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A broadband fixed-beam leaky-wave antenna based on transformation electromagnetics Al Noor, Asif 2016

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A Broadband Fixed-beamLeaky-wave Antenna Based onTransformation ElectromagneticsbyAsif Al NoorA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)October 2016© Asif Al Noor, 2016The undersigned certify that they have read, and recommend to the Collegeof Graduate Studies for acceptance, a thesis entitled: A Broadband Fixed-beamLeaky-wave Antenna Based on Transformation Electromagnetics sub-mitted by Asif Al Noor in partial fullment of the requirements of the degreeof Master of Applied ScienceDr. Loïc Markley, School of EngineeringSupervisor, Professor (please print name and faculty/school above the line)Dr. Kenneth Chau, School of EngineeringSupervisory Committee Member, Professor (please print name and faculty/school above the line)Dr. Thomas Johnson, School of EngineeringSupervisory Committee Member, Professor (please print name and faculty/school above the line)Dr. Zheng Liu, School of EngineeringUniversity Examiner, Professor (please print name and faculty/school above the line)October 18, 2016(Date Submitted to Grad Studies)iiAbstractA broadband xed-beam leaky-wave antenna is presented in this thesis. Theproposed antenna consists of a graded dielectric superstrate placed on top ofa closely-spaced thin slot array. The graded dielectric superstrate is designedusing transformation electromagnetics to couple the radiation from underlyingleaky slot-line into free space. Wave propagation in graded dielectric media,properties of leaky-wave antennas, and conformal transformation electromag-netics have been explored prior to the design. The behaviour of the proposedantenna has been subsequently improved through developing a technique thatexploits transformation electromagnetics. The technique adjusts the discrepancyin phase produced as a result of coordinate stretching at the boundary of trans-formed medium. Full-wave simulations are carried out to demonstrate the per-formance of the leaky-wave antenna. Broadband radiation characteristics areachieved from the antenna with peak radiation around 33o, 30% side-lobe level,53% back-lobe level, 30.6o beamwidth, and 11.8 directivity. Such performancemakes the antenna suitable for planar applications where a xed oblique beamis required over a broad bandwidth.iiiPrefaceThis work has been done under the guidance of Dr. Loïc Markley at theSchool of Engineering in The University of British Columbia.Portions of this thesis have also presented at the following conference:• “Achieving Linear Phase Through Geometrically-compensated Transfor-mation Domains for Leaky-wave Antenna Radiation", IEEE InternationalSymposium on Antennas and Propagation and North American Radio ScienceMeeting in Fajardo, Puerto Rico, June 26 - July 1, 2016.Portions of this thesis have also been presented at the following poster pre-sentation session:• “A slot line leaky-wave antenna using graded index materials" at the Schoolof Engineering’s 2nd Annual Engineering Graduate Symposium, The Uni-versity of British Columbia, Okanagan Campus, June 8, 2015.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Broadband Antennas . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Leaky-wave antennas . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Transformation electromagnetics . . . . . . . . . . . . . . . . . 61.5 Proposed design . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5.1 Motivation and objective . . . . . . . . . . . . . . . . . . 81.5.2 Design specications . . . . . . . . . . . . . . . . . . . . 81.6 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . 9Chapter 2: Background . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Electromagnetics for antenna analysis . . . . . . . . . . . . . . . 112.2 Antenna radiation parameters . . . . . . . . . . . . . . . . . . . 132.3 Leaky-wave antennas . . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 Fast and slow-wave radiation . . . . . . . . . . . . . . . 152.3.2 Radiation characteristics . . . . . . . . . . . . . . . . . . 202.4 Slot-line leaky-wave antenna . . . . . . . . . . . . . . . . . . . . 232.4.1 Principle of radiation . . . . . . . . . . . . . . . . . . . . 23vTABLE OF CONTENTS2.4.2 Fixed-beam leaky-wave antennas . . . . . . . . . . . . . 272.5 Transformation electromagnetics . . . . . . . . . . . . . . . . . 302.5.1 Form invariance property of Maxwell’s equations . . . . 302.5.2 Conformal mappings in transformation optics . . . . . . 322.5.3 Inverse transformation . . . . . . . . . . . . . . . . . . . 38Chapter 3: The Linear Gradient Design . . . . . . . . . . . . . . . . 393.1 Wave propagation in graded-index dielectrics . . . . . . . . . . . 393.2 Leaky-wave source - single slot vs slot array . . . . . . . . . . . 403.3 Antenna design principle . . . . . . . . . . . . . . . . . . . . . . 403.4 Antenna properties . . . . . . . . . . . . . . . . . . . . . . . . . 423.4.1 Leaky-wave radiation inside graded-index half-space . . 443.4.2 Obtaining radiation in free-space . . . . . . . . . . . . . 453.5 Antenna characterization and simulation results . . . . . . . . . 473.6 Limitation of the design . . . . . . . . . . . . . . . . . . . . . . . 50Chapter 4: Geometric Compensation Technique . . . . . . . . . . 534.1 Numerical conformal transformation . . . . . . . . . . . . . . . 534.1.1 Linking to the antenna design . . . . . . . . . . . . . . . 534.1.2 Validation of simulations . . . . . . . . . . . . . . . . . . 544.2 Geometric compensation technique . . . . . . . . . . . . . . . . 564.2.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2.2 Linear phase gradient at the radiating interface . . . . . 574.2.3 Determining the source domain geometry . . . . . . . . 594.2.4 Optically transform the source domain into target domain 624.3 Phase-discrepancies in the transformation process . . . . . . . . 644.4 Compensating the geometry . . . . . . . . . . . . . . . . . . . . 654.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Chapter 5: Broad Band Fixed-beam Leaky Wave Antenna . . . . . 725.1 Design principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Design considerations . . . . . . . . . . . . . . . . . . . . . . . . 745.3 Design of the superstrate . . . . . . . . . . . . . . . . . . . . . . 765.4 Antenna performance . . . . . . . . . . . . . . . . . . . . . . . . 825.4.1 Fixed-beam broad band performance . . . . . . . . . . . 835.4.2 Improvement of radiation pattern using geometric com-pensation technique . . . . . . . . . . . . . . . . . . . . 83viTABLE OF CONTENTS5.5 Center-fed design with a back reector . . . . . . . . . . . . . . 895.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Chapter 6: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 956.1 Summary of work . . . . . . . . . . . . . . . . . . . . . . . . . . 956.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98viiList of FiguresFigure 1.1 Original photograph and outline of the rst antenna usedby Marconi for radio communication in 1901. . . . . . . 2Figure 1.2 Planar representation of bow-tie and two-am spiral an-tennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Figure 1.3 Diagrams of a uniform and a periodic leaky wave an-tenna implemented on rectangular waveguides. . . . . . 4Figure 1.4 CAD drawing of the leaky lens antenna proposed by Netoet al. and prototype of a similar lens antenna proposedby Bruni et al.. . . . . . . . . . . . . . . . . . . . . . . . 6Figure 1.5 Coordinate transformation of a curved domain to a rect-angular one. . . . . . . . . . . . . . . . . . . . . . . . . . 7Figure 2.1 Illustration of nulls, secondary lobes and directivity ofan antenna radiation pattern. . . . . . . . . . . . . . . . 13Figure 2.2 Models of simple slow-wave structures including folded-back transmission line, zigzag transmission line, inter-digital transmission transmission line, corrugated trans-mission waveguide, and helix transmission line. . . . . . 16Figure 2.3 Transition of a bounded wave from a rectangular waveg-uide to a structure that supports leaky-wave mode. . . . 17Figure 2.4 Line-source model of a leaky-wave structure. . . . . . . 19Figure 2.5 Leaky-wave antennas based on microstrip structures in-cluding the comb line array and a series microstrip patcharray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.6 Beam-scanning performance of a typical leaky-wave an-tenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 2.7 Radiation patterns showing the variation of beamwidthwith the change of leakage constant of a leaky-wave an-tenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23viiiLIST OF FIGURESFigure 2.8 Complementary electric dipole in free-space and slot-line on an innite conductive plane. . . . . . . . . . . . 24Figure 2.9 Innitely stretched slot located at the interface of twodielectric half spaces. . . . . . . . . . . . . . . . . . . . . 25Figure 2.10 A radiation pattern comparing the leaky-wave and sourcewave radiation originating from a slot-line. . . . . . . . 26Figure 2.11 The prototype of Neto’s xed-beam leaky-wave lens an-tenna and a diagram demonstrating its radiation towardsbroadside. . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 2.12 Diagram of a microstrip line with a superluminal sub-strate that is capable to radiate leaky-wave radiation intofree-space. . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 2.13 Illustrations of slot-line leaky-wave antenna consistingof uniform dielectric prism and anisotropic inhomoge-neous metamaterial tranistion layers. . . . . . . . . . . . 30Figure 2.14 Full-wave 2D simulations of the leaky-wave mode radi-ating into a metamaterial slab designed to couple to free-space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 2.15 Coordinate mapping of region w = f(x, y) into a regiont = f(u, v). . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 2.16 Coordinate grid lines of constant x and y when confor-mally mapped from region w to region t. . . . . . . . . 33Figure 2.17 Transformation and inverse transformation between re-gion w and t. . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 3.1 Field plot of the slot-line radiation into denser dielectrichalf-space for a single slot and an innite slot array. . . 41Figure 3.2 Normalized absolute eld values along the direction ofpropagation for a single slot and an innite slot arraylocated underneath a dielectric half-space. . . . . . . . . 42Figure 3.3 A diagram demonstrating that a slot-generated leaky-wave radiation remains trapped when emitted into a rect-angular uniform dielectric superstrate. . . . . . . . . . . 42Figure 3.4 Bending of leaky-wave inside the graded dielectric slab. 43Figure 3.5 Bending of leaky-wave forming a sinusoidal envelope in-side the graded dielectric slab placed on top of a center-fed slot-line. . . . . . . . . . . . . . . . . . . . . . . . . . 43ixLIST OF FIGURESFigure 3.6 Field plot of leaky-wave radiation into a graded dielectrichalf-space. . . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 3.7 Simulation domain of the antenna in the preliminary de-sign process. . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 3.8 Broadside radiation from the antenna with a slab heightH = 0.57λ0. . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 3.9 Oblique radiation from the antenna with a slab heightH = 0.4λ0. . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 3.10 Normalized radiation patterns at dierent frequency stepsof the initially designed leaky-wave antenna. . . . . . . 49Figure 3.11 Characterization of the leaky-wave antenna having a gra-dient index slab. . . . . . . . . . . . . . . . . . . . . . . 51Figure 4.1 Validation of numerical transformation electromagneticsby transforming uniform dielectric into a region lledwith isotropic inhomogeneous dielectric medium. . . . 54Figure 4.2 Coordinate grid lines of the transformation shown in Fig.4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Figure 4.3 Normalized absolute value of the electric eld in the stand-ing waves in initial domain and transformed domain con-rming the numerical transformation process. . . . . . 56Figure 4.4 A diagram illustrating the process of geometric compen-sation compensation. . . . . . . . . . . . . . . . . . . . . 58Figure 4.5 The initial and target domain geometries excited with apoint source. . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 4.6 A diagram showing the process of choosing a geometryin source domain in (x,y) coordinate. . . . . . . . . . . 60Figure 4.7 Diagram of the simulation domain used in the primarystep of the geometric compensation technique. . . . . . 61Figure 4.8 Selection of a source domain geometry using a target lin-ear phase prole. . . . . . . . . . . . . . . . . . . . . . . 62Figure 4.9 The initial conformal mapping from a homogeneous sourcedomain to rectangular graded dielectric target domain. . 63Figure 4.10 Unwrapped phase proles along the upper boundary ofthe target domain. . . . . . . . . . . . . . . . . . . . . . 65Figure 4.11 Conformal mapping of the compensated homogeneoussource domain geometry to rectangular graded dielectrictarget domain. . . . . . . . . . . . . . . . . . . . . . . . 66xLIST OF FIGURESFigure 4.12 Permittivity distribution inside the geometrically com-pensated superstate in the target domain. . . . . . . . . 67Figure 4.13 Field plot showing the compensated slab excited with apoint source are absorbed into an inhomogeneous PMLregion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 4.14 Unwrapped phase proles along the upper boundaries ofthe initial and compensated geometry. . . . . . . . . . . 68Figure 4.15 Comparison of the radiation patterns of the point source-fed substrates before and after applying the geometriccompensation technique. . . . . . . . . . . . . . . . . . 69Figure 4.16 Algorithm of geometric compensation technique. . . . . 71Figure 5.1 A diagram showing how a slot emitted leaky-wave ra-diation from a corner-fed slot-line leaky-wave antennaarray remains trapped inside the uniform dielectric su-perstrate. A convex dielectric domain couples the gen-erated radiation into free-space. The same phenomenoncan be implemented with a rectangular graded-dielectricsuperstrate. . . . . . . . . . . . . . . . . . . . . . . . . . 73Figure 5.2 The structure and physical dimensions of the simulatedantenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 5.3 Simulation setup for the analysis of the proposed leaky-wave antenna. . . . . . . . . . . . . . . . . . . . . . . . 77Figure 5.4 Fields produced in a uniform dielectric half-space thatare used to determine the source domain geometry. . . . 78Figure 5.5 Conformal transformation of the initial geometry into arectangular domain. . . . . . . . . . . . . . . . . . . . . 79Figure 5.6 Radiation patterns of the uncompensated leaky-wave an-tenna at dierent frequencies. . . . . . . . . . . . . . . . 81Figure 5.7 Unwrapped phase proles along the upper boundary ofthe uncompensated geometry in the target domain. . . . 82Figure 5.8 Compensating the source domain geometry using thecorrected phase-prole. . . . . . . . . . . . . . . . . . . 83Figure 5.9 The compensated source domain geometry and its trans-formation to rectangular target domain. . . . . . . . . . 84Figure 5.10 Permittivity distribution inside the geometrically com-pensated superstate in the target domain. . . . . . . . . 84xiLIST OF FIGURESFigure 5.11 Simulated electric eld of the designed xed-beam leaky-wave antenna. . . . . . . . . . . . . . . . . . . . . . . . 85Figure 5.12 Radiation patterns of the compensated leaky-wave an-tenna at dierent frequencies. . . . . . . . . . . . . . . . 86Figure 5.13 Normalized radiation patterns for the initial and com-pensated (blue) corner-fed leaky-wave antenna at par-ticular frequencies. . . . . . . . . . . . . . . . . . . . . . 87Figure 5.14 Characterization plots comparing the angle of radiation,sidelobe level, backlobe level, 3dB beamwidth and direc-tivity of the corner-fed antenna over a broad bandwidth. 88Figure 5.15 Input impedance and the return loss of the geometricallycompensated leaky-wave antenna. . . . . . . . . . . . . 89Figure 5.16 Simulation domain of the center-fed leaky-wave antennawith a back-reector. . . . . . . . . . . . . . . . . . . . . 90Figure 5.17 Characterization plots comparing the Peak eld, Radi-ated beam angle, sidelobe level, 3dB beam width, andbroadside eld of the slot radiation into a two dieerenthalfspaces. . . . . . . . . . . . . . . . . . . . . . . . . . 91Figure 5.18 Permittivity distribution inside the inhomogeneous su-perstrate used for the center-fed design. . . . . . . . . . 92Figure 5.19 Normalized radiation patterns for the initial as well asthe compensated center-fed leaky-wave antenna with andwithout a backreector at dierent frequencies. . . . . . 93Figure 5.20 Characterization plots comparing the angle of radiation,sidelobe level, backlobe level, 3dB beamwidth and direc-tivity of the center-fed antenna over a broad bandwidth. 94xiiAcknowledgementsI am grateful to several individuals who have assisted me during the courseof this work. First, I would like to acknowledge my debt to my supervisor, Dr.Loïc Markley, for introducing me to the world of electromagnetism, and micro-mentoring me throughout my graduate studies. His dead-on comments on myideas and incisive responses to my many queries has critically facilitated my re-search. I consider myself to be only the rst of many students who will growthrough his admirable personality, white-hot brilliance, and intellectual guid-ance. My appreciation also goes to Dr. Kenneth Chau and Dr. Thomas Johnson,for seeing me through the whole course of this project. With their extensiveexpertise in the eld of electromagnetics, they provided many new perspectivesduring my initial course works that later became instrumental for this work.My tenure as a graduate student would not have been the same without thepointless corridor chats and laughs that I shared with my colleagues from theSchool of Engineering. A dept of gratitude is owed to my lab mates, Iman Ah-ganejad, Mohammed Al-Shakhs, Masoud Ahmadi and Max Bethune-Waddell, fortheir companionship during the pursuit of this goal. A warm word for my greatfriends Salamah Meherier, who has been a part of almost every episodes of mylife in Kelowna, and Abhinav Karanam, with whom I had the best coee breaks.On a more personal note, I am forever grateful to my parents, Zakir andAirin, who have always been a major source of encouragment. I am proud tohave your genes and look up to you more than anyone else. Thanks to my sister,Samira, for being my loyal ‘secret double agent’ in the family. The villainousdesire to start one more unnecessary argument with you kept me going for pasttwo years. I am also indebted my aunts for keeping tabs on me from time totime. Especially, thanks to aunts Raina and Rebeka, for aording me two fabulouswinter vacations. Last, but certainly not least, a special thank you to my ancée,Tabassum, for her good-humored friendship and covert support during the moststressful periods of this research. I look forward to our lifelong journey.xiiiDedicationTo the memory of my grandfather, Abdus Sattar,who would be the happest person to read this thesis.To my grandmother, Jahanara,who has shown me the essence of unconditional love.To my parents, Zakir and Airin,who have advertised the world in the best way possible.xivChapter 1IntroductionThis thesis describes the concept, design process, and performance of a broad-band xed-beam leaky-wave antenna. The antenna is designed through devel-oping a method that applies transformation electromagnetics to antenna design.The proposed leaky-wave antenna oers xed-beam performance over a broadrange of frequencies. This introductory chapter outlines the essential conceptsof electromagnetics that are subsequently mentioned in this thesis.1.1 AntennasElectromagnetic radiation is the process of electromagnetic energy transmis-sion over a distance, usually through free-space. It is the basis upon which mod-ern communication systems have developed. But how does electromagnetic en-ergy get radiated into free-space, and how does the radiation get collected by areceiving system? A special electric device called antenna makes the emissionand reception of electromagnetic