^-polarization and eq. ( 3.1.6 ) for H-polarization. Similarly the total far field of the half plane on the left side is given by replacing r2, 92 withr, #and rs2[m], 9s2[mJ with rs[m], 9s[m] in the elementary beam field Bm{r,9). In the far field limit (r»L0), L IK . ( 4 . U ) r2=r + -£-sin0, 92 = — + </> is substituted in the exponential terms of Bm (r, 9) and rx=r2=r is used in the amplitude terms. 42 The singly diffracted far fields of the slit can be calculated by with (4 .1 .2 ) ->(*r-ir/4) B. =• -e 2 l[ml cos\ W J) - 4 2 k r ' U m l C 0 S rir + <t>+e. J) + e 2 ,;«»ijf«;n"(^ -».jf»;) p\ (S. -per, sl[m} cos\ _ ei*,itmiM++9>tr-i)p -ftkrlilml cos\ J) J) (4.1.3) For //-polarization, Ht replaces Ez in eq. (4.1.2) and B. - j(b—x/4) -Jnkr COS] J) 2 JJ COS] + e 2 e frtltml ml) pi -42kr>Vml C°* 'i + t + O, ^ J) 'slim) J) (4.1.4) 4.2 Numerical Results 4.2.1 /^-Polarization Far Field Calculation The diffraction patterns of Figs. 4.2(a) and (b) are calculated for an in phase cosine-43 squared aperture distribution in apertures of width L0 = 5\, and 9\, respectively. The apertures are parallel to and at a height y0 = LJ2 above a slit with the same width as the aperture. The solid curves are non-interaction diffraction fields calculated from an array of complex beam sources located along the aperture plane with the same weighting factors from table 1 as in Figs. 2.2(b) and (c) to simulate the cosine-squared aperture distribution. For both cases (L0 = 5A-, and 9X), the total field diffraction patterns have the same number of sidelobes but with higher individual sidelobes levels than the total far field patterns with slit absent. For comparison, the normalized far field diffraction pattern of a slit excited by a single beam source with the same HPBW (kb = 46.85 and 152.5 for L0 = 5X, and 9X, respectively) as the cosine-squared aperture far field radiation pattern in isolation are also included (dashed curves). Clearly the single beam is a good approximation in the main lobe region, but shows lower sidelobes levels near the axis and higher sidelobes levels far from the axis than the beam series computation. By moving the aperture distribution away form the slit from y0 = LJ2 to 2La, an additional side lobe starts to appear between the main and the first side lobe (Fig. 4.3). A single beam is a good approximation in this case, as might be expected for a cosine-squared aperture distribution. When the aperture plane is further from the slit, more sidelobes appear and the sidelobe levels increase due to stronger illumination of the edge relative to the center of the slit. The pattern resembles that of plane-wave incidence on the slit for large separations (y0 » L0). The diffraction patterns by a slit when the aperture distribution is at ya = \0Lo are shown in Figs. 4.4(a) and (b) for slit widths Lc = 5\, and 9X respectively. The solid curves are non-interaction diffraction fields for cosine-squared aperture distribution and the dotted curves are the diffraction patterns of plane wave incidence. Also the far field patterns for a cosine-squared distribution with the slit absent are included (dashed curves) for comparison. 44 The normalized far field diffraction patterns of a slit excited by a uniform aperture distribution are shown in Fig. 4.5. The aperture plane is very close to the slit (y„ - L0 I 2 = L\ I10X). The solid curves and the dashed curves again are the non-interaction diffraction fields for the beam series solution and a single beam source with the same HPBW in the far field radiation patterns. Clearly the single beam is not a good approximation in this case, as expected. Notice that the slit is well within the Fresnel zone of the aperture. At this range, the single beam source cannot represent even the cosine-squared distribution well (see Figs. 4.2). In Fig. 4.6, by moving the aperture distribution away from the slit from (y0 = L0 I 2) to (y„ = 2 L a = 4 L ] I X ) , the single beam source is a good approximation in the cosine-squared case (Fig. 4.3) but cannot accurately predict the diffraction pattern of the uniform distribution at this range. However, by moving the aperture further from the slit, all results resemble plane wave incidence on the slit. 4.2.2 //-Polarization Far Field Calculation The //-polarization total far field diffraction patterns of a slit in an aperture of width L0 = 5A. excited by a cosine-squared and uniform aperture distribution have also been calculated from eq. ( 4.1.2 ) with proper weighting factors A m and basis element for //-polarization (eq. ( 4.1.4)) substitution and is shown in Figs. 4.7 and 4.8. There are similarities between the E and //-polarization results. When the aperture plane is very close to the slit (y0 = LJ2), a single beam source cannot accurately represent the diffraction patterns of a slit for both cosine squared and uniform aperture distributions. By moving the aperture plane away from the slit to ya = 2 L a , the single beam source gives a good approximation in cosine-squared distribution case but not for a uniform aperture distribution. 45 However, a single complex line source beam can accurately represent the diffraction far field of a slit for both eases, as expected, if the aperture plane is far from the slit. The calculations described above are accurate only for slits sufficiently wide that interaction between them is negligible. For more accurate results, inclusion of interaction between the edge is necessary for narrower slits. The fields singly diffracted from the edge of the conducting half plane in the direction of the opposite edge of the other conducting half plane is replaced by the field of a line source of equal amplitude located at the edge from which the singly diffracted field emanated. This procedure can be used to improve the accuracy by the including first-order interaction fields. However the edge diffraction fields are not omnidirectional as assumed for a line source, so the results are not made more accurate by further repeating the procedure. Higher order edge interaction might be accurately included by using the self consistent method of Karp and Russek [59], but with marginal advantage. Since higher order interaction is weak in the examples studied above as shown in Suedan and Jull (1978) [5], it is sufficient to include only non-interaction fields here. Since a slit in a conducting screen is the Babinet complement of a strip, solutions for high-frequency scattering by a strip can be easily obtained by the results from previous sections. It has been done for the reflected fields and for singly diffracted fields of the edges of a strip [28]. 46 Fig. 4.1 Geometry of an array of complex source beams for diffraction by a slit. 47 Slitco2.xls-slitco2 -10 + -20 4-I I £ -30 •3 -40 + -SO + -<J0 0 Cosine-squared kb = 46.65 + 30 60 <f> ( d e g r e e s ) 90 0 -10 + -20 + I I * •a I + -SO + -<J0 0 (b) Cosine-squared - «> = 152.5 30 60 iff ( d e g r e e s ) 90 Fig. 4.2 A comparison of normalized E-polarized far field diffraction patterns of a slit of width L0 excited by a cosine-squared distribution and a single complex line source with beam directivity kb at y0 behind the center of the slit. L = X/2,y0 = LJ2. (a) M= 6, L0 = 5X.(b)M=9,L0 = 9X. 48 Slitco2.xls-slitco2 2 <l> (degrees) <j> (degrees) Fig. 4.3 A comparison of normalized E-polarized far field diffraction patterns of a slit of width L0 excited by a cosine-squared distribution and a single complex line source with beam directivity kb at y0 behind the center of the slit. L = Xll, y0 = 1L0. (a) M = 6, L0 = 5X.Qo)M=9,L0 = 9X. 49 Slitco2.xls—slitco2_3 0 (degrees) <f> (degrees) Fig. 4.4 Normalized E-polarized far field diffraction patterns of a slit of width L0 excited by a cosine-squared distribution at y0 behind the center of the slit (solid curves). L = A/2, y0 = 10I o. (a) M=6,L0 = 5A. (b) M = 9, L0 = 9A. Aperture with slit, Aperture alone, • • • • • Plane wave incidence. 50 Slitco2.xls-slhUn 0 -5 + -10 + •15 + I -20 -25 1 -30 + -35 + -40 + -45 0 "HlllililllP* \ (a) ll ll u . . Ii — I i Uniform - - kb = 89.08 II 1 1 A '" W\ la \ \ \ >l l l •l ! V ll ll ""• ( — ' — ' — 1 — — , — , — , — 30 60 <p (degrees) 90 -10 + -15 + I §3 -20 I -J0 -35 + -40 + -45 (b) Uniform jfc& = 286.9 30 60 <f> (degrees) 90 Fig. 4.5 A comparison of normalized E-polarized far field diffraction patterns of a slit of width L0 excited by a uniform distribution and a single complex line source with beam directivity kb aty0 behind the center of the slit. L = X/2, y0 = LJ2. (a) M= 8, L0 = 5X. (b) M=\\,L0 = 9X. 51 Slhco2.xls-slitUn2 $ (degrees) $ (degrees) Fig. 4.6 A comparison of normalized E-polarized far field diffraction patterns of a slit of width L0 excited by a uniform distribution and a single complex line source with beam directivity kb at y0 behind the center of the slit. L = A/2, y0 = 2L0. (a) M= 8, L0 = 5 A. (b) M= 11,Z0 = 9A. 52 i (gp) J9M0J (qp) J3M0J 3A11VPH I 1 o "S C/5 C3 5 II o e •2 a t> 5 2 * " » w . T3 g 4> O o -c a to © 8 N a CN oo "St 3 o a <+-o G o en a 4 m i a O u r- O . t i <u •is o e h 2 o £ u o u 12 • S3 o X) ^ o II •B II -a x> J9M0J 9Atf0p)/ 53 u ec 3"* (SP) J3M0J 3A]tVJ9tf O § S •ti u •3 d CO M 3 «, •o << I fe • T3 T3 N 6 . O II oo o " o <U OS • < * > I I -p> ~ V—' S..I N CO '•a ^ 2 <2 a fe, o 3 g « to >> 6 & o o < u OO >—] ^ s «4-l o "S CJ u J 3 -«-» -O S3 •a A S o a I s S I I ••3 £ (gp) J3M0J SAflVPH 54 CHAPTER 5 APERTURE DIFFRACTION BY A CONDUCTING WEDGE 5.1 Analytical Expressions Scattering of electromagnetic waves by a two dimensional conducting wedge has received considerable attention. The asymptotic forms of its solution can be obtained by transforming the infinite series modal solution to an integral equation which is then evaluated by the method of steepest descents. The resulting terms of the integral evaluation can be identified as representing the incident and reflected geometrical optics fields and the diffracted fields. A uniform solution has been used to solve numerous practical scattering problem [1]. They are exact only for the half plane case but is very accurate for 90° wedges. These uniform solutions in terms of Fresnel integrals are convenient and easily adapted to the complex source point method to solve beam diffraction by wedge [4]. Consider a perfectly conducting wedge of exterior angle nit is illuminated by an array of complex line sources parallel to the edge, as shown in Fig. 5.1. from symmetry properties, we only consider the following conditions: 0<6 o <nn/2 0<6<n7t (5.1.1) 3 / 2 < n < 2 The reason we include only 0 < 60 < nn 12 is that 6 measured from the upper surface is the same as that nn I 2 < 6g < nn when 6 is measured from the lower wedge surface. We can use eq. ( A. 1 ) to get a two dimensional beam solution from an array of 55 complex line sources. The only thing that remains to be determined is the shadow boundary 9,u for reflection boundary from the upper surface 9„, and the reflection boundary from the lower surface 9,T' for each complex beam source. According to the definition of shadow and reflection boundaries [10], it can be shown that the shadow boundary position for each beam source is [4] y °<m> j (5.1.2) the reflection boundary position of the upper surface for each beam source is Qsrim] = n- 9ntw,, + cos'1 ""[ml 'o[mI Ba>^+e0[m, (5.1.3) and the reflection boundary position for the lower surface is dsr.[m,=(2n-\)jt-do[m]*cos-'o(ml - + (5.1.4) J with all angles measured from the upper wedge surface. Now the total field of a conducting wedge illuminated by an array of complex beam sources can be written as ^ M ) - < - 7 « ( 4 , - / ~e)+ErtlmlU(ewlml-e) < 5.1.5) + E[lmlu(d-dsi,ml}i(n7C-e)+E'z[ml where E[(mj, Erz[m, are, respectively, the far field of the source alone eq. ( A 2 ) and the reflected far field of the upper surface eq. ( A 3 ), and Ezlm], Ez[m] are, respectively, the field reflected form the lower wedge surface eq. ( A 4 ) and diffracted field eq. ( A 5 ) with r,[ml, 6s[m, replacing rs, d, respectively. 