radiation possible in practice. Antennas areessential to control and transfer electromagnetic energy when the use of wiredsystem is no way possible. Wireless radio communication was pioneered dur-ing the rst trans-Atlantic radio communication by Guglielmo Marconi in 1901[1]. The aerial antenna Marconi used during his experiment is presented in Fig.1.1. The experiment successfully transmitted wireless data over a large distanceand stimulated general attention in wireless communication and antennas. Sincethen, the world has witnessed an ever-growing research interest in antennas asan integral part of wireless communication.Antennas come in various size and shapes. There are generally numerousalternatives while choosing an antenna for a given application. Finding a suitableantenna type is a primary step in the design of a wireless system. A well-designedtransmitter antenna is expected to convert utmost input power to radiated power,while sending as much of it as possible in the direction of the receiver antenna.11.2. Broadband Antennas(a) (b)Figure 1.1: Original photo of the aerial that was used at the Poldhu station byMarconi for the rst radio communication (courtesy: Bodleian Libraries, Uni-versity of Oxford) (a). An outline of the aerial of 54 wires were put up, supportedby two masts 48 meters high and 60 meters apart (b). The aerial wires were ar-ranged fan shaped. This type of antenna is now called a fan monopole antenna[1].1.2 Broadband AntennasBroadband antennas belong to the category of antennas that provides com-paratively constant performance over a wide frequency range. Research interestin broadband antennas has grown steadily in recent years, in part due to the needfor devices that operate at multiple frequency bands. Broadband applications in-clude radar and tracking systems [2, 3, 4], electromagnetic compatibility (EMC)measurement [5][6], and broadband communications systems.The primary challenge for designing broadband antennas is to obtain a wide-impedance bandwidth while maintaining good radiation eciency through theentire frequency range. Broadband antennas oer negligible impedance and pat-tern variations over wide range of frequencies. The most widely used antennasgenerate radiation through the principle of resonance. Examples of such anten-nas are the dipole antenna, horn antenna, microstrip antenna. These antennasgenerate radiation by resonating at a designed frequency [7]. The resonatingcharacteristics make the these antennas unsuitable for most broadband appli-cations. Broadband antennas are frequently utilized by incorporating multipleresonance frequencies. For instance, a simple dipole antenna operates by res-onating at a length of half-a-wavelength at its design frequency. Therefore, there21.2. Broadband AntennasFeed(a)Arm 1Arm 2(b)Figure 1.2: Planar representation of a bow-tie antenna [8] (a) and a two-am spiralantenna [9] (b).is a narrow range of frequencies that provides resonance for such physical lengthof the dipole. In order to apply dipole antennas for broadband applications, itsshape can be modied to form a bow-tie antenna as shown in Fig. 1.2a [10].The structure of a bow-tie antenna allows it to resonate over a wider range offrequencies.Apart from multiple resonant structures, broadband antennas are often em-ployed through self-complimentary structures. If radiation properties of an an-tenna structure is determined by its dimensions in terms of wavelengths, its per-formance would strictly be a function of frequency. Therefore, an antenna de-scribed by its electrical length results in narrow bandwidth of operation. Theonly way to obtain frequency-independent properties of an antenna is by de-scribing its shape by angular specications alone. Such a design ensures that theshape of the antenna scales in proportion to the variation of frequency. Anten-nas that employ self-complimentary structures are called frequency independentantennas [11]. A spiral-shaped self-complimentary structure, illustrated in Fig.1.2b, is a commonly used self-complimentary structure. The antenna structure isfed between the closest ends of the two arms. Spiral antennas are generally im-plemented on planar or conical surfaces and achieve fractional bandwidth (high-est frequency / lowest frequency) of approximately 5 : 1 to 30 : 1.31.3. Leaky-wave antennas(a) (b)Figure 1.3: The rst leaky-wave antenna was implemented on a rectangularwaveguide by introducing a longitudinal aperture in the side wall of the structure(a). The same waveguide can be transformed into a periodic leaky-wave antennaby it with lling with a dielectric material and introducing an array of holes (b).The gray plane surrounding the aperture and holes is an innite ground plane.A broadband antenna is expected to demonstrate constant input impedancewith the frequency variations. Bow-tie, helical, and log-periodic are some ofmany other antennas that demonstrate wide input impedance bandwidth. An-other type of antenna which achieves a wide impedance bandwidth is the leaky-wave antennas.1.3 Leaky-wave antennasLeaky-wave antennas are a category of traveling-wave antennas [12]. traveling-wave antennas have a structure that allows an electromagnetic wave to travelalong a transmission line in one direction only. Being non-resonant in principle,traveling-wave antennas are well-suited for broadband applications. They canbe categorized into surface-wave antennas and leaky-wave antennas. Surfacewave antennas employ surface waves to generate radiation. Surface wave an-tennas radiate using surface waves and leaky wave antennas radiate using leakywaves. The rst leaky-wave antenna was published in 1946 [13] which was astructure based on a rectangular waveguide with a long uniform longitudinalslit (see Fig. 1.3a). A propagating mode exists within the waveguide which isleaks energy through the longitudinal slit. The antenna performed based on atraveling wave along the waveguide that produced radiation through the openslit into free-space.In 1957, Hines and Upson proposed the rst perturbed leaky-wave antenna by41.3. Leaky-wave antennasreplacing the uniform slit by a series of circular holes on a rectangular waveguide[14]. Introduction of an array of holes initiated a new category of leaky-waveantennas that utilized a periodic structure. Referred to as “holey waveguide", thisnew kind of leaky-wave antenna oered several advantages over the ones havinga uniform aperture. The structure had higher leakage of energy per unit length ofthe structure [15]. Hence it was physically smaller than the previous designs. Thestructures of these rst two leaky-wave antenna is presented in Fig. 1.3. In lateryears, focus was shifted towards array structures of leaky-wave antennas. In1980, Oliner, Lampariello, Shigesawa and Peng individually performed extensiveinvestigation on one-dimensional leaky-wave antenna arrays [16]. Concurrently,Alexopoulos and Jackson studied two-dimensional leaky-wave antennas [17, 18].Typical LWAs are beam scanning. Their primary beam sweeps adjacent spaceas a function of frequency for a wide range of scan-angles. This makes theantenna suitable for common applications like automotive collision avoidance[19] [20] and radar system [21]. However, some applications requires the radi-ation to be independent of frequency and directed at a certain location over theoperating frequency range. Such applications include wireless local area net-works [3], ultrawideband technology [4], satellite communications, and radio-metric eld sensing. Research on reducing the beam scanning characteristics hasbeen conducted by cascading negative-refractive-index transmission-line meta-material unit cells [22] and applying non-Foster articial transmission lines [23].These designs oered reduced beam-scanning performance from leaky-wave an-tennas; however, they demonstrated small scanning characteristics. In order toachieve xed-beam radiation from leaky-wave antennas, rather than reducingtheir beam scanning performance, Neto and Maci proposed uniques structurefor leaky-wave antennas. In 2003, they reported the existence of a leaky-wavemode in a slot-line printed on an innite ground plane by Neto and Maci [24, 25].The radiated beam from the leaky-wave mode generates a constant beam inde-pendent of the operating frequency. This investigation opened a new approach ofresearch on leaky-wave antennas that focused on xed-beam radiation, leadingto various novel designs. Neto et al. implemented the rst xed-beam leaky waveantenna by placing a hemispherical dielectric lens on top of a slot-line as shownin Fig. 1.4a [26]. The structure produced directive broadside radiation. Bruni etal. proposed an alternative of the lens-based structure in 2007 [27]. The struc-ture was implemented with a tilted slot-line so that the leaky-wave beams werenormal to the upper interface. The wavefronts of leaky-wave radiation emergedfrom the slot-line encountered the dielectric lens interface at normal incidence.A prototype of the design is presented in Fig. 1.4b. The slot-generated radiation51.4. Transformation electromagnetics(a) (b)Figure 1.4: CAD drawing of the leaky lens antenna proposed by Neto et al. [26](a) Prototype of a similar lens antenna proposed by Bruni et al. [27] (b).propagates through the denser dielectric lens above and couples into free-space.The leaky-wave radiation from the slot can directly radiate into free-space if thesubstrate below the slot-line has a relative permittivity below unity. In 2011,Sievenpiper used a non-Foster circuit substrate to design such an antenna struc-ture as a planar alternative to the designs of Neto and Bruni et al. [28]. Thisdesign had non-broadside radiation capabilities; however it could only be imple-mented by active circuits to implement the permittivity-below-unity substrateover a wide bandwidth.Most leaky-wave antenna applications are used for their beam scanning capa-bility. Slot line LWAs provide a way to produce broadband xed-beam radiation.This thesis presents a novel design of a broadband xed-beam leaky-wave an-tenna. The antenna is based of the works of Neto et al. [26] and Bruni et al. [27].It implements graded dielectric index and oers adjustable xed-beam directionperformance for planar applications.1.4 Transformation electromagneticsThe design concept of the antenna presented in this thesis employs trans-formation electromagnetics. Transformation electromagnetics, also known astransformation optics, is a technique that utilizes coordinate transformation tocontrol the propagation of light. The technique was introduced 2006 [29][30]. Itprovides exibility in designing media to control the propagation of electromag-netic waves. Optically transformed structures have been widely used in various61.4. Transformation electromagneticsyx uv vuInitial Domain Transformed DomainFigure 1.5: Coordinate transformation of a uniform dielectric initial domain torectangular one. The resulting inhomogeneous material distribution inside thetransformed domain is presented in the surface plot. ©IOP Publishing. Repro-duced with permission. All rights reserved [42].electromagnetic applications including cloaking [31][32], electromagnetic rota-tors [33][34], optical black holes or absorbers [35] [36], hyperlenses [37][38],electromagnetic concentrators [39][40], sensor cloaks [41] and so on.Transformation electromagnetics relies on the concept of coordinate trans-formation that allows one coordinate system to be mapped into another [43]. Itallows the transformation of one geometry into another while maintaining itselectromagnetic waveguiding properties. Figure 1.5 demonstrates transforma-tion optics where an arbitrary shaped region in rectangular (x, y) coordinates istransformed into a at region in (u,v) coordinate. The initial domain containsuniform dielectric. The rectangular coordinate grids of the initial region under-goes stretching and squeezing to map into the transformed space. The regions in(u,v) domain that experiences coordinate squeezing has higher dielectric indexthan the equivalent spaces in (x,y) domain . This leads to inhomogeneous distri-71.5. Proposed designbution of dielectric index in the transformed space . Due to the inhomogeneousdistribution of material parameters inside the medium, electromagnetic wavespropagate in a curved path. Thus, by employing transformation optics propaga-tion path of the waves can be manipulated. This has been useful in designinghigh-performance antennas [44] [45]. In particular, transformation electromag-netics is mostly utilized to transform bulky antennas to low prole ones. For in-stance, parabolic reector antennas are used for wide range of applications due totheir high directivity [46]. However, at low frequencies their structure becomesbulky. Transformation electromagnetics is used to design a planar alternativedesign of parabolic reector antennas [47] [48]. Directivity of a horn antenna isimproved by incorporating a graded index dielectric lens through transformationelectromagnetics [49].1.5 Proposed design1.5.1 Motivation and objectiveLeaky-wave antennas are commonly known for their beam scanning perfor-mance while maintaining a wide input impedance bandwidth. Oblique xed-beam leaky-wave antennas have received less attention as compared to beam-scanning ones. As will be shown in the next chapter, a few broadband xed-beamleaky-wave antenna have been proposed in the past decade. The rst designspresented by Neto et al. [26] and Bruni et al. [27] radiated only in the broadsidedirection. Sievenpiper’s design in 2011 implemented active circuits [28]. Theprinciple motivation for this research was to propose a passive broadband pla-nar leaky-wave antenna, with a capability to radiate at an oblique direction forthe entire bandwidth.1.5.2 Design specicationsGeometryLeaky-wave antennas are typically electromagnetically long as compared toother antenna categories. The reason is that the leaky-wave antenna structuremust be long enough to support a traveling wave that decays while propagating.Among other leaky-wave antennas, the one proposed by Neto et al. was 10.9λ0[26], Bruni et al. was 42.6λ0 [27], Sievenpiper was 10λ0 long at the maximum81.6. Thesis organizationoperating frequency [28], where λ0 is the free-space wavelength. The rectangu-lar antenna presented in this thesis has a length of 8.4λ0 at the design frequencyand a height of 5.8λ0.BandwidthFrequency bandwidth is the most signicant parameter for broadband an-tennas. The challenge of the proposed leaky-wave antenna was to maintain aconsistent input impedance for producing a desired oblique beam over a widerange of frequency. The leaky-wave antenna proposed by Neto et al. was ca-pable to radiate at a xed angle for a percentage bandwidth of 43% [26], whichwas later improved to 160% [27]. However, the design was limited to broadsideradiation only. In 2011, Sievenpiper solved the issue by employing non-Foster cir-cuits. The design had a bandwidth of 163.64%[28]. The xed-beam leaky-waveantenna proposed in this thesis is a passive device which is capable to radiate ata non-broadside direction with a percentage bandwidth of more than 120%.ContributionThe major contributions of the thesis to the eld of electromagnetics, in par-ticular, the antenna design, can be summarized in the following points:• The proposal, development, and validation of a technique to improve ra-diation characteristics of planar antennas that incorporates transforma-tion electromagnetics. The technique compensates for the discrepancy inexpected phase prole at the boundary of an electromagnetically trans-formed medium.• Study the inuence of dierent physical parameters of slot-line leaky-waveradiation.• Development of a broadband slot-line leaky-wave antenna for oblique ra-diation.1.6 Thesis organizationThis thesis aims to thoroughly describe the performance of the proposed an-tenna as well as to provide background details on electromagnetic wave theoryand leaky-wave antennas. In order to narrate the important aspects and dierent91.6. Thesis organizationstages of the research, this thesis is divided into ve chapters. The introductorychapter contains the groundwork of associated research with a brief overview ofthe fundamental concepts. A brief summary of the contents in the other chaptersis given below.• Chapter 2 introduces electromagnetic wave theory, in particular, Maxwell’sequations and their extension to wave equations. Dierent types of leaky-wave antennas and their radiation mechanism are reviewed. Specically,mechanism for xed-beam radiation from slot arrays is addressed in thischapter. This is followed by a brief review on transformation electromag-netics. The mathematical concepts behind coordinate mapping and deriva-tion of material parameters in the transformed medium is also presented.• Chapter 3 discusses the preliminary research procedure of this researchproject.• Chapter 4 demonstrates how an optically transformed medium comprisesof phase-discrepancies at the air-dielectric interface. Based on equationsof transformation electromagnetic, a geometric compensation technique isdemonstrated to compensate the phase-discrepancies.• Chapter 5 is devoted to the design and analysis of a broadband xed beamleaky-wave antenna using the technique introduced in chapter 4. Broad-band performance of the antenna including side-lobe and backlobe behav-ior, xed-beam characteristics and directivity of the designed antenna ispresented.• Finally, Chapter 6 summarizes this thesis with conclusions followed bypotential ideas for future work.The thesis is an outcome of the research experience of the author and im-mense guidance of his supervisor. The author hopes that this thesis will guidebeginner researchers to acquire general concepts of leaky-wave antennas as wellas the specialists to design antennas using transformation electromagnetics. Inaddition to the basic concepts of leaky-wave antennas and transformation elec-tromagnetics, the reader will also experience the decision process that the authorwent through during his investigation.10Chapter 2BackgroundIn order to understand the antenna design procedure as well as to interpretits performance outlined in the following chapters of this thesis, backgroundknowledge on electromagnetics is essential. Therefore, background materialsand mathematical concepts related to leaky-wave antennas and transformationelectromagnetics are included in this chapter. In particular, this chapter presentsa unied description of necessary concepts pertaining to antennas in general,dierent kinds of leaky-wave antennas and their design architecture, xed-beamleaky-wave antennas in literature and extension of coordinate mapping to trans-formation electromagnetics. This chapter prepares the reader for Chapters 4, 5and 6 in which the detailed design procedure and characterization of the pro-posed leaky-wave antenna is presented.2.1 Electromagnetics for antenna analysisElectric eld intensity E is produced by electric charges. In contrast, mag-netic eld intensity H is caused by charges in motion. Time-varying electricand magnetic eld intensities are coupled together and give rise to electromag-netic waves that propagates away from its source. This interaction of electricand magnetic eld intensities was rst realized by Scottish scientist James ClerkMaxwell. In 1861, Maxwell blended the individual works of Coulomb, Gauss,Faraday and appended his own intuition to demonstrate that time-varying elec-tric and magnetic eld intensities accompany each-other. Originally publishedin 1865 and later extended by Hertz, Heaviside, Lodge, and FitzGerald, Maxwell’sequations describe the behavior of electromagnetic waves in a medium [50]. Ex-pressed in terms of electric and magnetic eld intensities, the frequency-domainrepresentation of these equations in the absence of sources are∇ ·B = 0 (2.1a)∇ ·D = 0 (2.1b)∇× E+ jωµH = 0 (2.1c)112.1. Electromagnetics for antenna analysis∇×H− jωE = 0 (2.1d)where  and µ are the relative permittivity and permeability of the medium thatsupports the electromagnetic eld, respectively. D and B are derived electricand magnetic ux densities. For a simple medium, Maxwell’s equation can beextended into electromagnetic wave equations by taking the curl of the curl equa-tions 2.1c and 2.1d and substituting the divergence equations 2.1a and 2.1b intothe expression. The wave equations are expressed as∇2E+ ω2µE = 0 (2.2a)∇2H+ ω2µH = 0 (2.2b)The electromagnetic wave equations, also known as the Helmholtz equation, de-scribes the propagation of electromagnetic wave through a transmission mediumhaving a permittivity and permeability of  and µ, respectively. The solution ofthe wave Eq. 2.2a is the general plane waveE = E0e±jkz+jωt (2.3)where k is the wave number and is expressed ask =ω√µ(2.4)The impedance of the medium is represented byη =E0H0=√µ(2.5)The solution to Maxwell’s equations in Eq. 2.3 reveals that a bounded mediumor a transmission line supports both forward and backward traveling wave. Aforward traveling wave in a transmission line gives rise to reected and transmit-ted waves when it encounters a segment of transmission line with mismatchedimpedance. [51]. The reected wave or the backward traveling wave takes awaya portion of the transmitted power intended to be transferred to the load. Forthe sake of maximum power transfer, a transmission line that feeds an antennais generally expected to support forward wave only.122.2. Antenna radiation parametersMain lobeBack lobeNullSide lobeSiSD =SSiFigure 2.1: Illustration of nulls, secondary lobes and directivity of an antennaradiation pattern. The radiation pattern of the real antenna is shown by the solidline, while the dashed line represents an isotropic radiation. [53]2.2 Antenna radiation parametersAccording to IEEE standards, antennas is dened as “means for radiatingand receiving radio waves" [52]. Antennas are employed in the rst or last stageof a wireless system to couple guided electromagnetic waves from an electricalcircuit into free-space and vice versa [53, 10, 54]. Antennas come in variousshapes and sizes depending on their applications. Design principles and radiationmechanisms dier widely for dierent categories of antennas.In antenna engineering, the performance of an antenna is explained in termsof various antenna parameters. In order to interpret the results presented in thisthesis, it is necessary for the reader to understand the signicance of these widelyused antenna parameters. Some common parameters are summarized below:• Radiation pattern is a plot of the strength of the emitted electromagneticradiation in dierent directions. It describes the far-eld behavior of anantenna. A sample radiation pattern of an antenna is presented in Fig. 2.1.