56 If the edge of the wedge lies on the beam axis, then the complex source position . (rs[n,,> 0s[n}) can be simplified to rs[m] = r0[mj + jb0 and 9s[m] = Go[m]. Then one of the cotangent functions in the uniform diffraction coefficient eq. ( A.6 ) is singular on the shadow or reflection boundaries, but it becomes finite when multiplied by the corresponding modified Fresnel integral G(a>). In the cases we study below, only the central beam of the array of complex line source is directly incident upon the edge. It is noted that 6s[m] for the outer beam sources are complex number, therefore the diffraction coefficient is always finite, and the field uniform. 5.2 Numerical Results 5.2.1 £-Polarization Far Field Calculation The diffraction patterns of a right angled conducting wedge illuminated by an inphase cosine-squared and uniform aperture distribution in an aperture of width L0 are calculated and shown in Figs. 5.2. to 5.4. In Figs. 5.2a and 5.2b the diffraction far field is calculated for an inphase cosine-squared aperture distribution with aperture size in wavelengths L0 = 2.5A located at kr0^] = 16 and fcr0/o; = 32 respectively, from the edge of the right angled wedge and at an incidence angle 00[OJ = rt/3 respectively. A l l the beam sources are spaced L = A/2 apart and with the same beam directivity and direction (ba = 0.25A, B0 = 4rc/3). The relative weighting factors Am in eq.( 5.1.5 ) of each beam source are again calculated according to eq. ( 1.3.2 ) and are given in Table 1. Note that the conducting wedge is in the Fresnel zone of the aperture. In Fig. 5.2a we can see two beams in the diffraction partem appear, one located near the incident shadow boundary of the central beam at 9si[0] = 4rc/3 and another along the reflection boundary at 0si(oj = 27x73. 57 In the region between these two beams and the region close to the upper surface, interference between incident, reflected, and diffracted fields is observed. In this case, there is a reflected field from the upper wedge surface only. In Fig. 5.2b all the parameters are chosen to be the same as in Fig. 5.2a except the distance from the central beam to the edge of the wedge increases from kr0[o\ = 16 to krol0] = 32. Fig. 5.2b is similar to Fig. 5.2a. The differences are that the interference between incident, reflected and diffracted fields in the region between the forward scattered and reflection beams and the region close to the upper surface is more pronounced and the two main beams are sharper because a distant aperture distribution is more directive at the edge. Also included are the diffraction patterns for a single line source beam with the same half power beam width (HPBW) as the cosine-squared aperture far field radiation pattern. Clearly although a single beam predicts two main beams along the incident and reflection shadow boundaries, their angular locations do not coincide with the calculation for the cosine-squared aperture distribution in kr0[o) = 32 case. Also interference between incident, reflected and diffracted fields in the region between the two main beams and the region close to the upper surface is not accurately presented in the single beam source calculation. In Figs. 5.3a and 5.3b the diffraction far field is calculated for a uniform aperture distribution with aperture size in wavelength is L0 = 2.5A- located at kro[o] = 16 from the edge of the right angled wedge and at an incidence angle 0O[o] = 7t/3 and 90{o\ = 2rc/3 respectively (solid lines). Also included are the diffraction patterns for a single line source beam with the same HPBW as the uniform aperture far field radiation pattern (dashed lines). A single beam predicts the two main beam positions at the incident and 58 reflection shadow boundaries but shows large discrepancies for interference between incident, reflected and diffracted fields, This is to be expected for an aperture distribution significantly different from Gaussian. A comparison between normalized far field diffraction patterns of a right angled wedge excited by a cosine-squared aperture distribution in an aperture of width L„ = 5\ and a single complex source beam at kr0 = 16 from the edge with incidence angles 90 = rt/3 and 90 = 2TC/3 are shown in Fig. 5.4. In this case a single beam is a good approximation since the aperture plane is very close to the wedge (ro[0] = L2011CU) and the first side lobe in the near field pattern does not appear at this range. In Fig. 5.5 the normalized far field diffraction patterns have been calculated for an uniform distribution in an aperture of width L0 - 5\ and a single complex source beam at kr0 = 16 from the edge with incidence angle d0 = rc/3 and 90 = 2n/3 in Figs 5.5a and 5.5b respectively. From Figs. 5.5a and 5.5b it is apparent that a single beam cannot accurately represent the far field diffraction patterns of a uniform distribution near a wedge except for the two mam beams. In order to verify the accuracy and validity of the computer programs, the exact series solution was programmed according to eq. ( 3.2.1 ) with q = 3/2 for right angled wedge. The results are shown in Fig. 5.6 and found that 30 terms in eq. (3.2.1) provides results essentially identical with those calculated from eq. ( 5.1.5 ) for far field diffraction patterns of a right angled wedge excited by an uniform distribution 5.2.2 ^-Polarization Far Field Calculation The //-polarization diffraction patterns of a right angled conducting wedge 59 illuminated by an inphase cosine-squared and uniform aperture distribution in an aperture of width L0 = 2.5 A. are shown in Figs. 5.7. to 5.10. In Figs. 5.7a and 5.7b the diffraction far field is calculated for an inphase cosine-squared aperture distribution with aperture size in wavelength is L0 = 2.5X located at kr0[OJ = 16 with incidence angle 9o[Q] = TC/3 and 90[OJ = 2TC/3 from the edge of the right angled wedge respectively. In Fig. 5.7a we can see two beams in the radiation pattern appear, one located near the incident shadow boundary of the central beam at 9slf0/ -4v:/3 and another along the reflection boundary at 9sr[Q] = 2TC/3. In the region between the two main beams and the region close to the upper surface, interference between incident, reflected, and diffracted fields are observed. In this case, there is a reflected field from the upper wedge surface only. In Fig. 5.7b the reflected field from both the upper and lower wedge surfaces are included. Also included are the diffraction patterns for a single line source beam with the same HPBW as the cosine-squared aperture far field radiation pattern. Clearly although a single beam can predict the two main beams along the incident and reflection shadow boundaries, their angular location and magnitude do not coincide with the calculation of the cosine-squared aperture distribution especially in the 9o[0] = 2n/3 case. Also the interference between incident, reflected and diffracted field in the region between the two main beams and the region close to the upper surface is not accurately presented in the single beam source calculation. In Fig. 5.8 the calculation is shown for the aperture plane away from the wedge with kro[0] = 32. A single complex source beam diffraction patterns also included for 60 comparison. In this case the single complex beam gives a good approximation for the two main beams. In Figs. 5.9a and 5.9b the far field diffraction pattern is shown for an inphase uniform aperture distribution with an aperture size in wavelength is L0 = 2.5k located at kr0[Q] - 16 from the edge of the right angled wedge and at an incidence angle 90[OJ = n/3 and 9o[0] = 2n/3 respectively (solid lines). Also included are the diffraction patterns for a single line source beam (kb = 22.84) with the same half power beam width as the uniform aperture far field radiation pattern (dashed lines). A single beam source can predict the two main beams along the incident and reflection shadow boundaries but shows large discrepancies in the interference between incident, reflected and diffracted fields as expected for a uniform distribution. Fig. 5.10 shows a comparison of normalized far field diffraction patterns of a right angled wedge excited by an uniform aperture distribution and a single complex source beam at kro[0J = 32 from the edge. Again a single beam is a good approximation only in the two main beams along the incident and reflection boundaries of Figs. 10a is and not accurate in the interference region. In Fig. 5.10b, interference between the elements of the array appears on both reflection lobes as well. 61 62 Wedco2.xls-wedco2_l 0 (degrees) 0 (degrees) Fig. 5.2 A comparison of normalized E-polarized far field of a right angled wedge excited by a cosine-squared distribution and a single complex source beam (kb = 11.5) at kr0 from the edge. (M= 2, L0 = 2.5X, L = Xll, G0 = 7t/3). Cosine-squared distribution, kb = 11.5. (a) kr0 = 16, (b) kr0 = 32. 63 Wedco2 ,xls-wedun_ 1 0 90 180 270 0 90 180 270 0 ( d e g r e e s ) 0 ( d e g r e e s ) Fig. 5.3 A comparison of normalized E-polarized far field of a right angled wedge excited by a uniform aperture distribution and a single complex source beam (kb = 22.84) at kr0 = 16 from the edge. (M= 2, L0 = 2.5A., L = X/2,). Uniform distribution, kb = 22.84. (a) 0O = 7t/3, (b) 0O = 2n/3. 64 Wedco2.xls-wedco2 2 0 (degrees) 6 (degrees) Fig. 5.4 A comparison of normalized .E-polarized far field of a right angled wedge excited by a cosine-squared aperture distribution and a single complex source beam (to = 46.85) at kr0 = 16 from the edge. (M= 6, L0 = 5X,L~ XI2,). Cosine-squared distribution, to = 46.85. (a) 6>0 = 7i/3,(b) 0o = 2TC/3. 65 Wedco2. xls-wedUn_2 0 90 180 270 0 90 180 270 9 (degrees) 9 (degrees) Fig. 5.5 A comparison of normalized £-polarized far field of a right angled wedge excited by a uniform aperture distribution and a single complex source beam (kb - 89.08) at kr0 ^ 16 from the edge. (M= 8, L0 = 5X,L = X/2,). Uniform distribution, kb -89.08. (a) 90 = Tt/3, (b) 90 = 2n/3. 66 Wedco2.xls~seriesUn 0 (degrees) 0 (degrees) Fig. 5.6 Convergence of the exact series solution and the UTD solution for E-polarized far field of a uniform aperture distribution above a perfectly conducting wedge. 67 Wpi_2.xls-wedco2H 90 180 9 (degrees) 270 90 180 9 (degrees) 270 Fig. 5.7 A comparison of normalized //-polarized far field of a right angled wedge excited by a cosine-squared aperture distribution and a single complex source beam (kb = 11.5) at kr0 = 16 from the edge. (M= 2, L0 = 2.5X, L = X/2,). Cosine-squared distribution, kb = 11.5. (a) 90 = n/3, (b) 90 = 2n/3. 68 Wpi_2.xls-wedco2H_2 90 180 9 (degrees) 270 90 180 9 (degrees) 270 Fig. 5.8 A comparison of normalized /f-polarized far field of a right angled wedge excited by a cosine-squared aperture distribution and a single complex source beam (kb = 11.5) at kr0 = 32 from the edge. (M = 2, LQ = 2.5X, L = Xll,). Cosine-squared distribution, kb = 11.5. (a) 90 = nJ3, (b) 90 = 2TX/3. 69 Wpi_2. xls-wedunH 0 90 180 270 0 90 180 270 9 (degrees) 0 (degrees) Fig. 5.9 A comparison of normalized //-polarized far fields of a right angled wedge excited by a uniform aperture distribution and a single complex source beam (kb = 22.84) at kr0 = 16 from the edge. (M = 2, L0 = 2.5V L = XI2). Uniform distribution, kb = 22.84. (a) 90 = T C / 3 , (b) 90 = 2TC/3 70 Wpi_2.xls--wedco2H_2 0 90 180 270 0 90 180 270 9 (degrees) 9 (degrees) Fig. 5.10 A comparison of normalized //-polairzed far fields of a right angled wedge excited by a uniform aperture distribution and a single complex source beam (kb = 22.84) at kr 0 = 32 from the edge. (M = 2, L0 = 2.5X, L = Xll). Uniform distribution, kb = 22.84. (a) 0o = n/3, (b) 90 = 2TC/3 71 CHAPTER 6 APERTURE DIFFRACTION BY A CONDUCTING SQUARE CYLINDER 6.1 Analytical Expressions This chapter investigates the scattering patterns of a large rectangular conducting cylinder by a local complex source beam. The total field is the superposition of the solution for the far field of a beam source in the presence of four right angled wedges and the first order interaction between the edges is included when significant. This solution is expressed uniformly in terms of Fresnel integrals. The Fresnel integral arguments become complex but these integrals are complimentary error functions with complex arguments (see eq. ( 3.1.5 )), which can be computed with a standard subroutine. This approach was used for diffraction of an array of beam sources by a local conducting wedge (Chapter 5). It is also applied to an array of such beams constituting an antenna aperture distribution scattered by a rectangular cylinder. A perfectly conducting rectangular cylinder is illuminated by an array of complex line sources parallel to the axis of the cylinder extending along the z axis, as shown in Fig. 6.1. The uniform theory of diffraction (UTD) has been combined with the complex source point (CSP) method for solving the diffraction problem by a conducting rectangular cylinder to an array of beam sources in this chapter [50], [52]. Because of symmetry we only consider the field in region - 45° < 9 < 135° for inclined incidence at 90 = 135° (Fig. 6.1a) and the field in region - 90° < 9 < 90° for normal incidence on the face of a cylinder (9 0 = 90°) (Fig. 6.1b). In Fig. 6.1a the total far field diffraction pattern of an array of complex line sources 72 and rectangular cylinder combination is represented by a superposition of the diffraction patterns of discrete elementary beams linearly spaced by a distance L along the aperture plane between neighbouring beams. For the basis elements Bm here, we use the solution for the total field at r, 9 in cylindrical coordinates due to the beam sources at r , ., 9 , , and parallel to the edge with beam parameters b0, B0, thus m-M (6.1.1) Bm = E'fmJu(9slB[m/ -x + 9)+ Efmlu{9srB(ml - n + 9)u(9srA[m/ -&) + Ed:[m, E':(„j, E^l/, and Ed.[m] are respectively, the incident field, the field reflected from plane AB, and the total diffracted fields from edges A, B, and C. u(z) is a unit step function. 0s,B[mj md 9srB(mJ are the shadow and reflection boundaries due to edge B measured clockwise from plane AB and 9srA[mJ is the reflection boundary due to edge A measured counter-clockwise from plane AB. The incident far field with coordinates referred to edge A (see Fig. 6.2) is E'r r (6.1.2) :[m< JkTA The incident shadow boundary for incident field E':[m] due to edge B is ^ Re(rsB[mi) The far field reflected from the plane AB is > P: ^ + 0*1., (6 -1 .3 ) E™'-- ^ (6-1.4) The reflection boundaries for reflected fields E'zfmj due to edges A and B are 73 QsrBfml = K - Q o B [ m l ± COS -1 I roA[mj ) (6 .1 .5 ) The fields diffracted by edge A, B and C contribute to the total diffracted field Ed:[nl. The general expression for the field diffracted by edges A, B and C is given by g " jk{r!A.8.Ci *rst A B Cllm/) E;[m] ' = j , rr— ^{f(A.B.C)'^(A.B.C)'rs(A,B.C)[ml>^s(A.B,C)lml) (6.1.6) \l<r(A.B.C) \la's(A.B.C)[ml with a uniform diffraction coefficient D\[rfABC),9(ABC),rs(ABC)[ml,9sfABC}[ml) is defined as in eq. ( A.6 ) with r[ABC)[mj, 9{ABC), rs(ABC)[mJ, and 9s(ABC)[mJ replacing r, 9, r5 and 9S for edges A, B and C respectively and rs(ABC)[mj and 9s{ABC)[m] are calculated from eq. ( 2.1.2) with r0(ABC)[m], 90(ABC)[mj, andp<0ABC) replacing r0[mj, 90[mJ, and B0. Now the field diffracted by a complex source beam can be written as 7dB •rd pdA pdB ^zfmj + C .-/>;-0<9<7t/2 -x/4<9<0 (6.1.7) In above equations, r{ABC), ro(ABC)[mJ, and rs{ABC)[mj, are the distances of the observation point P, the real source point and the complex source point to edges A, B and C respectively. 9A, 9oA[mJ, 9sA[mJ, and ft are the directions of rA, roA[m/, rsA[mj and the beam vector jbQ measured counter-clockwise from plane AB or the x axis (see Fig. 6.2a). Similarly 9B, 9oB[m,, 9sBfm), and/?/ are the directions of rB, roB[mjl rsB[mj and the beam vector jb0 measured clockwise from plane AB or the -x axis (Fig. 6.2b), 9C, 9oC[mj, 74 0sc[mi> m d Po ^ t n e directions of rc, roC[mj< rsC[mj and the beam vector jb0, measured clockwise from plane AC or the y axis (Fig. 6.2c). Beam sources normally incident upon the rectangular cylinder side AB (Fig. 6.1b) can be treated similarly to the above. Thus K = E':[mju(95lB[mJ - n + 0) + E^mju(0srB[m/ - * + *) + Ed:[ml (6.1.8 ) and (6-1.9) E % M J , -KI2<9<Q. Analysis of the //-polarization diffraction problem can be treated in a similar way to £-polarization. For //-polarization, H replaces E in the above expressions, the (-) sign in the non exponential part of the reflected wave in eq. ( 6.1.4 ) becomes (+) and the diffraction coefficient D[r(A .B.O'^(A.B.c)'rs(A.B.c)(m]'^s(A.B.c)(mi) I S multiplied by a factor ( - l ) ' + l . This non-interaction solution is inadequate for //-polarization. In order to improve the accuracy it is necessary to include also first order interaction between adjacent edges. In the cases discussed there are six more terms to be added for inclined incidence on edge A and four more terms added for normal incidence on side AB. The singly diffracted field of an edge in the directions of an adjacent edge is replaced by the field of a line source of equal amplitude located at the edge. A similar technique was used to calculate beam diffraction by a wide slit [5]. For example the doubly diffracted field from edge B is produced by the singly diffracted field from edge A in the 9\ = 0 direction. This can be written as H%" = l l ' ^ ' L / L ^L^J'°'rsA[mj^sAfmjHrB,9B>a,0) (6.1.10) 75 where a is the width of the square cylinder, and LAB[mj is the distance parameter . arsA(mJ L*B(»I=— (6.1.11) U rsA[m I Similarly the doubly diffracted from the edge C due to a singly diffracted field from the edge A in the 9A = 3K / 2 is given by eq. (6.1.10 )with rc, 9C, and LAC replacing rB, 9B, and LAB respectively and the second argument in the first diffraction coefficient changed to 3TT/2. The doubly diffracted field from edges A and D due to a singly diffracted field from edge B in the 9B = 0 and 3TT/2 direction and from edges A and D due to a singly diffracted field from edge C in the 9B - 0 and 3rc/2 direction respectively can be obtained in a similar way. The incident field vanishes on the conductor and thus the first order doubly diffracted field will be zero in the £-polarization case. There is however a weaker diffraction based not on the magnitude of the incident field at the point of diffraction but rather on the rate of change of the incident field at the point of diffraction. It is referred to as slope diffraction [30] [31]. 6.2 Numerical Results 6.2.1 £-polarization Far Field Calculation Fig. 6.3 shows the total far field patterns at r, 9 for a single beam source (m = 0) with kb0 = 0, 2, 4, and 8, p* = 7nJ4, parallel to and at kroA[Q[ = 16, 9oA[0[ = 3«/4 from the upper left edge of a square cylinder of side 2X for £-polarization (Fig. 6.1a). The solid curve is the singly diffracted field calculated with basis element Bo from eq. ( 6.1.1 ) and A0 =1. The minor lobe in the forward (9= -7t/4) direction increases with increased beam 76 directivity. The two peaks around at 9 = 25° and 45° are identified as reflection boundaries of edges B and A respectively. Notice that the beam source acts like an aperture distribution and rays emanating from larger apertures are nearly parallel to each other, so that as the beam directivity increases the two reflection boundaries merge together at around 9 = 45°. The patterns have larger oscillation lobes for less a directive beam in the region 45° < 6 < 135° because of the interference between the incident wave from upper surface AB and diffracted waves from edges A, B, and C. In the region 9 < 0, the field increases with increasing beam directivity for small directive beam sources (kb = 0, 2, 4) because a larger portion of radiated power is intercepted by the cylinder. It decreases with increasing directivity for larger directive beam sources because there is more blockage by the square cylinder, i.e when edges B and C are no longer in the central portion of the main beam. The dotted curves in Fig. 6.3 are from a calculation by the moment method with a point matching technique. There is a small difference between UTD and moment method calculations in regions around 9 = n/2 and 9=0 with larger beam directivity due to a stronger Ulurnination of edges A, B, and C. This is because interaction between the edges is ignored in the UTD calculation. Note that the HPBW of the most directive beam (kb = 8) is around 34° , so the square cylinder of side 2X is just inside the angle subtended by the half power points in its far field pattern. It is apparent that even with a square cylinder of width as small as 2X, inclusion of interaction between the edges is not really necessary for E-polarization, especially for a beam of low directivity. Fig. 6.4 shows the total far field patterns at r, 9 for a single beam source (m = 0) with kb0 = 0, 2, 4, and 8, fi£ = lid A parallel to and at kroA[0] = 16, 6oA[0] =3TC/4 from edge A of a larger square cylinder of width 5X for E polarization. Again there are more oscillations in the angular region 45" < 6 < 135" for less directive beams. Also there is 77 only one dominant peak around the 8 = 30° direction because of a relatively stronger illumination of edge A and weaker reflection from faces AB and AC. When we increase the size of the square cylinder to a = 25X, the total far field patterns in the region 0 < 6 < 3n/4 become like those for a single beam source incident upon a right angled wedge. In Fig. 6.5 all the parameters are chosen to be the same as Fig. 6.2 except the distance from the source to the edge A is moved from kroA[0} = 16 to 32. It shows that the interference between incident, reflected, and diffracted fields is stronger for less directivity because of the relatively strong iUumination of edges A, B, and C. In the region 6 < 0, the field is stronger with increased directivity compared to Fig. 6.3 because there is less blockage by the square cylinder for a more distant source. Also included is the calculation by the moment method with a point matching technique (dotted curves). Again, except for the small difference between UTD and moment method calculation in region around 6 = id! and 0 = 0 , with larger beam directivity the two methods agree well with each other over the entire pattern. The discrepancy around 6 = n/2 and 6=0 between moment method and UTD is smaller in this case than in Fig. 6.3 because singly diffracted fields from edges A, B, and C are relatively stronger than the doubly diffraction field due to the interaction between these adjacent edges. Now we consider the total far field patterns at r, 6 for a single beam source (m = 0) with kb0 = 0, 1, 2, and 4, and fi0 = 3n/2 parallel to and at kro[0] = 32, 6o[0j = n/2 from the upper surface of a square cylinder of side 2X for ^-polarization (Fig. 6.1b). The bold solid curve in Fig. 6.6 is the diffracted field including single and double diffraction (slope diffraction). The light solid curves is the diffracted field including only single diffraction and the dotted curve is the moment method result with the point matching technique. Note that higher order diffraction (slope diffraction) is especially significant in this calculation 78 for the appearance of the forward (8 = - 90°) lobe, although it still does not represent the field there very accurately. However including slope diffraction fields also improves the accuracy over the remaining region. In Fig. 6.7 the beam source is moved to kro[0] = 16 with beam directivity kba = 0, 1, 2, and 4. The bold solid curve including single and double diffraction now gives a better agreement with the moment method results on the forward lobe. The reason the UTD result does not give an accurate prediction of the forward lobe region for the kro[0] = 32 case is that UTD fails to predict the diffracted field near the alignment of source and two diffracting edges and the observation field point. This occurs at 8= - 90° as kra —> «>. The curves in Fig. 6.8 compare the total far field patterns at r, 8 for a uniform aperture distribution kro[0] = 16 and 32 at from edge A of a square cylinder of side 2X and 5X calculated with basis elements Bm(r,8) from eq. ( 6.1.1) and appropriate values of Am form Table 1. The angle of incidence 6oA[0] of the axis of aperture with respect to edge A is 3TC/4. The aperture size in wavelengths is Lg = 2.5X which is simulated with 5 (Af = 2) complex source parallel beams spaced L = XI2. The central beam normally directed at the edge A (/?A =7TC/4) and each with beam directivity kb0 = n/2. For the same cylinder width 2X (Fig. 6.8a, c) and 5A, (Fig. 6.8b, d) with near and more distant aperture distributions from cylinder edge A, the scattered fields have more oscillations due to larger interaction between the direct wave, the reflected wave from different sources and the diffracted wave from cylinder edges A, B, and C for the distant aperture distribution case. With the same distance from the edge to the aperture, the far field patterns of the larger cylinder show a sharper peak around 8 = 45° corresponding to the position of the reflection boundary of edge A and less penetration into the shadow region 8 < 0. The beam directivity of the array 79 of source beams used is kb0 = nil and it is known that for this beam directivity the singly diffracted fields of a square cylinder alone give reliable results even for small cylinder. 