• Half power beam width describes the angle between the two directionswhere the radiation power of the main beam is half its maximum.• Main lobe is the region with the highest radiated power density. Figure 2.1presents dierent lobes of a radiation pattern from an antenna. Sidelobesare the largest of all undesired regions of the radiation pattern of the an-tenna. In contrast, backlobes are the lobes in the direction opposite to themain lobe. Sidelobe level and Backlobe level are the strengths of side lobe132.3. Leaky-wave antennasand backlobe radiation relative to the main lobe, respectively. Each lobesare separated by regions of zero eld strength which are called nulls.• Directivity provides a measure of how directive a radiation patten is. It isgenerally dened as the ratio of the radiation intensity in the direction ofmaximum radiation, to the radiation intensity averaged over all directions.In Fig. 2.1, radiation intensities for an isotropic radiator and a given an-tenna are labeled as Si and S, respectively. Therefore, the directivity canbe represented by D = S/Si.• Bandwidth of antenna is the range of frequencies over which the it demon-strates some given performance. In general, it is represented by the follow-ing equation:f(x) =2(fu − fl)fu + fl× 100%, if bandwidth < 100%.fufl: 1, if bandwidth > 100%.(2.6)where fu and fl are the upper and lower cut-o frequencies [55]. Band-width of an antenna can be dened in terms of impedance, radiation pat-tern, and polarization. Impedance bandwidth represents the frequencyrange that allows satisfactory energy to be transferred from a transmis-sion line to the antenna and vice versa. Pattern bandwidth is determinedby specifying any of the antenna pattern parameters as either minimum ormaximum according to the system requirements. Pattern bandwidth wasprimarily chosen to characterize the antenna presented in this thesis.The antenna parameters, such as the main beam direction, side and back lobe,beamwidth, etc, vary with frequency. Variations in the parameters result essen-tially from frequency-dependent distributions of electric and magnetic elds inthe antenna structure. The antenna presented in this thesis has been designedupon specifying allowable sidelobe and backlobe levels for a broad bandwidth.2.3 Leaky-wave antennasLeaky-wave antennas are a category of traveling-wave antennas. Traveling-wave antennas allow an electromagnetic wave to travel along a transmissionline only in the forward direction and radiate energy in the process [10][56].142.3. Leaky-wave antennasThe transmission line is long enough for the propagating wave so that it signi-cantly decays before reaching the transmission line end. The structure often hasa matched termination in order to prevent reections. Either way, a negligibleamount of power is left in the forward traveling wave at the terminating endof the traveling-wave structure. Traveling wave antennas can be divided intoslow and fast wave antennas depending on the type of traveling-wave it uses forradiation mechanism. The fast wave traveling antennas are usually referred toas leaky-wave antennas since the energy is continuously leaked into free-spacefrom the wave that propagates through the antenna structure [57]. Conversely,traveling-wave antennas that support slow waves are called slow-wave anten-nas. Understanding the theoretical concept of fast and slow traveling-waves isnecessary to analyze the radiation mechanism in a leaky-wave antenna.2.3.1 Fast and slow-wave radiationSlow-waveThe terms fast or slow refers to the phase velocity of a traveling-wave modeas compared to the speed of light. Slow wave travels with a phase velocity slowerthan the speed of light (vp < c) in that medium. Consequently, the propagationconstant of a slow wave is greater than that of light and expressed asβslow =ωvp> k0 =ωc(2.7)Some example of slow wave structures are presented in Fig. 2.2. Helix type is themost widely used slow-wave structure since it is suitable for broadband (morethan > 30% bandwidth) applications.The antenna proposed in the later chapters of this thesis is a leaky-wave an-tenna which utilizes fast waves as radiation mechanism. Therefore the followingdiscussions are restricted to reviewing concepts of fast-wave radiation.Fast-waveA fast wave has a phase velocity greater than the velocity of light in thatmedium (vp > c). Hence, the propagation constant of a fast wave is less thanthat of light as shown in the following equationβfast =ωvp< k0 =ωc(2.8)152.3. Leaky-wave antennas(a) (b)(c) (d)(e)Figure 2.2: Models of simple slow-wave structures. (a) Folded-back transmissionline. (b) Zigzag transmission line. (c) Interdigital transmission transmission line.(d) Corrugated transmission waveguide. (e) Helix transmission linewhere ω is the operating frequency, k0 and c is the wave constant and velocity oflight in that medium, respectively. The fundamental TE10 mode of a rectangularwaveguide and higher order modes in a parallel plate waveguide are example offast waves. A fast wave is capable of simultaneously leaking energy into free-space [58][59].Radiation from slow and fast wavesTo demonstrate how a propagating fast wave generates radiation, a guidedstructure located at z <= 0 is considered in Fig. 2.3. The structure is surroundedby free space. The region in the positive-x has a slot aperture along its upperwall in xy plane. The aperture introduced along the length of the transmissionline in positive-x region allows the propagating wave to continuously leak itsenergy provided that the its propagation constant is βfast < k0. The radiation162.3. Leaky-wave antennaszyxθDecay ofLW modeE = yˆe−jβxE(x, z) = yˆe−jβxe−j√k20−β2zFigure 2.3: Transition of a bounded wave from a rectangular waveguide to astructure that supports leaky-wave mode. A fast wave (dotted line with arrow-head) that propagates through a closed waveguide in the negative x region re-mains trapped inside the structure. The grey region along the upper wall of thewaveguide represents a slot aperture in the positive x region. The propagatingwave in this region is a leaky-wave mode that leaks energy through the lon-gitudinal aperture of the structure. The propagating leaky-wave mode decaysexponentially inside the waveguide.in z > 0 forms a conical beam. The normalized eld distribution of the boundwave on the aperture is given byE = yˆe−jβx (2.9)where, yˆ is the direction of electric eld, β is the wave number of the wave prop-agating in the x direction. The wave number β depends on the type of boundedwave inside the guided structure. It equals to the free-space wave number k0if the structure line is a parallel-plate waveguide lled with air. In contrast, βbecomes larger (slow wave) than k0 if the waveguide consists of dielectric mate-rials. β can also be smaller than k0 (fast wave) for higher order modes of a parallelplate waveguide. The radiating mode from a wave having a phase velocity fasterthan that of light is known as leaky wave mode.We are interested to study the radiation produced in the region z > 0 bythe planar aperture. We use the Fourier transform method demonstrated in [12],chapter 4. The solution for the electric eld in free-space (z > 0) is expressed asE(x, z) = yˆe−jβxe−j√k20−β2z (2.10)172.3. Leaky-wave antennasEq. 2.10 states that if a wave dened by Eq. 2.9 propagates along an innitelylong slot, the corresponding elds in free-space has both x and z components ofpropagation constant. If the propagating wave along the slot is slow (β > k0),the term√k20 − β2 becomes imaginary, which results in evanescent waves in theregion z < 0. In contrast, if the propagating wave along the slot is fast (β < k0), aplane wave is generated from the slot at an angle given by the following equationsinθ =βk0(2.11)The amplitude of the traveling-wave exponentially decays in the direction ofpropagation due to constant leakage of energy through the slot. The exponentialdecay of the wave is expressed by attenuation constant α which is related to thecross-sectional geometry of the transmission line. It is be noted that attenuationconstant α of a leaky-wave mode is not necessarily related to the material loss,but can appear due to radiation losses alone. As for the propagation constant βin Eq. 2.9, it represents the phase constant of the wave in the transmission line.β determines the nature of the traveling-wave and the amplitude constant α isthe leaked energy per unit length. Most of the characteristics of a leaky-waveantenna is governed by the two parameters α and β.To account for the decay of the leaky-wave mode, a decreasing exponentialterm e−αx is incorporated with the expression for the elds in Eq. 2.9.E = yˆe−jβxe−αx (2.12a)E = yˆe−jβxej2αx (2.12b)E = yˆe−j(β−jα)z (2.12c)where, β is the phase constant of the propagating wave prior to introducing theaperture and α is the leakage constant of the leaked power [15]. From the aboveequation, it can be concluded that the propagation constant of a leaky wave modeinto free space is represented bykLW = β − jα (2.13)The elds created by a leaky wave mode described in Eq. 2.9 correspondsto innitely long apertures. Finding the elds in free-space for a nite apertureis more complicated [60]. The process can be simplied by replacing the nite182.3. Leaky-wave antennasLxzθFigure 2.4: Line-source model of a leaky-wave structure.aperture by a line-source model extended in the positive x direction as demon-strated in Fig. 2.4[53]. With the line-source model, the leaky-wave structure canbe assumed as a lossless waveguide with a uniform cross section that supportsa traveling wave with a phase constant β in the x direction. Since leaky-waveradiates from a uniform cross-section, all leaky-wave antennas can be expressedby a line-source with a voltage distribution [61]:.V (x) = e−jβx (2.14)where kLW is the complex leaky-wave number as expressed by Eq. 2.13. Thesubstitution facilitates a straightforward approach using scalar quantities likefar-eld radiation pattern [62]. The far eld radiation pattern of the radiatedwave is given by:f(θ) =∫ L0e−jkLWxe−jk0 sin θxdx (2.15)It is known the the integral of such format can be express in terms of a sincfunction:f(θ) = sinc[(β − k0 sin θ)L2)]L (2.16)The peak of the radiation pattern f(θ) is evaluated at an angle θm:sin θm =βk0(2.17)192.3. Leaky-wave antennaswhich is the similar found in Eq. 2.11 through eld quantities.Eq. 2.16 can result in dierent kinds of radiation depending on the proportyof β. For a fast wave with β < k0 the case is quite simple. The direction ofradiation at an angle θm is given by the Eq. 2.11. As presented in Eq. 2.16,the far eld function f(θ) is represented by sinc which is a continuous function.Therefore, the far eld radiation pattern contains sidelobes. This is dierent fromradiation pattern of an innite structure which is free of sidelobes. All practicalleaky-wave antennas have a nite structure and consist of sidelobes originatedfrom reected backward waves in the transmission line.A periodic leaky-wave antenna uses a slow wave as radiation mechanism.Equation 2.16 suggest that for slow waves with β > k0 leads to a complex valueof radiation peak radiation angle θm (= arcsin(β/k0) > 1). Therefore, if a slot-line is introduced to a slow-wave structure, radiation is not achieved in free-space. However, it was found that a propagating slow-wave can radiate if theassociated aperture, rather than being longitudinally uniform, has periodic per-turbations. Periodic modulations at the aperture introduces innite number ofspace harmonics in the guided wave in the transmission line, provided that themain space harmonic is a slow wave [57][63]. The perturbations are introducedin such a way that one of the space harmonics is a fast wave having a wavenumber less than k0 [15]. Radiation is obtained through allowing the fast spaceharmonics to leak energy into free-space[62]. Such emission is known as slowleaky-wave radiation.2.3.2 Radiation characteristicsDirection of primary beamThe radiated beam from a leaky-wave antenna has a conical shape. The di-rection of maximum radiation is obtained from equation 2.11 is expressed asθ = arcsinβk0(2.18a)= arcsinλ0λg(2.18b)= arcsincvLW(2.18c)where λ0 and c is the free-space wavelength and velocity of the radiated wave,λg and vLW is the wavelength and phase-velocity of leaky-wave mode inside the202.3. Leaky-wave antennasE plane(a)E plane(b)Figure 2.5: Leaky-wave antennas based on microstrip structures: (a) the combline array and (b) series microstrip patch array [62] ©2008 IEEE.guiding structure and θ is the direction of maximum radiation measured frombroadside direction. Equation 2.18c shows that the angle of radiation from aleaky-wave antenna depends on free-space wavelength λ0, hence on operatingfrequency. As the operating frequency is increased, the angle of peak radiationθ increases from broadside towards endre. The beam-scanning performance aswell as the regions of radiation is demonstrated in Fig. 2.6. Uniform leaky-waveantennas can only operate in the forward quadrant. On the other hand, periodicleaky-wave antennas oer the exibility to obtain radiation both in forward andbackward quadrants. It is generally dicult to obtain radiation near endre re-gion (θ = 90o) for an air-lled waveguide. The scanning operation near endrerequires the antenna to operate at frequencies above cuto of the leaky-wavemode.Beam-widthAnother interesting feature of leaky-wave antenna is that the beamwidth (be-tween the −3dB points) of the main lobe varies very little with the variation of212.3. Leaky-wave antennasLWA guiding structure x−xzθForward quadrantBackward quadrantEndreBroadsideFrequency increaseFigure 2.6: Beam-scanning performance of a typical leaky-wave antenna. Theprimary beam scans the space from backward quadrant (only for periodic struc-tures) to forward quadrant (for periodic and uniform structures).frequency. While the beam direction in Eq. 2.18c depends on β, beamwidth iscontrolled by the attenuation constant α. The beamwidth for uniform leaky-wave antenna is given by the following equation [15]:∆θ ≈ 1Lλ0cos θm∝αk0cos θm(2.19)where ∆θ is the beamwidth, θm direction of maximum radiation and L is thelength of the aperture of the leaky-wave antenna. It is to be noted that the angleof radiation θm is dependent on frequency as shown in Eq. 2.18c. However, thebeamwidth ∆θ depends on the length of the aperture L in terms of wavelengthλ0. A large value of α indicates a short eective aperture, leading to a largebeamwidth of the radiated wave. On the other hand, smaller value of α leads toa longer eective aperture of the antenna structure, making the beam narrower.Radiation patterns in the scanning plane for two values of α/k0 are presented inFig. 2.7 [15] where radiation is achieved at an angle θm = 45o. In accordancewith Eq. 2.19, the pattern having larger α value has a much larger beamwidth.Note that the radiation patterns in Fig. 2.7 do not have side-lobes because theycorrespond to an innite uniform leaky-wave antenna.222.4. Slot-line leaky-wave antennaFigure 2.7: Radiation patterns for a leaky mode having β/k0 = 0.7071 and twodierent values of α/k0 : 0.1 (dashed line) and 0.01 (solid line) [15].Equations 2.18c and 2.19 suggests that as frequency is increased in a leaky-wave antenna, the main beam steers from broadside towards endre while main-taining a constant beamwidth.2.4 Slot-line leaky-wave antenna2.4.1 Principle of radiationA slot-line on an innite conductive plane and an electric dipole in free-spaceare complementary structures [64]. Illustrated in Fig. 2.8, antenna A is the com-plementary structure of antenna B. The shaded region in Fig. 2.8b is an inniteground plane. The eld radiation from an electric dipole is calculated using elec-tric currents since it radiates using electric currents. In contrast, eld calcula-tion for a slot antennas can also be done by considering nearby surface currents.However, evaluation of slot radiation becomes simpler if the source of radiationis considered to be a ctitious magnetic current. Using full-wave analytic tech-niques, existence of leaky-wave mode on a transversely excited slot-line was rstreported in 1988 [65]. Following a similar approach, Neto and Maci analyzed themagnetic currents of an innitely long slot-line placed between two homoge-neous media in 2003 [24][25]. The ctitious magnetic current corresponding tothe slot aperture was analyzed in order to obtain and solve the dispersion rela-tion of the slot mode. The electric dipole source excitation was expanded in thespectral domain, followed by solving the dispersion equation of the slot-line us-232.4. Slot-line leaky-wave antennaAirI(a)V(b)Figure 2.8: Complementary electric dipole in free-space excited with a current I(a) and slot-line on an innite conductive plane excited with a current V (b).ing Fourier transform. The detailed analytical expressions are beyond the scopeof this thesis. In order to explain the steps in brief, an innite slot etched in aperfect electric conductor (PEC) ground plane is illustrated in Fig. 2.9. The slotis excited by a transverse current element in y direction. The upper half (z > 0)and lower half (z < 0) of the slot consist of homogeneous dielectrics while with arelative dielectric permittivity r1 and r2, respectively. The normalized magneticcurrent (voltage) along the slot in spectral domain is expressed as [24]:V (kx) = − 2pi∫∞−∞G(kx, ky)J0(12wsky)dky(2.20)where, J0 is the Bessel function of zero order, ws is the thickness of the slot,G(kx, ky) is the Fourier transform of Green’s function of a slot placed in betweentwo dielectric half spaces is given byG(kx, ky) = − 1k0ζ02∑i=1k2i − k2x√k2i − k2x − k2y(2.21)where k0 is the free-space propagation constant, ζ0 is the characteristic impedance,and ki for i = 1, 2 are the wave numbers in medium 1 and 2 respectively. By ap-242.4. Slot-line leaky-wave antennaGround planeDielectrichalf-space (r1)Dielectrichalf-space (r2)zxyFigure 2.9: Innitely stretched slot located at the interface of two dielectric halfspaces. The slot is etched on a ground plane [24] ©2003 IEEE.plying the Green’s function from Eq. 2.21 in Eq. 2.20, the normalized voltagealong the slot-line can be obtained by applying inverse-Fourier transformation:v(x) =12pi∫ ∞−∞e−jkxxV (kx)dkx (2.22)The complex propagation constant of the leaky wave mode along the slotcan be obtained by solving Eq. 2.22 [26]. If the upper dielectric half-space hashigher dielectric permittivity than the one at the bottom (r2 > r1), the complexpropagation constant kLWx is given by the following equationkLWx ≈ β +k2d2β[1− j 4piln(γekdws8)] (2.23)where, γe = exp(γ) = 1.781 . . . is the exponential of Euler constant, β and kd,associated with the average permittivity of media 1 and 2, is expressed asβ =√k21 + k212(2.24a)kd =√k22 − k212(2.24b)252.4. Slot-line leaky-wave antennaǫr2ǫr1θkLWxk2kzFigure 2.10: (color inline) A sketch of a radiation pattern showing that the slot-emitted leaky wave (blue) radiates predominantly into the denser (upper) dielec-tric half-space. Two beams are generated since the slot-line is excited in thecenter. The spherical source wave (red) is competitively weaker in power.provided that medium 2 is the denser medium. Equation 2.23 suggests that if thewidth of the slot ws is very narrow, then the propagation constant of the fastwave mode along the slot (kLWx ) simplies to be:kLWx = β =√k21 + k212(2.25)which is an established expression stating the slot-line mode has a phase constantbetween that of the associated medium [66].The propagating leaky-wave mode is fast for the upper dielectric region andslow for the lower halfspace. Therefore, the slot generates leaky-wave radiationinto the upper dielectric halfspace. Radiation in the denser media is obtainedfrom Eq. 2.18b:θ = arcsinkLWxk2(2.26)where kLWx and k2 is the wave number of the leaky-wave mode and radiated262.4. Slot-line leaky-wave antennawave, respectively (see Fig. 2.10). Equation 2.26 can be simplied by applyingthe relation of Eq. 2.25 as below:θ = arcsinkLWxk2(2.27a)= arcsin√k21 + k212k2(2.27b)= arcsin√r1 + r22r2(2.27c)where, θ is measured from the broadside. The direction of radiation in Fig. 2.10is independent of frequency since it is a function of the permittivity values ofthe associated media. This implies that the leaky-wave mode in the slot pro-duces a beam which is directed at a xed angle irrespective of frequency. Apartfrom the leaky-wave radiation, the slot generates a spherical source wave fromthe excitation [25]. The source wave contributes to the elds in the upper di-electric half-space; however, is weak when the slot is narrow in terms of wavelength. Therefore it does not aect the xed-beam performance and wide inputimpedance bandwidth of the slot. The polar plot in Fig. 2.10 shows the leaky-wave (blue line) and spherical source wave radiation (red line) pattern originatingfrom a center-fed leaky-wave antenna.2.4.2 Fixed-beam leaky-wave antennasThe leaky-wave radiation emitted from a slot placed at the interface of twodierent dielectric media can be utilized to design a xed-beam leaky-wave an-tenna. There are three major designs available in the literature that involve xed-beam leaky-wave radiation. The design principles and radiation mechanism ofthese antennas are described in this section.Dielectric lensNeto et al. designed and built the rst xed-beam leaky-wave antenna uti-lizing the slot-generated leaky-wave radiation [26]. The actual structure of theantenna is presented in Fig. 2.11. The design consisted of a dielectric half circleplaced on top of a slot-line printed on a ground plane. The dielectric permittivityof the free-space region below the slot is lower than that of the dielectric lens.272.4. Slot-line leaky-wave antenna(a)r > 1(b)Figure 2.11: The prototype of Neto’s xed-beam leaky-wave lens antenna [67]©2010 IEEE. The dielectric lens is the white hemispherical medium (a). Cross-section of the elliptical dielectric hemisphere placed of top of a leaky-slot. Thegray region illustrates the dielectric lens with a permittivity  > 0. The slot isrepresented by the gap between ground plane at the bottom of the lens (b).The slot-line generates leaky-wave radiation into the elliptical lens. The outersurface of the lens couples the radiation into free-space, resulting in broadsideradiation. The antenna operated with a percentage bandwidth of 43% (−13 dBS11) and was later improved to 160% (−8.5 dB S11) [27] and 173% [67]. However,the overall antenna structure was electrically large.Active non-Foster circuitsThe previous design involved a dielectric lens with permittivity  > 0 thatcoupled leaky-wave radiation into free-space. As an alternative, coupling intofree-space can be achieved if the slot is located on top of a substrate with a permit-tivity 0 <  < 0. In that case, air would be denser as compared to the lower per-mittivity dielectric substrate. Therefore, the leaky-wave radiation would directlyradiate into free-space from the slot. The substrate with relative permittivity (r)below unity can be implemented through metamaterials. Metamaterials are arti-cially produced materials that exhibits unusual properties not found in nature.Metamaterials oer a broad range of exibility on engineering a medium that282.4. Slot-line leaky-wave antenna0 < r < 1Microstrip lineFigure 2.12: Diagram of a microstrip line with a superluminal substrate that iscapable to radiate leaky-wave radiation into free-space.requires exotic values of constitutive parameters ( and µ). However, their reso-nant nature limits bandwidth of operation of the antenna [68][69]. In contrast,active structures, achievable through non-Foster circuit elements, can be usedfor broadband applications. Sievenpiper proposed a structure for a microstripline using such motivation and achieved xed-beam leaky-wave radiation over abroad bandwidth [28]. The design is presented in Fig. 2.12. Although the struc-ture was electrically long (10λ0), it had comparatively small thickness (.05λ0at the highest operating frequency). The structure operated with a percentagebandwidth of 163.64%. Similar xed-beam leaky-wave antennas with a substratehaving a permittivity below 1 has been studied in [70] and [71].Metamaterial transition layerIn 2014, Markley et al. utilized transformation electromagnetics to propose adesign of xed-beam leaky-wave antenna [72]. The antenna radiated using slot-line leaky-wave mode and consisted of a rectangular superstrate to couple theslot-generated radiation into free-space.The design principle is demonstrated in Fig. 2.13. A uniform dielectric prismin Fig. 2.13 coupled the radiation into free-space. Using transformation elec-tromagnetics, the prism was transformed into a rectangular transition layer asshown in Fig. 2.13. Figure 2.14 shows full wave simulation of an anisotropicmetamaterial transition layer designed using transformation electromagneticswhich is placed on top of a slot. The superstrate couples the leaky-wave mode292.5. Transformation electromagneticsθ(a)θrad(b)Figure 2.13: A slot-line leaky-wave antenna with a uniform dielectric prism layer(a) and an anisotropic inhomogeneous metamaterial layer (b).into free-space at a desired direction.The design technique oered exibility in achieving a suitable radiation anglefrom the antenna. However, it lead to very high values of dielectric permittivityinside the superstrate. The transition layer presented in Fig. 2.14 resembles amatched medium with equal permittivity and permeability values. The magneticresponse of the anisotropic can be implemented using resonant metamaterials,which leads to narrow bandwidth of the antenna.2.5 Transformation electromagneticsTransformation electromagnetics is a method of manipulating electromag-netic elds through coordinate transformation. Although the idea behind trans-formation optics can be dated back to 1920s [73][74], it was not until 2006 whenit received great attention [29][30]. Transformation optics has been widely usedas a convenient tool to design novel devices such as cloaks, super lenses [75] andantennas [76, 77, 78, 79, 80, 81].2.5.1 Form invariance property of Maxwell’s equationsThe transformation electromagnetics procedure is rooted in the form invari-ance of Maxwell’s equations. The property allows them to stay in the same formunder coordinate transformations. Ward and Pendry studied the propagation ofelectromagnetic waves in a complex structure dened by coordinate transforma-tion [82]. The frequency domain source free Maxwell’s equations are presented302.5. Transformation electromagneticsFigure 2.14: (color inline) Full-wave 2D simulations of the traveling-wave moderadiating into a metamaterial slab designed to couple to free-space [72] ©2014IEEE.in equations 2.1a−2.1d which describes how electric and magnetic elds behavesin a linear isotropic dielectric media.Consider a coordinate transformation from the coordinate system (x, y, z) to(x′, y′, z′). The transformed medium parameters are related by a Jacobian ma-trix which denes the coordinate transformation. It can shown that equations2.1a−2.1d stay in the same form even if they are transferred from one set of co-ordinate system into another [83]. The transformed space would then be char-acterised by material parameters ′ and µ′, electric eld E′, magnetic eld B′.Maxwells’ equations in the transformed space can be expressed in the followingform:∇ ·B′ = 0 (2.28a)∇ ·D′ = 0 (2.28b)∇× E′ + jωµH′ = 0 (2.28c)∇×H′ − jωE′ = 0 (2.28d)Generally, the material parameters (′ and µ′) in the transformed medium coor-dinate system turn out as tensors, typically leading to non-isotropic and inho-mogeneous medium.312.5. Transformation electromagneticswxy(a) Region wtuv(b) Region tFigure 2.15: Coordinate mapping of region w in the xy plane into a region t inuv plane.2.5.2 Conformal mappings in transformation opticsCoordinate mappingConformal transformation maps a coordinate system into another withoutvarying the local angles of the coordinate grid intersections. Consider the regionw in Fig. 2.15a dened by the coordinate system (x,y) to be mapped into theregion t in Fig. 2.15b described by the coordinate system (u,v). The relationbetween the coordinate systems of the two regions is then given by(x, y)f−→ (u, v) (2.29)If w(x,y) is analytic (complex dierentiable at every point) everywhere in-side the domain, then u and v must satisfy the Cauchy-Reinmann conditions inequations 2.30a and 2.30b [84], for the mapping (f) to be a conformal.∂u∂x=∂v∂y(2.30a)∂u∂y= −∂v∂x(2.30b)Applying partial dierentiation to the equations above with respect to x andy and relating the resulting equation, following relationships are obtained:∂2u∂x2+∂2u∂y2= 0 (2.31a)322.5. Transformation electromagnetics−→dv−→du−→dx−→dyregion w(x, y) region t(u, v)Figure 2.16: Demonstration of conformal transformation. Coordinate grid linesof constant x and y is shown in domains w(x,y) and t(u,v).∂2v∂x2+∂2v∂y2= 0 (2.31b)Equations 2.31a and 2.31b are equivalent to Laplace’s equations for analytic func-tions of x(u, v) and y(u, v).Local orthogonalityIt can be shown that if a conformal mapping is executed by satisfying Laplace’sequation, the intersection angles of the u and v curves in region t would be thesame as that of x and y curves in region w [85]. In other words, conformal trans-formation preserves the local angles and aspect ratio between the intersectinggrids. To demonstrate the local orthogonality, consider region w as the simplestorthogonal coordinate system: the Cartesian system, as demonstrated in Fig.2.16.As demonstrated in Fig. 2.16, the coordinate lines of the region w(x,y) forma rectangular grid, intersecting orthogonally. There would always be a set ofu(x, y) and v(x, y) curves that is constant for certain values of x and y. Sinceu(x, y) is constant in region t, the dierential du would be zero.du = 0 (2.32a)from the chain rule,∂u∂xdx+∂u∂ydy = 0 (2.32b)or,∂u∂xdx = −∂u∂ydy (2.32c)332.5. Transformation electromagneticstherefore,dydx|u=constant = −∂u∂x∂u∂y(2.32d)A similar relationship can be obtained for the constant v(x, y) curves in regiont(u,v):dydx|v=constant = −∂v∂x∂v∂y(2.33)Taking the product of equations 2.32d and 2.33 and applying the Cauchy-Reinmanncondition in Eq. 2.30:∂u∂x∂u∂y∂v∂x∂v∂y= −∂u∂x∂u∂y∂u∂y∂u∂x= −1 (2.34)The product of the two tangents of the slope is −1. Which means that the twocurves x(u, v) and y(u, v) corresponding to constant values of u and v orthogo-nally intersect. Therefore it can be concluded that under conformal coordinatetransformation, the transformed coordinate system (region t(u,v) for this case)remains locally orthogonal. The angle between the two coordinate curves arepreserved at any given point if the mapping satises the Cauchy-Riemann con-dition and the plane contains constant u and v values corresponding to x andy.Material parametersConformal mapping can be performed by satisfying Laplace’s equation be-tween two domains provided that either of them is a rectangle. In fact, confor-mal map between any two domains can be found by mapping each other througha rectangle. Laplace’s equation which can be solved, either analytically or nu-merically, to describe the transformed domain t(u,v) in Fig. 2.15b. The Jacobianmatrix is used to obtain its constituent material parameters ((u, v) and µ(u, v))from the solution. The Jacobian matrix of the coordinate transformation is de-342.5. Transformation electromagneticsned asΛ =∂u∂x∂u∂y0∂v∂x∂v∂y00 0 1 (2.35)Jacobian matrix is used in coordinate transformation to transform a vector orfunction from one coordinate system to another. A vector function in two dif-ferent coordinate system is related though the Jacobian matrix as follows:E(u,v) = ([Λ]T )−1E(x,y) (2.36a)E(x,y) = [Λ]TE(u,v) (2.36b)where [Λ]T is the transpose of the Jacobian matrix Λ, E(x,y) and E(u,v) arerepresentations of o vector functions in regions w(x,y) and t(u,v), respectively.In addition, an operation, such as derivative, integral, convolution, Fourier trans-form, etc., can also be transformed into a new coordinate system using Jacobianmatrix. Transformation of an operation F () between two coordinates can beperformed through the following identity:[F (u, v)] =[Λ][F (x, y)][Λ]Tdet[Λ](2.37a)[F (x, y)] = det[Λ][Λ]−1[F (u, v)]([Λ]T )−1 (2.37b)Using the expressions in equations 2.36 and 2.37, it can be shown that the relativepermittivity r(u, v) and relative permeability µr(u, v) of a conformally trans-formed media t(u,v) is dened through Jacobian matrix [29].[r(u, v)] = r(x, y)ΛΛT|Λ| (2.38a)[µr(u, v)] = µr(x, y)ΛΛT|Λ| (2.38b)where r(x, y) and µr(x, y) is the relative permittivity and permeability of re-gion w(x,y), respectively, and |Λ| is the determinant of Jacobian matrix. ΛΛT is352.5. Transformation electromagneticsexpressed as:ΛΛT =(∂u∂x)2+(∂u∂y)2∂u∂x∂v∂x+∂u∂y∂v∂y0∂u∂x∂v∂x+∂u∂y∂v∂y(∂v∂x)2+(∂v∂y)200 0 1 (2.39)It should be noted that this two-dimensional transformation is applicable onlyfor transversely (in the direction out of the page) polarized electric or magneticelds. If Cauchy-Reinmann conditions in Eq. 2.30 are used in Eq. 2.39, the matri-ces can be simplied into the following equations to nd the material parametersin region t(u,v)[r(u, v)] = r(x, y)1 0 00 1 00 01|Λ| (2.40a)[µr(u, v)] = µr(x, y)1 0 00 1 00 01|Λ| (2.40b)Equations 2.40 describe an isotropic, inhomogeneous and matched medium, aninherent property of conformal transformation [30]. Conformal method has beenutilized to study optical cloaks [86], carpet cloaks [31], antennas [76], waveg-uides [87]. Inhomogeneous media can be dened by space-dependent index ofrefraction and are referred to as graded index media [88]. Gradient index are nonresonant and are therefore prefered over resonant metamaterials for broadbandimplementations.The material constituent parameters of a conformally transformed mediumis isotropic with equal permittivity and permeability, (see Eq. 2.40). However,designing a medium with wide-range of permeability values is dicult in prac-tice. Unity permeability value in the transformed medium can be achieved for athe right polarization of wave. Equation 2.40 suggests that both permittivity andpermeability tensors of the transformed medium contain z components (33 andµ33) in the diagonal entries. For a z polarized wave only 33 from Eq.2.40a andand µ33 from Eq.2.40b aects the electromagnetic behaviour of the transformed362.5. Transformation electromagneticsRegion w(x,y) Target region t(u,v)Transformation(f )r(x, y), µr(x, y)Inverse transformation(f−1)r(u, v), µr(u, v)Figure 2.17: Transformation and inverse transformation between region w andt.medium [89][30]. Therefore, Eqs. Eq.2.40a and Eq.2.40b can be simplied as fol-lows:r(u, v) =r(x, y)|Λ| (2.41a)=r(x, y)√(dudx)2+(dudy)2 (2.41b)=r(x, y)√dudxdvdy− dudydvdx(2.41c)=r(x, y)√(dvdx)2+(dvdy)2 (2.41d)andµr(u, v) = 1 (2.42)372.5. Transformation electromagnetics2.5.3 Inverse transformationInverse transformation follows the equations of transformation optics by map-ping the inhomogeneous transformed domain back to the homogeneous initialdomain [90]. That is, Laplace’s equation is solved for the transformed space todene the medium in an original space. The relationship between the two re-gions in Fig. 2.16 can be described by:(u, v)f−1−→ (x, y) (2.43)Note that the relation is the inverse of Eq. 2.29. The process of inverse trans-formation is illustrated in Fig. 2.17. The coordinate mapping relationship is rep-resented by f−1. Once the solution of the inverse transformation is known bysolving Laplace’s equation, the material parameters are modied by invertingthe Jacobian expression in equations 2.41d and 2.42 and expressed as:r(x, y) = r(u, v)√dvdx2+dvdy2(2.44a)µr(x, y) = 1 (2.44b)Equations 2.44 can be used to inversely transform a transformed media back intoits original domain. The technique presented in Ch. 4 of this thesis utilizes theinverse transformation method which is later used in Ch. 5 to develop a xed-beam leaky-wave antenna.38Chapter 3The Linear Gradient DesignAn inhomogeneous medium is characterized by a spatially varying refrac-tive index. For high frequency applications, the terms gradient-index or gradedindex is typically used to refer to isotropic, inhomogeneous, and non-magneticmedia. In the preliminary design process of this research, a xed-beam leaky-wave antenna was intended to be implemented with 1D graded-index materialslab. This chapter describes the technique that was followed during the initialdesign process. Simulation results of the initially designed antenna will also bepresented.3.1 Wave propagation in graded-indexdielectricsEvolution of an electromagnetic wave in a uniform media is well-known andcommon in antenna applications. However, analysis of optical bers, liquid crys-tals, weakly magnetized plasma, condensed matter physics, etc requires study ofelectromagnetic waves in anisotropic inhomogeneous media. The analytic so-lution of Maxwell’s equation in an inhomogeneous media is dicult to achieve.Since 1970s, wave propagation in inhomogeneous media has been analyzed usinggeometric optics, the study of waves through rays [91, 92, 93] [92]. Analytic so-lutions using geometric optics are highly case-specic and follow dierent tech-niques for linear, axial, radial, and spherical gradients [94] [95]. However, rayoptics can only describe wave propagation accurately within the geometric op-tics limit. That is, the propagating wave must have a very short wavelength ascompared to the other dimensions of the problem [96]. The analyses on thisthesis are performed on structures having dimensions comparable to the wave-length. Therefore, ray optics is not used for the investigations.393.2. Leaky-wave source - single slot vs slot array3.2 Leaky-wave source - single slot vs slot arrayThe physical length of the leaky-wave antenna is related to the leakage rateof the propagating leaky-wave mode along the slot-line. Being the source of theleaky-wave radiation, the slot is required to be long enough to radiate no lessthan 90% of input power into the dielectric for wide input impedance bandwidthand improved sidelobe level behavior of the antenna [15]. Therefore, a higherleakage rate of the leaky-wave mode or a matched termination is recommendedfor shorter antennas. Since the antenna proposed in this thesis is not terminatedwith matched resistors, the leakage rate of the leaky-mode inside the slot had tobe investigated to ensure it was sucient for the length of the proposed design.After observing the leakage rate for dierent slot apertures, it can be statedthat a slot array demonstrates higher decay relative to a single slot-line. A corner-fed