6.2.2 //-polarization Far Field Calculation Fig. 6.9 shows the total far field patterns at r, 9 for a single beam source (m = 0) with kb0 = 0, 2, 4, and 8, = 77t/4, parallel to and at kroA[0, = 16, 6oA[0J = 3TT/4 from the upper left edge of a square cylinder of side 2X for //-polarization (see Fig. 6.1a). The solid curves are include only the singly diffracted fields and the dashed curves include also first order interaction between adjacent edges of the square cylinder. It is seen that the discontinuity in the single diffraction calculation along the diffraction boundaries such as 6 = 0° and 90" is reduced by including of the doubly diffraction field, as shown by the dashed curves. Also it is noticed that large differences between the singly and doubly fields in //-polarization representing interaction between adjacent edges is important for small cylinder sizes, especially with a directive beam. Fig. 6.10 shows the total far field patterns at r, 9 for a single beam source (m = 0) with kb0 = 0, 2, 4, and 8, $ = ln/4 parallel to and at kroA[0] = 16, 6oA[0J =3n/4 from edge A of a larger square cylinder of side 5X for //-polarization. Although there is little difference between the calculation including single diffraction only (solid line) and with the inclusion of double diffraction (dashed line), the discontinuity along the diffraction boundaries for a directive beam is still evident and the double diffraction contribution is needed to remove it. In Fig. 6.11 all the parameters are the same as Fig. 6.9 except the distance from the source to the edge A is increased from kroAlQ] = 16 to 32. It shows that the interference between incident, reflected, and diffracted fields is stronger because of the relatively strong 80 illumination on edges A, B, and C. In the region 9 < 0, the field increases with increasing directivity because there is less blockage by the square cylinder for a distant source. The discontinuity along the 9 = n/2 direction is larger than that in the kroA[0] = 16 case, showing that higher interaction contributions are still important for directive (kb = 4, 8) beams incident upon on small cylinder (a = 2X). Now we consider the total far field patterns at r, 9 for a single beam source (m = 0) with kb0 = 0, 2, 4 and 8, and (30 = 3rc/2 parallel to and at kro[0] = 16 6o[oj = n/2 from the upper surface of square cylinder of side 2X for //-polarization (Fig. 6.1b). The solid curves in Fig. 6.12 include only singly diffracted fields, and the dashed curves include both single and double diffraction. The latter oscillates about the former in the lateral and forward angular regions of the patterns. Also it is noted that the discontinuity at 9=0° decreases with increasing beam directivity and that the total field in the forward direction (9 = -nl2) is almost entirely contributed by single diffraction, in contrast to the situation for E-polarization. Similar results are observed for a more distant source incident upon on the cylinder in Fig. 6.13. Figs. 6.14 to 6.17 shows a comparison of UTD results with the superposition of the singly and doubly diffracted fields and moment method solutions for a single beam source with kb0 = 0, 2, 4, and 8 and inclined and normally incident upon a square cylinder of side 2X. In all cases, the moment method calculation and UTD give similar results. The curves in Fig. 6.18 represent the total far field patterns r, 9 for the beam axis of a uniform aperture distribution directed at edge A of square cylinders of sides 2X and 5X. The aperture is located at distances from the edge A of kroA[0[ =16 and 32 and the angle of incidence 90A[OI of the beam axis of the aperture distribution with respect to side AB is 81 3n/4. The aperture size in wavelengths is La = 2.5A. A l l the line source beams are spaced L = A/2 apart and are normally directed at the edge A (/?0A =ln/4) with directivity kb0 = n/2. For the same cylinder width a = 2A (Fig. 6.8a, c) and 5A, (Fig. 6.8b, d) with near and distant aperture distribution from cylinder edge A, the patterns are smooth or have no discontinuity around 6 = 0 for the more distant aperture distribution case. With the same distance from the edge to the aperture, the far field patterns of larger cylinders show the sharper peaks around 6 = 45° corresponding to the reflection boundary of side AB and less penetration to the shadow region 6 < 0. A simple uniform solution for an aperture distribution near a square cylinder was obtained here. It can be made highly accurate with sufficient array elements. Notice that when the cylinder is close enough to intercept the first sidelobe of the near field radiation pattern of the aperture, a single beam source with the same HPBW as the aperture far field radiation pattern or a single Gaussian beam cannot accurately predict the diffraction pattern. For smaller cylinders (a = 2A,), higher order interaction is important, especially in the forward direction, for normal incidence upon the upper surface of the cylinder in E-polarization and for removal of the discontinuity along the shadow boundary with •re-polarization case. However, slope diffraction does not accurately predict the diffracted field for near grazing incidence with source and edges aligned for ^-polarization. An improvement discussed in [32] requires either source or receiver to be aligned within the two edges. In [33] this restriction is removed but the scattered far field is for plane wave incidence only. [34] uses a local source for double wedge diffraction. This technique might be applicable here. [35] compares results from different methods for plane wave scattering by rectangular cylinders.. 82 Fig. 6.1 Geometry of an array of complex source beams for diffraction by a conducting square cylinder. 83 Fig. 6.2 Coordinate systems of the diffraction problem. 84 1 T kb=0 kb =2 -45 0 45 90 135 0 ( d e g r e e s ) -45 0 45 90 135 9 ( d e g r e e s ) / T kb =4 kb =8 1 T -45 0 45 90 135 -45 9 ( d e g r e e s ) ' 1 I 1 1 I 1 1 I 0 45 90 135 9 ( d e g r e e s ) Fig. 6.3 E-polarized far field of a square cylinder for a single beam source. (kr0 = 16, 60 = 37C/4, a =2X), CSP/UTD solution (single diffraction only), • • • • • • Moment method. 85 r a ^ • -45 0 45 90 135 -45 0 45 90 135 9 (degrees) 9 (degrees) Fig. 6.4 E-polarized far field of a square cylinder for a single beam source. (krQ = 16, 60 = 371/4, a = 5X), CSP/UTD solution (single diffraction only). 86 RecE l.xls—rec2 r o \ a 1 T kb=2 0.5 4-1 T kb = 4 0.5 + 1 T 0 45 90 135 0 (degrees) kb =8 1 0 -45 0 45 90 135 -45 0 (degrees) 0 45 90 135 0 (degrees) Fig. 6.5 E-polarized far field of a square cylinder for a single beam source, (kr 0 = 32, e0 = 37t/4, a = 2X), CSP/UTD solution (single diffraction), Moment method. 87 -90 -45 0 45 90 -90 -45 0 45 90 9 (degrees) 9 (degrees) 9 (degrees) 9 (degrees) Fig. 6.6 .E-polarized far field of a squared cylinder for a single beam source at a distance r0 above the upper surface. (kr0 = 32, 90 = nil, a = Tk,h = 2X), CSP/UTD solution (single + double diffraction), CSP/UTD single diffractiononly, • • • • Moment method. 88 -90 -45 0 45 90 -90 -45 0 45 90 9 ( d e g r e e s ) 9 ( d e g r e e s ) -90 -45 0 45 90 -90 -45 0 45 90 9 ( d e g r e e s ) 9 ( d e g r e e s ) Fig. 6.7 E-polarized far field of a squared cylinder for a single beam source at a distance r0 above the upper surface. (kr0 = 16, 90 - n/2, a-2X,h = 2X), CSP/UTD solution (single + double diffraction), CSP/UTD single diffractiononly, • • • • Moment method. 89 Fig. 6.8 E-polarized far field of a square cylinder for a uniform aperture distribution. (60 = 3n/4, M = 2, L0 = 2.5A, L = Xll), CSP/UTD solution (single diffraction only). 90 RecH.xls-tecHl 4 y > X 1 T kb =0 / T kb = 2 -45 1 T 0 45 90 9 ( d e g r e e s ) 135 -45 kb =4 1 T 0 45 90 9 ( d e g r e e s ) kb =8 135 -45 0 45 90 135 -45 0 45 90 135 9 ( d e g r e e s ) 9 ( d e g r e e s ) Fig. 6.9 //-polarized far field of a square cylinder for a single beam source. (kr0 - 16, 90 = 3TC/4, a = 2X), single diffracted field, singly + doubly diffracted field. 91 RecH.xls-recHl y X 1 T kb = 0 1 T Aft =2 -45 1 T j 1 1 . . 1 1 ' 1 *— 0 45 90 9 ( d e g r e e s ) 135 -45 0 45 90 9 ( d e g r e e s ) 135 kb = 4 1 T kb =8 0 - P - 1 — ' — I — 1 — 1 — I — 1 — ' — I — 1 — 1 — I -45 0 45 90 135 -45 0 45 90 135 0 ( d e g r e e s ) 9 ( d e g r e e s ) Fig. 6.10 //-polarized far field of a square cylinder for a single beam source. (kr0 =16, 90 = 3TC/4, a = 5X), single diffracted field, singly + doubly diffracted field. 92 RecH.xls-rccH2 X 1 T -45 1 T kb = 0 1 T j — i — | — . — i — | — i — i — | o 0 45 90 135 -45 0 (degrees) 1 T to = 2 - ! 1 1 L _ _ . 1 1 1 1-0 45 90 6 (degrees) kb =8 135 0 45 90 135 0 (degrees) Fig. 6.11 //-polarized far field of a square cylinder for a single beam source. (kr0 = 32, 60 = 3TC/4, a = 2X), single diffracted field, _ singly + doubly diffracted field. 93 RecH_2.x ls - recH3 1 T 1 T / T \Hz\i 0.5 + •90 -45 0 45 0 (degrees) 90 -90 1 T -45 0 45 0 (degrees) 90 •90 -45 0 45 90 0 (degrees) -90 -45 0 45 90 0 (degrees) Fig. 6.12 //-polarized far field of a square cylinder for a single beam source at a distance r0 above the upper surface. (kr0 = 16, 0O = n/2, a = 2X,h = 2X), single diffracted field, singly + doubly diffracted field. 94 RecH_2.x ls - recH3 9 (degrees) 9 (degrees) Fig. 6.13 //-polarized far field of a square cylinder for a single beam source at a distance rQ above the upper surface. (kr0 = 32, 90 = TC/2, a = 2\,h = 2X), single diffracted field, singly + doubly diffracted field. 95 9 (degrees) 9 (degrees) Fig. 6.15 //-polarized far field of a square cylinder for a single beam source. (kr0 = 32, 90 = 3n/4, a = 2X), CSP/UTD solution (single +double diffraction), • • • • • Moment method. 96 y f r X 1 T kb = 0 1 T kb = 2 0 - i — 1 — 1 — I — 1 — 1 — I — 1 -45 0 45 90 135 0 (degrees) 1 T 0 H—1—1—I—1—1—h -45 0 45 90 755 9 (degrees) 1 T AA =8 0 45 90 JJ5 9 (degrees) Fig. 6.14 //-polarized far field of a square cylinder for a single beam source. (kr0 = 16, 90 = 3TC/4, a = 2X), CSPAJTD solution (single -fdouble diffraction), Moment method. 97 h 2 ,1 2 T -90 -45 0 45 90 -90 -45 0 45 90 6 (degrees) 0 (degrees) Fig. 6.16 //-polarized far field of a square cylinder for a single beam source at a distance rQ above the upper surface. (kra = 16, 0O = n/2, a = 2X, h = 2X), CSP/UTD solution (single + double diffraction), • • • • • • Moment method. 98 -90 -45 0 45 90 -90 -45 0 45 90 0 (degrees) 0 (degrees) 0 (degrees) 0 (degrees) Fig. 6.17 //-polarized far field of a square cylinder for a single beam source at a distance r0 above the upper surface. (kr0 = 32, 0O = 7C/2, a = 2X, h = 2X), CSP/UTD solution (single + double diffraction), • • • • • • Moment method. 99 RecUn.xls-recUn_H -45 0 45 90 135 -45 9 ( d e g r e e s ) -45 0 45 90 9 ( d e g r e e s ) 135 0 45 90 135 9 ( d e g r e e s ) -45 0 45 90 0 ( d e g r e e s ) 135 Fig. 6.18 Normalized //-polarized total far field diffraction pattern of a uniform aperture distribution at a distance of r0 from the edge of a perfectly conducting square cylinder. (0O = 3TC/4, A / = 2, L0 = 2.5*., L = X/2), singly diffracted field, singly +doubly diffracted field. 