slot with a width of λ0/50 under a uniform dielectric half-space with r = 2,as a single element and as an innite array element with a separation of λ0/10,was simulated using COMSOL Multiphysics’ nite element solver. The gener-ated transverse electric eld of the leaky-wave radiation from a single slot anda slot array is shown in Fig. 3.1. Figure 3.2 demonstrates that the leakage rateof the transverse electric eld for an innite array is more than that of a singleslot. A innite slot array is therefore selected as the leaky-wave source for thetheoretical design of the proposed antenna.3.3 Antenna design principleAccording to the analysis of slot-lines under a uniform dielectric media insection 2.4, the slot-line emits leaky-wave radiation into the upper dielectricmedium [24][25]. If the lower dielectric medium is free-space and the mediumabove is uniform dielectric, the leaky-wave radiation generates at an angle (frombroadside) much larger than that required to be coupled into free-space. The phe-nomenon is demonstrated in Fig. 3.3 where a uniform dielectric slab is placed ontop of a slot-line. The radiation from the slot is generated at an angle θ which islarger than the that required to couple into free-space. The leaky-wave radiationremains trapped inside the dielectric medium due to total internal reection. Theinitial stages of the research project was aimed to reduce the angle to below thecritical angle through the use of graded-index dielectric slab.For the sake of implementation of a xed-beam leaky-wave antenna, a non-magnetic graded dielectric slab can be placed on top of a slot-line. The distribu-403.3. Antenna design principlev(λ0)u(λ0)(a)v(λ0)u(λ0)λ0 (b)Figure 3.1: (color inline) Normalized eld plot of the slot-line radiation intodenser dielectric half-space for a single slot (a) and an innite slot array (b). Thedenser dielectric medium consists of a relative permittivity of 2 and the slot widthis λ0/50. The black line indicates one wavelength in free-space.tion of refractive index is considered to be linearly decreasing along the length,as presented in Fig. 3.4. The slab had a maximum index at x = 0 and minimumindex (unity) at the x = L, where L was the length of the slab. The leaky-waveradiation originated from underlying slot-line encounters gradual decrease in re-fractive index as it propagates through the slab. The change of refractive indexcauses the propagating wave to bend towards the region where the refractiveindex is higher, as presented in Fig. 3.4. The bending can be described by Snell’slaw which states that if a wave is incident into an optically lighter medium froma denser medium, the light bends away from the normal.At the interface of the graded-dielectric slab, the leaky-wave (solid line) isincident with an angle smaller than it originally emitted (dashed line) from theslab. The gradient of refractive index needs to be chosen in such a way that thebending of the wave not only couples into free-space, but also radiates at a de-sired angle. As presented in Fig. 3.4, the radiated wave creates a primary beamat an angle θrad into free-space. The slot-generated radiation is independent offrequency, assuming material parameters are not dispersive. Therefore, the pri-mary beam of the antenna is expected be xed with frequency variations. Themaximum refractive index nmax, permittivity gradient, and the height of the slabH were varied to nd the optimum height H for a suitable direction of radiationfrom the antenna.413.4. Antenna properties0 1 2 3 4 5 6 700. length (λ0)AbsoluteE-eld(V/m)single slotslot arrayFigure 3.2: (color inline) Normalized absolute value of the transverse electric eldalong the direction of propagation for a single slot (brown) and an innite slotarray (green) located underneath a dielectric half-space with relative permittivityr = 2.θUniform dielectric slabFigure 3.3: A diagram demonstrating that a slot-generated leaky-wave radiationremains trapped when emitted into a rectangular uniform dielectric superstrate.3.4 Antenna propertiesThe rectangular graded-index dielectric slab was the principal component ofthe antenna. The antenna performance depends on the dimension and materialproperty of the slab. As demonstrated in Fig. 3.4, the length and height of theslab is L and H , respectively with a linearly decreasing refractive index with theincrease of L. Given a specic permittivity gradient, the slab heightH is the keyparameter of the antenna since direction of radiation θrad directly depends onH .423.4. Antenna propertiesGraded dielectric slabθradxLSlot linenprole of indexnmax6?HFigure 3.4: Bending of leaky-wave inside the graded dielectric slab. The solid linepresents the propagation of the wave inside the inhomogeneous slab. The dashedline represents the propagation path if the slab was a homogeneous dielectricslab. The prole of refractive index is shown in the graph above.Point of excitationWave-frontsparallel tointerfaceSlot lineSlot lineAB-6xy6?HbSymmetryFigure 3.5: Bending of leaky-wave forming a sinusoidal envelope inside thegraded dielectric half-space placed on top of a center-fed slot-line. The solidand dashed lines represent two separate waves generated from the slot-line. Theinset presents the refraction of waves due to change in refractive index.433.4. Antenna propertiesAirAirPMLGradedDielectricxyzλ0Figure 3.6: (color inline) Field plot of leaky-wave radiation into a graded dielectrichalf-space. One wavelength in free-space is indicated by the black line.3.4.1 Leaky-wave radiation inside graded-index half-spaceIn order to design the rectangular slab as well as to understand the impactof H in its performance, consider the semi-innite graded-dielectric half-spaceplaced on top of a slot-line in Fig. 3.5. The half-space is innitely stretched inthe positive y direction. The slot-line is excited with a transverse line-current atx = y = 0. The length of the slot-line on either side of the point of excitationis L. The refractive index is maximum at x = 0 and decreases linearly with thedistance away from the point of excitation. Two separate leaky wave radiation(dashed and solid line) emits in positive y direction into the half-space from theslot-line, forming a symmetry line at x = 0. As will be shown in later sections,this symmetry line can be used to reduce simulation domains. The slot-emittedwaves goes through continuous refraction and bends away from the normal ofincidence, gradually bending towards the denser dielectric region. As the wavescontinues to refract, at a certain point they achieve an angle greater than critical443.4. Antenna propertiesangle through total internal refraction to entirely change the direction of prop-agation. The phenomenon is shown inset in Fig. 3.5. The wave denoted by solidline encounters total internal refraction at point A and changes the direction ofpropagation towards the denser dielectric region. After point A, the wave trav-els from lower index region to higher index region. The two leaky-wave beams,denoted by dashed and solid lines, intersects at B which is located at x = 0 inthe positive y direction. As it cross point B, the waves tend to bend back to thehigher index region. The propagation course of the wave appears to be a sinu-soidal path. The phenomenon was simulated using COMSOL Multiphysics’ RFmodule for a maximum index nmax = 5 and a slab length of 2λ0, where λ0 is thefree space wavelength for 1 GHz. The generated elds in the simulation domainare presented in Fig. 3.6. As observed in Fig. 3.5, the graded dielectric region issymmetrical in either side of the point of excitation. Perfect electric conductors(PEC) and perfect magnetic conductors (PMC) were used as symmetry planesto reduce the simulation domain and simulate half of the graded dielectric half-space, which is why Fig. 3.6 presents the graded dielectric medium in the posi-tive x direction. In order to simulate the graded dielectric medium as innitelyextended, material loss (conductivity, σ = 0.025 S/m) was incorporated. Alter-natively, the domain could be terminated with a PML layer; however, a gradedPML layer is more complicated to simulate and does not provide perfect match-ing in COMSOL. The generated leaky-wave radiation inside the medium decayedexponentially due to incorporated material loss, as shown in the inset in Fig. 3.6.The elds produce a roughly sinusoidal envelope in its path of propagation.3.4.2 Obtaining radiation in free-spaceIn Fig. 3.5, the wavefront of the slot-generated wave is parallel to the slot-axis at point A. If the dielectric half-space is terminated at A, at the height ofHmax, the wave will be normally incident into free-space. Under this condition,broadside radiation can be achieved from the antenna. This scenario was simu-lated using COMSOL for a maximum index nmax = 5 and a slab length of 2λ0,where λ0 is the free space wavelength for 1 GHz. One-half of the structure canbe simulated around the symmetry line shown in Fig. 3.5. Using image the-ory, perfect electric conductors (PECs) and perfect magnetic conductors (PMCs)were implemented where structural symmetry was observed. Simulating half ofthe overall structure allowed half of the computational resources to be spared.Figure 3.7 represents the 3D simulation domain of the antenna characterizationprocess. The graded blue represents the rectangular slab and the solid dark line453.4. Antenna propertiesHLPMLairairPMLPMCPMC⊗ PEC-6xy⊗zFigure 3.7: Simulation domain of the antenna in the preliminary design process.The blue region represents the graded-dielectric slab. The PEC is directed into-the-page.underneath is the slot innite array. Since the yz plane was parallel to the elec-tric eld, a PMC plane was used as a symmetry plane at x = 0 to simulate thedouble sided slot-line. In contrast, PEC planes were used in the xy plane sinceit lay normal to the electric eld. This allowed an innite slot to be eectivelysimulated. The dimensions of the simulation were L = 2λ0 for a dielectric gra-dient of 6/λ0, where λ0 represented the free-space wavelength at the designedfrequency. Such gradient led to a maximum relative permittivity of 6 at x = 0and minimum relative permittivity 1 at x = L. The height H was varied to ndthe optimum performance of the antenna over a broad frequency range.Analyzing Fig. 3.6, Hmax was found to be 0.57λ0. If a graded dielectricslab with a height 0.57λ0 is placed on top of a slot array, broadside radiationis achieved. Figure 3.8a shows that the leaky-wave radiation couples into free-space to produce broadside radiation. Figure 3.8b presents the normalized far-eld plot. If the graded index dielectric half-space in Fig. 3.6 is terminated at aheight less thanH , the antenna will generates a double-sided oblique beam. Thescenario is simulated in COMSOL and presented in Fig. 3.9. Figure 3.9a showsthe elds plot for a slab height H = 0.4λ0, which is less than Hmax. The cor-responding normalized far-eld plot is shown in Fig. 3.9b. The height of the463.5. Antenna characterization and simulation results-6xy⊗zλ0(a) (b)Figure 3.8: (color inline) Field plot of the leaky-wave antenna with a graded-dielectric slab for a height H = Hmax = 0.57λ0 (a) The grid dimensions are inmeters. The produced normalized far-eld polar plot with a primary beam atbroadside (b).graded index dielectric can be varied to to nd a suitable radiation angle fromthe structure. The length of the graded index dielectric depends on the gradientprole of the graded dielectric. The dielectric slab should be long enough for theelds inside to bend and create the sinusoidal envelope. For higher gradient ofthe refractive index Hb will be lower, resulting in a thinner dielectric slab. Dif-ferent gradients of refractive index, height, and slot dimension need to varied tond out a suitable radiation pattern.3.5 Antenna characterization and simulationresultsTo achieve proper broadband performance with minimum side-lobes, the an-tenna needs to be optimized in terms of its dependent variables. Parametricsweeps were carried-out using COMSOL Multiphysics’ full-wave simulator tocharacterize the relationship between the design parameters and the antennaperformance by considering one parameter at a time. Figure 3.10 represents thenormalized radiation patterns at dierent frequencies of the antenna during thepreliminary design process. The simulation domain in Fig. 3.7 was simulated473.5. Antenna characterization and simulation results-6xy⊗zλ0(a) (b)Figure 3.9: (color inline) Normalized eld plot of the leaky-wave antenna witha graded-dielectric slab for a height H = 0.4λ0 (a) The grid dimensions are inmeters. The produced normalized far-eld polar plot with an oblique beam (b).using COMSOL Multiphysics’ full wave solver. The radiation patterns plots atsome frequencies (1.1f0, 1.2f0, 1.8f0) demonstrate inspiring results with samedirection of primary beam and lower secondary lobes. However, the overall per-formance of the antenna is evidently unsatisfactory. The antenna contains a backlobe even stronger than and equal to the main lobe at 0.8f0 (Fig. 3.10b) and 1.4f0(Fig. 3.10f). The sidelobe level at 0.6f0 (Fig. 3.10a), 0.8f0 (Fig. 3.10b, 2f0 (Fig.3.10i) are more than 50%. In addition, the broadside radiation is unexpectedlylarge at most frequencies. The presented results belong to the leaky-wave struc-ture with L = 2λ0, where λ0 represents the free-space wavelength at the designfrequency. The height was H = 0.32λ0 and the permittivity gradient was 3-per-λ0, which lead to a maximum relative permittivity of 6 (nmax = 6). Othersimulations with dierent values of H and permittivity gradient suggested un-desired performance. In order to observe the antenna performance for dierentheights of the slab, the leaky-wave antenna was simulated using COMSOL for awide range of slab-height. Figure 3.11 demonstrates the variation of peak eld,directivity, beamwidth, angle of radiation (measured from endre), sidelobe leveland broadside eld for dierent heights of the slab at a xed frequency. It is fairlyuncommon to use broadside eld strength as a parameter to dene an antennaperformance. However, as shown in Fig. 3.10, the radiation pattern of the lin-ear gradient design at some frequencies consisted of signicant level of elds in483.5. Antenna characterization and simulation results0306090120150180210240 270 300330(a) Frequency = 0.6f00306090120150180210240 270 300330(b) Frequency = 0.8f00306090120150180210240 270 300330(c) Frequency = f00306090120150180210240 270 300330(d) Frequency = 1.1f00306090120150180210240 270 300330(e) Frequency = 1.2f00306090120150180210240 270 300330(f) Frequency = 1.4f00306090120150180210240 270 300330(g) Frequency = 1.6f00306090120150180210240 270 300330(h) Frequency = 1.8f00306090120150180210240 270 300330(i) Frequency = 2f0Figure 3.10: Normalized radiation patterns at dierent frequency steps of theinitially designed leaky-wave antenna.493.6. Limitation of the designbroadside. Therefore, broadside eld strength of the linear gradient design hasbeen studied along with other antenna parameters. The slab as well as the slotlength was specied to be 2λ0 and the relative permittivity varied linearly from1 to 6 over the slab length. Reections at the air-dielectric interface is minimumwhen the dielectric slab is matched to free-space, having equal values of r andµr. Since r varied linearly along the interface, the slab was simulated with dif-ferent values of µr. In order to obtain improved matching at the air-interfaceof the slab, relative permeability (µr) was varied from 2 to 6. It was expectedthat higher values of µr would lead to increased coupling of leaky-wave intofree-space.Figure 3.11 depicts that the leaky-wave antenna performance is very sensitiveto H . It can be observed that the angle of radiation varies widely with a slightchange of H with high sidelobe levels. Around a height H=0.4λ0 for µ = 5, thesidelobe level is low (around 50%), beam is tight (around 20o), and radiation angleis stable(around 80o); however, the bandwidth is not broad enough. It is alsoobserved that larger values of µ do not provide improved antenna performance.3.6 Limitation of the designObserving the antenna performance over a broad frequency range, the fol-lowing points were observed:• The antenna did not provide a xed-beam performance over a wide fre-quency range. The radiation angle varied unpredictably with the changeof frequency, which was not expected. The eld plots in Fig. 3.8a andFig 3.9a suggest that reected waves inside the rectangular slab might beresponsible for the irregular radiation pattern from the antenna.• The side-lobe level (SLL) and back-lobe level (BLL) of the antenna was verylarge. At some frequencies, the SLL and BLL were small; however, theirvariation was too unpredictable over a broad bandwidth.• At broadside, the eld strength was too high compared to typical leaky-wave antennas.The linear gradient index design resulted in unsatisfactory antenna perfor-mance. In order to improve the antenna performance more sophisticated re-fractive index gradient was required to be incorporated into the dielectric slab.Transformation electromagnetics was implemented in the next design process503.6. Limitation of the designFigure 3.11: (color inline) Peak eld (a), Directivity (b), Beamwidth (c), Angle ofradiation (d), Sidelobe Level (e) and Broadside Field (f) of the leaky-wave struc-ture for dierent heights of the slab. Five graphs represents in each plot repre-sents various permeability (µ) values, for a relative permittivity (r) gradient of3-per-λ.513.6. Limitation of the designwhich provided improved results for the leaky-wave antenna. The followingchapter describes a geometric compensation technique that uses conformal trans-formation to improve the antenna performance. The following chapter presentsthe antenna performance that are better than ones presented in this chapter.52Chapter 4Geometric CompensationTechniqueIt is shown in the previous chapter that the idea of implementing a broadbandantenna using a linearly graded dielectric was successful in coupling the leaky-wave radiation into free-space. However, it has high sidelobe levels and the mainbeam direction is inconsistent over broad bandwidth. In the second approachto the design of the antenna, conformal transformation has been employed totransform a region with a convex upper interface to a rectangular domain. It isdemonstrated in this chapter that the phase distribution at the interface of anoptically transformed medium is not identical from the initial domain, whichleads to undesired antenna performance. This chapter is dedicated to developinga technique that can eliminate the phase-deviations to improve the antenna per-formance. The technique is used to design the xed-beam leaky-wave antennain the following chapter.4.1 Numerical conformal transformationTransformation electromagnetics allows controlling the propagation of elec-tromagnetic waves in a medium by engineering its constituent parameters. Con-formal transformation is a special category of transformation optics that reducesmaterial complexity of the resultant medium by employing isotropic inhomoge-neous graded index structures.4.1.1 Linking to the antenna designIn the introductory section 2.5, it has been demonstrated how conformaltransformation can be employed to transform a curved geometry into an elec-tromagnetically equivalent rectangular domain. A Jacobian matrix Λ relates thematerial parameters of the intial region into the transformed region as demon-strated in Eqs. 2.41d and 2.42. This thesis denotes the initial geometry as the534.1. Numerical conformal transformationABDCInitial domain (w)abcdIntermediate domain (r)A′B′D′C ′Transformed domain (t)Figure 4.1: Optical transformation from region ABCD lled with uniform di-electric material into an isotropic inhomogeneous dielectric medium A′B′C ′D′.For the validation process, the domains were excited with a point source at pointsA and A′.source domain in (x,y) coordinates and the transformed geometry as the targetdomain in (u,v) coordinate.The conformal transformation was executed numerically using COMSOL Mul-tiphysics [87]. The numerical approach is implemented by solving Laplace’sequation to dene the rules of coordinate mapping between two domains. Thedielectric permittivity of transformed medium consists of two-dimensional vari-ations. The technique is used in section 4.2.4 for developing the geometric com-pensation technique. Therefore, the numerical transformation was restricted totwo-dimensional (2D) domain.4.1.2 Validation of simulationsThe numerical transformation electromagnetics for the development of thegeometric compensation technique has been carried out using COMSOL Mul-tiphysics. Prior to applying the technique for designing the antenna, the nu-merical process was validated by performing conformal transformation betweenarbitrary 2D geometries, as demonstrated in Fig. 4.1. The geometry ABCDlled with uniform dielectric material in the initial domain was transformed intothe isotropic inhomogeneous dielectric with a geometry A′B′C ′D′ in the trans-formed domain.Conformal transformation can only take place between two media providedthat one of the media is rectangular [97]. However, the geometries in Fig. 4.1 donot fulll the criteria. Therefore, the transformation was performed through anintermediate rectangular geometry. In other words, the transformation t −→ whas to be performed through an intermediate rectangular domain r. In the rststep, the initial domain w is transformed to the intermediate domain r using544.1. Numerical conformal transformationABDC(a)A′B′D′C ′(b)Figure 4.2: (color inline) Coordinate grids during the optical transformation pro-cess of the initial and transformed domain of Fig. 4.1.transformation equation r −→ w, as presented in Fig. 4.1. The ratio of height towidth of the intermediate domain represents the conformal module of the twotransformations. In the second step, the transformed domain t is transformedto the intermediate domain r using transformation equation r −→ t. The con-formal module for the both transformations are same. The coordinate grids ofthe initial and transformed domains are presented in Fig. 4.2. The relative per-mittivity distribution of transformed domain was obtained from the solution ofthe numerical conformal transformation process. The initial and transformeddomains were then excited with a point source at the bottom-left corners (A andA′) in order to verify that the two media had identical electromagnetic behav-ior. The rules of coordinate transformation and transformation optics is validonly inside initial and transformed domains. The transformation equations donot incorporate the interaction of associated domains with external media. Inorder to ensure validation of the transformation process, it is necessary to elim-inate the interaction of the transformation regions w and t with free-space. Theboundaries were enclosed with a perfect electric conductor (PEC) to neglect theinteraction with external media.The electromagnetic waves propagates within the media and completely re-ects from the PEC boundaries, creating standing waves throughout the domain.The electric eld values inside the two media have to be identical at the equiv-alent points if the transformation is accurate. Arbitrary points in initial domainwere selected that are test points over which the elds were compared withequivalent points in the transformed domain. The test point locations in trans-formed domain that corresponded to the ones in initial domain were determinedby applying the coordinate transformation to the test points. The normalizedelectric eld values of the standing waves at the test points for both media arepresented in Fig. 4.3. The brown lines represents the normalised eld values554.2. Geometric compensation technique0 0.1 0.2 0.3 0.4 0.5 0.600. distance in the initial domain (λ0)NormalisedElectriceld(V/m) Medium 1Medium 2Figure 4.3: (color inline) Normalized absolute value of the electric eld in thestanding waves in initial domain (brown) and transformed domain (green). Thepoints in the initial domain were arbitrarily selected along the vertical coordi-nate grid .14λ0. The test locations in transformed domain corresponding to theequivalent points in initial domain were selected from transformation results.of the standing waves in initial domain while the green line shows the sameat equivalent points of transformed domain. The gure shows that the standingwaves inside the two media creates identical eld values at equivalent test points.The overlapping lines in Fig. 4.3 indicates that the elds inside the initial andtransformed domains enclosed by PEC boundaries have identical standing wavepatterns. Therefore, it can be deduced that both domains are electromagneticallyidentical.4.2 Geometric compensation technique4.2.1 PrincipleIn order to achieve a xed angle of radiation, a linear phase gradient in thetarget domain is required. This corresponds to a linear phase gradient in thesource domain. Since the transformation equations are unknown prior to per-564.2. Geometric compensation techniqueforming the source-to-target domain transformation, a perturbation approachis required to nd a source domain geometry that transforms into a target do-main having the required linear phase. The idea is demonstrated in Fig. 4.4. Theperturbation approach, named the geometric compensation technique, begins bychoosing a linear phase gradient (red line) in the source domain and obtaining ageometric path s that possesses that phase gradient. This path then denes theupper interface of the target domain (see Fig. 4.4a). Using conformal transfor-mation optics, the source domain geometry is mapped into a rectangular targetdomain. Due to coordinate stretching, the phase distribution along the upper in-terface of the target domain becomes a function of u, as shown by the red line insee Fig. 4.4b. However, a linear phase is required in the target domain to achievea xed-beam radiation. In order to x this discrepancy in linear phase, the phaseerrors are determined and transformed back into the source domain using thetransformation equations. The inverse-transformed phase prole, shown by theblue line in Fig. 4.4c, is used to achieve an altered geometry s′. Further trans-formations are performed to nd the required linear phase in the target domain(see Fig. 4.4d). The procedure is used in the following chapter to achieve broad-band xed-beam performance from a leaky-wave antenna. Before applying thetechnique in antenna design, each step of the technique is demonstrated belowusing a point source excitation.4.2.2 Linear phase gradient at the radiating interfaceThe rst step of the technique is to determine a path through the sourcedomain along which the elds have a linear phase gradient. The geometry shownin the source domain in Fig. 4.5 comprises of a curved interface along which thephase fronts interact with a linear prole. The angle of the coupled radiation isθrad is related to the phase gradient along the upper interface by the followingequation:dφds= k0 sin θrad = constant (4.1)where k0 is the free-space wave number and ds is the change in arc length of theboundary of the air interface which is a function of the space. For a straight paths, a linear phase gradient in Eq. 4.1 would produce a radiated beam at an angleof θrad. We wish to nd the path in the source domain that corresponds to thesame gradient so that it can be transformed into the rectangular target domain.574.2. Geometric compensation techniquexuxuyvyvuu0φφφφss(x; y) =) (u; v)(x; y) =) (u0; v0)(a)(b)(c)(d)s0ssactualactualobtaineddesired (x; y domain)desired (u; v domain)ff 0SourceDomainTargetDomainSourceDomainTargetDomainFigure 4.4: A diagram illustrating the process of geometric compensation com-pensation. The source domain is represented by (x,y) coordinate, while the tar-get domain is in (u,v) coordinate. The left columns presents the phase distribu-tions along the upper interface of the geometries to their right. The compensatedand uncompensated geometries are presented by red and blue lines, respectively.584.2. Geometric compensation techniqueθradθradSource domain Target domainFigure 4.5: The initial and target domain geometries excited with a point source.The initial domain consists of uniform dielectric material, while the rectangu-lar target domain geometry contains graded index materials. The dotted linesrepresent phase fronts of the radiated wave. It is expected that both media willhave identical phase prole at the upper-interface, leading to the same angle ofradiation from broadside.4.2.3 Determining the source domain geometryThe initial geometry in the source domain consists of uniform dielectric, se-lected such that the source-generated radiation meets the interface with a linearphase delay. The curvature of upper interface of the source domain geometrydetermines the radiation angle in the target domain. The shape of the sourcedomain geometry is obtained from a linear phase gradient. The change of linearphase with respect to space is equal to a constant, which is a function of radiationangle θrad. The governing equation is given by:∇φ · sˆ = dφds(4.2)where ∇φ is the phase gradient and sˆ refers to the unit vector tangent to theboundary. ∇φ is given by:∇φ = φxxˆ+ φyyˆ (4.3)where φx and φy are the derivatives of φ with respect to x and y. dφ/ds in Eq.4.2 is a constant related to the free-space wave vector (k0) and radiation angleθrad by Eq. 4.1. Equation 4.2 can be expanded as follows:|∇φ| · 1 · cos(θs∇) = dφds(4.4)594.2. Geometric compensation techniqueθsrθrθs(negative gradient)φ(x; y)xypoint sourceFigure 4.6: A diagram showing the process of choosing a geometry in the sourcedomain in (x,y) coordinate.which leads to the following equation to determine the arc of the source domaingeometry:θs∇ = arccos( |dφ/ds||∇φ|)(4.5)where cos(θs∇) is the angle that the upper boundary forms with the phase gra-dient. From Fig 4.6, the angle of the upper boundary arc (blue line) with respectto x axis is denoted as θs and represented as:θs = θ−∇ − θs∇ (4.6)where, θ−∇ is the angle of the negative gradient of phase with respect to x axisand θs∇ is the angle between the arc and negative gradient. θ−∇ can be obtainedfrom the argument of∇φ in Eq. 4.3 and expressed as:θ−∇ = pi + θ∇ (4.7a)θ∇ = arctan(φyφx)(4.7b)From Eqs. 4.6, 4.5 and 4.7, the nal expression for determining the new curve bysatisfying the following equation:θs = pi + arctan(φyφx)− arccos( |dφ/ds||∇φ|)(4.8)604.2. Geometric compensation techniqueUniform dielectric halfspaceSimulation domainPMLPMC Symmetry Plane-6xyFigure 4.7: Diagram of the simulation domain used in the primary step of the geo-metric compensation technique. A uniform dielectric halfspace has been excitedwith a point source in the center. To simulated the innite dielectric halfspace,the boundaries were eliminated with PML. Eective simulation domain was re-duced by using PMC symmetry planes. The dark grey region is the simulationdomain that was actually simulated.The upper boundary of the source domain turns out to be convex since itcorresponds to the phase of eld being radiated obliquely leaving the medium ata xed angle. Using the curvature equation in Eq. 4.8, the geometry of the sourcedomain is cropped from a uniform dielectric half-space. Figure 4.7 presents theuniform dielectric half-space which is a basis for determining the source domaingeometry. The source of uniform dielectric half space is identical to point sourceof the desired source domain geometry. The idea is to ensure that the source do-main geometry (in Fig. 4.4) and uniform dielectric half space (in Fig. 4.7) possessidentical phase distributions since the former is a portion of the later. The pointsource is allowed to radiate into a uniform dielectric halfspace, forming a phasedistribution in the medium. Figure 4.8a shows the selection of the geometry fora radiation of 30◦ in the source domain. Corresponding phase delay along the614.2. Geometric compensation technique(a)0 1 2 3−25−20−15Arc Length (λ0)Unwrappedphase(degrees)(b)Figure 4.8: (color inline) Determination of initial geometry with a point sourcethat leads to 30◦ of radiation in the target domain. The surface plot in (a) repre-sents the unwrapped phase (degrees) of the transverse electric eld. The pointsource generating the elds is located at coordinate point (0, 0). The blue curverepresents the desired upper interface. The interface is determined by selectingthe points that corresponds to the linear phase prole (degrees) in (b).upper interface is presented in Fig. 4.8b.)4.2.4 Optically transform the source domain into targetdomainOnce the geometry in the source domain is obtained, the optical transforma-tion is applied. Consider the transformation f between the source domain in Fig.4.9a and the target domain in Fig. 4.9b. The convex uniform dielectric mediumand the rectangular inhomogeneous superstrate have a geometry in source do-main (Fig. 4.9a) in (x,y) coordinate and target domain (Fig. 4.9b) in (u,v) coor-dinate, respectively. The source domain ABCD belong to semi-innite homo-geneous dielectric half-space with a relative permittivity r, as described in Fig.4.7. The geometry of the target domain abcd contains graded dielectric material.A point source is placed at the bottom-left corner C in the source domain and cin the target domain. Since the geometry ABCD is lled with homogeneous di-electric, the point-source generates radiation that emits into free-space through624.2. Geometric compensation techniqueABC D-6xy(a)a bc d-6uv-ff L6?H ′(b)Figure 4.9: Conformal mapping from initial homogeneous source domainABCDin (x,y) coordinate (a) to a rectangular graded dielectric target domainA′B′C ′D′(u,v) coordinate(b).the curved upper boundary AB.The geometry in the target domain is described by the following equation:0 ≤u ≤ L (4.9a)0 ≤v ≤ LM= H (4.9b)where L represents the length of the rectangular domain in (u,v) coordinate andM is dened as the conformal module [30].Formal denition of the conformalmodule can be found in [98]. For the scope of this thesis it is sucient to considerthe conformal module as the aspect ration of a rectangular domain, dened bythe following equation [99]:Mrectangle = max{heightwidth,widthheight}(4.10)In order to preserve isotropy in the transformed domain in the w region, localorthogonality in the coordinate grids needs to be satised for the target domaincoordinate system (u,v). This condition can be ensured only if the target do-main shares the same conformal module as the source domain [43]. For confor-mal transformation, it is possible to construct a mapped region if M is known apriori. However, nding the conformal module of an arbitrary geometry is notstraightforward. Equation 4.10 suggests that conformal module M is dened bythe ratio of length to width of the rectangular target medium. For the sourcedomain that consists a convex geometry, M is calculated numerically after ob-taining the transformation equations for the source domain for unity conformalmodule (M = 1) [100, 101, 49, 102]. According to Cauchy-Reinmann condition634.3. Phase-discrepancies in the transformation processin Eq. 2.30, the ratio of ∂v/∂y and ∂u/∂xmust be 1 to ensure conformal transfor-mation. Since the initial transformation was carried out for M = 1, the ratio of∂v/∂y and ∂u/∂x must provide the actual conformal module M that is requiredfor conformal transformation. For the given source domain in Fig. 4.9, M wasfound to be 2.11. Then conformal transformation was carried out with a combi-nation of Dirichlet and Neumann boundary conditions in the source domain (seeSec. 2.5). Dirichlet boundary condition transforms the curved boundaries intoedges of the rectangle, while Neumann boundary condition enforces orthogonal-ity at the grid nodes [43]. Laplace’s equations for x(u, v) and y(u, v) are solvedfor the source domain using the following set of Dirichlet-Neumann boundaryconditions:u|bd = L u|ac = 0 nˆ · ∇u|ab,cd = 0 (4.11a)v|ab = L/M v|cd = 0 nˆ · ∇v|bd,ac = 0 (4.11b)Equation 2.34 provides the boundary condition for an isotropic transformed do-main:∂x∂u∂x∂v+∂y∂u∂y∂v= 0 (4.12)The relative permittivity of the target domain can be obtained from Eq. Phase-discrepancies in the transformationprocessWhen the source domain is projected to the target domain, the coordinatesystem is subjected to compression and elongation. The arc length of the upperboundary AB in the source domain undergoes change as it maps into ab in thetarget domain. Therefore, the linear phase gradient along AB no longer trans-forms to a linear phase gradient along ab. Due to the non-linear phase-prolealong the interface ab, the elds generated from the graded index superstrate hasundesired direction of the main beam, higher sidelobe level and lower directivityof the far eld radiation pattern. The red line in Fig. 4.10 represents phase pro-le along the boundary ab when the designed superstrate contains a slot arraygenerating leaky-wave radiation at the boundary cd. The phase prole along abis clearly not linear. This phase prole was determined through full-wave sim-ulations of the graded index target domain using COMSOL Multiphysics [103].A perfectly matched layer was placed along the radiating boundary to eliminate644.4. Compensating the geometry0 1 2 3−8−6−4−20Arc length (λ0)Unwrappedphase(radians)Achieved phaseIdeal linear phaseFigure 4.10: Unwrapped phase proles along the upper boundary of the targetdomain.reections due to mismatch along the air-dielectric interface [104]. Reectionsfrom the air-dielectric interface would create interference inside the slab, addingto the existing discrepancies in the linear phase.4.4 Compensating the geometryIn order to achieve an improved radiation pattern from the leaky-wave an-tenna, the upper interface of the inhomogeneous rectangular target domain musthave a more linear phase-prole. As mentioned in the previous section, the dis-crepancies in the linear phase arise from the coordinate stretching during thetransformation process. Improved linearity in phase prole along ab can beachieved by compensating the phase-discrepancies prior to the coordinate trans-formation. A modied geometry in the initial source domain geometry contain-ing a uniform dielectric with r = 2 is determined which would translate to alinear phase-prole in the target domain. Section 4.2.3 illustrates the method ofselecting the arc of the a geometry from the phase gradient. The blue line in Fig.4.6 is the arc that is to be determined. The dashed line represents the negativegradient of phase of the elds emitted from the point source. Equation 4.6 pro-654.4. Compensating the geometryA′B′C ′ D′-6xy(a)a′ b′c′ d′-6u′v′-ff L6?H ′(b)Figure 4.11: Conformal mapping of the compensated homogeneous source do-main geometry (a) to rectangular graded dielectric target domain (b).vides the angle of the arc with respect to the x axis and Eq. 4.7 expresses theangle of phase gradient in terms of the phase components. θs∇ can be written as:θs∇ = arccos( |dφ/ds||∇φ|)(4.13)Therefore, the compensated curve is calculated by ensuring the followingequation is satised:θs = pi + arctan(φyφx)− arccos( |dφ/ds||∇φ|)(4.14)Equation 4.14 for determining the compensated geometry is identical to Eq.4.8that was used to determine the initial arc. However, both equations follow dif-ferent phase gradients to determine corresponding arcs. The reformed geometryA′B′C ′D′ is illustrated in Fig. 4.11a. The curved upper interface A′B′ is de-termined by transforming the phase-errors (in Fig. 4.10) along ab back to thesource domain using inverse transformation, providing a new target phase gra-dient dφ(s)/ds along the modied arc. The new curve A′B′ is then calculatedbased on the expression 4.14. The geometrically-compensated source domainA′B′C ′D′ (Fig. 4.11a) described by the (x,y) coordinates is once again mappedinto the rectangle abcd in the target domain (Fig. 4.11b) described by the (u′,v′)coordinates using new equations of transformation. The mapping is representedby f ′. The target domain geometry a′b′c′d′ has a length L = 3λ and a heightH ′ = 1.4λ and contains inhomogeneous dielectric material, dened by Eq. 2.41.The relative permittivity distribution inside the geometrically compensated tar-get domain is shown in Fig. 4.12.664.4. Compensating the geometryFigure 4.12: Permittivity distribution inside the geometrically compensated su-perstate in the target domain.