100 CHAPTER 7 APERTURE DIFFRACTION BY A CIRCULAR CYLINDER 7.1 Analytical Expressions Diffraction by a circular cylinder, being a simple case among two-dimensional structures, has interested many investigators. Measurement and calculation of the complete field when the electric field is parallel to the axis of the cylinder have been made by Kodis [36] and by King and Wu [37]. Computation of the magnetic field when this is parallel to the axis of the circular cylinder have been made by King and Wu [38]. There are solutions for local isotropic line sources calculated by Pathak, Burnside, and Marhefka [39], and Faran [40]. Two dimensional beam diffraction by a circular conducting cylinder is considered in this chapter. By assigning a complex value to the location of a source point, one can convert a time harmonic isotropic or line source in a homogeneous unbounded medium into a highly collimated wave field. With fields of a large aperture distribution incident, the computation of the scattered field can be done by decomposing the aperture field into basis elements (eg. Gaussian beams) and then summing all the contributions from the individual beam sources. High frequency analyses of the reflection and diffraction of a Gaussian beam by a perfectly conducting parabolic surface with an edge and by a conducting cylinder are presented in [54], [55], and [56], [57], respectively. Sometimes, it is not necessary to determine either the total field or the scattered field in complete detail at all points. It may often suffice to determine only certain average characteristics of a scattering obstacle instead of the entire diffracted field for many practical purposes. The 101 total scattering cross section is one such important quantity in the study of the general problem of scattering from obstacles of complex shape, and of the circular cylinder. A. E-polarized single line source Fig. 7.1 shows the circular cylinder of radius a with center located at the origin illuminated by an array complex line sources parallel to the cylinder axis (£) . For single . line source, only the central (m = 0) beam located at (r0, </)0) is considered. Then the field directly contributed by the line source intensity at any observation point (r, (f>) can be written in terms of a superposition of cylindrical waves as E[ =H™(kR)= t,Jm(kr0)H™(kr)eim^) (7.1.1 ) for r > ro. The circular coordinates (r, are related to rectangular coordinates (x, y) by the transformation x = rcos<p, y = rsin0, where R is the separation distance between the observation point and the source, ' R = jr2+r?-2rr0cos(<p-<P0) (7.1.2) Then the scattered field intensity is given by El = - X ^^H^(kr0)H^(kr)e^ ( 7.1.3 ) In the far field (r —> <*>), the asymptotic forms of the Bessel and Hankel function may be approximately written as I 7 / / ( 2 ) (kR) ~ .-Hkr-tcIA,) .ikr„ cos( f>-&) 0 V Ttkr \nkr (7.1.4) 102 Appling eq. ( 7.1.4) to eqs. (7.1.1 ) and ( 7.1.3 ) the total electric far field is K ~ i n k r e X ±J / / f ( t e ) m l J J ( } On the surface r = a, the magnetic field intensity is H,=H;+H;=-f-^ „ 0 ) , , \ « " c ^ ) (7-1.6) If the electric line source is located on the surface (r = a), it is easy to show the total electric field is identically zero everywhere according to eqs. (7.1.1) and (7.1.3 ). As is pointed out earlier, it is convenient for many practical purposes to evaluate only certain average quantities to characterise the radiating properties of obstacles in an incident field. Scatterers are usually characterized by a scattering cross section defined as the ratio of scattered power to incident power density. The total scattered power per unit length in the z-direction P s is the real part of the radial component of the complex Poynting vector integrated over a cylindrical surface of large radius surrounding the scattering cylinder. This is P< =±RejsE*z(-H;)dSl ,2 (7.1.7) -21 f ' k ±L Jm(ka) and the scattering cross section is a =P S IS', where the incident power density on the cylinder is - R e f Eim'adfi 9 ha z * r S ' = - (7.1.8) a A a where Aa is the angular range on its surface over which the cylinder intercepts the 103 incident field. We know the geometrical optics scattering cross section is equal to twice the projected area of a unit length of the scattering cylinder. Therefore we define the normalized scattering cross section an=o I (2aAa). According to this definition the normalized scattering cross section for plane wave incidence becomes Jp{ka) H?\ka) (7.1.9) which corresponds to aT I Aa where aT is given by [41, eq. 2.8]. The surface density of current Kz on a perfectly conducting cylinder in an incident field of eq. (7.1.1 ) is equal to the tangential component of the magnetic field Hp at the surface (r = a). That is readily obtained from eq. (7.1.6). B. //-polarized line source The solution for an //-polarized line source located at (rB, 0O) incident upon a circular cylinder may be obtained in a similar manner to the E-polarized case, such that H[ =Hl 2)(m = J,Jm{kro)HmV(kr)eM^) (7.1.10) for r > r0. Then the scattered field is given by K B-l4rrT^feW,(*r>WH1 (7.1.11 ) The prime indicates partial derivative with respect to the entire argument of the Bessel or Hankel function. In the far field limit (r -» © o ) , the total magnetic field intensity is H< ^ ^ e - ' 1 ™ 4 ^ * 0 * ™ ~ I Jmif^H-fa>M™] (7.1-12) On the cylinder surface r = a 104 (7.1.13) The definition of total scattering cross section for an //-polarized line.source is the same as for the E-polarized case. But now - — X kYomtL aAa C. Beam source By making the real source coordinates complex fs = f0 - jb , the omnidirectional wave becomes a directive beam uniform in the z direction. Thus the complex coordinates rs, (j)s calculated from eq. ( 1.2.1) replace r0, </)0 in the above equations from ( 7.1.1) to ( 7.1.14 ) for an electric or magnetic beam source parallel to the cylinder axis and located at (r0, 0O) incident upon a perfectly conducting cylinder [49], [51], [53]. For large aperture distributions, we can synthesize antenna patterns from arrays of basis elements. Such a generalization is represented by the Gabor representation as discussed in previous chapters. The radiation field is presented by a discrete collection of beams linearly shifted with respected to one another at distances L along the aperture and linearly phase shifted between neighbouring beams. Thus the fields are M Ez,Ht=^AmBm (7.1.15) m = - M with proper basis elements Bm substituted. Jm(ka) H^'ika} H, (2) (7.1.14) 105 7.2 Numerical Results 7.2.1 Scattering Patterns / A. ^-polarization. The normalized total far field diffraction patterns and scattered field pattern in the first row of Fig. 7.2 were calculated from eqs. ( 7.1.5 ) and ( 7.1.3 ) with a circular cylinder of radius ka = 2.5rt for the single E-polarized local omnidirectional line source (kb = 0) located at (kxa = 5it, ky0 = 0). The calculations for the beam source with beam directivity kb = 2, 5, and 11.5 are also presented in Fig. 7.2. The real source position ( r o ,0 o ) in eqs. ( 7.1.5 ) and ( 7.1.3 ) is then replaced by the complex source position {rs,<t>s) according to eq. ( 1.2.2 ). The scattered field patterns in the right hand side of Fig. 7.2 show the forward (<j> = 7t) and backward (<p = 0) direction lobe peaks become more pronounced with increasing kb or decreasing incident beamwidth. The total far field in the left hand side of Fig. 7.2 is the sum of incident fields which exist if the circular cylinder was absent and the fields scattered by the cylinder surface. The interference between incident and scattered fields is stronger for a less directive source because there is uniform amplitude in all directions for an omnidirectional incident field. The lobe peak of the total field in the forward direction (<p = ri) represents constructive interference of the two diffracted fields from opposite sides of the cylinder. Fig. 7.3 shows the total far field and scattered field of a circular cylinder with a distant line source located at (kx0 = 25TC, ky0 = 0). From the scattered fields in the right hand side of Fig. 7.3, it is evident that they are essentially the same for all incident beam 106 directivities used here. This is because the line source is located in the far field region of the cylinder and therefore only the central portion of the incident wave is scattered by the cylinder. Again, stronger interference occurs for a less directive incident wave. On the illuminated side of the cylinder (-%I2 < 0 < n/2), the total field looks omnidirectional for a directive line source. The numerical analysis is now extended from one local beam source to an array of complex sources with appropriate choice of amplitude, phase, and orientation to simulate large aperture distributions. The arrangement for a cosine-squared aperture distribution with aperture size L0 = 2.5A, incident upon a circular cylinder of radius ka = 2.5TT centred at origin is shown in Fig. 7.1. The normalized total far field patterns and scattered field of a circular cylinder with incident field E[ = cos2(/Ty I L0)rect(y I L 0 ) shown in Fig. 7.4 are calculated from eq. ( 7.1.15 ) with eqs. ( 7.1.5 ) and ( 7.1.3 ) for basis element Bm substitutions. Here the separation distance between beam sources is L = A/2, which is in line with the guidelines proposed before. Evaluation of the corresponding amplitudes for different line sources is according to eq. ( 1.3.2 ) (Gaussian window function is used). The beam sharpness is ba = L2IX and the beam orientation fi0 = n. Five (M = 2) line sources are used for the calculation of the total far field and the scattered field in Fig. 7.4. It is evident that the scattered field and the total field for a cosine-squared aperture distribution incident upon a circular cylinder from distance kx0 = 57t (Fig. 7.4) are nearly the same as the corresponding field for a beam source (kb = 11.5) incident from this distance (Fig. 7.2). The far field HPBW of the cosine-squared distribution with aperture size L0 = 2.5A and the beam source (kb = 11.5) are the same. When the distance between 107 the aperture and the cylinder is sufficiently large (kx0 = 50%, and I O O T T ) the scattered fields are independent of range and aperture distribution. In Fig. 7.5 the configuration differs from Fig. 7.4 only by replacing the cosine-squared aperture distribution by a uniform distribution E[ = rect(yIL0). For this aperture sufficiently close to the cylinder (kx0 = 5%, and lOrt), the forward and backward lobes of the scattered field for uniform distribution are narrower than those of the corresponding cosine-squared distribution. When the aperture is moved away from the cylinder (kxa = 50%, and 100%) the scattered field for the uniform distribution is the same as for the cosine-squared distribution. The interference of the scattered fields and the incident fields from different sources in the uniform distribution case is stronger than the cosine-squared distribution case because of the relatively higher amplitudes of each complex line source in the uniform distribution. B. //-polarization. The normalized total far field diffraction patterns and scattered fields of a circular cylinder of radius ka = 2.5TC with an //-polarized local beam source at (kx0 = 5%, ky0 = 0) of directivity kb = 0, 2, 5, and 11.5 are shown in Fig. 7.6 as calculated from eqs. (7.1.12) and (7.1.11). The scattered fields in the right hand side show standing wave patterns due to two creeping waves originating at two shadow boundaries and travelling around the cylinder in opposite directions with less decrease in amplitude than the E-polarized case. It is evident that the total field on the left hand side shows stronger interference patterns between scattered wave and incident wave than in Fig. 7.2. Also it is noted that the amplitude of the forward scattered lobe in the center of the shadow region (tp = it) is larger for the //-polarized case than the E-polarized case. 108 Fig. 7.7 shows the scattering patterns as the magnetic line source is moved away from the cylinder to kx0 = 25K, ky0 = 0. Similar to the E-polarized case, the scattered fields alone are the same for various directive beam sources because the source is located in the far field of the cylinder. Interference between incident and scattered waves appears in all regions of the total field pattern for an omnidirectional incident wave. In Figs. 7.8 and 7.9 respectively, the far field diffraction patterns and scattered fields of a circular cylinder with an //-polarized cosine-squared and uniform aperture distribution are shown. The pattern characteristics are similar to the E-polarized cases of Fig. 7.4, 7.5. The total far field and the scattered field of a cosine-squared incident field with aperture size L0 = 2.5X at position kx0 = 5n, ky0 = 0 in Fig. 7.8 are the same as for single beam source incidence (kb = 11.5) from the same position (Fig. 7.6). Again, both the scattered field for cosine-squared and uniform aperture distribution located at large distances (kxa = 5071, and lOOrt) are the same. Interference between the scattered field and the incident field from an array of complex sources in the aperture plane is stronger for an uniform aperture distribution than for a tapered distribution. 7.2.2 Scattering Cross Section The numerical results for the normalized total scattering cross section in this section are calculated by using the appropriate incident power and scattered power equations; i.e. eqs (7.1.7 ) and ( 7.1.8 ), for the E-polarized incident field and eq. ( 7.1.14 ) for //-polarized field. For the beam source solution, calculation of scattered power is straightforward. The only change required is replacement of the real source coordinates (r0, <p0) by 109 complex coordinates (rs,<ps) in; eqs. ( 7.1.7 ) and ( 7.1.14 ). However, the evaluation of the incident power of a complex source needs to be considered carefully because we must account for the curvature of the ray paths diverging from the source for local directive sources. For an incident beam field H?>(kR.)-]--^e-*K-+*'" (7.2.1) where Rs = -yjr2 + r2 - 2rrs cos(<p -<ps), the direction of the ray path at any point is found from evaluation of the gradient of the wave front at that point, i.e. V(Re( Rs)). Rays from the source tangentially incident upon the cylinder define the angular sector Aa in eqs. ( 7.1.8 ) and ( 7.1.14) over which integration of the incident power density should occur. The tangent points are determined from r • V{Re( Rs j) = 0, where f is the unit normal to the cylinder. Fig. 7.10 shows the normalized total scattering cross section i.e. an = Ps / {laAaS1) where Ps and Sl are given by eqs. ( 7.17 ) and ( 7.1.8 ) of an E-polarized beam source located at kx0 =16, ky0 = 0 incident upon a circular cylinder with increasing radius from ka = 0.1 to ka = 10. The bold solid, dashed, and dotted lines represent the total scattering cross section for different beam source directivities kb = 0, 5, and 11.5. For small ka, the source is located in the far field of the cylinder and the total scattering cross section is independent of beam source directivity. The thin solid curves of Figs. 7.10a and b are from eq. ( 7.1.9 ) and represent the normalized scattering cross section for plane wave incidence; i.e. for kx0 —> °°. They correspond to familiar earlier results (eg. Fig. 2.9 of [41]). At the high frequency limit ( £ a - » ° ° ) the scattering cross 110 section approaches the geometrical cross section (on = 1). The difference between sources with different beam directivity is clear when the radius of the cylinder is enlarged. Fig. 7.11 has the same parameters as in Fig. 7.10 but is for an //-polarized beam source instead of an E-polarized beam source. Again, the normalized total scattering cross section is independent of beam directivity for small ka but differs with increasingly larger ka. The total scattering cross section for a fixed cylinder radius increases with increasing directivity for small directive beam sources and decreases with increasing directivity for large directive beam sources and converges to the value of the total scattering cross section for plane wave incidence. Fig. 7.12 shows the total scattering cross section for E and //"-polarized beam source incidence on a cylinder of radius ka = 3.5 as a function of kx0. With increasing distance between the cylinder and the beam source, all curves for different directivity beam sources converge to plane wave total scattering cross section value for ka = 3.5. It can be seen that when the ratio of source distance to cylinder radius is fixed (Fig. 7.13a), the normalized total scattering cross section first decreases and then increases with increasing beam directivity. This can be understood by appreciating that increasing directivity at a fixed range is essentially the same as decreasing the range from an aperture of fixed directivity. The cylinder is in the far field of an omnidirectional line source and the phase paths of the incident field are radial. Increasing source directivity implies increasing the aperture width (2b) which, at a fixed distance from the cylinder, means that the cylinder is increasingly in its near field. The scattering cross section of Fig. 7.13a is reduced, for E-polarization, because of destructive interference of the 111 incident field from the aperture. At a range where this destructive interference is a maximum the scattering cross section is a minimum. In Fig. 7.13a a minimum in the scattering cross section occurs for kb ~ 16 or an aperture at about the same distance from the cylinder axis as the aperture width (2b), and a cylinder radius half this distance. The cylinder is well in the Fresnel zone of the aperture. Thereafter increasing directivity tends to increase the is-polarization scattering cross section towards the result for plane wave incidence as the phase paths become the essentially parallel lines. paraxial to a highly directive source. Similar effects occur for //-polarization but the scattering cross section first increases with small but increasing kb (Fig. 7.13b) and then falls towards the plane wave result at the cylinder is increasingly in the near field. 7.2.3 Current Density The calculation of the surface current Kz and for a cylinder of radius ka = 6 with electric and magnetic fields parallel to the cylinder axis respectively are shown in Fig. 7.13. When the incident field E\ is parallel to the axis of the cylinder, the surface current Kz=Hlj>, eq. ( 7.1.6 ). When the magnetic field is parallel to the axis of the cylinder the incident field is given by H[, the surface density of current on the cylinder Kt=Hz,eq. (7.1.13) Fig. 7.14 suggests travelling waves of current exist on the shadow side of the cylinder. The two waves travelling around the cylinder in opposite directions originating from the two shadow boundaries of the incident field interfere and produce a standing wave. This standing wave phenomenon is observable in the vicinity of the centre of the 112 shadow side of the cylinder where the wave are in phase. The surface current density on a conducting cylinder is similar for electric and magnetic field parallel to the axis. However, there are interesting differences. When the electric field is parallel to the axis, the current is also parallel to the cylinder axis. This means that the travelling waves around the cylinder surface are transverse waves because of the direction of the motion of the charge in the £ direction is perpendicular to the direction of the wave travelling in the <p direction. When the magnetic field is parallel to the axis of the cylinder, the current and the direction of the travelling wave is in the 0 direction. This current wave may be considered as a longitudinal wave. A comparison of E and //-polarized cases in Fig. 7.14 reveals that the standing wave patterns are more extensive in the case of the magnetic field parallel to cylinder axis than in the case of the electric field parallel to this axis. That is because the amplitude of decreases much less slowly around the cylinder than does Kz. This results in a superposition of the two travelling waves originating at the two shadow boundaries and interacting with each other to produce more extensive standing wave patterns. This can be seen especially in the forward direction (<p =TZ). 113 Fig. 7.1 Geometry of an array of complex source beams diffraction by a conducting circular cylinder. 114 Cyco2_4E 1 .xls~cyco2_4 Fig. 7.2 Normalized total far field diffraction patterns (a) 20*log 1 0(|E z |) in dB and scattered field (b) \EZ s | of a circular cylinder for E -polarized incidence, ka = 2.5n, kx0=5n. 115 Cyco2_4E 1 .xls~cyco2_4 Fig. 7.3 Normalized total far field diffraction patterns (a) 20*log10(|£21) in dB and scattered field (b) \EZ s | of a circular cylinder for E -polarized incidence, ka = 2.5n, kx0= 25TC, ky0 = 0. 116 Cyco2_4E 1 .xls-cyco2_4 Fig. 7.4 Normalized total far field diffraction patterns (a) 20*log10(|E21) in dB and scattered field (b) \EZ" | of a circular cylinder for E -polarized cosine-squared aperture distribution incidence, ka = 2.5n, ky 0 = 0, M = 2, N = 0, L 0 = 2.5X, L = XI2. 117 Cyco2_4El.xls~Cyun Fig. 7.5 Normalized total far field diffraction patterns (a) 20*log 1 0 ( |£ z |) in dB and scattered field (b) \E z s | of a circular cylinder for E -polarized uniform aperture distribution incidence, ka = 2.5n, ky0 = 0, M = 2, N = 0, L0 =2.5X,L =X/2. 118 Cyco2_4magl xls-cymagl Fig. 7.6 Normalized total far field diffraction patterns (a) 20*log 1 0(|// z |) in dB and scattered field (b) \HZS \ of a circular cylinder for //-polarized incidence, ka = 2.5n, kx0 = 5n,ky0 =0. 119 Cyco2_4mag 1 .xls-cymag 1 Fig. 7.7 Normalized total far field diffraction patterns (a) 20*log 1 0(|// z |) in dB and scattered field (b) \HZS | of a circular cylinder for H -polarized incidence, ka = 2.5n, kx0= 25TT, ky 0 = 0. 120 Cyco2_4magl .xls-cymag2 Fig. 7.8 Normalized total far field diffraction patterns (a) 20*log1 0(|/f z |) in dB and scattered field (b) \HZ s | of a circular cylinder for //-polarized cosine-squared aperture distribution incidence, ka = 2.5TC, ky 0 = 0, M = 2, N = 0,Lo =2.5X,L = XI2. 121 Cyco2_4magl .xls-cymag2 Fig. 7.9 Normalized total far field diffraction patterns (a) 20*log 1 0(|// z |) in dB and scattered field (b) \HZS | of a circular cylinder for //-polarized uniform aperture distribution incidence, ka = 2.5n, ky 0 =0,M = 2,N=0, L0 =2.5X,L =XI2. 122 123 124 Crosskxo.xls-crosskxo 1.22 T («) / — — — - . — - •—•- •—<-' _i -• • • • ' 20 -1 L_ 40 kx0 60 _, 1 , u 80 100 \ (b) —i i i i i i i i i i + + • • • 0 20 40 kx0 60 80 100 Fig. 7.12 Normalized total scattering cross sections for a conducting cylinder with radius ka = 3.5. {a ) E -polarization, (b) //-polarization. = 11.5. kb =0, kb =5, kb 125 Crosskb.xls-crosskb 1.16 T 1.08 + 1.04 T 0.96 + 0.88 0.8 0 10 20 kb 30 40 50 Fig. 7.13 Normalized total scattering cross sections for conducting cylinders with the source at a distance kx0 =\6.(a)E-polarization, (b) H-polarization. ka =6, ka = 8, ka= 10. 