-6u(λ0)v(λ0)(a) (b)Figure 4.13: (color inline) The compensated slab excited with a point source at acorner (a). The generated elds are being absorbed into an inhomogeneous PMLwhich has a relative permittivity distribution as shown in (b).The elds in the target domain in Fig. 4.11c were re-simulated with a pointsource along the corner c′. The generated elds are shown in Fig. 4.13a. In-homogeneous PML layers were placed at the boundaries of the target domainto eliminate the reected waves from the interface. In order to reduce numeri-cal errors, the direction of relative permittivity distribution of the PML was setaligned with the direction of radiation (see Fig. 4.13b). Figure 4.14 demonstratesthe phases along the upper dielectric-PML interface of the uncompensated andgeometrically compensated target domain. The PML is placed to absorb the eldsand eliminate reections from mismatch. It can be observed how the phase pro-674.5. Remarks0 1 2 3−8−6−4−20Arc length (λ0)Unwrappedphase(radians)Initial designCompensated geometryIdeal linear phaseFigure 4.14: Unwrapped phase proles along the upper boundaries of the initialand compensated geometry.le associated with the geometrically-compensated superstrate is more linear ascompared to the initial design.4.5 RemarksIt can be noted that the phase along the upper interface of the geometricallycompensated target domain in Fig. 4.14 is not exactly linear. This is because theoptical transformations f and f ′ in Fig. 4.4 are not identical. The mapping f isused to determine the phase-discrepancies which are used to compensated theinitial geometry in the source domain. On the other hand, f ′ is used to trans-form the geometrically compensated source domain in the target domain. Thetransformation equations of f and f ′ are certainly not identical because theytransform two dierent source domains into target domains. Therefore, the ge-ometric compensation technique only applies to optical transformations wherethe phase-deviation is minor and one transformation can be approximated by theother. The applied compensation contributes to achieving a desired direction ofradiation. Figure 4.15 demonstrates the improvement in radiation pattern afterapplying the geometric compensation technique. The radiation pattern (red line)684.6. Summary0306090Initial designCompensated designFigure 4.15: (color inline) Quarter radiation patterns (linear) are plotted for boththe initial domain geometry and the phase-compensated domain geometry. Theslab is designed for an oblique radiation of 30 degrees from broadside.generated from the initial geometry is o from the designed 30o from broadside.The compensated radiation pattern (dashed red line) produces a radiation thatis directed closed to the designed 30o with lower sidelobes and more directivebeams. Also, the sidelobe levels of the compensated design is lower than that ofthe initial design.4.6 SummaryConformal transformation can be used to design the xed-beam leaky-waveantenna. A transformed domain is electromagnetically equivalent to the initialdomain due to laws of coordinate transformation. However, the laws of transfor-mation only applies within the source and target media. Their interaction withthe external medium is not included in the transformation process. The trans-formed domain interacts dierently with the external medium as compared tothe initial domain due to their dierence in dielectric permittivity. A geometriccompensation technique was presented in this chapter that attempts to eliminatethe dissimilarity. The approach is based on achieving a target phase prole atthe interface with the interface of the transformed medium. It has been demon-strated that by compensating the initial domain geometry in the transformation694.6. Summaryprocess, a target phase prole can be achieved which eventually contributes toachieving the desired performance from the transformed domain. The overallgeometric compensation technique is summarized in the ow chart 4.16.704.6. SummaryGenerate radiationin a homogeneousdielectric halfspaceSelect a geometry out ofthe phase distribution inthe (x, y) domain whichhas a linear phase delayalong the radiating-interfaceUsing quasi-conformaltransformation equa-tions to nd the phase-discrepancies along theradiating-interface in therectangular (u, v) domainDetermine the newphase prole from thephase-discrepanciesin the (x, y) domainUsing inverse quasi-conformal transformationequations, modify the initialgeometry in the (x, y)domain to compensate forthe phase-discrepanciesin the (u, v) domainTransform the compen-sated geometry into arectangular geometryin the (u, v) domainFigure 4.16: Algorithm of geometric compensation technique.71Chapter 5Broad Band Fixed-beam LeakyWave AntennaThe leaky-wave antenna presented in this thesis consists of two components:a slot-line array to generate the leaky-wave radiation and a rectangular gradeddielectric superstrate placed on top of it. The superstate receives the generatedleaky-wave radiation from the underlying slot array and couples it to free-space.The inhomogeneous distribution of permittivity inside the superstrate guides thewave to produce refraction at the upper interface at a specic angle. The geo-metric compensation technique presented in Sec. 4.4 is employed in this chapterto design the rectangular slab. The chapter begins by describing the context ofthe xed-beam radiation for the slot-array and proceeds to describe the designprocedure. It is demonstrated that the geometric compensation technique decid-edly improves the radiation characteristics of the antenna. A few variations ofthe proposed design are also presented in the nal section of this chapter.5.1 Design principleThe design of the proposed leaky-wave antenna is inuenced by the anal-ysis of an innitely long and optically narrow slot-line located at the interfacebetween two dierent dielectric media [24][25]. The radiation mechanism fromsuch slot-lines is demonstrated in section 2.4.1. Figure 5.1a is a similar illustra-tion, showing a slot aperture extending in the positive x direction and etchedon a metallic ground plane in the xy plane. A dielectric slab with a dielectricpermittivity r is located on top of the slot-line while the region underneath isfree-space. If the slot aperture is transversely excited with a current element,a leaky-wave mode will propagate longitudinally in the direction of x axis andcontinuously generate leaky-wave radiation into the dielectric slab. However,the generated leaky-wave radiation remains conned inside the superstrate dueto total internal reection at the air-dielectric interface, as illustrated by the ray-diagram in Fig. 5.1a. The incident angle (measured from broadside) of the leaky-725.1. Design principleUniform dielectric(a)Uniformdielectricθrad(b)θradGraded indexdielectric-63xyz-63HxHyEz(c)Figure 5.1: The slot emitted leaky-wave radiation from a corner-fed slot-lineleaky-wave antenna array remains trapped inside the uniform dielectric super-strate (a). The phase-mismatch at the interface is reduced at a uniform dielectricconvex layer which couples the generated radiation (b). The same phenomenoncan be implemented with a rectangular graded-dielectric layer (c).wave at the upper interface of the uniform dielectric slab needs to be reducedin order to be coupled into free-space. This could be achieved by replacing therectangular slab with a convex dielectric superstrate to couple the radiation intofree-space through its curved upper interface, as shown in Fig. 5.1b. The leaky-wave antenna reported in [26] functioned with similar conguration. Here weseek to reduce the prole of the antenna and enable radiation in directions otherthan broadside.Using transformation electromagnetics, the convex dielectric superstrate shownin Fig. 5.1b is reshaped into an inhomogeneous rectangular one as shown Fig.5.1c. In contrast to the curved upper interface of the uniform dielectric mediumin Fig. 5.1b, the air-interface of the rectangular superstrate in Fig. 5.1c is paral-lel to the slot-line. The inhomogeneous superstrate mimics the uniform dielec-735.2. Design considerationstric hemisphere in Fig. 5.1a and bends the slot-generated radiation to reduce itsinident angle at the interface. The phase-prole of the wave along the upperinterface of the superstrate determines the radiation angle of the leaky-wave an-tenna. The design process of the antenna involves obtaining the dimensions andpermittivity distribution of the superstrate through transformation electromag-netics.5.2 Design considerationsThe leaky-wave antenna presented in this thesis consists of a slot-line ar-ray with a rectangular superstrate placed on top of it, as demonstrated in Fig.5.1c. The graded index dielectric superstate receives the generated leaky waveradiation from underlying the slot array and couple into free-space. The designof the antenna was conducted with an innite slot array. The superstrate wasconsequently considered innitely stretched in the direction transverse to theslot length, which aided saving computational resources by reducing the overallsimulation domain.The essential parameters considered for designing the leaky-wave antennaare:• The radiation angle: The xed-beam leaky-wave antenna proposed byNeto et al. was limited to broadside radiation [26][27]. The primary de-sign criterion for the leaky-wave antenna is to produce a main beam at anoblique direction. The design procedure and simulation results presentedin this chapter aims for a xed radiation angle θrad = 30◦ from broadside.• Constituent material: The superstrate of the leaky-wave antenna consistsof non-magnetic graded dielectric index material. The permittivity () ofthe slab is spatially distributed with unity permeability (µ) value. The per-mittivity variation of the slab is two-dimensional with no variation in thedirection transverse (the z direction) to the slot-line. The superstrate wasdesigned such that the value of relative permittivity ranges from 1 to 9. Amaximum relative permittivity value of 9 was specically chosen to limitthe r values to those of commonly available dielectrics. A maximum valueof relative permittivity of 9 limits the refractive index value to 3. The min-imum relative permittivity value was chosen to be unity because a relativepermittivity below one would require articial dielectrics which are eithernarrowband or require active elements. For instance, the antenna proposed745.2. Design considerationsby Sievenpiper consists of a dielectric substrate with relative permittivitybelow unity [28]. However, the design utilizes active non-foster circuits toachieve xed-beam radiation from the antenna. The design process of theantenna presented in this thesis was limited to passive devices only.• Length of the antenna: In order to keep the antenna structure from be-ing bulky, the length of the superstrate as well as the slot array was ini-tially specied to be L = 6λ0, where λ0 is the free-space wavelength. Anantenna of such dimension is electromagnetically large for general appli-cations. However, leaky-wave antennas are typically electromagneticallymuch long, typically around 10-12 wavelengths. It will be shown later thatthe desired antenna performance can be achieved from a lengthL = 4.2λ0,which is smaller than previous designs proposed by Neto et al. and Sieven-piper.• Frequency bandwidth: The objective was to achieve a percentage band-width over 100%. While operating over a broad band, the antenna as well asthe slot array becomes electrically smaller with the decrease of frequency.At lower frequencies, the leaky wave mode in the slot fails to radiate suf-cient energy due to reduced length for propagation. This introduces re-ections inside the slot that creates longitudinal reections and interfer-ence inside the slot. As a result, the far-eld radiation at lower frequenciescontains higher sidelobe and wider beam at lower frequencies, as will beshown in later in section 5.4. One way to get around this problem is toterminate the slot with a matched load so that the reected waves do notinterfere with the leaky-wave mode. In contrast to operation at lower fre-quencies, at higher frequencies, the antenna demonstrates improved per-formance in terms of sidelobe level and directivity because the antennabecomes electrically longer. We consider 2f0 for the upper frequency limitof the antenna, which leads to a length of 12λ at the maximum frequency.The nal design was simulated to have a fractional bandwidth of 5:1 wherethe sidelobe level remained below 30% and the directivity was above 11.Hence the primary design criterion for the leaky-wave antenna was to pro-duce a main beam at a xed angle θrad = 30◦ over a wide-frequency range from0.5f0 to 2f0. The designed antenna structure as well as the dimensions obtainedthrough transformation electromagnetics is presented in Fig. 5.2. The slot widthof the innite slot array was specied to be 0.02λ0 with an array spacing of755.3. Design of the superstrate2.9λ04.2λ00.1λ00.02λ0vuzFigure 5.2: The structure and physical dimensions of the simulated antenna. Alllengths are represented in terms of free-space wavelength λ0 where frequencyis c. The yellow substrate represents an inhomogeneous dielectric slab.0.1λ0. As will be shown, the antenna length and height was 4.2λ0 and 2.9λ0,respectively.5.3 Design of the superstrateThe rectangular superstrate of the leaky-wave antenna is designed usingtransformation optics along with the geometric compensation technique demon-strated in section 4.4. According to the technique, the rectangular shape of thesuperstrate is considered in the target domain. The graded permittivity distribu-tion inside the superstrate is achieved through a two-step numerical conformaltransformation process.In the rst step, an initial hypothetical geometry in the source domain isobtained through the technique shown in 4.2.3. Using Eq. 4.2, a linear phasegradient is obtained which is a function of the radiation angle θrad from broadsidein the target domain. The achieved phase gradient is used to crop a shape outof a uniform dielectric half-space having a relative permittivity r = 2 is placedon top of a thin slot array. The scenario is simulated in COMSOL multiphysics.765.3. Design of the superstrateL=6λairdielectricPECExcitationff67yzx(a)Ldielectricair(b)Figure 5.3: Simulation setup for the analysis of the proposed leaky-wave antenna.Half of a unit cell is simulated with the PEC boundary conditions that extendingthe structure innitely in the z direction. The dashed region represents PMLabsorbing layer while the solid region is the actual simulation domain.Figure 5.3 presents the simulation setup. An innite array of slot-line etchedon a ground plane placed at the interface of air and a dielectric was simulatedusing PEC/PMC symmetry planes and PML layers to mimic an innite arraylocated at the interface of two innitely stretched half spaces. The solid linesrepresent the actual region of consideration while the dotted lines show the PMLabsorbing layers. They grey surface portrays the ground plane that holds a slot-line extended in the positive ±x direction. The slot-line is excited at the cornerwith a current source transverse to the length. The ground plane and the slotextends into the PML layer in the positive ±x direction. The PML layer absorbsthe propagating leaky-wave mode in the slot-line and prevents reections fromboundary [104]. Standing waves originated from the reected elds would cause775.3. Design of the superstrate-6xy⊗zy(λ0)x(λ0)λ0(a) (b)(c)Figure 5.4: (color inline) The elds produced in the structure demonstrated inFig. 5.3 (a). The unwrapped phase (degrees) of the elds are plotted and usedto determine the source domain geometry for the transformation optics procees.The blue lines correspond to the upper interface that corresponds to the phaseprole being linear (b). The linear phase prole along the upper interface (c).error in determination of the source domain geometry. In the negative x directionthere is an air gap between the PML boundary and round plane. The PML layeron the left of air gap innitely extends the free-space region in in the negative xdirection, allowing simulation of a corner-fed leaky-wave antenna. The groundplane as well as the slot perfect electric conductor (PEC) boundaries repeat thedomain in the ±z direction, extending the structure into an innite array [51].785.3. Design of the superstratezyxCut-line(a)ABC D-6xy(b)a bc d-6uv-ff L6?H(c)Figure 5.5: The initial geometry (a) that leads to a linear phase prole in thesource domain. The conformal mapping from initial homogeneous source do-main (b) to initial graded dielectric target domain (c).The transverse current source excitation of the slot-line generates a leaky-wave mode propagate in x direction and radiates primarily into the upper di-electric half-space. Generated waves form a phase distribution in the dielectrichalf-space. The elds originated from the structure and the corresponding un-wrapped phase are presented in Fig. 5.4a and 5.4b. The source domain geometryfor the geometric compensation technique is selected using linear variation ofphase with space. The electric eld transverse to the length of the slot is con-sidered to nd the phase gradient from the dielectric half-space. The blue linesin Fig. 5.4b represent the upper interface that is selected using the linear phaseprole in Fig. 5.4c. The upper interface of the source domain geometry designedwith the intention of creating a linear phase prole, corresponding to a xedradiation θrad from broadside. Figure 5.5a presents the geometry in the sourcedomain which is determined for a radiation angle of θrad = 30◦. The overall de-sign is carried out for a two-dimensional conformal transformation. Therefore,region under the upper interface (blue line) can be innitely stretched in the ±z795.3. Design of the superstratedirection to nd the domain in Fig. 5.5a. The geometry with an innite slot arrayat the bottom interface is presented in Fig. 5.5a. The linear phase gradient alongthe upper interface of the geometry can be observed along a cut-line in the upperinterface.For the two-dimensional conformal transformation, the geometries in gures5.5b and 5.5c are considered. The shape ABCD in 5.5b is a portion of the uni-form dielectric in the xy plane along the cut-line indicating in Fig. 5.5a. Usingthe description in section 4.2.3, the geometryABCD is numerically mapped intoa rectangle abcd in (u, v) domain presented in Fig. 5.5c. The initial curved upperboundary is generated by setting the phase gradient along the curve equal to aconstant. The constant is a function of the radiation angle from ab in the targetdomain. The length of the target domain was specied to be L = 6λ0. Its heightH , limited by the conformal module was 2.9λ0. The permittivity distribution in-side the rectangular medium is obtained from the transformation process. If theinhomogeneous rectangular superstrate is placed on top of the slot array used inthe initial transformation process, generated leaky-wave radiation couples intofree-space. The radiation angle depends on the phase-gradient along the upperinterface ab. Due to the change of length of the upper boundary (AB to ab),the phase gradient along ab deviates from the expected linear prole. Figure5.7 presents the phase gradient along the radiating interface ab in the target do-main, which is clearly non-linear with repect to an ideal linear phase-prole. Thenon-linearity in phase prole along the interface leads unsatisfactory radiationpatterns where the primary lobe deviates from the desired 30◦ radiation angleand radiated power is signicantly distributed in the side and back lobes. Thelinear radiation patterns from the structure are shown in Fig. 5.6 for a frequencyrange of .5f0 to 2f0. The plot shows that although the the structure has been de-signed from a source domain for 30◦ radiation, the primary beam deviates fromthe angle over the frequency range.In order to achieve a solid xed-beam performance, the phase gradient alongthe upper interface of the superstrate needs to be precisely linear. In order to xthe discrepancies in the phase gradient, conformal transformation is required tobe carried out. Using Eq. 4.14, a modied source domain is selected that leadsto a dierent target domain medium. As described in the previous chapter, themodied geometry is selected from the unwrapped phase of the elds radiatedinto a uniform dielectric half-space. The red line in Fig. 5.8a presents upperinterface of the compensated geometry, while the blue line is the initially selectedinterface (Fig. 5.4).The modied geometry with the slot array is presented in Fig. 5.9a. A slice of805.3. Design of the superstrate0306090120150180210240270300330f = 0.4f0f = 0.6f0f = 0.8f0f = 1.0f0f = 1.2f0f = 1.4f0Figure 5.6: (color inline) Radiation