126 Kphi.xli 0 -\ ' 1 1 1 ' \ 0 45 <f> 90 135 180 Fig. 7.14 Curent density on a conducting cylinder of ka = 6 for (a) E -polarized line sources and (b) H-polarized line sources at a range he 0=16. kb =0, kb =5, Plane wave 127 CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS 8.1 Conclusions The complex source point technique is known as a very efficient means of extending point or line source diffraction solutions to beam solutions. Gaussian beams in a Gabor series have also been established as a complete representation of the fields of radiating apertures and it was suggested [6], [7] that the two techniques could be effectively combined. Here this combination is used together with canonical diffraction solutions and high frequency asymptotic solutions. This is done for diffraction by half planes, slits, wedges rectangular cylinders and circular cylinders in the presence of two dimensional radiating apertures. Only symmetrical aperture distribution were chosen here as the simplest and most common examples, but the method may be extended to arbitrary shaped aperture distributions on non planar surfaces. An efficient solution has a minimum number of beams. This depends on the situation under consideration and warrants a demonstration of the accuracy of the beam arrangement used to represent the aperture field in isolation. While this was investigated previously [6], [7] with Gaussian beams, in chapter 2 complex source beams were used instead to demonstrate the accuracy of the numerical calculations and provide a useful guide for synthesis of antenna patterns. It is necessary to show the differences introduced by using the complex source beam. These may be significant with small apertures represented by few sources and occur mainly at wide angles off the pattern main beam axis. The advantage of using CSP beams is not only that they represent an exact solution 128 to the wave equation, but also that they allow the use of rigorous and convenient elementary beam diffraction solutions. For the half plane this solution is exact and rather simple. Determination of the beam amplitudes in chapter 2 from the aperture distribution can be simplified by using many, closely spaced beams and determining the amplitude directly from the profiles of the aperture distribution. Since closer spacings imply broader beams, in the limit, there is no need for beams. Omnidirectional sources suffice for representing radiating aperture fields. However beams permit the use of fewer, more widely separated sources, an important practical consideration when diffraction by local obstacles is to be calculated. Beam spacings of a half wavelength seem to minimize the number of beams required, at least for smaller apertures, as larger spacings require both aligned and tilted beams. Of course the beam amplitude, once determined, apply for any scattering problem at almost any range. The synthesis of aperture patterns with far field sidelobes such as uniform and cosine-squared aperture distributions, was given by the numerical results in chapter 2. Such arrays of complex sources of appropriate amplitude and phase are used to calculate the scattering patterns of a local obstacle such as a conducting half plane in chapter 3. The solution of beam diffraction by a half plane is exact in the far field, uniform everywhere and valid for all beam directivities and directions. Usually shadow and reflection boundaries must be located in applying the geometrical theory of diffraction, but this is not needed here because a uniform total field solution is used. The numerical results show that single beam diffraction represents an aperture with a highly tapered 129 aperture distribution well but that beam diffraction is required for a uniform distribution. This applies to most of the examples in the following chapters also. By using the results for beam diffraction by a half plane, we solve the problem of beam diffraction by a wide slit in a conducting plane. Two dimensional non-interaction fields for aperture diffraction by a wide slit were calculated in chapter 4. For narrower slits, inclusion of interaction between the edges is necessary for more accurate results. Since a slit in a conducting screen is the Babinet complement of a strip, solutions for high-frequency scattering by a strip can be easily obtained from the results of the slit diffraction calculation. The ordinary UTD was used in solving the problem of the conducting wedge in chapter 5. In order to verify the accuracy and validity of the UTD results, the exact series solution for the total field due to a set of complex line source diffracted by a conducting right angled wedge. A l l the numerical results show that a single beam with the same half power beam width as the radiation patterns of cosine-squared aperture distribution can give a good approximation in most cases. However it is apparent that a single beam cannot accurately represent the far field diffraction patterns of half planes, slits and wedges illuminated by a local uniform distribution except for the main reflected beam. Chapter 6 investigates the scattering patterns of a local rectangular cylinder by arrays of complex sources. The total scattered field is the superposition of the solution for the far field of a beam source in the presence of four right angled wedges and first order interaction between the edges is included when significant. The moment method solution also is calculated to check the accuracy of this solution for a square cylinder of side 2k. For the choice of a square cylinder side of 5k, single diffraction is sufficient for 2s-130 polarization and first order interaction between adjacent edges is needed for H-polarization, since diffracted wave interaction along the conducting surface is much weaker for E-polarization that for //-polarization. Consequently there is more field penetration into the shadow region for //-polarization, especially for inclined incidence and this penetration can be substantially enhanced by using a more directive source. A comparison of the scattering patterns of circular cylinders (Chapter 7) for near and far sources with uniform and tapered aperture distributions shows some expected results. There are more pronounced interference oscillations in the pattern for far field uniform source distribution than for tapered distribution and stronger evidence of creeping wave lobes with //-polarization incidence than for ^-polarization incidence. The apparently unexpected occurs in the behavior of the scattering cross section. For an E-polarized local source it first decreases with increasing source directivity and then increases. For //-polarization the reverse occurs: first an increase in scattering cross section with increasing directivity and than a decrease. This behavior is a result of the cylinder effectively moving from the far field to the near field as the source directivity increases. For a local source uniformity of illumination first decreases with increasing directivity, and then increases as the phase paths straighten in to the parallel straight lines of plane wave incidence of an infinitely directive source. The minimum scattering cross section for ^-polarization and the maximum for //-polarization occur when the aperture width 2b and distance x0 are twice the cylinder radius a. That the scattering cross section variation with local source directivity for E-polarization is the opposite to that for //-polarization appears to be entirely related to the well known difference for scattering for small cylinders and plane wave incidence. The 131 above analysis can be extended to larger cylinders through a Watson transformation of the cylindrical mode series of through application of the uniform geometrical theory of diffraction. However these small cylinder results are possibly the more interesting. 8.2 Future Possibilities It has been shown that an array of complex source diffracting beams arranged in a Gabor series is a powerful technique for calculating the effects of local obstacles on antenna patterns. This approach also can be applied to a variety of situations and the following may be considered. 1) Calculation of diffraction of two dimensional aperture distributions by structures with combinations of the canonical shapes used here. Examples are cylinder-tipped half planes and wedges, thick half planes and slits or strips, etc. 2) Extensions to three dimensional radiating apertures for scalar fields. Single beam diffraction by a half-plane has been dealt with in this way [58]. Although it will be analytically and computationally much more laborious, no basically new phenomena or difficulties are anticipated in extending this to beam array diffraction in three dimensionals. For vector fields ray tracing techniques in three dimensions generally are very complicated. 3) In contrast to plane wave spectrum formulation which may represent conveniently only planar aperture, the beam series constitutes a superposition of highly localized contributions which may represent the field distribution specified on a arbitrary curved surface. This constitutes a significant extension which can be considered [42]. 4) Here we have dealt only with problems that assume perfect conductors. Solutions for beam diffraction by simple shapes with impedance boundary conditions 132 may be obtained by using the corresponding solutions for omnidirectional sources, but these solutions will be less convenient. 5) Rising communication requirements and limited usable spectrum result in increasing frequency reuse in communication systems. This has led to increasing problems of interference. The use of artificial barriers for shielding antennas is one method for interference reduction. The techniques described in this thesis may be applied to the placement of antennas and to the design of site shielding of ground station antennas. 133 REFERENCES. [I] R. G. Kouyoumjian, and P. H. Pathak, " A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface", Proc. IEEE, vol. 62, pp. 1448-1461, Nov. 1974. [2] G. A. Deschamps, "Gaussian beam as a bundle of complex rays", Electron. Lett., vol. 7, pp. 684-685, 1971. [3] L. B. Felsen, "Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams", Geophys. J. R. Astr. Soc. pp. 77-88, 1984. [4] G. A . Suedan and E. V . M l , "Beam diffraction by half planes and wedges: uniform and asymptotic solutions", Journal of Electromagnetic Waves and Applications, vol. 3, no. 1, pp. 17-26, 1989. [5] G. A . Suedan and E. V . M l , "Two dimensional beam diffraction by a half-plane and wide slit", IEEE Trans. Antennas Propagat., vol. AP-35, no. 9, pp. 1077-1083, Sep. 1987. [6] J. J Maciel and L. B. Felsen, "Systematic study of fields due to extended apertures by Gaussian beam discretization", IEEE Trans. Antennas Propagat., vol. 37, pp. 884-892, Jul. 1989. [7] P. D. Einziger, S. Raz, and M . Shapira, "Gabor representation and aperture theory", J. Opt. Soc. Am. A, vol. 3, no. 4, pp. 508-522, Apr. 1986. [8] L. B. Felsen, "Evanescent waves", J. Opt. Soc. Am. A, vol. 66, no. 8, pp. 751-760, 1962. [9] M . Couture, and P. Belanger, "From Gaussian beam to complex source point spherical wave", Phys. Review A, vol. 24, no. 1, pp. 355-359, 1981. [10] A . C. Green, H , L, Bertoni, and L. B. Felsen, "Properties of the shadow cast by a half-screen when illvirninated by a Gaussian beam", J. Opt. Soc. Am., vol. 69, no. 11, pp. 1503-1508,1979. [II] D. Gabor, "Theory of communication", J. Inst. Elec. Eng. 93 (III). P. 429, 1946. [12] M . J. Bastiaans, " A sampling theorem for the complex spectrogram and Gabor's expansion of a signal in Gaussian elementary signals", Opt. Eng., vol. 20, pp. 594-598, 1981. [13] J. B. Keller, and W. Streifef, "Complex rays with an applicatiion to Gaussian beams", J. Opt. Soc. Am. A, vol. 61, no. 1, pp. 40-43,1971. 134 [14] M. Hashimoto, "Beam waves with sources at complex location", Electron. Lett. vol. 21, no. 23, pp. 1006-1007, 1985. [15] P. D. Einziger and S. Raz, "Beam-series representation and the parabolic approximation: the frequency domain", J. Opt. Soc. Am. A, vol. 5, no. 11, pp. 1883-1892, 1988. [16] A. J. E. M. Jansen, "Gabor representation of generalized function", J. Math. Anal. Appl., vol 83, pp. 377-394, 1981. [17] J. B. Keller, "Geometrical Theory of Diffraction", / . Opt. Soc. Am. A, vol. 52, no. 2, pp. 116-130, 1962. [18] R. F. Harrington, Field computation by moment methods. New York: MacMillan, 1968 [19] R. F. Harrington et al., Matrix Methods for Solving Field Problems, final report for Contract AF30(602)-3724 with Rome Air Development Center, Griffiss Air Force Base, Rome, New York: DDC No. AD 639744, August 1966. [20] G. A. Suedan, "High frequency beam diffraction by apertures and reflectors", Ph. D. thesis, Department of Electrical Engineering, University of British Columbia, June, 1987. [21] Y. Dezhong, "Complex source representation of time harmonic radiation from a plane aperture", IEEE Trans. Antennas Propagat., vol AP- 43, no. 7, pp. 720-723, 1995. [22] Y. Z. Ruan, J. Lin and H. J. Zhou, "Complex source beam simulation of antenna patterns", International Symposium on Antenna and Propagation, Seattle, voL 3, pp. 1888-1891, 1994. [23] E. V. Jull, "Aperture antennas and diffraction theory", IEE Electromagnetic Wave Series, voL 10, Peter Peregrinus Ltd, Stevenage, UK and New York, 1981. [24] A. Sommerfeld, "Mathematische theorie der diffraction", Math. Ann., 47, pp. 317-374, 1986. [25] P. C. Clemmow, "A method for the exact solution of a class of two-dimensional diffraction problems", Proc. Roy. Soc, A205, pp. 286, 1951. [26] M. Born and E. Wolf, "Principle of optics", Pergamon Press, Oxford, 5* edn., 1976. 135 [27] P. C. Clernmow, "A note on the diffraction of a cylindrical wave by a perfectly conducting half-plane", Quart, Journ. Mech. and Appl. Math., vol. 3, pp. 377-384, 1950. [28] G. A. Suedan and E. V. Jull, "Beam diffraction by planar and parabolic reflectors", IEEE Trans. Antennas Propagat., vol. AP-39, no. 9, pp. 521-527, Apr. 1991. [29] S. N. Karp, and J. B. Keller, "Multiple diffraction by an aperture in a hard screen", Optica Actra, vol. 8, pp. 61-72, Jan. 1961. [30] R. G. Kouyoumjian, "The geometrical theroy of diffraction and its application", in Numerical and Asymptotic Techniques in Electromagnetics, R. Mittra (Ed.), Springer, New York, 1975, Chapter 6. [31] C. A. Mentzer, L. Peter, Jr. , and R. C. Rudduck, "Slope diffraction and its application to horns," IEEE Trans. Antennas Propagat., vol. AP-23, No. 2, March 1975, ppl53-159. [32] R. Tiberio and R. G. Kouyoumjian, " Calculation of the high-frequency diffraction by two nearby edges illuminated at grazing incidence", IEEE Trans. Antennas Propagat., vol. AP-34, pp. 323-336, Nov. 1984. [33] R. Tiberio, G. Manara, G. Pelosi, and R. G. Kouyoumjian, "High-frequency electromagnetic scattering of plane waves form double wedges", IEEE Trans. Antennas Propagat., vol. AP-37, pp. 1172-1180, Sep. 1989. [34] M. Schneider and R. J. Luebbers, "A general, uniform double wedge diffraction coefficient", IEEE Trans. Antennas Propagat., vol. AP-39, pp. 8-14, Jan. 1991. [35] L. A. Polka, C. A. Balanis, and A. C. Polycarpou, "High-frequency methods for multiple diffraction modelling: application and comparison", Journal of Electromagnetic Waves and Applications, vol. 8, no. 9/10, pp. 1223-1246,1994. [36] Kodis, "An Experimental Investigation Of Microwave-Diffraction", Technical Report No. 105, Cruft Laboratory, Harvard University (June 1950); J. Appl. Phys. 23, pp. 249,1952. [37] R. King and T. T. Wu, "The Reflection of Electromagnetic Waves from Surfaces of Complex Shape", Scientific Report No. 13, Cruft Laboratory, Harvard University (December 1957) [38] R. King and T. T. Wu, The Scattering and Diffraction of Waves, Harvard University Press, Cambridge, Massachusetts 1959. 136 [39] P. H. Pathak, W. D. Burnside, and R. J. Marhefka, " A uniform GTD analysis of the diffraction of electromagnetic waves by a small convex surface", IEEE Trans. Antennas Propag., vol. AP-28, No. 5, pp. 631, September 1980.Cylin-12 [40] J. J. Faran, "Scattering of Cylindrical Waves by a Cylinder, " /. Acoust. Soc. Am. 25, pp. 155, 1973. [41] J. J. Bowman, T. B. A. Senior and P. L. E. Uslenghi, "Electromagnetic and Acoustic Scattering by Simple Shapes", Chap. 2, North-Holland, Amsterdam, 1969. [42] N . N . Andrew and T. B. Hansen, "Exact complex source representation of time-harmonic radiation", Wave Motion, 25, pp. 127-141, 1997. [43] T. Oguzer, A. I. Nosich, and A. Altintas, "Accurate simulation of reflector antennas by complex source-dual series approach", IEEE Trans. Antennas Propagat., vol. AP-43, pp. 793-801, Aug. 1995. [44] A . I. Nosich, V . B. Yurchenko, and A. Altintas, "Numerically exact analysis of a two-dimensional variable-resistivity reflector fed by a complex-point source", IEEE Trans. Antennas Propagat., vol. AP-45, pp. 1592-1601, Nov. 1997. [45] M . R. Barclay and W. V . Rusch, "Moment-method analysis of large, axially symmetric reflector antennas using entire-domain function", IEEE Trans. Antennas Propagat., vol. AP-39, pp. 491-496, Apr. 1991. [46] P. D. Einziger and S. Raz, "Beam-series representation and the parabolic approximation: the frequency domain", / . Opt. Soc. Am A, vol. 5, no. 11, pp. 1883-1892, 1988. [47] B. Z. Steinberg, E.Heyman and L. B. Felsen, "Phased-space beam summation for time-harmonic radiation from large apertures", J. Opt. Soc. Am. A, vol. 8, no. 1, pp. 41-59, 1991. [48] H . D. Cheung and E. V . Jull, 'Two dimensional diffraction of an antenna pattern by a local half-plane", Proc. IEEE AP-S Internal Symp., June, 1997, Montreal, pp. 1776-1779. [49] H . D. Cheung and E. V . Jull, "Distortion of antenna patterns by local cylinders", Proc. URSI Symp. on Electromagnetic wave theory. Thessaloniki, pp. 348-350, May, 1998. [50] H . D. Cheung and E. V . Jull, "Antenna pattern diffraction by a local rectangular cylinder", Proc. IEEE AP-S Internat. Symp., Atlanta, pp. 1274-1277, Jun, 1998. [51] H . D. Cheung and E. V . Jull, "Cylinder scattering cross sections for directive local sources", Proc. URSI Symp. on Electromagnetic wave theory. Toronto, 1999. 137 [52] H. D. Cheung and E. V . Jull, "Beam diffraction by rectangular cylinder", 1999 International Symp. on Electromagnetic Compaibility, Tokyo, May, 1999. [53] H . D. Cheung and E. V . Jull, "Scattering of Antenna Beams by Local Cylinders", Journal of Electromagnetic Waves and Applications, vol. 13, pp. 1315-1331, 1999. [54] H. Anastassiu, and P. Pathak, "High frequency analysis of Gaussian beam scattering by a 2-D parabolic contour of finite width", Radio Science, vol. 30 #3, pp. 493-503, May-June 1995. [55] H-T Chou, and P. Pathak, "Uniform asymptotic solution for the E M reflection and diffraction of an arbitrary Gaussian beam by a smooth surface with an edge", Radio Science, vol. 32 #4, pp. 1319-1336, May-June 1997. [56] S. Kozaki, " A new expression for the scattering of a Gaussian beam by a conducting cylinder", IEEE Trans. Antennas Propag., vol. AP 30, pp. 881-887, Sept. 1982. [57] S. Kozaki, "High frequency scattering of a Gaussian beam by a conducting cylinder", IEEE Trans. Antennas Propag., vol. AP 31, pp. 795-799, Sept. 1983. [58] G. A. Suedan and E. V. Jull, " Three-dimensional scalar beam diffraction by a half plane", Computer Physics Communications, vol. 68, pp. 346-452, 1991. [59] S. H. Karp and A. Russek, "Diffraction by wide slit", J. Appl. Phys., vol. pp. 886-894, 1955. 138 C/J C •o 2 cS • 3 sr • u a '53 o U ON 0 0 139 APPENDIX A UNIFORM THEORY OF DIFFRACTION (UTD) FOR A WEDGE. The Uniform Theory of Diffraction [1] is a method that is developed within the context of GTD where the emphasis is placed on finding a compact diffraction coefficient for plane, cylindrical, or spherical wave incidence on a curved edge. The solution is to remain valid in the transition region across shadow and reflection boundaries, where Keller's GTD fails [17]. UTD has been combined with the complex source point method by making the source coordinates complex to a line source locates r0, 90 from the edge of a perfectly conducting wedge of exterior angle mt, the total field of a directive beam in real space to be determined at r, 9 is [4] Ez = E'u(9si -$)+ Erzu(9sr -9)+ Erzu{9-9sr) + Edz (A . l ) where u is a unit step function. 9si, 9sr are the shadow boundary and reflection boundary of the upper surface and 9sr, is the reflection boundary of the lower surface. The far field of the source alone is g-y*[r-r,cos( e-e,)\ £ ; -^ r~ (A-2) The reflected far field of the upper wedge surface is - E'Jklr-r'cos(8+e^\ The field reflected from the lower wedge surface is _ -jk[r-r, cos(2/»r-( 6-0,))] E'-- ( A' 4 ) The diffracted field is 140 kTJkT. D(r,e,rs,ds) with a uniform diffraction coefficient -ji74 4 where ( A.5) T = M.2 *-(***,) _ _n + {e + d,) ' 3.4 — " 2n ' 2* ( A - 7 ) with the upper and lower signs corresponding to the first and second subscripts. In the eq. ( A.6 ), G(co) is a modified Fresnel integral G((o) = ±2j<oejeoi F(± co), Re(± co) ~ 0 J±a> coxl = -JlkL, cos e y 3 4 = -^2kLs cos -(0 + 0,)^ (A.8) A / ' , and N* are the integers which most nearly satisfy the following equations 2nnN* Re(0,)) = -7r 2n^rV2? -(0TRe(0,)) = +^ and Ls is a distance parameter which is detennined for different types of illumination. For cylindrical wave incidence, (A.9) L. = rr. r + r. (A.10) 141 APPENDIX B MOMENT METHOD (MM). The GTD and Moment Methods have been applied to antenna and scattering problems for many years. GTD has limitations when the radius of curvature is too small in wavelengths or when edges are too close together in wavelengths. On the other hand, MM techniques can be applied to arbitrarily shaped objects provided that the overall size is small in terms of wavelengths. Thus a combination of both methods may lead to an approach to handle a wide variety of new problems. The basic idea of MM is to reduce a functional equation to a matrix equation, and then solve the matrix equation by known techniques. The following discussions are based on a book by Harrington [18]. Consider a perfectly conducting scatter excited by an impressed electric field E\, the field induces surface currents Jz on the conducting scatterer, which produces a scattered field E'. The field due to Jt is given by E;=^-\j,(r')H?{k\r-r])ds' (B .U) where the integration contour is over the cross section of the scatterer and r\ is the intrinsic impedance of free space. From the boundary condition, the total tangential electric field E\ + E\ vanishes on C. Hence, we have integral equation Ei ^\jt{r)H^{k\r-r\)ds' (B.12) c where the incident tangential electric field E\ is known and Jz is the unknown to be 142 determined. By using the pulse function f„ for a basis and point matching for testing to evaluate the numerical solution of ( B.2 ), we obtain the matrix equation K - = (B.13) where the elements of \an ] are the c& coefficients of Jt = X &nf*. the elements of are 8m=E't(xm,ym) (B.14) and the elements of [/„ „ ] are ^ - T J ^ 2 , N ( ^ - ) 2 + ( / - J ' - ) 2 ^ AC. L dl' (B.15) By treating an element J,ACn as a filament of current when the field point is not on AC„ the approximation of eq. ( B.5 ) can be written as (B.16) when m*n. For the diagonal element, the Hankel function has an singularity and the integral must be evaluated analytically. Using the small argument formula # i 2 ) ( z ) = l -y- lnM^|, z « l (B.17) where /= 0.5772157 is Euler's constant. Then an evaluation of eq. ( B.5 ) then gives kn 1- ; —In — — K \ 4e (B.18) It is noted that the approximation made above will not converge to the exact solution as the numbers of segments is increased, because the /m,„, n, are not exact in the limit. The series will converge to the exact solution when the approximation in eq. ( B.6 ) is 143 replaced by more accurate approximation. It has been found that the rate of convergence is almost twice as fast if a piecewise linear approximation is used instead of the step approximation. Therefore, a more convenient way of obtaining a better approximation may be obtained by using three-stepped function approximates a triangle function [19].. For the //-polarization case, the procedure is similar to the /^ -polarization above. The scattered field due to the induced current on the scatterer's surface is -Akr*3K/4) H'2 = , k \ J(r')(n • R)eik^^'^dst ( B.19 ) where R is a unit vector from the source point to the field point, n is a unit vector normal to the surface, and 0 is the angle between R and n. The elements of \lmn ] are /„ „ =1/2, m = n, lm,n-jkACn{n-R)H^[k^(xm-xnf +(ym - y „ ) 2 } , m*n ( B ' 2 ° } The moment method technique is useful provided that the scatterer is not electrically large. It can also be used to verify the accuracy of GTD solutions for scatterers with dimensions of no more than a few wavelengths. One way to extend the moment-method technique to large bodies is to use entire domain basis functions [45]. The matrix size can be made smaller, but the matrix filling time increases. 144 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0065036/manifest

Comment

Related Items