patterns (linear) of the uncompensated leaky-wave antenna at dierent frequencies. The six radiation patterns are for 0.5f0-2f0, where f0 = c. Higher gain patters represents higher frequencies.the modied source domain along a cut-line is presented by geometryA′B′C ′D′in Fig. 5.9b. The shape is mapped to the rectangular geometry a′b′c′d′ in Fig. 5.9cthrough a two-dimensional conformal transformation. The geometry a′b′c′d′ iseventually used as the superstrate of the leaky-wave antenna superstrate. Thedomain has the same length L as compared to the uncompensated one, however,has a height H ′ = 2.9λ0.The blue curve in Fig. 5.8a demonstrates how the phase prole along theupper boundary ab is more linear as compared to the initial design. The plot il-lustrates how the linearity in phase-prole along the radiating interface improvesafter applying the geometric compensation technique. As will be shown in sec-tion 5.4, the geometrically compensated target domain a′b′c′d′ demonstrates im-proved antenna perforamnce as cormpared to the initially designed medium abcd.It is observed that the relative permittivity inside the superstrate varies from0.0790 to 8.8748, as shown in the countour plot of Fig. 5.10. Naturally occurringdielectric materials generally have a refractive index greater than unity. Mediahaving a relative permittivity bellow unity needs to be implemented using reso-815.4. Antenna performance0 1 2 3 4 5 6 7−20−100Arc length (λ0)Unwrappedphase(degrees)Initial designIdeal linear phaseFigure 5.7: Unwrapped phase proles along the upper boundary of the initiallytransformed geometry. The dashed blue line presents the ideal linear phase gra-dient required.nant materials which are narrowband. In order to avoid resonant materials, theregion with relative permittivity below 1 were approximated to be 1. The regionwith unity relative permittivity was distributed along the last end of the super-strate, as illustrated in Fig. 5.10. This eectively trimmed the physical length ofthe superstrate from L = 6λ0 to L = 4.2λ0, causing an drop in directivity andincrease in sidelobe levels.The elds in the target domain a′b′c′d′ in Fig. 5.9c are re-simulated in thenext section with a slot array placed at the bottom interface c′d′. The array isspecied by a slot spacing of λ0/10, slot width of λ0/50, and slot length of 6λ0,where λ0 is the free-space wavelength.5.4 Antenna performanceThe leaky-wave antenna is designed as described in section 5.3 and simulatedusing COMSOL Multiphysics. The simulated antenna consisted of a corner-fedinnite slot array with a graded index dielectric superstrate on top, similar to thestructure shown in Fig. 5.1c. The rectangular superstrate is designed throughtransformation optics to linearize the phase along the air-dielectric interface.825.4. Antenna performancey(λ0)x(λ0)(a)0 1 2 3 4 5 6 7−20−100Arc length (λ0)Unwrappedphase(degrees) Initial designCompensated geometryIdeal linear phase(b)Figure 5.8: (color inline) Determining the compensated geometry from the sur-face plot of phase (degrees) corresponding to the elds in an uniform dielectrichalf-space(a). Unwrapped phase proles along the upper boundaries of the initialand compensated geometry (b).5.4.1 Fixed-beam broad band performanceFig. 5.11 depicts how the elds bend inside the graded dielectric index sub-strate and subsequently couple into free-space. Interference of elds is visibleinside the superstrate which is originated from the reection at the air-dielectricinterface. The oblique radiation of the leaky-wave is visible from the coupledwaves. Figure 5.12 illustrates linear radiation patterns of the antenna over broadbandwidth. The six curves represent the radiation patterns from .5f0 to 2f0(f0 = c). The plot shows that the structure generates a primary beam directedapproximately at the same angle, at 30◦ from broadside in this case.5.4.2 Improvement of radiation pattern using geometriccompensation techniqueTo compare the performance of the antenna at dierent frequencies, nor-malized radiation patterns at dierent frequencies need to be compared. Nor-malized radiation patterns at dierent frequencies comparing the initial designand the compensated design is presented in Fig. 5.13. The blue and red curves835.4. Antenna performancewvuCut-line(a)A′B′C ′ D′-6xy(b)a′ b′c′ d′-6uv-ff L6?H ′(c)Figure 5.9: The compensated geometry (a) that leads to a linear phase prolein the target domain. The conformal mapping from geometrically-compensatedhomogeneous source domain (b) to geometrically compensated graded dielectrictarget domain (c).Figure 5.10: Permittivity distribution inside the geometrically compensated su-perstate in the target domain.presents the far-eld radiation pattern for the geometrically compensated andinitial design, respectively. The normalized radiation patterns illustrates howthe performance of the antenna increases with the increase of frequency. At845.4. Antenna performancev(λ0)u(λ0)Figure 5.11: (color inline) Simulated electric eld transverse to the slot. The slotgenerated leaky-wave radiation bends towards the higher permittivity regionand couples into free-space.higher frequencies, the electrical antenna structure is long enough to allow theleaky-wave mode in the slot to decay. However, the antenna structure becomeselectrically smaller at lower frequencies. The propagating leaky-wave mode isstrong enough at the end of the structure is strong enough to cause signicantstanding wave components in the slot. Therefore the antenna performance de-grades at lower frequencies. Regardless of higher sidelobe and backlobe levels atlower frequencies, the primary beam remains at the designed angle (θrad = 30◦from broadside).Fig. 5.14 presents the parameters of the simulated leaky-wave antenna withthe geometrically compensated superstrate and the initial design. Simulated ra-diation angle, 3dB beamwidth, percentage sidelobe level, percentage backlobelevel and directivity of the antenna at dierent frequencies for both initial andgeometrically compensated design. As illustrated in Fig. 5.14a, the radiation an-gle for the compensated superstrate remain consistent as compared to the initialdesign, uctuating around the designed 30 degrees over the desired bandwidth.According to in Fig. 5.14b, 3dB beamwidth for the compensated design is rela-tively less than the initial design and follows a downward trend with the increaseof frequency. The sidelobe level (Fig. 5.14c) and backlobe level (Fig. 5.14d) graphsshow the percentage sidelobe and backlobe level for the compensated design is855.4. Antenna performance0306090120150180210240270300330f = 0.4f0f = 0.6f0f = 0.8f0f = 1.0f0f = 1.2f0f = 1.4f0Figure 5.12: (color inline) Radiation patterns (linear) of the compensated leaky-wave antenna at dierent frequencies. The six radiation patterns are for 0.5f0-2f0, where f0 = c. Higher gain patters represents higher frequencies.reasonably less than that for the initial design throughout the frequency range.At the lower cut-o frequency the simulated sidelobe level and backlobe level are37.3% and 79.8%, respectively for the compensated superstrate while results forthe same antenna parameters initial design are 46.4% and 91.6%, respectively.The percentage sidelobe level and backlobe level marginally decrease to 30% and46%for the compensated superstrate, being lower than the initial design over thefrequency range. As for the directivity of the leaky-wave antenna in Fig. 5.14e,it shows an upward trend with the increase of frequency for both designs. Thecompensated design generates more directive beam over the range of frequency,evaluated to be 4.6 and 15 at the lower and upper cut-o frequencies, respec-tively.Real and imaginary part of the input impedance has been calculated usingEq. 2.5. The impedance as a function of frequency for the simulated leaky-waveantenna is presented in Fig. 5.15a. It is observed that the input impedance isfairly consistent over the desired range of frequency, which indicates that theantenna can be fairly matched to an optimized network over the entire operatingbandwidth. Figure 5.15b presents the return loss of the structure when matched865.4. Antenna performance0306090120150180210240 270 300330(a) Frequency = 0.5f00306090120150180210240 270 300330(b) Frequency = 0.8f00306090120150180210240 270 300330(c) Frequency = 1.2f00306090120150180210240 270 300330(d) Frequency = 1.4f00306090120150180210240 270 300330(e) Frequency = 1.6f00306090120150180210240 270 300330(f) Frequency = 2f0Figure 5.13: (color inline) Normalized radiation patterns for the initial (red) andcompensated (blue) leaky-wave antenna at dierent frequencies.875.4. Antenna performance0.65f 0 f 01.35f 01.65f 0 2f020304050θ rad(deg)(a)0.65f 0 f 01.35f 01.65f 0 2f01020303dBBW(deg)(b)0.65f 0 f 01.35f 01.65f 0 2f020406080SLL(%)(c)0.65f 0 f 01.35f 01.65f 0 2f0406080100BLL(%)(d)0.65f 0 f 01.35f 01.65f 0 2f001020FrequencyDirectivity(linear)(e)Compensated: Corner-fedUncompensated: Corner-fedFigure 5.14: (color inline) (a) Radiated beam angle, (b) 3dB beam width, (c) side-lobe level, (d) backlobe level and (e) directivity of the leaky-wave antenna for thegeometrically compensated superstrate (blue) and initially designed superstrate(red). The dashed black line in (a) represents the average beam angle of 33o.885.5. Center-fed design with a back reector0.65 1 1.35 1.65 250100150Frequency (f0)Inputimpedance(Ohms) Real(Zin)Imag(Zin)(a)0.65 1 1.35 1.65 2−30−20−100Frequency (f0)ReturnLoss(dB)(b)Figure 5.15: (color inline) Real (brown) and imaginary (green) input impedanceof the leaky-wave antenna with the geometrically-compensated superstrate (a).Return loss (dB) matched to the average input impedance (Z0=69.367 + 73.64jOhms)(b).to the average value of the input impedance (Z0=69.367 + 73.64j Ohmss).5.5 Center-fed design with a back reectorFor certain applications, a xed-beam antenna may be required to producetwo beams instead of a single beam. The corner-fed design explained in pre-vious section can be extended into a center-fed leaky-wave antenna. It will belater shown that at some frequencies, the backlobe level of the antenna is high.A back reector was incorporated can the structure to eliminate the backlobes;however, the eect of the back reector on primary beam direction, the sidelobesand directivity is unknown.In order to observe the variation of the leaky-wave radiation in presence of aback reector, a slot-line was placed under a uniform dielectric half-space with aPEC back reector. The simulation domain is presented in Fig. 5.16. The slot-lineis positioned under a dielectric half-space (grey region). PEC and PMC symmetryplaces were implemented to simulated the structural periodicity of the center-fed design. The distance of the back reector was varied from 0.5λ0 to 1.5λ0to observe the radiation inside the dielectric. The peak eld, angle of radiation(measure from broadside), percentage sidelobe level and beamwidth is observedand presented in Fig. 5.17. Two dierent dielectric half spaces, having relativepermittivity values of 2 and 4, were simulated for each position of back-reector.895.5. Center-fed design with a back reectorUniform dielectricLine currentGround plane (PEC)PMCPECPMLPMLPMLxyzFigure 5.16: Simulation domain of the center-fed leaky-wave antenna with aback-reector.It is observed that the position of the back reector less than 0.5λ contains zeroside lobes and contains decent directivity. Therefore, the back reector is to beplaced in that particular region.The center-fed design was simulated using COMSOL with and without aback reector. The graded-dielectric superstrate was design using transforma-tion optics and improved by the geometric compensation technique. Figure 5.18presents the dielectric distribution of the superstrate, designed using the geo-metric compensation technique. To simulated the antenna as a center-fed designwith two primary beams, is extended in the positive and negative x direction.In that way, the structure of the antenna doubles in size, however, two primarybeams are generated from the structure. The superstrate is designed using the ge-ometric compensation technique. A back reector is placed at a distance 0.25λ0below the slot-line. Figure 5.19 represents the normalized far-eld plots of thecenter-fed design. The red curves present the patterns for the aced uncompen-sated design, while the blue and red curve shows the pattern for the geometri-cally compensated design without and with a back reector, respectively. Fig.5.20 illustrates the antenna parameters of the simulated center-fed design for ge-ometrically compensated superstrate with and without a back reector as well905.5. Center-fed design with a back reector00.25 0.50.75 11.25 1.501,0002,0003,0004,000Peakeld(V/m)(a)00.25 0.50.75 11.25 1.5020406080θ rad(deg)(b)00.25 0.50.75 11.25 1.5050100150SLL(%)(c)00.25 0.50.75 11.25 1.5020406080100BW(deg)(d)00.25 0.50.75 11.25 1.501,0002,0003,0004,000Distance from ground plane (λ0)Broadsideeld(V/m)(e)Relative permittivity, r =2Relative permittivity, r =4Figure 5.17: (color inline) Peak eld, Radiated beam angle, sidelobe level, 3dBbeam width, and broadside eld of the slot radiation into a halfspace havingr = 2 (red) and r = 4 (violet). The x axis represents the distance of a groundplane from the slot plane.915.6. SummaryFigure 5.18: (color inline) Permittivity distribution inside the inhomogeneoussuperstrate used for the center-fed design.as for the initial design. It can be observed that although the back reector elim-inates the backlobe of the antenna, it increases the overall sidelobe level andbeam variation over the beamwidth. However, it increases the directivity of theantenna.5.6 SummaryUsing the geometric compensation technique presented in Ch. 4, a leaky-wave antenna with xed beam characteristics has been demonstrated in thischapter. Couple of variations, included a corner-fed as well as a center-fed withand without a back reector, in the design process has been shown with simu-lations results and analysis. The results depict that the design using geometriccompensation technique indeed oers improved radiation from the antenna ascompared to the preliminary design shown in Ch. 3.925.6. Summary0306090120150180210240 270 300330(a) Frequency = 0.5f00306090120150180210240 270 300330(b) Frequency = 0.8f00306090120150180210240 270 300330(c) Frequency = 1.2f00306090120150180210240 270 300330(d) Frequency = 1.4f00306090120150180210240 270 300330(e) Frequency = 1.6f00306090120150180210240 270 300330(f) Frequency = 2f0Figure 5.19: (color inline) Normalized radiation patterns for the initial (red) aswell as the compensated center-fed leaky-wave antenna with (green) and with-out (blue) a backreector at dierent frequencies.935.6. Summary0.65f 0 f 01.35f 01.65f 0 2f020304050θ rad(deg)(a)0.65f 0 f 01.35f 01.65f 0 2f01020303dBBW(deg)(b)0.65f 0 f 01.35f 01.65f 0 2f020406080SLL(%)(c)0.65f 0 f 01.35f 01.65f 0 2f0406080100BLL(%)(d)0.65f 0 f 01.35f 01.65f 0 2f001020FrequencyDirectivity(linear)(e)Compensated: Centre-fedCompensated: Centre-fed with Back ReectorUncompensated: Centre-fedFigure 5.20: (color inline) (a) Radiated beam angle, (b) 3dB beam width, (c) side-lobe level, (d) backlobe level and (e) directivity of the leaky-wave antenna forthe geometrically compensated superstrate (blue) the back reector (green) andinitially designed superstrate (red).94Chapter 6Conclusion6.1 Summary of workLeaky-wave antennas are known to have wide input impedance bandwidth;however, their emitted beam scans with frequency variations. Chapter 1 raiseda question of whether or not an obliquely radiating xed-beam leaky-wave an-tenna can be designed for broadband applications. This question has been ad-dressed through the development of this thesis. A leaky-wave antenna is pro-posed in this work that eliminates the conventional beam scanning performancewhile maintaining its wide input impedance. The antenna consists of a gradeddielectric superstrate placed on top of a slot array. The superstrate functions as atransition layer to couple the slot-generated leaky wave radiation into free space.The radiation mechanism of previously proposed leaky-wave antennas hasbeen presented in Ch. 2. Broadband xed-beam radiation from leaky-wave an-tennas has received less attention as compared to conventional beam-scanningcharacteristics. The rst xed beam leaky-wave antenna was implemented bya dielectric lens to generate particularly broadside radiation. Oblique radiationfrom this antenna can be achieved by simply tilting the antenna structure. How-ever, the lens structure of the antenna made it electromagnetically large which isunsuitable for planar applications. In another planar approach, xed-beam radi-ation was achieved using non-foster circuits. However non-foster circuits haveto be implemented using active elements which increases the power consump-tion of the overall structure. The project reected in this thesis directed towardsthe design of a planar xed-beam broadband leaky-wave antenna.As presented in Ch. 2, an innitely long slot aperture located at the interfaceof two dierent dielectric half-space produces leaky-wave mode along the slot-line. The propagating mode generated leaky wave radiation which, in contrastto conventional one, remained constant with the change of operating frequency.The properties of the generated radiation was characterized as a function of dif-ferent slot-width and dielectric materials. A leaky-wave antenna was designedwith a simple 1D graded index material superstrate. A 1D dielectric distribution956.2. Future directionswas not sucient enough to provide an agreeable broadband antenna perfor-mance.The method was implemented on the rectangular superstrate to design thedesired xed-beam wide-band leaky-wave antenna. The design process is pre-sented in Ch. 5. Simulation results presented in the same chapter demonstrateshow the antenna radiation improves by applying the proposed technique. Theslot-line leaky-wave antenna that is capable to radiate at a xed-angle, which isdesigned for a bandwidth of 123%. As compared to existing antennas, the pro-posed design consists of a slot-line array placed underneath an inhomogeneoussuperstrate, which is purely a passive device.To date, a number of transformation electromagnetics-based antenna hasbeen proposed. The geometric compensation technique presented in this thesisopens the door for further improvement in antenna design inspired by transfor-mation electromagnetics, in particular, lens antennas. In addition, the methodcan be applied into at lenses where oblique radiation is required.6.2 Future directionsFuture directions of this research include improving the existing antenna de-sign, fabricating and testing the prototype. The major challenge in the implemen-tation of the proposed antenna is the fabrication of graded dielectric materials.Although the analysis of radiation pattern presented in this thesis is done for anupper limit of 2f0, the antenna is expected to operate at even higher frequen-cies. It is anticipated the proposed antenna to be very useful for xed-beam ap-plications. Fabrication of graded-index inhomogeneous medium has been stud-ied since 1900s for weak gradients. In 1969, Pearson utilized diusive ion ex-change technique to report a fabrication technique for gradient-index materialsfor image relays using glass rods [105]. However, the method was not applica-ble for waveguide applications due to high absorption and scattering losses. Thetechnique was extensively used for manufacturing graded-index dielectrics inthe 1970s. In 1975, Martin demonstrated a fabrication technique that applied towaveguide applications [106]. In later years, gradient-index materials have beenfabricated using co-evaporation technique, physical and chemical vapor deposi-tion, pulsed laser deposition, glancing angle deposition (GLAD) techniques [107].These techniques are highly applicable for specic applications and sensitive toparticular frequency bands and angle of incidence of wave.For millimeter wave applications, the additive manufacturing process that966.2. Future directionsemploys denser dielectric powders of higher permittivity to merge into a sheetof less denser substrate [108]. Variation of the density of the powder creates aslab of spatially graded permittivity. The density of the powder dispensed de-termines the eective relative permittivity of the composite. 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